# Accurate ab initio calculations of RaF electronic structure indicate the
need for more laser-spectroscopical measurements

Andrei Zaitsevskii<EMAIL_ADDRESS>NRC ”Kurchatov Institute” -
PNPI, Orlova Roscha, 1, 188300 Gatchina, Russia
Chemistry Dept., M. Lomonosov Moscow State University, Moscow 119991, Russia
Leonid V. Skripnikov  skripnikov_lv<EMAIL_ADDRESS>http://www.qchem.pnpi.spb.ru  NRC ”Kurchatov Institute” - PNPI, Orlova Roscha,
1, 188300 Gatchina, Russia
Saint Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg,
199034 Russia  Nikolai S. Mosyagin<EMAIL_ADDRESS>NRC ”Kurchatov
Institute” - PNPI, Orlova Roscha, 1, 188300 Gatchina, Russia  Timur Isaev
<EMAIL_ADDRESS><EMAIL_ADDRESS>NRC ”Kurchatov Institute” -
PNPI, Orlova Roscha, 1, 188300 Gatchina, Russia  Robert Berger
<EMAIL_ADDRESS>Fachbereich Chemie, Philipps-Universität Marburg,
Hans-Meerwein-Str 4, 35032 Marburg, Germany  Alexander A. Breier
<EMAIL_ADDRESS>Laboratory for Astrophysics, Institute of
Physics, University of Kassel, 34132 Kassel, Germany  Thomas F. Giesen
<EMAIL_ADDRESS>Laboratory for Astrophysics, Institute of Physics,
University of Kassel, 34132 Kassel, Germany

###### Abstract

Recently a breakthrough has been achieved in laser-spectroscopic studies of
short-lived radioactive compounds with the first measurements of the radium
monofluoride molecule (RaF) UV/vis spectra. We report results from high
accuracy _ab initio_ calculations of the RaF electronic structure for ground
and low-lying excited electronic states. Two different methods agree
excellently with experimental excitation energies from the electronic ground
state to the ${}^{2}\Pi_{1/2}$ and ${}^{2}\Pi_{3/2}$ states, but lead
consistently and unambiguously to deviations from experimental-based adiabatic
transition energy estimates for the ${}^{2}\Sigma_{1/2}$ excited electronic
state and show that more measurements are needed to clarify spectroscopic
assignment of the ${}^{2}\Delta$ states.

###### pacs:

31.50.-x, 33.20.Tp, 33.70.Ca, 37.10.Mn

##  I Introduction

It is recognised that molecules with heavy nuclei are versatile tools to study
fundamental symmetries of physical laws and the interactions and properties of
subatomic particles [1, 2, 3, 4]. In such molecules effects resulting from
both parity violation ($\cal P$-odd ) and time-reversal violation ($\cal
T$-odd ) can be considerably enhanced with respect to atomic systems [5, 6, 4,
7]. The molecules of radium monofluoride (RaF) containing different isotopes
of Ra nuclei are predicted to have very high sensitivity for effects that are
$\cal P$-odd or simultaneously $\cal P$-odd and $\cal T$-odd [8, 9, 10, 6, 11,
12, 13] and recently an experimental breakthrough has been achieved in laser-
spectroscopic studies of RaF [14, 15]. It was found that experimental values
for adiabatic transition energies $T_{\mathrm{e}}$ and harmonic vibrational
wavenumbers $\tilde{\omega}_{\mathrm{e}}$ for the ground and the first excited
electronic states are consistent within the claimed theoretical accuracy of
values reported in Refs. [8, 9, 11]. Recently extended calculations of
transition energies to low-lying electronic states of RaF have been performed
in [16], using a number of basis sets and methods available in the dirac
program package, with the authors of Ref. [16] claiming good agreement of
their theoretical values with the reported experimental data.

The accuracy for the predicted $T_{\mathrm{e}}$ in RaF was estimated
conservatively as 1200 cm-1 in paper [9], although accompanying calculations
of the same quality for BaF, that could be compared to available experimental
data, allowed to suggest that this accuracy can be at the level $\sim$500
cm-1. One can expect that a large part of the uncertainty of $T_{\rm e}$ in
all the mentioned calculations (including [16]) arises from the neglect of
correlations involving the $5d$ shell of Ra; the uncertainty can be larger
than that from the $4d$ shell for BaF because of a stronger secondary
relativistic destabilization of $5d$ levels. Taking into account the Ra+F--
like electronic structure of RaF at the equilibrium internuclear distance and
the localization of low-energy excitation on the Ra+, the uncertainty in
excitation energies can be roughly estimated from the corresponding
uncertainty for the Ra+ atomic ion, which exceeds 500 cm-1 for $7s-6d$ and 400
cm-1 for $7s-7p$ (see Supplementary materials for details). Nevertheless, for
some transitions a reasonable agreement with experimental data is still
achieved due to partial error compensation from the lack of correlations
involving the $5d$ shell and insufficient basis set flexibility (especially in
what concerns the description of angular correlations in the outer core region
essential for the stabilization of $6d$-like states).

