Improving the Performance of Tao–Mo Non-empirical Density

Article Cite This: J. Phys. Chem. A 2019, 123, 6356−6369

pubs.acs.org/JPCA

Improving the Performance of Tao−Mo Non-empirical Density Functional with Broader Applicability in Quantum Chemistry and Materials Science Subrata Jana,*,† Kedar Sharma,†,‡ and Prasanjit Samal*,† †

School of Physical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar 752050, India School of Physics, Indian Institute of Science Education and Research, Maruthamala, Vithura, Thiruvananthapuram 695551, India

‡

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S Supporting Information *

ABSTRACT: A revised version of the semilocal exchange-correlation functional [Tao, J.; Mo, Y. Phys. Rev. Lett. 2016, 117, 073001] (TM) is proposed by incorporating the modifications to its correlation content obtained from the full high-density second-order gradient expansion as proposed in the case of the revised Tao−Perdew−Staroverov−Scuseria (revTPSS) [Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Constantin, L. A.; Sun, J. Phys. Rev. Lett. 2009, 103, 026403] functional. The present construction improves the performance of the TM functional over a wide range of quantum chemical and solid-state properties (thermochemical and structural). More specifically, the cohesive energies, jellium surface exchange-correlation energies, and real metallic surface energies are improved by preserving the accuracy of the solid-state lattice constants and bulk moduli. The present proposition is not only physically motivated but also enhances the applicability of the TM functional. New physical insights with the proper exemplification of the present modification, which is presented here, can further help in constructing more realistic non-empirical density functionals.

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first one is known as non-empirical or semi-empirical density functional approximations, which are practically useful for both quantum chemists and solid-state physicists. However, heavily parametrized density functionals19,20,31 are also proposed, but those functionals perform well within the parametrized test set and practically not so stable for solid-state calculations. There are various ways of constructing the non-empirical density functionals. Some functionals are constructed from constraint satisfaction7,15,21−30 or the exchange hole model,18,30 or by satisfying both.30 Starting from the local density approximation (LDA), 3 the higher rungs of XC density functional approximations are constructed by including the gradient of density and incorporating the Kohn−Sham (KS) kinetic energy dependency. The gradient-dependent functionals are known as generalized gradient approximations (GGAs).3−17

INTRODUCTION 1,2

Density functional theory is visualized as an extremely simplified version of the complicated many-electron Schrödinger equation. In this, all the quantum many-electron effects are embedded into an effective one-electron-like potential comprising an unknown exchange-correlation (XC) density functional. As the exact analytic form of the XC functional is not known, the central task of DFT is to approximate the XC energy/potential functional. Several approximations of the XC functionals with a broad range of applications in quantum chemistry, solid-state physics, and materials science are thus proposed.3−33 However, due to the reliable, quick, and accurate output, the approximate semilocal XC functionals are widely used, and the corresponding successes in quantum chemistry and condensed matter physics are undisputed.34−52 In fact, approximations of the semilocal exchange-correlation functionals are proposed from various physical viewpoints. In density functional semilocal approximations, there are mainly two classes of approximations that have been widely used. The © 2019 American Chemical Society

Received: March 28, 2019 Revised: June 6, 2019 Published: June 26, 2019 6356

DOI: 10.1021/acs.jpca.9b02921 J. Phys. Chem. A 2019, 123, 6356−6369

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The Journal of Physical Chemistry A Meanwhile, those obtained by incorporating the KS kinetic energy functionals as an extra ingredient are recognized as the meta-generalized gradient approximations (meta-GGAs).18−33 In this way, developments and subsequent attempts to construct very accurate approximations of XC functionals make DFT practically a very appealing and widely used theory to extract the wealth of several observable quantities, such as thermochemistry and kinematics of molecules, bond lengths and angles, reaction barrier heights, dynamics of molecules, dipole moments, polarizabilities, infrared intensities, lattice constants of solids, bulk moduli, cohesive energies, surface energies, and so on.34−52 However, recent advances in functional developments and their applications indicate that the desired accuracy of both quantum chemistry and solid-state physics is achievable by satisfying more exact constraints by the non-empirical functionals. One such functional was proposed recently by Tao and Mo.30 The motivation of the present work follows from the alluring features of the TM functional. The TM semilocal functional is designed using the density matrix expansion (DME) techniques, which are accurate for compact density, that is, atoms and molecules. To perform accurately in the case of solids, the slowly varying fourth-order gradient approximation is also included within the functional form. In the original TM functional, it is shown that the TM exchange performs differently for several solid-state properties with both the TPSS22 and TM correlation energy30 (designed by modifying the TPSS correlation). However, the TM correlation obeys a more exact constraint than TPSS for the low-density limit,30 and the TM exchange functional coupled with TM correlation performs more accurately than its TMTPSS counterpart for most of the molecular and solidstate properties.41,42,45 Interestingly, for jellium surface XC energy, the TPSS correlation performs better than TM correlation.45 In this work, we seek modification to the TM correlation energy prompted by the improvement achieved by the revTPSS correction23 over TPSS correlation. It is done by implementing the revTPSS-like modifications into the correlation energy of the TM functional. It is shown that the modified correlation coupled with TM exchange keeps all the good features of the TM functional intact. Additionally, it leads to noticeable improvement of results for most of the thermochemical test sets, cohesive energies, jellium surface XC energies, and surface energies of real metals by retaining the accuracy of lattice constants and bulk moduli. This improvement clearly indicates that the change in correlation energy is necessary for it to perform equally well for the thermochemical, bulk, and surface properties of solids. To present all these modifications and functional performances, we organize this paper as follows. In the following section, we will present the underlying physical motivations and relevant modifications to the TM functional. Following it, we will assess the performance of the revised functional thus obtained in the context of thermochemical accuracy, solid-state lattice constants, bulk moduli, cohesive energies, jellium surface XC energies, and surface energies of the real metals. We will conclude by discussing the results and future prospects of the proposed revision of the TM functional.

ExTM = −

TM 3 dr ∫ ρ(r)ϵunif x (r)Fx

(1)

where ϵunif is the exchange energy density in the uniform x electron gas approximation and Fx is the TM exchange enhancement factor30 given by FxTM = wFxDME + (1 − w)Fxsc

(2)

In this, the DME-based exchange enhancement factor FDME is x given by FDME = 1/f 2 + 7R/(9f4) where R = 1 + 595(2λ − x 1)2p/54 − [τ − 3(λ2 − λ + 1/2)(τ − τunif − | ∇ ρ|2/72ρ)]/τunif, occup 1 τ = ∑σ ∑i |∇ψiσ | is the KS kinetic energy density for the 2 occupied states ψiσ, τ unif = 3kF 2ρ /10 is the kinetic energy density of a uniform electron gas, p = s2 = | ∇ ρ|2/(2kFρ)2 is the square of the reduced density gradient (s), kF = (3π2ρ)1/3 is the Fermi wave vector, f = [1 + 10(70y/27) + βy2]1/10, y = (2λ − 1)2p, λ = 0.6866, and β = 79.873. Meanwhile, Fsc x is the slowly varying fourth-order gradient approximation (GE4), which is 2 given by Fsc x = {1 + 10[(10/81 + 50p/729)p + 146q̃ /2025 − 1/10 (73q̃/405)[3τW/(5τ)](1 − τW/τ)]} where q̃ = (9/20)(α − τ−τ 1) + 2p/3 and α = unifW is the meta-GGA ingredients with τ

τW =

|∇ρ|2 8ρ

being the von Weizsäcker kinetic energy density. In

the original construction of the TM functional, extrapolation is done between the compact density (i.e., DME) and slowly varying fourth-order density correction (sc) through a function w. As for solids, the slowly varying bulk valance region is important. Therefore, it is necessary to recover the correct fourth-order density gradient approximation of exchange. The function w is given by w=

z 2 + 3z 3 , (1 + z 3)2

(3)

τW

where z = τ is the meta-GGA ingredient. Due to different behaviors of z, w switches from the molecular or atomic systems to the slowly varying solid-state system; that is, it switches from DME to the slowly varying density limit. It is noteworthy to mention that near the bond center of the molecules, z ≈ 0 implies that w ≈ 0. In the core and density tail region, the systems become effectively one or two electron-like, that is, τ ≈ τW, which implies w ≈ 1. In bulk solids where the semi-classical gradient expansion of the kinetic energy density is valid, w < 1. Therefore, both the DME exchange enhancement factor and slowly varying fourth-order density gradient expansion contribute to the slowly varying density approximation. Regarding the correlation content of the TM functional,30 it is based on one-electron self-interaction free TPSS correlation. It includes the modified TPSS correlation satisfying a (nearly) exact constraint in the low-density or strong-interaction limit of the meta-GGA correlation.30 Also, this modification actually reduces the errors of the lattice constant of bulk solids compared to its corresponding TPSS correlation. The modified TM correlation proposed by Tao and Mo is given as follows30

