THE JOURNAL OF CHEMICAL PHYSICS 124, 141103 共2006兲
Optimized effective potentials yielding Hartree–Fock energies and densities
Viktor N. Staroverov and Gustavo E. Scuseria
Department of Chemistry, Rice University, Houston, Texas 77005
Ernest R. Davidson
Department of Chemistry, University of Washington, Seattle, Washington 98195
共Received 1 March 2006; accepted 17 March 2006; published online 14 April 2006兲
It is commonly believed that the exchange-only optimized effective potential 共OEP兲 method must
yield total energies that are above corresponding ground-state Hartree–Fock 共HF兲 energies except
for one- and two-electron systems. We present a simple procedure for constructing local
共multiplicative兲 exchange potentials that reproduce exactly the HF energy and density in any finite
basis set for any number of electrons. For any finite basis set, no matter how large, there exist
infinitely many such OEPs, which questions their suitability for practical applications. © 2006
American Institute of Physics. 关DOI: 10.1063/1.2194546兴
Kohn–Sham 共KS兲 density functional schemes in which
the exchange energy is treated exactly as an orbital functional play a central role in the development of a new generation of density functional methods.1–3 The exact
exchange-only KS scheme employs the same total energy
functional as the Hartree–Fock 共HF兲 method,
occ.
E=
兺兺
␴=↑,↓ i
+
冕
冕
冉 冊
1
␾i*␴共r兲 − ⵜ2 ␾i␴共r兲dr
2
␳共r兲vext共r兲dr +
1
2
冕
␳共r兲vJ共r兲dr + Ex ,
共1兲
兩 ␾i␴兩2, vext is the external
where ␴ is spin, ␳ = 兺␴␳␴, ␳␴ = 兺occ.
i
potential, vJ共r兲 = 兰dr⬘␳共r⬘兲 / 兩r − r⬘兩, and
Ex = −
1
兺
2 ␴=↑,↓
冕 冕
dr
兩␳␴共r,r⬘兲兩2
dr⬘
兩r − r⬘兩
共2兲
is the exchange energy written in terms of the density matrix
*
␳␴共r , r⬘兲 = 兺occ.
i ␾i␴共r兲␾i␴共r⬘兲, which is invariant with respect
to unitary transformations of the orbitals ␾i␴共r兲. The HF density matrix ␳␴HF共r , r⬘兲 is normally constructed from eigenfunctions of the Fock operator
F̂␴ = − 21 ⵜ2 + vext共r兲 + vJ共r兲 + K̂␴ ,
共3兲
where K̂␴ is the nonlocal HF exchange potential defined by
the property K̂␴␾i␴ = ␦Ex / ␦␾i*␴, whereas the KS density matrix ␳␴KS共r , r⬘兲 is built from eigenfunctions of the exchangeonly KS Hamiltonian
Ĥ␴ = − 21 ⵜ2 + vext共r兲 + vJ共r兲 + vx␴共r兲,
vx␴共r兲 = ␦Ex / ␦␳␴共r兲
共4兲
where
is a local 共multiplicative兲 exchange
potential which is the same for all the electrons. Thus, the
difference between the HF and exchange-only KS schemes
resides solely in the exchange operator.
Since Ex does not depend explicitly on ␳␴共r兲, the corresponding KS exchange potential cannot be obtained directly
0021-9606/2006/124共14兲/141103/4/$23.00
by functional differentiation. The most-used alternative route
is to find vx␴共r兲 as the optimized effective potential 共OEP兲.
The OEP of the exchange-only KS problem 共xOEP兲 is
defined4,5 as that vx␴共r兲 in Eq. 共4兲 which yields the KS density
matrix minimizing the energy functional 共1兲. Starting from
this definition, one can derive the integral OEP equation
which relates vx␴共r兲 to K̂␴ and the static density response
function.1 Practical implementations of the OEP method either attack the OEP equation6–9 or minimize the total energy
with respect to vx␴共r兲.10
It is a widely held belief that the xOEP method
must yield total energies that are above HF except for
one- and two-electron closed-shell systems 共N 艋 2兲, where
K̂␴ = −vJ共r兲 / N. This view originates from the fact that in the
complete 共infinite兲 basis set limit, the action of K̂␴ on a set of
orbitals is irreproducible by a multiplicative operator.11,12
Most practical implementations of the OEP method, however, employ a finite basis set. To the best of our knowledge,
no calculations where ExOEP = EHF for N ⬎ 2 have been reported in the literature, although several workers did observe
that xOEP energies obtained with their methods can be extremely close to the respective finite-basis-set HF values.13–15
In this Communication, we show that for any finite basis set,
no matter how large, there always exist infinitely many multiplicative exchange potentials that reproduce the HF energy
and density in that basis exactly. We also show how to construct these potentials in practice.
