# Variational analysis of HF dimer tunneling rotational spectra using an ab
initio potential energy surface

Oleg L. Polyansky111To whom correspondence should be addressed; email:
<EMAIL_ADDRESS>Roman I. Ovsyannikov  Jonathan Tennyson  Sergei P. Belov
Mikhail Yu. Tretyakov  Vladimir Yu. Makhnev  Nikolai F. Zobov  Department of
Physics and Astronomy, University College London, Gower Street, London WC1E
6BT, United Kingdom  Institute of Applied Physics, Russian Academy of
Sciences, 46 Ulyanov Street, Nizhny Novgorod, 603950, Russia

###### Abstract

A very accurate, (HF)2 potential energy surface (PES) due to Huang et al.(J.
Chem. Phys., 150, 154302 (2019)) is used to calculate the energy levels of the
HF dimer by solving the nuclear-motion Schrödinger equation using variational
program WAVR4. Calculations on an extended range of rotational states show
very good agreement with experimental data. In particular the known empirical
rotational constants for the ground and some observed excited vibrational
states are reproduced with an accuracy of about 50 MHz. This level of accuracy
is shown to extend to higher excited inter-molecular vibrational states $v$
and higher excited rotational quantum numbers $(J,K_{a})$. These calculations
allow the assignment of 3 new $J$-branches in an HF dimer tunneling-rotation
spectra recorded 30 years ago. These branches belong to excited $K_{a}$ = 4
state of the ground vibrational state, and $K_{a}$ = 0 states of excited
inter-molecular vibrational states.

Keywords: HF dimer; tunneling-rotation-vibration spectra; subterahertz
experimental data; variational calculations.

##  1 Introduction

The study of weakly bound molecular complexes with hydrogen bonds and Van der
Waals bonding accounts for a large sub-branch of molecular spectroscopy [1].
High resolution spectroscopy can provide very accurate information on these
complicated quantum systems but often requires theoretical support to provide
interpretation and understanding. High precision study of hydrogen bonded
species also extends beyond molecular physics to the study of biological
systems, while still relying on a quantum mechanical interpretation.

Quantum chemical calculations of the HF dimer potential energy surface (PES)
started in 1974 [2]. Already the 1991 review by Quack and Suhm [3] cites 35
papers reporting ab initio PESs. Further work on the ab initio (HF)2 PESs is
presented, for example, by Klopper, Quack and Suhm [4, 5]. The PES used here
is due to Huang et al.[6] and is based on about 100 000 points calculated
using a very high level of theory. The accuracy of this PES is demonstrated
below in Section 2.

Most molecular complexes can only be observed experimentally in cold jets or
molecular beams. In the gas phase at room temperature these complexes
dissociate into monomers or condense into polymers on the gas cell walls. The
HF dimer is one of the rare molecular complexes possessing a large
dimerisation constant which therefore results in sufficient collisionally-
formed complexes at pressures appropriate for high resolution spectroscopy at
higher temperatures than achievable in molecular jets and beams. Other such
examples include HF-H2O [7] and HF-HCN [8]. A major reason for studying such
complexes is that thermal occupation allows one to observe spectra involving
highly excited vibrational and rotational states. That is the main reason why
HF dimer spectra are so well studied [9]. On the other hand these gas phase
observations highlight significant incompleteness in the theory of HF dimers.
As discussed below, observations made of HF dimer spectra in highly excited
vibrational states remain unassigned due to the limitations of theory.

This paper presents a significant step forward in the accurate representation
of highly excited energy levels of the HF dimer. We present first principles
variational calculations of rotationally-excited HF dimer states and compare
them with the high $J$ energies, derived from line frequencies obtained from
high resolution HF dimer spectra. Moreover, we use these calculations to
provide first assignments to lines in the tunneling-rotation-vibration spectra
of the HF dimer, recorded 30 years ago [10]. At the time of our recording this
sub-millimeter spectrum of HF dimer at equilibrium in the gas phase, the
theoretical tools for assigning high resolution HF dimer spectra were rather
limited. The microwave spectrum of the HF dimer was first observed by Dyke et
al. [11], and then Pine and Lafferty studied its mid infrared (IR) spectrum in
the region of the HF monomer fundamental stretch [12]. Subsequently, a
thorough study of the gas phase HF dimer millimeter region spectrum was
performed by Lafferty et al. [13]. The spectrum of the HF dimer behaves in
many respects like that of a linear molecule, as for certain vibrational
states and $K$ values the rotation-tunneling spectrum consists of the
characteristic quasi-equidistant branches with increasing values of the total
rotational quantum number $J$. This behavior can be described by a simple
formula (see Eq. (1) of Section II). So although the HF dimer is an asymmetric
top and, in principle, the asymmetric top quantum number $K_{a}$ should be
used, it is actually almost a symmetric top and one can use the designation
$K$ instead.

When in 1990 we recorded the spectrum of the HF dimer in what at the time was
the more exotic and difficult to work in sub-millimeter wave region around 10
cm-1 [10], the first obvious thing to do was to continue the branches observed
in the lower part of the millimeter region [13] to higher frequencies and, as
we dealt mostly with the R-branches, to higher $J$s of the same $K$ rotational
quantum numbers. We did that and assigned the $K=0,1$ and 2 R-branches of the
tunneling-rotation spectrum. As the HF dimer is a floppy, quasi-linear system,
the extrapolation to higher $K$’s, first of all to $K=3$, proved impossible
using the standard effective Watson Hamiltonian. Fortunately, by that time the
method of one-dimensional approximation of effective Hamiltonians in a Padé-
Borel form had already been developed by one of us [14] and it was applied to
the extrapolation to higher $K$’s of the ground and first excited bending
state of water monomer [15, 16]. Before that approximants in Padé form were
used by Pine et al. [17] to assist assignment of the $K=4$ branch in the
$\nu_{1}$ band of (HF)2. The use of approximants allowed us to extrapolate
[10] the $B$ constants for $K=0,1$ and 2 to the $B$ constant for $K=3$. Using
this prediction we identified the $K=3$ branch of the tunneling rotation
spectrum. However, since the Q-branch of this spectrum was in the millimeter
region, beyond the capability of the spectrometer we used to record the
spectrum, we had to ask our colleagues at the National Institute of Standards
and Technology (NIST) to observe the $K=3$ Q-branch we predicted, as well the
continuation of our R-branch series of lines. Our colleagues at NIST
successfully observed the Q-branch using our predictions. Thus, our
extrapolation from $K=0,1$ and 2 to $K=3$ was confirmed as being correct and
the assignment of the $K=3$ branch established [10].

Nevertheless, at that time we were unable to go further. We observed many
unassigned lines belonging to the HF dimer, but could not do anything with
them because of limitations in the effective Hamiltonian technique, even in
its more sophisticated Padé-Borel form. Although the Padé approximation
efficiently sums the divergent perturbation series of the effective
Hamiltonian and extrapolates well to higher rotational quantum numbers, it can
do nothing with the vibrational spectrum; the majority of unassigned lines in
our spectrum were assumed to be due to the vibration-tunneling-rotation
transitions in four low-lying inter-molecular vibrational states $\nu_{3}$,
$\nu_{4}$, $\nu_{5}$ and $\nu_{6}$. The values of the $J=0$ energy levels of
these bands are respectively about 475 cm-1 [5], 125 cm-1 [5], 160 cm-1 [5]
and 370 cm-1 [3]. These values are comparable with the ground state values of
116 cm-1 (for $K=J=2$) and 235 cm-1 (for $K=J=3$). It means that the Boltzmann
factor of the low $K$ ($K=0$ and 1) lines in these excited vibrational states
is comparable to the Boltzmann factor of the excited $K$ lines in the ground
vibrational state, so we should have been able to see them. This suggests that
many of the observed unassigned lines belonged to these excited vibrational
states and not, for example, to impurities.

