Asymptotic wave functions for Be, alkaline earths, and their isoelectronics
S. H. Patil and G. S. Setlur
Citation: J. Chem. Phys. 95, 4245 (1991); doi: 10.1063/1.460780
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Asymptotic wave functions for Be, alkaline earths, and their isoelectronics
s. H.
Patil and G. S. Setlur
Physics Department. Indian Institute o/Technology. Bombay 400 076. India
(Received 7 January 1991; accepted 29 May 1991)
We have developed equations for obtaining asymptotic wave functions and used them to
construct simple, model wave functions for Be, alkaline earths, and their isoelectronics. The
general approach is also useful for getting perturbed wave functions. It allows us to obtain
reliable expressions for many properties of these systems, such as diamagnetic susceptibilities,
polarizabilities, van der Waals constants, hyperpolarizabilities, higher order Zeeman effects,
etc.
I. INTRODUCTION
Wave functions of many electron atoms and ions have a
very complex structure. The mathematical complexity of the
SchrOdinger equation makes it very difficult to deduce exact
or reliable results for their properties. The problem is somewhat simpler but still quite formidable when the number of
electrons is small.
A. Motivation to our work
The analyses of many-electron wave functions are per
force approximate. They incorporate some of the general,
exact properties of the wave functions, such as exchange
symmetry, and hope that the approximation is sufficiently
good that the resulting wave functions are reliable. Most of
the approximations are based on variational approaches in
some form or other, for example, the coupled Hartree-Fock
equation, configuration interaction, etc. Some approaches
are based on other ideas such as Coulomb approximation, I
pseudopotentials,2,3 etc.
There is one property of the atomic wave functions
which has not received adequate attention. It has been
known for some time,4-1O that the leading asymptotic behavior of an atomic wave function is governed by the ionization
energy. This property has been discussed at a level of considerable mathematical sophistication.s,1O However, the interest has been focused on the leading term and that too mainly
on verifying the existence of such a term rather than exploiting the behavior by incorporating the behavior in model
wave functions. We believe that this behavior is particularly
relevant for atomic properties such as polarizabilities and
susceptibilities which depend strongly on the asymptotic behavior of the electron wave functions.
II. ASYMPTOTIC WAVE FUNCTIONS
We start with the Schrodinger equation for an N-electron atom or ion (we use atomic units)
H(N)</J(N)(r l ,r2 ,· .. , rN) =E(N)</J(N)(rl>r2 , ... , rN)'
B. An outline of our work
In a recent paper I I we had pointed out that atomic wave
functions develop a simple and precise structure in the
asymptotic domain defined in a specific way. This asymptotic wave function can serve as a good model wave function.
This approach has been analyzed 12 in considerable detail for
two-electron systems and shown to lead to simple but reliable expressions for susceptibility, polarizabilities, van der
Waals constants, etc. This has encouraged us to consider
asymptotic wave functions for more complicated systems.
J. Chem. Phys. 95 (6), 15 September 1991
Our present work is a follow-up of our earlier work.I1,12
Here we develop a more general framework for asymptotic
wave functions and apply it to describe the properties of Be,
alkaline earths, and their isoelectronics.
We first develop a set of equations for the N-electron
atom, which in principle, can yield the asymptotic series for
the wave function. These equations are solved for the leading
terms to develop model wave functions for Be, alkaline
earths, and their isoelectronic sequences. The wave function
is subjected to what is known as the local energy test and
then used to calculate diamagnetic susceptibilities.
The multipolar polarizabilities are related to the atomic
wave functions perturbed by multipolar fields. We again obtain the asymptotic solutions for the perturbed wave functions and test their accuracy. These solutions are then used
to calculate mUltipolar polarizabilities, van der Waals constants, etc., for alkaline earths and their isoelectronic sequences.
The alkaline earths and their isoelectronic sequences are
relatively simple systems in the sense that the outermost two
electrons are dominant in determining properties such as
polarizabilities. However there is a complication due to their
excited p-state energies being close to the ground-state energies. The wave functions and their predictions based on the
asymptotic expansions, do take this into account in terms of
the correlation terms, and are shown to provide a good description of the properties of Be, alkaline earths, and their
isoelectronic sequences.
(2.1)
where
N
H(N)
=
L (!P7 -
N
Zri-
I
)
+ L rij
(2.2)
1
i~
i=1
with Z being the nuclear charge and r ij = 'ri
-
rj ,.
A. Formal solutions
To be specific, we consider solutions in the ordered subspace
0021-9606/91/184245-13$03.00
@ 1991 American Institute of Physics
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4245
4246
S. H. Patil and G. S. Setlur: Asymptotic wave functions
(2.3)
Solutions in other parts of the space can be obtained from the
solution in R, except for a possible change in sign, by appropriate permutations of coordinates.
The equations are simplified by first separating out the
usual asymptotic exponential factor. 4 ,9,10 Regarding
¢(N)(r l ,r2, ... , r N) as a function of (r2,r3,... , r N), we expand it in terms of the complete set of wave functions
¢~N - I) (r2,r3 , ... , r N ) and write it as
order to exploit this property, we separate the t
Eq. (2.10) and develop an iterative solution
In (r l
) =
[E6 N -
E~N-I) + Q, ] -
I) -
+ I' Vnl (r 1 ) [ E 6N-
I{ V na(r l )
I) - E ~ N - I)
+ QI ] - I
I
(2.14)
where the primed summation excludes the (N - 1)-partic1e
ground state. Substituting this in Eq. (2.9), one gets
¢(N)(r"r2,.. ·, r N )
(2.4)
Qllo(r l ) = {voo(r l
n
)
+,f' VOt(rl)[EaN-I)
where
H(N-I)¢~N-I)(r2,r3"'"
rN)
= E~N-1)¢~N-I)(r2,r3"'" r N
a l = 2112[E6 N - I ) - E(N)r 12 ,
-
(2.5)
),
(2.6)
with E 6N - I) being the ground state energy of the (N - 1)electron system. Of course, E 6N - I) - E (N) is the ionization
energy of the N-electron system. Substituting expression
(2.4) for ¢(N) into Eq. (2.1), and projecting out state n, we
get
E~N- I) + QI ] -
= [E6N-I)-E~N-I)]f,.(rl)'
(2.7)
Zi =Z-N+i
(2.8)
with LI being the angular momentum operator for particle
1, and the expectation values are taken with respect to functions¢~N-I) (r2,r3, ... , r N ).
Formally, one may write Eq. (2.7) for n = 0, and n fO,
as
(2.9)
Vnt (r l )j, (r l )
for n fO,
(2.10)
I
where
IV,O (r l )
+ "'}Io (r l ).
(2.15 )
Finally, using Eq. (2.14), we can write ¢(N)(rl ,r2'"'' r N ) as
¢(N)(r l ,r2"'" r N
)
= e_a,r'{¢6N-I)(r2,r3"'" r N )
- E ~ N - I)
xI
= 0 term in
+ QI ] -
I VtO (r I )
+ "'}fo (r I ).
(2.16 )
Equations (2.15) and (2.16) provide us with formal solutions for ¢(N)(r l ,r2 ,... , r N ). In principle, one may solve Eq.
(2.15) for Io(r l ) and use it in Eq. (2.16) for obtaining
¢(N)(r l ,r2,· .. , r N)·
B. Asymptotic solutions
In view of the property in Eq. (2.13), it follows that Eqs.
(2.15) and (2.16) are very convenient for developing
asymptotic solutions. The procedure is clear. We first solve
Eq. (2.15) forlo(r l ), iteratively, in the asymptotic region
r l ~ (1,r2,r3, ... , r N ), and then use thisfo (r l ) in Eq. (2.16)
to obtain an asymptotic expression for ¢(N) (rl ,r2"'" rN)'
We observe that for r l ~ (1,r2 ,r3, ... , r N ),1o (r l ) in Eq.
