- Email: [email protected]
Van der Waals forces, scattering functions and charge density susceptibility. II. Application to the HeHe interaction potential
Chemical Physics 62 (1981) 423438 North-Ho!land Publishing Company
VAN DER WAALS IFORCES, SCATTERING FUNCTIONS AND CHARGE DENSITY SUSCEPTIBILITY. II. APBLHCATION TO THE He-He INTERACTION POTENTIAL+ P. MALINOWSKI*, of Chnistry,
Department
A. C. TANNER, Florida State Unicerrity,
K.F. LEE and B. LINDER Taliahassee. Florida 32306,
USA
Received 2 April 198I
Previous work relating the dispersion energy to the static form factorsof X-ray scatrering is extended to the pokization energy. For interacting heliumatomswe calculateboth those contributions(krhout recourse to the mulripok expansion), as well as the first order Coulomb and exchange forces. For large separations (R as.0 au), the shape of our potential agrees well with previous. accurate results. Since we use Uns6ld’s approximation, we discuss several choices for the mean excitation energy, s, which is required to put our potential on an absolute scale. A fixed %? suffices for large separations. In the region of overlap, a R-dependent E(R) is needed for accurate determination of the potential.
4. Introduction The calculation of long range intermolecular forces is often treated by second order perturbation theory. Such a calculation must confront the usual difficulty of evaluating the infinite summation and integration over discrete and continuum states. An early method for performing those summations in studies of intermolecular forces was London’s [l]. He used U&M’s [2] method of approximately summing the second order perturbation energy. By also approximating the Coulomb potential by the dipole term alone, London related the dispersion energy to the static polarizability LY,a quantity available from experiment. Despite the clever simplicity and accuracy of his method, it is a dipole approximation and so is limited to small systems. In this paper we continue our earlier work [3] the aim of which is the development of more widely applicable computational methods which nevertheless retain the spirit of London’s theory, namely the use of readily available data to calcuIate long range intermolecular potentials in a simple way. The method used is based on the response-perturbation technique of intermolecular forces. Of importance in this treatment is the charge-density susceptibility, in terms of which the second and higher order terms may be developed. The susceptibility takes account of spatial and temporal correlations; the ordinary polarizability a is independent of spatial variables and may be obtained as a special case from the charge-density susceptibility. The formulation of the interaction energy in terms of the stisceptibility uses the full Coulomb potential, thus avoiding the disadvantages of the multipole expansion, and in particular the limited precision of the dipole approximation. It was shown earlier [3] that the charge-density susceptibility in momentum space is related to generaliied scattering functions and an approximation to the susceptibility is expressible in terms of static form factors. In the case of He it is easy enough to evaluate analytically these form factors and
t Supported by the Natior.aI Institute of Fe&h, Grant No. GM23223. * Permanent address: Institute of Physics, Nicholas Copernicus Univenity,
0301-0104/81/0000-0000/$02.75
@ 1981 North-Holland
87-100 Tort&
Poland.
P. hlalimowski ct al. f He-He interaction potential
424
the interaction energy from self-consistent field wavefunctions and this was done here for convenience. A prescription for evaluating the interaction energy from readily available computed and experimental form factors [4] will be given elsewhere. We calculate the second order dispersion and polarization energies of two He atoms by applying the UnsSId approximation to the susceptibilities. This has the effect of introducing a mean excitation energy for the dispersion energy which is twice that for the polarization energy. It is shown in section 4.2 that the same result can be obtained by applying the Unsiild approximation directly to the energy provided that the energy is first decomposed into its dispersion and polarization parts and the Unsiild approximation is applied to each term separately. Although the polarization energy is small, it is not negtigible at short distances, an observation surprising perhaps to those accustomed only to the dipole approximation for which the polarization energy is zero. We also calcuiate the total potential energy by adding the first order Coulomb and exchange energies to the second order energy. The susceptibility formulation of the problem provides a direct link between intermolecular potentials and scattering phenomena. The development in momentum space allows considerable mathematical simplification and other authors have used the same technique in similar calculations
PI. 2. Outline of the method 2.1.
The second order interaction
As mentioned
where: and n p,,(r) and
energy ir, terms of the susceptibility
in the introduction
m, n are the states
the system
of the system,
p,,_(r)
is characterized
is the transition
by the susceptibility
charge
densi&
[6]
between
states
m
= (mHr)ln);
(2)
p’(rj is the charge density operator,
p^(r)=e
g ZJ(r--r,)-e v=l
defined
as
f S(r-q), ;=1
(3)
in which e is the absolute value of the electron charge, A is the number of nuclei, N is the number electrons and 2, is the atomic number of the uth atom. w”,,, is connected with the unperturbed energies E,, and E,,, of the system by =E,,-E,,
hGn o
of
(4)
is the frequency of the applied potential and cr,, is a matrix element of the statistical operator 6:
e = exp (-PIkjlQ, where fz, is the hamiltonian or partition function defined
Q = 2 exp (-BE). I
(5) of the unperturbed as
system,
p the Boltzmann
factor
and Q the state sum
(6)
et al. / He-He itzreractioft potential
P. hfalinowski
425
The Et are the energies of the unperturbed system. Such susceptibilities can be utilized to calculate the free energy of the system. In this paper we will use these quantities to obtain the second order dispersion and polarization energies. Let us consider molecule “a” characterized by susceptibility x.&i, r,, W) and molecule “b” by Xb(rzr r;, w). Let these molecules interact via the Coulomb potential V(rl, rz) =
(7)
Ir2-r:l-‘.
As described in the previous paper [6] the second order dispersion energy can be expressed as Ej;:,(T)=-(h/4;i)ReiSdr,Idr;
f
do ,y&:,
rl, w)Kf’
i8)
(rI, r\, o) coth (h&/2).
-33 The quantity Kf’(r,,
Kg’ (rl, r;,
r;, O) is the real part of the reaction potential
dr$ V(ri, rz) X&2,&
o) =
susceptibility
w) V(rS, 4).
This expression for the second order dispersion energy describes the interaction of two fluctuating charge distributions. The charge distribution on molecule “a” can also polarize molecule “b”, giving rise to the poIarization energy? (10) The charge distribution on molecule “b” gives a similar contribution polarization energy
to the second order
(11) The total polarization energy is of course the sum of these two contributions E$, (T) = “‘bE~,(T)+b-=E~*(T).
(12)
These formulas for the second order dispersion and polarization contributions are however very difficult to evaluate in concrete calculations due to the infinite summations, including the continuum. In practice we consider only T = 0 which enables us to put n = 0 and s = 0 in eqs. (10) and (ll), and to simplify (8) with lim coth (h&~/2) = 1, 1-O
and
lim qnnn= lim a,, = 1. T-0
T-d
Our key quantities can be written as follows E$:,(O)=E~&,
= -(A/45i) Re i
(13) -m
t This, in fact, is the induction energy. The terms polarization energy and induction energy are often used interchangeably.
previous publication.ref. [71. a distinction was made between the two. There the term polarization potential was used to denote an auxiliary function, which when multiplied by l/2 and averaged over the rotational states yielded the induction energy (see, for example, eqs. (25) and (32) of ref. [7]). In the absence of rotational states the induction energy reduces to expression (10) of the present paper. We shall refer to this expression as polarization energy.
In a
(154
(15b) Let US fOcUS OUTattention on the dispersion energy of the two interacting molecules. The System of coordinates used in this paper is show in fig. 1. This choice of coordinate system centred on molecule “a” enables us to rewrite the formula for EyLP as
E~~,=-ih/d;r)ReiIdr,Jdr:Idr25dr;
i
dw x,(6,
rr. 0)
Vh,
r?)
xb(rZ-R,
ri
-i?,
co)
V(r&
r;),
--z (W
where r, and r; now refer to the common center on --a”, and R is the vector from “a” to “b”. It is Useful to define susceptibilities that are Fourier transformed to momentum space (k-Space) x&;,
rl,
w)
= II-’
5
dkl
J
dk2 e-ik;‘r; x&C:, kl, O) eikl-rl
(17)
and
where R = (2rr)‘. Substitution of (17) and (1X) into eq. (16) and integration
Thus by going to momentum
space one has reduced
over rr, r;, rz and r; yields
drastically
the dimension
Fig. 1. Systemof coordinates.
of integration.
P. Malinowski et al. j He-He inferaction potential
Making use of eqs. (15a) and (1%) and the definition of the Fourier transformation (18)] we can express the susceptibilities in momentum space P~~(k’)P:0(-k) o -m,o-ib
x.(k, k’, 0) = Fr$ fi-’ ;(
_ &,(-k)PL(k’) o+o,o-ib
427
[eqs. (17) and
I’
(20)
where P&(k) = (016”(k)ln~,
(21)
b”(k) = I em”“p^“(r)dr.
(22)
The susceptibility &k, k’, W) can be expressed similarly in terms of transition form factors p&(k). In order to simplify the expression for the second order energy in momentum space let us express the susceptibilities along the negative imaginary frequency axis. Assuming that the functions characterizing both molecules are reaI, one can rewrite eq. (20) as Eo!lows ~~(k’, k, -iy) = -2fi-’
1 +%p&(k’)p~o(-k). n On0 +y-
(23)
Thus the second order dispersion energy can be expressed E$?;, = - 185r5 jdyIdkjdk’~~,(k.,k-iY)Xe(%L.,iy).
(24)
0
Following a similar procedure,
one can rewrite expressions for the second order polarization energy as &k’)
'+b&d,_-
&A--k)
X&r
k’, 01,
(25a)
85r4
b-ilEg,
2.2.
=
&
,I J dk
e”k-k”‘R b pm(k) &A-k’) xdk’, k, 0).
dk,
kzk,z
(25b)
UnsGId’s approximation
In order to nse eqs. (24) and (25) we apply Unsiild’s approximzdon [2] to the expression for the susceptibilities, since it is the simpiest and most straightforward approximation. Thus we replace on0 and oSO by mean excitation energies C&Oand &a in the expressions for susceptibilities x1 and xb. After this substitution we add and subtract the terms with n = 0 and s = 0 respectively. Thus we obtain xz(k’,
k, -iy)
=
-2fi-’
X,,( k, k’, -iy) = -2h-’
on0
7 WW(k’G”(-k)lO) C%;o+y-
+%
us0 + y-
- ~Olp^“~k’~lO>~Olp^“~-k)(O)},
~~Ol~b~k~~b~-k’~lO~-~OI~b~k~lO~~Ol~b~-k’~lO~~.
