# Complex angular momentum theory of state-to state integral cross sections:
resonance effects in the F+HD$\to$ HF($v^{\prime}$=3)+D reaction111Dedicated
to Prof. Tullio Regge (1931-2014)

D. Sokolovskia,b E. Akhmatskayab,c C. Echeverra-Arrondoa D. De Faziod a
Departmento de Química-Física, Universidad del País Vasco, UPV/EHU, Leioa,
Spain,  b IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3,
48013, Bilbao, Bizkaia, Spain,  c Basque Center for Applied Mathematics
(BCAM), Alameda de Mazarredo, 14 48009 Bilbao, Bizkaia, Spain,  d Istituto di
Struttura della Materia, CNR, 00016 Roma, Italy

###### Abstract

State-to-state reactive integral cross sections (ICS) are often affected by
quantum mechanical resonances, especially at relatively low energies. An ICS
is usually obtained by summing partial waves at a given value of energy. For
this reason, the knowledge of pole positions and residues in the complex
energy plane is not sufficient for a quantitative description of the patterns
produced by a resonance. Such description is available in terms of the poles
of an $S$-matrix element in the complex plane of the total angular momentum.
The approach was recently implemented in a computer code ICS_Regge , available
in the public domain [Comp. Phys. Comm. 185 (2014) 2127]. In this paper, we
employ the ICS_Regge package to analyse in detail, for the first time, the
resonance patterns predicted for integral cross sections (ICS) of the
benchmark F+HD$\to$ HF($v^{\prime}$=3)+D reaction. The $v=0$, $j=0$,
$\Omega=0$ $\to$ $v^{\prime}=3$, $j^{\prime}=0,1,2$, and
$\Omega^{\prime}=0,1,2$ transitions are studied for collision energies from
$58.54$ to $197.54$ $meV$. For these energies, we find several resonances,
whose contributions to the ICS vary from symmetric and asymmetric Fano shapes
to smooth sinusoidal Regge oscillations. Complex energies of metastable states
and Regge pole positions and residues are found by Padé reconstruction of the
scattering matrix elements. Accuracy of the ICS_Regge code, relation between
complex energies and Regge poles, various types of Regge trajectories, and the
origin of the $J$-shifting approximation are also discussed.

34.10,+x 34.50 Lf,

##  I introduction

Recently there has been much interest in chemical reactivity in a cold
environment, largely because of the spectacular progress in production and
trapping of translationally cold molecules using a wide variety of
experimental methods Krems09 . At low temperatures, reaction rates are often
strongly influenced by quantum mechanical resonances and tunneling, which may
cause large deviations of the rate constants from the Arrhenius law JPC14
-AR2 . Modelling of resonance phenomena, in which the reactants form
intermediate complex before breaking up into products, requires large scale
dynamical calculations on highly accurate potential surfaces. No less
important is the task of interpreting the resonance patterns observed in the
differential (DCS) and integral (ICS) cross sections. The DCS are more
sensitive to the presence of rotating metastable complexes, which lead to
observable nearside-farside oscillations in the angular distributions. A
recent discussion of the interference effects in reactive DCS can be found,
for example, in Refs. DCS1 -DCS2 .

Integral cross sections can also be affected by resonances. A resonance
manifests itself in two types of singularities of a scattering matrix element
${\bf S}(E,J)$, which depends on two conserved quantities, the energy $E$ and
the total angular momentum $J$. (We use bold symbols for complex valued
quantities.) A pole of ${\bf S}$ as function of ${\bf E}$ for a fixed $J$
determines the energy and the lifetime of the metastable state. (In the
following we will use bold symbols when we refer to complex mathematical
quantities, and use normal characters otherwise.) In a similar way, a pole of
${\bf S}$ in the complex ${\bf J}$-plane at a fixed $E$ determines the angular
momentum and the mean angle of rotation before the decay (lifeangle) of the
complex formed at this energy. These complex angular momentum (CAM) or Regge
poles REG1 -REG2 , are more useful for quantifying the resonance effects in
observables, such as the ICS, which are given by sums over partial waves at a
given energy. This was first realised by Macek and co-workers, who related
low-energy oscillations in the ICS for single-channel proton scattering by
hydrogen atoms Macek1  and inert gases Macek2 , and gave a simple expression
(known as the Mulholland formula Mull ) for the resonance contribution to an
ICS. Their approach has proven to be useful for the studies of chemical
reactions where resonances abound, and in Ref. JCP2  the Mulholland formula
was extended to the multichannel case.

