Approaching the basis set limit in Gaussian-orbital-based periodic calculations with transferability: Performance of pure density functionals for simple semiconductors

Abstract

Simulating solids with quantum chemistry methods and Gaussian-type orbitals (GTOs) has been gaining popularity. Nonetheless, there are few systematic studies that assess the basis set incompleteness error (BSIE) in these GTO-based simulations over a variety of solids. In this work, we report a GTO-based implementation for solids and apply it to address the basis set convergence issue. We employ a simple strategy to generate large uncontracted (unc) GTO basis sets that we call the unc-def2-GTH sets. These basis sets exhibit systematic improvement toward the basis set limit as well as good transferability based on application to a total of 43 simple semiconductors. Most notably, we found the BSIE of unc-def2-QZVP-GTH to be smaller than 0.7 mE_h per atom in total energies and 20 meV in bandgaps for all systems considered here. Using unc-def2-QZVP-GTH, we report bandgap benchmarks of a combinatorially designed meta-generalized gradient approximation (mGGA) functional, B97M-rV, and show that B97M-rV performs similarly (a root-mean-square-deviation of 1.18 eV) to other modern mGGA functionals, M06-L (1.26 eV), MN15-L (1.29 eV), and Strongly Constrained and Appropriately Normed (SCAN) (1.20 eV). This represents a clear improvement over older pure functionals such as local density approximation (1.71 eV) and Perdew-Burke-Ernzerhof (PBE) (1.49 eV), although all these mGGAs are still far from being quantitatively accurate. We also provide several cautionary notes on the use of our uncontracted bases and on future research on GTO basis set development for solids.

I. INTRODUCTION

Condensed phase simulations using quantum chemistry tools originally developed for molecules have gained popularity over many years,^1–9 with the hope of enabling development of new systematically improvable tools that can go beyond standard density functional approaches,¹⁰ as well as existing Green’s function methods^11,12 in the field. These simulations can be broadly categorized into two classes: (1) large Γ-point calculations to describe spatial inhomogeneity as found in gas, liquid, and surface simulations and (2) calculations with a relatively small unit cell and a large number of k-points as relevant for simulations of solids. The former category resembles large cluster calculations that are routinely performed in the molecular community and the use of Gaussian-type orbitals (GTOs) as a computational basis is not uncommon and numerically well-behaved. The use of GTOs to reach the thermodynamic limit (TDL) of (dense) solids often faces numerical difficulties associated with overcompleteness of GTOs that leads to a severe linear dependency among basis functions toward the TDL.^13–16 Nonetheless, many studies have employed Gaussian basis sets using those developed for molecular calculations, or those developed for periodic mean-field calculations,^15–18 or those optimized systems specifically without much in the way of transferability guarantees.^19–21 The use of GTOs for solid-state calculations has been growing as well exemplified by many existing GTO-based quantum chemistry programs with the periodic boundary condition capability.^22–30

The development of compact GTO basis sets^31,32 has a long history in molecular quantum chemistry.^33,34 Since McWeeny’s first proposal³⁵ and Boys’s early attempt³⁶ to use GTOs for molecular systems, many developments on contracted Gaussian basis sets such as atomic natural orbital,³⁷ correlation-consistent,³⁸ and polarization-consistent^39,40 basis sets have made high-accuracy quantum chemistry calculations practical. However, these highly optimized contracted basis sets are usually not considered applicable to solids due to emerging linear dependencies.¹⁶ In the early days of basis set development, even-tempered^41,42 and well-tempered⁴³ bases were explored as a means to obtain high-quality results using only primitive GTOs, reducing the complications in sophisticated optimization procedures for exponents and contraction coefficients. In the even-tempered bases, one employs three parameters for each angular momentum shell l to define a set of “even-tempered” primitive GTOs by

$${\varphi }_{lmk}(\mathbf{r})\propto \exp (-{\zeta }_{lk}{r}^2)r^l{S}_{lm}(\mathrm{\Omega }),$$ (1)

where ϕ_lmk is an atomic orbital, l and m are angular momentum quantum numbers, S_lm(Ω) are the real spherical harmonics at a solid angle Ω, k sets the total number of primitive GTOs for l, m, and ζ_lk is parameterized by a geometric series,

$${\zeta }_{lk}={\alpha }_l{\beta }_l^{k-1},{\alpha }_l,{\beta }_l>0,{\beta }_l\ne 1.$$ (2)

In the well-tempered variants, a more sophisticated form is used for ζ_lk. In the even-tempered basis, one needs to pick a total of three parameters k, α_lk, and β_lk. The appropriate values may be obtained by looking at atoms and small molecules, although finding these values can generally be tedious.⁴² Even-tempered basis sets are generally much larger than contracted GTOs, and thus, they are rarely used in modern quantum chemistry calculations. Nonetheless, these bases have not yet been explored in the context of solid-state applications.

In this work, we propose an even simpler basis set generation protocol than that of even-tempered bases, which does not involve any optimizations. Our procedure is to generate large uncontracted GTO bases that yield density functional theory (DFT) total energies per cell within 0.7 mE_h per atom in the unit cell from the complete basis set limit obtained by a planewave (PW) basis. The idea is to take two existing GTO bases (one from the def2-series⁴⁴ and SZV-MOLOPT-SR-GTH¹⁵), uncontract these bases, and take the union of the resulting primitive GTOs while removing core orbitals that are treated by the underlying GTH pseudopotential. Like the even-tempered bases, our sets are much larger than typical contracted GTOs available in the literature, but they are not optimized for specific systems and/or mean-field methods, so they should naturally bear transferability.

As an application of these bases, we focus on the basic goal of quantifying the basis set error of Gaussian-based DFT calculations. This goal is even more important to reach when considering correlated wavefunction calculations. However, the basis set incompleteness error (BSIE) in correlation energies can be quantified and characterized only after the underlying mean-field energy is converged to the basis set limit. The BSIE was directly quantified by employing the same pseudopotential proposed by Hutter and co-workers (called the GTH pseudopotential)^45,46 in both the new Gaussian-based program developed in this work and a PW-based code, Quantum Espresso (QE).⁴⁷

Furthermore, we also apply our basis set to validating the performance of ten selected pure exchange-correlation (XC) functionals. These ten XC functionals consist of one local density approximation (LDA) functional,^48,49 five generalized gradient approximation (GGA) functionals (PBE,⁵⁰ PBEsol,⁵¹ revPBE,⁵² BLYP,^53,54 and B97-D⁵⁵), and four meta-GGA (mGGA) functionals (SCAN,⁵⁶ M06-L,⁵⁷ MN15-L,⁵⁸ and B97M-rV^59,60). Our benchmark set has a total of 43 semiconductors where 40 of them were taken from the SC40 set⁶¹ and the remaining three (LiH,^62–64 LiF,^65,66 and LiCl^65,66) were taken from other places. The performance of LDA and Perdew–Burke–Ernzerhof (PBE) on the majority of these systems using GTOs was already documented in Ref. 61 though the underlying BSIE of the associated GTO basis sets is unclear. Many PW-based codes including QE have LDA, GGA, and SCAN functionals available, so it is not very difficult to assess their performance using PW-based codes.⁴⁷ In fact, the performance of LDA and GGA functionals, as well as the SCAN mGGA, is relatively well understood for bandgap problems.^67,68 However, the recently developed functionals that were combinatorially optimized for main group molecular chemistry, ωB97X-V,⁶⁹ ωB97M-V,⁷⁰ and B97M-V,^59,60 have rarely appeared in condensed phase studies^71–75 and are relatively less common and less used in PW-based codes. The same is true for the Minnesota functionals (M06-L and MN15-L). Replacing the -V tail with the -rV tail [the rVV10 van der Waals (vdW) correction⁷⁶ instead of the VV10 vdW correction⁷⁷], an efficient implementation of the -rV tail is now available in some planewave-based codes.⁴⁷ Aside from the computational cost associated with the long-range exact exchange, an efficient implementation of these functionals should be readily possible. These combinatorially optimized functionals were found to be statistically the best XC functionals at each rung of Jacob’s ladder for main group chemistry problems,⁷⁸ and they have performed very well in other molecular benchmarks also.^79,80 In the condensed phase, the mGGA (B97M-rV) appears to describe the properties of liquid water as accurately as far more computationally demanding hybrid functions.⁷² However, the performance of B97M-rV for bandgap problems is largely unknown at present. Motivated by this, we report the performance of B97M-rV for bandgaps here.

