PHYSICAL REVIEW A
VOLUME 58, NUMBER 1
JULY 1998
Boundary-condition-determined wave function for the ground state of positronium ion ēee
S. H. Patil
Department of Physics, Indian Institute of Technology, Bombay 400 076, India
~Received 9 December 1997!
We have developed a one-parameter, model wave function for the positronium ion ēee, which incorporates
the correct cusp and coalescence properties when two particles are close to each other, and the asymptotic
property when one of the electrons is far away. The predicted values for the energy and other properties are
close to the exact values and generally superior to the values from a 13-parameter wave function.
@S1050-2947~98!03307-1#
PACS number~s!: 31.10.1z
I. INTRODUCTION
Ever since the discovery of positronium, the negative positronium ion ēee has attracted a great deal of attention. It
was originally shown by Wheeler @1#, and by Hylleraas @2#,
that the electron is bound to the positronium with a binding
energy of at least 0.19 eV. Since then many elaborate calculations have been carried out to obtain accurate values for the
binding energy and for other properties of the system such as
the average value of the distance between the two electrons.
These calculations are generally based on the variational approach with a large number of variational parameters. Some
of these follow: by Kolos et al. @3# with 50 terms, by
Schroeder @4# with six parameters, by Frost et al. @5# with 50
terms, by Cavaliere et al. @6# with six and ten parameters, by
Bhatia and Drachman @7# with 203 terms, by Ho @8# with 125
terms, and Frolov and Yeremin @9# with 700 terms. These
calculations yield for the binding energy of the last electron
a value of 0.012 005 a.u. It may be mentioned that Frolov
and Yeremin @9# have obtained the energy to an accuracy of
13 significant digits. However, the variational wave functions do not emphasize the structural properties of the system. It is of considerable interest to develop wave functions
which illustrate some of the important properties such as the
behavior of the wave function when two particles are close to
each other or when they are far away from each other. Wave
functions which provide a clear illustration of these properties provide a deeper understanding of the structure of the
system.
Recently @10#, we have developed parameter-free wave
functions for the ground state of two-electron atoms and
ions, based on some general local properties of the exact
wave functions. These wave functions, in addition to being
very simple, provide accurate values for the binding energies
and expectation values of r 2n . The predictions are particularly striking for H2 where they are far better than the predictions of two-parameter wave functions, and quite close to
the essentially exact, variational values of the energy. Here
we extend the analysis to the case of the positronium ion,
ēee. Since the mass of ē is equal to the mass of the electron,
unlike the situation in H2 , the details are modified in a significant way. In particular, the center of mass not being at the
positive charge brings in essential changes in the asymptotic
behavior. Based on the local properties, we develop a oneparameter wave function which clearly exhibits the structural
1050-2947/98/58~1!/728~4!/$15.00
58
properties of the positronium ion in different regions. It gives
a value of 20.261 03 a.u. for the total energy and also reliable values for other properties such as ^ r i j & .
II. SOME LOCAL PROPERTIES
OF THE WAVE FUNCTIONS
Here we briefly discuss some general, local properties of
the wave functions. We use atomic units.
A. The Hamiltonian
The Hamiltonian for a three-particle system is given by
3
H5
(
i51
3
1 2
q iq j
pi 1
,
2m i
i, j r i j
(
~1!
where rW i j 5rW i 2rW j . After separating out the center of mass
term, the kinetic energy can be written as
T5
1
1
p 2i j 1
p2 ,
2m i j
2m i j,k i j,k
~2!
m im j
,
m i 1m j
~3!
where
mi j5
m i j,k 5
~ m i 1m j ! m k
.
m i 1m j 1m k
~4!
Here, p i j is the relative momentum of the particles i and j,
and p i j,k is the relative momentum of particle k and the
center of mass of particles i and j. The kinetic energy can
also be written as
T5
1
1
1
p2 1
p 2 1 pW •pW .
2m i j i j 2m k j k j m j i j k j
~5!
The form in Eq. ~2! is convenient for analyzing the
asymptotic behavior of the wave function, and the form in
Eq. ~5! is convenient for the calculation of the expectation
values of the energy.
728
© 1998 The American Physical Society
PRA 58
BRIEF REPORTS
729
B. Asymptotic behavior
The ground-state eigenfunction of the Hamiltonian H in
Eq. ~1! can be expanded in terms of the one-particle energy
eigenfunctions:
H c 5E c ,
~6!
c 5 ( u n ~ rW i j,k ! f n ~ rW i j ! ,
~7!
n
S
D
1
q iq j
¹ 2i j 1
f n 5E ~n1 ! f n .
2m i j
rij
2
~8!