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THEORY As our starting point for the proposed modification to the TM functional, we consider the TM exchange energy functional having the form30

EcTM[ρ↑ , ρ↓ ] =

∫ d3rρϵcrevPKZB[1 + d ϵcrevPKZB(τ W /τ)3]

(4)

where 6357


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The Journal of Physical Chemistry A

increases with the reduced density gradient (s) (shown in Figure 1). Regarding the parameter b, it is chosen to be 0.40 in

ϵcrevPKZB = ϵcPBE(ρ↑ , ρ↓ , ∇ρ↑ , ∇ρ↓ )[1 + CTM(ζ , ξ)(τ W /τ )2 ] n − [1 + C(ζ , ξ)](τ W /τ )2 ∑ σ ϵ̃c (5) σ n

with CTM(ζ , ξ) =

0.1ζ 2 + 0.32ζ 4 {1 + ξ [(1 + ζ )−4/3 + (1 − ζ )−4/3 ] /2}4 2

(6)

and ϵ̃c = max[ϵGGA (ρσ , 0, ∇ρσ , 0), ϵGGA (ρ↑ , ρ↓ , ∇ρ↑ , ∇ρ↓ )] c c (7)

Here, ζ = (ρ↑ − ρ↓)/n, and ξ = | ∇ ζ | /2(3π ρ) . The parameter d = 2.8 hartree−1 is chosen to accurately predict the jellium surface correlation energy.22 The TM functional is proven to be very accurate in predicting the energetic and structural properties of atoms, molecules, and solids.41,42,45 This is due to the exact constraint satisfaction of the exchange hole, which is incorporated through FDME . This is the first ever functional of its kind that x extrapolates between the compact density and slowly varying fourth-order gradient expansion. It is noteworthy to mention that the prime contribution for the atomic and molecular exchange energy comes from the DME exchange term. However, there is still room for improving the performance of the TM functional. The motivation of the present paper follows from the works elsewhere41,42,45 where it is shown that the TM exchange combined with the TM and TPSS correlations performs differently for the structural and energetic properties of solids. Nevertheless, it is also common practice to improve the functional performance based on simple modifications on the exchange and correlation form. Different modifications based on various physical motivations are done in this direction. For example, based on the Perdew− Kurth−Zupan−Blaha (PKZB)21 meta-GGA, the TPSS metaGGA functional was proposed, and later a simple modification to the TPSS exchange and correlation was done through revTPSS.23 Also, we want to acknowledge several works on the recent modifications of the revTPSS and SCAN functionals.24,25,28,32 Now, we propose two simple modifications of the TM exchange-correlation functional, which improves the molecular, bulk, and surface properties of solids, retaining all the good properties of the TM functional. In doing so, (i) we adopt the flexible choice for the meta-GGA ingredient q̃ by replacing it 2p 9(α − 1) with q̃b where qb̃ = 1/2 + 3 . This modification of 2

1/3

Figure 1. Slowly varying enhancement factor Fsc x of the TM and revTM functionals with α = 2.0, α = 3.0, and α = 4.0.

the TPSS- and revTPSS-based functional so that Fx becomes a monotonically increasing function of reduced density gradient (s). However, in our present case, we find no reason to change this value for modification of the TM functional. Therefore, this modification is done in the same spirit as is proposed from the PKZB to TPSS meta-GGA functional. Also, note that for the molecular test set, b = 0 produces nearly the same results.22 Having refined the TM exchange, we now focus on the TM correlation. The TM correlation is based on the TPSS correlation where the parameter β = 0.066725 is obtained from the high-density limit (rs → 0) of the slowly varying second-order gradient approximation of the correlation energy functional. The TPSS correlation, which is based on the PBE correlation, adopts the same value of the parameter β. Later, Perdew et al.23 adopted a density-dependent β parameter derived by Hu and Langreth.58 The β parameter of revTPSS has two limiting values: Firstly, β = 0.066725 which is obtained by satisfying the high-density limit (rs → 0) of the slowly varying second-order gradient approximation of correlation energy, and secondly, β = 0.0375 which satisfies the local density linear response criterion in the low-density limit (rs → ∞). Note that this second-order gradient expansion of the correlation energy functional, that is, β = 0.066725, is used in the PBE7 and TPSS functionals.22 In addition, the rsdependent β of the revTPSS23 functional satisfies the local density linear response criterion in the low-density limit (rs → ∞) of the correlation energy functional. In the low-density limit (rs → ∞), the second-order gradient expansion for exchange (which is 10/81) cancels with the second-order gradient expansion of the correlation energy functional (where β = 0.0375), which retains the local density linear response criterion in this limit.11,23 In the present modification of the TM exchange, we incorporate this excellent feature of the revTPSS correlation in the functional form of revTM. Therefore, in our modification of the correlation, (ii) we adopt the proposition made by Perdew et al.23 in the revTPSS functional and have modified β TPSS to β revTPSS (r s ) =

20[1 + bα(α − 1)]

q̃ is used in the TPSS and revTPSS functionals but not in the TM functional. By construction, q̃b becomes q̃ at b = 0. Both q̃b and q̃ follow closely the reduced Laplacian gradient (q). For 0 ≤ α ≤ 1 (single orbital toward slowly varying density ‑ sc region52), the FrevTM (exchange enhancement factor of x slowly varying density correction of the revised TM (revTM) ‑ sc functional with q̃b) and FTM (exchange enhancement factor x of slowly varying density correction of the TM functional with q̃) essentially follow the same behavior. Meanwhile, for α > > 1 ‑ sc (regions of overlapping closed shells52) and s ≈ 0, FTM and x ‑ sc FrevTM are quite different, which is relevant in the middle of x bonds. On the other hand, when α is closer to 1 (slowly varying density region), these enhancement factors are ‑ sc practically the same. In both cases, FrevTM monotonically x

0.066725(1 + 0.1rs)/(1 + 0.1778rs) where rs =

1/3

( ) 3 4πρ

is

the Seitz radius. Apart from this modification, we keep the 6358


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The Journal of Physical Chemistry A form of the TM correlation intact, which is revised from the TPSS correlation as30 C(ζ , 0) = 0.1ζ 2 + 0.32ζ 4

(8)

This modification (eq 8) is incorporated to improve the correlation energy functional in the low-density or stronginteraction limit in which the XC energy is spin-independent.30 Also, as the TM exchange obeys the exact fourth-order gradient expansion (GE4), the choice of βrevTPSS in principle satisfies the linear response of the spin-unpolarized uniform electron gas in the low-density limit (rs → ∞) of the correlation through the error cancellation between secondorder gradient expansion of exchange and correlation as explained before. Therefore, modification of the β parameter from the high-density second-order gradient expansion (as is done in TPSS) to the full second-order gradient expansion (which is incorporated in revTPSS) keeps conditions (i)−(iii) intact for correlation in ref 30, and additionally, in the lowdensity limit, the second-order gradient terms of the correlation cancel with the second-order gradient expansion of the exchange in order to retain the local density linear response criterion of the revTM functional. Also, note that all important constraints of the meta-GGA correlation functional follow the present modification of the correlation energy functional. Essentially, in the uniform scaling to the high-density limit (r → λr where λ is the coordinate scaling, dr → (1/λ3)d(λr), ρ(r) → λ3ρ(λr), rs → 0, t → ∞ as λ → ∞), the correlation energy of the TM or revTM functional follows from the Görling−Levy56,59,60 second-order perturbation theory (GL2) where the form of ϵGL2TM or c ϵGL2revTM can be derived from its TPSS form.61 Thus, in the c uniform scaling to the high-density limit, the gradient terms of GGA or meta-GGA cancel the logarithmic term of the uniform density limit.7,56,61 In this limit, the rs-dependent β becomes βrevTPSS(rs → 0) = βTPSS. Most importantly, as mentioned before, due to the rs-dependent β form, the second-order gradient terms for exchange and correlation cancel for the spinunpolarized uniform electron gas in the low-density limit (rs → ∞) of the slowly varying density correction (t → 0) of the revTM functional for which the revTM functional behaves as ϵrevTM → ϵLDA + β(rs)ϕ(ζ)3t2 (in the slowly varying density c c correction) 1 [(1 2

