Asymptotic behavior of twoelectron atomic wave functions
S. H. Patil
Citation: J. Chem. Phys. 80, 2689 (1984); doi: 10.1063/1.447065
View online: http://dx.doi.org/10.1063/1.447065
View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v80/i6
Published by the American Institute of Physics.
Additional information on J. Chem. Phys.
Journal Homepage: http://jcp.aip.org/
Journal Information: http://jcp.aip.org/about/about_the_journal
Top downloads: http://jcp.aip.org/features/most_downloaded
Information for Authors: http://jcp.aip.org/authors
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
Asymptotic behavior of two-electron atomic wave functions
s. H. Patil
Department of Physics, Indian Institute of Technology, Bombay 400076, India
(Received 20 September 1983; accepted 29 September 1983)
The relation between the asymptotic behavior of atomic wave functions and their energies is used
to determine model wave functions and their energies, for two-electron atoms H - , He, Li + , and
Be++.
The helium-like atoms have been subjected to a large
number of investigations over the years. Of the many approaches to the problem, variational calculations are the
most common. They have the advantage that their accuracy
can be systematically improved by introducing additional
parameters. A high point in such calculations was reached
by Pekeris I who used variational functions containing 1078
terms. However, with the increase in the number of terms,
one quickly loses a simple physical interpretation. Hence
there is still a considerable effort 2-6 directed towards finding
simpler, physically meaningful wave functions.
Recently, Wu 2 has discussed a three-parameter wave
function
(1 )
for the two-electron atoms. Here the, 12 term describes the
correlation between the two electrons and the different coefficients for, > and r < incorporate the fact that the electron
which is farther away sees a smaller charge. This expression
is the generalization of the expression used by Srivastava and
Bhaduri4 who consider the wave function in Eq. (1) but with
A = O. In fact the observation of different charges by the two
electrons is implicit in an earlier paper by Shull and Lowdin 7
who use a symmetrized wave function with different exponents for'l and '2' In all these cases, the predicted binding
energies are in good agreement with those of accurate variational calculations. I
Though the wave function of a two-electron atom is
quite complicated in structure, it has a simple behavior in the
region'l or '2--+00. In terms of this asymptotic behavior, the
coefficient of, < is related to the nuclear charge and that of
, > is related to the ionization energy of the atom. Therefore,
if we demand that the wave function has the correct asymptotic behavior, then the exponents in the wave functions are
essentially determined. We exploit this property to obtain
simple wave functions not only for the ground state of the·
two-electron atoms but also for some of the excited states.
The Hamiltonian for a two-electron atom (in atomic
units) is
=! {PI2
2
+P2)-Z
(1-+-1) +-.
1
rl
I "'n (rdrn ('zl,
n=O
(3)
where
(~pi - ~)"'n(r.) =
(4)
En"'n(rtJ·
Since we have the relation
• Z
2
Z
Z-1
H --+ Z(PI +P2)------,
"~oo
']
r2
(5)
we can operate by H on t/' and project out the nth state to get
Z-l)
2
(
~P2
for
--'-2'2--+ 00
fn('2) = (E-E nlfn('2)
(6)
.
Thus we have the asymptotic condition
()
f n'2
Bn --+ D n'2 e
an"
,
(7)
an = [2(En -EW /2 ,
(8)
Pn
(9)
= Z -1 -1.
an
This implies that the n = 0 term corresponding to the
ground state dominates the asymptotic region,
(10)
In some cases, we may have reasons to include higher energy
terms, i.e., n = 1,2, ... , etc., along with the asymptotic forms
for f n (,) given in Eq. (7). The above arguments are also valid
for,.--+ 00. In the following discussion, we will impose the
condition in Eq. (10) as a consistency requirement. Though
the asymptotic behavior in terms of the ionization energy is
well known, 8 these conditions have not so far been used in
the determination of two-electron wave functions. In order
to keep the integrations elementary, we will incorporate only
the asymptotic exponential behavior.
