Canceling spin-dependent contributions and systematic shifts in precision spectroscopy of the molecular hydrogen ions

Canceling spin-dependent contributions and systematic shifts in precision spectroscopy of the molecular hydrogen ions

S. Schiller Institut für Experimentalphysik, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany    V. I. Korobov Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Abstract

We consider the application of a basic principle of quantum theory, the tracelessness of a certain class of hamiltonians, to the precision spectroscopy of the molecular hydrogen ions. We show that it is possible to obtain the spin-averaged transition frequencies between states from a simple weighted sum of experimentally accessible spin-dependent transition frequencies. We discuss the cases and , which are distinct in the multipole character of their rovibrational transitions. Inclusion of additional frequencies permits canceling also the electric quadrupole shift, the Zeeman shift and partially the Stark shift. In this context, we find that measuring electric quadrupole transitions in is advantageous. The required experimental effort appears reasonable.

The precision spectroscopy of isolated molecules is making strong advances thanks to novel techniques of trapping, cooling and manipulation. One family of molecules, the molecular hydrogen ions, is particularly attractive because their transition frequencies can be calculated ab initio with a precision that challenges current experimental approaches Korobov2017a ().

The comparison of theoretical with experimental frequencies allows extracting the values of certain fundamental constants, such as the electron-to-proton mass ratio and the Rydberg constant Koelemeij2007 (); Bressel2012 (); Biesheuvel2016 (); Alighanbari2018 (). In this endeavor, one is faced with the problem that the theoretical frequencies are not just given by the solution of the Schrödinger three-body problem, but include relativistic and QED contributions, as well as spin-dependent contributions. In addition, the experimentally measured frequencies are perturbed by external fields whose strength cannot always be determined accurately enough.

The spin-dependent contributions can be calculated and, in fact, the accuracy of the calculation has made impressive progress in recent years Korobov2016 (). Nevertheless, it will become harder to push the accuracy further. Thus, it is worthwhile to consider whether there are experimentally viable approaches to determine the (not directly observable) spin-averaged transition frequency from a combination of measured transition frequencies. The solution to this query makes use of the mathematical property of a class of hamiltonians, the tracelessness. In atomic physics, this approach is well-known; here the computed “center of gravity” of the fine-structure or of the hyperfine structure of a level is often considered. In the field of metrology it is for example used to characterize the helium isotope shift Pachucki2015 ().

We further consider an important subset of systematic shifts affecting the transition frequencies. When the shifts are small, so that first-order and second-order perturbation is applicable, their mathematical structure is such that an appropriate average over them vanishes exactly. This principle was recently implicitly used by Karr et al. Karr2016 () in an analysis of promising transitions for the precision spectroscopy of . Here we illuminate the principle from a broader perspective and show the close relationship to the concept of the spin-averaged transition frequency determination.

Composite frequencies being at the focus of the present work, it complements our previous discussion Schiller2014 (), in which we considered the combination of different rovibrational transitions for the cancellation of systematic shifts. Here, individual rovibrational transitions are considered.

The paper is structured as follows. In Sec. I we derive cancellation conditions from the tracelessness of the hamiltonian. In Sec. II it is shown that they lead to the possibility of determining the spin-averaged frequency via the combination of a set of electric-dipole transitions. In Sec. III the conditions for cancellation of three types of systematic shifts are derived: linear Zeeman shift, quadratic Zeeman shift, and electric quadrupole shift. A fourth systematic shift, the Stark shift, can be partially canceled, since its tensor part is canceled whenever the quadrupole shift is. In Sec. IV we state how spin contributions and shifts can be simultaneously canceled. In Sec. V we apply the general discussion of spin structure cancellation to the molecular hydrogen ions and . In Sec. VI we show that an obvious extension leads to cancellation of the linear Zeeman shift as well. Further shifts can also be canceled by including more transitions. Subsequently, the general method from Sec. IV is applied. For each approach, the systematic shifts stemming from the un-cancelled perturbations are evaluated and discussed, leading to estimates for the achievable residual shifts. In Sec. VII we discuss a more efficient approach for canceling both spin-structure and systematic shifts. Sec. VIII draws the conclusions.

