JAMPJournal of Applied Mathematics and Physics2327-4352Scientific Research Publishing10.4236/jamp.2022.1011225JAMP-121523ArticlesPhysics&Mathematics Exchange-Correlation Functional Comparison of Electronic Energies in Atoms Using a Grid Basis AnthonyD. Ryan1AndresGama1FrankFelerski1WilliamD. Parker1*Department of Mathematics and Physics, University of Wisconsin-Parkside, Kenosha, WI, USA0911202210113392340722, October 202226, November 2022 29, November 2022© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Calculation of total energies of the electronic ground states of atoms forms the basis for the frozen-core pseudopotentials used in atomistic calculations of much larger scale. Reference values for these energies provide a benchmark for the validation of new software to calculate such potentials. In addition, basic atomic-scale electronic properties such as the (first) ionization energy provide a simple check on the approximation used in the calculation method. We present a comparison of the total energies and ionization energies of atoms Z = 1 - 92 calculated in density functional theory with several levels of exchange-correlation functional and the Hartree-Fock method, comparing ionization energies to experiment. We also investigate the role of relativistic treatment on these energies.

Density Functional Theory Hartree-Fock Theory Electronic Energies Exchange-Correlation Potential Exchange and Correlation Functional Ionization Energy1. Introduction

Density functional theory (DFT) is a quantum mechanical method widely used in chemistry [1] and materials science [2] to calculate system properties from first principles within its one significant approximation: the exchange-correlation functional. DFT calculations of atomic total electronic energies are important for calibrating advancements in exchange-correlation functionals among codes [3] and producing reliable pseudopotentials [4]. Recent advances in exchange-correlation functionals continue to show reduced error in molecular [5] [6] and solid-state [6] [7] test sets compared and call for an investigation of the functionals' effect on energies calculated in isolated atoms.

Non-empirical pseudopotentials constructed for use with plane waves in DFT simulations of systems with periodic boundary conditions rely on atomic calculations in which the electrons are represented through Kohn-Sham states constituting the electron density on a radial grid [8]. Existing research has explored the effects of exchange-correlation functional at the local-density [8] and generalized-gradient [9] [10] levels of approximation, including the effects of incorporating spin-polarization and relativistic effects [8] on atomic density functional total energies and ionization energies.

In this work, we compare the published values of these total and (first) ionization energies of all-electron atoms to two existing codes in order to establish a baseline of comparison. We then extend these studies to the meta-generalized gradient and exact-exchange and hybrid levels of exchange-correlation approximation, investigating the effects of these additional levels of complexity on the total energies and ionization energies of atoms calculated using a grid basis, comparing ionization energy to experiment. Finally, we also investigate the effect of combining relativistic effects in both scalar-relativistic and fully relativistic calculations to these energies using the same levels of exchange-correlation approximation.

2. Method

Quantum ESPRESSO [11] [12] is “an integrated suite of Open-Source computer codes for electronic-structure calculations and materials modeling at the nanoscale, based on density-functional theory, plane waves, and pseudopotentials” that includes an atomic density-functional code atomic OPIUM [13] is a stand- alone pseudopotential generation code that implements both density functional and Hartree-Fock theory to calculate electronic total energies. Both atomic and OPIUM use a radial grid to represent the single-particle Kohn-Sham states that is of the form:

r n = a ( e b ( n − 1 ) − 1 ) (1)

where the parameters a and b determine the relationship of the N points that are indexed from n = 1 to N + 1 . In atomic, a by default is set to 1/Z where Z is the atomic number of the atom being simulated, and, in OPIUM, a defaults to ( 10 / Z ) 1 3 / 10 . In both codes, b is an exponential grid spacing parameter.

As of Quantum ESPRESSO version 6.3, atomic includes support for a variety of exchange-correlation functionals at the local-density approximation (LDA) and generalized-gradient approximation (GGA) levels while, at version 4.1, OPIUM implements exchange-correlation functionals at the LDA and GGA levels as well as the Hartree-Fock [14] (HF) and hybrid (PBE0) [15] functional levels. Yao and Kanai [16] implemented the TPSS [17] and SCAN [18] meta-GGA exchange-correlation functionals in a modified version of atomic. In this investigation, electronic orbital occupation of the Kohn-Sham states was chosen to match the ground-state electron configurations of the atoms established by experiment [19].

