Interaction Between Two Closed Shell Atoms - Sociedade Brasileira ...

Revista Brasileira de Física, Vol. 17, n9 2, 1987
Interaction Between Two Closed Shell Atoms
IRINEU LUIZ DE CARVALHO
Departamento de F~síca e Química, CEG, Universidade Federal do Espírito Santo, 29CK)O, Vitória,
ES, Brasil
and
FERNANDO S. DA PAIXÃO
Instituto de Fhica, Universidade Estadual de Campinas, Caixa Postal 1170, Campinas, 131a0, SP, Brasil
Recebido em 16 de junho de 1986
Abstract We use the theory of Boehm-Yaris and Jacobi-Csanak to calculate
the di pele-d i pole, dipole-quadrupole, quadrupole-di pole and quadrupol e-
-quadrupole contributions to the dispersion energy between two different
closed shell atoms. To this energy we add one of the Born-Meyer type wrresponding
to valence effects. In this way we find a finite total interaction
energy for any interatomic distance, whose asymptotic behavior
reproduces the usual dispersion energy. The results are compared to experimental
data and to some theoretical values found in the literature.
The interaction potential between atoms and/or molecules is of
fundamental importante to understand several static and dynamic pro-
prieties in gases, liquíds and solidsl.
Since the pioneer work of Slater in 1928~, a number of s impl i-
f ied potent ials have been suc~~ested~'~~, mai nl y based on asymptot ic sol-
ut ions of the Schrod i nger equat ion.
I n more recent papers one has tr ied to obta i n rel iabl e wliversal
formulae for the intermolecular potential in certain gas types,
mostly
by means of ab initio calculations. Particularly 8arkanZ6 suggested re-
cently that the Kiara potential, wirh careful ly calculated
yields a self-consistent description of several macroscopic
of inert gases.
parameters,
proper t i es
Even though it is artificial, it is convenient to divide the in-
teraction potentia 11 into two types: short range potential (also cal led
This work was part ia1 ly supported by CAPES (Brazi l ian government agency).
273

Revista Brasileira de Física, Vol. 17, no 2, 1987
valence or chemical potential) and long range potential (or Van der
Waals potential).
For two closed shell atoms separated by a distance R, the
valence potential can be expressed by the ul tra-simpl if ied form (Born-
-Meyer type)
where A and b are characteristic parameters of the a tom i c pa i r under
study.
The long range interaction between two non-polar sys tems in
their respective ground states (as in the case of this paper) is characterized
by a dispersion (sometimes cal led ~ondon) potential . This potential
is due to the correlation between electrons in distinct atoms. Here
the expansion of the inverse internuclear distance (R-') forms a problematic
detai 1 . Second order perturbat ion theory, by means of a mul ti polar
expansion, gives for the dispersion potential between two atoms the expans
ionZ7
where the coefficient C6 represents the dipole-dipole interaction, C,
the dipol e-quadrupole interaction and C1 o
drupole and dipole-octupol e interactions.
refers to the quadrupole-qua-
From eq. (2) i t fol lows that V(R) -t -- when R -+ O. Nevertheless,
it is generally desirable to obtain damped potentials for
intermediate
and small values of R. Severa1 authors have already treated the damping
of interaction dispersion for decreasing values of
R. Bucki ngham and
corner7 were the first to work in this direction multiplying the dis-
persion terms by a damping function dependent on R. Later, Musher and
mos", introduci ng terms formed by mul tipl y ing pol ynomial s by exponen-
tial functions, obtained convergent dispersion series.
More recentl y, analytical fosrmulae were proposed for the damp-
ing of the London potential in ground state atoms. The first of them
suppl ies expl icitly the principal term of the dispersion potential be-
tween two identical atoms and was publ ished by Jacobi and csanak2'. The
technique introduced by those authors is of basic importance
for the

