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Revista <strong>Brasileira</strong> de Física, Vol. 17, n9 2, 1987<br />
<strong>Interaction</strong> <strong>Between</strong> <strong>Two</strong> <strong>Closed</strong> <strong>Shell</strong> <strong>Atoms</strong><br />
IRINEU LUIZ DE CARVALHO<br />
Departamento de F~síca e Química, CEG, Universidade Federal do Espírito Santo, 29CK)O, Vitória,<br />
ES, Brasil<br />
and<br />
FERNANDO S. DA PAIXÃO<br />
Instituto de Fhica, Universidade Estadual de Campinas, Caixa Postal 1170, Campinas, 131a0, SP, Brasil<br />
Recebido em 16 de junho de 1986<br />
Abstract We use the theory of Boehm-Yaris and Jacobi-Csanak to calculate<br />
the di pele-d i pole, dipole-quadrupole, quadrupole-di pole and quadrupol e-<br />
-quadrupole contributions to the dispersion energy between two different<br />
closed shell atoms. To this energy we add one of the Born-Meyer type wrresponding<br />
to valence effects. In this way we find a finite total interaction<br />
energy for any interatomic distance, whose asymptotic behavior<br />
reproduces the usual dispersion energy. The results are compared to experimental<br />
data and to some theoretical values found in the literature.<br />
The interaction potential between atoms and/or molecules is of<br />
fundamental importante to understand several static and dynamic pro-<br />
prieties in gases, liquíds and solidsl.<br />
Since the pioneer work of Slater in 1928~, a number of s impl i-<br />
f ied potent ials have been suc~~ested~'~~, mai nl y based on asymptot ic sol-<br />
ut ions of the Schrod i nger equat ion.<br />
I n more recent papers one has tr ied to obta i n rel iabl e wliversal<br />
formulae for the intermolecular potential in certain gas types,<br />
mostly<br />
by means of ab initio calculations. Particularly 8arkanZ6 suggested re-<br />
cently that the Kiara potential, wirh careful ly calculated<br />
yields a self-consistent description of several macroscopic<br />
of inert gases.<br />
parameters,<br />
proper t i es<br />
Even though it is artificial, it is convenient to divide the in-<br />
teraction potentia 11 into two types: short range potential (also cal led<br />
This work was part ia1 ly supported by CAPES (Brazi l ian government agency).<br />
273
Revista <strong>Brasileira</strong> de Física, Vol. 17, no 2, 1987<br />
valence or chemical potential) and long range potential (or Van der<br />
Waals potential).<br />
For two closed shell atoms separated by a distance R, the<br />
valence potential can be expressed by the ul tra-simpl if ied form (Born-<br />
-Meyer type)<br />
where A and b are characteristic parameters of the a tom i c pa i r under<br />
study.<br />
The long range interaction between two non-polar sys tems in<br />
their respective ground states (as in the case of this paper) is characterized<br />
by a dispersion (sometimes cal led ~ondon) potential . This potential<br />
is due to the correlation between electrons in distinct atoms. Here<br />
the expansion of the inverse internuclear distance (R-') forms a problematic<br />
detai 1 . Second order perturbat ion theory, by means of a mul ti polar<br />
expansion, gives for the dispersion potential between two atoms the expans<br />
ionZ7<br />
where the coefficient C6 represents the dipole-dipole interaction, C,<br />
the dipol e-quadrupole interaction and C1 o<br />
drupole and dipole-octupol e interactions.<br />
refers to the quadrupole-qua-<br />
From eq. (2) i t fol lows that V(R) -t -- when R -+ O. Nevertheless,<br />
it is generally desirable to obtain damped potentials for<br />
intermediate<br />
and small values of R. Severa1 authors have already treated the damping<br />
of interaction dispersion for decreasing values of<br />
R. Bucki ngham and<br />
corner7 were the first to work in this direction multiplying the dis-<br />