3. Results3.1. Total Energy

To corroborate the existing implementations of exchange-correlation in atomic and OPIUM against the published values, Table 1 presents the mean absolute relative error of the DFT total energies produced by the atomic and OPIUM codes from the published values for LDA/LSD (local spin density) [8] [20] and LDA & GGA [22] exchange-correlation functionals. The atomic data use the internal exchange-correlation functional implementation in Quantum ESPRESSO, and tests using the libxc [27] implementations of exchange-correlation in Quantum ESPRESSO produce statistically indistinguishable differences. Table 2 shows the mean absolute relative error of the HF total energies produced by the OPIUM code from prior published values [28] [29].

The NIST data are reported to 10−6 Ha so a difference of 9 × 10−7 Ha for the smallest total energy (E(H) [LDA] = −0.445671 Ha) would produce the largest possible absolute relative error of 2 × 10−6. The calculated LDA MAREs for both atomic and OPIUM are below this threshold, and the atomic LSD MARE also

Mean absolute relative error (MARE) for the data in each reference for atomic code in Quantum ESPRESSO [<xref ref-type="bibr" rid="scirp.121523-ref11">11</xref>] [<xref ref-type="bibr" rid="scirp.121523-ref12">12</xref>] and the OPIUM code [<xref ref-type="bibr" rid="scirp.121523-ref13">13</xref>], for varying exchange-correlation functionals and spin-polarization treatment. The NIST [<xref ref-type="bibr" rid="scirp.121523-ref8">8</xref>] [<xref ref-type="bibr" rid="scirp.121523-ref19">19</xref>] [<xref ref-type="bibr" rid="scirp.121523-ref20">20</xref>] data use the Vosko-Wilk-Nusair [<xref ref-type="bibr" rid="scirp.121523-ref21">21</xref>] (VWN) parameterization of the local density approximation (LDA) while the Lee and Martin [<xref ref-type="bibr" rid="scirp.121523-ref22">22</xref>] data use the Perdew-Zunger [<xref ref-type="bibr" rid="scirp.121523-ref23">23</xref>] parameterization for LDA. Lee and Martin compare two generalized-gradient approximations (GGAs), Perdew-Wang 1991 [<xref ref-type="bibr" rid="scirp.121523-ref24">24</xref>] (PW91) and Perdew-Burke-Ernzerhof [<xref ref-type="bibr" rid="scirp.121523-ref25">25</xref>] [<xref ref-type="bibr" rid="scirp.121523-ref26">26</xref>] (PBE)
atomicOPIUM
NISTLee & MartinNISTLee & Martin
LDA-VWN0.000000000.000032 (51)0.000000020.000035 (51)
LSD-VWN0.00000000---
GGA-PW91-0.000019 (50)-0.000019 (50)
GGA-PBE-0.000008 (50)-0.000008 (50)
 Mean error (ME), mean absolute error (MAE), mean relative error (MRE), and mean absolute relative error (MARE) for the total energy using the Hartree-Fock method in the OPIUM code with respect to the values reported in Davidson et al. [<xref ref-type="bibr" rid="scirp.121523-ref28">28</xref>] for elements Z = 3 - 10
ME0.0217
MAE0.0217
MRE0.0004
MARE0.0004

lies below this threshold (the spin-polarized VWN functional is not implemented in OPIUM).

The data of Lee and Martin are reported to 10−3 Ry so a difference of 9 × 10−4 Ry for the total energy of hydrogen would produce the largest possible absolute relative error of 1 × 10−3. The calculated LDA, PW91, and PBE MAREs for both atomic and OPIUM lie below this threshold.

Davidson et al. [2] report Hartree-Fock energies to 10−6 Ha calculated using Slater-type orbitals with an 11s, 10p, 9d, 8f, 6g, 4h, 2i basis set. A comparison with Hartree-Fock calculations using the exponential radial grid in OPIUM shows mean relative and mean absolute relative errors of 0.04%, greater than the 0.0001% difference that would be produced by 9 × 10−7 Ha difference on the smallest total energy in this set (E(Li) [HF] = -7.432727 Ha). The values for Z = 6 - 8 deviate in the 0.05% 0.2% range while the values for Z = 3 - 5, 9 - 10 deviate on the order of 1 × 10−6 %, implying that the only numerically significant difference lies in the total energies for C, N, and O.