This paper is organized as follows: (1) We first review basic formalisms of periodic mean-field calculations, the Gaussian planewave (GPW) density fitting scheme, and an efficient implementation of rVV10, (2) we then describe our strategies for generating transferable GTO bases for simulating solids toward the TDL, (3) we discuss computational details, (4) we present results for basis set convergence of DFT total energies and bandgaps using the proposed bases, (5) we assess the performance of pure XC functionals comparing against experimental bandgaps, (6) we deliver cautionary notes on using our bases and on the future basis set development for solids featuring striking failures of existing GTH bases, and (7) we then conclude.

II. THEORY

Periodic mean-field calculations using a linear combination of atomic orbitals have been well-documented in many places.^81,82 Nonetheless, we aim to give a pedagogical review of the relevant theories on periodic DFT calculations within the GPW implementation and the implementation of rVV10 since these are the key compute kernels in our new implementation. Experienced readers may skip Secs. II A–II C and start from Sec. II D.

A. Periodic mean-field calculations

As a consequence of real-space translational symmetry, crystal momentum (k) is a good quantum number. Periodic mean-field (PMF) calculations with GTOs are hence done with crystalline molecular orbitals (CMOs), $\{{\psi }_i^{\mathbf{k}}\}$,⁸³

$${\psi }_i^{\mathbf{k}}(\mathbf{r})=\sum _{\mu }{C}_{\mu i}^{\mathbf{k}}{\varphi }_{\mu }^{\mathbf{k}}(\mathbf{r}),$$ (3)

where crystalline atomic orbitals (CAOs) are defined with a lattice summation,

$${\varphi }_{\mu }^{\mathbf{k}}(\mathbf{r})=\sum _{\mathbf{R}}{\varphi }_{\mu }^{\mathbf{R}}(\mathbf{r})e^{i\mathbf{k}\cdot \mathbf{R}},$$ (4)

where R denotes a lattice vector represented by a sum of integer multiples of primitive vectors of the direct lattice. In PMF calculations, analogously to their molecular counterparts, the PMF energy is minimized when the CMO coefficient matrix obeys a self-consistent Roothaan–Hall equation,

$${\mathbf{F}}^{\mathbf{k}}{\mathbf{C}}^{\mathbf{k}}={\mathbf{S}}^{\mathbf{k}}{\mathbf{C}}^{\mathbf{k}}{ϵ}^{\mathbf{k}},$$ (5)

where F^k is the Fock matrix at k, S^k is the overlap matrix of CAOs at k defined as

$$S_{\mu \nu }^{\mathbf{k}}=\sum _{\mathbf{R}}⟨{\varphi }_{\mu }^{\mathbf{0}}|{\varphi }_{\nu }^{\mathbf{R}}⟩e^{i\mathbf{k}\cdot \mathbf{R}}=\sum _{\mathbf{R}}{S}_{\mu \nu }^{\mathbf{0}\mathbf{R}}{e}^{i\mathbf{k}\cdot \mathbf{R}},$$ (6)

and ϵ^k is the band energy at k.

In periodic calculations with GTOs, it is very common to observe linear dependencies of the CAOs, which makes the metric (overlap) matrix S^k poorly conditioned. Within finite precision computer arithmetic, the resulting truncation error in the inverse metric can lead to numerical instability, convergence issues, and non-trivial errors in the PMF energies. Therefore, handling exact and near linear dependencies is crucial in GTO-based periodic calculations especially when one attempts to get to the basis set limit where linear dependencies become progressively severe. In this work, we adopted the canonical orthogonalization procedure.⁸⁴ The canonical orthogonalization procedure is defined as follows:

• 1.
  The diagonalization of S^k is performed for each k,

  $${\mathbf{S}}^{\mathbf{k}}={\mathbf{U}}^{\mathbf{k}}{\mathbf{s}}^{\mathbf{k}}{({\mathbf{U}}^{\mathbf{k}})}^{†}.$$ (7)
• 2.For a given threshold ϵ_lindep, one retains the $N_{\text{CMO}}^{\mathbf{k}}$ eigenvalues in s^k above ϵ_lindep along with their corresponding eigenvectors. We refer to these subsets of eigenvalues and eigenvectors to as ${\tilde{\mathbf{s}}}^{\mathbf{k}}$ and ${\tilde{\mathbf{U}}}^{\mathbf{k}}$, respectively.
• 3.
  We then define the orthogonalization matrix X^k,

  $${\mathbf{X}}^{\mathbf{k}}={\tilde{\mathbf{U}}}^{\mathbf{k}}{({\tilde{\mathbf{s}}}^{\mathbf{k}})}^{-1/2}.$$ (8)

  The dimension of X^k is N_CAO-by-$N_{\text{CMO}}^{\mathbf{k}}$, and $N_{\text{CMO}}^{\mathbf{k}}$ is the dimension of the effective variational space after removing numerical linear dependencies. We note that we then have

  $${({\mathbf{X}}^{\mathbf{k}})}^{†}{\mathbf{S}}^{\mathbf{k}}{\mathbf{X}}^{\mathbf{k}}={\mathbf{I}}_{N_{\text{CMO}}^{\mathbf{k}}}.$$ (9)

  The choice of ϵ_lindep should be made so as to balance between the numerical stability (i.e., removing enough basis functions to avoid excessive roundoff error and precision loss) and quality of the resulting basis set (i.e., keeping as many basis functions as possible). We picked ϵ_lindep to be 10⁻⁶, which is the default value of our molecular computations in Q-Chem.^85,86 We note that this linear dependency threshold is chosen to be reasonable for double precision and could be tightened up if one could afford quadruple or higher precision arithmetic.