Revista Brasileira de Física, Vol. 17, n? 2, 1987
present paper and wi 11 be discussed in the fol lowing section. ~ichardson~'
proposed a completely different technique supplying analytical
results
for the d i pol e-di pol e, di pol e-quadrupol e and quadrupol e-quadrupol e con-
tributions to the dispersion interaction between two atoms. This is
semi-classical formulation where every atom is treated as a harmonic
oscillator. ~oide~l, too, introduced a method supplying a convergent
series for the dispersion energy. In this method physical properties of
every atom appear separately. Koide stud'ied particularly the H2 system,
obtaining analyt ical formulae for the dipole-dipole'and dipol e- quadru-
pole contributions, in the interaction dispersion. The Koide
is valid only for spherically symmetrical systems and is nearly
a
expansion
equiv-
alent to that of Jacobi-Csanak. Finally, Battezzati and ~a~nasco~~,also,
developed an analytical formula for the dispersion energy
by a method
inspired in works by ~ o n ~ u e t - ~ i and ~ ~ ~ i nc s ~ ~ e~
e n For ~ ~ the ~ . dipole
-dipole contribution the results obtained in references 32 and 29 are
coincident.
2. THE JACOBICSANAK TECHNIQUE
This technique represents essentially an improvement introduced
by these authors to that part of the second quantization
Boehm and yaris3' treating the dispersion energy. This
formalism of
forma 1 i sm de-
scribes the interaction between two systems by a linear response theory
based on themany body Green's function techniques of Martin and
~chwin~er~~. Instead of using a multipolar expansion leading to eq. (21,
Jacobi and Csanak adopted the strategy of introducing in the
formalism
of Boehm and Yaris the analytical representation of the Born ampl i tudes
obtained by Csanak and ~a~lor~'.
The basic equation of the work by Jacobi-Csanak can be written
where (RlR2L;000) are Clebsch-Gordan coeff icients,

Revista Brasileira de Física, Vol. 17, n? 2, 1987
(vnR = exci tation energy)
and
In eq. (5) FBR(q) is the radial part of a Born ampl i tude which,
atomic system, can be factorized in the form3'
for an
with Y
h
(c) denoting a spherical harmonic of order R and
I n eq. (7) Ji- and $; are, respect ivel y, the wave funct ions of
n
the exci ted and ground states of an atom wi th N electrons and q the mo-
mentum absorbed by i t during the exci tation process. By x = (xl ,r2,.
. ,x ) we are denoting the set of coordinates of the N electrons.
"j'" + N
r = (r.,~.) refers to the four coordinates of the j-th electron, three
i 3 3
spatial (r.) and one of spin (w .) . i and o indicate the set of quantum
3 3
numbers respectively defining the excited and ground states.
. . ,
The inte-
gration ranges over a1 1 the atomic coordinates (including summing over
the spin coordinates) and d~ is the volume element.
We wi 11 adopt the Jacobi-Csanak a~proximation~~, replacing the
exact Born ampl i tudes [eq. (7)] by simpl if ied forms of the corresponding
series of ~sanak-~aylor~~. We refer to the expressions (in atomic uni-
ties):

Rwista Brasileira de Física, Vol. 17, n9 2, 1987
and
where
In eq. (12) I is the ionization energy and Fi one exci tation energy.
Equation (3) was applied by its authors only in the calculation
of the principal term (Ri=R2=1) of the Van der Waals potential between
two helium atoms, separated by intermediate and largekelative to the
atomic diameter) distances. It was used alço by J.C.
to calculate
the dipole-quadrupole term of the dispersion potential between
two helium atoms.
3. DISPERSION INTERACTION BETWEEN TWO DIFFERENT CLOSEDSHELL ATOMS
a) Introductory cons iderat ions
We wi l l use equations (3), (8) and (9) to obtain anal ytical express
ions for the dipole-dipole, dipol e-quadrupole, quadrupole- d i po 1 e
and quadrupole-quadrupole terms of the dispersion potential between two
different closed shell atoms. In particular we will numerically calculate
the interact ion between an hel ium atom and neon one. We think that
this work is a fair generalization of the calculation made by Jacobi-
-csanakZ9 and J.C. Antonio 3 ', mainly because our study of the interaction
between two distinct atoms demanded an analytical calculation,with
very 1 ittle aid of tables, of quite complex integral5 (see Appendix).
We express the dispersion energy In the form

Revista Brasileira de Física, Vol. 17, n? 2, 1987
and calculate each of these four contributions separately
b) Dipole-dipole contribution
Taking in eq. (3) Ri = R2 = 1 and considering that the Clebsch
-Gordan coeff icients (R1R2~;000) are different f rom zero only if the
conditions IRi-R2( á L S (R1+!&) and (R1+R2+1;) = even number are simul-
taneous fullfilled, we will have
vd,d(R)
(0,O) (2,2)
r gn1l,n2l kn,l,n2~ (R) + 2 1 nll ,n21(R)] (14)
2n5 n, ,nZ
'-L
Assuming the approximationfl indicated in eqs. (8) and (9)
where, for each one of the systemç@and@, the parameter a defined by
(12) is independent of the principal quantum number n, introducing (8)
in (5) (withL =L' = O and L =Lf = 2) and substituting the results
into eq. (]h), we obtain
where
(in a.u.)
On the other hand, defining the oscillator strength in an atom
as
ii O
and compar ing wi th eq. (1 O) , we have

