Adiabatic potentials for alkali–inert gas systems in the ground state
S. H. Patil
Citation: J. Chem. Phys. 94, 8089 (1991); doi: 10.1063/1.460091
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Adiabatic potentials for alkali-inert gas systems in the ground state
s. H. Patil
Department 0/ Physics, Indian Institute o/Technology, Bombay 400 076, India
(Received 26 December 1990; accepted 19 February 1991)
We have developed perturbative expressions for the adiabatic potential for alkali-inert gas
systems. The predictions of the model for the equilibrium separation Rm and the minimum
potential em are in good agreement with the experimental values. The model provides a unified
approach to the interaction potential which indicates a conformal structure for some groups,
implying a useful relation between R m , em' and van der Waals constant C6 •
I. INTRODUCTION
Alkali-inert gas systems are of considerable experimental and theoretical interest. For understanding their interactions, it is essential to have accurate interatomic potentials.
As such, a great deal of effort has been directed towards
understanding and developing these potentials.
A. A brief review
Experimental determination of alkali-inert gas potentials are generally based on using appropriate model potentials to describe various scattering experiments. Fairly reliable ground-st,ate potentials were obtained for Ar, Kr, Xe
inert gases interacting with alkali atoms, by Buck and Pauly1 and later for Li interacting with He, Ne, Kr, and Xe, by
Dehmer and Wharton. 2 Several other groups2-6 have deduced potentials for other systems or confirmed earlier results.
A large number of theoretical attempts have been made
to deduce alkali-inert gas potentials, most of them based on
pseudopotentials. For example, Baylis7 used semiempirical,
statistical pseudopotentials to calculate alkali-inert gas potentials which are reasonably good for K, Rb, and Xe but are
poor for He and Ne. More recently, Czuchaj et al. s have
used semilocal, i-dependent pseudopotentials to obtain adiabatic potentials for alkali-neon systems. Their results are
particularly good for excited states. In most of these calculations, the loosely-bound electron of the alkali atom, interacts
with the alkali ion and the inert-gas atom via empirical pseudopotentials.
B. An outline of our work
We analyze the interaction between alkali atoms and
inert-gas atoms in the ground state in terms ofthe exchange
perturbation theory. Here the attractive forces are provided
mainly by the long-range, van der Waals terms, whereas the
short-range repUlsion is provided by exchange effects. Such
an approach is reasonable for the ground state in which the
equilibrium separation between the atoms is quite large, but
not for excited states in which the inert-gas atom is quite
close to the alkali ion. Weare encouraged to pursue this
approach since (a) quite accurate van der Waals coefficients
are now available9 •10 for alkali-inert gas atom systems, (b)
fairly reliable techniques have been developed 11.12 for evaluating short-range, exchange-Coulomb terms.
We consider the interaction energy as a sum of first and
J. Chern. Phys. 94 (12). 15 June 1991
second-order terms in the exchange perturbation theory.
The evaluation of the first-order terms requires the knowledge of one-and two-electron densities, particularly in the
asymptotic domain for the alkali atoms. For these we use the
wave functions of the valence electron in the alkali atoms,
given by Bates and Damgaard. 13 These are appropriate since
their two leading asymptotic terms conform to the general
asymptotic requirements. We also modify the van der Waals
terms by including suitable damping terms as discussed in
Ref. 11.
Using the exchange Coulomb terms, and damped van
derWaals terms, we obtain suitable expressions for alkaliinert gas potentials for the ground states. They allow us to
calculate well depths and separation at minimum potentials
for all 25 systems, which are in very good agreement with
empirical values. We conclude with a critical discussion of
the model, and its implications.
II. THE INTERACTION POTENTIAL
In this section we deduce the first- and second-order
terms in the perturbation series for the interaction energy,
using suitable expressions for electron densities.