Variational calculations on the HF dimer with $J=0$ have a long history. Most
ab initio PESs have been produced by Suhm, Quack and their co-workers, see
Refs. [3, 4, 5, 18, 19] and references therein. These were full dimensional 6D
calculations. There were also reduced dimensional (frozen monomer)
calculations by Bunker and Jensen [20] and a (4,2)D calculation by Quack, Suhm
and others [4]. So far all full 6D variational calculations only deal with
$J=0$ states; highly excited $J$s have been studied only using less accurate
Quantum Monte Carlo (QMC) calculations [21]. The $J=0$ calculations of Quack
and Suhm augmented by their QMC excited $K$ calculations helped them to assign
many spectra in the $\nu_{5}$ and $\nu_{6}$ bands in the IR region [3].
However, in the tunneling rotation sub-millimeter region, where all the HF
dimer lines belonging to various vibrational states are mixed together, $J=0$
only calculations are not enough.

In this paper we present variational calculations on (HF)2 for the excited $J$
states; these are described in detail in the next section. Section 3
summarizes the 1990 experimental sub-millimeter spectrum of HF dimer. Section
4 describes the application of our calculations to the analysis of this
spectrum. The first information on tunneling-rotational high resolution
spectra in the $v_{4}=1$ and $v_{3}=1$ states is given. We also assigned the
$K=4$ branch of rotation-tunneling spectrum in the ground vibrational state.
Section 5 presents our conclusions.

##  2 Variational calculations of HF dimer tunneling rotation-vibration energy
levels

Variational calculations of excited rovibrational tunneling energy levels of
the HF dimer are performed using the ab initio PES calculated by Huang et al.
[6]. This PES is both of a high theoretical level, using CCSD(T) with an aug-
cc-pvQZ basis set, and built on a very large number of calculated ab initio
points – 100 000, which makes the resulting surface very smooth. Use of this
gives highly accurate predictions of rovibrational-tunneling energy levels. In
particular, the discrepancies between calculated and empirical values (Tables
1 and 2 below) are at least as accurate as the best empirical ($i.e.$
spectroscopically determined or, in other words, adjusted to experimental
data) PESs. The HF dimer empirical PES of Klopper, Quack and Suhm [5]
reproduced the available experimental data on $K=0$ and excited $K$ energy
levels better than any of the then available ab initio surfaces. This work
used QMC calculations of excited rotational $J$ and $K$ levels to fit the
(HF)2 PES to experimental energy levels with $K>0$ [5]. However, QMC
calculations of nuclear motion energy levels lack accuracy; for example, this
work only computed rotational $B$ constants with 1 % accuracy. We are unaware
of any previous variational calculations of highly excited $J$ and $K$ states;
such calculations are necessary to obtain accurate energy levels beyond band
origins. These energies are needed for many purposes, in particular to provide
the more numerous theoretical counterparts to experimental measurements for
accurate characterisation of the PES and also to assist assignment of the
spectra.

For the variational solution of the nuclear motion problem we use the WAVR4
[22] program suite, developed especially for the calculation of energy levels
and wavefunctions of tetratomic molecules with large amplitude internal
motions. WAVR4 uses an exact, body-fixed kinetic energy operator [23, 24]. The
program offers a choice of orthogonal (or polyspherical) coordinates. For the
present work we used diatom – diatom Jacobi coordinates and discrete variable
representation (DVRs) based on Morse-oscillator-like functions to represent
the three radial coordinates. Rotationally excited states are treated using a
two-step procedure which involves solving an initial set of effective
vibrational problems which excludes Coriolis coupling and then using the
results as a basis to solve the full problem [25]. The original implementation
of this procedure in WAVR4 [26] proved to be rather inefficient for higher
rotational states [27] but has been improved so that for the present
calculations the computer time taken only depends linearly on $J$. This new
version of WAVR4, which includes the capability of computing dipole transition
intensities, is currently being prepared for publication [28].

Calculations were performed for $J$ up to 9. This allows us to calculate both
$B$, the rotational constant, and $D$, the distortion constant, for
corresponding branches of the HF dimer spectrum as well as many theoretical
line frequencies directly (up to $J=9$) without resorting to simplified
treatments of rotational excitation, using formulae of the type

$E=BJ(J+1)-DJ^{2}(J+1)^{2}.$ (1)

However, energy levels with $J>9$, if needed, could be obtained using this
formula.

Before describing our excited $J$ calculations, we consider the $J=0$
calculations. These calculations provide the first step in the high $J$
calculations and also illustrate the accuracy with which our adopted PES
reproduces the empirical-determined frequencies of all four inter-molecular
vibrations.

After the matching of the PES [6] with the program WAVR4, by transformation of
coordinate systems and the units used, calculation of the vibration-rotation
levels of the HF dimer consisted of the following steps:

  1. 1.

Choice of an integration region where the potential is well defined, namely,
for the HF monomer stretches $1.48\ a_{0}<r_{1},r_{2}<2.4\ a_{0}$ and for the
HF – HF stretch $3.5\ a_{0}<R<40\ a_{0}$ (where $r_{1}$ and $r_{2}$ are
internuclear distances of the two HF monomers, and $R$ is the distance between
centers of mass of the monomers). To select a mesh extending to $R>20$ a0, it
was necessary to modify WAVR4 by increasing the number of iterations in the
LAGUER algorithm which selects the location and weights of integration points.
The LAGUER algorithm in WAVR4 is an improved version of a LAGUER algorithm
from DVR3D [29]. We used the following Morse-like oscillator parameters which
are designed to give DVR grids over the appropriate integration range: for the
monomer stretches we used $r_{e}$ = 1.03 Å, $w_{e}$ = 7500 cm-1, $D_{e}$ =
43000 cm-1 and for the inter-molecular stretch $R_{e}$ = 7.7 Å, $w_{e}$ = 16
cm-1, $D_{e}$ = 5000 cm-1.

  2. 2.

To ensure convergence, a sufficiently large basis must be used to represent
the vibrational-rotational states of interest. Only 10 DVR points were needed
for the stiff monomer $r_{1}$ and $r_{2}$ coordinates, while the inter-
molecular stretch required 90 DVR points to obtain a good representation. For
bending modes, $l_{max}=16$ basis functions were used, and $k_{max}$ = 6 basis
functions for internal rotation, where $l_{max}$ determines sizes of H1F1F2
and H2F2F1 bending basis sets and $k_{max}$ gives the size of H1F1-H2F2
internal rotation basis set. The quality of convergence becomes apparent from
the shift in energy levels with increasing basis parameters. Thus, with an
increase in $l_{max}$ to 20, the energies up to 500 cm-1 changed by no more
than $10^{-4}$ cm-1. With an increase in $k_{max}$ to 10, the energies up to
500 cm-1 changed by no more than 10-3 cm-1. Increasing the DVR grid along
$r_{1}$ and $r_{2}$ to 16 points (with a change in the basis parameter
$w_{e}=12000$ cm-1 to maintain the same integration interval), the energies of
fundamental inter-molecular vibrations changed by no more than 1 cm-1. While
increasing the DVR grid along $R$ to 100 points (with a change in the basis
parameter $w_{e}=18$ cm-1 to maintain the same integration interval), the
energies of fundamental inter-molecular vibrations changed by no more than 0.5
cm-1. A final vibrational Hamiltonian matrix of dimension 7200 was used. This
size comes from truncation at 80 basis function for each of the 90 radial
points. Increasing the truncation from 80 up to 120 basis functions leads to a
test vibrational Hamiltonian matrix of dimension 10800, and shows the
following convergence of the main Hamiltonian: $6\times 10^{-5}$ cm-1 for
energies below 500 cm-1 and $2\times 10^{-4}$ cm-1 for energies below 1000
cm-1.

