Susceptibility and polarizability of atoms and ions
S. H. Patil
Citation: J. Chem. Phys. 83, 5764 (1985); doi: 10.1063/1.449654
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Susceptibility and polarizability of atoms and ions
s. H.
Patil
Department of Physics, Indian Institute of Technology, Bombay-400 076, India
(Received 29 August 1984; accepted 25 June 1985)
With the help of some relations for the oscillator strengths and semiclassical relations, we have
calculated diamagnetic susceptibilities and dipole polarizabilities for a large number of atoms and
ions. We have also obtained simple expressions for susceptibility and polarizability in terms of
ionization energy.
I. INTRODUCTION
The response of an atom to an external, weak static electric field is described mainly by its electric dipole polarizability. The average polarizability of an atom is given by (in
atomic units)
~~ L 1(01 ~+)I'
a
p
3 k
Ek -Eo
(Ll)
The calculation of this quantity encounters formidable difficulties because (a) the states required for the evaluation of the
matrix elements are many-electron states, (b) the expression
requires a summation over an infinite number of intermediate states. As a result, even with very elaborate computations, large uncertainties still remain in the theoretical estimations of atomic polarizabilities.
Another property of interest, especially in the case of
inert gases and their isoelectronic sequences, is the diamagnetic susceptibility. It is given by
(1.2)
It is intimately related to dipole polarizability and can play
an essential role in the determination of polarizability.
A. A very brief review
The state of experimental and theoretical progress in the
determination of atomic polarizabilities has been comprehensively reviewed by Miller and Bederson. 1 They also list
some recommmended values for the polarizabilities along
with estimated errors. Here we mention a few of the more
recent theoretical efforts.
The coupled Hartree-Fock approximation first proposed by Dalgarn0 2 has been used by Markiewicz, McEachran, and Stauffer3 to obtain accurate values for the polarizabilities of some atoms and ions with two electrons in the
outermost shell. Using the self-consistent field method, Voegel, Hinze, and Tobin4 have calculated the polarizabilities of
the atoms in the He-Ne series. Density dependent potentials
have been used by Mahan 5 to calculate multipole polarizabilities of several atoms and ions with closed shells. For the
alkali metals, accurate values have been obtained by using
configuration interaction approach, 6 pseudopotential approach, 7 and effective quantum number approach. 8.9
More directly of interest to us are the calculations based
on statistical approaches. It was shown by Bruch and Lehnen lO that the Thomas-Fermi-Dirac model gives polariza5764
J. Chem. Phys. 83 (11), 1 December 1985
bilities for inert gases, which are many times larger than the
observed values although they do show improvement for
larger atomic numbers. On the other hand, the calculated
values for alkaline earths are smaller than the experimental
values. Similar calculations have also been carried out by
Shevelko and Vinogradov.ll
Diamagnetic susceptibilities have been calculated 12 by
using Hartree-Fock wave functions, for inert gases, to within about 10% accuracy. Recently, efforts have also been
made 13 •14 to calculate susceptibilities from extensions of the
Thomas-Fermi model.
B. An outline of our approach
The polarizability of an atom depends quite significantlyon the large r behavior of atomic wave functions. Therefore, statistical approaches do not readily yield accurate values for atomic polarizabilities. It has, however, been shown 14
that iflarge rbehavior is incorporated in the Thomas-Fermi
model, one gets reliable values for diamagnetic susceptibilities of inert gases, values which are comparable in accuracy
to those from Hartree-Fock calculations. With this encouragement, we have analyzed atomic polarizabilities and susceptibilities using a semiclassical approach. The analysis is
presented in three parts.
(a) We consider the well-known function Stu) defined
as 15-22
Stu) =
L (Ek -
EotlkO'
k
(1.3)
IkO = (Ek - Eo) 1 (01
~ r Ik )
i
2
1
,
in terms of which one has
ap
=~S(
(1.4)
- 2).
It is known that Stu) is relatively easier to evaluate for
u = - 1,0, 1, 2, and has a pole at u = 5/2. The values ofS (0)
and S(2) are particularly simple to estimate, while S( - 1)
and S (1) are related to the diamagnetic susceptibility and the
energy of the system, and the two-particle probability function.
(b) We obtain an expression for the two-particle probability function within a semiclassical approximation, which
allows us to estimate S ( - 1) and S (1). The value of S ( - 1)
depends on the diamagnetic susceptibility which is calculated within the framework of a modified Thomas-Fermi model which incorporates the correct asymptotic behavior. 14
(c)WiththehelpofthevaluesofS(u)foru = - 1,0,1,2,
0021-9606/85/235764-08$02.10
@ 1985 American Institute of Physics
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S. H. Patil: Susceptibility and polarizability
we develop a suitable expression for S (u), which is then used
for extrapolation to u = - 2 and hence obtain dipole polarizabilityap-
5765
(2.7)
In obtaining Eq. (2.6), we have made use ofVinti's relation 24
(2.8)
c. Results
The main results are summarized here.
