Alkali ion–inert gas potentials
S. H. Patil
Citation: J. Chem. Phys. 86, 7000 (1987); doi: 10.1063/1.452348
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Alkali ion-inert gas potentials
s. H. Patil
Department ofPhysics, Indian Institute of Technology, Bombay 400 076, India
(Received 24 October 1986; accepted 3 February 1987)
Analytic expressions are obtained for the Coulomb and exchange energies for alkali ion-inert
gas interaction, incorporating the correct asymptotic behavior of electron densities. We have
also introduced suitable damping factors for the polarizability and van der Waals terms. This
allows us to obtain accurate values for the interaction energies.
I. INTRODUCTION
Recently several attempts have been made to understand the properties of alkali ion-inert gas systems in terms
of their interaction potentials.
A. A brief review
There are two major experimental approaches for deducing alkali ion-enert gas potentials. In the approach based
on the mobility measurements, the inversion technique developed by Viehland and Mason l has been used to obtain2-4
the potentials over a wide range of separations for K + , Rb +,
and Cs + interacting with Ar, Kr, and Xe. In the alternate
approach, potentials for K + interacting with Ne, Ar, Kr,
and Xe have been obtained5 by using beam scattering measurements. Though the two approaches give similar results,
there are some significant differences in their predictions,
well outside the mutual error estimates.
There has been only one major theoretical attempt to
deduce the alkali ion-inert gas potentials. Based on the electron-gas model, Waldman and Gordon 6 have obtained the
potentials for Na +, K + , and Rb + interacting with Ne, Ar,
Kr, and Xe. The theoretical predictions are in reasonably
good agreement with the experimental results though there
are some differences. In view of the differences in the various
experimental and theoretical results, it is important to obtain
reliable values for these potentials.
The alkali ion-inert gas interaction energy can be obtained from the perturbation expansion of Murrell and
Shaw7 or by a modification of Rittner's arguments 8 for alkali
halides. It can be written as (in atomic units)
-4 -
(!{3b
+ C6 )R -6 + B(R),
(1.1 )
where R is the separation between the alkali ion A + and the
inert gas atom D, a b and{3 b are the dipolar and quadrupolar
polarizabilities of D, C6 is the van der Waals coefficient, and
B(R) is the Born-Mayer repulsion energy. The repulsion
energy is difficult to evaluate and is the main source of error
in the estimation of E(R).
It has been found that one and two electron densities in
atoms or ions can be determined9-12 in terms oftheir susceptibilities by imposing certain asymptotic constraints. These
densities have led to accurate values for the energies ofRydberg states in alkali atoms, 9 for quadrupolar moments, 10 and
for interaction potentials of alkali halides II and of inert gas7000
II. PRELIMINARIES
We first give a summary of the quantum mechanical
results of Murrell and Shaw, 7 and then describe a method for
obtaining simple but reliable expressions for the electron
densities in alkali ions and inert gas atoms.
A. Perturbative expression for E(R)
In the quantum-mechanical exchange perturbation theory of Murrell and Shaw,? an expansion is carried out in
orders of electron exchange. It leads to energy in successive
orders,
E=E(1)+E(2)+ "',
E(1)= Voo +
I
E(2) =
Voo -Soo Voo + "',
(2.1 )
(2.2)
2
lVo,I +"',
(2.3)
,¥oEo-E,
B. An outline of our work
E(R) = - ¥tbR
es. 12 We use these densities to evaluate the first order Coulomb and exchange terms for the alkali ion-inert gas energy.
We also carry out a simple analysis of the damping of the
polarizability and van der Waals terms and deduce suitable
expressions for the damping factors. This allows us to obtain
accurate expressions for the interaction energy which are
used to resolve some of the differences between the different
experimental and also theoretical results.
