Download One-Particle Density Matrix Functional for Correlation in Molecular

One-Particle Density Matrix Functional
for Correlation in Molecular Systems
MARIO PIRIS, PETER OTTO
Institute of Physical and Theoretical Chemistry, Friedrich-Alexander-University
Erlangen-Nuremberg, Egerlandstrasse 3, D-91058 Erlangen, Germany
Received 16 October 2002; accepted 8 May 2003
DOI 10.1002/qua.10707
ABSTRACT: Based on the analysis of the general properties for the one- and twoparticle reduced density matrices, a new natural orbital functional is obtained. It is
shown that by partitioning the two-particle reduced density matrix in an
antisymmeterized product of one-particle reduced density matrices and a correction ⌫c
we can derive a corrected Hartree–Fock theory. The spin structure of the correction
term from the improved Bardeen–Cooper–Schrieffer theory is considered to take into
account the correlation between pairs of electrons with antiparallel spins. The analysis
affords a nonidempotent condition for the one-particle reduced density matrix. Test
calculations of the correlation energy and the dipole moment of several molecules in the
ground state demonstrate the reliability of the formalism. © 2003 Wiley Periodicals, Inc.
Int J Quantum Chem 94: 317–323, 2003
Key words: reduced density matrix; density matrix functional; natural orbital
functional; electron correlation; dipole moment
Introduction
T
he most accurate methods available today for
calculating the ground-state energy in molecular systems, e.g., the configuration interaction (CI)
method [1] or the coupled cluster (CC) method [1],
are computationally too expensive to be applied to
large systems. Rather than dealing directly with the
many-body wave function, we can alternatively de-
Correspondence to: M. Piris; e-mail: [email protected]
M. Piris’ present address is Instituto Superior de Ciencias y
Tecnologı́a Nucleares, Ave. Salvador Allende y Luaces, La Habana 10600, Cuba.
International Journal of Quantum Chemistry, Vol 94, 317–323 (2003)
© 2003 Wiley Periodicals, Inc.
termine the ground-state energy by minimizing an
energy functional with respect to the electronic
charge density [2]. Unfortunately, this functional is
not known exactly and attempts to construct it have
not been successful due to the strong nonlocality of
the kinetic energy term.
Because the interactions between electrons are
pairwise within the Hamiltonian, the energy may
be determined exactly from a knowledge of the
two-particle reduced density matrix (2-RDM). Although this functional is known, the 2-RDM has not
replaced the wave function as the fundamental
quantity of electronic structure because a simple set
of necessary and sufficient conditions for ensuring
PIRIS AND OTTO
that the 2-RDM corresponds to an N-particle wave
function is not known (the N-representability problem) [3]. The minimization of the energy with respect to a 2-RDM constrained by the well-known D,
Q, and G necessary conditions [3, 4] has been recently presented [5] with accurate results. The contracted Schrödinger equation (CSE), also known as
the density equation, offers a nonvariational approach to constrain the 2-RDM to be approximately
N-representable [6]. The difficulty with this method
lies in the need for the 3- and 4-RDMs that have to
be reconstructed from the 2-RDM. The number of
elements in the 2-RDM in general is still large.
In this article we consider a theory focused on
the one-particle reduced density matrix (1-RDM).
The properties of the total energy functional of the
1-RDM are well known [7]. A major advantage of a
density matrix formulation is that the kinetic energy and the exchange energy are explicitly defined
using the one-matrix and do not require the construction of a functional. The unknown functional
in a 1-RDM theory only needs to incorporate electron correlation. It does not rely on the concept of a
fictitious noninteracting system. Consequently,
density matrix schemes are not expected to suffer
from the numerous problems of Kohn–Sham (KS)
methods [8] based upon approximate exchange correlation density functionals [9].