We begin by recalling that HF and xOEP states are
completely determined by their density matrices and that
HF
␴
= 兺occ.
兩 ␾iHF
the HF density matrix ␳ˆ HF
i
␴ 典具␾i␴ 兩 minimizing the
energy functional 共1兲 commutes with the Fock operator,
␴
兴 = 0. Similarly, the optimal density matrix of the
关F̂␴ , ␳ˆ HF
KS
␴
= 兺occ.
兩 ␾iKS
xOEP problem, ␳ˆ KS
i
␴ 典具␾i␴ 兩, satisfies the equation
␴
␴
␴ ˆ␴
关Ĥ , ␳KS兴 = 0. Clearly, if ␳ˆ KS = ␳ˆ HF at convergence, then
ExOEP = EHF. The two density matrices will in fact be the
same if vx␴共r兲 is such that
124, 141103-1
© 2006 American Institute of Physics
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J. Chem. Phys. 124, 141103 共2006兲
141103-2
␴
兴 = 0.
关Ĥ␴, ␳ˆ HF
共5兲
This requires only that the subspaces spanned by the occupied KS and HF orbitals be the same and does not imply
Ĥ␴ = F̂␴, which is a more stringent condition. Multiplying Eq.
␴
␴
兲 from the right and recalling that ␳ˆ HF
is idem共5兲 by 共1 − ␳ˆ HF
␴
␴
␴
potent, we obtain ␳ˆ HFĤ 共1 − ␳ˆ HF兲 = 0 or, equivalently,
␴ HF
具␾iHF
␴ 兩Ĥ 兩␾a␴ 典 = 0,
共6兲
where i runs over all occupied and a over all virtual orbitals.
Equation 共6兲 is our fundamental working condition for the
xOEP method to yield the HF energy and density.
With an infinity of orbitals, Eq. 共6兲 cannot in general be
satisfied by a local potential vx␴共r兲 for all occupied-virtual
共o-v兲 orbital pairs.11 In a finite basis, however, this is possible
as we now proceed to show. We expand vx␴共r兲 in a set of M ␴
auxiliary basis functions
vx␴共r兲
M␴
= 兺 bt␴ f t共r兲,
共7兲
t=1
and use the fact that canonical HF orbitals 共i.e.,
eigenfunctions of F̂␴兲 satisfy the Brillouin condition
HF
␴
具␾iHF
␴ 兩 F̂ 兩 ␾a␴ 典 = 0. Subtracting this result from Eq. 共6兲 we
HF
␴
␴
obtain 具␾iHF
␴ 兩 vx − K̂ 兩 ␾a␴ 典 = 0 or
M␴
HF ␴
␴
具␾iHF
兺
␴ 兩f t兩␾a␴ 典bt = Kia ,
t=1
FIG. 1. Various local exchange potentials yielding the HF energy and density of the Ne atom in the aug-cc-pVTZ basis. Each vx共r兲 was obtained as an
expansion in a set of 9 primitive s-type Gaussians with exponents ␣k
= ␣max / ␥k−1 共k = 1 , 2 , . . . , 9兲 by solving the linear system of Eqs. 共8兲. Note
the scale factors for vertical axes. See Table I for more detail.
共8兲
HF
␴
␴
= 具␾iHF
where Kia
␴ 兩 K̂ 兩 ␾a␴ 典. The number of such equations is
␴
,
the same as the number of nonzero matrix elements Kia
ov
which we will denote by M ␴ .