Making use of the definition of the charge-density operators, eq. (3), we obtain for atom “a” the matrix elements involved in eqs. (24), (25a) and (25b)
(26a)
t26b)
428
P. Malinowski et af./ He-He inreracrion porenzial
_N,(o]
3
e-ik~rj~)+(~l
?
;=*
eik”’
3 eiW’-k)-,lo)),
e-‘k-,lo)+(q
i#j
(27)
i-l
These matrix elements can be expressed in terms of one- and two-particle coordinate space [S],
density matrices in
(01 z e”“f\O) = N,(Oleik“‘[O) = J dr eik’-r-&r)=F,(k’),
(28)
i=1
(o/$
p-r, e-ik-rilO)= N,(N,-
l)(Ol J*‘+ e-ik-rIO)= 2 J dr J dr’ eiL’+ r,(r, r’lr, r*) e-ik.r ~~=(kl,
k).
i*i
(29)
The function Fa(k’) is simply the Fourier transform of the one particle density matrix ~~(r]r). The quantity L,(k, k’) is the Fourier transform of the two-particle density matrix Pa(r, r’jr, r’) of molecule “a” and for totally uncosrelated functions (e.g. a simple Hartree product): ) F,(-k’)
L,(k, k’) = (1 -N,’
F,(k).
(30)
For a single determinant two electron function eq. (30) also holds. We restrict ourselves to approximation (30). Putting (28) and (29) into (27) we obtain (ojp^a(k’) ,5”(-k)]O) = e’[N: -N,Fa(k’) -N,F,(-k)+(l Using eq. (31) and eq. (28) the susceptibility expressed as: x,(k’,
)~~(-k)F,(k’)+F,(k’-k)}.
for molecule “a” in Unsiild’s approximation
(31) can be
+y’)X~H(k’-k)-N,‘F,(k’)F,(-k)}.
k,-iy) =[-2e’c&/h(&&
Thus, by applying UnsSl’s over y:
-IV,’
approximation
(32)
we are able to integrate the formulas for EFA, and Eg\
a 1 I
0
(&
+ y’)(&l
+ y’)
dy=
_
_
"
2w@J&J,o+~so)
(33)
Final!y the expressions for these two contribtttions take forms which conveniently factor into an energy dependent part and a distance dependent part. Lettting b&, = E and h&, =hEb we have E:dp =A;~p(R)/(~,+=b),
(34)
with ei(k-k').R AE,UG=-$
JdkJdk’
2k k
Wct&k, k’)
and b-rEz, = “aA;:,(R)/=a,
(35)
with ei(k-k’kR
“aAg,(R)
= - $
1,
J d-4:J dk’ 2 k k,
b-=W&k, k’).
P. &fQlinowski er al. / I-k-Ffe
hreracrion
porential
429
The integrands in the expressions for the A”’ quantities are: W,,,(k,
k’) =F,(k’-
-Nb’Fb(k)
k) Fb(k - k’) -A’;‘&(-k)F,(k’)F&
Fb(-k’) F,(k’-k)+N;‘N;‘F,<-k)
-k’)
F,(k’) F,(k) Fb(-k’)
(36)
and ““W,,(k,
k’) = F,(k’-
k) F,(k) Fb(-k’) -f&?a(k’-
+N;F,(k’-
k)-N~F,(-k)F,(k’)IrY,+NbFa(-k)
+f=v,F,(-k)
F,(k’) F,,(-k’)/N,-F,(-k)
k) &(k) - Nd=a(k’- k) a’=,(-k? Fa(k’)Fb(k)/N,
F,(k’) Fb(k) Fb(-k’)/&
(37)
In the case of two identical interacting molecules the integrands defined by eqs. (36) and 37) take simpIer forms: Wd;s,(k,k’) =F(k -k’) F(k’-k)-F(k) -F(k -k’) F(-k) “‘“W&k;
F(k’)/N+F(k)
K)=N’F(K-
F(-k’) F(-k)
F(k’-k)lN
F(k’) F(-k’)/N’,
k)-~(K)F(-k)-NF(k)F(K-
(38) k);F(k)F(K)F(-k)-NF(-K)F~K-
tF(-k’)F(k’)F(-k);F(-k’)F(k)F(k)(k’-k)-F(-k’)F(k)F(k’)F(-k)/N.
k) (39)
The total second order Coulomb energy between identical systems will take the form (in atomic units) EgAul= (2m-‘(A$$
i40)
t2A:$),
where A’” E PO1
acbAf;I+b+a&2)
(41)
POI.
3. Application to the helium-helium interaction In order to use eqs. (34) and (35) for obtaining the second-order dispersion and polarization contributions to the total interaction energy between two helium atoms we must construct the quantities F(k) defined in eq. (28). TO do SO, we need to obtain the first order density matrix v(r]r). In our calculations we have used functions of the general form Y(L2)
= 2-“2$h)Q(rd[aO)
p(2)--P(l)
a(2)!.
(42)
The 4’s are expressed as sums of normalized Slater ls-orbit& Q(rl)=;
CiSi(Ji);
Si( r) = IV; e-“~‘Yao(f)
(43)
Making use of this function we get the Fourier transform of the first order density matrix: F(k) = 4 C CiCj i.i
NiNi(qi + vi! k21p
[(vi + vi)’ +
The integrals needed in eq. (40) have been obtained analytically [9]; the definitions are presented in the appendix. We performed the calculations using: (a) the double-zeta function reported by Ciementi [lo], (b) the best analytical Hartree-Fock function presented by Clementi and Roetti [ll].
P.Malinowski et al. / He-He irireracrim pofenriai
430
4. Numerical
results and discussion
We performed
our calculations
of the interaction
energy of two helium atoms at distances from 3 to
15 au. In addition to the calculation of the second-order dispersion and polarization energies based on the formalism presented in section 2 we have also obtained the first-order ener,7 using a standard technique 4.1.
(see section 4.2).
The second-order
energy results
EL&, and E$, have been calculated using the helium functions mentioned in section 3. In Table 1 we present the vatues of the auxiliary integraIs (defined in the appendix) for all terms which appear in the integrals of Wdi,p(k,k’) and W&k, k’) caIcuIated with the j-term Clementi-Roetti function [ll]. For distances greater than 5 au the doubIe-zeta function gives very similar results. Because the value of E is arbitrary, we prefer to deal with the “A” quantities of eqs. (34) and (35). To make some comparison of those quantities with other calculations we can expand the dispersion term in the following
multipole
expansion
valid for large R:
A;~:,=~~;~/R~~C~IR~~C~~~R'~~....
where cE = 2ECfi,
and the C, are the usual multipole
EL:;, = C-,/R6 +CC,/RS+ClofR’o+.
coefficients:
__ _
The en obtained --- from Alp
for R greater than 8 au are presented in table 2. The ratio of these coefficients is Cs/ CJC,O = l/9.54/121.66. The ratios C6/Cs/C10 obtained from the results of Maeder and Kutzelnigg [12] are l/9.54/121.64; those from Tang et al. [13] are l/9.53/115.51; and finally those of Luyckx et al. [14] are l/9.64/124.9. The agreement is obviously good. Below 8 au the limited expansion of our results is no longer highly accurate because higher order multipoles are neglected and because the charge densities begin to penetrate each other. Note that we have expanded only A:&, in a multipole series. As seen from tabie 3, beginning at 5.6 au (generally Table 1 Contributionsto the second-order interactionenergy for helium atoms calculatedfrom a Hartree-Fock function”. All quantities in atomic units * 6j2;rZ ;_j46 tJ4R E3;25;R A,&~P~ Ail/4-= 3 4 5 5.6 6 7 8 9 10 11 12 13 14 15
0.500895 0.266268 0.165986 0.131199 0.113824 0.083049 0.063314 0.049885 0.040327 0.033279 0.027934 0.023781 0.020492 0.017841
‘) Expressions
0.924636 0.513552 0.325504 0.258547 0.224813 0.164640 0.125797 0.099259 0.080322 0.066335 0.055710 0.047449 0.040899 0.035618
1.328320 0.999622 0.799972 0.714280 0.666665 0.571428 0.499999 0.444444 0.399999 0.363636 0.333333 0.307692 0.285714 0.266666
for the integais are given in the appendix.
0.666231 0.499976 0.339998 0.357143 0.333333 0.285714 0.249999 0.222222 O.lQQQQ9 0.181818 0.166666 0.153846 0.142857 0.133333
0.235161 0.128647 0.081393 0.064640 0.056204 0.041160 0.031449 0.024815 0.020080 0.016584 0.013927 0.011862 0.010225 0.008905
0.465674 0.257000 0.162766 0.t29276 0.112407 0.082320 0.062898 0.049629 0.040161 0.033167 0.027855 0.023725 0.020449 0.017809
P. Mdinowski
et al. / He-He interaction potential
431
Table 2 Coetiicients in the muhipole expansion of A$&,“’ Wavefunction
G
4
60
Double-E
3.751
35.792
456.393
Hartree-Fock
3.744
35.723
455.507
a’ See text. accepted as the position of the van der Waals minimum) the polarization contribution is about one percent of the dispersion term; for distances greater than 10 au, polarization is totally negligible. Though the relative shape of our potential is good at large R as discussed above, we must of course determine a value for B in order to obtain absolute values for our potential. A commonly cited choice for hE [15,16] proposed by London is the first ionization potential but it is recognized as leading to poor results [lfi]. Using the experimental ionization energy for He WC have BP = 0.91795 au, where the superscript “IP” denotes the method used to obtain the mean excitation energy. Another method for fixing -E involves the dipole polarizability tensor a(o) which can be calculated from the susceptibility [6]
a(o) = -
dr
I
I
dr’ r’ ,y(r, r’, w)r.
