The hyperfine structure of the hydrogen molecular ion

CfA No. 4476

The hyperfine structure of the hydrogen molecular ion*

J. F. Babb

Institute for Theoretical Atomic and Molecular Physics,

Harvard-Smithsonian Center for Astrophysics,

60 Garden Street, Cambridge, Massachusetts 02138

Abstract

Theoretical investigations of the hyperfine structure of the hydrogen molecu-

lar ion (one electron and two protons) are discussed. The nuclear spin-rotation

interaction has been found to be of the same sign as in the hydrogen molecule

and the hyperfine transition frequencies can be accurately predicted. With

measurements of the hyperfine structure of the deuterium molecular ion (or

of HD +) it should be possible to obtain a value of the deuteron quadrupole

moment that could be compared with the values obtained from the deuterium

molecule and from nuclear theory.

________

*Invited talk given at the Inaugural Conference of the Asia Pacific Center for Theoretical Physics,

Seoul, Korea, June 4-10, 1996, to be published by World Scientific.

INTRODUCTION

The calculation of hyperfine structure (hfs) frequencies presents a substantial challenge

to theory, in part because the experimental spectroscopic data available are so precise. For

the hydrogen atom in the 1s state, with electron spin S = 1_2and nuclear spin I = 1_2, the

transition frequency between the F = 0 and F = 1 states, where F = I + S, is known

to be 1 420:405 751 766 7(9) MHz [1], in comparison to the most complete theoretical result

of 1 420:405 1(8)MHz [2], which includes reduced mass, QED, and hadronic (mainly proton

recoil and size) effects. Indeed, a simple "Fermi" theory of the hfs (see, for example, [3],

p. 74) with a fixed proton and the electron and proton g-factors of, respectively, 2:002 319

and 5:585 694, gives about 1 422 MHz and with the correction due to the reduced mass

of the proton-electron system it yields 1 420:5 MHz . Further inclusion of the known QED

corrections yields a value of about 1 420:452 MHz [4] and an additional ~ -45 kHz and

the theoretical error comes from hadronic effects [2]. Needless to say, the difficulty of the

calculations increases substantially at each stage [5].

The "hydrogen atom" of molecular physics is the hydrogen molecular ion H 2+ consisting

of two protons bound by a single electron. From a theoretical point of view this molecule

offers a good starting point for investigations of molecular hfs since the electronic wave func-

tion can be calculated to high accuracy. Surprisingly, there have been only a few experiments

on the hfs of H 2+ [6- 8], one experiment on HD + [9,10] , and no experiments on D 2+.

With the deuterated ions it should be possible to determine the electric quadrupole mo-

ment of the deuteron, once the hfs transition energies are measured, and a further discussion

of the prospects of using D 2+ to do this will be given below.

THEORY OF H2+ HFS

As with the hydrogen atom, the hfs in H 2+ arises primarily from interactions between

the electron spin and nuclear spin, but there are also smaller magnetic effects arising from

the electron and nuclear spins interacting with the magnetic field of the rotating nucleus.

The hfs energy can be described by the effective spin Hamiltonian

Hhfs= bI . S + cIzSz + dS . N + f I . N; (1)

where I, S, and N are, respectively, the total nuclear spin, electron spin, and rotational

angular momenta, and b, c, d, and f are the coupling constants. Transition frequencies

are obtained by diagonalizing the matrix elements of (1) with constants calculated from

theoretical expressions or if given measured frequencies the constants can be determined

by fitting using matrix elements of (1); usually the bfiS [11] angular momentum coupling

scheme is used in which the intermediate vector F2 = I + S is formed, and then the total

F = F2 + N.

Early theoretical studies of H 2+ hfs were motivated by a need for predictions of transi-

tion frequencies of the N = 1 rotational level of the v = 0 vibrational state for astrophysical

searches (it is now believed that there is little prospect for detection, cf. [12]), because no

experimental data were available. The researchers calculated coupling constants and tran-

sition frequencies using (1) with f = 0. Transition frequencies were estimated by Burke [13]

using calculations of b, c, and d, and by Mizushima [14] who used values of b and c calculated

by Stephens and Auffray [15]. A significant advance in the theory was the derivation by

Dalgarno, Patterson, and Somerville [16] of the b, c, and d terms of (1) from the Dirac eq

for H 2+ by a nonrelativistic reduction within the Breit-Pauli approximation. They demon-

strated that d could be written d = d1 + d2, where d1 is the electron spin-rotational magnetic

interaction and d2 is a second-order term in the electronic wave function. They calculated

values for b, c, and d1, estimated d2, and calculated transition frequencies and the constants

b and c were used by Richardson, Jefferts, and Dehmelt to make the first experimental esti-

mate of d [6]. Further improvements in calculations of b, c, and d1, and improved estimates

of d2 were presented by Somerville [17,18] and by Luke [19]. The second-order constant

d2 was calculated accurately using a variation-perturbation method by Kalaghan and Dal-

garno [20] and by McEachran, Veenstra, and Cohen [21], who also calculated b, c, and d1

and accurate transition frequencies.