J. Chem. Phys. 86 (12), 15 June 1987
where
V's = (tP,lVltPs)'
V;s = (PabtP,lVltPs)'
S;s = (PabtP,ltPs)'
(2.4)
(2.5)
(2.6)
Here one has tP, = ¢a¢b with ¢a and ¢b being eigenfunctions
of the alkali ion A + and inert gas atom D, respectively, t = 0
is the ground state, Pab is the sum of all permutations
between the electrons of A + and D with appropriate signs,
and V is the interaction potential between A + and D,
(2.7)
(2.8)
(2.9)
where R is the separation between A + and B, i electrons
0021-9606/87/127000-07$02.10
@ 1987 American Institute of Physics
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7001
S. H. Patil: Alkali ion-inert gas potentials
energy, we take the leading Thomas-Fermi expression
belong to A + andj electrons to B,
r bj =rj -R,
(2.10)
+ R,
(2.11 )
raj = rj
(2.12)
rij=R+rj-rj
with rj being the positions of i electrons with respect to the
nucleus of A + and rj being the positions ofj electrons with
respect to the nucleus of B. The separation of V into VI and
V2 is on the basis that if A + is regarded as an inert gas atom
with an additional positive charge at the nucleus, then VI is
the interaction between the neutral atoms and V2 represents
the interaction between the remaining positive charge and B.
B. Electron densities
The evaluation of the interaction energy requires the
knowledge of electron wave functions and densities in A +
and B. Since the separation between A + and B is generally
large, it is essential that the wave functions and densities
satisfy the correct asymptotic behavior.
The asymptotic behavior of electron wave functions is
given b y9,13
t/J(r ,r ,00.) _r't,/2e l 2
(2E')"'r,~,l2e -
-
Ea,b = 0.769 (Na,b )
4/3
(2.19)
,
where N a•b is the number of electrons in A + or B.
The densities must satisfy the conditions
f
3
(2.20)
pA,B(r)d r=Na,b'
3
JPA,B (r)rd r
= 6Xa,b'
(2.21)
where X is the diamagnetic susceptibility. We also observe
from Eq, (2.13), thati = 1,2,00' (orj = 1,2,00') terms correspond to the successive electron wave functions in Eq.
(2.13) so that their contributions to Na (or N b ) may be
expected to be equal
Alai =A2a~ =A 3aL
(2.22)
Blbi =B2b~ =B3b~.
(2.23)
We can solve for Aj and Bj by using Eqs. (2.20)-(2.23), to
obtain
(2.24)
(2E,) "'r,
(2.13)
(2.25)
(2.14)
These expressions can be used in Eq. (2.20) to determine A
and B whose contribution in our calculations, however, is
negligibly small.
For our analysis, we also need the correlation function
for the calculation of exchange effects. The outer electrons in
the inert gases are all in the I = 1 state so that for the long
range properties we may write for ion A +,
with
Un
=
(Z - N
+ n)(2/En )1/2 -
2,
where Z is the nuclear charge, N is the number of electrons,
E} is the separation energy of the last electron, E2 is the
separation energy of the next electron (after the last electron
is removed), etc. A reliable electron density should incorporate this behavior of the wave functions. The numerical values of Un are quite small for A + and B. We therefore propose
for the density of electrons in A + ,
PA (r)
=L
fA (r,r') =
L t/J':(r)t/Ja (r')
a
It/Ja (r) 12
a
(2.26)
3
~
--
~
j=1
A je -rla'+A e
-rIa
,
(2.15)
where
aj
=
(SE j
I' (
)-1/2,
JB
a = (SEa) -1/2
(2.16)
with E j being the separation energy of the ith successive electron, and Ea is the average energy in A +. In Eq. (2.15), the
i = 1,2,3 terms represent the contributions from outer electrons and the last term represents the contribution from the
inner electrons. Similarly for the density of electrons in atom
B, we have
3
~
( )-
PBr-~
Bje - rlbj
+B e
- rIb
,
(SE.)-1/2,
J
b = (SEb )-1/2
') _
-
~B
~ j
j
e -(r+")/2b,pI (A.A,)
rr .
Similary, for
(2.27)
We are now in a position toevaluateE(l)inEq. (2.1) by using
the densitiesPA.B (r) andfA.B (r,r').
III. FIRST ORDER INTERACTION ENERGY
The main contribution to the first order energy is given
by the first two terms in Eq. (2.2).
A. Penetration energy
where
=
r,r
= r'.
(2.17)
j=1
b.1
which reduces to the correct limit for r
atom B, one has
(2.1S)
with Ej being the separation energy of thejth successive electron, and Eb is the average energy in B. For the average
The first order penetration or Coulomb energy V00 is
evaluated by using the densities in Eqs. (2.15) and (2.17).