Accuracy of ab initio electronic structure calculations of heavy-atom
compounds is constantly increasing [17, 18, 19, 20, 21, 22, 23], together with
growing power of available supercomputers. For small molecules, the
theoretical accuracy of calculated electronic transition energies approaches
the level of 100 cm-1 $hc$ and for certain cases even better [22, 24]. In Ref.
[22] a method to take into account contributions of quantum electrodynamics
(QED) effects to transition energies in molecules has been proposed and
implemented. This formulation of the model QED Hamiltonian is closely related
to the formulation of model QED operator in Ref. [25], that is now widely used
for atomic calculations [25, 26, 27].

Thus it is timely to perform precise calculations of the electronic structure
of RaF, providing transition energy estimates with an uncertainty well below
the values of vibrational quanta. To assess the reliability of the values of
obtained molecular parameters we use two different schemes to compute
spectroscopic parameters for the ground and excited electronic states. Special
care is taken to study systematically the uncertainties introduced by
different approximations.

##  II Calculation methods

###  II.1 Scheme 1: Fock-space relativistic coupled cluster calculations

The present Fock-space relativistic coupled cluster (FS RCC) excited-state
studies were performed using two basic electronic structure models. One series
of calculations employed accurate shape-consistent relativistic
pseudopotentials (RPP) derived from the valence solutions of atomic four-
component Dirac–Fock–Breit equations with Fermi nuclear charge distributions
[28, 29, 30]. The Ra pseudopotenial replaced the inner core shells
$1\\!-\\!4s$, $2\\!-\\!4p$, $3\\!-\\!4d$, $4f$; relativistic and finite-
nuclear-size effects for the fluorine atom were described by the ‘empty-core’
pseudopotential leaving all electrons for explicit treatment [31].
Alternatively, a Dirac–Coulomb–Gaunt Hamiltonian was used to solve the closed-
shell SCF problem and then converted to the two-component all-electron
Hamiltonian by means of the X2C technique within the molecular mean field
approximation (X2C MMF, [32]).

The employed FS RCC scheme of correlation treatment closely resembles that
used in our previous study on RaCl [33]. The Fermi vacuum was defined by the
ground-state determinant of the positive molecular ion, while the target
neutral states were considered as belonging to the one-particle $(0h1p)$
sector. The FS RCC active space normally comprised 9 Kramers pairs of lowest-
energy virtual spinors of RaF+ arising from the $7s$, $7p$ and $6d$ spinors of
Ra+. Electronic transition energies as functions of the internuclear
separation $R$ were evaluated within the singles-and-doubles approximation for
the cluster operator (FS RCCSD). In most cases, 7 electrons of F and 19
electrons of Ra+ (including the $5d$ shell) are correlated. The
$[10s\,9p\,9d\,7f\,4g\,3h\,2i]$ Ra basis set compatible with the RPP model is
taken from Ref. [33]; its essential feature consists in using ANO-type high-
angular-momentum $(g,\,h,\,i)$ functions optimized for Ra and Ra+ within the
scalar relativistic approximation. Its counterpart for all-electron
calculations was obtained by combining the same $[4g\,3h\,2i]$ ANOs and
diffuse $f$ functions with the primitive $(28s\,25p\,18d\,12f)$ Gaussian sets
from Ref. [34]. The fluorine basis used with both models of relativistic
Hamiltonians was the aug-cc-pvQZ one [35] adapted to the relativistic
treatment [36].

Excited-state potential energy curves were constructed by adding the FS RCCSD
electronic transition energies as functions of the internuclear separation $R$
to the accurate ground state potential which was computed by means of the
single-reference coupled cluster method with perturbative account for the
contribution from triple excitations (RCCSD(T) scheme) and counterpoise
corrections of the basis set superposition errors (cf. Ref. [37]). Calculated
potential energy curves are given in Fig. 1.

The contributions from the correlations involving the core $5s5p$ subshells of
Ra were estimated in a single-point ($R$=2.249 Å) FS RCCSD calculations
performed with an appropriately modified basis: the $[4g\,3h\,2i]$ ANO set of
Ra was replaced by $[5g\,4h\,3i]$ ANOs optimized to describe the correlations
of all explicitly treated electrons of Ra. One can also expect certain
contributions arising from higher-rank terms in the cluster operator
expansion. Unfortunately, full nonperturbative treatment even of connected
triple excitations (FS RCCSDT) remains unfeasible whereas the Fock space
analogues of efficient single-reference schemes with perturbative triples
(like the famous CCSD(T) one) are not reliable [17]. To estimate the effect of
connected triples on the computed excitation energies, we computed FS RCCSD
and FS RCCSDT effective Hamiltonians ($H^{\rm eff}_{\rm SD,\,r}$ and $H^{\rm
eff}_{\rm SDT,\,r}$ respectively) correlating only 17 electrons of RaF and
rejecting the cluster operator components involving one-electron levels above
a certain threshold (up to 2.2 $E_{\mathrm{h}}$). Approximate FS CCSDT
energies are then obtained as eigenvalues of the operator

$H^{\rm eff}_{\rm SDT}\approx H^{\rm eff}_{\rm SD,\,full}+H^{\rm eff}_{\rm
SDT,\,r}-H^{\rm eff}_{\rm SD,\,r}.$

where $H^{\rm eff}_{\rm SD,\,full}$ is the FS RCCSD effective Hamiltonian
calculated with no restriction imposed on single and double excitations in the
cluster operator. The $T_{\rm e}$ values incorporating the resulting
corrections for triple excitations and $5s5p$ correlations are marked as
“final” in Table 1.