For completeness, we present the total energies for elements Z = 1 - 92 in Tables A1-A7 of the Appendix calculated using the DFT exchange-correlation functionals LDA-PW, GGA-PBE, GGA-PBEsol, metaGGA-TPSS, metaGGA-SCAN, and hybrid-PBE0 together with Hartree-Fock.

Following Kotochigova et al., we plot −Z7/3Etotal against Z to contrast the DFT exchange-correlation functionals with Hartree-Fock (Figure 1). At the global level, all of these methods follow the same trend. Zooming in on the Z = 2 - 10 region, we see the LDA energies lowest followed by the GGA-PBEsol energies, and the GGA-PBE, metaGGA-TPSS, metaGGA-SCAN, and hybrid PBE0 energies converging at this scale for Z > 5.

In addition to the atomic number-scaled total energies, we plot the relative difference from the corrected Thomas-Fermi energy (Figure 2). For this quantity, the differences between the functionals appear most starkly in low Z elements. In Period 3, the metaGGA functions group with the largest relative difference, followed by GGA-PBE and hybrid-PBE0. The reduced gradient expansion coefficient in both the exchange and correlation components of GGA-PBEsol

compared to GGA-PBE makes its relative differences from the corrected Thomas-Fermi energy closer to the LDA-VWN values.

3.2. Ionization Energy

The (first) ionization energy of an atom is calculated from the difference between the total energies of the all-electron atom and the singly ionized cation [30]. The density functional theory values for ionization energy reproduce the qualitative trends of the experimentally determined values [8], and including gradient and higher-order corrections does not significantly alter these trends (see Figures 3-5). Ionization energies rise across each period of elements, starting low in Group 1 and rising to a peak in Group 18, before dropping for the Group 1 element in the subsequent period. For increasing Z across Groups 4 - 12, ionization energy shows fluctuations of up to nearly 5 eV reflecting the complexities of the d- and f-shell orbital energies in both the neutral atom and the singly ionized cation.

3.2.1. Comparison to Experiment

For more detailed consideration of the calculated ionization energies, we investigate the difference of the calculated ionization energy from the experimental values with increasing atomic number (Figures 6-8) as reported in the NIST Computational Chemistry Comparison and Benchmark Database [31]. We collate into Figure 9 the comparison of the differences using all of these levels of theory. The mean-field approximation of density functional theory shows up with peak differences occurring at Z = 2, 27, 28, 78, 79 and Group 16 & 17 elements, the difference from experiment trending downward with increasing Z without any attention to relativistic effects in the calculation (we consider these effects in Section 3.2.2).

3.2.2. Relativistic Effects

The effects of special relativity on the electrons can be accounted for by transforming the Kohn-Sham equations into Dirac-like equations with a spinor containing two components, each indexed by the quantum numbers n, l , and j where j is the total angular momentum. These equations are coupled first-order equations of the spinor components containing the Dirac quantum number κ , and the charge density is constructed by summing the sum of the squares of the

Mean absolute relative error (MARE) for the ionization energy with respect to the experimental values for elements Z = 1 - 36, varying relativistic treatment and exchange-correlation functional
NoneScalarFull
LDA-PW0.0730.0730.070
GGA-PBE0.0290.0240.065
GGA-PBEsol0.0290.0250.067
 Mean absolute relative error (MARE) for the ionization energy with respect to the experimental values for elements Z = 37 - 92, varying relativistic treatment and exchange-correlation functional
NoneScalarFull
LDA-PW0.0680.0810.065
GGA-PBE0.0730.0630.046
GGA-PBEsol0.0740.0660.049

spinor components, multiplied by the occupancies and divided by the spherical surface area, over the quantum numbers n, l , and j. This is the so-called fully relativistic treatment. A simplification reduces the coupled first-order equations into a single second-order equation for the large spinor component and averages over the spin-orbit components—this is the scalar relativistic case.

In Figures 10-12, we show the effects of the two models of relativity together with the non-relativistic calculations on the difference of calculated and experimental ionization energies for elements in periods four, five, and six (Z = 19 - 36, 37 - 54, and 55 - 86, respectively) using the GGA-PBE functional. The qualitative ordering of the three calculations on each particular element remain the same in LDA-PW and GGA-PBEsol as in GGA-PBE.

