Density functional theory (DFT) and beyond-DFT methods are often used in combination with photoelectron spectroscopy to obtain physical insights into the electronic structure of molecules and solids. The Kohn–Sham eigenvalues are not electron removal energies except for the highest occupied orbital. The eigenvalues of the highest occupied molecular orbitals often underestimate the electron removal or ionization energies due to the self-interaction (SI) errors in approximate density functionals. In this work, we adapt and implement the density-consistent effective potential method of Kohut, Ryabinkin, and Staroverov [J. Chem. Phys. 140, 18A535 (2014)] to obtain SI-corrected local effective potentials from the SI-corrected Fermi–Löwdin orbitals and density in the Fermi–Löwdin orbital self-interaction correction scheme. The implementation is used to obtain the density of states (photoelectron spectra) and HOMO–LUMO gaps for a set of molecules and polyacenes. Good agreement with experimental values is obtained compared to a range of SI uncorrected density functional approximations.

Density functional theory1–3 (DFT) and beyond-DFT methods are often used in combination with photoelectron spectroscopy to obtain physical insights into the electronic structure of molecules and solids. The Kohn–Sham (KS) eigenvalues are not electron removal energies except for the highest occupied one,4–8 but they often, though not always, provide good approximations to electron binding energies (EBEs).9–12 Eigenvalues of the range separated hybrid functionals13 using tuned separation parameters generally provide good approximations to EBEs due to mitigation of self-interaction (SI) errors.13–15 SI error has a large role in the underestimation of the magnitude of the KS eigenvalues. For excitation energies, the HOMO–LUMO orbital difference in time-dependent KS theory plays a significant role. The first ionization energy and electron affinity in the exact KS theory are determined by the HOMO (highest partly occupied molecular orbital) energies of the system before and after the addition of a fraction of an electron. The HOMO–LUMO gap at the fixed electron number is not equal to the first-excitation energy but is close enough to it in a molecule to enable an accurate calculation in TDDFT (time-dependent density functional theory) of the charge transfer excitation of donor and acceptor molecules. The standard implementations of self-interaction correction (SIC) methods in the generalized KS scheme16,17 correct only the occupied orbitals, while the unoccupied orbitals see only the mean-field DFA (density functional approximation) potential. We here seek a way to make the unoccupied orbitals (including the Rydberg states) see a SI-corrected potential. For that purpose, we adapt and implement the density-consistent effective potential (DCEP) method of Kohut, Ryabinkin, and Staroverov18,19 to obtain effective local potentials from the Perdew–Zunger20 (PZ) self-interaction-corrected orbitals and density.

The Perdew–Zunger self-interaction correction (PZSIC) approach is closer to the SIC schemes20–22 that remove SI error on an orbital-by-orbital basis. The PZ energy results in an energy functional that is orbital dependent. In the KS-DFT scheme, implementing such orbital-dependent functionals requires computing the multiplicative potential vXC(r)=δEXCδρ(r). Here, EXC is the exchange–correlation functional and ρ(r) is the total electron density. When the functional EXC is not an explicit functional of the density, the potential cannot be obtained by a straightforward evaluation. Traditionally, an effective potential has been obtained by solving the optimized effective potential (OEP) integral equation.23–25 The OEP approach is often not well suited to routine calculations due to numerical instabilities and difficulties in solving it in finite basis sets.26–30 The PZSIC method has also been implemented within the Krieger–Li–Iafrate (KLI) approximation31–36 and within OEP using the real space approach.37 However, as mentioned earlier, OEP implementations using finite Gaussian basis sets are usually fraught with numerical instabilities. The recently developed method by Kohut, Ryabinkin, and Staroverov provides a practical alternative to the OEP method to obtain KS potentials in a straightforward manner.18,19,38,39 Kohut, Ryabinkin, and Staroverov in Ref. 19 have laid out a hierarchy of successively more accurate approximations, ending in DCEP. In this work, we adapt the DCEP method and apply it to the PZSIC using the Fermi–Löwdin orbital self-interaction correction scheme40 (FLOSIC) to obtain the SI-corrected eigenvalues and corresponding orbitals. The density of states (photoelectron spectra) and HOMO–LUMO gaps obtained using this approach are compared with experimental values for several molecules.

In Sec. II, we describe the DCEP method and validate our implementation in the UTEP-NRLMOL code.41 In Sec. III, we present our adaptation of the DCEP to the FLOSIC method. The computational details and results are presented in Sec. IV.