The calculated constants discussed so far were all determined in the Born-Oppenheimer

approximation in which the electron is assumed to move in the field generated by averag-

ing over the electronic wave function. Nonadiabatic effects on the ground electronic state

through the excitation of other electronic states due to the coupling of electronic and nuclear

motion contribute significantly to the constant b and the effects of the reduced mass of the

electron on the constants have been treated [22,23].

The nuclear spin-rotation term f was introduced into the effective spin Hamiltonian (1)

on phenomenological grounds by Jefferts [7], who measured the hfs transition frequencies

for the v = 4-8 vibrational states. For the N = 1 rotational level for each v, he obtained

values of the constants by fitting to (1). Varshalovich and Sannikov [12] obtained different

fits to Jefferts' data, notably, they found the value of f to be negative. The first (and to

date the only) measurements of the hfs of the N = 1 level of v = 0 state were carried out

by Fu, Hessels, and Lundeen [8] through spectroscopy of high angular momentum Rydberg

states of the hydrogen molecule H 2 and by fitting the measured Rydberg transition levels to

an effective spin Hamiltonian including (1) they determined values for b, c, d, and f .

A recent theoretical study of the nuclear spin-rotation interaction in H 2+ found the sign

of f to be negative [24]. The nuclear spin-rotation interaction can be written

f (R) = f1(R) + f2(R); (2)

and the major contributions [25] to the energy hf are from the interaction of each nuclear

magnetic moment with the magnetic field generated by the other rotating nucleus,

4gp2N
hf1(R) = - _______; (3)
R3

and with the magnetic field generated by the orbiting electron [26],

12gp2N
hf2(R) = - ________oehf(R); (4)
ff2R2

where in atomic units R is the internuclear distance, N is the nuclear magneton, and the

dimensionless quantities ff, gp, and oehf are, respectively, the fs constant, the proton g-factor

defined previously, and the high-frequency component of the magnetic shielding constant as

defined in Eq. (3) of [24].

In Table I values of the hfs constants calculated as in [23] and the nuclear spin-rotation

constants calculated as in [24], averaged over the vibrational wave functions, are compared

with constants measured [8] for the N = 1 level of the v = 0 state and with constants

determined [12] from the measurements [7] on the N = 1 level of the vibrational state v = 4.

The agreement between theory and experiment is impressive_the theoretical constants in

Table I reproduce all measured transition frequencies to within 175 kHz. The formalism

of [23] accounts for reduced mass effects on the hfs constants but not fully for radiative and

relativistic effects.

DEUTERON MOLECULAR ION

The electric quadrupole moment Q of the deuteron can serve as a sensitive test of models

of the neutron-proton nuclear force [27]. The most accurate determination Q = 0:2860

0:0015 x 10-26 cm 2 was obtained semiempirically from measurements [28] of the magnetic hfs

of the deuterium molecule D 2 through a theoretical value for the electric field gradient at the

nucleus due to the other constituent charges of the molecule [29,30]. It should be possible

to measure Q semiempirically using D 2+, which has the theoretical advantage of a simpler

electronic structure than D 2, thereby facilitating accurate computation, but to date there

have been no measurements of the hfs of D 2+. (An experiment that measured hfs levels of

HD + was reported, but the quadrupole effects were not resolved [9].)

For D 2+ , the effective spin Hamiltonian is (1) plus an additional nuclear quadrupole

interaction [31] proportional to eqQ, where e is the proton charge and q is the electric field

gradient. Predictions of the hfs constants and electric quadrupole coupling constant for the

N = 1 level of the v = 0 state of D 2+ are given in Table II. The constants b, c, and d

have been calculated as in [23] with the inclusion of nonadiabatic effects in b and c, and f

has been calculated using the theory of [24]. The term eqQ was estimated by calculating

q in the Born-Oppenheimer approximation and using the semi-empirical value of Q from

D 2. More detailed calculations of the hfs constants, quadrupole coupling constants, and

transition frequencies will be reported [32].

