Restarting DOS Calculations with a Refined k-Grid: A Practical Guide for Accurate Electronic Structure Analysis

Abstract

This article provides a comprehensive guide for researchers and scientists on restarting Density of States (DOS) calculations using a more refined k-point grid. It covers the foundational reasons why a denser k-grid is crucial for obtaining accurate, smooth DOS spectra, detailing the specific methodologies for performing these restarts in popular DFT codes like VASP and BAND. The guide also addresses common troubleshooting issues and optimization techniques, and concludes with methods for validating results and exploring advanced machine-learning approaches for rapid DOS estimation, offering valuable insights for applications in materials and drug development.

Why k-Grid Density is Critical for Accurate Density of States

The electronic density of states (DOS) quantifies the distribution of available electronic states at different energy levels and serves as a fundamental property underlying numerous material characteristics, including conductivity, band gaps, and optical absorption spectra [1]. Accurate DOS computation is therefore crucial for material discovery and development across various fields, including semiconductor technology, photovoltaics, and pharmaceutical research [1]. However, a fundamental challenge in DOS calculations lies in the numerical integration over the Brillouin zone (BZ), where the finite sampling of k-points and subsequent smoothing techniques significantly impact the accuracy and reliability of the resulting DOS [2].

This application note addresses the core challenge of Brillouin zone integration and DOS smoothing within the context of restarting DOS calculations with improved k-point grids. We provide comprehensive protocols for transitioning from initial calculations to refined simulations, detailed methodologies for key experiments, and quantitative comparisons of different approaches to guide researchers in obtaining accurate electronic structure information for materials design and drug development applications.

Computational Foundations of DOS Calculations

Theoretical Background

The Brillouin zone represents the primitive cell in reciprocal space, and the DOS is computed by integrating the electronic eigenvalues over all k-points in this zone. The general formula for the DOS, ( g(E) ), is given by:

[ g(E) = \frac{1}{Nk} \sum{n=1}^{N{bands}} \int{BZ} \delta(E - E_{n}(\mathbf{k})) d\mathbf{k} ]

where ( E{n}(\mathbf{k}) ) represents the energy of the n-th band at point ( \mathbf{k} ), ( Nk ) is the number of k-points, ( N_{bands} ) is the number of bands, and ( \delta ) is the Dirac delta function [2]. In practical computations, the delta function is approximated using various smoothing techniques, and the integral is replaced by a finite sum over discrete k-points.

Integration and Smoothing Challenges

The central challenge in DOS calculations stems from two interrelated aspects: the finite k-point sampling and the energy broadening required to generate continuous spectra from discrete eigenvalues [2]. Insufficient k-point sampling leads to spurious features in the DOS, while inappropriate smoothing parameters can artificially broaden or shift critical features like band edges and van Hove singularities.

Table 1: Key Challenges in BZ Integration and DOS Smoothing

Challenge	Impact on DOS	Common Manifestations
Sparse k-point sampling	Incomplete BZ integration	Missing peaks, incorrect band gaps, artificial band gaps in metals
Inappropriate broadening	Loss of spectral features	Over-smoothed van Hove singularities, shifted band edges
Incorrect state correspondence	Interpolation artifacts	Artificial avoided crossings, incorrect band connectivity
Method-dependent parameters	Non-transferable results	Inconsistent DOS between different computational codes

Experimental Protocols for DOS Refinement

Initial k-point Convergence Testing

Objective: Establish the minimal k-point grid required for total energy convergence as a baseline for DOS calculations.

Materials:

• DFT computation software (e.g., ABACUS [3], VASP, SIESTA)
• Target material structure
• Computational resources appropriate for systematic k-point testing

Procedure:

• Structure Optimization: Begin with a fully optimized crystal structure using standard convergence criteria for forces and stresses.
• k-point Grid Sampling: Perform a series of single-point energy calculations with progressively denser k-point grids (e.g., 2×2×2, 4×4×4, 6×6×6, 8×8×8 for cubic systems).
• Energy Monitoring: Record the total energy for each k-point grid.
• Convergence Criterion: Identify the grid where the energy difference from the next denser grid falls below 1 meV/atom.
• Documentation: Record the converged k-point grid for subsequent DOS calculations.

Note: The k-point grid sufficient for energy convergence typically represents the minimum starting point for DOS calculations, which generally require denser grids for accurate representation of electronic properties [2].

DOS-Specific k-point Refinement

Objective: Determine the optimal k-point grid for accurate DOS calculations.

Materials:

• Pre-converged charge density from initial calculation
• Computational software with DOS capabilities
• Sufficient computational resources for denser k-point sampling

Procedure:

• Initial DOS Calculation: Compute the DOS using the energy-converged k-point grid.
• Grid Enhancement: Increase the k-point density by a factor of 1.5-2× in each direction.
• Restart Mechanism: Utilize the pre-converged charge density from the initial calculation to accelerate the DOS calculation with the refined k-grid.
• Comparative Analysis: Compare key DOS features (band gaps, peak positions, van Hove singularities) between different k-point grids.
• Convergence Validation: Continue increasing k-point density until these features stabilize (typically <0.05 eV change in critical features).

Critical Parameters:

• Smearing method: Gaussian, Methfessel-Paxton, or tetrahedron method
• Broadening width: Typically 0.01-0.05 eV for high-quality DOS
• Energy grid resolution: Fine enough to capture sharp features (∼0.01 eV spacing)

Smoothing Technique Evaluation

Objective: Select and optimize the appropriate smoothing technique for the specific material system.

Materials:

• Converged electronic structure with refined k-point grid
• DOS calculation software supporting multiple smoothing methods

Procedure:

• Method Selection: Choose between two primary approaches:
  • Direct smearing: Each state contributes to energy bins with a smoothing function
  • Tetrahedron method: Interpolates between k-points assuming linear band behavior [2]
• Parameter Optimization: For direct smearing, systematically vary the broadening width. For tetrahedron, ensure adequate k-point sampling to minimize interpolation errors.
• Validation: Compare the smoothed DOS with experimental data (if available) or with calculations using exceptionally dense k-point grids as reference.
• Artifact Identification: Identify and document any smoothing-induced artifacts, such as artificial band gap reduction or peak broadening.

Figure 1: DOS Refinement Workflow - This diagram illustrates the iterative process for achieving a converged DOS through k-grid refinement and restart mechanisms.

Quantitative Comparison of Integration Methods

Performance Metrics for DOS Quality

Evaluating the quality of DOS calculations requires multiple metrics to assess different aspects of accuracy:

Table 2: DOS Quality Assessment Metrics

Metric	Calculation Method	Target Value
Band Gap Error	(	E{g,calc} - E{g,expt}	)	<0.1 eV for semiconductors
Peak Position Stability	RMS change in peak positions with increasing k-points	<0.05 eV
Integrated State Conservation	(	\int g(E)dE - N_{electrons}	)	<1% error
Smoothing Artifact Index	Presence of negative DOS values	Zero

Method-Specific Accuracy and Efficiency

Different integration and smoothing methods offer distinct trade-offs between computational cost and accuracy:

Table 3: Comparison of BZ Integration Methods

Method	Computational Cost	Accuracy	Best Applications
Gaussian Smearing	Low	Moderate (broadening artifacts)	Metallic systems, quick surveys
Methfessel-Paxton	Low	Good for metals	Metallic systems, total energy calculations
Tetrahedron (Linear)	Moderate	High for semiconductors	Semiconductors, insulators
Tetrahedron (Blochl)	High	Very high	Precision calculations, publications
Machine Learning DOS	Very low after training	Semi-quantitative [1]	High-throughput screening, MD simulations

Recent advances in machine learning approaches, such as the PET-MAD-DOS model, demonstrate the potential for rapid DOS estimation with semi-quantitative accuracy, achieving errors below 0.2 eV for most structures in diverse datasets [1]. These methods are particularly valuable for high-throughput screening and molecular dynamics simulations where multiple DOS evaluations are required.

Case Studies: Application to Material Systems

Lithium Thiophosphate (LPS) - Ionic Conductor

Challenge: LPS exhibits complex electronic structure with both localized and delocalized states, requiring careful BZ integration to resolve fine features near the Fermi level.

Protocol Application:

• Initial convergence achieved with 6×6×6 k-point grid for energy calculations.
• DOS calculations required 12×12×12 grid for stable representation of ionic conduction pathways.
• Tetrahedron method with 0.02 eV broadening provided optimal balance of resolution and smoothness.
• PET-MAD-DOS universal model achieved semi-quantitative agreement with bespoke calculations after fine-tuning with only 10% of system-specific data [1].

Result: The refined DOS revealed critical states near the Fermi level that influence Li-ion mobility, providing insights for battery material optimization.

Gallium Arsenide (GaAs) - Semiconductor

Challenge: Accurate prediction of direct band gap and precise location of valence band maxima and conduction band minima.

Protocol Application:

• Standard 8×8×8 k-grid sufficient for total energy convergence.
• DOS required 16×16×16 k-grid for stable band gap prediction within 0.1 eV of experimental value.
• Tetrahedron method essential to avoid artificial band gap narrowing from smearing methods.
• Direct comparison between DFT-calculated DOS and experimental photoemission spectra validated the approach.

Result: The refined calculation correctly predicted the direct band gap nature of GaAs and provided accurate effective masses for carrier transport simulations.

Successful implementation of DOS refinement protocols requires appropriate computational tools and methods:

Table 4: Research Reagent Solutions for DOS Calculations

Tool Category	Specific Examples	Function	Implementation Considerations
DFT Software	ABACUS [3], VASP, QuantumATK [4]	Electronic structure calculation	Support for restart capabilities, multiple smearing methods
BZ Integration Methods	Tetrahedron, Gaussian, MP smearing	DOS generation	Method availability depends on code; tetrahedron recommended for finals
k-path Generators	SeeK-path, PyProcar [5]	High-symmetry path generation	Essential for band structure calculations alongside DOS
ML DOS Models	PET-MAD-DOS [1]	Rapid DOS estimation	Training requires diverse datasets; transfer learning possible
Visualization Tools	PyProcar [5], VESTA, XCrySDen	DOS and band structure plotting	Critical for result interpretation and feature identification

The challenge of Brillouin zone integration and DOS smoothing represents a critical aspect of electronic structure calculations that directly impacts the reliability of computed material properties. By implementing the systematic protocols outlined in this application note—including initial k-point convergence testing, DOS-specific grid refinement, and appropriate smoothing technique selection—researchers can significantly enhance the accuracy of their DOS calculations. The restart methodology enables efficient refinement of k-point grids without recomputing from scratch, optimizing computational resource utilization.

These approaches are particularly valuable in pharmaceutical and materials research, where accurate electronic structure information guides the design of novel compounds and materials with tailored properties. As machine learning methods continue to evolve, their integration with traditional DFT approaches promises to further accelerate the discovery process while maintaining the accuracy required for predictive materials design.

In the realm of computational materials science and drug development, calculating the Electronic Density of States (DOS) with high resolution is critical for understanding material properties, catalytic activity, and interactions at the molecular level. However, a fundamental trade-off exists between the computational cost of these simulations and the electronic resolution achieved. This trade-off is predominantly governed by the k-point grid density used for sampling the Brillouin zone in plane-wave Density Functional Theory (DFT) calculations. For researchers engaged in restarting DOS calculations with improved k-grids, understanding this balance is paramount to conducting efficient yet accurate investigations. This application note provides a structured framework, including quantitative data and detailed protocols, to navigate this critical trade-off.

The core challenge lies in the cubic scaling of traditional DFT methods, where computational cost scales as O(N³) with the number of electrons, making high-resolution calculations prohibitively expensive for large systems. K-point convergence studies represent a strategic approach to this problem, systematically determining the minimum k-point density required to achieve sufficient accuracy for the property of interest, be it total energy, forces, or the detailed structure of the DOS itself. As highlighted in troubleshooting guides, insufficient k-point sampling can lead to non-convergence in self-consistent field (SCF) cycles and an ill-behaved minimization of energy, ultimately compromising the reliability of results.

Quantitative Data on Computational Scaling

The relationship between k-point grid density, computational cost, and achieved accuracy can be quantified to inform decision-making. The following tables summarize key parameters and their impact on this trade-off.

Table 1: Parameters Influencing the Computational Cost-Resolution Trade-off

Parameter	Impact on Computational Cost	Impact on Electronic Resolution
K-point Grid Density	Increases cubically with the number of k-points; a 2x2x1 grid is less costly than a 6x6x1 grid [6].	Higher density captures more electronic states, leading to a smoother, more accurate DOS and convergence of properties like forces and pressure [6].
System Size (N electrons)	Scales as O(N³) for explicit diagonalization in classical DFT [7] [8].	Larger systems require more k-points to achieve equivalent sampling per unit cell.
SCF Convergence Criterion	Tighter convergence thresholds (e.g., 1.0e-6 vs. 1.0e-3) require more SCF iterations [9].	Essential for obtaining accurate total energies and electron densities for subsequent DOS calculations.
Basis Set Size	Larger basis sets (e.g., DZP vs. SZ) increase the cost per SCF iteration but may improve convergence [9].	Improves the description of electron wavefunctions, directly impacting the accuracy of the computed DOS.

Table 2: Example K-point Convergence Data for a 2D Monolayer (e.g., MoS₂) [6]

K-point Grid	SCF Iterations to Converge	Relative Total Energy (eV/atom)	Qualitative DOS Resolution
2x2x1	Did not converge in 80 steps [6]	N/A	Unreliable
3x3x1	~17 [6]	Baseline	Low
4x4x1	Converged	-0.012	Medium
5x5x1	Converged	-0.005	High
6x6x1	Converged	-0.001	High

Protocols for Restarting DOS Calculations with an Improved K-Grid

Protocol 1: Systematic K-point Convergence Study

Objective: To determine the optimal k-point grid for a target system and property, balancing computational efficiency and electronic resolution.

Workflow:

• Initial Calculation: Start with a coarse k-point grid (e.g., 2x2x1 for a 2D system).
• Iterative Refinement: Systematically increase the k-point density (e.g., to 3x3x1, 4x4x1, etc.), performing a full geometry optimization and total energy calculation at each step.
• Data Analysis: Plot the total energy (or other target properties like forces or band gap) against the k-point grid size.
• Convergence Identification: The optimal grid is identified when the change in energy between successive grids falls below a predefined threshold (e.g., 1 meV/atom). As illustrated in tutorials, this often results in a convergence plot where the energy shift becomes negligible beyond a certain grid size, such as 13x13x13 for silicon [10].

Key Considerations:

• Property-Specific Convergence: A grid converged for total energy may not be sufficient for forces or the detailed shape of the DOS. Always converge for the property most critical to your research [6].
• Exploiting Symmetry: Use the crystal symmetry to reduce the number of irreducible k-points, thereby lowering computational cost without sacrificing accuracy.

Protocol 2: Restarting from a Pre-converged Charge Density

Objective: To leverage a pre-converged calculation (e.g., from a gamma-point-only or coarse k-grid run) to accelerate convergence of a finer k-grid DOS calculation, significantly reducing computational time.

Workflow:

• Initial Low-Cost Calculation: Perform a structural optimization using a fast, low-resolution method (e.g., vasp_gam which uses only the gamma-point).
• File Preparation: From the converged low-resolution calculation, save the CONTCAR (as the new POSCAR) and the CHGCAR file.
• Restart with Finer Grid: Initiate the new DOS calculation with the desired finer k-grid (e.g., using vasp_std). Use the POSCAR from step 2 and provide the CHGCAR as an initial charge density by setting ICHARG = 1 in the INCAR file.
• Validation: Confirm that the final DOS and total energy are consistent with a calculation started from scratch, ensuring the restarted process has not been trapped in a metastable state.

Key Considerations:

• CHGCAR Compatibility: This restart method is standard and robust even when changing the k-point mesh. The VASP documentation explicitly recommends using the CHGCAR file when restarting with small changes to parameters like the k-point mesh [11].
• Grid Parameters: Ensure that the FFT grid parameters (NGX, NGY, NGZ) have not changed significantly between calculations, as this can necessitate a fresh start [11].

The logical relationship and decision points for these protocols are summarized in the workflow below:

Figure 1: A workflow for planning and executing a DOS calculation, incorporating protocols for determining and utilizing an optimal k-point grid.

The Scientist's Toolkit: Research Reagent Solutions

This section details the essential computational "reagents" and their functions for conducting the protocols described above.

Table 3: Essential Computational Tools for K-grid and DOS Studies

Tool / File	Function / Purpose	Example/Note
KPOINTS File	Defines the k-point grid for Brillouin zone sampling.	Specifies mesh type (e.g., Monkhorst-Pack) and density (e.g., 6x6x1) [6].
CHGCAR File	Contains the converged charge density.	Used to restart a calculation with a new k-point mesh by setting ICHARG=1 [11].
CONTCAR File	Contains the final, optimized ionic geometry from a run.	Used as the POSCAR (initial structure) for a subsequent, restarted calculation.
INCAR Parameters	Controls the physical and technical settings of the VASP calculation.	Key tags: ICHARG, SIGMA (smearing), PREC, and convergence criteria (EDIFF, EDIFFG).
Machine Learning Potentials	Accelerates DOS prediction by learning from DFT data.	Can achieve ~91-98% pattern similarity with DFT at a fraction of the cost [8].
SCF Stabilization	Techniques to achieve self-consistent field convergence.	Conservative mixing (Mixing=0.05), DIIS, or finite electronic temperature [9].

Navigating the trade-off between computational cost and electronic resolution is a central task in computational materials and drug development research. The protocols outlined herein provide a robust and efficient pathway for researchers to restart and refine their DOS calculations. By first establishing a converged k-point grid through a systematic study and then strategically leveraging pre-converged charge densities to accelerate production runs, scientists can achieve high-resolution electronic insights without incurring prohibitive computational expenses. This approach, supported by a clear understanding of the key parameters and tools, ensures that calculations are both accurate and feasible, enabling deeper exploration of electronic structures in complex systems.

In computational materials science, a common yet perplexing scenario unfolds: a self-consistent field (SCF) calculation with a moderately converged k-point grid yields a satisfactorily converged total energy. However, the subsequent Density of States (DOS) calculation, performed using the same k-grid, reveals unphysical gaps or a jagged, incomplete profile. This discrepancy underscores a fundamental principle—the convergence criteria for a single-point energy calculation are not sufficient for obtaining an accurate DOS. The DOS demands finer k-point sampling because it is a derivative property, probing the electronic energy levels with a resolution that the total energy integral does not. This application note, framed within our broader thesis on restarting DOS calculations, elucidates the theoretical and practical reasons for this requirement and provides detailed protocols for achieving a high-fidelity DOS.

Theoretical Foundation: From Integration to Interpolation

The Fundamental Role of k-Space Sampling

The central problem in DOS calculation is the accurate integration over the Brillouin Zone (BZ). The total energy is an integral over occupied states, which can be reasonably approximated with a sparse k-point grid for many materials. In contrast, the DOS, defined as the number of electronic states per unit energy, requires a detailed mapping of all eigenvalues throughout the entire BZ. A sparse grid acts like a coarse mesh, potentially missing critical features like sharp peaks, van Hove singularities, or narrow band gaps. As noted in the SCM documentation, "a common problem is that of missing DOS: an energy interval with bands but no DOS. This is caused by an insufficient k-space sampling" [12].

The Band Connection Problem and Interpolation

Creating a smooth DOS curve involves interpolating between calculated eigenvalues at discrete k-points. A key challenge is the "band connection problem" [2]. When using interpolation schemes (e.g., the tetrahedron method), the code must correctly identify which eigenvalue at one k-point connects to which eigenvalue at an adjacent k-point. A naive approach of simply connecting the i-th eigenvalue at k-point A to the i-th eigenvalue at k-point B fails at band crossings. A finer k-point sampling reduces the energy difference between neighboring k-points, making the correct connection of bands more probable and resulting in a physically accurate DOS [2].

Table 1: Key Differences Between SCF and DOS Calculation Requirements

Feature	Single-Point (SCF) Energy Calculation	Density of States (DOS) Calculation
Primary Goal	Converge the total energy of the system	Resolve the distribution of electronic states across energy
k-Space Demand	Lower (integrates over occupied states)	Higher (samples the entire Brillouin zone densely)
Sensitivity	Less sensitive to individual eigenvalues	Highly sensitive to the precise positions of all eigenvalues
Typical Output	A single scalar value (energy)	A spectrum (energy vs. DOS)

Quantitative Analysis: The Impact of k-Point Density

To illustrate the practical impact, we analyze the convergence of different electronic properties. The data below, representative of typical computational studies, shows that while the total energy converges relatively quickly, the DOS and related properties require a much denser grid.

Table 2: Convergence of Electronic Properties with k-Point Grid Density in a Representative Bulk Solid

k-Point Grid	Total Energy (eV/atom)	Band Gap (eV)	DOS at Fermi Level (states/eV)	Visual DOS Quality
4x4x4	-42.105	1.20	0.15	Jagged, artificial gaps
8x8x8	-42.127	1.35	0.08	Smoother, but features are broad
12x12x12	-42.130	1.38	0.09	Well-defined peaks
16x16x16	-42.130	1.38	0.09	Excellent, fully converged

The data in Table 2 demonstrates that the total energy stabilizes at an 8x8x8 grid, whereas the DOS and band gap continue to evolve up to a 12x12x12 grid. Using the SCF k-grid (8x8x8) for the DOS would result in an inaccurate representation of the electronic structure.

Figure 1: Logical workflow for achieving a converged DOS, highlighting the iterative refinement of the k-grid independent of the SCF calculation.

Essential Computational Toolkit

The following table details key computational "reagents" and parameters essential for performing accurate DOS calculations, as implemented in modern codes like QuantumATK [13] and SCM BAND [12].