A. Perturbative expression for E
The perturbation in the Hamiltonian for an alkali atom
A and inert-gas atom B is
V( ritrj
)
=l
IJ
(..!.. - _1___1_ + ..!..),
R
r bi
raj
(2.1 )
rij
where R is the separation between the atoms, i electrons
belong to A, j electrons belong to B,
(2.2)
fbi = f i -R,
faj
= fj + R,
(2.3)
fij=R+rj-ri'
(2.4)
with r i being the positions of i electrons with respect to A,
and rj being the positions of j electrons with respect to B.
Then, the first and second order energies of interaction are
given by
(.25)
(2.6)
where
0021-9606/91/128089-07$03.00
@ 1991 American Institute of Physics
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8089
8090
S. H. Patil: Potentials for alkali-inert gas systems
Vts = (1ftIVI1fs),
V;s = (Pob 1ft IV 11fs),
(2.7)
(2.8)
and 1ft = ¢a¢)b with ¢o and ¢b being the properly symmetrized eigenfunctions of A and B respectively, t = 0 is the
ground state, and Pob is the sum of all permutations between
the electrons of A and B with appropriate signs.
B. Electron densities
The evaluation of interaction energies requires the
knowledge of electron densities and correlation functions. In
particular, we need them in the asymptotic domain for the
alkali atom.
The asymptotic behavior for the wave function of an
alkali atom in the ground state is given byl4
A-.(N)(12
'/'0
" ••• , N)
--+r,U[l
+.vr -I + O( r -2)] e -r,/o
X¢6 N -')(2, ... ,N) for r,
--+ 00,
(2.9)
PB ()
r
£..
(rn) B = 41T(2n
(2.18 )
.
+ 2)!2: BI (b;l2)2n + 3.
(2.19)
C. First-order interaction energy
The evaluation of the first-order interaction energy is
greatly simplified by the observation that the electrons in the
inert-gas atoms are much more tightly bound than the valence electron in the alkali atoms, so that a ~ bI' For evaluating the Coulomb energy
Voo
=
J
3
d rd 3r'[R
-I
-IR - rl- I
-
I.R+ r'l
+ IR-r+r'I-I]PA(r)PB(r'),
we use the representation
1 IJ
2ff2
(2.20)
,k r
'
d 3 ke k2
(2.21)
(2.10)
(2.11)
v=-!a2 (a-1),
(2.12)
E, being the first ionization energy of the atom. The Bates-
(2.22)
with
Damgaard wave function 13 for the valence electron in alkali
atoms has the asymptotic behavior
1fBD (r) --+ (81T) -1I2rr(a + 1)] -1(2/a)0+ 1I2rU
I
X [1 + vr- + 0(r-2) ]e-
r o
/
which is consistent with Eq. (2.9). Furthermore, the normalization has been shown 15 to lead to accurate results for dipolar polarizabilities of alkali atoms. We can therefore write
the correlation function for the electrons in an alkali atom as
= 2: ¢:(r)¢a (r') --+1fBD (r)1fBD (r')
gA(k 2 )
= Jd3rpA(r)[l-e-,k.r],
gB (k 2)
=
J
d 3r' PB (r') [1 _ e 'k'r '],
for r--+ C!J,
(2.13)
for r,r'--+ 00.
o
(2.14)
Since PB ( r') is localized in a small region around r'
expand exp (zk'r') about r' = 0 to get
Voo
= - 4<f
-
1
n=I(2n+1)!
(rn) B (V~ )n~ IpA (R)
(2.23)
(2.24)
= 0 we
(2.25)
for R #0.
Since a~bl' for R ~ 1 we will generally include only the first
few leading terms in Eq. (2.25) and usepA (R) given in Eq.
(2.15).
