The dipole polarizability of the hydrogen molecular ion
J. M. Taylor, A. Dalgarno, and J. F. Babb
Institute for Theoretical Atomic and Molecular Physics,
Harvard-Smithsonian Center for Astrophysics,
60 Garden Street, Cambridge, MA 02138
Abstract
A procedure is described for the precise nonrelativistic evaluation of the
dipole polarizabilities of H 2+ and D 2+ that avoids any approximation based
on the size of the electron mass relative to the nucleus mass. The proce-
dure is constructed so that sum rules may be used to assess the accuracy of
the calculation. The resulting polarizabilities are consistent with experiment
within the error bars of the measurements and are far more precise than values
obtained by other theoretical methods.
PACS numbers: 33.15.Kr, 33.15.-e, 31.15.Ar
Typeset using REVTEX
1
The separation of nuclear and electronic motion is the underlying principle of the theory
of molecular structure. The theory is challenged by recent measurements of Jacobson et
al. [1] of the electric dipole polarizabilities of H 2+ and D 2+ which have a precision beyond
that obtained in the Born-Oppenheimer approximation. The measurements stimulated the
introduction of methods [2- 5] that take into account the diabatic coupling omitted in the
earlier calculations and they led to polarizabilities that agree with the measured values
within the combined experimental and theoretical uncertainties. We present here new the-
oretical predictions of much greater accuracy which in turn pose a significant challenge to
experiment. The accuracy of our method can be assessed by the use of sum rules and we
predict nonrelativistically the polarizabilities of H 2+ and D 2+ to a precision well beyond
that achieved by the experiments. The method is general and it should be possible to apply
it to many-electron diatomic molecules.
Separating out the center of mass motion we may write for the Hamiltonian of H 2+ or
D 2+ in an electric field F = F ^nlying along the Z-axis of the space-fixed frame
H = - 1__2Mr2R- 1_2(1 + _1_2M)r2r+ V (r; R) + (1 + ffl)F ^n. r; (1)
where R is the vector joining the nuclei, r is the position vector of the electron measured
from the midpoint of R, M is the mass of the proton or deuteron, V (r; R) is the electrostatic
interaction potential and (1 + ffl) = [1 + (1 + 2M )-1 ]. We use atomic units throughout. The
change in energy of the system for small values of the applied field is given by E = - 1_2ffdF 2,
where ffd is the polarizability. Thus if (0)(r; R) is the eigenfunction of the unperturbed
system with Hamiltonian H0 and E0 is the eigenvalue, the polarizability can be written
ffd = -2<(1)|(1 + ffl)^n. r|(0)>; (2)
where
(H0 - E0)(1)(r; R) + (1 + ffl)^n. r(0)(r; R) = 0: (3)
Alternatively (1) can be determined from the stationary value of the functional
2
J = <(1)|H0 - E0|(1)> + 2(1 + ffl)<(1)|^n. r|(0)>: (4)
If we write (1)(r; R) as an expansion over some chosen basis set n(r; R),
XN
(1)(r; R) = Qn n(r; R); (5)
n=1
assumed to diagonalize the unperturbed Hamiltonian H0 so that < n|H0| n0> = Enffinn0, the
polarizability may be written
XN |<(0)|^n. r| >|2
ffd = 2(1 + ffl)2 _____________n____: (6)
n=0 En - E0
This expression for the polarizability is stationary with respect to first order errors in (1)
and is bounded from below.
The completeness of the set n(r; R) can be assessed by inspecting other sum rules.