If M ␴ = M ␴ov and the auxiliary functions are such that the
HF
products ␾iHF
␴ f t␾a␴ have no exact linear dependencies 共which
is conceivable for any finite set of functions兲, then the linear
system of Eq. 共8兲 has a solution. We now use this solution, a
set of expansion coefficients 兵bt␴其, to construct the KS Hamiltonian via Eqs. 共7兲 and 共4兲. Diagonalization of Hˆ␴ yields a set
KS
␴
of KS orbitals which satisfy 具␾iKS
␴ 兩 Ĥ 兩 ␾a␴ 典 = 0. Because of
Eq. 共6兲, the occupied KS orbitals will differ from the occupied HF orbitals by no more than a unitary transformation, so
the HF and xOEP density matrices will be identical, as intended.
One unusual twist on the xOEP procedure is that the KS
orbitals minimizing the functional of Eq. 共1兲 are those that
span the occupied HF space, not necessarily the N␴ lowesteigenvalue orbitals. By forcing the Aufbau principle, one can
miss the global minimum of the xOEP problem. To determine which KS orbitals should be occupied, we test them
␴
.
with the projector ␳ˆ HF
Note that we place no restrictions on the nature of auxiliary functions except that they be of proper spatial symmetry and cause no exact linear dependencies. In fact, one can
construct infinitely many xOEPs for a given finite set of
canonical HF orbitals by using different sets of M ␴ov auxiliary
functions. As shown in Fig. 1, these potentials can differ
dramatically. If M ␴ ⬍ M ␴ov, then the system of linear Eq. 共8兲
has no solution and the HF energy minimum is not achievable. If M ␴ ⬎ M ␴ov, the number of xOEPs yielding ␳␴HF共r , r⬘兲
is again infinite.
As the number of virtual orbitals increases, Eq. 共8兲 approach exact linear dependence, so that a solution 兵bt␴其 may
be hard to obtain in practice. Whether or not we can actually
solve Eq. 共8兲, a vx␴共r兲 satisfying Eq. 共6兲 for a given finite set
of HF orbitals still exists.
To summarize, our procedure for constructing effective
exchange potentials that reproduce the HF energy and density in a given finite basis set 兵␹␮其 is as follows:
1.
2.
3.
4.
5.
6.
Solve the HF equations in 兵␹␮其 to get the Fock matrix and canonical HF orbital coefficients ckHF
␴.
ov
Pick a set of at least M ␴ auxiliary functions 兵f t其 and
evaluate the matrix elements Q␮t ␯ = 具␹␮ 兩 f t 兩 ␹␯典.
Transform the HF exchange matrix K␴ and each Qt
to the basis of canonical HF orbitals and solve the
system of Eq. 共8兲 for 兵bt␴其.
Form the KS Hamiltonian matrix H␴ using 兵bt␴其.
Diagonalize H␴ to get the KS orbital coefficients ckKS
␴
␴
and compute c̃k␴ = PHF
SckKS
␴ , where S␮␯ = 具␹␮ 兩 ␹␯典 and
N␴ HF
KS
␴
*
兲␮␯ = 兺i=1
c␮i␴共c␯HF
共PHF
i␴兲 . If c̃k␴ = ck␴ , then orbital k
should be occupied; if c̃k␴ = 0, virtual.
␴
Construct PKS
and evaluate the total energy to verify
␴
␴
that PKS = PHF and ExOEP = EHF.
Our basic method requires only that the o-v block of H␴
vanish in the basis of canonical HF orbitals. If desired, one
can also demand that the o-o or v-v block of H␴ be equal to
the respective block of F␴ in that basis. These conditions
give rise to additional equations similar to Eq. 共8兲 and call
for more than M ␴ov auxiliary functions to represent the xOEP.
If the o-v and o-o blocks of the Fock and KS matrices are
equal, then the solution of the combined system of equations
for 兵bt␴其 will yield an xOEP whose occupied orbitals will be
identical with those of the HF solution. If the entire KS and
Fock matrices are the same, then the xOEP will reproduce
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141103-3
J. Chem. Phys. 124, 141103 共2006兲
Optimized effective potentials yielding H-F energies
TABLE I. Finite-basis-set Hartree–Fock energies of many-electron systems
which we have reproduced exactly with the xOEPs of Eqs. 共7兲 and 共11兲. The
former requires at least M auxiliary functions, the latter contains M̃ terms.