Until now, only a few density matrix functionals
have been proposed so far. Goedecker and Umrigar
(GU) [10] proposed a simple functional without
adjustable parameters. Holas [11], Cioslowski and
Pernal [12] proposed different generalizations of
this functional. The GU functional satisfies the Hermiticity and particle permutation conditions but
violates the nonnegativity condition for the diagonal elements of the 2-RDM. Moreover, the GU functional gives a wrong description of the occupation
numbers for the lower occupied levels [12]. Csanyi
and Arias (CA) [13] proposed another functional
from the condition that the 2-RDM is a tensor product of one-particle operators and that it satisfies the
Hermiticity and particle permutation constraints.
The CA functional gives almost vanishing correlation energies in contrast to the GU functional. Yasuda [14] obtained a correlation energy functional
Ec of the 1-RDM from the first- and second-order
density equations together with the decoupling approximations for the 3- and 4-RDMs. The Yasuda
functional is capable of properly describing a highdensity homogeneous electron gas [15] and encouraging results have also been reported for atoms and
molecules [14]. Some shortcomings of this func-
318
tional are also pointed out [16]. Mazziotti [17] proposed a geminal functional theory (GFT) where an
antisymmetrical two-particle function (geminal)
serves as the fundamental parameter. The 1-RDM–
geminal relationship allowed him to define a
1-RDM theory from GFT [18]. Improved techniques
for correcting the 1-RDM GFT functional are at
present being explored.
The aim of this article is to propose a new density matrix functional in terms of the natural orbitals satisfying the most general properties of the
2-RDM: proper normalization, nonnegativity of the
diagonal elements (D-condition), particle permutational sysmmetry, fermionic antisymmetry, and
Hermiticity. We keep the spin structure that we
used in the improved Bardeen–Cooper–Schrieffer
(IBCS) approximation [19], but introduce a new
spatially dependence in the correction term of the
2-RDM. The reliability of this natural orbital functional (NOF) is then established by applying it to
several molecules.
Functional
For a system of N electrons, the exact energy
functional of the 1- and 2-RDMs is
E⫽
冘
具q ␴ 兩ĥ兩q⬘ ␴ ⬘典 1 ⌫ q⬘ ␴ ⬘;q ␴ ⫹
q, ␴ ,q⬘, ␴ ⬘
冘
q 1 , ␴ 1 ,q 2 , ␴ 2 ,q⬘1 , ␴1⬘,q⬘2 , ␴2⬘
⫻ 具q 1 ␴ 1 , q 2 ␴ 2 兩q⬘1 ␴ ⬘1 , q⬘2 ␴ ⬘2 典 2 ⌫ q⬘1␴1⬘,q⬘2␴2⬘;q 1␴ 1,q 2␴ 2,
(1)
in which 具q␴兩ĥ兩q⬘␴⬘典 is the matrix element of the
kinetic energy and nuclear attraction terms and
具q1␴1, q2␴2兩q⬘1␴1⬘ , q⬘2␴2⬘ 典 are the electronic repulsion
integrals. The states 兩q␴典 constitute a complete orthonormal set of single-particle wave functions,
where q denotes the orbital and ␴ is the sign of the
spin projection (it takes two values: ⫹1 and ⫺1).
The matrices 1⌫q⬘␴⬘;q␴ and 2⌫q1⬘␴1⬘,q2⬘␴2⬘;q1␴1,q2␴2 are the 1and 2-RDM, respectively, both in the (q␴)-representation.
The RDMs satisfy important sum rules. The trace
of the 1-RDM equals the number of electrons
冘⌫
1
q ␴ ;q ␴
⫽N
(2)
q, ␴
and the trace of the 2-RDM gives the number of
electron pairs in the system
VOL. 94, NO. 6
ONE-PARTICLE DENSITY MATRIX FUNCTIONAL
冘
2
⌫ q⬘ ␴ ⬘,q ␴ ;q⬘ ␴ ⬘,q ␴ ⫽
q, ␴ ;q⬘, ␴ ⬘
N共N ⫺ 1兲
.
2
(3)
Their diagonal elements are always nonnegative
because 1⌫q␴;q␴ is related to the probability of finding one electron at (q␴) and 2⌫q⬘␴⬘,q␴;q⬘␴⬘,q␴ is related
to the probability of finding one electron at (q␴) and
another at (q⬘␴⬘).