Table 1 presents a comparison of our $J=0$ calculations with values obtained
empirically. Drawings of all six HF dimer fundamental vibrational modes are
given, for example, in Fig. 2 of the paper by Quack et al. [3]. There are no
accurate high resolution data on the frequencies of these modes. However,
there have been several attempts to obtain relatively accurate indirect
estimates of their values. In particular, Nesbitt and co-authors [30, 31] used
high resolution combination band spectra to estimate the fundamental
frequencies. In Table I of their paper [31] $\nu_{4}$ and $\nu_{5}$
fundamental frequencies are given with an estimated accuracy of 0.1 cm-1. For
$\nu_{4}$ the frequency was given as 125 cm-1, whereas for $\nu_{5}$ it was
estimated as 160 cm-1. These two frequencies derived from extrapolation of
high resolution spectra of combination bands $\nu_{1}+\nu_{4}$,
$\nu_{2}+\nu_{4}$, $\nu_{1}+\nu_{5}$ and $\nu_{2}+\nu_{5}$. Less accurate
extrapolations have been performed for the higher energy inter-molecular modes
$\nu_{3}$ and $\nu_{6}$. Table II of paper [30] gives $\nu_{3}=475$ cm-1 and
$\nu_{6}=395$ cm-1 with an estimated extrapolation accuracy of 3 and 8 cm-1,
respectively. These extrapolations were made using accurate values of
$\nu_{1}+\nu_{3}$, $\nu_{2}+\nu_{3}$, $\nu_{1}+\nu_{6}$ and $\nu_{2}+\nu_{6}$.
Since there are more than one estimated frequency for both $\nu_{3}$ and
$\nu_{6}$ [5, 30], we present both values for comparison in Table 1.

Table 1: Calculated and empirical values of inter-molecular vibrations with
$(J,K)=0$ given in cm-1. Uncertainties in units of the final digit are given
in parenthesis.

| calc. | empirical
---|---|---
$\Delta_{K=0}$ | 0.679 | 0.6587(1)a
$\nu_{4}$ | 124.8 | 125.1(1)b
$\nu_{5}$ | 162.7 | 160.6(6)b
$\nu_{6}$ | 413.0 | 395(8)c
$\nu_{6}$ | 413.0 | 420(5)d
$\nu_{3}$ | 485.0 | 475(3)c
$\nu_{3}$ | 485.0 | 480(10)d

a Ref. [32]
b Ref. [31]
c Ref. [30]
d Ref. [5]

Table 2: Calculated and experimentally determined values of band origins with
$K>0$ in cm-1.

$(v_{3}v_{4}v_{5}v_{6})$ | calc. | exp.
---|---|---
(0000) $K=1$ | 36.4 | 35.4a
(0010) $K=1$ | 37.5 | 36.5a
(0000) $K=2$ | 118.0 | 116.1a
(0010) $K=2$ | 120.0 | 118.1a
(0000) $K=3$ | 235.4 | 232.6a
(0010) $K=3$ | 239.3 | 236.5a
(0010) $K=4$ | 390.5 | 386.7a
(0020) $K=3$ | 394.0 | 393.6b
(0001) $K=1$ | 399.4 | 399.8c
(0011) $K=1$ | 401.3 | 400.8c
(0001) $K=2$ | 467.2 | 465.3c
(0011) $K=2$ | 471.7 | 468.8c
(0101) $K=1$ | 507.5 | 508.1d
(0111) $K=1$ | 511.8 | 507.3d

a Ref. [32]
b Ref. [33]
c Ref. [34]
d Value obtained from energy of $\nu_{6}+\nu_{4}-\nu_{4}$, $K=1$ state [35]
and $\nu_{4}$

The quality of the HF dimer PES we use is illustrated by the discrepancies
between the inter-molecular vibrations energy levels and the results of our
calculations described above. All these discrepancies are close to the error
bars in the extrapolation of the high resolution values of the experimental
band origins. There is no way to improve further the PES based only on the
$J=0$ comparisons with experiment; therefore further developments need to use
calculations that consider excited $J$s.

Table 2 presents a comparison between variationally calculated and
experimentally-derived band origins of the $J$-branches of excited $K$ levels
of the ground and vibrationally excited inter-molecular modes. The values for
the ground state were taken from our 1990 study [10] and the excited $K$
values of the excited vibrational states are taken from various works by Quack
and Suhm. The agreement between calculated and experimentally derived values
shown in Table 2 is very good. It again confirms the high quality of the ab
initio PES used. Even more so, since the information in Table 2 is directly
derived from experiment, unlike the estimated values of Table 1, the much
better agreement between calculations and experiment in Table 2 emphasises
once more the importance of high $K$ and thus excited $J$ calculations. The
comparisons given in Table 2 could not be made in the original paper reporting
the ab initio PES [6] since that work only performed $J=0$ calculations.

To calculate the rotational energy levels of the (HF)2 dimer, we used
parameters $J_{\rm max}=9$ and $K_{\rm max}=9$. We had to increase from 10 to
500 the number of levels written out by the program for each symmetry at the
fourth stage, that is, when the Coriolis-decoupled $J>0$ energy levels are
calculated. Additionally, the angular vibrational basis set parameter with a
rotational momentum $l_{max}$ had to be increased from 16 to 20. The final ro-
vibrational calculations involved diagonalisation of Hamiltonian matrices of
dimension $40\times(J+1)$. This is sufficient to converge the energy levels of
interest to better than 0.1 cm-1. Our calculated (HF)2 vibrational-rotational
energy levels with $J\leq 9$ are given in Table A1 of the Supplementary
Materials.

Experimental information on the excited $J$ states of (HF)2 is usually
compressed by fitting values of rotational constants (such as $B,D$ and $H$,
see Eq. (1) and Eq. (2)) to the observed line positions. This compression is
possible because the HF dimer behaves like a nearly linear molecule and for a
certain vibrational and $K_{a}$ states, the lines of the HF dimer are grouped
in regular progressions, similar in structure to the rotational series
observed for diatomic or linear molecules. The observed rotational constants
are obtained from experimental line positions and calculated constants from
variational calculations using excited $J$ energy levels. We determined B and
D constants from the fit of calculated levels with $J$ from 1 to 9 using Eq.
(1).

Table LABEL:tab:obs.calc.const presents values for our computed $B$ and $D$
constants and, where possible, compares them with empirical values. We will
refer to these collectively as ’spectroscopic constants’ here and elsewhere.
For convenience and completeness Table LABEL:tab:obs.calc.const also presents
experimental values for $K=4$ of the ground state and some excited inter-
molecular vibrational states obtained in Section 4 of this paper; other
experimental values are taken from the literature, mostly from the papers by
Quack and Suhm.