Substituting the expression for c into Eq. ~6!, and projecting
out the state f n , we get for r i j,k →`
S
D
1
1
p 2 1q k ~ q i 1q j !
u 52 e n u n
2m i j,k i j,k
r i j,k n
for
c→
S
1 2ar 2 ~ 1/2! r
1 rW 13•rW 23
23 11
a
e 13
r 13
2
r 13
~10!
r 13→`.
~19!
C. Cusp and coalescence conditions
When two particles i, j with masses m i ,m j and charges
q i ,q j approach each other, r i j →0, two terms in the Hamiltonian dominate. These are the Coulomb interaction between
these particles, and the kinetic energy of the two particles. In
the center of mass frame of these two particles, one has
S
~9!
~1!
for
The presence of the second term in the large parentheses is
just a consequence of the fact that when Wr 13 is parallel to Wr 23 ,
electron 2 is closer to the center of mass of particles 1 and 3
than when rW 13 is antiparallel to rW 23 .
r i j,k →`,
e n 5E n 2E.
D
2
D
1
q iq j
2
¹i j1
c 5O ~ 1 !
2m i j
rij
for
r i j →0,
~20!
where m i j is the reduced mass of the particles i and j. Expanding the wave function in terms of spherical harmonics,
The asymptotic form of u n is given by
[2q ~ q i 1q j !~ m i j,k /2e n ! 1/221] 2 ~ 2m i j,k e n ! 1/2r i j,k
u n →r i j,k k
e
.
Clearly, the leading behavior is from the smallest value of
e n , i.e., corresponding to n50. Therefore we have
c →r bi j,k e 2ar i j,k f 0 ~ r i j ! ,
r i j,k →`
a5 ~ 2m i j,k e 0 ! 1/2,
b52
q k ~ q i 1q j ! m i j,k
21.
a
rW i j,k 5rW k j 2
mi
rW ,
m i 1m j i j
c 5 ( g lm ~ r i j ! Y ml ~ u i j , f i j ! ,
~11!
~12!
~13!
and projecting out an lm state, we get
d2
dr 2i j
@ r i j g lm ~ r i j !# 2
5O ~ r l11
ij !
r i j →0.
~15!
into Eq. ~22!, one obtains
W
For the case of the positronium ion ēee, we use the indices 1,2 to characterize the two electrons and 3 to characterize
the positron. Then when electron 1 is far away, we get
1
W W
c → e 2ar 131 ~ 1/2! a ~ r 13•r 23! /r 13e 2 ~ 1/2! r 23, r 13→` ~17!
r 13
a5 ~ 4 e 0 /3! 1/2,
for
Substituting
~18!
where e 0 is the binding energy of the last electron. The second term in the exponent would be negligibly small if the ē
is replaced by a proton. This is the major modification introduced by the positron mass being comparable to the mass of
the electron. To understand clearly the implications of this
term, we note that e 0 '0.012, and 21 a'0.06, so that expanding the second exponential, one obtains
~22!
g lm ~ r i j ! 5r li j ~ a 0 1a 1 r i j 1••• !
a 15
with e 0 being the binding energy of particle k. Expanding
r i j,k in inverse powers of r k j , we get
W
l ~ l11 !
g lm ~ r i j ! 22m i j q i q j g lm ~ r i j !
rij
~14!
c →r bk j e 2ar k j 1a[m i / ~ m i 1m j ! ][ ~ r k j •r i j ! /r k j ] f 0 ~ r i j ! , r k j →`.
~16!
~21!
l,m
~23!
m i jq iq j
a .
~ l11 ! 0
~24!
This is essentially a realization of the Kato condition @11#.
For the case of two electrons in the singlet state, the leading term is the l50 term so that
1
a 1 5 a 0 , ee in singlet state,
2
~25!
and the corresponding g 00 has the behavior
g 00→a 0 ~ 11 21 r i j !
for
r i j →0, singlet state.
~26!
For ē getting close to an electron, one has
1
a 1 52 a 0 ,
2
g 00→a 0 ~ 12 21 r i j !
for
r i j →0. ~27!
We now consider a model wave function for the ground
state of the ēee system incorporating the asymptotic condition in Eq. ~19! and the cusp and coalescence conditions in
Eqs. ~26! and ~27!.
730
BRIEF REPORTS
PRA 58
S
For the description of the wave function of the negative
positronium ion ēee, we treat ē as the point of reference and
use the notation rW 1 and rW 2 to describe the positions rW 13 and
rW 23 of the two electrons with respect to the positron. Then the
Hamiltonian corresponding to Eq. ~5! is given by
2
2
W 1 •¹
W 22
H52¹ 1 2¹ 2 2¹
S
D
1
1
1
1 2
. ~28!
r 1 r 2 u rW 1 2rW 2 u
For r 12→0, this leads to
¹ 212 f 2
S
S
H c 5E c ,
~29!