4. Conclusion

We present density functional atomic total energies for an exponential radial grid basis using exchange-correlation approximations ranging from local density through generalized gradient and meta-generalized gradient to exact-exchange hybrid levels of complexity in comparison with the same energies calculated using the Hartree-Fock method. We demonstrate that, while these increased levels of complexity change the absolute energies and first ionization energies of the atoms, the relative trends with increasing atomic number and difference from experiment remain largely the same. Adding scalar and spinor relativistic corrections to the density-functional Hamiltonian can alter significantly the difference of the first ionization energy from experiment, changing the sign of the difference, but the mean absolute relative error in the first ionization energy for elements Z = 37 - 92 reduces only from 7% to 5%.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Ryan, A.D., Gama, A., Felerski, F. and Parker, W.D. (2022) Exchange-Correlation Functional Comparison of Electronic Energies in Atoms Using a Grid Basis. Journal of Applied Mathematics and Physics, 10, 3392-3407. https://doi.org/10.4236/jamp.2022.1011225

Appendix—Total Energies Total spin-polarized energies (in Ha) for atoms in the first two periods of the periodic table (Z = 1 - 10) with varying exchange-correlation functionals (as well as DFT to HF) and no relativistic correction. Values are depicted to 10<sup>−</sup><sup>4</sup>, indicative of the digit where no grid basis error is evident
ElementsPWPBEPBEsolTPSSSCANHFPBE0
H−0.4787−0.5000−0.4887−0.4978−0.4981−0.5000−0.4723
He−2.8345−2.8929−2.8577−2.9063−2.9096−2.8616−2.8952
Li−7.3433−7.4622−7.3976−7.4836−7.4881−7.4327−7.4606
Be−14.4465−14.6300−14.5323−14.6655−14.6711−14.5730−14.6366
B−24.3525−24.6054−24.4723−24.6473−24.6539−24.5291−24.6079
C−37.4683−37.7937−37.6235−37.8458−37.8530−37.6597−37.7644
N−54.1344−54.5358−54.3270−54.6034−54.6110−54.2962−54.4387
O−74.5248−75.0010−74.7492−75.0723−75.0799−74.7692−74.9615
F−99.1112−99.6657−99.3701−99.7508−99.7583−99.4093−99.6628
Ne−128.2299−128.8665−128.5258−128.9725−128.9798−128.5471−128.8718
 Total spin-polarized energies (in Ha) for atoms in the third period of the periodic table (Z = 11 - 18) with varying exchange-correlation functionals (as well as DFT to HF) and no relativistic correction. Values are depicted to 10<sup>−</sup><sup>4</sup>, indicative of the digit where no grid basis error is evident
ElementsPWPBEPBEsolTPSSSCANHFPBE0
Na−161.4436−162.1732−161.7843−162.2979−162.2967−161.8589−162.1800
Mg−199.1353−199.9557−199.5177−200.0925−200.0907−199.6146−199.9708
Al−241.3166−242.2326−241.7460−242.3799−242.3780−241.8767−242.2483
Si−288.2178−289.2327−288.6965−289.3935−289.3916−288.8346−289.2323
P−340.0001−341.1164−340.5298−341.2942−341.2921−340.6489−341.0811
S−396.7382−397.9468−397.3101−398.1362−398.1337−397.4785−397.9523
Cl−458.6655−459.9715−459.2838−460.1754−460.1723−459.4821−460.0026
Ar−525.9398−527.3470−526.6072−527.5691−527.5654−526.8175−527.3884
 Total energies for atoms (in Ha) in the fourth period of the periodic table (Z = 19-36) with varying exchange-correlation functionals (as well as DFT to HF) and no relativistic correction
ElementsPWPBEPBEsolTPSSSCANHFPBE0
K−598.1992−599.7111−598.9171−599.9402−599.9368−599.1648−599.7514
Ca−675.7353−677.3495−676.5013−677.5859−677.5821−676.7582−677.3925