persion terms by a damping function dependent on R. Later, Musher and<br />
mos", introduci ng terms formed by mul tipl y ing pol ynomial s by exponen-<br />
tial functions, obtained convergent dispersion series.<br />
More recentl y, analytical fosrmulae were proposed for the damp-<br />
ing of the London potential in ground state atoms. The first of them<br />
suppl ies expl icitly the principal term of the dispersion potential be-<br />
tween two identical atoms and was publ ished by Jacobi and csanak2'. The<br />
technique introduced by those authors is of basic importance<br />
for the
Revista <strong>Brasileira</strong> de Física, Vol. 17, n? 2, 1987<br />
present paper and wi 11 be discussed in the fol lowing section. ~ichardson~'<br />
proposed a completely different technique supplying analytical<br />
results<br />
for the d i pol e-di pol e, di pol e-quadrupol e and quadrupol e-quadrupol e con-<br />
tributions to the dispersion interaction between two atoms. This is<br />
semi-classical formulation where every atom is treated as a harmonic<br />
oscillator. ~oide~l, too, introduced a method supplying a convergent<br />
series for the dispersion energy. In this method physical properties of<br />
every atom appear separately. Koide stud'ied particularly the H2 system,<br />
obtaining analyt ical formulae for the dipole-dipole'and dipol e- quadru-<br />
pole contributions, in the interaction dispersion. The Koide<br />
is valid only for spherically symmetrical systems and is nearly<br />
a<br />
expansion<br />
equiv-<br />
alent to that of Jacobi-Csanak. Finally, Battezzati and ~a~nasco~~,also,<br />
developed an analytical formula for the dispersion energy<br />
by a method<br />
inspired in works by ~ o n ~ u e t - ~ i and ~ ~ ~ i nc s ~ ~ e~<br />
e n For ~ ~ the ~ . dipole<br />
-dipole contribution the results obtained in references 32 and 29 are<br />
coincident.<br />
2. THE JACOBICSANAK TECHNIQUE<br />
This technique represents essentially an improvement introduced<br />
by these authors to that part of the second quantization<br />
Boehm and yaris3' treating the dispersion energy. This<br />
formalism of<br />
forma 1 i sm de-<br />
scribes the interaction between two systems by a linear response theory<br />
based on themany body Green's function techniques of Martin and<br />
~chwin~er~~. Instead of using a multipolar expansion leading to eq. (21,<br />
Jacobi and Csanak adopted the strategy of introducing in the<br />
formalism<br />
of Boehm and Yaris the analytical representation of the Born ampl i tudes<br />
obtained by Csanak and ~a~lor~'.<br />
The basic equation of the work by Jacobi-Csanak can be written<br />
where (RlR2L;000) are Clebsch-Gordan coeff icients,
Revista <strong>Brasileira</strong> de Física, Vol. 17, n? 2, 1987<br />
(vnR = exci tation energy)<br />
and<br />
In eq. (5) FBR(q) is the radial part of a Born ampl i tude which,<br />
atomic system, can be factorized in the form3'<br />
for an<br />
with Y<br />
h<br />
(c) denoting a spherical harmonic of order R and<br />
I n eq. (7) Ji- and $; are, respect ivel y, the wave funct ions of<br />
n<br />
the exci ted and ground states of an atom wi th N electrons and q the mo-<br />
mentum absorbed by i t during the exci tation process. By x = (xl ,r2,.<br />
. ,x ) we are denoting the set of coordinates of the N electrons.<br />
"j'" + N<br />
r = (r.,~.) refers to the four coordinates of the j-th electron, three<br />
i 3 3<br />
spatial (r.) and one of spin (w .) . i and o indicate the set of quantum<br />
3 3<br />
numbers respectively defining the excited and ground states.<br />
. . ,<br />
The inte-<br />
gration ranges over a1 1 the atomic coordinates (including summing over<br />
the spin coordinates) and d~ is the volume element.<br />
We wi 11 adopt the Jacobi-Csanak a~proximation~~, replacing the<br />
exact Born ampl i tudes [eq. (7)] by simpl if ied forms of the corresponding<br />