I Cancellations: Elementary Considerations

Consider a complete set of basis states, where denotes the set of quantum numbers uniquely identifying a state, and a generic traceless operator . The traceless property implies,

(1)

The trace can be evaluated in any basis. In particular, we can choose a set of basis states that diagonalizes the operator, its eigenstates.

Consider now the operator to be a contribution to the total hamiltonian . The tracelessness of then yields a relationship between the energy contributions :

(2)

where is the degeneracy factor for those states having the same energy contribution , and denotes the set of quantum numbers identifying the different energy levels. The expression implies the existence of positive and negative energy contributions.

Often, one is concerned with states that are characterized by a total angular momentum quantum number, , and a magnetic quantum number, . We may choose these states as basis states. In case of rotational invariance of the hamiltonian , its expectation values are independent of . In this case the above expression takes the form

(3)

where is the magnetic degeneracy factor, and denotes all other quantum numbers necessary to identify the state.

The typical situation in atomic and molecular physics is that the system is described by a total hamiltonian which is the sum of a “dominant” term, (spin-independent or spin-averaged) and several traceless perturbations, each denoted by . The energy of an eigenstate of the total hamiltonian is approximated by first-order perturbation theory,

(4)

The expectation values are taken for the unperturbed states , the eigenstates of the dominant hamiltonian . In fact, the quantum numbers for a state can be written more explicitly as , where refers to the set describing the state space of the dominant hamiltonian, and to the space of the perturbation hamiltonian. The trace condition is then

(5)

When we perform spectroscopy, we measure energy differences . The question we pose is whether a combination of such energy differences, provided they are experimentally accessible, allows to determine the dominant contribution , which may be of interest. To this end, we consider a linear combination of transition frequencies, the “traceless” frequency , of the form

(6)

with (positive or negative) weights . This ansatz is successful, if the three conditions hold:

  • all transition frequencies are experimentally measurable (necessarily, the transitions must be allowed by selection rules),

  • is satisfied for all possible values ,

  • is satisfied for all possible values .

Here is the total number of states for given , etc. (II) and (III) simply mean that in the set of selected transition frequencies each initial state should occur with equal total weight and each final state should occur with equal total weight .

Then, eq. (6) simplifies to

(7)

where we have made use of the tracelessness, .

Ii Cancellation of Spin Structure Contributions

Generally, the effective spin hamiltonian, comprising fine-structure, hyperfine structure, and interaction of the electron with the nuclear quadrupole moment, may be expressed as a sum of traceless operators,

(8)

where and are some irreducible tensors of spin or orbital operators and dot is a tensor scalar product

A first example is the spin-orbit interaction , where is the operator of the total orbital angular momentum and is the spin operator of particle . A second example is the tensor quadrupole interaction,

This form is provided by the spin-dependent part of the Breit-Pauli hamiltonian. Higher-order corrections enter either as corrections to the coefficients of the already existing interactions, or as new, more complicated irreducible tensor (traceless) interactions, or they contribute to the spin-averaged part of the energy of a state.

If the system is composed by three particles with spin, we introduce the two-particle spin operator , and the three-particle spin operator , which generally do not commute with the total hamiltonian. The total angular momentum, , does commute with , and are good quantum numbers. As basis states we can take pure angular momentum states or the eigenstates of the total hamiltonian . In both cases we can denote them by . In the latter case, the numbers are chosen as the integers closest to the numbers resulting from the expectation values and in the given state. In absence of magnetic field, the eigenstates are degenerate in . We denote by the perturbation energy of state .

The following sum rule holds for the traceless spin hamiltonian:

(9)

The sum is over all spin states. The proof is straightforward because it is fulfilled for each individual term in eq. (8) Schwarz1955 ().

We now consider transitions and show how the traceless frequency eq. (6) can be implemented.