The simplest approximation to the OEP relies on the idea developed by Slater42 to construct an orbital-averaged potential weighted by |ϕi|2/ρ, where |ϕi|2 and ρ are, respectively, the density of the ith orbital and total electron density. In the context of the Hartree–Fock (HF) approximation, this results in the so-called Slater potential,

vS(r)=1ρ(r)i=1Nϕi*K̂ϕi=12ρ(r)|γ(r,r)|2|rr|dr,
(1)

with the Fock exchange operator K̂ and reduced density matrix γ(r,r)=iNϕi(r)ϕi*(r), where N is the number of occupied orbitals. Kohut et al.19 in 2014 defined a methodology to obtain higher-order approximations to the OEP for Hartree–Fock calculations. The method begins by rearranging the Fock equations as given by

122+vext(r)+vHHF(r)+K̂ϕiHF=εiHFϕiHF.
(2)

Multiplying both sides in the above equation by ϕiHF, summing over i from 1 to N, and then dividing both sides by ρHF(r) give

τLHF(r)ρHF(r)+vext(r)+vHHF(r)+vSHF(r)=ĪHF(r),
(3)

where τLHF(r) is the HF kinetic energy density in the Laplacian form and ĪHF(r) is the HF average local ionization energy. Likewise, repeating these steps for the KS equations gives

τLKS(r)ρKS(r)+vext(r)+vHKS(r)+vX(r)=ĪKS(r).
(4)

Here, the Laplacian kinetic energy density for a given wavefunction (WF) (e.g., HF or KS) is

τLWF(r)=12i=1NϕiWF*2ϕiWF,
(5)

and the average local ionization energy (sometimes ε̄WF) is defined by

ĪWF(r)=1ρWF(r)i=1NεiWF|ϕiWF|2.
(6)

The highest level of approximations defined by Kohut, Ryabinkin, and Staroverov is DCEP. This approximation relies on the assumption that the ground state densities in two schemes are equal. This equates to imposing the constraint ρKS(r)=ρHF(r). Subtracting Eq. (4) from Eq. (3) leads to

vXDCEP(r)=vSHF(r)+ĪHFĪKS+τHF(r)ρHF(r)τKS(r)ρKS(r),
(7)

where τ is the positive-definite form of the kinetic energy density such that τHF(r)=12i=1N|2ϕiHF|2.

In the DCEP method, the eigenvalues are then shifted so that ϵHOMOKS=ϵHOMOHF to ensure the correct behavior of the asymptotic potential in the limit of r. A full self-consistent calculation is performed subsequently. In this work, we define the tolerance for self-consistent calculations such that self-consistency is reached when the relative difference between the potentials in the two successive iterations is less than 10−8, that is, VnVn1Vn<108.

We validate our DCEP implementation in the UTEP-NRLMOL code41 by comparing our results against the HF results from the original DCEP work.19 The NRLMOL code,43–45 on which UTEP-NRLMOL is based, is a pure density functional code. The HF exchange was introduced in the UTEP-NRLMOL code using a semi-analytic scheme similar to what is used for calculating the Coulomb energy.46Figure 1 shows the DCEP potentials for the Ar atom and Li2 molecule obtained from the HF densities using the NRLMOL basis set.47 These potentials compare very well with the OEP exchange potentials reported by Kohut et al.,19 thus validating the present implementation of the DCEP method. There is a slight difference around 0.1 a.u. where the first bump in the exchange potential is somewhat less conspicuous in the present result. We find that this is a basis set artifact and uncontracting the basis reproduces the bump in the exchange potential closely.

FIG. 1.

The exchange potential curves obtained from the Hartree–Fock densities for (a) the argon atom plotted radially and (b) Li2 plotted along the molecular axis. The red curves are OEP from Ref. 19, and the black curves are obtained from the implementation in this work. The green circles represent grid points where the data points are calculated.

FIG. 1.

The exchange potential curves obtained from the Hartree–Fock densities for (a) the argon atom plotted radially and (b) Li2 plotted along the molecular axis. The red curves are OEP from Ref. 19, and the black curves are obtained from the implementation in this work. The green circles represent grid points where the data points are calculated.

Close modal

We propose to extend the DCEP method to obtain a multiplicative potential from PZSIC orbitals and density. To do so, we start with the PZSIC equations

122+vext(r)+vH(r)+vXC(r)+VSICi(r)ϕi=εiϕi.
(8)

Here, i is the orbital index, vext is the external potential, vH is the Coulomb potential, and VSICi is the self-interaction-correction term for the ith orbital. VSICi consists of the self-Coulomb potential U[ρi] term and the self-exchange–correlation potential vxc[ρi, 0] term as follows:

VSICi={U[ρi]+vxc[ρi,0]}
(9)
=d3rρi(r)|rr|vxc[ρi,0].
(10)

In our case, we use the Fermi–Löwdin orbital (FLO) implementation of PZSIC17,48 to determine VSICi such that ϕi=ϕiFLO and ρi=ρiFLO=|ϕiFLO|2. To form FLOs, first, a set of Fermi orbitals Fi is constructed with the density matrix and 3N positions called Fermi orbital descriptor (FOD) positions given as