B. Review of the GPW algorithm

The GPW density fitting algorithm was first proposed by Lippert et al.⁸⁷ and has been popularized via the implementation in CP2K.^29,88 The central idea of the algorithm is that one employs planewaves as the auxiliary basis set for density fitting while using GTOs as the primary computational basis set. This strategy is particularly well-suited for solid-state calculations since periodic boundary conditions are naturally imposed and planewave density fitting can be done efficiently. While applying GPW to three-dimensional (3D) systems is the most straightforward, lower-dimensional systems (0D, 1D, and 2D) need special attention to remove spurious image–image interactions. The application of GPW was successfully carried out by Füsti-Molnár and Pulay^89,90 for molecules (i.e., 0D) where spurious image–image interactions were removed exactly by using a truncated Coulomb potential. A similar idea can be generalized to 1D and 2D.⁹¹

Among various terms in F^k, in this work, we focus on the Coulomb matrix, J^k because this contribution is typically the computational bottleneck in pure DFT calculations. We want to compute the Coulomb matrix element between a basis function ϕ_μ located in a unit cell R = 0 (denoted as ${\varphi }_{\mu }^{\mathbf{0}}$) and a basis function ϕ_ν located in a unit cell R (denoted as ${\varphi }_{\nu }^{\mathbf{R}}$),

$$\begin{array}{ll}\hfill J_{\mu \nu }^{\mathbf{0}\mathbf{R}}& \equiv {\int }_{\mathbf{r}}{\varphi }_{\mu }^{\mathbf{0}}(\mathbf{r})V_J(\mathbf{r}){\varphi }_{\nu }^{\mathbf{R}}(\mathbf{r})\hfill \\ \hfill & =\sum _{{\mathbf{R}}^{\prime }}{\int }_{\mathbf{r}\in {\mathbf{R}}^{\prime }}{\varphi }_{\mu }^{\mathbf{0}}(\mathbf{r})V_J(\mathbf{r}-{\mathbf{R}}^{\prime }){\varphi }_{\nu }^{\mathbf{R}}(\mathbf{r}),\hfill \end{array}$$ (10)

where V_J(r) is the Coulomb potential defined as

$$V_J(\mathbf{r})={\int }_{{\mathbf{r}}^{\prime }}\frac{\rho ({\mathbf{r}}^{\prime })}{\mathbf{r}-{\mathbf{r}}^{\prime }},$$ (11)

and we used the fact that V_J(r) is periodic in the unit cell displacements. We note that r ∈ R′ implies that the domain of the integration is restricted to the unit cell centered at R′.

The evaluation of V_J(r) can be done with $\mathcal{O}(N_g\log N_g)$ complexity via the fast Fourier transform (FFT) algorithm for discrete Fourier transform where N_g is the number of grid points within the simulation cell. In reciprocal space,

$$V_J(\mathbf{G})=\frac{4\pi }{|\mathbf{G}{|}^2}\tilde{\rho }(\mathbf{G}),$$ (12)

where

$$\tilde{\rho }(\mathbf{G})=\frac{1}{\mathrm{\Omega }}{\int }_{\mathbf{r}}\rho (\mathbf{r})e^{i\mathbf{G}\cdot \mathbf{r}},$$ (13)

with Ω being the volume of the computational unit cell. Using these, the GPW algorithm computes J^k as follows:

• 1.
  Compute ρ(r) within a unit cell via

  $$\rho (\mathbf{r})=\frac{1}{N_k}\sum _{\mathbf{k}}\sum _i\sum _{\mu \nu }{C}_{\mu i}^{\mathbf{k}}{(C_{\nu i}^{\mathbf{k}})}^{*}{\varphi }_{\mu }^{\mathbf{k}}(\mathbf{r}){({\varphi }_{\nu }^{\mathbf{k}}(\mathbf{r}))}^{*},$$ (14)

  where N_k is the number of k-points.
• 2.Fourier transform ρ(r) to obtain $\tilde{\rho }(\mathbf{G})$. This is the “density fitting” step using a planewave auxiliary basis set.
• 3.Compute the Coulomb potential in reciprocal space via Eq. (12) and inverse Fourier transform to obtain V_J(r). Note that we ignore the |G| = 0 component.
• 4.
  Compute J^k via

  $$J_{\mu \nu }^{\mathbf{k}}={\int }_{\mathbf{r}\in \text{U.C.}}{({\varphi }_{\mu }^{\mathbf{k}}(\mathbf{r}))}^{*}{V}_J(\mathbf{r}){\varphi }_{\nu }^{\mathbf{k}}(\mathbf{r}),$$ (15)

  where the quadrature is performed only within the unit cell (U.C.).

Our implementation computes ${\varphi }_{\mu }^{\mathbf{k}}(\mathbf{r})$ once in the beginning and stores these in memory. Therefore, our GPW implementation for the J-build has $\mathcal{O}(N_k{N}_g)$ storage cost [due to storing ${\varphi }_{\mu }^{\mathbf{k}}(\mathbf{r})$] and $\mathcal{O}(N_k{N}_g+N_g\log N_g)$ computational cost assuming sparsity of CAOs. Since N_g scales with the unit cell volume, while N_k does not, this algorithm approaches $\mathcal{O}(N)$ scaling. Diagonalization is performed by dense linear algebra with $\mathcal{O}(N^3)$ scaling.

C. Summary of implementation of rVV10

Some of the more modern density functionals use the VV10 vdW correction, but the cost of evaluating VV10 scales quadratically with system size. Using ideas from the work of Román-Pérez and Soler,⁹² Sabatini and others proposed an alternative functional form called rVV10,⁷⁶ which can be implemented efficiently with linear complexity for planewave codes while retaining similar accuracy as VV10. Subsequently, the use of rVV10 was verified for combinatorially optimized density functionals (B97M-V, ωB97X-V, and ωB97M-V), leading to B97M-rV, ωB97X-rV, and ωB97M-rV.⁶⁰ We are interested in investigating the performance of these combinatorially optimized functionals for bandgaps, so an efficient implementation of rVV10 is highly desirable.

The rVV10 energy functional reads^76,77

$$E_{\text{rVV}10}=E_{\text{rVV}10}^{\text{local}}+E_{\text{rVV}10}^{\text{non}-\text{local}},$$ (16)

where the local part can be absorbed into the local density approximation terms and the non-local part poses implementational challenges with a naíve quadratic scaling cost. The non-local contribution is defined as

$$E_{\text{rVV}10}^{\text{non}-\text{local}}=\frac{1}{2}{\int }_{\mathbf{r}}{\int }_{{\mathbf{r}}^{\prime }}\rho (\mathbf{r})\kappa {(\mathbf{r})}^{-3/2}\rho ({\mathbf{r}}^{\prime })\kappa {({\mathbf{r}}^{\prime })}^{-3/2}\mathrm{\Phi }({\mathbf{r},\mathbf{r}}^{\prime }),$$ (17)

where ρ(r) is the electron density and the kernel Φ(r, r′) is

$$\mathrm{\Phi }({\mathbf{r},\mathbf{r}}^{\prime })=\frac{-1.5}{(q(\mathbf{r})R^2+1)(q({\mathbf{r}}^{\prime })R^2+1)(q(\mathbf{r})R^2+q({\mathbf{r}}^{\prime })R^2+2)}$$ (18)

with R = |r − r′|. The remaining terms are

$$q(\mathbf{r})=\kappa {(\mathbf{r})}^{-1}\sqrt{C|\frac{\nabla \rho (\mathbf{r})}{\rho (\mathbf{r})}{|}^4+\frac{4}{3}\pi \rho (\mathbf{r})}$$ (19)

and

$$\kappa (\mathbf{r})=1.5b\pi {\left(\frac{\rho (\mathbf{r})}{9\pi }\right)}^{1/6}.$$ (20)

The fixed parameters b and C are a part of the definition of each XC functional that includes the rVV10 contribution. The evaluation of this leads to an overall quadratic scaling in N_g due to its six-dimensional double integral in Eq. (17).