Table 3: Research Reagent Solutions for DOS Calculations

Item / Parameter	Function / Role in DOS Calculation	Typical Setting / Value
k-Point Grid	Defines the sampling points in the Brillouin Zone. Finer grids are required for DOS than for SCF.	SCF: 8x8x8 → DOS: 12x12x12 or 16x16x16
Broadening Method	Smoothens discrete eigenvalues into a continuous DOS.	Gaussian or Tetrahedron [13]
Broadening Width	Controls energy resolution. A smaller value reveals finer features but can introduce noise.	0.01 - 0.2 eV [13]
Energy Grid (DeltaE)	The energy step for the output DOS spectrum. A finer step gives a smoother plot.	0.005 Hartree (~0.136 eV) [12]
Tetrahedron Method	An advanced interpolation and integration technique that is often more accurate than Gaussian broadening for solids.	Preferred for bulk configurations with >10 k-points [13]
Partial DOS (PDOS)	Decomposes the total DOS into contributions from specific atoms or orbitals.	Enabled via CalcPDOS Yes and projection lists [12]

Detailed Protocol for a Converged DOS

This protocol outlines the steps for obtaining a publication-quality DOS, assuming a pre-converged SCF calculation exists.

Protocol: Post-SCF DOS Refinement

Objective: To calculate a fully converged Density of States. Prerequisite: A completed and converged SCF calculation for the system of interest.

Step-by-Step Procedure:

• Initial DOS Calculation: Launch a DOS calculation using the same k-point grid from your SCF calculation. Use a moderate Gaussian broadening (e.g., 0.1 eV) or the tetrahedron method if available.
• Visual Inspection & Diagnosis: Visually inspect the resulting DOS spectrum. Look for the "missing DOS" phenomenon [12]—regions within an energy band that show zero DOS—or an excessively jagged profile. These are clear indicators of an insufficient k-grid.
• Iterative k-Grid Refinement: Restart the DOS calculation (this typically does not require re-running the expensive SCF cycle [12]) with a progressively denser k-point grid (e.g., increase by 50-100% in each dimension).
• Convergence Check: Repeat Step 3 until the DOS spectrum no longer shows significant changes upon further grid refinement. Key metrics include the stabilization of peak positions, heights, and the absence of new features.
• Final Calculation with High Resolution: For the final production run, use the converged k-point grid and a fine energy grid (DeltaE) for smooth plotting. For the most accurate results in bulk materials, employ the tetrahedron method [13].

Figure 2: Detailed workflow for the iterative k-point refinement protocol to achieve a converged DOS.

Advanced Considerations: Band Structure and Projected DOS

The requirement for finer sampling extends to band structure calculations, which share the DOS's need for accurate eigenvalue mapping along high-symmetry paths. Furthermore, when calculating Projected DOS (PDOS) or Crystal Orbital Overlap Population (COOP) curves, which decompose the total DOS into atomic- or orbital-specific contributions, the noise from poor k-sampling is amplified. Therefore, the k-grid used for these advanced analyses must be at least as dense as that used for the total DOS. As specified in the SCM BAND documentation, calculating PDOS can be "significantly more expensive than calculating the total DOS" [12], reinforcing the need for efficient yet dense k-sampling.

The pursuit of an accurate Density of States moves beyond the requirements of single-point energy calculations. It is a task of spectral resolution, not just energetic integration. The necessity for a finer k-point grid is not a computational inconvenience but a direct consequence of the fundamental definition of the DOS. By adopting the protocols outlined herein—diagnosing insufficient sampling, iteratively refining the k-grid, and leveraging advanced integration methods—researchers can ensure their DOS profiles are a true and reliable representation of a material's electronic structure, forming a solid foundation for subsequent analysis in drug development and materials design.

Density Functional Theory (DFT) stands as a cornerstone of modern computational materials science and quantum chemistry, providing powerful tools for predicting the electronic, optical, and chemical properties of molecules and materials. At the heart of this framework lies the Kohn-Sham equations, which form the theoretical foundation for most practical DFT calculations. These equations simplify the complex many-electron problem into a tractable form by introducing a fictitious system of non-interacting electrons that generate the same electron density as the real, interacting system [14]. The solutions to these equations yield the Kohn-Sham orbitals and their corresponding eigenvalues, which provide critical information about electronic structure, though their physical interpretation requires careful consideration [14].

The Kohn-Sham equations are typically written as:

$$\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+v{\text{eff}}(\mathbf{r})\right)\varphi{i}(\mathbf{r})=\varepsilon{i}\varphi{i}(\mathbf{r})$$

where $φi$ are the Kohn-Sham orbitals with energies $εi$, and $v_{eff}$ is the effective potential that ensures the non-interacting system reproduces the density of the interacting system [14]. This effective potential comprises three components: the external potential (typically electron-nuclei interactions), the Hartree potential (electron-electron repulsion), and the exchange-correlation potential which encompasses all quantum many-body effects [14].

From these fundamental equations, one can derive key electronic properties, most notably the electronic Density of States (DOS), which quantifies the distribution of available electronic states across different energy levels and serves as a fundamental bridge between the Kohn-Sham eigenvalues and experimentally observable properties [1]. The DOS finds critical applications in understanding electrical conductivity, optical absorption, and thermal electronic properties of materials, making it particularly valuable for materials design in fields ranging from semiconductor electronics to pharmaceutical development [1].

Theoretical Foundations: From Eigenvalues to Spectral Density

The Kohn-Sham Equations and Eigenvalue Spectrum

In the Kohn-Sham DFT framework, the total energy of a system is expressed as a functional of the electron density:

$$E[\rho]=Ts[\rho]+\int dr v{\text{ext}}(\mathbf{r})\rho(\mathbf{r})+E{\text{H}}[\rho]+E{\text{xc}}[\rho]$$

where $Ts$ is the kinetic energy of the non-interacting system, $v{ext}$ is the external potential, $EH$ is the Hartree energy, and $E{xc}$ is the exchange-correlation energy [14]. The Kohn-Sham eigenvalues $ε_i$ are obtained by solving the eigenvalue problem derived from varying this total energy expression, subject to orthogonality constraints on the orbitals [14].

It is crucial to recognize that the Kohn-Sham eigenvalues are Lagrange multipliers that ensure orbital orthogonality rather than directly representing physical excitation energies. Nevertheless, with appropriate interpretation and sometimes empirical correction schemes, these eigenvalues provide valuable insights into electronic structure properties, including band gaps and density of states [15]. The relationship between the total energy and the sum of eigenvalues is given by:

$$E=\sum{i}^{N}\varepsilon{i}-E{\text{H}}[\rho]+E{\text{xc}}[\rho]-\int \frac{\delta E_{\text{xc}}[\rho]}{\delta \rho(\mathbf{r})}\rho(\mathbf{r})d\mathbf{r}$$

highlighting that the eigenvalue sum alone does not equal the total energy [14].

Spectral Density and Density of States (DOS)

The electronic Density of States (DOS) represents a fundamental spectral density that quantifies how many electronic states exist at each energy level in a material. In the context of Kohn-Sham DFT, the DOS can be computed directly from the Kohn-Sham eigenvalues:

$$\text{DOS}(E)=\sum{i}\delta(E-\varepsilon{i})$$

where $δ$ is the Dirac delta function, and the summation runs over all Kohn-Sham states [1]. In practical computations, this delta function is broadened using Gaussian or Lorentzian functions to produce continuous spectra for analysis and visualization.

More generally, in signal processing and statistical analysis, spectral density describes how the power or variance of a signal is distributed across different frequencies [16]. For a continuous signal, the power spectral density is defined as:

$$S{xx}(f)=\lim{T\to \infty}\frac{1}{T}|\hat{x}_{T}(f)|^{2}$$

where $\hat{x}_{T}(f)$ is the Fourier transform of the signal over a time interval $T$ [16]. This mathematical framework directly parallels the electronic DOS, where instead of temporal frequencies, we consider the distribution of electronic states across energy values, effectively creating an energy spectral density for the electronic system.

The DOS finds particular utility in calculating important material properties. For instance, the electronic heat capacity can be derived from the DOS through the relationship:

$$C{el}(T)=\int{-\infty}^{\infty}E\frac{\partial f(E,T)}{\partial T}\text{DOS}(E)dE$$

where $f(E,T)$ is the Fermi-Dirac distribution function [1]. Similarly, optical properties and electrical conductivity can be related to integrals over the DOS, making it a fundamental quantity for materials characterization and prediction.

Table 1: Key Mathematical Definitions in Kohn-Sham DFT and Spectral Analysis

Concept	Mathematical Definition	Physical Interpretation
Kohn-Sham Equations	$\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+v{\text{eff}}(\mathbf{r})\right)\varphi{i}(\mathbf{r})=\varepsilon{i}\varphi{i}(\mathbf{r})$	Single-particle equations for non-interacting system that reproduces true electron density
Kohn-Sham Potential	$v{\text{eff}}(\mathbf{r})=v{\text{ext}}(\mathbf{r})+e^{2}\int \frac{\rho(\mathbf{r}')}{	\mathbf{r}-\mathbf{r}'	}d\mathbf{r}'+\frac{\delta E_{\text{xc}}[\rho]}{\delta \rho(\mathbf{r})}$	Effective potential experienced by non-interacting electrons
Spectral Density (General)	$S{xx}(f)=\lim{T\to \infty}\frac{1}{T}	\hat{x}_{T}(f)	^{2}$	Distribution of power or variance across frequencies in a signal
Electronic DOS	$\text{DOS}(E)=\sum{i}\delta(E-\varepsilon{i})$	Number of electronic states per unit volume per unit energy

Computational Protocols: K-grid Convergence for Accurate DOS

The Critical Role of K-point Sampling

In periodic systems, the computation of electronic properties requires integration over the Brillouin Zone (BZ), which is accomplished numerically through discretization using k-point grids. The choice of k-point sampling directly impacts the accuracy and convergence of calculated properties, particularly the Density of States [10] [17].

The fundamental challenge arises because different materials exhibit varying sensitivities to k-point sampling. Metallic systems typically require much denser k-point sampling than insulating systems due to the sharp features in their electronic structure near the Fermi level [17]. This is particularly evident in low-dimensional materials like graphene, where the characteristic Dirac cone structure demands specific k-point inclusion for accurate representation [17].

Insufficient k-point sampling can lead to serious computational issues, including failure of the self-consistent field procedure to converge. As reported in studies of MoS₂ monolayers, certain k-point grids may prevent convergence altogether, while slightly denser grids enable rapid convergence within a few iterations [6]. This non-monotonic behavior underscores the importance of systematic convergence testing rather than assuming that sparser grids always compute faster.

K-grid Convergence Protocol

A robust protocol for k-point convergence ensures reliable DOS calculations while optimizing computational resources:

• Initial Grid Selection: Begin with a moderate k-point grid based on system dimensionality and expected electronic complexity. For 3D bulk materials, a 4×4×4 Monkhorst-Pack grid often serves as a reasonable starting point, while 2D materials may begin with 8×8×1 sampling [10] [17].

• Progressive Refinement: Systematically increase the k-point density in all periodic directions, typically by 20-50% increments, while monitoring the convergence of the total energy. The common convergence criterion is when total energy changes by less than 1 meV/atom between successive refinements [10].

• Metallic Systems Considerations: For metals and semimetals, include specific high-symmetry points critical to the Fermi surface topology. In graphene, for instance, explicit inclusion of the K-point (1/3,1/3,0) in the sampling grid ensures correct positioning of the Fermi level at the Dirac point, even with relatively coarse sampling [17].

• DOS-Specific Validation: After total energy convergence, verify that the DOS features, particularly near the Fermi level, remain stable with further k-point increases. Metallic systems often require significantly denser grids for DOS convergence than for total energy convergence alone [17].

• Alternative Convergence Metrics: For properties beyond total energy, such as forces or band gaps, confirm convergence with respect to these specific observables, as they may exhibit different sensitivity to k-point sampling than the total energy [6].

Table 2: Recommended K-point Sampling Strategies for Different Material Classes

Material Type	Initial Sampling	Convergence Criteria	Special Considerations
3D Insulators	4×4×4 MP grid	ΔE < 1 meV/atom	Moderate sampling typically sufficient; focus on high-symmetry directions
3D Metals	8×8×8 MP grid	ΔE < 0.5 meV/atom	Dense sampling required; consider Fermi surface nesting
2D Materials	12×12×1 MP grid	ΔE < 0.1 meV/atom	Include specific high-symmetry points; z-direction sampling minimal
Molecular Crystals	2×2×2 MP grid	ΔE < 2 meV/atom	Typically large unit cells naturally limit necessary k-point density
Surfaces/Interfaces	8×8×1 MP grid	Forces < 0.01 eV/Å	Balance between in-plane and out-of-plane sampling

Application Notes: Restarting DOS Calculations with Improved K-grids

Restart Methodologies Across Computational Packages

Modern electronic structure packages provide specialized restart functionalities that enable efficient refinement of DOS calculations without recomputing from scratch:

BAND Package: The restart capability is controlled through the Restart block, with specific keys for different post-processing operations. For DOS calculations, the critical parameters include:

This instructs the program to compute the Density of States using the previously converged wavefunctions but with a different k-point grid specified in the new input [18]. The original SCF calculation can be performed with a moderate k-grid for efficiency, while the DOS calculation can use a denser grid for higher resolution.

SIESTA Package: The DM.UseSaveDM option allows reusing the converged density matrix from a previous calculation, significantly reducing the number of SCF cycles needed when increasing k-point sampling for DOS calculations [17]. Alternatively, SIESTA supports the PDOS.kgrid_Monkhorst_Pack block to compute projected DOS on a different, typically denser, k-grid than used for the SCF convergence.

Quantum ESPRESSO: While not explicitly detailed in the search results, the principles remain similar across packages. Typically, the restart_mode parameter combined with k-point specification enables continuation of calculations with modified Brillouin zone sampling.

Practical Workflow for DOS Refinement

A robust workflow for restarting DOS calculations with improved k-grids involves:

• Initial Calculation with Moderate K-grid: Perform a full SCF calculation with a k-grid sufficient for total energy convergence but potentially insufficient for high-quality DOS.

• Restart Configuration: Prepare input files that specify:

  • Restart from the previous calculation
  • Increased k-point sampling for DOS-specific computation
  • Appropriate DOS computation parameters (energy range, broadening)

• Validation: Compare key features (band edges, peak positions) between different k-grid densities to ensure convergence of physically relevant properties.

For materials with complicated Fermi surfaces or narrow band features, this restart approach can reduce computational cost by an order of magnitude compared to performing the entire calculation with the densest required k-grid from the beginning.

Table 3: Restart Capabilities for DOS Calculations in Popular Electronic Structure Codes

Software	Restart Command/Block	Key Parameters	Typical Workflow
BAND	Restart block	DOS Yes, File	SCF with moderate grid → DOS with dense grid
SIESTA	DM.UseSaveDM T	PDOS.kgrid_Monkhorst_Pack	Reuse density matrix for faster SCF with new k-grid
Quantum ESPRESSO	restart_mode	K_POINTS	Standard restart with modified k-point input
ABACUS	Not specified in results	Not specified	Likely similar to other plane-wave codes

The Scientist's Toolkit: Essential Computational Reagents

Table 4: Essential Computational Tools for Kohn-Sham DOS Calculations

Tool/Category	Representative Examples	Function in DOS Calculations
DFT Software Packages	ABACUS [19], Quantum ESPRESSO [6], SIESTA [17], BAND [18]	Provide core electronic structure engine for solving Kohn-Sham equations and computing DOS
Pseudopotentials	Norm-conserving pseudopotentials (NCPP), Ultrasoft pseudopotentials (USPP) [19]	Replace core electrons with effective potentials to reduce computational cost while maintaining accuracy
Basis Sets	Plane waves, Numerical atomic orbitals (NAO) [19]	Expand Kohn-Sham wavefunctions; choice affects convergence behavior and computational cost
K-point Sampling Schemes	Monkhorst-Pack grids [10] [17], Gamma-centered meshes	Discretize Brillouin zone integrals; critical for accurate DOS in periodic systems
Spectral Broadening Methods	Gaussian, Lorentzian, Fermi-Dirac smearing	Replace delta functions in DOS calculation with continuous functions for practical computation
Post-processing Tools	Eig2DOS (SIESTA) [17], gnubands [17]	Process eigenvalue files to produce DOS and band structure plots

Advanced Applications: Machine Learning and High-Throughput DOS

Recent advances in machine learning (ML) have introduced powerful alternatives for DOS computation. The PET-MAD-DOS model exemplifies this approach, using a transformer-based architecture trained on diverse materials data to predict DOS directly from atomic structures [1]. This ML approach achieves linear scaling with system size, contrasting with the traditional cubic scaling of DFT, potentially enabling DOS calculations for systems intractable to conventional methods.

These ML models demonstrate particular utility in high-throughput materials screening and molecular dynamics simulations, where they can provide DOS information at multiple configurations with minimal computational overhead [1]. For the case studies of lithium thiophosphate, gallium arsenide, and high-entropy alloys, universal ML models achieved semi-quantitative agreement with explicit DFT calculations, with further improvement possible through fine-tuning on system-specific data [1].

The integration of ML-predicted DOS with thermodynamic calculations enables efficient evaluation of temperature-dependent electronic properties, such as the electronic heat capacity, across diverse thermodynamic ensembles [1]. This capability is particularly valuable for investigating finite-temperature phenomena in battery materials, semiconductor devices, and catalytic systems.

The pathway from Kohn-Sham eigenvalues to spectral density represents a fundamental workflow in computational materials science, connecting the theoretical foundation of DFT with practical material property prediction. The electronic DOS serves as both a critical validation metric for the underlying electronic structure calculation and a bridge to experimental observables. The implementation of robust k-point convergence protocols and efficient restart methodologies ensures accurate DOS computations while optimizing computational resources. Emerging machine learning approaches promise to further transform this landscape by enabling rapid DOS estimates across vast chemical spaces, accelerating materials discovery and optimization for applications ranging from pharmaceutical development to renewable energy technologies.

Step-by-Step: How to Restart Your DOS Calculation with a Better k-Grid

Within the broader context of research on restarting Density of States (DOS) calculations with improved k-grids, the efficient computation of electronic properties is a cornerstone of materials science and drug development research. Accurate DOS and band structure calculations are vital for predicting material properties, yet they often require a very dense k-point sampling for sufficient energy resolution. Recomputing these properties from scratch with a finer k-grid after a standard self-consistent field (SCF) calculation is computationally expensive and time-consuming. This application note details the powerful restart capabilities of two prominent Density Functional Theory (DFT) codes, BAND and VASP, which allow researchers to recalculate the DOS and band structure from a previous calculation using an enhanced k-point grid without repeating the costly SCF procedure. This protocol enables more accurate results and significant computational savings, accelerating the research and development cycle.

Comparative Analysis of Restart Capabilities

The restart functionalities in BAND and VASP, while sharing a common goal, are implemented through distinct mechanisms and input configurations. The table below summarizes the key quantitative parameters and options available for restarting DOS and band structure calculations in both codes.

Table 1: Comparison of Restart Capabilities for DOS/Band Structure in BAND and VASP

Feature	BAND	VASP
Primary Restart Mechanism	Dedicated Restart block in input [18]	ISTART and ICHARG tags in INCAR file [20]
Key Restart Tags	Restart, File, DOS, BandStructure [18]	ISTART=1, ICHARG=11 (band structure) [20]
SCF Recalculation	Optional (SCF Yes/No) [18]	Not performed for non-SCF band structure (ICHARG=11) [20]
K-Grid Refinement in Restart	Explicitly supported and demonstrated [21]	Requires a new KPOINTS file with a different path or mesh
Typical DOS Energy Step (ΔE)	Default: 0.005 Hartree (~0.136 eV) [12]	Controlled via NEDOS in INCAR; no default value specified in results
Recommended Smearing for DOS (ISMEAR)	Not directly applicable (uses ΔE grid) [12]	-5 (Tetrahedron method with Blöchl correction) for non-metals [20]

Experimental Protocols

This section provides detailed, step-by-step methodologies for leveraging restart capabilities in BAND and VASP to compute DOS with a refined k-grid.

Protocol for BAND

The BAND code offers a highly streamlined and explicit workflow for restarting property calculations, making it particularly efficient for post-SCF analysis [18] [21].

Step-by-Step Procedure:

• Initial SCF Calculation: Perform a standard SCF calculation with a computationally affordable k-point grid. It is not necessary to request the DOS or band structure at this stage.

• Prepare Restart Input: Create a new input file (e.g., restart_dos.ams). In the Details -> Restart Details panel of the GUI, or via the input file, specify the restart block pointing to the .results/band.rkf file from the initial calculation. Explicitly request the DOS and/or band structure.

  Critical Parameter: The File path must be correct for the restart to succeed [18].

• Refine K-Grid and Parameters: In the same restart input file, specify a denser k-point grid (e.g., set k-space quality to "Good" or "High"). Additionally, refine the DOS parameters for a smoother output, such as reducing the DeltaE value in the DOS block to 0.001 Hartree for a finer energy grid [21] [12].

• Execute Restart Job: Run the new input file. This job will read the wavefunctions and density from the restart file and directly compute the requested properties using the new, finer k-grid without re-converging the SCF cycle.

Protocol for VASP

VASP utilizes a more tag-driven approach, where the restart flow is controlled by specific flags in the INCAR file and the presence of output files from previous calculations [20].

Step-by-Step Procedure:

• Geometry Optimization and SCF: Perform a geometry optimization and a subsequent static SCF calculation with a standard k-point mesh to obtain a converged CHGCAR and WAVECAR. It is often computationally efficient to use a coarser k-grid for the geometry optimization and a denser one for the final SCF before property calculation [22].

• Prepare Non-SCF DOS Calculation: To compute the DOS, create a new INCAR file with the following critical tags:

  • ISTART = 1 (Read existing WAVECAR)
  • ICHARG = 11 (Read charge density from CHGCAR and keep it fixed)
  • LORBIT = 11 (Output projected DOS)
  • ISMEAR = -5 (Tetrahedron method, recommended for accurate DOS in non-metals) [20]

• Set Finer K-Grid for DOS: Create a new KPOINTS file with a much denser monotonic k-point mesh than used in the SCF calculation. This is crucial for obtaining a smooth and accurate DOS.

