# The electric quadrupole moment of molecular hydrogen ions and their potential for a molecular ion clock

## Abstract

The systematic shifts of the transition frequencies in the
molecular hydrogen ions are of relevance to ultra-high-resolution
radio-frequency, microwave and optical spectroscopy of these
systems, performed in ion traps. We develop the ab-initio
description of the interaction of the electric quadrupole moment
of this class of molecules with the static electric field
gradients present in ion traps. In good approximation, it is
described in terms of an effective perturbation hamiltonian. An
approximate treatment is then performed in the Born-Oppenheimer
approximation. We give an expression of the electric quadrupole
coupling parameter valid for all hydrogen molecular ion species
and evaluate it for a large number of states of ,
, and . The systematic shifts can be evaluated as
simple expectation values of the perturbation hamiltonian. Results
on radio-frequency (M1), one-photon electric dipole (E1) and
two-photon E1 transitions between hyperfine states in are
reported. For two-photon E1 transitions between rotationless
states the shifts vanish. For a large subset of rovibrational
one-photon transitions the absolute values of the quadrupole
shifts range from 0.3 to 10 Hz for an electric field gradient of
10^{8} V/m^{2}. We point out an
experimental procedure for determining the quadrupole shift which
will allow reducing its contribution to the uncertainty of
unperturbed rovibrational transition frequencies to the
^{-15} relative level and, for selected
transitions, even below it. The combined contributions of
black-body radiation, Zeeman, Stark and quadrupole effects are
considered for a large set of transitions and it is estimated that
the total transition frequency uncertainty of selected transitions
can be reduced below the level.

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## I Introduction

One of the fascinating aspects of the ion trap invented by W. Paul and its later variants is the suitability for trapping a wide variety of particles. While atomic ions are the most \changestartfrequently \changeendstudied particle types, today, cold molecular ions are being studied in an increasing number of laboratories world-wide. The molecular ion most intensely studied \changestartso far \changeendfrom a spect\changestartr\changeendoscopic point of view is the molecular hydrogen ion , for which significant progress has been made in the last decade, both on the experimental (1); (2) and on the ab-initio theory front (see Ref. (3) and references therein). Combined studies of and of the isotopologue molecules ( (4), (5), , etc.), may in the near future lead to the determination of several fundamental physical constants, such as the ratios of proton, deuteron and triton mass relative to the electron mass, and the Rydberg energy, etc.(6); (7); (1); (8) with potentially competitive accuracy and with a different experimental approach than in atomic laser spectroscopy and Penning trap spectroscopy. A first step in this direction has been performed with two laser-spectroscopic measurements on (1); (2), from which the ratio of the electron mass to the reduced nuclear mass can be inferred with a relative experimental inaccuracy of approximately 4 and 2 parts in , respectively.

Moreover, the molecular hydrogen ions may be suited to investigate the question whether the mentioned dimensionless fundamental constants are independent of time (7) and of location in space, a postulate made by the principle of local position invariance of General Relativity.

These possibilities are only feasible if the experimental uncertainty in the measurement of transition frequencies can be reduced to a level necessary for the particular application. For example, in order to make competitive determinations of the fundamental constants, (currently) uncertainties of or less are desirable, while for the investigation of their time-independence, \changestartor less \changeendis desirable. A series of systematic effects needs to be carefully taken into account, including the effects of the external electric and magnetic fields in the volume occupied by the molecular ions. The Zeeman shift of the transition frequency induced by the weak magnetic fields usually present in experiments was thoroughly investigated in (9); (10). Various aspects of the Stark effect of the molecule have been studied in (11); (12), and recently in (13); (14).

In the present paper we determine theoretically the energy shifts caused by the interaction of the permanent electric quadrupole moment of the molecular ion with the inhomogeneities of the electric field of the ion trap. For atomic ions used in optical clocks, this is a well-known systematic effect, but for molecular ions, this effect has not been treated before for any molecule, to the best of our knowledge.

Concerning related work, we mention that the electric quadrupole transitions of the molecular hydrogen ions have been of some theoretical interest. The transition matrix elements for have first been treated by Bates and Poots (15) and later more extensively in Refs. (16); (17); (18); the value of the permanent quadrupole moment in the vibrational ground state is reported in Refs. (15); (16); (19); (20). To our knowledge, there is only a single calculation concerning , namely of its permanent quadrupole moment in the level , in Ref. (21). Recently, the quadrupole transition moments for have been reported (22).

