139
Progress of Theoretical Physics, Vol. 18, No. 2, August 1957
The Repulsive Potential between Two Normal Helium Atoms
Sigeru HUZINAGA
General Education Department, Kyushu Uniyersity, Fukuoka
(Received May 9, 1957)
The repulsive potential between two nonnal helium atoms is calculated ·using the LCAO MO
method. The main idea is that we use two ·different effective charges for O'g and 0',. orbirals, dropping the restriction of a common charge for the both orbirals. It is found that remarkable improvements are achieved for the theoretical calculation but there still remains a large discrepancy between
theory and experiment. A critical discussion of the present status of theoretical calculatioo8 is attempted
in some detail.
§ I. Introduction and general remarks
The liquid helium and the atomic nucleus present interesting common features in
several respects as the quantum mechanical many-body problems. They are both " condensed " systems and the behaviors of the constituent particles are believed to correlate
·each other in some very complicated way. Yet, strange to say, it seems that some gaslike models or independent particle models can claim their rights as meaningful first
approximations of real situation for both cases, that is, the Bose-gas model for the liquid
helium and the Fermi-gas model and the so-called shell model for the nucleus. To find
1:he reason for this is one of the important problems awaiting a clear-cut answer from
1:heoretical physicists.
In the problem of the nuclear matter one of the most serious difficulties is that
we lack the exact knowledge of the interaction between nucleons. The present meson
1:heory provides only a partial and incomplete answer to it and the difficulty is closely
related to the fundamental problem of the quantum field theory in general. The twobody force might be determined through the analysis of scattering experiments but there
is no positive evidence that the many-body effect could be negligible in the nuclear matter.
As for the interaction between helium atoms, there is no obstacle, in principle, that
prevents us to reach the final answer by theoretical calculations. The fact is, however,
1:hat we do not know the interaction between helium atoms with sufficient accuracy, just
like in the problem of the nucleon interaction. One of the reasons for this situation
is probably that the two helium atoms do not form a stable molecule, hence they are
devoid of apparent chemical interest. However, the He-He interaction is the simplest
case of the interaction between two closed-shell electron clouds, which must be quite
important also for chemistry.
About thirty years ago, Slate~> calculated the repulsive potential between two normal
helium atoms and obtained the following approximate expression,
140
S. Huzinaga
0.
(R in A)
V(R)=7.7XI0- 10 exp(-4.60R) erg
(1·1)
for the potential as a function of the internuclear distance R. Later, the inverse six-power
dispersion energy was calculated separately 2l and added to the above potential :
0
[770exp(-4.60R)-(1.49/R6)]X10-12 erg
(R in A).
(1·2)
This is so-called Slater-Kirkwood potential and has been widely used.
In Slater's calculation of the repulsive valence force a very accurate atomic wave
function was used, but certain multiple exchange integrals were neglected and several
integrals were only approximately calculated. Rosen3l criticized this point and performed
his calculation with a simpler but less accurate atomic wave function so that all integrals.
could be evaluated. He used, for the spatially unsymmetrized wave function for the
unperturbed helium atom,
IJl".-...exp[- (2.15r1 +1.19r2)/a0]
(a0
:
(1·3)
Bohr radius)
and obtained results which can be summarized in the form,
0
V(R)=9.25XI0- 10exp(-4.40R) erg (R in A).
(1·4)
This formula gives considerably higher values of inter~ction energy than Slater's formula
(1·1). Both Slater and Rosen performed their calculation using the valence bond method.
As for the experimental data, rather an accurate potential at small internuclear distances is provided by the recent work of Amdur and Harkness 4 (A-H). The result is:
2.884/~" 79 eV
4.713/R5"94 eV
for
for
0.52<R< 1.02
1.27<R< L59
0
(R in A)
(1· 5)
When we compare the experimental values calculated with the above formulas with Rosen's.
theoretical result, we find serious disagreement between them, especially at small separations.
At R=l.O au (atomic units) (1·4) gives 56 eV while (1·5) yields only 9 eV and
so the discrepancy amounts to 47 eV.