The integrations are simplified by using the representations
1 -2fd3k k
r - I =-1T
2
-2
ejk'r .
J. Chern. Phys .• Vol. 86. No. 12, 15 June 1987
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(3.1)
S. H. Patil: Alkali ion-inert gas potentials
7002
In this region, the PI functions in/A and/B may be replaced
by unity. Then the exchange interaction can be written as a
sum of contributions from l/R, l/rbi' l/raj , and l/rij terms
One then obtains
J
V(r,r')PA (r)PB (r')d 3 rd 3 r'
=
Voo
= (VI)oo
+ (V2 )oo
(3.2)
with
VtJo = V R + Vb + V~ + V~b'
The first term is given by
V R=
(3.9)
-~(l +Na-I)R - I IA,Bj [I(x,y,R)]2,
2
i,j
(3.10)
I
(V2 )OO = 1T'R 2
Bj (1 + x + y) (x + y) -3e -2(x+Yl,
where the overlap integral is
j
I(x,y,R) = !1T'R 3[G(X,y)
(3.4)
+ G(x, -
y)],
where
(3.12)
+ y)] -3[2xy(x - y - 4xy)
+ (x + y)2(X - Y - 2xy) ]e- 2(x- yl,
x = !R(b j-I + 0,-1),
y = !R(b j - I - ai-I).
=
F(x,y)
[xy(x
(3.5)
The second term is
Vb
(3.6)
= ~R 2 I
4
(3.7)
AiBJ(x,y,R) (xy) -2
',j
+ (2xy + y -
X [(x - y)e- (x+yl
and the third term is
The first order exchange energy can be written as
VtJo = -
~
J
V~
V(r,r')/A (r,R + r')
where the factor of 1/2 is because of spin and/A,B are given in
Eqs. (2.26) and (2.27), To evaluate this expression we first
note that for given values of rand r', the exponents in/A and
IB have the smallest magnitudes for r and r' parallel to ± R.
=
~(1 + N a- I )1T'R 2 I
4
AiBjI(x,y,R) (xy)-2
i,j
X [(x + y)e- (x-yl - (2xy + y + x)e- (X+Yl].
(3.8)
a
x)e- (x-Yl],
(3.13)
B. Exchange energy
V'b
(3.11 )
G(x,y) = (xy)-3[(X+ l)y2+ (y_l)x 2]e-(X- Yl.
(3.14)
The last term is evaluated 14 by using the Neumann expansion for l/rij and elliptic coordinates, and is given to a high
accuracy by (see the Appendix)
=~rR5 IA.B.x-6[(~x4+x3+~x2-~x)e-2x+2S( -x)S( -x)EI (4x)
10
i,j'
9
J
12
24
- 4S(x)S( - x)E I (2x) - 2S(x)S(x)(C + Inx) ]
T'
ab
+ T~b'
(3.15)
=_~R5"A.B."(il+1){[U(X)(1+/2+/)+4/(/+1)+3][(21'_g)2_~15
-~15]
16
t;' J"'7
x
8x
"I
1
9 10
225 12
(3.16)
where C = 0.5772 is the Euler constant, and
S(z)
= (jz1 +z+
E(z)
=
1
l)e- z ,
_
(3.17)
2
gl - 21 + 1
[(/+ 1)(/+2) J;
21 + 3
1+ 2
00
t- I e-tdt,
u(x) =!(C+lnx)
dl =
I
(3.18)
-dl ,
(3.19)
(3.23 )
1
m - 1,
(3.20)
do = 0,
m=1
J;+
I
2
(21 + 1)(2/ + 2/- 1) J; + l(l- 1) J;
].
/ 2
+
(21+3)(2/-1)
1
2/_1
-
= (l/y) (21 + 1)J;
to =y-I sinhy,
+J;- 1>
(3.21)
II =y-I(1o - coshy),
(3.22)
Since theJ; andgl functions decrease very rapidly with I, we
retain only the first four terms in the summation over 1in Eq.
(3.16).