2/3

where

t=

3π 2 16

1/3

( )

s ϕ(ζ )rs1/2

Figure 2. Difference of the TM and revTM correlation energy densities with respect to LDA.

strong-interaction limit (small ω) to the high-density limit (large ω). From Figure 3 (left panel), it is evident that the revTM correlation performs better than its TM counterpart due to the satisfaction of a more exact constraint and due to the error cancellation in the low-density limit (small ω), and the revTM XC also performs slightly better in this limit compared to TM (shown in Figure 3 (right panel)). However, a more improved β parameter for this type of model system is also proposed in the TPSSloc functional,24 and later it is used in the BLOC functional.25 To complete our analysis, we also consider the exchange and correlation energy of noble gas atoms. The results are summarized in Table 1 where we calculate the exchange and correlation energy contributions as obtained from the exchange and correlation part of the TM and revTM functionals. In Table 2, we calculate the atomization energy as obtained from the exchange-only functional and Hartree− Fock exchange together with different meta-GGA correlations. This is done to encapsulate the behavior of change in the exchange and correlation part of the functional. Inspecting Tables 1 and 2, it is clearly evident that the modification in the exchange part has no effect on the atomization energy although the atomic exchange energies differ moderately. However, the change in the correlation improves atomization energy. The more extensive benchmark test is presented in the Results section.

and

2/3

ϕ(ζ ) = + ζ ) + (1 − ζ ) ]. Since in the low-density limit, the effects of correlation become stronger in nature compared to the exchange, error cancellation between revTM exchange and revTM correlation is expected. Note that all these conditions can be proven and derived from the coordinate scaling relation.61 Next, to stress on the improvement due to the change in the correlation energy functional, we consider the correlation energy densities with respect to LDA as a functional of distance (z) of the jellium surface at the bulk parameter rs = 2 in Figure 2. Unlike the TM correlation, the revTM correlation slightly de-enhanced inside the bulk solids and in the vacuum. This is logical because the revTM correlation uses the parameter β, which is modified from TPSS, and the revTM correlation more exactly satisfies the low-density and highdensity limits.23,30 Also, the improvement in the correlation part is clearly evident from Figure 3 where we show the revTM correlation for Hooke’s atom. This model system is used to predict the functional performance from the low-density or

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RESULTS Computational Details. To assess the performance of the revTM functional with other popular GGA and meta-GGA functionals, we implement the revTM functional in the NWChem62 and Vienna ab initio simulation package (VASP) 63−66 codes for the molecular and solid-state calculations, respectively. The molecular calculations are performed using the 6-3114++G(3df,3pd) basis set. Meanwhile, the plane-wave basis set is used for the solid-state calculations. For molecular calculations, we consider Minnesota 2.035 and G2/14867 test sets. The reference values of the G2/148 test set were taken from ref 30. In particular, our test set includes the following: (i) AE17, atomic energies of 17 atoms (H−Cl); (ii) AE6, atomization energies of 6 molecules; (iii) G2/148, atomization energies of the G2/148 molecular test set; (iv) EA13, 13 electron affinities; (v) IP21, 21 ionization potentials; (vi) PA8, 8 proton affinities; (vii) BH6, 6 6359


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Figure 3. Errors for the (a) correlation and (b) exchange-correlation energies of Hooke’s atom for the TM and revTM at different values of the classical electron distances r0 = (ω2/2)−1/3 where ω is the frequency of the isotropic harmonic potential. Here, Ec and Exc are the exact correlation and exchange-correlation of Hooke’s atom.

Table 1. Exchange and Correlation Energies (in hartree) of TM and revTM Functionals of Noble Gas Atoms As Obtained Using the Spin-Restricted Hartree−Fock Orbitals and Densitiesa53 Ex

Ec

atoms

HF

TM

revTM

expt

TM

revTM

He Ne Ar Kr Xe

−1.026 −12.109 −30.190 −93.892 −179.169

−1.032 −12.154 −30.109 −93.204 −177.555

−1.032 −12.149 −30.095 −93.193 −177.534

−0.0420 −0.3905 −0.7222 −2.0664

−0.0291 −0.3153 −0.6522 −1.6883 −2.8195

−0.0301 −0.3208 −0.6615 −1.7046 −2.8420

a

HF exact exchange values are taken from ref 54, and exact correlation values are taken from refs 54−56.

Table 2. Atomization Energies (in kcal/mol) for the AE6 Test Set As Computed Self-Consistently with Exchange-Only Functionals (Hartree−Fock, TMx, and revTMx) and Hartree−Fock Exchange Together with meta-GGA Correlation Functionalsa exchange only

HF exchange + meta-GGA correlation

atoms

HF

TMx

revTMx

ref 57

HF + TMc

HF + revTMc

SiH4 SiO S2 C3H4 C2H2O2 C4H8 MAE

256.1 107.6 50.2 523.2 420.6 870.3

255.2 149.8 74.9 547.3 492.5 875.4 28.1

253.6 149.5 74.1 547.8 492.6 875.0 28.3

322.4 192.1 101.7 704.8 633.4 1149.0

317.7 145.2 86.4 679.3 566.1 1144.6 27.4

319.3 145.7 86.4 681.3 567.4 1148.0 25.9

a

The last row reports MAE (in kcal/mol). The 6-311++(3df,3pd) basis set is used.

barrier heights; (viii) HB6, 6 hydrogen bonding; (ix) DI6, 6 dipole bonding; and (x) CT7, dissociation energies of the 7 charge transfer molecules. Meanwhile, our solid-state test set contains equilibrium lattice constants, bulk moduli, and cohesive energies of 29 solids, which includes the following: simple metals (Li, Na, K, Ca, Sr, Ba, Al), transition metals (V, Ni, Cu, Rh, Pd, Ag, Pt), ionic solids (LiF, LiCl, NaF, NaCl, MgO), insulators (BN, BP, AlN), and semiconductors (C, Si, Ge, SiC, GaN, GaP, GaAs). These test sets have been considered in ref 68. Along with this test set, we consider interlayer and intralayer binding energies of two layer materials. The details of all these test sets with the space group are presented in Table 3. All the bulk calculations are performed with the 16 × 16 × 16 Γ-centered k points with an energy cutoff of 700 eV for the smooth convergence with respect to the plane wave and energy convergence. All solid-

Table 3. Test Set of the 29 Strongly and 2 Layered Material Solids Considered in this Worka strongly bound solids Li (Im3̅m), Na (Im3̅m), K (Im3̅m), Ca (Fm3̅m), Sr. (Fm3̅m), Ba (Im3̅m), Al (Fm3̅m) V (Im3̅m), Ni (Fm3̅m), Cu (Fm3̅m), Rh (Fm3̅m), Pd (Fm3̅m), Ag (Fm3̅m), Pt (Fm3̅m) LiF (Fm3̅m), LiCl (Fm3̅m), NaF (Fm3̅m), NaCl (Fm3̅m), MgO (Fm3̅m) BN (F4̅3m), BP (F4̅3m), AlN (F4̅3m) C (Fd3̅m), Si (Fd3̅m), Ge (Fd3̅m), SiC (F4̅3m), GaN (F4̅3m), GaP (F4̅3m), GaAs (F4̅3m) layered solids graphite (P63/mmc), h-BN (P63/mmc) a

6360

Space groups of individual solids are given in parenthesis.