For a single- or few-channel scattering problem, Regge poles can be found
numerically by integrating the Schroedinger equation for complex-valued ${\bf
J}$’s SE1 ; REG2 ; Macek1 ; SE2 ; INT2 , or by semiclassical methods Macek2 .
This is no longer the case for a realistic reactive system with dozens of open
channels, where evaluating the scattering matrix for physical integer values
of $J$ is already a challenging task. Alternatively, it is possible to use the
computed values of the ${\bf S}$-matrix elements in order to construct a Padé
approximant, which will analytically continue $\bf S$ into the complex ${\bf
J}$-plane PADE1 -PADE2 , and allow us to find the relevant Regge pole
positions and residues. The method was implemented in the Fortran code PADE_II
described in PADE2 , to which we refer the reader for more technical details.
More recently, a code ICS_Regge designed specifically for analysing the ICS in
a multichannel problem was reported in CPC2 ; SPRING . The code, which uses
the Padé reconstruction just outlined, identifies the trajectory, described by
a Regge pole as energy varies, and evaluates its resonance contribution using
the Mulholland formula of Ref.JCP2 . This allows us to decompose an often
complicated pattern in the ICS into contributions from several Regge
trajectories, corresponding to known resonance states. In this work, we use
the ICS_Regge code in order to perform this analysis on the F+HD$\to$
HF($v^{\prime}$=3)+D reaction at collision energies between $58.54$ and
$197.54$ $meV$.

The F+HD$\to$ HF+D reaction (together with its isotopic variant F+H${}_{2}\to$
HF+H) has earned its reputation as a benchmark case for the study of chemical
kinetics after the 1985 molecular beam experiment of Ref. EXP1 , followed by
molecular beam studies of higher resolution EXP2 -EXP9 . Most of these works
provide evidence of the main resonance of the system (the resonance A in this
work) affecting, in different ways, the total EXP2  and the state-to-state ICS
EXP7 , as well as the rotational distributions EXP9 , and the DCS EXP2 ; EXP3
of the HF($v^{\prime}$=2) product. However, the presence of sharp forward and
backward peaks in the DCS for the $v^{\prime}=3$ vibrational manifold EXP2 ;
EXP3 , suggested that an important role may be played by one or several
resonance mechanisms also for this vibrational channel. Availability of
detailed experimental data prompted theoretical interest in the F+HD system,
further encouraged by the fact that the ab initio potential energy surface
(PES) obtained by Stark and Werner (SW) PES1  was able to reproduce, at least
qualitatively, some of the resonance features observed EXP2 . The SW PES was
later modified in the entrance channel region PES2 , in order to take into
account the spin-orbit and the long-range interactions IJQC01 , in the light
of previous elastic and inelastic molecular beam experiments Candori . This
surface, called PES III PES3 , was able to achieve a better agreement JCP06
in the height and shape of the resonance peak observed in EXP2 , as well as
predict quantitatively the cold (few Kelvin) behaviour of the F + H2 reaction,
recently measured in S57 . Dynamical calculations for the F+HD system on the
PES III are reported, for example, in JCP08 ; JPC09 ; D1 . Analysis of JCP08 ;
JPC09 ; D1  suggested a need for modifying the PES also in the exit channel
region, in order to achieve better agreement with experimental data, and, in
particular, a better description of various resonance features. Further
progress in this direction was made by Fu, Xu and Zhang (FXZ) PES4  obtaining
a new ab initio PES able to reproduce with an impressive accuracy EXP7 ; D2
the resonance features observed. Moreover, the analysis reported in D2 ; JCP13
suggests a rich resonance pattern for the $v^{\prime}$=3 manifold also
predicted in the PES III calculations D1  and not yet fully explored in the
experiments. Minor inaccuracies in the entrance and transition state regions
of the PES, leading to small discrepancies with experimental data, especially
in the DF reactive channel D2 , have been partially corrected recently in S63
. In this work, we will use numerical S-matrix elements obtained in the
dynamical calculations on the FXZ PES performed by De Fazio and co-workers D2
, as the input data for our analysis of resonance effects in the title
reaction.

