Download hidden symmetry and explicit spheroidal eigenfunctions of the

Hidden symmetry and explicit spheroidal elgenfunctions
of the hydrogen atom
Stella M. Sung and Dudley R. Herschbach
Department of Chemistry, Harvard University, Cambridge, Massachusetts 02138
(Received 17 June 1991; accepted 30 July 1991)
The Schrödinger equation for a hydrogenic atom is separable in prolate spheroidal coordinates,
as a consequence of the “hidden symmetry” stemming from the fixed spatial orientation of the
classical Kepler orbits. One focus is at the nucleus and the other a distance R away along the
major axis of the elliptic orbit. The separation constant a is not an elementary function of Z or
R or quantum numbers. However, for given principal quantum number n and angular
momentum projection m, the allowed values of a and corresponding eigenfunctions in
spheroidal coordinates are readily obtained from a secular equation of order n m. We
evaluate a ( n,m;ZR) and the coefficients g, (a) that specify the spheroidal eigenfunctions as
hybrids of the familiar Inim) hydrogen-atom states with fixed n and m but different 1 values.
Explicit formulas and plots are given for a and g, and for the probability distributions derived
from the hybrid wave functions, E,g, (a) Inlm), for all states up through n = 4. In the limit
these hybrids become the solutions in parabolic coordinates, determined simply by
R
geometrical Clebsch—Gordan coefficients that account for conservation of angular momentum
and the hidden symmetry. We also briefly discuss some applications of the spheroidal
eigenfunctions, particularly to exact analytic solutions of two-center molecular orbitals for
special values of R and the nuclear charge ratio Z /Zb.
—
-.
I. INTRODUCTION
2 degeneracy of the nonrelativistic
The extraordinary n
hydrogen-atom energy levels manifests a “hidden” but wellknown symmetry. In addition to the Hamiltonian H and
orbital angular momentum 1, the Lenz vector a is a constant
of the motion.’ Classically, for bound states the a vector
points along the major axis of the Kepler elliptic orbit and its
length is proportional to the eccentricity. Although 1 and a
(1/2) (1 + a)
do not commute, the hybrid operators k +
and k = (1/2) (1 a) do. Furthermore, the operators
k ± and k obey the commutation relations for angular
momentum. This allows bound state eigenfunctions speci
fied by the complete set of commuting constants of the mo
tion to be constructed simply as angular momentum eigen
states. The quantum Kepler problem is thereby solved by
purely geometrical means. The familiar nim) eigenfunc
tions in spherical polar coordinates correspond to a coupled
representation which simultaneously diagonalizes H, k,
k, (k + kr), and the projection (k÷
2 +k
); the
2
latter two operators coincide with 12 and l, respectively. The
equivalent but distinct eigenfunctions in parabolic coordi
nates correspond to an uncoupled representation which si
multaneously diagonalizes H, k
÷, k, and the separate
2
projections k + and k
(tantamount to l and as); these
eigenfunctions are linear combinations of the familiar jnlm)
hydrogen-atom states with fixed n and m but different! val
ues, determined by Clebsch—Gordan coefficients.
In addition to spherical and parabolic coordinates, the
hydrogenic Schrödinger equation is separable in prolate
spheroidal coordinates, with one focus at the nucleus and the
other located along the Lenz vector at a distance R away.
These coordinates, ordinarily used for two-center problems
—
—
—
J. Chem. Phys. 95 (10), 15 November 1991
such as H
, serve to “dress the atom in molecular clothing.”
2
In view of the myriad applications of hydrogenic eigenfunc
tions, it is curious that the spheroidal solutions have received
scant attention; e.g., graphs of them seem not to exist else
where. Previously, general features have been explored by
Coulson and Robinson,
5 who noted that the limits R -.0 and
R co yield the spherical and parabolic solutions, respec
tively. Robinson
6 also showed that the spheroidal solutions
provide the correct zero-order basis states for treating the
interaction of a point charge or a point dipole with the hy
7 used the spheroidal eigenfunctions
drogen atom. Demkov
to construct analytic solutions (exact for the Born—Oppen
heimer problem) for an electron interacting with two nuclei
for certain special values of the internuclear distance R and
the charge ratio ZQ/Zb. Other aspects of the spheroidal ei
genfunctions were elucidated by Judd,
8 particularly the con
nection to the four-dimensional spherical harmonics.
Here, we apply and extend these results to evaluate ex
plicitly the spectrum of the separation constant a ( n,m,ZR)
for the hydrogen atom in spheroidal coordinates and the
eigenfunctions. These are obtained from a secular equation
of order n m which provides coefficients g, (a) for expan
sion of the spheroidal eigenfunctions in the usual I nim)
states. In effect, the g, (a) functions interpolate between the
coupled and uncoupled representations and thereby play the
role of generalized Clebsch—Gordan coefficients. We plot
these coefficients and the probability distributions for the
hybrid wave functions, E,g, (a) I nim), for all states up thor
ough n = 4. We also briefly discuss some prospective appli
cations. These include driven oscillator states created by in
teraction with an intense laser field,
9 recently treated in a
two-center spheroidal basis;’° planetary excited states of
.-.
—
0021-9606/91/227437-1 2$03.00
© 1991 American Institute of Physics
7437
S. M. Sung and D. R. Herschbach: Explicit spheroidal eigenfunctions
7438
two-electron atoms;” computing tunneling splittings and
electronic exchange energy for excited states of the H mol
2 and further exact solutions akin to those of
ecule-ion;’
7 for special diatomic molecular orbitals.