(a) Of general interest are the expressions we have obtained for S ( - 1) and S (1) based on a semiclhssical approximation.
(b) We have calculated polarizabilities of a large number
of atoms and ions. The agreement betweeen the calculated
values and the experimental values is good over the range
Ne-Ba. Especially striking is the agreement between the calculated values for the ten-electron isoelectronic sequence
and the empirical values of Edlen. 23
(c) We have also calculated diamagnetic susceptibilities
for these atoms and ions. Since the susceptibilities play an
important role in the analysis, we have obtained a semiempirical expression for them. The agreement between the calculated values, those from the semiempirical expression, and
the available experimental values is quite good.
(d) Combining our results with the well-known relation
a p = 16 X2/N, we have obtained two approximate relations
for susceptibility and polarizability,
x = N /4E;(lm + 1),
a p =N/E~(lm + W,
(1.6)
II. EXPRESSIONS FOR S(u)
It is known l5- 22 that while S( - 2) is quite difficult to
evaluate,S(u) is relatively easier to analyze for u = - 1,0,1,
2. We first simplify S (u) for these values.
A. S(u) for u = - 1, 0, 1, 2
The well-known 15-22 expressions for S (u) are
(ol( ~ r;Ylo),
=
(2.1)
S(O)=~,
S(2)
(2.2)
(ol( ~ p;Ylo),
(2.3)
= 21TZ (01 ~ 8(r;) 10),
(2.4)
S(l) =
where N is the number of electrons in the atom or ion and Z
is the nuclear charge. In terms of the electron density PI(r)
and the two-electron density P12(r,r'), the expressions for
S( - 1), S(l), and S(2) reduce to
S( - 1) =
J
PI(r)r d 3r
(2.9)
with Vtot being the total potential energy and Veo being the
energy due to electron-electron interaction. Now the virial
theorem implies that I Vtot I = 21E, I where E, is the total energy. Also I Vee I is expected to be relatively small. Assuming
an electron density of D exp ( - 2r Z 1/3) as suggested by the
Thomas-Fermi model, IVeo I is estimated to be about 20% of
IVtot I for neutral atoms, and smaller for positive ions. One
may therefore write
S(1):::::2.4IE, 1 +Z
+
J
pdr,r')r r' d 3rd 3r', (2.5)
0
f
pdr,r')
r~r' d 3rd 3r'.
(2.10)
Proceeding directly with Eq. (2.3), Dehmer et a/. 22 have obtained a somewhat different expression,
Sill
=
IE,I
+ L (Olp;· PjIO).
(2.11)
;#j
(1.5)
where N is the number of electrons, E; is the ionization energy and /m is the highest / value of the occupied states. The
predictions ofEqs. (1.5) and (1.6) are in quite good agreement
with the observed values for those neutral atoms which do
not have a low-lying excited state.
S( -1)
The first term on the right-hand side of Eq. (2.6) represents the negative of the energy due to interaction of the
electrons with the nucleus. Designating it by IVen lone has
The advantage of using Eq. (2.6) is that S (1) can be expressed
in terms of the density functions in the r space.
One can estimate S (2) by noting that most of the contribution to PI(O) comes from the s-wave electrons in the
n = 1,2 states. This leads to
(2.12)
S(2):::::4Z [(Z - O.4f +!(Z - 2.4)3],
where we have used23 a screening charge of 0.4 for the n = 1
electrons and 2.4 for the n = 2 electrons.
B. S(u) near u = 5/2
It is known 22 that S (u) diverges at u = 5/2. This divergence must come from the summation over the large values
of Ek in Eq. (1.3). The matrix elements involving such states
emphasize the behavior at small r. One therefore expects to
get a good estimation for S (u) near the singular point by
considering the innermost electrons.
For Ek very large, one may ignore the interaction and
use free-particle wave functions. Then for n = 1 states, we
have the large-Ek matrix element
(0Ixlk):::::(Z3/1T)1/2
J
x exp(ik o r-rZ)d 3 r.
(2.13)
Hence the singular part of Stu) is given approximately by
Sd(u):::::20481TZ
5
d 3k (k 2 + Z 2)U +
(21T)3
2
J
I
k2
(k2+Z2)6'
(2.14)
where the subscript d indicates that we are considering the
divergent part. It is observed that this integral diverges at
u = 5/2. For later reference, we note that
Sd(2)/Sd(1) = 3Z 2.