The purpose of this paper is, therefore, to perform a detailed analysis of the
effects produced by quantum mechanical resonances in the HF($v^{\prime}$=3)
ICS’s of the benchmark F+HD reaction. We will also check the efficiency and
accuracy of numerical techniques employed by the ICS_Regge .

The rest of the paper is organised as follows. In Section II we introduce the
ICS’s for the transitions of interest at collision energies close to the
reactive thresholds and in the higher energy region, and obtain complex
energies of the relevant metastable states as functions of the total angular
momentum. In Sect. III we describe the Mulholland formula of Ref.JCP2 , and
discuss two of its limiting cases. In Sect. IV we show that in the higher
energy region the ICS’s are affected by Regge oscillations produced by a
single resonance. In Section V we discuss, in the simplest case, the relation
between the complex energy of a metastable state and the corresponding Regge
trajectory. In Sect. VI we analyse the behaviour of the ICS’s in the
neighbourhood of the reactive thresholds, where they are affected by several
resonances. Section VII contains our conclusions.

##  II The integral cross sections

A state-to-state reactive ICS is given by a partial wave sum (PWS) over the
integer values of the total angular momentum $J$ (we are using $\hbar=1$)

$\displaystyle\sigma_{\nu^{\prime}\leftarrow\nu}(E)=\frac{2\pi}{k_{\nu}^{2}}\sum_{J=J_{min}}^{\infty}(J+1/2)|{\bf
S}_{\nu^{\prime}\leftarrow\nu}(E,J)|^{2},$ (1)

where ${\bf S}_{\nu^{\prime}\leftarrow\nu}(E,J)$ is the ${\bf S}$-matrix
element for the reactants, prepared in the quantum state $\nu$, to form
products in the state $\nu^{\prime}$. For an atom-diatom collision, multi-
index $\nu=(v,j,\Omega)$ includes the vibrational and rotational quantum
numbers of the diatomic, $v$ and $j$, and the absolute value of the diatomic’s
angular momentum projection, $\Omega$, onto atom-diatom relative velocity,
also known as the helicity. Since this projection cannot exceed $J$, the
summation in Eq.(1) starts from $J_{min}=max(\Omega,\Omega^{\prime})$.
Finally, $k_{\nu}$ is the reactant’s wave vector.

In this work we will analyse the resonance effects in the reactive transitions
$(3,j^{\prime}=0,1,2,\Omega^{\prime})\leftarrow(0,0,0)$ in the interval of
collision energies $58.54$ $meV\leq E\leq 197.54$ $meV$. Somewhat
artificially, we divided this interval into ’lower’ and ’higher’ energy
ranges, comprising $58.54$ $meV\leq E\leq 78.44$ $meV$, and $78.54$ $meV\leq
E\leq 197.54$ $meV$, respectively. The reason for such a division becomes
obvious from Fig.1 showing the ICS in both regions (note the change of the
energy scale from Fig.1a to Fig1b). In the higher range shown in Fig.1b, all
six ICS’s exhibit similar slowly varying oscillatory structures superimposed
on a smooth background. In the lower range shown in Fig.1a, the structure of
the ICS’s is much more complicated, with sharp peaks and saw-like features.
This difference suggests that the two regions should be treated separately, as
we will do in the following.

Figure 1: Six ICSs for the transitions
$(v^{\prime}=3,j^{\prime}=0,1,2,\Omega^{\prime}=0,1,2)\leftarrow(v=0,j=0,\Omega=0)$
in the lower (a) and higher (b) energy regions. Vertical dashed lines at
$E=62.72$ and $71.77$ $meV$ indicate threshold energies beyond which the
$j^{\prime}=1$ and $j^{\prime}=2$ transitions become possible. Note the
difference in energy scales between the panels (a) and (b).