Fi(r)=jNψj(ai)ψj(r)ρi(ai).
(11)

Then, the set of Fi’s is orthogonalized using Löwdin’s scheme to obtain FLOs. Similarly, as with the previous case in Sec. II, we multiply both sides of Eq. (8) with ϕi, sum over i from 1 to N, and divide both sides with ρSIC(r). This yields averaged-over quantities given as

τ(r)ρSIC(r)+vext(r)+vH(r)+vXC(r)+V̄SIC(r)=ĪSIC(r),
(12)

where ρSIC = iρi. Here, V̄SIC averages over the orbital SIC potentials using

V̄SIC(r)=i=1NvSICi(r)ρi(r)ρSIC(r).
(13)

Finally, subtracting Eq. (4) from Eq. (12) subject to the constraint in Eq. (7), we obtain the density-consistent effective potential

vXCDCEP(r)=vXCSIC(r)+V̄SIC(r)+ĪSICĪKS+τSIC(r)ρSIC(r)τKS(r)ρKS(r).
(14)

These SIC terms are obtained from a self-consistent FLOSIC calculation with optimized sets of FODs. Once they are determined, a self-consistent calculation can be performed to obtain the SIC effective potential.

The DCEP method was implemented in a highly scalable version of the UTEP-NRLMOL41 code developed at UTEP. The code uses a Gaussian orbital basis47 and an accurate numerical integration grid scheme.43 SI-corrected inputs were obtained from the FLOSIC code,49,50 which is based on the NRLMOL code. The default grid in the FLOSIC code requires a much higher grid density than standard DFT calculations. The mesh generated with the FLOSIC code was reused for the DCEP method for each system. All calculations use the default NRLMOL47 basis set and the PBE exchange–correlation functional.51,52

In the FLOSIC scheme, self-consistency is obtained using the Jacobi-like iterative scheme17 and only the occupied orbitals are affected directly by SIC. The DCEP method allows us to examine the effect of SI-corrected wavefunctions on unoccupied states. A study by Zhang and Musgrave53 compares the HOMO, LUMO, and HOMO–LUMO gap of 11 functionals and compares those with experimental ionization potential (IP) and the lowest excitation energy for a set of molecules. Following Zhang and Musgrave, we define the experimental HOMO energy as the negative of the experimental ionization energy and the experimental LUMO energy as the difference between the experimental lowest excitation energy and the HOMO energy. All the DFT and TDDFT (with adiabatic kernels) in our tables and figures other than the DCEP-SIC-PBE are from Zhang and Musgrave. Allen and Tozer54 utilized the procedure of Zhao, Morrison, and Parr55 to determine the KS eigenvalues and HOMO–LUMO gaps from coupled-cluster BD (Brueckner Doubles) electron densities. For the sake of clarity, we mention that our goal is not to assess various exchange–correlation functionals, which now run into several hundreds, but to test how good the DCEP-SIC unoccupied orbitals are. For this purpose, we restricted the comparison with the results reported by Zhang and Musgrave53 and Allen and Tozer.54 We have chosen C2H2, CO, H2, HF, C6H6, C10H8, C14H10, H2O, and NH3 and C2H4 molecules. The HOMOs, LUMOs, and HOMO–LUMO gaps calculated with DCEP-SIC are compared with the experimental values and with the results from Ref. 53 in Fig. 2. The mean absolute errors (MAEs) of the DCEP-SIC method for the HOMO, LUMO, and HOMO–LUMO gaps are compared with the MAEs of the 11 functionals used in Ref. 53 in Table I. We find that the DCEP-SIC eigenvalues perform well for all three categories of HOMOs, LUMOs, and HOMO–LUMO gaps. The HOMO eigenvalues of the FLOSIC scheme have previously shown good agreement with vertical ionization potentials from coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] and experiments.36,56–60 As detailed in Sec. II, the eigenvalues in DCEP-SIC calculations are shifted such that the DCEP HOMO matches with the FLOSIC HOMO values. Thus, the HOMO levels in DCEP-SIC are the same as those in the FLOSIC method. For the chosen systems, the MAE of the FLOSIC HOMO eigenvalues is 1.09 eV with respect to the experimental ionization energies. This MAE is relatively small when compared with the 11 functionals, with only the KMLYP functional that has about 55.7% of the HF exchange providing better HOMO prediction (MAE, 0.83 eV) than DCEP-SIC (FLOSIC HOMO).

FIG. 2.

The calculated DCEP-SIC HOMO, LUMO, and HOMO–LUMO gaps (in eV) against experimental IPs and lowest excitation energies: (top left) HOMO energies, (top right) LUMO energies, (bottom left) HOMO–LUMO gaps, and (bottom right) TDDFT excitation energies. For comparison, values for the other functionals from Ref. 53 are shown. The solid line indicates the ideal agreement with experiments.