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The Journal of Physical Chemistry A Table 4. Summary of Deviations Using Different Methods in Terms of ME and MAEa functional

parameter

L(S)DA

Atomic energy (hartree) AE17 ME −0.673 MAE 0.673 Atomization energy (kcal/mol) AE6 MAE 76.4 G2/148 MAE 83.4 Electron affinity (kcal/mol) EA13 ME −5.60 MAE 5.71 Ionization potential (kcal/mol) IP13 ME −4.48 MAE 5.31 Proton affinity (kcal/mol) PA8 ME 4.99 MAE 4.99 Barrier heights (kcal/mol) BH6 ME −17.9 MAE 17.9 Hydrogen bond dissociation energy (kcal/mol) HB6 ME 4.50 MAE 4.50 Dipole bond dissociation energy (kcal/mol) DI6 ME 2.92 MAE 2.92 Charge transfer dissociation energy (kcal/mol) CT7 ME 6.64 MAE 6.64 RMAE 7.27

PBE

TPSS

revTPSS

SCAN

TM

revTM

−0.029 0.038

−0.220 0.251

−0.054 0.054

−0.054 0.054

−0.076 0.076

0.028 0.028

15.0 18.2

5.6 6.1

6.0 4.4

3.4 3.5

5.1 6.4

4.3 6.1

−1.05 2.20

0.76 2.37

1.42 2.69

1.01 3.30

3.20 3.72

2.39 3.31

−2.01 3.49

−1.69 3.02

−0.61 2.91

−3.80 4.47

0.40 2.99

−0.01 2.98

−0.16 1.42

−2.81 2.81

−2.92 2.92

−1.11 1.20

−1.60 1.88

−1.52 1.61

−9.42 9.42

−8.34 8.34

−7.43 7.43

−7.49 7.49

−7.45 7.45

−7.65 7.65

0.12 0.32

−0.43 0.43

−0.40 0.40

0.93 0.93

0.13 0.20

−0.12 0.18

−0.33 0.40

−0.35 0.51

−0.30 0.47

0.60 1.17

0.50 0.50

0.17 0.37

2.85 2.85 1.68

2.10 2.10 1.09

2.24 2.24 1.05

2.84 2.84 1.00

2.97 2.97 1.04

2.72 2.72 0.94

1

a

M

The lowest MAE between the TM and revTM is in bold font. The relative mean absolute error (RMAE = M ∑i = 1 MAEi /MAEi ,SCAN ) of all test sets (excluding the AE17 test set) in the last row is taken with respect to the SCAN functional. M is the total number of benchmark tests.

However, in the atomization energies test sets AE17, G2/148, and IP13, the performance of TM and revTM is qualitatively the same. The ME of IP13 is predicted to be much better in revTM than those of other functionals considered here. In the case of BH6, the TM shows better performance than revTM. It is noteworthy to mention that accurate prediction of barrier heights is quite difficult for semilocal functionals because of the many-electron self-interaction problems in the transition state of those molecules. The range-separated hybrid functionals perform better in such cases because of the proper incorporation of the non-locality information.69−73 We also observe that for hydrogen (HB6), dipole (DI6), and charge transfer (CT7) bond dissociation energies, revTM improves the performance of TM as those are non-covalent interactions and improvement of revTM compared to TM clearly indicates that the change in correlation is important. It is also a very interesting observation that even though the revTPSS correlation does not improve dramatically over the TPSS correlation as far as the thermochemical accuracy is concerned, the change in correlation however in the revTM suits perfectly for most of the thermochemical test sets. Now, concerning the performance of TM-based functionals with other meta-GGAs and GGAs, the SCAN performs very well for the AE6 atomization energies. However, we observe that the MAE of revTM for the EA13 matches closely with SCAN for which PBE performs better than others. For IP13, TPSS, revTPSS, TM, and revTM perform similarly. In this case, the SCAN functional gives an MAE of 4.47 kcal/mol. In particular, the SCAN functional is quite accurate in the case of

state calculations are done considering non-magnetic phases and ambient-condition crystal structures except Ni for which the magnetic calculations are taken into account. The antisymmetric box size of 18 × 19 × 20 Å3 is considered for the atomic calculations. Besides all these fundamental tests, we also calculate the jellium surface XC energies and surface energies of real metallic systems to incorporate the functional performances. Regarding the surface energy calculations, it is done with 16 × 16 × 1 Γ-centered k points with an energy convergence criterion of 1.0 × 10−6 eV and an energy cutoff of 700 eV. The dipole corrections are also taken into consideration in this calculation along with >20 Å vacuum to avoid the interaction between the periodic surfaces. The functionals considered for comparison with the revTM are LDA, Perdew−Burke− Ernzerhof (PBE),7 PBE reparametrized for solids (PBEsol),11 TPSS,22 revTPSS,23 strongly constrained and appropriately normed (SCAN),29 and TM.30 The accuracy in the performance of all the functionals is assessed by calculating the mean error (ME), mean absolute error (MAE), and mean absolute relative percentage error (MARPE). Thermochemical Accuracy. Let us start with the thermochemical accuracy of each functional. The results of all tests are summarized in Table 4 where we report the ME and MAE of the individual test set for each functional. Inspection of Table 4 shows that the revised version of TM, that is, revTM, performs better than TM for most of the thermochemical test sets. In particular, for AE6, EA13, PA8, HB6, DI6, and CT7 test cases, revTM shows improvement. 6361


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quite accurate for the solid-state structural properties, and its comparison with other functionals is necessary to check the robustness of the functional performances in a more competitive manner. Here, we report only the ME, MAE, and MARPE of the test set. The individual values are given in the Supporting Information (Tables S1−S3). The overall statistics are presented in Table 3. Strongly Bound Solids. Lattice Constants. First, we focus on the performance in the case of lattice constants. The lattice constant is one of the fundamental properties of solids, and the accuracy of several solid-state structural properties depends on the accuracy of the lattice constants. In Table S1 in the Supporting Information, we summarize the performances of all the functionals. The zero-point an-harmonic expansion (ZPAE)-corrected experimental lattice constant values are taken from different studies in the literature, and they are listed in the Table S1 of the Supporting Information. In Figure 5, we also plot the percentage deviation of the different solids given in Table S1 of the Supporting Information. Concerning the overall performance of the relevant functionals, as usual, the LDA functional underestimates the lattice constants, whereas PBE overestimates the lattice constants. The overestimation of the PBE functional is reduced by restoring the exact second-order gradient correction as applied in the PBEsol functional. Regarding the meta-GGA functionals, the TPSS functional overestimates the lattice constants, which are improved by the revised version of the TPSS, that is, revTPSS. As mentioned before, the revTPSS functional has been proposed by modifying the exchange enhancement factor such that the slowly varying density gradient approximation is satisfied for a wide range of the reduced density gradients (s). Also, modification in the correlation energy functional is incorporated in order to take care of the full linear response. Regarding the most advanced meta-GGA functionals, SCAN improves the meta-GGA performance and results in the lowest MAE. Concerning the overall performances of TM and revTM functionals, revTM slightly improves the performance of TM.

AE6, PA8, and BH6 test sets. From the improvement of revTM over the TM functional, it is quite clear that the change in correlation results in the balance of the description of different thermochemical test sets and thus the change in correlation part is relevant. To put our results in a more competitive manner, we also plot the ratios of the MAE of the individual test set with respect to the recent most popular and accurate meta-GGA functionalSCAN. In the last row of Table 4, we calculate the total relative mean absolute error (RMAE) of all the functionals. From Table 4 and Figure 4, it is clear that the

Figure 4. Shown is the MAE of different test sets with respect to the SCAN functional using different functionals. The L(S)DA functional and AE17 test set are not considered.

TM and revTM functionals perform in a quite balanced way for all the test sets. Overall, the RMAE of revTM is slightly better than those of SCAN and TM functionals. However, in these test cases, we do not include the dispersion-bonded or weakly bonded systems because those require van der Waals correction to properly take the binding energies. Solid-State Performances. Having established the thermochemical accuracy of the revTM functional, we now focus on the solid-state performance of the functional. Unlike the thermochemical test set, in the present solid-state performance, we include the PBEsol functional, which is

Figure 5. Shown is the MARPE of the lattice constants obtained from different functionals. 6362


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Table 5. Error Statistics (ME, MAE, and MARPE) of the Test Set of 29 Strongly Bound Solids for the Lattice Constant (a0), Bulk Modulus (B0), and Cohesive Energy (Ecoh) Using Different Functionalsa a0

B0

Ecoh

functional

ME

MAE

MARPE

ME

MAE

MARPE

ME

MAE

MARPE

LDA PBE PBEsol TPSS revTPSS SCAN TM revTM

−0.089 0.040 −0.013 0.029 0.010 −0.003 −0.023 −0.008

0.089 0.053 0.036 0.042 0.034 0.027 0.037 0.034

1.952 1.154 0.778 0.920 0.768 0.615 0.830 0.767

10.0 −11.2 0.5 −5.4 −0.6 2.0 2.1 0.5

12.3 14.9 9.2 12.5 11.5 9.0 7.5 8.6

11.3 11.2 6.4 10.1 8.8 7.6 6.0 5.9

0.711 −0.105 0.272 −0.053 0.063 0.089 0.297 0.202

0.711 0.160 0.289 0.169 0.218 0.176 0.300 0.235

16.803 4.327 6.746 4.344 5.481 4.978 8.554 6.252

a

Detailed performance of the individual functional for individual solids is given in the Supporting Information. The units of ME and MAE are Å, GPa, and eV/atom for a0, B0, and Ecoh, respectively, while for MARPE, it is in %. The lowest and largest errors of all functionals are in bold font and underlined.