The closure approximation ,f(r,
r’,
0)
dr’
a(o) = Table 3 Values of Ag,, atomic units)“’
3 4 5 5.6 6 7 8 9 10 11 12 s3 14 15 ‘I The
to the coordiante
= [-20,~/(~3’no-02)][r(r)
and so the corresponding
A&
and A”’
0.50964 (-2) 0.27717 (-3) 0.15286 (-4) 0.26928 (-5) 0.84648 (-6) 0.46925 (-7) 0.26021(-g) 0.14430 (-9)
r’ jj(r,
(45)
Sk-r’)
polarizability r’,
w)
space susceptibility -T(r,
is (46)
r’)],
is
r.
(47)
as a function of internuclear separation R for two helium wavefunctions.
0.17424 (-1) 0.25226 (-2) 0.46921(-3) 0.20017 (-3) 0.12098 (-3) 0.41372 (-4) 0.17155 (-4) 0.80821(-5) 0.41700 (-5) 0.23060 (-5) 0.13477 (-5) 0.82432 (-6) 0.52378 (-6) 0.34381(-6)
0.27617 (-1) 0.30770 (-2) 0.49978 (-3) 0.20555 (-3) 0.12268 (-3) 0.41466 (-4) 0.17160 (-4) 0.80823 (-5) 0.41700 (-5) 0.23060 (-5) 0.13477 (-5) 0.82432 (-6) 0.52378 (-6) 0.34381(-6)
number in parenthesesis the exponent of ten.
0.51460 0.28203 0.16038 0.28969 0.92?63 0.54058 0.31610 0.18515
<1o-to
(-2) (-3) (-4) (-5) (-6) (-7) (-8) (-9)
0.17368 0.25267 0.47121 0.20103 0.12144 0.41467 0.17175 0.50875 0.41718 0.23066 0.13480 0.82444 0.52383 0.34383
(-1) (-2) (-3) (-3) (-3) (-4) (-4) (-5) (-5) (-5) (-5) (-6) (-6) (-6)
(A11 quantities are in
0.27660 (-1) 0.30908 (-2) 0.50329 (-3) 0.20682 (-3) 0.12330 (-3) 0.41575 (-4) 0.17181(-4) 0.80879 (-5) 0.41718 (-5) 0.23066 (-5) 0.13450 (-5) 0.82444 (-6) 0.52383 (-6) 0.34383 (-6)
P. Maiinowski et al. / He-He interaction potential
432
Assuming a single determinant wavefunction we simplify the two particle density function to obtain: z(r, r’, 0) = [-26,,/(0:,
-0~)][5(r
--c’j y(r) -4~~1
y(r’)].
(48)
Since the helium atom is spherically symmetric we calculate just the zz component of the polarizability tensor. (49) Using Slater type orbitals we find
For the Hartree-Fock wavefunction I=, = 0.790. We now choose zso that our static polarizability a=,(O) agrees with the experimental value, 1.383 au 1171. We thus obtain hE= 1.145, about 20% higher than EtP, which clearly shows that hELP is not adequate for the polarizability; nor is it adequate for the interaction energy. To see this we determine other values of z by requiring our interaction energy to agree at large R with other, absolute potentials i.e. E=
A’“(R)/2E&(R),
(51)
where A”‘~&?+2A’$,.
(52)
Note that %? can now be- considered a variable, R-dependent scaling factor. In table 4 we present 4.E values obtained frcm MCSCF [18] CI [19,20], multipoie expansion (ME) [X2,13] - type calculations and from the Burgmans et al. [2i] potential which is a synthesis of
Table 4 Mean excitation energies from different Type of talc.
CI
types of calcuIations”“-c’ IMCSCF
ME
Liu and
McLaughlin
Bertoncini
Tang
McLean Cl91
and Schaefer 1201
and Wahl [W
et al. [I31
R
z-E
15 14 13 12 11 IO 9
1.34 -
8 7 6
1.37 I.51 2.25
Experimental
Maeder and Kutzelnigg WI
Burgmans et al. [Zl]
KE-=
_ _ 1.23 1.24 1.27 1.37 1.90
1.25 1.25 1.29 1.41 2.04
=’ See eq. (51) of text. ‘I All quantities in atomic units. =’ A”’ in eq. (51) was calculated from the Hartree-Fock fmxtion.
1.27 1.27 1.27 1.27 1.28
1.28
1.28 1.29 1.30 1.33 1.42
1.29 1.30 1.31 1.35 1.44
1.28 1.28 1.29 1.29
I.28 1.28 1.28
1.28 1.29 1.29 1.29 1.32
1.47 2.06
433
P. Malinowski et al. / He-He ixteractinn potential
accurate experimental and theoretical results. Notice that although these six AE values are different they are almost constant within the range of R between 15 and 8 au. For ? au there is a visible increase in these values and for 6 au there is generally a great jump in the magnitude of z. Because of the almost identical values of A~I and A&, for the double-zeta function (see table 2) the mean excitation energies for this function are the same. We now enquire whether the variation in E is due to the inadequacy of the closure approximation, to overlap inter- and intra-corre!ation effects, or short range Coulomb interactions. We do so by comparison with_reliable results for the total interaction potential; we thus need to estimate the total inte-radian energy by augmenting our second order results with first order Coulomb and exchange energies. 4.2.
The first-order energy conrribution.
The total interaction
energy
We have calculated the first-order Coulomb and exchange energies using the Murrell-ShawMusher-Amos [22] perturbation theory. The results are presented in table 5. Because of almost identical results for the double zeta and Hartree-Fock functions we may say that the quality of both functions is the same. Therefore we will limit our discussion to the Hartree-Fock results. We modify (51) to accommodate our first order results. =
= A”‘/2[&
(53)
- &“I,
where
One observes in table 6 that now the AE values are less variable for R > 5.6 au. The changes of and pxp for R > 5.6 au are within 24, 20, 20 and 21% respectively. hEz’, -KE, LPcsCF These deviations of a from a constant value stress that for distances less than 7 au either the Unssld approximation breaks down, or overlap and correlation effects become important or the potential is not well approximated by first and second order perturbation theory. In order to investigate those possibilities let us analyze the results collected in table 7. Our total interaction energy listed there was obtained by adding the first-order and second-order contributions with an assumed fixed AE of 1.28 au. One sees that our predicted minimum is too deep and occurs at too short a distance. Also the repulsive part of the potential curve is not steep enough. Obviously in the
Table 5 First-order Coulomb, atomic units) *’
3 4 5 5.6 6 7 8 9
0.33803 (-2) 0.28104 (-3) 0.21900 (-4) 0.46164 (-5) 0.16209 (-5) 0.11551(-6) 0.79976 (-8) 0.54135 (-9)
exchange and total energies for two helium atoms calculated from two wavefunctions.
0.18159 0.16929 0.14436 0.31859 0.11498 0.86943 0.63139 0.44384
(-1) (-2) (-3) (-4) (-4) (-6) (-7) (-8)
0.14779 (-1) 0.14119 (-2) 0.12246 (-3) 0.27243 (-4) 0.98769 (-5) 0.75392 (-6) 0.55142 (-7) 0.38971(-9)
0.33698 (-2) 0.28358 (-3) 0.2X41(-4) 0.48543 (-5) 0.17324 (-5) 0.12841(-6) 0.92882 (-8) 0.65884 (-9)
0.17758 (-1) 0.16959 (-2) 0.14997 (-3) 0.33965 t-4) 0.12488 (-4) 0.99261(-6) 0.76012 (-7) 0.56463 (-8)
(All quntities
0.14388 0.14124 0.12733 0.29100 0.10755 0.86420 0.66724 0.49875
(-1) (-2) (-3) (-4) (-4) (-6) (-7) (-8)
ir.
P. Mafinowski er al. / He-He inferaction poremid
434 Table 6 Corrected
mean excitation en:rgies.
(~11
quantitiesin atomicunitsFb! MCSCF
B&on&i and
Experimental Burgmans
Wahl [lS]
et al. [21]
~ICSCF
~jg.“’
1.30
1.33
1.38
i.43 1.52 i.76
1.50 1-S 1.97
1.52 1.63 1.94
T>TJe of talc.
Cl Liu and McLean [19]
McLaughlin and Schaefer [20]
R
z?’
ssc
7
1.43
5.6 5
1.61 1.76 2.13
6
=’ See table 4.
” Calculatrd u&g
2q. (53) of text.
region of overlap either the second order contribution is too negative or the first-order contribution is not positive enough, or both. Our first-order results differ from the Murrell and Shaw [23) and Claverie [24] values by about 10%) and from the Murrell and Varandas [25] values by 6%. Correlation can change the first-order energy by approximately 10% [26]; we thus conclude that our first-order energy is accurate to within 20%. This inference seems to be confirmed by the recent studies of Szalewicz and Jeziorski [27]. That possible error in our first-order calculation obviously cannot account for the total discrepancy in the interaction potential. In recent years it has been found that neglect of the penetration of the interacting charge distributions at small distances leads to divergence of the usual multipole expansion of Es&,. A more complete calculation yields a finite dispersion energy at all separations and SO damping functions have been introduced to compensate for the overly large negative values of Eiz, obtained from the multipole expansion [se, 5f, 281. Since we use the full Coulomb potentiai, there is no justification for introducing those damping functions in our work. However, we note that if we use a R-dependent Table 7 Total He-He R
3 4 5 5.6 6 7 8 9 10 11 12 13 14 15
potential. (All quantities in atomic units)“’
This work”
0.35892 (-2) 0.20573 (-3) -0.69159 (-4) -0.51646 (-4) -O-37381(-4) -0.15367 (-4) -5.66411 (-5) -0.31526 (-5) -0.16287 (-5) -0.90052 (-6) -0.52627 (-6) -0.32187 (-6) -0.20451 (-6) -0.13424 (-6)
‘) See Table 2.
“B=
1.28.
