The Hartree–Fock (HF) approximation has been an important tool for quantum-chemical calculations since its earliest appearance in the late 1920s and remains the starting point of most single-reference methods in use today. Intuition suggests that the HF kinetic energy should not exceed the exact kinetic energy; but no proof of this conjecture exists, despite a near century of development. Beginning from a generalized virial theorem derived from scaling considerations, we derive a general expression for the kinetic energy difference that applies to all systems. For any atom or ion, this trivially reduces to the well-known result that the total energy is the negative of the kinetic energy and, since correlation energies are never positive, proves the conjecture in this case. Similar considerations apply to molecules at their equilibrium bond lengths. We use highly precise calculations on Hooke’s atom (two electrons in a parabolic well) to test the conjecture in a nontrivial case and to parameterize the difference between density functional and HF quantities, but find no violations of the conjecture.

This paper addresses the simple question: In a typical electronic structure problem, can the Hartree–Fock1–3 (HF) kinetic energy ever exceed the exact kinetic energy? If not, can we prove that it cannot? This is an intriguingly simple question of principle that is surprisingly unaddressed, despite the use of HF as a starting point for many modern quantum-chemical methods, such as MP24 and coupled-cluster single double triple [CCSD(T)],5 or Green’s function calculations in materials, such as GW.6 Perhaps even more surprising is that, within density functional theory (DFT), where the definition of correlation is subtly different, it has long been known that the Kohn–Sham (KS) kinetic energy can never exceed the exact kinetic energy. The simplicity of the proof in this case relies on both quantities being defined as density functionals, so the comparison is made on the same density.7,8 In HF theory, the HF density differs from the exact density, which complicates matters considerably.

Here, we limit consideration to spin-unpolarized nonrelativistic, nonmagnetic electronic calculations. Moreover, we consider only those cases where symmetries are not broken, and the HF minimizer is a single Slater determinant. Nonetheless, we are unable to show that the HF kinetic energy cannot exceed the exact kinetic energy.

There are two key differences between HF quantities and KS quantities. While their definition in terms of orbitals is identical, the former is evaluated on those orbitals that minimize the HF energy for a given potential, while the latter are those orbitals that minimize only the kinetic energy on a given density. In the special case of just two electrons, the one occupied orbital is trivially determined by the density, so only the difference in densities remains. Even in this simpler case, we are unable to prove our conjecture. However, we do derive a virial relation for the HF kinetic energy difference that applies to all cases. Our result shows that the difference vanishes as the error in the HF density vanishes, i.e., in the high-density (or weakly correlated) limit.

In the absence of a proof, the next best thing is to look for a counterexample. This is less trivial than it at first appears. For atoms and ions, and for molecules at equilibrium, it is trivial to show that, because correlation energies are negative, the HF kinetic energy is never greater than the exact value. Moreover, the differences, especially those with DFT values, where the conjecture is true, are extremely small, making them very difficult to determine with sufficient accuracy using standard atom-centered basis sets.

Instead, we turned to the two-electron Hooke’s atom. The quadratic external potential ensures that the conjecture is not trivially true, allowing the possibility of a counterexample. Very precise numerical calculations demonstrate that the conjecture is correct for all values of the oscillator frequency for which the restricted HF solution is valid.

Why might this question be of practical importance? Exact conditions, such as the positivity of the kinetic correlation energy, are used in density functional theory all the time.9 Relationships between the total energy and kinetic energy (exact and approximate) have aided in the development of correlated basis sets for quantum-chemical calculations.10 Exact conditions are built in to approximations, or their violation is checked in approximations that do not automatically satisfy such conditions. Apart from intellectual curiosity, it may prove useful in the future of wavefunction theory to prove analogs. Although here we have failed, we hope our first steps might allow others to succeed.