Revista Brasileira de Física, Vol. 17, n9 2, 1987
So, from eqs. (4) and (18) it follows that
Thereforqdef ining the dynamic polar izabi 1 i ties in an atom as4'
and the dispersion coefficients C as 2 '
R13 R2
eq.
(1 5) takes the form
where C is obtained through eq. (20).
c) Dipole-quadrupol e contr ibution
Taking in eq. (3) R1 = 1 and R2 = 2 and considering that (1 2. L;
O O O) is different from zero only for L = 1 and L = 3 we obtain
Now, introducing eqs. (8) and (9)ineq(5) (with L -- L' = 1 and L =L1=3)
and substi tuting the resul ts into eq. (22) we have

Revista Brasileira de Física, Vol. 17, n? 2, 1987
where the fS(li) are funct ions to be obta ined from eq. (16).
m
On the other hand from eqs. (1 I ) and (1 7) we obtain
Further, f rom eqs. (41, (18) and (24) i t fol lows that
C
C
Final ly, using relations (19) and (20) successively in (25) and
ducing the resul t thus obtained in (231, we have
intro-
d) Quadrupole-dipole contribution
Taking in eq.(3) R1 = 2 and R2 = 1 and performing the same
stages as we did in the preceding sub-section, we find
where the coeff icient C is calculated through eq. (20) and the par-
231
ameters cik (k = 1,2) are obtained through equation (12). The functions
z13(R) and z~~(R) resul t from two of the integrations indicated in eq.
4 3 4 3
(16).
e) Quadrupol e-quadrupole cont r ibution
Taking in eq. (3) R, = Ri = 2 and using the procedures and
f ini tions of sub-sections b) and c), we obtain
de-

Revista Brasileira de Física, Vol. 17, n? 2, 1987
f) Resul ts
Introducing eq.(A18) in eqs. (211, (261, (27) and (28) and substituting
the results thus obtained into eq.(13), the usual dispers ion
energy W(K) is reproduced
On the other side, adding the valence energy given by eq.(l) to
our d ispers ion energy [eq. (1311 we obtain the total interact ion energy
For a nurnerical appl ication we choose the He-Ne system, for which A =
= 57.00 a.u. and b = 2.43 a.u. As dispersion coefficients we use
those of reference 27, that is, C = 3.13 a.u., C = 17.5 a.u.,C,
l, ,
= 15.2 a.u and C = 15.7 a.u..
2Y2
In figures 1 and 2 we show a surnrnary of our results, together
wíth equivalent resuls obtained by other authors. As for the heliumatorn
we first take a1 = 1.67508 a.u. (corresponding with the I'S + 3 ' tran- ~
si tion) and subsequentely al = 2.48535 a.u. (corresponding wi th the
auerage energy ezcitatZon calculated by Victor et a2. I). As for the
neon atorn we remain with the transitions (lsI2 ( 2 ~ (2p)6 ) ~ (Is ) -+ 3s'
1 o
o
(2)1 ('P~), corresponding to a, = 1.84720 a.u..
4. ANALYSIS OF THE RESULTS
Our equations and graphs show that:
a) In the asymptotic region (R > 4.0 8) our resul ts reproduce the usual
dispersion energy, being therefore practically independent from the
parameters a, and a,.
o
b) In the interrnediate region (2.0 A < R 6 4.0 2) our curves decrease
quite more slowly than in the corresponding usua 1 dispersion energy
graph. In this way the dependence of our results
a, does not rernain negligible any longer. The sh
nounced as R becomes smaller. (See in fig. 1
the
relative to a, and
if t becomes more proregion
R < 4.0 a).

Revista Brasileira de Física, Vol. 17, r)? 2, 1987
Fig.1 - (a) our dispersion
energy for a1 = 1.67508 a.u.
and a2 = 1 .84720 a.u. ; (b)
our dispers ion energy for a,
= 2.48535 a.u. and a;! =
= 1.84720 a.u.; (c) usual
dispersion energy [eq. (2911.
Fig.2 - (d) total interaction
energy for A = 57.0 a.u,
b = 2.43 a.u., a, = 1.67508
a.u. and a2 = 1.84720 a.u.
(ao = 0.529 8) ;(e) the same as
in (d) for ai = 2.48535 a.u.
and a2 = 1.84720 a.u.; (f)
total interaction energy calculated
for ~ a; (g) e exper- ~ ~
imental results of by Chen
et UZ.'~