Figure 2: Complex energy poles found in the higher (c and d) and lower (a and
b) energy ranges by Padé reconstruction of the $(3,0,0)\leftarrow(0,0,0)$
${\bf S}$-matrix element in the complex ${\bf E}$-plane for integer values of
the total angular momentum $J$.

As was mentioned in the Introduction, below we will use mostly the ${\bf S}$
matrix elements reported in Ref. D2 . However, to get an accuracy of about 1
$\%$ in the modules and phases of the single state-to-state ${\bf S}$-matrix
elements, quantum reactive scattering calculations in the lower energy range
were repeated using the improved code described in Refs. He12 ; DDF14 . Also
for better convergence, the value of $\rho_{max}=20$ $a.u.$ and $jmax=25$ were
used in these additional calculations. In particular, we will analyse in
detail the ICS’s for three reactive transitions,

$\displaystyle(3,0,0)\leftarrow(0,0,0),$ (2)
$\displaystyle(3,1,1)\leftarrow(0,0,0),$
$\displaystyle(3,2,2)\leftarrow(0,0,0),$

each representative of its rotational manifold. The behaviour for other
allowed helicity states, $\Omega^{\prime}=0$ for $j=1$, and
$\Omega^{\prime}=0,1$ for $j=2$, is very similar, and would contribute little
new information.

##  III The complex energy poles

We expect the structures in the ICS’s in Fig. 1 to be signatures of a
resonance mechanism, or mechanisms, and look for a further confirmation. One
usually starts by looking for poles of the ${\bf S}$-matrix element in the
complex energy (CE) plane, at physical values of the total angular momentum,
$J=0,1,2....$. There is an ongoing discussion (see DCS2  and Refs. therein)
about when a particular pole, or a combination of poles, should be considered
a resonance. We will avoid the controversy by directly identifying a pole or
poles, and quantifying their effect on the integral cross sections of
interest. Accurate pole positions can be obtained, for instance, by evaluating
Smith’s lifetime ${\bf Q}$-matrix Smith  using numerical differentiation of
the ${\bf S}$-matrix elements DARIO04 , or by a direct approach described in
Ref. JCP05 . Here we use a different method, based on Padé reconstruction of
the chosen ${\bf S}$-matrix element. The Padé reconstruction analytically
continues ${\bf S}_{\nu^{\prime}\leftarrow\nu}(E,J)$, evaluated for a set of
real energies, into the complex ${\bf E}$-plane (see, for example, PADE2 ).

Plotting the CE pole positions for different values of $J$ we obtain CE
trajectories, shown in Fig. 2 for the $(3,0,0)\leftarrow(0,0,0)$ reactive
transition. We note that there is only one CE pole in the higher energy
region, while in the lower energy range there are many CE trajectories. While
a CE pole at ${\bf E}_{0}$ is the same for all the elements of the ${\bf
S}$-matrix Land , the extent to which it affects the reaction probability
$P_{\nu^{\prime}\leftarrow\nu}(E,J)=|{\bf
S}_{\nu^{\prime}\leftarrow\nu}(E,J)|^{2}$ depends on the corresponding
residue, $\boldsymbol{\rho}_{\bf E}=\mbox{lim}_{{\bf E}\to{\bf E}_{0}}({\bf
E}-{\bf E}_{0}){\bf S}_{\nu^{\prime}\leftarrow\nu}({\bf E},J)$, specific to
each transition. Thus, certain transitions may be affected by a particular CE
pole, while some may not, and a different choice of $\nu$ and $\nu^{\prime}$
may result in different sets of trajectories.