As discussed in Ref. 92, we first use cubic splines to interpolate Φ such that

$$\mathrm{\Phi }({\mathbf{r},\mathbf{r}}^{\prime })\approx \sum _{\alpha ,\beta }\mathrm{\Phi }(q_{\alpha },q_{\beta },R)p_{\alpha }(q(\mathbf{r}))p_{\beta }(q({\mathbf{r}}^{\prime })),$$ (21)

where q_α and q_β are interpolation points and p_α and p_β are interpolating polynomials. This makes the evaluation of Φ computationally convenient because Φ becomes a function of only |r − r′|. Its dependence on q_α and q_β is easy to handle as q_α and q_β are fixed interpolation points. The number of the interpolation points is also very manageable as it is typically set to 20.⁷⁶ We now define an intermediate,

$${\theta }_{\alpha }(\mathbf{r})=\rho (\mathbf{r})\kappa {(\mathbf{r})}^{-3/2}{p}_{\alpha }(q(\mathbf{r})),$$ (22)

and use it to recast the non-local energy contribution into a convolution form,

$$\begin{array}{ll}\hfill E_{\text{rVV}10}^{\text{non}-\text{local}}& =\frac{1}{2}\sum _{\alpha ,\beta }{\int }_{\mathbf{r}}{\int }_{{\mathbf{r}}^{\prime }}{\theta }_{\alpha }(\mathbf{r}){\theta }_{\beta }({\mathbf{r}}^{\prime }){\mathrm{\Phi }}_{\alpha \beta }(|{\mathbf{r}-\mathbf{r}}^{\prime }|)\hfill \\ \hfill & =\frac{1}{2}\sum _{\alpha ,\beta }{\int }_{\mathbf{G}}{\tilde{\theta }}_{\alpha }(\mathbf{G}){\tilde{\theta }}_{\beta }(\mathbf{G}){\tilde{\mathrm{\Phi }}}_{\alpha \beta }(|\mathbf{G}|).\hfill \end{array}$$ (23)

Since Φ_αβ(|r − r′|) = Φ_αβ(R) is spherically symmetric, its Fourier transform can be computed by one-dimensional Fourier-sine transformation. The values of ${\tilde{\mathrm{\Phi }}}_{\alpha \beta }(|\mathbf{G}|)$ on a pre-defined set of |G| points can be precomputed and tabulated. These tabulated values are then used for interpolation to perform the convolution in Eq. (23) for a specific set of |G|. We note that the convolution in Eq. (23) can be performed in $\mathcal{O}(N_g\log N_g)$ time as opposed to the quadratic-scaling runtime of the naíve algorithm. A similar approach can be used to compute the Fock matrix contribution associated with rVV10.

D. Strategies for assessing the basis set error for simple solids and generating transferable GTOs

Our goal in this work is to access the near basis set limit of pure density functionals for solids using GTOs. For this purpose, it is critical to have well-defined basis set limit reference values. A popular planewave code, Quantum Espresso (QE), also implements the GTH pseudopotential⁴⁵ developed by Goedecker et al., which was originally developed for use in the CP2K program.²⁹ We have adopted the same GTH pseudopotential for use in our code as well. This allows for a direct comparison between QE and our code, which is particularly useful because QE can converge the total energy to the basis set limit almost completely by increasing the planewave cutoff.

We considered the 40 semiconductors benchmark set (SC40) first proposed by Heyd et al.⁶¹ along with three rocksalt solids (LiH,^62–64 LiF,^65,66 and LiCl^65,66). For these compounds, all-electron GTO basis sets have been proposed, but their accuracy remains largely unknown.⁶¹ Moreover, to be used with the GTH pseudopotential, we need a basis set without core electrons. Unfortunately, the standard GTH basis set series does not have a broad coverage of the Periodic Table beyond its minimal basis set (SZV-GTH).¹⁵ To access the basis set limit for a variety of solids considered in this work, we propose a simple way to generate a large basis set that yields the total energy per cell close to the basis set limit (errors smaller than 0.7 mE_h per atom for DFT calculations performed here, as will be shown later). We also note that this same strategy of uncontracting existing GTO bases can be applied to the generation of all-electron bases as well.

To generate the basis set, we follow a straightforward procedure:

• 1.We take the existing def2-bases and uncontract the contracted GTOs therein. We then remove GTOs with an exponent greater than 20 since they correspond to core electrons that are already covered by the GTH pseudopotential. This cutoff of 20 was empirically determined, and we expect that the results are not sensitive to the precise value of the cutoff given the large size of our final basis set (see below for more discussion).
• 2.We take the union of these uncontracted def2 bases and the uncontracted SZV-MOLOPT-SR-GTH basis set to enhance the resolution within the minimal basis set space defined by the GTH pseudopotential. This final basis set will be referred to as unc-def2-X-GTH where X can be SVP, TZVP, QZVP, etc.

One may think that having a fixed cutoff of 20 for all elements could be unphysical because increasing the atomic number tends to increase all of GTO exponents. In our case, however, the GTOs from def2 bases with an exponent larger than the largest exponent in SZV-MOLOPT-SR-GTH belong to the core region that is already treated by the GTH pseudopotential. Inspecting the range of exponents in a SZV-MOLOPT-SR-GTH basis, one finds that the largest ones are smaller than 20 with the exception of Na (23.5) and Mg (30.7) up to atomic number 86. Based on our results on solids involving Mg, a cutoff of 20 works well for this element as well. Overall, the contraction of electron density due to the increase in the nuclear charge is reflected appropriately, and there is no sensitivity stemming from this cutoff. We also note that when taking the union of two bases, some of the exponents can be very close in value, but for simplicity we do not remove those obvious near linear dependencies. Instead, we let the canonical orthogonalization procedure take care of them. We report these unc-def-GTH bases (unc-def2-SVP-GTH, unc-def2-SVPD-GTH, unc-def2-TZVP-GTH, unc-def2-TZVPD-GTH, unc-def2-TZVPP-GTH, unc-def2-TZVPPD-GTH, unc-def2-QZVP-GTH, unc-def2-QZVPD-GTH, unc-def2-QZVPP-GTH, and unc-def2-QZVPPD-GTH) through the Zenodo repository,⁹³ as well as in the text files included in the final publication.

With regard to the existing GTH-based contracted GTO basis sets, at present, neither the range of Gaussian exponents nor the contraction coefficients have been specifically optimized to approach the basis set limit: Rather, they have been designed to offer a good trade-off between computational cost and accuracy for solid-state applications. The use of uncontracted basis functions in this work is an attempt to probe the suitability of using a broad range of Gaussian exponents and angular momenta while obtaining the contraction coefficients via variational energy minimization (i.e., the MO coefficients are the contraction coefficients in our case). As a consequence of decontraction, our proposed basis sets range from quite large to very large and are heavily linearly dependent. Nonetheless, this brute force approach will permit us to assess systematic convergence of our total energies toward the basis set limit energies obtained through QE. We emphasize that potential numerical instability issues are quite well handled by the simple canonical orthogonalization procedure.

Last but not least, we note that our Gaussian basis set generation procedure does not utilize any system-dependent parameters or optimization protocols. As evidenced by even-tempered bases,^41,42 this is particularly important for ensuring transferability. Our exponents retain both tight exponents that are effective for condensed phase and relatively diffuse exponents that are effective for atomic (or molecular) limits. Therefore, we expect that the BSIE is relatively insensitive to the underlying geometry. Nonetheless, when atoms come too close together, GTOs are expected to perform more poorly due to the higher degree of linear dependence, as shown in Sec. V C.