J. Chem. Phys., Vol. 86, No. 12, 15 June 1987
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S. H. Patil: Alkali ion-inert gas potentials
IV. SECOND ORDER TERMS
The second order terms are given in terms of multipolar
polarizabilities of the inert gas atom B, and the van der
Waals coefficients.
7003
Taking the density to be ofthe form exp( - rlt):
2L+ 2 1
S(L,R) = 1- e- R1t
(R It)".
L -
(4.6)
n!
Since the main contribution to S(L,R) comes from the first
11=0
two terms ofthe density in Eq. (2.17), we take
A. Dipolar and quadrupolar potentials
The leading contributions of V2 to the second order interaction energy are given in terms of dipolar moment ab
and quadrupolar moment /3 b of B:
(4.1 )
While the dipolar contribution is dominant, the quadrupolar
contribution also is significant. We had recentlylO examined
the various predictions for /3 b and recommended suitable
values for them. These values are expected to be accurate to
within about 10%.
B. van der Waals potential
The leading contribution of VI to the second order interaction energy is given in terms of the van der Waals constant
C6 :
(4.2)
(4.7)
The expansion of the potential leading to the van der
Waals potential is valid for R > ri + rj • The corrections may
be incorporated 12 by taking an additional R dependence in
the van der Waals coefficients,
F 6(R) - SPA (r)PB (r')rr'2()(R - r - r')d 3rd 3 r'.
(4.8)
Using for PA (r) andPB (r), a term of the form exp( - rlq)
we find that the van der Waals term needs to be multiplied by
a damping factor:
F 6 (R) = 1 -
e- R1q
±~
,,=0
n!
(R /q)".
(4.9)
Since the first two terms in the densities in Eqs. (2.15) and
(2.17) give the main contributions, we take
(4.10)
The values of C6 have been calculated by Mahan 15 for Na + ,
K+, Rb+ interacting with Ne, Ar, Kr. For the others, we
take suitable extrapolations using the ratios from inert gas
coefficients. Overall, these values are expected to be accurate
to within about 10%.
The damping factor in Eq. (4.9) is similar to the empirical
expression suggested by Tang and Toennies,16 though there
are differences in the number of terms and the behavior for
R---O.
C. Corrections at short distances
V.RESULTS
The polarizability and van der Waals potentials are
terms which represent the correct form of the asymptotic
potential. For finite separations they require corrections.
The polarizability terms come from an expansion of
IR - rl- I for R > r. The corrections may be incorporated by
taking R -dependent polarizabilities II
In this section we collect the different terms in the potential and discuss the consequences.
= as(1,R),
(4.3)
/3(R) =/3S(2,R),
(4.4)
a(R)
where
S(L,R) =
!'f"
foR p(r)r Ld 3
p(r)r2L d 3r.
(4.5)
A. The Interaction potential
The first and second order terms together give us
E(R) = Voo
+ Voo -
!abS(1,R)R
- !/3b S (2,R)R
-6 -
-4
CJi'6(R)R
-6,
where Voo is given inEq. (3.2), Voo inEq. (3.9),S(L,R) in
Eq. (4.6), and F 6 (R) in Eq. (4.9). For the choice of Xa
needed to determine Ai> we note that there are several 17- 21
calculations and some empirical deductions for the suscepti-
TABLE I. Input values of a, or bi' a or b, susceptibility X. coefficients A, or Bi' and polarizabilitiesa andp, all
in atomic units.
a. orb.
a2 or b2
03 or b3
oorb
X
A.orB.