Introduce the oscillator strength
fn = 2[(En - E0)=(1 + _1_2M)]|<(0)|^n. r| n>|2 (7)
and define the sum
X1
S(p) = [(En - E0)=(1 + _1_2M)]pfn (8)
n=0
so that
ffd = (1 + ffl)2(1 + _1_2M)-1 S(-2): (9)
Then provided the n form a complete set,
S(-1) = 2_3<(0)|r2|(0)> (10)
and
S(0) = 1: (11)
The eigenfunctions (0)(r; R) and n(r; R) can be written as sums of products of nuclear
and electronic wave functions of the form
3
2N + 1 1=2 N*
s(N M ) = ________ DM (; ; 0)OEs (r; R)Os (R); (12)
4ss
where (; ) are angles specifying the orientation of the internuclear axis in the space-
fixed frame, N is the total angular momentum quantum number, M is the projection on to
the space-fixed Z-axis, is the projection of the electronic angular momentum on to the
internuclear axis and D is the rotation matrix [6]. For the ground state of H 2+ or D 2+ ,
N = M = = 0 and the electronic wave function has +g symmetry. The perturbed state
is a superposition of states with N = 1; M = 0 and = 0 and 1, the electronic wave
functions having +u and u symmetry.
To calculate the matrix elements of the Hamiltonian and of the electric dipole oper-
ator we transform from the space-fixed frame to the body-fixed frame following standard
procedures [6,7]. The nuclear kinetic energy operator may be written
r2R 1 @ 2 @
- ____= - ___________R ____+ Hrot; (13)
2M 2M R2 @R @R
where Hrot is given by
1 2
Hrot = _______(N - L)
2M R2
1 2 2 - + + - 2
= _______(N + L - N L - N L - 2 ); (14)
2M R2
in which L is the electronic angular momentum and indicates angular momentum raising
and lowering operators. These are the operators that couple and states. The elec-
tronic wave functions for H 2+ and D 2+ are separable in prolate spheroidal coordinates and
we expressed the electronic basis functions OEff (r; R) in terms of these. The corresponding
formulas for the matrix elements of Hrotare given by Moss and Sadler [8]. A detailed descrip-
tion of the representation of the nuclear and electronic eigenfunctions and the construction
of the unperturbed eigenfunction (0)and the basis functions n together with a discussion
of the convergence properties is given by Taylor et al. [9].
The electric dipole operator must also be transformed to the body-fixed axis. The neces-
sary procedures are described by Lefebvre-Brion and Field [7]. For matrix elements of ^n. r
connecting +g states to +u states
4
<N; = 0|^n. r|N + 1; = 0>
= [(N + 1)=3]1=2< = 0|z| = 0> (15)
and connecting +g states to u states
<N; = 0|^n. r|N + 1; = 1> = [(N + 2)=3]1=2
x < = 0|2-1=2(x iy)| = 1>; (16)
where r = (x; y; z). The calculation of <0|z|0> and <0|x iy| 1> in prolate spheroidal
coordinates is straightforward.
Calculations of S(p) were carried out with basis sets n comprised of electronic and
vibrational functions [9,10]. The converged values of S(0) and S(-1) obtained using 121
electronic and 11 vibrational basis functions are given in Table I. The convergence of the
sum rules with basis set size is approximately logarithmic. Errors were determined for each
sum S(p) by finding A and c such that Ae-cn is the difference between the values obtained
with basis sets of sizes n x n x n and (n + 1) x (n + 1) x (n + 1). The total error given in
P 1 -ct -cn -c -1
Table I for each entry is A t=n e = Ae [1 - e ] .
The values of the calculated sums S(0) and S(-1) agree with the exact values [11,12] to
better than 2 parts in 108. Table I also lists the values of S(-2), S(-3), and S(-4). We
anticipate no loss of accuracy in evaluating S(-2) since the summation Eq. (6) is stationary
with respect to first order errors. The corresponding values of the dipole polarizabilities ffd
are given in Table II and Fig. 1. The sums S(-3) and S(-4) are related to quantities occur-
ring in the determination of the polarizabilities [1,13- 15]. S(-3) enters in the combination
B6 3_2S(-3) - _1_10C0, where C0 is the scalar quadrupole polarizability. With C0 = 23:99
for H 2+ and 23:24 for D 2+ [16], we predict that B6 = 7:77 for H 2+ and 7:24 for D 2+ . The
empirical value for H 2+ derived by Jacobson et al. [1] is 7:8(5).