(2.15) satisfies the equation
(a
a
l --
arl
ZI-alfc
rl
o(r l
)
=0
for r l ~(I,r2,r3"'" r N ),
so that
(2.17)
10 (r l ) --+ /z, - a, )/a, Y~' (f)l>¢1 )
Equations (2.9) and (2.10) are particularly convenient for
writing asymptotic solutions. We observe that
for r l ~(1,r2,r3"'" r N ).
(2.18)
This is essentially a known result, 4,9,10 except for the angular
dependence. However, we had recently argued, II that the
angular dependence is defined precisely for a large number
of atoms and ions. We then consider an asymptotic series
solution
for rl~(r2' r3"'" r N ).
(2.13)
Using this property in Eqs. (2.9) and (2.10), we deduce that
asymptotically, 10 (r I ) dominates over f,. (r I ) for n f O. In
(2.19)
and
Vnl(rl)=O(l/ri)
J. Chem. Phys., Vol. 95, No.6, 15 September 1991
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4247
S. H. Patil and G. S. Setlur: Asymptotic wave functions
This series may be asymptotic, indeed we expect it to be.
However, an asymptotic series can be quite useful in developing approximate solutions. Having stated this, we proceed to substitute this expression into Eq. (2.15) and solve
for coefficients C i •
We note that V00 (r l ) is of the order of O( lIrt)
or smaller for r l ..... 00, exponentially small if
ifJ~N- I) (r2,r3 , ... , rN) has vanishing angular momentum.
Also, Vo,(r l ) V,o(r l ) is ofthe order of O(1/rt). We also
note that QI is made up ofterms which are either derivatives
and/or which contain inverse powers of'l' Therefore operation by QI on a power term in'l lowers the power, and one
may regard QI as being small for'l -> 00. We can then expand (E6 N- I ) - E~N-I) + QI] -I in powers ofQI' Taking the leading coefficient Co to be 1 and using Eqs. (2.19)
and (2.15), we obtain for the first four coefficients
(2.20)
Co = 1,
CI = - -
1
2at
1
(ZI - al )Z\ + - I I (/1
2a l
+ 1),
(2.21)
(2.22)
+_1_ /1 (11
6a 1
+ l)C2
a(N-I)
__
1__
(2.23)
6a l
where a\N - 1) is the dipolar polarizability of the (N - 1)electron system in its ground state. The series expression for
10 (r t ), obtained in this way, can then be used in Eq. (2.16)
to obtain the N-electron wave function tPN(rl ,r2 ,... , r N). Of
course, for this, in addition to 10 (r\), we also need
ifJ'oN-I) (r 2 ,r 3 , ... , r N) and related functions in Eq. (2.16).
But then, the procedure we have used for obtaining
tP(N)(r 1 ,r2 , ••• , r N) can be repeated for obtaining
tP'oN-l) (rZ ,r3 , ••• , rN)' The corresponding equations for
tP~N-I) (r2 ,rJ , ••• , rN) are
tP~/-
\)(r2 ,r3 , ... , r N)
-
E~N-2) + Q2] -tV;o(r2 ) + ... }fo(r 2 ),
QJo(rz ) =
[voo(rz ) +
-
+' Vo,(r2)(E~N-2)
(2.24)
E~N-2) + Q2] -IV;o(r\) + " ']fo(f2 ),
(2.25)
where
a
a
1 B2
1
Z2 Q2 = - - - a2 -+--+
2 ari
B'2'2 B'2
02
'2
V~tCr2) = {tP~N-2)1
Nf (_1
-1-)lifJi
r
rz
J=3
a2
- L-~,
(2.26)
2ri
2
2j
)\'
I
= 211Z[ E //-2) _ E'oN- I)] 112.
(2.27)
(2.28)
We can again use an expression of the type given in Eq.
(2.19), but for fa Cr 2 ) and index 1 replaced by index 2. The
coefficients can be evaluated as before to obtain expressions
(2.20)-(2.22) but withZ\ ,0\, II replaced by Z2,02' 12 , The
procedure can be repeated for the successive particles to obtain the complete expansion for, \ ~,2 ~,3 ~ ... ~,N'
C. Local energy test for wave functions
The asymptotic wave functions have the correct behavior for'l ~ (l,r2 ,'3 , ••• , rN)' However, before one can usefully utilize them for useful calculations, we must estimate the
region where they are accurate. For this we consider what is
known as the local energy test. 13 We note that an eigenfunction t/J of H with eigenvalue E will satisfy the equation
_1_ (E - H)¢ = O.
(2.29)
E¢
The value of the left-hand side of this equation, for a model
wave function will give a measure of the local fractional deviation of the wave function from satisfaction of the Schrodinger equation.
We apply the local energy test for the asymptotic wave
function for the ground state of Be. Retaining only the first
two terms in Eq. (2.19) for !oCr l ), and observing that
/1 = 0, we have
tP64 ) (r l ,r2 ,r3 ,r4 )
= e - at't,\
(Z, -
a,l/a'e
1 + hi rl- I )tP'o3) (r 2 ,r 3 ,r4 )
(2.30)
in subspace R defined in Eq. (2.3), where
1
hI = - - ( Z \ -a\)Z\,
2a~
H
(3)cP~3) (f2 ,r3 ,r4 )
= E ~3)tP~3) (r 2 ,r3 ,r4 ).
(2.31)
(2.32)
Since tPb ) (r 2,r 3,r4 ) is taken to be the exact eigenfunction of
the ground state of Be + , the deviation of (E - H) ¢ from
zero would be of the order of [E (4) - E '03 )] ¢ and hence
should be compared with [E (4) - E b3 )] ¢. Therefore for a
more appropriate estimation of the fractional deviation, we
replace Ein the denominator ofEq. (2.29) by E (4) - E 63 ).
With this change, we have evaluated the left-hand side ofEq.
(2.29) for tP64 ) in Eq. (2.30) in the region cos ()\2 = 0 and
('3,r4 ) ~ 1 with E in the numerator being the total energy
E (4) of Be. The values are given in Table I. It is clear that the
asymptotic wave function in Eq. (2.30) is good for rl >3r2 ,
r2 ::::: 1, but requires corrections for smaller values of r l . The
situation is similar for Mg, Ca, Sr, and Ba.
3
III. MODEL WAVE FUNCTIONS FOR BE AND ALKALINE
EARTHS
From our earlier discussion, it is clear that the asymptotic wave function in Eq. (2.30), while good at larger values
of'l' requires corrections at 'I :::::2r2 • This we incorporate by
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S. H. Patil and G. S. Setlur: Asymptotic wave functions
4248
TABLE I. Fractional deviation of the approximate wave function for Be, as
measured by the left-hand side ofEq. (2.29) but with Ein the denominator
replaced by E(4) _E~3). The values are for (r3 ,r.) =0,
r2 = l/a 2 =0.8463,r l ·r2 = O,thefirstcolumn being the valueofrJr2 ,the
second column being the deviation for the function in Eq. (2.30), the third
column being the deviation for the function in Eq. (3.8) with d l = - 0.313
obtained from Eq. (3.11).
[E(')-H]I{II
[E(4) -H]I{I2
r1 /r2
[E(') -Ef,3)]I{I1
[E(4) - Ei/)]1/12
2
3
-1.71XIO- 1
- S.33X 10- 2
- 1.12 X 10- 2
- 3.81X 10- 3
- 1.14x 10- 3
- 3.69X 10- 2
3.19x 10- 3
8.36X 10- 3
6.01 X 10- 3
3.61 X 10- 3
5
7
10
¢6N- I )(r2,r3,... , r N )
=e-a2'2r~Z;,-a2)/a2(1
X¢6 N -
¢(r l ,r2,· .. , r N
t
Vto(r l ) = \¢:N-l)
we have
I r ;21¢6N-I)).
l
(3.3)
H(N-I)]",(
'i"
)
2a2
ri
(3.7)
which has the same two leading terms in r2 as Eq. (3.6).
Substituting this in Eq. (3.1) we get
¢6 N )(r l ,r2 ,· .. , r N
)
_ -a,', r (z,-a,)/a'[1
-e
l
r2 +d)]
e
2a 2 ri
X (1 + dlrl-I)¢6N-I)(r2,r3,'''' r N ).
r l 'r2 (
---
= !Z2 (3 + Z2a2-I)a2-2 + d 2.