Table 3: Calculated and experimental rotational constants (powers of ten are given in parenthesis). State | Units | $B_{calc}$ | $B_{exp}$ | $D_{calc}$ | $D_{exp}$ | Exp. source
---|---|---|---|---|---|---
(0000) |  |  |  |  |  |
$K=0$ | cm-1 | 0.2154 | 0.2167 | 2.26($-6$) | 2.06($-6$) | Ref. [32]
| MHz | 6457.5 | 6496.9 | 0.0633 | 0.0618 |
$K=1$ | cm-1 | 0.2158 | 0.2179 | 1.92($-6$) | 1.99($-6$) | Ref. [32]
| MHz | 6469.5 | 6531.3 | 0.0577 | 0.0596 |
$K=2$ | cm-1 | 0.2174 | 0.2186 | 1.89($-6$) | 1.94($-6$) | Ref. [32]
| MHz | 6517.4 | 6553.7 | 0.0567 | 0.0582 |
$K=3$ | cm-1 | 0.2169 | 0.2181 | 1.47($-6$) | 1.76($-6$) | Ref. [32]
| MHz | 6502.4 | 6538.2 | 0.0441 | 0.0529 |
$K=4$ | cm-1 | 0.2150 | 0.2175 | 6.14($-5$) | -6.01($-5$) | This work
| MHz | 6444.6 | 6498.6 | 1.84 | -1.8 |
(0010) |  |  |  |  |  |
$K=0$ | cm-1 | 0.2153 | 0.2166 | 2.41($-6$) | 2.04($-6$) | Ref. [32]
| MHz | 6451.5 | 6492.8 | 0.0625 | 0.0612 |
$K=1$ | cm-1 | 0.2177 | 0.2177 | 2.08($-6$) | 1.97($-6$) | Ref. [32]
| MHz | 6525.5 | 6526.8 | 0.0625 | 0.0591 |
$K=2$ | cm-1 | 0.2173 | 0.2185 | 1.86($-6$) | 1.95($-6$) | Ref. [32]
| MHz | 6514.3 | 6551.1 | 0.0558 | 0.0586 |
$K=3$ | cm-1 | 0.2176 | 0.2188 | 1.60($-6$) | 1.87($-6$) | Ref. [32]
| MHz | 6522.7 | 6558.9 | 0.0479 | 0.0561 |
$K=4$ | cm-1 | 0.2181 | 0.2194 | 1.74($-6$) | 2.28($-6$) | Ref. [32]
| MHz | 6539.1 | 6576.9 | 0.052 | 0.068 |
(0100) |  |  |  |  |  |
$K=0$ | cm-1 | 0.2119 | 0.2140 | 3.04($-6$) | 4.34($-6$) | This work
| MHz | 6353.0 | 6415.5 | 0.0910 | 0.130 |
(0110) |  |  |  |  |  |
$K=0$ | cm-1 | 0.2104 | 0.2161 | 2.71($-6$) | 5.07($-6$) | This work
| MHz | 6306.3 | 6478.5 | 0.0813 | 0.152 |
(1000) |  |  |  |  |  |
$K=0$ | cm-1 | 0.2052 | 0.2129 | 3.24($-6$) | 1.88($-4$) | This work
| MHz | 6350.2 | 6383.9 | 0.0971 | 5.63 |
(1010) |  |  |  |  |  |
$K=0$ | cm-1 | 0.2099 | 0.2002 | 2.97($-5$) | 1.52($-4$) | This work
| MHz | 5922.3 | 6000.4 | 0.890 | 4.57 |
(0001) |  |  |  |  |  |
$K=0$ | cm-1 | 0.2154 |  | 1.02($-5$) |  |
| MHz | 6460.1 |  | 0.306 |  |
$K=1$ | cm-1 | 0.2132 | 0.2105 | 1.13($-5$) | 2.5($-6$) | Ref. [34]
| MHz | 6393.7 | 6310.6 | 0.34 | 0.07 |
$K=2$ | cm-1 | 0.2127 | 0.2113 | 1.03($-5$) | 2.1($-6$) | Ref. [34]
| MHz | 6375.4 | 6334.6 | 0.31 | 0.06 |
(0011) |  |  |  |  |  |
$K=0$ | cm-1 | 0.2157 |  | 3.09($-6$) |  |
| MHz | 6466.9 |  | 0.0926 |  |
$K=1$ | cm-1 | 0.2132 | 0.2103 | 1.08($-5$) | 2.5($-6$) | Ref. [34]
| MHz | 6385.9 | 6304.6 | 0.324 | 0.07 |
$K=2$ | cm-1 | 0.2140 | 0.2129 | 0.99($-6$) | 2.05($-6$) | Ref. [34]
| MHz | 6415.9 | 6382.5 | 0.30 | 0.0615 |
(0101) |  |  |  |  |  |
$K=1$ | cm-1 | 0.2080 | 0.2133 | 7.67($-6$) | 3.74($-6$) | a
| MHz | 6236.5 | 6334.6 | 0.23 | 0.112 |
(0111) |  |  |  |  |  |
$K=1$ | cm-1 | 0.2072 | 0.2114 | 9.54($-6$) | 5.03($-6$) | a
| MHz | 6211.3 | 6337.6 | 0.286 | 0.151 |
(0020) |  |  |  |  |  |
$K=3$ | cm-1 | 0.2227 | 0.2217 | 4.5($-6$) | 1.67($-6$) | Ref. [33]
| MHz | 6679.4 | 6646.3 | 0.13 | 0.050 |
aConstants calculated from $\nu_{4}$ constants determined
in this work and values for $\Delta{X}$ ($X=B,D$) in Ref. [34]

Table LABEL:tab:obs.calc.const shows that the ground state $B$ constants taken
from the Belov et al. [10] and calculated variationally in the present paper
coincide to within about 30 – 50 MHz. This agreement between $B_{obs}$ and
$B_{calc}$ is unexpectedly good.

The impressive accuracy of prediction of $B$ and $D$ constants associated with
the tunneling rotation spectrum of the ground vibrational state of the HF
dimer and also of the $B$ values of some excited vibrational states taken from
the papers by Quack and Suhm, paves the way for us to apply the results of
these calculations to an analysis of unassigned lines in the HF dimer
tunneling-rotation spectra. These lines belong to excited inter-molecular
vibrations and higher $K$ states of the ground vibrational state; for
instance, the high resolution spectra of tunneling-rotational lines in
$v_{4}=1$ and $v_{3}=1$ vibrational states are assigned for the first time
here (Section 4). For other vibrational states, high resolution spectra have
been analysed by Quack and Suhm [35, 34, 33, 19].

Figure 1: Difference between the ab initio energies and ones obtained from the
fit for $(v_{3}v_{4}v_{5}v_{6})=(0~{}0~{}0~{}0)$ band of (HF)2, $K=0$ Figure
2: Difference between the ab initio energies and ones obtained from the fit
for $(v_{3}v_{4}v_{5}v_{6})=(0~{}0~{}1~{}0)$ band of (HF)2, $K=0$

We also performed a least-squares fit to the ab initio energies using the
formula of Belov et al. [10]

$\begin{split}E(J,K,\nu)=\Delta_{K}\delta_{\nu,1}+F_{K}+B_{J}^{\nu}J(J+1)\\\
-D_{J}^{\nu}J^{2}(J+1)^{2}+H_{J}^{\nu}J^{3}(J+1)^{3}.\end{split}$ (2)

Results of this fit are presented in Figures 1 and 2. Small residuals (less
than 100 kHz) confirm that the formula 1 used for the fit is appropriate and
we can thus safely use the results of this fit to represent the energies
obtained by our ab initio calculations. In other words the values of $B$ and
$D$ given in Table LABEL:tab:obs.calc.const give an excellent representation
of the values calculated when used with Eq. (2) and effect on energy levels
related to H-constants may be omitted.

##  3 Experimental details of the observation of sub-millimeter spectrum of HF
dimer in the gas phase

Thirty years ago some of us published a sub-millimeter study [10] of the (HF)2
spectrum. At the time it was not possible to completely assign this spectrum.
Before discussing new assignments we have been able to make using our
variational calculations (Section 4), we first give some experimental details.