Table II and Fig. 1 contain a comparison of our calculated values of ffd with experi-
ment and with the results of other theoretical methods. We leave aside calculations of the
polarizability corresponding to an electric field along the body-fixed axis [17,18]. Moss [4]
5
employed a variational method and an artificial channel method, with a classical descrip-
tion of the rotation. We are able to reproduce his results with our procedure if we take
N = 0 for the intermediate states with the consequent neglect of - coupling, the error
introduced by ignoring rotational coupling being accordingly one in the fourth decimal place
in the calculated polarizability. The calculations of Bhatia and Drachman [5] and Shertzer
and Greene [2] make no approximations other than in the numerical applications of their
methods and yield values consistent to within the precision they claim with our results.
We have determined the non-relativistic electric dipole polarizabilities of the lowest ro-
tational state of H 2+ and D 2+ to a precision, we believe, of one part in 108. We expect that
relativistic corrections will enter at the level of one part in 105 based on known corrections
for the hydrogen atom [19]. Other effects arising from the finite size of the nucleus and
nuclear spin will be still smaller. A new analysis of the experimental data [1] incorporating
our values of the sum rules may yield improved estimates of other properties that enter the
interpretation.
This work was supported in part by the U.S. Department of Energy, Division of Chemical
Sciences, Office of Basic Energy Sciences, Office of Energy Research. The Institute for
Theoretical Atomic and Molecular Physics is supported by a grant from the National Science
Foundation to the Smithsonian Institution and Harvard University.
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TABLES
_________TABLE_I._Nonrelativistic_evaluation_of_the_sum_S(p),_Eq._(8),_for_H_2+_and_D_2+.__________*
*__
___________________________________________________________________________________________________*
*__
p H 2+ D 2+
___________________________________________________________________________________________________*
*__
0 1.000 000 0(1) 1.000 000 0(2)
-1 1.653 650 96(2) 1.635 744 78(6)
-2 3.167 000 94(1) 3.071 152 0(2)
-3 6.780 745 959(7) 6.375 365 3(3)
-4 1 5.889 406 225(5) 1 4.325 799 4(6)
___________________________________________________________________________________________________*
*___
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TABLE II. Comparison of theoretical nonadiabatic values of the electric dipole polarizability
for the ground states of H 2+ and of D 2+ with experimental values. The results from Refs. [1] and
[4]_have_been_multiplied_by_the_factor_(1_+_ffl)2._________________________________________________*
*__
___________________________________________________________________________________________________*
*__
H 2+ D 2+ Ref.
___________________________________________________________________________________________________*
*__
3.168 0+0:0018-0:0001 3.067 1+0:0016-0:0020 [5], variational
3.168 2(4) 3.071 4(4) [2], finite element
3.168 5 3.071 87 [4], artificial chan*
*nel
3.168 3 3.071 78 [4], variational
3.168 725 6(1) 3.071 988 7(2) This work
3.168 1(7) 3.071 2(7) [1], experiment
___________________________________________________________________________________________________*
*___
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FIGURES
FIG. 1. Polarizabilities of H 2+ and D 2+ in their ground states. For each of the two calculati*
*ons
from the present work the error bar is within the vertical line crossing through the data point.
9
REFERENCES
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[3]W. Clark, Ph.D. thesis, Univ. of Colorado, 1998.
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in press; preprint: physics/98011043.
[10] The basis functions utilized here depend on nonlinear scaling parameters ff, fi, and fl.
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was replaced in the present work by fln = (2n + fi + 1)-1 fl, where n is the order of the
Laguerre polynomial. Parameters used in the present calculations for both H 2+ and
D 2+ are ff = 2:7, fi = 77, and fl = 3500 for the u functions and ff = 2:8, fi = 77, and
fl = 3500 for the u functions. They differ from those determined in Ref. [9], where the
emphasis was on obtaining minimum values for the energies. In the present work we
demanded maximum values of the polarizabilities in optimizing the u and u states.
The values of the nonlinear parameters for the ground electronic g state are identical
with those of Ref. [9].
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10
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