Griffing ~and Wehner5l (G-W) were the first who adopted the antisymmetrized
molecular orbital (ASMO) method for the present problem. They used simple Slatertype Is-functions, Xa and Xb> as AO's and constructed two LCAO MO's:
(1· 6)
where
Xa= (Z 3/n) 112exp( -Zra)
(1· 7)
and similarly for Xb with Z=27 /16. Using these molecular orbitals they calculated theHe-He interaction energy by ASMO method and could obtain a better result than previous
calculations. It is well known that, so long as we use an atomic orbital of the form
(1· 7), the ASMO method is just equivalent to the simple Heitler-London theory ~
the present problem. Comparing the result of G-W with that of Rosen, we notice rather
The R.epulsive Potential between Two Normal Helium Atoms
141
a curious fact : Rosen started with a wave function for an isolated He atom which is
better than the one of G-W, and the result is that his energy values are considerably
higher than the values obtained by the simpler calculation of G-W.
Recently Sakamoto and Ishiguro6J (S-I) performed an interesting calculation on the
present subject. They attempted to take into account the deformation of atomic orbitals
due to the approach of another atom by making use of the modified atomic orbital of
Inui7l and Nordsieck8l, which was successful in the treatment of the ground state of H 2
molecule. The repulsive interaction energy of the He-He system obtained by S-I is lower
than that of G-W but it is to be noted that the difference between these two results is
rather small, namely within 1 eV over nearly the whole range of the internuclear distance
in discussion and if we compare the so far best theoretical result of S-I with the experiment of A-H, we still find a large discrepancy which amounts to about 30 eV at the
internuclear distance of 0.5
A.
If the experimental result of Amdur and Harkness is
reliable, this discrepancy must be attributed to some defects in the theoretical estimations.
In this connection the present author has attempted a simple united-atom treatment
of the problem9l at small internuclear distances where the discrepancy between theory and
experiment remains quite large. In place of (/! 1 and (/! 2 given in (1 · 6) and (I· 7), we
here use the following united-atom wave functions,
(/! 1
= (Iso-;
() =N(I, ()exp( -(r) Y 00 ; o-n
(/! 2
= (2po-;
7)) =N(2, 7J)r exp( -7Jr) Y10
(I· 8)
;
O""u
(2(}"+(1/2) / [ (2n) !J 12 and the coordinates of electrons are measured from
the middle point between two nuclei. ( and 7J are variable parameters. It is clear that
where N(n, ()
=
this approximation is a very crude one, especially in lp10 because it is unable to describe
the concentration of the electron cloud at the positions of two nuclei. In spite of ·this
apparent crudeness, this simple treatment gives a better result at the internuclear distance
R= 0.8 au than the elaborate valence bond theory of S-I does. Even at R= 1.0 au our
result is definitely better than that of Rosen. A slight improvement on (/!1 yields 34.7
eV. as the repulsive interaction energy at R= 1.0 au, which is 3.7 eV lower than the
corresponding value of S-I, 38.4 eV. This is rather an unexpected situation and it may
not be unreasonable to infer that, at the internuclear distance of about I au, the total
wave functions used by various authors in their more or less elaborate calculations would
bear some definite defects.
Griffing and Wehner used the molecular orbitals given in (I· 6) and (I· 7) and
took the same fixed value of the effective charge for
orbitals. This procedure seems to be too restrictive.
Xa+xb
(Iso-g) and xa-xb (Iso-u)
It is instructive for us to remind
that in the Hi ion problem there exists a remarkable difference between the manners of
variations of effective charges for (Iso- 0 ) and (Iso-u) orbitals of LCAO type:. at smaller
internuclear distances the value for (Iso-9 ) rises up steeply and the one for (Iso-u) falls
down rapidly. This is displayed, for example, in Fig. 4.6 on P. 85 of Coulson's text
S. Huzinaga
142
book10l. In addition, for the present He-He system we must take account of the exclusion
-effect of the Pauli principle and this should reinforce the requirement of the separate
variation of effective charges for u 0 and u" orbitals. In short, u 0 and u,. orbitals must
reduce to ls and 2p atomic orbitals, respectively, of Be in the united-atom limit and to
cope with this physical requirement we should, at least, drop the severe restriction of equal
·effective charges.
Thus we are naturally led to the adoption. of two different values of effective charges
for ( 1 su0 ) and ( 1 su,.) orbitals. This simple modification --of the calculation of Griffing
and Wehner brings us a significant improvement of the theoretical values of the He-He
repulsive interaction, which will be described in the following.