1
1
1e 2 ~ 1/2! r 2 e 2ar 1
11br 2
11br 1
D
D
rW 1 •rW 2
1
3 11 a 2 2 1/2 f ~ r 12! ,
2 ~ r 1 1r 2 !
f 5a 0 1a 1 r 121a 2 r 2121•••
FS
DG
1/2
1
b5 2a,
2
~31!
~38!
S D
~39!
3
1E a 0 50,
4
which lead to
1
a 15 a 0 ,
2
a 25
,
~37!
2a 1 2a 0 50,
6a 2 2a 1 1
~30!
where
4
1
a5
2 2E
3
4
~36!
Expanding f in powers of r 12 ,
Based on the asymptotic condition in Eq. ~19! and the
cusp and coalescence conditions in Eqs. ~26! and ~27!, we
propose for the ground-state wave function,
c 5C
S D
3
1
f 1 1E f 5O ~ r 12! .
r 12
4
~35!
and substituting in Eq. ~36!, we get
A. Model wave function
e 2 ~ 1/2! r 1 e 2ar 2
D
rW 1 rW 2
1
1
W 12 f 1
2
•¹
c.
1 f0
2
r1 r2
r 12
III. MODEL WAVE FUNCTION AND ITS IMPLICATIONS
~40!
S
D
1 1 3
2 2E a 0 .
6 2 4
Here, the first relation is the cusp condition in Eq. ~26!. Now
for the positronium ion, energy E'20.2610, so that a 2 is
indeed very small and we take it to be zero:
a 2 '0.
~32!
and f (r 12) is a correlation function. The choice of a in Eq.
~31!, with positronium ground-state energy of 21/4, ensures
that the correct asymptotic behavior given in Eq. ~19! is incorporated. The choice of b in Eq. ~32! ensures that the
coalescence condition in Eq. ~27! is satisfied for r 1 or r 2
→0. The function f (r 12) is the correlation function which is
introduced to incorporate the cusp condition in Eq. ~26!.
~41!
~42!
Based on these conditions, we take for the correlation function f (r 12),
f ~ r 12! 512
S DS
1
11l
11
D
l
r e 2lr 12,
2 12
~43!
which satisfies the conditions in Eqs. ~40! and ~42!. The
parameter l is treated as a variational parameter.
C. Calculation of the energy
B. Correlation function f„r 12…
The correlation function f (r 12) has been analyzed @10# for
the two-electron atoms and ions, in an essentially perturbative approach. We extend this analysis to the case of the
positronium ion.
Consider a wave function of the form
c 5 f 0 ~ r 1 ,r 2 ! f ~ r 12! ,
~33!
f 0 ~ r 1 ,r 2 ! 5e 2 ~ 1/2!~ r 1 1r 2 ! .
~34!
This has the correct behavior for r 1 or r 2 →0. Substituting
the function c in the Schrödinger equation in Eq. ~29!, we
get
S D
1
1 rW 1 •rW 2
2
E c 52 c 2
c 2 f 0 ¹ 12 f
2
4 r 1r 2
The wave function c in Eq. ~30! is used to calculate the
energy of ēee as
E5
^cuHuc&
,
^cuc&
~44!
with the correlation function f (r 12) in Eq. ~43!, and the
Hamiltonian in Eq. ~28!. It should be noted that the parameter a in Eq. ~31! depends on E and is therefore determined
iteratively so as to yield a self-consistent E. The parameter l
is determined by minimizing the energy.
The calculation of the energy E in Eq. ~44! is greatly
simplified by the use of the identity
E
W i •¹
W j !~ f f ! d t 5
~ f f !~ ¹
E
W i •¹
W jf
@ f 2f ¹
W i f !•~ ¹
W j f !# d t , ~45!
2 f 2~ ¹
PRA 58
BRIEF REPORTS
TABLE I. The values of the variational parameter l, normalization constant C in Eq. ~30!, and the predicted values of the total
energy E and average values of different functions of the ēe and ee
separations Wr 1 p and Wr 12 .
l
C
E
^ r 1p &
^ r 21p &
^ r 21
1p &
^ r 12&
^ r 212&
^ r 21
12 &
^ d (rW 1 p ) &
^ r 41p &
^ r 412&
Our results
~one parameter!
13-parameter results
~Ref. @8#!
Exact results
~Refs. @7,8#!
