J. Phys. Chem. A 1998, 102, 3151-3156
3151
Connections between High-Density Scaling Limits of DFT Correlation Energies and Second-Order Z-1 Quantum Chemistry Correlation Energy Stanislav Ivanov and Mel Levy* Department of Chemistry and Quantum Theory Group, Tulane UniVersity, New Orleans, Louisiana 70118 ReceiVed: September 29, 1997; In Final Form: February 4, 1998
In the continuing search for ever-better approximations to the full density-functional correlation energy functional Ec[n], we established the link between the second-order component of the correlation energy, (2) 3 E(2) c [n] [which occurs through uniform scaling, Ec [n] ) lim Ec[nλ], where nλ(x,y,z) ) λ n(λx,λy,λz)], and λf∞ QC,(2) . Except when the known result for the second-order Z-1 quantum chemistry correlation energy, Ec QC,(2) certain degeneracies occur, E(2) [n] e E , with an equality only for two electrons. On the other hand, the c c correlation energy functional HFEc[n], whose functional derivative is meant to be added to the Hartree-Fock non-local effective potential to produce, via self-consistency, the exact ground-state density and ground-state QC,(2) HF energy, satisfies the equality HFE(2) , where HFE(2) Ec[nλ], for any number of electrons, c [n]) Ec c [n] ) lim λf∞ (2) except when some degeneracies occur. Because quantities 2Ec [n] and 2HFE(2) c [n] are the initial slopes in the adiabatic connection formulas for Ec[n] and HFEc[n], respectively, the presented equalities involving HF (2) Ec [n] are especially significant. Five numerical tests are presented for closed- and open-shell densities obtained from hydrogenic orbitals. These tests are applied to widely used approximations to correlation energies.
I. Introduction To arrive at the very best approximations to the exact densityfunctional correlation energies, one needs knowledge of as many conditions as possible that reflect their properties. The idea is to modify approximate functionals so they satisfy newly discovered conditions. With this in mind, the study in this paper was undertaken. We shall first consider the correlation energy functional Ec[n]. This functional1-5 is meant to be employed as part of the full exchange-correlation functional, Exc[n], for variational calculations. Although Ex[n] possesses simple dimensionality in that it scales homogeneously as Ex[nλ] ) λEx[n], where nλ(x,y,z) ) λ3n(λx,λy,λz), Ec[n] is the part of Exc[n] that contains the complicated dimensionality in that its scaling is not homogeneous.5 One of the reasons that Ec[n] is used instead of the traditional quantum chemistry (QC) correlation energy, EQC c , for the correlation part of Exc[n] is that the simple scaling for the exchange only occurs with Ec[n]. Thus, accurate approximations for the exchange component of Exc[n] are more easily obtained with the use of Ec[n] than with the use of EQC c . The simple homogeneous scaling for Ex[n] stems from the fact that Ex[n] is defined through the wavefunction that minimizes only the kinetic energy operator and yields density n(r).5 The scaled wavefunction, which yields the scaled density nλ(r), also minimizes the kinetic energy operator. The HartreeFock (HF) exchange energy is defined through the single determinant that minimizes the Hamiltonian for the system of interest, and yields the HF density. The scaled version of the HF wavefunction does yield the scaled HF density but does not minimize the same interacting Hamiltonian.5 There are very few known values of Ec[n] for use for comparisons with approximations to Ec[n]. Examples include