A practical problem of recent interest where this is very relevant is in the use of HF densities as inputs to hybrid density functionals, which depend on the KS orbitals.11,12 In principle, one should perform a KS inversion to extract the KS orbitals for the HF density, but in practice it is infinitely easier to use the HF orbitals themselves. This introduces small differences in the kinetic energy and the vital question is as follows: Are these small relative to the differences produced by the density? The answer appears to be yes they are, but accuracy is limited by the limitations of KS inversions in atom-centered basis sets.13,14

Recently, it has been shown that there are certain regions of the Hubbard dimer where the HF kinetic energy becomes greater than the exact value.15 The authors report that the Hubbard dimer HF kinetic energy exceeds the exact value approximately when the on-site interaction strength is comparable to the external potential difference between the sites. Our conjecture only pertains to real-space Hamiltonians, unlike the case of the dimer.16,17 Discrete lattice systems, such as the Hubbard model and its variants, require separate considerations as many basic theorems do not hold as they would in real space.18 

Consider a Hamiltonian of the form

Ĥ=T̂+V̂ee+V̂,
(1)

where T̂ is the kinetic energy, V̂ee is the electron–electron interaction, V̂ is a one-body multiplicative external potential V̂=j=1Nv(rj), with atomic units used throughout. Consider a scaled wavefunction Ψγ = γ3N/2Ψ(γr1, …, γrN), whose γ = 1 value corresponds to the exact ground state wavefunction of Ĥ. From the variational principle, the scaled wavefunction Ψγ must obey the following minimization condition:

ddγΨγ|Ĥ|Ψγ|γ=1=0.
(2)

Using scaling properties of the energy operators,19,20 the minimization condition can be expressed as

ddγγ2T+γVee+d3rv(r)nγ(r)γ=1=0,
(3)

where nγ(r) = γ3nv(γr) is the scaled ground state one-body density. Computing the derivative and rearranging yields the virial theorem for the ground state energy,

E=T+I[v,n],
(4)

where

I[v,n]=d3r1+rv(r)n(r)
(5)

is the virial of the potential v(r) with the ground state density n(r). The virial theorem holds for all potentials v(r) whose ground state density n(r) vanishes sufficiently rapidly at infinity.

An analogous virial relation to Eq. (4) holds for the Hartree–Fock quantities because the HF solution is a variational minimum and the energy scales identically to the exact solution. Thus,

EHF=THF+I[v,nHF],
(6)

where nHF(r) is the HF ground state density, i.e., the density produced by the single Slater determinant ΦHF that minimizes the expectation value of the Hamiltonian Ĥ. Now, Ecconv.=EEHF is the conventional quantum-chemical definition of the correlation energy, originally credited to Löwdin.21 To distinguish Löwdin’s definition from the DFT correlation energy, we refer to Ecconv. simply as the conventional correlation energy. We define the kinetic contribution to the conventional correlation energy via

Tcconv.=TTHF,
(7)

and our conjecture is that this quantity is nonnegative. Taking the difference of Eqs. (4) and (6) yields

Tcconv.+Ecconv.=I[v,nnHF].
(8)

Equation (8) can be considered from several different perspectives. Since Ecconv. is nonpositive (by the variational theorem), only if I becomes more negative than Ecconv. will our conjecture be violated. However, we do not know if I in Eq. (8) always has a definite sign. We can expect it to involve tremendous cancellations inside the real-space integral as correlation energies are small and kinetic correlation energies are of the same order, while the inputs to the virial integral are not.

Next, we list several special cases where I[v, nnHF] vanishes. First, in the limit of weak correlation (such as the large-Z limit of nondegenerate ions of fixed N22–24), nHF(r) → n(r) sufficiently fast to make I on the right vanish, while Ecconv. remains finite. Thus, in such a limit, Tcconv.|Ecconv.|, guaranteeing its nonnegativity. Second, for any potential v(r) = −Z/r, such as an atom or ion, the virial itself identically vanishes, no matter what density it is evaluated on, and the same conclusion can be drawn. Finally, for any molecule at equilibrium (but not otherwise), treating the nuclei as classical,25 the virial for the electronic energy matches the atomic result, yielding the same conclusion.

