Download Generalized Statistical Approach to the Study of Interatomic Interactions M. E.

INTERNATTONALJOURNAL OF QUANTUM CHEMISTRY, VOL. XXXV, 267-275 (1989)
Generalized Statistical Approach to the Study of
Interatomic Interactions
R. SANTAMAR~A,M. BERRONDO, AND 0. NOVARO
Instituto de Fi'sica, UNAM, Apdo. Postal 20-364, Mgxico, D.F. 01000, Mkxico
E. S . KRYACHKO
Institute of Theoretical Physics, Academy of Sciences of the Ukrainian SSR, 252130, Kiev-130, USSR
Abstract
We present a simple model for calculating the interatomic interaction energies in the electron gas approximation. We use a generalization of the supermolecular electronic density which includes a density
overlap term. We present numerical calculations for the He-He interaction as an illustration of the method.
1. Introduction
The problem of evaluating the exact energy of interaction between two molecules
is far from solved. The interatomic case also becomes too complex as the atomic
number increases. As a result, a number of different approximate schemes have been
suggested and developed to calculate intermolecular interactions [ 1,2]. A method
which is particularly appealing because of the simple picture it provides is the statistical approach, based on the original Thomas-Fermi model which starts off with an
electron gas. Although the model has evolved to different expressions of the energy
as a functional of the electronic density, in the particular case of interatomic and intermolecular interactions it has been firmly established that a good starting point for
the total density is to take it as a simple superposition of the isolated fragment densities [ 1-11]. For short interatomic distances, the approximation obviously fails, so in
the present paper we propose to include a correction due to the density overlap.
In general terms, we can say that the statistical approach is based on the idea that
the total energy of a given atomic or molecular system is a functional of the total (effective) one-electron density p(r) [ 121
r
In the electron gas approximation, the functional is given as the sum of the kinetic
E,, Coulomb E,, exchange E,,, and correlation contributions, with the well known
expressions
0 1989 John Wiley & Sons, Inc.
CCC 0020-7608/89/020267-09$04.00
268
SANTAMARI~ET AL.
where z, is the nuclear charge of the ath nucleus, and R, is its position.
Different approximations have been given for the correlation energy as a functional
of the density [ I , 2, 111, but in the present paper, we shall simply neglect it.
Within this framework the interaction energy of two systems A and B can be calculated by taking the difference
where
E[(p(r)l
=
Ekb(r)l
+ Eex[p(r)l
+ Ec[p(r)l
(4)
In the case of intermolecular interactions, further development of the statistical approach is related with introducing additional correction terms in formulas (2) for the
kinetic, exchange, and correlation density functionals [ 11, 13-21]. However, in most
of these developments, the main emphasis is directed to improving the form of E [ p ]
in the frame of the zero-order perturbation treatment for the supermolecular density
pAB(r),i.e., the supermolecular density still remains as a sum of the unperturbed oneelectron densities of the isolated subsystems A and B [22]:
PAB(~)
=
PA(^) + P B ( ~ )
(5)
where all the densities are assumed to be normalized to the corresponding number of
electrons.
It is evident that, as emphasized by Claverie [ 11, the dispersion and polarization interactions cannot be described in the zero-order perturbation treatment of the real supermolecular density PAB. Nevertheless, some authors [ 17,211 have attempted to take
such terms into account by introducing additional effective functionals under the
frame of the representation of pABgiven by Eq. (5).
Some attempts to modify and improve the statistical approach for calculating intermolecular interactions have been directed to the construction of better expressions for
the kinetic, exchange, and correlation energy functionals while preserving the approximation of the supermolecular density (5).
KoXos and Radzio [ 181 and Radzio-Andzelm [21] have, however, established the
foundations for studying corrections to p A B ( r ) . Considering the He dimer as an example, KoXos and Radzio showed that the zero-order approximation for pAB(r)given
by Eq. (5) is rather unreliable. These authors used SCF calculations for the oneelectron, supermolecular density PAB. It is hence of great interest to determine the
higher-order corrections to pABin terms of the corresponding one-electron densities.