The composition of relativistic states in terms of their scalar relativistic
counterparts was determined by the projection technique described in Ref.
[38]. The requisite scalar relativistic states were obtained within the same
computational scheme by switching off the spin-orbit parts of the
pseudopotentials. Since the projection analysis was restricted to the model-
space parts of the wavefunctions, it was considered reasonable to extend the
model space for this task, augmenting the number of active spinor pairs to 34.
To suppress the effect of intruder states normally encountered for large model
spaces, the technique of simulated imaginary shifts of energy denominators
[39] was employed.

The construction of one-electron spinors and molecular integral evaluation, as
well as the single-reference RCCSD(T) ground state calculations, were
performed with the DIRAC 19 code [40, 41] whereas the EXP-T program [42, 17]
was used for FS RCC calculations. Vibrational energy levels were evaluated
with the help of the program VIBROT [43].

###  II.2 Scheme 2: single-reference calculations

Another series of calculations employed the single reference approach as a
base. For this, we have followed the scheme which was developed in Ref. [22]
(see also [24]) and applied to calculation of excitation energies for low-
lying electronic states of Ra+ and transition energy of the first excited
state of RaF. This scheme included the following steps: The main correlation
calculation has been performed within the CCSD(T) method using the Dirac-
Coulomb Hamiltonian. All electrons were included in the correlation treatment
and the virtual energy cutoff has been set to 10000 $E_{\mathrm{h}}$. Such
cutoff ensures that correlation contributions of the inner-core electrons are
described correctly [44, 45] which is important for the case of all-electron
calculation. The basis set for Ra optimized in Ref. [22] was used. It
corresponds to the modified uncontracted Dyall’s AEQZ [46] basis set augmented
by diffuse functions of $s$-, $p$-, $d$-, and $f$-types. Functions of $g$-,
$h$-, and $i$\- types were partly replaced by uncontracted natural-like
functions constructed using the procedure and code developed in Refs. [47, 48,
49]. In total, the basis set for Ra included
$[42s\,38p\,27d\,17f\,11g\,3h\,2i]$ functions. The uncontracted AETZ [46]
basis set has been used for F. Contribution of the Gaunt interelectron
interaction to transition energies of RaF has been treated at the FS-CCSD
level within the X2C MMF approach [32]. To consider the basis set extension
contribution, the basis set on F has been increased up to the uncontracted
AAEQZ [46] one and basis set on Ra up to $[42s\,38p\,27d\,27f\,13g\,9h\,6i]$
[22]. This basis set extension contribution has been calculated within the FS-
CCSD method with excluded $1s\ldots 3d$ electrons of Ra. To take into account
more functions with L$\leq$6 we have performed calculations within the
37-electron EOM-EA-CCSD [50] approach using the scalar-relativistic variant of
the RPP operator [51, 28, 29, 30]. Such approximation has been tested in Ref.
[22] (see also [23]). In particular, such an approach allowed us to take into
account the contribution of $[15g\,15h\,15i]$ functions which was practically
impossible within the Dirac-Coulomb calculations. Following Ref. [22] we have
also added extrapolated correction on harmonics with L$>$6\. The contribution
of iterative triple and perturbative quadruple cluster amplitudes has been
obtained within the CCSDT(Q) method [52] using the two-component RPP
Hamiltonian [51, 28, 29, 30]. The basis set consisting of natural compact
contracted $(20s\,20p\,15d\,10f)/[6s\,6p\,7d\,4f]$ functions [48, 23, 47] has
been used for Ra, while the aug-cc-pvDZ-DK basis set [53, 36] has been
employed for F. In the correlation calculation on the CCSDT(Q) level, 27 outer
electrons of RaF have been included and the virtual energy cutoff has been set
to 5 $E_{\mathrm{h}}$. Finally, we have calculated the contribution of the
vacuum-polarization and self-energy quantum-electrodynamics effects to the
electronic energies of molecular terms. For vacuum polarization operator we
have used the model Uehling potential approximate formula from Ref. [54]. The
self-energy contribution has been calculated within the model QED Hamiltonian
using the expression suggested and developed in Ref. [22]. This formulation is
close to the expression developed and applied for atomic calculations in Ref.
[25].