The situation is eerily similar, but importantly different, for Kohn–Sham DFT. In this case, all energies are defined as density functionals and are therefore evaluated on the exact density for a given system. For any given density, the KS kinetic energy is defined to be the minimum over all possible wavefunctions,

TS[n]=minΨnΨ|T̂|Ψ.
(9)

Typically, the minimizer is a single Slater determinant, i.e., the KS wavefunction of KS orbitals. Since T is the exact interacting kinetic energy, by construction

Tc=TTS0,
(10)

for any density, including the exact density of our system. The scaling behavior of the correlation energy results in an analogous virial theorem,26 

Ec+Tc=J[vc,n],
(11)

where vc(r) is the exact correlation potential and

J[v,n]=d3rrv(r)n(r)
(12)

is just the virial of the potential. As we argue (and show) below, we expect this expression to be quantitatively close to its HF analog, Eq. (8), but its smallness arises in a very different way. Here, the potential is already small, but there is no density difference. Finally, we note that, in the few cases with sufficiently precise evaluations, J has never been found negative, i.e., Tc < |Ec|, but this has also never been proven to be generally true.

Like the HF case, in the weakly interacting limit, the virial term vanishes, and Ec → −Tc. Unlike the HF case, for atoms, ions, and molecules at equilibrium, the virial does not require J to vanish, and in general it does not. Thus, Ec is −42.1 mhartree for the He atom, but Tc is 36.6 mhartree.27 

Finally, we discuss differences between HF and KS-DFT quantities. First, we note that

ΔEc=EcEcconv.0
(13)

because the HF energy minimizes the expectation value of the Hamiltonian over all orbitals, while the KS scheme restricts orbitals to those from a single multiplicative KS potential.28 However, typically, ΔEc is far smaller in magnitude than Ec. In the weakly interacting limit, Ecconv.Ec, so that ΔEc vanishes. For the He atom, ΔEc is only −63 μhartree.28 The equivalent kinetic quantity is

ΔTc=TcTcconv.=THFTS,
(14)

which also vanishes in the weakly interacting limit. Finally, we might expect in general ΔEc + ΔTc to be even smaller in magnitude than any object discussed so far. However, this is not true when the virial makes Tcconv.=Ecconv., such as for the He atom. In that case, ΔTc is −5.5 mhartree, the difference being entirely due to the difference between the exact and approximate densities.

In Sec. III, we give exact results for a two-electron system for which the virial of the potential does not vanish.

Hooke’s atom consists of two electrons with Coulomb repulsion in a parabolic potential well v(r) = ω2r2/2. The presence of a parabolic potential does not permit a simplification of the virial result into the familiar atomic expression and it remains to be seen whether Tcconv. is positive. Closed form solutions exist for certain values of ω,29,30 but for arbitrary values, the solutions must be found numerically. The ground state can be expressed formally as an infinite series that truncates whenever a closed form solution exists.31 No closed form expression exists for the Hartree–Fock density, and it must be found numerically as well, but the result converges rapidly when using Gaussian basis functions. We produce the relevant Hartree–Fock quantities using the basis functions of O’Neill and Gill.32 When plotted, the exact and HF densities coincide closely for large ω (as seen in Fig. 1), but their difference is vital to our question.

FIG. 1.

Radial ground state densities for moderate and small oscillator frequencies of Hooke’s atom.

FIG. 1.

Radial ground state densities for moderate and small oscillator frequencies of Hooke’s atom.

Close modal

The following is an analysis of the kinetic energies and the terms that appear in the virial expression for the conventional correlation energy Eq. (8). In the high-density (ω) limit, the asymptotic behavior of the exact and HF kinetic energies is derivable from second order perturbation theory,33,34

E(ω)=3ω+2ωπc+O1ω,
(15)

where all energies are taken to be in hartree units and the exact value of the constant is given by

c=2π1+ln2177.9mhartree.
(16)

For the Hartree–Fock problem, the constant is exactly

cHF=4π1+ln(843)1328.2mhartree.
(17)

The Hellmann–Feynman theorem33,35,36 yields

I(ω)=E(ω)+T(ω)=3ω2dE(ω)dω,
(18)

and it applies exactly and within HF.37 Taking the difference between the exact and HF quantities to O(ω1/2),

ΔI(ω)dω,
(19)

where ΔI = I[v, nnHF]. The constant d has a closed form expression and is determined from the perturbation theory of the exact and HF energies. The numerical value can be found to many digits and is approximately d ≈ 7.03 mhartree.38 In the high-density limit, we get the expected result that Ecconv.Tcconv., and by extension T > THF.