(3.8)
(3.9)
This asymptotic wave function was subjected to the local
energy test described in Sec. II C for r l 'r 2 =
and
(r3 ,r4 , ... ) ~ 1, withd l equal to the asymptotic value hi given
in Eq. (2.31). It was found to be satisfactory for r l ;> r2 but
not so satisfactory for r l < 3r2 •
To improve the range of usefulness of the wave function
in Eq. (3.8), we determine d l by demanding that the wave
function should satisfy the local energy test for a smaller
value of rl' In particular, we choose
r l =2/a p r2 = l/a 2,
Clearly, ¢ (r l ,r2 , ... , r N ) satisfies the equation
[E o(N-I) -
(3.6)
°
N
¢~N-1)(r2,r3'"'' r N )[E6 - I )
),
I ]¢(N- 2) (r 3 ,r4"'"
= _ rl.:2 (-I_){r2 +HZ2(3+Z2a2-1)
de
For r l ;>r2 ;> (r3,... ,rN
),
xa2-2+2d2]}¢6N-I)(r2,r3,'''' r N ),
For the development of reliable wave functions for Be
and alkaline earths, we consider the first two terms in Eq.
(2.16). For the estimation of the second term, we can regard, as discussed in Sec. II B that QI is asymptotically
small. We also include only the first two terms info (r l ) and
write
I'
rN
for r l ;>r2 ;>(r3,... , rN ).
This we prefer to write in the form
A. Wave functions for Be and alkaline earths
¢(r l ,r2,.. ·, r N ) =
)(r3 ,r4"'"
Substituting this expression for ¢6 N-1)(r 2 ,r3,'''' r N ), into
Eq. (3.4), we can easily obtain the first two terms in the
asymptotic series,
+ d2 a2modifying the nonleading asymptotic term in the wave function so as to satisfy the local energy constraint. We will confine our discussion to Be, alkaline earths, and their isoelectronics.
2
+d2r2-1)
f l 'f2
=0,
(r3 ,r4 , ... ) = 0,
r l ,r2,.. ·, rN )
(3.4)
For solving this equation, we use the asymptotic expression
for ¢6 N - I ) (r2 ,r3 , ••• , rN) from Eq. (2.24). Retaining the
first two terms info (r 2 ), we have
(3.10)
as the point at which the local energy test should be satisfied.
While the choice of r l and r2 is somewhat arbitrary, this is
the region where the wave function is important and where
we can still regard r l ;> r2 and the asymptotic form of the
wave function is still meaningful. Substituting Eq. (3.8) in
Eq. (2.29), and evaluating it at the point defined in Eq.
(3.10), we find that
(3.11 )
J. Chern. Phys., Vol. 95, No. 6,15 September 1991
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4249
S. H. Patil and G. S. Setlur: Asymptotic wave functions
Using this value of d l in Eq. (3.11) evaluated at the point
defined in Eq. (3.10), we subject the wave function in Eq.
(3.8) to the local energy test. The values of the left-hand side
of Eq. (2.29) with E in the denominator replaced by
E(N) - EbN-I), for fl"f2 = 0, r2 = 1/a2 , (r3 ,r4 , ... ) = 0,
are given in the last column of Table I for Be. It is observed
that the fractional deviation is quite small for the range of
values of r l = (2r 2 , 1Or2 ). The deviations are of similar order for alkaline earths and their isoelectronic sequences.
Overall, we believe that Eq. (3.8) gives a very good representation of the dependence of the wave function ¢bN) on f l , in
region R defined in Eq. (2.3).
We can now proceed with the development of the wave
function ¢b N - I) (f2 ,f 3 , ... ,fN)' Since, in region R, the dependence of the total wave function on r2 is less important than
on r l , we use only the first two terms in the asymptotic
expression for ¢b N - I) (f2 ,f3 , ... ,rN ) given in Eq. (3.5)
X¢bN-2)(f3,r4, ... ,rN)'
(3.12)
For ¢bN-2)(f3,f4, ... ,fN), it is adequate to retain only the
leading asymptotic terms
whereE= I,
.1. (
'1'0
x¢b N tPl (f l ,f2 , ... ,rN )
where function g depends only on angles. For Be and its
isoelectronics, function g is a constant, but for alkaline
earths it is given byll
(3.14 )
where r,. are unit vectors in the direction of i\.
Collecting all the terms, we obtain for Be, alkaline
earths, and their isoelectronics, the wave function in subspace R,
N
,.= I
where E = 1,
a,. = 2112[Eb N-
ail/a;
g
«(),/..
)
3 ''1'3 ,... ,
l
(f2 ,r2 , ... ,rN
fl"f2
(3.18)
),
+ de )tPo (f l ,f2 , ... ,rN ),
-----:3 (r2
2a 2 r l
where a,. are defined in Eq. (3.16), Z,. in Eq. (2.8), d l in Eq.
(3.11), de in Eq. (3.9), and ¢bN - I ) in Eq. (2.5) with its
asymptotic expression given in Eqs. (3.12) and (3.13).
Though there is an overall normalization constant in the
wave function, since all our results will be expressed as ratios
of integrals, we do not write it explicitly.
B" Susceptibilities and related quantities
The wave function in Eq. (3.15) can be used for calculating
/3L =2
L [E~N) -EbN)]I(¢bN)ILrfPL(COS(),.)I¢~N»12.
n
,.
(3.20)
It is well known l4,Is that/3L reduces to
= L (¢bN) I L r;L - 21¢bN)·
(3.21 )
,.
In particular one observes that /3 2 is related to diamagnetic
susceptibility K by
/32
= 12K.
(3.22)
We can evaluate /3L in Eq. (3.18) by using the wave
function in Eq. (3.15). We repeat, that though the wave
function in Eq. (3.15) is defined in subspace R, the wave
function in other regions is obtained, except for a possible
change in sign, by permuting the coordinates f,.. With the
wave function in Eq. (3.15), the expression for /3L to first
order in E, reduces to
/3L
X II e - airir,.(Z, -
= -
I)
+ d I r -I)
( 3.19)
/3L
(3.13)
r l ,r2 ,· .. ,f N ) = e -alrl r(ZI-al)/al(l
l
+ 2d 1J(2L - 3,0)
+ diJ(2L - 4,0) + J(0,2L - 2)
+ 2dJ( - 1,2L - 2) + diJ( - 2,2L -
= N~L
(3.15 )
[J(2L - 2,0)
2)],
(3.23)
2
N o- =J(0,0) +2d I J( -1,0) +diJ( -2,0),
O
-Eb N-,.+I)]I12,
(3.16 )
d l is obtained by evaluating the expression in Eq. (3.11) at
r l = 2/a\l r2 = 1/a 2, d 2 is given in Eq. (3.5), de in Eq.
(3.9), and Z,. in Eq. (2.8). Since the rl"r2/~ term is smaller
than the leading term by order ( 1/ri ) for large r I , we will do
all our calculations to only first order in E. The angle dependent function g is, as mentioned before, a constant for Be and
its isoelectronics, but is given by Eq. (3.14) for alkaline
earths and their isoelectronics.
It is convenient to write the wave function in Eq. (3.8),
in the form
¢bN)(f l ,f2""~N) = tPo (r l ,f2''''~N)
+ EtPI (fl ,r2""~N)'
( 3.17)
(3.24)
where
+ 2d2l(l\l/2
J(I\l/2 ) = l(ll ,/2 )
- 1)
+ dU(I\l/
2 -
2),
(3.25)
with
N
X
II e
-
2a;r; 2z/a;()(
r,.
r,. _ I
-
r,. ) ,
(3.26)
;=2
where ()(x) = 1 for x;;;. 0 and ()(x) = 0 for x < O. The integrations are carried over the region R, and may be multiplied by
N! to obtain the total integral. Finally, the integrals l(n,m)
are simplified as described in the Appendix. The values of
J. Chem. Phys., Vol. 95, No.6, 15 September 1991
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S. H. PatH and G. S. Setlur: Asymptotic wave functions
4250
TABLE II. Input valuesofionization energiesE, = E~N- ') - E~N),E2 = E~N- 2) - E~N-l) taken from the
Moore tables, number of electrons n. in the inner shell defined in Eq. (A 1), average ionization energy Hin Eq.