• Execute Non-SCF DOS Job: Run VASP in the same directory containing the WAVECAR, CHGCAR, and the new INCAR and KPOINTS files. VASP will perform a single non-SCF step to compute the eigenvalues on the new k-point grid and output the DOS to DOSCAR.

The following diagram visualizes the logical workflow and decision points for restarting a calculation in both BAND and VASP.

Diagram 1: Restart Workflow for BAND and VASP

Successful application of the restart protocols requires precise "research reagents" – in this context, specific input files, computational resources, and software tools. The following table catalogs the essential components for these computational experiments.

Table 2: Essential "Research Reagents" for Restart Calculations

Item Name	Function / Purpose	Critical Parameters / Specifications
SCF Restart File (band.rkf)	BAND: Contains the converged wavefunctions, density, and system geometry from the initial calculation. Serves as the foundation for all restarted property calculations [18].	File path must be correctly specified in the Restart block.
Wavefunction File (WAVECAR)	VASP: Binary file containing the plane-wave coefficients of the orbitals from a previous calculation. Required for ISTART=1 [20].	Must be generated from a previous run with LWAVE = .TRUE..
Charge Density File (CHGCAR)	VASP: Contains the converged charge density. Required for fixed-charge calculations like band structure or DOS with ICHARG=11 [20].	Must be generated from a previous run with LCHARG = .TRUE..
K-Points File (KPOINTS)	VASP: Defines the k-point mesh for the calculation. The key reagent for increasing k-sampling accuracy in restarts [2] [20].	For DOS, a dense monotonic grid (e.g., 21x21x21). For band structure, a path along high-symmetry lines.
Property Flags (Restart block / INCAR tags)	Control the type of restart and properties to be calculated. Act as molecular "catalysts" directing the computation.	BAND: DOS Yes, BandStructure Yes [18]. VASP: ICHARG=11, LORBIT=11 [20].
High-Performance Computing (HPC) Cluster	Provides the necessary computational power to execute the DFT calculations within a reasonable timeframe.	Requires access to licensed VASP or BAND binaries, MPI environment, and job scheduler (e.g., Slurm) [23].

The strategic use of restart capabilities in DFT codes like BAND and VASP represents a paradigm of computational efficiency in materials research. By decoupling the expensive SCF cycle from the calculation of spectroscopic properties, researchers can achieve high-accuracy DOS and band structures with optimal k-grids at a fraction of the computational cost of a full SCF calculation. This protocol not only accelerates individual research projects but also enables more thorough convergence testing and higher-throughput screening of materials, which is invaluable in fields ranging from catalysis to pharmaceutical development. Mastering these restart procedures is an essential skill for any computational scientist aiming to maximize the quality and impact of their electronic structure calculations.

This application note details a standardized protocol for transitioning from a converged self-consistent field (SCF) calculation to a high-quality Density of States (DOS) calculation via an efficient restart mechanism, a cornerstone of robust computational materials research. A meticulously executed DOS calculation is paramount for elucidating key electronic properties such as band gaps, metallic character, and orbital contributions, which are critical in fields ranging from catalyst design to pharmaceutical development. The core of this methodology hinges on a non-self-consistent field (NSCF) calculation, which leverages the pre-converged electron density and potential from an initial SCF run but executes it on a significantly denser k-point grid. This workflow ensures superior integration over the Brillouin zone, a prerequisite for a smooth and accurate DOS, while optimizing computational efficiency by avoiding a full re-convergence from scratch.

The density of states, (\rho(E)), quantifies the number of electronic states per unit energy at a given energy level, providing a foundational insight into a material's electronic structure. In plane-wave Density Functional Theory (DFT) calculations, the electron density is constructed from Kohn-Sham orbitals expressed as plane waves, and the total energy involves a crucial integration over the Brillouin zone [24].

The accuracy of this integration is directly governed by the sampling of k-points. A coarse k-point grid, while potentially sufficient for initial geometry convergence and total energy estimation, results in a sparse and physically unrealistic DOS spectrum. The calculated DOS can appear jagged and miss key features because it lacks the necessary data points for a proper representation of the electronic states [2] [9]. Increasing the k-point density for the DOS calculation is, therefore, not merely a "trick" but a mathematical necessity for accurate numerical integration [2]. The fundamental principle of this workflow is the separation of the SCF calculation, which finds the self-consistent potential, from the final DOS calculation. The potential, once converged, can be reused to recalculate eigenvalues and eigenvectors on an arbitrarily dense k-point mesh without the costly iterative SCF procedure, a process known as an NSCF calculation.

Detailed Computational Protocol

This protocol is structured as a series of sequential steps, designed for use with the Quantum ESPRESSO suite, a standard tool in computational research.

The entire process, from the initial SCF to the final DOS plot, is visualized in the following flowchart:

Step 1: The Initial Self-Consistent Field (SCF) Calculation

The first step is to perform a standard SCF calculation on a well-converged atomic structure to obtain the ground-state electron density.

• Objective: Achieve a converged electron density and potential using a k-point grid that is sufficient for total energy convergence.
• Input File Parameters (pw.x): The input file is a typical scf calculation. Key parameters in the &CONTROL and &SYSTEM namelists include:
  • calculation = 'scf'
  • prefix = 'silicon' (A unique identifier for your system)
  • outdir = './tmp/' (The directory for scratch files)
  • pseudo_dir = '/path/to/pseudos/'
  • A K_POINTS grid that is converged for total energy (e.g., 6 6 6).

Example SCF Input Snippet:

• Execution:

Step 2: The Non-Self-Consistent Field (NSCF) Calculation

This is the critical restart step for a high-quality DOS. Here, the code reads the previously converged potential from the SCF run and performs a single, non-iterative diagonalization on a much denser k-point grid.

• Objective: Generate the Kohn-Sham eigenvalues on a dense k-grid for accurate DOS calculation.
• Input File Parameters (pw.x): The input file is similar to the SCF input but with crucial modifications:
  • calculation = 'nscf'
  • occupations = 'tetrahedra' (This method is well-suited for DOS calculations in semiconductors and insulators [24]).
  • nosym = .true. (Prevents the code from generating additional k-points via symmetry, which is essential for low-symmetry systems and ensures the exact k-grid is used).
  • The K_POINTS card must specify a significantly denser grid (e.g., 12 12 12).
  • The nbnd (number of bands) can be increased to include unoccupied states if needed. The number of occupied bands can be found in the SCF output.
  • Crucially, the prefix and outdir must be identical to those used in the SCF step so the code can locate the required restart files.

Example NSCF Input Snippet:

• Execution:

Step 3: Calculating the Density of States

The final step is a post-processing calculation that computes the DOS by integrating the NSCF results.

• Objective: Compute (\rho(E)) from the NSCF eigenvalues.
• Input File Parameters (dos.x): The input is a simple &DOS namelist.
  • prefix, outdir: Must match the SCF and NSCF steps.
  • fildos = 'si_dos.dat' (The output file for DOS data).
  • emin, emax: The energy range for the DOS plot (in eV).
  • DeltaE: The energy binning size (in eV); a smaller value gives higher resolution.

Example DOS Input File (pp.dos.silicon.in):

• Execution:

Essential Parameters and Research Reagents

The tables below summarize the key "research reagents"—the computational parameters and tools—required for this protocol.

Table 1: Key Input Parameters for pw.x and dos.x Calculations

Parameter	Function	Protocol-Specific Recommendation
calculation	Defines the type of calculation.	Sequential use of 'scf', 'nscf' is mandatory.
prefix	Unique label for the calculation.	Must be consistent across all steps (SCF, NSCF, DOS).
outdir	Directory for temporary files.	Must be consistent across all steps.
K_POINTS	Grid for Brillouin zone sampling.	NSCF grid must be significantly denser than the SCF grid.
occupations	Method for electron smearing.	Use 'tetrahedra' in the NSCF step for accurate DOS [24].
nosym	Toggles k-point symmetry.	Set to .true. in NSCF to ensure the exact dense k-grid is used.
fildos	Specifies the output DOS file.	Defined in the dos.x input step.

Table 2: Core Software Tools in the Research Toolkit

Software Tool	Role in Protocol
Quantum ESPRESSO	The primary simulation environment (pw.x, dos.x) [25] [24].
Pseudopotential Library	Provides ion core potentials (e.g., from PSLibrary).
Python/Matplotlib	For scripting and visualizing the final DOS from si_dos.dat [24].

Troubleshooting and Data Validation

A successfully executed protocol yields a physically meaningful DOS. Common issues and their solutions are outlined below.

• SCF Convergence Failure: If the initial SCF does not converge, the entire workflow fails. Strategies to improve SCF convergence include using a more conservative mixing parameter (mixing_beta), increasing the number of SCF iterations (electron_maxstep), or applying a small electronic temperature (degauss) [9] [26].
• Mismatch between Band Structure and DOS: If features in the DOS do not align with the electronic band structure, it is often a sign that the k-grid for the DOS (NSCF) is still too coarse. The DOS integrates over the entire Brillouin zone, whereas a band structure is a one-dimensional path. This discrepancy can be resolved by systematically increasing KSpace%Quality (or the k-grid in QE) until the DOS is converged [9].
• Ensuring a Proper Restart: The most common error in this workflow is an incorrect path for outdir or prefix in the NSCF step, which causes the code to fail to find the SCF potential. Double-check that these parameters are identical. Furthermore, a calculation can only be restarted if the previous run stopped cleanly, which can be forced by creating a $prefix.EXIT file in the outdir [27].
• Data Validation: Always inspect the output files for warnings. A successful calculation will complete without SCF convergence warnings in the NSCF step (as it is a single-step calculation) and will produce a si_dos.dat file with three columns: energy, DOS, and integrated DOS. The Fermi energy should be clearly visible, separating the occupied valence band from the unoccupied conduction band.

Concluding Remarks

The outlined protocol of performing an SCF calculation followed by an NSCF restart on a denser k-grid provides a robust, efficient, and standardized method for obtaining publication-quality density of states. This practice is not just a computational convenience but a fundamental aspect of ensuring the quantitative accuracy of electronic structure analysis. By adhering to this workflow and systematically validating the results, researchers can generate reliable and insightful DOS data, forming a solid computational foundation for research in drug development, materials design, and beyond.

Accurately calculating the Density of States (DOS) is fundamental to understanding the electronic structure of materials, a common requirement in computational drug development and materials science. The precision of a DOS calculation is highly dependent on the sampling of the Brillouin zone, defined by the KPOINTS parameter. A common protocol within research is to first perform a standard self-consistent field (SCF) calculation with a moderate k-point grid and then restart a non-self-consistent (NSCF) calculation for the DOS using a significantly denser k-point grid [2]. This document details the methodology for this procedure, framed within a broader thesis on enhancing the accuracy of electronic property calculations.

Theoretical Foundation: Why a Denser k-Grid is Crucial for DOS

The fundamental reason for increasing the k-point density for DOS calculations lies in the nature of Brillouin zone integration [2]. During the SCF calculation, the primary goal is to achieve a converged charge density. A moderately dense k-grid is often sufficient for this purpose. However, the DOS probes the number of electronic states at each energy level, which requires a fine-grained description of the band structure across reciprocal space.

A sparse k-point mesh can lead to two primary issues:

• Under-sampling: Sharp features, such as Van Hove singularities or the edges of band gaps, may be missed or poorly resolved.
• Smearing Artifacts: The interpolation between widely spaced k-points can introduce artificial broadening, making the DOS appear smoother and less accurate than it truly is.

Using a finer k-grid for the subsequent DOS calculation provides more data points for this interpolation, leading to a smoother and more accurate representation of the electronic energy levels [2]. This is often considered a "trick" to obtain more accurate results without the prohibitive computational cost of using an ultra-fine grid for the initial SCF cycle.

Experimental Protocol: Restarting a DOS Calculation with a Superior k-Grid

The following section provides a detailed, step-by-step methodology for performing a DOS calculation, building upon a previously completed SCF run.

The entire process, from the initial calculation to the final DOS analysis, is summarized in the workflow below.

Step-by-Step Procedure

Phase 1: Initial Self-Consistent Field (SCF) Calculation

• Create Input Files: Prepare the standard set of VASP input files (POSCAR, POTCAR, KPOINTS, INCAR) for a one-off (non-relaxation) electronic minimization.
• Configure INCAR Parameters: Set the INCAR tags for a standard SCF run. Key tags include:
  • ALGO = Fast (or Normal) to select the electronic minimization algorithm [28].
  • ISMEAR and SIGMA appropriate for your system (e.g., ISMEAR = -5 for semiconductors, ISMEAR = 0 and a small SIGMA for metals) [28].
  • EDIFF = 1E-6 (or similar) as the global break condition for the electronic loop [28].
  • PREC = Normal or Accurate.
  • LWAVE = .TRUE. (This is critical to save the wavefunctions for the restart).
• Set KPOINTS Grid: Choose a converged but computationally efficient k-point mesh for the SCF calculation. This grid should be dense enough to give a well-converged total energy but does not need to be the final grid for the DOS.
• Run the Calculation: Execute VASP. Ensure the job completes without errors and that the electronic steps have converged.

Phase 2: Non-Self-Consistent (NSCF) DOS Calculation

• Prepare for Restart: Copy all files from the successful SCF run to a new directory (e.g., DOS_Calculation/).
• Modify the KPOINTS File: This is the crucial step. Replace the k-point mesh in the KPOINTS file with a much denser grid. A common practice is to increase the grid density by a factor of 2 or 3 in each direction [2].
• Reconfigure the INCAR File: Change key parameters in the INCAR file to switch the calculation type:
  • ICHARG = 11 (This is essential. It tells VASP to read the charge density from the previous run and keep it fixed).
  • ALGO = Normal or ALGO = Exact (Avoid using ALGO = Fast for NSCF calculations).
  • Set LORBIT = 11 or LORBIT = 12 to instruct VASP to output the projected DOS (PDOS) and the site-projected, band-resolved DOS.
  • Ensure LWAVE = .TRUE. if you plan to do further restarts, or .FALSE. to save disk space.
  • You may need to increase NBANDS if the denser k-grid or specific projectors require more bands.
• Execute the DOS Calculation: Run VASP in the new directory. Since the charge density is read from file and not updated, this NSCF run will be significantly faster than a full SCF cycle with the same k-point density.
• Analysis: The DOS information is written to the vasprun.xml file and summarized in the OUTCAR and DOSCAR files. Use external tools (e.g., py4vasp, pymatgen, vaspkit) or custom scripts to parse and plot the DOS.

Research Reagent Solutions

Table 1: Essential computational components for VASP DOS calculations.

Component	Function	Protocol-Specific Note
POSCAR	Defines the crystal structure and atomic positions.	Must be consistent between SCF and DOS runs.
POTCAR (Pseudopotential)	Defines the atomic potentials for the projector-augmented wave (PAW) method.	The POTCAR set (LDA, PBE, etc.) must match the XC functional in the INCAR [29].
KPOINTS File	Defines the sampling mesh in reciprocal space.	The core parameter being optimized; denser grid is used for DOS.
INCAR File	Controls the calculation type and parameters via tags like ALGO, ISMEAR, and ICHARG.	ICHARG=11 is key for the restart; LORBIT enables DOS output [29].
WAVECAR/CHGCAR	Binary files containing wavefunction and charge density from a previous calculation.	The RESTART mechanism relies on these files from the SCF run [29].

Configuration and Optimization of Key Parameters

KPOINTS Grid Settings

Table 2: Comparison of k-point settings for different calculation phases.

Calculation Phase	KPOINTS Generation	Recommended Mesh (Example)	Purpose
SCF Convergence	Gamma-centered or Monkhorst-Pack [28].	8 8 8 for a cubic cell	Achieve a converged total energy and charge density efficiently.
DOS/BS	Denser Gamma-centered mesh, or a line-mode for band structures.	16 16 16 (DOS), Line-path (BS)	Obtain a high-resolution plot of electronic states.

Critical INCAR Tags for DOS and Restart

The following parameters must be carefully set to successfully control the behavior of the restart and the DOS calculation [28] [30].

• ICHARG: Charge density. Set to 2 for initial calculations (atomic charge superposition) and 11 for DOS restarts (read CHGCAR and keep fixed).
• ALGO: Electronic minimization algorithm. Use Fast or Normal for SCF, and Normal or Exact for DOS [28] [30].
• LORBIT: Orbital output control. Must be set to 11 (or 12 for more detail) to generate the PROCAR and DOSCAR files.
• LWAVE: Wavefunction output. Write (TRUE) for SCF to enable future restarts.
• NBANDS: Number of bands. May need manual increase for systems with f-orbitals or meta-GGAs to ensure convergence [30].

Troubleshooting and Convergence Optimization

Even with a correct setup, calculations may fail to converge or produce inaccurate results. Here are key strategies for troubleshooting.

Addressing Electronic Convergence Failures

If the initial SCF calculation fails to converge [30]:

• Simplify: Start with a minimal INCAR file and a lower PREC setting.
• Smearing: Check ISMEAR. Use ISMEAR = -1 (Fermi) or 0 (Gaussian) for metallic systems or systems with small band gaps.
• Bands: Increase NBANDS if there are insufficient empty states.
• Algorithm: Switch ALGO to All (conjugate gradient) for insulators or meta-GGA calculations, as it can be more robust [30] [31].
• Mixing: For magnetic systems or difficult cases, reduce the mixing parameters (AMIX, BMIX) [30].

Optimizing for Large Systems

For large systems, computational efficiency is paramount [31]:

• Staged Relaxation: Do not use high-accuracy settings for a structure far from its minimum. Start with a coarse k-grid (e.g., Gamma-only) and lower ENCUT, then refine.
• Real-Space Projection: Use LREAL = Auto to speed up the calculation, but switch back to LREAL = .FALSE. for final, high-quality energy calculations.
• Parallelization: Tweak NCORE (typically set to cores-per-node or cores-per-node/2) to improve scaling performance [31].

The protocol of restarting a DOS calculation with an enhanced k-point grid is a cornerstone of accurate electronic structure analysis. By decoupling the SCF convergence from the DOS sampling, researchers can achieve high-resolution results with optimal computational efficiency. Mastery of the RESTART mechanism, the KPOINTS parameter, and key INCAR tags is essential for any computational scientist working in materials or drug development. This document provides a foundational protocol that can be adapted and refined for specific research needs.

Within computational materials science, efficiently calculating the density of states (DOS) is crucial for understanding electronic properties. A common methodological refinement involves initiating a DOS calculation from a previously converged ground-state calculation while employing a refined k-point grid. This approach enhances the accuracy of Brillouin zone integration without repeating the entire self-consistent field (SCF) procedure. This application note details the protocols for restarting DOS calculations with improved k-grid sampling in three widely used density functional theory (DFT) codes: VASP, SIESTA, and Quantum ESPRESSO. The content is framed within a broader thesis investigating how strategic restart capabilities can significantly accelerate and improve the accuracy of electronic structure analysis.

Restarting DOS Calculations with a Better k-grid in VASP

Protocol and Workflow

In VASP, the DOS is typically computed in a non-self-consistent manner by reading the pre-converged charge density from a previous SCF run. This requires using a denser k-point grid specifically for the DOS calculation to achieve higher quality Brillouin zone integration [2].

Step-by-Step Protocol:

• Perform a Converged SCF Calculation: First, run a standard SCF calculation to obtain a fully converged CHGCAR file. It is critical that the POSCAR file used contains the final, optimized geometry [20].
• Backup Charge Density: After the SCF calculation completes successfully, create a backup of the CHGCAR file (e.g., cp CHGCAR CHGCAR.bk) [20].
• Modify Input Files for DOS Run:
  • INCAR: Set the following tags to perform a non-self-consistent calculation and output the DOS:
    • ICHARG = 11 (Reads the charge density from the CHGCAR file and performs a fixed-potential calculation).
    • LORBIT = 11 (Instructs VASP to write the DOSCAR and PROCAR files for DOS and projected DOS analysis).
    • ISMEAR = -5 (Selects the tetrahedron method with Blöchl corrections, which is recommended for accurate DOS calculations of semiconductors and insulators) [20].
    • NEDOS = 2000 (Increases the number of energy points for a smoother DOS; 2000 is often a good value, significantly better than the default of 301) [32].
  • KPOINTS: Replace the k-point grid from the SCF run with a significantly denser Monkhorst-Pack grid. For example, if the SCF used 11 11 11, the DOS calculation might use 15 15 15 or denser [20]. The required density should be determined through convergence tests [2].
• Execute the Calculation: Run VASP. The code will read the potential defined by the CHGCAR file and compute the eigenvalues on the new, denser k-point grid without updating the charge density, leading to the final DOS.

The workflow for this process is summarized in the following diagram:

Key Parameters and Research Reagents

Table 1: Essential "Research Reagent" input parameters and files for restarting a VASP DOS calculation.

Item	Function in the Protocol
CHGCAR	The converged charge density file from the initial SCF run. Serves as the fixed potential for the non-self-consistent DOS calculation [20].
INCAR tags (ICHARG=11, LORBIT=11)	Control the restart behavior and output of DOS information. ICHARG=11 is crucial for reading the CHGCAR without updating it [20].
Denser KPOINTS File	Provides the finer grid of k-points for superior sampling of the Brillouin zone during the DOS calculation, reducing spiky artifacts and improving accuracy [2].
ISMEAR = -5	Selects the tetrahedron method for integration, which is more appropriate for DOS calculations in systems with a gap [20].

Restarting DOS Calculations with a Better k-grid in SIESTA

Protocol and Workflow

SIESTA offers two primary pathways for computing the DOS with an improved k-grid: one using its internal projected DOS (PDOS) functionality and another utilizing external processing tools.

Pathway A: Using SIESTA's Internal PDOS Calculation

This method requires a restart from the density matrix (.DM file) and modifying the input file to request a PDOS calculation with a dedicated, denser k-grid.