The broad-band (180-380 GHz) spectrum of the HF dimer was recorded at the end
of eighties using a spectrometer with a backward wave oscillator (BWO) as a
radiation source [36] and Radio-Acoustic Detection of absorption (RAD), which
we call the RAD-3 spectrometer. A detailed description of the instrument can
be found on the Laboratory website222http://mwl.ipfran.ru and in our report on
a similar study of the H2O-HF dimer [7]. A brief description of the experiment
considered was given in the corresponding section of Ref. [10]. For the
reader’s convenience we repeat here the most important details with some other
complementary information which was used in the present analysis of the
spectrum.

Equilibrium gas-phase HF dimer spectra were observed at temperatures of about
200 - 220 K in a static pressure stainless steel cell at pressures between 0.5
– 1.5 Torr. HF vapor was obtained by thermal decomposition of KFHF. The
intensity of the observed spectrum depended on pressure and temperature. The
prominent spectral lines appeared and grew up only with cooling the cell down
and their intensity also increased with increased pressure as expected from
the general pressure and temperature dependence characterized by the
dimerisation constant. The optimal temperature was found to be near 210 K.
Neither the HF gas generator nor the cell thermostat allowed active control of
experimental conditions. Because of this the spectral intensity in the
repeated recordings varied from day to day but relative intensity of lines
within one recording was preserved.

The spectrum was recorded continuously (without gaps) covering the range from
180 up to 385 GHz. The continuity is crucial for unambiguous assignment of
observed lines. At the time of the experiment, such broadband continuous
recordings were not possible in this spectral range as the radiation frequency
was controlled using a phase locking loop systems with poor scanning ability
(see review [37] and references therein). Frequency measurements in the
spectrometer were calibrated using the lines from a SO2 reference spectrum
[38]. The mean square value of the frequency uncertainty was determined in
preliminary tests as 0.5 MHz for strong isolated monomer lines. The
uncertainty was somewhat larger in the case of weak and pressure broadened HF
dimer lines. However, we consider a value of a few MHz as a reasonable
estimate of the possible error in the frequency measurements for most lines.
Figure 3 presents a portion of the observed HF dimer spectrum recorded
simultaneously with a SO2 reference spectrum. Unfortunately, only half of the
recorded spectrum has survived the transition from large specialized counting
machines to personal computers. Only recordings in the range of 268 – 385 GHz
are now available. Analysis of this spectrum is given in the next section.

It is our view that repeating the experiment using a modern, more sensitive
and more accurate apparatus alongside extension of the spectral range to
higher frequencies, where the characteristic Q-branches are located, is now
important for obtaining data for comparison with high accuracy ab initio
calculations. However, this would have to be the subject of a specialized
project with adequate funding.

Figure 3: Small portion of recorded in the late 80-ties tunneling-rotation
spectra of (HF)2 in violet (lower spectrum) and calibration SO2 in light brown
(upper spectrum)

##  4 Analysis of the sub-millimeter spectrum

Accurate ab initio prediction of highly rotationally excited $J$ and $K$
energy levels are essential for the analysis of the sub-millimeter tunneling-
rotation spectrum of the HF dimer in the limited region between 268 and 385
GHz. Quack and Suhm were able to assign the lines of the far IR HF dimer
spectrum, [35, 34, 3] since the IR experimental spectra with much wider
coverage provided information on all the P, Q and R branches. Moreover, ro-
vibrational lines belonging to the different fundamental bands of the
different inter-molecular modes, $\nu_{3}$, $\nu_{4}$, $\nu_{5}$ and
$\nu_{6}$, occur in distinct, non-overlapping regions of the spectrum. Let us
briefly summarise their assignments. Von Puttkammer and Quack [35] studied the
region between 350 and 550 cm-1, where the observation of two Q-branches
helped them to assign the $\nu_{6}$, $K=1\leftarrow 0$ band with a Q-branch at
around 400 cm-1. In the same paper they observed another clearly resolved $P,\
Q,\ R$ structure with a Q-branch at 383 cm-1. In a subsequent paper [3], Quack
and Suhm attributed this structure to the $\nu_{6}+\nu_{4}-\nu_{4}$ hot band.
Von Puttkammer et al. [34] analyzed the $\nu_{6}$, $K=2\leftarrow 1$ subband
at 465 cm-1, and Quack and Suhm [33] investigated $K=2$ and 3 subbands of the
$\nu_{5}$ band and predicted the 6.83 cm-1 sub-millimeter rotation-vibration
tunneling band between excited vibrational states. Figure 4 of their paper
[33] is a diagram of all known and estimated high $K$ energy levels of
$\nu_{4}$, $\nu_{5}$ and $\nu_{6}$ bands of the HF dimer. This figure
facilitates understanding of the energy levels presented in Tables 1 and 2.

To summarize the previously published work, mostly by Quack, Suhm and co-
authors, experimental information on inter-molecular vibrations with $J=0$ and
excited $K$ the following is known (Tables 1 and 2). $K=0$ is unknown from
analysis of high resolution spectra for all inter-molecular states. However,
different extrapolation procedures have allowed frequencies of all four low-
lying vibrations to be estimated with differing levels of accuracy (Table 1).
In particular, for the $\nu_{4}$ band indirect experimental information on the
125 cm-1 band origin, derived from $\nu_{1}+\nu_{4}$ and $\nu_{2}+\nu_{4}$
data extrapolation by Anderson et al. [31], and from an analysis of the $J\geq
30$ resonance between $K=2$ of the ground state and $K=0$ of the $v_{4}=1$
state by Quack and Suhm [3].

Comprehensive information on the $K\leftarrow K-1$ bands for $K=1$ and 2 was
obtained by Quack and Suhm for $\nu_{5}$ [33] and $\nu_{6}$ [34] states.
However, no high resolution information on $\nu_{3}$ and $\nu_{4}$ has been
published. In what follows we present the first high resolution experimental
assignment of lines belonging to these two states.