Demkov
5,
n ±
rr
rr
.y
0
m =
rr
rrrr
•2
vsr vr -v
3
2’
IT
0
IT
°
‘Vif
II. COUPLED AND UNCOUPLED REPRESENTATIONS
Three consequences of the dynamical symmetry’ of
the Kepler problem lead immediately to a geometrical solu
tion,
(i) The pair of hybrid operators, k ± = (l/2)(I ± a)
commute with H and with each other and behave as angular
momentum vectors.
(ii) The angular momentum and Lenz vectors are per
pendicular, l•a = a•l = 0; hence
is‘yr
n
0
3
/
1
[
rs[T
Y
rvr rVr
2V5
I
I
S
IS-
yr
n =
5,
m
=1
i[i
2V5
I
I
S
I]
‘VT
5
5
3
/
[
1
r
r it
2V5
2V5
÷ =k 2
2
k
+
=(l
)
ask
a
2V5
so these operators have the same eigenvalues.
(iii) The Hamiltonian can be written in terms of other
constants of the motion,
H= —(2k +2k’
= 4, m =
‘Vi
+1y’,
2V5
I
I
s
n
=
I
1
3,
in =
n
0
= 4, m =
I
n =
5, m
I2V
(2.1)
with energy in hartree units, angular momenta in units.
These properties imply that simultaneous eigenstates exist
for k, k + k, and k ; if these are denoted by
t’mlkm+ ,km_),
2/
[
/[/
—
kØ=k(k+l)b,
k±b=m±b.
n, m = n
(2.2)
I
?i’
k,
The eigenvalues k =0, 1/2, 1, 3/2,... and m ± =
—k + l,...,k— 1,k.NowEq.(2.l)yie1dsH=Eb,with
.
2
fl
‘V
—
(2.3)
so the principal quantum number n as (2k + 1) = 1,2,3
For a given value of k, there are 2k + 1 different values of
both m ÷ and m ,and therefore a total of (k + 1)2 =
different states with the same energy. Accordingly, the ex
ceptional n
2 degeneracy results from the presence of the
Lenz vector as a constant of the motion.
The eigenstates b of Eq. (2.2) pertain to the uncoupled
representation. Hence they are not eigenfunctions of the or
bital angular momentum, 12 = (k. + k. ), although they
are eigenfunctions of the projection, l = (k + + k
with eigenvalue m = m + + m , always an integer. In the
coupled representation, simultaneous eigenstates exist for
÷ , k , 12, and 4; if these are denoted by ‘I’ = urn),
2
k
k’P=k(k+1)’V, 12Il=l(l+l)4I, 4’V=m’I’,
—
—
(2.4)
where 1 ranges in integer steps between 1= 0 to
l=2k=(n—1)andm=—1,—l+1,...,l—l,LTheunitary transformations relating the two equivalent descrip
tions are
C(kkl;rn+m...rn)Ikrn+,krn_) (2.5)
Ilrn)=
m
+
,m
—
and
Ikrn+ ,krn ) =2C(kkl;m÷ rn_ m)Ilm),
(2.6)
where the C’s are Clebsch—Gordan coefficients with two
4 Aside from specifying notation, this
”
3
equal arguments.’
FIG. 1. Matrices of Clebsch—Gordan coefficients for the transformations
between coupled and uncoupled representations, Eqs. (2.5) and (2.6).
Here, k=1(n 1) and m = m+ + m_. Rows correspond to coupled
states Iln), in increasing order of!; columns correspond to the uncoupled
states (m + ,m
ranging from (k,m k on the left-hand side to
(m k,k on the right-hand side.
—
—
,
—
—
outline serves to emphasize how simple matters become
when the hybrid quasiangular momentum vectors, k + and
k , are regarded as the basic operators.
Figure 1 gives matrices of Clebsch—Clordan coefficients,
sufficient to treat all hydrogenic states up through n = 5;
those for higher states can likewise be evaluated from stan
dard tables.’
5 Rows correspond to the coupled states urn),
in increasing order of!; columns correspond to the uncou
pled states (rn + ,m
ranging from (k,m k on the lefthand side to (rn k,k on the right-hand side. Subsequent
ly, we will examine both the coupled and uncoupled states in
explicit coordinate representations.
—
—
,
—
—
III. TWO-CENTER SPHEROIDAL EIGENSTATES
For an electron at distances r and r from two fixed
Coulomb centers with charges Z
0 and Zb a distance R apart,
the wave function is separable in the form L(2)M(u)
exp( ±irnc5),with2= (r +rb)/R andp=(r. rb)/R
the spheroidal coordinates and qf’ the azimuthal angle about
the line between the centers. The separated equations for the
L (2) and M(u) factors are
J. Chem. Phys., Vol.95, No. 10,15 November1991
7439
S. M. Sung and D. R. Herschbach: Explicit spheroidal eigenfunctions
n nm Degeneracy
22
]L(2) = 0
L(2) + [A + (Za + Zb )R2 —p
2
L
(3.1)
003
and
.i) + [_ A
1
LM(
—
(Za
—
]M(
u
2
+p
u
) = 0,
Zb )RAU 4
(3.2)
where A is the separation constant and p
2=
energy parameter. The operator
—
ER 2 is an
2
m
d
(33)
1’
22_i’
‘d2
and L, has the same form with the factor (22 1) replaced
by(1 _2). The pairofequations forL(2) andM(is) are
6 For a
commonly referred to as the two-center equations.’