System
Nec
LiHd
H 2O e
Basis
aug-cc-pVDZ 关4s,3p兴
aug-cc-pVTZ 关5s,4p兴
aug-cc-pVQZ 关6s,5p兴
aug-cc-pV5Z 关7s,6p兴
aug-cc-pV6Z 关8s,7p兴
HF limit
Numerical OEP
cc-pVDZ
cc-pVTZ
cc-pVQZ
HF limit
6-31G*
6-311+ G*
6-311+ + G**
HF limit
E 共Eh兲
Ma
M̃ b
−128.496349731
−128.533272825
−128.543755937
−128.546785545
−128.547062087
−128.5470981
−128.54542
−7.983618612
−7.986634147
−7.987178253
−7.9873524
−76.010529976
−76.036894349
−76.052846042
−76.067488
6
9
12
15
18
10
15
20
25
30
14
28
46
14
28
46
27
44
57
27
44
57
FIG. 2. Optimized effective potentials yielding the HF energy and density of
the Ne atom in various basis sets. Each vx共r兲 was obtained in the form of
Eq. 共11兲 by solving the linear system of Eqs. 共12兲. Note the scale factors for
vertical axes.
a
M is the number of distinct nonzero matrix elements Kia.
M̃ = 兺⌫N⌫o N⌫v , where N⌫o and N⌫v are the numbers of occupied and virtual
orbitals belonging to irrep ⌫.
c
HF limit is from Ref. 30. The numerical OEP value is from Ref. 20.
d
r共LiH兲 = 3.015 bohr, HF limit is from Ref. 31.
e
r共OH兲 = 0.9572 Å, ␪共HOH兲 = 104.52°, HF limit is from Ref. 32.
occ. vir.
b
vx␴共r兲
= 兺 兺 B␴jb
j
b
冕
HF
␾HF
j␴ 共r⬘兲␾b␴ 共r⬘兲
dr⬘
兩r − r⬘兩
共11兲
and leads to the following linear inversion problem
occ. vir.
the full HF orbital spectrum. In that case, there should be no
violation of the Aufbau principle.
Now we will discuss a particular choice of the auxiliary
basis set which lends a physical interpretation to our method.
Let us represent the exchange potential as
vx␴共r兲 =
兺 B␮␴ ␯
␮艋␯
冕
␹␮共r⬘兲␹␯共r⬘兲
dr⬘ ,
兩r − r⬘兩
共9兲
where ␹␮ is some finite basis used for the HF calculation.
The expansion coefficients 兵B␮␴ ␯其 of the xOEP in this basis
are determined by the equation
共␬␭兩␮␯兲B␮␴ ␯ = K␬␴␭ = − 兺 共␬␮兩␭␯兲P␮␴ ␯ ,
兺
␮艋␯
␮,␯
共10兲
where 共␬␭ 兩 ␮␯兲 are standard two-electron repulsion integrals
and P␮␴ ␯ are elements of the HF density matrix. If 兵␹␮其 is a
finite set of, say, even-tempered primitive Gaussians, then all
products ␹␮␹␯ will be strictly linearly independent. The
square matrix of integrals 共␬␭ 兩 ␮␯兲 will have no exact zero
eigenvalues and therefore Eq. 共10兲 will formally have a solution corresponding to ␳␴KS共r , r⬘兲 = ␳␴HF共r , r⬘兲. Equation 共10兲
also reveals that, essentially, we represent the nonlocal HF
exchange potential K̂␴ by a local Coulomb potential that is
created by the effective charge ˜␳␴共r兲 = 兺␮艋␯B␮␴ ␯␹␮共r兲␹␯共r兲
and has the same matrix elements as K̂␴ in the 兵␹␮其 basis.
In double-precision calculations with Gaussian basis
sets, only the smallest sets form linearly independent
products.16–19 For larger sets, the matrix of 共␬␭ 兩 ␮␯兲 is very
ill-conditioned which makes solving Eq. 共10兲 difficult. From
a practical point of view, it is highly advantageous to transform Eqs. 共9兲 and 共10兲 to the basis of canonical HF orbitals
and use only the equations for the o-v block. This results in
a shorter expansion
兺j 兺b 共ia兩jb兲B␴jb = Kia␴ .
共12兲
The system of Eq. 共12兲 can be solved in practice even for
relatively large basis sets 共see Table I and Fig. 2兲.
