A variational model for the hyperfine resolved spectrum of VO in its ground electronic state

A variational model for the infra-red spectrum of VO is presented which aims to accurately predict the hyperfine structure within the VO $\mathrm{X}\,^4\Sigma^-$ electronic ground state. To give the correct electron spin splitting of the $\mathrm{X}\,^4\Sigma^-$ state, electron spin dipolar interaction within the ground state and the spin-orbit coupling between $\mathrm{X}\,^4\Sigma^-$ and two excited states, $\mathrm{A}\,^4\Pi$ and $\mathrm{1}\,^2\Sigma^+$, are calculated ab initio alongside hyperfine interaction terms. Four hyperfine coupling terms are explicitly considered: Fermi-contact interaction, electron spin-nuclear spin dipolar interaction, nuclear spin-rotation interaction and nuclear electric quadrupole interaction. These terms are included as part of a full variational solution of the nuclear-motion Schr\"odinger equation performed using program DUO, which is used to generate both hyperfine-resolved energy levels and spectra. To improve the accuracy of the model, ab initio curves are subject to small shifts. The energy levels generated by this model show good agreement with the recently derived empirical term values. This and other comparisons validate both our model and the recently developed hyperfine modules in DUO.

A variational model for the infra-red spectrum of VO is presented which aims to accurately predict the hyperfine structure within the VO X 4 Σ − electronic ground state.
To give the correct electron spin splitting of the X 4 Σ − state, electron spin dipolar interaction within the ground state and the spin-orbit coupling between X 4 Σ − and two excited states, A 4 Π and 1 2 Σ + , are calculated ab initio alongside hyperfine interaction terms. Four hyperfine coupling terms are explicitly considered: Fermi-contact interaction, electron spin-nuclear spin dipolar interaction, nuclear spin-rotation interaction and nuclear electric quadrupole interaction. These terms are included as part There are two reasons for this. First, the spectra of VO and TiO are heavily overlapped making them very hard to disentangle at low resolution. Secondly, while the availability of a high-resolution TiO line list suitable for high-resolution spectroscopic studies 9 has led to the confirmation of TiO in exoplanetary atmospheres, 10-12 the corresponding VO line list 13 is not of sufficient accuracy to be used in similar studies. 14 Both the TiO and VO line lists cited were produced using similar methodology by the ExoMol project 15 but a major difference between them is due to the underlying atomic physics. While 16 O and 40 Ti both have nuclear spin, I, equal to zero, the dominant isotope of vanadium, 51 V, has I = 7/2. The interaction between the spin of unpaired electrons and the nuclear spin yields a very pronounced hyperfine structure which manifests itself at even moderate resolution.
This hyperfine structure reduces parts of the 51 V 16 O spectra to "blurred chaos at Dopplerlimited resolution" 16 . Progress in identifying VO in exoplanetary atmospheres using high resolution spectroscopy requires the development of a model which includes a treatment of these hyperfine effects. These effects were not considered in the ExoMol VOMYT line list. 13 A full survey of available high resolution spectroscopic data for VO has recently been completed by Bowesman et al. 17 as part of a MARVEL (measured active rotation vibration energy levels) study of the system. The nuclear hyperfine structure of 51 V 16 O has been measured [18][19][20][21][22] and modeled by effective Hamiltonians. 22,23 However, for the the X 4 Σ − ground electronic state, the experiments only gave the hyperfine constants for the lowest (v = 0) vibrational level and therefore provide limited information for the observations of hot VO spectra involving higher vibrational levels.
Hyperfine structure in molecular spectra are usually treated using perturbation-theory based effective Hamiltonians; these are usually accurate enough to reconstruct the energy levels using the assumption that hyperfine effects arise from small perturbations. shows that interactions between the electronic states reshape the line positions and intensities of VO. Although we focus on the X 4 Σ − electronic ground state of VO in this paper, the spin-orbit couplings between the low-lying X 4 Σ − and 1 2 Σ + states as well as the X 4 Σ − and A 4 Π states are also included in our model with the aim of obtaining the correct spin splittings for the X 4 Σ − state. This allows us to construct a full, predictive spectroscopic model of the ground state which can be used as input to the variational, diatomic spectroscopic program Duo 24 which we have recently extended to give a full variational treatment of hyperfine effects. 25 This paper presents the development of this model.