FIG. 2.

The calculated DCEP-SIC HOMO, LUMO, and HOMO–LUMO gaps (in eV) against experimental IPs and lowest excitation energies: (top left) HOMO energies, (top right) LUMO energies, (bottom left) HOMO–LUMO gaps, and (bottom right) TDDFT excitation energies. For comparison, values for the other functionals from Ref. 53 are shown. The solid line indicates the ideal agreement with experiments.

Close modal
TABLE I.

Mean absolute errors (in eV) of calculated HOMO, LUMO, and HOMO–LUMO gaps and TDDFT excitation energies with respect to experimental ionization potentials and the first-excitation energies.

MethodHOMOLUMOHOMO–LUMOTDDFT
DCEP-SIC-PBE 1.09 0.73 1.01 ⋯ 
SVWNa 3.69 3.74 0.67 1.05 
BLYPa 4.41 4.30 0.66 0.76 
BP86a 4.20 4.29 0.66 1.03 
BPW91a 4.30 4.41 0.66 1.07 
PW91a 4.24 4.21 0.64 0.99 
PBEa 4.30 4.04 0.65 1.00 
B3LYPa 3.13 4.93 1.80 0.85 
KMLYPa 0.83 5.96 5.15 1.52 
BH and HLYPa 1.58 6.04 4.49 1.49 
O3LYPa 3.67 4.67 1.04 1.05 
B1B95a 2.90 5.44 2.52 1.20 
MethodHOMOLUMOHOMO–LUMOTDDFT
DCEP-SIC-PBE 1.09 0.73 1.01 ⋯ 
SVWNa 3.69 3.74 0.67 1.05 
BLYPa 4.41 4.30 0.66 0.76 
BP86a 4.20 4.29 0.66 1.03 
BPW91a 4.30 4.41 0.66 1.07 
PW91a 4.24 4.21 0.64 0.99 
PBEa 4.30 4.04 0.65 1.00 
B3LYPa 3.13 4.93 1.80 0.85 
KMLYPa 0.83 5.96 5.15 1.52 
BH and HLYPa 1.58 6.04 4.49 1.49 
O3LYPa 3.67 4.67 1.04 1.05 
B1B95a 2.90 5.44 2.52 1.20 
a

Errors obtained using the data provided in Ref. 53.

Zhang and Musgrave53 found that all 11 functionals provide rather a poor estimate of the experimental LUMO eigenvalues. We find that the DCEP-SIC provides good agreement with experiment with an MAE of 0.73 eV. Although local-density approximation (LDA) and generalized gradient approximation (GGA) functionals do not perform well for HOMOs and LUMOs, they all give a much better estimate of HOMO–LUMO gaps when compared with the experimental lowest excitation energies. The MAEs for the HOMO–LUMO gaps for these functionals range from 0.64 to 0.67 eV. These functionals benefit from error cancellation while taking the difference between the HOMO and LUMO energies. In contrast, the errors are much larger for the hybrid functionals. The hybrid functionals are typically implemented in the generalized KS scheme. The mixing of HF exchange with DFA in hybrid functionals improves occupied eigenvalues due to mitigation of self-interaction errors, and as a result, HOMO eigenvalues are more accurate in hybrid functionals, but the LUMO eigenvalues remain poor. The MAE for the HOMO–LUMO gaps for the hybrid functionals range from 1.04 to 5.15 eV. The present DCEP-SIC method performs much better with an MAE of 1.01 eV. Zhang and Musgrave have also compared the HOMO–LUMO gaps with (the lowest) TDDFT excitation energies. In the bottom right plot of Fig. 2, we show the comparison of the TDDFT excitation energies reported by Zhang and Musgrave with experiment to facilitate a comparison of the HOMO–LUMO gaps shown in the bottom left panel against the TDDFT excitation energies. The plots show that the DCEP-SIC HOMO–LUMO gaps compare well to the TDDFT excitation energies (cf. Table I) for these molecules. Overall, the results show that the functionals that perform best for HOMO eigenvalues (KMLYP, BH, and B1B95) perform the worst for gaps since their LUMO predictions are rather poor. At the same time, the functionals that perform best for gaps such as PBE and BLYP are among the worst for predicting HOMO eigenvalues accurately. In contrast, the DCEP-SIC method gives reliable results for all three quantities. It, by construction, retains the accuracy of the PZSIC HOMO eigenvalues and yields accurate LUMO eigenvalues and thereby yields accurate HOMO–LUMO gaps, as borne out by the comparison of MAEs in Table I.