McLaughlin and
Bertoncini and
Maeder and
Schaefer [20]
Wahl [IS]
Kutzelnigg [12]
0.11610 (-1) 0.8611 I(-3) -0.11147 (-4) -0.38002 (-4) -0.32079 (-4) -0.15106 (-4) -0.67769 (-5) -0.32586 (-5) -0.17006 (-5) -
0.3930 -0.3604 -0.2993 -0.1472 -0.6620 -0.3210 -0.1660
(-5) (-4) (-4) (-4) (-5) (-5) (-5)
-0.15423 (-4) -0.65516 (-5) -0.31153!-5) -0.16137 (-5) -0.89411(-6) -0.52313 (-6) -0.32018 (-6) -0.20353 (-6) -0.13364 (-6)
Burgmans et al. [Zl] 0.12708 0.87222 0.19566 -0.33466 -0.29661 -0.14094 -0.64703 -0.31324 -0.16220 -0.89852 -0.52560 -0.32165 -0.20445 -0.13423
(-1) (-3) (-5) (-4) (-4) (-4) (-5) t-5) (-5) (-6) (-6) (-6) (-6) (-6)
I’, Malinowski er a!. / lie-He
inreracrion porenrial
433
mean excitation energy instead of a fixed one, we can get greatly improved rest&s. The notion of a R-dependent mean excitation energy was first introduced by Dalgarno and Lynn [29] who proposed a mean excitation energy of the following form (see also Kreek and Meath [30]) z(R)
(SSj
= AE(m) + CS(R),
where D(co) is the mean excitation energy at infinite separation, S(R) the overlap integral and C a constant determined empirically by fitting the second order energy to the difference between the experimental [21] and first order energy at a particular R, say R = 5.6 au (the equilibrium distance). We find that for He-He E(R)
= 1.28075 +52.0340
S(R).
(56) Table 8 lists the modified potential energy and M(R) obtained by using (56). The results are in excellent agreement with the experimental values of Burgmans et al. [21] as shown in table 7. Comparison of the calculated and experimental curves is shown in fig. 2. Throughout this paper we have applied the Unsold approximation to the charge-density susceptibility. We now proceed to show that this is equivalent to applying the Unsijld approximation directly to the dispersion and polarization energies. To do SO we switch to r-space and using eqs. (13), (14a), (14b), (15a), (lSb), (9) and (7) we obtain (see also ref. [6] section 5)
(57) The first term is the dispersion energy and the last two terms are the polarization energies. Eq. (57) is the formulation for the second order Coulomb energy obtained by Longuet-Higgins [31] on the basis of ordinary time-independent perturbation theory. It is easy to see that by writing the second order Coulomb energy in this fashion the Unsold approximation can be applied separately to each of the terms and when this is done the mean excitation energies for the three terms are respectively Table 8 Modified He-He R 3 4 5 5.6 6 7 s 9
10 11 12 13 14
I5
potential, z(R)
and overlap integral S(R)
S(R)
en*
s(R)”
1.208 C-1) 4.236 (-2) 1.414 (-2) 7.151(-3) 4.507 (-3) I.391 (-3) 4.189 (-4) 1.236 (-4) 3.583 (-5) 1.025 (-5) 2.894 (-6) 8.087 (-7)
1.256 (-2) 9.703 (-4) 2.543 (-6) -3.347 (-5) -2.993 (-5) -1.450 (-5) -6.529 (-6) -3.137 (-6) -1.626 (-6) -9.001(-7) -5.262 (-7) -3.219 (-7)
7.566 3.495 2.017 1.653 1.5 16 1.353 1.303 1.287 1.283 1.282 1.281 1.281
2.239 (-7) 6.151(-S)
” ~(R)=l.28075+52.0340
-2.045 (-7) -1.342 (-7)
S(R).
1.281 I.?81
P. Malinowki
436
er at. / He-He Weracrion porenrial
-e 3-
I--_I3 4
5
6
7 R
8
* 9
:b
I,
I !2
Fig. 2. He-He total potentials. Solid curve: the ESMMSV parametric potential of ref. [21]. Dashed curve: thii work with E(R)=1.28075+52.0340 S(R), where S(R) is the overlap integral. Dotted curve: this work with E=l.ZS.
h(&o+&o), hi& and AL&..~,which are the same as in the susceptibility formulation. If the Unsijld approximation is applied directly to the second order Coulomb energy without first decomposing it, as is often done, then a single mean excitation energy is obtained which is characteristic of dispersion but inappropriate for polarization. Although we have dwelt at some length on the first order energy and total potential, our main goal and interest has been the development of reliable techniques for treating the second order Coulomb ener,q. The approximation which uses static form factors appears to give a good description of the He-He long range potential. Although we have used analytic expressions for the static form factors and for the integrals involved in eqs. (34) and (35), the same could have been accomplished by using numerical values of F(k) [32]. Indeed the numerical approach becomes more apppealing the more complex the system is. A purely numericai approach circumvents the use of wavefunctions and is independent of the size of the system and the so&e of the scattering functions.
Acknowledgement The authors are grateful to Professor William J. Meath for valuable discussions and helpful suggestions.
Appendix In this appendii we list the integrals needed in the evaluation of the quantities Asp and A$ eqs. (34) and (35)]. Anaiytic evaluation of the integrals will be presented elsewhere [9]-
[see
437
Integrals for A'?' Am
(I).
Nk-L’PR
A&?)’
J
dk
We -g-pT
J
F(R)
F(4)
F(C)
F(-k’).
This integral factors -4_,G3
=9(R)
B(-R),
where S-R
B(R)=
dk+F(k)
I
F(-k).
The other two integrals needed for A;& are: Sk-k’)-R A’s=
J
dk
J
r&----
k2k,z
F(k
-k’)
F(k’-
k)
and i(k-k9.R
A&R)
=
I
dk
I
dk’ +(k
-k’) F(-k)
F(k’).
2. Integralsfor AZ1 Besides ALF and ALL defined above, the following integrals are needed for AZ,:
A similar integral arises with k replaced by -k’ and k’ replaced by -k in the arguments of F. The next integral is: r(R)-IdkJdk’GF(k’-k).
Another integral which factors is i[k-k’1.R
J
dk
J
F(-k)
dk’+
F(k’) = b(R) 6(-R),
where ik*R b(Rf+k+
F(k).
Finally i(k -k’)_R
J
dk
J
dk’+
F(-k)
F(k) F(k’) = b(-R)B(R).
A similar integral arises which is the product b(R)B(-R). of R for Am, As, B, b, 5 and r.
In iable 1 values are given at several values
References [!I [2] [3] [Ij
F. London. Z. Physik, Chem. Bll (1930) 222. A. Uns6ld. Z. Phgsik 43 (1927) 563. B. Lindrr, K.F. Lee. P. Maiinowski and A.C. Tanner, Chem. Phys. 52 11980) 353. (ai J. H. Hubbell. N’.J. \‘cig&. E.A. Briggs. R. T. Brown, D.T. Cramer and R.J. Howerton, J. Phys. Chem. Ref. Data 4 t19751 Gil: tbj J.H. Hubbell and I. Ovsrbo. J. Phys. Chem. Ref. Data 8 (1979) 69; (cJ K.F. Lee, unpubIished resuIts. is] (ai N. Jacobi and G. Csanak, Chem. Phys. Letters 30 (1975) 367; fib) P.W. Langhot?. Chem. Phys. Letters 20 (1973) 33; ic) T.B. SIacRury and 5. Linder, J. Chem. Phys. 58 (1973) 5388; Id! R. Boehm and R. Saris. J. Chem. Phys. 55 (1975) 2620: (e) .4. Koidr, J. Phys. B9 (19761 3173; (i) A. Koide, W. J. hleath and A. R. Allnart, Chem. Phys. 38 (19813 105. [6] B. Linder and D. A. Rabenold, Advan. Quantum Chem. 6 (1972) 203. [7] R. van Bergen, Y. van Bergen and B. Linder, J. Chem. Phys. 67 (1977) 3878. [P] P.O. L;iwdin. Phys. Rev. 97 (1955) 1471. [9] P. Malinowski, A.C. Tnnner, K.F. Lee and B. Linder. Int. J. Quantum Chem., to be published. [IO] E. Clementi. IBM J. Res. Dev. ISuppl) 9 (1965) 2. [ll] E. Clrmenti and C. Roetti, At. Data Sucl. Data Tables 14 (197-1) 177. j12] F. ,Maeder and W. Kutzelnigg, Chem. Phys. 42 (1979) 95. [13] K.T. Tang. J.!& Norbeck and P.R. Certain, J. Chem. Phys. 64 (1976) 3063. [la! R. Luyckx. P. Coulon. and H.N.W. Lekkerkerker. J. Chem. Phys. 69 (1978) 2424. 1151 M.J. Huron and P. Claverie, Chem. Phys. Letters 4 (1969) 429. [16] A. Datgarno and W.D. Davison, Advan. At. Mol. Phys. 2 (1966) 1. [17] P.W. Langhoff and M. Karplus, J. Opt. Sot. Am. 59 (1969) 863. [18] P. Bertoncini and A.C. Wahl, Phys. Rev. Letters 25 (1970) 991. [19] B. Liu and A.D. McLean, J. Chem. Php. 59 (1973) 4557. [20] D.R. McLaughiin and H.F. Schaefer III, Chem. Phys. Letters 12 (1971) 244. [21] A.L.J. Burgmans, J.M. Farrar, and Y. T. Lee, J. Chem. Phys. 64 (1976) 1345. [t2] (a) J. N. Xfurrell and G. Shaw, J. Chem. Phys. 46 (1967) 1768; (bj I.J. Musher and A.T. Amos, Phys. Rev. 164 (1967) 31. [Z3] J.N. Murrell and G. Shaw, Mol. Phys. 12 (1967) 375. [13] F. Clavrrie, in: IntermoIecuIar interactions: from diaromics to biopolymers, ed. B. Pullman (Wiley and Sons. New York, 1978). [X] J.N. ~Murrelf aEd A.J.C. Varandas, Mol. Phys. 30 (1975) 223. [X] A. Conway and J.N. Munell, Mol. Phys. 23 (1972) 1143. [27] K. Szalewia and Et. Jeziorsk:, Mol. Phys. 38 (1979) 191. [2S] (a) M. Krauss, D.B. Neumann, and W.J. Stevens, Chem. Phys. Letters 66 (1979) 29; (b) M. Krauss, W.J. Stevens, D.9. Neumann, Chem. Phys. Letters 71 (1980) 500. [29] A. Dalgarno and N. Lynn, Proc. Phys. Sot. A69 (19.56) 821. [30] H. Kreek and W.J. Meath. Mol. Phys. 22 (1971) 915. [31] H.C. Longuet-Higgins, Proc. Roy. Sot. (London) A235 (1956) 537. [32] K.F. Lee, Ph.D. Dissertation, Florida State University (1981).