=
Z
and
we
take
Z
hydrogenic atom,
Zb = 0, and since
bound
states
of interest here,
Z2
/n for the
E=
R2
/n The pair of two-center equations then be.
2 = AZ 2
p
come the same equation with different ranges for the vanables,
L
—
d (22
1
2
6
300
2
Whenx=2(rangeltoaz),F(x)=L(2),andwhenx=/A
(range
ito + 1), F(x) = M(
u); in either case L has the
1
form of Eq. (3.3). For simplicity, henceforth we write
mmlml. Forp = 0, ifwetake,u = cos 8or2 = cosh 6, then
Eq. (3.4) reduces to the eigenvalue equation for P7’ (x), the
associatedLegendrepolynomials,withA =
1(1 + 1).Itis
only necessary to extend Rodriques’s formula,
i+ m
2 m/2
fL )
(1 2) for Lu I s i,
P7’ (p) =
2’!!
du’ ± m
(3.5)
—
—
j
,,
—
—
) factors with (2 2 1) for the
2
by replacing the (1 p
range 2 I I. Therefore, for p O a solution of either two6
”
5
center equation has the form
1
0
1 2 0
0 3 0
0 0 2
3
r
V
2 —(
0
0
1
—
is
lj
2
4
2 0 o1
1 1 0
0 2
3
0 0 1
2
0
&
0 1
0)
1
2s÷2
1
1
p+3d
(3.4)
1F(x)=0.
x
2
LF(x)+[A+2pnx—p
2
0 0 0
1
=
FIG. 2. Degenerate energy eigenstates for each n sorted according torn 0,
1, 2,... (u,ir,S,,...) and according to eigenvalues of the separation constant
A (designated by the spheroidal quantum numbers nA and n,,), thereby
specifying spheroidal eigenstates as hybrids of the usual spherical eigen
states.
—
—
F(x)=exp(—px)c,P7’(x),
(3.6)
where the summation runs from 1= m to 1= n 1. On sub
stituting this into Eq. (3.4) and simplifying with the aid of
identities linking polynomials of different 1, we obtain a
three-term recursion relation for the coefficients,
—
C,c, + C, , c,, + C,_ , c,_,
= 0,
(3.7)
with
c, = A —p
2 + 1(1+ 1),
c,., = 2p(1+n+1)(1+m+l)
(21+3)
(l—n)(l—m)
= —2p
(21— i)
As another consequence of the hidden dynamical symmetry
of the hydrogen atom, Eq. (3.7) is invariant to interchang
ing the quantum numbers n and m. This striking property
8 and related to the Lenz vector.
has been analyzed by Judd
TABLE I. Secular equations for Lenz vector eigenvalues.
m
Secular equation
2
3
4
0
—
—
—
02
2 0 i’)
1 I 1
0 2
4
—
n
1
2
1 =4p2
a,a,,,÷
a
a,,,a,,,+ia,,,+
=
(
a
3
+
1
a
2
fl,,,+
,,,a,,,+
,,,a,,,.
$,,,+
,,,+
4p
4
l44p
)
3
+8,,,+,am+za,,,+
—
=
Notation: p
R2
/n with Z the nuclear charge, R the internuclear distance, and n the principal quan
,
2 Z2
tum number. a,,, ma + rn(rn + 1); amA p2, where
A is theeigenvalue ofthe Lenz vector [cf. Eq. (1)1.
—
fl,=,8,(n,rn)m[(1
)
—
)
2
]/[(21—
(n
m
! 1)(21+ 1)].
—
—
J. Chem. Phys., Vol.95, No. 10, 15 November1991
S. M. Sung and D. A. Herschbach: Explicit spheroidal elgenfunctions
7440
spheroidal quantum numbers n and nh,, which specify the
number of nodes in the 2 and theu coordinate, respectively.
Table I gives the secular equations in terms of a = A —
and Fig. 3 plots the eigenvalues as functions of ZR.
The secular determinants for the spheroidal eigenstates
8 such that c, Q,g,, where
can be symmetrized
3
sU
4pci
—
=
—
—l
=
( — l)’[(21+ l)(l—m)!/
(1 + m)!(l + n)!(n —1— 1)!] 1/2
—3
(3.8)
The recursion relation for the new coefficients g, is then
(3.9)
,+G,÷,g,, +G,_,g,_, =0,
G
g
1
2
with
—
0
{
m (1+ 1)2]
[n
—
l)
]
2
[(1+ —
(21+ l)(21+ 3)
11/2
1’
1
[n
m
}”
_
J[1
1
—
]
2
(2!— l)(2l+ 1)
1 coefficients and Fig. 4 plots
Table II gives formulas for the g
to unity) as functions of ZR
sum
(normalized
to
values ofg
4.
n
up
through
all
eigenstates
for
Simple results are obtained for both the small-R and
large-R limits. For R —0, where a — 1(1 + 1), only oneg,
coefficient is nonzero for each eigenstate (so the normalized
1), and the corresponding values of n, 1, and m are used
where
For
R —, ,
state.
label
that
to
m ), the eigenvalue becomes propor
a—’ — 2p(m +
3 and
tional to that for the z component of the Lenz vector’
the g, become the Clebsch—Gordan coefficients of Eq. (2.6)
and Table I. Accordingly, in Fig. 3 we have scaled the eigen
values by a factor, 2p + is, in order to have finite limits for
both R —.0 and R m. In both Figs. 3 and 4, the limiting
values are marked on the ordinate axis.