We also examine the effect on all occupied eigenvalues by comparing the SIC and DCEP-SIC results to the photoelectron spectra of polyacenes obtained through Ultraviolet Photoelectron Spectroscopy (UPS).61,62 The SIC eigenvalues, with LDA and PBE, shown in the top left plot for benzene in Fig. 3 are in qualitative agreement with experimental values but on a much broader energy scale. Additionally, SIC introduces an additional peak around 13.0 eV not seen in the experimental spectra. DCEP-SIC compresses the eigenvalue spectrum and provides a much closer fit to experimental results in terms of the number of peaks and energy spacing between the eigenvalues. The LDA calculations result in slightly higher energies in comparison to PBE but retain the similar spacing for both SIC and DCEP-SIC. Similar behavior for SIC against DCEP-SIC and DCEP-SIC-LDA against DCEP-SIC-PBE eigenvalues is found also for the other three polyacenes (naphthalene, antracene, and tetracene), and so only experimental and DCEP-SIC-PBE results are plotted in Fig. 3. For larger polyacenes, DCEP-SIC shows good agreement with experimental spectra for low- to mid-level energy states. For anthracene, the spectrum displays a rigid shift higher than the experiment spectrum. In all cases, DCEP-SIC over-compresses the spectra, so high-energy (core) states are underestimated.

FIG. 3.

Calculated spectra of polyacenes: (top left) benzene, (top right) naphthalene, (bottom left) anthracene, and (bottom right) tetracene. The calculated spectra are shifted to match the HOMO eigenvalue with the IP of experiments. The DCEP-SIC-PBE spectra are shown using black solid lines and DCEP-SIC-LDA results are shown using dashed lines. Experimental UPS results from Refs. 61 and 62 are shown in blue. For benzene, SIC results with PBE are shown at the top of the plot using red solid lines and SIC-LDA results are shown using red dashed lines.

FIG. 3.

Calculated spectra of polyacenes: (top left) benzene, (top right) naphthalene, (bottom left) anthracene, and (bottom right) tetracene. The calculated spectra are shifted to match the HOMO eigenvalue with the IP of experiments. The DCEP-SIC-PBE spectra are shown using black solid lines and DCEP-SIC-LDA results are shown using dashed lines. Experimental UPS results from Refs. 61 and 62 are shown in blue. For benzene, SIC results with PBE are shown at the top of the plot using red solid lines and SIC-LDA results are shown using red dashed lines.

Close modal

We have implemented the DCEP method to generate multiplicative effective potentials from FLOSIC orbitals and densities. While the FLOSIC method (within the generalized Kohn–Sham scheme) can provide accurate HOMO energies due to explicit removal of SI error, the unoccupied states are essentially the same as those of the uncorrected functional. The same behavior can be seen for many hybrid functionals as well. The use of a multiplicative effective potential results in a much improved description of properties related to unoccupied orbitals and so gives an accurate description of the HOMO and LUMO as well as the HOMO–LUMO gap, whereas the 11 functionals tested by Zhang and Musgrave fail for one or more of these quantities. We also find the HOMO–LUMO gap of DCEP-SIC to be comparable to TDDFT excitation energies of GGAs and hybrid functionals. The present results show that the DCEP-SIC eigenvalues provide a much better description of the experimental spectroscopic results compared to the standard PZSIC eigenvalues obtained with the FLOSIC formalism. For a set of polyacenes, we find that the DCEP-SIC eigenvalues provide a better approximation of the photoelectron spectra than the standard FLOSIC eigenvalues as DCEP-SIC corrects the broadened spectra of standard FLOSIC calculations.

R. R. Z. acknowledges discussions with Prof. Viktor Staroverov. The authors thank Dr. Po-Hao Chang for discussions. This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0018331. Computational time at the Texas Advanced Computing Center through NSF Grant No. TG-DMR090071 and the NERSC is gratefully acknowledged.

The data that support the findings of this study are available within the article.