(A3) for the inner shell, the valueofjin Eq. (A7), value of d, in Eq. (3.11), value of d2 in Eq. (3.5), and the
normalization N 0- 2 defined in Eq. (3.24); predictions for the valence shell expectation values (:I.r. L - 2) v.'
total expectation value (:I.r.), for Be and alkaline earths. The numbers in the brackets are from other calculationswith (a) Ref. 3; (b) Ref. 16; (c) Ref. 17; (d) Ref. 20; (e) Ref. 21.
E,
E2
n.
E
Z
j
d,
d2
N O- 2
(:I.i r.}v.
(+r.t,
(:I.i 1}vs
(:I.i1)vs
(:I.,~) ..
Be
Mg
Ca
Sr
0.3426
0.6693
2
6.83
3.5
3
-0.313
-0.544
6.83
1.43 X 10'
(1.58 X 10')"
0.2810
0.5526
6
6.0
5.5
3
-0.493
-0.817
1.33 X 10'
2.26X 10'
(2.34 X 10')"
0.2247
0.4364
6
3.5
5.5
4
- 0.789
- 1.307
3.44 X 10'
4.23x 10 1
(3.59X 10')"
(4.09 X 1O')b
0.2093
0.4054
6
3.0
5.5
4
-0.907
- 1.506
4.68X 10'
5.34X 10'
(4.76X 10')"
1.48 X 10'
2.76X 10'
5.66X 10'
8.22X 10'
1.18 X 10'
(1.64 X 10 1 )d
1.96 X 102
(2.38x J02)"
4.60X 103
1.68 X 10'
(2.97 X 10')d
4.22 X 102
(4.85 X 10')"
1.31 X 10"
6.17X 10'
(5.67 X 1O')d
1.16xW
(1.06x 103 ).
4.98X 10'
3.17x 106
(7.64X 10')·
1.65 X 103
(1.83XW)"
7.91 X 10'
5.53X 10"
(1.09 X 10')·
2.80X 103
(2.70XW)"
1.51 X 10'
1.19 X 107
relevant input quantities and the predictions are given in
Tables II and III.
The main contribution to (~i rTL - 2) in subspace R
comes from the i = 1,2 electrons, i.e., the valence shell electrons. To this we have added the contribution from Be + + ,
Mg + +, etc. which have been estimated l8 •19 for L = 2. For
higher L values, these contributions are quite small and may
be neglected at the level of our estimation. It is observed that
Ba
0.1915
0.3677
6
2.5
5.5
4
- 1.074
- 1.812
5.85X 10'
7.78X 10'
(5.86 X 10')"
the predictions are in very good agreement with the results of
other calculations.
IV. POLARIZABILITIES
The asymptotic approach described in Sec. II, can easily
be extended to obtain atomic wave functions perturbed by
multipolar fields. These wave functions may then be used to
calculate multipolar polarizabilities.
TABLE III. Input values of ionization energies E, = E ~ N- ') - E ~N), E, = E ~ " - 2) - E ~ N - ') taken from
the Moore tables, number of electrons n, in the inner shell defined in Eq. (AI), average ionization energy Hin
Eq. (A3) for the inner shell, the value ofjin Eq. (A7), value of d, in Eq. (3.11), value of d, in Eq. (3.5), and
the normalization N 0- 2 defined in Eq. (3.24); predictions for the valence shell expectation values (:I.r. L - 2)",
total expectation value (:I.r.) , for some ions. The numbers in the brackets are from (a) Ref. 21.
E,
E,
ne
H
Z
j
d,
d,
N o-'
(:I.r.)v.
p:r;)
'01
('i, 1)..
(2:. 1)
,
B+
Al+
Sc+
C++
Si + +
0.9246
1.3940
2
11.0
4.5
3
- 0.337
- 0.429
1.99X 10-'
7.0
7.3
(7.98)"
0.6920
1.046
6
8.0
6.5
4
- 0.606
-0.771
4.39X 10-'
1.43 X 10'
1.84X 10'
(1.74 X 10')"
0.4739
0.910
6
4.4
6.5
4
- 1.151
- 1.009
2.57
2.23X 10'
3.43 X 10'
(2.3X 10')'
1.760
2.370
2
16.21
7.5
3
- 0.307
- 0.353
1.36 X 10 -,
4.28
4.5
(4.64)"
1.230
1.659
6
11.2
7.5
4
- 0.603
-0.721
2.90X 10-'
1.04 X 10'
1.36 X 10'
(1.20XIO')
4.39X 10'
1.42 X 10'
3.72X 10'
J.52x 10'
7.03X 10'
4.47 X 10'
2.28X 103
9.69X 10'
8.78X 10'
6.98X 10'
6.81 X 103
5.15XIO·
3.66X 10'
7.46 X 10'
9.80X 103
vs
(+~t
J. Chem. Phys .• Vol. 95, No.6, 15 September 1991
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S. H. PatH and G. S. Setlur: Asymptotic wave functions
- (r )
Q2 F
2
A. Perturbation due to multipolar fields
= -
4251
2 Ia2p
(
0 )(1
rL+q-2+Z
L cos z
2
+ dzrz-I) ,
(4.9)
For evaluating multipolar polarizabilities, we need solutions to the perturbed equation
[H(N) - E(N)]171 (f l ,f 2 , ... ,
fN)
j
where the unperturbed solution tPo is given in Eq. (3.18). As
before, we analyze the solutions in subspace R defined in Eq.
(2.3 ).
For deducing the asymptotic solutions, we again expand
171 (f l ,f2 ,. •• , fN) in terms of the complete set of eigenfunc.I.(N-I) ( fz,r3"'" fN )<
·
t 10nS'l'n
_
The asymptotic solutions to Eqs. (4.4) and (4.9) are
obtained by using expansions
Fo(r l ) =rl-I+ZllattoAi(L,q)
+ ~: :t~Ai(L,q-l)]
xrf + q- iPL (cos 0 1 ),
F(r z )
-1+z,la2 [Q
i~O Bi (L,q)
= rz
(4.11)
q
I
dz
+;;
i~O Bi (L,q -
1)
]
(4.12)
n
(4.2)
Substituting Eq. (4.2) into Eq. (4.1) and projecting out
state n, one gets
[E~N-I) - E~t-I) - QI ]Fn(r l )
+L/¢~N-I)lf
t
) = 2
\
l
l
N
X
)
t) _
E 6N -
I)] -
I
XP L (cos 192 )1¢6 N - I »rl-
<¢~N - I) l,f + q t
+
Z,la, (1
+ ZI/a l ) (2L + q - i + ZI/a l )
2a l (L
+q -
i-I)
I
+ dIrt-I).
( 4.5)
In Eg. (4.5), we have dropped ";-PL (cos 19;), i >3, and also
the QI operator which gives rise to terms which are smaller
by at least a factor of 1/11. Substituting Eq. (4.5) into Eq.
(4.2), we get
(4.6)
Bo (L,q) = [a z (L
+ q)] - 1,
[H(N-I)-Eft- I )]F(r2,r3,... , r N )
=,f+q-IPdcosez)¢6N-i)(rZ,r3"'" r N ).
(4.7)
We proceed as we did in the analysis ofEq. (4.1), and obtain
F(r 2,r3 , ... , r N ) =e-a2r2¢ft-Z)(r3,r4"'" rN)F(r z ),
(4.8)
( 4.15)
Bi+ I (L,q)
= Bi(L,q)
X
(q - i - I
+ Z2/a2) (2L + q - i + Z2/a2)
.