Sc−758.6778−760.3916−759.4911−760.6306−760.6267−759.7357−760.4263
Ti−847.2954−849.1098−848.1569−849.3527−849.3488−848.3701−849.1210
V−941.7335−943.6496−942.6440−943.8973−943.8932−942.8037−943.6192
Cr−1042.2086−1044.2318−1043.1756−1044.4882−1044.4841−1043.1418−1044.0567
Mn−1148.6341−1150.7570−1149.6448−1151.0155−1151.0115−1149.6259−1150.5819
Fe−1261.2130−1263.4331−1262.2665−1263.6892−1263.6850−1262.2909−1263.3215
Co−1380.1565−1382.4750−1381.2542−1382.7323−1382.7278−1381.3084−1382.4160
Ni−1505.6038−1508.0221−1506.7470−1508.2821−1508.2776−1506.8158−1508.0017
Cu−1637.7812−1640.2999−1638.9720−1640.5665−1640.5615−1638.8261
Zn−1776.5615−1779.1840−1777.7991−1779.4510−1779.4528−1777.8481−1779.1917
Ga−1921.8390−1924.5717−1923.1302−1924.8402−1924.8345−1923.2610−1924.5950
Ge−2073.8164−2076.6615−2075.1637−2076.9344−2076.9285−2075.3404−2076.6819
As−2232.5731−2235.5331−2233.9783−2235.8126−2235.8067−2234.1722−2235.5330
Se−2398.1209−2401.1849−2399.5729−2401.4663−2401.4598−2399.8432−2401.2316
Br−2570.6124−2573.7847−2572.1159−2574.0707−2574.0637−2572.4413−2573.8628
Kr−2750.1333−2753.4175−2751.6911−2753.7104−2753.7028−2752.0550−2753.5126
 Total energies (in Ha) for atoms in the fifth period of the periodic table (Z = 37 - 54) with varying exchange-correlation functionals (as well as DFT to HF) and no relativistic correction
ElementsPWPBEPBEsolTPSSSCANHFPBE0
Rb−2936.3271−2939.7267−2937.9402−2940.0142−2940.0063−2938.3575−2939.8257
Sr−3129.4380−3132.9502−3131.1038−3133.2324−3133.2241−3131.5457−3133.0557
Y−3329.5096−3333.1348−3331.2302−3333.4091−3333.4004−3331.6842−3333.2392
Zr−3536.7411−3540.4818−3538.5186−3540.7494−3540.7403−3538.9687−3540.5734
Nb−3751.2785−3755.1461−3753.1248−3755.4104−3755.4009−3753.4914−3755.1528
Mo−3973.1449−3977.1347−3975.0527−3977.3965−3977.3867−3975.3687−3977.0887
Tc−4202.3077−4206.4089−4204.2647−4206.6637−4206.6532−4204.6068−4206.3803
Ru−4439.0262−4443.2398−4441.0368−4443.4919−4443.4816−4441.4559−4443.2971
Rh−4683.3162−4687.6451−4685.3816−4687.8918−4687.8813−4685.8370−4687.7389
Pd−4935.3493−4939.7953−4937.4708−4940.0419−4940.0305−4909.4362−4939.9225
Ag−5195.0177−5199.5836−5197.1976−5199.8231−5199.8116−5197.6985−5199.7176
Cd−5462.3711−5467.0536−5464.6051−5467.2847−5467.2819−5465.1331−5467.1988
In−5737.2935−5742.0973−5739.5870−5742.3212−5742.3085−5740.1692−5742.2507
Sn−6019.9514−6024.8778−6022.3066−6025.0973−6025.0841−6022.9142−6025.0248
Sb−6310.3978−6315.4485−6312.8160−6315.6650−6315.6513−6313.4262−6315.5752
Te−6608.6290−6613.7934−6611.0995−6614.0024−6613.9883−6611.7627−6613.9567
I−6914.7563−6920.0375−6917.2830−6920.2414−6920.2273−6917.9809−6920.2243
Xe−7228.8342−7234.2354−7231.4198−7234.4357−7234.4210−7232.1384−7234.4337
 Total energies (in Ha) for atoms in the sixth period of the periodic table without the lanthanoids (Z = 55 - 56 & 72 - 86) with varying exchange-correlation functionals (as well as DFT to HF) and no relativistic correction
ElementsPWPBEPBEsolTPSSSCANHFPBE0
Cs−7550.5395−7556.0620−7553.1835−7556.2439−7556.2341−7553.9337−7556.2604
Ba−7880.0891−7885.7318−7882.7904−7885.8984−7885.8879−7883.5438−7885.9309
Hf−14317.4817−14324.9934−14321.0739−14324.8088−14324.7918−14321.2240−14325.0883
Ta−14795.8960−14803.5317−14799.5493−14803.3298−14803.3121−14799.7544−14803.6232
W−15283.4952−15291.2573−15287.2113−15291.0392−15291.0206−15287.4496−15291.3311
Os−16286.3431−16294.3520−16290.1777−16294.1022−16294.0826−16290.5398−16294.4628