1.
W.
Kohn
and
L. J.
Sham
, “
Self-consistent equations including exchange and correlation effects
,”
Phys. Rev.
140
,
A1133
A1138
(
1965
).
2.
R. O.
Jones
and
O.
Gunnarsson
, “
The density functional formalism, its applications and prospects
,”
Rev. Mod. Phys.
61
,
689
746
(
1989
).
3.
R. O.
Jones
, “
Density functional theory: Its origins, rise to prominence, and future
,”
Rev. Mod. Phys.
87
,
897
(
2015
).
4.
C.-O.
Almbladh
and
U.
von Barth
, “
Exact results for the charge and spin densities, exchange-correlation potentials, and density-functional eigenvalues
,”
Phys. Rev. B
31
,
3231
3244
(
1985
).
5.
M.
Levy
,
J. P.
Perdew
, and
V.
Sahni
, “
Exact differential equation for the density and ionization energy of a many-particle system
,”
Phys. Rev. A
30
,
2745
2748
(
1984
).
6.
J. P.
Perdew
,
R. G.
Parr
,
M.
Levy
, and
J. L.
Balduz
, “
Density-functional theory for fractional particle number: Derivative discontinuities of the energy
,”
Phys. Rev. Lett.
49
,
1691
1694
(
1982
).
7.
J. P.
Perdew
and
M.
Levy
, “
Comment on ‘Significance of the highest occupied Kohn-Sham eigenvalue’
,”
Phys. Rev. B
56
,
16021
16028
(
1997
).
8.
M. K.
Harbola
, “
Relationship between the highest occupied Kohn-Sham orbital eigenvalue and ionization energy
,”
Phys. Rev. B
60
,
4545
4550
(
1999
).
9.
N. O.
Jones
,
S. N.
Khanna
,
T.
Baruah
,
M. R.
Pederson
,
W.-J.
Zheng
,
J. M.
Nilles
, and
K. H.
Bowen
, “
Magnetic isomers and local moment distribution in Mn5O and Mn6O clusters
,”
Phys. Rev. B
70
,
134422
(
2004
).
10.
G.
Liu
,
A.
Pinkard
,
S. M.
Ciborowski
,
V.
Chauhan
,
Z.
Zhu
,
A. P.
Aydt
,
S. N.
Khanna
,
X.
Roy
, and
K. H.
Bowen
, “
Tuning the electronic properties of hexanuclear cobalt sulfide superatoms via ligand substitution
,”
Chem. Sci.
10
,
1760
1766
(
2019
).
11.
T.
Körzdörfer
,
S.
Kümmel
,
N.
Marom
, and
L.
Kronik
, “
When to trust photoelectron spectra from Kohn-Sham eigenvalues: The case of organic semiconductors
,”
Phys. Rev. B
79
,
201205
(
2009
).
12.
M.
Mundt
,
S.
Kümmel
,
B.
Huber
, and
M.
Moseler
, “
Photoelectron spectra of sodium clusters: The problem of interpreting Kohn-Sham eigenvalues
,”
Phys. Rev. B
73
,
205407
(
2006
).
13.
H.
Iikura
,
T.
Tsuneda
,
T.
Yanai
, and
K.
Hirao
, “
A long-range correction scheme for generalized-gradient-approximation exchange functionals
,”
J. Chem. Phys.
115
,
3540
3544
(
2001
).
14.
R.
Baer
,
E.
Livshits
, and
U.
Salzner
, “
Tuned range-separated hybrids in density functional theory
,”
Annu. Rev. Phys. Chem.
61
,
85
109
(
2010
).
15.
L.
Kronik
,
T.
Stein
,
S.
Refaely-Abramson
, and
R.
Baer
, “
Excitation gaps of finite-sized systems from optimally tuned range-separated hybrid functionals
,”
J. Chem. Theory Comput.
8
,
1515
1531
(
2012
).
16.
R. A.
Heaton
,
J. G.
Harrison
, and
C. C.
Lin
, “
Self-interaction correction for density-functional theory of electronic energy bands of solids
,”
Phys. Rev. B
28
,
5992
6007
(
1983
).
17.
Z.-h.
Yang
,
M. R.
Pederson
, and
J. P.
Perdew
, “
Full self-consistency in the Fermi-orbital self-interaction correction
,”
Phys. Rev. A
95
,
052505
(
2017
).
18.
I. G.
Ryabinkin
,
S. V.
Kohut
, and
V. N.
Staroverov
, “
Reduction of electronic wave functions to Kohn-Sham effective potentials
,”
Phys. Rev. Lett.
115
,
083001
(
2015
).
19.
S. V.
Kohut
,
I. G.
Ryabinkin
, and
V. N.
Staroverov
, “
Hierarchy of model Kohn-Sham potentials for orbital-dependent functionals: A practical alternative to the optimized effective potential method
,”
J. Chem. Phys.
140
,
18A535
(
2014
).
20.
J. P.
Perdew
and
A.
Zunger
, “
Self-interaction correction to density-functional approximations for many-electron systems
,”
Phys. Rev. B
23
,
5048
5079
(
1981
).
21.
I.
Lindgren
, “
A statistical exchange approximation for localized electrons
,”
Int. J. Quantum Chem.
5
,
411
420
(
1971
).
22.
M. S.
Gopinathan
, “
Improved approximate representation of the Hartree-Fock potential in atoms