The first-order exchange term is much harder to evalu:.ate. We start with the expression
The corresponding electron density has the behavior
PA(r)
2r/b/
I
r
u=a-1,
fA (r,r')
Ie
With this density we get
-=-
= (2E, ) - '12,
B -
i
where
a
~
=
= 2:1¢a(r) 12
Voo =
a
-
~Jd3rd3r' fA(r,R+r')fB(r',r-R)
(2.15)
X[R -1-IR-rl-I-IR+r'I-1
(2.16)
For the inert-gas atoms, the correlation function and the
density have a much more complicated structure since there
are eight electrons in the outermost shell, with different angular momenta. However, we will need only the general
form of these quantities and we take
(2.17)
+ IR-r+r'I-I],
(2.26)
where the factor of! comes from spin. This exchange term
gives rise to the repulsion between the two atoms at short
distances. Since fB (r',r - R) is localized mainly around
small values ofr' and r-R, the important region of integration is where the valence electron is near the inert-gas atom
B. Since the wave function of the valence electron varies
rather slowly near B, it may be regarded as having I = 0 with
respect to B. Therefore the exchange repulsion arises mainly
from the I = 0 terms in fB (r',r - R). The corresponding
contributions from I #0 terms are smaller and may, together
J. Chern. Phys., Vol. 94, No. 12, 15 June 1991
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S. H. Patil: Potentials for alkali-inert gas systems
with the Coulomb terms, provide rather small short-range
contributions. This then will be our model: we will consider
short-range terms arising only from the I = 0 terms. Though
we are guided by the preceding qualitative argument in formulating this model, its ultimate validity is in terms of the
agreement of its predictions with experimental observations.
Using expressions in Eqs. (2.14) and (2.17) for fA and
fB' we have
Voo~ -~A(R)~BJd3ud3r'exp( -r"Rla-u'Rla
- r' /b i
-
u/bi ) [
~+
-
u
1],
Ir' -ul
(2.27)
(2.28)
u=r-R.
Since a>;>b" we expand in inverse powers of a, and get
V!Jo
= 8ff2PA (R)"2:, Bi[~b i + E..b ?a~2
4
i
. where a is defined in Eq. (2.10) and b is the corresponding
value for the inert-gas atom,
b = (2Ebl
(2.36)
) -112,
Eb I being the first ionization energy of the inert-gas atom.
These damping factors were found II to be quite useful for
the description of inert-gas potentials.
III. RESULTS
In this section, we collect the different potential terms
and discuss the consequences.
A. Adiabatic potential
The adiabatic potential for alkali-inert gas systems in
the ground state is taken to be
E(R)
= j1TPA (R) [(r) B + §6(r4 ) Ba -2 + ~(r6) Ba -4]
12
- CJ" (R)R -
+~b~a-4+
... J.
10
6 -
CJg (R)R -
8,
(3.1)
(2.29)
I
where the asymptotic density PA (R) is given in Eq. (2.16),
(rn) B are the sums of the expectation values for the two swave electrons in the outermost shell of the inert-gas atom,
and the damping factorsh.L+4 (R) are given in Eq. (2.34).
Using Eq. (2.19) one can write
V!Jo
8091
= 21TPA(R)[(r)B +~(r4)Ba-2
135
+~(r6)
525
a- 4 + ... ].
(2.30)
B
Combining this with Eq. (2.25) for the Coulomb term, we
obtain for the first-order interaction energy
E(\) =j1TPA (R) [(r) B + ~(r4) Ba- 2
+ t!s(r )Ba- + ... ].
6
4
(2.31 )
The important point to be noted in view of our earlier discussion is that the expectation values in Eq. (2.31) are to be
taken for only the s-wave electrons in the inert-gas atom,
major contribution to which comes from the s-wave electrons in the outermost shell. Good estimations have been
obtained for these from relativistic Dirac-Fock calculations. 15 We will retain only the first three terms in the series
expansion.
B. Numerical results
For the evaluation of the interaction energy, we have
taken (rn) B for the s-wave electrons in the outermost shell
of the inert-gas atoms, from relativistic Dirac-Fock calculationsY There are two compilations of van der Waals constants for alkali-inert gas systems, Refs. 9 and 10. While they
are consistent with each other for Li, Na, K, and Rb, the
values given in Ref. 10 are smaller by about 10% for Cs. We
have taken the average values of the lower and upper bounds
given in these sources. For Na, the upper bound given in Ref.