§ 2. Calculation and result
The system consists of doubly charged helium nuclei a and b separated by a distance
Our problem is to construct a suitable wave function which makes
1:he energy of the system as low as possible within limited labor. In the present calcula1:ion we are going to use, as the total wave function, an antisymmetrized product of
orthonormal molecular spin orbitals, which are products of MO' s with the usual spin
eigenfunctions. As stated in the last part of the preceding section, the MO's assume the
following forms :
R with four electrons.
([Ju=NvCXa+·'b) ; uu
([J,.=N,.(x,.'-xb');
(2 ·1)
u,.
where
N0 = 1/[2 + 2 CxaiXb) T12,
N,. = 1/[2-2 CXa'IXb') ]1'2,
J
CxaiXb) = XaXbdV,
X and
x'
(Xa'lxb')
=J
xa'x/dV.
are simple Is-functions;
x= (C3/nY 12 exp c-r;,r),
x' = (('3/7!)
1/2
exp ( -('r).
(2·2)
The point is that we assign different values of the effective charges ( and (' for ug and
-u,. orbitals separately. In the calculation of G-W (=('=27 /16.
The total wave function is thus :
¢= (1/24r 12~
( -1) J-p>-[({} (1) a (1) ({Jg(2) ,8(2) ([),.(3) a (3) ([),. ( 4) .8 (4)]
).
0
(2 · 3)
"
where P>- is the operation permuting the electron coordinates, and a and ,8 are the usual
-orthonormal spin eigenfunctions. The energy operator for the atomic units*
* We
use the following atomic units :
Length: a 0 =0.529171 X I0-8cm (Bohr radius),
Energy: e2fao=27.2100eV (twice the ionization energy of hydrogen).
The Repulsive Potential between Two Normal Helium Atoms
..
4
.
H= 2.; H(i)
4
+ 2.;
(1/r,j)
i>j=1
i=l
+ (4/R),
143
(2·4)
where
Then the total energy of the system is
where
J
H,.= J<p,.
H0=
] 09
=
},...=
Juu=
<p0 (1)H(1)<p9 (1)dV1,
(1) H(1) <p... (1) dVn
JJ <p (1)<p (2) (1jr12)<p (1)<pg(2)dV1dV2,
0
0
0
H
<p,.(1)<p,.(2) (1jr12)<pu(l)<pu(2)dV1dV2 ,
H
H
K9 u=
<p0 (1)<p,.(2) (1/r12)<p 9 (1)<pu(2)dV1dV2,
<{Jg (1) <fu (2)
(1/r12) <fu (1)<p0 (2) dV1dV2.
Next we decompose these integrals into more basic integrals expressed in X and X' and
1:hen start the routine work of evaluating a number of integrals.
It seems convenient to introduce the following notations and definitions according to
the work of Roothaan and Rudenberi 1>,
JXaXbdV= (XalXb) (saJsb),
JXa(1/ra)XbdV= (Xal1/raJXb)
(2 ·6)
=
J
= (!aJ1:/raJsb),
Xb(1/ra)XbdV= (Xbl1/ralXb) = (sbJ1/raJsb),
(2· 7)
(2·8)
and the similar formulas for the case of X'. TlJ.e above are the one-electron integrals.
As for the two-electron integrals, Coulomb, exchange and hybrid types of integrals appear
in the present calculation. These integrals can be expressed generally in the form,
(2·9)
144
S. Huzinaga
where Q and Q are charge distributions consisting of the product of two atomic orbitals.
For example,
[sa(1)sa(1) lsb(2)sb(2)]=
[sa (1) Sb (1) ISa (2) sb (2)] =
[s,(1)sa(1) lsa(2)sb(2)]=
JJ
JJ
Xa(l);(a(1) (1/r12)xb(2);(b(2)dV1 dV2 ,
(2 ·10)
(Coulomb)
Xa (1)xb (1) (1/r12)xa (2)xb (2) dV1 dV2 ,
(2 ·11)
(exchange)
H
Xa(1)x,(1) (1/r 12 );(,(2);(b(2)dV1 dV2
,
(2 ·12)
(hybrid)
and others may be inferred according to the above examples.