series of ~sanak-~aylor~~. We refer to the expressions (in atomic uni-<br />
ties):
Rwista <strong>Brasileira</strong> de Física, Vol. 17, n9 2, 1987<br />
and<br />
where<br />
In eq. (12) I is the ionization energy and Fi one exci tation energy.<br />
Equation (3) was applied by its authors only in the calculation<br />
of the principal term (Ri=R2=1) of the Van der Waals potential between<br />
two helium atoms, separated by intermediate and largekelative to the<br />
atomic diameter) distances. It was used alço by J.C.<br />
to calculate<br />
the dipole-quadrupole term of the dispersion potential between<br />
two helium atoms.<br />
3. DISPERSION INTERACTION BETWEEN TWO DIFFERENT CLOSEDSHELL ATOMS<br />
a) Introductory cons iderat ions<br />
We wi l l use equations (3), (8) and (9) to obtain anal ytical express<br />
ions for the dipole-dipole, dipol e-quadrupole, quadrupole- d i po 1 e<br />
and quadrupole-quadrupole terms of the dispersion potential between two<br />
different closed shell atoms. In particular we will numerically calculate<br />
the interact ion between an hel ium atom and neon one. We think that<br />
this work is a fair generalization of the calculation made by Jacobi-<br />
-csanakZ9 and J.C. Antonio 3 ', mainly because our study of the interaction<br />
between two distinct atoms demanded an analytical calculation,with<br />
very 1 ittle aid of tables, of quite complex integral5 (see Appendix).<br />
We express the dispersion energy In the form
Revista <strong>Brasileira</strong> de Física, Vol. 17, n? 2, 1987<br />
and calculate each of these four contributions separately<br />
b) Dipole-dipole contribution<br />
Taking in eq. (3) Ri = R2 = 1 and considering that the Clebsch<br />
-Gordan coeff icients (R1R2~;000) are different f rom zero only if the<br />
conditions IRi-R2( á L S (R1+!&) and (R1+R2+1;) = even number are simul-<br />
taneous fullfilled, we will have<br />
vd,d(R)<br />
(0,O) (2,2)<br />
r gn1l,n2l kn,l,n2~ (R) + 2 1 nll ,n21(R)] (14)<br />
2n5 n, ,nZ<br />
'-L<br />
Assuming the approximationfl indicated in eqs. (8) and (9)<br />
where, for each one of the systemç@and@, the parameter a defined by<br />
(12) is independent of the principal quantum number n, introducing (8)<br />
in (5) (withL =L' = O and L =Lf = 2) and substituting the results<br />
into eq. (]h), we obtain<br />
where<br />
(in a.u.)<br />
On the other hand, defining the oscillator strength in an atom<br />
as<br />
ii O<br />
and compar ing wi th eq. (1 O) , we have
Revista <strong>Brasileira</strong> de Física, Vol. 17, n9 2, 1987<br />
So, from eqs. (4) and (18) it follows that<br />
Thereforqdef ining the dynamic polar izabi 1 i ties in an atom as4'<br />
and the dispersion coefficients C as 2 '<br />
R13 R2<br />
eq.<br />
(1 5) takes the form<br />
where C is obtained through eq. (20).<br />
c) Dipole-quadrupol e contr ibution<br />
Taking in eq. (3) R1 = 1 and R2 = 2 and considering that (1 2. L;<br />
O O O) is different from zero only for L = 1 and L = 3 we obtain<br />
Now, introducing eqs. (8) and (9)ineq(5) (with L -- L' = 1 and L =L1=3)<br />
and substi tuting the resul ts into eq. (22) we have
Revista <strong>Brasileira</strong> de Física, Vol. 17, n? 2, 1987<br />
where the fS(li) are funct ions to be obta ined from eq. (16).<br />
m<br />
On the other hand from eqs. (1 I ) and (1 7) we obtain<br />
Further, f rom eqs. (41, (18) and (24) i t fol lows that<br />
C<br />
C<br />
Final ly, using relations (19) and (20) successively in (25) and<br />
ducing the resul t thus obtained in (231, we have<br />
intro-<br />
d) Quadrupole-dipole contribution<br />
Taking in eq.(3) R1 = 2 and R2 = 1 and performing the same<br />
stages as we did in the preceding sub-section, we find<br />
where the coeff icient C is calculated through eq. (20) and the par-<br />
231<br />
ameters cik (k = 1,2) are obtained through equation (12). The functions<br />