II. COMPUTATIONAL DETAILS
The electronic structure of VO has been investigated in numerous works. [26][27][28][29][30][31][32][33][34][35][36] Among them, the results for excited states represented by multi-reference configuration interaction (MRCI) wavefunctions are more accurate. [33][34][35][36] The most recent one by McKemmish et al. 35 laid the basis of the ExoMol VO linelist, VOMYT. 13 We also perform MRCI level calculations in this work to get the potential energy curves (PECs) and spin-orbit coupling curves for the electronic states of interest. The electron spin-dipolar interaction and hyperfine coupling curves of X 4 Σ − were obtained at the complete active space self consistent field (CASSCF) level.

A. Quartet states
In this work, the potential energy and spin-orbit coupling curves are calculated using MOLPRO 2015 37 at the MRCI level. The energies are also improved by adding a Davidson correction (+Q).
First, the ground X 4 Σ − state was calculated on its own to avoid effects from other electronic states. The active space used is larger than employed by McKemmish et al., 35 as the work of Miliordos et al. 33 shows that the occupation of 4p orbitals of vanadium is not negligible. In this work, the 1s orbital of oxygen and the 1s, 2s, 2p, 3s, 3p orbitals of vanadium were treated as doubly occupied. The active space includes the 2s, 2p orbitals of oxygen and 4s, 3d, 4p orbitals of vanadium. In the four irreducible representations of C 2v group, viz. a 1 , b 1 , b 2 , a 1 , the numbers of occupied orbitals are (12,5,5,1) while the default setup was used to specify the closed, core orbitals as (6, 2, 2, 0). We used the the internally contracted MRCI algorithm (icMRCI) implemented in MOLPRO. The basis set used in our calculation is aug-cc-pVnZ n = 3, 4, 5 38,39 so that we can estimate the potential energy curve at the complete basis set (CBS) limit by extrapolation.
According to Miliordos et al., 33 ionic avoided crossings are expected around 2.75Å, while we found a discontinuity in the dipole moment around 1.9Å. We tried to add a second 4 Σ − state but failed to find an avoided crossing structure in that region.
Off-diagonal spin-orbit interaction between the X 4 Σ − and A 4 Π states contributes to the spin splitting of X 4 Σ − . As A 4 Φ and A 4 Π have the same irreducible representations in the C 2v group, it is impossible to omit the A 4 Φ in MRCI calculations. Therefore, we calculated the A 4 Π and A 4 Φ states together with the X 4 Σ − states using the same active space but only with the aug-cc-pVQZ basis set.
B. Interaction of doublet states with X 4 Σ − Previous studies 13,35 show that the spin splitting of the X 4 Σ − state of VO is dominated by the off-diagonal spin-orbit interaction between its X 4 Σ − and 1 2 Σ + states.
The 1 2 Σ + state of VO, designated a 2 Σ + in the experimental work of Adam et al., 21 is easily obtained in a CASSCF calculation with MOLPRO when its LQUANT (i.e. the projection of orbit angular momentum on the internuclear axis) is assigned. However, a MOLPRO MRCI calculation may converge to the 1 2 Γ state, which has degenerate A 1 and A 2 representations. The 1 2 ∆ state also has the same irreducible representations and is lower than 1 2 Σ + . In principle, the three states 1 2 Σ + , 1 2 Γ and 1 2 ∆ should be optimized simultaneously in the 2 A 1 symmetry block. Our calculation therefore included these three low-lying doublets states of VO together with its ground state. The two higher 2 Π states were also included in the work of McKemmish et al. 35 but are not considered here.
We must provide a reasonable CASSCF reference for the MRCI calculations. The 1 2 Σ + and 1 2 Γ states have the same electron configuration as X 4 Σ − while 1 2 ∆ has a different one. 40 Thus, we initially calculated only the 1 2 ∆ and ground state, and then subsequently added one 2 Γ state and one 2 Σ + state. Nonetheless, we could not obtain the correct 1 2 ∆ state in a state-average CASSCF calculation including 4 Σ − , 2 Γ, 2 ∆ and 2 Σ + when the closed orbitals were set to (6, 2, 2, 0). To make the reference wavefunctions physically appropriate, we closed more orbitals, (8, 2, 2, 0), in CASSCF calculation, while we still used the closed (6, 2, 2, 0) space in the subsequent icMRCI calculation. Again we used an aug-cc-pVQZ basis set.