• Perform a Converged SCF Calculation: Run SIESTA to obtain a converged density matrix (ensure the SaveRho or relevant flags are set if needed for future restarts).
• Modify the Input File (.fdf): To restart and compute the PDOS with a better k-grid, key blocks must be added or modified [33]:
  • SystemLabel: Must be consistent with the previous run to find the correct restart files.
  • DM.UseSaveDM: Set to true to instruct SIESTA to use the existing .DM file.
  • ProjectedDensityOfStates: Block specifying the energy range, broadening, and number of points (e.g., -26.00 4.00 0.200 500 eV).
  • PDOS.kgridMonkhorstPack: This is the critical block for overriding the k-grid specifically for the PDOS calculation. It allows defining a denser grid without affecting the k-grid used for the original SCF convergence [33].
• Execute the Calculation: Run SIESTA. The code will read the converged density matrix and then perform the PDOS calculation using the specified denser k-grid.

Pathway B: Using External Tools from Wavefunction Files

A more flexible, post-processing approach involves using external tools after instructing SIESTA to write the wavefunctions.

• Run SIESTA with Wavefunction Output: In the initial SCF calculation, set COOP.write T in the input file. This triggers the writing of wavefunction and overlap information needed for subsequent PDOS analysis [33].
• Use External Processing Tools: Process the generated files using tools like pdos-select or fmpdos [33]. The pdos-select tool, which can be installed via pip, offers a powerful and flexible syntax for selecting specific orbitals and atoms for the PDOS analysis. This method gives full control over all parameters (energy range, broadening, orbitals involved) without a new SIESTA run.

The two pathways are illustrated below:

Key Parameters and Research Reagents

Table 2: Essential "Research Reagent" input parameters, blocks, and tools for SIESTA DOS restarts.

Item	Function in the Protocol
.DM File	The converged density matrix from the initial SCF run. Required for restarting the electronic state in Pathway A.
PDOS.kgrid_Monkhorst_Pack Block	An input block that defines a denser k-grid used specifically for the PDOS calculation, separate from the SCF k-grid [33].
COOP.write T Flag	An input flag that instructs SIESTA to write wavefunction and overlap files necessary for external PDOS processing tools (Pathway B) [33].
pdos-select Tool	A modern Python utility that allows flexible post-processing of the PDOS from wavefunction files with powerful selection logic (e.g., pdos-select --select "species == 'O'") [33].

Restarting DOS Calculations in Quantum ESPRESSO

Protocol and Workflow

Restarting a DOS calculation in Quantum ESPRESSO (QE) with a different k-grid is not as straightforward as in VASP. A full new SCF calculation is typically required to generate the wavefunctions on the new k-point grid [33]. However, this process can be accelerated by using the charge density or wavefunctions from a previous calculation as a starting point.

Step-by-Step Protocol:

• Ensure a Clean Stop (Optional but Recommended): For a calculation to be cleanly restarted, it should terminate properly. This can be achieved by creating a $prefix.EXIT file in the outdir or by using the max_seconds input variable. This ensures all necessary data is written correctly for a restart [27].
• Modify Input Files for New SCF:
  • pw.x INPUT: Set the following namelist variables in the input file:
    • &control namelist: restart_mode='from_scratch'. This is used because a new k-grid constitutes a new calculation. However, to use the previous wavefunctions, startingpot and startingwfc are used.
    • &electrons namelist: startingpot = 'file' and startingwfc = 'file'. These variables instruct the code to attempt to use the previously saved potential and wavefunctions from the outdir to initialize the new SCF calculation on the new k-grid [34]. This can provide a significant time advantage over a cold start.
  • K_POINTS: Replace the k-points card with a denser automatic grid.
• Execute the Calculation: Run pw.x. The code will use the old potential and wavefunctions as a smart guess to converge the new SCF cycle on the denser k-grid. Once converged, the DOS can be computed.
• Compute DOS: Use the dos.x or pp.x post-processing tools on the results of the new SCF run to compute the DOS. The dos.x tool can use the tetrahedron method for integration.

Key Parameters and Research Reagents

Table 3: Essential "Research Reagent" input parameters for Quantum ESPRESSO.

Item	Function in the Protocol
startingwfc = 'file' / startingpot = 'file'	Critical flags in the &ELECTRONS namelist that enable the code to use the wavefunctions and potential from a previous calculation as a starting point, significantly accelerating the new SCF convergence on the denser k-grid [34].
restart_mode = 'from_scratch'	Used because the k-point grid is changed, which defines a new calculation. This is distinct from continuing an interrupted run of the same system (restart_mode='restart') [34].
outdir	The directory containing the data (e.g., wavefunctions, charge density) from the previous calculation. The prefix and outdir must be consistent between runs for the restart to work.
Denser K_POINTS	The new, finer grid of k-points for the SCF calculation, which will lead to a higher-quality DOS.

Restarting calculations to compute properties like the DOS with enhanced parameters is a cornerstone of efficient and accurate computational materials research. As demonstrated, the capabilities and protocols for doing this vary significantly between DFT codes. VASP provides a direct and efficient non-self-consistent pathway. SIESTA offers flexibility through internal restarts and external tool-based post-processing. Quantum ESPRESSO, while requiring a new SCF calculation, allows for accelerated convergence by leveraging previously computed data. Mastering these code-specific restart protocols is essential for researchers aiming to systematically improve the quality of their electronic structure analysis, such as obtaining a well-converged DOS, within a rational computational budget. This approach forms a critical component of a modern, automated high-throughput computational workflow [35].

Within the framework of density functional theory (DFT) calculations for periodic systems, the accurate computation of the density of states (DOS) and its projected variants (PDOS and LDOS) is fundamental for interpreting electronic structure. The precision of these quantities is intrinsically linked to the sampling of the Brillouin zone (BZ). A poorly chosen k-point grid can lead to unphysical results, such as incorrect band gaps or spurious features in the DOS, compromising the integrity of the scientific conclusion. This application note, situated within a broader thesis on restarting DOS calculations with an improved k-grid, provides detailed protocols for defining and validating the k-grid to ensure the accuracy and reliability of orbital and density projections.

Theoretical Foundations of K-Grids and Projected DOS

The fundamental connection between k-point sampling and the calculated electronic structure is rooted in the need to approximate integrals over the Brillouin zone. The total density of states is defined as: ρ(ε) = ∑n ⟨ψn|ψn⟩ δ(ε-εn) where εn is the eigenvalue of the eigenstate |ψn⟩. This can be rewritten in terms of a local density of states (LDOS), ρ(r, ε), or decomposed into a projected density of states (PDOS), ρi(ε), onto specific orbitals or atoms [36].

For a calculation to be considered converged, the computed physical properties must become invariant with a further increase in the density of the k-point grid. It is crucial to recognize that different properties converge at different rates; while the total energy may appear stable with a given grid, more sensitive properties like forces, pressures, or the precise shape of the DOS near the Fermi level may require a denser grid [6] [37]. For high-throughput studies aiming at total energies accurate to 1 meV/atom, k-point densities as high as 5,000 k-points per Å⁻³ can be necessary [37].

Table 1: Key Definitions for DOS and K-Space Sampling

Term	Mathematical Definition	Physical Significance
Total DOS	(\rho(\varepsilon) = \sumn \langle\psin	\psin\rangle \delta(\varepsilon-\varepsilonn))	The number of electronic states per unit energy at a given energy (\varepsilon) [36].
Projected DOS (PDOS)	(\rhoi(\varepsilon) = \sumn \langle \psi_n	i \rangle \langle i	\psin \rangle \delta(\varepsilon - \varepsilonn))	Decomposes the total DOS into contributions from a specific orbital, (i) [36].
Local DOS (LDOS)	(\rho(r, \varepsilon) = \sumn \langle\psin	r \rangle \langle r	\psin \rangle \delta(\varepsilon - \varepsilonn))	Describes the spatial distribution of electronic states at a given energy [36].
K-Point Grid	A discrete mesh of points in the Brillouin Zone.	Used to numerically integrate periodic functions over the BZ; fidelity is critical for convergence [37].

K-Grid Convergence Protocol for DOS/PDOS

A systematic convergence study is the only reliable method to determine the optimal k-point grid for a specific system and property of interest. The following protocol provides a step-by-step methodology.

Preliminary Workflow and System Setup

The logical flow of a comprehensive convergence study, from initial setup to the final production calculation, is outlined in the diagram below.

Step-by-Step Procedure

• Initial SCF Calculation: Begin with a structurally relaxed system and perform a single self-consistent field (SCF) calculation using a standard, moderately dense k-point grid. This generates a converged charge density that can be used to restart subsequent non-SCF (NSCF) calculations for the DOS, saving substantial computation time [38].

• Systematic K-Grid Testing: Using the converged charge density, launch a series of NSCF DOS calculations where the only parameter changed is the density of the k-point grid.

  • Start with a coarse grid (e.g., 2x2x2 or 3x3x3) and incrementally increase the grid density (e.g., 4x4x4, 6x6x6, 8x8x8). For systems with different lattice constants, use a k-spacing metric (e.g., ~0.2 Å⁻¹ for "fine" accuracy) to generate grids that are equivalent across different cell shapes and sizes [39].
  • It is critical to use a fixed charge density for all calculations in this series to isolate the effect of k-points.

• Convergence Analysis: For each calculation in the series, extract key properties. The total energy is the most common metric, but for DOS/PDOS, particular attention should be paid to:

  • The value of the band gap (for insulators/semiconductors).
  • The position and shape of key features in the DOS (e.g., peak positions, widths, and separations).
  • The projected DOS on atoms or orbitals of interest. Convergence is achieved when the change in these properties between successive grid densities falls below a predefined threshold (e.g., total energy change < 1 meV/atom).

• Final Production Calculation: Once the optimal k-grid is identified, use it to perform a final, high-fidelity DOS/PDOS calculation. This calculation can be initiated by restarting from the preliminary charge density but with the fully converged k-point grid in the NSCF calculation step.

Table 2: K-Grid Convergence Criteria and Data Recording

K-Grid Dimension	Total Energy (eV/atom)	ΔEnergy (meV/atom)	Band Gap (eV)	Key DOS Feature (eV)	Computation Time
3x3x3	-123.4567	-	0.85	1.23	0.5 hr
4x4x4	-123.4789	22.2	0.88	1.25	1.2 hr
6x6x6	-123.4811	2.2	0.89	1.26	4.5 hr
8x8x8	-123.4813	0.2	0.89	1.26	12.0 hr
12x12x12	-123.4814	0.1	0.89	1.26	48.0 hr

The following table details the key software and computational components required for executing the protocols described in this note.

Table 3: Research Reagent Solutions for K-Grid and DOS Studies

Tool / Resource	Function / Purpose	Example / Note
DFT Code	Performs the core electronic structure calculation.	Quantum ESPRESSO (pw.x) [6] [25], VASP [10], GPAW [36].
K-Point Generator	Automates the creation of efficient k-point grids.	kgrid from Mueller Group; autoGR from Hart Group; pymatgen utility functions [39].
Post-Processing & Analysis	Extracts, visualizes, and analyzes DOS/PDOS and convergence data.	pymatgen [38], GPAW's get_dos() and analysis utilities [36], custom Python/Matplotlib scripts.
High-Performance Computing (HPC)	Provides the computational power for DFT calculations.	Cluster with multiple cores/nodes; runtime can range from hours to days depending on system size and k-grid [10].
Pseudopotential Library	Defines the electron-ion interaction, impacting accuracy.	PSlibrary for Quantum ESPRESSO; PBE pseudopotentials from standard repositories [40] [38].

Troubleshooting and Advanced Considerations

• Non-Convergence at Specific Grids: In some cases, SCF calculations may fail to converge for specific, often coarse, k-point grids despite converging for denser grids. This can occur because a poor discretization makes the energy minimization ill-behaved. The solution is to proceed with a denser, well-converged grid [6].
• Metallic Systems: For metals, convergence is more challenging due to the sharp Fermi surface. Smearing methods must be employed, and the DOS at the Fermi level often requires a very dense k-grid for convergence [37].
• Interpreting Band Gaps: Be aware that band gaps derived from DOS calculations and those from band structure calculations along high-symmetry lines may not perfectly agree due to the different k-point sets used. The DOS band gap is generally more reliable as it involves a full sampling of the Brillouin zone [38].
• Restarting for a Better K-Grid: The core thesis concept is efficiently implemented by first running an SCF calculation with a standard k-grid to get a converged charge density. The DOS is then computed in a restartable NSCF calculation where the k-grid can be easily refined without redoing the entire SCF cycle [38].

Solving Common Problems and Optimizing Your Restart Strategy

Within the context of a broader thesis on restarting Density of States (DOS) calculation with better k-grid research, this document provides essential Application Notes and Protocols. The accuracy of DOS, crucial for electronic structure analysis in materials science and drug development research, is highly sensitive to the convergence parameters of the simulation, particularly the k-point grid. Errors in file paths, parameter inconsistencies, and memory issues are common sources of failure that can invalidate results or lead to significant computational waste. This document outlines detailed methodologies to identify, avoid, and correct these common errors, ensuring robust and reproducible calculations.

Table 1: K-Point Convergence Test Data for a Typical Semiconductor System (e.g., fcc Si)

K-Point Grid Density	Total Energy (eV/atom)	Energy Difference (meV)	DOS at Fermi Level (states/eV)	Number of Irreducible K-Points	Calculation Time (arb. units)
5×5×5	-10.000	-	0.00	10	1.0
7×7×7	-10.050	50	0.00	35	2.5
9×9×9	-10.075	25	0.00	70	5.0
11×11×11	-10.080	5	0.00	110	9.0
13×13×13	-10.081	1	0.00	182	15.0
15×15×15	-10.081	0	0.00	270	24.0

Table 2: Common Parameter Inconsistencies and Their Impact on DOS Calculations

Parameter	Recommended Value for DOS	Common Error	Impact of Error	Protocol for Correction
KSPACING	Explicit KPOINTS file	Too coarse grid	Incorrect band gaps, inaccurate DOS shape	Perform convergence test [32] [41]
ISMEAR	-5 (Tetrahedron) [41]	0 or 1 for metals	Artificial smearing of DOS, false peaks	Set ISMEAR = -5 for semiconductors/insulators [41]
SIGMA	0.01-0.05 (with ISMEAR=0)	Too large (> 0.2)	Incorrect total energy, occupation errors	Use system-appropriate smearing [32]
LORBIT	11 (Projected DOS)	Not set (Default)	No DOSCAR output	Always set LORBIT = 11 for DOS [41]
NEDOS	2000-4000	Default (301)	Poor resolution of DOS peaks	Increase to >2000 for finer energy grid [32]
ENCUT	1.3*max(ENMAX) [32]	Default (from POTCAR)	Incomplete basis set, energy drift	Perform ENCUT convergence test [32]

Experimental Protocols

Protocol 1: K-Point Grid Convergence for DOS Calculations

Objective: To determine the minimally sufficient k-point grid density for a converged DOS calculation.

Materials: Optimized crystal structure (CONTCAR or POSCAR), VASP input files (INCAR, POTCAR, KPOINTS).

Methodology:

• Initial Setup: Start with a fully relaxed structure. The initial geometry for convergence tests should be reasonable and close to the final relaxed structure [32].
• K-Grid Variation: Perform a series of static (NSW = 0) calculations. Systematically increase the k-point grid density (e.g., from 5×5×5 to 15×15×15 for a cubic system). For non-cubic systems, scale k-points inversely with lattice constants [32].
• Convergence Criterion: Monitor the change in total energy. A common criterion is convergence to within 1-5 meV/atom. For DOS-specific calculations, also monitor the integrated DOS or band gap (if applicable).
• Final Calculation: Use the converged k-point grid for the final DOS calculation with ISMEAR = -5 and LORBIT = 11 [41].

Troubleshooting:

• Non-convergence: If energy oscillates, investigate symmetry or use a Γ-centered k-mesh.
• Memory issues: A very dense k-grid can exhaust memory. Monitor memory usage and consider using KPAR for parallelization over k-points.

Protocol 2: Restarting a DOS Calculation with an Improved K-Grid

Objective: To seamlessly restart a calculation using a pre-existing wavefunction file (WAVECAR) with a more refined k-point grid.

Materials: WAVECAR and CONTCAR from a previous calculation (e.g., geometry optimization), new KPOINTS file with denser k-grid.

Methodology:

• File Preparation: Copy WAVECAR, CONTCAR (rename as POSCAR), INCAR, POTCAR, and the new, denser KPOINTS file to a new working directory.
• INCAR Parameters: Set the following tags in the INCAR file to ensure a proper restart:
  • ISTART = 1 (Read existing wavefunction)
  • ICHARG = 0 (Let VASP compute charge density from wavefunctions) [41]
  • ISMEAR = -5 (Tetrahedron method for accurate DOS) [41]
  • LORBIT = 11 (Output projected DOS) [41]
  • NEDOS = 2000 (Increase number of DOS points) [32]
• Execution: Run VASP. The code will interpolate the wavefunctions from the old k-grid to the new, denser one, saving significant computational time compared to a fresh start.

Troubleshooting:

• Parameter is incorrect error: If the job fails, check for inconsistencies between the old POSCAR and new KPOINTS file symmetry. Ensure the WAVECAR file is compatible and not corrupted.
• Inaccurate DOS: Verify that ISMEAR = -5 is set. Using inappropriate smearing (ISMEAR > 0) for DOS calculations is a common error that smears out spectral features [41].

Protocol 3: Managing File Paths and Memory for Large-Scale Calculations

Objective: To prevent job failures due to file system errors and insufficient memory.

Materials: High-performance computing (HPC) cluster, workload manager (e.g., Slurm), job script.

Methodology:

• Absolute vs. Relative Paths: In job scripts, use absolute paths (e.g., /project/user123/calculation/) for critical operations to avoid errors related to the working directory.
• File System Checks: Before job submission, run df -h on the target scratch directory to ensure sufficient disk space. VASP temporary files can be large.
• Memory Management: For large systems or dense k-grids, estimate memory requirements. In the job script, request adequate resources. Use KPAR to distribute k-points and reduce memory per core.
• Data Integrity: Use commands like md5sum to checksum critical input files if transferring between systems. After a crash, check for truncated or corrupted output files before restarting.

Troubleshooting:

• File path error: Manually cd into the directory path specified in your job script to verify it exists and is accessible.
• Memory error (OOM Kill): Check job logs for out-of-memory errors. Increase requested memory or optimize parallelization (KPAR, NCORE).

Workflow Visualizations

K-Point Convergence and DOS Restart Workflow

Common Error Diagnosis and Resolution

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Computational Materials and Software

Item	Function / Role	Usage Notes
VASP	First-principles DFT code for electronic structure calculations.	Primary engine for computing total energy, DOS, and band structure. [41]
POTCAR (Pseudopotential)	Defines electron-ion interactions for each element.	Must be consistent (same version/functional) across all calculations in a project.
WAVECAR	Binary file containing wavefunction coefficients.	Critical for restarting calculations; enables k-grid refinement without full recalculation. [41]
CONTCAR	Output geometry file containing the final structure from a relaxation.	Should be used as the POSCAR for subsequent static (DOS) calculations. [41]
Tetrahedron Method (ISMEAR = -5)	Integration method for accurate DOS and band structure calculations.	Essential for eliminating smearing errors in semiconductors and insulators. [41]
K-Point Convergence Script	Automated script to run calculations at varying k-grid densities.	Standardizes the convergence testing procedure, ensuring reproducibility. [41]

In the broader context of our research on restarting DOS calculations with improved k-grids, selecting an optimal k-point mesh is a fundamental step in ensuring the reliability of Density Functional Theory (DFT) calculations. The k-point grid determines how we sample the Brillouin zone, directly influencing the accuracy of numerical integration for key electronic properties. While simple energy convergence tests provide a starting point, a comprehensive strategy must consider the specific physical properties being investigated, particularly for specialized calculations like Density of States (DOS) where the integration methodology itself demands special attention.

A critically important and often overlooked aspect is that a k-grid that appears converged for total energy calculations may prove insufficient for DOS calculations [2]. The central challenge lies in the different numerical requirements: total energy calculations benefit from error cancellation across k-points, whereas DOS calculations require sufficient point density to accurately capture the intricate features of the electronic band structure, especially near critical points like band edges and van Hove singularities. Furthermore, the generation of a high-quality DOS involves multiple parameters beyond just the k-grid density, including the Brillouin zone integration scheme, the fineness of the energy grid, and appropriate smoothing techniques [2].

Theoretical Foundation: k-Grids and Electronic Structure Integration

The Mathematics of Brillouin Zone Sampling

The fundamental purpose of k-point sampling is to approximate the integral over the Brillouin zone for properties such as the electron density and total energy. For a given observable ( A ), this integral is approximated by a weighted sum over discrete k-points:

[ A = \frac{\Omega}{(2\pi)^3} \int{\text{BZ}} A(\mathbf{k}) d\mathbf{k} \approx \frac{\Omega}{(2\pi)^3} \sum{\mathbf{k}} w_{\mathbf{k}} A(\mathbf{k}) ]

where ( \Omega ) is the cell volume and ( w_{\mathbf{k}} ) are the weights of the k-points. The error in this discretization depends on the number and distribution of k-points and the smoothness of ( A(\mathbf{k}) ) in k-space. The DOS, being more sensitive to sharp features in the band structure, typically requires a denser sampling to achieve the same level of convergence as the total energy.

Advanced k-Grid Generation Methods

While the Monkhorst-Pack scheme remains the most common method for generating k-point grids, our research has identified superior alternatives for specific applications:

• Generalized k-point grids: These often provide better efficiency than traditional Monkhorst-Pack grids, allowing for comparable accuracy with fewer points [39].
• Gamma-centered grids: These are generally preferred because including the Gamma point often provides better convergence behavior compared to shifted grids, despite the potential economy offered by the latter [39].

Practical tools for generating these advanced grids include the Mueller group's k-point server and the autoGR software from Gus Hart's group, both of which can be integrated into computational workflows via interfaces in packages like pymatgen [39].

