J. PhyS. Chem. 1982, 86, 3513-3517
3513
ARTICLES
Hellum Atom in the Momentum Representation
John R. Lombard1
Downloaded by CITY COLLEGE OF NEW YORK on August 24, 2009
Published on September 1, 1982 on http://pubs.acs.org | doi: 10.1021/j100215a006
~~t
of t%mbby, Ct& Co/h?geof New York, New Yo& New York 10031 (Recehred: July 28, lS81)
The ground state of the helium atom is examined in momentum space. The transform used is one in which
the momentum variables are conjugate to corresponding position space variables. An intqpl equation is derived
which is solved by a simple self-consistentprocedure. The trial functions chosen are the momentum space
analogues of a simple product of one-electron Slater orbitals. One-electron equations are obtained by averaging
over the momentum distribution of the second electron. The optimized one-electron energy and total energy
are identical with the analogous position space calculation. The calculations are shown to be vastly simplified
by considering the momentum variables to be complex. This enables the use of the theory of residues to evaluate
the resulting integrals. In addition, the mathematical as well as physical significance of singularities in the
complex plane are examined.
Introduction
The problem of determining the momentum space eigenfunctions for atomic systems was initially examined
over 50 years ago by Pauling and Podolsky,' who used a
Fourier transform on position space functions for the hydrogen atom. Later Fock2 solved momentum space
equations directly to obtain the same function. Bethe and
SalpeterS present a rather thorough discussion of their
results.
Since Pauling and Podolsky, a variety of authors have
attempted to determine the momentum space functions
for the helium atom. Kirkpatric, Ross, and Ritland4 used
the transform of Podolsky and Pauling on Slater orbitals
to obtain momentum distributions in several of the lighter
elements. Hickss applied the transform of Podolsky and
Pauling to approximations to the correlated functions
obtained by Hylleraas.6 McWeeny and Coulson7 introduced the effective nuclear charge as a variational parameter into the functions of Podolsky and Pauling. These
functions serve as the momentum analogue of Slater orbitals. They proceeded to obtain still better functions by
use of an iterative technique, which resulted in momentum-correlated functions. An extensive review of early
work is given by Cooper.* More recent work has been
conducted by Bayard and co-workers?JO who used the
same transform on Hartree-Fock functions as well as the
configuration interaction functions of Weiss.'l
In a recent work12 we have shown that the transform
used by Podolsky and Pauling does not lead to a true
(1) G. Podoleky and L. Pnuling, Phys. Rev., 34, 109 (1929).
(2) V. Fock, 2.Phys., 98,146 (1936).
(3) H. A. Bethe and E. E. Selpeter, yQuantum Mechanics of One and
Two Electron Systems",Plenum-Rosetta,New York, 1977.
(4) P. Kirkpatric, P.A. Roes, and H. 0. Ritland, Phys. Reu., 50,928
(1936).
(6) B. Hicks, Phys. Rev., 62, 436 (1937).
(6) E. A. Hvllerm. 2.Phvs.. 64.347 (1929).
A.
(7) R. McWeeny Ad C. A: Couhn, &oc. Phys. SOC.,London, Sect.
(1949).
_ _ ._
,.
(8) M. Cooper, Adu. Phys., 20, 463 (1971).
(9) K. E. Bayard and J. C. Moore, J. Phys. B, 10,2781 (1977).
(10) K. E. Bayard and C. E. Reed, J. Phys. B, 11,2957 (1978).
(11) A. W . We&, Phys. Rev.! 122,1826 (1961).
(12) J. R. Lombardi, Phys. Rev. A, 22,797 (1980).
609
.-, 62.
__,
-.
.
0022-3654/82/2086-3513$01.25/0
momentum representation in that the momentum space
variables used were not chosen properly conjugate to their
poeition space variables. Instead a transform was obtained
involving momentum variables chosen to be Hermitian,
and so that a commutation relation with the appropriate
spatial variable exists leading to a Heisenberg uncertainty
relation. This transform is not a Fourier transform, but
it preserves the reciprocal relation between r and r-l in the
momentum space. The resulting momentum space equations are solved to give eigenfunctions which differ considerably from those of Podolsky and Pauling. Instead of
angular functions which are products of Legendre polynomials and exponentials, we obtain Bessel functions and
Dirac 6 functions. For radial functions, instead of Gegenbauer polynomials, we obtain functions of the radial
momentum in the complex plane which are linear combinations of simple poles.