Ir−16801.6903−16809.8191−16805.5815−16809.5544−16809.5347−16806.0392−16809.9879
Pt−17326.5569−17334.8107−17330.5104−17334.5384−17334.5197−17331.0655−17335.0387
Au−17860.7629−17869.1433−17864.7784−17868.8584−17868.8392−17865.4001−17869.3994
Hg−18404.2404−18412.7448−18408.3140−18412.4409−18412.4234−18408.9915−18413.0201
Tl−18956.9279−18965.5609−18961.0646−18965.2388−18965.2180−18961.8248−18965.8529
Pb−19518.9760−19527.7388−19523.1781−19527.4010−19527.3793−19523.9912−19528.0332
Bi−20090.4187−20099.3129−20094.6875−20098.9607−20098.9386−20095.5302−20099.5965
Po−20671.2387−20680.2541−20675.5636−20679.8835−20679.8609−20676.4807−20680.5789
At−21261.5242−21270.6638−21265.9091−21270.2770−21270.2537−21266.8817−21271.0170
Rn−21861.3113−21870.5778−21865.7588−21870.1760−21870.1519−21866.7722−21870.9481
 Total energies for lanthanoid atoms in the sixth period of the periodic table (Z = 57-71) with varying exchange-correlation functionals (as well as DFT to HF) and no relativistic correction
ElementsPWPBEPBEsolTPSSSCANHFPBE0
La−8217.5565−8223.3205−8220.3173−8223.4711−8223.4601−8221.0667−8223.5150
Ce−8563.3496−8569.2294−8566.1658−8569.3575−8569.3463−8566.8484−8569.4036
Pr−8917.6918−8923.6851−8920.5618−8923.7868−8924.7249−8921.0634−8923.7892
Nd−9280.2979−9286.5632−9283.3036−9286.4577−9287.3806−9283.7003−9286.5391
Pm−9651.6112−9657.8410−9654.5956−9657.7232−9658.6603−9654.8642−9657.8147
Sm−10031.4581−10037.8069−10034.5001−10037.5908−10038.5452−10034.6311−10037.6932
Eu−10419.9970−10426.4656−10423.0972−10426.1353−10427.1106−10423.0761−10426.2490
Gd−10816.9890−10823.5771−10820.1467−10823.5837−10823.5704−10820.1271−10823.3250
Tb−11223.0809−11229.7743−11226.2853−11710.0870−11230.5828−11226.3007−11229.6909
Dy−11637.9502−11644.7573−11641.2077−12158.4238−11645.6381−11641.2295−11644.7255
Ho−12061.8042−12068.7258−12065.1154−12068.6458−12068.6311−12065.1352−12068.7345
Er−12494.7175−12501.7546−12498.0831−12501.6519−12501.6367−12498.0921−12501.7918
Tm−12936.7644−12943.9179−12940.1850−12943.7930−12943.7767−12940.1744−12943.9711
Yb−13388.0190−13395.2897−13391.4951−13395.1432−13395.1241−13391.4562−13395.3453
Lu−13848.2048−13855.5951−13851.7380−13855.4286−13855.4122−13851.8080−13855.6783
 Total energies (in Ha) for atoms in the seventh period of the periodic table up to U (Z = 87-92) with varying exchange-correlation functionals and no relativistic correction
ElementsPWPBEPBEsolTPSSSCANHFPBE0
Fr−22470.2876−22479.6839−22474.7978−22479.2553−22479.2304−22475.8587−22480.0581
Ra−23088.6521−23098.1757−23093.2231−23097.7207−23097.6948−23094.3037−23098.5550
Ac−23716.4644−23726.1165−23721.0987−23725.6353−23725.6087−23722.1921−23726.4962
Th−24353.8106−24363.5942−24358.5107−24363.0880−24363.0609−24359.6011−24363.9642
Pa−25001.2830−25011.1905−25006.0428−25010.6519−25010.6243−24963.6722−24968.0673
U−25658.4367−25668.4741−25663.2607−25667.9087−25667.8805−25664.2094−25668.7856
ReferencesKotochigova, S., Levine, Z.H., Shirley, E.L., Stiles, M.D. and Clark, C.W. (1997) Local-Density-Functional Calculations of the Energy of Atoms. Physical Review A, 55, 191-199. https://doi.org/10.1103/PhysRevA.55.191Kraisler, E., Makov, G. and Kelson, I. (2010) Ensemble v-Representable ab Initio Density-Functional Calculation of Energy and Spin in Atoms: A Test of Exchange-Correlation Approximations. Physical Review A, 82, Article ID: 042516. https://doi.org/10.1103/PhysRevA.82.042516Lehtola, S. (2019) A Review on