Figure 6. Shown is the MARPE of the bulk moduli of different functionals.

modification in the exchange enhancement factor quite reasonably improves the lattice constants of those solids for which TM has the tendency to underestimate the lattice constants slightly. Note that in the alkali metals and ionic solids, the long-range van der Waals interaction between the semi-core states shrinks the lattice constants, which is correctly captured by the SCAN, TM, and revTM functionals because SCAN includes the intermediate vdW interaction and TMbased functionals include the long-range vdW interaction due to the oscillation of the exchange hole. However, slight elongation in revTM is observed compared to that in TM because the revTM exchange enhancement factor is now deenhanced compared to that in TM in the density overlap region as shown in Figure 1. Through the modifications of the TM functional, we observe a small but systematic improvement in the lattice constants of revTM over TM (Table 5). Bulk Moduli. The accuracy of bulk moduli depends on the accuracy of the lattice constants. The performances of functionals for the bulk moduli are presented in Table S2 of the Supporting Information, and MRPE is plotted in Figure 6. Here, we use the third-order Birch−Murnaghan equation of state to fit the energy−volume curve. Regarding the overall performances, the TM functional gives the lowest MAE

Now, we focus on the performance of TM and revTM for the individual group of solid-state structures. For the simple metals and ionic insulators, we observe a clear improvement in the lattice constants by the revTM functional. The TM functional only performs better than revTM for the semiconductors. However, for transition metals and insulators, both TM and revTM perform almost equivalently. The systematic improvement in the performances of revTM can be understood as the change in the correlation functional and the change of q̃ → q̃b, which slightly reduces the effect of the exchange enhancement factor. More physically, in the intershell region (where α may be larger even for small s (as ‑ sc enhances the exchange energy shown in Figure 1)), the FTM x density. Thus, the exchange hole becomes more centered in the intershell region than in the outer shell region, which effects a reduction in the lattice constants. On the other hand, the modification of q̃ → q̃b in the revTM functional actually reduces this effect and results in extension in the lattice constants. This is clearly evident from Table 1 of the Supporting Information where we observe that the lattice constants obtained using the revTM functional for simple metals and ionic solids extend a little bit compared to its TM functional counterpart. Therefore, we can say that this small 6363


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Figure 7. Shown is the MARPE of the cohesive energies of different functionals.

Table 6. Intralayer (a) and Interlayer (c) Lattice Constant for Graphite and h-BN As Obtained Using SCAN, TM, and revTM Functionalsa ref 74

SCAN

TM

revTM

solids

a

c

a

c

a

c

a

c

graphite h-BN

2.46 2.51

6.70 6.54

2.45 2.50

6.86 6.84

2.46 2.51

6.64 6.47

2.46 2.51

6.93 6.69

a

Note that here, we do not consider other functionals for the sake of simplicity.

However, within meta-GGA functionals, the lowest MAE is observed using the TPSS functional. Also, the SCAN functional is quite good in predicting the cohesive energies compared to TM and revTM. The SCAN functional is better suited for the simple metals, transition metals, and insulators in comparison to the revTM. Regarding the GGA-based functionals, as usual, PBE performs better compared to PBEsol. This is due to the improved atomization energies of PBE compared to those of PBEsol. Layered Solids. To encapsulate the impact of the change in the exchange functional to the solid state properties, we consider the interlayer (c) and intralayer (a) lattice constants of two popular layered solids: graphite and h-BN. We choose the layered solids for a realistic reason. The interlayer lattice constant of the layered solids depends on the vdW interactions. Meanwhile, the intralayer lattice constant is related to stronger covalent-type bonding.74−78 To date, the SCAN + revised Vydrov and Van Voorhis non-local correlation (rVV10)76 is the most popular for predicting accurately both the interlayer and intralayer lattice constants in equal footing. However, in ref 76, it was shown that the SCAN functional overestimates c, which is modified upon addition of the rVV10 correlation. From Table 6, it is evident that the interlayer lattice constants of revTM are quite close to those of the SCAN functional. Meanwhile, the TM functional results are close to the reference values. Also, all the functionals are quite close to the reference value for predicting the intralayer lattice constant. Physically, the different behavior of the TM and revTM originates from the change in q̃ → q̃b. In the regions of

followed by the revTM and SCAN functionals. The revTM functional improves the performance of the TM functional for simple metals and ionic solids. This is because the lattice constants of the revTM functionals are obtained to be better for those solids. For the very same reason, the semiconductor bulk moduli are underestimated by revTM compared to the TM functional. Concerning the performance of other functionals, the performances of bulk moduli follow the tendency in the lattice constants. Except the SCAN- and TM-based functionals, other GGA and meta-GGA functionals show underestimation in results for most of the solids. The most pronounced underestimation is observed for the PBE functional, while LDA, as usual, overestimates the bulk moduli and shows a tendency opposite to that of PBE. Cohesive Energies. Next, we assess the performance of revTM along with other functionals for the cohesive energies of solids. The performance of functionals is presented in Table S3 of the Supporting Information and plotted in Figure 7. Here, we also observe interesting performance for revTM compared to TM functionals as revTM improves the performance over TM for most of the solids. Individual consideration shows that revTM improves the performance for simple metals, transition metals, insulators, and semiconductors. For ionic solids, the performance of TM is slightly better than that of revTM. This is quite natural because the change in correlation energy actually suits more revTM than the TM correlation. We observe that for most of the solids, the revTM reduces the cohesive energies of the TM and therefore improves the overall performance of the cohesive energies. 6364


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The Journal of Physical Chemistry A Table 7. Jellium Surface Exchange Energies (σx) (in erg/cm2)a rs

σLDA x

σPBE x

σPBEsol x

σTPSS x

σrevTPSS x

σSCAN x

σTM x

σrevTM x

exact

2 3 4 6 ME MAE MARE

3036 669 224 43.5 160.9 160.9 45.8

2436 465 128 11.8 −72.1 72.1 20.9

2666 540 162 22.9 15.5 15.5 2.9

2553 498 141 15.5 −30.4 30.4 11.9

2657 532 157 20.6 9.4 10.1 2.2

2632 489 127 6.3 −18.7 22.7 22.6

2640 528 155 20.2 3.6 5.5 2.6

2640 527 155 20.1 3.3 5.2 2.7

2624 526 157 22

a

Exact values are taken from ref 23. The LDA, PBE, PBEsol, and revTPSS values are taken from ref 23. The SCAN values are taken from ref 51. The rest of the functional values are calculated in this work.

Table 8. Jellium Surface Exchange-Correlation Energies (σxc) (in erg/cm2) of Different Functionalsa rs

σLDA xc

σPBE xc

σPBEsol xc

σTPSS xc

σrevTPSS xc

σSCAN xc

σTM xc

σrevTM xc

DMC

2 3 4 6 ME MAE MARE

3354 764 261 53.0 −10.5 10.5 0.4

3265 741 252 52.0 −41.0 41.0 3.1

3374 774 267 56.7 −0.6 8.4 2.6

3380 772 266 55.5 −0.1 5.9 1.9

3428 783 268 55.4 15.1 15.1 2.5

3422 788 274 58.9 17.2 17.2 4.9

3517 824 291 64.7 55.7 55.7 11.1

3468 803 280 60.3 34.3 34.3 6.9

3392 ± 50 768 ± 10 261 ± 8 53.0

a

LDA, PBE, PBEsol, and 4revTPSS values are taken from ref 23. The SCAN values are taken from ref 51. The DMC values are taken from ref 51. The rest of the functional values are calculated in this work.

and varies rapidly on the surface. Because in solids, the valence electron density is slowly varying, the analysis of jellium surface XC energies helps to understand the accuracy of a particular XC energy for the solid-state systems. The underlying jellium surface exchange-correlation energy is defined as13

overlapping density of the layered materials (overlapping density of the monolayers of layered material), which is recognized as α > > 1, the TM functional is more attractive than revTM. Therefore, TM shrinks the lattice constants and performs close to the experimental one. We expect that the performance of revTM can be improved for the interlayer lattice constants by incorporating an appropriate long-range vdW correction. However, without further elaborate investigation, we cannot say whether TM or revTM+vdW is more accurate.