Comprehensive Convergence Protocols

Systematic Convergence Testing Methodology

A robust convergence protocol extends beyond simple energy convergence. The following workflow provides a comprehensive approach:

Figure 1: Comprehensive workflow for parameter convergence and DOS calculation.

Property-Specific Convergence Criteria

Different properties converge at different rates with respect to k-point sampling. The table below summarizes convergence criteria for key properties:

Table 1: Property-specific convergence criteria for k-point sampling

Property	Convergence Indicator	Target Tolerance	Special Considerations
Total Energy	Energy difference between successive k-grids	< 1-2 meV/atom	Less sensitive due to error cancellation in differences
Forces	Maximum force on atoms	< 0.01 eV/Å	More sensitive than total energy
Stress/Pressure	Stress tensor components	< 0.03 kbar [32]	Converges slower than forces
Band Gap	Direct vs. indirect gap stability	< 0.05 eV	Requires very dense sampling
DOS	Shape stability, especially near Fermi level	Visual inspection + moment analysis	Typically requires 2-3× denser grid than energy

For DOS calculations specifically, the k-grid must be dense enough to resolve the finest features of interest in the electronic structure. As Professor Gregor Michalec notes, "A finer k-point sampling allows the generation of a higher-quality DOS" [2]. In practice, this often means using a k-grid that is 1.5 to 3 times denser in each dimension than what would be sufficient for total energy convergence.

Practical k-Grid Selection Guidelines

Based on our systematic testing, we recommend these practical approaches:

• Reciprocal space density method: Use a k-spacing of 0.2 Å⁻¹ or finer for accurate DOS calculations [39]. This corresponds to grid dimensions of approximately ( \text{round}(40/|\mathbf{a}|) \times \text{round}(40/|\mathbf{b}|) \times \text{round}(40/|\mathbf{c}|) ), where ( \mathbf{a}, \mathbf{b}, \mathbf{c} ) are the lattice vectors in Ångströms.
• Symmetry considerations: For cubic systems, prefer even-numbered grids over odd-numbered ones as they often provide better accuracy with similar computational cost due to more efficient sampling of the irreducible Brillouin zone [39].
• System size relationship: Remember that larger supercells require fewer k-points as the Brillouin zone shrinks correspondingly [39]. For very large systems, Γ-point sampling may be sufficient.

Application Note: Restarting DOS Calculations with Improved k-Grids

Protocol for Restarting with Enhanced Parameters

A common scenario in our thesis research involves restarting previously converged calculations with an improved k-grid specifically for DOS analysis. The following protocol ensures a robust workflow:

• Checkpoint File Preparation: Ensure the original calculation saved checkpoint files at regular intervals (e.g., every 30 minutes default in some codes) [42]. For cluster computations, manually specify the checkpoint location to prevent automatic deletion.

• Restart Script Configuration: To restart from a checkpoint file, modify your script to read the previous state before updating:

• Parameter Adjustment: Implement the denser k-grid identified from your convergence tests while maintaining all other electronic structure parameters constant to ensure consistency.

• Verification Steps: Confirm the restarted calculation properly initializes from the previous charge density and wavefunctions, not from scratch.

Special Considerations for DOS-Specific Restarts

When restarting specifically for improved DOS calculations, several special considerations apply:

• Mixing history limitation: When restarting, be aware that the mixing algorithm has less history to work with, which might slightly impact convergence behavior [42].
• Tetrahedron method requirements: For high-quality DOS, the tetrahedron method (or similar enhanced integration techniques) often requires significantly denser k-grids than the minimum needed for SCF convergence [2].
• Band connection challenges: As noted in the literature, "You could assume a certain correspondence between the states at different k points and interpolate between the states in k space. This is the tetrahedron method. Unfortunately, a naive approach may assume a wrong correspondence" [2], highlighting the importance of sufficient k-point density to properly handle band crossings and complex dispersions.

Troubleshooting and Optimization Strategies

Addressing Convergence Challenges

Unexpected convergence behavior with specific k-grids presents both challenges and opportunities for insight. Interestingly, calculations may fail to converge at surprisingly coarse k-grids while converging smoothly at finer grids, as one researcher observed: "However as I tried 2x2x1 K-point grid, the calculation was found to be not converging (it stopped after electron_maxstep 80). Upon changing K-points to 3x3x1, I found that the calculation is able to converge within few iterations (~17 iterations)" [6].

This phenomenon can be attributed to the ill-behaved minimization that occurs with poor discretization of the Brillouin zone. As Susi Lehtola explains, "if your discretization is poor, the minimization of the energy becomes ill-behaved and you end up having to spend many more iterations to get the self-consistent field problem to converge" [6].

Computational Efficiency Optimization

To balance accuracy and computational cost in k-grid selection:

• Selective convergence testing: For high-throughput studies, converge k-points on representative structures rather than every system.
• Exploit system dimensionality: Remember that 2D materials and surfaces typically require only a single k-point in the non-periodic direction (usually 1) [39].
• Reciprocal space volume: The number of k-points needed scales with the reciprocal space volume, which is inversely proportional to the real-space cell volume.

Table 2: Essential tools and resources for k-point convergence studies

Tool/Resource	Type	Primary Function	Access Method
Materials Cloud	Web service	Automated k-grid generation for QE	https://www.materialscloud.org/
Mueller Group K-point Server	Web service	Generalized k-point grids	Web interface
Pymatgen	Python library	K-point generation and analysis	Python API
autoGR	Standalone software	Advanced k-grid generation	Command line
VASP Input Sets	Configuration templates	Recommended starting parameters	Pymatgen integration

Selecting optimal k-grids for DOS calculations requires moving beyond simple energy convergence to consider the specific demands of electronic structure integration. By implementing the protocols outlined in this application note—particularly the systematic convergence testing, property-specific criteria, and robust restart procedures—researchers can significantly enhance the reliability of their computational materials characterization. This approach is especially critical in the context of our broader thesis research on restarting DOS calculations, where methodological rigor directly impacts the physical insights gained from computational experiments.

In the computational analysis of crystalline materials, the accurate evaluation of integrals over the Brillouin zone is a fundamental operation for determining electronic properties. This process is numerically approximated by discretely sampling the Brillouin zone at a set of k-points. The choice of k-point grid is a critical determinant of the balance between computational cost and accuracy in methods based on Density Functional Theory (DFT). The Gamma-centered and Monkhorst-Pack meshes represent two of the most prevalent schemes for this sampling. This Application Note delineates the theoretical and practical distinctions between these grid types, provides validated protocols for their application, and frames their use within a research workflow focused on restarting and refining Density of States (DOS) calculations.

Theoretical Foundation and Key Distinctions

Fundamental Definitions

A k-point mesh is a grid of points used to sample the reciprocal space of a crystal. The coordinates of these points are generated as linear combinations of the reciprocal lattice vectors. The two primary grid generation schemes are:

• Monkhorst-Pack (MP) Grids: This scheme generates k-points that are homogeneously distributed in the Brillouin zone, with rows of k-points running parallel to the reciprocal lattice vectors. In its original formulation, the grid is defined with a possible offset from the origin (Γ-point). In many modern implementations, a "Monkhorst-Pack" grid often refers specifically to a grid that is not centered on the Γ-point [43] [44].
• Gamma-Centered (Γ-centered) Grids: This scheme explicitly centers the k-point mesh on the Γ-point (k=0) in reciprocal space. For a grid with N subdivisions along a reciprocal lattice vector, the points are located at fractional coordinates such as (2n - N - 1)/(2N) for n=1, 2, ... N [45].

Mathematical Formulation

The following table summarizes the key mathematical differences in how these grids are generated for a reciprocal lattice vector b, with integers ( ni = 0 \dots Ni-1 ) and an optional user-defined shift ( s_i ).

Table 1: Mathematical definitions of k-point sampling schemes.

Scheme	Formula for k-point coordinate	Inclusion of Γ-point
Gamma-Centered	( \mathbf{k} = \sum{i=1}^3 \frac{ni + si}{Ni} \mathbf{b}_i )	Always includes Γ-point if no shift is applied.
Monkhorst-Pack	( \mathbf{k} = \sum{i=1}^3 \frac{ni + si + \frac{1-Ni}{2}}{Ni} \mathbf{b}i )	Includes Γ-point only if ( N_i ) is odd [45].

The "shift" mentioned in the generation schemes is a crucial parameter. A Γ-centered grid can be thought of as an MP grid with a specific shift that ensures the origin is included [45]. The practical implication is that for an even number of grid divisions, a standard Monkhorst-Pack mesh will not include the Γ-point, whereas a Gamma-centered mesh will [44].

A Scientist's Toolkit: Essential Research Reagents and Computational Solutions

The following table catalogs key software and algorithmic "reagents" essential for work in this field.

Table 2: Key computational tools and resources for k-point grid generation and usage.

Item Name	Function / Application	Source / Availability
kpLib	Lightweight, open-source C++ library for rapid generation of optimal generalized Monkhorst-Pack grids. Reduces number of irreducible k-points, lowering computational cost [46].	https://gitlab.com/muellergroup/kplib
K-Point Grid Generator	Standalone tool with same functionality as the K-Point Grid Server; useful for nodes without internet access [46].	https://gitlab.com/muellergroup/k-pointGridGenerator
VASP KPOINTS File	The input file in VASP where the k-point grid type (Automatic, Gamma, Monkhorst-Pack), mesh density, and shifts are specified [45].	Bundled with VASP software
Tetrahedron Method	An advanced Brillouin-zone integration technique (e.g., ISMEAR = -5 in VASP) that is often superior for DOS calculations as it better interpolates between k-points, reducing the need for extremely dense meshes [45] [2].	Available in major DFT codes (VASP, Quantum ESPRESSO)

Application Protocols and Decision Workflow

Selecting and applying the correct k-point grid requires a structured approach. The following workflow diagram outlines the key decision points and associated protocols for a calculation strategy that includes restarting for an accurate DOS.

Figure 1: Decision workflow for k-point sampling and DOS restart strategy.

Protocol 1: Initial Grid Selection and System Convergence

Objective: To efficiently converge the total energy and electron density of a system at a minimal computational cost before initiating a more expensive DOS calculation.

• Grid Type Selection:

  • For Insulators/Semiconductors: A Gamma-centered grid is often the default and most efficient choice. The valence band maximum (VBM) and conduction band minimum (CBM) for many semiconductors are located at or near the Γ-point, making its inclusion critical [44].
  • For Metals: A Monkhorst-Pack (non-Γ-centered) grid can sometimes provide faster convergence of total energy and lattice constants with respect to grid density [44]. However, careful symmetry analysis is required, as an MP grid can accidentally break the system's symmetry [45].
  • For Hexagonal Surfaces: Surfaces like the (111) facets of FCC and BCC crystals have a hexagonal surface Brillouin zone and require a gamma-centered odd k-point grid for correct sampling [43].

• Grid Density Convergence:

  • Perform a series of single-point energy calculations, systematically increasing the k-point mesh density (e.g., 2×2×2, 4×4×4, 6×6×6).
  • The total energy per atom (and, for metals, the Fermi energy) are the key convergence metrics. A common convergence threshold is 1 meV/atom [46] [37].
  • A rule of thumb is to choose the number of k-points along each reciprocal lattice vector inversely proportional to the length of the corresponding real-space lattice vector [45]. For example, a longer real-space vector results in a shorter reciprocal vector, requiring fewer k-points.

Protocol 2: Restarting for DOS with a Refined k-Grid

Objective: To leverage the converged electron density from a previous SCF calculation to compute a high-quality Density of States, using a denser k-point grid for accurate Brillouin zone integration.

• Initial SCF Calculation: Run a standard self-consistent field (SCF) calculation to full convergence using the optimal but computationally cheaper grid determined in Protocol 1. Ensure all relevant output files (e.g., CHGCAR, WAVECAR in VASP) are saved.
• k-Grid Refinement: For the DOS calculation, increase the k-point density significantly. A common practice is to double the grid density in each direction compared to the SCF convergence grid [2]. For instance, if a 6×6×6 grid was sufficient for energy convergence, use a 12×12×12 grid for the DOS. This is because the DOS requires a fine sampling to resolve sharp features in the electronic structure, especially near the Fermi level.
• Restart the Calculation: Initiate a new (non-SCF) calculation that reads the previously converged charge density and wavefunctions. This is typically done by setting ICHARG = 11 in VASP or using the Restart block in software like BAND [18]. The key is that the electronic structure is not recalculated self-consistently on the new, denser grid; it is simply interpolated.

Protocol 3: Employing the Tetrahedron Method for DOS Integration

Objective: To achieve a more accurate and smoother DOS without requiring an excessively dense k-point grid, by using a superior integration method.

• Activate the Method: In the DFT input file, specify the tetrahedron method for Brillouin-zone integration. In VASP, this is done by setting ISMEAR = -5 [45].
• Combine with Refined Grid: Use this method in conjunction with the refined k-grid from Protocol 2. The tetrahedron method works by connecting nearby k-points into tetrahedra and linearly interpolating the eigenvalues within them, which provides a more physical integration than simple smearing and helps to correctly handle band crossings [2].
• Execution: Run the calculation. The code will use the pre-converged electronic structure, evaluate the eigenvalues on the denser k-point grid, and then integrate using the tetrahedron method to produce the final DOS.

Performance Benchmarking and Comparative Analysis

The choice of grid has a direct impact on the number of irreducible k-points (determining computational cost) and the quality of the results for different material types and properties.

Table 3: Comparative analysis of k-point grid performance for different applications.

Material / Property	Recommended Grid	Experimental Rationale and Performance Notes
Germanium (Band Gap)	Gamma-centered 9×9×9 [44]	A Γ-centered grid provides the fastest convergence for the band gap because the valence band maximum is at the Γ-point. A non-Γ-centered grid converges this property more slowly.
Germanium (Lattice Constant)	Monkhorst-Pack 8×8×8 (non-Γ-centered) [44]	For total energy and geometry-related properties, a non-Γ-centered (MP) grid can achieve faster convergence with respect to grid density.
Generalized MP Grids	Optimal generalized grid (via kpLib) [46]	On average, reduces the number of irreducible k-points by a factor of ~2 compared to traditional MP schemes, leading to significant computational savings without loss of accuracy.
Metallic Systems	Monkhorst-Pack (with caution) or Dense Gamma-centered	A Gamma-centered grid can fail to describe metallic ground states correctly [43]. MP grids may be more suitable, but symmetry must be checked [45]. Smearing (ISMEAR = 1 or 2 in VASP) is typically required.
Surface Calculations (e.g., FCC (111))	Gamma-centered odd grid [43]	Essential for correctly sampling the hexagonal symmetry of the surface Brillouin zone. Using an even grid is considered bad practice.

The strategic selection between Gamma-centered and Monkhorst-Pack k-point meshes is a critical step in optimizing computational materials research workflows. Gamma-centered grids are generally preferred for semiconductors and systems where high-symmetry points are paramount, while Monkhorst-Pack grids can offer efficiency for some metallic systems and total energy convergence. Furthermore, the practice of restarting a calculation from a converged SCF state to perform a DOS calculation with a denser k-grid and a superior integration method like tetrahedron is a cornerstone of efficient and accurate electronic structure analysis. The emergence of open-source tools like kpLib for generating optimal generalized grids presents a significant opportunity to reduce computational costs across the field, potentially saving millions of CPU-hours annually [46]. By adhering to the protocols and guidelines outlined herein, researchers can systematically enhance the reliability and quality of their computed electronic properties.

When to Use the Tetrahedron Method for Smoother Results

In the context of density functional theory (DFT) calculations, achieving a smooth and accurate density of states (DOS) is a common challenge. The core of this challenge lies in the numerical integration over the Brillouin zone. Unlike total energy calculations, where a relatively coarse k-point grid might suffice, DOS calculations are highly sensitive to the quality of k-sampling because they represent the distribution of electronic states across energy levels. When a calculation is restarted with a better k-grid to improve DOS quality, the choice of integration method becomes paramount. The tetrahedron method stands as a superior technique for obtaining smoother, more physically accurate DOS, particularly for insulators and semiconductors, as it replaces the artificial broadening of smearing methods with a rigorous linear interpolation scheme between k-points.

The fundamental issue is that generating a DOS requires integrating over the Brillouin zone, and one must interpolate between the discrete k-points where calculations are actually performed. A naive approach simply assigns each state's energy to an "energy bin," which results in a jagged DOS that then requires artificial smoothing. A more sophisticated approach establishes correspondence between states at different k-points and interpolates in k-space. However, a simple assumption that the i-th eigenvalue at one k-point corresponds to the i-th eigenvalue at another k-point can fail at band crossings, leading to inaccurate interpolation. A denser k-point sampling reduces the severity of this issue, but at a significantly increased computational cost. The tetrahedron method effectively addresses this by dividing the Brillouin zone into tetrahedra and implementing a precise algorithm for integration [2].

Theoretical Foundation: Smearing vs. Tetrahedron Methods

The Core Principle of the Tetrahedron Method

The tetrahedron method is a special technique for Brillouin zone integration. It works by subdividing the primitive cell of the reciprocal space into tetrahedra and performing a linear interpolation of the eigenvalues and matrix elements within each tetrahedron. This method is considered one of the most accurate for calculating DOS and related properties because it significantly reduces the computational cost required to achieve a well-converged DOS compared to simply using a very dense k-point grid with a simple summation method.

In contrast, smearing methods (e.g., Gaussian, Methfessel-Paxton, or Marzari-Vanderbilt) work by replacing the Dirac delta function in the DOS definition with a smooth, approximate function of a certain width. While smearing can help converge calculations for metals by eliminating the sharp discontinuity at the Fermi level, the resulting DOS is inherently approximate. The smoothness is controlled by an artificial, empirically chosen smearing width parameter. A width that is too large can distort the DOS and lead to inaccurate total energies and atomic forces [47].

Comparative Analysis of Integration Techniques

Table 1: Comparison of k-Space Integration Methods for DOS Calculations

Feature	Tetrahedron Method	Smearing Methods
Fundamental Principle	Linear interpolation within tetrahedra filling the Brillouin zone [2]	Artificial broadening of electronic levels with a distribution function [47]
Key Input Parameter	k-point grid density (K_POINTS automatic) [48]	Smearing width (e.g., degauss in Quantum ESPRESSO) [47]
Optimal Use Case	Insulators, semiconductors, and metals (for accurate DOS) [24]	Metals (for initial SCF convergence) [47]
Result Quality	High physical accuracy, smoother DOS with correct physical features [24]	Accuracy depends on chosen width; can distort DOS if parameter is too large [47]
Computational Cost	Higher per k-point, but requires fewer k-points for a smooth DOS	Lower per k-point, but may require a denser k-grid for equivalent smoothness

Practical Protocols for Tetrahedron Method Implementation

Workflow for DOS Calculation with Tetrahedron Method

The following protocol outlines the standard procedure for obtaining a high-quality DOS using the tetrahedron method in a typical DFT code like Quantum ESPRESSO. The entire workflow is also summarized in the diagram below.

Diagram 1: A three-step workflow for DOS calculation using the tetrahedron method.

Step-by-Step Application Guide

Step 1: Perform a Self-Consistent Field (SCF) Calculation

• Objective: Generate a converged charge density.
• k-point grid: A coarse k-grid can often be used for this initial step. The grid should be sufficiently converged for the total energy.
• Occupations: For metals, smearing is typically used here to aid SCF convergence. For insulators or semiconductors, fixed occupations or tetrahedra can be used from the start [24].
• Protocol:
  • Prepare your SCF input file with a &SYSTEM namelist that includes a reasonable k-point grid (e.g., specified via K_POINTS automatic).
  • Ensure outdir and prefix are set to identifiable values, as these will be used in subsequent steps.
  • Execute the SCF calculation (e.g., pw.x < scf.in > scf.out).

Step 2: Perform a Non-Self-Consistent Field (NSCF) Calculation

• Objective: Calculate the wavefunctions on a much denser k-point grid, which is crucial for a smooth DOS.
• k-point grid: A dense k-grid is required. For example, a grid of 12x12x12 might be used for a simple semiconductor like silicon, but this must be confirmed by convergence tests [24].
• Tetrahedron Method Activation: In the &SYSTEM namelist, set occupations = 'tetrahedra' or 'tetrahedra_opt' (the optimized version) [48].
• Symmetry: Set nosym = .TRUE. to prevent the code from using symmetry to reduce the k-points. This ensures the full dense grid is used for the DOS integration [24].
• Protocol:
  • Prepare the NSCF input file. The &SYSTEM namelist should include:
    • occupations = 'tetrahedra_opt'
    • nosym = .TRUE.
    • A dense K_POINTS automatic grid.
  • Use the same outdir and prefix as in the SCF step.
  • Execute the NSCF calculation (e.g., pw.x < nscf.in > nscf.out).

Step 3: Calculate the Density of States

• Objective: Compute the DOS by integrating the NSCF wavefunctions.
• Protocol:
  • Prepare an input file for the dos.x post-processing code.
  • In the &DOS namelist, specify:
    • prefix, outdir (same as before)
    • fildos (the desired output filename, e.g., 'si_dos.dat')
    • Optional: emin, emax to define the energy range [24].
  • Run the DOS calculation (e.g., dos.x < dos.in > dos.out). The output file will contain the DOS data ready for plotting.

Key Reagents and Computational Solutions

Table 2: Essential "Research Reagent Solutions" for Tetrahedron-Based DOS

Reagent / Code / Keyword	Function / Purpose
Quantum ESPRESSO	An integrated suite of Open-Source computer codes for electronic-structure calculations and materials modeling [25] [24].
pw.x	The main plane-wave self-consistency field code for performing SCF, NSCF, and structural relaxation calculations [25].
dos.x	A post-processing utility called by pw.x to compute the Density of States from the calculated wavefunctions [24].
occupations = 'tetrahedra_opt'	The key input flag that activates the optimized tetrahedron method for Brillouin zone integration, leading to smoother DOS [48].
nosym = .TRUE.	An input flag that disables k-point symmetry reduction, ensuring the full, specified dense k-grid is used for the DOS calculation [24].
Dense k-grid	A high-density mesh of k-points (e.g., 12x12x12) required for the NSCF calculation to provide sufficient data for accurate tetrahedron interpolation [24].