2a (L + q - i - 1)
2
( 4.16)
We have retained only the (q + 1) leading terms in the
asymptotic series expansion. Higher order terms require a
more detailed analysis and are not included in the calculation of polarizabilities. Indeed, in order to improve the range
of applicability of the asymptotic solution, we determine the
last term inFo (r l ), i.e., Aq (L,q) by imposing the local energy test in some suitable region. For this, we note that the first
term in Eq. (4.6) is the solution to Eq. (4.1) for the j = 1
term which is the most important term in the asymptotic
domain r l >r2 >(r3 ,r4 ,. .. , rN ). We test the accuracy of this
solution by evaluating the fractional difference in the two
sides of Eq. (4.1) with only the j = 1 term, namely
-SI}/SI'
SI =rt+q-IPL(COSOI)tPO(rl,r2"'" r N ),
with
where
,
(4.14 )
(4.3)
t
ll) ( 1 + d 1 r - I) ,
= - r Ll + q - 2 + Z,/a,pL ( cos (71
l
= [E ~ N -
(q - i - I
-I))
(4.4)
Fn (f I
(4.13)
=Ai(L,q)
where Qt is the operator defined in Eg. (2.11).
For the asymptotic solutions with r l >r2 > (r3 , .. ·, rN ),
we can neglect the second term on the left-hand side in Eq.
(4.3), which is smaller than the right-hand side by at least a
factor of 1/rf + I (for q> 1), and obtain
QI F.0 (r )
l
= [a l (L+q)]-1,
l
(¢~N- Itt! 1+ q-IPL (cos e) I¢6
x (1 + d r I)r I + Z,/a"
=
Ao(L,q)
Ai+ I (L,q)
(_1
-~)I¢:N-I»)Ft(rl)
r
r
lj
Substituting these expressions into Eqs. (4.4) and ( 4. 9), one
gets
( 4.17)
(4.18 )
with tPo given in Eq. (3.18). The values of t:. for Be, with
Fo (r l ) in Eg. (4.11) and Ai in Eqs. (4.13) and (4.14), are
given in the second column of Table IV. To improve the
range of usefulness of the solution, we determineA q (L,q) so
that t:. vanishes at a suitable point. We choose this point at
which ll. = 0, to be the same as in Eg. (3.10), i.e., r l = 2/a l ,
r2 = 1/a 2 , and r 1 'r2 = 0, except that in some cases the zero
J. Chem. Phys., Vol. 95, No.6, 15 September 1991
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S. H. Patil and G. S. Setlur: Asymptotic wave functions
4252
TABLE IV. Fractional deviation A in Eq. (4.17), for the first term on the
right-hand side in Eq. (4.6) with Fo(rl ) given in Eq. (4.11), which is a
solution to Eq. (4.1) with only the j = 1 term and 1/10 given in Eq. (3.1S).
The results are for Be, withdl = - 0.313. The second column is for L = I,
q = I, with Ao (1,1) = 0.6040, Adl,l) = 1.479 as given by Eqs. (4.13)(4.14). The third column is for Ao (1,1) = 0.6040 given by Eq. (4.13), and
AI (1,1) = 2.071 as given by Eq. (4.19). The fourth column is for L = 3,
q=I, with Ao (3,1)=0.3020 as .~given by Eq. (4.13), and
AI (3,1)=0.5S02 as given by Eq. (4.19). The values are for
r. = l/a, = 0.8463, rl'r, =0, (r3 ,r., ... ) =0.
rl/r,
2
3
5
7
10
AI (L = I, q = I)
- 3.88XIO- 1
-1.51xIO- I
- 4.78 X 10- 2
- 2.29X 10- 2
-1.06XIO- 2
A. (L = I, q = I)
- 1.67 X 10- 1
1.70 X 10- 2
6.01 X 10- 2
5.58X 10-'
4.51 X 10- 2
A(L= 3, q= I)
-7.89X 10- 2
8.16X 10- 3
2.97X 10- 2
2.79x 10- 2
2.28x 10- 2
oft,bo(r r 2 , ••• , r N ) at'l = -d , is close to 2/a , . In this
"
the point at which A = to be'l = - 2.5d
case we choose
"
'2 = 1/a2 , and r 1 o r 2 = 0. We then have
1 + ~r,:- dAi (L,q) GU),: - i
G(q),l-q
,
(4.19)
Aq(L,q) = -
'I
where U1 =(ZI-al)/a l ,
= max (2/a , , -5d, /2),
= 1/a2 • Of course, for q = 0, there is only 1 in the numerator ofEq. (4.19). The solution in Eqs. (4.6) and (4.11),
with Ai(L,q), i<q given in Eqs. (4.13) and (4.14), and
Aq (L,q) given in Eq. (4.19), is reliable over a wider range of
values of" and '2' The corresponding values of the fractional deviation A defined in Eq. (4.17), for Be, are given for
some values of" and'2 in the third and fourth columns of
Table IV. It is observed that these deviations are generally
small. The deviations are of similar order for other systems
as well.
'2
B. Multlpole polarlzabilltles
For the evaluation of polarizabilities, we consider the
more general expression
SL(-m-l)=2
°
I
' I<il~j 1PL(COSOj )IO)1
i
[E}N> - EaN>]m
2
'
(4.21 )
where m = 1 corresponds to the 2L -polar polarizability. For
evaluating S L ( - m - 1), we first deduce solutions to equations
GU)=!'1- 2 [(L+q+u l -i)(L+q+u 1 +1-i)
- L(L + 1)]
+ 1-
i)] -
+ '1-1 [Z2 - a
(,-1 + ~ ) - 112,
l
(L
+q +U
1
=
(4.20)
I
rfPdcos OJ)t,bo(r l ,r2 ,· .. , r N
)
j
TABLE V. Some input values of D,(L,I), and predictions for 2L_pole polarizabilitiesau andSL ( - m). The
values in brackets are from some other calculations (a) Ref. 3; (b) Ref. 1.
Be
Do (1,1)
D I (1,1)
al
SI (-4)
SI (- 6)
Do (2,1)
DI (2,1)
a2
S2 (- 4)
S, (- 6)
Do (3,1)
DI (3,1)
a3
S3( - 4)
S3 ( - 6)
D o (4,1)
DI (4,1)
2.071
0.604
3.14X 10 1
(3.67X 10 1 )"
6.78Xlo'
1.54 X IQ4
0.913
0.403
2.26X1O'
(3.03X 10')"
(1.68X 102)b
2.15X103
2.24xlQ4
0.580
0.302
3.20X103
(4.13X IO')"
2.14X 1Q4
1.60 X 10'
0.424
0.242
a.
8.21Xl~
S.( -4)
4.45 X 10'
2.71 X 10"
0.334
0.201
3.29xl0"
1.55 X 10'
8.16X 10'
S. ( - 6)
Do (5,1)
DI (5,1)
a,
S, (- 4)
S, ( - 6)
Mg
3.185
0.667
6.60XIO I
(7.05 X lo l )a
2.97X 103
1.4OX 10'
1.313
0.445
6.60X 102
(8.28X 10')"
(5.60X 102)b
1.12 X IQ4
2.06xlO'
0.815
0.334
1.20X 104
(1.47X IQ4)"
1.37X 10'
1.70X 10"
0.589
0.267
3.92X 10'
3.51 X 10"
3.46 X 10'
0.460
0.222
1.99 X 10'
1.51xI08
1.27 X 109
Ca
5.683
0.746
1.62 X 10'
(1.53 X 10')"
1.94 X 104
2.50X 10"
2.057
0.497
2.61 X 103
(2.72X IO')"
(2.61 X 103)b
9.07Xl~
3.34XIO"
1.227
0.373
6.36X IQ4
(6.15X I~)'
1.37 X 10"
3.12XIO'
0.868
0.298
2.74X 106
4.45 X 10'
7.71 X 108
0.670
0.249
1.82X 108
2.44 X 109
3.53X 10 10
Sr
Ba
7.0S1
0.773
2.19X 10'
(2.06X 1o')b
3.82X IQ4
7.27X 10"
2.405
0.515
4.22 X 103
(3.67X 103)b
9.74
0.80S
3.08X 102
(2.95 X 1o')b
8.9SX IQ4
3.02X 10'
2.97
0.539
8.31 X IO'
(7.53X 103)b
1.87X lO'
8.79X 10"
1.411
0.386
1.13 X 10'
5.06 X 10'
3.30X 10'
1.694
0.404
2.47XIO'
3.00X 10·
8.39X 10'
0.989
0.309
5.31 X 10"
1.05 X lOs
2.19X 109
0.759
0.258
3.84X 108
6.20X 109
1.07 X 1011
8.65X 10"
3.16X 108
1.173
0.323
1.30X 10'
3.29X lOs
8.75X 109
0.893
0.269
1.04 X 109
2.14x 10 10
4.62X 1011
J. Chem. Phys., Vol. 95, No.6, 15 September 1991
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L;>I,
(4.22)
S. H. Patil and G. S. Setlur: Asymptotic wave functions
4253
where tPo is the unperturbed solution in Eq. (3.18). Clearly,
one has
F o , F, and Fgiven in Eqs. (4.11), (4.8), and (4.12), and the
[H(N) -Elt)]17m(rl>r 2 , ... , r N)
=17m_l(r l ,rz ,'''' rN) for m>2.