A 2 0r B2
A 3 0r B3
a
P
(5.1)
K+
Rb+
Cs+
Ne
Ar
Kr
Xe
0.327
0.272
0.236
0.055
3.05
7.35
12.78
19.56
0.352
0.292
0.249
0.036
4.9
8.20
14.37
23.18
0.368
0.282
0.244
0.028
7.8
11.32
25.15
38.83
0.397
0.288
0.231
0.087
1.42
1.48
3.88
7.23
2.7
9.2
0.465
0.351
0.288
0.059
4.12
1.92
4.46
8.08
11.1
54.4
0.492
0.372
0.304
0.037
6.06
2.12
4.92
9.01
16.8
92.4
0.529
0.401
0.326
0.028
9.24
2.26
5.19
9.65
27.3
174.7
J. Chern. Phys., Vol. 86, No. 12, 15 June 1987
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S. H. Patil: Alkali ion-inert gas potentials
7004
TABLE II. Input values of C6 • and predictions for well depth~, equilibrium position Rm. and some values of W(R 1Rm) =
K+-Ne
C6
~
Rm
W(0.85)
W(0.92)
W(0.97)
W(0.99)
W( 1.01)
W( 1.03)
W(1.08)
W( U5)
W( 1.27)
W(1.47)
12.7
1.48 X 10- 3
5.39
0.40
-0.727
-0.974
- 0.996
- 0.997
- 0.983
- 0.897
- 0.752
-0.513
-0.279
K+-Ar
K+-Xe
K+-Kr
38.6
4.20X 10- 3
5.79
0.26
- 0.750
-0.976
-0.997
-0.997
- 0.985
-0.904
-0.764
-0.529
-0.292
53.5
5.31 X 10- 3
6.03
0.22
- 0.761
-0.977
- 0.997
-0.997
-0.984
-0.903
-0.764
- 0.531
-0.294
bilities. We generally favor the adjusted Hartree-Fock results 17• 18 where we mUltiply the calculated susceptibility of
A + by the ratio of the experimental susceptibility to the calculated susceptibility of the corresponding inert gas. These
are expected to be accurate to within about 5%. The quadrupolar polarizabilities ofNe, Ar, Kr are taken to be the values
recommended in Ref. 10, while that of Xe is taken to be
slightly larger than the one recommended in Ref. 10. The
various input values are listed in Table I. Using these, the
interaction energy can be obtained from Eq. (5.1).
B. Numerical results
The predictions for some values of the potential are given in Tables II and III.
As mentioned in the Introduction, there are significant
differences between the potentials obtained from mobility
measurements,2-4 and those obtained from beam experiments, 5 as also the theoretical potentials6 obtained from the
electron gas model. Our predictions for the potentials can be
used to select reliable values from the different results. We
have summarized the different predictions for the equilibrium distances Rm and well depths € of different alkali ioninert gas systems, in Table IV.
K + -inert gas: Our results are in close agreement with
73.0
6.81X 10- 3
6.34
0.22
- 0.758
- 0.975
-0.996
- 0.997
- 0.985
-0.908
-0.772
- 0.539
-0.300
Rb+-Ne
C6
~
Rm
W(0.85)
W(0.92)
W(0.97)
W(0.99)
W( 1.01)
W(1.03)
W(1.08)
W( 1.15)
W( 1.27)
W( 1.47)
82.9
4.44x 10- 3
6.39
0.428
- 0.719
- 0.973
- 0.995
- 0.998
-0.984
-0.901
- 0.757
-0.520
- 0.285
Rb+-Xe
59.5
3.42x 10- 3
6.2
0.49
- 0.731
-0.977
- 0.993
- 0.997
-0.980
-0.894
-0.749
-0.512
- 0.279
those of electron gas calculations. 6 The theoretical results
agree well with those of beam experiments for Ne and Ar but
the mobility experiments2give a much larger Rm and a much
smaller € for Ar. All the four predictions are in essential
agreement with each other for Kr. For Xe, the predictions
for Rm are in agreement with each other, but the theoretical
predictions support the prediction of the mobility experiment4 for€.
Rb + -inert gas: Our predictions are again in good agreement with those of electron gas calculations. The predictions
of the mobility experiment2·3 are in reasonable agreement
with the theoretical predictions for Ar and Kr, but the experimental predictions4 for Xe give a much larger Rm and a
much smaller €.
Cs + -inert gas: Our predictions and those of the mobility
experiments2-4 are in good agreement for Ar, and reasonable
agreement for Kr. The large value of Rm and the small value
of € predicted by the mobility experiments for Xe (interacting with Cs + or Rb +) are against the expected trend, and
difficult to understand.
C. Discussion
Our predictions for the equilibrium distance Rm and
well depth € are generally in good agreement with the predic-
Cs+-Ne
115.0
5.67X 10- 3
6.73
0.346
- 0.743
- 0.976
-0.997
-0.998
-0.984
-0.901
- 0.759
-0.524
-0.289
all in atomic units.