The relationship between the eigenstates in spheroidal
coordinates, designated as nam), and the customary states
in spherical coordinates, designated by nim), is extremely
simple,
1
G,.
da
=
=
—
+
4
p
—
—
.1
+
—I
+
—.
—‘
ZR
2 of two-center equa
FIG. 3. Eigenvalues for separation constant a A p
tions, as functions of ZR. The lower panel pertains to states with
in
2; the middle panel to n in = 3; the upper panel to ii m 4.
n
In accord with Fig. 2, eigenvalues are labeled by the Inim) states to which
the spheroidal Inam) reduces for R —.0, where a— 1(1 + 1). Indicated on
limit, where
in ) which govern the R —
the right are values of (m +
1n ). The ordinate is scaled by l/(2p + n) to remain
2p(m +
a—.
=
=
—
—
—
—
—
—
g,nIm),
(3.10)
In1m)=g,Inam).
(3.11)
Inam)
=
=
—
and
—
finite in both limits.
Forgiven n and m, the sum in Eq. (3.10) extends from! m
tol= n — land thatinEq. (3.ll)overthen mdistincta
values. The symmetrized set of coefficients {g,} thus defines
a unitary transformation between the spheroidal and spheri
cal eigenstates, and has the role of generalized Clebsch—Gor
dan coefficients,
=
—
For each pair of values of n and m, the recursion relation
gives a secular determinant of order n m. The roots deter
mine the eigenvalues of the separation constant A and the
corresponding {c, } which specify the eigenfunctions.
Figure 2 shows the resulting pattern ofspheroidal eigen
states up through n = 4. For each energy level there are n
degenerate spheroidal states with different m (denoted by a,
ir, ô, form = 0, 1,2,...), each comprised of a linear combi
nation of the n — m terms in Eq. (3.6) for 1= m to
1= n — 1. Also listed for each of the eigenstates are the
—
...
(3.12)
namInlm) (nlmlnam),
in analogy to the angular momentum recoupling of Eqs.
(2.5) and (2.6). In Appendix A we derive this relationship.
8 that in the
It follows from the fact, as pointed out by Judd,
limit of large 2 the spheroidal coordinates become polar
spherical coordinates.
=
J. Chem. Phys., Vol. 95, No. 10, 15 November 1991
=
7441
S. M. Sung and D. R. Herschbach: Explicit spheroidal eigenfunctions
TABLE H. Spheroidal hybridization coefficients.
n
—
g,(a,p)
m
Any
g,,,
=
=
(n!mlnam>
const
2
3
/g,,, =l[(2m+3)/(m+2)]”(a,,,/p)
1
g,,,..
/g,,,
2
g,,,
=
(a,,,/a,,,÷
l)/(m+2)]”
)
[(m+ 2
a,,,/p)
=[4(2m+3)/(2m+5)]”
(
g,,,/g,,, 2
4
+5)]
+3)/3]”
12p
(2m+3)—(2m
(a,,,a,,,÷
/g,,,=(m+1)(m
g,,,÷
[
2
/
1
)
I [(2m+3)(Zm+5)1v2[(2m+3)
611
—
3 ,,r—[
g,,,
J Li2m+5
(m+l)(m+3)
P
2
)
°1
3
a÷
Notation as in Table I.
IV. EXPLICIT EIGENFUNCTIONS AND PROBABILITY
DISTRIBUTIONS
The eigenstates in spheroidal coordinates (A,p,) are
given explicitly by
Inam) =exp[
— l)(l
2
—p(2+p)]{(A
_,12)}m/2
=
—
2
1x
(X),
lImhZNmC
)
2
(4.2)
-
(x),
=g,Q,C
)
2
(4.3)
aside from normalization. Again, for given n and rn, there
are n m distinct values of a, and the sum extends from
1. Table III gives these canonical polyno
1= m to 1= n
mials for all eigenstates up through n = 4. Appendix B lists
the roots of the polynomials for these eigenstates for several
values of ZR.
Polar spherical coordinates (r,8,) and parabolic co
are related to the prolate spheroidal co
ordinates
ordinates by
—
—
p=2p(2+u), pcos8=2p(l+2s),
(4
2_
l)(l _2)]1/2,
p sin 8= p
2
[’:A
wherep = 2Zr/n with (rr), and by
=p(2—l)(l—p),
i=p(2+l)(l+p),
(4.5)
with 2p = ZR In. With these substitutions, we find that the
eigenfunctions I nim) in spherical coordinates and I nrm) in
7 contain the same factors as Eq.
parabolic coordinates’
(3.13) except that fnam (2 )fnan, (ku) is replaced by
pl_n1L
(p)C
where L (x) is an associated Laguerre polynomial. In the
spherical case, for each 1, the Laguerre factor has n 1— 1
radial nodes, and the Gegenbauer factor has 1— m angular
nodes. The total number of nodes thus is it m 1, inde
pendent of the I value. In the parabolic case,
s+ t= a= n—rn—I is again the total number of nodes.