C. Electron spin dipolar coupling and nuclear hyperfine coupling curves
The electron spin-spin coupling was treated as an empirical fine tuning factor by McKemmish et al.. 13 Using the quantum chemistry program ORCA, 41 we calculated the electron spin-spin dipolar contribution to the zero-field splitting D tensor of the ground state at the CASSCF level with eleven electrons distributed in ten active orbitals.
Fully-resolved hyperfine splittings have been observed in the v = 0 vibrational levels of the X 4 Σ − state. We calculated the nuclear hyperfine A tensor and the nuclear electric quadrupole coupling constant in ORCA, 41 with the aim of predicting the hyperfine structure in vibrationally-excited levels of VO.
The zero field splitting tensor was calculated with an aug-cc-pVTZ basis set. The nuclear magnetic A-tensor and electric quadrupole coupling constant were calculated with an augcc-pwCVQZ basis set.
The nuclear spin-rotation coupling constants were calculated with another quantum chemistry program, DALTON 42 2020.0, at the CASSCF level with an aug-cc-pVQZ basis set. The active space is the same as used in ORCA.
We failed to find a quantum chemistry program which calculates the electron spin-rotation constant γ and therefore used the constant empirical value determined for v = 0 instead (See Table IV).

III. AB INITIO RESULTS
A. X 4 Σ − potential energy curve The dashed curves in Fig. 1 are the ab initio potentials of the X 4 Σ − state of VO. We estimated its potential energies at the CBS limit using the formula and obtained the solid potential energy curve shown in the left panel of Fig. 1.
The ab initio curves were calculated to build the spectroscopic model of VO. For numerical stability purposes, we fitted the discrete points with continuous curves. The extrapolated potential energy curve at the CBS limit was fitted to a second-order extended Morse oscillator (EMO) function: 24 where R and R e is the internuclear distance and its value at the equilibrium point and A e is the asympotic energy relative to the minimum of the ground electronics state. β EMO is expressed as where y(R) is given by: Only the points given as crosses in the righthand panel of Fig. 1 were included in the fit to give a better approximation of the lower vibrational levels. Although the calculated potential energies marked by circle were excluded, they are still well represented by the fitted curve.
The EMO parameters are listed in Table I.
The fitted PEC is not sensitive to the extrapolation formula in the region of interest (i.e. E ≤ 10 000 cm −1 ). Figure 2 compares the fitted EMO PECs of two extrapolation formulae: corresponding to E (n) are listed in Table I too.  Parameter has the same energy zero. The potentials of A 4 Π and 1 2 Σ + were fitted with second-order EMO functions whose parameters are listed in Table II.

C. Spin-orbit couplings
The calculated spin-orbit coupling curves are shown in the left panel of Fig. 4. Note that the spin-orbit coupling constant has a phase of i as MOLPRO uses a Cartesian representa- The polynomial coefficients a i are given in Table III.
where S = (S x , S y , S y ) is the spin vector operator and D is a dipolar interaction tensor. In principle axes, D is diagonal and As a dipolar interaction tensor, D is traceless and thus H ZFS only has two degrees of freedom.
In electron spin resonance spectroscopy, it is usual to define two constants, D and E, to describe zero-field splitting: The Hamiltonian can be rewritten as with the principle axis chosen such that For the X 4 Σ − state of VO, E = 0, and hence D xx = D yy .
The calculated zero-field splitting curve is shown in the right panel of Fig. 4. The two points marked by circles were excluded from the fit. The other points were fitted with a parabolic curve whose coefficients are given in Table IV.
We used the constant experimental value 22 for the spin-rotation coupling curve, as shown in the last column of Table IV.
The hyperfine coupling tensor can be divided into an isotropic term A iso and a dipolar term A dip : A iso is also known as the Fermi-contact interaction constant. The isotropic hyperfine coupling constant is given by The calculated curve A iso are shown in the left panel of Fig. 5. The points were fitted with a linear function, whose coefficients are given in Table V.
In the principle axis representation, the off-diagonal matrix elements of the dipolar interaction tensor A dip vanish. Since A dip is also traceless, we obtain Moreover, for the X 4 Σ − state. Thus, there is only one independent parameter for A dip . The calculated A dip zz term, which is plotted in the right panel of Fig. 5, was fitted with a parabolic curve whose coefficients are given in Table V.  The nuclear electric quadrupole coupling and nuclear spin-rotation coupling are relatively weak for the X 4 Σ − state as shown in Fig. 6. They were fitted by polynomials, see eq. 4, whose coefficients are listed in Table V.