(2.1S)
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5766
S. H. Patil: Susceptibility and polarizability
III. SEMICLASSICAL APPROACH
In Sec. II we discussed the expressions for Stu) for
- 1,0, 1, and 2. In particular, it was shown that S ( - 1)
and S (1) can be expressed in terms of one- and two-electron
density functions. Here. we discuss these functions within a
semiclassical approach and obtain closed expressions for
S( - 1) andS(!).
u=
where 8 12 is the angle between r and r'. This term may be
expected to be significant mainly for r near r'. Using the
identity
(3.9)
sin WI sin W2 = ![COS(WI - W2) - cos(w} + w2)]
for the sin functions, and neglecting the second term which
oscillates rapidly one gets
1
F 12(r,r'):::::-)'
A. One- and two-electron densities
81T
We consider wave functions of the form
l/I(rl,r2' ... ) = (N!)-1/2£ij ••• ~;(rl)tPj(r2) .•• ,
= 2 2: ItPj(rW,
(3.2)
j
where the factor of 2 is for the spin states of the electrons.
Similarly, the two-electron density function is given by
pdr,r') = PI(rlol(r') - 2 [Fdr,r')F,
(3.3)
2: ~r(r)tPi(r')
rr'p,.(R)
P,(cos 8u)
(3.10)
where we have used the approximations
f
(3.4)
p,,(r")dr" :::::p,.(R)(r - r'),
[p,. (r)p" (r')] 1/2:::::p,,(R),
(3.11)
with R = (r + r')l2. The summation over n may now be carried out using Eq. (3.6) to yield
,
F I2 (r,r) =
k/(R)
where
Fdr,r') =
tf
X cos [p,,{R l(r - r')],
(3.1)
where ~j(r) are orthonormal functions and N is the number
of electrons. The one-electron density function is then given
by
PI(r)
IC,. 12(21 + 1)
=
1
~ )' (21
+ l)p/(cos 8121
sin [k/(R )(r- r')]
471rr' ."
[2~ - (I + 1I2)2/R 2pn.
r- r
"
(3.12)
This expression and Eq. (3.7) can be used in Eq. (3.3) to obtain the two-electron density function.
I
and the factor of 2 takes into account the spin states of the
electrons.
B. WKB wave functions and densities
The single-particle wave functions may be evaluated
within the WKB approximation. They are given by25
~n./.m(r)=
C
n
r[Pn (r)]
PrJ (r) = [2(E"
1/2sin[fpn(r')dr'+.E::.]Yi,
"
4
(3.5)
C. Expressions for S( - 1) and S(1)
The one- and two-electron density functions are substituted in Eq. (2.5) to simplify the expression for S ( - 1). Using
r' r'
= rr' cos 8 12,
and carrying out the angular integrations, we get
S(-1)=~f2:(21+
l)k/(r)rdr
1T
,
+~) _ (I + ~/2)2] 1/2,
4
--
r
where En is the energy and tP is the effective potential and rj
are the turning points. The quantization condition and the
normalization condition lead t025
Using the wave functions in Eq. (3.5), and Eq. (3.6), one obtains25 for the one-electron density
PI(r) =
2~r
++
(21
1)[ ~ - (I +~/2)2r/2.
(3.7)
One may also carry out the summation over I by integrating
over I but this is not necessary for our purpose.
For obtaining the two-electron density function, we
start with
F
r'
1"
dr, ) = 41T
f.t
-2R dtJ
0
."
tJ
2 -
¥1
2
2
(3.14)
),
wheretJ = r - r' and we have changed the integration variables from rand r' to R and 1:1. Now since the main contribution comes from the large R region (as in the case of diamagnetic susceptibility). one can take limits of 1:1 integration to
± 00 and neglect the last term 1:1 2 , to get
S(
-1):::::~f2:(2/+
l)k,(r)rdr
1T
,
-! +
f
2Ik,(R)R 2 dR.
k/(R):::::+(2/+ 1)k/(R)/+(21+ 1).
p,.(r")dr"
+ 1T/4]
One then obtains
xSin[i~ p,.(rH)dr" + 1T/4].