After developing the general theory in Sec. II, as in a previous work (14) the numerical calculations are performed in the Born-Oppenheimer approach, introduced in Sec. III, which provides rovibrational energy levels and matrix elements with relative accuracy of approximately , but is entirely sufficient for the evaluation of the electric quadrupole effect in ion traps. This will be justified a posteriori by the small size of the calculated corrections. The detailed study of a large number of transitions in is given in Sec. IV. The discussion (Sec. V) shows that Zeeman, electric quadrupole, and Stark shifts can be controlled to a sufficient level even in spectroscopy aiming for\changestart\overstrikeon \overstrikeoff\changeend high accuracy.

## Ii Electric quadrupole shift in three-particle bound systems

In this section we derive the general expressions for the quadrupole interaction effect in a three-body bound system.

We use the Jacobi coordinate vectors of the three-body system, , and , which are related to the individual particle position vectors by means of

(1) |

where are the masses of the particles. In the ion, labels the deuteron, the proton and the electron, respectively. Note that is defined as the radius vector of the electron reckoned from the center of mass of the two nuclei. In terms of the Jacobi vectors, the non-relativistic Hamiltonian splits into the sum of the free Hamiltonian of the system “as a whole” and the Hamiltonian of the internal degrees of freedom:

(2) |

(3) |

(4) |

where , and are the momenta conjugate to , and , respectively, and are the particle charges in units of .

In an external electric potential , the non-relativistic Hamiltonian acquires an additional term: with

(5) |

being the electrostatic energy of the particles. For external fields that vary slowly in space and time, is approximated with the truncated multipole expansion

(6) | |||||

where is the electric dipole moment of the system with respect to , , is the irreducible tensor of rank 2 of the quadrupole moment with Cartesian components

(7) |

(8) |

is the external electric field at the center point, and

(9) |

, together with , are dropped because of being related to the degrees of freedom of the 3-body system “as a whole”. Different aspects of the second-order perturbation contribution of the dipole term have been evaluated in (11); (13); (14). In what follows, we focus our attention on the contribution of the quadrupole interaction term in first order of perturbation theory.

The Cartesian components in terms of the Cartesian components of the vectors , and (in the center-of-mass frame ) are

(10) |

(11) |

Note the factor in the definition of that is not present in the analogous expressions in Refs. (15); (21). In evaluating the matrix elements of in the angular momentum representation, similar to Refs. (23); (24) we use the expansion of the non-relativistic three-body wave function of the bound state with the orbital momentum quantum number , the projection of on the space-fixed -axis equal to , the vibrational quantum number and the parity in the basis of the symmetrized Wigner functions ,

(12) |

where is the angle between the vectors and : , while , and are the Euler angles of the rotation that transforms the space-fixed into the body-fixed reference frame with -axis along and in the plane.

The amplitudes are normalized by the condition .

The normalized symmetrized Wigner functions are linear combinations with definite parity of the complex conjugated standard Wigner functions:

(13) |

Next, the cyclic components of the quadrupole moment (labeled with the tilde to distinguish from the Cartesian components) are put in the form of a sum of terms with factorized dependence on the sets of angular and radial variables:

(14) | |||||

where are the “small” Wigner -matrices given in (25). The are the zero-th cyclic components of irreducible tensor operators of rank 2 acting on the angular variables:

(15) |

The reduced matrix elements of in the angular basis of Eq. (13) have the form:

(16) | |||||

where , and are the Clebsch-Gordan coefficients.

Thus, the matrix elements of in the basis Eq. (12) become

(17) |

(18) |

where are the contravariant cyclic components of and denote the following integrals,

(19) |

Eqs. (14-18), after the appropriate changes of variables in Eqs. (14) and (19) can be used with any alternative choice of the arguments of the radial amplitudes in the expansion Eq. (12), e.g. the variables or their linear combinations (24), but need be reworked for alternative basis sets in the space of functions of the angular variables, such as the expansion in bi-harmonics of Refs. (6); (26).

The quadrupole term in the expansion Eq. (6) couples, in the general case, states with different values of the orbital momentum and its projection and shifts the energy levels of the three-body states by amounts that depend on .

Previous studies of the effects of external magnetic fields (10) had demonstrated the advantages of considering the various perturbations to the dominating Coulomb interactions due to relativistic effects, particle spin and external fields on the same footing. An efficient implementation of these calculations in first order of perturbation theory is the use of an “effective Hamiltonian” . We remind that the “effective spin Hamiltonian” of an atomic system is the projection of the spin interaction operator on the finite dimensional space of eigenstates of the non-relativistic Hamiltonian of the system with definite values of the orbital angular momentum and the remaining non-relativistic quantum numbers, in which couplings to different are neglected.