Attempts to determine the energy by minimizing
expression (1) are complicated by the fact that the
N-representability conditions for the 2-RDM are not
known. Consequently, the energy obtained may not
be an upper bound to the true energy. The 2-RDM
can be partitioned into an antisymmeterized product of the 1-RDMs, which is simply the Hartree–
Fock (HF) approximation, and a correction ⌫c to it,
2
⌫ q⬘1␴1⬘,q⬘2␴2⬘;q 1␴ 1,q 2␴ 2 ⫽
1 1
共 ⌫ q 1⬘␴ 1⬘;q 1␴ 11 ⌫ q 2⬘␴ 2⬘;q 2␴ 2
2
c
⫺ 1 ⌫ q⬘1␴1⬘;q 2␴ 21 ⌫ q⬘2␴2⬘;q 1␴ 1 ⫺ ⌫ q⬘
兲.
1 ␴1⬘,q⬘
2 ␴2⬘;q 1 ␴ 1 ,q 2 ␴ 2
(4)
This decomposition of the 2-RDM is well known
from the cumulant theory. ⌫c is the cumulant matrix [20] or the connected part [14, 21] of the 2-RDM
but with the opposite sign. Note that ⌫c cannot be
decomposed into terms that depend only on
1-RDM elements because it arises from interactions
in the Hamiltonian.
The first two terms on the right side of Eq. (4)
together satisfy general properties of the 2-RDM: the
Hermiticity, the particle permutational sysmmetry,
and the fermionic antisymmetry [22]. Therefore, our
correction ⌫c to the HF approximation should satisfy
these relations.
There is an important contraction relation between 1- and 2-RDMs that is in agreement with the
previous normalization:
1
⌫ q⬘ ␴ ⬘;q ␴ ⫽
2
N⫺1
冘⌫
2
q⬘ ␴ ⬘,is;q ␴ ,is
.
(5)
i,s
It can be shown from Eq. (5), taking into account
the normalization condition for the 1-RDM (2), that
the corrected 1-RDM satisfies then the following
nonidempotent condition:
冘共⌫
1
i,s
q⬘ ␴ ⬘;is
1
c
1
⌫ is;q ␴ ⫹ ⌫ q⬘
␴ ⬘,is;q ␴ ,is 兲 ⫽ ⌫ q⬘ ␴ ⬘;q ␴ .
(6)
This condition guarantees also the normalization
condition (3) for the 2-RDM if the normalization
condition for the 1-RDM is imposed. Equation (6)
generalizes our previous used nonidempotent condition [19] and is equivalent to the partial trace
relation involving the two-particle cumulant matrix
and the 1-RDM obtained by Mukherjee and Kutzelnigg [23].
It is well known that the HF approximation accounts for most of the correlation effects between
electrons with parallel spins. The unknown functional in a corrected RDM theory only needs in
principle to incorporate correlation effects between
electrons with antiparallel spins. Therefore, we can
obtain a suitable approximation for ⌫c in the IBCS
form:
⌫ qc 1⬘␴ 1⬘,q 2⬘␴ 2⬘;q 1␴ 1,q 2␴ 2 ⫽ ␴ 1 ␴ ⬘1 ⌫ qc 1⬘,q 2⬘;q 1,q 2␦ ⫺ ␴ 2, ␴ 1␦ ⫺ ␴2⬘, ␴ 1⬘,
(7)
where the spinless correction ⌫q⬘c1,q⬘2;q1,q2 must be a Hermitian matrix and satisfies the following symmetry
properties:
c
⫽ ⌫ qc 2⬘,q 1⬘;q 1,q 2 ⫽ ⌫ qc 1⬘,q 2⬘;q 2,q 1.