Rb+-Ar
19.1
1.24x 10- 3
5.74
0.58
-0.700
- 0.973
- 0.997
- 0.998
- 0.983
-0.892
-0.740
-0.498
- 0.267
TABLE III. Input values ofC6 • and predictions for well depth~, equilibrium position Rm, and some values of W(R 1Rm) =
Rb+-Kr
E(R)/~.
27.0
1.10X 10- 3
6.05
0.921
-0.688
-0.974
- 0.993
-0.997
-0.980
-0.883
-0.727
- 0.483
-0.255
Cs+-Ar
89.0
3.08 X 10- 3
6.50
0.430
-0.720
- 0.972
- 0.995
-0.997
-0.979
- 0.884
- 0.732
- 0.493
-0.265
E(R)/~.
Cs+-Kr
122.0
3.87XIO- 3
6.72
0.700
-0.700
-0.970
-0.994
- 0.997
- 0.982
-0.895
-0.747
-0.507
- 0.275
all in atomic units.
Cs+-Xe
180.0
4.89X 10- 3
7.08
0.572
-0.697
- 0.975
- 0.996
- 0.998
- 0.985
-0.900
-0.754
- 0.515
-0.281
J. Chem. Phys .• Vol. 86, No. 12, 15 June 1987
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S. H. Patil: Alkali ion-inert gas potentials
7005
TABLE IV. Predictions for the equilibrium distance Rrn and well depth €, by the present calculations, electron
gas calculations (Ref. 6) (WG), beam experiments (Ref. 5), and mobility experiments (Refs. 2-4), all in
atomic units.
K+-Ne
Rrn
€
K+-Ar
Rrn
€
K+-Kr
Rrn
K+-Xe
Rrn
Rb+-Ne
Rrn
Rb+-Ar
Rrn
€
€
€
€
Rb+-Kr
Rrn
Rb+-Xe
Rm
Cs+-Ne
Rrn
Cs+-Ar
Rrn
€
€
€
€
Cs+-Kr
Rrn
Cs+-Xe
Rrn
€
€
Present
WG
Beamexpt.
Mobility expt.
5.39
1.48 X 10- 3
5.79
4.20X 10- 3
6.03
5.31 X 10- 3
6.34
6.81 X 10- 3
5.74
1.24 X 10- 3
6.20
3.42X 10- 3
6.39
4.44 X 10- 3
6.73
5.67X 10- 3
6.05
1.10X 10- 3
6.50
3.08X 10- 3
6.72
3.87X 10- 3
7.08
4.87X 10- 3
5.54
1.43 X 10- 3
5.90
4.08 X 10- 3
6.04
5.40 X 10- 3
6.35
6.88X 10- 3
5.92
1.25 X 10- 3
6.24
3.6OX 10- 3
6.41
4.67X 10- 3
6.69
6.03 X 10- 3
5.29
1.43 X 10- 3
5.61
4.41 X 10- 3
5.94
4.85XIO- 3
6.39
5.85X 10- 3
6.28
3.13X 10- 3
6.20
5.llXIO- 3
6.52
6.88X 10- 3
6.63
3.IOXIO- 3
6.60
4.45 X 10- 3
7.67
4.00 X 10- 3
tions of the electron gas calculations, 6 also in reasonable
agreement with the beam data,S and except for Rb+ -Xe,
Cs+ -Xe with the mobility results. 2 ,4 We have plotted our
potential for K+ -Kr in Fig. 1, and compared it with the
mobility predictions 2 and the N-6-4 potentials from the
beam data. This comparison is quite instructive. It is observed that for large R, our potential agrees very well with
the mobility potential indicating the correct choice for the
van der Waals and quadrupolar terms. It is also observed
.002
.001
\
E(R)
6.5
3.23X 10- 3
6.7
4.20X 10- 3
7.4
4.54X 10- 3
0
R
- .001
- .002
-.003
-.004
-.005
-,oos'---_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _......J
FIG. 1. Plot of interaction energy as a function of R, for K + -Kr, in atomic
units, solid line for the present calculation, long dashed line for the beam
experiment (Ref. 5), and short dashed line for the mobility experiment
(Ref. 2).
that in the N-6-4 potential obtained from the beam experiments,S the power representation of the short-range repulsion is inadequate, and one should use exponential terms for
its description.