Four-component calculations have been performed within the dirac code [40,
41]. High order correlation effects were calculated using the mrcc [55, 56]
code. Scalar relativistic correlation calculations to ensure the basis set
completeness and to generate compact basis sets were performed using the cfour
[57] code. The code developed in Ref. [22] has been employed to calculate the
QED contribution to molecular and atomic transition energies.

##  III Results and discussion

In presented calculations of the electronic structure of RaF we analyzed the
main sources of possible theoretical uncertainties. Within the scheme 1,
systematic errors are primarily due to the basis set incompleteness with
certain contributions from incomplete account for triples and the neglect of
higher cluster amplitudes in the cluster operator as well as the neglect of
QED effects. Since the electronic structure of RaF in all states under study
roughly correspond to ionic configurations, Ra+F-, it seems reasonable to
derive the corrections from the comparison of computed excitation energies at
the Ra+ \+ F- dissociation limit with the experimental data on Ra+. The _ab
initio_ energies, corresponding to $7s-6d$ and $7s-7p$ excitations of the free
Ra+, are systematically overestimated (by 125 cm-1 to 235 cm-1).
Unfortunately, as follows from rather large transition moments to the ground
state arising mainly from the $7s$ state of Ra+ (Table 1), low-lying molecular
states which can be formally associated with the $6d$ states of Ra+ receive a
significant $7p$ contribution, so that the common practice of shifting each
potential curve to fit exactly the corresponding experimental dissociation
limit seems not well founded. However, one can hope to improve $T_{\rm e}$
estimates by shifting uniformly all excited-state energies to minimize the
overall error for all $6d$ and $7p$ limits. The corresponding $T_{\rm e}$
values are referred to as “shifted” in Table 1.

As in Ref. [22] the main uncertainty of scheme 2 is the remaining basis set
incompleteness, neglect of the retardation part of the Breit interaction and
interference between different contributions such as high order correlation
effects and contribution of high harmonics in the basis set. The theoretical
uncertainty of the prediction within scheme 2 is expected to be about 5 meV
(40 cm-1) as in Ref. [22]. The resulting energies of scheme 2 and the
“shifted” $T_{\rm e}$ energies obtained within scheme 1 are in perfect
agreement. Note, that we did not apply any empirical energy shifts to
molecular transition energies within scheme 2.

The energies of the excited electronic states of the BaF molecule
corresponding to states considered here for RaF were studied in Ref. [24].
Application of the calculation approach similar to scheme 2 [22] to BaF
resulted in the agreement of all considered theoretical transition energies
with the experimental ones within $\approx$20 cm-1 [24].

It can be seen from Tables 1 and 2 that the first excited electronic state can
be reliably identified as ${}^{2}\Pi_{1/2}$ and the theoretical and
experimental values for $T_{\mathrm{e}}$ are in excellent agreement. The next
state (${}^{2}\Delta_{3/2}$) has to be in the region of 14000 cm-1, a
wavenumber region that was not investigated in the experiment [14] though. The
state at $\sim$15100 cm-1 assigned tentatively in [14] as
${}^{2}\Delta_{3/2}$, is rather of ${}^{2}\Delta_{5/2}$ type and transitions
to this state from the ground electronic state are possible due to rovibronic
coupling effects (estimates in [58] suggested $\sim$0.05% transition intensity
of $X\rightarrow\>^{2}\Delta_{5/2}$ transition in units of
$X\rightarrow\>^{2}\Delta_{3/2}$ transition intensity). For the next
${}^{2}\Pi_{3/2}$ state we again see excellent agreement with the experiment,
but for the next ${}^{2}\Sigma_{1/2}$ state theory and experiment disagree on
the level of 450 cm-1. On purely energetical grounds, a possible explanation
might be an incorrect vibrational quanta assignment of the electronic
transitions into the excited ${}^{2}\Sigma_{1/2}$ state starting instead from
a vibrational hot level of the electronic ground state into the vibrational
ground state of the excited state. Re-evaluation of the four observed
transitions into the excited ${}^{2}\Sigma_{1/2}$ state [14], assuming them,
now, as hot band transitions
($X(\nu^{\prime\prime})\rightarrow\>^{2}\Sigma_{1/2}(\nu^{\prime}):1\rightarrow
0,2\rightarrow 1,3\rightarrow 2,\text{ and }4\rightarrow 3$), leads to an
experimental $T_{\rm e}$ value of 16620.8(2) cm${}^{\text{-1}}$, agreeing
excellently with the theoretical value. The observed vibronic profile for this
transition would, however, be at variance with the Franck–Condon profile
expected by virtue of the small change in equilibrium distance upon electronic
excitation. New experimental RaF-studies in the spectral region around 16620
cm${}^{\text{-1}}$ might, thus, provide valuable information about the
fundamental transition from $X\rightarrow\>^{2}\Sigma_{1/2}$, confirming the
assignment of vibrational quanta.