The low-density (ω → 0) behavior of the total energy can be found by expanding the external potential about its classical equilibrium position and then computing the perturbation theory. At order O(ω4/3), the energy is approximately33 

E(ω)3ω42/3+2(3+3)ω4+79ω44/3,
(20)

with analogous expressions for T and I.

Determining the small ω dependence of THF and I[v, nHF] has several complications. Throughout all calculations, we assumed that the Hartree–Fock wavefunction remains a single Slater determinant, but in the strongly correlated limit, the minimizing Hartree–Fock wavefunction might be spin unrestricted or multi-determinantal in nature. For small values of ω, the HF eigenvalue problem becomes numerically unstable, but we can show that Ecconv. closely agrees with the form of Eq. (20),

Ecconv.(ω)ω2/3j=05ajωj/3(ω0),
(21)

and a similar form exists for Tcconv., where all coefficients have been determined numerically. According to perturbation theory in the high-density limit, Ecconv. can be closely approximated by a series in powers of ω−1/2,

Ecconv.(ω)j=04(1)jbjωj/2(ω),
(22)

again, with a similar form for Tcconv. and where the coefficients are determined numerically. The coefficients for Tcconv. and ΔI were determined from the fit of Ecconv. in both the low- and high-density limits via the Hellmann–Feynman theorem.

What follows are plots of the virial quantities for various values of the frequency. The curves change dramatically from small to large ω; in Fig. 2, we plot against log10(ω) to demonstrate the overall effectiveness and trends. The fitting functions plotted are piecewise defined, taking the low-density form for ω ≤ 1/2 and the high-density form for ω > 1/2. Figure 3 contains the rescaled low- and high-density fits to show their effectiveness in each respective limit. We choose ω = 1/2 as the switching point between fits since the energy is known analytically here.

FIG. 2.

Calculated correlation energies (points) and their approximate fits (solid) in hartrees.

FIG. 2.

Calculated correlation energies (points) and their approximate fits (solid) in hartrees.

Close modal
FIG. 3.

Energies and fits in low (left) and high (right) density limits, in hartrees: Ecconv. (green), Tcconv. (red), and ΔI (blue); dashed orange line denotes ω = 1/2.

FIG. 3.

Energies and fits in low (left) and high (right) density limits, in hartrees: Ecconv. (green), Tcconv. (red), and ΔI (blue); dashed orange line denotes ω = 1/2.

Close modal

We also produce the corresponding DFT counterparts and their approximate forms in the high- and low-density limits for comparison in Fig. 4. For two electrons in spin singlet, the noninteracting kinetic energy is given by the von Weizsäcker kinetic energy density functional39 and the exchange energy is simply negative of half the Hartree energy. For ω = 1/2, the exact expression for the kinetic energy can be produced in closed form. Below, we detail the errors of the fits in the low- and high-density limits in Tables IIII.

FIG. 4.

Difference between HF (dark) and DFT (light) components as a function of ω; same scheme as Fig. 3.

FIG. 4.

Difference between HF (dark) and DFT (light) components as a function of ω; same scheme as Fig. 3.

Close modal
TABLE I.

Coefficients of low-density fit Eq. (21), in hartrees.

a0a1a2a3a4a5
Ecconv. 0.618 2.190 3.900 3.956 2.197 0.518 
Ec 0.663 2.501 4.774 5.227 3.112 0.780 
Tcconv. 1.095 3.900 5.934 4.394 1.295 
Tc 0.030 1.055 3.411 4.943 3.539 1.020 
a0a1a2a3a4a5
Ecconv. 0.618 2.190 3.900 3.956 2.197 0.518 
Ec 0.663 2.501 4.774 5.227 3.112 0.780 
Tcconv. 1.095 3.900 5.934 4.394 1.295 
Tc 0.030 1.055 3.411 4.943 3.539 1.020 
TABLE II.