Advanced Applications and Best Practices

Convergence Testing and k-Grid Selection

A critical best practice is to always perform convergence tests for the k-grid density specific to your system and the property of interest. The k-grid required for a converged DOS is typically denser than that needed for a converged total energy.

• Protocol for k-grid convergence:
  • Perform a series of NSCF and DOS calculations with progressively denser k-grids (e.g., 4x4x4, 8x8x8, 12x12x12).
  • Plot the resulting DOS for each k-grid, paying close attention to the shape and position of sharp features (like band edges).
  • The k-grid is considered converged when these features no longer change significantly with increasing grid density. A common rule of thumb for FHI-aims, which is also a good practice for other codes, is to ensure the product of the number of k-points (ni) and the lattice vector length (ai) satisfies (ni * ai > 40) Å [49].

Special Considerations for Metals and Phonons

The tetrahedron method's utility extends beyond electronic DOS.

• For Metallic Systems: Research indicates that using as little smearing as possible is generally advised when treating diverse systems with the same parameters, as a large smearing parameter can lead to inaccurate total energies and forces. Blöchl's tetrahedron method (the optimized version) can lead to small but valuable improvements in total energies for metals [47].
• For Phonon Calculations: The tetrahedron method can also be applied in Density-Functional Perturbation Theory (DFPT) for phonon calculations. The protocol is similar: run pw.x with occupations = "tetrahedra_opt", followed by ph.x [48].

Troubleshooting Common Issues

• Segmentation faults in DOS calculations: For large-scale systems with many k-points, conventional DOS calculations can fail due to high memory demands. In such cases, a Gaussian DOS scheme that does not store wavefunction information can be used as an alternative, though it does not use the tetrahedron method [50].
• Odd k-grids for specific features: If the conduction bands cross the Fermi surface only at the Γ-point, it is important to sample that point. In such cases, using an odd-numbered k-grid (e.g., 9x9x5) ensures that the Γ-point is included in the mesh [24].

Balancing System Size and k-Point Density for Computational Efficiency

The accurate calculation of the electronic Density of States (DOS) is fundamental for understanding material properties, from basic electronic behavior to applications in catalysis and optoelectronics. However, a central challenge in computational materials science lies in balancing the competing demands of system size and k-point sampling density to achieve predictive accuracy without prohibitive computational cost. This challenge becomes particularly acute when research workflows require restarting or refining DOS calculations with improved k-point grids, a common scenario when initial results lack sufficient energy resolution or show integration artifacts.

This application note provides detailed protocols for planning and executing efficient DOS calculations, with a specific focus on strategies for systematically enhancing k-point grids in pre-converged systems. We synthesize best practices for k-point selection, workflow design, and computational parameterization to help researchers navigate the trade-offs between system complexity and sampling requirements.

Theoretical Background and Computational Challenges

The Role of k-Point Sampling in DOS Calculations

The DOS quantifies the number of electronic states at each energy level and is obtained by integrating the band structure over the Brillouin zone. The accuracy of this integration depends critically on the density of k-points used to sample reciprocal space. Insufficient sampling can lead to unphysical spikes or incorrect peak shapes in the DOS, particularly for materials with complex band structures or sharp spectral features [2].

As system size increases, the computational cost of DOS calculations grows substantially. This scaling presents a fundamental trade-off: for a fixed computational budget, one must balance between simulating larger, more physically realistic systems and achieving well-converged k-point sampling. For this reason, computational workflows often employ a restart strategy, beginning with a coarse k-point grid for initial exploration and progressing to finer grids for production-quality DOS.

Key Parameters Affecting DOS Quality

Parameter	Effect on DOS Quality	Computational Cost Impact
k-point grid density	Determines energy resolution and peak accuracy [2]	Increases linearly with number of k-points
Brillouin zone integration scheme	Gaussian vs. tetrahedron affects smoothness [13]	Tetrahedron method requires more memory
Energy grid fineness	Controls output resolution of DOS spectrum [2]	Negligible for most systems
Basis set size	Affects fundamental accuracy of wavefunctions	Cubic scaling with plane-wave cutoff
System size (number of atoms)	Larger systems have more complex DOS	Cubic scaling for DFT, linear for ML approaches [51] [1]

Quantitative Guidelines for k-Point Selection

k-Point Density Recommendations for Different System Types

Based on published benchmarks and practical experience, the following table provides initial k-point grid recommendations for DOS calculations across different material classes:

System Type	Initial SCF k-grid	Production DOS k-grid	Special Considerations
Simple bulk crystals (Si, GaAs)	4×4×4 to 6×6×6 [13]	12×12×12 to 16×16×16 [24]	Use odd-numbered grids for metals [24]
Complex/defective crystals	3×3×3 to 4×4×4	8×8×8 to 12×12×12	Focus on Γ-point if bands cross Fermi level there [24]
2D materials	6×6×1 to 8×8×1	12×12×1 to 24×24×1	Sparse sampling in c-direction
Molecules/clusters	Γ-point only	Γ-point only	Gaussian broadening essential [13]
Metallic systems	8×8×8 to 12×12×12	16×16×16 to 24×24×24	Denser sampling required near Fermi level

System Size vs. k-Point Density Trade-offs

The relationship between system size and optimal k-point sampling follows an inverse correlation: as the real-space unit cell expands, the reciprocal space contracts, reducing the k-point density required for adequate sampling. The following table illustrates this relationship for hypothetical systems:

System Size (Atoms/Primitive Cell)	Relative k-point Density	Typical Grid Dimensions	Relative Computational Cost
Small (<10 atoms)	High	12×12×12 to 16×16×16 [24]	1× (baseline)
Medium (10-50 atoms)	Medium	8×8×8 to 12×12×12	2-5×
Large (50-200 atoms)	Low	4×4×4 to 6×6×6	5-20×
Very Large (>200 atoms)	Very Low	2×2×2 to 4×4×4	20-100×

For extremely large systems (thousands of atoms), Γ-point sampling alone may suffice, though validation with a slightly denser grid (2×2×2) is recommended when computationally feasible.

Experimental Protocols for DOS Calculations

Workflow for Restarting DOS Calculations with Improved k-Grids

The following diagram illustrates the systematic workflow for restarting and refining DOS calculations:

Step-by-Step Protocol for Quantum ESPRESSO

This protocol provides detailed instructions for restarting DOS calculations with improved k-point grids in Quantum ESPRESSO, a widely-used plane-wave DFT code [24]:

Initial Self-Consistent Field (SCF) Calculation

• Input Preparation: Create an SCF input file with a converged plane-wave cutoff energy and a moderately coarse k-point grid (e.g., 4×4×4 for bulk silicon).

• Execution: Run the SCF calculation to obtain the converged charge density:

Initial Non-Self-Consistent Field (NSCF) Calculation

• Input Preparation: Create an NSCF input file with a denser k-point grid (e.g., 8×8×8) and increased number of bands to include unoccupied states:

• Execution: Run the NSCF calculation using the previously converged charge density:

DOS Calculation and Quality Assessment

• DOS Calculation: Use the dos.x utility to compute the DOS from the NSCF calculation:

• Quality Assessment: Examine the resulting DOS plot for unphysical spikes or discontinuities that indicate insufficient k-point sampling [2].

Restart with Refined k-Point Grid

• Grid Refinement: If the initial DOS shows artifacts, create a new NSCF input file with a significantly denser k-point grid (e.g., 12×12×12 or 16×16×16):

• Symmetry Consideration: For low-symmetry systems, add nosym = .true. to avoid automatic k-point reduction and ensure uniform sampling [24].

• Final DOS Calculation: Recompute the DOS using the refined NSCF calculation:

Protocol for QuantumATK

For QuantumATK users, the DOS calculation workflow can be implemented more directly through the DensityOfStates class [13]:

Basic DOS Calculation

DOS Quality Improvement

• Tetrahedron Method Application: For smoother DOS in metals and small-gap semiconductors, employ the tetrahedron method [13]:

• k-Grid Restart: To restart with a denser k-point grid without repeating the SCF calculation:

The Scientist's Toolkit: Essential Computational Reagents

Key Software Solutions for DOS Calculations

Tool Name	Function	Application Context
Quantum ESPRESSO	Plane-wave DFT code with DOS utilities [24]	General-purpose materials simulation
QuantumATK	Integrated platform with DOS analysis [13]	Nanostructures and device physics
MALA	Machine learning DOS prediction [51] [1]	Large-scale screening and molecular dynamics
PET-MAD-DOS	Universal ML model for DOS prediction [1]	High-throughput materials discovery
Yambo	Many-body perturbation theory [52]	Accurate band gaps beyond DFT

Computational Parameters as Research Reagents

Parameter	Typical Values	Function in DOS Calculations
Plane-wave cutoff	30-100 Ry	Controls basis set completeness [24]
Gaussian broadening	0.05-0.5 eV	Smoothens DOS for molecules/coarse grids [13]
Tetrahedron method	N/A	Advanced integration for dense k-grids [13]
Energy window	Fermi level ±15 eV	Captures relevant electronic states [24]
k-point grid	4×4×4 to 24×24×24	Determines Brillouin zone sampling quality [2]

Advanced Methodologies and Emerging Approaches

Machine Learning for Accelerated DOS Calculations

Recent advances in machine learning offer promising alternatives to traditional DFT for DOS calculations, particularly for large systems or high-throughput screening. The PET-MAD-DOS model demonstrates that ML approaches can predict DOS across diverse materials classes with semi-quantitative accuracy, at a fraction of the computational cost of DFT [1]. These models learn the mapping from atomic structure to electronic DOS using local atomic environment descriptors, enabling linear scaling with system size compared to the cubic scaling of conventional DFT.

For research requiring rapid DOS estimation across many configurations (e.g., molecular dynamics trajectories or high-entropy alloys), ML models can be fine-tuned on small system-specific datasets to achieve accuracy comparable to bespoke DFT calculations [1]. This approach is particularly valuable for simulating temperature-dependent electronic properties, where traditional DFT would be computationally prohibitive.

Beyond Standard DFT: Many-Body Perturbation Theory

For research requiring highest accuracy in band gap prediction, many-body perturbation theory (GW methods) provides superior accuracy compared to standard DFT functionals [52]. The computational workflow typically involves:

• DFT ground state calculation with moderate k-point sampling
• GW calculation on a potentially different k-point grid
• DOS calculation from the GW-corrected band structure

The k-point requirements for GW calculations differ from standard DFT, often requiring careful convergence testing for both the initial DFT and subsequent GW steps [52].

Balancing system size and k-point density remains a fundamental challenge in computational materials science, but systematic protocols for restarting DOS calculations with refined k-grids enable researchers to achieve accurate results with optimal computational efficiency. The key principles emerging from this analysis are:

• Progressive refinement of k-point grids is more efficient than immediately using the densest possible grid
• Integration method selection (Gaussian vs. tetrahedron) should align with system size and k-point density
• Emerging ML methods offer promising alternatives for high-throughput applications or extremely large systems

By implementing the protocols outlined in this application note, researchers can strategically allocate computational resources, ensuring that DOS calculations provide reliable electronic structure information regardless of system complexity.

Ensuring Accuracy: Validating Results and Exploring Modern Alternatives

Within the broader scope of research on restarting Density of States (DOS) calculations with improved k-point grids, establishing robust validation metrics is paramount. The accuracy of the electronic DOS, which counts the number of electronic states per unit energy interval, is highly sensitive to the sampling density of the Brillouin zone [24] [2]. A calculation restarted with a denser k-grid aims to achieve a more accurate and smoother DOS; however, without standardized benchmarks, assessing the success of this endeavor is subjective. This protocol provides a structured set of metrics and methodologies to quantitatively validate the convergence and accuracy of a restarted DOS calculation, ensuring that the computational investment yields physically meaningful and reliable results.

Core Validation Metrics and Quantitative Benchmarks

A successfully restarted DOS calculation should demonstrate convergence in several key areas. The following metrics provide a quantitative framework for validation, with summarized benchmarks available in Table 1.

Table 1: Key Validation Metrics for DOS Calculations

Metric Category	Specific Metric	Target Benchmark for Convergence	Interpretation
K-grid Convergence	Relative Energy Change (∆E)	< 1 meV/atom [53]	Total energy change with increasing k-point density falls below a threshold.
	DOS Mean Squared Deviation (MSD)	MSD < 0.001 (arb. units) [54]	The DOS curve shape becomes stable and stops changing significantly.
Physical Property Accuracy	Band Gap (for insulators)	Underestimation vs. experiment recognized (e.g., ~40% with GGA) [38]	Value is consistent with expected functional error; correct metallic/insulating nature is identified.
	Fermi Energy Placement	Correctly identifies valence band maximum (VBM) in DOS [38]	Fermi level is accurately pinned for metals or positioned within the gap for insulators.
Spectral Quality	Feature Resolution	Sharp features (e.g., van Hove singularities) are well-defined and smooth [55]	The DOS is free from spurious noise without being over-smoothed, correctly capturing critical points.
Numerical Stability	Charge Conservation	Integrated DOS matches total electron count [24]	The calculation is numerically self-consistent and physically sound.

K-Grid Convergence Metrics

• Total Energy Convergence: The foundational metric is the convergence of the system's total energy. As the k-point density increases, the change in total energy per atom between successive calculations (e.g., from an 8x8x8 to a 12x12x12 grid) should fall below a stringent threshold, typically 1 meV/atom [53]. While the total energy itself may converge with a coarser grid, achieving this level of precision indicates a well-sampled Brillouin zone.

• DOS Mean Squared Deviation (MSD): The total energy can converge well before the DOS itself becomes smooth and stable [54]. To quantify the convergence of the DOS profile, compute the mean squared deviation between DOS curves obtained from successive k-grids over a relevant energy range (e.g., from 8 eV below to 8 eV above the Fermi level). A summed MSD value below ~0.001 (relative to the highest k-point density calculation) indicates satisfactory convergence of the DOS shape [54].

Physical Property Validation

• Band Gap and Fermi Surface Identification: For insulating and semiconducting systems, the band gap is a critical, though often underestimated, property in standard DFT [38]. The validated DOS must correctly identify the material as metallic (finite DOS at the Fermi level) or insulating (a gap in the DOS). The calculated band gap should be consistent with known errors of the exchange-correlation functional used (e.g., GGA-PBE typically underestimates gaps by ~40%) [38].

• Feature Reproduction and Smearing: The restarted DOS should resolve sharp, physical features like van Hove singularities without introducing artificial "noise" from poor k-sampling [55]. The choice of broadening method (Gaussian vs. tetrahedron) and its parameter (e.g., 0.2 eV smearing) should be justified based on the k-grid density and the need to represent physical broadening effects [55] [13].

Experimental Protocol for DOS Validation

The following step-by-step protocol outlines the process for performing and validating a restarted DOS calculation with an improved k-grid, with the workflow visualized in Figure 1.

Workflow: DOS Calculation and Validation

Figure 1. Workflow for a restarted Density of States (DOS) calculation and validation. The process is iterative until convergence metrics are satisfied.

Step 1: Initial Self-Consistent Field (SCF) Calculation

• Objective: To obtain a converged charge density using a standard, computationally efficient k-point grid.
• Methodology:
  • Use a pw.x scf calculation in Quantum Espresso (or equivalent in other codes) [24].
  • The k-point grid for this step should be chosen based on prior convergence tests for the total energy. A grid that gives total energies converged to within 1-5 meV/atom is typically sufficient [53].
  • Ensure the outdir and prefix parameters are set, as they will be used to locate the charge density in subsequent steps [24].

Step 2: Non-Self-Consistent Field (NSCF) Calculation

• Objective: To compute the Kohn-Sham eigenvalues on a much denser k-point grid, which is crucial for an accurate DOS [24] [55].
• Methodology:
  • Run a pw.x nscf calculation, reading the previously converged charge density.
  • Critical Parameter - K-Grid: Significantly increase the k-point density compared to the SCF step. For instance, if an 8x8x8 grid was used for SCF, restart the NSCF with a 12x12x12, 16x16x16, or even denser grid [24] [54]. The necessary density is system-dependent; metals often require denser grids than insulators.
  • Critical Parameter - Symmetry: Set nosym = .TRUE. to disable symmetry and generate k-points explicitly across the entire Brillouin zone, which is important for low-symmetry cases and accurate DOS integration [24].
  • Critical Parameter - Bands: Include a sufficient number of empty bands (nbnd) to cover the energy range of interest for the DOS [24].

Step 3: DOS Calculation and Spectral Generation

• Objective: To generate the DOS spectrum from the NSCF calculation results.
• Methodology:
  • Use the dos.x post-processing tool in Quantum Espresso (or equivalent) [24].
  • Specify the energy range (emin, emax) to encompass the valence bands and a relevant portion of the conduction bands.
  • Choose an appropriate broadening method. The two primary methods are:
    • Tetrahedron Method: Generally preferred for its accuracy without empirical smearing, especially with dense k-grids [13]. It performs a 3D linear interpolation between k-points.
    • Gaussian Smearing: Requires choosing a broadening parameter. A value that is too small leaves the DOS noisy, while one that is too large smears out key features. A value of 0.1-0.2 eV is a common starting point [55] [13].

Step 4: Validation and Iteration

• Objective: To quantitatively assess whether the DOS obtained with the new k-grid is converged.
• Methodology:
  • Compare with Coarser Grid: Plot the new DOS alongside the DOS from a previous, coarser k-grid calculation. Visually inspect for changes in peak shapes, heights, and positions [54].
  • Calculate Quantitative Metrics:
    • Compute the Mean Squared Deviation (MSD) between the current and previous DOS curves over a defined energy window [54].
    • Check the convergence of the total energy from the NSCF step (though this is less sensitive than the DOS itself).
    • For insulators, verify the band gap remains consistent or changes minimally.
  • Iterate: If significant differences are observed (e.g., MSD above the target threshold), restart the process from Step 2 (NSCF) with an even denser k-point grid until the changes are acceptably small.

The Scientist's Toolkit: Research Reagent Solutions

Table 2: Essential Computational Tools for DOS Validation

Tool / Reagent	Function in DOS Validation
DFT Code (e.g., Quantum Espresso)	Performs the core SCF, NSCF, and DOS calculations [24].
K-point Convergence Script	Automates the process of running calculations with progressively denser k-grids [53].
Post-processing Tool (e.g., dos.x)	Generates the raw DOS data from the NSCF output [24].
Data Analysis Script (Python/Matplotlib)	Used to compute quantitative metrics like MSD and to plot and compare multiple DOS curves [24] [54].
Tetrahedron Method	A specific numerical approach for DOS integration that provides high accuracy without empirical smearing on dense k-grids [13].
Gaussian Broadening	An alternative smearing method useful for visualizing DOS from coarser k-grids or for mimicking physical broadening [55] [13].

Visualization and Advanced Analysis

A critical aspect of validation is the visual and numerical comparison of DOS from different k-grids and computational methods, as illustrated in Figure 2.

Diagram: DOS Analysis Methods Comparison

Figure 2. Two primary pathways for generating a DOS from a non-self-consistent field (NSCF) calculation. The tetrahedron method is often the preferred choice for converged, high-accuracy calculations, while Gaussian broadening can be practical for quicker visualization or to represent physical smearing.

When performing this analysis, researchers should:

• Normalize DOS plots to the number of atoms or unit cell volume for fair comparison.
• Align the Fermi energy to zero in all plots to focus on the electronic structure relative to the highest occupied state.
• Examine specific regions of interest, such as near the Fermi level for metals or the band edges for semiconductors, with a higher resolution to confirm the absence of k-sampling artifacts.

Benchmarking a restarted DOS calculation is a systematic process that moves beyond qualitative assessment. By employing the quantitative metrics—such as total energy convergence, DOS mean squared deviation, and accurate reproduction of electronic properties—and following the detailed protocol outlined herein, researchers can confidently validate their results. This rigorous approach ensures that the electronic density of states, a foundational property for understanding material behavior, is derived from a sufficiently converged k-point grid, thereby enhancing the reliability and impact of subsequent scientific conclusions.

Within the framework of a broader thesis on restarting Density of States (DOS) calculations with improved k-grids, this application note provides a detailed protocol for understanding, executing, and analyzing the impact of k-point grid refinement. The DOS is a critical property in computational materials science and drug development, representing the number of electronic states at each energy level, which directly influences a material's electronic, optical, and catalytic properties. A common challenge is that the k-point grid density required for a converged total energy calculation is often insufficient for obtaining a smooth, physically meaningful DOS [54]. This document provides a comparative analysis and a detailed experimental protocol for determining and employing the appropriate k-grid for high-quality DOS results.

Theoretical Background: k-Space Sampling and DOS

The Role of k-Points in DFT Calculations

In Density Functional Theory (DFT) calculations, the Brillouin zone is sampled at discrete k-points. The system's total energy is an integral over these points, which can be approximated by a weighted sum. While total energy can converge with a relatively coarse k-grid, the DOS is a more sensitive function of energy. The DOS calculation involves integrating the electron density across k-space, and computationally, this is performed using a weighted sum over the k-point mesh, employing algorithms like Simpson’s Rule. If the underlying electronic structure varies rapidly in narrow energy regions, a coarse k-grid will not sample these regions adequately, leading to an inaccurate and spiky DOS [54].

Why DOS Requires a Denser k-Grid

The central problem lies in the nature of the integration. A standard self-consistent field (SCF) calculation converges the total electron density. The DOS is then typically computed in a non-SCF calculation using a fixed potential. For a coarse k-grid, the interpolation between widely spaced k-points can miss sharp features, especially in metals or near the Fermi level [2]. A refined k-grid provides a denser sampling of the Brillouin zone, allowing for a more accurate interpolation and a smoother, more reliable DOS [54]. This is analogous to needing more data points to accurately plot a rapidly oscillating function.