⌫ qc 1⬘,q 2⬘;q 1,q 2 ⫽ ⌫ q⬘
2 ,q 1⬘;q 2 ,q 1
(8)
A large number of choices for the correction ⌫c is
possible. It has a dependence of four indices and
direct computation with such magnitudes is too
expensive to be applied to large systems. We want
to maximize the physical content of ⌫c to a few
number of terms. Considering the symmetry properties (8), we express ⌫c by means of a two-index
Hermitian matrix ␥q1q2:
⌫ qc 1⬘,q 2⬘;q 1,q 2 ⫽ ␥ q 1q 2共 ␦ q 1⬘,q 1␦ q 2⬘,q 2 ⫹ ␦ q 1⬘,q 2␦ q 2⬘,q 1兲.
(9)
Let us now consider that our single-particle
wave functions {兩q␴典} are the natural spin orbitals:
1
⌫ q⬘ ␴ ⬘;q ␴ ⫽ n q ␦ q⬘q ␦ ␴ ⬘ ␴ .
(10)
According to expression (6) matrix ␥ must satisfy
the relation
␥ qq ⫹
冘␥
qi
⫽ n q 共1 ⫺ n q 兲.
(11)
i
The dependence of ␥qq⬘ on the indices q and q⬘ is
perhaps a difficult one. We propose a simple dependence where indices are separated, satisfying
always the above nonidempotent set of conditions
(11),
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
319
PIRIS AND OTTO
120000⫹10 ⫺13 ␧ q12
1
␥ qq⬘ ⫽ f q f q⬘ ⫺ g q g q⬘ ⫹ ␦ qq⬘
2
f q ⫽ ⫺n q
⫻ 关n q 共1 ⫺ n q 兲 ⫹ g ⫺ f ⫺ S 䡠 f q 兴
2
q
2
q
(12)
共1 ⫺ n q 兲 0.22 ,
g q ⫽ 关n q 共1 ⫺ n q 兲兴 0.9 ,
f q ⫽ n q⫺20␧ q共1 ⫺ n q 兲 0.22 ,
冘f
S⫽
q
(13)
q
and the functions gq are compelled to obey the
constraint
冘 g ⫽ 0.
q
(14)
q
The electronic energy functional will then be
E关兵兩q典其, 兵n q 其]
冘 具q兩ĥ兩q典n ⫹ 冘 关2具qq⬘兩qq⬘典
⫺ 具qq⬘兩q⬘q典]n n ⫺ 冘 关具qq⬘兩qq⬘典
⫽2
q
q
q,q⬘
q
q⬘
q,q⬘
⫹ 具qq⬘兩q⬘q典]关 f q f q⬘ ⫺ g q g q⬘ 兴
冘 具qq兩qq典关 f
2
q
⫹ S 䡠 f q ⫺ n q 共1 ⫺ n q 兲 ⫺ g q2 兴.
q
(15)
In this equation, the first term is the sum of the
kinetic and electron-nuclei potential energies. It can
be seen that the electron correlation modifies this
term directly because the value of nq may now be
fractional. The second term is the well-known sum
of the Coulomb and exchange energies. The last
two terms are energies that adjust the correlation of
particles with antiparallel spins.
Concerning the choice of the type of the correlation functions fq and gq, we think that they should
depend on the product of the level occupation nq
and the level vacancy 1 ⫺ nq to annihilate the correlation corrections for the HF case. The exact functional forms of them are probably complicated, requiring a distinction between occupied and virtual
HF orbitals [22]. Moreover, we consider a different
functional form for the lower occupied levels. To be
specific we propose the following potential dependences for the occupied HF levels:
320
(16)
g q ⫽ ⫺关n q 共1 ⫺ n q 兲兴 0.9 ,
␧ q ⬎ ␧ cf ,
in which
⫹
␧ q ⬍ ␧ cf
(17)
where ␧q is the one-particle orbital energy of the
level q and ␧cf is the value of the energy where the
functions fq and gq change their signs. For the case
of virtual HF orbitals the exponent values of function fq must be set equal to unity:
f q ⫽ n q 共1 ⫺ n q 兲,
g q ⫽ 关n q 共1 ⫺ n q 兲兴 0.9 .