the terms in (2 · 5) by using the integrals defined above:
Now we can write down_
Hg=2N/ { (1/2) (ajR) 2-2 (ajR) -2 (sbl1/rah) +[ (ajR) -4] (sal1/r,lsb)
- (1/2) (ajR) 2 (s.. lsb)},
Hu=2N,} { (1/2) ({i/R),2 -2 ((i/R)- 2 (sb'l1/ralsb')
+ [ ((i/ R) -4] (s..'l1/ralso
1)
- (1/2) ((i/R) 2 (sa' lsr!)},
]gg=2N/ {[sasalsasa] + 4 [s,sal sasb] + [sasaisosb] + 2[sasbl sasb]}·,
fuu=2N/Nu 2{[sas,ls,' sa']+ [sasalsbb']- 2 ([sasa Isa' sb']- [sa' sa'ls,sb]) - 2[sasbls'..sr/]},
Kgu=2N/Nu2{[sasa' lsasa']- [sasa'lsbsb1] - 2 ([sasa' lsasb1] - [s,sa'lsa' s~])
+ [Sasb ISasr/]- [sasb Isa' sb]} ,
1
Nu2 =1/[2+2(salsb)],
N,}=1/[2-2(s,'ls/)],
a=(,R,
1
(i=(,'R.
The explicit analytical expressions for the numerical evaluation of these integrals are found·
in the Appendix. Because all of them are the integrals between 1s atomic orbitals, theintegrations can be performed without difficulty. Some of them can be checked both
in the analytical forms and in the numerical values with the results of Hirschfelder and
Linnett12l. A little troublesome are [saso'ls,s/] and [sasb'lsa'sb] in Kuu because they are
expressed in the form of some infinite series, while all of others are expressed in closed
analytical forms in terms of several auxiliary functions. Fortunately the convergence ot
the infinite series is sufficiently rapid, especially in the combined form of [sasb'lsasb']
- [sasb' Isa' sb], and only the first three or four terms are necessary for the present purpose.
It is desirable to determine the best values of a and {i through the continuous'
variation of these variable parameters, but we did not do that. The first reason is that:
we entirely rely upon the numerical table of Kotani, Amemiya, Ishiguro and Kimura:tal
(KAIK) , where the auxiliary functions are tabulated . with the step of 0.25 of ZR.
Besides, while the present theoretical calculation yields a result much better than the-
The Repulsive Potential between Two Normal Helium Atoms
145
previous calculations as will be shown presently, there still remains a large discrepancy
between theory and experiment, and we want to reserve our labor for a more improved
treatment we are planning now. This is the second reason or excuse why we did not
try to look for the strictly best values of a and {1. The merit ofthe present calculation
is that it reveals to us what was wrong in the previous theoretical treatments rather than
its numerical result itself.
Table 1. The numerical results of the present calculation.
Actual calculations were
performed at three values of
R(au)
1.0
1.5
2.0
the internuclear distance R,
2.25
3.0
3.5
a=t:R
1.0, 1.5 and 2.0 au, and the
{j=t:/R
1.25
2.0
3.0
results are summarized in Table
1. E is the total energy of
E(au)
-4.70584
-5.32781
-5.58131
the He-He system and V(R)
V(R)
0.98947 au
0.36750 au
0.11400 au
is the repulsive potential energy
(26.92 eV)
(10.0 eV)
(3.10 eV)
defined by E (R) -E (co).
We took E (co)=- 5,69531 au in accordance with Sakamoto-Ishiguro6J and GriffingW ehner"J.
The result of the present calculation may be expressed in the following
approximate formula :
V(R)=8.58817exp(-2.16097R) au, 1<R<2 (R in au),
(2·13)'
This is compared with the corresponding formula of S-I:
V(R) = 15.02222 exp( -2.36478R) au, (R in au).
The potential curves due to these two formulas are found in Fig. 1.
§ 3.
Discussion
Table 2 and Fig. 1 are prepared for the purpose of comparison of various theoretical
estimations between them and with the experimental result reported by Amdur and
Harkness4 J.
Table 2.