Table 4: Observed frequencies (in MHz) for the $K=0$ branch of the tunneling-rotation spectrum of HF dimer in the $\nu_{4}$ excited vibrational state. Frequency |  $J$’ |  $J$” | obs.-fit.
---|---|---|---
274791.3 | 15 | 14 | -0.1
285878.5 | 16 | 15 | -0.5
296872.7 | 17 | 16 | 1.5
307772.9 | 18 | 17 | -0.7
318591.5 | 19 | 18 | 0.1
329329.8 | 20 | 19 | -1.0
339995.3 | 21 | 20 | -0.1
350590.3 | 22 | 21 | 0.4
361118.4 | 23 | 22 | 1.8
371574.7 | 24 | 23 | -2.1
381970.6 | 25 | 24 | 0.6
Table 5: Values (in MHz) of the effective rotational $B$ and $D$ constants as well as the tunneling splitting constant $\Delta_{K}$ (Eq.(2)) of the $K=0$ tunneling-rotation spectrum of $\nu_{4}$. | Calc. | Obs.
---|---|---
$\Delta_{0}$ | 88659 | 94494
Upper
$B$ | 6306.3 | 6433.9
$D$ | 0.0813 | 0.167
Lower
$B$ | 6353.0 | 6488.8
$D$ | 0.091 | 0.172
Table 6: $K=4$ branch of tunneling-rotation spectrum in the ground vibrational state in MHz. Frequency |  $J$’ |  $J$” | obs.-fit
---|---|---|---
280614.3 | 5 | 4 | -0.1
294326.6 | 6 | 5 | 1.2
308144.8 | 7 | 6 | -3.4
322151.6 | 8 | 7 | 4.9
336419.3 | 9 | 8 | -3.7
351119.4 | 10 | 9 | 1.1
366409.2 | 11 | 10 | 0.2
382504.2 | 12 | 11 | 0.1
Table 7: Values (in MHz) of effective rotational $B$ and $D$ constants as well as the tunneling splitting constant $\Delta_{K}$ (Eq.(2)) of the $K=4$ tunneling-rotation spectrum of HF dimer in the ground vibrational state. | Calc. | Obs.
---|---|---
$\Delta_{4}$ | 225051 | 213545
Upper
$B$ | 6539.1 | 6576.9
$D$ | 0.052 | 0.068
Lower
$B$ | 6444.6 | 6498.6
$D$ | 1.84 | -1.8
Table 8: $K=0$ branch of tunneling-rotation spectrum of HF dimer in the $\nu_{3}$ state (in MHz)  Frequency |  $J$’ |  $J$” | obs.-fit.
---|---|---|---
325666.6 | 1 | 0 | 1.5
336768.3 | 2 | 1 | -6.3
346933.0 | 3 | 2 | 4.3
356077.7 | 4 | 3 | 8.1
364146.9 | 5 | 4 | -14.2
371203.5 | 6 | 5 | 5.7
377198.7 | 7 | 6 | 0.8
382203.6 | 8 | 7 | -0.9
Table 9: Values (in MHz) of effective rotational $B$ and $D$ constants as well as the tunneling splitting constant $\Delta_{K}$ (Eq.(2)) of the $K=0$ tunneling-rotation spectrum in the $\nu_{3}$ state. | Calc. | Obs.
---|---|---
$\Delta_{0}$ | 317824 | 313681
Upper
$B$ | 5957.4 | 6000.5
$D$ | 0.89 | 4.57
Lower
$B$ | 6232.7 | 6383.91
$D$ | 0.097 | 5.63

As an example of the use of calculated HF dimer tunneling splittings, band
origins and rotational constants, we assigned part of the $K=0$ R-branch
tunneling-rotational transitions within the exited fundamental inter-molecular
vibration $\nu_{4}$. The $K=0$, $\nu_{4}$ state lies at about 125 cm-1, which
is close to the ground state $K=1$ (35 cm-1) and $K=2$ (116 cm-1) states. So
the Boltzmann factor suggests it should be the strongest of the remaining
unassigned bands. Manually we have a set of 11 lines covering all the
available experimental region of 268–385 GHz. Using seven constants ($B$, $D$
and $H$ rotational constants for upper and lower tunneling split states and
the value of this splitting) as fitting parameters, the lines are described
with a standard deviation of 1.7 MHz corresponding to an estimated
experimental accuracy of 2-5 MHz. If we omit the two $H$ constants, the
standard deviation increases to 4.5 MHz which is still inside experimental
accuracy. As the number of fitted lines is not much larger than the number of
constants when H constants are included, the $H$ constants are poorly
determined. For this reason only fit with $H$ constants fixed to zero are
presented

The fit was made assuming that the inversion splitting of the $K=0$, $\nu_{4}$
state is close to the calculated value of about 3 cm-1 (90 GHz). This value,
plus the values of the $B$ rotation constants, determine the upper and lower
$J$’s of the measured transitions as in Eq. (2). Changing the splitting by 12
GHz leads to a change in the $J$’s by one. The correct $J$ values for the
observed transitions could not be determined from available experimental
transitions in the region 268–385 GHz. Fits to the experimental frequencies
with $J$’s differing by one give practically the same standard deviation, and
similar values for the rotation constants. To empirically determine the
correct $J$ values one has to observe either the Q-branch or the beginning of
the R-branch in the centimeter wavelength region. Fit residuals for the
tunneling-rotation $K=0$ branch lines in the $v_{4}=1$ state are presented in
Table 4. A comparison of experimentally derived $B$ and $D$ constants with
those obtained variationally is given in Table 5. Table
LABEL:tab:obs.calc.const presents a corresponding comparison for all known $v$
and $K$ states. The residuals of ab initio calculations for the observed
frequencies are of order 0.1 cm-1 (3000 MHz).

This first determination of $B$ and $D$ constants for the $\nu_{4}$ state
helped us to obtain another important piece of experimental information. In
their study on the first IR hot spectrum of combination band
$\nu_{6}+\nu_{4}-\nu_{4}$ Von Puttkamer and Quack [35] could determine only
$\Delta B$ and $\Delta D$ constants as they did not of a know the $B$ and $D$
constants of the lower state $\nu_{4}$, so only differences between upper and
lower state constants could be determined. Our determination of the $\nu_{4}$
$B$ and $D$ constants allows us to determine the $B$ and $D$ constants of the
upper combination state of $\nu_{6}+\nu_{4}$. These newly determined values
are used in Table LABEL:tab:obs.calc.const for comparison with the
variationally determined $B$ and $D$ constants.

Let us now consider the assignment of tunneling-rotation lines in the
$\nu_{3}$ state. This mode is the highest energy inter-molecular vibration
(about 480 cm-1) and its lines should be among the weakest in the recorded
spectrum. The inversion splitting of the $v_{3}=1$ state is calculated to be
about 318 GHz.We were able to find a series of lines starting at 325 GHz. Fits
to this series of 8 lines with 5 constants ($B$ and $D$ constants for upper
and lower states and the tunneling splitting) gives a standard deviation of
6.7 MHz. Observed values of the frequencies and the residuals of the fit are
given in Table 8 and constants obtained from the fit and our variational
calculations are given in Table 9.

The last series of lines which we assign in the 268 - 385 GHz tunneling-
rotation spectrum was the $K=4$ branch of the ground state. Table 7 gives
lines of the branch for $J$ ranging from 4 to 11 together with the residuals
between observed and fitted frequencies using the $B$, $D$ and $\Delta_{K}$
model. $B$ and $D$ constants of the upper state given in Table 7 were fixed to
the values reported by to Quack and Suhm [32], obtained from the
$K=4\leftarrow 3$ band. $\Delta_{K}$, $B$ and $D$ constants are determined in
the fit of Eq.(2) to the data of Table 6. The standard deviation of this fit
is 4.7 MHz.

To summarize our new assignments of the tunneling rotation sub-millimeter
spectrum: lines belonging to previously unobserved $\nu_{3}$ and $\nu_{4}$
states are found and assigned for their $K=0$ branches. The $K=4$ branch of
the ground vibrational state had only a single known tunneling state [34] and
now both tunneling states involved in the tunneling rotation band have been
observed, assigned and $B$, $D$ and $\Delta_{K}$ constants determined. The
variationally calculated $B$ and $D$ constants together with the constants
obtained from the fit are given in the tables of Section 2 and this section.
It transpires that after assignment and analysis of these branches, the values
of the experimental $B$ and $D$ constants are reproduced very well, with
similar accuracies of tens of MHz, by our variational calculations. The
planned further improvement of the ab initio PES should lead to the further
improvement in the agreement between variational and experimental $B$ and $D$
constants as well as the agreement between calculated and observed band
origins.

What remains to be observed in the HF dimer spectrum within the sensitivity of
the existing experimental setup and the Boltzmann factor of the states to be
observed is the following: $K=5$ and 6 of the ground vibrational state, $K=1$
and 2 of $\nu_{3}$ and $K=1,2,3$ and 4 of $\nu_{4}$ as well as $K=0$ of
$\nu_{6}$. These observations could be made using active thermal stabilization
of an experimental gas cell and HF generator improving the signal-to-noise
ratio of the observed spectra and an expansion of the region of observation to
180 - 260 GHz and 370 - 535 GHz (accessible with corresponding radiation
sources [36]) in order to observe Q-branches and expand the $J$ ranges of the
P and R branches. The resulting set of experimental $\Delta_{K}$, $B$ and $D$
constants would serve as an excellent source of data for comparison with an
improved ab initio HF dimer PES.