the recent two-electron results of Umrigar and Gonze.6,7 On the other hand, we do know a number of exact values for the traditional QC correlation energy, EQC (see, for instance, the c results of Ivanova and Safronova8 and Davidson and coworkers9A,9B). Consequently, in this paper we shall utilize a few of these known values of this latter familiar correlation energy. Specifically, our main goal is to establish a link between the known numbers obtained by means of the asymptotic 1/Z expansion and the previously unknown density functional theory (DFT) correlation energies, Ec[n], of hydrogen-like densities. We introduce relationships that connect known results for the , in the wellsecond-order QC correlation energy, EQC,(2) c studied 1/Z expansion as Zf∞,8-12 and the second-order component of the DFT correlation energy, E(2) c [n] , which is a result of uniform scaling of the density (i.e., lim Ec[nλ] ) λf∞ 13-16 Previous DFT-Z-1 connections have been made E(2) c [n]). by Perdew, McMullen, and Zunger17 and by Chakravorty and Davidson.9A Here the n densities shall be ground-state densities of atomic Hamiltonians without the electron-electron repulsion operator. For all nondegenerate cases, and for certain degenerate QC,(2) cases, we shall show that E(2) , with an equality for, c [n] e Ec and only for, two electrons. There is an important definition for a correlation energy for which an equality actually generally occurs. In particular, we QC,(2) shall show that HFE(2) for any number of c [n] equals Ec HF electrons, where Ec[n] is defined slightly differently from Ec[n], and lim HFEc[nλ] ) HFE(2) c [n]. The functional derivative λf∞ of HFEc[n] is meant to be added to the HF nonlocal potential leading to HF-like equations. The resultant modified HF equations allow one, in principle, via self-consistency, to calculate the exact ground-state density and the exact groundstate energy. For a detailed discussion of HFEc[n], see ref 14.
S1089-5639(97)03141-1 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/15/1998
3152 J. Phys. Chem. A, Vol. 102, No. 18, 1998
Ivanov and Levy
As we shall discuss, HFE(2) c [n] is especially significant because (2) HF 2 Ec [n] is the initial slope in the adiabatic connection formula for HFEc[n].
The corresponding series for the wavefunctions ΨGS R and are
ΦHF R
∞
(0) 1 (1) 2 (2) 3 (3) ΨGS R ) ΨGS + R ΨGS + R ΨGS + R ΨGS ... )
II. Definitions, Notation, and Theoretical Results In atomic units, let us define the Hamiltonian operator H ˆ R as N
H ˆ R ) Tˆ + RVˆ ee +
RjΨ(j) ∑ GS j)0 (10)
and
Vo(ri) ∑ i)1
(1)
∞
(0) 1 (1) 2 (2) 3 (3) ΦHF R ) ΦHF + R ΦHF + R ΦHF + R ΦHF ... )
with Tˆ being the kinetic energy operator
RjΦ(j) ∑ HF j)0 (11)
N
1 Tˆ ) - ∇i2 2 i)1
∑
(2)
Vˆ ee is the operator of the electron-electron repulsion N-1
Vˆ ee )
N
1
∑ ∑ i)1 j)1+i |r - r | i
(3)
When H ˆ o has a nondegenerate ground state, both series begin with the same Φo, which is the ground-state solution to eq 9, k (0) ) 0. In other words, Ψ(0) GS ) ΦHF ) Φo. Unless otherwise stated, this nondegeneracy shall be assumed. The second and third terms in the corresponding expansions HF for EGS R and ER , with nondegenerate Eo, are
j
and Vo(r) is a local spin-independent multiplicative operator. (Later we shall utilize numbers for which Vo(r) ) -1/r.) The ground-state energy of H ˆ R, EGS R , is given by GS ˆ R|ΨGS EGS R ) 〈ΨR |H R 〉
(4)
(5)