(4.23)
The solution for 171 (r l ,r2,... , r N ) is given in Eq. (4.6) with
Aq which is given in Eq. (4.19). We can then obtain
17m(r l ,r2,... , r N ) by solving Eq. (4.23) successively. Fol-
17m ( r l ,r2 , .. ·, rN )
coefficientsA j and B j given in Eqs. (4.13 )-( 4.16) except for
I
lowing the same procedure as the one used for obtaining
171 (rl,rz ,"" rN),weget
= e -aIr, r l-1+Z,/a, e -a2 2r2-1+z,la2.1.(N_2)(
'f'0
r 3 ,r4 , .. ·,
r
XPL (cos (}I)
+ (1 + d l r l-
I
)
"to
rN
)[(1
~ D n (L ,m )_L+n
+ d 2 r2-I) n~O
'1
(4.24)
En (L,m)rf+ nPL (cos (}2) ],
with
1
D,.(L,m) =
ns
2 - "t
L L'" L
"1
=0 n2==O
E,.(L,m)
1
Z - ",
Nt =
0 n2 = 0
[A",(L,l)+o",.ldIAo(L,0)]An2(L,2-nl)"·Anm(L,ns)on.n,_nm'
(4.25)
[Bn, (L,I) +on,.ld2Bo(L,0)]Bn2(L,2-nl)·"Bnm(L,ns)on,n,_nm'
(4.26)
nm=O
n,
= L L'" L
nm = 0
m-I
n. = m -
L
(4.27)
ni , m> I,
;=1
where An and Bn are defined in Eqs. (4.13)-( 4.16) except for Aq (L,q) which is given in Eq. (4.19).
The expressions for S L ( - m - 1) are then given by
SL ( - m - 1) = 2{f [tPo (r l ,r2,· .. , r N )
+ 2E"'1 (r l ,r2'"''
r N )] jtl rfPL (cos {}j )17m (r l ,r 2 , ... , rN)d 3rl d 3 r2 "·d 3 rN }
(4.28)
"'0
"'I
and
given in Eqs. (3.18) and (3.19). Using the asymptotic expressions for t/Jf/ correct to first order in E, with
tP6 N - 2 ) given in Eqs. (3.12) and (3.13), we get
2
m
Sd - m - 1) = - - N~
{Dn (L,m) [J(2L + n,O) + dJ(2L + n - 1,0)] + En (L,m)[K(0,2L + n)
2L
+1
L
n=O
+ d 2 K(O,2L
+n -
E
1)] - -OL,ID" (I,m) [J(n - 1,3)
3a 2
E
+ dedJ(n - 2,2)] - OL,I En (l,m) [K( - l,n
3a 2
+ ded 2 K( -
l,n
+ 2d l l(/1
+dU(/1 -2'/2),
+ dlJ(n -
+ 3) + d
2
2,3)
K( - l,n
+ dJ(n -
1,2)
+ 2) + dcK( -
l,n
+ 2)
+ I)]},
(4.29)
where J(/I,/2 ) is defined in Eq. (3.25), d l , d 2 , dc, a 2 are
given in Eqs. (3.11), (3.5), (3.9), and (2.28), respectively,
and
K(lI'/2) = /(11 '/2)
I) and
A. Van der Waals coefficients
Van der Waals coefficients for two atoms A and Bare
given by23
C6 = C(l,l),
- 1'/2)
(4.30)
with 1(/1 ,/2 ) defined in Eq. (3.26). The predictions for some
of these quantities are given in Tables V and VI for Be, alkaline earths, and some of their isoelectronics, along with the
values of some other calculations. The agreement is generally good. We believe that our predictions are reliable, particularly for L > 1 results, to within -10%-20%.
(5.1)
+ C(2,l),
C(2,2) + C(1,3) + C(3,l),
Cs = C(l,2)
(5.2)
(5.3 )
C IO =
where C(lA,/B ) are given by23
C(/A.lB) =
(2/A + 2/B )!
21T(2/A )!(2/B )!
Sa'" al
A'
0
B'
(zw)a/ (zw)dw,
A
B
(5.4)
with a1,B(w) being the 2L-polar dynamic polarizabilities of
atomAorB
V. SOME OTHER APPLICATIONS
Here we consider some other applications of our asymptotic wave functions, and end with a few concluding re-
.
I
~, 1(0 Il:j rfPL (cos (}j) iW(Ej
j
(Ei _EO)2+W 2
aL{Iw) = 2 L
marks.
-
Eo)
(5.5)
J. Chem. Phys., Vol. 95, No.6, 15 September 1991
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S. H. Patil and G. S. Setlur: Asymptotic wave functions
4254
TABLE VI. Predictions for 2 L _pole polarizabilities a L and SL ( - m). The values in brackets are from some
other calculations (a) Ref. 21; (b) Ref. I; (c) Ref. 22.
B+
a,
AI+
8.69
(10.3 )a
(12.6)b
5.14X 10'
3.22X 102
2.35X 10'
(1.97X 1O')b
(3.03 X 1O')C
4.59X 10'
9.40XIO'
1.36 X W
(1.92X 102 )C
1.67 X 102
2.18X 102
1.42 X 10'
1.32 X 10'
1.32 X 103
2.27X 10'
1.76 X 10'
1.48 X 10'
S,( -4)
S, (- 6)
a2
S2( -4)
S2( - 6)
a3
S3( -4)
S3( - 6)
a.
S.( -4)
S.( - 6)
a,
S,(-4)
S, ( - 6)
3.44x 10'
(2.70XIO,)a
(3.98X 1O')C
1.17 X 103
4.46 X 10'
1.33 X 102
(9.92X 1O')b
(1.50X W)C
7.58X W
4.50X 103
1.11 X 10'
(1.44 X 103 )C
3.38X 103
1.06 X 10"
1.65 X 104
3.52X 10'
7.73X 10'
3.75X 10'
6.33X 10'
1.11 X 106
Sc+
1.23 X W
3.55
(3.38)'
(4.41)C
9.38
2.68X 10'
4.87
(4.12)b
(5.46)"
3.34
2.37
1.55X 10',
(1.83 X 1O')C
6.20
2.58
8.77X 10'
2.57X 10'
7.88
7.65 X 102
1.81XWZ
4.57X 10'
2.68X 10'
6.36X 106
6.09 X 102
1.24 X 10'
2.63X 10'
8.14X 10'
7.67X 10'
7.34X 10'
1.96 X 10'
1.20X 106
7.43 X 106
7.21X 106
3.32XW
1.56x 108
where Ei are the energies of the atom.