A transition frequency between two particular spin states of two rovibrational levels is

(10)

In E1-allowed transitions, do not change, but the values must nevertheless be included to uniquely identify the state.

The following quantity vanishes,

(11)

where the sum is over all E1-allowed transitions between levels and .

The proof is as follows. The E1 selection rule is . If , then .

We use the obvious relation

or for : . That proves that the l.h.s. of eq. (11) may be written as

which vanishes on account of tracelessness, eq. (9). Thus, the traceless frequency constructed from all E1-allowed transition frequencies

(12)
(13)

is equal to the spin-averaged transition frequency . Here, is a normalization factor. In Sec. V.2 below, we show an example of the set of transitions that are included in the sum.

When E1 transitions do not exist, the treatment must be modified, see Sec. V.3 below.

Iii Cancellation of Systematic Shifts

iii.1 Linear Zeeman shift and electric quadrupole shift: a simple model

We now consider a simple example: a quantum system possessing angular momentum and an electric quadrupole moment, exposed to an external magnetic field pointing along the -axis and to an external electric field gradient . Assume the magnetic and electric quadrupole perturbation hamiltonian to be of the form

(14)

Both contributions are traceless. The coefficients and may in general differ in different states of the unperturbed hamiltonian . The state quantum numbers summarized by include , which is assumed fixed in this example. The angular momentum projection quantum number now plays the role of the quantum number . The linear Zeeman (LZ) shift and the electric quadrupole (EQ) shift are computed in 1st-order perturbation theory. The tracelessness of is expressed by the sum rule,

(15)

which holds independently of the specific values of (integer or half-integer), , (and thus of the specific level ) and of the field strengths , and .

The following examples illustrate how to include this sum rule into the traceless frequency. In Fig. 1 (a-c) we show schematically some energy levels of the total hamiltonian, a lower level having or 2, and an upper level having or 3. The energy of a state is . A number of electric dipole (E1) transitions between the two levels, obeying the selection rules , , have been selected and assigned specific weights satisfying the conditions (II) and (III). These weights can easily be found by solving appropriate conditions. In the case Fig. 1 (b), the 9 transitions , which altogether address every state of the lower level with the same weight (1/3) and every state of the upper level with the same weight (1/5), serve to null the effect of four interactions, proportional to , , and ,

The specific cases (a-c) can easily be generalized to higher values of and .

The panels (d-e) in the same figure show cases of half-integer angular momenta, , , and electric-quadrupole (E2) transitions. A suitable set of transitions, satisfying the selection rule for E2 transitions, and appropriate weights are found by inspection. For example, panel (d) shows the case where the measurement of a total of 4 individual transitions allows canceling the effect of Zeeman and EQ interactions in both lower and upper level, defined by four interactions. In panel (e) is larger, and now 6 transitions are required for canceling again four interactions. However, in these particular cases having , the EQ interaction in the lower level vanishes, so effectively only three interactions are canceled.

The cases are trivial, therefore only one example is shown, panel (a), right-hand side.

We emphasize that the presented scheme for nulling the effects of the Zeeman and quadrupole shifts is not unique. Other types of combinations of transition frequencies are possible. Specifically for , Karr et al. Karr2016 () have considered a combination of frequencies involving two different angular momentum sub-spaces in one of the two levels (see also below). Such other combinations can be more efficient, requiring a smaller number of transitions to be measured.

iii.2 Zeeman shift and electric quadrupole shift: realistic case

The EQ interaction between an external field gradient and the electronic plus nuclear charge distribution has the form

where the r.h.s. is an irreducible tensor operator of rank 2. This hamiltonian is traceless. Its effects are evaluated in first-order perturbation theory, due to the smallness of the field gradient occurring in experiments. The hamiltonian can be replaced by Bakalov2014includingcorrigendum ()

The first-order energy shifts of a given state are , the same as for the simple model above.