III. COMPUTATIONAL DETAILS

We implemented a GPW-based periodic DFT code in a development version of Q-Chem.^85,86 Our implementation assumes the overall $\mathcal{O}(N^2)$ memory storage that amounts to storing Fock, density, and CMO coefficient matrices. For the systems examined in this work, our memory storage is dominated by keeping the GTO basis function values evaluated on the FFT grid despite its formal linear scaling based on sparsity. In the future, this practical memory bottleneck can be removed by computing these on the fly. We also note that our GPW algorithms scale linearly with the system size to produce the Fock matrix and our SCF program scales cubically with system size due to linear algebra functions such as matrix diagonalization. We control the resolution of the PW density fitting basis with a single parameter: the kinetic energy cutoff (E_cut). Each auxiliary basis PW can be indexed by three integers (n₁, n₂, n₃), which reside on a $(2n_1^{\max }-1)\times (2n_2^{\max }-1)\times (2n_3^{\max }-1)$ grid where each integer $n_i\in \{-n_i^{\max },\dots ,n_i^{\max }\}$ with

$$n_i^{\max }=\frac{\sqrt{8E_{\text{cut}}}}{\Vert {\mathbf{b}}_i\Vert },$$ (24)

where b_i denotes one of the reciprocal vectors. For our GPW calculations, we used E_cut of 1500 eV for everything except those that contain Ba (2000 eV) and Mg (4500 eV). The resulting density fitting error was found to be smaller than 100 μE_h per atom in the unit cell, which is negligible for the purpose of this paper.

The reference planewave basis calculations were all performed with QE where we used E_cut (for the wavefunction itself) of 1200 Ry for total energy calculations. For the band structure calculations, we used E_cut of 1200 Ry for systems containing Mg and E_cut of 750 Ry for everything else.

The lattice constants were fixed at experimental values,⁶¹ and experimental bandgaps for the SC40 set were taken from Ref. 61. The experimental bandgaps and lattice constants of LiH, LiF, and LiCl were taken from Refs. 62–66. We used the GTH-LDA pseudopotential in all calculations for both GPW and PW (through QE) calculations to enable direct comparison of total energies. We used the Monkhorst–Pack⁹⁴ k-mesh to sample the first Brillouin zone and ensured the convergence of the total energy per cell to the TDL for all solids examined here. We found that a 6 × 6 × 6 k-mesh is enough to converge the total energy per cell to an error of smaller than 0.1 mhartree for all solids considered. Therefore, for band structure calculations and cold curve calculations, we used a 6 × 6 × 6 Monkhorst–Pack k-mesh. Since the GPW implementation is also available in other open-source packages such as CP2K²⁹ and PySCF,⁹⁵ we also used these two packages to validate our implementation in the initial stage of this work.

We examined a total of ten XC functionals, LDA (Slater exchange⁴⁸ and PZ81 correlation⁴⁹), PBE,⁵⁰ PBEsol,⁵¹ revPBE,⁵² BLYP,^53,54 B97-D,⁵⁵ SCAN,⁵⁶ M06-L,⁵⁷ MN15-L,⁵⁸ and B97M-rV.^59,60 For 11 solids in our benchmark set (C, Si, SiC, BN, BP, AlN, MgO, MgS, LiH, LiF, and LiCl), widely used GTH basis sets¹⁵ are available: DZVP-GTH, TZVP-GTH, TZV2P-GTH, QZV2P-GTH, and QZV3P-GTH. We therefore assessed the accuracy of those existing bases only over a smaller subset of our benchmark set, but our proposed basis sets were examined for all 43 solids considered in this work. The basis set convergence study against PW was carried out only for LDA and PBE, while the overall bandgap accuracy was examined for all ten functionals.

IV. RESULTS AND DISCUSSION

A. Basis set convergence of total DFT energies

We first examine the subset of 11 solids for LDA and PBE functionals as presented in Fig. 1. In particular, Fig. 1 shows the root-mean-square-deviation (RMSD) of total energies compared to QE total energies (namely, total energies in the basis set limit) as a function of the size of the k-mesh. N_k = 216 (6 × 6 × 6) is enough to reach the TDL. While smaller k-mesh calculations such as the Γ-point only calculations are unphysical, we are interested in how the basis set error changes as a function of the k-mesh size. Since both PW and our GTO results share the same Hamiltonian for each size of k-mesh, we can assess the basis set error of GTO calculations as long as the reference PW energies are fully converged to the basis set limit. For all k-mesh sizes, the GTH basis set series shows systematically more accurate results relative to the basis set limit as the cardinality (and the size of the basis set) increases. We note that N_k = 1 (just including the Γ-point) shows the largest basis set error in all examples considered here. This is because the local expressive power of GTOs also increases as one increases the size of k-mesh due to the non-orthogonality of GTOs. While the systematic improvement of GTH bases is very appealing, we note that the residual basis set error with QZV3P-GTH is still about 5 mE_h, which is quite large considering how small the simulation cells are (2 or 4 atoms in total).

FIG. 1.

Root-mean-square-deviation (RMSD) of DFT total energies (mE_h) per cell with respect to that of QE over 11 solids as a function of the number of k-points for (a) LDA and (b) PBE functionals using GTH and unc-def2-GTH bases.

The unc-def2-GTH basis series also shows a systematic improvement with cardinal number, with much smaller errors than the corresponding contracted GTH basis results. As an example, the performance of unc-def2-SVP-GTH is nearly on par with TZV2P-GTH except at the Γ-point. The larger bases, unc-def2-TZVP-GTH and unc-def2-QZVP-GTH, both perform excellently on this set, including the Γ-point result. In particular, unc-def2-QZVP-GTH is able to deliver total energies in the TDL that are all within 1 mE_h of the basis set limit. This shows the completeness of our proposed bases, although, of course, these are much bigger in size than standard GTH bases and therefore far more computationally demanding. We provide more detailed information on selected elements in Sec. V A. Finally, we note that Fig. 1(a) for LDA and (b) for PBE show virtually no difference, which suggests that our conclusions do not depend on functionals (of course, functionals that depend particularly strongly on fine details of the density may be far harder to converge to the basis set limit using GTOs⁹⁶).

In Fig. 2, we repeat the same analysis but over the entire benchmark set of 43 solids. As before, unc-def2-GTH bases struggle for N_k = 1 but work well for larger k-meshes. RMSD systematically decreases as we increase the size of the basis set. With the largest basis set, unc-def2-QZVP-GTH, we achieve better than 1 mE_h accuracy in the TDL for the LDA and PBE functionals, as measured by the RMSD values. Systems with the largest error in the TDL are SrSe (1.2 mE_h) in the case of LDA and GaP (1.4 mE_h) in the case of PBE. As observed in the case of even-tempered bases, we expect that the result can be systematically made better by adding more exponents and increasing the maximum angular momentum.^41,42 For instance, in the case of SrSe/LDA, employing unc-def2-QZVPP-GTH (adding two additional f functions to both Sr and Se), we observe an error of 0.4 mE_h, which is three times smaller than that of unc-def2-QZVP-GTH. While we can obtain overall better results by using unc-def2-QZVPP-GTH, we will mainly focus on the use of the unc-def2-QZVP-GTH basis set for the rest of the paper for simplicity. In summary, these benchmark calculations suggest that unc-def2-GTH basis sets can achieve near basis set limit DFT total energies reliably toward the TDL. This result implies that the range of exponents and angular momenta in our bases is quite appropriate for solids.

FIG. 2.

Root-mean-square-deviation (RMSD) of DFT total energies (mE_h) per cell with respect to that of QE over 43 solids as a function of the number of k-points for (a) LDA and (b) PBE functionals using unc-def2-GTH bases.

B. Basis set convergence of DFT bandgaps

In many materials applications, DFT calculations are used not just to compute the ground state energy but also to obtain spectral information through Kohn–Sham orbital energies.⁶⁷ In doing so, one uses information from virtual orbitals in addition to that from occupied orbitals. In the case of total energies presented in Sec. IV A, only occupied orbitals affect the results. Here, we are assessing the quality of the difference between the lowest energy virtual orbital (i.e., the conduction band minimum) and the higher energy occupied orbital. It is possible that some BSIEs may cancel when taking energy differences.