where ΦHF ˆ R〉. R is the single determinant that minimizes 〈H The QC correlation energy EQC c,R is defined as the difference HF 10,13 namely, between EGS R and ER ; QC Ec,R
≡
EGS R
-
EHF R
)
〈ΨGS ˆ R|ΨGS R |H R 〉
-
〈ΦHF ˆ R|ΦHF R |H R 〉
∞
1 (1) 2 (2) 3 (3) EGS R ) Eo + R EGS + R EGS + R EGS ... ) Eo +
RjE(j) ∑ GS j)1
1 1 (1) E(2) ˆ ee|Ψ(1) ˆ ee|Φo〉 GS ) 〈Φo|V GS〉 + 〈ΨGS|V 2 2
(13)
1 1 (1) E(2) ˆ ee|Φ(1) ˆ ee|Φo〉 HF ) 〈Φo|V HF〉 + 〈ΦHF|V 2 2
(14)
To preserve the general character of eqs 13 and 14, the explicit complex forms of the wavefunctions are kept, even though the energies are real. By subtracting eq 8 from eq 7, along with making use of eqs 12-14, one obtains11 an expression for 2 QC,(2): EQC c,R that begins with R Ec ∞
2 QC,(2) + R3EQC,(3) ... ) EQC c,R ) R Ec c
(6)
We first follow Linderberg and Shull11 and discuss the perturbation expansion for EQC c,R , for small enough R. By applying the standard Rayleigh-Schro¨dinger perturbation theory, HF we develop expansions for EGS R and ER and their respective GS HF wavefunctions ΨR and ΦR . The energy expressions read as follows:
(12)
and
where ΨGS ˆ R. The HF R is the ground-state wavefunction of H is defined as energy EHF R HF EHF ˆ R|ΦHF R ) 〈ΦR |H R 〉
(1) ˆ ee|Φo〉 E(1) GS ) EHF ) 〈Φo|V
1 (1) 2 (2) 3 (3) EHF R ) Eo + R EHF + R EHF + R EHF ... ) Eo +
RjE(j) ∑ HF j)1 (8)
In eqs 7 and 8, Eo is the ground-state energy of H ˆ o (i.e., H ˆ R at R ) 0). The eigenvalue problem corresponding to H ˆ o is
H ˆ oΦk ) EkΦk
Eo < E1 e E2 e...e Ek e...
(9)
where we shall concern ourselves with situations where Eo is nondegenerate. (Note that H ˆ o is a Kohn-Sham Hamiltonian and that Φo is a Kohn-Sham determinant.) For k >0, Φk are the excited-state wavefunctions of H ˆ o.
(15)
with (j) ≡ E(j) EQC,(j) c GS - EHF
(16)
When the ground state of H ˆ o is degenerate, the expansion for 11 with term linear in R EQC c,R is expected, in general, to begin (1) (1) because EGS < EHF. The second-order energy EQC,(2) , given in terms of eigenc states of H ˆ o, reads
(7) ∞
RiEQC,(j) ∑ c j)2
∞
EQC,(2) c
)
∑
k)1 s DE
|〈Φk|Vˆ ee|Φo〉|2 Eo - Ek
(17)
In eq 17, DE signifies double excitations. In other words, the summation in eq 17 goes over those eigenstates Φk of H ˆ o that are obtained by exciting two electrons from Φo. There is no contribution coming from singly excited states (see the Appendix). Next, we turn our attention to the correlation energy Ec[n] defined by
ˆ + Vˆ ee|ΦKS Ec[n] ≡ 〈Ψn|Tˆ + Vˆ ee|Ψn〉 - 〈ΦKS n |T n 〉 (18)
DFT Correlation Energies
J. Phys. Chem. A, Vol. 102, No. 18, 1998 3153
According to the constrained-search formulation of DFT2, Ψn is, among all wavefunctions that yield the density n(r), the one that minimizes the expectation value 〈Tˆ + Vˆ ee〉. Similarly, ΦKS n , the Kohn-Sham (KS) wavefunction, is among all wavefunctions yielding n(r), the one that minimizes just 〈Tˆ 〉 (i.e., the wavefunction with the lowest kinetic energy). Ec[n] is meant to be used in a full traditional density functional calculation. (Note that Φo ) ΦKS n .) The correlation energy functional HFEc[n] is defined by
Ec[n] ≡ 〈Ψn|Tˆ + Vˆ ee|Ψn〉 - 〈ΦHF ˆ + Vˆ ee|ΦHF n |T n 〉 (19)
HF R Vc [n]
) 2RHFE(2) c [n]
(24)
Hence, for arbitrary n(r), the quantity 2HFE(2) c [n] is especially important because it is the initial slope (slope at zero R) of the integrand HFVRc [n] for the correlation energy, HFEc[n], in eq 22. Further, when approximating HFVRc [n] for use in eq 22, we can HFE [n ]; so it is approximate HFE(2) c λ c [n] by approximating lim λf∞ important to know the exact values for the latter limit, as reported in this paper.