∞

σxc =

∫−∞ dz ρ(z)[ϵxc[ρ; z] − ϵunif xc [ρ ̅ ]]

(9)

In Tables 7 and 8, we present the jellium surface X and XC energies of the functionals under consideration. Again from Table 7, we want to stress that the change in exchange has not much impact on the jellium surface X energies. However, the change in correlation improves the jellium surface XC energies. However, for these simple jellium systems, the LDA, GGA, and other meta-GGA functionals are also quite accurate. The improvement in the jellium surface energies is a well-known fact from the change in correlation as it is already mentioned in a revTPSS paper.23 In the work, it is mentioned that the reduction of β with rs actually decreases surface energies, which in fact is reflected in the performance of the revTM functional. Therefore, the change in the correlation is important and well suited to the TM functional. As a real test and assessment of the revTM functional, we consider the surface energy of the real metals compiled in ref 51. It is defined as the energy required to construct a surface from an infinite crystal. Computationally, it is measured by the formula

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REAL METALLIC SURFACE ENERGIES Investigating any new XC functional performance for the real metallic surfaces is particularly interesting because those analyses have industrial applications. Several meta-GGAs are developed along the Jacob ladder with increasing accuracy in metal surface energy. Recent progress in this direction shows that the non-local functional SCAN+rVV1051 (SCAN functional plus revised Vydrov−Van Voorhis non-local correlation functional) and quasi-2D-GGA functionals17 most accurately predict metallic surface energies. However, in this work, we assess the metallic surface energies of the semilocal XC functionals without including any non-local correlation. The main aim of this paper is to assess the accuracy of LDA, PBE, PBEsol, TPSS, revTPSS, SCAN, TM, and revTM functionals for the metal surface energies. However, in our comparison, we do not include the SCAN+rVV10 results, and to the best of our knowledge, the comparison of SCAN and TM functionals for real metallic surfaces was done for the first time in this paper. However, before going into the real application of the metallic surface energies, we consider the surface exchange and surface exchange-correlation energy of the jellium model.13 The jellium serves as a model for the metallic systems. It consists of a homogeneous electron gas where the electron charges are compensated for with the positive background charge. The bulk density of the jellium model remains constant

σ=

1 slab bulk (EN − N ϵatom ) 2A

(10)

where Eslab N is the total energy of the relaxed surface slab with N atoms, ϵbulk atom is the energy of the bulk with one atom, and A is the slab surface area. In practice, it is suggested to measure the mean of all surfaces to compare with the experimental surface energies. Therefore, we compare the surface energies, taking the mean of the contribution of the values obtained from (100), (110), and 6365


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The Journal of Physical Chemistry A Table 9. Metallic Surface Energy (J/m2) Computed Using the Considered Functionalsa metals Al

Cu

Ru

Rh

Pd

Ag

Pt

Au

ME (J/m2) MAE (J/m2) MARPE (%)

surfaces

LDA

PBE

PBEsol

TPSS

revTPSS

SCAN

TM

revTM

100 110 111 σ̅ 100 110 111 σ̅ 100 110 111 σ̅ 100 110 111 σ̅ 100 110 111 σ̅ 100 110 111 σ̅ 100 110 111 σ̅ 100 110 111 σ̅

1.15 1.09 0.99 1.08 1.99 2.13 1.81 1.98 3.34 3.42 2.81 3.19 3.04 2.86 2.67 2.86 2.43 2.25 1.88 2.19 1.16 1.32 1.13 1.20 2.35 2.46 1.98 2.26 1.39 1.61 1.24 1.41 0.04 0.15 7.21

0.95 0.96 0.77 0.89 1.48 1.63 1.33 1.48 3.02 2.27 2.14 2.48 2.77 2.55 2.09 2.47 1.79 1.61 1.36 1.59 0.81 0.93 0.78 0.84 1.88 1.94 1.56 1.79 0.86 0.99 0.75 0.87 −0.43 0.43 23.58

1.08 1.11 0.99 1.06 1.76 1.88 1.59 1.74 3.25 2.94 2.49 2.89 2.97 2.77 2.40 2.71 2.15 1.93 1.63 1.90 1.04 1.19 1.00 1.08 2.21 2.31 1.85 2.12 1.13 1.26 1.10 1.16 −0.15 0.17 9.16

1.07 1.20 0.96 1.08 1.75 1.89 1.59 1.74 3.36 3.17 2.73 3.09 2.71 2.76 2.34 2.60 1.80 1.92 1.59 1.77 1.03 1.13 0.94 1.03 2.13 2.25 1.74 2.04 1.10 1.16 0.92 1.06 −0.18 0.20 11.12

1.15 1.28 1.04 1.16 1.96 2.11 1.79 1.95 3.56 3.39 2.92 3.29 2.94 2.99 2.55 2.83 2.00 1.89 1.78 1.97 1.19 1.30 1.09 1.19 2.35 2.48 1.94 2.26 1.27 1.34 1.07 1.23 0.00 0.15 7.42

1.08 1.09 0.91 1.03 1.71 1.84 1.49 1.68 3.11 2.81 2.39 2.77 2.71 2.76 2.33 2.60 2.03 1.83 1.54 1.80 1.00 1.12 0.97 1.03 2.04 2.08 1.64 1.92 1.05 1.20 0.93 1.06 −0.25 0.25 13.40

1.31 1.44 1.19 1.31 2.08 2.23 1.99 2.10 3.58 3.42 2.95 3.32 2.97 3.03 2.59 2.86 2.07 2.22 1.87 2.05 1.29 1.40 1.19 1.29 2.43 2.57 2.03 2.34 1.38 1.46 1.18 1.34 0.09 0.17 8.99

1.21 1.33 1.10 1.21 2.02 2.15 1.86 2.01 3.52 3.36 3.02 3.30 2.91 2.97 2.53 2.80 2.00 2.14 1.79 1.98 1.22 1.32 1.12 1.22 2.36 2.49 1.96 2.27 1.31 1.38 1.11 1.27 0.02 0.15 7.55

expt

1.14

1.79

3.04

2.66

2.00

1.25

2.49

1.51

a

LDA, PBE, PBEsol, and SCAN functional results, and reference values are taken from ref 51. The best values are in bold, and the most deviating values are underlined.

Figure 8. Shown are the surface energies (J/m2) for the selected metals using different functionals.

error in exchange and correlation. PBE underestimates the surface energies due to the inability to recover the exact second-order gradient approximation, which is indeed necessary for good surface energies. By recovering the exact second-order gradient approximation, the PBEsol improves the performance over PBE. While improving the correlation and more conveniently obeying the fourth-order gradient approximations over a broader range of density gradient approximations, revTPSS improves the surface energies over TPSS.

(111) surfaces. In Table 9, we summarize the contribution of each surface energy. Here, the LDA, PBE, PBEsol, and SCAN values are taken from ref 51. The TPSS, revTPSS, TM, and revTM surface energies are calculated in this work. In Figures 8 and 9, we plot the surface energy contribution from each surface and MARPE, respectively. For the investigation of the contribution of the surface energies obtained from different surfaces using semilocal functionals, we remark that LDA gives quality surface energies due to the well-known cancellation of 6366


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motivated β form can also be derived for the TM exchange functional which is a matter of our future study.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.9b02921. Individual values as obtained using different functionals for the 29 bulk solids (PDF)

■

AUTHOR INFORMATION

Corresponding Authors

Figure 9. Shown is the total MARPE of different functionals calculated using mean surface energies for selected metals.

*E-mail: [email protected]; [email protected] (S.J.). *E-mail: [email protected] (P.S.).

For the very same reason (as applied in the correlation), we observe the improvement in the surface energies for revTM over the TM functional. Overall consideration shows that the LDA, revTPSS, and revTM functionals agree very well with the experimental values. An interesting observation is that the TM functional overestimates the surface energies for most of the metals, which are actually corrected by its revTM counterpart.

ORCID

Subrata Jana: 0000-0002-3736-1948 Prasanjit Samal: 0000-0002-0234-8831 Notes

The authors declare no competing financial interest.

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■

ACKNOWLEDGMENTS S.J. and P.S. would like to acknowledge and thank Dr. Lucian A. Constantin for providing some computational resources and useful comments regarding the work. S.J. would also like to acknowledge the financial support from the Department of Atomic Energy, Government of India. K.S. would like to acknowledge the financial support from the Department of Science and Technology, Government of India, during his summer internship in NISER under the supervision of P.S.