Other interesting corrections to the additive density hhve been given by Gordon and
collaborators 123-251 and by Wood and Piper [26] and by Parr and collaborators
[27,28].
INTERATOMIC INTERACTIONS
269
The present paper is devoted to calculating the supermolecular density pAB(r)taking into account the exchange of electrons between two closed-shell subsystems A
and B. The study is carried out starting from a Slater determinant for each subsystem
constructed from the corresponding Hartree-Fock orbitals. The paper is organized as
follows. Section I1 analyzes the asymptotic representation of the one-electron density
matrix y(r; r’). The corrected PAB for homonuclear and heteronuclear atoms is derived
in Section 111. A generalization of the statistical approach and its application to studying rare gas interactions are included there. Discussion of the results obtained for the
He dimer and comparison with similar results using the Hartree-Fock, perturbation,
and Gordon-Kim methods are presented in Section IV. A general discussion is included in Section V.
11. Asymptotic Properties of the One-particle Density Matrix
As is well known, the 1 matrix in terms of the natural orbitals [29,30] is represented in the following way:
where i runs over all the natural orbitals $;(r), and n, is the occupation number. The
corresponding one-electron density p(r) = y(r; r) is assumed to be normalized to the
total number of electrons N.
As a result of the exponential tail of all the natural orbitals [313, Tal and Bader
[32], March and Pucci [33], and March [34] proved that, for large r and r ’ , the
asymptotic behavior of y(r; r’) takes the form
In the particular case of a single Slater determinant, Eq. (7) follows directly from
the results obtained by Ludefia [35]. This form of y was also exploited by Alonso and
Girifalco [36] and recently by Kryachko and Kryachko and Ludefia [37].
Based on the asymptotic representation of y(r; r’) given by (7), we can assume that
the quantity [p(r)p(r’)]”2should be the basic one for constructing different generalizations of the statistical theory and its applications to the intermolecular interactions.
In particular, the term takes into account the overlap effect of the two subsystems A
and B which is of importance for evaluating the intermolecular exchange energy.
This point of view is slightly different from the one proposed recently by Kim et al.
and by Nyeland and Toennies [38].
111. Corrections to the Supermolecular Density
The wave function of a system with N electrons can be approximated by an antisymmetrized product of the two wave functions $, and $B of subsystems A and B
with N A and NB electrons, respectively:
$AB<~~>E“=
I)
= A ~ [ s ~ < { ~ i > ~ ~ ) $ B ( ~ . ~ i > ~ ~ ~ (8)
~~~)l
where xi = (si,ri) are the spin and coordinate of the ith electron, and AN is the anti-
SANTAMARIAET AL.
270
symmetrization operator for N electrons.
and I,!J~ can be further assumed to be
single Slater determinants constructed from the natural orbitals of A and B.
In order to include corrections to the density ( 5 ) , we shall start from the particular
case of two He atoms interacting with each other. This will suggest to us an ansatz
for the interacting electron density in the more general case of the ground state of two
closed-shell subsystems.
The He atom is hence described as a Slater determinant,
JIHe(X,, .2)
=
(b(rJ(bb-2)[a(s,)P(sz) - a ( s J P ( s J 1
(9)
so that the atomic spin-free electron density is given by
PHe(r) = 2[(b(r)I2
( 10)
It is obvious that, in this particular case, we can obtain the (nodeless) atomic orbital
explicitly from the density.