10 is rather poor. In this case we take the average value to
bear the same ratio to the lower bound as for K.
The various values used for evaluating the adiabatic potentials are given in Tables I-V along with the input values l5
of (rn) B for the s-wave electrons in the outermost shell.
The values of the interaction potential are presented in
Tables I-V, for the scaled function
D. Van der Waals terms
(3.2)
The second-order interaction energy in Eq. (2.6) leads
to the long-range van der Waals terms:
E(2)(R) = -C6 R -6-Cg R -8+ ....
(2.32)
Quite reliable estimations are available9 ,lo for the van der
Waals coefficients. These terms require modifications for
smaller values of R. Earlier ll we had suggested that the van
der Waals coefficients be multiplied by damping factors
C2L + 4 --+C2L+~L+4 (R),
(2.33 )
where
h.L+4(R)
2L+7 1
-(R/a)n.
= l_e-R/a L
n!
Guided by our earlier considerations, II we take
(2.34)
n=O
a=!(a+b),
(2.35)
where Rm is the position of the minimum potential Em'
C. Discussion
The calculated values of Em and Rm given in Tables I-V
are seen to be in good agreement with the experimental values. These tables also contain the values for the scaled function W(R 1Rm) defined in Eq. (3.2), at several values of the
scaled variable R IR m , which can be used to obtain the potential in the region of interest. It is interesting to observe
that the scaled function W(R 1Rm) is approximately conformal for He (Table I) and Ne (Table II) interacting with
different alkali atoms. The conformal property is observed
for Ar, Kr, and Xe also but to a lesser extent, the variation
for different alkali atoms being about 10%. This conformal
structure implies that since the asymptotic behavior of the
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8092
S. H. Patil: Potentials for alkali-inert gas systems
TABLE I. Input parameters for the interaction potential, position Rm of the minimum potential Em' and some
values of W(R /Rm) = E(R)/Em for Li-He, Na-He, K-He, Rb-He, and Cs-He. Experimental values for
R ~Pt and f",,';pt, for Li-He are from Ref. 2.
E,
a
b
A
(r)B
(r4) B
(I') B
C6
Cs
Rm
R:pt
Em
~~Pt
W(0.774)
W(0.888)
W(0.972)
W(1.Q28)
W(1.112)
W(1.226)
W(1.416)
W(1.640)
Li-He
Na-He
K-He
Rb-He
Cs-He
0.1982
1.588
0.744
0.0519
2.37
7.78
5.05X 10 1
2.25x10 1
1.06 X 103
11.71
11.4
- 6.09 X 10- 6
-5.18XlO- 6
- 8.71
0.1889
1.627
0.744
0.0450
2.37
7.78
5.05XlO '
2.47 X 10 1
1.29XW
12.11
0.1595
1.771
0.744
0.0259
2.37
7.78
5.05Xl0 '
3.89XlO '
2.66XW
13.57
0.1535
1.805
0.744
0.0226
2.37
7.78
5.05X10 1
4.46Xl0 1
3.18XW
13.86
0.1431
1.869
0.744
0.0173
2.37
7.78
5.05XlO '
5.12X 10 '
4.34X10 3
14.61
- 5.48 X 10- 6
- 4.46 X 10- 6
- 4.48 X 10-"
-3.82XlO- 6
-8.70
0.0
0.926
0.975
0.765
0.474
0.205
0.082
-8.60
-1.0XlO- 2
0.960
0.976
0.764
0.473
0.204
0.082
- 8.40
-1.0XlO- 2
0.960
0.977
0.766
0.474
0.203
0.082
- 8.50
-1.0xlO- 2
0.962
0.975
0.762
0.469
0.200
0.080
-1.0XlO-~
0.962
0.977
0.775
0.480
0.207
0.083
scaled functions is the same for a given inert-gas atom interacting with different alkali atoms, one has
(3.3 )
where k6 depends only on the inert gas. The values of the
dimensionless constant k6 are given in Table VI for different
systems and are indeed found to vary by less than 5% from
the average value for a given inert-gas atom.