Non-Relativistic, Fully Numerical Electronic Structure Calculations on Atoms and Diatomic Molecules. International Journal of Quantum Chemistry, 119, e25968. https://doi.org/10.1002/qua.25968Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Chiarotti, G.L., Cococcioni, M., Dabo, I., et al. (2009) Quantum ESPRESSO: A Modular and Open-Source Software Project for Quantum Simulations of Materials. Journal of Physics: Condensed Matter, 21, Article ID: 395502. https://doi.org/10.1088/0953-8984/21/39/395502Giannozzi, P., Andreussi, O., Brumme, T., Bunau, O., Buongiorno Nardelli, M., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Cococcioni, M., et al. (2017) Advanced Capabilities for Materials Modelling with Quantum ESPRESSO. Journal of Physics: Condensed Matter, 29, Article ID: 465901. https://doi.org/10.1088/1361-648X/aa8f79https://iopscience.iop.org/article/10.1088/1361-648X/aa8f79Opium—Pseudopotential Generation Project. https://opium.sourceforge.net/Al-Saidi, W.A., Walter, E.J. and Rappe, A.M. (2008) Optimized Norm-Conserving Hartree-Fock Pseudopotentials for Plane-Wave Calculations. Physical Review B, 77, Article ID: 075112. https://doi.org/10.1103/PhysRevB.77.075112Yang, J., Tan, L.Z. and Rappe, A.M. (2018) Hybrid Functional Pseudopotentials. Physical Review B, 97, Article ID: 085130. https://doi.org/10.1103/PhysRevB.97.085130Yao, Y. and Kanai, Y. (2017) Plane-Wave Pseudopotential Implementation and Performance of SCAN Meta-GGA Exchange-Correlation Functional for Extended Systems. The Journal of Chemical Physics, 146, Article ID: 224105. https://doi.org/10.1063/1.4984939Tao, J., Perdew J.P., Staroverov, V.N. and Scuseria, G.E. (2003) Climbing the Density Functional Ladder: Nonempirical Meta-Generalized Gradient Approximation Designed for Molecules and Solids. Physical Review Letters, 91, Article ID: 146401. https://doi.org/10.1103/PhysRevLett.91.146401Sun, J., Ruzsinszky, A. and Perdew, J.P. (2015) Strongly Constrained and Appropriately Normed Semilocal Density Functional. Physical Review Letters, 115, Article ID: 036402. https://doi.org/10.1103/PhysRevLett.115.036402Kotochigova, S., Levine, Z., Shirley, E., Stiles, M. and Clark, C. (2003) Atomic Reference Data for Electronic Structure Calculations. Physical Measurement Laboratory, NIST, Gaithersburg. http://physics.nist.gov/DFTdataKotochigova, S., Levine, Z.H., Shirley, E.L., Stiles, M.D. and Clark, C.W. (1997) Erratum: Local-Density-Functional Calculations of the Energy of Atoms. Physical Review A, 56, 5191-5192. https://doi.org/10.1103/PhysRevA.56.5191.2Vosko, S.H., Wilk, L. and Nusair, M. (1980) Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: A Critical Analysis. Canadian Journal of Physics, 58, 1200-1211. https://doi.org/10.1139/p80-159Lee, I.-H. and Martin, R.M. (1997) Applications of the Generalized-Gradient Approximation to Atoms, Clusters, and Solids. Physical Review B, 56, 7197-7205. https://doi.org/10.1103/PhysRevB.56.7197Johnson III, R.D. (2022) NIST Computational Chemistry Comparison and Benchmark Database. NIST Standard Reference Database Number 101, NIST, Gaithersburg. https://doi.org/10.18434/T47C7ZMcNaught, A.D. and Wilkinson, A. (1997) The IUPAC Compendium of Chemical Terminology. 