II. ZERO PARAMETER WAVE FUNCTION FOR THE
GROUND STATE
I. ASYMPTOTIC BEHAVIOR
H
00
t/'('."z) =
'2
(11)
(2)
'12
An eigenstate of this Hamiltonian with eigenvalue E can be
expanded in terms of the eigenstates of the one-electron
atom:
J. Chern. Phys. 80 (6),15 March 1984
We first consider a simple wave function of the form
for the ground state, where the condition in Eq. (10) demands
that
1/a = Z,
0021-9606/84/062689-04$02.10
(12)
(13)
© 1984 American Institute of PhySics
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
2689
2690
S. H. Patil: Two-electron atomic wave functions
TABLE I. Predictions for - E of the ground state of H -, He, Li +, and
Be+ +, along with those of Refs. 1,2, and 4.
Zero-parameter (present)
One-parameter (present)
Two-parameter (Ref. 4)
Three-parameter (Ref. 2)
Exact (Ref. 1)
H-
He
Li+
Be++
0.5226
0.506
0.5213
0.5278
2.867
2.895
2.873
2.899
2.904
7.235
7.269
7.246
7.276
7.280
13.607
13.643
13.621
13.651
13.656
(7) and (8),
J... = ( _j Z2 _
2E)1/2.
In these expressions one uses the self-consistent value of E
obtained by calculating the expectation value of the Hamiltonian in Eq. (2) with the wave function in Eq. (17).
The expectation value of the Hamiltonian in Eq. (2) is
given by
2(¢I H d¢)
1
+ (¢1_
I¢)
E=
The expectation value of the Hamiltonian in Eq. (2) with this
wave function is
E =A
+
2~[aJb(1 -
2Zb) + b 3a(l- 2Za)
r l2
(¢I¢)
where HI is the one-particle Hamiltonian,
(22)
I
z
Z
H 1 =zPl
--.
r1
128 x6 (1_Za_Zb)
ab
+ 2(a2 + b 2)x3 + 6(a + b)x4 + 4Ox5 ] ,
The matrix elements are given in the Appendix. We determine E iteratively. To illustrate the procedure, we start with
an input value of
(14)
(23)
E in = - 2.895
for He, which gives
where
a
ab ...
X=--;
a+b
A
(21)
C
2
(15)
= (2~a3b 3 + 128~X6)-I.
(16)
Since b isrelated to E by Eq. (13), Eq. (14) gives an implicit
equation for E which is determined iteratively. We find that
for Z = 1, there is no bound state satisfying Eq. (14). The
predicted energies for Z = 2, 3, 4 are given in Table I. Our
zero-parameter values are an improvement over the usual
one-parameter solutions, the parameter being the average
screened charge.
III. ONE-PARAMETER WAVE FUNCTION FOR THE
GROUND STATE
The essential point of the asymptotic conditions is that
they help us to determine the exponents in the variational
functions. We now try to incorporate corrections to the
expression in Eq. (11), by including a term corresponding to
the first excited level in Eq. (3), viz. n = 1 term.
Consider a ground-state wave function of the form
rp(r l ,r2 ) = u(r l )v(r2 ) + v(r l )u(r2 ) + g~w(rJlw(r2)
c
(17)
with g being a variational parameter and
u(r) = e -
ria,
vIr) = e -
rib,
w(r)
= e-
(18)
= 0.5,
b = 0.747 435,
c = 0.456 912.
(24)
Using the matrix elements in the Appendix, one gets for
these values
5.571 906 + 5.886627 g + 1.387 226g2.
1.943 696 + 1.966 136 g + 5.388 209 g2
Varying g, we find that E is minimum at
E
= _
(25)
g= 0.564.
The minimum energy is
E min
= -
(26)
2.895,
consistent with the input value in Eq. (23). The minimum
energies are given in Table I, for Z = 1 (g = 2.48), 2
(g = 0.564), 3 (g = 0.347), and 4 (g = 0.250). These values are
consistently better than the two-parameter values of Srivastava and Bhaduri.4
It is interesting to note that while our one-parameter
energies are slightly higher than the three-parameter values
of Wu 2 for Z = 2, 3, and 4, our value for H- is distinctly
superior to that ofWu. Since the electrons in H- are loosely
bound, this is reasonable, and the asymptotic part of their
wave function may be expected to be important. Our approach is specifically oriented towards incorporating the appropriate asymptotic behavior.