A. Spectroscopic model
A spectroscopic model considering the X 4 Σ − , A 4 Π and 1 2 Σ + states of VO was developed for the diatomic variational nuclear motion program Duo. 24 The equilibrium bond length of the X 4 Σ − ab initio PEC was shifted by about 0.009 A so that R e = 1.5894809 A, (15) resulting in the correct rotational constant. The coupling constants used in Duo follow the definitions generally adopted in experimental studies. 25 Some constants have the same definition as those given by quantum chemistry programs. For example, the Fermi-contact coupling constant is just Definitions of others are different and we give the relevant interconversion formulae below.
In a Cartesian representation, the Hamiltonian of a diagonal electron spin-spin dipolar interaction of diatomic molecule is where S is the electron spin angular momentum and S z is its z component. Comparing H SS with H ZFS , we have In a Cartesian representation, the Hamiltonian of the nuclear spin-electron spin dipolar interaction is given by e (S − I z + S z I − ) exp(iφ) where c, d and e are three nuclear spin-electron spin dipolar interaction constants; I is the nuclear spin angular momentum; I z , I + and I − are the components of I; S z , S + and S − are the components of S; φ is the variable of spherical harmonics, see Eq. 4 of Slotterback et al. 44 . Comparing the Hamiltonian with the matrix elements of I T A dip S, we have: For the ground state, we have A dip xx = A dip yy . The only non-vanishing constant is There are four states (shown as red circles in the lefthand panel) whose calculation errors are greater than 0.1 cm −1 , so outside the range of the righthand panel of Fig. 8. The energy levels between 100 to 200 cm −1 have larger uncertainties than the others, as shown in the right panel. As discussed previously, 19,21,22 this behavior arises from the internal perturbations near N = 15, resulting in an avoided crossing structure as shown the lefthand panel of Fig. 9. The righthand panel of Fig. 9 illustrates the interactions of states in the F 2 series of X 4 Σ − . The interactions mix energy levels which makes it difficult to assign quantum number to these states. The globally J-dependent systematic error can be attributed to inaccurate spin-orbit, spin-spin and spin-rotation coupling curves. We plan to refine these curves in our future work.

D. Transition intensities and lifetime
The hyperfine resolved VO line list was used to generate spectra of the X 4 Σ − band using the program ExoCross. 46 The left panel of Fig. 10  several other curves were also shifted to reproduce the coupling constants given in Table 4    are not necessarily distributed around the non-hyperfine transitions, as the transitions near 9.8 cm −1 indicate. We emphasize again that in this paper the word 'non-hyperfine' is used as shorthand notation for the terms given without considering nuclear hyperfine interactions.
The word has a different meaning from 'hyperfine unresolved' which is used to describe blended hyperfine transitions.
As J is no longer a good quantum number for hyperfine structure, the J and J values here

V. CONCLUSION
In this work, we investigate the hyperfine-resolved infra-red spectra of VO X 4 Σ − electronic state. The fine and hyperfine coupling curves required to construct the spectroscopic model were calculated ab initio where possible but then scaled to reproduce the observed hyperfine structure. The hyperfine splitting of X 4 Σ − is mainly determined by the Fermicontact and electron spin-nuclear spin dipolar interactions. Nevertheless, we also included the nuclear spin-rotation and nuclear electric quadrupole coupling curves in our calculation.
The hyperfine resolved and unresolved cross sections show good consistency with each other when using wide line broadening parameters. The comparison between calculated and and empirical energy levels reveals the inaccuracy of our ab initio fine and hyperfine coupling curves even when computed using state-of-the-art methods and hence the need for empirical refinement. We plan to refine these curves and use them to generate a full, hyperfine-resolved line list for VO in future work.

SUPPLEMENTARY MATERIAL
The Duo input file used in this work is given as supplementary material; our potential energy curves are included as part of this input file. Two tables, which lists the sample states and transitions calculated from the input, are given as supplementary materials.