z13(R) and z~~(R) resul t from two of the integrations indicated in eq.<br />
4 3 4 3<br />
(16).<br />
e) Quadrupol e-quadrupole cont r ibution<br />
Taking in eq. (3) R, = Ri = 2 and using the procedures and<br />
f ini tions of sub-sections b) and c), we obtain<br />
de-
Revista <strong>Brasileira</strong> de Física, Vol. 17, n? 2, 1987<br />
f) Resul ts<br />
Introducing eq.(A18) in eqs. (211, (261, (27) and (28) and substituting<br />
the results thus obtained into eq.(13), the usual dispers ion<br />
energy W(K) is reproduced<br />
On the other side, adding the valence energy given by eq.(l) to<br />
our d ispers ion energy [eq. (1311 we obtain the total interact ion energy<br />
For a nurnerical appl ication we choose the He-Ne system, for which A =<br />
= 57.00 a.u. and b = 2.43 a.u. As dispersion coefficients we use<br />
those of reference 27, that is, C = 3.13 a.u., C = 17.5 a.u.,C,<br />
l, ,<br />
= 15.2 a.u and C = 15.7 a.u..<br />
2Y2<br />
In figures 1 and 2 we show a surnrnary of our results, together<br />
wíth equivalent resuls obtained by other authors. As for the heliumatorn<br />
we first take a1 = 1.67508 a.u. (corresponding with the I'S + 3 ' tran- ~<br />
si tion) and subsequentely al = 2.48535 a.u. (corresponding wi th the<br />
auerage energy ezcitatZon calculated by Victor et a2. I). As for the<br />
neon atorn we remain with the transitions (lsI2 ( 2 ~ (2p)6 ) ~ (Is ) -+ 3s'<br />
1 o<br />
o<br />
(2)1 ('P~), corresponding to a, = 1.84720 a.u..<br />
4. ANALYSIS OF THE RESULTS<br />
Our equations and graphs show that:<br />
a) In the asymptotic region (R > 4.0 8) our resul ts reproduce the usual<br />
dispersion energy, being therefore practically independent from the<br />
parameters a, and a,.<br />
o<br />
b) In the interrnediate region (2.0 A < R 6 4.0 2) our curves decrease<br />
quite more slowly than in the corresponding usua 1 dispersion energy<br />
graph. In this way the dependence of our results<br />
a, does not rernain negligible any longer. The sh<br />
nounced as R becomes smaller. (See in fig. 1<br />
the<br />
relative to a, and<br />
if t becomes more proregion<br />
R < 4.0 a).
Revista <strong>Brasileira</strong> de Física, Vol. 17, r)? 2, 1987<br />
Fig.1 - (a) our dispersion<br />
energy for a1 = 1.67508 a.u.<br />
and a2 = 1 .84720 a.u. ; (b)<br />
our dispers ion energy for a,<br />
= 2.48535 a.u. and a;! =<br />
= 1.84720 a.u.; (c) usual<br />
dispersion energy [eq. (2911.<br />
Fig.2 - (d) total interaction<br />
energy for A = 57.0 a.u,<br />
b = 2.43 a.u., a, = 1.67508<br />
a.u. and a2 = 1.84720 a.u.<br />
(ao = 0.529 8) ;(e) the same as<br />
in (d) for ai = 2.48535 a.u.<br />
and a2 = 1.84720 a.u.; (f)<br />
total interaction energy calculated<br />
for ~ a; (g) e exper- ~ ~<br />
imental results of by Chen<br />
et UZ.'~
Revista <strong>Brasileira</strong> de Física, Vol. 17, n? 2, 1987<br />
c) Our total interaction energy, given in eq. (30), shows a good agree-<br />
ment with the theoretical work of ~ a e and ' ~ with the experimental re-<br />
sults of Chen et a~~~ (see fig.2).<br />
It is interesting to see that in<br />
the problematic region of the Van der Waals minimum our calculations<br />
show a better agreement wi th the experimental data than Rae's resul ts.<br />
Further, it is simple to obtain still better results choos ing ad-<br />
equate values for the parameters ai and ae.<br />
d) Our resul ts, differentely from the usual dispersion energy, do not<br />
diverge for small<br />
R-values. Is this way, they may be useful in the<br />
study of problems such as atom-atom scattering.<br />
The au thors wish to thank Raquel<br />
for assistance in the execution of numerica 1 calculations and Dr. Karel<br />
Frans Van den Bergen for his generous help<br />
wor k .<br />
Regis Azevedo de<br />
Carvalho<br />
in the last stage of this<br />