The asymptotic conditions in Eq' (10) can be used to
calculate the ionization energies of the excited states as well.
The calculations are especially simple if the excited states are
orthogonal to the ground state.
rl<.
For simplicity we have used a symmetric product form for
the correction term. If vIr} governs the asymptotic behavior,
we have from Eq. (10),
1/a =Z,
1/b = ( - Z2 - 2E)1/2.
(19)
(20)
Since w(r) describes the asymptotic behavior when the remaining electron is in the first excited state, we get from Eqs.
IV. IONIZATION ENERGY OF THE 2 3S STATE
For the 2 3S state, we consider a wave function of the
form
¢(r l ,r2 ) = D [e - "la(I _ gr2)e -
r2/b
(27)
where a and b are given by Eqs. (19) and (20), and g is a
J. Chem. Phys., Vol. 80, No.6, 15 March 1984
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
2691
S. H. Patil: Two-electron atomic wave functions
TABLE II. Predictions for - E of some low-lying excited states of He, Li + ,
and Be + + compared with the exact values of Ref. 8, given in brackets.
He
Li+
Be++
2'S
2.1738
(2.1753)
5.1090
(5.1109)
9.2953
(9.2976)
2 'p
2.1223
(2.1239)
4.9898
(4.9935)
9.1058
(9.1111)
2'P
2.1302
(2.1332)
5.0215
(5.0279)
9.1670
(9.1753)
E:::::2.1223
+
1:8x
2
6
E:::::2.1302
(1 - 3gX)2
- Za 3b 2(2 - 4bg + 3b 2g2) + a3b (1 _ bg + b 2g2)
+ 2a 2(ab 2 _ ax 2 _ x 3 ) + 2a 2g( _ 2ab 3 + 2ax 3 + 3x4 )
+ 3a 2g2(ab 4 _ ax4 _ 2x5 ) - 8x 5 (5 - 25xg + 33x2g2)], (28)
with x = ab /(a
+ b ) and
D 2 = [2rra 3b 3(1 - 3bg + 3b 2g2) - 128rrx6 (1 - 3xg)2] - I .
(29)
Since the parameters a and b depend on E, they are determined iteratively. In the case of helium, we find that for an
input value of E = - 2.1738, the expression in Eq. (28) for
the energy has an extremum at
g:::::2.03.
(30)
The corresponding minimum energy is
E min :::::
-
2.1738,
(35)
for the 2 I P !ltate, corresponding to an ionization energy of
3.327 eV, and
variational parameter. The expectation value of the energy is
given by
E = rrD2[ - ab 3(1 - 3bg + 3b 2g2)
Since b is a function of E, the above equation is solved iteratively. For helium, we get
(31)
consistent with the input energy. This energy implies an ionization energy of 4.727 eV which is in quite good agreement
with the exact ionization energy9 of 4. 767 eV. The minimum
energies for Z = 2(g = 2.03), Z = 3(g = 2.13), and Z = 4(g
= 2.55) are given in Table II, along with the accurate values
of Accad et al. 9 The agreement between the two sets of values is quite encouraging.
(36)
for the 2 3P state, corresponding to an ionization energy of
3.541 eV. These are to be compared with the exact ionization
energies9 of 3.368 and 3.623 eV, respectively. The energies
for Z = 2,3, and 4 are given in Table II. The agreement with
the accurate calculations of Accad et al. 9 is in general quite
satisfactory considering that there are no variational parameters in these calculations.
VI. DISCUSSION
The asymptotic behavior of two-electron atomic wave
functions relates the exponents of the model wave functions
to the energies of the atom and the charge of the nucleus. The
model wave functions which incorporate these relations,
provide simple and physically meaningful approximations
to the exact wave functions. Their predictions for the energies are in good agreement with the experimental values.
They are particularly useful for H- in which the electrons
are loosely bound and the asymptotic part of the wave function may be expected to be important. Many of these considerations can be extended to atoms with three or more electrons.