Since all of the above-mentioned work on the helium
atom utilizes, in one way or another, the transform of
Podolsky and Pauling, it is worthwhile to reconsider helium in the light of the new hydrogenic results. In this
work we shall carry out a simple self-consistent field calculation on the ground state of the helium atom. We begin
by determining the appropriate integral equation in momentum space, which involves mainly the determination
of the proper form of the electron-electron repulsion. We
may choose an appropriate trial function using simple
products of the momentum analogue of Slater orbitals. We
then determine an effective one-electron equation by averaging over the momentum of the second electron. The
resulting equation may be solved by the variational principle, yielding one-electron orbital energies, as well as
optimized functions. The energies obtained are the same
as the corresponding position space calculation.
Momentum Space Equations for the Ground State
of Helium
We choose the coordinate system in position space with
an origin at the atomic nucleus. The resulting equations
may be completely specified by using six variables, the
three spherical polar coordinates of each of the two electrons. We may write
0 1982 American Chemical Soclety
8514
The Journal of Physical Chemistry, Vol. 86, No. 18, 1982
However, simple geometric considerations6indicate that
the problem may be reduced to three dimensions since the
potential energy depends only on three variables, the two
radial positions rl and rz as well as the angle between the
electrons y. If we further restrict consideration to
ground-state trial functions which are simple products of
radial orbitals (Is, Slater type), it is easy to show that in
the repulsion term which may be expanded
1
r12
-=
-
E
n=O
r<n
r.>n+l
y)
----P,(COS
r> = Max (r1,r2)
Downloaded by CITY COLLEGE OF NEW YORK on August 24, 2009
Published on September 1, 1982 on http://pubs.acs.org | doi: 10.1021/j100215a006
r< = Min (r1,r2)
all terms in the sum vanish upon integration over the
angular variables except for n = 0. This may be demonstrated in momentum space as well.
Additionally the above restrictions lead us to
v12 =
a2
2 a
rl ar,
-+ - dr12
Lombard1
the next few integrals. We shall also require a number of
integrals which have been obtained from a compilation by
Gradshteyn and Ryshik.13 For convenience we shall refer
to them as GR along with the appropriate reference number. We wish to evaluate
This integral, however, diverges unless we extend the
range of integration to [-a,-]and consider its principal
value. This procedure, requires that we extend the values
considered for p in the complex plane to -p as well. Thus,
for example, if we carry out contour integrals in the complex p plane, our contours must be symmetric with respect
to inversion through the origin.
It is also important to note that this extension no longer
ensures that the variable r12remains positive. In fact, we
must choose either r1 or r2 to integrate over initially, introducing a dissymmetry into the process. The resulting
integral is no longer symmetric with respect to interchange
of p1 and pz. To remedy this, we must carry out the integration twice, integrating first over the other variable,
and average the two results.
1-
e-iP;;drl
with a similar expression for r2. Then eq 1 becomes
-a2
+ - - +2 - a+ - arl2
r1 dr,
a2
r2 drz
"-z[;+
ar22
If we let $(p1,p2) be the momentum function in terms
of the radial momenta of the electrons p1 and p2,we may
write
+
-i(;
rl-l
;)
+P2
- -isp1
-
rZ-l
where we have referred to GR(8.2). The second term is
the cosine integral function.
dplr = I1
GR(3.941,2; 6.224).
Integrating a second time, first over r2 and then over rlt
gives
-ispz
dp,' = Iz
where the last three expressions are regarded as integral
operators, which act on the functions pi).
v(Pl-Plr,P2-Pzr) =
P1+ Pz
@2
- Pl)(Pl + Pz)
Pz2
Taking one half the sum of these expressions, we obtain
(5)
S is the appropriate two-electron transform of radial
functions between position space and momentum space,
analogous to that of ref 12. In order to find the correct
integral equation in momentum space, we must evaluate
the function V. Since this is a function of pl-pl', p2-p<,
we shall use only p1 and p2for simplicity in notation in
This is the appropriate transform of the n = 0 term of r12-l.
See Appendix I for the transform of the general term of
(13) I. S. Gradshteyn and I. M. Ryahik, "Tables of Integrals, Series,
and Products", 4th ed., Academic Press, New York, 1965.
(14) See, for example, M. Karplw and R. N. Porter, 'Atoms and
Molecules", W. A. Benjamin, New York, 1970.
(15) W. M.Huo and E. N. Lassettre, J. Chem. Phys., 72,2374 (1980).
Hellum Atom in the Momentum Representatkn
The Journal of phvslcal Chemistry, Vol. 86, No. 18, 1982 3515
eq 2. We may then write the equation in momentum
space.