We therefore include the effects of the quadrupole interaction in the form of an additional term in the effective spin Hamiltonian , introduced in (27) (denoted by there) in the calculation of the hyperfine structure and completed to by terms that describe the Zeeman shifts in (10). That is, we set

(20) |

where is the tensor square of the orbital momentum operator - the only irreducible tensor operator of rank 2 acting in the space of states with definite value of . In Eq.(20) we have shown explicitly the dependence of the effective Hamiltonian and its various terms on the quantum numbers of the non-relativistic state to which they refer. From the next section on, in order to simplify the notations we shall omit these quantum numbers while keeping in mind the dependence on them. The advantage of using the effective Hamiltonian is that the integrals of the 3-body wave functions of Eqs. (28) or (12) over , and are encoded in the single constant , so that the electric quadrupole shift of each individual quantum state is calculated by standard angular momentum algebra.

The expression for reads:

(21) | |||||

## Iii Born-Oppenheimer approximation

The gradient of the electric field acting on an ion in a
quadrupole ion trap is of the order of
V/m^{2} and in what follows it will be
shown that this magnitude gives rise to energy level shifts not
exceeding 100 Hz, significantly below the Zeeman shifts of most
levels for the typical fields that are applied in ion traps
(9); (10). This situation softens the requirements to the
numerical and theoretical accuracy of the treatment, and allows
for using the Born-Oppenheimer wave functions instead of the
highly accurate variational wave functions of Ref. (28).

The Born-Oppenheimer approximation assumes that instead of the molecular ion’s “motion as a whole” is associated with the nuclear center-of-mass position vector and its conjugate momentum . then takes the form

(22) |

where is that part that depends only on the internal degrees of freedom. Separation of external and internal degrees of freedom occurs by neglecting the cross term . This neglect limits a priori the relative accuracy of the results to the magnitude of the omitted terms, of order . The inaccuracy due to the replacement of in the denominator of in Eq. (2) by in is smaller.

In order to further separate the degrees of freedom of the electron from the relative motion of the nuclei we expand the wave function of the eigenstates of in the basis of eigenfunctions of the electronic Hamiltonian:

(23) |

(24) |

(25) |

We solved Eq.(24) numerically using its separability in the prolate spheroidal coordinates

(26) |

Their definition ranges are . These coordinates are related to and of Eq. (12) by means of

(27) |

We reproduced the results for of Ref. (29).

The calculations of the dipole polarizabilities of the lower ro-vibrational states of in Ref. (14) have shown (by comparison with the high precision variational results of Ref. (30)) that a relative accuracy of in the computation of the energy values and of the dipole moments may be reached by keeping only the first term in the expansion Eq.(24) and by neglecting the diagonal correction term . We therefore adopted this approximation in the evaluation of the quadrupole shift as well and took the wave functions of “normal” parity (the index is omitted in what follows) in the form:

(28) |

with normalization conditions

(29) |

(30) |

We calculated numerically the as solutions of the radial Schrödinger equation.

One could then obtain by using Eq. (21) and evaluating the integrals in Eq. (19) with the wave functions of Eq. (28). Instead, we re-expand of Eq. (6) around the “Born-Oppenheimer central point” so that the quadrupole interaction term takes the form

(31) |

The tensor differs from of Eq. (10) by terms of order or smaller:

(32) |

and the error due to replacing by is within the adopted accuracy limits. Note that for the homonuclear ions , , but differs for the heteronuclear ones.

Similar to Eq. (14), we expand the cyclic components over the set of irreducible tensor operators , but keep only the terms involving since the matrix elements of vanish in the -term approximation with , adopted in Eq. (28):

(33) |

In order to facilitate comparison with the results of earlier papers on the subject, instead of using the more general notations of Eq. (19), we put the reduced matrix elements of in the form:

(34) | |||||

We have made use of the symmetry of the wavefunction squared with respect to , so that only terms with even powers of contribute.

The function may also be expressed as (the angular brackets refer to the averaging over the electronic coordinates with ). Note that is independent of the molecular species and is the same for all homonuclear species: . The function , which gives the correction to the asymptotic behavior of , was introduced in (16). In Fig. 1 we plot it.

### iii.1 Comparison with previous work

The values of for homonuclear ions, calculated with our numerical values of the function , agree with the results of Refs. (15) (Table 1 therein), (20) (Table 2 therein) and (31). Also, the values of the function are essentially identical to those extracted from Ref. (16) in their Table II.

For , our value at.u. is in agreement with Ref. (19) (Table 1 therein) and the numerically less accurate, older value of Ref. (15) (Table 3 therein). Our values ,, agree within 0.001 atomic units with the more accurate values of Ref. (20) (Table 3 therein) computed with the adiabatic potential.