Revista Brasileira de Física, Vol. 17, n? 2, 1987
c) Our total interaction energy, given in eq. (30), shows a good agree-
ment with the theoretical work of ~ a e and ' ~ with the experimental re-
sults of Chen et a~~~ (see fig.2).
It is interesting to see that in
the problematic region of the Van der Waals minimum our calculations
show a better agreement wi th the experimental data than Rae's resul ts.
Further, it is simple to obtain still better results choos ing ad-
equate values for the parameters ai and ae.
d) Our resul ts, differentely from the usual dispersion energy, do not
diverge for small
R-values. Is this way, they may be useful in the
study of problems such as atom-atom scattering.
The au thors wish to thank Raquel
for assistance in the execution of numerica 1 calculations and Dr. Karel
Frans Van den Bergen for his generous help
wor k .
Regis Azevedo de
Carvalho
in the last stage of this
To illustrate the analytical technique used in calculating the
integrals fS def ined by eq. (16) let us take
mn
In eq.(A3)
the integrand is an even function; so

Revista Brasileira de Física, Vol. 17, n? 2, 1987
The integral (A41 can be calculated with the help of residue
theory. To do that let us define the functions f(z) of a complex vari-
able z as follows.
Consider the functions
and use the contour shown in fig. AI.
Fig.AI - Contour to be used in the calculation of integral (A4).
Accord ing to the res idue theory
where Ri and RI are the residues of the poles of f(z)
ly i'ng within the
integration contour (see f ig. AI). We know that those residues are given
by

Revista Brasileira de Física, Vol. 17, nQ 2, 1987
where m is the
pole's order. In the 1 imi t R + rn, J f (z) dz + O and so
c2
I(R) =$f(q)dq =2iR1 +2iR2 . (A81
-m
From eqs.
(As) and (A7) if follows (for rn = 3) that
and
Applying these two resul ts in eq. (A8) and executing the operations shown
in eq. (~2) we obtain:
where
P~(R) = + E,~]R , P ~(R) = El12 + ElI3R and P (R) =
6
3 (a: -0.2 ) 'R
9
wi th

Revista Brasileira de Física, Vol. 17. nP 2, 1987
By an direct generalisation of this method we obtain the following
results for the remaining integrals pointed out in eqs. (21),(26),
(27) and (28). We have
2 2
1
Z (R) = -{L
3 3 ala2
-a2R
+ [P,, (R) - P, (R)] e P, (R) + P, (R)) e
},
with
and

Revista Brasileira de Física, Vol. 17, n? 2, 1987
We have
where
We have
where
wi th
p1 (R) = (IW~R)R~/(~~-~:) ' and P*, (R) = (~+a~~)~~/(a~-a~)' ,

Revista Brasileira de Fisica, Vol. 17, n? 2, 1987
We have
We have

Revista Brasileira de Fisica, Vol. 17, n? 2, 1987
where
1 1
Pl (R) = E21 5(1 +alR +- 2 a2R2
1 ió
a;R3) + E,, ,R4 + E,~,R~ + E,, ,R6 ,
PZo
1 1
(R) = E,, ,(1 + a2R + a ' ~ + ~ a2R3) + E,] o ~ + 4 E2 i ,
2
P2
(1 + ~,R)R~ (1 + a2~.)R2
(R) =
(a: - a:)
(R) = and PZ2
= -[ a a I - +
a: (ai-a:) '
+
+ a: (a:-a:) a: (a:-ai>
We have
(A1 5)
where

Revista Brasileira de Fhica, Vol. 17, nP 2, 1987
and
- 1
- and E =
E224 - - 1 6a2 ($-a:) (a:-a:) 22 48aZ (a:-a:) *
1
We have
where
and
1
P (R) = E2210(l + a R +-a:R2) + E R3 + E R' + E R'
2 7 2 2 2211 2212 2213

Revista Brasileira de Física, Vol. 17, nQ 2, 1987
Finally we have
where
(A1 7)
and

Revista Brasileira de F isica, Vol. 17, no 2, 1987
It is interesting to note that the functions (A9) - (A171 are
analytical and that, for large values of R, we have the following behavior
15 1
, 240: + O , zZ4 + O and + ---- - .
233 +- -
105 1
43 a8a6 ~4 rr 4
1 2
44 (ala2)' R5
(A181
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Revista Brasileira de Física, Vol. 17, n? 2, 1987
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Resumo
O formal ismo de Bohem-Yaris e Jacobi-Csanak 6 usado no cálculo
dos termos dipolo-dipolo, dipolo-quadrupolo, quadrupolo-dipolo e quadrw
polo-quadrupolo da energia de dispersão entre dois diferentes átomos de
camadas fechadas. A esta energia foi adicionada uma energia de valência
do tipo Born-Meyer. Assim foi obtida uma energia total de integração f i-
nita para todas as distâncias inter-atômicas, cuja forma assintõtica reproduz
a energia de dispersão usual. Os resultados foram comparados com
dados experimentais e outros valores teõr icos disponíveis na l i teratura.