Next we must evaluate, once again using residues
VEff(P1) =
1
P1+ P2 - is;
~ C +Z ip1
2t.z - iP1
One-Electron SCF Calculation in Momentum
Space
We shall select as a trial function a simple product of
functionswhich depend on the momenta of electrons 1and
2, respectively. It is convenient to choose the analogue of
Slater orbitals in momentum space (Appendix 11) so that
2ai3i[6(p1 - 2iJi) - 6 b 1 + 2iS;)I
We may then write our effective one-electron equation in
terms of the orbital energy el as
[f/2Pl2- 231 + Ieff - elI41b1) = 0
Downloaded by CITY COLLEGE OF NEW YORK on August 24, 2009
Published on September 1, 1982 on http://pubs.acs.org | doi: 10.1021/j100215a006
We may determine the trial energy by
where we take C1 and to be variable parameters. Initially
we shall allow them to have distinct values. Cl and tz in
position space may be interpreted as the effective nuclear
charge due to screening of the bare nucleus by the other
electron. In momentum space we see that they represent
a pole of order 2 in the complex momentum plane. By
analogy with the hydrogen atom, we see that l1and have
units of momentum and represent the value of the momentum at which ionization occurs.
We seek an effective one-electron equation which may
be obtained by averaging the two-electron interaction
operator Ilzover the coordinates of electron 2.
I,&l) = (42(P2)II12142(P2))
The inner integral is facilitated by the substitution x =
pz - pi. It then becomes
2a2
In keeping with the requirements of the previous section,
we must extend the range of integration to include the
negative real axis. We choose a contour along the real axis
(-R,R), counterclockwise along a quarter circle radius R,
along the h"ryaxis (+iR,-iR) and clockwise along the
quarter circle in the third quadrant ending at -R on the
negative real axis. R is chosen large enough to enclose the
singularities at *(Pz - i{J. It is a simple matter to show
that the integrals along the curved portions vanish as R
---c
and that the integral along the h " r y a x i s is equal
to the integral along the real axis. Evaluating the residues,
we then obtain
(Pz - is; - X I 2
a032
- ii-Zl2
P1+ Pz - it-2
P1 -Pz + irz
The integral over the first term in bracketa may be
shown to be zero by simple integration. The second term
is the same as the integral over I,; the result is S;. By the
substitution x = p1- pl' the integral over the third term
becomes
Once again the inner integral must be extended to
and a contour which encloses singularities at f(pl
- ill) must be chosen. We may utilize the same contour
as used in the evaluation of V., Evaluating the residues
and carrying out the second integration, we obtain
[--OD,-]
9516
The Journal of Physicai Chemistry, Voi. 86, No. 18, 1982
The integral over the 6 functions is quite simple and may
be shown to be
Collecting all of the terms, we have
which we wish to minimize with respect to f1. Differentiating with respect to and setting the result equal to
zero, we may at this point let 5; = 3; = {. Solving, we find
rl,
r=z-
5/16
and substituting we obtain for the one-electron energy, in
terms of Z
€1
= -y2.??
+ Y8.Z - 75/512
The total energy of the He atom may be shown to be
Lombard1
iterated, resulting in functions which include momentum
correlation. These functions are not simple transforms of
the correlated position space functions used so successfully
by Hylleraas.6 It is possible that iterating the functions
obtained in the present work will give improved momentum-correlated functions. This will be the topic of further
work.
Acknowledgment. I am indebted to the City University
of New York PCS-BHEfaculty award program as well as
Sociometrics for financial assistance in this work.
Appendix I
We seek a general expression for the operator rl2-l in
momentum space. We need only consider the three coordinates r', r2, and y and y is the angle between rl and
r2. The corresponding momentum variables will be designated PI, PZ,and pr.
l/r12
Downloaded by CITY COLLEGE OF NEW YORK on August 24, 2009
Published on September 1, 1982 on http://pubs.acs.org | doi: 10.1021/j100215a006
dPl' dPz' dP,' V(P1-Pl',P2-Pz',Py-Pyl)
JPIJPPJP'
E = 2c1 - (41421I121dJldJ2) = 2%- 5/8r
For Z = 2 we obtain a value of -2.8476 au, in agreement
with the analogous position space calculation.'*
-
V(Pl-Pl',P2-P2/,Py-Py')
Discussion
We have carried out an approximate solution to the
ground state of the helium atom in the momentum representation. The representation is that of a recent work
on hydrogen in which we have chosen momentum variables
which are conjugate to the sperical polar variables in
position space. This latter choice distinguishes it from the
earlier work of Podolsky and Pauling' upon which all
previous work on the helium atom is based. We have
shown that a simple self-consistent oneelectron calculation
analogous to the corresponding position space calculation
may be carried out, with the same results. In so doing we
have illustrated the value of considering the momentum
variables in the complex plane despite the fact that the
interval of interest for observable quantities involves only
real values of the momentum. The integrations required
were simplified by the use of residue theories, which depend on the fact that the value of functions in the complex
plane depend on the location and nature of their singularities. The trial functions chosen were Slater orbitals,
which in momentum space were shown to have poles of
order 2 along the imaginary axis. The location of these
poles were i{ where {was the variational parameter. By
varying the location of these poles, we were able to obtain
optimized one-electron energies.