VAN DER WAALS IFORCES, SCATTERING FUNCTIONS AND CHARGE DENSITY SUSCEPTIBILITY. II. APBLHCATION TO THE He-He INTERACTION POTENTIAL+ P. MALINOWSKI*, of Chnistry,
Department
A. C. TANNER, Florida State Unicerrity,
K.F. LEE and B. LINDER Taliahassee. Florida 32306,
USA
Received 2 April 198I
Previous work relating the dispersion energy to the static form factorsof X-ray scatrering is extended to the pokization energy. For interacting heliumatomswe calculateboth those contributions(krhout recourse to the mulripok expansion), as well as the first order Coulomb and exchange forces. For large separations (R as.0 au), the shape of our potential agrees well with previous. accurate results. Since we use Uns6ld’s approximation, we discuss several choices for the mean excitation energy, s, which is required to put our potential on an absolute scale. A fixed %? suffices for large separations. In the region of overlap, a R-dependent E(R) is needed for accurate determination of the potential.
4. Introduction The calculation of long range intermolecular forces is often treated by second order perturbation theory. Such a calculation must confront the usual difficulty of evaluating the infinite summation and integration over discrete and continuum states. An early method for performing those summations in studies of intermolecular forces was London’s [l]. He used U&M’s [2] method of approximately summing the second order perturbation energy. By also approximating the Coulomb potential by the dipole term alone, London related the dispersion energy to the static polarizability LY,a quantity available from experiment. Despite the clever simplicity and accuracy of his method, it is a dipole approximation and so is limited to small systems. In this paper we continue our earlier work [3] the aim of which is the development of more widely applicable computational methods which nevertheless retain the spirit of London’s theory, namely the use of readily available data to calcuIate long range intermolecular potentials in a simple way. The method used is based on the response-perturbation technique of intermolecular forces. Of importance in this treatment is the charge-density susceptibility, in terms of which the second and higher order terms may be developed. The susceptibility takes account of spatial and temporal correlations; the ordinary polarizability a is independent of spatial variables and may be obtained as a special case from the charge-density susceptibility. The formulation of the interaction energy in terms of the stisceptibility uses the full Coulomb potential, thus avoiding the disadvantages of the multipole expansion, and in particular the limited precision of the dipole approximation. It was shown earlier [3] that the charge-density susceptibility in momentum space is related to generaliied scattering functions and an approximation to the susceptibility is expressible in terms of static form factors. In the case of He it is easy enough to evaluate analytically these form factors and
t Supported by the Natior.aI Institute of Fe&h, Grant No. GM23223. * Permanent address: Institute of Physics, Nicholas Copernicus Univenity,
0301-0104/81/0000-0000/$02.75
@ 1981 North-Holland
87-100 Tort&
Poland.
P. hlalimowski ct al. f He-He interaction potential
424
the interaction energy from self-consistent field wavefunctions and this was done here for convenience. A prescription for evaluating the interaction energy from readily available computed and experimental form factors [4] will be given elsewhere. We calculate the second order dispersion and polarization energies of two He atoms by applying the UnsSId approximation to the susceptibilities. This has the effect of introducing a mean excitation energy for the dispersion energy which is twice that for the polarization energy. It is shown in section 4.2 that the same result can be obtained by applying the Unsiild approximation directly to the energy provided that the energy is first decomposed into its dispersion and polarization parts and the Unsiild approximation is applied to each term separately. Although the polarization energy is small, it is not negtigible at short distances, an observation surprising perhaps to those accustomed only to the dipole approximation for which the polarization energy is zero. We also calcuiate the total potential energy by adding the first order Coulomb and exchange energies to the second order energy. The susceptibility formulation of the problem provides a direct link between intermolecular potentials and scattering phenomena. The development in momentum space allows considerable mathematical simplification and other authors have used the same technique in similar calculations
PI. 2. Outline of the method 2.1.
The second order interaction
As mentioned
where: and n p,,(r) and
energy ir, terms of the susceptibility
in the introduction
m, n are the states
the system
of the system,
p,,_(r)
is characterized
is the transition
by the susceptibility
charge
densi&
[6]
between
states
m
= (mHr)ln);
(2)
p’(rj is the charge density operator,
p^(r)=e
g ZJ(r--r,)-e v=l
defined
as
f S(r-q), ;=1
(3)
in which e is the absolute value of the electron charge, A is the number of nuclei, N is the number electrons and 2, is the atomic number of the uth atom. w”,,, is connected with the unperturbed energies E,, and E,,, of the system by =E,,-E,,
hGn o
of
(4)
is the frequency of the applied potential and cr,, is a matrix element of the statistical operator 6:
e = exp (-PIkjlQ, where fz, is the hamiltonian or partition function defined
Q = 2 exp (-BE). I
(5) of the unperturbed as
system,
p the Boltzmann
factor
and Q the state sum
(6)
et al. / He-He itzreractioft potential
P. hfalinowski
425
The Et are the energies of the unperturbed system. Such susceptibilities can be utilized to calculate the free energy of the system. In this paper we will use these quantities to obtain the second order dispersion and polarization energies. Let us consider molecule “a” characterized by susceptibility x.&i, r,, W) and molecule “b” by Xb(rzr r;, w). Let these molecules interact via the Coulomb potential V(rl, rz) =
(7)
Ir2-r:l-‘.
As described in the previous paper [6] the second order dispersion energy can be expressed as Ej;:,(T)=-(h/4;i)ReiSdr,Idr;
f
do ,y&:,
rl, w)Kf’
i8)
(rI, r\, o) coth (h&/2).
-33 The quantity Kf’(r,,
Kg’ (rl, r;,
r;, O) is the real part of the reaction potential
dr$ V(ri, rz) X&2,&
o) =
susceptibility
w) V(rS, 4).
This expression for the second order dispersion energy describes the interaction of two fluctuating charge distributions. The charge distribution on molecule “a” can also polarize molecule “b”, giving rise to the poIarization energy? (10) The charge distribution on molecule “b” gives a similar contribution polarization energy
to the second order
(11) The total polarization energy is of course the sum of these two contributions E$, (T) = “‘bE~,(T)+b-=E~*(T).
(12)
These formulas for the second order dispersion and polarization contributions are however very difficult to evaluate in concrete calculations due to the infinite summations, including the continuum. In practice we consider only T = 0 which enables us to put n = 0 and s = 0 in eqs. (10) and (ll), and to simplify (8) with lim coth (h&~/2) = 1, 1-O
and
lim qnnn= lim a,, = 1. T-0
T-d
Our key quantities can be written as follows E$:,(O)=E~&,
= -(A/45i) Re i
(13) -m
t This, in fact, is the induction energy. The terms polarization energy and induction energy are often used interchangeably.
previous publication.ref. [71. a distinction was made between the two. There the term polarization potential was used to denote an auxiliary function, which when multiplied by l/2 and averaged over the rotational states yielded the induction energy (see, for example, eqs. (25) and (32) of ref. [7]). In the absence of rotational states the induction energy reduces to expression (10) of the present paper. We shall refer to this expression as polarization energy.
In a
(154
(15b) Let US fOcUS OUTattention on the dispersion energy of the two interacting molecules. The System of coordinates used in this paper is show in fig. 1. This choice of coordinate system centred on molecule “a” enables us to rewrite the formula for EyLP as
E~~,=-ih/d;r)ReiIdr,Jdr:Idr25dr;
i
dw x,(6,
rr. 0)
Vh,
r?)
xb(rZ-R,
ri
-i?,
co)
V(r&
r;),
--z (W
where r, and r; now refer to the common center on --a”, and R is the vector from “a” to “b”. It is Useful to define susceptibilities that are Fourier transformed to momentum space (k-Space) x&;,
rl,
w)
= II-’
5
dkl
J
dk2 e-ik;‘r; x&C:, kl, O) eikl-rl
(17)
and
where R = (2rr)‘. Substitution of (17) and (1X) into eq. (16) and integration
Thus by going to momentum
space one has reduced
over rr, r;, rz and r; yields
drastically
the dimension
Fig. 1. Systemof coordinates.
of integration.
P. Malinowski et al. j He-He inferaction potential
Making use of eqs. (15a) and (1%) and the definition of the Fourier transformation (18)] we can express the susceptibilities in momentum space P~~(k’)P:0(-k) o -m,o-ib
x.(k, k’, 0) = Fr$ fi-’ ;(
_ &,(-k)PL(k’) o+o,o-ib
427
[eqs. (17) and
I’
(20)
where P&(k) = (016”(k)ln~,
(21)
b”(k) = I em”“p^“(r)dr.
(22)
The susceptibility &k, k’, W) can be expressed similarly in terms of transition form factors p&(k). In order to simplify the expression for the second order energy in momentum space let us express the susceptibilities along the negative imaginary frequency axis. Assuming that the functions characterizing both molecules are reaI, one can rewrite eq. (20) as Eo!lows ~~(k’, k, -iy) = -2fi-’
1 +%p&(k’)p~o(-k). n On0 +y-
(23)
Thus the second order dispersion energy can be expressed E$?;, = - 185r5 jdyIdkjdk’~~,(k.,k-iY)Xe(%L.,iy).
(24)
0
Following a similar procedure,
one can rewrite expressions for the second order polarization energy as &k’)
'+b&d,_-
&A--k)
X&r
k’, 01,
(25a)
85r4
b-ilEg,
2.2.
=
&
,I J dk
e”k-k”‘R b pm(k) &A-k’) xdk’, k, 0).
dk,
kzk,z
(25b)
UnsGId’s approximation
In order to nse eqs. (24) and (25) we apply Unsiild’s approximzdon [2] to the expression for the susceptibilities, since it is the simpiest and most straightforward approximation. Thus we replace on0 and oSO by mean excitation energies C&Oand &a in the expressions for susceptibilities x1 and xb. After this substitution we add and subtract the terms with n = 0 and s = 0 respectively. Thus we obtain xz(k’,
k, -iy)
=
-2fi-’
X,,( k, k’, -iy) = -2h-’
on0
7 WW(k’G”(-k)lO) C%;o+y-
+%
us0 + y-
- ~Olp^“~k’~lO>~Olp^“~-k)(O)},
~~Ol~b~k~~b~-k’~lO~-~OI~b~k~lO~~Ol~b~-k’~lO~~.