The Stark shift arises from light fields and trap fields. In general, it has a scalar, vector, and tensor contribution Schiller2014a () (see eq. (5) in ref. Bakalov2015 () for the formal expression for the static Stark shift). The tensor contribution has the same dependence on the angular momentum quantum numbers as the EQ shift. Therefore cancellation of EQ shift implies cancellation of the tensor Stark shift.

In real systems, the interaction with the external magnetic field is not of the form eq. (14). Instead, in general,

This hamiltonian is traceless, and furthermore commutes with . We can take advantage of the structure of the magnetic shifts that occur in first order and in second order in and incorporate them into the traceless frequency.

In first-order perturbation theory,

The LZ shift is proportional to by virtue of the Wigner-Eckhart theorem.

At today’s desired precision levels, it is insufficient to consider only the LZ shift. It is necessary to also consider the quadratic Zeeman (QZ) shift, which according to second-order perturbation theory is

(16)

for a state state . The sum goes over all spin-structure states but is limited to states in the same level . Also, since commutes with the operator . Because the denominator is anti-symmetric under “state exchange” , while the numerator is symmetric, it follows that for any given

(17)

The sum over the quadratic Zeeman shifts of all spin states in a given rovibrational level and having a given is zero. This sum rule does not contain any degeneracy factor since is fixed.

Note that there exists only one state having and one having (stretched states). These states therefore do not exhibit a QZ shift.

Iv Combining spin structure cancellation with systematic shift cancellation

The standard situation in the description of the molecular hydrogen ions is to consider the spin structure contributions and systematic shifts for each rovibrational level , independently of the others. This is a good approximation because both types of contributions are very small compared to the energy difference to neighboring rotational levels of the same vibrational level and even smaller compared to the energy difference to neighboring vibrational levels. The hamiltonians of the perturbations therefore are effective hamiltonians, i.e. they contain parameters that depend on the concrete level : . Given an arbitrary basis of spin states for the particular level , one can set up the hamiltonian matrix

(18)

and diagonalize it in order to find the eigenstates and the eigenenergies . Because each contribution in is traceless, we have the sum rule

(19)

We emphasize that this sum rule refers to the “exact” total perturbation shifts, to all orders in any perturbation parameter, e.g. the magnetic field strength or electric field gradient strength.

Since it is permissible to consider various orders of a particular perturbation, e.g. the linear Zeeman shift and the quadratic Zeeman shift, there necessarily follow separate sum rules for these orders. Some of these have been already given above. Indeed, for practical reasons it is useful to take this point of view: in precision spectroscopy, one strives experimentally to make as many of the perturbations as small as possible, so that the dominant contribution to is the spin structure . The other perturbations are then treated in first-order perturbation theory with respect to the eigenstates of , and only occasionally also second-order perturbation theory is applied. The advantage of this approach is that the perturbation energy of a state can then be written as a sum of contributions, and the total energy is

Here, corresponds to . The spin interaction is treated exactly (by diagonalization), while the Zeeman interaction is treated to second-order perturbation theory (LZ, QZ), and the EQ interaction only to first-order perturbation theory. All perturbations come with their own sum rules, which have been presented above. They do differ in form:

  • the LZS can be nulled, for any particular , by summing over all , or pairwise over and .

  • the EQS can be nulled, for any particular , by summing over all .

  • the QZ shift can be nulled, for any fixed , by summing over all states containing this Zeeman state.

  • The spin structure contributions can be nulled by summing over all states , or over all states with .

Although these sum rules differ, not surprisingly they can nevertheless be incorporated together in a traceless frequency as in eq. (6), by defining

(20)

A sum over all possible transitions is taken, with the following restrictions:

(1) Only those transitions are included that allow the spin structure cancellation as in the case of absence of external fields. Depending on the particular transition , either each upper state is associated with one lower state only, or vice-versa (for the whole set of lower states). Here one disregards the concrete Zeeman components.