Coefficients of high-density fit Eq. (22), in mhartrees.

b0b1b2b3b4
Ecconv. 49.70 93.68 1.051 0.044 1.098 × 10−03 
Ec 49.70 93.68 1.114 0.069 5.819 × 10−03 
Tcconv. 49.70 16.40 2.628 0.141 4.395 × 10−03 
Tc 49.70 19.34 3.944 0.038 4.306 × 10−03 
b0b1b2b3b4
Ecconv. 49.70 93.68 1.051 0.044 1.098 × 10−03 
Ec 49.70 93.68 1.114 0.069 5.819 × 10−03 
Tcconv. 49.70 16.40 2.628 0.141 4.395 × 10−03 
Tc 49.70 19.34 3.944 0.038 4.306 × 10−03 
TABLE III.

Errors of the fitting functions, in μhartrees. The first two columns give the difference between the fit and its respective ω = 1/2 value. The last column gives the difference of the derivatives between the high- and low-density fits at ω = 1/2.

ω12+ω12Deriv.
Ecconv. 2.076 −0.063 14.73 
Ec 2.246 −3.333 222.1 
Tcconv. 4.845 −4.066 279.7 
Tc 1.099 −0.143 261.2 
ΔEc 0.170 −3.270 207.4 
ω12+ω12Deriv.
Ecconv. 2.076 −0.063 14.73 
Ec 2.246 −3.333 222.1 
Tcconv. 4.845 −4.066 279.7 
Tc 1.099 −0.143 261.2 
ΔEc 0.170 −3.270 207.4 

In the low- and high-density limits, the behavior of the correlation and the kinetic correlation energies is analogous to that of the conventional values. The fitting functions are taken to be the same as for Ecconv. and Tcconv., with the only exception being that the coefficients of −Tc were fit separately from Ec and cannot be determined from the Hellmann–Feynman theorem.

From Fig. 5, we find that ΔEc ≤ 0, as we would expect, and this quantity is accurately reproduced by our fits. Differences between the high-density fits of Ec and Ecconv. first begin at O(ω1). For all values of ω, we find that TTHF from the positivity of the conventional kinetic correlation energy. Since we find Tcconv.>Tc, the stronger condition TS > THF is satisfied for Hooke’s atom.

FIG. 5.

Difference between conventional and DFT correlation energies in the low-density (left) and high-density (right) limits and their fits, in hartrees. Dashed orange line denotes ω = 1/2.

FIG. 5.

Difference between conventional and DFT correlation energies in the low-density (left) and high-density (right) limits and their fits, in hartrees. Dashed orange line denotes ω = 1/2.

Close modal

If NA classical nuclei are assumed within the Born–Oppenheimer approximation, the molecular virial integral difference simplifies considerably due to the Hellmann–Feynman theorem,

ΔI({Rα})=α=1NARααEcconv.({Rα}),
(23)

where Ecconv.({Rα}) is the conventional correlation energy for an arrangement of nuclei {Rα}. For this case, the kinetic energy difference can also be cast into a simpler expression,

Tcconv.({Rα})=β=1NARβRβEcconv.({Rα}).
(24)

We can see that for the case of molecules, a sufficient condition to guarantee T > THF is the requirement that RβEcconv.({Rα}) be monotonically increasing as a function of its radial coordinates Rβ.

For homonuclear diatomic molecules, the atomic form of the virial theorem holds in both the united atom limit (zero nuclear separation) and at the dissociation limit (infinite nuclear separation).40 However, HF will usually break symmetry at sufficiently large separations. For classical nuclei in a diatomic molecule, the virial yields

Ecconv.+Tcconv.=RdEcconv.dR,
(25)

where R is the internuclear separation. Below equilibrium, the conventional H2 correlation energy is relatively insensitive to bond length (at equilibrium, it is −42 mhartree, the same as the united atom limit), we expect Tcconv.Ecconv. and so remains positive. In Fig. 6, we plot values of the conventional correlation energy and the kinetic energy difference near the united atom limit of the hydrogen molecule. In the united atom limit, the conventional correlation energy can be seen to approach a negative constant, where eventually Ecconv.Tcconv.. The correlation energies were determined using PySCF,41 where the exact values were computed from CCSD; all calculations were done using cc-pVTZ basis functions.