CONCLUSIONS In this work, we have proposed a revised form of the TM (revTM) functional by improving the slowly varying density correction of the exchange and incorporating the revTPSS-like correlation energy in the functional form. The proposed modification to the TM functional is assessed using the solidstate lattice constants, bulk moduli, cohesive energies, jellium surface exchange-correlation energies, and surface energies of the metals. The performance of the revTM functional shows its accuracy over a broad range of molecular and solid-state systems. Specifically, it improves the cohesive energies, jellium surface energy-correlation energies, surface energies of real metals, and molecular atomization energies by maintaining the overall accuracy of the lattice constants and bulk moduli. The revTM functional works in a more balanced way for both the molecular (localized) and solid-state (delocalized) systems. It is because the density matrix expansion-based exchange hole is localized by nature and proper incorporation of the slowly varying fourth-order gradient expansion. The present modification retains all the good properties of the TM functional. It is also noticed that the present modifications treat both the quantum chemical and solid-state properties in a more balanced way than other accurate and widely appreciated meta-GGAs like the SCAN functional. As a concluding remark, we want to again stress the influence of the modification in the exchange and correlation on molecular and solid-state test cases. The change in the exchange has not much effect on the molecular test set, but it has an impact on the solid-state lattice constants, especially on the interlayer lattice constants of the layered materials. However, further investigation is needed for layered solids. Regarding the influence of the modification of the correlation energy functional, it has an impact on both the molecular and solid-state test cases. We observe improved correlation energy in Hooke’s atom and AE6 atomization energies. Considering the solid-state physics, cohesive energies, jellium surface exchange-correlation energy, and surface energy of the real metallic surfaces show improvements. Lastly, we want to conclude that the more conventionally and quite physically

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REFERENCES

(1) Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864−B871. (2) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133− A1138. (3) Perdew, J. P.; Zunger, A. Self-interaction correction to densityfunctional approximations for many-electron systems. Phys. Rev. B 1981, 23, 5048−5079. (4) Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A 1988, 38, 3098−3100. (5) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Phys. Rev. B 1988, 37, 785−789. (6) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 1992, 46, 6671−6687. (7) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868. (8) Armiento, R.; Mattsson, A. E. Functional designed to include surface effects in selfconsistent density functional theory. Phys. Rev. B 2005, 72, No. 085108. (9) Wu, Z.; Cohen, R. E. More accurate generalized gradient approximation for solids. Phys. Rev. B 2006, 73, 235116. (10) Zhao, Y.; Truhlar, D. G. Construction of a generalized gradient approximation by restoring the density-gradient expansion and enforcing a tight Lieb-Oxford bound. J. Chem. Phys. 2008, 128, 184109. (11) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Vydrov, O. A.; Scuseria, G. E.; Constantin, L. A.; Zhou, X.; Burke, K. Restoring the Density-Gradient Expansion for Exchange in Solids and Surfaces. Phys. Rev. Lett. 2008, 100, 136406.

6367


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The Journal of Physical Chemistry A (12) Constantin, L. A.; Perdew, J. P.; Pitarke, J. M. Exchangecorrelation hole of a generalized gradient approximation for solids and surfaces. Phys. Rev. B 2009, 79, No. 075126. (13) Constantin, L. A.; Chiodo, L.; Fabiano, E.; Bodrenko, I.; Sala, F. D. Correlation energy functional from jellium surface analysis. Phys. Rev. B 2011, 84, No. 045126. (14) Fabiano, E.; Constantin, L. A.; Della Sala, F. Two-Dimensional Scan of the Performance of Generalized Gradient Approximations with Perdew-Burke-Ernzerhof-Like Enhancement Factor. J. Chem. Theory Comput. 2011, 7, 3548−3559 PMID: 26598253 . (15) Constantin, L. A.; Terentjevs, A.; Della Sala, F.; Cortona, P.; Fabiano, E. Semiclassical atom theory applied to solid-state physics. Phys. Rev. B 2016, 93, No. 045126. (16) Cancio, A.; Chen, G. P.; Krull, B. T.; Burke, K. Fitting a round peg into a round hole: Asymptotically correcting the generalized gradient approximation for correlation. J. Chem. Phys. 2018, 149, No. 084116. (17) Chiodo, L.; Constantin, L. A.; Fabiano, E.; Della Sala, F. Nonuniform Scaling Applied to Surface Energies of Transition Metals. Phys. Rev. Lett. 2012, 108, 126402. (18) Becke, A. D.; Roussel, M. R. Exchange holes in inhomogeneous systems: A coordinate-space model. Phys. Rev. A 1989, 39, 3761− 3767. (19) Van Voorhis, T.; Scuseria, G. E. A novel form for the exchangecorrelation energy functional. J. Chem. Phys. 1998, 109, 400−410. (20) Zhao, Y.; Truhlar, D. G. A new local density functional for main-group thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. J. Chem. Phys. 2006, 125, 194101. (21) Perdew, J. P.; Kurth, S.; Zupan, A.; Blaha, P. Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation. Phys. Rev. Lett. 1999, 82, 2544−2547. (22) Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Climbing the Density Functional Ladder: Nonempirical Meta− Generalized Gradient Approximation Designed for Molecules and Solids. Phys. Rev. Lett. 2003, 91, 146401. (23) Perdew, J. P.; Ruzsinszky, A.; Csonka, G. I.; Constantin, L. A.; Sun, J. Workhorse Semilocal Density Functional for Condensed Matter Physics and Quantum Chemistry. Phys. Rev. Lett. 2009, 103, No. 026403. (24) Constantin, L. A.; Fabiano, E.; Sala, F. D. Semilocal dynamical correlation with increased localization. Phys. Rev. B 2012, 86, No. 035130. (25) Constantin, L. A.; Fabiano, E.; Della Sala, F. Meta-GGA Exchange-Correlation Functional with a Balanced Treatment of Nonlocality. J. Chem. Theory Comput. 2013, 9, 2256−2263 PMID: 26583719 . (26) Sun, J.; Haunschild, R.; Xiao, B.; Bulik, I. W.; Scuseria, G. E.; Perdew, J. P. Semilocal and hybrid meta-generalized gradient approximations based on the understanding of the kinetic-energydensity dependence. J. Chem. Phys. 2013, 138, No. 044113. (27) Sun, J.; Perdew, J. P.; Ruzsinszky, A. Semilocal density functional obeying a strongly tightened bound for exchange. Proc. Natl. Acad. Sci. U. S. A. 2015, 112, 685−689. (28) Ruzsinszky, A.; Sun, J.; Xiao, B.; Csonka, G. I. A meta-GGA Made Free of the Order of Limits Anomaly. J. Chem. Theory Comput. 2012, 8, 2078−2087. (29) Sun, J.; Ruzsinszky, A.; Perdew, J. P. Strongly Constrained and Appropriately Normed Semilocal Density Functional. Phys. Rev. Lett. 2015, 115, No. 036402. (30) Tao, J.; Mo, Y. Accurate Semilocal Density Functional for Condensed-Matter Physics and Quantum Chemistry. Phys. Rev. Lett. 2016, 117, No. 073001. (31) Wang, Y.; Jin, X.; Yu, H. S.; Truhlar, D. G.; He, X. Revised M06-L functional for improved accuracy on chemical reaction barrier heights, noncovalent interactions, and solid-state physics. Proc. Natl. Acad. Sci. U. S. A. 2017, 114, 8487−8492.