We consider the following form 24 for a L (iliJ):
•
adlliJ)
=
Gj(L)
2
L
2
j=lliJ
+ [liJj(L)]
G (L)
2
=
adO) - P3liJi
2
P2
-
2
(liJ l -
1.09 X W
2.93X 10'
2.16X 102
(2.09X 102 )C
2.68X 102
3.37X WZ
1.94 X 103
l.60x 103
1.33 X 103
2.65x 10'
1.66 X 10"
1.06 X 10'
2
liJ2
(5.9)
- 4P2 ) 112] ,
(5.10)
{3LSL ( - 6) - a L (O)SL ( - 4)
{3
LSL( -4) -
adO)SL( -6) -
[adO)]
2
[SL( _4)]2
= --------=-...:.::....--~
(5.8)
2
2'
liJ 2 )
1 [ PI - (2
PI - 4P2 ) 112] ,
2P2
P2
PI =
P2
3
-
1 [PI + (2
= -2
PI
(5.7)
2'
P2 (liJ 2 - liJ l )
a (0)
P liJ2
L
5.01 X 102
1.42 X 10"
4.23 X 10'
(4.0IXIO')C
2 _
The parameters Gj and liJj can be determined by requiring
appropriate limits for liJ -+ 0 and liJ -+ 00. This yields
G I (L) =
2.03 X 10'
( 1. 74 X 1O')C
liJ l -
(5.6)
2 •
Si+ +
C++
{3L S L( -4) -
He-Sr
Sr-He
He-Ba
Ba-He
Be-Sr
Sr-Be
Be-Ba
Ba-Be
Mg-Sr
Sr-Mg
Mg-Ba
Ba-Mg
Ca-Sr
Sr-Ca
Ca-Ba
Ba-Ca
Sr-Sr
Sr-Ba
Ba-Sr
Ba-Ba
5.05X 10'
7.27X 10'
6.96XW
9.20xW
1.28X 10'
1.67 X 10'
2.60X 10'
3.41 X 103
3.31 X 103
4.39X 103
5.91 X 103
C(2,2)
C(I,3)
C6
C.
3.08X 10
2.30X 102
5.35x 103
3.34X 102
4.63 X 104
l.44x 10'
8.33X 104
1.94x 10'
8.27X 104
3.80X 10'
1.50 X 10'
5.04X 10'
1.61 X IO'
1.30X 10'
2.95X 10'
1.69 X 10'
2.55X 10'
5.05X 10'
3.31 X 103
7.27X 10'
5.68X 103
6.96x W
6.07 X 104
9.20X 102
1.03 X 10'
1.28 X 103
1.21 X 10'
1.67 X 103
2.DDx 10'
2.60X 103
2.91 X 10'
3.41 X 10'
4.64X 10'
2.DDx 10'
2.46 X 107
4.47 X 107
1.84X 10'
1.91 X 103
3.60X 10'
2.79x 103
2.64X 106
4.06X 10'
5.35x 106
5.52X 10'
4.64 X 106
1.38 X 106
9.49 X 106
1.85X 106
8.87x 106
6.47 X 106
1.83 X 107
8.51x106
1. lOX 107
2.28X 107
1.44 X 107
2.97X 107
3.31XIO'
4.39X 103
4.DDX 10'
5.91 X 103
9.54X 10'
3
3.68X 10'
2.61X 10'
4.77 X IO'
4.41 X 104
1.81 X 106
3.22X 106
4.76x 106
8.54X 106
1.61 X 107
2.91 X 107
8.17x10 7
( 5.12)
(5.13 )
P3 =P2{3L'
C(I,2)
(5.11)
[adO)r
TABLE VII. Some of the multipolar interaction coefficients C(/A,IB) defined in Eqs. (5.1)-(5.4), and C6 ,C.
for systems involving Sr and Ba.
C(I,I)
'
6.29X 10'
J. Chem. Phys., Vol. 95, No.6, 15 September 1991
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S. H. Patil and G. S. Setlur: Asymptotic wave functions
wherePL is defined in Eq. (3.21). Substitution ofEq. (5.6)
into Eq. (5.4) leads to
(21A
B. Hyperpolarizabilities
The energy of an atom in an electric field F may be written as 27
+ 2IB )!
C(lA ,Is) = 4(21 )!(2/ )!
A
X
±±
B
E= Eo _.la.F2 __I_rF4+ ... ,
2
24
where r is the hyperpolarizability given by
G~(lA )GJ(ls)
i= • j=. ali
(lA
)alj
(iB)
[ali (iA)
4255
+ alj (is)]
(5.16)
1
1
-24'r="4a1Sd -3)
(5.14)
The values of C(lA,IS) for different values Of/A and IB are
given in Table VII. For Be, Mg, Ca systems, generally our
results do not differ significantly from those of Standard and
Certain25 especially if the differences in the input values of
polarizabilities are taken into consideration. We have therefore given values for only those systems which have not been
considered by Standard and Certain. 25 We also note that
since our input values for the dipolar polarizability of Be are
somewhat smaller than those used by Standard and Certain,25 our predictions for systems including Be are probably
slightly underestimated. We have also calculated the threebody coefficients Z( 1,1,1) which describe26 the three-body
long-range potential for three identical particles
+
' (°!V.li)(i!Vdi)<i!V.lk )(k !V. 10) ,
iJ,
(Eo -Ei)(Eo -Ej)(Eo -Ek)
6
( 5.17)
with
v. = L riP. (cos 0;).
(5.18 )
i
To evaluate the second term in Eq. (5.17), we define
(H - Eo)r m (r.,r2,... , r N ) = V. r
m-. (r. ,r2'"'' r N),
m> 1,
(5.19 )
with
(5.20)
Z(1,I,1) = - 1
1T
L"" [a.
where 7]. isgiveninEq. (4.6).
In subspace R, the i = 1 term in VI is the most important term. Retaining only this term, we get
(5.15)
(ial)] 3dal.
0
These values are included in Table VIII.
r 2 (r ..r 2 , .. ·,
•
r N)=
2-n,
L
L
[A n,(1,I)+8n,.• d.A o (1,Q)]
n, = 0
[jAn,(O,4-nl)PO(cosO.)
n, =0
0)1 r4-n,-n, r -I+Z,/a, e -a,r'A,,(N-I)(
n. )P2 ( cos.
'/'0
r 2 ,r 3 , .. ·, rN ) ,
l
l
2-n,
[A n,(1,l) +8n"ld.A o (1,0)] L [jAn,(0,4-:-n l ) +Mn,(2,2-n.)]
2.4 (22
+ Y'n,
'
I
r 3 (rl,r2,· .. ,
rN
)
=
L
nl
=0
n2
(5.21)
=0
(5.22)
with tPo given in Eq. (3.12), and A; (L,q) for i< q, given in Eqs. (4.13) and (4.14), and Aq (L,q) given in Eq. (4.19). In Eq.
(5.22) we have left out the term in P 3 (cos 0 1 ), which does not contribute to r. We then have for the last term in Eq. (5.17),
I
2-n,
~ r2 ~ (,po Irl p. (cos 0. ) Ir 3) = jN~
[An, (1,1) + 8 n,,1 dlAo (1,0)]
[jAn, (0,4 - n l ) + 1s An, (2,2 - n l )]
L
nl
5-
X
L
=0
n 2 =0
n2
nl -
L
n3=0
A n3 (1,5-n l -n 2 )[J(7-n l -n 2 -n 3 ,0) +d.J(6-n l -n2 -n 3 ,0)].
TABLE VIII. Our predictions for three-body coefficients Z( 1,1,1), functions S, ( - 3), hyperpolarizability r,
and the coefficient K, of - B4 in the Zeeman energy.