It is clear that, if we chose to improve our trial function
by adding higher configurations of Slater orbitals, additional singularities would be added along the imaginary
axis. Similar considerations have been utilized by Huo and
to examine momentum space functions of Ne
Las~ettre'~
and Ar. Using the transformation of Podolsky and
Pauling, they show that a set of singular points along the
imaginary axis are determined by the location of orbital
energies (diagonal Lagrangian multipliers) for a closed-shell
Hartree-Fock function.
It would also be worth exploring possible improvements
of the functions utilizing an iterative technique similar to
that employed by McWeeny and C o u l s ~ n .They
~ showed
that the integral equations in momentum space could be
(16) "Handbookof Mathematical Functions with Formulas, Graphs,
and Mathematical Tabla", M. Abramowitz and I. A. S t e p , Eds., National Bureau of Standard Applied Mathematice Series 55, Washingtan,
DC,1966.
=
S @ g z ~ ~ ; r , r ~ y ) r , ~sin
r 2y2 dr, dr, dy
We may expand
where in the text we confined ourselves to the n = 0 term.
Considering the nth term, we may readily integrate over
y (GR(7.321))
ei@~-P,')cosyP,(cos
y) d cos y = p;1/2Jn+l,2(pr)
In carrying out the integration over the radial variables,
we must once again take principal values, extending the
range to (-a,-). We must also remember to integrate
again interchanging the order of integration. First integrating over r1
Making the substitution p = r1/r2, we may utilize the
integrals16
We find
S,-
.
e-cr91-
r<n drl
=
We must now integrate over r2
Helium Atom in the Momentum Representatlon
The Jocnmel of phvslcal Chemktfy, Vol. 86, No. 18, 1982 3517
En+l(i~lr2)+ Bn(i~lr2)ldr2
The first two terms are given in ref 16, p 230, and the
third may be obtained with the aid of GR(2.111-3). The
result is
k1)"Pl"
0 1
+ P2)(P1 - P2)
P12
+-(-1)"Pz"
Pl"+l
In
Repeating the integration, f i i t over r2 and then over rl
and averaging, combining all of the results, we obtain
Requiring
Downloaded by CITY COLLEGE OF NEW YORK on August 24, 2009
Published on September 1, 1982 on http://pubs.acs.org | doi: 10.1021/j100215a006
we obtain normalized functions
We see immediately several important features of these
functions. Firstly, note that, in the complex pr plane, they
represent poles of order n + 1 with singularities at p r =
if. We can thus make use of Cauchy's theorem
It is easy to see that this reduces to the result obtained
in the text for n = 0.
Appendix I1
In a recent paper12 we have shown that the Schrainger
equation for the hydrogen atom could be solved in a momentum space in which the variables pr, r,, and p* are
chosen appropriately conjugate to the spherical polar
and u (=ei+). Since in either space
variables r, t (=ei-,
the functions are separable, we shall here only be concerned with the radial variables;
p, = -i(d/ar + l/r)
bnrl = i
A transformation S(r,p,) was obtained relating the
functions in position space with those in momentum space
such that
where it was shown
s(P,,~)= (fi/r)e-irpr
We wish here to find the analogue of Slater-type orbitals
in momentum space. These may be written
$&)
= NF-le-12
where f is usually chosen as a variation parameter.
Transforming to momentum space, we obtain
where the path of integration surrounds the singularity Q.
All quantum-mechanical integrals of interest involve the
interval (-a,-)along the real pr axis. By choosing a semicircle in the upper half-plane and the real axis for a
contour, one may evaluate integrals easily by summing
residues in the upper half-plane. It is also worthwhile
comparing these functions to the radial hydrogenic functions (I = 0)
+ iZ/n)"-'
4Hbr) =
(5)
(P, - iZ/n)"+'
where we have chosen atomic units in which p = e = h =
1. Note that these functions also have poles of order n +
1, located at p r = iZ/n = 47 where I is the ionization
potential of the hydrogen atom. Thus, the singularity of
4(Pr)in momentum space represents i times the momentum at which ionization occurs, enabling an analagous
interpretation of f. In position space f is usually interpreted as an effective nuclear charge, screened by electrons
which spend some time closer to the nucleus. In momentum space if is mathematically the point at which the
wave function has a pole, and is physically a momentum
at which ionization is achieved. Note also that the hydrogenic functions have a node at p r = iZ/n, while the
Slater functions are nodeless. This feature contributes to
their utility in evaluation of integrals, a simplicity also
obtained in position space.