0.262
0.014 921
20.261 03
5.452
48.31
0.3369
8.513
94.31
0.1591
0.019 39
20.261 01
5.114
38.83
0.3429
7.842
74.81
0.1639
0.019 43
20.262 00
5.489
48.39
0.3398
8.548
93.13
0.1556
0.020 73
1.0763104
2.3263104
731
asymptotic exponent a of the wave function in Eq. ~30! depends on E and our binding energy is slightly smaller, the
exact results for ^ r 41p & and ^ r 412& are expected to be smaller
than our predictions by about 5%. In a general way, the
asymptotic properties and cusp and coalescence properties
are of great importance in developing simple model wave
functions for the positronium ion, which not only provide
physical insight into the physical structure in different regions, but also allow us to obtain accurate values for different properties of the system.
APPENDIX
Some of the terms required for the evaluation of the average energy are as follows:
¹ 2 e 2 ~ 1/2 ! r 5
¹ 2 e 2ar
which follows from integration by parts. This identity is a
generalization of the identity for the special case of i5 j,
which was obtained by Siebbles et al. @12#. Some of the
details of the calculation are given in the Appendix.
@1# J. A. Wheeler, Ann. ~N.Y.! Acad. Sci. 48, 219 ~1946!.
@2# E. A. Hylleraas, Phys. Rev. 71, 491 ~1947!.
@3# W. Kolos, C. C. J. Roothan, and R. A. Sack, Rev. Mod. Phys.
32, 178 ~1960!.
@4# U. Schroeder, Z. Phys. 173, 221 ~1963!.
@5# A. A. Frost, M. Inokuti, and J. P. Low, J. Chem. Phys. 41, 482
~1964!.
@6# P. Cavaliere, G. Ferrante, R. Gercitano, and L. Lo Cascio, J.
Chem. Phys. 63, 624 ~1975!.
~A1!
S DS
1 rW 1 •rW 2
2 r 1r 2
S
a1
D S D
1
2ab
e 2ar
,
11br
11br
W 1 •¹
W 2 ! e 2 ~ 1/2! r 1 2ar 2
~¹
5
1 1 2 ~ 1/2! r
2 e
,
4 r
2b 2
1
2a
2b
1
5 a 22
2
2
11br
r
r
11br
!
~
~ 11br !
1
D. Results
The results of the calculation are given in Table I, along
with the results from 13-parameter calculations @8#, and essentially exact results @7,8# using 125 and 203 terms. Our
results with a one-parameter wave function incorporating
some local properties are close to the exact results and are
generally superior to the results using 13 parameters. This
emphasizes the importance of the local properties of the
wave function, when two particles are close to each other or
when they are far away from each other. In particular, we
note that if the term in Eq. ~30! proportional to Wr 1 •rW 2 is not
included, the energy changes from the present value of
20.2610 to about 20.2562 which is a poor result.
We also note that since our wave function incorporates
the correct asymptotic behavior, its predictions for the average values of higher powers of separation r 1p between an
electron and the positron and r 12 between the two electrons,
are expected to be quite reliable. However, since the
S DS
S D
S
1
11br 2
D
D
S
~A2!
D
1
b
e 2 ~ 1/2! r 1 2ar 2
,
11br 2
11br 2
D
~A3!
rW 1 •rW 2
1
W 1 11 a
¹
f ~ r 12!
2
2 ~ r 1 1r 22 ! 1/2
S
D
rW 1 •rW 2
1
W 1 f ~ r 12!
5 11 a 2 2 1/2 ¹
2 ~ r 1 1r 2 !
1
S
D
rW 2
a
~ rW 1 •rW 2 ! rW 1
2
f ~ r 12! ,
2 ~ r 21 1r 22 ! 1/2 ~ r 21 1r 22 ! 3/2
W 1 f ~ r 12! 5
¹
rW 12
l
~ 11lr 12! e 2lr 12.
2 ~ 11l !
r 12
~A4!
~A5!
@7# A. K. Bhatia and R. J. Drachman, Phys. Rev. A 28, 2523
~1983!.
@8# Y. K. Ho, J. Phys. B 16, 1503 ~1983!.
@9# A. M. Frolov and A. Ya Yeremin, J. Phys. B 22, 1263 ~1989!.
@10# U. Kleinekathofer, S. H. Patil, K. T. Tang, and J. P. Toennies,
Phys. Rev. A 54, 2840 ~1996!.
@11# T. Kato, Commun. Pure Appl. Math. 10, 151 ~1957!.
@12# L. D. A. Siebbles, D. P. Marshall, and C. Le Sech, J. Phys. B
26, L321 ~1993!.