HF
where ΦHF is the single determinant that minimizes the n expectation value 〈Tˆ + Vˆ ee〉 and yields the density n(r). The functional derivative of HFEc[n] is meant to be added to the HF nonlocal potential to produce HF-like equations, whose solution upon self-consistency, in principle, leads to the exact density and to the exact ground-state energy.14 Without this added functional derivative, the familiar HF density and energy are obtained upon self-consistency. Note that HFEc[n] is different from traditional QC correlation energy. In the HF theory, one is interested in the single determinant that minimizes the Hamiltonian of the system under investigation without any constraints on the density. According to eq 19, ΦHF n can be viewed as the single determinant that minimizes the same Hamiltonian but yields the ground-state density. We shall derive relationships between the leading secondorder components for Ec[n] and HFEc[n] and the second-order energy EQC,(2) , because published numbers are already aVailc able for the latter, and we shall present these numbers. Here n(r) is density of Φo; namely, we shall derive QC,(2) lim Ec[nλ] ≡ E(2) c [n] e Ec
λf∞
(20A)
with an equality for only the two-electron case. And, we shall also derive QC,(2) lim HFEc[nλ] ≡ HFE(2) c [n] ) Ec
λf∞
(20B)
The latter equality holds for any number of electrons, with the caveat of nondegeneracy for Φo. The uniformly scaled density nλ(r) in eq 20 is given by
nλ(r) ) λ3 n(λr) ) λ3 n(λx,λy,λz)
(21)
where n(r) is the ground-state density of the Kohn-Sham H ˆ o in eq 9. The adiabatic connection formula for HFEc[n] given by Stoll and Savin is18
Ec[n] )
HF
∫01 HFVRc [n] dR
(22)
where HF R Vc [n]
) 〈Ψmin,R |Vˆ ee|Ψmin,R 〉 - 〈ΦHF,R |Vˆ ee|ΦHF,R 〉 (23) n n n n
where Ψmin,R is the wavefunction that minimizes 〈Tˆ + RVˆ ee〉 n is the single determinant that and yields n(r), and where ΦHF,R n minimizes 〈Tˆ + RVˆ ee〉 and yields the same density n(r). Approximations to HFVRc [n] have stimulated the development of successful hybrid schemes for approximating HFEc[n] (see ref 19 and references within). For small enough coupling constant R, it has been shown that14
III. Numerical Results The Hamiltonian given by eq 1, for the special test case when Vo(r) ) -1/r, is the same as the one that appears in the well-known 1/Z expansion of the QC correlation energy. Consequently, we are able to use known second-order results, EQC,(2) , to test approximations to the DFT correlation energies c Ec[n] and HFEc[n]. We shall consider numbers for EQC,(2) c associated with HF energies resulting from restricted HF calculations. In other words, the calculation of EQC,(2) correc sponding to N
H ˆ R ) Tˆ (r1,...,rN) + RVˆ ee(r1,...,rN) -
1
∑ i)1 r
R ) Z-1
i