The other natural quantum number is s t = r =
a,
—o•+ 2,..., a— 2,a thisindexrremainsagoodquantum
number in the presence of a uniform electric field along the z
7 Hence for given n and rn there are n m parabolic
axis.’
eigenstates related to spherical eigenstates in the same fash
ion as the spheroidal states designated in Fig. 2. Note that
interchange of2 andi leaves Eqs. (4.6) and (4.7) invariant,
are invariant to this interchange. We
since p, cos 8 and
thus have three options for constructing the polynomial fac
tors in the spheroidal eigenfunctions: from Eq. (4.3) as a
product of two identical Gegenbauer functions; or from Eq.
(4.6) as products of Laguerre and Gegenbauer functions,
summed overt according to Eq. (3.10); or from Eq. (4.7) as
products of two Laguerre functions, summed over the r
quantum number.
In the limit R —, 0, each spheroidal eigenfunction narn)
reduces to a particular spherical function I nlm), specified by
a—. — 1(1 + 1). Likewise, in the limit R—. , each Inam)
becomes a particular parabolic function Inrm), which in
turn can be obtained from Eq. (3.10) as a linear combination
of spherical functions with the g, given by the Clebsch—Gor
dan coefficients of Eq. (2.6) and Fig. 1. Other aspects of the
transition to these limits have been examined and illustrated
5
by Coulson and Robinson.
For the probability distributions, a format analogous to
that customary for spherical functions can be obtained from
Eq. (3.10). In spherical coordinates the joint distribution
obtained from the squared modulus (nam I nam) is not sep
arable,
—
—
—
—
—
2 mr ( m + ), which replaces the associat
where Nm ir”
ed Legendre function with a Gegenbauer polynomial,
(x). Accordingly, the polynomial factors in
2
C”
narn) are given by
fnam(X)
(4.7)
—
(4.1)
Xfnam(2)fnam(I1) exp( ±imcb),
5 This form is obtained from Eq.
aside from normalization.
(3.6) with the substitution
P7’(x)
L’()L”(s),
1/2)
(cos 8)
(4.6)
Pncxrn
(p,O,c)
=
or by
,.pR (p)
p)pR,.
g
1
g
(
(4.8)
J. Chem. Phys., Vol. 95, No. 10,15 November 1991
111
14
0,
.
0,
p
till
.
,..
.—
8—
i—
1
e__________
.4
a
8
0
I—
8
i—
0
1
uI
0
—
i-
0
.
p.
I—
.4—
ie
oP
.4
_____
_____
_____
______
_
________
7443
S. M. Sung and 0. R. Herschbach: Explicit spheroidal eigenfunctions
TABLE III. Canonical polynomials.
n
m
f,,,Jx)
1
2
2
3
3
3
0
0
1
0
1
2
1
x—(2p/a)
1
x’—l[(a+6)/p]x+[1+(8/a)j
x—[2p/(a+2)]
1
+a(a+18) (a+12)(52_a(a+2)]
2
X3__F(+)lX2_ 12p
6pd(p,a)
d(p,a)
J
2 1
p
_l[(a+12)/plx+[(a+14)/(a+2))
2
x
x—[2p/(a+6)l
1
4
1
2
3
4
4
4
Notation as in Tables I and H, with
)
8
(
m
P,,
d(p,a)
=
20
—
a(a + 2).
(4 10)
,
2
=gIY,m(O,.b)I
2
where R, (p) is the usual normalized radial wave function
5 Figure 5 shows the
and Y,,,, (,ç) is a spherical arm
radial probability distributions for all states up through
Of course the small-R
n = 4 for ZR = 0, 5, 15, 50, and
and large-R limits represent the familiar spherical and para
bolic results, respectively. Figure 6 shows corresponding an
gular probability distributions for ZR = 0, 15, and on. The
various knobs and lobes that emerge or retreat as ZR is var
ied reflect the shifting pattern of nodes as the weighting fac
tors g change. Particularly when the number of nodes,
I, is large such plots resemble the polyhedra
u=n m
known as “stellations,” but with rounded rather than spiky
8 However, the probability distributions do not take
lobes.’
their simplest or most revealing form in these spherical co
ordinates, since p and 0 involve sums and products of the
spheroidal coordinates, according to Eq. (4.4). In particu
lar, in Pfla,,, (6) the symmetry about the plane 0= 900 con
ceals the asymmetry about the p = 0 plane which exists in
the actual spheroidal eigenstates.
In terms of spheroidal coordinates, the probability dis
tribution as obtained directly from Eq. (3.13) is separable,
.
—
—
(4.11)
)Pnarn(I)’
2
I) = Pnczrn(
Pnarn(
’
2
with
t)IX
)1 (4.12)
1Im[fnam(X
nam(X)P(_2P
_
2
,
1,1) and rep
where again x represents p in the interval (
Figure
a few exam7
shows
resents 2 in the interval (1, on).
—
ples for both x = p and x = 2. Others may be readily con
structed from Table III, augmented by the list of zeros of the
0m (x) polynomials given in Appendix B. In these coordi
f
nates, the nodal structure is rather simple and stable. Each
eigenstate nam) corresponds to particular values of the
, np,, m, listed in Fig. 2),
2
spheroidal quantum numbers (n
and thus the number of nodes in each of these coordinates
remains the same as R is varied.