(3.15)
Note that the first term is equal to 6x [see Eqs. (2.5)]. In the
second term we approximate k/(R ) by its weighted average
2
ICn I (2/+ 1)
P
8
rr' [prJ (r)pn (r')] 1/2 /(cos 12)
xSin[L
ioo dR f2R
X)' I sin[k,(R)tJ ]sin[k/_dR)tJ]
x(R
(3.6)
(3.13)
(3.8)
S( - 1):::::6X [ 1
+Y+
(I + 112)],
J. Chern. Phys., Vol. 83, No. 11, 1 December 1985
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(3.16)
(3.17)
5767
S. H. Patil: Susceptibility and polarizability
where the summation is over the occupied I values. The second term inside the bracket is equal to Im/(lm + 1) where 1m
is the highest value of I for the occupied states, so that
S( - 1)::::;6X/(/m
+ 1).
(3.18)
The analysis for S (1) is similar. Starting from Eq. (2.10)
and going through the same steps as in the derivation of Eq.
(3.15), we get
S(I)=2.4IEt l -2Z
1T
i
oo
0
dR
2Ik /(R).
-L
R /
(3.19)
If we substitute Eq. (3.7) in the first term of Eq. (2.6), one
observes that it is similar to the second term in Eq. (3.19)
except for the factor of 21 + 1 in place of 2/. Since the first
term in Eq. (2.6) was shown to lead to 2.4 lEt I, we have
S(I)
= 2.4[IEt 1- )' _2/_ IE (I)I],
~
(3.20)
2/+ 1
where E (I) is the contribution to energy from the angular
momentum I states, and the summation is over the occupied
I levels. The contribution of the second term in Eq. (3.20) is
relatively small, being about 10%-25% of the first term.
with a and b given in Eq. (3.24), and
(3.26)
'T/(O)=N.
What one does is to start with a suitable value of A and the
asymptotic behavior in Eq.(3.25), and trace the solution to
smaller values of r by using Eq. (3.24). We then choose that
value of A for which the condition in Eq. (3.26) is satisfied.
Empirically, we find that A for neutral atoms is given quite
accurately, within about 10%, by the expression
A = 21.4N 1/2E :.35IE)12
for Z = N.
(3.27)
The susceptibility of an atom is given in terms of electron density PI(r) by the relation
X=i
J
rpI(r)d 3 r .
(3.28)
It is evaluated by using the density
PI(r) =
3~ (2'T//r )3/2 + :1T
rOe-
br
(3.29)
which is obtained by solving Eq. (3.24).
D. Diamagnetic susceptibility
Since S ( - 1) in Eq. (3.17) is given in terms ofX' we need
to develop a reliable method for calculating X. Experimentally, we have accurate values of X for only the inert gases.
The diamagnetic susceptibility depends quite sensitively
on the large-r behavior of atomic wave functions, so that a
reliable approach to determine X should include the correct
asymptotic behavior. It is known that the asymptotic behavior of electron density in atoms has the form 14,26
p(r)_rOe- br for r> 1,
a = (Z - N
+ 1)12/E; 11/2 -
2,
(3.21)
b = 212E; 11/2 ,
where E; is the ionization energy. We had recently proposed 14 a modification of the Thomas-Fermi model which
includes this asymptotic behavior, and gives quite satisfactory values for susceptibility.
The proposed equation for the potential was l4
!!:.- (rt,6) = (...!..)
(2rt,6)3/2 + Aya + Ie - br,
(3.22)
dr
31T
r l/2
where the right-hand side is 41T r PI(r) and A is a constant. In
this equation, the last term dominates the equation for large
r. However for ions, one has
rt,6--+Z -N for r--+
00,
(3.23)
so that the first term in Eq. (3.22) dominates at large r and the
density would be incorrectly represented. We instead propose an equation
(...!..)
d 2'T/ =
(2'T/)3/2
dr
31T
r l/2
'T/(r) = rt,6 - (Z - N),
+ Ar ° + Ie - br,
(3.24)
which is acceptable for neutral atoms as well as ions. The
boundary conditions are
(3.25)
IV. DIPOLE POLARIZABILITIES
We are now in a position to estimateS (u) for u = - 1,0,
1, and 2, from Eqs. (3.18), (2.2), (3.20), and (2.12). This provides us with sufficient information to obtain a reliable parametric expression for stu).
A. Parametric expression for S(u)
For a normal atom, the summation over the intermediate states involves roughly three characteristic energies.
These are the ionization energy of the atom, average energy
of the electrons in the atom, and the energy of the innermost
electrons. This would correspond to Ek - Eo being equal to
the ionization energy E;, often used in the single term approximations, average energy Eo and the energy of the innermost electrons which may be represented by Z2. We therefore consider a representation
where B, C, D are constants for a given atom. The last term
represents the singular part given in Eq. (2.14). Note that it is
consistent with the ratio of the singular parts for u = 2 and 1,
given in Eq. (2.15).