In order to place our results in a proper perspective, it is
necessary to analyze all the uncertainties and errors in the
calculated values and experimental numbers. To start with,
we note that while the experimental values for Rm are quite
reliable, generally accurate to within about 5%, the values
for em show a considerable variation. For example, in the
TABLE II. Input parameters for the interaction potential, position Rm of the minimum potential Em' and some
values of W(R /Rm) = E(R)/Em for Li-Ne, Na-Ne, K-Ne, Rb-Ne, and Cs-Ne. Experimental values for
R ~Pt and f",,';pt are the average values from Ref. 2 for Li-Ne, Ref. 6 for Na-Ne, Ref. 4 for K-Ne.
Li-Ne
E,
a
b
A
(r)B
(r4) B
(I') B
C6
Cs
Rm
R~Pt
Em
E~Pt
W(0.774)
W(0.888)
W(0.972)
W(1.Q28)
W(1.112)
W(1.226)
W(1.416)
W(1.640)
0.1982
1.588
0.794
0.0519
1.93
3.70
1.27 X 10 '
4.40X 10 '
2.14X10 3
9.22
9.46
- 4.22X 10- 5
- 4.23 X 10- 5
-3.70
0.410
0.975
0.982
0.818
0.548
0.255
0.105
Na-Ne
0.1889
1.627
0.794
0.0450
1.93
3.70
1.27 X 10 1
4.8X10 1
2.59X 103
9.55
10.0
-3.78XlO- 5
-3.7XlO- 5
-3.80
0.410
0.976
0.982
0.815
0.545
0.255
0.105
K-Ne
0.1595
1.771
0.794
0.0259
1.93
3.70
1.27 X 10 1
7.69X WI
5.32XW
10.60
10.2
- 3.17X 10- 5
-2.8XlO- S
- 3.40
0.430
0.970
0.982
0.818
0.547
0.256
0.105
Rb-Ne
Cs-Ne
0.1535
1.805
0.794
0.0226
1.93
3.70
1.27 X 10 1
8.90X101
6.34XW
10.77
0.1431
1.869
0.794
0.0173
1.93
3.70
1.27 X 10 '
1.02 X 102
8.62X 103
11.39
- 3.27X 10- 5
- 2.74X 10- 5
- 3.10
0.450
0.976
0.983
0.820
0.555
0.260
0.108
-3.10
0.450
0.977
0.983
0.821
0.554
0.260
0.108
J. Chern. Phys., Vol. 94, No. 12, 15 June 1991
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S. H. Patil: Potentials for alkali-inert gas systems
8093
TABLE III. Input parameters for the interaction potential, position Rm of the minimum potential Em' and
some values of W(RIR m ) = E(R)IEm for Li-Ar, Na-Ar, K-Ar, Rb-Ar, and Cs-Ar. Experimental values
for R :Pt and pt are from Ref. 1.