FIG. 6.

H2 conventional correlation energy (green), negative of the conventional kinetic correlation energy (red), and their difference (blue), in mhartrees, as a function of nuclear separation. The equilibrium bond length is given by orange line at R ≈ 0.74 Å.

FIG. 6.

H2 conventional correlation energy (green), negative of the conventional kinetic correlation energy (red), and their difference (blue), in mhartrees, as a function of nuclear separation. The equilibrium bond length is given by orange line at R ≈ 0.74 Å.

Close modal

We have conjectured, for cases where a single Slater determinant is the minimizer in a Hartree–Fock calculation, that the HF kinetic energy is always less than the true kinetic energy, i.e., that of the exact solution of the Schrödinger equation. We have explored this topic in depth, making use of the virial theorem, but have been unable to find a proof of this conjecture. We have shown why the Kohn–Sham kinetic energy is always less than the true kinetic energy, by construction, in density functional theory.

We numerically calculate several nontrivial examples (the Hooke’s atom of two electrons in an oscillator potential and the H2 molecule away from equilibrium). In each case, we find no violations of our conjecture.

Our conjecture, if true, might prove to be a useful constraint on wavefunction approximations. Our conjecture is limited to Hamiltonians that are in the real-space continuum and where the inter-electron repulsion is Coulombic. The case of greatest practical interest is when the external potential is a sum of Coulombic attractions, i.e., the nonrelativistic, Born–Oppenheimer limit of molecules and solids in the absence of external fields. We hope the publication of this work will lead to either a proof or a counterexample.

See the supplementary material (1) for table of computed conventional and DFT correlation energies in the case of a Hooke’s atom system at various oscillator frequencies and (2) for the conventional correlation values of H2 at various nuclear separations.

S.C. and K.B. were supported by NSF Award No. CHE-2154371. M.L. was supported by the Julian Schwinger Foundation.

The authors have no conflicts to disclose.

S. Crisostomo: Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). M. Levy: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). K. Burke: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