(18)
The exponential values are critical to fulfill the
2-RDM nonnegativity condition (D-condition). We
fixed their values to obtain the CI total energy with
inclusion of single, double, triple, and quadruple
excitations (CI-SDTQ) for the water molecule. It is
well known that this kind of calculation on this
molecule yields more than 99% of the basis set
correlation energy [1]. The huge values for the exponents in Eq. (16) are necessary to guarantee that
occupations for the lower occupied levels approach
to unity.
The functions fq and gq obtained with these exponents are appropriate ones. They fulfill all properties of the RDMs imposed before and give a good
description of the occupation numbers. We recall
that all occupation numbers must be fractional to
guarantee that all orbitals have the same chemical
potential.
Results
Applying the well-known procedure of taking
molecular orbitals as linear combination of atomic
orbitals (MO-LCAO), Eq. (15) becomes the functional E[{cq}, {nq}]. One has to calculate the gradient
of the functional both with respect to natural orbitals coefficients {cq} and the occupation numbers
{nq}. Because the minimization with respect to occupations is much cheaper than with respect to the
orbitals, one can adopt the procedure used by us in
the IBCS method. In the inner loop we find the
optimal occupation numbers for a given set of orbitals. In the outer loop we minimize with respect
to the orbital coefficients. The iterative procedure is
repeated until convergence not only for {cq} but also
for {nq} is reached.
VOL. 94, NO. 6
ONE-PARTICLE DENSITY MATRIX FUNCTIONAL
2. The N-representability condition of the
1-RDM [3]: 0 ⱕ nq ⱕ 1.
3. The consequence of the nonidempotent condition ¥q gq ⫽ 0.
4. The orthonormality condition C†S C ⫽ I (S is
the overlap matrix).
FIGURE 1. Occupations of the molecular orbitals of
the H2O molecule.
The constraints that have to be satisfied simultaneously in the minimization process are:
1. The constant number of particles 2 ¥q nq ⫽ N.
Figure 1 shows a typical solution for {nq} of the
variational problem. As we can see, due to the
correlation, the electrons partially occupy molecular orbitals outside of the Fermi surface. The valence orbitals closer to the Fermi level have the
largest contribution to the electronic correlation because their occupations differ more from the HF
values. For the lower occupied levels, occupations
approach to unity whereas for the higher virtual
levels they approach to zero.
In Table I we report the values obtained for the
total energies of several molecules, employing the
experimental geometry [24]. These values were
computed using contracted Gaussian basis sets
6-31G** [25]. For comparison, we included in this
table the total energies calculated with the Gaussian
TABLE I ______________________________________________________________________________________________
Total energies (Etotal) in Hartrees.
Molecule
HF
H2O
NH3
CH4
LiF
BeO
C2
BF
CO
N2
HCN
C2H2
C2H4
H2CO
H2S
PH3
HCl
CH3F
F2
ClF
HFa
CCDb
NOFc
G3d
⫺100.0098
⫺76.0226
⫺56.1950
⫺40.2014
⫺106.9337
⫺89.4066
⫺75.3787
⫺124.1006
⫺112.7360
⫺108.9408
⫺92.8752
⫺76.8206
⫺78.0377
⫺113.8679
⫺398.6733
⫺342.4522
⫺460.0643
⫺139.0378
⫺198.6696
⫺558.8160
⫺100.1959
⫺76.2256
⫺56.3960
⫺40.3851
⫺107.1222
⫺89.6330
⫺75.6828
⫺124.3431
⫺113.0188
⫺109.2520
⫺93.1661
⫺77.0937
⫺78.3461
⫺114.1898
⫺398.8287
⫺342.6018
⫺460.2215
⫺139.3735
⫺199.0316
⫺559.1394
⫺100.2145
⫺76.2271
⫺56.3965
⫺40.3921
⫺107.2523
⫺89.7498
⫺75.7647
⫺124.4966
⫺113.1716
⫺109.3896
⫺93.2914
⫺77.2255
⫺78.4644
⫺114.3857
⫺399.0667
⫺342.8911
⫺460.4863
⫺139.5431
⫺199.3710
⫺559.6950
⫺100.4011
⫺76.3820
⫺56.5070
⫺40.4576
⫺107.3675
⫺89.8465
⫺75.8896
—
⫺113.2674
⫺109.4840
⫺93.3754
⫺77.2760
⫺78.5074
⫺114.4310
⫺399.2384
⫺342.9785
⫺460.6546
⫺139.6496
⫺199.4261
⫺559.7701
a
HF total energies.