Table I shows that a finite-basis-set xOEP energy can be
arbitrarily close to the HF basis set limit and well below the
fully numerical xOEP value.20 Note also that our xOEP of
the Ne atom in Fig. 2共a兲 resembles the numerical xOEP6 but
shows more oscillations in Figs. 2共b兲–2共d兲. Presumably, this
reflects the increasing nonlocality of the HF exchange operator in larger basis sets.21
So far, we have discussed our method as a non-iterative
procedure which requires solving the HF equations in the
first step. With a few caveats, local exchange potentials that
yield finite basis set HF energies can be obtained by the
existing OEP methods7–10 without prior knowledge of the HF
solution. First, one must use at least M ␴ov auxiliary functions
to represent vx␴共r兲. If the expansion is not long enough, the
HF energy can never be found. Second, even if the expansion
has enough terms, the global minimum of the finite-basis
xOEP problem may not correspond to filling the N␴ lowest
KS orbitals. In the absence of a HF solution defining the
occupied and virtual spaces, one can test all possible sets of
N␴ orbitals and choose the one that yields the lowest total
energy. Finally, OEP methods minimizing the energy with
respect to vx␴共r兲 have to explore remote regions of parameter
space because the optimal values of coefficients 兵bt␴其 can be
very far from the initial guess. In our experience, an appropriately modified OEP method of Yang and Wu22 easily finds
the HF energy and density of atoms and small molecules in
common finite basis sets.
Although we have focused on the xOEP problem, our
procedure can be generalized to any orbital-dependent
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J. Chem. Phys. 124, 141103 共2006兲
141103-4
exchange-correlation functional. This implies that in a finite
basis set, the OEP method for an orbital functional should
yield the same global energy minimum as the minimization
of that functional with respect to orbitals.
OEPs obtained by our basic method may lack certain
properties of the true KS exchange potential. For example,
the highest occupied molecular orbital 共HOMO兲 of the
exchange-only KS determinant should have exactly the same
eigenvalue as the HF HOMO.23 This can be achieved by
adding the equation 具␾NKS 兩 vx␴ 兩 ␾NKS典 = KN␴ N to the system of
␴ ␴
␴
␴
Eq. 共8兲. Further, the KS exchange potential should decay
asymptotically as −1 / r. This can be accomplished by forcing
the slowest-decaying function f t共r兲 to have this property.
One can probably satisfy all necessary and sufficient
conditions24 on vx␴共r兲 and still reproduce a given finite-basisset HF density matrix.
Our conclusions may appear paradoxical. For any finite
basis set, no matter how large, there exist infinitely many
xOEPs that deliver exactly the ground-state HF energy in
that basis, however close it may be to the HF limit. Nonetheless, in the complete basis set limit, the xOEP is
unique25,26 and ExOEP is above EHF.11,12
If the KS Hamiltonian matrix is identical with the full
Fock matrix, not just the o-v block, then the KS and HF
orbitals and energies will also be the same. This leads to a
second paradox. In a complete basis set, all HF orbitals decay exponentially at the same rate determined by the HOMO
energy,27 whereas each KS orbital decays exponentially at a
rate determined by its eigenvalue. Thus, in the complete basis set limit, a HOMO can be both HF and KS 共as it should兲,
yet the rest of the orbitals cannot be simultaneously KS and
HF. In a finite basis set, however, there is no such contradiction, because each HF and KS orbital decays at a rate determined by its most diffuse basis function.
Finally, the fact that there can exist infinitely many KS
potentials yielding the same ground-state energy and density
seems to contradict the Hohenberg–Kohn theorem. The resolution of this paradox lies in the fact that the unique densityto-potential mapping does not exist in a finite basis set.28,29
The absence of such mapping is of no consequence if the KS
potential is calculated by explicit differentiation of the density functional, but it manifests itself in paradoxes of
the OEP method. The non-uniqueness of OEPs in a finite
basis set raises doubt about their usefulness in practical
applications.
V.N.S. and G.E.S. acknowledge discussions of the OEP
method with Aron J. Cohen, John P. Perdew, Andreas Savin,
and James D. Talman. This research was supported by the
National Science Foundation under Grant CHE-0457030 and
the Welch Foundation.
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