c.:
Li-Ar
E,
a
b
A
(~)8
(r4 ) 8
(I') 8
C6
Cs
Rm
R ~:pt
Em
e«=~Pt
W(0.774)
W(0.888)
W(0.972)
W(1.028)
W(1.112)
W( 1.226)
W( 1.416)
W(1.640)
0.1982
1.588
0.929
0.0519
4.66
1.85X 10'
1.20X 1Q2
1.75 X 10'
9.16X103
8.86
9.36
- 2.09 X 104
-1.95Xl0- 4
- 3.34
00430
0.974
0.984
0.825
0.558
0.262
0.109
Na-Ar
0.1889
1.627
0.929
0.0450
4.66
1.85x 10'
1.20 X 10'
1.90X 10'
1.11 X 104
9.14
9047
-1.91xlO- 4
-2.04XlO- 4
-3.30
0.440
0.976
0.983
0.820
0.5.51
0.258
0:107
K-Ar
0.1595
1.771
0.929
0.0259
4.66
1.85x 10'
1.20XW
2.99X 10'
2.24XI04
9.92
10.1
-1.76XI0- 4
-1.93XlO- 4
-2.71
00494
0.978
0.984
0.830
0.570
0.273
0.114
Rb-Ar
Cs-Ar
0.1535
1.805
0.929
0.0226
4.66
1.85 X 10'
1.20 X 10'
3040 X 10'
2.67 X 104
10.04
0.1431
1.869
0.929
0.0173
4.66
1.85X 10 1
1.20 X 10'
3.92XW
3.61 X 104
10046
lOA
-1.83Xl0- 4
- 2045
0.520
0.978
0.984
0.833
0.574
0.277
0.lf6
case of Li-He, Dehmer and Wharton2 give three values for
em (one of them is given in our Table I) which differ by
about 35% from the mean value. Furthermore, the semiempirical and ab initio calculations also differ quite significantly,rangingfrom -1.18Xl0- s inRef.l6,to -4.8XlO- 6
in Ref. 17. The variation is much less in the values for R m ,
between 11.4 and 12.0. The situation for other systems is
-1.64XlO- 4
-2.06XI0- 4
-2.20
0.545
0.979
0.985
0.836
0.579
0.281
0.118
somewhat better, particularly for Na-Ne, where the experimental value of Ref. 6 may be accurate to within 10% for em'
In this connection, we also note that the relation
(3.4)
for Na-Ne, given in Ref. 18, is satisfied quite satisfactorily by
the experimental numbers of Ref. 6, and by our predictions,
TABLE IV. Input parameters for the interaction potential, position Rm of the minimum potential Em' and
some values of W(R 1Rm) = E(R)/Em for Li-Kr, Na-Kr, K-Kr, Rb-Kr, and Cs-Kr. Experimental values
for E:pt and e",;:pt are from Ref. 2 for Li-Kr, Ref. 1 for Na-Kr, K-Kr, Rb-Kr, and Cs-Kr.
Li-Kr
E,
a
b
A
(,:')8
(r') B
(I') B
C6
C.
Rm
R~pt
Em
e:
pt
W(0.774)
W(0.888)
W(O.972)
W(1.028)
W( 1.112)
W(1.226)
W(1.416)
W(1.640)
0.1982
1.588
0.985
0.0519
5.88
2.80X 10'
2.08X 102
2.59X 10'
1.37X 104
8.84
9.08
- 3.17X 10- 4
- 3.28X10- 4
- 3.10
0.449
0.976
0.9.82
0.820
0.554
0.260
0.107
Na-Kr
0.1889
1.627
0.985
0.0450
5.88
2.80X 10 1
2.08X 1Q2
2.83X 10'
1.68 X 104
9.07
9.38
-2.98XIO- 4
-3.19XI0- 4
- 3.15
0.445
0.976
0.983
0.821
0.555
0.261
0.108
K-Kr
0.1595
1.771
0.985
0.0259
5.88
2.80X 10'
2.08 X 10'
4.42 X 10'
3.39X 104
9.78
9.91
-2.84xl0- 4
- 3.26X 10- 4
-2.55
0.520
0.977
0.984
0.832
0.572
0.274
0.115
Rb-Kr
0.1535
1.805
0.985
0.0226
5.88
2.80X10'
2.08XIO'
5.02X 10'
4.04X104
9.87
10.0
-2.97XI0- 4
- 3.33X 10- 4
-2.30
0.522
0.977
0.985
0.835
0.580
0.280
0.118
Cs-Kr
0.1431
1.869
0.985
Q.0173
5.88
2.80X10 1
2.08 X 10'
5.80XIO'
5047 X 104
10.22
10.3
-2.76XIO- 4
-3.37XIO- 4
-2.01
0.560
0.980
0.985
0.839
0.587
0.287
0.121
J. Chern. Phys., Vol. 94, No. 12, 15 June 1991
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S. H. Patit: Potentials for alkali-inert gas systems
8094
TABLE V. Input parameters for the interaction potential, position Rm of the minimum potential Em' and some
values of W(R IRm) = E(R)/Em for Li-Xe, Na-Xe, K-Xe, Rb-Xe, and Cs-Xe. Experimental values for
R ~Pt and E",,';pt are from Ref. 1.