Coupled cluster doubles total energies.
c
Natural orbital functional total energies computed in this work.
d
G3 total energies.
b
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY
321
PIRIS AND OTTO
TABLE II ______________________________________
Dipole moments (␮) in Debyes.
Molecule
HF
H2O
NH3
LiF
CO
HCN
H2CO
H2S
PH3
CH3F
HCl
ClF
HFa
CCDb
NOFc
Exp.d
1.97
2.19
1.88
6.20
0.32e
3.23
2.75
1.39
0.80
2.06
1.45
1.26
1.88
2.11
1.83
5.96
0.01
2.93
2.69
1.30
0.79
1.96
1.38
1.05
1.82
2.04
1.75
5.76
0.06
3.08
2.45
1.22
0.65
1.92
1.28
1.04
1.82
1.85
1.47
6.33
0.11
2.98
2.33
0.97
0.58
1.85
1.08
0.88
a
HF dipole moments.
Coupled cluster doubles dipole moments.
c
Natural orbital functional dipole moments computed in this
work.
d
Experimental dipole moments.
e
This value has an opposite sign relative to the experimental
value.
b
94 system of programs [26] at the coupled cluster
doubles (CCD) level. We also included in Table I
the total energies obtained with the Gaussian-3 (G3)
theory [27]. This composite technique is designed to
arrive at a total energy of a given molecular system.
According to Table I, the values we obtained are
in general lower than those obtained with the CCD
method. Our NOF calculations on 10-electron molecules (HF, H2O, NH3, CH4) give total energies
close to the CCD ones, which are accurate results
for the basis set correlation energies. Note that the
percentage of the correlation energy obtained by
CCD decreases as the size of the molecule increases,
whereas our functional keeps giving a substantial
portion of the correlation energy (see, e.g., the HCL
molecule). Note that the energy improvement is not
lower than the G3 theory.
For molecules with reported experimental values
[28] of the dipole moment, we also evaluated this
property (Table II). The results clearly underline the
importance of the correlation effects. An important
case is the CO molecule, for which the HF approximation gives a dipole moment in the wrong direction, whereas correlation methods approach it to
the experimental value. In the case of LiF, we have
to mention that with all methods the dipole moment is decreased with respect to the HF result,
whereas the experiment yields a higher value.
322
Conclusions
A natural orbital functional (15) that satisfies the
general properties of reduced density matrices has
been proposed. The approach presented is a wellcomprehensible density matrix theory that does not
require the introduction of a noninteracting kinetic
energy functional. It is shown that by partitioning
the two-particle reduced density matrix into an antisymmeterized product of one-particle reduced
density matrices and a correction ⌫c we can derive
a corrected HF theory.
The nonidempotency of the one-particle reduced
density matrix is a key aspect of many correlated
systems that all exhibit occupation numbers that
deviate from one or zero. The set of these conditions, Eq. (6), has been reduced to only one constraint (14) that may help in future calculations for
large systems.
The resultant two-particle reduced density matrices satisfied the D condition, while the one-particle reduced density matrices were exactly N representable. Complementary research on Q and G
conditions for the 2-RDM should be make. Although these necessary conditions are not sufficient, they will be useful to eliminate unphysical
solutions.
The results are encouraging: A simple functional
yields correlation energies as good as those obtained from the density functionals.
ACKNOWLEDGMENT
The authors are grateful for the generous financial support of the Alexander von Humboldt (AvH)
Foundation (AvH ID IV KUB 1071038 STP).
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