(25) as well as higher-order energy terms has been of interest for quite a long time,8-12 and there are many available numbers of very high accuracy, mainly by Ivanova and Safronova8 and recently by Davidson and co-workers.9A,9B To test approximations of Ec[nλ] and HFEc[nλ], as λf∞, we take the exact values for the second-order energy EQC,(2) c associated with eq 25, for five different hydrogenic densities.8,9A The two- and 10-electron densities are generated from the nondegenerate ground-state wavefunctions of the noninteracting H ˆ o in eq 25. For the three-, nine-, and 11-electron densities, the ground state of H ˆ o in eq 25 is degenerate, but the expansion for EQC still begins with EQC,(2) because of symmetry, accordc,R c ing to Linderberg and Schull.11 The three-, nine-, and 11electron densities are obtained from Φo wavefunctions that correspond to configurations 1s22s, 1s22s22p5 , and 1s22s22p63s, respectively. We compare, in Table 1, the values of EQC,(2) with those of c APP [n ] refers to three approximate Ec [nλ], as λf∞ , where EAPP λ c correlation energy functionals with gradients; that is, the recently derived GGA of Perdew, Burke, and Ernzerhof (PBE),20 the one of Lee, Yang, and Parr (LYP),21 and the one of Levy and Wilson.22 The second column in the table is the value corresponding to lim HFEc[nλ] ) EQC,(2) when n(r) is the hydrogenic groundc λf∞ state density of H ˆ o in eq 25. For Ec[n], as required by eq 20A for more than two electrons, the tested approximations give more negative values than the respective values for EQC,(2) . For c HFE(2)[n] ) a two-electron density, the equality E(2) c [n] ) c EQC,(2) is desired. c The functionals give reasonable to good values, depending on the functional and the number of electrons. When the adiabatic connection formula (eq 22) is employed to generate a new approximation to HFEc[n], there is room for improvement with all three functionals when these functionals are used to segment of the integrand, which is the approximate the 2EQC,(2) c initial slope in this adiabatic connection formula.
3154 J. Phys. Chem. A, Vol. 102, No. 18, 1998
Ivanov and Levy
TABLE 1: Comparison of EcQC,(2) for H ˆ rQC in Equation 25 with EcAPP[nλ]a density
EcQC,(2)
2-electron 3-electron 9-electron 10-electron 11-electron
-0.0467b
b
lim EPBE c [nλ]
lim ELYP c [nλ]
λf∞
λf∞
-0.0479 -0.0584 -0.3856 -0.4577 -0.4753
-0.0537c -0.3694c -0.4278c -0.4534d
lim EWL c [nλ]
λf∞
-0.0565 -0.0991 -0.4648 -0.5275 -0.5868
-0.0480 -0.0568 -0.3826 -0.4504 -0.4718
N
∞
QC,(2) E(2) + c [n] ) Ec
∑
Go¨rling and Levy14 have recently introduced an adiabatic DFT perturbation theory. The effective potential and the electronelectron interaction along the coupling constant path, which connects a noninteracting and a fully interacting system with the same electron density, have been used for this DFT perturbation theory.14,15 By using their perturbation theory, they have derived the following high-λ expansion for Ec[n]: -1 (3) -2 (4) Ec[nλ] ) E(2) c [n] + λ Ec [n] + λ Ec [n] + ...