The main difference of the high-accuracy quantum chemical calculations
reported in the present work as compared to our earlier theoretical
predictions reported in Ref. [9], however, pertains to the systematic
energetical lowering of the ${}^{2}\Delta$ manifold of states by about 650
cm-1, suggesting the low-lying ${}^{2}\Delta_{3/2}$ state to be as of yet
experimentally unidentified. This level remains energetically well above the
lowest excited ${}^{2}\Pi_{1/2}$ state as predicted earlier when studying the
prospects for laser-coolability of RaF [8, 9].

Figure 1:  Potential energy functions for low-lying electronic states of RaF.
Excited state functions are derived from FS RCC excitation energies computed
within the pseudopotential model (curves and empty circles) and all-electron
X2C MMF approximation (crosses). Table 1:  FS RCC molecular constants for the
low-lying electronic states of RaF. The meaning of abbreviations are as
follow: RPP - calculations within the pseudopotential model; AE - all-electron
X2C MMF calculations; $\Delta T({\rm T})$ \- estimated contribution of triple
cluster amplitudes and $\Delta T(5s5p)$ \- contributions from correlations of
core-like $5s5p$ subshells to relative term energies. The equilibrium
distances $R_{e}$ are in Å, adiabatic transition energies $T_{\mathrm{e}}$ and
harmonic vibrational quanta $\omega_{\rm e}$ are in cm-1, FS RCC model-space
estimates of squared transition dipole moments to the ground electronic state
$d^{2}$ are in $e^{2}\,a_{0}^{2}$.

State | (2)1/2 | (1)3/2 | (1)5/2 | (2)3/2 | (3)1/2
---|---|---|---|---|---
| (${}^{2}\Pi_{1/2}$) | (${}^{2}\Delta_{3/2}$) | (${}^{2}\Delta_{5/2}$) | (${}^{2}\Pi_{3/2}$) | (${}^{2}\Sigma_{1/2}$)
Molecular parameters, scheme 1
$R_{e}$ (RPP) | 2.248 | 2.258 | 2.253 | 2.243 | 2.263
(AE) | 2.247 | 2.258 | 2.253 | 2.243 | 2.262
$\omega_{\rm e}$ (RPP) | 436.8 | 430.3 | 433.8 | 436.3 | 431.9
(AE) | 436.9 | 430.8 | 433.9 | 436.4 | 432.1
(Exptl)[14] | 435.5 |  |  |  |
Transition energies, scheme 1:
$T_{\rm e}$ (RPP) | 13412 | 14509 | 15327 | 15434 | 16754
(AE, 27$e$) | 13396 | 14522 | 15345 | 15435 | 16751
$\Delta T({\rm T})$ | +49 | $-86$ | $-74$ | +58 | +16
$\Delta T(5s5p)$ | +23 | +62 | +62 | +38 | +30
$T_{\rm e}$(RPP, final) | 13484 | 14485 | 15315 | 15530 | 16800
(RPP, final, shifted by $-164~{}\mathrm{cm}^{-1}$) | 13320 | 14321 | 15151 | 15366 | 16636
$d^{2}$ (RPP) | 7.8 | 0.33 | 0 | 7.0 | 8.4
Transition energies, scheme 2:
CCSD(T), DC, 97e | 13381 | 14603 | 15402 | 15463 | 16746
High harmonics, CBS (L) | 2 | -98 | -98 | 1 | -15
CCSDT(Q)-CCSD(T), 2c-RPP, 27e | -28 | -15 | -5 | -40 | -26
Gaunt | 5 | -65 | -78 | -18 | -11
QED | -56 | -73 | -70 | -57 | -50
$T_{e}$, final, scheme2 |  13303 [22] | 14352 | 15151 | 15348 | 16644
$T_{\rm e}$(Exptl) [14] | 13288 | 15148(?) |  | 15355 | 16181
|  |  |  |  | 16620.8(2)111When re-assigned, for details see text.
Data from [9]
$T_{e}$, AE(DC), 17$e$, FS-CCSD/$10^{4}$ | 1.33 | 1.50 | 1.58 | 1.54 | 1.67
Data from [16]
$T_{e}$, RPP, 19$e$, FS-CCSD | 13298 | 14978 | 15740 | 15332 | 16614

Ground state: $R_{e}$=2.244 Å, $\omega_{e}$=440.6 cm-1 (exptl 441.8(1) cm-1)

Table 2: Composition of full relativistic states of RaF ($R$(Ra–F)=4.25 $a_{0}$) in terms of scalar relativistic states. $X(1)1/2$ | 100% $(1)^{2}\Sigma^{+}$
---|---
$(2)1/2$ | 86% $(1)^{2}\Pi$, 13% $(2)^{2}\Sigma^{+}$
$(1)3/2$ | 96% $(1)^{2}\Delta$, 4% $(1)^{2}\Pi$
$(1)5/2$ | 100% $(1)^{2}\Delta$
$(2)3/2$ | 96% $(1)^{2}\Pi$, 3% $(1)^{2}\Delta$
$(3)1/2$ | 87% $(2)^{2}\Sigma^{+}$, 13% $(1)^{2}\Pi$

##  IV Conclusion

We have calculated molecular parameters and transition frequencies between the
ground and five low-lying excited electronic states of the RaF molecule on a
new level of accuracy. We used two different high-accuracy calculation schemes
and achieved good agreement between these two theoretical studies, but not
with experiment for the adiabatic transition energy from the electronic ground
to the excited ${}^{2}\Sigma_{1/2}$ state, whereas excellent agreement between
theory and experiment is found for transitions to the states of approximate
${}^{2}\Pi_{1/2}$ and ${}^{2}\Pi_{3/2}$ character. Our results also indicate
that more spectroscopic measurements are needed to clarify the spectroscopic
assignment of the ${}^{2}\Delta_{3/2}$ state.