Next we consider the helium dimer for which we can obtain the supermolecular 1
matrix explicitly. For this purpose let us now distinguish the spin orbitals corresponding respectively to subsystems A and B in the following form:
A, = (bAa
Ap
=
B,
=
&a
Bp = # B P
(11)
where letters A or B indicate the orbital and the subindexes ( a or P ) indicate the spin
state of the subsystem with which we are dealing. Making use of this notation, the
wave function associated to the total system is
$.dl
9
2,334) = A,{[Ae(1)Ap(2) - Aa(2)Ap(1)1[B,(3)Bp(4> - Ba(4)Bp(3>1)
(12)
Applying the antisymmetrizer A , it is possible to show that one has 24 permutations
of the form
(13)
where the set { i ,j , k, I} can be {1,2,3,4} in any order. Considering the orthononnality of the spin orbitals, the nonvanishing terms in the spinless density matrix assume,
by using Eq. (10) and then Eq. (8), the following form:
A,(i)Ap(j)Ba(m$3(0
where
INTERATOMIC INTERACTIONS
27 1
and C is a normalization constant. The expression for pAB(r)follows directly:
where C is given by
)
c = -1 ( 1 - 0iB-,
2
NANB
so that
In the homonuclear case N A = NB, this expression reduces to Eq. (5) asymptotically, since the overlap OABgoes to zero. Equation (18) can thus be interpreted as an
improved expression for the supermolecular density which includes exchange effects,
which become important at small interatomic distances. For atoms larger than He, the
atomic orbitals are no longer nodeless, so that the approximation (14) for the
1-density matrix, calculated from the atomic densities only, is not very good. The
density pAB(r),however, is smoother because the nodal structure of the orbitals is lost
in the diagonal elements of the density matrix. Hence, Eq. (18) should be a much
better approximation than Eq. (14).
The heteronuclear case NA # NB also has the correct asymptotic behavior and differs from the simple superposition [7- 101 by the corresponding normarization factors
for each atom. This can be understood also by taking a simple Hartree product of
atomic wave functions, which is appropriate for large interatomic distances:
+AB
=
+~+g
(19)
and
hence it follows that
Before evaluating the energy terms given in Eqs. (2), we should notice that Eq. (7)
provides the exact expression for a helium atom, but certainly not for a helium dimer:
272
SANTAMARIA ET AL.
The approximation (22) does not even give the correct asymptotic result as R A B
which is given by
-+ m,
IV. Results
The different contributions to the statistical interaction energy given in Eqs. (2)
have been calculated according to the model presented here for the helium dimer.
Calculations using the correct functional form were carried out also, using the following expressions:
(24)
where Z, and R, mean the same in Eqs. (2).
In all the calculations the 1 matrix for the system AB is given by Eq. (14). The
atomic densities were obtained from the analytic Hartree-Fock wave functions given
by Clementi [39], and the numerical integrations were computed using Gaussian
quadratures.
Table I presents the contribution of the kinetic energy to the interaction energy. It
includes results from Ref. 18. A E ; , A E F , and AEL refer to results obtained from
first-order perturbation theory, SCF, and statistical calculations, respectively.
From Table I we can conclude that the SCF results carried out using the 1 matrix
proposed here (see second column) are very close to the correct ones (see first and
third columns) found by KoXos and Radzio [18]. Statistical results agree among themselves, except those computed using Gordon and Kim’s method. The proposed approximation for pAB(r)works considerably better than Eq. (5) (compare columns five
and seven). Most of the statistical results are not close to the ones obtained from SCF
and first-order perturbation theory.
Table I1 presents the contribution of the potential energy, i.e., the contribution of
the sum of the Coulomb and exchange energies to the interaction energy. It includes
results from Ref. 18 for comparison.
The conclusions for this Table are similar to the previous ones. The worst results
are those which use Eq. (5) as the density.
213
INTERATOMIC INTERACTIONS
TABLEI. Kinetic energy contribution (in a.u.) to the interaction energy in He-He.