##  5 Discussion and Conclusions

We present calculations on the tunneling-rotation-vibration spectrum of the HF
dimer performed using the variational nuclear motion program suite WAVR4 [22,
28] and an ab initio HF dimer PES [6]. We used the results of these
calculations to analyze a 30-years old gas phase tunneling-rotation spectrum
of the HF dimer, recorded using a RAD-3 spectrometer in the sub-millimeter
region [10]. Excellent reproduction of the observed $B$ constants of the
previously assigned [10] $K_{a}=0,1,2$ and $3$ branches encouraged us to
assign further branches of the HF dimer spectrum. In particular, the higher
$K_{a}$ = 4 branch of the ground vibrational state and the two $K_{a}$ = 0
branches of the exited $\nu_{4}$ and $\nu_{3}$ fundamental vibrational inter-
molecular modes were assigned for the first time. No previous high resolution
study had identified transitions in the $\nu_{4}$ and $\nu_{3}$ states; we
were able to identify lines of these states in our sub-millimeter spectrum on
the basis of predictions obtained using high accuracy excited $J$ variational
calculations.

The present study paves the way for the further improvements of the ab initio
HF dimer PES and application of it to the accurate calculation of HF dimer
spectrum as well as the assignment of the further energetically higher lying
branches. This should help fully characterise the HF dimer spectrum up to
dissociation, provided the corresponding experimental tools are available.
Further improvement of the ab initio calculations are necessary to achieve
agreement with experiment. This work currently is in progress.

The drive for obtaining a more complete understanding of HF dimer spectra is
not limited to an interest in the HF dimer itself. The HF dimer is similar in
many respects to the water dimer which enhances the motivation for studying
highly excited rovibrational states of (HF)2. The ability to accurately
calculate highly excited $J$ levels of the water dimer remains an unsolved
problem but one which is important for both fundamental molecular spectroscopy
and atmospheric physics. Developing viable theoretical methods for this is a
prerequisite for the calculation of accurate water dimer line lists which
would provide an important step towards a solution of various physical
problems, not the least of which is the problem of the water continuum in the
Earth’s atmosphere [39, 40] and atmospheres of exoplanets. The unambiguous
observation of rotational features of water dimers in equilibrium water vapor
at close to atmospheric conditions [41, 42, 43] paves the way to a fuller
understanding of the water dimer spectrum via a complete theoretical line list
and further experimental observations, see [44, 45]. However, a reliable
interpretation of such experimental observations requires an accurate
theoretical line list, which includes treatment of highly excited vibrational
and rotational states. This problem remains very challenging.

The HF dimer system has strong similarities with the water dimer: It has the
same number of electrons and is similarly floppy. The tunneling-rotational
spectrum of these two dimers are quite similar in terms of the adiabatic
relations between the low energy and high energy motions. However, there are
also a few significant differences. The water dimer is of much higher
importance for applications to the atmospheric studies of the Earth and
exoplanets, and to an understanding of liquid water. The second difference is
the number of nuclear degrees of freedom: 6 for (HF)2 and 12 for (H2O)2.
Because of this it is much harder to calculate accurately both the PES and
tunneling-rotation-vibration energy levels of (H2O)2, than in the case of
(HF)2. These similarities and differences open wide opportunities for the
study of both dimers. Studies of (HF)2 are of increased importance as a
similar but easier to study system, which will teach us how to deal with the
more important and more complicated water dimer. The third big difference
between these two dimers is the relative ease of observing HF dimer spectra
under equilibrium and relatively warm gas phase thermodynamic conditions, as
demonstrated by numerous studies in both the microwave and the IR regions, and
the utter impossibility of observing spectra of similar high resolution for
the water dimer. This fact makes studies of the HF dimer even more important,
as high accuracy high resolution experimental data facilitates detailed
comparison with theory. Such comparisons give guidance on how to achieve
comparable accuracy of calculations for the water dimer, without having to
actually compare these calculations with highly excited vibrational and
rotational water dimer energy levels. Nevertheless, extensive data on the
water dimer obtained in beams and jets allows such comparisons to be made for
low-lying rotations and vibrations. These comparisons can confirm ideas about
the similar accuracy of calculations of energy levels and line intensities for
the water- and HF-dimers; this should allow one to extrapolate our
understanding of the accuracy of highly excited HF dimer calculations to
practical calculations of highly excited states of the water dimer.

The resulting observed minus calculated residuals for the HF dimer based on
experimental and semi-experimental data, presented here and the corresponding
variationally calculated values, provide valuable information for estimating
the accuracy of a corresponding water dimer ab initio calculations of energy
levels and line positions. Clearly this will be the case when ab initio
calculations on the water dimer are performed at the same level of electronic
structure theory. Thus our HF dimer study provides useful underpinning
information for ab initio studies on the water dimer supporting our objective,
which is to calculate a very accurate water dimer ab initio PES whose accuracy
can be benchmarked by studies of the HF dimer.

There are two alternative approaches to further improving our HF dimer PES and
hence spectrum calculations. The first is to produce an improved PES by
fitting to experimental data, as Klopper et al. have done [5]. However, this
approach will not guide us to improved water dimer calculations. The other
alternative is to increase the levels of electronic structure theory. Work
along these lines is currently in progress paved by impressive results
reported in this paper.

## Acknowledgments

We thank Tony Lynas-Gray for helpful comments on our manuscript. We also thank
the UK Natural Environment Research Council (NERC) grant NE/T000767 and ERC
Advanced Investigator Project 883830 for supporting aspects of this project.
RF State Project 0030-2021-0016 is also acknowledged.

## Supplementary material

A text files giving the calculated HF dimer energy levels is given as
supplementary data.

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T)^{0.25}T^{\frac{1}{2}(1-\alpha)}C+2LL_{0}R$ $\displaystyle\leq
O\left(d\sqrt{\log T}T^{\alpha}+\sqrt{d}(\log
T)^{\frac{1}{4}}T^{\frac{1}{2}(1-\alpha)}C+dC\right),$

which completes the proof.

##  D Additional Experiments

In this section, we list all the experiment details. At the beginning, we
introduce the regret order fitting method, different types of corruption that
we simulate in the experiments, and baseline algorithms for comparison.

Fitted Order of Regret. We introduce the methodology that we use to compute
the order of $\text{Reg}_{T}$ in terms of the number of iteration, $T$. To fit
the order of the regret, we first convert it into $\log$ scale. Then we input
the last $1\%$ of data, run linear regression, and use ordinary least squares
to estimate the slope of the line, which is the _fitted order_ of
$\text{Reg}_{T}$.

Corruption. In the experiments, we consider two types of corruption:
corruption induced by $\rho$-_imperfect_ user and _arbitrary_ corruption. For
corruption induced by $\rho$-_imperfect_ user, we simulate
$c_{t}(a_{t},a^{\prime}_{t})$ according to Definition A.1. We set $\lambda=2$
and $C_{0}=0.1(\max_{a\in\mathcal{A}}\mu(a)-\min_{a\in\mathcal{A}}\mu(a))$.
Moreover, we add $c_{t}(a_{t},a^{\prime}_{t})$ to the utility difference of
$a_{t}$ and $a^{\prime}_{t}$ adversarially. In particular, if
$\mu(a_{t})>\mu(a^{\prime}_{t})$, then $\hat{\mathbb{P}}(a_{t}\succ
a^{\prime}_{t})=\sigma(\mu(a_{t})-\mu(a^{\prime}_{t})-c_{t}(a_{t},a^{\prime}_{t}))$,
then vice versa. For _arbitrary_ corruption, we force the user to submit her
least preferred item each round to the algorithm over the first $C$ rounds.