Making use of the definition of the charge-density operators, eq. (3), we obtain for atom “a” the matrix elements involved in eqs. (24), (25a) and (25b)
(26a)
t26b)
428
P. Malinowski et af./ He-He inreracrion porenzial
_N,(o]
3
e-ik~rj~)+(~l
?
;=*
eik”’
3 eiW’-k)-,lo)),
e-‘k-,lo)+(q
i#j
(27)
i-l
These matrix elements can be expressed in terms of one- and two-particle coordinate space [S],
density matrices in
(01 z e”“f\O) = N,(Oleik“‘[O) = J dr eik’-r-&r)=F,(k’),
(28)
i=1
(o/$
p-r, e-ik-rilO)= N,(N,-
l)(Ol J*‘+ e-ik-rIO)= 2 J dr J dr’ eiL’+ r,(r, r’lr, r*) e-ik.r ~~=(kl,
k).
i*i
(29)
The function Fa(k’) is simply the Fourier transform of the one particle density matrix ~~(r]r). The quantity L,(k, k’) is the Fourier transform of the two-particle density matrix Pa(r, r’jr, r’) of molecule “a” and for totally uncosrelated functions (e.g. a simple Hartree product): ) F,(-k’)
L,(k, k’) = (1 -N,’
F,(k).
(30)
For a single determinant two electron function eq. (30) also holds. We restrict ourselves to approximation (30). Putting (28) and (29) into (27) we obtain (ojp^a(k’) ,5”(-k)]O) = e’[N: -N,Fa(k’) -N,F,(-k)+(l Using eq. (31) and eq. (28) the susceptibility expressed as: x,(k’,
)~~(-k)F,(k’)+F,(k’-k)}.
for molecule “a” in Unsiild’s approximation
(31) can be
+y’)X~H(k’-k)-N,‘F,(k’)F,(-k)}.
k,-iy) =[-2e’c&/h(&&
Thus, by applying UnsSl’s over y:
-IV,’
approximation
(32)
we are able to integrate the formulas for EFA, and Eg\
a 1 I
0
(&
+ y’)(&l
+ y’)
dy=
_
_
"
2w@J&J,o+~so)
(33)
Final!y the expressions for these two contribtttions take forms which conveniently factor into an energy dependent part and a distance dependent part. Lettting b&, = E and h&, =hEb we have E:dp =A;~p(R)/(~,+=b),
(34)
with ei(k-k').R AE,UG=-$
JdkJdk’
2k k
Wct&k, k’)
and b-rEz, = “aA;:,(R)/=a,
(35)
with ei(k-k’kR
“aAg,(R)
= - $
1,
J d-4:J dk’ 2 k k,
b-=W&k, k’).
P. &fQlinowski er al. / I-k-Ffe
hreracrion
porential
429
The integrands in the expressions for the A”’ quantities are: W,,,(k,
k’) =F,(k’-
-Nb’Fb(k)
k) Fb(k - k’) -A’;‘&(-k)F,(k’)F&
Fb(-k’) F,(k’-k)+N;‘N;‘F,<-k)
-k’)
F,(k’) F,(k) Fb(-k’)
(36)
and ““W,,(k,
k’) = F,(k’-
k) F,(k) Fb(-k’) -f&?a(k’-
+N;F,(k’-
k)-N~F,(-k)F,(k’)IrY,+NbFa(-k)
+f=v,F,(-k)
F,(k’) F,,(-k’)/N,-F,(-k)
k) &(k) - Nd=a(k’- k) a’=,(-k? Fa(k’)Fb(k)/N,
F,(k’) Fb(k) Fb(-k’)/&
(37)
In the case of two identical interacting molecules the integrands defined by eqs. (36) and 37) take simpIer forms: Wd;s,(k,k’) =F(k -k’) F(k’-k)-F(k) -F(k -k’) F(-k) “‘“W&k;
F(k’)/N+F(k)
K)=N’F(K-
F(-k’) F(-k)
F(k’-k)lN
F(k’) F(-k’)/N’,
k)-~(K)F(-k)-NF(k)F(K-
(38) k);F(k)F(K)F(-k)-NF(-K)F~K-
tF(-k’)F(k’)F(-k);F(-k’)F(k)F(k)(k’-k)-F(-k’)F(k)F(k’)F(-k)/N.
k) (39)
The total second order Coulomb energy between identical systems will take the form (in atomic units) EgAul= (2m-‘(A$$
i40)
t2A:$),
where A’” E PO1
acbAf;I+b+a&2)
(41)
POI.
3. Application to the helium-helium interaction In order to use eqs. (34) and (35) for obtaining the second-order dispersion and polarization contributions to the total interaction energy between two helium atoms we must construct the quantities F(k) defined in eq. (28). TO do SO, we need to obtain the first order density matrix v(r]r). In our calculations we have used functions of the general form Y(L2)
= 2-“2$h)Q(rd[aO)
p(2)--P(l)
a(2)!.
(42)
The 4’s are expressed as sums of normalized Slater ls-orbit& Q(rl)=;
CiSi(Ji);
Si( r) = IV; e-“~‘Yao(f)
(43)
Making use of this function we get the Fourier transform of the first order density matrix: F(k) = 4 C CiCj i.i
NiNi(qi + vi! k21p
[(vi + vi)’ +
The integrals needed in eq. (40) have been obtained analytically [9]; the definitions are presented in the appendix. We performed the calculations using: (a) the double-zeta function reported by Ciementi [lo], (b) the best analytical Hartree-Fock function presented by Clementi and Roetti [ll].
P.Malinowski et al. / He-He irireracrim pofenriai
430
4. Numerical
results and discussion
We performed
our calculations
of the interaction
energy of two helium atoms at distances from 3 to
15 au. In addition to the calculation of the second-order dispersion and polarization energies based on the formalism presented in section 2 we have also obtained the first-order ener,7 using a standard technique 4.1.
(see section 4.2).
The second-order
energy results
EL&, and E$, have been calculated using the helium functions mentioned in section 3. In Table 1 we present the vatues of the auxiliary integraIs (defined in the appendix) for all terms which appear in the integrals of Wdi,p(k,k’) and W&k, k’) caIcuIated with the j-term Clementi-Roetti function [ll]. For distances greater than 5 au the doubIe-zeta function gives very similar results. Because the value of E is arbitrary, we prefer to deal with the “A” quantities of eqs. (34) and (35). To make some comparison of those quantities with other calculations we can expand the dispersion term in the following
multipole
expansion
valid for large R:
A;~:,=~~;~/R~~C~IR~~C~~~R'~~....
where cE = 2ECfi,
and the C, are the usual multipole
EL:;, = C-,/R6 +CC,/RS+ClofR’o+.
coefficients:
__ _
The en obtained --- from Alp
for R greater than 8 au are presented in table 2. The ratio of these coefficients is Cs/ CJC,O = l/9.54/121.66. The ratios C6/Cs/C10 obtained from the results of Maeder and Kutzelnigg [12] are l/9.54/121.64; those from Tang et al. [13] are l/9.53/115.51; and finally those of Luyckx et al. [14] are l/9.64/124.9. The agreement is obviously good. Below 8 au the limited expansion of our results is no longer highly accurate because higher order multipoles are neglected and because the charge densities begin to penetrate each other. Note that we have expanded only A:&, in a multipole series. As seen from tabie 3, beginning at 5.6 au (generally Table 1 Contributionsto the second-order interactionenergy for helium atoms calculatedfrom a Hartree-Fock function”. All quantities in atomic units * 6j2;rZ ;_j46 tJ4R E3;25;R A,&~P~ Ail/4-= 3 4 5 5.6 6 7 8 9 10 11 12 13 14 15
0.500895 0.266268 0.165986 0.131199 0.113824 0.083049 0.063314 0.049885 0.040327 0.033279 0.027934 0.023781 0.020492 0.017841
‘) Expressions
0.924636 0.513552 0.325504 0.258547 0.224813 0.164640 0.125797 0.099259 0.080322 0.066335 0.055710 0.047449 0.040899 0.035618
1.328320 0.999622 0.799972 0.714280 0.666665 0.571428 0.499999 0.444444 0.399999 0.363636 0.333333 0.307692 0.285714 0.266666
for the integais are given in the appendix.
0.666231 0.499976 0.339998 0.357143 0.333333 0.285714 0.249999 0.222222 O.lQQQQ9 0.181818 0.166666 0.153846 0.142857 0.133333
0.235161 0.128647 0.081393 0.064640 0.056204 0.041160 0.031449 0.024815 0.020080 0.016584 0.013927 0.011862 0.010225 0.008905
0.465674 0.257000 0.162766 0.t29276 0.112407 0.082320 0.062898 0.049629 0.040161 0.033167 0.027855 0.023725 0.020449 0.017809
P. Mdinowski
et al. / He-He interaction potential
431
Table 2 Coetiicients in the muhipole expansion of A$&,“’ Wavefunction
G
4
60
Double-E
3.751
35.792
456.393
Hartree-Fock
3.744
35.723
455.507
a’ See text. accepted as the position of the van der Waals minimum) the polarization contribution is about one percent of the dispersion term; for distances greater than 10 au, polarization is totally negligible. Though the relative shape of our potential is good at large R as discussed above, we must of course determine a value for B in order to obtain absolute values for our potential. A commonly cited choice for hE [15,16] proposed by London is the first ionization potential but it is recognized as leading to poor results [lfi]. Using the experimental ionization energy for He WC have BP = 0.91795 au, where the superscript “IP” denotes the method used to obtain the mean excitation energy. Another method for fixing -E involves the dipole polarizability tensor a(o) which can be calculated from the susceptibility [6]
a(o) = -
dr
I
I
dr’ r’ ,y(r, r’, w)r.
The closure approximation ,f(r,
r’,
0)
dr’
a(o) = Table 3 Values of Ag,, atomic units)“’
3 4 5 5.6 6 7 8 9 10 11 12 s3 14 15 ‘I The
to the coordiante
= [-20,~/(~3’no-02)][r(r)
and so the corresponding
A&
and A”’
0.50964 (-2) 0.27717 (-3) 0.15286 (-4) 0.26928 (-5) 0.84648 (-6) 0.46925 (-7) 0.26021(-g) 0.14430 (-9)
r’ jj(r,
(45)
Sk-r’)
polarizability r’,
w)
space susceptibility -T(r,
is (46)
r’)],
is
r.