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

1.
D. R.
Hartree
, “
The wave mechanics of an atom with a non-coulomb central field. Part I. Theory and methods
,”
Math. Proc. Cambridge Philos. Soc.
24
,
89
110
(
1928
).
2.
J. C.
Slater
, “
The self consistent field and the structure of atoms
,”
Phys. Rev.
32
,
339
348
(
1928
).
3.
V.
Fock
, “
Näherungsmethode zur lösung des quantenmechanischen mehrkörperproblems
,”
Z. Phys.
61
,
126
148
(
1930
).
4.
D.
Cremer
, “
Møller–Plesset perturbation theory: From small molecule methods to methods for thousands of atoms
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
1
,
509
530
(
2011
).
5.
H. G.
Kummel
, “
A biography of the coupled cluster method
,” in
Recent Progress in Many-Body Theories
(
World Scientific Publishing Company
,
2002
), pp.
334
348
.
6.
L.
Hedin
, “
On correlation effects in electron spectroscopies and the GW approximation
,”
J. Phys.: Condens. Matter
11
,
R489
R528
(
1999
).
7.
W.
Kohn
and
L. J.
Sham
, “
Self-consistent equations including exchange and correlation effects
,”
Phys. Rev.
140
,
A1133
A1138
(
1965
).
8.
W.
Kohn
, “
Nobel Lecture: Electronic structure of matter-wave functions and density functionals
,”
Rev. Mod. Phys.
71
,
1253
(
1999
).
9.
L. O.
Wagner
,
Z.-h.
Yang
, and
K.
Burke
, “
Exact conditions and their relevance in TDDFT
,” in
Fundamentals of Time-Dependent Density Functional Theory
, edited by
M. A.
Marques
,
N. T.
Maitra
,
F. M.
Nogueira
,
E.
Gross
, and
A.
Rubio
(
Springer
,
Berlin, Heidelberg
,
2012
), pp.
101
123
.
10.
R. N.
Hill
, “
Rates of convergence and error estimation formulas for the Rayleigh–Ritz variational method
,”
J. Chem. Phys.
83
,
1173
1196
(
1985
).
11.
P. M. W.
Gill
,
B. G.
Johnson
,
J. A.
Pople
, and
M. J.
Frisch
, “
An investigation of the performance of a hybrid of Hartree-Fock and density functional theory
,”
Int. J. Quantum Chem.
44
,
319
331
(
1992
).
12.
S.
Song
,
S.
Vuckovic
,
E.
Sim
, and
K.
Burke
, “
Density-corrected DFT explained: Questions and answers
,”
J. Chem. Theory Comput.
18
,
817
827
(
2022
).
13.
S.
Nam
,
S.
Song
,
E.
Sim
, and
K.
Burke
, “
Measuring density-driven errors using Kohn–Sham inversion
,”
J. Chem. Theory Comput.
16
,
5014
5023
(
2020
).
14.
S.
Nam
,
R. J.
McCarty
,
H.
Park
, and
E.
Sim
, “
Ks-pies: Kohn–Sham inversion toolkit
,”
J. Chem. Phys.
154
,
124122
(
2021
).
15.
S.
Giarrusso
and
A.
Pribram-Jones
, “
Comparing correlation components and approximations in Hartree–Fock and Kohn–Sham theories via an analytical test case study
,”
J. Chem. Phys.
157
,
054102
(
2022
).
16.
C. A.
Coulson
and
I.
Fischer
, “
XXXIV. Notes on the molecular orbital treatment of the hydrogen molecule
,”
Philos. Mag.
40
,
386
393
(
1949
).
17.
D. J.
Carrascal
,
J.
Ferrer
,
J. C.
Smith
, and
K.
Burke
, “
The Hubbard dimer: A density functional case study of a many-body problem
,”
J. Phys.: Condens. Matter
27
,
393001
(
2015
).
18.
M.
Penz
and
R.
van Leeuwen
, “
Density-functional theory on graphs
,”
J. Chem. Phys.
155
,
244111
(
2021
).
19.
J. O.
Hirschfelder
and
J. F.
Kincaid
, “
Application of the virial theorem to approximate molecular and metallic eigenfunctions
,”
Phys. Rev.
52
,
658
661
(
1937
).
20.
P.-O.
Löwdin
, “
Scaling problem, virial theorem, and connected relations in quantum mechanics
,”
J. Mol. Spectrosc.
3
,
46
66
(
1959
).
21.
P.-O.
Löwdin
, “
Quantum theory of many-particle systems. III. Extension of the Hartree-Fock scheme to include degenerate systems and correlation effects
,”
Phys. Rev.
97
,
1509
1520
(
1955
).
22.
E. H.