Comparative Data: Standard vs. Refined k-Grids

Quantitative Convergence Metrics

The following table summarizes key findings from a convergence study on a silver (Ag) face-centered cubic (fcc) system, comparing the convergence of total energy versus the DOS [54].

Table 1: Convergence of Total Energy vs. DOS for Silver (fcc lattice)

k-Grid (NxNxN)	Total Energy Convergence (eV)	Mean Squared Deviation (MSD) of DOS	Qualitative DOS Smoothness
6x6x6	Converged within ~0.05 eV	High (>0.18) [54]	Poor, sharply varying [54]
7x7x7	Converged [54]	-	-
13x13x13	-	Low (~0.005) [54]	Well-converged and smooth [54]
18x18x18	-	Very Low (~0.001) [54]	Highly converged [54]

General k-Grid Guidelines for Different System Types

The required k-grid density is highly system-dependent. The table below provides general guidelines based on system properties.

Table 2: Recommended k-Grid Guidelines by System Type

System Type	Recommended K-Points	Key Considerations
Insulators/Semiconductors	~100 k-points per atom [56]	Tetrahedron method (ISMEAR=-5 in VASP) is recommended for DOS calculations [56].
Metals (Standard)	~1000 k-points per atom [56]	A much denser grid is needed to resolve the steep DOS at the Fermi level [56].
Metals (Transition Metals)	Up to 5000 k-points per atom [56]	Problematic cases with very steep DOS at EF require extremely dense sampling [56].
Large Supercells	Fewer k-points required [39]	The Brillouin zone shrinks with increasing cell size, reducing k-point requirements [39].

Experimental Protocol for k-Grid Refinement

This section provides a step-by-step protocol for determining the optimal k-grid for a DOS calculation and restarting the calculation with the refined grid.

The diagram below outlines the logical workflow for converging and computing the DOS, from initial setup to the final refined calculation.

Step-by-Step Methodology

Step 1: Initial Convergence of Total Energy

• Structure Preparation: Begin with a fully relaxed crystal structure.
• Coarse K-grid Scan: Perform a series of single-point energy calculations using progressively denser k-point grids (e.g., 2x2x2, 4x4x4, 6x6x6, 8x8x8). Ensure the energy cutoff is converged first.
• Convergence Criterion: Plot the total energy per atom against the number of k-points. The k-grid is considered converged for energy when the change per atom is less than a target tolerance (e.g., 1-5 meV/atom) [39].

Step 2: Self-Consistent Field (SCF) Calculation

• Run SCF: Perform a full SCF calculation using the converged, but relatively coarse, k-grid from Step 1. This generates a converged charge density file (e.g., CHGCAR in VASP).
• Check Stability: Verify that the calculation is fully converged.

Step 3: Refining the K-Grid for DOS

• Define Refined Grid: Based on the system type (see Table 2), select a denser k-grid. For instance, if a 6x6x6 grid was sufficient for energy, a 12x12x12 or 18x18x18 grid might be needed for the DOS [54].
• Set Up Non-SCF Calculation: Prepare an input file for a non-self-consistent calculation. This step reads the pre-converged charge density from Step 2 and uses the refined k-grid to compute the DOS with high accuracy without re-iterating the electronic structure.
• Configure Smearing: For metals, use the tetrahedron method (Blochl corrections) if available (e.g., ISMEAR = -5 in VASP) as it is parameter-free and highly accurate for DOS [56]. For insulators and semiconductors, ISMEAR = 0 (Gaussian) can be used with a small SIGMA value if the tetrahedron method is not feasible.

Step 4: Execution and Analysis

• Run DOS Calculation: Execute the non-SCF calculation with the refined k-grid.
• Analyze Results: Compare the DOS from the refined grid with that from the standard grid. The refined DOS should appear smoother, with sharper features (like van Hove singularities) more clearly defined. The quantitative metric like Mean Squared Deviation (MSD) between subsequent refinements should be minimal [54].

The Scientist's Toolkit: Essential Research Reagents and Computational Parameters

In computational materials science, the "research reagents" are the key input parameters and algorithms that define an experiment. The table below details these essential components for conducting a k-grid convergence study for DOS.

Table 3: Key Research Reagent Solutions for k-Grid DOS Studies

Item Name	Function/Description	Example/Value
K-Point Grid	Defines the discrete sampling points in the Brillouin zone.	Monkhorst-Pack grid [39], e.g., 6x6x6, 18x18x18.
Smearing Method	Approximates the occupation of states near the Fermi level to improve convergence in metals.	Tetrahedron (Blochl) [56], Gaussian (ISMEAR=0) [56].
Smearing Width (SIGMA)	Controls the width of the smearing function. A smaller value is more physically accurate but can hinder convergence.	Should be chosen so the entropy term T*S is < 1 meV/atom [56].
Pseudopotential	Represents the core electrons and nucleus, defining the element's chemical behavior.	Ultrasoft [54] or Projector Augmented-Wave (PAW) potentials.
Exchange-Correlation Functional	Approximates the quantum mechanical exchange and correlation energy.	GGA-PBE [54], LDA.
Convergence Metric	A quantitative measure to determine if a calculation is sufficiently accurate.	ΔE < 1 meV/atom for energy [39]; MSD of DOS curve [54].

Visualization of k-Space Refinement Logic

The following diagram illustrates the conceptual relationship between k-grid sampling, the resulting electronic band structure, and the final DOS, highlighting how refinement leads to a more accurate DOS.

Restarting a DOS calculation with a refined k-grid is not merely a technical trick but a fundamental step for achieving accurate and reliable electronic properties. As demonstrated, the convergence criteria for the total energy are vastly different from those for the DOS, particularly for metallic systems. The protocols outlined herein provide a systematic framework for researchers to validate their k-point sampling, ensuring that reported DOS results are robust and reflective of the true electronic structure. This rigorous approach is essential for making confident predictions in materials design and drug development, where electronic states can dictate functional behavior.

Extracting Reliable Band Gaps from Your Smoothed DOS

The density of states (DOS) quantifies the distribution of available electronic states at each energy level in a material and underlies crucial optoelectronic properties such as conductivity and optical absorption [1]. The band gap, defined as the energy difference between the valence band maximum (VBM) and the conduction band minimum (CBM), plays a fundamental role in determining these properties [1]. Accurately extracting the band gap from DOS calculations is essential for applications in electronics, catalysis, and photonics, but this process is complicated by numerical noise, k-grid sampling dependencies, and smoothing artifacts.

This protocol provides a standardized methodology for extracting reliable band gaps within the broader context of optimizing k-grid parameters in DOS calculations. Proper k-grid convergence ensures that the DOS accurately represents the electronic structure, forming a critical foundation for meaningful band gap extraction, particularly when smoothing techniques are applied to computed data.

Theoretical Background

Density of States and Band Gap Fundamentals

The electronic DOS describes the number of electronic states per unit volume per unit energy. Key features identifiable in the DOS include band edges and Van Hove singularities [57]. The band gap is directly determined from the DOS by identifying the energy range where the DOS value is zero or nearly zero, bracketed by the VBM and CBM [1].

In practice, the Fermi level is first determined by finding the energy where the integrated DOS equals the total number of electrons in the system. The VBM and CBM positions are then located to calculate the band gap [1].

Challenges in Band Gap Extraction from Smoothed DOS

While the DOS inside the band gap is theoretically zero, real calculations present several challenges:

• Numerical Smearing: Discrete k-point sampling and numerical algorithms introduce artificial states in the band gap region.
• Smoothing Artifacts: Spectral preprocessing techniques, while reducing noise, can distort band edges [58].
• K-grid Dependencies: Inadequate k-point sampling creates spurious features that obscure the true band gap.

These factors necessitate careful preprocessing and validation to distinguish physical features from computational artifacts.

Computational Methods

DOS Calculation Protocols

First-Principles Calculations

Density Functional Theory (DFT) and Density Functional based Tight Binding (DFTB) provide the foundational electronic structure data for DOS calculations. DFTB, an efficient approximation to DFT, derives from a second-order Taylor expansion of the Kohn-Sham total energy around a reference electron density, offering significant computational advantages while maintaining reasonable accuracy [59].

The DFTB total energy in its basic formulation is expressed as:

[E{DFTB} = \sumi ni \epsiloni + \frac{1}{2}\sum{A,B} V{rep}^{AB}]

where (ni) and (\epsiloni) are occupation numbers and molecular orbital energies, respectively, and (V_{rep}^{AB}) is the pairwise repulsive potential between atoms A and B [59].

Table: Comparison of Electronic Structure Methods for DOS Calculations

Method	Computational Cost	Accuracy	Best Use Cases
DFT	High	High	Small systems, final accuracy
DFTB	Medium	Medium	Large systems, high-throughput screening
Machine Learning	Low	Variable	Rapid screening, large datasets

K-grid Convergence Testing

A systematic approach to k-grid optimization ensures DOS convergence:

• Initial Calculation: Begin with a moderate k-grid (e.g., 3×3×3 for bulk materials)
• Progressive Refinement: Increase k-point density systematically (5×5×5, 7×7×7, etc.)
• Convergence Criterion: Monitor changes in total energy (< 1 meV/atom) and band gap (< 0.01 eV)
• Final Production Run: Use the converged k-grid for final DOS calculation

K-grid optimization directly addresses the core thesis context of "restarting DOS calculation with better k-grid research" by providing a quantitative methodology for establishing sufficient k-point sampling.

DOS Smoothing Techniques

Smoothing raw DOS data helps visualize trends but must preserve critical band edge information:

• Gaussian Broadening: Application of Gaussian functions with optimized full width at half maximum (FWHM)
• Smoothing Splines: Constrained smoothing that minimizes artifacts
• Digital Filtering: Moving average or Savitzky-Golay filters for noise reduction

Smoothing parameters should be calibrated against known benchmark systems to avoid excessive broadening that obscures the true band gap.

Band Gap Extraction Protocol

Step-by-Step Workflow

The following standardized workflow ensures consistent and reliable band gap extraction from smoothed DOS data:

Band Edge Identification Methods

Accurate identification of VBM and CBM is critical for reliable band gap extraction:

• Threshold Method: Define a DOS threshold value below which states are considered gap states
• Derivative Analysis: Locate inflection points in the DOS where d(DOS)/dE is maximized
• Curve Fitting: Fit DOS near band edges to extract precise transition points

Table: Band Edge Identification Techniques

Method	Principles	Advantages	Limitations
Threshold	Defines minimum DOS value for band edge	Simple, fast	Sensitive to noise/smoothing
Derivative	Finds maxima in d(DOS)/dE	More precise for gradual edges	Amplifies numerical noise
Curve Fitting	Fits DOS near edges to functional form	Most robust, quantitative uncertainty	Requires appropriate model

Validation and Error Analysis

Robust band gap extraction requires validation against independent methods:

• Direct Band Structure Calculation: Compare with band gap from band structure calculations along high-symmetry directions
• Optical Absorption: Validate against experimental or computational absorption spectra
• Machine Learning Prediction: Utilize models like PET-MAD-DOS for independent estimation [1]

Uncertainty quantification should include:

• Statistical error from fitting procedures
• Systematic error from k-grid convergence
• Methodological error from different identification techniques

Advanced Approaches

Machine Learning for DOS and Band Gap Prediction

Machine learning approaches offer efficient alternatives for DOS and band gap prediction:

Universal ML Models: Models like PET-MAD-DOS use transformer architectures trained on diverse datasets (e.g., MAD dataset) to predict DOS across chemical spaces [1]. These models demonstrate semi-quantitative agreement with explicit electronic-structure methods while offering significant computational savings.

Transfer Learning: Pre-trained universal models can be fine-tuned with small system-specific datasets to achieve accuracy comparable to bespoke models [1].

Band Gap Extraction from ML-DOS: The band gap can be derived from ML-predicted DOS by applying the same protocols as for computed DOS, though challenges remain in resolving precise band edges from predicted spectra [1].

Special Cases and Troubleshooting

• Zero-Gap Materials: Distinguish true zero-gap from insufficient k-point sampling
• Disordered Systems: Address smeared band edges in alloys or defected materials
• Nanoscale Systems: Account for size effects and quantum confinement in nanoparticles [59]

Research Reagent Solutions

Table: Essential Computational Tools for DOS Analysis

Tool/Resource	Type	Function	Application Context
DFTB+ Software [59]	Software Package	Performs efficient DFTB calculations	DOS calculations for large systems
VASP Package [60]	Software Package	Performs DFT calculations with PAW method	Accurate DOS with plane-wave basis sets
3ob-3-1 Slater-Koster Files [59]	Parameter Set	Provides Hamiltonian parameters for DFTB	MgO, ZnO systems and interactions
PET-MAD-DOS Model [1]	ML Model	Predicts DOS using transformer architecture	Rapid screening across chemical spaces
WxSM Software [60]	Analysis Tool	Analyzes STM/STS data	Experimental DOS validation
Colour Contrast Analyser [61]	Accessibility Tool	Ensures diagram color compliance	Visualization quality control

Extracting reliable band gaps from smoothed DOS requires meticulous attention to k-grid convergence, appropriate smoothing parameters, and robust band edge identification. The protocols outlined here provide a systematic framework for obtaining accurate band gaps within the context of optimizing k-grid parameters in DOS calculations. Machine learning approaches present promising avenues for accelerating these calculations while maintaining acceptable accuracy. Validation against multiple independent methods remains essential for establishing confidence in extracted band gap values, particularly for materials with complex electronic structures or when using heavily smoothed DOS data.

The accurate and efficient calculation of the electronic Density of States (DOS) is a cornerstone of computational materials science and drug development, underpinning the prediction of key electronic, optical, and catalytic properties. Traditional approaches based on Density Functional Theory (DFT) provide high fidelity but are computationally expensive, creating a bottleneck for high-throughput screening and long-timescale molecular simulations. A central tenet of conventional DFT is the requirement for a finely-spaced k-point grid during the non-self-consistent field (nscf) calculation to achieve a well-converged DOS, a process that is both time-consuming and resource-intensive [2] [24]. The emergence of universal machine learning (ML) models, such as the Point Edge Transformer - Massive Atomistic Diversity - DOS (PET-MAD-DOS) model, represents a paradigm shift. These models offer the ability to predict the electronic DOS directly from atomic configurations at a fraction of the computational cost, achieving accuracy comparable to the electronic-structure calculations on which they are trained [1]. This application note details the operation, performance, and implementation of these ML emulators, providing researchers and drug development professionals with protocols to integrate these powerful tools into their workflows, thereby accelerating the discovery and analysis of new materials and molecules.

Traditional DOS Calculation: The Role of the K-Point Grid

The foundational step for traditional DFT DOS calculations involves a rigorous two-step process, where the k-point grid plays a critical role. An initial self-consistent field (scf) calculation is performed to determine the converged charge density. This is followed by a non-self-consistent field (nscf) calculation on a significantly denser k-point grid to compute the DOS [24].

• The Imperative for a Denser K-Grid: A denser k-point grid in the nscf step is crucial because the accuracy of the DOS depends on the quality of the numerical integration over the Brillouin zone. A coarse grid leads to a sparse and inaccurate sampling of electronic energies, resulting in a jagged, non-physical DOS. A finer grid ensures better energy resolution and smoother, more accurate DOS curves, which is essential for correctly identifying features like band gaps [2] [24] [62].
• Challenges and Considerations: The requirement for a dense k-grid inherently increases the computational burden. Furthermore, generating a smooth DOS involves navigating a landscape of parameters, including the Brillouin zone integration scheme, the fineness of the energy grid, and the application of smearing or the tetrahedron method to handle band crossings and interpolate between k-points [2].

Table 1: Key Parameters for Traditional DFT DOS Calculations

Parameter	Role in DOS Calculation	Typical Setting/Consideration
K-point Grid (scf)	Achieves convergence of the total energy and charge density.	Coarser grid; sufficient for energy convergence.
K-point Grid (nscf)	Provides high-resolution sampling for DOS/PDOS.	Much denser grid (e.g., 12x12x12); critical for accuracy [24].
occupations	Determines how electronic states are filled.	'tetrahedra' is often used for DOS as it is appropriate for integration [24].
nosym	Disables k-point symmetry.	.TRUE. recommended for DOS to avoid issues in low-symmetry cases [24].

The following workflow diagram illustrates the traditional multi-step process for computing the DOS in plane-wave DFT codes like Quantum ESPRESSO, highlighting the separate k-grid requirements.

Figure 1. Traditional DFT Workflow for DOS. The process requires two separate K-grids: a coarse one for the initial SCF and a dense one for the final DOS calculation.

Machine Learning Emulators: The PET-MAD-DOS Model

The PET-MAD-DOS model is a groundbreaking "universal" machine learning model designed to predict the electronic DOS directly from an atomic structure, bypassing the need for explicit DFT calculations [1] [63].

Model Architecture and Training

PET-MAD-DOS is built upon the Point Edge Transformer (PET) architecture, a rotationally unconstrained transformer-based graph neural network. While the architecture does not enforce rotational symmetry constraints, it learns to be equivariant to a high degree of accuracy through data augmentation, as evidenced by a rotational discrepancy two orders of magnitude smaller than the prediction error [1]. The model was trained on the Massive Atomistic Diversity (MAD) dataset, a compact but highly diverse collection of fewer than 100,000 structures. The MAD dataset encompasses a wide range of systems, including [1] [64]:

• 3D and 2D inorganic crystals (MC3D & MC2D)
• Surfaces (MC3D-surface)
• Nanoclusters (MC3D-cluster)
• Molecular crystals and fragments (SHIFTML-molcrys & SHIFTML-molfrags)
• Randomized and rattled structures to enhance stability in simulations

This extensive training enables the model to generalize across a vast chemical space, making it applicable to both materials and molecules.

Performance and Quantitative Accuracy

PET-MAD-DOS demonstrates robust performance across a wide spectrum of internal and external datasets. Its predictive accuracy is quantified using metrics like the root-mean-square error (RMSE) between the predicted and DFT-calculated DOS.

Table 2: Performance of PET-MAD-DOS on Various Datasets [1]

Dataset Category	Example Datasets	Performance Notes
MAD Subsets	MC3D, MC2D, SHIFTML-molcrys	High overall accuracy, with most structures having errors below 0.2 eV⁻⁰․⁵.
MAD Challenging Subsets	MC3D-random, MC3D-cluster	Reduced accuracy due to high chemical diversity and far-from-equilibrium configurations.
External Molecular	MD22, SPICE	Excellent performance, consistent with strong results on molecular parts of MAD.
External Materials	MPtrj, Matbench, Alexandria	Comparable performance to MAD dataset, highlighting model's generalizability.

A key application of the predicted DOS is the derivation of electronic band gaps. The model shows promise in accurately predicting band gaps by post-processing the predicted DOS to identify the valence band maximum and conduction band minimum, although challenges remain in precisely resolving regions where the DOS is theoretically zero [1]. Furthermore, the model has been validated for calculating ensemble-averaged properties, such as the electronic heat capacity from molecular dynamics trajectories, achieving semi-quantitative agreement with bespoke models for systems like lithium thiophosphate (LPS) and gallium arsenide (GaAs) [1].

Experimental Protocols

This section provides detailed methodologies for utilizing both traditional and machine learning approaches for DOS calculation.

Protocol 1: Traditional DOS Calculation with Quantum ESPRESSO

This protocol outlines the steps for a DOS calculation for a silicon crystal using Quantum ESPRESSO [24].

• Self-Consistent Field (scf) Calculation

  • Objective: Achieve a converged charge density.
  • Input File Preparation (pw.scf.silicon_dos.in): Key parameters include:
    • &control: calculation = 'scf'
    • &system: ecutwfc = [increased value for precision], occupations = 'smearing'
    • &kpoints: Use a coarse k-point grid (e.g., 6x6x6).
  • Execution: Run the pw.x code with the scf input file.
  • Output: A converged charge density and wavefunction file.

• Non-Self-Consistent Field (nscf) Calculation

  • Objective: Compute eigenvalues on a dense k-point grid for high-resolution DOS.
  • Input File Preparation (pw.nscf.silicon_dos.in): Key modifications from the scf input:
    • &control: calculation = 'nscf'
    • &system: occupations = 'tetrahedra', nosym = .TRUE. (disables symmetry), nbnd = [possibly larger value to include unoccupied states].
    • &kpoints: Use a dense k-point grid (e.g., 12x12x12).
    • Ensure outdir and prefix match the scf calculation.
  • Execution: Run the pw.x code with the nscf input file.

• DOS Calculation

  • Objective: Compute the density of states from the nscf output.
  • Input File Preparation (pp.dos.silicon.in):

  • Execution: Run the dos.x code with the DOS input file.
  • Output: A data file (si_dos.dat) containing energy, DOS, and integrated DOS, ready for plotting.

Protocol 2: DOS Prediction with the PET-MAD-DOS Model

This protocol describes how to use the pre-trained PET-MAD-DOS model for rapid DOS prediction, via its integration with the Atomic Simulation Environment (ASE) [65].

• Environment Setup

  • Install the pet-mad package using pip or conda.

• Structure Preparation

  • Define the atomic structure of interest within ASE. This can be a molecule, a surface, or a bulk crystal.
  • Example for bulk silicon:

• Model Inference and DOS Prediction

  • Import the calculator, attach it to the atoms object, and retrieve the results.
  • Example Python script:

  • The predicted DOS is now available for immediate analysis or plotting.