R(au)
(1)
He-He repulsive potential (in eV)
(2)
(3)
(4)
51.84
49.94
0.8889
55.58
72.96
1.0
42.45
56.24
1.1858
27.08
36.53
L2
21.12
35.30
1.4
16.08
22.16
-
I
(5)
(6)
10.96
34.23
9.011
26.92
24.78
6.647
18.02
-
23.94
6.502
17.48
-
14.92
4.934
11.34
24.86
38.41
1.6
9;892
-
9.294
3.885
1.77788
6.423
9.261
6.212
6.115
3.216
5.013
1.8
6.088
8.730
5.792
3.147
4.778
2.0
3.747
5.481
-
3.609
2.606
3.102
13.91
7.363
S. Huzinaga
146
(1)
(2)
(3)
( 4)
(5)
Slater
Rosen
Griffing and Wehner
Sakamoto and Ishiguro
Experimental result of
Amdur and Harkness
(6)
Present calculation according to the formula (2 ·13) of this paper:
(according to Table V of S-I)
V(R) =8.58817 exp(-2.16097 R.) au, (R. in au)
V(R) in eV
70
60
50
40
30
20
10
0.8
1.0
1.5
2.0
Fig. 1.
It is observed that at R.=2.0 au there is no serious discrepancy between various calculations but at R= 1.0 au there arise remarkable diversities and differences. At R= 1.0 au
the value 27 eV of the present calculation is about 11 eV better than the value 3 8 eV
of S-I and is a half of the very large value 56 eV of Rosen. ·
In this connection, Fig. 2 may be helpful to realize the reason for this significant
improvement of the theoretical value at small separation. As stressed before, the essential
The Repulsive Potential between Two Normal Helium Atoms
147"
difference between G-W's treatment and the present one is that we drop the severe restriction of equal effective charge both for u 9 and u u orbitals in G-W' s treatment. In
their calculation (=(1 =27/16=1.6875 for all values of internuclear distance R, while
in our present treatment ( = 2.25 and ( 1 = 1.25 at R = 1.0 au. (See Table 1) . The
situation is shown schematically in the middle part of Fig. 2. The necessity of the
separate variations of ( and ( 1 may be recognized most directly when we imagine the
united-atom litnit, a Be atom, of the present He-He system (right-hand side of Fig. 2).
(G-W)
(H)
6·
R: large
R: small
R-o
Fig. 2.
In a Be atom, it is usually conceived that two of four electrons occupy the innermost 1s
orbital and the electron cloud is tightly bound to the nucleus under the influence of bare
nuclear charge. The other two electrons are forced to occupy other states because of
Pauli exclusion principle. Thus, the ground state configuration of Be is (1s) 2 (2s) 2 and
the level of (1s) 2 (2p) 2 lies several eV above the ground level. For 2s or 2p electrons
the nuclear charge is screened considerably by the inner two 1s electrons. Now let us
return to the He-He system. Imagine that we build up the whole system by dividing
the procedure into two steps. First, we allot two electrons to the bare field of two nuclei
to form, say, a (He-He)++ system. This is analogous to the well-examined H 2 problem_
In the case of H 2, it is well-known that, as the nuclei approach each other, the effective
charge of the 1s atomic orbitals increases quite appreciably from the original value 1 at
smaller values of the internuclear distance R. It would be natural to e~Cpect that th~
same would happen for the present (He-He)++ system. If it is true, the assumption
of (=27 /16 for all values of R would not be the most reasonable value but a little
larger values should be taken for small separations. Nextl suppose we add the remaining
two electrons to the (He-He) ++ system. Except for too small values of R, these two
S. Huzinaga
"148
electrons are expected to occupy 2pcr state. Then, even if a 2pcr electron moves in full
attraction ·of the bare nuclei, the effective charge (' woUld decrease considerably from the
original value (' = 2 at Stnall separations, as exemplified in the case of H.{ problem10l.
In the present problem this tendency will be reinforced because of ·the screening of the
hare nuclear field by the ·inner electrons and of the Pauli exclusion principle. Thus, the
use_ of a fixed value ('=27 /16 is inadequate also for the 2pcr orbital and, contrary to
the case of the 1scr orbital, we should take smaller values of (' for small separations.
Table 3 may help us to understand the situation explained above a little more quantitatively.
Table 3.
The values of various energy terms at R=1.0 au.