(47)
as a function of internuclear separation R for two helium wavefunctions.
0.17424 (-1) 0.25226 (-2) 0.46921(-3) 0.20017 (-3) 0.12098 (-3) 0.41372 (-4) 0.17155 (-4) 0.80821(-5) 0.41700 (-5) 0.23060 (-5) 0.13477 (-5) 0.82432 (-6) 0.52378 (-6) 0.34381(-6)
0.27617 (-1) 0.30770 (-2) 0.49978 (-3) 0.20555 (-3) 0.12268 (-3) 0.41466 (-4) 0.17160 (-4) 0.80823 (-5) 0.41700 (-5) 0.23060 (-5) 0.13477 (-5) 0.82432 (-6) 0.52378 (-6) 0.34381(-6)
number in parenthesesis the exponent of ten.
0.51460 0.28203 0.16038 0.28969 0.92?63 0.54058 0.31610 0.18515
<1o-to
(-2) (-3) (-4) (-5) (-6) (-7) (-8) (-9)
0.17368 0.25267 0.47121 0.20103 0.12144 0.41467 0.17175 0.50875 0.41718 0.23066 0.13480 0.82444 0.52383 0.34383
(-1) (-2) (-3) (-3) (-3) (-4) (-4) (-5) (-5) (-5) (-5) (-6) (-6) (-6)
(A11 quantities are in
0.27660 (-1) 0.30908 (-2) 0.50329 (-3) 0.20682 (-3) 0.12330 (-3) 0.41575 (-4) 0.17181(-4) 0.80879 (-5) 0.41718 (-5) 0.23066 (-5) 0.13450 (-5) 0.82444 (-6) 0.52383 (-6) 0.34383 (-6)
P. Maiinowski et al. / He-He interaction potential
432
Assuming a single determinant wavefunction we simplify the two particle density function to obtain: z(r, r’, 0) = [-26,,/(0:,
-0~)][5(r
--c’j y(r) -4~~1
y(r’)].
(48)
Since the helium atom is spherically symmetric we calculate just the zz component of the polarizability tensor. (49) Using Slater type orbitals we find
For the Hartree-Fock wavefunction I=, = 0.790. We now choose zso that our static polarizability a=,(O) agrees with the experimental value, 1.383 au 1171. We thus obtain hE= 1.145, about 20% higher than EtP, which clearly shows that hELP is not adequate for the polarizability; nor is it adequate for the interaction energy. To see this we determine other values of z by requiring our interaction energy to agree at large R with other, absolute potentials i.e. E=
A’“(R)/2E&(R),
(51)
where A”‘~&?+2A’$,.
(52)
Note that %? can now be- considered a variable, R-dependent scaling factor. In table 4 we present 4.E values obtained frcm MCSCF [18] CI [19,20], multipoie expansion (ME) [X2,13] - type calculations and from the Burgmans et al. [2i] potential which is a synthesis of
Table 4 Mean excitation energies from different Type of talc.
CI
types of calcuIations”“-c’ IMCSCF
ME
Liu and
McLaughlin
Bertoncini
Tang
McLean Cl91
and Schaefer 1201
and Wahl [W
et al. [I31
R
z-E
15 14 13 12 11 IO 9
1.34 -
8 7 6
1.37 I.51 2.25
Experimental
Maeder and Kutzelnigg WI
Burgmans et al. [Zl]
KE-=
_ _ 1.23 1.24 1.27 1.37 1.90
1.25 1.25 1.29 1.41 2.04
=’ See eq. (51) of text. ‘I All quantities in atomic units. =’ A”’ in eq. (51) was calculated from the Hartree-Fock fmxtion.
1.27 1.27 1.27 1.27 1.28
1.28
1.28 1.29 1.30 1.33 1.42
1.29 1.30 1.31 1.35 1.44
1.28 1.28 1.29 1.29
I.28 1.28 1.28
1.28 1.29 1.29 1.29 1.32
1.47 2.06
433
P. Malinowski et al. / He-He ixteractinn potential
accurate experimental and theoretical results. Notice that although these six AE values are different they are almost constant within the range of R between 15 and 8 au. For ? au there is a visible increase in these values and for 6 au there is generally a great jump in the magnitude of z. Because of the almost identical values of A~I and A&, for the double-zeta function (see table 2) the mean excitation energies for this function are the same. We now enquire whether the variation in E is due to the inadequacy of the closure approximation, to overlap inter- and intra-corre!ation effects, or short range Coulomb interactions. We do so by comparison with_reliable results for the total interaction potential; we thus need to estimate the total inte-radian energy by augmenting our second order results with first order Coulomb and exchange energies. 4.2.
The first-order energy conrribution.
The total interaction
energy
We have calculated the first-order Coulomb and exchange energies using the Murrell-ShawMusher-Amos [22] perturbation theory. The results are presented in table 5. Because of almost identical results for the double zeta and Hartree-Fock functions we may say that the quality of both functions is the same. Therefore we will limit our discussion to the Hartree-Fock results. We modify (51) to accommodate our first order results. =
= A”‘/2[&
(53)
- &“I,
where
One observes in table 6 that now the AE values are less variable for R > 5.6 au. The changes of and pxp for R > 5.6 au are within 24, 20, 20 and 21% respectively. hEz’, -KE, LPcsCF These deviations of a from a constant value stress that for distances less than 7 au either the Unssld approximation breaks down, or overlap and correlation effects become important or the potential is not well approximated by first and second order perturbation theory. In order to investigate those possibilities let us analyze the results collected in table 7. Our total interaction energy listed there was obtained by adding the first-order and second-order contributions with an assumed fixed AE of 1.28 au. One sees that our predicted minimum is too deep and occurs at too short a distance. Also the repulsive part of the potential curve is not steep enough. Obviously in the
Table 5 First-order Coulomb, atomic units) *’
3 4 5 5.6 6 7 8 9
0.33803 (-2) 0.28104 (-3) 0.21900 (-4) 0.46164 (-5) 0.16209 (-5) 0.11551(-6) 0.79976 (-8) 0.54135 (-9)
exchange and total energies for two helium atoms calculated from two wavefunctions.
0.18159 0.16929 0.14436 0.31859 0.11498 0.86943 0.63139 0.44384
(-1) (-2) (-3) (-4) (-4) (-6) (-7) (-8)
0.14779 (-1) 0.14119 (-2) 0.12246 (-3) 0.27243 (-4) 0.98769 (-5) 0.75392 (-6) 0.55142 (-7) 0.38971(-9)
0.33698 (-2) 0.28358 (-3) 0.2X41(-4) 0.48543 (-5) 0.17324 (-5) 0.12841(-6) 0.92882 (-8) 0.65884 (-9)
0.17758 (-1) 0.16959 (-2) 0.14997 (-3) 0.33965 t-4) 0.12488 (-4) 0.99261(-6) 0.76012 (-7) 0.56463 (-8)
(All quntities
0.14388 0.14124 0.12733 0.29100 0.10755 0.86420 0.66724 0.49875
(-1) (-2) (-3) (-4) (-4) (-6) (-7) (-8)
ir.
P. Mafinowski er al. / He-He inferaction poremid
434 Table 6 Corrected
mean excitation en:rgies.
(~11
quantitiesin atomicunitsFb! MCSCF
B&on&i and
Experimental Burgmans
Wahl [lS]
et al. [21]
~ICSCF
~jg.“’
1.30
1.33
1.38
i.43 1.52 i.76
1.50 1-S 1.97
1.52 1.63 1.94
T>TJe of talc.
Cl Liu and McLean [19]
McLaughlin and Schaefer [20]
R
z?’
ssc
7
1.43
5.6 5
1.61 1.76 2.13
6
=’ See table 4.
” Calculatrd u&g
2q. (53) of text.
region of overlap either the second order contribution is too negative or the first-order contribution is not positive enough, or both. Our first-order results differ from the Murrell and Shaw [23) and Claverie [24] values by about 10%) and from the Murrell and Varandas [25] values by 6%. Correlation can change the first-order energy by approximately 10% [26]; we thus conclude that our first-order energy is accurate to within 20%. This inference seems to be confirmed by the recent studies of Szalewicz and Jeziorski [27]. That possible error in our first-order calculation obviously cannot account for the total discrepancy in the interaction potential. In recent years it has been found that neglect of the penetration of the interacting charge distributions at small distances leads to divergence of the usual multipole expansion of Es&,. A more complete calculation yields a finite dispersion energy at all separations and SO damping functions have been introduced to compensate for the overly large negative values of Eiz, obtained from the multipole expansion [se, 5f, 281. Since we use the full Coulomb potentiai, there is no justification for introducing those damping functions in our work. However, we note that if we use a R-dependent Table 7 Total He-He R
3 4 5 5.6 6 7 8 9 10 11 12 13 14 15
potential. (All quantities in atomic units)“’
This work”
0.35892 (-2) 0.20573 (-3) -0.69159 (-4) -0.51646 (-4) -O-37381(-4) -0.15367 (-4) -5.66411 (-5) -0.31526 (-5) -0.16287 (-5) -0.90052 (-6) -0.52627 (-6) -0.32187 (-6) -0.20451 (-6) -0.13424 (-6)
‘) See Table 2.
“B=
1.28.