Baseline Algorithms. We consider three baseline algorithms for comparison,
Doubler (Ailon et al., 2014), Sparring (Ailon et al., 2014), and Versatile-DB
(Saha and Gaillard, 2022).

Doubler is the first approach that transforms a dueling bandit problem into a
standard multi-armed bandit (MAB) problem. It operates in epochs of
exponentially increasing length: in each epoch, the left arm is sampled from a
fixed distribution, and the right arm is selected using a MAB algorithm to
minimize regret against the left arm. The feedback received by the MAB
algorithm is the number of wins the right arm achieves compared to the left
arm. Under linear link assumption, Doubler has been proven to experience
regret as the same order as underlying MAB algorithm. For continuous _action_
space and general concave utility, we choose Bandit Gradient Descent (BGD,
(Flaxman et al., 2004)), with regret $O(T^{3/4})$, as the underlying MAB
algorithm.

Sparring initializes two MAB instances and lets them compete against each
other. It is a heuristic improvement over Doubler. Although it does not come
with a regret upper bound guarantee, it is reported to enjoy better
performance compared to Doubler (Ailon et al., 2014). We also choose BGD as
the underlying MAB algorithm.

Versatile-DB applies novel reduction approach which converts dueling bandit to
MAB. Essentially, it designs simplex over $K$ arms using the idea of follow-
the-regularized-leader and updates the simplex using information of whether
the arm wins. It has been proved that Versatile-DB is robust to corruption
defined as the number of flips of duels and Versatile-DB enjoys regret linear
in $C$ (Saha and Gaillard, 2022).

###  D.1 Experiments on Synthetic Data

####  D.1.1 DBGD Lower Bound Simulation

Experiment Setup. To validate Corollary 1, we construct the following problem
instance. Consider the linear utility function $\mu(a):=\theta^{\top}a$.
Choose $d=2,\theta=[\frac{1}{2},\frac{1}{2}],T=10^{5}$. Let _action_ set
$\mathcal{A}:=\\{(a_{1},a_{2}):a_{1}\geq 0,a_{2}\geq
0,\frac{1}{2}a_{1}+a_{2}-\frac{1}{4}\leq 0\\}$. We repeat the experiments for
$50$ times, each time with a different seed.

Results and Discussion. Each grey-colored dotted line in Figure 2 indicate the
cumulative regret for each simulation. The blue line is the average cumulative
regret over 50 repeats. The error bar indicates $\pm$ one standard deviation
of the regret. The fitted order over the last $1\%$ of the average cumulative
regret is 0.76, which is close $0.75$ and supports Corollary 1.

####  D.1.2 Regret Upper Bound in Theorem 1

Experiment Setup. To validate regret upper bound in Theorem 1, we consider a
strongly concave utility
$\mu_{\theta}(a):=\theta^{\top}a-\frac{1}{2}\|a\|^{2}_{2}$, a logistic link
function $\sigma(x)=\frac{1}{1+\exp(-x)}$. We choose $d=5$, and $T=10^{5}$.
Our _action_ space $\mathcal{A}$ is a $d$-dimensional ball with radius $R=10$.
The preference parameter $\theta$ is randomly sampled from the surface of
$\mathcal{A}$. In our problem setting, the optimal action $a^{*}$ is $\theta$
and $\mu_{\theta}(a^{*})=50$. We simulate corruption induced by
$\rho$-_imperfect_ user and conduct experiments for $\rho\in[0.5,1.0]$. For
each value of $\rho$, we repeat the experiments for $5$ times, each under a
different seed. We set the learning rate $\eta_{\rho}$ according to
Proposition C.1.

Figure D.1: Performance under Known C

Results and Discussion. Figure D.1 shows that when $\rho$ is known, NC-SMD can
tolerate linear $\rho$-_imperfect_ user corruption, which confirms theoretical
analysis in Proposition C.1 and implies that lower bound in Theorem 1 is
tight.

####  D.1.3 Robustness under Corruption induced by Imperfect User

In Figure D.2, we simulated the performance of DBGD, NC-SMD, Sparring, Doubler
in the same experiment setting as in Section 6.1 expect using corruption
induced by $\rho$-Imperfect User. We notice that the experiment results also
align with theoretical prediction (Theorem 2 and Proposition 2), which
predicts when $\alpha=0.25$ for DBGD and $\alpha=0.5$ for NC-SMD, they can
tolerate $O(T^{0.75})$ agnostic corruption.

Figure D.2: Robustness under Corruption induced by $\rho$-Imperfect User

####  D.1.4 Efficiency-Robustness Tradeoff in Theorem 2

To validate efficiency-robustness trade-off in Theorem 2 and Proposition 2, we
adopt the same experiment setup as in Section 6.1 except different $\alpha$
values. In the first row of Figure D.3, we consider $\alpha=0.05$ for DBGD and
$\alpha=0.9$ for NC-SMD. We consider $\rho\in[0.5,0.95]$. According to the
theoretical prediction, both DBGD and NC-SMD can tolerate at most
$O(T^{0.95})$ agnostic corruption, which aligns with experiment results. In
the second row of Figure D.3, we consider $\alpha=0.1$ for DBGD and
$\alpha=0.8$ for NC-SMD. According to the theoretical prediction, both DBGD
and NC-SMD can tolerate at most $O(T^{0.9})$ agnostic corruption. From the
figure, we can see they have smaller fitted order of regret when $\rho=0.5$
while at a cost of tolerating smaller magnitude of agnostic corruption (it has
linear regret when $\rho=0.95$), which reveals intrinsic tradeoff between
efficiency and robustness.

Figure D.3: Efficiency-Robustness Tradeoff

###  D.2 Experiments on Spotify Recommendation Data

Evaluation Setup. We evaluate our approach on Spotify recommendation data
(Spotify, 2020). The objective is to recommend songs to incoming users. This
dataset includes $17\times 10^{4}$ songs ($|\mathcal{A}|=17\times 10^{4}$),
each is described by 15 distinct features (d = 15).

User Profile Identification. In order to model different types of user, we
first use Standard Scaler to standardize the dataset. Then we use Kmeans to
conduct clustering (Pedregosa et al., 2011).

User Utility. We use the average of the song embedding within each group as
the preference vector. Our aim is to recommend songs to users that have the
highest cosine similarity (utility function) with their preference vectors. We
rescale the cosine similarity to $[-100,100]$.

####  D.2.1 Performance of Versatile-DB on Spotify Recommendation Data

To test the performance of Versatile-DB, we randomly sample $K=2000$ songs
from the Spotify Recommendation Dataset. We identify 2 different user types by
clustering the dataset into 2 groups. We use the average of the song embedding
within each group as the preference vector and compute the user utility for
each recommendation. We consider corruption induced by $\rho$-_imperfect_ user
for $\rho\in[0.5,0.75]$. We set $\alpha=0.25$ for DBGD and run both algorithms
for $T=100$ iterations. This is because Versatile-DB is extremely
computationally intensive and running $K=2000,T=100$ takes around 20 CPU
hours. For each user type and for each value of $\rho$, we repeat the
experiment over 5 times, each under a different seed. From Figure D.4, we
notice that DBGD outperforms Versatile-DB in all $\rho$, which highlights
DBGD’s robustness and applicability in real-world data.

Figure D.4: Performance Comparison between DBGD and Versatile-DB