,”
Phys. Rev. A
15
,
2135
2142
(
1977
).
23.
R. T.
Sharp
and
G. K.
Horton
, “
A variational approach to the unipotential many-electron problem
,”
Phys. Rev.
90
,
317
(
1953
).
24.
J. D.
Talman
and
W. F.
Shadwick
, “
Optimized effective atomic central potential
,”
Phys. Rev. A
14
,
36
40
(
1976
).
25.
J. B.
Krieger
,
Y.
Li
, and
G. J.
Iafrate
, “
Systematic approximations to the optimized effective potential: Application to orbital-density-functional theory
,”
Phys. Rev. A
46
,
5453
5458
(
1992
).
26.
S.
Ivanov
,
S.
Hirata
, and
R. J.
Bartlett
, “
Finite-basis-set optimized effective potential exchange-only method
,”
J. Chem. Phys.
116
,
1269
1276
(
2002
).
27.
W.
Yang
and
Q.
Wu
, “
Direct method for optimized effective potentials in density-functional theory
,”
Phys. Rev. Lett.
89
,
143002
(
2002
).
28.
T.
Heaton-Burgess
,
F. A.
Bulat
, and
W.
Yang
, “
Optimized effective potentials in finite basis sets
,”
Phys. Rev. Lett.
98
,
256401
(
2007
).
29.
S.
Kümmel
and
L.
Kronik
, “
Orbital-dependent density functionals: Theory and applications
,”
Rev. Mod. Phys.
80
,
3
(
2008
).
30.
A.
Heßelmann
,
A. W.
Götz
,
F.
Della Sala
, and
A.
Görling
, “
Numerically stable optimized effective potential method with balanced Gaussian basis sets
,”
J. Chem. Phys.
127
,
054102
(
2007
).
31.
J.
Garza
,
J. A.
Nichols
, and
D. A.
Dixon
, “
The optimized effective potential and the self-interaction correction in density functional theory: Application to molecules
,”
J. Chem. Phys.
112
,
7880
7890
(
2000
).
32.
S.
Patchkovskii
,
J.
Autschbach
, and
T.
Ziegler
, “
Curing difficult cases in magnetic properties prediction with self-interaction corrected density functional theory
,”
J. Chem. Phys.
115
,
26
42
(
2001
).
33.
X.-M.
Tong
and
S.-I.
Chu
, “
Density-functional theory with optimized effective potential and self-interaction correction for ground states and autoionizing resonances
,”
Phys. Rev. A
55
,
3406
3416
(
1997
).
34.
J.
Messud
,
P. M.
Dinh
,
P.-G.
Reinhard
, and
E.
Suraud
, “
Improved Slater approximation to SIC–OEP
,”
Chem. Phys. Lett.
461
,
316
320
(
2008
).
35.
C. D.
Pemmaraju
,
S.
Sanvito
, and
K.
Burke
, “
Polarizability of molecular chains: A self-interaction correction approach
,”
Phys. Rev. B
77
,
121204
(
2008
).
36.
C. M.
Diaz
,
T.
Baruah
, and
R. R.
Zope
, “
Fermi-Löwdin-orbital self-interaction correction using the optimized-effective-potential method within the Krieger-Li-Iafrate approximation
,”
Phys. Rev. A
103
,
042811
(
2021
).
37.
T.
Körzdörfer
,
S.
Kümmel
, and
M.
Mundt
, “
Self-interaction correction and the optimized effective potential
,”
J. Chem. Phys.
129
,
014110
(
2008
).
38.
E.
Ospadov
,
I. G.
Ryabinkin
, and
V. N.
Staroverov
, “
Improved method for generating exchange-correlation potentials from electronic wave functions
,”
J. Chem. Phys.
146
,
084103
(
2017
).
39.
I. G.
Ryabinkin
,
A. A.
Kananenka
, and
V. N.
Staroverov
, “
Accurate and efficient approximation to the optimized effective potential for exchange
,”
Phys. Rev. Lett.
111
,
013001
(
2013
).
40.
M. R.
Pederson
, “
Fermi orbital derivatives in self-interaction corrected density functional theory: Applications to closed shell atoms
,”
J. Chem. Phys.
142
,
064112
(
2015
).
41.
C. M.
Diaz
,
L.
Basurto
,
Y.
Yamamoto
,
T.
Baruah
, and
R. R.
Zope
, UTEP-NRLMOL code: This is a Gaussian based code based on the old NRLMOL code that has several new features, including Hartree-Fock exchange used for the validation of the Ryabinkin-Kohut-Staroverov method presented in this work. This scalable code, developed with an emphasis on being optimized for memory, can simulate systems containing several thousand basis functions.
42.
J. C.
Slater
, “
A simplification of the Hartree-Fock method
,”
Phys. Rev.
81
,
385
390
(
1951
).
43.
M. R.
Pederson
and
K. A.
Jackson
, “
Variational mesh for quantum-mechanical simulations