Elntat=E0 +E,.+4}0 u-2K0 u+4, E0 =2H0 + ] 00, E,.=2H,.+ ]uu·
2.25
2.5
1.75
1.25
1.0
1.75
-4.316344
-4.345730
-4.040239
-2.489686
-2.275115
-2.661907
+1.149554
+1.239626
+0.950354
+0.745940
+0.614447
+0.985012
+0.862256
+0.767899
+0.935962
+0.219149
+0.166879
+0.256689
-7.483134
-7.451834
-7.130124
-4.233432
-3.935783
-4.338802
+3.010726
+2.737838
+3.230470
Etntal
-4.705840
-4.649779
-4.238456
V(R=1)
26.92 eV
28.45 eV
39.64 eV
a=cR=c
f1=c'R=c'
Hu
H ..
luu
}uu
lou
Kgu
Eu=2Hu+Tuu
E,.=2H,.+}uu
4}0 u-2K0 ,.
Here E 0 =2H0 +]00 and E .. =2H.. +],.u are the energies of two electrons occupying 1scr(cr0 )
and 2pcr(cr,.) orbitals respectively. The case of (=('=1.75 may serve as a substitute
for the case of (=('=27/16=1.6875. In fact the value V=39.6 eV in this case is
very close to those of G-W and S-I. Griffing and Wehner attempted to improve the
calculation by minimizing the energy with respect to the effective charge for each distance.
But their conclusion was that " the variation of effective nuclear charge is unimportant
in the He 2 problem from the view-point of decreasing the energy." This probably due
to the cancellation of the effects of opposite direction on crP and cr" orbitals as may be
seen in Table 3. Of course, it is quite another problem why the value (=('=27 /16,
determined by a simple variational calculation on an isolated He atom, happened to be
nearly the best one under the restriction of equal effective charges.
In the treatment of Sakamoto and Ishiguro6 l, the total wave function is
rp-
a(1)a(1)
a(1)~(1)
b(1)a (1)
b(1)~(1)
a (2) a
(2)
a(2)~(2)
b(2)a(2)
b(2)~(2)
a(3)a(3)
a(3)~(3)
b(3)a(3)
b(3){1(3)
a(4)a(4)
a(4)~(4)
b(4)a(4)
b(4)~(4)
(3 ·1)
The Repulsive Potential between Two Normal Helium Atoms
149
-Here ·a ·and f3 ·are the usual spin functions and a and bare the modified AO's introduced
:by Inuf> and Nordsieck8>, namely
(3 ·2)
where ~ and 7J :are the prolate spherical coordinates defined by
and conversely
After simple manipulations of addition and subtraction between four columns we obtain,
;as described in Seitz~s . text book14l,
¢--
·9'0 (1) a (1)
9'0 (1) fi (1)
9'u (1) a (1)
9'.. (1)fi(1)
9'0 (2)a(2)
{/Jg (2) fi(2)
9',.(2)a(2)
9'u (2) fi(2)
'9'u(3)a(3)
9'u(3)fi(3)
9',.(3)a(3)
9'u (3) fi (3)
·9'0 (4)a(4)
9'u ( 4) fi( 4)
9',.(4) a (4)
9'u ( 4) fi (4)
(3 '3)
where
fir;,
'9'.. =a-b=e-""-~'11-e-"H~'fl= -2e-"" sinh fir;.