Li-Xe
E,
a
b
A
(7'")8
(1')8
(I') 8
C6
C.
Rm
E:pt
Em
~~Pt
W(0.774)
W(0.888)
W(0.972)
W(1.028)
W(1.112)
W(1.226)
W( 1.416)
W( 1.640)
Na-Xe
0.1982
0.1889
.. 1.627
1.588
... 1.059
1.059
0.0519
0.0450
8.22
8.22
5.26x 10'
5.26X 10'
5.04X IQ2
5.04 X 102
4.06x 102
4.43 X 102
2.28X 10'
2.73 X 10'
9.12
9.38
9.26
9.57
- 4.30X 10-' - 3.99X 10-'
- 4.84X 10-' - 4.77 X 10-'
- 3.70
-3.60
0.40
0.40
0.974
0.973
0.983
0.984
0.817
0.820
0.547
0.550
0.253
0.255
0.104
0.104
K-Xe
Rb-Xe
0.1595
1.771
1.059
0.0259
8.22
5.26X 10'
5.04X 102
0.1535
1.805
1.059
0.0226
8.22
5.26X 10'
5.04x102
7.78X 10'
6.50X 10'
10.22
..
6.90 X 102
5.48 X 10'
10.10
9.92
- 3.85X 10-'
- 5.05xlO- 4
-3.00
0.45
0.975
0.984
0.828
0.563
0.267
0.110
-3.97XlO-'
-2.70
0.49
0.976
0.985
0.831
0.570
0.272
0.113
the values for the product being 3.7X 10- 4 and
3.61 X 10 - 4, respectively. In spite of the reservations about
the accuracy of the experimental numbers, the agreement of
the predictions of our model with the experimental values
for 15 sets of Rm and Em values ranging over two orders of
magnitude, is an indication that the model is meaningful and
deserves consideration.
Coming to the uncertainties in the inputs of our model,
it is obvious that the predictions are sensitive to the values of
the van der Waals coefficients C6 and Cs . Fortunately, the
values given in Refs. 9 and 10 are quite accurate, which allow
us to be confident about the asymptotic part of our potential.
We have not included higher order van der Waals terms,
which are smaller than the leading terms in the region under
consideration but which may not be negligible. Again, we
have considered only the leading order short-range exchange-Coulomb terms. The higher order terms are expected to be smaller in the region under consideration, but may
not be negligible. We also note that in order to get a compact,
analytic expression for the short-range exchange-Coulomb
terms, we have carried out an expansion essentially in pow-
TABLE VI. The values of the dimensionless constant kb defined in Eq.
(3.3), for different systems.
He
Ne
Ar
Kr
Xe
Li
Na
K
Rb
Cs
Average
1.43
1.70
1.73
1.71
1.64
1.43
1.69
1.71
1.71
1.64
1.40
1.11
1.78
1.80
1.40
1.74
1.81
1.83
1.72
1.38
1.70
1.82
1.84
1.75
1.41
1.71
1.77
1.78
1.69
·1.69
Cs-Xe
0.1431
1.869
1.059
0.0173
8.22
5.26Xl0'
5.04X102
8.98X IQ2
8.80X 10'
10.56
10.3
-3.71XlO-'
- 5.00X 10-'
-2.48
0.51
0.977
0.986
0.836
0.576
0.277
0.115
. ..