(2) For each of these transitions, one measures the components as shown in Fig. 1, multiplies the components’ frequencies with the weights indicated, and further multiplies each with the factor . The resulting weights are the of eq. (20).

is free from LZ shift and EQ shift by construction, already at the level of each individual transition , because according to Fig. 1, (i) equal weight are given to the and transitions, (ii) all and sub-states contribute with equal weight. Furthermore, is free of spin structure because of the definition of weights as given in (2). Finally, with this procedure, the total QZ shift also cancels: because each state enters with the same (total) weight and similarly for each state, eq. (17) applies. As an example, consider the states in Fig. 1 (a,b,c): in each panel, their total weight is always 1/3. Thus,

(21)

Thus, the traceless frequency defined in this way is equal to the unperturbed (spin-averaged) frequency . This scheme is generally applicable, but variations can be more efficient in terms of minimizing the number of transitions to be measured, and will be discussed below.

iv.1 Special cases

For systems with integer , there exist the Zeeman components. Measuring only these for the traceless frequency leads to zero total LZ shift.

Similarly, for systems with half-integer spin, the two Zeeman components and exist. In the mean frequency of these two components the LZ effect vanishes, for each hyperfine transition .

Such traceless frequencies will have a well-defined, nonzero EQ shift and QZ shift. We shall take up this again in Sec. VI.1.

Figure 1: Schematic energy diagrams comprising a lower level havingangular momentum and an upper level . (a-e) show different cases . The green arrows denote transitions between specific magnetic sub-levels to be measured. Each assigned coefficient multiplies the corresponding transition frequency so that in the weighted sum the external-field perturbations average to zero. (a -d): E1 or E2 transitions; (e): E2 transitions. The case is not shown; it can easily be obtained by generalization of (a). The coefficients for the cases where are obtained from the above cases by symmetry.

V Application to the Molecular Hydrogen Ions

v.1 Motivation

The scope of the following discussion is to present in a detailed manner the application of the traceless frequency to the elimination of the spin-dependent energies in and .

Consider a single spin component within the spin structure of a transition between the rovibrational levels , (: vibrational quantum number, : rotational quantum number). The lower and upper spin state are enumerated by quantum numbers denoted collectively as and . The frequency of an individual transition is computed as a sum of two contributions:

(22)

The spin-averaged frequency (corresponding to above) depends only on , , , . Currently, it can be computed with a fractional inaccuracy due to theory at the level, for both vibrational Korobov2017a () and rotational transitions Alighanbari2018 (). The second contribution is the spin-structure (hyperfine) shift due to spin interactions. The effective spin hamiltonian has been derived within the Breit-Pauli approximation Bakalov2006 (); Korobov2006 (). It contains a set of - dependent coefficients , The number of coefficients in the set is up to 9 for and up to 5 for . The inaccuracy of the ab-initio calculation of the set of coefficients is the dominant source of the theoretical inaccuracy of , at present Korobov2016 ().

In general, the spin energies of a state are obtained numerically by diagonalizing the spin hamiltonian. For , only a few particular states have energies expressible in explicit form. This is the case for the stretched states (states of maximum total angular momentum, , and maximum total angular momentum projection, ), for which (see eq. (6) in Bakalov2011includingerratum ()),

(23)

where are the coefficients of the effective spin Hamiltonian Bakalov2006 (); Bakalov2011includingerratum (). This expression is helpful in showing how any theoretical inaccuracy of the will affect the inaccuracy of the overall transition frequency .

Since the present considerations are independent of the particular vibrational levels, we shall often omit the mention of in the state designations when they are not essential.

v.2 The molecular ion

Each state of is uniquely defined by the quantum numbers and . As introduced in Sec. II, is (approximate or exact) spin of the electron - proton pair, is the (approximate or exact) total spin of the three particles, and is the (exact) total angular momentum quantum number of the molecule.