##  V Acknowledgement

The authors are grateful to A.N. Petrov for early estimates of non-adiabatic
effects for ${}^{2}\Delta_{5/2}$ state.

This work has been carried out using computing resources of the federal
collective usage centre Complex for Simulation and Data Processing for Mega-
science Facilities at NRC “Kurchatov Institute”, http://ckp.nrcki.ru/, and
computers of Quantum Chemistry Lab at NRC “Kurchatov Institute” - PNPI.

L.V.S. acknowledges support by the Russian Science Foundation under Grant No.
19-72-10019 for the electronic structure calculations within the scheme 2 and
RFBR according to research project No. 20-32-70177 for the calculations of QED
and Gaunt contributions. The work on the RPP generation for the light elements
was supported by the personal scientific fellowship for N.S.M. from the
governor of Leningrad district. T.I. and A.Z. are grateful for financial
support of FS RCC studies to RSF-DFG grant N 21-42-04411.

## References

  * DeMille et al. [2008] D. DeMille, S. B. Cahn, D. Murphree, D. A. Rahmlow, and M. G. Kozlov, Phys. Rev. Lett. 100, 023003 (2008).
  * Andreev et al. [2018] V. Andreev, D. G. Ang, D. DeMille, J. M. Doyle, G. Gabrielse, J. Haefner, N. R. Hutzler, Z. Lasner, C. Meisenhelder, B. R. O’Leary, et al., Nature 562, 355 (2018).
  * Cairncross et al. [2017] W. B. Cairncross, D. N. Gresh, M. Grau, K. C. Cossel, T. S. Roussy, Y. Ni, Y. Zhou, J. Ye, and E. A. Cornell, Physical Review Letters 119, 153001 (2017).
  * Safronova et al. [2018] M. S. Safronova, D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, Rev. Mod. Phys. 90, 025008 (2018).
  * Flambaum [2019] V. V. Flambaum, Physical Review C 99, 035501 (2019).
  * Isaev and Berger [2014] T. Isaev and R. Berger, J. Molec. Spectrosc. 300, 26 (2014), ISSN 0022-2852.
  * Kozlov and Labzowsky [1995] M. G. Kozlov and L. N. Labzowsky, J. Phys. B 28, 1933 (1995).
  * Isaev et al. [2010] T. A. Isaev, S. Hoekstra, and R. Berger, Phys. Rev. A 82, 052521 (2010).
  * Isaev and Berger [2013] T. A. Isaev and R. Berger, ArXiv e-prints physics.chem-ph:1302.5682 (2013), eprint 1302.5682.
  * Borschevsky et al. [2013] A. Borschevsky, M. Iliaš, V. A. Dzuba, V. V. Flambaum, and P. Schwerdtfeger, Phys. Rev. A 88, 022125 (2013).
  * Kudashov et al. [2014] A. D. Kudashov, A. N. Petrov, L. V. Skripnikov, N. S. Mosyagin, T. A. Isaev, R. Berger, and A. V. Titov, Phys. Rev. A 90, 052513 (2014).
  * Sasmal et al. [2016] S. Sasmal, H. Pathak, M. K. Nayak, N. Vaval, and S. Pal, Phys. Rev. A 93, 062506 (2016).
  * Petrov and Skripnikov [2020] A. N. Petrov and L. V. Skripnikov, Phys. Rev. A 102, 062801 (2020).
  * Garcia Ruiz et al. [2020] R. F. Garcia Ruiz, R. Berger, J. Billowes, C. L. Binnersley, M. L. Bissell, A. A. Breier, A. J. Brinson, K. Chrysalidis, T. E. Cocolios, B. S. Cooper, et al., Nature 581, 396 (2020), ISSN 1476-4687, URL https://doi.org/10.1038/s41586-020-2299-4.
  * Udrescu et al. [2021] S. Udrescu, A. Brinson, R. G. Ruiz, K. Gaul, R. Berger, J. Billowes, C. Binnersley, M. Bissell, A. Breier, K. Chrysalidis, et al., Physical Review Letters 127, 033001 (2021).
  * Osika and Shundalau [2021] Y. Osika and M. Shundalau, Spectrochim. Acta A 264, 120274 (2021).
  * Oleynichenko et al. [2020a] A. V. Oleynichenko, A. Zaitsevskii, L. V. Skripnikov, and E. Eliav, Symmetry 12, 1101 (2020a).