Our major input has been the use of densities and correlationfunctionsinEqs. (2.15), (2.17), (2.26), and (2.27) in
evaluating the first order Coulomb and exchange terms.
They have the necessary asymptotic behavior and also give
the correct susceptibility and charge. The importance of the
correct asymptotic behavior may be seen from the fact that
even a 1% error in the exponent of the leading term introduces more than 10% error in the potentials. It may also be
noted that since the last term in the densities gives a negligibly small contribution to the susceptibility as well as to the
potential, the results do not depend significantly on the precise value of the average energy (for simplicity we have used
the Thomas-Fermi value).
The reliability of our densities can be assessed by comparing them with the results of other calculations. For example, the Hartree-Fock wave functions of Mann 22 give for the
densities of Ar at r = 4.29, 6.45, and 8.18, values
2.54X 10- 4 , 1.96x 10- 6 , and 4.18X 10- 8 , whereas our expressions give values 2.14X 10-4, 1.86x 10- 6 , and
4.43 X 10- 8 , respectively. A further support to our densities
is provided by a modified Thomas-Fermi model23 which incorporates the correct asymptotic behavior, which predicts
densities of 2.31 X 10-4, 2.08x 10- 6 , and 4.87X 10- 8 , respectively, for the above values of r. We also note that the use
of our densities has yielded good results for Rydberg levels of
alkali atoms, 9 quadrupolar polarizabilities,1O and potentialsY·12 We therefore believe that our Coulomb-exchange
J. Chern. Phys., Vol. 86, No. 12, 15 June 1987
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S. H. Patil: Alkali ion-inert gas potentials
7006
terms in Eq. (5.1) are quite reliable, probably to the extent of
10%. Since the Coulomb-exchange terms are about 1/2 of
the total potential at equilibrium separation, this introduces
an uncertainty of about 5% in the potential. In the second
order terms, the dipolar term which is quite accurately
known is the most important term, and the errors in the
remaining terms are not very significant. Overall, we expect
our predictions to be accurate to within about 5% for E and
3% for Rm'
Our approach emphasizes the importance of surface
properties of atoms and ions (correct asymptotic behavior
and susceptibility). As such it would not be expected to give
a good description of Na + which is a relatively small and
compact ion. We therefore have not given our predictions for
Na + -inert gas systems though we have found that our predictions are in reasonable agreement with those of electron
gas calculations. 6
ACKNOWLEDGMENT
I thank Mr. G. S. Shah for his help in numerical calculations.
APPENDIX
In terms of the elliptic coordinates
P = IR - rl + r,
q = IR - rl - r,
p'=r'+ IR+r'l, q'=r'-IR+r'l.
(AI)
The Neumann expansion 14 for l/r 12 is
1
2 '"
-=-
r 12
L L/
(-l)m(21+ I)
R/=om=-/
x[(/-lmj}!I(/+ Iml)!)2Plm '(p)
xP .lml(q)Q lm'( p')P lm'(q')eim(tPt - tP2),
(A2)
where Pi and Q i are the associated Legendre functions of
the first and second kind, respectively. This expression is
valid for p' > p, and one must interchange p and p' for p' <p.
SubstitutingEq. (A2) inEq. (3.8) and carrying out some of
the elementary integrations, we get
X
i
'"
'P'
dP'l dp e-x(p+p')W( p',p,y) ,
(A3)
W( p',p,y) = QJ (p')p/ (p) [2p2}; (y) - g/ (y»
X [2p':!;(y) -g/(y)],
(A4)
where}; andgl are defined in Eqs. (3.21)-(3.23). Sinceyis
relatively small, we separate W into two parts:
W(p',p,y)
=
W(p',p,O)
+ [W(p',p,y) -
W(p',p,O»),
(A5)
where the first term gives the usual Sugiura result, 14 i.e., Eq.
(3.15) without T ~b' For evaluating the contribution of the
second term, we note that x is quite large and expand the
second term about p = p' = 1. Retaining the first two terms
in the expansion inp as also inp', we obtainT~b'
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J. Chern. Phys., Vol. 86, No. 12, 15 June 1987
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