To illustrate the analytical technique used in calculating the<br />
integrals fS def ined by eq. (16) let us take<br />
mn<br />
In eq.(A3)<br />
the integrand is an even function; so
Revista <strong>Brasileira</strong> de Física, Vol. 17, n? 2, 1987<br />
The integral (A41 can be calculated with the help of residue<br />
theory. To do that let us define the functions f(z) of a complex vari-<br />
able z as follows.<br />
Consider the functions<br />
and use the contour shown in fig. AI.<br />
Fig.AI - Contour to be used in the calculation of integral (A4).<br />
Accord ing to the res idue theory<br />
where Ri and RI are the residues of the poles of f(z)<br />
ly i'ng within the<br />
integration contour (see f ig. AI). We know that those residues are given<br />
by
Revista <strong>Brasileira</strong> de Física, Vol. 17, nQ 2, 1987<br />
where m is the<br />
pole's order. In the 1 imi t R + rn, J f (z) dz + O and so<br />
c2<br />
I(R) =$f(q)dq =2iR1 +2iR2 . (A81<br />
-m<br />
From eqs.<br />
(As) and (A7) if follows (for rn = 3) that<br />
and<br />
Applying these two resul ts in eq. (A8) and executing the operations shown<br />
in eq. (~2) we obtain:<br />
where<br />
P~(R) = + E,~]R , P ~(R) = El12 + ElI3R and P (R) =<br />
6<br />
3 (a: -0.2 ) 'R<br />
9<br />
wi th
Revista <strong>Brasileira</strong> de Física, Vol. 17. nP 2, 1987<br />
By an direct generalisation of this method we obtain the following<br />
results for the remaining integrals pointed out in eqs. (21),(26),<br />
(27) and (28). We have<br />
2 2<br />
1<br />
Z (R) = -{L<br />
3 3 ala2<br />
-a2R<br />
+ [P,, (R) - P, (R)] e P, (R) + P, (R)) e<br />
},<br />
with<br />
and
Revista <strong>Brasileira</strong> de Física, Vol. 17, n? 2, 1987<br />
We have<br />
where<br />
We have<br />
where<br />
wi th<br />
p1 (R) = (IW~R)R~/(~~-~:) ' and P*, (R) = (~+a~~)~~/(a~-a~)' ,
Revista <strong>Brasileira</strong> de Fisica, Vol. 17, n? 2, 1987<br />
We have<br />
We have
Revista <strong>Brasileira</strong> de Fisica, Vol. 17, n? 2, 1987<br />
where<br />
1 1<br />
Pl (R) = E21 5(1 +alR +- 2 a2R2<br />
1 ió<br />
a;R3) + E,, ,R4 + E,~,R~ + E,, ,R6 ,<br />
PZo<br />
1 1<br />
(R) = E,, ,(1 + a2R + a ' ~ + ~ a2R3) + E,] o ~ + 4 E2 i ,<br />
2<br />
P2<br />
(1 + ~,R)R~ (1 + a2~.)R2<br />
(R) =<br />
(a: - a:)<br />
(R) = and PZ2<br />
= -[ a a I - +<br />
a: (ai-a:) '<br />
+<br />
+ a: (a:-a:) a: (a:-ai><br />
We have<br />
(A1 5)<br />
where
Revista <strong>Brasileira</strong> de Fhica, Vol. 17, nP 2, 1987<br />
and<br />
- 1<br />
- and E =<br />
E224 - - 1 6a2 ($-a:) (a:-a:) 22 48aZ (a:-a:) *<br />
1<br />
We have<br />
where<br />
and<br />
1<br />
P (R) = E2210(l + a R +-a:R2) + E R3 + E R' + E R'<br />
2 7 2 2 2211 2212 2213
Revista <strong>Brasileira</strong> de Física, Vol. 17, nQ 2, 1987<br />
Finally we have<br />
where<br />
(A1 7)<br />
and
Revista <strong>Brasileira</strong> de F isica, Vol. 17, no 2, 1987<br />
It is interesting to note that the functions (A9) - (A171 are<br />
analytical and that, for large values of R, we have the following behavior<br />
15 1<br />
, 240: + O , zZ4 + O and + ---- - .<br />
233 +- -<br />
105 1<br />
43 a8a6 ~4 rr 4<br />
1 2<br />
44 (ala2)' R5<br />
(A181<br />
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Resumo<br />
O formal ismo de Bohem-Yaris e Jacobi-Csanak 6 usado no cálculo<br />
dos termos dipolo-dipolo, dipolo-quadrupolo, quadrupolo-dipolo e quadrw<br />
polo-quadrupolo da energia de dispersão entre dois diferentes átomos de<br />
camadas fechadas. A esta energia foi adicionada uma energia de valência<br />
do tipo Born-Meyer. Assim foi obtida uma energia total de integração f i-<br />
nita para todas as distâncias inter-atômicas, cuja forma assintõtica reproduz<br />
a energia de dispersão usual. Os resultados foram comparados com<br />
dados experimentais e outros valores teõr icos disponíveis na l i teratura.