V. DISCUSSION
Beyond providing an unconventional molecular per
spective for the hydrogen atom, the spheroidal eigenfunc
tions have practical utility. As is well known, the parabolic
eigenstates provide the correct zeroth-order linear combina
tions of the degenerate spherical eigenstates for treating the
Stark effect of the hydrogen atom in a uniform electric
7 Although little known, the spheroidal eigenstates
field.’
likewise provide the correct zeroth-order hybrids, specified
by the g, coefficients, for treating the Stark effect induced by
a point charge or a point dipole.
6 In the R
limit, this
perturbation becomes equivalent to a uniform field and the
spheroidal functions indeed reduce to the parabolic eigen
states.
Other applications arise when an hydrogenic atom is
subject to a two-center perturbation. Such a situation in opti
cal physics is exemplified in recent work of Pont and Gavrila
on hydrogen in a circularly polarized, high-intensity and
high-frequency laser field.
9 The coupling of the radiation
field with the atom produces a potential containing an addi
tional time-dependent center, displaced from the nucleus by
a distance proportional to the square root ofthe light intensi
ty. The electron oscillates between the nucleus and this cen
ter, driven by the radiation. Consequently, the spheroidal
J. Chem. Phys., Vol.95, No. 10, 15 November1991
—.
S. M. Sung and D. R. l-ferschbach: Explicit spheroidal eigenfunctions
7444
o
3jw
/4dir
3scr
9(\4P7t
i/i
ä
•°f
2so
,f4ds
3pir
10
10
0
5
10
15
20
0
5
190
15
20
0
5
l,O
15
20
1520
0
I,D
5
15
20
15 (-••-), 50 (•••),
FIG. 5. Radial probability distributions for the spheroidal hybrids up through n = 4, as specified in Eq. (4.9), for ZR = 0 (—), 5
and
(---). The lowest row shows the four eigenstates of Fig. 2 with n — m = 1; the next row shows the three pairs of eigenstates with n — = 2; the next
the two triplets with n — m = 3; the top row the quartet with n — m =4. The abscissa scale in each case pertains top = 2Zr/n, for the range p = 0 to 20.
(-S-),
J. Chem. Phys., Vol. 95, No. 10, 15 November1991
S. M. Sung and D. R. Herschbach: Explicit spheroidal elgenfunctions
H
eigenfunctions provide an appropriate basis for treating such
a system.
Similarly, spheroidal eigenstates are appropriate for the
exchange or tunneling of an electron between a pair of pro
tons. This process has been treated extensively for the g,u
pairs of electronic states that stem from separated atoms
with n = 1 and for all excited states with maximal values of
m =l=n — I aswe11.’ Thesecorrespondtothe lsu, 2pir,
3d5, 4f,... states, as depicted in Figs. 2, 5, and 6, that do not
hybridize with others of the same n when subject to a twocenter perturbation; in effect, such states remain spherical.
The spheroidal eigenstates offer a natural basis for treating
the many other excited states that have less than maximal m
values and strong hybridization. The spheroidal basis is im
plicit in asymptotic expansions in hR developed by Dam° for the eigenvalues
2
burg and Propin” and by Cizek eta!.
of Hj states.
The spheroidal eigenstates also offer a convenient
means to construct “elliptic” states with maximum localiza
’ These involve a sum
2
tion on the classical Kepler orbits.
22
over m, in contrast to Rydberg atoms in “circular” states,
which correspond in the classical limit to an electron in a
circular orbit and have the maximal value of m = 1= n — 1.
Another natural application pertains to “planetary” states
In such states, for which the princi
of two-electron
pal quantum numbers of the electrons differ markedly
(n, n
2 ), the nodal structure of the eigenfunctions is found
to match closely that for the hydrogenic spheroidal eigen
states with n = n,. With allowance for the perturbation
, the spheroi
2
23 with n = n
from the more distant electron,
dal eigenstates should provide an efficient route to calculat
ing emission lifetimes and other properties of the planetary
states.
The hydrogenic spheroidal eigenstates can be used to
construct exact analytic solutions for special configurations
of the general two-center Coulombic system comprised of a
charged particle interacting with a pair of fixed charges Z,
6 AsseeninEqs. (3.1) and (3.2),
6 adistanceR apart.’
andZ
for this general problem the L (2) and M(p) factors of the
eigenfunctions each satisfy similar equations, with the same
values forp
2 and A but different parameters, (Za + Zb) and
0 — Zb), respectively. Thus, as noted by Coulson and
(Z
5 both factors may have hydrogenic forms if for
Robinson,
some pair of principal quantum numbers n and n,, the ener
gy eigenvalues are equal and the nodal structures correspond
to hydrogenic states. The energy condition requires
E=
FIG. 6. Polar plots of angular probability distributions for the spheroidal
hybrids up through n = 4, as specified in Eq. (4.10), for ZR = 0 (—), 15
(),and
(---). The layout corresponds to Fig. 5. The z axis is horizon
tal.
7445
—
6)
/n = — (Z — Zb )
2
/n.
2
(Za + Z
(5.1)
The nodal condition for L (2) requires that the number of
nodesdoesnotexceednA — m — lintheinterval(l,co)and
for M(p) that it not exceed
— m — 1 in the interval
1,
1).