In Fig. 3, we present the RMSD of bandgaps using GTO bases compared to the basis set limit results for LDA and PBE functionals. To compare unc-def2-GTH bases with GTH bases, we limit ourselves to the subset of 11 solids for the time being. Somewhat surprisingly, we observe almost no improvement in the bandgap when going from DZVP-GTH to TZVP-GTH. By contrast, Fig. 1 shows a reduction in the RMSD of total energies of about 8 mE_h when increasing the basis set size from DZVP-GTH to TZVP-GTH. However, this total energy improvement does not result in any bandgap improvement. Nonetheless, past TZVP-GTH, the GTH bases do show systematic improvement in the bandgap estimation. With the largest GTH basis set (QZV3P-GTH), RMSD in the bandgap is 18–20 meV depending on the XC functional. Consistent with the total energy benchmark presented in Sec. IV A, unc-def2-GTH bases also exhibit systematic improvement. While the quality of unc-def2-SVP-GTH was on par with TZV2P-GTH in Fig. 1, its bandgap is clearly worse than that of TZV2P-GTH highlighting favorable error cancellation in TZV2P-GTH. Nonetheless, unc-def2-TZVP-GTH is similar to QZV3P-GTH and unc-def2-QZVP-GTH has RMSD of 5.8 and 4.2 meV for LDA and PBE, respectively, showing its ability to converge bandgaps to the basis set limit.

FIG. 3.

Root-mean-square-deviation (RMSD) of DFT bandgaps (meV) with respect to those of QE over 11 solids (a) LDA and (b) PBE functionals using GTH and unc-def2-GTH bases.

Encouraged by these results, we also analyzed the BSIEs in bandgaps over all 43 solids using unc-def2-GTH bases as presented in Fig. 4. With unc-def2-SVP-GTH, the RMSD value is about 40 meV and it becomes less than 20 meV when using unc-def2-TZVP-GTH. Finally, with unc-def2-QZVP-GTH, the RMSD value becomes 6.9 and 6.3 meV, respectively, for LDA and PBE. However, we note that the largest deviation is about 20 meV in both functionals, which corresponds to the bandgap of SrSe. SrSe is the system with the largest total energy error for LDA as noted in the discussion of Fig. 2 in Sec. IV A. Again, this remaining error can be further reduced by adding more GTOs to unc-def2-QZVP-GTH (e.g., using unc-def2-QZVPP-GTH), but we do not pursue this here. The central message of this section is that the BSIE in the bandgap reported in this paper using unc-def2-QZVP-GTH is smaller than 20 meV based on the numerical data. This is about 50 times smaller than the intrinsic errors in standard functionals for bandgaps, so unc-def2-QZVP-GTH should be suitable for benchmarking purposes.

FIG. 4.

Root-mean-square-deviation (RMSD) of DFT bandgaps (meV) with respect to those of QE over 43 solids: (a) LDA and (b) PBE functionals using unc-def2-GTH bases.

C. Performance of pure DFT functionals

Having established the accuracy of unc-def2-GTH bases, we assess the performance of pure DFT functionals over these simple solids. Unfortunately, some of the 43 solids considered here do not have experimental bandgaps. These solids are BSb, CaS, CaSe, CaTe, SrS, SrSe, and SrTe. Leaving aside these seven cases, we have a total of 36 experimental bandgaps. unc-def2-QZVP-GTH is used with all XC functionals considered in this section. The DFT bandgaps over 43 solids along with the available experimental gaps are presented in Table I. For an overall summary, it may be more instructive to look at statistics of the bandgap results, as shown in Fig. 5. Looking at the mean-average-deviation (MAD), it is immediately evident that all pure functionals examined here exhibit the infamous bandgap underestimation problem of pure functionals.⁶⁷ Nonetheless, one can still find systematic improvement for going from the simplest functional, LDA, to more modern meta-GGA functionals, SCAN, M06-L, MN15-L, and B97M-rV in terms of the root-mean-square-deviation (RMSD) values. While the performance of B97M-rV is not great for those bandgaps, it still stays as one of the more accurate pure functionals for these problems. This is encouraging because B97M-rV is statistically the most accurate pure XC functional in main group chemistry applications.⁷⁸ Overall, all pure functionals perform poorly in this benchmark study and the inclusion of exact exchange seems necessary. This is not a new observation on its own and has been well-documented even for nearly the same benchmark set that we study here.^61,97 In Ref. 97, the authors consider many different ways to analyze the statistical data of LDA, PBE, and PBEsol along with other hybrid functionals and revealed that hybrid functionals always perform the best in nine out of ten statistical analyses. It will be interesting to revisit this benchmark set with modern hybrid functionals in the future.

TABLE I.

Experimental and theoretical bandgaps (or fundamental gaps) (eV) from various functionals over 43 solids. N/A means “not available.” RMSD, MAD, and MAX denote the root-mean-square-deviation, mean-average-deviation, and maximum deviation, respectively, in reference to experimental values. We took experimental references for three rocksalt solids (LiH, LiF, and LiCl) from Refs. 62–66, respectively. The rest of experimental values were taken from Ref. 61.