,”
Phys. Rev. B
41
,
7453
7461
(
1990
).
44.
K.
Jackson
and
M. R.
Pederson
, “
Accurate forces in a local-orbital approach to the local-density approximation
,”
Phys. Rev. B
42
,
3276
3281
(
1990
).
45.
M. R.
Pederson
,
D. V.
Porezag
,
J.
Kortus
, and
D. C.
Patton
, “
Strategies for massively parallel local-orbital-based electronic structure methods
,”
Phys. Status Solidi B
217
,
197
218
(
2000
).
46.
C. M.
Diaz
,
L.
Basurto
, and
R. R.
Zope
, Implementation of Hartree-Fock exchange in the UTEP-NRLMOL code using semi-analytic approach.
47.
D.
Porezag
and
M. R.
Pederson
, “
Optimization of Gaussian basis sets for density-functional calculations
,”
Phys. Rev. A
60
,
2840
2847
(
1999
).
48.
M. R.
Pederson
and
T.
Baruah
,
Self-Interaction Corrections within the Fermi-Orbital-Based Formalism
(
Academic Press
,
2015
), Chap. 8, pp.
153
180
.
49.
R. R.
Zope
,
T.
Baruah
, and
K. A.
Jackson
, FLOSIC 0.2, Based on the NRLMOL code of M. R. Pederson.
50.
Y.
Yamamoto
,
L.
Basurto
,
C. M.
Diaz
,
R. R.
Zope
, and
T.
Baruah
, “
Self-interaction correction to density functional approximations using Fermi-Löwdin orbitals: Methodology and parallelization
” (unpublished).
51.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
, “
Generalized gradient approximation made simple
,”
Phys. Rev. Lett.
77
,
3865
3868
(
1996
).
52.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
, “
Generalized gradient approximation made simple [Phys. Rev. Lett. 77, 3865 (1996)]
,”
Phys. Rev. Lett.
78
,
1396
(
1997
).
53.
G.
Zhang
and
C. B.
Musgrave
, “
Comparison of DFT methods for molecular orbital eigenvalue calculations
,”
J. Phys. Chem. A
111
,
1554
1561
(
2007
).
54.
M. J.
Allen
and
D. J.
Tozer
, “
Eigenvalues, integer discontinuities and NMR shielding constants in Kohn-Sham theory
,”
Mol. Phys.
100
,
433
439
(
2002
).
55.
Q.
Zhao
,
R. C.
Morrison
, and
R. G.
Parr
, “
From electron densities to Kohn-Sham kinetic energies, orbital energies, exchange-correlation potentials, and exchange-correlation energies
,”
Phys. Rev. A
50
,
2138
(
1994
).
56.
S.
Akter
,
Y.
Yamamoto
,
C. M.
Diaz
,
K. A.
Jackson
,
R. R.
Zope
, and
T.
Baruah
, “
Study of self-interaction errors in density functional predictions of dipole polarizabilities and ionization energies of water clusters using Perdew-Zunger and locally scaled self-interaction corrected methods
,”
J. Chem. Phys.
153
,
164304
(
2020
).
57.
Y.
Yamamoto
,
C. M.
Diaz
,
L.
Basurto
,
K. A.
Jackson
,
T.
Baruah
, and
R. R.
Zope
, “
Fermi-Löwdin orbital self-interaction correction using the strongly constrained and appropriately normed meta-GGA functional
,”
J. Chem. Phys.
151
,
154105
(
2019
).
58.
J.
Vargas
,
P.
Ufondu
,
T.
Baruah
,
Y.
Yamamoto
,
K. A.
Jackson
, and
R. R.
Zope
, “
Importance of self-interaction-error removal in density functional calculations on water cluster anions
,”
Phys. Chem. Chem. Phys.
22
,
3789
3799
(
2020
).
59.
S.
Akter
,
Y.
Yamamoto
,
R. R.
Zope
, and
T.
Baruah
, “
Static dipole polarizabilities of polyacenes using self-interaction-corrected density functional approximations
,”
J. Chem. Phys.
154
,
114305
(
2021
).
60.
S.
Adhikari
,
B.
Santra
,
S.
Ruan
,
P.
Bhattarai
,
N. K.
Nepal
,
K. A.
Jackson
, and
A.
Ruzsinszky
, “
The Fermi-Löwdin self-interaction correction for ionization energies of organic molecules
,”
J. Chem. Phys.
153
,
184303
(
2020
).
61.
M.
Yamauchi
,
Y.
Yamakita
,
H.
Yamakado
, and
K.
Ohno
, “
Collision energy resolved Penning ionization electron spectra of polycyclic aromatic hydrocarbons
,”
J. Electron Spectrosc. Relat. Phenom.
88-91
,
155
161
(
1998
).
62.
S.-Y.
Liu
,
K.
Alnama
,
J.
Matsumoto
,
K.
Nishizawa
,
H.
Kohguchi
,
Y.-P.
Lee
, and
T.
Suzuki
, “
He I ultraviolet photoelectron spectroscopy of benzene and pyridine in supersonic molecular beams using photoelectron imaging
,”
J. Phys. Chem. A
115
,
2953
2965
(
2011
).