McLaughlin and
Bertoncini and
Maeder and
Schaefer [20]
Wahl [IS]
Kutzelnigg [12]
0.11610 (-1) 0.8611 I(-3) -0.11147 (-4) -0.38002 (-4) -0.32079 (-4) -0.15106 (-4) -0.67769 (-5) -0.32586 (-5) -0.17006 (-5) -
0.3930 -0.3604 -0.2993 -0.1472 -0.6620 -0.3210 -0.1660
(-5) (-4) (-4) (-4) (-5) (-5) (-5)
-0.15423 (-4) -0.65516 (-5) -0.31153!-5) -0.16137 (-5) -0.89411(-6) -0.52313 (-6) -0.32018 (-6) -0.20353 (-6) -0.13364 (-6)
Burgmans et al. [Zl] 0.12708 0.87222 0.19566 -0.33466 -0.29661 -0.14094 -0.64703 -0.31324 -0.16220 -0.89852 -0.52560 -0.32165 -0.20445 -0.13423
(-1) (-3) (-5) (-4) (-4) (-4) (-5) t-5) (-5) (-6) (-6) (-6) (-6) (-6)
I’, Malinowski er a!. / lie-He
inreracrion porenrial
433
mean excitation energy instead of a fixed one, we can get greatly improved rest&s. The notion of a R-dependent mean excitation energy was first introduced by Dalgarno and Lynn [29] who proposed a mean excitation energy of the following form (see also Kreek and Meath [30]) z(R)
(SSj
= AE(m) + CS(R),
where D(co) is the mean excitation energy at infinite separation, S(R) the overlap integral and C a constant determined empirically by fitting the second order energy to the difference between the experimental [21] and first order energy at a particular R, say R = 5.6 au (the equilibrium distance). We find that for He-He E(R)
= 1.28075 +52.0340
S(R).
(56) Table 8 lists the modified potential energy and M(R) obtained by using (56). The results are in excellent agreement with the experimental values of Burgmans et al. [21] as shown in table 7. Comparison of the calculated and experimental curves is shown in fig. 2. Throughout this paper we have applied the Unsold approximation to the charge-density susceptibility. We now proceed to show that this is equivalent to applying the Unsijld approximation directly to the dispersion and polarization energies. To do SO we switch to r-space and using eqs. (13), (14a), (14b), (15a), (lSb), (9) and (7) we obtain (see also ref. [6] section 5)
(57) The first term is the dispersion energy and the last two terms are the polarization energies. Eq. (57) is the formulation for the second order Coulomb energy obtained by Longuet-Higgins [31] on the basis of ordinary time-independent perturbation theory. It is easy to see that by writing the second order Coulomb energy in this fashion the Unsold approximation can be applied separately to each of the terms and when this is done the mean excitation energies for the three terms are respectively Table 8 Modified He-He R 3 4 5 5.6 6 7 s 9
10 11 12 13 14
I5
potential, z(R)
and overlap integral S(R)
S(R)
en*
s(R)”
1.208 C-1) 4.236 (-2) 1.414 (-2) 7.151(-3) 4.507 (-3) I.391 (-3) 4.189 (-4) 1.236 (-4) 3.583 (-5) 1.025 (-5) 2.894 (-6) 8.087 (-7)
1.256 (-2) 9.703 (-4) 2.543 (-6) -3.347 (-5) -2.993 (-5) -1.450 (-5) -6.529 (-6) -3.137 (-6) -1.626 (-6) -9.001(-7) -5.262 (-7) -3.219 (-7)
7.566 3.495 2.017 1.653 1.5 16 1.353 1.303 1.287 1.283 1.282 1.281 1.281
2.239 (-7) 6.151(-S)
” ~(R)=l.28075+52.0340
-2.045 (-7) -1.342 (-7)
S(R).
1.281 I.?81
P. Malinowki
436
er at. / He-He Weracrion porenrial
-e 3-
I--_I3 4
5
6
7 R
8
* 9
:b
I,
I !2
Fig. 2. He-He total potentials. Solid curve: the ESMMSV parametric potential of ref. [21]. Dashed curve: thii work with E(R)=1.28075+52.0340 S(R), where S(R) is the overlap integral. Dotted curve: this work with E=l.ZS.
h(&o+&o), hi& and AL&..~,which are the same as in the susceptibility formulation. If the Unsijld approximation is applied directly to the second order Coulomb energy without first decomposing it, as is often done, then a single mean excitation energy is obtained which is characteristic of dispersion but inappropriate for polarization. Although we have dwelt at some length on the first order energy and total potential, our main goal and interest has been the development of reliable techniques for treating the second order Coulomb ener,q. The approximation which uses static form factors appears to give a good description of the He-He long range potential. Although we have used analytic expressions for the static form factors and for the integrals involved in eqs. (34) and (35), the same could have been accomplished by using numerical values of F(k) [32]. Indeed the numerical approach becomes more apppealing the more complex the system is. A purely numericai approach circumvents the use of wavefunctions and is independent of the size of the system and the so&e of the scattering functions.
Acknowledgement The authors are grateful to Professor William J. Meath for valuable discussions and helpful suggestions.
Appendix In this appendii we list the integrals needed in the evaluation of the quantities Asp and A$ eqs. (34) and (35)]. Anaiytic evaluation of the integrals will be presented elsewhere [9]-
[see
437
Integrals for A'?' Am
(I).
Nk-L’PR
A&?)’
J
dk
We -g-pT
J
F(R)
F(4)
F(C)
F(-k’).
This integral factors -4_,G3
=9(R)
B(-R),
where S-R
B(R)=
dk+F(k)
I
F(-k).
The other two integrals needed for A;& are: Sk-k’)-R A’s=
J
dk
J
r&----
k2k,z
F(k
-k’)
F(k’-
k)
and i(k-k9.R
A&R)
=
I
dk
I
dk’ +(k
-k’) F(-k)
F(k’).
2. Integralsfor AZ1 Besides ALF and ALL defined above, the following integrals are needed for AZ,:
A similar integral arises with k replaced by -k’ and k’ replaced by -k in the arguments of F. The next integral is: r(R)-IdkJdk’GF(k’-k).
Another integral which factors is i[k-k’1.R
J
dk
J
F(-k)
dk’+
F(k’) = b(R) 6(-R),
where ik*R b(Rf+k+
F(k).
Finally i(k -k’)_R
J
dk
J
dk’+
F(-k)
F(k) F(k’) = b(-R)B(R).
A similar integral arises which is the product b(R)B(-R). of R for Am, As, B, b, 5 and r.
In iable 1 values are given at several values
References [!I [2] [3] [Ij
F. London. Z. Physik, Chem. Bll (1930) 222. A. Uns6ld. Z. Phgsik 43 (1927) 563. B. Lindrr, K.F. Lee. P. Maiinowski and A.C. Tanner, Chem. Phys. 52 11980) 353. (ai J. H. Hubbell. N’.J. \‘cig&. E.A. Briggs. R. T. Brown, D.T. Cramer and R.J. Howerton, J. Phys. Chem. Ref. Data 4 t19751 Gil: tbj J.H. Hubbell and I. Ovsrbo. J. Phys. Chem. Ref. Data 8 (1979) 69; (cJ K.F. Lee, unpubIished resuIts. is] (ai N. Jacobi and G. Csanak, Chem. Phys. Letters 30 (1975) 367; fib) P.W. Langhot?. Chem. Phys. Letters 20 (1973) 33; ic) T.B. SIacRury and 5. Linder, J. Chem. Phys. 58 (1973) 5388; Id! R. Boehm and R. Saris. J. Chem. Phys. 55 (1975) 2620: (e) .4. Koidr, J. Phys. B9 (19761 3173; (i) A. Koide, W. J. hleath and A. R. Allnart, Chem. Phys. 38 (19813 105. [6] B. Linder and D. A. Rabenold, Advan. Quantum Chem. 6 (1972) 203. [7] R. van Bergen, Y. van Bergen and B. Linder, J. Chem. Phys. 67 (1977) 3878. [P] P.O. L;iwdin. Phys. Rev. 97 (1955) 1471. [9] P. Malinowski, A.C. Tnnner, K.F. Lee and B. Linder. Int. J. Quantum Chem., to be published. [IO] E. Clementi. IBM J. Res. Dev. ISuppl) 9 (1965) 2. [ll] E. Clrmenti and C. Roetti, At. Data Sucl. Data Tables 14 (197-1) 177. j12] F. ,Maeder and W. Kutzelnigg, Chem. Phys. 42 (1979) 95. [13] K.T. Tang. J.!& Norbeck and P.R. Certain, J. Chem. Phys. 64 (1976) 3063. [la! R. Luyckx. P. Coulon. and H.N.W. Lekkerkerker. J. Chem. Phys. 69 (1978) 2424. 1151 M.J. Huron and P. Claverie, Chem. Phys. Letters 4 (1969) 429. [16] A. Datgarno and W.D. Davison, Advan. At. Mol. Phys. 2 (1966) 1. [17] P.W. Langhoff and M. Karplus, J. Opt. Sot. Am. 59 (1969) 863. [18] P. Bertoncini and A.C. Wahl, Phys. Rev. Letters 25 (1970) 991. [19] B. Liu and A.D. McLean, J. Chem. Php. 59 (1973) 4557. [20] D.R. McLaughiin and H.F. Schaefer III, Chem. Phys. Letters 12 (1971) 244. [21] A.L.J. Burgmans, J.M. Farrar, and Y. T. Lee, J. Chem. Phys. 64 (1976) 1345. [t2] (a) J. N. Xfurrell and G. Shaw, J. Chem. Phys. 46 (1967) 1768; (bj I.J. Musher and A.T. Amos, Phys. Rev. 164 (1967) 31. [Z3] J.N. Murrell and G. Shaw, Mol. Phys. 12 (1967) 375. [13] F. Clavrrie, in: IntermoIecuIar interactions: from diaromics to biopolymers, ed. B. Pullman (Wiley and Sons. New York, 1978). [X] J.N. ~Murrelf aEd A.J.C. Varandas, Mol. Phys. 30 (1975) 223. [X] A. Conway and J.N. Munell, Mol. Phys. 23 (1972) 1143. [27] K. Szalewia and Et. Jeziorsk:, Mol. Phys. 38 (1979) 191. [2S] (a) M. Krauss, D.B. Neumann, and W.J. Stevens, Chem. Phys. Letters 66 (1979) 29; (b) M. Krauss, W.J. Stevens, D.9. Neumann, Chem. Phys. Letters 71 (1980) 500. [29] A. Dalgarno and N. Lynn, Proc. Phys. Sot. A69 (19.56) 821. [30] H. Kreek and W.J. Meath. Mol. Phys. 22 (1971) 915. [31] H.C. Longuet-Higgins, Proc. Roy. Sot. (London) A235 (1956) 537. [32] K.F. Lee, Ph.D. Dissertation, Florida State University (1981).