(26)
and they have identified the second-order energy E(2) c [n] as
lim Ec[nλ] )
λf∞
E(2) c [n]
)
∞
|〈Φk|∆Vˆ |Φo〉|2
k)1
Eo - Ek
∑
QC,(2) E(2) c [n] e Ec
(27)
N
(28)
and where
with an equality only for two-electron systems, because only for two electrons does each term vanish in the summation over single excitations in eq 33. For the high-density scaling limit of HFEc[n], Go¨rling and Levy14 arrived at -1 HF (3) Ec[nλ] ) HFE(2) Ec [n] + λ-2 HFEc [n] + ... c [n] + λ (35) (4)
HF
with
lim HFEc[nλ] ) HFE(2) c [n] )
λf∞
∞
|〈Φk|Vˆ ee -
∑
2 {Vˆ HF ∑ x ([Φo];ri) + u([n];ri)}|Φo〉| i)1
u([n];ri) )
∫
(29)
In eq 28, the Hartree potential u([n];r) and the exchange potential Vx([n];r) are local multiplicative potentials obtained by taking functional derivatives of U[n] and Ex[n] with respect to the density n(r). The density n(r) is obtained from the noninteracting Φo, the Kohn-Sham single determinant. The Hartree electron-electron repulsion energy U[n] and the exchange energy Ex[n] are defined as follows:
U[n] )
1 2
∫∫
n(r1)n(r2) dr dr |r1 - r2| 1 2
Ex ) 〈Φo|Vˆ ee|Φo〉 - U[n]
(30) (31)
By utilizing the fact that Vˆ ee is a two-body operator, and u([n];r) and Vx([n];r) are one-body operators, we separate the infinite summation in eq 27 in two parts: summation over single excitations and summation over double excitations. For simplicity of the notation, we define Vxu([n];r) ) u([n];r) + Vx([n];r). As a result, eq 27 is re-expressed as ∞
E(2) c [n]
)
∑
k)1 s SE
|〈Φk|∆Vˆ |Φo〉|2 Eo - Ek
∞
+
By making use of eq 17, we obtain
∑
k)1 s DE
|〈Φk|Vˆ ee|Φo〉|2 Eo - Ek
(32)
(36)
Eo - Ek
In the eq 36 formula, Vˆ HF x ([Φo];r) is the familiar nonlocal Fock exchange operator, but here built from the one-particle orbitals of Φo instead of from the HF orbitals. By applying identity (eq A9) to eq 36 and invoking eq 17, we express HFE(2) c [n] as ∞
n(ri) dr |ri - r|
(34)
N
with
{u([n];ri) + Vx([n];ri)} ∑ i)1
(33)
Because of the nonpositive contributions from all single excitations in eq 33, it follows that
k)1
∆Vˆ ) Vˆ ee -
Vxu([n];ri)|Φo〉|2 ∑ i)1 Eo - Ek
k)1 s SE
a Determined as λ f ∞, where n is the ground-state density of H ˆ oQC. From refs 8, 9A, and 16. c From ref 8. d From ref 9A.
IV. Derivations of Equations 20A and 20B
|〈Φk|Vˆ ee -
HF (2) Ec [n]
)
∑ k)1
|〈Φk|Vˆ ee|Φo〉|2 Eo - Ek
) EQC,(2) c
(37)
s DE
The combination of eqs 33, 34, and 37 leads to eqs 20A and 20B. V. Concluding Comments We have established the link between the second-order component of the unknown DFT correlation energy, E(2) c [n], and the known result for the second-order quantum chemistry correlation energy, EQC,(2) . We have also shown that, except c for the existence of certain degeneracies, the leading secondorder term in the high-density expansion for the correlation energy functional HFEc[n] satisfies the equality HFE(2) c [n] ) HFE(2)[n] is especially important beEQC,(2) . The quantity 2 c c cause it is the initial slope of the adiabatic connection formula for the correlation energy HFEc[n], for all densities, not just for those in the high-density limit. The functional derivative of HFE [n] is meant to be added to the HF nonlocal effective c potential to produce, via self-consistency, the exact ground-state density and ground-state energy. Five numerical tests on viable approximations were presented for closed- and open-shell densities obtained from hydrogenic orbitals. These tests should be used to assess approximations to correlation energies. Moreover, it is possible to generate data for different sets of densities if one uses the corresponding Vo(r) for which the eigenfunctions and eigenvalues, corresponding to H ˆ o with this
DFT Correlation Energies
J. Phys. Chem. A, Vol. 102, No. 18, 1998 3155
Vo(r), are determined. For example, if Vo(r) ) r2 , then one has the Harmonium (Hooke’s) Atom.23-25 Recently, Huang and Umrigar25 have numerically studied the Z-1 expansion for the DFT correlation energy for two-electron densities and found that the respective series begins with the same second-order energy as the similar Z-1 expansion for traditional QC correlation energy. A formal proof of their observation as well as a generalization to more than two electrons have very recently been derived by Ivanov and Levy.26 Acknowledgments. Discussions with Professors Ernest Davidson and Cyrus Umrigar are greatly appreciated. This research was supported by the National Institute of Standards and Technology.