  * Knecht et al. [2018] S. Knecht, H. J. A. Jensen, and T. Saue, Nature Chemistry 11, 40 (2018).
  * Skripnikov [2020a] L. V. Skripnikov, The Journal of Chemical Physics 153, 114114 (2020a).
  * Zaitsevskii et al. [2020] A. Zaitsevskii, A. V. Oleynichenko, and E. Eliav, Symmetry 12, 1845 (2020).
  * Mosyagin et al. [2020] N. S. Mosyagin, A. V. Zaitsevskii, and A. V. Titov, International Journal of Quantum Chemistry 120, e26076 (2020), eprint https://onlinelibrary.wiley.com/doi/pdf/10.1002/qua.26076.
  * Skripnikov [2021] L. V. Skripnikov, J. Chem. Phys. 154, 201101 (2021).
  * Skripnikov [2016] L. V. Skripnikov, J. Chem. Phys. 145, 214301 (2016).
  * Skripnikov et al. [2021] L. V. Skripnikov, D. V. Chubukov, and V. M. Shakhova, J. Chem. Phys. 155, 144103 (2021).
  * Shabaev et al. [2013] V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Phys. Rev. A 88, 012513 (2013).
  * Pašteka et al. [2017] L. F. Pašteka, E. Eliav, A. Borschevsky, U. Kaldor, and P. Schwerdtfeger, Phys. Rev. Lett. 118, 023002 (2017).
  * Kaygorodov et al. [2021] M. Y. Kaygorodov, L. V. Skripnikov, I. I. Tupitsyn, E. Eliav, Y. S. Kozhedub, A. V. Malyshev, A. V. Oleynichenko, V. M. Shabaev, A. V. Titov, and A. V. Zaitsevskii, Phys. Rev. A 104, 012819 (2021).
  * Mosyagin et al. [2010] N. S. Mosyagin, A. Zaitsevskii, and A. V. Titov, Int. Rev. At. Mol. Phys. 1, 63 (2010), URL https://www.auburn.edu/cosam/departments/physics/iramp/1_1/mosyagin_zaitsevskii_titov.pdf.
  * Mosyagin et al. [2016] N. S. Mosyagin, A. V. Zaitsevskii, L. V. Skripnikov, and A. V. Titov, Int. J. Quantum Chem. 116, 301 (2016).
  * [30] N. S. Mosyagin and A. V. Titov, _Generalized relativistic effective core potentials_ , http://www.qchem.pnpi.spb.ru/recp.
  * Mosyagin et al. [2021] N. Mosyagin, A. Oleynichenko, A. Zaitsevskii, A. Kudrin, E. Pazyuk, and A. Stolyarov, J. Quant. Spectrosc. Radiat. Transfer 263, 107532 (2021), ISSN 0022-4073.
  * Sikkema et al. [2009] J. Sikkema, L. Visscher, T. Saue, and M. Iliaš, J. Chem. Phys. 131, 124116 (2009), eprint https://doi.org/10.1063/1.3239505, URL https://doi.org/10.1063/1.3239505.
  * Isaev et al. [2021] T. A. Isaev, A. V. Zaitsevskii, A. Oleynichenko, E. Eliav, A. A. Breier, T. F. Giesen, R. F. Garcia Ruiz, and R. Berger, J. Quant. Spectrosc. Radiat. Transfer 269, 107649 (2021), ISSN 0022-4073, URL https://www.sciencedirect.com/science/article/pii/S0022407321001424.
  * Roos et al. [2005] B. O. Roos, R. Lindh, P.-Å. Malmqvist, V. Veryazov, and P.-O. Widmark, Chem. Phys. Lett. 409, 295 (2005).
  * Woon and Dunning Jr. [1993] D. E. Woon and T. H. Dunning Jr., J. Chem. Phys. 98, 1358 (1993).
  * de Jong et al. [2001] W. A. de Jong, R. J. Harrison, and D. A. Dixon, J. Chem. Phys. 114, 48 (2001).
  * Pazyuk et al. [2015] E. A. Pazyuk, A. V. Zaitsevskii, A. V. Stolyarov, M. Tamanis, and R. Ferber, Russ. Chem. Rev. 84, 1001 (2015).
  * Zaitsevskii et al. [2017] A. Zaitsevskii, N. S. Mosyagin, A. V. Stolyarov, and E. Eliav, Phys. Rev. A 96, 022516 (2017).
  * Oleynichenko et al. [2020b] A. V. Oleynichenko, L. V. Skripnikov, A. Zaitsevskii, E. Eliav, and V. M. Shabaev, Chem. Phys. Lett. 756, 137825 (2020b).