The
requirement
that
the separation constant
(— +
must be the same for both equations, A, = A, determines
the specific internuclear distance .1? for which this hydro
genic solution of the two-center problem holds. Such solu
tions have been obtained for several electronic states by
7 These solutions are simply products of two differ
Demkov.
ent hydrogenic spheroidal eigenfunctions, with the same val
ues for a = A — p
, m, and R but different values for the
2
J. Chem. Phys., Vol.95, No.10, 15 November1991
S. M. Sung and D. R. Herschbach: Explicit spheroidal elgenfunctions
7446
........4
o
40
20
0
2040
0!\
[‘\
3po
3w
II
10
LL0
..________
20
40
0
40
20
40
3d
/Sf \
0
20
40
i1\
o.:
20
s factors are plotted
1
FIG. 7. Probability distributions for them = 0 spheroidal hybrids of Fig. 2, as specified in Eq. (4.12). In each case, the A and
B.
in
Appendix
tabulated
are
factors
polynomial
of
the
Zeros
100
and
(••),
(•--).
for ZR = 1 (—), 15
J. Chem. Phys., Vol. 95, No. 10, 15 November 1991
separately,
7447
S. M. Sung and D. R. Herschbach: Expflcit spheroidal eigenfunctions
principal quantum numbers, nA and ng, related by
(Za +Zb)/n = (4 —Z)/n,.
ACKNOWLEDGMENTS
We have enjoyed discussions with John Briggs, Don
Frantz, Sabre Kais, Mario Lopez, and particularly with Jan
Rost. We are grateful for an NSF Fellowship (to S. M. S.)
and for support received from the Venture Research Unit of
the British Petrolelum Company.
APPENDIX A: RELATION OF SEPARATION CONSTANT
TO LENZ VECTOR
24 have evaluated in a convenient
Coulson and Joseph
form the operator FA corresponding to the separation con
stantA of the two-center equations, Eqs. (3.1 )—( 3.2); for the
case Za = Z and Z,, = 0, the hydrogenic atom, this gives
the eigenvalue relation
(Al)
with W=L(2)M(u),
FA’I’=AW
dal eigenstates Inam) must be related to the usual spherical
eigenstates nim) by a unitary transformation of the form of
Eqs. (3. l0)—(3 1.2); it is only necessary to demonstrate that
the {g,} obtained from Eq. (Al) are the appropriate coeffi
5 that
cients. However, in the limit 2—. o, it is readily shown
the spheroidal coordinates become polar coordinates:
2—2r/R and 4
u—cos 0. Then L(2)M(
u) for a given a be
4
comes a linear combination of the R, (p )P 7’ (0), summed
over 1. From this it follows that the {g,} coefficients indeed
specify the unitary transformation.
8 has shown that a recursion relation identical to
Judd
Eq. (3.9) appears when a operates on a four-dimensional
spherical harmonic.
25 Consequently, an equivalent proce
dure for solving Eq. (Al) is to expand 4’ in four-dimensional
spherical harmonics, and the same {g, } coefficients specify
this expansion. Other elegant properties that arise because 1
and a are generators of the four-dimensional rotation group
have been amply ’
413
discussed.
2
6
7
where
APPENDIX B: NODES OF CANONICAL POLYNOMIALS
FA =
—
[12
+ R(
—
2E) “
a
2
—
(A2)
E],
R2
Since the fnam (x) polynomials of Eq. (4.3) depend on
the coefficients {g, (a) }, or in the explicit form of Table III
on the eigenvalues a (n,m,ZR), these must be obtained by
solving the secular determinant, Eq. (3.9). In order to facili
tate quick estimates ofthe spheroidal eigenfunctions, we give
in Table IV the zeros of these polynomials for a few values of
ZR. From the zeros x
,x
1
,..., the polynomial can of course
2
be obtained as 1(x) = (x x
1 )(x x
).... Interpolation
2
of the zeros thus offers an efficient means to estimate the
polynomials for a wide range of ZR. As in Figs. 2—7, each of
the spheroidal eigenstates Inam) islabeled by the quantum
numbers (n,1
0 .m) for the spherical state to which it reduces
and 1 is the orbital angular momentum, a the Lenz vector,
and E =
Z2
/n the bound-state energy. In terms of the
energy parameter p
ER this becomes
2=
—
,
—
(A3)
+2pa—p
—(1
)
.
FA— 2
for
the
In the text, we set up the secular determinant
eigenvalues
the
spheroidal eigenstates, Eq. (3.9), to obtain
1 } coeffi
a=A p
2 of the separation constant and the {g
of
(Al) in
Eq.
cients, by expanding the eigenfunctions 4’
(3.6).
according
to
Eq.
polynomials,
Legendre
associated
spheroi
numbers,
the
quantum
remain
good
and
m
Since n
—
—
—
TABLE IV. Zeros off,,,,,, (x) polynomials.