Z(1,1.1)
S, (- 3)
1.58 X 10'
3.53X IQl
1.45 X 10'
4.39XW
1.75 X IQl
2.83X 10'
5.08X IQl
2.09 X 10'
1.95 X 10'
5.67
Be
Mg
Ca
Sr
Ba
B+
Al+
C+ +
r,
7.77 X 10'
4.14XW
3.46X 10"
7.34X 10"
2.IOX 107
1.77 X 103
3.77 X 10'
I.40X 10'
r
5.04X 104
2.40X lOS
1.76X 10"
3.62X 106
1.16 X IQ7
6.80XW
- 2.55X 103
1.92 X 10'
K,
5.08
1.41 X 10'
5.04X 10'
7.83X 10'
1.44 X 10'
4.77
2.33
9.3XIO-'
J. Chem. Phys., Vol. 95, No.6, 15 September 1991
Downloaded 01 Mar 2012 to 14.139.97.79. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
(5.23 )
4256
S. H. Patil and G. S. Setlur: Asymptotic wave functions
The predictions for r2 and r are given in Table VIII. A
particularly interesting point about these results is that there
is a high degree of cancellation between the two terms in Eq.
(5.17) so that r is rather small. Indeed, r is found to be a
small but negative number for Al +, Sc +, and Si + +. It
would certainly be interesting to verify this point.
C. Higher order Zeeman effect
The asymptotic wave functions allow us to calculate the
energy levels of atoms in the presence of strong magnetic
fields which may be encountered by donor impurities28 and
excitons,29 and in astrophysics. 30 The energy of an atom or
ion with zero angular momentum, in a magnetic field B, is
given by
E=Eo +!KB2_KIB4+ ....
(5.24)
The coefficients K and KI are related to the perturbation of
the energy of an atom or ion due to the interaction
V2=
i1J2 L r: sin2 0i
i
= -6.B 2 L r: [Po (cos 0i)
(5.25)
- P2 (cos 0i) ] .
;
Susceptibility K is given by
(5.26)
and can be obtained from Table 1. For the evaluation of K 1 ,
we need solutions TI to the perturbed equation
[H(NJ-E 6NJ]TI(fpfz,""
fN)
Vz ¢6 )(f l ,fz ,"" f N )·
Following the same steps as before, we get
N
=
TI
(fl,fz ,""
=-6.B2
fN)
I
L
nl
[A n ,(0,3) +8 n ,.ld I A o (0,2) -A n ,(2,1)
=0
XPz (cos ( 1)
3-n,
X 'I
(5.27)
- 8 n ,.1 dlAo (2,0)Pz (cos ( 1)]
-I+Z,/a,
'1
e
-a",,,/,(N-I)(
'f'0
,r3 , .. ·,
fz
)
APPENDIX
fN ,
(5.28)
where A; (L,q) fori<qaregiveninEqs. (4.13) and (4.14),
and Aq(L,q) is given in Eq. (4.19). This is the solution in
subspace R, where we have retained only the leading asymptotic term corresponding to i = 1. From this, one obtains
KI
=_1_2
12B
erties. We have considered the first few terms in the asymptotic series for Be, alkaline earths, and their isoelectronics, in
the subspace R: '1> 'z > .. "N' These are used to develop a
simple, model wave function which allows us to calculate
various properties of these systems such as susceptibilities,
polarizabilities, van der Waals coefficients, hyperpolarizabilities. The local energy test is used to increase the range of
validity of these wave functions.
While it is difficult to be precise about the accuracy of
the results, we expect our predictions to be current to within
-10%-20%. We do have a control over the wave function
in terms of the local energy test which for our solutions implies an accuracy of -10%. However, the main source of
our error is the treatment of the electrons in the inner shell.
The use oftheir asymptotic wave functions and the asymptotic expressions for the integral in Eq. (A2) is found to be
adequate for Be, Mg, and marginally so for Ca, but is less
accurate for Ba. For Ba, the error from this source may be as
large as 15%.
Another source of error might be the existence of excited states with L = 1 whose eigenvalues are close to E 6N ).
This means that in solving for the perturbed wave function in
Eq. (4.1), a large change in 7] I may when operated upon by
H (N) - E 6N ) give rise to a small term. This may be the reason why our prediction for the dipolar polarizability of Be is
-15% smaller than the value from other calculations. 3
The general accuracy and reliability of our results can be
improved by introducing additional, asymptotically subdominant terms, and applying the local energy test over a wider
range of the variables. Of course, this will reduce the basic
simplicity of the asymptotic wave functions. Nevertheless
we believe that this approach will provide accurate representations of wave functions which can lead to reliable evaluations of atomic and ionic properties. Finally we make use of
this opportunity to note two typographical errors in our earlier work l2 : the P 3 (cos 0l2) terms in the third line of Eq.
(2.27) should have a factor of '1- 4, and the first variable of I
in Eq. (4.39) should be 5 - n l •
Here we simplify the evaluation of I(n,m). We first observe that a; and Z;la; and i> 2, vary rather slowly as i
increases. We may therefore replace these a; and Z; by the
average values for the inner shell. This allows us to write
1
I( n,m ) ;:::::-
ne!
(¢6 lrf[I -Pz(cosO\)]IT
NJ
I )
d rle -
Za,"
ZZ,/a,
r "+
l
0
(AI)
1
I
=--N~
[A n ,(0,3)+8",.ldI A o (0,2)
144
", =0
L
+ !An, (2,1) + !8 ,.1 d l Ao (2,0) ]
X [J(5 - nl,O) + d 1 J(4 - nl,O)].
l""
where ne is the number of electrons in the next shell, 2 for Be
and 6 for Mg, Ca, Sr, and Ba, and their isoelectronics,
n
(5.29)
u(,z) =
1'2
e - ZaryZZlad,;:::::
re
X3
+ 1)(20) -
The calculated values of KI are given in Table VIII.
X [ 1 - (1
D. Concluding remarks
+ X3/ra + X3 (x 3 -
X3 -
I
1 )/~)
We have tried to demonstrate the importance of asymptotic wave functions in the analysis of atomic and ionic propJ. Chem. Phys .• Vol. 95. No.6, 15 September 1991
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(A2)
S. H. Patil and G. S. Setlur: Asymptotic wave functions
where
X3
'a =2'li'2'
=2Zra, a= (2£)112,
(A3)
~ith
£ being the average value of Ei and
Z = Z - N + (ne + 5)/2 is the average value of Zi for the
inner shell. Here, as our interest is primarily in the region
'a> I, we have retained only the first three terms in the
asymptotic expression.
Since ais quite large, U('2) is a slowly varying function
in the region of interest, i.e., '2::::: 1Ia2 , we may expand
[U('2)] n, about some point,o to be selected appropriately,
[u('2)r':::::bo
where
bi
=
+ ('2 -'o)bl>
(d~Ji [U('2) r'l
'2
=
(A4)
(AS)
'0'
We also use the interpolation
2
_(2Z
1+ (.J+ I - 2Z2)..i
- - - J')..i+
'-2
- '-2'
2Z2Ia 2
'2
-
a2
(A6)
a2
where integer j is chosen so that
j<2Z2 /a 2 ~ + 1.
This leads us to
(A7)
+b
[(n,m):::::boSo(n,m)
l
[SI (n,m) - 'oSo(n,m)], (AS)
with
S,,(n,m) =
2Z2
)
( -;;;-j M(n,m +j + k
+ (j + 1 where
2Z2) M(n,m
-;;;
+ I)
+ j + k),
_l_l""
ne'
dr
x(
2 )ircnl + i + I + 2Z /a
M(n I 'n 2 ) - ,
I
0
al
X(2a l
e-
2a"'rn , + 2Z,/a,
I
L"
d
(A9)
r2 e -
2a2'2 n2
r2
0
a
I
+a2
+2a2 )-n,-1-2z,/a,].
l )
(AlO)
4257
Finally, we optimize the convergence in Eqs. (A4) and
(AS), by choosing
'0
(All)
[(n,m) :::::So (n,m) [u(ro)] n ••
(AI2)
= SI (n,m)/So (n,m),
which results in
The procedure for evaluating I(n,m) is then as follows:
(l) determinejfrom Eq. (A7), (2) determine S" (n,m) defined in Eq. (A9) using Eq. (AlO), (3) determine,o from
Eq. (All), (4) determine I(n,m) as in Eq. (AI2) using
Eqs. (A9) and (A2).
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J. Chern. Phys., Vol. 95, No.6, 15 September 1991
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