The CE trajectories for the F+HD reaction have been studied earlier D1 ; D2 .
In D1  the authors have obtained such trajectories for a different PES, PES-
III PES3 , by fitting the peaks in the highest eigenvalue of the ${\bf
Q}$-matrix to a Lorentzian form. A preliminary study for the PES used in this
work (the FXZ PES PES4 ) was reported in D2 , where the authors followed some
of the peaks in the reaction probability, thus approximating only the real
parts of the corresponding CE trajectories, with the results similar to those
shown in Figs. 2 a and b. The five poles shown in Fig. 2 were labelled in
Refs. D1 ; D2 ; DARIO04 ; JCP05 , as A, B, C, D and E, a notation originally
introduced by Manolopoulos and co-workers mano96 , who were the first to
discuss their effect on the J=0 reaction probability of the F+H2 system. We
will continue to use the same nomenclature in what follows. Analysing the
build up of the scattering wave function in a particular spacial region D1 ,
or studying the adiabatic curves DARIO04  often allows one to identify the
physical origin of a particular CE pole. Thus, the pole A corresponds to
trapping the reactants close to the transition state of the PES, while the
poles B, C, D, and E describe capture in the metastable states of the van der
Waals well in the exit channel D1  corresponding to different bending and
symmetric stretching modes JCP05 .

The complex energy poles shown in Fig. 2 may suggest that the patterns seen in
Fig.1 are related to the resonances A, B, C, D and E. Complex energies,
however, are not particularly useful for our task of quantifying the resonance
effects produced in the ICS’s. Indeed, poles are most useful when there is an
integration and, by changing the contour and applying the Cauchy theorem, one
can isolate a pole’s contribution. Thus, the CE poles would be useful, if we
were to evaluate integrals over energy. In Eq.(1), however, $E$ is a fixed
parameter, and the summation is over $J$’s. Sums over $J$ can be transformed
into integrals, e.g., by applying the Poisson sum formula, and to evaluate
such integrals we would need the other kind of poles.

##  IV Regge pole decomposition of an integral cross section

The poles we need are the poles of the ${\bf S}$-matrix element in the complex
${\bf J}$-plane at a fixed real energy, also known as complex angular momentum
(CAM) or Regge poles. The elements of the scattering matrix diverge whenever
the solution of the corresponding Schroedinger equation contains only outgoing
waves at large interatomic distances. At a given real energy, this can only
happen in special cases, when the complex valued centrifugal potential is of
emitting kind. Thus, the positions in the first quadrant of the complex ${\bf
J}$-plane, are given by the values ${\bf J}_{n}$, $n=1,2,..$ for which ${\bf
S}_{\nu^{\prime}\leftarrow\nu}(E,{\bf J}_{n})$ becomes infinite. Following
Macek1 ; JCP2 ; CPC2 ; SPRING  we rewrite Eq.(1) in a different form,
separating the resonance contributions from the slowly varying ’background’
part,

$\displaystyle\sigma_{\nu^{\prime}\leftarrow\nu}(E)=\sum_{n}\sigma_{n}^{res}(E)+\sigma^{back}(E),$
(3)

where the summation is over all resonance poles. Each pole, e.g., the pole
number $n$ whose position is ${\bf J}_{n}$, contributes to the ICS

$\displaystyle\sigma_{n}^{res}(E)=\frac{8\pi^{2}}{k_{\nu}^{2}}\mbox{I}m\frac{\boldsymbol{\lambda}_{n}\boldsymbol{\rho}_{n}(E){\bf
S}^{*}_{\nu^{\prime}\leftarrow\nu}(E,\boldsymbol{\lambda}_{n}^{*})}{1+\exp(-2i\pi\boldsymbol{\lambda}_{n})},$
(4)

where

$\boldsymbol{\lambda}_{n}\equiv{\bf J}_{n}+1/2,$

$\boldsymbol{\rho}_{n}$ is the residue,

$\displaystyle\boldsymbol{\rho}_{n}(E)=lim_{{\bf J}\to{\bf
J}_{n}}(\boldsymbol{\lambda}-\boldsymbol{\lambda}_{n}){\bf
S}_{\nu^{\prime}\leftarrow\nu}(E,\boldsymbol{\lambda}),$ (5)

and ${\bf S}^{*}_{\nu^{\prime}\leftarrow\nu}(E,\boldsymbol{\lambda}_{n}^{*})$
the complex conjugate of ${\bf S}_{\nu^{\prime}\leftarrow\nu}$, evaluated at
the point conjugate to the poles own position, ${\bf J}_{n}^{*}$.