In this section we will present a derivation of eq 17. By subtracting eq 14 from eq 13, one obtains an expression for (1) EQC,(2) in terms of Ψ(1) c GSand ΦHF:
1 (1) EQC,(2) ) 〈Φo|Vˆ ee|{Ψ(1) c GS - ΦHF}〉 + 2 1 (1) 〈{Ψ(1) ˆ ee|Φo〉 (A1) GS - ΦHF}|V 2
∑
∞
Ψ(1) GS )
∑ k)1
(A2)
HF To find Φ(1) HF, we shall consider the equation to which ΦR is an eigenfunction. From HF theory it is known that for every R, ΦHF R is solution to
HF HF Vˆ HF([ΦHF h HF ∑ R ];ri) + ∑Vo(ri)}ΦR ) E R ΦR i)1 i)1
N
∞
(1) Ψ(1) GS - ΦHF )
〈Φk|Vˆ ee -
∑
HF HF ˆ HF Vˆ HF([ΦHF R ];ri) ) V x ([ΦR ];ri) + u([nR ];ri)
(A3)
(A4)
For small enough R, E h HF R is expanded in the following way ∞
) Eo +
+
+
R3E h (3) HF
... ) Eo +
RjE h (j) ∑ HF j)1 (A5)
Upon substituting eqs 11 and A5 in eq A3 and by equating all terms of order one, we obtain N
[H ˆ o - Eo]Φ(1) h (1) HF ) [E HF -
Vˆ HF ∑ x ([Φo];ri) + u([n];ri)]Φo i)1 (A6)
In terms of eigenfunctions of H ˆ o, Φ(1) HF is given by
Vˆ HF([Φo];ri)|Φo〉 ∑ i)1
Φk )
Eo - Ek
k)1 N
〈Φk|Vˆ ee -
∞
Vˆ HF([Φo];ri)|Φo〉 ∑ i)1
Φk +
Eo - Ek
k)1 s SE
∞
∑ k)1
〈Φk|Vˆ ee|Φo〉 Φk (A8) Eo - Ek
s DE
Because for every k corresponding to singly-excited determinant,
Vˆ HF([Φo];ri)|Φo〉 ) 0 ∑ i)1
(A9)
the summation over all single excitations drops out and the (1) difference {Ψ(1) GS - ΦHF} becomes ∞
(1) Ψ(1) GS - ΦHF )
∑ k)1
〈Φk|Vˆ ee|Φo〉 Φk Eo - Ek
(A10)
s DE
HF Note that E h HF R is different from ER in that the former is the sum of HF orbital energies. The potential Vo(r) in eq A3 is a local multiplicative one. With Vo(r) ) -Z/r and R ) 1, eq A3 leads to the familiar HF equations. In formula A3, Vˆ HF([ΦHF R ];r) is the HF nonlocal effective potential that has two contributions: the nonlocal exchange potential HF Vˆ HF x ([ΦR ];r) built from the occupied one-particle orbitals of HF ΦR , and the local Hartree potential u([nHF R ];r], associated with HF the density nHF R (r) obtained from ΦR :
R2E h (2) HF
(A7)
By subtracting eq A7 from eq A1, one finds
N
R1E h (1) HF
Φk
Eo - Ek
k)1
〈Φk|Vˆ ee -
〈Φk|Vˆ ee|Φo〉 Φk Eo - Ek
N
Vˆ HF([Φo];ri)|Φo〉 ∑ i)1
N
where Ψ(1) GS is expanded in the usual way
E h HF R
∞
Φ(1) HF )
∑
Appendix
{Tˆ + R
N
〈Φk|
contains a summation over Hence, the expression for EQC,(2) c doubly-excited states only: ∞
EQC,(2) c
)
∑
k)1 s DE
|〈Φk|Vˆ ee|Φo〉|2 Eo - Ek
(A11)