Our representation is similar in spirit to the one used by
Langhoff 27 and Cummings.21 In their scheme, the energies
also are taken as parameters and are determined from various physical requirements, e.g., 1S ( - 2) should be equal to
the observed polarizability. It is to be noted that our expression for S (u) is dominated by the first term for negative u, and
by the remaining two terms for positive u. Therefore, since
E; < Eo one expects a rapid changeinS(u) between u = - 1
and O. Th.e change is expected to be particularly sharp for
"loose" atoms which have a small ionization energy. This is
consistent with the behavior described by Dehmer et al. 22
The constants B, C, and D are to be determined from the
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5768
S. H. PatH: Susceptibility and polarizability
conditions discussed earlier and summarized here,
S( - 1) = 6X/(lm
S(O)=~,
+ 1),
(5.3)
(4.2)
(4.3)
S(I)=2.4[IEt l -
+
IE(l)ll/(I+ 112)].
S(2) = 4Z [(Z - 0.4)3 + ilZ - 2.4)3],
(4.4)
(r); =
(4.5)
where for simplicity, we have written the approximate relations as equalities. Actually we need only three of these equations to determine the constants B, C, and D. Equation (4.4)
will be used only as a consistency check for the validity of the
parametric form used for S(u).
B. Polarizabillty a p
It is found that the last term in Eq. (4.1) contributes a
negligible amount for u = 0, - 1, - 2. For example, in the
case of Xe, S(0)::::81 whereas the contribution of the last
term in Eq. (4.1) is less than 1. Therefore, we may determine
Band C from the relations
+ C::::6X/(/m + 1)
BE; + CEa ::::~N.
where the summation is over all the electrons. The (r); are
evaluated with suitably screened Coulomb wave functions to
give
X
X
(4.7)
X
Determining Band C allows us to obtain S ( - 2) and hence
electric polarizability a p as
X
4(Ea
+ EJX/(lm + 1) -
N
(4.8)
E;Ea
The parametrization we have described is not valid for
atoms in which there are excited states whose energies are
close to the ground state energies, e.g., alkali metals.
ap =
V. ISOELECTRONIC SEQUENCES
Within an isoelectronic sequence, the electronic structure is not expected to change drastically when we go from
one member to another. Therefore one expects simple relations to exist between the properties of different members of
a given sequence.
4
(10) _
1
- (Z _ 0.4)2
(18) -
-
(54) -
44
(Z _ 4.41)2 '
+ (Z _24910.0)2 '
(18) + 210
816
X
(Z _ 10)2 + (Z _ 23.6)2
- X
(36) -
+
- X
(10)
(36)
840
23.6f
+ (Z _
+
l
(5.1)
Since the number of contributing electrons is the same for
the different members of the isoelectronic sequence, the ratio
of diamagnetic susceptibilities of two different members is
given approximately by
Xq
(b q
-X q' - (b q•
+ 4)(bq + 3)
+ 4)(bq• + 3)
'
for N = 10, 18,36, and 54, respectively.
C. Polarizabillties
Polarizabilities can be calculated by using the above susceptibilities and Eq. (4.8). The ionization energy required for
the use of Eq. (4.8) can be obtained either from experiments
or from a parametrization
E; = 2: 2 (Z -
O'f
(5.6)
with n being the total quantum number for the outermost
shell and
(5.7)
Let the wave function of an electron in the outermost
shell be given by the square root of the density in Eq. (3.21).
Then the contribution to susceptibility Xe by this electron is
given by
+ 3~~b + 4)
(5.5)
2025
(Z _ 37.2)2 '
A. Ratio of susceptibilities
Xe::::i [(b
(5.4)
The screening charge for the outermost shell is critical in
determining X and may be determined by requiring that the
value given by Eq. (5.3) should agree with the observed value
of X for the neutral member of the sequence. For the inert
gases we take 0' = 0.4 for the n = 1 shell, and find 0' = 4.41
for n = 2, 0' = 10.0 for n = 3, 0' = 23.6 for n = 4, and 37.2
for n = 5. The susceptibility for the different sequences is
then given by
(4.6)
B
1) - 113]}.
n
{~_1. [/(1 + 2
(Z -of 2
2
n
(a- .)2
q
aq
(5.2)
,
where q and q' characterize the different values of Z - N.
where ao and a 1 are determined by requiring that Eq. (5.6)
gives the observed values of E; for, say, Z - N = 0 and 1.
Similarly the average value of Ea required for the use ofEq.