Concerning the only previous calculation known to us is Ref. (21) (Table I). There, the definition of the quadrupole moment is , the same as for the homonuclear ions. Thus, the expression was used, which involves a different dependence on the nuclear masses as compared with our Eq. (34). Our definition of the quadrupole moment for is . Accordingly, our value of at.u. differs from at.u. (table I in Ref. (21)). However, if we compute with the functions calculated in the present work, we obtain a similar result, at.u., which clearly indicates that the discrepancy is due to the different analytical expressions used, not to different wave functions.

### iii.2 The quadrupole coupling coefficients in the effective Hamiltonian

Eqs. (21, 34) lead to the following expression of the quadrupole coupling coefficients of the effective Hamiltonian for the ro-vibrational state in the adopted approximation:

(35) |

Tables III.2, III.2 and III.2 list the values of for 99 ro-vibrational states of , and , respectively, calculated using Eq. (35). Note the slow increase of with and the stronger decrease with . The entries with are not of relevance in the following, but are given for completeness since they are proportional to the normalized quadrupole moment of the states,

## Iv The quadrupole shift in

### iv.1 Generalities

We denote by the energy of the hyperfine state of in a magnetic field and in an electric field gradient . Because of the spin interactions these states are not in general eigenstates of the operators and and the quantum numbers and associated with them are not exact quantum numbers, but for weak fields may be considered as approximate quantum numbers. For given values of and , the number of\changestart\overstrikeon \overstrikeoff\changeend eigenstates is equal to the number of the combinations of quantum numbers in the spin coupling scheme of Ref.(10) allowed by angular momentum algebra. We therefore use the index , which enumerates the possible combinations of , to label the spin content of and associate each value of with the set of values of the approximate quantum numbers: . Note that is exact in absence of external fields, and is exact if the axial symmetry is conserved.

We consider in the following three levels of perturbation calculations, which are all restricted to a given level : (1) diagonalizing the whole effective Hamiltonian between angular momentum basis states; (2) diagonalizing the matrix of the quadrupole interaction between eigenstates of the effective Hamiltonian that includes magnetic, but not the quadrupole interaction; (3) compute the expectation value of the quadrupole interaction. We then show that the latter approximation is sufficient.

### iv.2 Diagonalization of the effective hamiltionian in a state

The energies are defined as eigenvalues of the matrix of the effective spin Hamiltonian of Eq. (20) in the subspace of states with fixed values of and . This matrix has dimension squared (, , being the spins of the three particles), i.e. . The matrix elements of the spin interaction operators (the first 9 terms of ) were computed in (27) I changed “give” to “computed” **** actually, we did not give the matrix elements, just the hamiltonian ***, and those of the interactions with external magnetic field (next 4 terms) in (10). With account of Eqs. (34) and (35), the matrix elements of the projection of on the subspace of a state with fixed values of and have the form:

(36) | |||||

Note that they vanish for levels.

### iv.3 Diagonalization of the electric quadrupole hamiltionian in the space of Zeeman hyperfine states

Comparison of the values of with the values of the
coefficients of the effective
Hamiltonian for the Zeeman effect (10) shows that, for the
electric field gradients and magnetic fields of interest here, the
quadrupole shift
is for the majority of levels much smaller than the Zeeman shift
.
Even the hyperfine states least sensitive to magnetic fields,
those with (having only a quadratic Zeeman shift),
exhibit at 1 G a typical shift of a few kHz or more, occasionally
only tens of Hz, while the electric quadrupole shift, in a
10^{8} V/m^{2} gradient, is of the
order of 100 Hz. Therefore, for sufficiently large magnetic fields
the electric quadrupole shift can conveniently be evaluated as a
perturbation to the Zeeman-shifted hyperfine energy levels by
diagonalizing the matrix of , Eq. (20),
in the basis of the Zeeman-shifted hyperfine states
, calculated as eigenvectors of
the spin and magnetic interaction part of the effective
Hamiltonian of Eq. (20):

(37) | |||||

where are the expansion coefficients of the hyperfine states in presence of a magnetic field in the field-free basis set (10):

(38) |

Note that in Eq. (38) there is no summation over the angular momentum projection , since it remains a good quantum number in a homogeneous magnetic field. The computational advantage of evaluating the electric quadrupole shift by diagonalizing the matrix of in Eq. (37) instead of is that no precision is lost in the subtraction . Note again, that the matrix element in Eq. (37) vanishes for .

### iv.4 First-order perturbation theory

In first order of perturbation theory the quadrupole shift is given by the diagonal matrix element of ,

(39) | |||||