	LDA	PBE	PBEsol	revPBE	BLYP	B97-D	SCAN	M06-L	MN15-L	B97M-rV	Expt.
C	4.12	4.33	4.16	4.38	4.60	4.57	4.64	4.84	4.24	4.67	5.48
Si	0.49	0.66	0.52	0.72	0.94	0.91	0.93	1.12	0.96	0.92	1.17
Ge	0.00	0.00	0.00	0.00	0.00	0.00	0.20	0.46	0.43	0.26	0.74
SiC	1.33	1.50	1.35	1.54	1.85	1.95	1.81	1.83	1.92	2.07	2.42
BN	4.36	4.64	4.42	4.71	5.03	5.07	5.03	4.94	4.98	5.28	6.22
BP	1.20	1.38	1.23	1.42	1.66	1.62	1.67	1.95	1.61	1.70	2.4
BAs	1.16	1.34	1.19	1.40	1.60	1.59	1.57	1.85	1.55	1.66	1.46
BSb	0.76	0.91	0.78	0.96	1.15	1.17	1.06	1.21	1.08	1.15	N/A
AlP	1.47	1.67	1.50	1.75	1.98	2.04	1.99	2.20	2.12	2.16	2.51
AlAs	1.36	1.58	1.40	1.66	1.89	1.95	1.86	2.00	1.99	2.03	2.23
AlSb	1.17	1.36	1.20	1.45	1.57	1.58	1.56	1.78	1.63	1.63	1.68
bGaN	1.61	1.79	1.68	1.85	1.86	1.96	1.86	1.88	1.43	1.86	3.3
GaP	1.44	1.66	1.50	1.75	1.70	1.72	1.85	1.89	1.84	2.05	2.35
GaAs	0.29	0.51	0.41	0.58	0.44	0.46	0.62	0.92	0.72	1.01	1.52
GaSb	0.00	0.17	0.08	0.23	0.08	0.08	0.21	0.50	0.36	0.64	0.73
InP	0.42	0.61	0.52	0.68	0.57	0.56	0.57	0.86	0.36	0.88	1.42
InAs	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.03	0.41
InSb	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.13	0.23
ZnS	1.80	2.12	1.94	2.24	2.15	2.24	2.38	2.56	2.00	2.36	3.66
ZnSe	1.01	1.33	1.15	1.45	1.35	1.43	1.60	1.80	1.31	1.67	2.7
ZnTe	1.07	1.38	1.22	1.49	1.38	1.43	1.60	1.83	1.40	1.79	2.38
CdS	0.85	1.15	0.98	1.27	1.16	1.22	1.22	1.34	0.77	1.23	2.55
CdSe	0.34	0.64	0.48	0.76	0.66	0.70	0.72	0.88	0.34	0.82	1.9
CdTe	0.52	0.81	0.66	0.93	0.81	0.82	0.84	1.07	0.52	1.06	1.92
MgS	3.27	3.57	3.39	3.73	3.71	3.88	4.14	4.03	4.06	4.17	5.4
MgTe	2.30	2.59	2.43	2.75	2.72	2.89	3.16	3.18	2.99	3.23	3.6
MgO	4.68	4.93	4.79	5.06	5.17	5.49	5.59	5.01	5.69	5.55	7.22
MgSe	1.70	2.01	1.89	2.16	2.10	2.47	2.58	2.66	3.20	2.70	2.47
CaS	2.17	2.41	2.27	2.51	2.52	2.64	2.92	2.56	3.03	3.05	N/A
CaSe	1.90	2.14	2.00	2.24	2.27	2.38	2.63	2.28	2.73	2.78	N/A
CaTe	1.42	1.65	1.51	1.74	1.80	1.90	2.08	1.74	2.19	2.27	N/A
SrS	2.22	2.49	2.33	2.61	2.62	2.74	2.92	2.57	2.86	2.95	N/A
SrSe	2.01	2.28	2.12	2.40	2.43	2.54	2.69	2.35	2.63	2.74	N/A
SrTe	1.57	1.83	1.66	1.94	2.00	2.10	2.20	1.87	2.17	2.29	N/A
BaS	2.01	2.26	2.11	2.38	2.36	2.44	2.58	2.25	2.38	2.51	3.88
BaSe	1.83	2.07	1.92	2.19	2.19	2.26	2.39	2.07	2.21	2.35	3.58
BaTe	1.49	1.74	1.58	1.85	1.87	1.93	2.03	1.74	1.90	2.03	3.08
LiH	2.64	3.01	2.78	3.15	3.44	3.69	3.61	3.87	4.52	4.39	4.9
LiF	8.92	9.33	9.11	9.56	9.49	9.92	10.08	9.64	10.27	9.77	14.2
LiCl	6.01	6.40	6.18	6.61	6.56	6.83	7.21	7.08	7.48	7.22	9.4
AlN	4.25	4.38	4.25	4.43	4.66	4.78	4.87	4.75	4.94	5.24	6.13
GaN	1.86	2.05	1.94	2.12	2.12	2.21	2.12	2.13	1.68	2.11	3.5
InN	0.00	0.00	0.00	0.01	0.00	0.04	0.00	0.00	0.00	0.00	0.69
RMSD	1.72	1.50	1.64	1.41	1.39	1.28	1.20	1.26	1.27	1.17	N/A
MAD	−1.46	−1.23	−1.37	−1.14	−1.10	−1.02	−0.95	−0.90	−0.99	−0.84	N/A
MAX	5.28	4.87	5.09	4.64	4.71	4.28	4.12	4.56	3.93	4.43	N/A

FIG. 5.

Bandgap (eV) comparison over 36 solids between DFT (ten different functionals) and experiments: blue—root-mean-square-deviation (RMSD) of DFT bandgaps (eV) with respect to those of experiments and red—mean-average-deviation (MAD) of DFT bandgaps (eV) with respect to those of experiments.

For simplicity and due to the unavailability of functional-specific GTH pseudopotentials for most XC functionals considered here, we employed the GTH-LDA pseudopotential for all functionals in this section. Since this is not ideal, we checked the sensitivity of our conclusions with respect to the choice of the pseudopotential by testing GTH-LDA, GTH-PBE, and GTH-BLYP pseudopotentials with the BLYP functional. In all cases, the RMSD and MAD are affected by less than 0.1 eV, which is a smaller energy scale than that of the bandgap errors by roughly a factor of 10. For completeness, we provide the relevant numerical data in the supplementary material (see Table S1). In the future, all-electron calculations could be done with all-electron basis sets generated via a similar protocol presented here. Alternatively, one could generate functional-specific GTH pseudopotentials for the modern XC functionals considered here.

V. OUTLOOK FOR FUTURE BASIS SET DESIGN

In this section, we would like to deliver cautionary notes on using our proposed bases and some discussion on future research in basis set design for solids.

A. Our basis set is accurate but very large

While our proposed unc-def2-GTH bases are of high quality, these bases are very large due to the decontraction from the original contracted GTO bases. This large size carries a significant computational cost. This is the major drawback of even-tempered and well-tempered bases, and it is the one that our unc-def2-GTH bases also share. To be more concrete, we provide the number of basis functions for selected elements (Si, C, O, and Mg) in Table II. unc-def2-SVP-GTH is about three times bigger than DZVP-GTH while our unc-def2-TZVP-GTH is roughly three times bigger than TZV2P-GTH. Similarly, our largest basis set unc-def2-QZVP-GTH is about 2.5–3 times larger than QZV3P-GTH.

TABLE II.

Number of basis functions in the basis sets used in this work for selected elements (Si, C, O, and Mg).

	Si	C	O	Mg
SZV-GTH	4	4	4	5
DZVP-GTH	13	13	13	14
TZVP-GTH	17	17	17	18
TZV2P-GTH	22	22	22	23
QZV2P-GTH	26	26	26	27
QZV3P-GTH	31	31	31	32
unc-def2-SVP-GTH	40	41	40	53
unc-def2-TZVP-GTH	62	58	57	68
unc-def2-QZVP-GTH	90	83	81	86

Because of computational cost and memory demand, there is a need to compress these bases for practical calculations. Perhaps, the most difficult (but most effective if done correctly) way to compress them is to obtain transferable contraction coefficients. One could start by inspecting the molecular orbitals (or Bloch orbitals) that our calculations produce for those simple solids. Another strategy is to take these mean-field molecular orbitals and compress the virtual space for subsequent correlation calculations, for instance using the random phase approximation (RPA). The use of natural orbitals to compress the virtual space was shown to be effective, and would be a good starting point for making our basis more compact.⁹⁸ We note that it is also unclear whether our proposed bases exhibit any scaling properties, which will allow for higher accuracy by using basis set extrapolation for correlation energy calculations, which could be further investigated in the future.

B. Even low-lying virtual orbitals can be difficult to describe well

Basis sets that are optimized for mean-field calculations such as GTH bases often behave erratically in correlated calculations.¹⁹ Since these bases tend to yield good occupied orbitals, the poor performance of correlation calculations can be attributed to virtual orbitals. Furthermore, low-lying virtual orbitals play important roles in describing optical properties and related excited states. Therefore, high-quality basis sets should produce qualitatively accurate virtuals. As an example, we present the band structure of MgO using QZV3P-GTH and unc-def2-QZVP-GTH and compare them against that of QE. MgO has a total of eight occupied orbitals, and we computed up to the 16th band in QE for comparison purposes. We note that the challenge of MgO conduction bands for GTOs was noted before in Ref. 99, but we focus on a wider range of conduction bands here. The pertinent band structures are presented in Fig. 6.

FIG. 6.

LDA band structure of MgO: (a) comparing QZV3P-GTH against QE and (b) comparing unc-def2-QZVP-GTH against QE. The band energies are shifted such that the highest valence band energy is located at zero. The red circle in (a) highlights the qualitative failure of QZV3P-GTH virtual orbitals.