AEEcF
R
AE;
A E ; ~ ~ ~AE;
AEic
AEid
AE;
2.117(-2)
0.285(-3)
0.760(-4)
0.332(-5 )
8.862(-2)
1.557(-3)
4.120(-4)
1.513(-5)
7.845(-2)
1.396(-3)
4.670(-4)
1.665(-5)
~
3.0
5.0
5.6
7.0
7.963(-2)
1.408(-3)
3.695(-4)
1.234(-5)
9.038(-2)
1.428(-3)
3.899(-4)
1.505(-5 )
9.224(-2)
1.451(-3)
3.753(-4)
1.289(-5)
7.915(-2)
1.539(-3)
4.167(-4)
1.424(-5)
~~~
calculations using y as in Eq. (14).
= p%.
‘pABis given in Eq. (5).
= pDAB (see Ref. 18).
‘Present approximation; pABis given in Eq. (18).
a~~~
‘PA6
TABLE11. Potential energy contribution (in a.u.) to the interaction energy in He-He.
A
R
3.0
5.0
5.6
7.0
E
~
-6.579(-2)
- 1.277(-3)
-3.492(-4)
- 1.148(-5)
aSCF Calculations using
b
AEya
BE:,
AE;
-8.308(-2)
- 1.450(-3)
-3.885(-4)
- 1.629(-5)
-7.576(-2)
- 1.319(- 3 )
-3.453(-4)
- I . 190(- 5 )
-1.291(-2)
-0.334( -3)
-1.136(-4)
-0.929( -5)
AE;‘
-7.909(-2)
- 1.495(-3)
-4.180(-4)
- 1.984(- 5)
AE;~
-7.439(-2)
- 1.404(-3)
-4.126(-4)
-2.095( - 5)
y as in Eq. (14).
.
pABis given in Eq. ( 5 ) .
‘pAB=
(see Ref. 18).
dPresent approximation; pABis given in Eq. (18).
piB
TABLE
111. Interaction energy (in a x . ) in He-He.
R
3.0
5.0
5.6
7.0
AE~;
1.384(-2)
1.275(-4)
0.294(-4)
0.086(-5)
AEga
AELT
0.730(-2)
-0.220(-4)
0.014(-4)
-0.124(-5)
1.515(-2)
1.324(-4)
0.308(-4)
0.098(-5)
A E ~ , ~
0.825(-2)
-0.490(-4)
-0.376(-4)
-0.600(-5)
W N T C
0.953(-2)
0.615(-4)
-0.060(-4)
-0.471(-5)
A
E
~
0.406(-2)
-0.080(-4)
0.544(-4)
-0.430(-5)
Note: see Table I1 for footnotes.
Total interaction energies are listed in Table 111. They are obtained simply adding
the kinetic and potential terms that appear in Tables I and 11, respectively.
From Table I11 we can see that columns four, five, and six differ among themselves and they fail around the equilibrium distance (5.6 a.u.) of the He, system, i.e.,
in the region of the potential well. We also note that the results of column six
[derived from Eq. (18)] are similar to those of column two, i.e., the SCF calculations
derived using Eq. (14). We conclude that the model developed in this paper gives
~
~
274
SANTAMARIA
ET AL.
good results for each component of the interaction energy but not good enough for
the total interaction energy in the region of the potential well.
V. Discussion
Statistical calculations were performed for the interaction energy of a He-He system at different interatomic distances, and comparison with calculations using the
1 matrix were also carried out. From these calculations we can conclude that the
supermolecular density which includes an overlap correction yields much better
results than the simple superposition of atomic densities, in particular for the separate
energy contributions. As remarked by Nikulin and Tsarev [ 5 ] , for the latter case the
total interaction energy improves with respect to the individual terms due to a fortuitous cancellation of errors.
On the other hand the generalized statistical approximation presented in this paper
yields very good results for each separate component of the interaction energy. In the
region of the potential well, however, the cancellation of errors is insufficient to give
the correct behavior. This is indeed a very sensitive region, and even the correlation
energy contribution might be important for obtaining the correct results.
The definite advantage of the present method is that the use of Hartree-Fock atomic
orbitals allows for better results than the direct calculations using SCF for the dimer.
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Received May 18, 1988
Accepted for publication July 8, 1988