2nd Edition, Blackwell Scientific Publications, Oxford.https://doi.org/10.1351/goldbookChakravorty, S.J., Gwaltney, S.R., Davidson, E.R., Parpia, F.A. and Fischer, C.F. (1993) Ground-State Correlation Energies for Atomic Ions with 3 to 18 Electrons. Physical Review A, 47, 3649-3670. https://doi.org/10.1103/PhysRevA.47.3649Davidson, E.R., Hagstrom, S.A., Chakravorty, S.J., Umar, V.M. and Fischer, C.F. (1991) Ground-State Correlation Energies for Two- to Ten-Electron Ions. Physical Review A, 44, 7071-7083. https://doi.org/10.1103/PhysRevA.44.7071Lehtola, S., Steigemann, C., Oliveira, M.J. and Marques, M.A. (2018) Recent Developments in Libxc—A Comprehensive Library of Functionals for Density Functional Theory. SoftwareX, 7, 1-5. https://doi.org/10.1016/j.softx.2017.11.002https://www.sciencedirect.com/science/article/pii/S2352711017300602Perdew, J.P., Burke, K. and Ernzerhof, M. (1997) Generalized Gradient Approximation Made Simple. Physical Review Letters, 78, 1396-1396. https://doi.org/10.1103/PhysRevLett.78.1396Perdew, J.P., Burke, K. and Ernzerhof, M. (1996) Generalized Gradient Approximation Made Simple. Physical Review Letters, 77, 3865-3868. https://doi.org/10.1103/PhysRevLett.77.3865Burke, K., Perdew, J.P. and Wang, Y. (1998) Derivation of a Generalized Gradient Approximation: The PW91 Density Functional. In: Dobson, J.F., Vignale, G. and Das, M.P., Eds., Electronic Density Functional Theory, Springer, Boston, 81-111. https://doi.org/10.1007/978-1-4899-0316-7_7Perdew, J.P. and Zunger, A. (1981) Self-Interaction Correction to Density-Functional Approximations for Many-Electron Systems. Physical Review B, 23, 5048-5079. https://doi.org/10.1103/PhysRevB.23.5048Kingsbury, R., Gupta, A.S., Bartel, C.J., Munro, J.M., Dwaraknath, S., Horton, M. and Persson, K.A. (2022) Performance Comparison of r<sup>2</sup>SCAN and Scan MetaGGA Density Functionals for Solid Materials via an Automated, High-Throughput Computational Workflow. Physical Review Materials, 6, Article ID: 013801. https://doi.org/10.1103/PhysRevA.55.191Furness, J.W., Kaplan, A.D., Ning, J., Perdew, J.P. and Sun, J. (2022) Construction of Meta-GGA Functionals through Restoration of Exact Constraint Adherence to Regularized Scan Functionals. The Journal of Chemical Physics, 156, Article ID: 034109. https://doi.org/10.1063/5.0073623Santra, G. and Martin, J.M.L. (2022) Pure and Hybrid Scan, Rscan, and r<sup>2</sup>SCAN: Which One Is Preferred in Ks- and Hf-Dft Calculations, and How Does D4 Dispersion Correction Affect This Ranking? Molecules, 27, Article No. 141. https://doi.org/10.3390/molecules27010141 https://www.mdpi.com/1420-3049/27/1/141Pickett, W.E. (1989) Pseudopotential Methods in Condensed Matter Applications. Computer Physics Reports, 9, 115-197. https://doi.org/10.1016/0167-7977(89)90002-6http://www.sciencedirect.com/science/article/pii/0167797789900026Lejaeghere, K., Bihlmayer, G., Bj&#246;rkman, T., Blaha, P., Blügel, S., Blum, V., Caliste, D., Castelli, I.E., Clark, S.J., et al. (1989) Reproducibility in Density Functional Theory Calculations of Solids. Science, 351, aad3000. https://doi.org/10.1126/science.aad3000Neugebauer, J. and Hickel, T. (2013) Density Functional Theory in Materials Science. WIREs Computational Molecular Science, 3, 438-448. https://doi.org/10.1002/wcms.1125https://wires.onlinelibrary.wiley.com/doi/pdf/10.1002/wcms.1125Verma, P. and Truhlar, D.G. (2020) Status and Challenges of Density Functional Theory. Trends in Chemistry, 2, 302-318.https://doi.org/10.1016/j.trechm.2020.02.005https://www.sciencedirect.com/science/article/pii/S2589597420300411