Before proceeding we note that the tracelessness property of the Breit-Pauli interaction hamiltonian and the sum rule for the QZ shifts are relationships that can be verified on computed energy shifts in order to check for correctness of the computation 111For example, an analytical calculation of the diagonal matrix elements of the Breit-Pauli Hamiltonian can be performed in the basis of the states , , , with , , . Taking the sum over these elements must yield zero. One can also verify tracelessness for many rovibrational levels computed in previous work, by performing the weighted average, eq. (2), over the numerical values of the spin energies given in Table III of ref. Bakalov2006 (). The sum rule for the QZ shifts can also easily be verified numerically as follows. By summing over all entries in the middle lines of Table 2 in ref. Bakalov2011includingerratum () (the values ) one verifies the case . The cases can be verified by appropriate sums over the middle and lower lines together, omitting those entries belonging to states having . The sum rule for the QZ shifts can also easily be verified numerically as follows. By summing over all entries in the middle lines of Table 2 in ref. Bakalov2011includingerratum () (the values ) one verifies the case . The cases can be verified by appropriate sums over the middle and lower lines together, omitting those entries belonging to states having ..

Let us initially ignore the external-field shifts. A general argument allows us to find the weights for the traceless frequency. For simplicity, we assume zero magnetic field and electric field gradient, so that the states are degenerate in , and we omit this quantum number in the following. The appropriate type of transitions are electric-dipole (E1), which allow and . Strong E1 transitions are those for which the conditions and are fulfilled.

For concreteness, we discuss the case of transitions from a lower level to an upper level . There is no restriction on and . Two transitions of this type have already been measured with - level fractional inaccuracy Bressel2012 (); Alighanbari2018 (). One relevant case is the rotational transition between the two lowest-energy rovibrational levels, , with (). The spin structure of this transition has been discussed previously Bakalov2011includingerratum (); Shen2012 (); Alighanbari2018 () and Fig. 2 reports the detailed energy diagram with the actual spin energies .

Figure 2: The energies , of the spin states of relevant for the fundamental rotational transition . The colored lines indicate the transitions to be measured in order to compute the traceless transition frequency. The numbers above the colored lines are the weights . For both levels, the zero of the vertical scale corresponds to the respective spin-averaged energy . Adapted from ref. Shen2012 ().

Each spin state of the lower level having , can be excited to three spin states (forming a “spin group”) of the upper level, namely , , , by strong (allowed) transitions. To these we assign weights proportional to the degeneracies of the respective upper states, . In the weighted sum over all of these transitions in the traceless frequency, each state of the upper level occurs once and therefore the total contribution of the spin-averaged energies of the upper states yields . The total contribution of their spin energies however average to zero because of the tracelessness of the spin hamiltonian. The state is special: it has only a single strong transition, to the state (green line in the figure). Nevertheless, the above weight assignment is suitable.

As a consequence of this weight assignment, the total weight of each lower spin state in the sum is , since three transitions start from each state. This total weight naturally turns out to be the degeneracy of the lower state, up to a constant factor. Here again, is the number of states in the upper level. The special state is again taken into account correctly, even if only a single transition starts from it. The total contribution of the 4 lower states yields

(24)
(25)

Here, the number of states in the lower level is .

Explicitly, the traceless frequency is (with the arbitrary vibrational quantum numbers reintroduced)

(26)

The summation is limited to the 10 strong transitions (the colored lines in Fig. 2). The traceless frequency eliminates the contributions from 11 spin structure coefficients (9 of the upper level, 2 of the lower level), for any choice of vibrational levels . For given , the transition frequencies lie in a range of several 10 MHz, and thus only a single radiation source is sufficient for their measurement.

Transitions between levels whose rotational quantum numbers are both nonzero require a generalization of eq. (26). The situation is now richer in the sense that in the lower level there will typically be more than just one spin state for a given quantum number pair in the lower level (see the right-hand side of Fig. 2 for the case ). This situation can easily be treated by setting up a set of equations for the unknown weights , requiring that the sum of weights of the strong spin transitions connecting to any particular state or be equal to the normalized Zeeman degeneracy of that state, or ’, respectively. The set contains one equation for every state in the lower and in the upper level. The solution of the set of equations shows that for a transition, 18 spin transitions (5, 1, 5, 7 for the four spin groups, respectively) must be measured, and for the number increases to 20 spin transitions (5, 1, 5, 9 for the four spin groups, respectively). It is found that not all strong transitions necessarily must contribute to . The overall result is that in the traceless frequency, the influence of 18 coefficients of the Breit-Pauli interaction is cancelled.

v.3 The molecular ion

exhibits some important differences compared to because it is homonuclear. States are denoted by , where is the (exact) total nuclear spin quantum number, and is the (approximate or exact) total particle spin angular momentum quantum number.

