The manuscript has been accepted by The Journal of Chemical Physics A variational model for the hyperne resolved spectrum of VO in its ground electronic state

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The manuscript has been accepted by The Journal of Chemical Physics

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

electronic state

Qianwei Qu, Sergei N. Yurchenko and Jonathan Tennyson1, a)

Department of Physics and Astronomy, University College London,

London WC1E 6BT, United Kingdom

(Dated: 25 October 2022)

A variational model for the infra-red spectrum of VO is presented which aims to ac-

curately predict the hyperﬁne structure within the VO X 4Σ−electronic ground state.

To give the correct electron spin splitting of the X 4Σ−state, electron spin dipolar

interaction within the ground state and the spin-orbit coupling between X 4Σ−and

two excited states, A 4Π and 1 2Σ+, are calculated ab initio alongside hyperﬁne inter-

action terms. Four hyperﬁne coupling terms are explicitly considered: Fermi-contact

interaction, electron spin-nuclear spin dipolar interaction, nuclear spin-rotation inter-

action 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)j.tennyson@ucl.ac.uk

arXiv:2210.02896v2 [physics.chem-ph] 23 Oct 2022

I. INTRODUCTION

Vanadium monoxide (VO) is an open shell diatomic molecule which absorbs strongly in

the near infrared and visible region of the spectrum. These absorptions are of importance

for astrophysics where VO is known to be an important component of the atmosphere of

cool stars.1Recently attention has turned to the possible role of VO in the atmospheres

of exoplanets where it has been suggested that alongside TiO, VO absorption can change

the temperature proﬁle of the planet’s atmosphere.2Some tentative detections of VO in

exoplanet atmospheres have been reported3–8 but none of these can be regarded as secure.

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 studies9has led

to the conﬁrmation of TiO in exoplanetary atmospheres,10–12 the corresponding VO line

list13 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 project15 but a

major diﬀerence between them is due to the underlying atomic physics. While 16O and

40Ti both have nuclear spin, I, equal to zero, the dominant isotope of vanadium, 51V, 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 51V16O spectra to “blurred chaos at Doppler-

limited 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 51V16O has been

measured18–22 and modeled by eﬀective Hamiltonians.22,23 However, for the the X 4Σ−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. Thus,

eﬀective Hamiltonians are widely used for ﬁtting measured hyperﬁne-resolved energies or

transitions, see Refs.22 and 23 for examples involving VO. However, the VOMYT line list13

shows that interactions between the electronic states reshape the line positions and inten-

sities of VO. Although we focus on the X 4Σ−electronic ground state of VO in this paper,

the spin-orbit couplings between the low-lying X 4Σ−and 1 2Σ+states as well as the X 4Σ−

and A 4Π states are also included in our model with the aim of obtaining the correct spin

splittings for the X 4Σ−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 spectro-

scopic program Duo24 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.26–36 Among

them, the results for excited states represented by multi-reference conﬁguration interaction

(MRCI) wavefunctions are more accurate.33–36 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 4Σ−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 201537 at the MRCI level. The energies are also improved by adding a Davidson

correction (+Q).

First, the ground X 4Σ−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 C2v

group, viz. a1, b1, b2, a1, 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-pVnZn= 3,4,538,39 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 2.75 ˚

A, while

we found a discontinuity in the dipole moment around 1.9 ˚

A. We tried to add a second 4Σ−

state but failed to ﬁnd an avoided crossing structure in that region.

Oﬀ-diagonal spin-orbit interaction between the X 4Σ−and A 4Π states contributes to the

spin splitting of X 4Σ−. As A04Φ and A 4Π have the same irreducible representations in the

C2vgroup, it is impossible to omit the A04Φ in MRCI calculations. Therefore, we calculated

the A 4Π and A04Φ states together with the X 4Σ−states using the same active space but

only with the aug-cc-pVQZ basis set.

B. Interaction of doublet states with X4Σ−

Previous studies13,35 show that the spin splitting of the X 4Σ−state of VO is dominated

by the oﬀ-diagonal spin-orbit interaction between its X 4Σ−and 1 2Σ+states.

The 1 2Σ+state of VO, designated a 2Σ+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 2Γ state, which has degenerate A1and

A2representations. The 1 2∆ state also has the same irreducible representations and is

lower than 1 2Σ+. In principle, the three states 1 2Σ+, 1 2Γ and 1 2∆ should be optimized

simultaneously in the 2A1symmetry block. Our calculation therefore included these three

low-lying doublets states of VO together with its ground state. The two higher 2Π 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 2Σ+

and 1 2Γ states have the same electron conﬁguration as X 4Σ−while 1 2∆ has a diﬀerent

one.40 Thus, we initially calculated only the 1 2∆ and ground state, and then subsequently

added one 2Γ state and one 2Σ+state. Nonetheless, we could not obtain the correct 1 2∆

state in a state-average CASSCF calculation including 4Σ−,2Γ, 2∆ and 2Σ+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 McK-

emmish et al..13 Using the quantum chemistry program ORCA,41 we calculated the electron

spin-spin dipolar contribution to the zero-ﬁeld splitting Dtensor 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 4Σ−state. We calculated the nuclear hyperﬁne Atensor 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 aug-

cc-pwCVQZ basis set.

The nuclear spin-rotation coupling constants were calculated with another quantum

chemistry program, DALTON42 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 γand therefore used the constant empirical value determined for v= 0 instead

(See Table IV).

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ThemanuscripthasbeenacceptedbyTheJournalofChemicalPhysicsAvariationalmodelforthehyperneresolvedspectrumofVOinitsgroundelectronicstateQianweiQu,SergeiN.YurchenkoandJonathanTennyson1,a)DepartmentofPhysicsandAstronomy,UniversityCollegeLondon,LondonWC1E6BT,UnitedKingdom(Dated:25October2022)Avariational...

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The manuscript has been accepted by The Journal of Chemical Physics A variational model for the hyperne resolved spectrum of VO in its ground electronic state

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