The background part, $\sigma^{back}(E)$, consists of two terms ($\lambda\equiv
J+1$)

$\displaystyle\sigma^{back}(E)=\frac{2\pi}{k_{\nu}^{2}}\int_{J_{min}}^{\infty}|{\bf
S}_{\nu^{\prime}\leftarrow\nu}(E,\lambda)|^{2}\lambda
d\lambda+I_{\nu^{\prime}\leftarrow\nu}(E).\quad\quad\quad$ (6)

The first one is just what one would get by replacing the sum in (1) by an
integral over $J$. The second one contains integrals along the imaginary
$\boldsymbol{\lambda}$-axis, and is expected to be both small and smooth.
Normally there is no need to evaluate it explicitly, since, if needed, it can
be obtained by subtracting $\sum_{n}\sigma_{n}^{res}(E)$ and the integral in
Eq.(6) from the exact ICS (1) .

The usefulness of the Mulholland formula (3-6) can be illustrated by the
following example. Like a CE pole, a CAM pole cannot vanish suddenly. Rather,
it moves in a continuos manner as the energy changes, describing what is known
as Regge trajectory SE1 . Consider a pole ${\bf J}_{0}(E)$ with a small
residue and so close to the real ${\bf J}$-axis, that it makes to ${\bf
S}_{\nu^{\prime}\leftarrow\nu}(J)$ a contribution, whose width is much smaller
than $1$. If at an energy $E$ the pole happens to lie close to an integer
value, say, of $\mbox{R}e{\bf J}_{0}(E)\approx 5$, the fifth partial wave in
Eq.(1) will be enhanced, and the value of the ICS will increase as a result.
Change the energy a little, so that the narrow shape now fits between $J=5$
and $J=6$, and neither of the two partial waves is enhanced. The value of the
ICS will fall back, and will increase again as the pole passes near the next
integer value of $\mbox{R}e{\bf J}_{0}(E)\approx 6$.

The first term in Eq.(6) is an integral over $J$, rather than a sum. It will
always receive a similar contribution from the Regge pole, and, therefore, be
a smooth function of energy. The rapid variations of the ICS are now contained
in the first term in Eq.(3), which will be enhanced each time the pole passes
near an integer, where its denominator has a minimum Macek1 . Thus, a Regge
trajectory passing near the real ${\bf J}$-axis, $\mbox{I}m{\bf J}_{n}<<1$,
produces in the ICS a series of sharp features, centred at the energies
$E_{n}^{K}$ at which the trajectory is close to a real integer value of $J$,
$\mbox{R}e{\bf J}_{n}(E_{n}^{K})=K$, $K=0,1,2,...$ JCP2 ,

$\displaystyle\sigma_{n}^{res}(E)\approx\frac{-2\pi}{k_{\nu}^{2}}\mbox{R}e\sum_{K}\frac{\boldsymbol{\gamma}_{n}^{K}}{E-E_{n}^{K}+i\Gamma^{K}_{n}/2}\equiv\sum_{K}\sigma_{n,K}^{res}(E),$
(7)

where

$\displaystyle\boldsymbol{\gamma}_{n}^{K}\equiv\boldsymbol{\lambda}_{n}\boldsymbol{\rho}_{n}{\bf
S}^{*}_{\nu^{\prime}\leftarrow\nu}(E,\boldsymbol{\lambda}_{n}^{*})/\partial_{E}\mbox{R}e\boldsymbol{\lambda}_{n}|_{E=E_{n}^{K}},$
(8)

and the peak’s width at the $K$-th energy, $\Gamma_{n}^{K}$, is given by

$\displaystyle\Gamma_{n}^{K}\equiv
2\mbox{I}m\boldsymbol{\lambda}_{n}/\partial_{E}\mbox{R}e\boldsymbol{\lambda}_{n}|_{E=E_{n}^{K}}.$
(9)

We note that $\sigma_{n,K}^{res}(E)$ may have either sign, and is not, in
general symmetric around the energies $E_{n}^{K}$. Equation (7) is a variant
of the celebrated Fano line shape formula FANO1 ; FANO2 ; FANO . Like the Fano
result, $\sigma_{n,K}^{res}(E)$ in Eq.(7) reduces to a Breit-Wigner Lorentzian
shape for $\mbox{R}e[\boldsymbol{\gamma}_{n}^{K}]=0$, and is antisymmetric
about $E=E^{K}_{n}$ for $\mbox{I}m[\boldsymbol{\gamma}_{n}^{K}]=0$.