In both bases, the valence bands and the first few conduction bands are in excellent agreement with those of QE. However, the higher-lying virtuals of QZV3P-GTH [in Fig. 6(a)] start to deviate significantly from those of QE. The most striking failure is the lack of the 5th virtual orbital highlighted under a red circle in Fig. 6(a). On the other hand, the virtuals from unc-def2-QZVP-GTH have visually indistinguishable energies when compared to QE, highlighting its potential utility for correlated calculations as well. We also tried a smaller unc-def2-GTH basis set, namely, unc-def2-TZVP-GTH. It turns out that even unc-def2-TZVP-GTH misses the same virtual that QZV3P-GTH misses as well. With further investigations, we found that the 5th virtual orbital is missed by basis sets without any f function on the Mg atom. To be more concrete, we added one f function to Mg in the QZV3P-GTH basis where an exponent of 0.16 was taken from unc-def2-QZVP-GTH. With this basis set, we recover the missing band at the Γ-point. This additional f function introduces only a 0.2 mE_h energy lowering in the ground state, but it is essential to capture one of the low-lying conduction bands.

This example emphasizes that more attention to the low-lying virtual orbitals should be paid when designing GTO basis sets for applications such as conduction band structure, time-dependent DFT and correlated methods such as RPA. Existing GTO bases designed primarily to describe the occupied space may likewise exhibit qualitative failures like this case.

C. Transferability across different lattice constants is challenging

Cold curves of solids are often of great interest for experimentalists. Cold curves are analogous to potential energy curves (PECs) in molecular quantum chemistry. Similar to PECs, as one shrinks the lattice constant and brings atoms close to one another, a larger number of near linear dependencies occur, and the quality of the underlying GTO basis degrades because of discarding such functions by canonical orthogonalization. Furthermore, system-dependent optimization strategies can struggle for cold curves because basis sets are usually optimized for one specific geometry (usually equilibrium geometries).^19,20 As a result of this, varying lattice constants can be challenging using GTO basis sets as the system approaches its high-pressure configuration (shorter lattice constants) or atomic limits (longer lattice constants).

As an example to illustrate this point, we computed a cold curve of SiC using PBE and the GTH-LDA pseudopotential with TZV2P-GTH, QZV3P-GTH, unc-QZV3P-GTH, unc-def2-TZVP, and unc-def2-QZVP. The Brillouin zone was sampled with 6 × 6 × 6 k-points via the Monkhorst–Pack algorithm. Here, unc-QZV3P-GTH is the basis obtained from decontracting QZV3P-GTH. Using unc-QZV3P-GTH, we can quantify the error coming from the contraction coefficients of QZV3P-GTH. As before, the QE results (the same functional and pseudopotential) serve as the basis set limit reference values. The pertinent results are presented in Fig. 7.

FIG. 7.

Investigation of a SiC cold curve using PBE: (a) Basis set error with respect to QE as a function of lattice constant for various basis sets. (b) Average number of linearly independent AOs as a function of lattice constant for various basis sets.

Figure 7(a) shows that TZV2P-GTH, QZV3P-GTH, and unc-QZV3P-GTH bases make a large error especially when compressing the lattice. It is also instructive to quantify the nonparallelity error (NPE) over the range of lattice constants examined here as a means to measure error cancellation. The NPE is 25.7 mE_h, 10.6 mE_h, 8.3 mE_h, 3.0 mE_h, and 0.9 mE_h for TZV2P-GTH, QZV3P-GTH, unc-QZV3P-GTH, unc-def2-TZVP-GTH, and unc-def2-QZVP-GTH, respectively. Interestingly, unc-QZV3P-GTH reduces the basis set error by only a small amount, which implies that the contraction coefficients of QZV3P-GTH for those elements are transferable over a wide range of lattice constants. It also suggests that the range of exponents in QZV3P-GTH becomes inappropriate for smaller lattice constants. Comparing the exponents of unc-QZV3P-GTH and unc-def2-TZVP-GTH, we find that unc-def2-TZVP-GTH has more compact exponents for spd shells and has an f shell for Si that is not present in unc-QZV3P-GTH. These more compact GTOs likely become more important at closer distances and hence explain the differences between two bases.

The main cause of these generally large NPEs is the fact that at closer distances the quality of those GTO bases degrades as shown in Fig. 7(b), which quantifies the number of orthogonalized basis functions retained after canonical orthogonalization. Since each k-point has a slightly different number of linearly independent AOs, we present the average values over 216 k-points as a function of lattice constant. Evidently, the number of linearly independent AOs decreases as the lattice constant decreases, which in turn increases the basis set incompleteness error. Nonetheless, the largest basis set, unc-def2-QZVP-GTH, is able to achieve a satisfying NPE in this case, which highlights the utility of this basis set for accurate cold curve simulations.

VI. CONCLUSIONS

In this paper, we discussed strategies for generating large and accurate uncontracted Gaussian bases (unc-def2-GTH bases), which do not resort to system- or method-specific optimizations. Using a new implementation of the Gaussian atomic orbital plus planewave density fitting approach in Q-Chem, the basis set incompleteness error in our proposed bases was then assessed over 43 prototypical semiconductors by comparing the pure density functional theory total energies per cell and bandgaps against those from fully converged planewave results. We found that the basis set incompleteness error in total energy and bandgap with our largest GTO basis (unc-def2-QZVP-GTH) is smaller than 0.7 mE_h per atom in the unit cell and less than 20 meV, respectively, verifying the validity of the range of exponents and angular momenta in the proposed bases.

In the application of our bases, we focused on the assessment of ten pure density functionals for predicting the bandgaps of 36 semiconductors whose experimental gaps are well-documented. Not surprisingly, we found that all examined pure functionals (LDA, PBE, PBEsol, revPBE, BLYP, B97-D, SCAN, M06-L, MN15-L, and B97M-rV) significantly underestimate the bandgaps of these materials. The combinatorially optimized mGGA functional, B97M-rV, performs as well as do other modern mGGA functionals. Our work suggests that combinatorially optimized range-separated hybrid functionals such as ωB97X-rV and ωB97M-rV will be highly interesting to study since they may also exhibit better accuracy compared to other relatively older range-separated hybrid functionals or even short-range hybrid functionals.

We also made several cautionary remarks on our GTO bases as well as on the future research in GTO basis design for solids:

• 1.Our basis sets are accurate but large, so there is a need for a way to compress our basis sets further for both mean-field and correlation calculations.
• 2.The widely used GTH bases may qualitatively fail for describing low-lying virtual orbitals, which will affect the subsequent correlation and optical calculations. At much greater computational cost, our unc-def2-QZVP-GTH basis set was shown to accurately capture all of the low-lying virtual orbitals of MgO including the one missed by QZV3P-GTH.
• 3.Reducing the non-parallelity error of the basis set incompleteness error is challenging particularly due to the high pressure region of cold curves, which exhibits a higher number of near linear dependencies.

In the future, we will test several ways (e.g., finding universal contraction coefficients and frozen natural orbitals⁹⁸) to compress our unc-def2-GTH bases and investigate the basis set convergence of correlation and optical methods with these bases. Furthermore, simple solids such as LiH have many numerical data of the total Hartree–Fock energies toward the basis set limit,^100–104 which could be a good testbed for our basis sets in the future.

SUPPLEMENTARY MATERIAL

The supplementary material contains the test of the impact of different GTH pseudopotentials in the bandgaps.

AUTHOR DECLARATIONS

Conflict of Interest

E.E. and M.H.-G. are part-owners of Q-Chem, Inc.

DATA AVAILABILITY

The data that support the findings of this study are available within the article and its supplementary material.