References and Notes (1) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864. Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133. (2) Levy, M. Proc. Natl. Acad. Sci. U.S.A. 1979, 76, 6062; Levy, M. Bull. Am. Phys. Soc. 1979, 24, 626. (3) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University: Oxford, 1989; references therein. (4) Dreizler, R. M.; Gross, E. K. U. Density Functional Theory; Springer: Berlin, 1990; references therein. (5) Levy, M.; Perdew, J. P. Phys. ReV. A 1985, 32, 2010. (6) Umrigar, C. J.; Gonze, X. Phys. ReV. A 1994, 50, 3827. (7) Filippi, C.; Gonze, X.; Umrigar, C. J. In Recent DeVelopments and Applications of Density Functional Theory, Theoretical and Computational Chemistry, Volume 4; Seminario, J., Ed.; Elservier: Amsterdam, 1996. (8) Ivanova, E. P.; Safronova, U. I. J. Phys. B: At. Mol. Opt. Phys. 1975, 8, 1591 and references therein. (9) (A) Chakravorty, S. J.; Davidson, E. R. J. Phys. Chem. 1996, 100, 6167; (B) Chakravorty, S. J.; Gwaltney, S. R.; Davidson, E. R.; Parpia, F. A.; Fisher, C. F. Phys. ReV. A 1993, 47, 3649; Davidson, E. R.; Hagstrom, S. A.; Chakravorty, S. J.; Umar, V. M.; Fisher, C. F. Phys. ReV. A 1991, 44, 7071. (10) Lo¨wdin, P.-O. AdV. Chem. Phys. 1959, 2, 207. (11) Linderberg, J.; Shull, H. J. Mol. Spectrosc. 1960, 5, 1 and references therein. (12) For a review see Baker, J. D.; Freund, D. E.; Hill, R. N.; Morgan, III, J. D. Phys. ReV. A 1990, 41, 1247.
3156 J. Phys. Chem. A, Vol. 102, No. 18, 1998 (13) Levy, M. Phys. ReV. A 1991, 43, 4637 and references therein. (14) Go¨rling, A.; Levy, M. Phys. ReV. B 1993, 47, 13105. (15) Go¨rling, A.; Levy, M. Int. J. Quantum Chem. Symp. 1995, 29, 93; Go¨rling, A.; Levy, M. Phys. ReV. A 1995, 51, 2851. (16) Ivanov, S.; Lopez-Boada, R.; Go¨rling, A.; Levy, M., accepted to J. Chem. Phys. (17) Perdew, J. P.; McMullen, E. R.; Zunger, A. Phys. ReV. A 1981, 23, 2785. (18) Stoll, H.; Savin, A. In Density Functional Methods in Physics; Dreizler, R. M., daProvidencia, J., Eds.; Plenum: New York, 1985.
Ivanov and Levy (19) Go¨rling, A.; Levy, M. J. Chem. Phys. 1997, 106, 2675. (20) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865; Phys. ReV. Lett. 1997, 78, 1396. (21) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (22) Wilson, L. C.; Levy, M. Phys ReV. B 1990, 41, 12930. (23) Kestner, N. R.; Sinanoglou, O. Phys. ReV. 1962, 128, 2687. (24) Filippi, C.; Umrigar, C. J.; Taut, M. J. Chem. Phys. 1994, 100, 1290 and references therein. (25) Huang, C. J.; Umrigar, C. J. Phys. ReV. A 1997, 56, 290. (26) Ivanov, S.; Levy, M., unpublished results.