APPENDIX
The matrix elements in Eq. (22) are given by the following expressions:
g2
+ 2'(wlwzlrizlwIW2)'
c
("'IHII"') =
(ulu)(vIHllv)
(37)
+ (vlv)(uIHllu)
+ 2(ulv)(vIH l lu) + 2~(wlw2IrI2Hllulvz)
c
V.IONIZATION ENERGY OF THE 2PSTATES
For the 2P states, we use wave functions of the form
/a 2p
.,,(
r I( cos 0)
-"Ib
'l'r l,r 2 ) = F[ e -" b
2 e
rip (
0) - "ib] ,
+ Ee - "Ia-Icosle
b
(32)
where E = + 1 for the 2 Ip state and E = - 1 for the 2 3p
state. As before, a and b are given by Eqs. (19) and (20). The
expectation value of the energy is
E = rrF2[ - ab 3 + a3 (b _ Zb 2)
b
+ b 4X3 + 3b 3X4 + 6b 2X 5 + lObx6 ) + ~~]
3
with x = ab /(a
b2
(33)
+ b) and
3
F2 = (2ra b 3)-1.
+ ~(ulw)(vlw) + ~(wlwzlrn!wlwZ)
c
c
where the subscript is the particle index, and
2x 5 + lOax 6
+ _1_(2a
2
(34)
(39)
(ulu) = 11'a 3 ,
(40)
(ulv) = 11'b 3,
(41)
(ulv) = 11'Xj,
(42)
(ulw) = 11'xL
(43)
J. Chern. Phys., Vol. 80, No.6, 15 March 1984
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
2692
S. H. Patil: Two-electron atomic wave functions
(vlw)
= 11'xi,
(44)
(u1vzlrIZiw1w Z ) = 16r I( XI,X Z ),
(w 1wzlri2Iw IW 2)
(uIHdu)
=
-
= Wc
8
(45)
(46)
,
!11'a,
(47)
(vIH1Iv) =!11'b(I-2Zb),
(VIHllu)=11'X~[(~
(wlw2IrI2HlluIV2)
-Z):3
= -
with
2bc
2ca
2ab
x 1 =--, x z = - - , x 3 = - - ,
b+c
c+a
a+b
I( xl,xz) = :fzx6tvi + 3y~ + ZPyi + '!fYI + Y;
-~2]'
8r
-2I( X1,x2)'
a
(49)
+ ZPyi + '!fYz),
rx~ [Yltvi + 3YI + 8)
+ 3yi
(48)
h (x\,x z) =
+ !Yz(3yi
+ 6y~ +
lOyz + 16)],
(50)
(56)
(57)
(58)
(59)
(60)
+ (~ (wlw2IrJ2HlrdwlwZ)
(u 1v2[_I_[u 1vz)
r 1z
= ~
Z
)h
(X I ,X2 ),
rc 6 (4 - 9cZ),
=~r(8a3bZ-2a3xj -aZx~),
(u 1vzl_l_lv\u z ) =irxL
r lz
(w1wzlrdw1w z ) = 1irc7 ,
(51)
(52)
(53)
(54)
(55)
'c. L. Pekeris, Phys. Rev. 115, 1216 (1959).
2M._S. Wu, Phys. Rev. A 26,1764 (1982).
3S. G. Lie, Y. Nogami, and M. A. Preston, Phys. Rev. A 18, 787 (1978).
4M. K. Srivastava and R. K. Bhaduri, Am. J. Phys. 45, 462 (1977).
'M. K. Srivastava, R. K. Bhaduri, and A. K. Dutta, Phys. Rev. A 14, 1961
(1976).
6R. K. Bhaduri and Y. Nogami, Phys. Rev. A 13, 1986 (1976).
7H. Shull and P. O. Lowdin, J. Chem. Phys. 25,1035 (1956).
sR. Ah1richs, M. Hoffmann-Ostenhoff, T. Hoffmann-Ostenhoff, and J. D.
Morgan, Phys. Rev. A 23,2106 (1981).
<>Yo Accad, C. L. Pekeris, and B. Schiff, Phys. Rev. A 4,516 (1971).
J. Chem. Phys., Vol. 80, No.6, 15 March 1984
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