  * [40] DIRAC, a relativistic ab initio electronic structure program, Release DIRAC19 (2019), written by A. S. P. Gomes, T. Saue, L. Visscher, H. J. Aa. Jensen, and R. Bast, with contributions from I. A. Aucar, V. Bakken, K. G. Dyall, S. Dubillard, U. Ekström, E. Eliav, T. Enevoldsen, E. Faßhauer, T. Fleig, O. Fossgaard, L. Halbert, E. D. Hedegård, B. Heimlich–Paris, T. Helgaker, J. Henriksson, M. Iliaš, Ch. R. Jacob, S. Knecht, S. Komorovský, O. Kullie, J. K. Lærdahl, C. V. Larsen, Y. S. Lee, H. S. Nataraj, M. K. Nayak, P. Norman, G. Olejniczak, J. Olsen, J. M. H. Olsen, Y. C. Park, J. K. Pedersen, M. Pernpointner, R. di Remigio, K. Ruud, P. Sałek, B. Schimmelpfennig, B. Senjean, A. Shee, J. Sikkema, A. J. Thorvaldsen, J. Thyssen, J. van Stralen, M. L. Vidal, S. Villaume, O. Visser, T. Winther, and S. Yamamoto (available at http://dx.doi.org/10.5281/zenodo.3572669, see also http://www.diracprogram.org).
  * Saue et al. [2020] T. Saue, R. Bast, A. S. P. Gomes, H. J. A. Jensen, L. Visscher, I. A. Aucar, R. Di Remigio, K. G. Dyall, E. Eliav, E. Fasshauer, et al., J. Chem. Phys. 152, 204104 (2020).
  * Oleynichenko et al. [2020c] A. V. Oleynichenko, A. Zaitsevskii, and E. Eliav, in _Supercomputing_ , edited by V. Voevodin and S. Sobolev (Springer International Publishing, Cham, 2020c), vol. 1331 of _Commun. Comput. Inf. Sci._ , pp. 375–386, ISBN 978-3-030-64616-5.
  * [43] D. Sundholm, _VIBROT_ , http://www.chem.helsinki.fi/$\sim$sundholm/software/GPL/.
  * Skripnikov et al. [2017] L. V. Skripnikov, D. E. Maison, and N. S. Mosyagin, Phys. Rev. A 95, 022507 (2017).
  * Skripnikov and Titov [2015] L. V. Skripnikov and A. V. Titov, J. Chem. Phys. 142, 024301 (2015).
  * Dyall [2012] K. G. Dyall, Theor. Chem. Acc. 131, 1217 (2012).
  * Skripnikov [2020b] L. V. Skripnikov, J. Chem. Phys. 153, 114114 (2020b).
  * Skripnikov et al. [2013] L. V. Skripnikov, N. S. Mosyagin, and A. V. Titov, Chem. Phys. Lett. 555, 79 (2013).
  * Skripnikov and Titov [2016] L. V. Skripnikov and A. V. Titov, J. Chem. Phys. 145, 054115 (2016).
  * Nooijen and Bartlett [1995] M. Nooijen and R. J. Bartlett, J. Chem. Phys. 102, 3629 (1995), eprint https://doi.org/10.1063/1.468592.
  * Titov and Mosyagin [1999] A. V. Titov and N. S. Mosyagin, Int. J. Quantum Chem. 71, 359 (1999).
  * Kállay and Gauss [2005] M. Kállay and J. Gauss, J. Chem. Phys. 123, 214105 (pages 13) (2005).
  * Kendall et al. [1992] R. A. Kendall, T. H. Dunning, and R. J. Harrison, J. Chem. Phys. 96, 6796 (1992), eprint https://doi.org/10.1063/1.462569.
  * Ginges and Berengut [2016] J. S. M. Ginges and J. C. Berengut, J. Phys. B 49, 095001 (2016).
  * [55] M. Kállay, P. R. Nagy, D. Mester, Z. Rolik, G. Samu, J. Csontos, J. Csóka, P. B. Szabó, L. Gyevi-Nagy, B. Hégely, I. Ladjánszki, L. Szegedy, B. Ladóczki, K. Petrov, M. Farkas, P. D. Mezei, and á. Ganyecz: The mrcc program system: Accurate quantum chemistry from water to proteins, J. Chem. Phys. 152, 074107 (2020).” “mrcc, a quantum chemical program suite written by M. Kállay, P. R. Nagy, D. Mester, Z. Rolik, G. Samu, J. Csontos, J. Csóka, P. B. Szabó, L. Gyevi-Nagy, B. Hégely, I. Ladjánszki, L. Szegedy, B. Ladóczki, K. Petrov, M. Farkas, P. D. Mezei, and á. Ganyecz. See www.mrcc.hu.
  * Kállay and Surján [2001] M. Kállay and P. R. Surján, J. Chem. Phys. 115, 2945 (2001).
  * Stanton et al. [2011] J. F. Stanton, J. Gauss, M. E. Harding, P. G. Szalay, et al. (2011), cfour: a program package for performing high-level quantum chemical calculations on atoms and molecules, http://www.cfour.de .
  * Petrov [2019] A. Petrov (2019), private communication.