The quadrupole coupling constant is about 57 kHz, see Table II. The experimental

precision achieved in the ion trap experiments on H 2+ was better than 3 kHz [7], in the

Rydberg H 2 experiment it was around 20 kHz [8], and in a laser-rf double-resonance ion

beam study of N 2+ it was better than 10 kHz [33]. It would appear that the prospects are

good for measuring the deuteron quadrupole moment using D 2+.

ACKNOWLEDGMENTS

The author is grateful to Y. M. Cho and the International Organizing Committee for

support to attend the Inaugural Conference of the APCTP and to S. D. Oh, H. W. Lee,

and W. Jhe for their hospitality. This work was supported in part by the National Science

Foundation through a grant for the Institute for Theoretical Atomic and Molecular Physics

at the Smithsonian Astrophysical Observatory and Harvard University.

REFERENCES

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[7]K. B. Jefferts, Phys. Rev. Lett. 23, 1476 (1969).

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[9]W. H. Wing, G. A. Ruff, W. E. Lamb Jr., and J. J. Spezeski, Phys. Rev. Lett. 36, 1488

(1976).

[10] There have been measurements of hfs in highly-excited vibrational states of HD + car-

ried out by Carrington and collaborators, cf. A. Carrington, I. R. McNab, and C. A.

Montgomerie, J. Phys. B 22, 3551 (1989), but the present paper is concerned with

the low-lying vibrational states for which hyperfine levels have been measured most

accurately.

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[12] D. A. Varshalovich and A. V. Sannikov, Astron. Lett. 19, 290 (1993).

[13] B. F. Burke, Ap. J. 132, 514 (1960).

[14] M. Mizushima, Ap. J. 132, 493 (1960).

[15] M. J. Stephen and J. P. Auffray, J. Chem. Phys. 31, 1329 (1959).

[16] A. Dalgarno, T. N. L. Patterson, and W. B. Somerville, Proc. Roy. Soc. London 259A,

100 (1960).

[17] W. B. Somerville, Mon. Not. R. Astr. Soc. 139, 163 (1968).

[18] W. B. Somerville, Mon. Not. R. Astr. Soc. 147, 201 (1970).

[19] S. K. Luke, Ap. J. 156, 761 (1969).

[20] P. M. Kalaghan and A. Dalgarno, Phys. Lett. 38A, 485 (1972).

[21] R. P. McEachran, C. J. Veenstra, and M. Cohen, Chem. Phys. Lett. 59, 275 (1978).

[22] J. F. Babb and A. Dalgarno, Phys. Rev. Lett. 66, 880 (1991).

[23] J. F. Babb and A. Dalgarno, Phys. Rev. A 46, R5317 (1992).

[24] J. F. Babb, Phys. Rev. Lett. 75, 4377 (1995).

[25] Ref. [3], pp. 162-166, 208-213.

[26] N. F. Ramsey, Phys. Rev. 78, 699 (1950).

[27] M. A. Preston and R. K. Bhaduri, Structure of the nucleus (Addison-Wesley, Reading,

Massachusetts, 1975), pp. 123-130.

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[32] J. F. Babb and A. Dalgarno, in preparation.

[33] N. Berrah Mansour, C. Kurtz, T. C. Steimle, G. L. Goodman, L. Young, T. J. Scholl,

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TABLES

TABLE I. Comparison of H 2+hfs constants from theory with those obtained from experimental

measurements of hfs levels, for the N = 1 rotational levels of the v = 0 and v = 4 vibrational

states. All values are given in MHz and numbers in parenthesis represent errors quoted on the last

digit._____________________________________________________________________________________________*
*__
___________________________________________________________________________________________________*
*__
______________________________b________________c________________d________________f____________Ref._*
*__

Theory, v = 0 880.163 128.482 42.421 -0:042 [23,2*
*4]

Experiment, v = 0 880.187(22) 128.259(26) 42.348(29) -0:003(15) [8]

Theory, v = 4 804.104 98.008 32.658 -0:036 [23,2*
*4]

Experiment,_v_=_4________804.087(2)_______97.930(2)_________32.649(2)_______-0:034(2)_________[12]_*
*__
___________________________________________________________________________________________________*
*__

TABLE II. Predictions of the hfs constants and the electric quadrupole coupling constant for

the molecular ion D 2+ in the N = 1 rotational state of the v = 0 vibrational level. All values are

given_in_MHz.______________________________________________________________________________________*
*__
___________________________________________________________________________________________________*
*__
____b______________________c______________________d_______________________f____________________eqQ_*
*__

135.739__________________19.944_________________21.461_________________-0:002__________________0.05*
*7_
___________________________________________________________________________________________________*
*__