(4.8) can be estimated by
1 L (Z - O';f
(5.8)
Ea - N
2'
;
2n;
where the summation is over all the electrons, with a suitable
choice of values for the screening charge 0';. The precise values of 0'; in this case are not very important since a p does not
depend sensitively on Ea.
For Z _ 00, and N = constant, we can neglect 0' and 0'/
in Eqs.(5.6) and (5.8), use Eq. (5.3) to determine X and hence
determine polarizability a p from Eq. (4.8) in terms Z, N, n,
and I values. For example we get for Z - 00
(5.9)
B. Susceptibilities
One can also obtain a simple expression for susceptibility by using the standard relation
withB = 770, 12 528,819712/13,320 778.4 for N = 10,18,
36, 54, respectively.
J. Chern. Phys., Vol. 83, No. 11. 1 December 1985
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5. H. Patti: Susceptibility and polarizability
TABLE I. Calculated values of diamagnetic susceptibilities and electric dipole polarizabilities for some neutral atoms, along with experimental values
given in brackets. The letters a, b, c represent a = Ref. 28, b = Ref. 1,
c=Ref.29.
Z
Ei
E.
A
X
ap
10
0.793
12.7
44.0
14
0.300
20.1
4.1
1.61
(1.42)"
5.06
15
0.404
22.3
11.9
4.2
16
0.381
24.6
10.1
4.5
17
0.478
26.6
21.0
3.9
18
0.579
28.8
35.5
31
0.221
61.9
1.52
32
0.290
64.6
5.4
8.9
33
0.361
67.3
12.5
7.6
34
0.359
70.0
12.5
7.9
35
0.435
72.8
24.2
7.1
36
0.515
75.5
40.0
42
0.261
92.8
52
53
0.331
0.384
126.8
130.0
11.4
19.9
10.9
10.1
54
0.446
133.3
32.4
9.14
(9.24)"
3.32
(2.67)b
31.9
(36.3)b
19.5
(24.5)b
22.3
(19.6)b
15.3
(14.7)b
11.3
(11.1)b
58.9
(54.9)b
39.4
(41.0)b
26.9
(29.1)b
28.1
(25.5)b
20.8
(20.6)b
15.6
(16.8)b
55.6
(61)"
42.8
34.1
(36)C
26.5
(27.3)b
3.88
3.51
(4.12)"
10.1
6.34
(6.06)"
11.2
VI. NUMERICAL RESULTS
In this section, we present the numerical results which
follow from our analysis. The polarizability calculations are
based on Eq. (4.8). The susceptibilities required in using
these equations and which are of independent interest are
obtained from Eq. (3.28) by solving Eq. (3.24).
A. X and a p for neutral atoms
The values of diamagnetic susceptibility X calculated
from Eq. (3.28) and polarizability a p calculated from Eq.
(4.8) are given in Table I, for some neutral atoms, along with
the values of the relevant parameters. The values of E; are
taken from experiments,30 while those of Ea are taken either
from experiments or are estimated from Eq. (5.8). The agreement between calculated values and experimental values is
quite good over the entire range of Z values. Even in the case
ofZ = 10, if we take the experimental valueofx = 1.42, one
gets a p = 2.80 which is close to the observed polarizability.
B. Isoelectronic sequences
The diamagnetic susceptibilities and polarizabilities of
isoelectronic sequences are of particular interest and our cal-
5769
culated values are listed in Table II, along with the experimental or empirically derived values.
The agreement between our calculated values for the
dipole polarizabilities and the empirically derived values of
Refs. 23 and 32, for the N = 10 isoelectronic sequence, is
excellent. There do not appear to be accurate empirical values of the polarizabilities for the other sequences. We urge
that accurate experiments be done to determine the polarizabilities of the ions of the N = 18, 36, 54 isoelectronic sequences.
With the susceptibilities tabulated in Table II, one can
verify the accuracy of the approximate relation for the ratio
of the susceptibilities given in Eq. (5.2). This relation gives,
for example,
xlo1xl1 = 1.56,
(6.1)
X17IXI8 = 1.17,
xlo1x42 = 68.1,
for the N = 10 series, where the SUbscript is the value of Z.
The corresponding ratios from the calculated values in Table
II are 1.41,1.16, and 46.9, respectively. For the higher N, Eq.
(5.2) is not that accurate. For example, we have from Eq.
(5.2),
= 1.15
X3~X37 = 1.26
X2~X25
for N
for N
= 18,
= 36,
(6.2)
X551X56 = 1.30 for N = 54,
whereas the corresponding values calculated from Table II
are 1.12, 1.15, and 1.14, respectively. The decrease in the
accuracy of Eq. (5.2) for larger N values may be expected
since the electronic structure becomes more complicated for
higher values of N.