9' 0 =a+b=e-""-~'~+·e-"<+~'11=2e-"'cosh
(3 ·4)
This is to be c-ompared with the corresponding functions, 9'1 and 9'2 , in G-W:
%"'e-Zra+e-z'D=e-<ZR/2)(H'fjl +e-(ZR/2)(1;-'I'jl = 2e-(ZRf2)1; cosh (ZR/2) Y),
'9'2"-e-zr,._e-z'D=e-<ZR/2)(H'fl) -e-(ZR/2)(';-'fl) = - 2e-<ZR/2 ); sinh (ZR/2) 7),
(3. 5)
and, further, with ([1 0 and 9',. in the present treatment:
·9'0 -...e-'r;r,. +e-'!;rb =e-('!;R/2)(H'fl) + e-l1:R/2)(';-'I'j) = 2e-('!;R/2)'< cosh ((R/2) 7),
({! .. -i~'!;lr;._e-t1rb=e-<1:'Rf2)(1;+'11) -e-('r;IRJ2Hl;-'fl) = - 2e-<VR/2)'; sinh
(C:' R/2) "/}·
(3. 6)
At R= 1:0 au, a and p in (3 · 4) are determined to be 0.81747 and 0.92350 respectively
;according -to S-I, while (ZR/2) =27 /32=0.84375 in (3 · 5) and (C:R/2) = 1.125 and
(C:'R/2) =·0.-625 in (3 ·6). Now it is clear that there is no significant difference between
the total wave function of G-W and of S-I. Then it follows that both of these calcula-tions yield ·almost the same result in regard to the energy values. The above comparison
.is also helpful to understand why Sakamoto and Ishiguro were not so successful in spite
of their inclusion of two variable parameters, while the present calculation is more successful with the :same number of variable parameters. In S-I' s treatment the deformation of
the electron -cloud happened to be small, not because the deformation is really small but
.because their _parameters, a and fi, could not afford to produce the necessary deformations
150
S. Huzinaga
explained above for (J"g and O",. orbitals. One of the simplest ways to improve S'I's calculation would be to take two different values of f3 in the manner that fJ >a> /3', instead
of one. This modification creates no troubles in the calculation of molecular integrals
because f3 and /3' are the coefficients of the variable 'TJ· We cannot, however, expect
much for this modification because it is a, the coefficient of ~. which determines the
degrees of spread of the G"g and O",. orbitals. We cannot expect to obtain a drastically
improved result as long as we assign the common a to both G"u and O",. orbitals.
We have seen that the present calculation yields a significantly improved result and
the reason for this achievement has been discussed. However, if we compare our present
result directly with the experiment of Amdur and Harkness4l we find still a large discrepancy remaining between them, especially at small internuclear distances. Then our next
problem will be to find how to diminish this large gap. We shall try to count out
some of possible causes responsible for this discrepancy, assuming that such discrepancy
really exists. The first is that our efforts in looking for the best values of the variational
parameters were not exhaustive as mentioned in § 2, but this could be a source of error
of 2 or 3 eV at best. The second is that the treatment of the correlation between two
inner electrons remains unsatisfactory because we used simple LCAO MO' s. The possibility
of improvement on this point remains open. The third is the neglect of contributions.
from other configurations, especially (lsO") 2 (2sG") 2 and (lsO") 2 (2prr±) 2 • At small separations it is probable that the (lsO") 2 (2sG") 2 configuration plays an important role, because
it is the ground state configuration of a Be atom, the united-atom limit of the He-He
system. In the present paper we are exclusively discussing the exchange force region of
the He-He interaction because we find there a large disagreement between theory and
experiment. However, our final aim is a consistent quantum mechanical treatment of the
whole range of the interaction force, from the repulsive exchange force region to the
attractive van der Waals force region. This has been accomplished for H 2 by Hirschfelder
and Linnete2J. For this purpose the inclusion of the (I sO") 2 (2prr±) 2 configurations will
be necessary. We are attempting to examine these three possibilities ..
Finally we should like to spend a few words about the problem of the non-additivity
of the force between helium atoms in the exchange force region. Recently Shostak15>
examined the problem in the case of the linear configuration of three normal helium
atoms. His conclusion is that the non-additivity is considerably large at small internuclear
distances. However, it is to be noted that his argument is based on the two-body force
due to the calculation of Griffing and W ehner5l. It has been shown in the present
paper that their calculation is not of sufficient accuracy, especially at small internuclear
distances. The problem of the non-additivity must be reexamined carefully by using wave
functions of sufficient accuracy.
Acknowledgement
The author would like to acknowledge his indebtedness to Prof. G. Araki for his
kind interest and to Prof. T. Murai for ~he helpfql discussions.
The Rtpulsive Potential behveen Two Normal Helium Atoms
lSI.
Appendix
The expressions are- given for the evaluation of numerical values of various integrals.
in § 2. In the following a=(R and f1=C'R.
1 a3
1
·
(sbll/r.. lsb) =--[A1 (a)B0 (a) +A0 (a)B1 (a) ]=-[1- (1 +a)e- 2"],
R 2
R
(s.. ll/r.. lsb)
=-..!:....a A (a) =_!_a(l+a)e-'",
3
R
1
R
These are the formulas for H 9 and } 99 • The corresponding formulas for H,. and };_ can
be obtained similarly by replacing s~s' and a~/1 in the above formulas .