ers of (b /a)2Ieading to Eq. (2.31). Now (b /a)2 <0.44, so
that the higher order terms neglected in this expression are
smaller. We have estimated that this expression is correct to
about 15% which is acceptable at the level of approximation
involved in estimating other terms. Finally, the most significant point of our model is that we have considered only the
contributions of s-wave electrons in the inert-gas atoms to
the short-range interaction. Of course, in the case of He this
is not an approximation. But what is surprising is that the
same expression provides a satisfactory potential for other
inert-gas atoms as well. While we have given some qualitative arguments as to why the contributions of 11'0 electrons
may be small, that it is so much smaller than the contribution
of s-wave electrons is unexpected. In a sense one may regard
this as an empirical result. Overall, though the analysis is
based on several theoretical assumptions which are notjustified with sufficient rigor, at the least, it provides a useful
empirical representation of the potential with no free parameters. It is also worth mentioning that the approach based on
the short-range exchange-Coulomb interaction, and the
long-range van der Waals interaction has been found 11 to be
very accurate for describing inert gas-inert gas potentials.
However, for the inert gas-inert gas system, since the outer
electrons for both the atoms have similar wave functions, it is
possible to obtain fairly reliable expressions II for the shortrange exchange-Coulomb interaction.
In conclusion, the model with long-range, damped van
der Waals interaction and short-range exchange-Coulomb
interaction of s-wave electrons in the inert-gas atoms with
the valence electron in alkali atoms, provides a unified description of the adiabatic potential for alkali-inert gas systems in the ground state. In addition to making reliable predictions for the equilibrium separation Rm and the
J. Chern. Phys., Vol. 94, No. 12, 15 June 1991
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S. H. Patil: Potentials for alkali-inert gas systems
minimum potential Em' the model suggests a conformal
structure for groups of potentials, and a useful relation between Rm,Em , and C6 given in Eq. (3.3).
I U. Buck and H. Pauly, Z. Physik 208,390 (1968).
2p. Dehmer and L. Wharton, J. Chern. Phys. 57, 4821 (1972).
lR. Diiren, G. P. Raabe, and Ch. Schlier, Z. Phys. 214, 410 (1968).
·Ch. Schlier, Ann. Rev. Phys. Chern. 20, 191 (1969).
sG. M. Carter, D. E. Pritchard, M. Kaplan, and T. W.Ducas, Phys. Rev.
Lett. 35, 1144 (1975).
6W. P. Lapatovich, R. Ahmad-Bitar, P. E. Moskowitz, 1. Renhorn, R. A.
Gottscho, and D. E. Pritchard, J. Chern. Phys. 73,5419 (1980).
8095
7W. E. Baylis, J. Chern. Phys. 51, 2665 (1969).
.. _
8E. Czuchaj, F. Rebentrost, H. Stoll, and H. Preuss, Chern. Phys. 136, 79
(1989).
9K. T. Tang, J. M. Norbeck, and P. R. Certain, J. Chern. Phys. 64,3063
(1976).
10 J. M. Standard and P. R. Certain, J. Chern. Phys. 83, 3002 (1985).
liS. H. Pati!, J. Phys. B 20,3075 (1987).
12S. H. Patil, J. Chern. Phys. 86, 7000 (1987).
13 D. R. Bates and A. Darngaard, Philos. Trans. R. Soc. London, Ser. A 242,
101 (1949).
14S. H. Patil, J. Phys. B 22,2051 (1989).
IS J. P. Desclaux, At. Data Nuc. Data Tables 12, 311 (1973).
16 J. Pascale, Phys. Rev. A 28,632 (1983).
17M. Jungen and V. Staernrnler, J. Phys. B 21, 463 (1988).
18R. Diiren, A. Frick, and Ch. Schlier, J. Phys. B 5,1744 (1972).
J. Chern. Phys., Vol. 94, No. 12, 15 June 1991
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