The value of X is not only of independent interest, but is
of critical importance in our evaluation of a p [see Eq. (4.8)].
Therefore, we would like to verify the accuracy of our calculated values by using the expressions in Eq. (5.5). Some of the
extrapolated values from Eq. (5.5) are
X12
= 0.77 (0.84),
= 0.117
X42 = 0.032
X24
(0.128),
(6.3)
(0.034),
for the N = 10 sequence, with the calculated values from
Table II given in brackets. The agreement between the two
sets of values is quite satisfactory. Even the small difference
between the two is ascribed to the fact that the first expression in Eq.(5.5) is normalized to givexlO = 1.42, whereas our
calculated value is X 10 = 1.61. Similar agreement is observed
for the other isoelectronic sequences as well. For example,
Eq. (5.5) gives
X26 = 1.07 (1.09) for N = 18,
X38 = 4.58 (4.83) for N = 36,
(6.4)
X57 = 6.87 (6.14) for N = 54,
where the calculated values from Table II are given in brackets. The generally satisfactory agreement between the calculated values and the values extrapolated from Eq. (5.5) gives
us confidence in the essential correctness of our approach.
J. Chem. Phys., Vol. 83, No. 11, 1 December 1985
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5770
S. H. Patil: Susceptibility and polarizability
TABLE II. Calculated values of diamagnetic susceptibilities and electric
dipole polarizabilities for some members of isoelectronic series, along with
experimental values given in brackets. The letters b, c, d, f, h represent
b = Ref. 1, c = Ref. 29, d = Ref. 31, f= Ref. 32, h = Ref. 23.
TABLE II. (continued).
ap
N
Z
Ei
E.
A
10
10
0.793
12.7
44.0
11
1.74
15.9
179
1.61
(1.42)
1.14
12
2.95
19.5
530
0.84
13
4.41
23.4
1.31 X 1Q3
0.66
14
6.13
27.7
2.85X 103
0.53
15
8.10
32.3
5.63X 103
0.44
16
10.33
37.3
1.02 X 10"
0.36
17
12.81
42.6
1.76 X 10"
0.31
18
15.48
48.3
2.89 X 10"
0.268
19
18.52
54.3
4.53 X 10"
0.230
X
20
21.76
60.9
6.9X 10"
0.202
21
25.24
68.0
1.01 X lOS
0.179
22
28.99
75.5
1.44 X 105
0.159
23
32.98
8304
2.04X WS
0.142
24
37.25
91.7
2.81 X lOS
0.128
25
41.76
10004
3.79X lOS
0.1l6
26
46.54
109.5
5.05XWS
0.105
27
51.58
119.0
6.58X 105
0.096
3.32
(2.67)b
1.09
(1.002)f
0.48
(0.47)h
0.259
(0.253)h
0.152
(0.15)h
0.0977
(0.095)h
0.0630
(0.063)h
0.0446
(0.0438)h
0.0323
(0.0313)h
0.0234
(0.0230)h
0.0177
(0.0 I 74)h
0.0136
(0.01 34)h
0.0106
(O.OI04)h
0.0084
(0.0083)h
0.0067
(0.0067)b
0.0055
(0.0054)h
0.0045
(O.OO44)h
0.0037
(0.0037t
18
36
54
36
108.2
222.5
4.61xl(f
0.0484
42
157.1
309.5
1.23 X 107
0.0343
35.5
3.51
18
0.579
28.8
19
1.17
32.6
120
2.90
20
1.88
36.6
342
2.42
21
22
23
24
25
2.72
3.67
4.75
5.93
7.21
41.0
45.6
50.4
55.5
60.8
840
1.88X loJ
3.90X 1Q3
7.55X loJ
1.38 X 10"
2.06
1.78
1.56
1.36
1.21
26
8.59
66.4
2.40X 10"
1.09
27
10.10
72.2
4.13X 10"
0.99
28
36
11.70
0.515
78.2
75.5
37
1.011
80.5
124
5.50
38
1.61
85.7
340
4.83
39
2.30
91.2
850
4.30
40
3.05
97.0
1830
3.83
54
0.446
133.3
55
0.923
138.8
114
7.83
56
1.50
144.7
343
6.85
57
2.14
150.7
853
6.14
6.7X 10"
40.0
32.4
0.90
6.34
9.14
0.00091
(0.00093)h
0.00045
(O.OOO46)b
11.3
(ll.l)b
4.66
(5.47)d
2.45
(3.l2)d
1.45
0.94
0.64
0.45
0.334
(0.29 ± 0.15)d
0.255
(0. 37)d
0.20
(0.26 ± O.04)d
0.157
15.6
(16.8)h
6.9
(9.5-12.2)<
3.81
(6.6)C
2.38
(3.71)C
1.61
(2.50)·
26.5
(27.3)b
11.0
(16.3-21.2)C
5.9
(1IA)C
VII. SUMMARY AND DISCUSSION
Here we summarize the main results of our analysis and
discuss some of their implications.