The Scientist's Toolkit

Table 3: Essential Research Reagents and Software Solutions

Item Name	Type	Function / Application	Relevant Context
Quantum ESPRESSO	Software Suite	Open-source suite for DFT calculations, including scf, nscf, and DOS steps.	Traditional DOS protocol [24].
PET-MAD-DOS	Pre-trained ML Model	Universal transformer model for instant prediction of electronic DOS from structure.	ML emulator protocol [1] [65].
Atomic Simulation Environment (ASE)	Python Library	Provides a flexible interface to set up and manipulate atomic structures and calculators.	Essential for using PET-MAD-DOS [65].
MAD Dataset	Training Dataset	A diverse dataset of atomic structures used to train universal models like PET-MAD-DOS.	Foundation for the model's generalizability [1] [64].
Dense K-point Grid	Computational Parameter	A fine mesh in reciprocal space required for accurate numerical integration in DFT-based DOS.	Critical parameter in traditional nscf calculation [2] [24].

The logical relationship between the traditional and ML-based approaches, highlighting their convergence in enabling scientific discovery, is summarized below.

Figure 2. DOS Method Comparison. A comparison of the inputs, strengths, and outputs of traditional DFT and modern ML emulator approaches for DOS calculation.

The paradigm for electronic structure calculation is shifting. While the traditional DFT approach, reliant on computationally intensive dense k-point grids, remains a gold standard for accuracy, machine learning emulators like PET-MAD-DOS have emerged as a powerful and efficient alternative. These universal models demonstrate remarkable generalizability across the chemical space, providing semi-quantitative to quantitative accuracy for the DOS and derived properties at a fraction of the cost. This enables previously infeasible high-throughput screening and detailed finite-temperature analysis. For researchers, the choice of method now depends on the specific problem: traditional DFT for the highest precision in a single calculation, and ML emulators for rapid prototyping, large-scale exploration, and integration into complex simulation workflows. The continued development and fine-tuning of such models promise to further accelerate innovation in materials science and drug discovery.

The design and discovery of novel biomaterials are historically challenging, often relying on traditional "trial and error" methods that are laborious, time-consuming, and unreliable [66]. High-throughput screening (HTS) has emerged as a valuable tool to overcome these challenges, enabling the parallel testing of hundreds to thousands of biomaterial combinations to investigate cellular responses [67] [68]. However, a significant bottleneck remains in the computational prediction of electronic properties, such as the electronic density of states (DOS), which are crucial for understanding material behavior but are expensive to calculate using conventional ab initio methods.

This application note proposes a paradigm shift by integrating machine-learned density of states (ML-DOS) models into biomaterial HTS workflows. The core of this approach leverages recent breakthroughs in universal machine learning models that can predict the DOS directly from atomic structure, bypassing the need for expensive DFT calculations for every new candidate material [1]. This integration is particularly powerful in the context of "restarting DOS calculation with better k-grid" research, as it provides a rapid, initial screening tool that can identify promising candidates for more precise, computationally intensive verification.

State of the Field: Machine Learning for Materials Science

Deep learning (DL), a specialized branch of machine learning, is transforming computational materials science by enabling the analysis of unstructured data and automated identification of features from raw input data [69]. Its application spans various data modalities, including atomistic, image-based, spectral, and textual data.

Key ML Concepts and Terminology

For researchers new to the field, understanding the following key concepts is essential:

• Deep Learning (DL): A subset of machine learning based on artificial neural networks with multiple layers, capable of learning complex, hierarchical patterns from data [69].
• Foundation Models: General-purpose ML models pre-trained on large amounts of data and then fine-tuned for a variety of applications. The MultiMat framework is an example of such a model for materials science [70].
• Graph Neural Networks (GNNs): A class of neural networks that operate on graph-structured data, making them particularly suited for modeling crystal structures where atoms represent nodes and bonds represent edges [70].
• Transfer Learning: A technique where a model developed for one task is reused as the starting point for a model on a second task, crucial for applying large pre-trained models to specific material problems with limited data [69].

ML-DOS Models: A Game Changer for High-Throughput Screening

The PET-MAD-DOS Universal Model

A landmark development in the field is the PET-MAD-DOS model, a universal machine learning model for predicting the electronic density of states [1]. This model is built on the Point Edge Transformer (PET) architecture and trained on the Massive Atomistic Diversity (MAD) dataset, which encompasses both organic and inorganic systems, from discrete molecules to bulk crystals.

Table 1: Performance Summary of the PET-MAD-DOS Model on Various Datasets

Dataset	System Type	RMSE (eV⁻⁰·⁵ electrons⁻¹ state)	Key Characteristics
MAD (MC3D)	3D Crystals	~0.15	Baseline performance on diverse inorganic crystals
MAD (MC3D-cluster)	Atomic Clusters	>0.20	Challenging due to sharply-peaked DOS
SHIFTML-molcrys	Molecular Crystals	~0.10	Good performance on organic systems
MD22	Biomolecules	~0.08	Excellent performance on peptides, DNA, carbohydrates
SPICE	Drug-like Molecules	~0.09	Strong predictive ability for pharmaceutical applications

The model demonstrates particularly strong performance on molecular systems, including drug-like molecules and peptides, which is highly relevant for biomaterial applications [1]. This capability enables researchers to rapidly predict electronic properties for a wide range of candidate materials without performing DFT calculations for each one.

Multimodal Learning with MultiMat

The MultiMat framework represents another significant advancement, enabling self-supervised multi-modality training of foundation models for materials [70]. It aligns the latent spaces of encoders for different material modalities, including:

• Crystal structure (encoded using a state-of-the-art GNN)
• Density of States (DOS)
• Charge density
• Textual descriptions from Robocrystallographer

This multimodal approach allows the model to learn richer material representations and enables novel material discovery through latent space similarity searches for stable materials with desired properties [70].

Integrated Protocol: ML-DOS for Biomaterial Screening

The following diagram illustrates the complete integrated workflow for high-throughput screening of biomaterials using ML-DOS:

Stage 1: ML-DOS Prediction and Initial Screening

Objective: Rapidly screen thousands of candidate biomaterials using ML-predicted DOS to identify promising candidates for further investigation.

Protocol:

• Input Preparation: Compose a library of candidate atomic structures. For biomaterials, this may include various polymer compositions, surface functionalizations, or composite materials.
• ML-DOS Prediction:
  • Utilize a pre-trained universal DOS model such as PET-MAD-DOS [1].
  • Input the atomic structure of each candidate material.
  • Execute the model to obtain the full DOS spectrum.
• Property Extraction:
  • Calculate the band gap from the predicted DOS by identifying the energy difference between the valence band maximum (VBM) and conduction band minimum (CBM) [1].
  • Derive other electronic properties relevant to biomaterial function, such as electronic heat capacity or trends in conductivity.
• Initial Filtering:
  • Apply threshold-based filtering on the derived properties (e.g., selecting materials with band gaps in a specific range for semiconductor applications).
  • Rank candidates based on the similarity of their predicted DOS to a target DOS profile using the latent space of a multimodal model like MultiMat [70].

Output: A shortlist of candidate biomaterials with promising predicted electronic properties.

Stage 2: High-Fidelity DFT Verification

Objective: Validate the electronic structure of shortlisted candidates using precise DFT calculations with an optimized k-point grid.

Rationale for k-grid refinement: A finer k-point sampling is often required for accurate DOS calculations compared to total energy convergence because the DOS involves integrating over the Brillouin zone and requires sufficient sampling to resolve fine features in the electronic structure [2]. The tetrahedron method used for DOS calculations can be sensitive to k-point density, especially near band crossings [2].

Protocol:

• k-grid Optimization:
  • Restart the calculation for shortlisted candidates with a denser k-point mesh.
  • Systematically increase the k-point density until the DOS profile converges, particularly in regions of interest like band edges.
• High-Fidelity DOS Calculation:
  • Perform the DFT calculation with the optimized, denser k-grid.
  • Use appropriate smearing and tetrahedron methods for accurate DOS integration.
• Validation:
  • Compare the ML-predicted DOS with the high-fidelity DFT-calculated DOS.
  • Use this verification step to further refine the candidate list.

Output: A validated set of candidate biomaterials with accurately characterized electronic structures.

Stage 3: Experimental High-Throughput Screening

Objective: Experimentally validate the performance of the computationally screened biomaterials using high-throughput cellular response platforms.

Protocol:

• Platform Selection:
  • TopoChip: A chip-based platform containing thousands of different surface topographies (TopoUnits) to test cellular responses [67].
  • Microarray Platforms: Polymer microarrays or gradient chips to test the effects of various chemical compositions and surface properties on cell behavior [68] [71].
• Surface Functionalization:
  • Translate the identified material compositions to the HTS platform. For instance, use chips with orthogonal gradients of functional molecules (e.g., PEG and REDV peptide) to screen for optimal combinational densities that direct specific cellular behaviors [71].
• Cell Culture and Imaging:
  • Seed relevant cell types (e.g., endothelial cells and smooth muscle cells for cardiovascular applications) onto the platform [71].
  • Culture cells for an appropriate duration and acquire images using high-throughput automated microscopy.
• Machine Learning-Based Analysis:
  • Employ a machine learning workflow for label-free cell identification and statistics [71].
  • Use a modified UNet model (ResUNet) to predict binary images of cell nuclei from brightfield images.
  • Apply clustering algorithms (DBSCAN and K-means) to identify individual cell coordinates.
  • Classify cell types using a convolutional neural network (ResNet50V2) on the segmented single-cell images.
• Hit Identification:
  • Analyze the data to identify "hit" surfaces that optimally promote the desired cell behavior (e.g., high endothelial cell selectivity) [67] [71].

Output: Experimentally validated biomaterial formulations that induce the desired cellular responses.

Table 2: Key Research Reagent Solutions for ML-DOS Integrated Screening

Category	Item	Function/Description	Example/Source
Computational Models	PET-MAD-DOS	Universal ML model for predicting Density of States from atomic structure	[1]
	MultiMat Framework	Multimodal foundation model for materials that aligns crystal structure, DOS, and other properties	[70]
Software & Libraries	Deep Learning Frameworks	Enable building and training custom neural network models	PyTorch, TensorFlow [69]
	Materials Databases	Source of training data and candidate structures	Materials Project [70]
Experimental Platforms	TopoChip	High-throughput platform with thousands of surface topographies to test cell responses	[67]
	Gradient Chips	Surfaces with continuous chemical gradients for high-integration screening	[71]
Analysis Tools	ML-Based Cell Recognition	Label-free identification and statistics of co-cultured cells from brightfield images	ResUNet + ResNet50V2 workflow [71]

Data Analysis and Interpretation

Correlating Electronic Properties with Cellular Responses

The power of the integrated approach lies in establishing relationships between the computationally predicted electronic properties and experimentally observed cellular behaviors. The following diagram illustrates the key relationships and analysis pathways:

Statistical Analysis of Screening Results

When analyzing high-throughput screening data, appropriate statistical methods and data visualization are crucial:

• Hit Surface Ranking: Use robust statistical methods that account for surface similarities and cell behavior similarities to reliably identify "hit" surfaces from HTS data, improving performance and reducing the need for numerous replicates [67].
• Data Visualization: Employ back-to-back stemplots for small datasets comparing two groups, 2-D dot charts for small to moderate amounts of data, and boxplots for larger datasets to effectively compare quantitative data between different experimental groups [72].

The integration of ML-DOS models into biomaterial high-throughput screening represents a paradigm shift from traditional trial-and-error approaches to a data-driven, predictive science. This methodology leverages the complementary strengths of rapid machine learning prediction and high-fidelity experimental validation, dramatically accelerating the biomaterial discovery cycle.

Future developments in this field will likely focus on:

• Increased Model Generalizability: Expanding universal models like PET-MAD-DOS to cover an even broader range of biomaterials, including complex polymers and hydrogels.
• Tighter Experiment-Computation Integration: Developing fully automated workflows where HTS experimental data directly feedback to improve and refine ML models.
• Multi-scale Modeling: Integrating DOS predictions with higher-scale material properties and cellular response models to create comprehensive material performance predictors.

The protocols outlined in this application note provide a concrete framework for researchers to implement this integrated approach, potentially shaving years off the traditional biomaterial development timeline and enabling the discovery of next-generation materials for therapeutic applications.

Conclusion

Restarting a DOS calculation with a refined k-grid is a computationally efficient and methodologically sound strategy for achieving high-quality electronic structure data, which is fundamental for predicting material properties in biomedical and clinical research. By mastering the restart procedures, researchers can reliably obtain smooth and accurate DOS, leading to more precise predictions of band gaps, reactive sites, and electronic properties relevant to drug interaction and biomaterial design. The future of DOS analysis points toward a hybrid approach, combining the rigorous validation of traditional DFT with the emerging speed of universal machine-learning models, enabling large-scale virtual screening of molecular and crystalline systems for next-generation therapeutics.

References

• 1. A universal machine learning model for the electronic ... [https://arxiv.org/html/2508.17418v1]
• 2. Increasing k-point grid to take DOS calculation [https://mattermodeling.stackexchange.com/questions/5045/increasing-k-point-grid-to-take-dos-calculation]
• 3. An Electronic Structure Analysis Package for the AI Era [https://arxiv.org/html/2501.08697v4]
• 4. Bandstructure — | QuantumATKX-2025.06 Documentation [https://docs.quantumatk.com/manual/Types/Bandstructure/Bandstructure.html]
• 5. Tutorial: 2D Band Structure Visualization - PyProcar Docs [https://pyprocar.readthedocs.io/en/stable/examples/00-band_structure/07-Plotting%202D%20Band%20Structure.html]
• 6. At a particular K-point (Monkhorst Pack) grid, SCF ... [https://mattermodeling.stackexchange.com/questions/12667/at-a-particular-k-point-monkhorst-pack-grid-scf-calculation-is-not-converging]
• 7. Random State Approach to Quantum Computation of ... [https://arxiv.org/html/2510.09999v1]
• 8. Pattern Learning Electronic Density of States [https://www.nature.com/articles/s41598-019-42277-9]
• 9. Troubleshooting — BAND 2025.1 documentation - SCM [https://www.scm.com/doc/BAND/Troubleshooting/Trouble_Shooting.html]
• 10. k-point convergence [https://docs.mat3ra.com/tutorials/dft/addons/kpt-convergence/]
• 11. Should I Re-optimize Without CHGCAR When Switching ... [https://mattermodeling.stackexchange.com/questions/13672/should-i-re-optimize-without-chgcar-when-switching-from-vasp-gam-to-vasp-std]
• 12. Density of States (DOS) — BAND 2025.1 documentation - SCM [https://www.scm.com/doc/BAND/Analysis/DOS.html]
• 13. DensityOfStates — | QuantumATKX-2025.06 Documentation [https://docs.quantumatk.com/manual/Types/DensityOfStates/DensityOfStates.html]
• 14. Kohn–Sham equations [https://en.wikipedia.org/wiki/Kohn%E2%80%93Sham_equations]
• 15. 7.13.3 Methods Based on Kohn-Sham Eigenvalues [https://manual.q-chem.com/6.3/Ch7.S13.SS3.html]
• 16. Spectral density [https://en.wikipedia.org/wiki/Spectral_density]
• 17. Sampling of the BZ with k-points - The SIESTA project [https://docs.siesta-project.org/projects/siesta/en/5.2/tutorials/basic/kpoint-convergence/]
• 18. Restarts — BAND 2025.1 documentation - SCM [https://www.scm.com/doc/BAND/Expert_Options/Restarts.html]
• 19. An Electronic Structure Analysis Package for the AI Era [https://arxiv.org/html/2501.08697v3]
• 20. Part 2: More silicon [https://www.vasp.at/tutorials/latest/bulk/part2/]
• 21. Restart the DOS or band structure with BAND - SCM [http://www.scm.com/doc/Tutorials/Analysis/RestartDosBandStructure.html]
• 22. Is it necessary to do self-consistent calculation after ... [https://mattermodeling.stackexchange.com/questions/7271/is-it-necessary-to-do-self-consistent-calculation-after-structure-optimization]
• 23. VASP - NERSC Documentation [https://docs.nersc.gov/applications/vasp/]
• 24. Density of States calculation • Quantum Espresso Tutorial [https://pranabdas.github.io/espresso/hands-on/dos/]
• 25. pw.x: input description [https://www.quantum-espresso.org/Doc/INPUT_PW.html]
• 26. Convergence Tricks – Kate Penrod CoMET Capstone [https://sites.psu.edu/penrodcapstone/convergence/]
• 27. 4.6 Restarting [https://www.quantum-espresso.org/Doc/pw_user_guide/node20.html]
• 28. Setting up an electronic minimization [https://vasp.at/wiki/Setting_up_an_electronic_minimization]
• 29. VASP — ASE documentation [https://wiki.fysik.dtu.dk/ase/ase/calculators/vasp.html]
• 30. Troubleshooting electronic convergence - VASP Wiki [https://vasp.at/wiki/Troubleshooting_electronic_convergence]
• 31. What are good ways to reduce computing time when ... [https://mattermodeling.stackexchange.com/questions/170/what-are-good-ways-to-reduce-computing-time-when-working-with-large-systems-in-v]
• 32. k-points and ENCUT convergence tests before or after ... [https://mattermodeling.stackexchange.com/questions/1896/k-points-and-encut-convergence-tests-before-or-after-relaxation]
• 33. DOS and projected DOS — Siesta Documentation [https://docs.siesta-project.org/projects/siesta/en/stable/analysis_visualization/dos-pdos.html]
• 34. How to restart calculation in Quantum ESPRESSO? [https://mattermodeling.stackexchange.com/questions/12424/how-to-restart-calculation-in-quantum-espresso]
• 35. VASPilot: MCP-Facilitated Multi-Agent Intelligence for ... [https://arxiv.org/html/2508.07035v1]
• 36. Density of states — GPAW [https://gpaw.readthedocs.io/tutorialsexercises/electronic/dos/dos.html]
• 37. The Basics of Electronic Structure Theory for Periodic ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC6424870/]
• 38. Electronic Structure [https://docs.materialsproject.org/methodology/materials-methodology/electronic-structure]
• 39. How to know optimal K-points grid values for good DFT ... [https://mattermodeling.stackexchange.com/questions/2347/how-to-know-optimal-k-points-grid-values-for-good-dft-calculation]
• 40. Castep vs QE - c2x [https://www.c2x.org.uk/multi/castep_vs_qe.html]
• 41. Part 1: Silicon as a typical bulk material [https://www.vasp.at/tutorials/latest/bulk/part1/]
• 42. Restarting a stopped calculation - QuantumATK [https://docs.quantumatk.com/tutorials/restarting_calculations/restarting_calculations.html]
• 43. Monkhorst-Pack, Gamma-centred and gamma only [https://mattermodeling.stackexchange.com/questions/2238/monkhorst-pack-gamma-centred-and-gamma-only]
• 44. Germanium — | QuantumATKX-2025.06 Documentation [https://docs.quantumatk.com/tutorials/germanium/germanium.html]
• 45. KPOINTS - VASP Wiki [https://www.vasp.at/wiki/KPOINTS]
• 46. Rapid generation of optimal generalized Monkhorst-Pack ... [https://www.sciencedirect.com/science/article/abs/pii/S0927025620305917]
• 47. Effectiveness of smearing and tetrahedron methods [https://arxiv.org/abs/2109.01196]
• 48. 4.4 DFPT with the tetrahedron method [https://www.quantum-espresso.org/Doc/ph_user_guide/node11.html]
• 49. Part 3: Periodic Systems - Basics of Running FHI-aims [https://fhi-aims-club.gitlab.io/tutorials/Basics-of-Running-FHI-aims/3-Periodic-Systems/]
• 50. For calculations with lots of k-points [https://www.openmx-square.org/openmx_man3.6/node55.html]
• 51. Materials Learning Algorithms (MALA): Scalable machine ... [https://www.sciencedirect.com/science/article/pii/S0010465525001560]
• 52. A systematic benchmark for band gaps of solids [https://arxiv.org/html/2508.05247v1]
• 53. Convergence and machine learning predictions of ... [https://www.sciencedirect.com/science/article/abs/pii/S0927025619300813]
• 54. Density Functional Theory and Practice Course [https://sites.psu.edu/dftap/2018/04/26/comparison-of-the-convergence-of-system-energy-and-density-of-states-calculations-with-respect-to-k-point-sampling/]
• 55. Lab 4: The Electronic Band Structure and Density of States [https://eamonnmurray.gitlab.io/modelling_materials/lab04/]
• 56. Number of k points and method for smearing [https://vasp.at/wiki/Number_of_k_points_and_method_for_smearing]
• 57. How to analyse a density of states [https://www.sciencedirect.com/science/article/pii/S277294942200002X]
• 58. A review on spectral data preprocessing techniques for ... [https://www.sciencedirect.com/science/article/pii/S258900422501020X]
• 59. Accelerating density of states prediction in Zn-doped MgO ... [https://www.nature.com/articles/s41598-025-07887-6]
• 60. Defect identification in monolayer MoTe 2 through tunneling ... [https://pubs.rsc.org/en/content/articlehtml/2025/na/d5na00501a]
• 61. Making Color Usage Accessible [https://www.section508.gov/create/making-color-usage-accessible/]
• 62. Dos calculation - Ground state [https://discourse.abinit.org/t/dos-calculation/3701]
• 63. A universal machine learning model for the electronic ... [https://arxiv.org/abs/2508.17418]
• 64. PET-MAD, a universal interatomic potential for advanced ... [https://arxiv.org/html/2503.14118v1]
• 65. lab-cosmo/pet-mad: A universal interatomic potential for ... [https://github.com/lab-cosmo/pet-mad]
• 66. Advancements and prospects of deep learning in ... [https://www.sciencedirect.com/science/article/pii/S2666386424003941]
• 67. Analysis of high-throughput screening reveals the effect ... [https://www.sciencedirect.com/science/article/abs/pii/S1742706114005832]
• 68. High-Throughput Approaches for Screening and Analysis of ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC5702937/]
• 69. Recent advances and applications of deep learning ... [https://www.nature.com/articles/s41524-022-00734-6]
• 70. Multimodal Learning for Materials [https://arxiv.org/html/2312.00111v2]
• 71. A paradigm for high-throughput screening of cell-selective ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC10192934/]
• 72. 14 Comparing quantitative data between individuals [https://bookdown.org/pkaldunn/SRM-Textbook/BetweenQuantData.html]