1
mZR=
0
nl
2
4
3
n
5
—
m
=
10
15
50
100
2 states
2scr
2pcr
4.2361
0.2361
2.4142
—0.4142
1.8685
—0.5352
1.6180
—0.6180
1.4770
—0.6770
1.2198
—0.8198
1.1422
—0.8755
1.0408
—0.9608
1.0202
—0.9803
3pir
3dir
2.0828
0.0828
6.1623
—0.1623
4.2361
—0.2361
3.3028
—0.3028
2.7621
—0.3620
1.7662
—0.5662
1.4770
—0.6770
1.1272
—0.8872
1.0618
—0.9418
4d6
4J
24.0416
—0.4159
12.0828
—0.0828
8.1231
—0.1231
6.1623
—0.1623
5.0000
—0.2000
2.7621
—0.3621
2.0806
—0.4806
1.2684
—0.7884
1.1272
—0.8872
n
—
=
3 states
3scr
14.3768
4.0433
7.4201
2.3200
5.1563
1.8061
4.0485
1.5709
3.3961
1.4389
2.1364
1.1995
1.7379
1.1281
1.2106
1.0362
1.1039
1.0178
3po-
11.9265
—0.2351
5.8940
—0.4095
3.9032
—0.5254
2.9408
—0.6034
2.3948
—0.6581
1.4871
—0.7904
1.2764
—0.8458
1.0658
—0.9450
1.0314
—0.9713
0.1487
0.7904
—0.4010
0.8955
—0.8017
0.9659
3dc,
—
0.5172
0.6286
—
0.4476
0.6722
—
0.3691
0.7093
—
0.2841
0.7409
—
0.1962
0.7678
—
J. Chem. Phys., Vol.95. No. 10, 15 November1991
—
—
—0.8992
0.9827
—
S. M. Sung and D. R. Herschbacft Explicit spheroidal elgenfunctions
7448
TABLE IV. (Continued.)
1
nZR=
nl
i
0
4pir
4dir
29.0677
11.1413
24.0000
0.8269
—
0.4129
0.4795
4fir
12.0027
0.1618
8.0086
0.2345
—
—
0.3767
—0.5099
0.3387
—0.5382
—
6.0187
0.2996
—
0.2587
—0.5892
0.2993
—0.5646
n
—
m
4.8331
0.3568
=
—
2.5492
0.5492
—
1.8822
0.6522
100
1.1965
1.0527
1.4060
1.1093
2.5019
1.4192
3.3565
1.6790
6.0725
2.5816
7.4715
3.0746
9.8305
3.9282
14.5969
5.6953
50
15
10
5
4
3
2
—
1.1885
0.8625
—
1.0868
0.9260
0.0516
—0.6871
—0.1313
—0.7533
—0.6544
—0.9069
—0.8186
—0.9513
4 states
4so-
31.2034
13.4038
3.9831
15.8167
6.9349
2.2909
10.7385
4.8336
1.7868
8.2216
3.8068
1.5563
6.7231
3.2028
1.4270
3.8545
1.9399
1.2130
2.8135
1.6725
1.1234
1.5201
1.1902
1.0345
1.2562
1.0935
1.0170
p
4
28.8605
10.9840
—0.2348
14.3433
5.4266
—0.4078
9.5127
3.5972
—0.5220
7.1200
2.7177
—0.5981
5.7075
2.2229
—0.6511
3.0337
1.4150
—0.7777
2.2400
1.2317
—0.8310
1.3059
1.0529
—0.9330
1.1448
1.0249
—0.9636
4dy
23.9859
0.5169
0.6285
11.9706
0.4465
0.6716
7.9537
0.3666
0.7082
5.9360
0.2802
0.7393
4.7195
0.1915
0.7658
2.3138
—0.1714
0.8490
—
1.6330
—0.3729
0.8883
—
1.1002
—0.7584
0.9583
1.0445
—0.8715
0.9779
0.6651
—0.2022
—0.8402
0.4877
—0:3748
—0.8815
0.2612
—0.5075
—0.9088
0.5213
—0.8247
—0.9682
4fo
—
—
—
0.7570
—0.0416
—0.7904
0.7375
—0.0829
—0.8047
0.7159
—0.1236
—0.8177
—
0.6918
—0.1634
—0.8295
—
—
—
—
0.7539
—0.9102
—0.9837
—
See Appendix B.
for R -.0, where a —.‘.l (l + 1). The zeros located at values
I <x< I per
of x> I pertain to the A coordinate; those at
tain to the p coordinate. The number of zeros in 2 or p is
given by the spheroidal quantum numbers, n,t or n, respec
tively, and remains the same as ZR is varied.
—
‘W. Pauli, Z. Phys. 36, 336 (1926).
2. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York,
L
1968), pp. 236—239.
G. Baym, Lectures on Quantum Theory (Benjamin/Cummings, Reading,
MA, 1969), pp. 175—179.
‘M. I. Englefield, Group Theory and the Coulomb Problem (Wiley-Interscience, New York, 1972).
C. A. Coulson and P. D. Robinson, Proc. Phys. Soc. London 71, 815
5
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P. D. Robinson, Proc. Phys. Soc. London 71, 828 (1958).
6
Yu. N. Demkov, Pis’ma Zh. Eksp. Teor. Fix. 7, 101 (1968) [JETP Lett. 7,
7
76(1968)].
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8
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9
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4
‘
R. N. Zare, Angular Momentum (Wiley, New York, 1988).
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E Teller and H. L. Sahlin, in Physical Chemistry—An Advanced Treatise,
6
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8
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20
ris, S. Nakai, J. Paldus, R. K. Propin, and H. J. Silverstone, Phys. Rev. A
33, 12 (1986).
21
3.-C. Gay, D. Delande, and A. Bommier, Phys. Rev. A 39, 6587 (1989).
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22
P. A. Braun, V. N. Ostrosvsky, and N. V. Prudov, Phys. Rev. A 42, 6537
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(1990).
C. A. Coulson and A. Joseph, Proc. Phys. Soc. London 90, 887 (1967).
24
2S
J Avery, Hyperspherical Harmonics: Applications in Quantum Theory
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26
D. R. Herrick, Phys. Rev. A 26, 323 (1982).
27
S.
12
J. Chem. Phys., Vol.95, No. 10, 15 November1991