Rovibrational levels with even are para levels with zero total nuclear spin . The total particle spin angular momentum is . The spin hamiltonian reduces to the spin-rotation interaction, . Rovibrational levels are therefore split into two if . The energies of the and levels are (if and , respectively.

In the case of odd the molecule is in an ortho () state and the total particle spin angular momentum is or . The number of spin levels is higher, 5 for , and 6 for . Fig. 3 shows the spin structure of the lowest rovibrational levels.

As a homonuclear molecule, cannot be interrogated by one-photon electric-dipole (E1) transitions. Therefore, compared to the - case, an adapted discussion is required. Accessible transitions are two-photon transitions and electric-quadrupole (E2) transitions, already discussed in detail Karr2008a (); Korobov2018a (). We consider here only E2 transitions, because they show greater potential than two-photon transitions. Fig. 3 shows E2 transitions relevant for the following discussion. The crucial issue are the selection rules for E2 transitions. The total molecular angular momentum can change by . In particular, the fact that are allowed transitions is important in the context of the present discussion. Such transitions also have an unsuppressed strength, if Korobov2018a (), therefore they are experimentally accessible. An additional selection rule is that is forbidden. Such transitions could hypothetically only occur in the case , see also Fig. 3. This case must be treated carefully. We now discuss the cases of transitions between para states and between ortho states separately.

v.3.1 Para-

The spin states of para levels () are simple, pure angular momentum states,

For transitions with , the traceless frequency is the weighted sum of two frequencies () corresponding to , - transitions, which are shown as orange arrows in Fig. 3:

(27)

Here, . The weights of and are and , respectively. The traceless frequency eliminates the effect of the two relevant spin structure coefficients, for the lower level and for the upper level.

If one wants to address levels with one must take into account that transitions with are forbidden. In this case, the traceless frequency is the weighted sum of two transitions with (or ), which start or end at a common single state. Now, the two transitions, denoted by have and , respectively. We consider the case , which is indicated as green arrows in Fig. 3. The spin energy of the - level is zero, therefore

(28)

The weights of and are 4/10 and 6/10, respectively. The traceless frequency eliminates the effect of the single spin coefficient present in the problem, for the upper level.

Figure 3: Schematic of the spin structure of the lowest rovibrational levels of . The number pairs in parentheses are . The thick lines represent the spin-averaged energies of the levels. Energy splittings are not to scale. The colored arrows denote transition frequencies to be measured. The weights are not indicated but are discussed in the text.

v.3.2 Ortho-

For transitions between ortho levels () we shall limit ourselves to the case , which we consider the most experimentally relevant at this time. Thus, the spin structure is the same in the initial and final levels. The number of states is . The traceless frequency is the weighted sum over all transitions with , ,

(29)

The sum includes 5 transitions if and 6 transitions if The traceless frequency eliminates the effect of 10 spin coefficients, 5 for the lower level, and 5 for the upper level.

As mentioned, the case is special because both the initial and the final rovibrational levels include two spin states having total angular momentum and the - transitions between these are forbidden. Thus, two frequencies in the first sum in eq. (29), and , cannot be experimentally accessed.This problem is solved by determining these via a combination of allowed transitions. Several such combinations are possible; one of them is:

(30)

These three transitions are the two brown arrows and the brown-magenta dashed arrow in the figure. Similarly, can be determined via the two blue arrows and the blue-magenta dashed arrow in the figure. The remaining three frequencies