. (b)
]g...
[s,.s,.js..'s,.']
(('
(C+C')a
CCZ+3C'('+C''2)
-
-
=_!_
af1
(a 2 +3af1+{12),
R (a+f1)a
[s,.s,.jsb1sb1] =_!_ f13[2 {A1 (/1) B0 _({1) +A0 ({1) B1 ({1)}
R4
+2 {A0 (a+ {1) B1 (a-{1) -A1 (a+/1) B0 (a-{1)}
+a {A0 (a+f1)B2 (a-{1) -A2 (a+f1)B0 (a-{1)} ].
[s,.s,.js,.'sb']=_!_ pa[4A1 ({1)
R 4
+2 {A0 (a+ {1) B1 (a) -A1 (a+ {1) B0 (a)}
+a {A0 (a+f1)B2 (a) -A2 (a+f1)B0 (a)} ],
152
S. Huzinaga
1 a3
[sa's,.'isas6 ]=--[ 4A1 (a) +2 {A0 (a+{3) B1 ((3) -A1 (a+ (3) B0 ((3)}
R4
+(3 {A0 (a+(3) B2 ((3) -A2 (a+ (3) B0 ((3)}],
[sasbis,/sl]=_!__ (a(3) 3 [45W0°(2, 2; a, (3) -15W0°(2, 0; a, (3)
R
90
-15W0° (0, 2 ; a, (3) + 5W0°(0, 0 ; a, (3) + 4W2° (0, 0 ; a, (3)].
(c)
Kgu
r= (1/2) (a+(3),
i3= (1/2) (a-(3),
1 20 (a(3) 3
R (a+(3) 5 '
[sasa'lsbs/]
=
1 4(a(3) 3
(3 [ A1 (r) B0 (r) +Ao (r) B1 (r) -2A1 (27)
R (a+ ) 3
+r{ (1/3)A0 (2r) -A2(27)} ],
[sasa' lsasb'] = 1 2 (a(3) 3 [2 {AI (r) Bo ( i3) -Ao (r) Bl ( i3)}
R (a+(3) 3
+2 {A0 (2T) B1 (a) -A1 (27) B0 (a)} +r {A0 (27) B2 (a) -A2 (2r) B0 (a)}],
. 1 2 (afdr [2 {Al Cr> Bo (i3) +Ao Cr) Bl (i3)}
R (a+ ) 3
+ 2 {A0 (27) B1 ((3) - A 1 (2r) B0 ((3)} +r {A0 (2r) B2 ((3) - A2 (27) B0 ((3)}],
-2G't 0 (0, i3)G't 0 (2, i3)W't 0 (2, 0; 7) +G't 0 (2, i3) 2W't 0 (0, 0; 7) ],
1
[sasb'lsa'sb]= R
(a(3) 3 ~(-1)'t(2r+1)[G't 0 (0,
8
't-O
i3) 2 W't 0 (2, 2 ;r)
-2G't 0 (0, i3)G't 0 (2, i3)W't 0 (2; o; r) +G't 0 (2, i3) 2W't 0 (0, o; r)J.
Here An(a), B,.(a), G/(m, a), w't~(m, n; a) and w't~(m, n; a, (3) are the functions
defined in KAIK's table13l, namely
w't~(m,
n; a)=W/(m, n; a, a),
The
R~[si,e
Potential bet-ween Two Nurmal Helium Atoms
153
where ~> is the larger of ~1 and ~2 and ·~ < the smaller. A.,. (a), B.,. (a), G~" (m, a)
and W/ (m, n; a) are tabulated in KAIK's table. The evaluations of W~" (m, n ; a, ~)
are made by using a series of recurrence formulas found also in the above mentioned
table. [ s..s&' Is,.sl] and [ s,.slJ sa' sb] in Kgu are expressed in infinite series. Fortunately the
convergence of these series is of sufficient rapidity for the present purpose, especially in
the combined form ;
[s,.slJs,.sl]- [s,.sb' Js,.' sb]
r
=__!_ (a~) 8 ±(4s+3) G2B+Ho, a) 2W2o+H2, 2; r)
R 4 •-0
L
References
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