(32) Mezei, P. D.; Csonka, G. I.; Kállay, M. Simple Modifications of the SCAN Meta-Generalized Gradient Approximation Functional. J. Chem. Theory Comput. 2018, 14, 2469−2479. (33) Della Sala, F.; Fabiano, E.; Constantin, L. A. Kinetic-energydensity dependent semilocal exchange-correlation functionals. Int. J. Quantum Chem. 2016, 116, 1641−1694. (34) Cramer, C. J.; Truhlar, D. G. Density functional theory for transition metals and transition metal chemistry. Phys. Chem. Chem. Phys. 2009, 11, 10757−10816. (35) Peverati, R.; Truhlar, D. G. Quest for a universal density functional: the accuracy of density functionals across a broad spectrum of databases in chemistry and physics. Philos. Trans. R. Soc., A 2014, 372, 20120476. (36) Staroverov, V. N.; Scuseria, G. E.; Tao, J.; Perdew, J. P. Comparative assessment of a new nonempirical density functional: Molecules and hydrogen-bonded complexes. J. Chem. Phys. 2003, 119, 12129−12137. (37) Zhao, Y.; Schultz, N. E.; Truhlar, D. G. Design of Density Functionals by Combining the Method of Constraint Satisfaction with Parametrization for Thermochemistry, Thermochemical Kinetics, and Noncovalent Interactions. J. Chem. Theory Comput. 2006, 2, 364−382. (38) Goerigk, L.; Grimme, S. Efficient and Accurate Double-HybridMeta-GGA Density FunctionalsEvaluation with the Extended GMTKN30 Database for General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions. J. Chem. Theory Comput. 2011, 7, 291−309. (39) Hao, P.; Sun, J.; Xiao, B.; Ruzsinszky, A.; Csonka, G. I.; Tao, J.; Glindmeyer, S.; Perdew, J. P. Performance of meta-GGA Functionals on General Main Group Thermochemistry, Kinetics, and Noncovalent Interactions. J. Chem. Theory Comput. 2013, 9, 355−363. (40) Goerigk, L.; Hansen, A.; Bauer, C.; Ehrlich, S.; Najibi, A.; Grimme, S. A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions. Phys. Chem. Chem. Phys. 2017, 19, 32184−32215. (41) Mo, Y.; Tian, G.; Car, R.; Staroverov, V. N.; Scuseria, G. E.; Tao, J. Performance of a nonempirical density functional on molecules and hydrogen-bonded complexes. J. Chem. Phys. 2016, 145, 234306. (42) Mo, Y.; Tian, G.; Tao, J. Performance of a nonempirical exchange functional from density matrix expansion: comparative study with different correlations. Phys. Chem. Chem. Phys. 2017, 19, 21707− 21713. (43) Haas, P.; Tran, F.; Blaha, P. Calculation of the lattice constant of solids with semilocal functionals. Phys. Rev. B 2009, 79, No. 085104. (44) Tran, F.; Stelzl, J.; Blaha, P. Rungs 1 to 4 of DFT Jacob’s ladder: Extensive test on the lattice constant, bulk modulus, and cohesive energy of solids. J. Chem. Phys. 2016, 144, 204120. (45) Mo, Y.; Car, R.; Staroverov, V. N.; Scuseria, G. E.; Tao, J. Assessment of the Tao-Mo nonempirical semilocal density functional in applications to solids and surfaces. Phys. Rev. B 2017, 95, No. 035118. (46) Mattsson, A. E.; Armiento, R.; Paier, J.; Kresse, G.; Wills, J. M.; Mattsson, T. R. The AM05 density functional applied to solids. J. Chem. Phys. 2008, 128, No. 084714. (47) Sun, J.; Marsman, M.; Csonka, G. I.; Ruzsinszky, A.; Hao, P.; Kim, Y.-S.; Kresse, G.; Perdew, J. P. Self-consistent meta-generalized gradient approximation within the projector-augmented-wave method. Phys. Rev. B 2011, 84, No. 035117. (48) Csonka, G. I.; Perdew, J. P.; Ruzsinszky, A.; Philipsen, P. H. T.; Lebègue, S.; Paier, J.; Vydrov, O. A.; Á ngyán, J. G. Assessing the performance of recent density functionals for bulk solids. Phys. Rev. B 2009, 79, 155107. (49) Jana, S.; Patra, A.; Samal, P. Assessing the performance of the Tao-Mo semilocal density functional in the projector-augmentedwave method. J. Chem. Phys. 2018, 149, No. 044120. (50) Jana, S.; Sharma, K.; Samal, P. Assessing the performance of the recent meta-GGA density functionals for describing the lattice 6368


Article

The Journal of Physical Chemistry A constants, bulk moduli, and cohesive energies of alkali, alkaline-earth, and transition metals. J. Chem. Phys. 2018, 149, 164703. (51) Patra, A.; Bates, J. E.; Sun, J.; Perdew, J. P. Properties of real metallic surfaces: Effects of density functional semilocality and van der Waals nonlocality. Proc. Natl. Acad. Sci. U. S. A. 2017, 114, E9188− E9196. (52) Sun, J.; Xiao, B.; Fang, Y.; Haunschild, R.; Hao, P.; Ruzsinszky, A.; Csonka, G. I.; Scuseria, G. E.; Perdew, J. P. Density Functionals that Recognize Covalent, Metallic, and Weak Bonds. Phys. Rev. Lett. 2013, 111, 106401. (53) Clementi, E.; Roetti, C. Roothaan-Hartree-Fock atomic wavefunctions: Basis functions and their coefficients for ground and certain excited states of neutral and ionized atoms, Z≤d54. At. Data Nucl. Data Tables 1974, 14, 177−478. (54) Perdew, J. P.; Tao, J.; Staroverov, V. N.; Scuseria, G. E. Metageneralized gradient approximation: Explanation of a realistic nonempirical density functional. J. Chem. Phys. 2004, 120, 6898− 6911. (55) Chakravorty, S. J.; Gwaltney, S. R.; Davidson, E. R.; Parpia, F. A.; p Fischer, C. F. Ground-state correlation energies for atomic ions with 3 to 18 electrons. Phys. Rev. A 1993, 47, 3649−3670. (56) Constantin, L. A. Correlation energy functionals from adiabatic connection formalism. Phys. Rev. B 2019, 99, No. 085117. (57) Lynch, B. J.; Truhlar, D. G. Small Representative Benchmarks for Thermochemical Calculations. J. Phys. Chem. A 2003, 107, 8996− 8999. (58) Hu, C. D.; Langreth, D. C. Beyond the random-phase approximation in nonlocal-density-functional theory. Phys. Rev. B 1986, 33, 943−959. (59) Görling, A.; Levy, M. Exact Kohn-Sham scheme based on perturbation theory. Phys. Rev. A 1994, 50, 196−204. (60) Seidl, M.; Perdew, J. P.; Kurth, S. Density functionals for the strong-interaction limit. Phys. Rev. A 2000, 62, No. 012502. (61) Perdew, J. P.; Staroverov, V. N.; Tao, J.; Scuseria, G. E. Density functional with full exact exchange, balanced nonlocality of correlation, and constraint satisfaction. Phys. Rev. A 2008, 78, No. 052513. (62) Valiev, M.; Bylaska, E. J.; Govind, N.; Kowalski, K.; Straatsma, T. P.; Van Dam, H. J. J.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T. L.; de Jong, W. A. NWChem: A comprehensive and scalable opensource solution for large scale molecular simulations. Comput. Phys. Commun. 2010, 181, 1477−1489. (63) Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 1993, 47, 558−561. (64) Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 1996, 54, 11169−11186. (65) Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 1999, 59, 1758− 1775. (66) Kresse, G.; Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mater. Sci. 1996, 6, 15−50. (67) Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. Gaussian-2 theory for molecular energies of first- and second-row compounds. J. Chem. Phys. 1991, 94, 7221−7230. (68) Fabiano, E.; Constantin, L. A.; Terentjevs, A.; Della Sala, F.; Cortona, P. Assessment of the TCA functional in computational chemistry and solid-state physics. Theor. Chem. Acc. 2015, 134, 139. (69) Vydrov, O. A.; Scuseria, G. E. Assessment of a long-range corrected hybrid functional. J. Chem. Phys. 2006, 125, 234109. (70) Patra, B.; Jana, S.; Samal, P. Long-range corrected density functional through the density matrix expansion based semilocal exchange hole. Phys. Chem. Chem. Phys. 2018, 20, 8991−8998. (71) Jana, S.; Samal, P. A meta-GGA level screened range-separated hybrid functional by employing short range Hartree-Fock with a long range semilocal functional. Phys. Chem. Chem. Phys. 2018, 20, 8999− 9005.

(72) Jana, S.; Samal, P. Screened hybrid meta-GGA exchange− correlation functionals for extended systems. Phys. Chem. Chem. Phys. 2019, 21, 3002−3015. (73) Jana, S.; Patra, A.; Constantin, L. A.; Myneni, H.; Samal, P. Long-range screened hybrid-functional theory satisfying the localdensity linear response. Phys. Rev. A 2019, 99, No. 042515. (74) Björkman, T.; Gulans, A.; Krasheninnikov, A. V.; Nieminen, R. M. van der Waals Bonding in Layered Compounds from Advanced Density-Functional First-Principles Calculations. Phys. Rev. Lett. 2012, 108, 235502. (75) Björkman, T. Testing several recent van der Waals density functionals for layered structures. J. Chem. Phys. 2014, 141, No. 074708. (76) Peng, H.; Yang, Z.-H.; Perdew, J. P.; Sun, J. Versatile van derWaals Density Functional Based on a Meta-Generalized Gradient Approximation. Phys. Rev. X 2016, 6, No. 041005. (77) Björkman, T. van der Waals density functional for solids. Phys. Rev. B 2012, 86, 165109. (78) Terentjev, A. V.; Constantin, L. A.; Pitarke, J. M. Dispersioncorrected PBEsol exchange-correlation functional. Phys. Rev. B 2018, 98, 214108.

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