3.71
accurate for the ions in the N = 36 and 54 isoelectronic sequences and we strongly urge that accurate experiments
with modem techniques be carried out for these ions.
B. Susceptibility and polarizablllty as functions of EI
A.Summary
The analysis is based on the expressions for S ( 1), S (0),
S(l), and S(2) given in Eqs. (4.2)-(4.5). We have used these
expressions to obtain polarizability a p as
a = 4(E;
p
+ Ea)l'/(lm + l)-N
E.E
I
a
(7.1)
'
whereE; is the ionization energy, Ea is the average energy, N
is the number of electrons, and 1m is the highest I value of
occupied states. The susceptibilities are calculated by using
an extension of the Thomas-Fermi equation given in Eq.
(3.24) which incorporates the correct asymptotic behavior.
Of particular interest are the susceptibilities and polarizabilities of isoelectronic sequences for which one has the
simple relations given in Eqs. (5.2), (5.5), and (5.9). We find
that the experimental polarizabilities available are not very
There is an approximate relation 1 between dipole polarizability and diamagnetic susceptibility,
a p = 16l'2 1N.
(7.2)
This relation is known to be fairly good for inert gases. We
find this relation to be reasonably accurate for the atoms
given in Table I as well. For example, in the case of As
(Z = 33), using our value of X = 7.6, Eq. (7.2) gives
a p = 28.0, whereas the experimental valuel is 29.1. However, Eq. (7.2) is generally unsatisfactory for atoms and ions
with resonant states as also for highly ionized atoms. As an
example note that for the ionized Ca (Z = 20, N = 10) with
ten electrons, Eq. (7.2) on using our value ofl' = 0.202 gives
a p = 0.065, whereas the experimental value 23 is 0.0174.
We can combine Eq. (7.2) with our relation in Eq. (7.1) to
obtain a p and X independently for atoms with no resonant
J. Chem. Phys., Vol. 83, No. 11, 1 December 1985
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S. H. Patil: Susceptibility and polarizability
states. On equating Eqs. (7.1) and (7.2) one gets
X=
N(Ea +Ej ) {
4EaEj(/m +
[
1+ 12
SEaEj(/m + 1)
(Ea +Ej )
W]1I2}
5771
higher multipole polarizabilities, van der Waals constants,
etc. Some of these will be considered separately.
I thank Srinivas Krishnagopal and Shobhana Narasimhan for their help in numerical computations.
(7.3)
and
a =
p
{I [1-
+ E j)2
+
4E~E~(/m + W
N(Ea
W]1I2}2,
4EaEj(/m +
(Ea + E j)2
(7.4)
where E j is the ionization energy, Ea is the average energy,
and 1m is the highest I value of occupied states. They give
quite satisfactory values for susceptibility and polarizability
of neutral atoms with no resonant transitions. For example,
they predict X = 3.63, 5.49, 9.S1 for A, Kr, and Xe respectively, whereas the experimental values are 4.12,6.06,9.24,
respectively. Similarly, they predict a p = 35.3, 11.7, 39.2,
13.4, 65.4, 2S.5 for Z = 14, IS, 32, 36,42, 54, respectively,
whereas the corresponding experimental values given in Table I are 36.3, 11.1,41.0, 16.S, 61, 27.3, respectively.
In most situations, especially the heavier atoms, one
finds that E j is much smaller than Ea' In these cases Eqs.
(7.3) and (7.4) simplify to
N
4Ej(/m
X~,
+ 1)
N
a -----p-
E~(/m
+ W'
(7.5)
(7.6)
whose predictions are in reasonable agreement with the experimental values. For example, in the case of Xe, they predict X = 10.1 and a p = 30.2, whereas the experimental values are 9.24 and 27.3, respectively. It may be noted that a
linear, parametric relation between In X and In E j has been
found to be quite useful 33 in relating the energies of isoelectronic sequences, consistent with Eq. (7.5).
C. Other applications
The techniques of extrapolation and semiclassical ap·
proach, and the resulting simplifications, can be used in the
analysis of other atomic properties as well. For example,
they can be used for analyzing dynamic dipole polarizability,
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J. Chern. Phys., Vol. 83, No. 11,1 December 1985
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