Lieb
and
B.
Simon
, “
The Thomas-Fermi theory of atoms, molecules and solids
,”
Adv. Math.
23
,
22
116
(
1977
).
23.
E. H.
Lieb
, “
Thomas-Fermi and related theories of atoms and molecules
,”
Rev. Mod. Phys.
53
,
603
641
(
1981
).
24.
J. G.
Conlon
, “
Semi-classical limit theorems for Hartree-Fock theory
,”
Commun. Math. Phys.
88
,
133
150
(
1983
).
25.
W. L.
Clinton
, “
Forces in molecules. I. Application of the virial theorem
,”
J. Chem. Phys.
33
,
1603
1606
(
1960
).
26.
M.
Levy
and
J. P.
Perdew
, “
Hellmann-Feynman, virial, and scaling requisites for the exact universal density functionals. Shape of the correlation potential and diamagnetic susceptibility for atoms
,”
Phys. Rev. A
32
,
2010
(
1985
).
27.
C.-J.
Huang
and
C. J.
Umrigar
, “
Local correlation energies of two-electron atoms and model systems
,”
Phys. Rev. A
56
,
290
296
(
1997
).
28.
E. K. U.
Gross
,
M.
Petersilka
, and
T.
Grabo
, “
Conventional quantum chemical correlation energy versus density-functional correlation energy
,” in
Chemical Applications of Density-Functional Theory
(
ACS Publications
,
1996
), Vol. 629, pp.
42
53
.
29.
N. R.
Kestner
and
O.
Sinanoḡlu
, “
Study of electron correlation in helium-like systems using an exactly soluble model
,”
Phys. Rev.
128
,
2687
2692
(
1962
).
30.
M.
Taut
, “
Two electrons in an external oscillator potential: Particular analytic solutions of a Coulomb correlation problem
,”
Phys. Rev. A
48
,
3561
3566
(
1993
).
31.
S.
Ivanov
,
K.
Burke
, and
M.
Levy
, “
Exact high-density limit of correlation potential for two-electron density
,”
J. Chem. Phys.
110
,
10262
(
1999
).
32.
D. P.
O’Neill
and
P. M. W.
Gill
, “
Wave functions and two-electron probability distributions of the Hooke’s-law atom and helium
,”
Phys. Rev. A
68
,
022505
(
2003
).
33.
J.
Cioslowski
and
K.
Pernal
, “
The ground state of harmonium
,”
J. Chem. Phys.
113
,
8434
8443
(
2000
).
34.
P. M. W.
Gill
and
D. P.
O’Neill
, “
Electron correlation in Hooke’s law atom in the high-density limit
,”
J. Chem. Phys.
122
,
094110
(
2005
).
35.
R. P.
Feynman
, “
Forces in molecules
,”
Phys. Rev.
56
,
340
343
(
1939
).
36.
R. E.
Stanton
, “
Hellmann‐Feynman theorem and correlation energies
,”
J. Chem. Phys.
36
,
1298
1300
(
1962
).
37.
E.
Yurtsever
and
J.
Hinze
, “
The Hellmann–Feynman theorem for open-shell and multiconfiguration SCF wave functions
,”
J. Chem. Phys.
71
,
1511
(
1979
).
38.
R. J.
White
and
W. B.
Brown
, “
Perturbation theory of the Hooke’s law model for the two-electron atom
,”
J. Chem. Phys.
53
,
3869
3879
(
1970
).
39.
C. F.
von Weizsäcker
, “
Zur theorie der kernmassen
,”
Z. Phys. A: Hadrons Nucl.
96
,
431
458
(
1935
).
40.
J. C.
Slater
, “
The virial and molecular structure
,”
J. Chem. Phys.
1
,
687
(
1933
).
41.
Q.
Sun
,
X.
Zhang
,
S.
Banerjee
,
P.
Bao
,
M.
Barbry
,
N. S.
Blunt
,
N. A.
Bogdanov
,
G. H.
Booth
,
J.
Chen
,
Z.-H.
Cui
,
J. J.
Eriksen
,
Y.
Gao
,
S.
Guo
,
J.
Hermann
,
M. R.
Hermes
,
K.
Koh
,
P.
Koval
,
S.
Lehtola
,
Z.
Li
,
J.
Liu
,
N.
Mardirossian
,
J. D.
McClain
,
M.
Motta
,
B.
Mussard
,
H. Q.
Pham
,
A.
Pulkin
,
W.
Purwanto
,
P. J.
Robinson
,
E.
Ronca
,
E. R.
Sayfutyarova
,
M.
Scheurer
,
H. F.
Schurkus
,
J. E. T.
Smith
,
C.
Sun
,
S.-N.
Sun
,
S.
Upadhyay
,
L. K.
Wagner
,
X.
Wang
,
A.
White
,
J. D.
Whitfield
,
M. J.
Williamson
,
S.
Wouters
,
J.
Yang
,
J. M.
Yu
,
T.
Zhu
,
T. C.
Berkelbach
,
S.
Sharma
,
A. Y.
Sokolov
, and
G. K.-L.
Chan
, “
Recent developments in the PySCF program package
,”
J. Chem. Phys.
153
,
024109
(
2020
).

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