For a pole passing at a larger distance from the real ${\bf J}$-axis, the
sequence of Fano-like features in the ICS is replaced by smoother sinusoidal
oscillations whose maxima do not necessarily coincide with $E_{n}^{K}$, and we
have JCP2

$\displaystyle\sigma_{n}^{res}(E)\approx\frac{4\pi^{2}}{k_{\nu}^{2}}|\boldsymbol{\lambda}_{n}\boldsymbol{\rho}_{n}{\bf
S}^{*}_{\nu^{\prime}\leftarrow\nu}(E,\boldsymbol{\lambda}_{n}^{*})|$ (10)
$\displaystyle\times\exp(-2\pi\mbox{I}m\boldsymbol{\lambda}_{n})\sin[2\pi\mbox{R}e\boldsymbol{\lambda}_{n}(E)+\phi(E)],$

where $\phi(E)$ is the phase of the product between the moduli signs,

$\displaystyle\phi(E)\equiv\mbox{Arg}\\{\boldsymbol{\lambda}_{n}\boldsymbol{\rho}_{n}{\bf
S}^{*}_{\nu^{\prime}\leftarrow\nu}(E,\boldsymbol{\lambda}_{n}^{*})\\}.$ (11)

Equation (10) helps illustrate a general trend: a resonance contribution
increases with the increase of the residue $\boldsymbol{\rho}$, and is rapidly
quenched, owing to the presence of
$\exp(-2\pi\mbox{I}m\boldsymbol{\lambda}_{n})$, if a trajectory moves deeper
into the complex ${\bf J}$-plane.

Finally, for a pole further away from the real axis, $\mbox{I}m{\bf
J}_{n}\gtrsim 1$, the resonance term will vanish, unless its residue is
extremely large. Such pole may, however, continue to influence angular
distributions, more sensitive to the presence of resonances (see, for example,
G1 ).

To use the Mulholland formula in Eq.(3) one needs to know the behaviour of the
${\bf S}$-matrix element in the complex ${\bf J}$-plane. For this purpose we
construct a Padé approximant from the known values of ${\bf
S}_{\nu^{\prime}\leftarrow\nu}$ at integer values of $J$ using the package
ICS_Regge CPC2 . The approximant analytically continues the ${\bf S}$ matrix
element into the complex ${\bf J}$-plane, and correctly represents it in the
vicinity of the integral $J$ values used in its construction. For resonance
poles sufficiently close to the real ${\bf J}$-axis we can, therefore,
evaluate all relevant quantities in Eqs.(3)-(6). The poles further away from
the real axis may not be represented correctly. But this does not matter,
since the resonance contributions of such poles vanish due to the rapidly
growing factor $\exp(-2i\pi\boldsymbol{\lambda}_{n})$ in Eq.(4). For more
information about Padé reconstruction of numerical scattering matrix we refer
the reader to Refs. G1  and PADE2 .

##  V The higher energy range: the resonance A and its Regge oscillations

Next we apply the analysis just described to the three reactive transitions in
(2) in the higher energy range $78.5$ $meV<E<198.5$ $meV$. In this range, all
three ICS’s show oscillatory structures superimposed on a slowly varying
background (see Figs. 3a, 3c, and 3e).

Figure 3:  a) The ICS for the $(3,0,0)\leftarrow(0,0,0)$ reactive transition
(solid). Also shown is the difference $\sigma(E)-\sigma^{res}(E)$ (dashed); c)
same as a) but for the $(3,1,1)\leftarrow(0,0,0)$ transition; e) same as a)
but for the $(3,2,2)\leftarrow(0,0,0)$ transition ; b) the resonance
contribution $\sigma^{res}(E)$ to the transition in a), exact (solid) and
given by Eq.(10) (dashed); d) same as b) but for the transition in c); f) same
as b) but for the transition in e).

With the help of the ICS_Regge program CPC2  we found three significant Regge
trajectories in the chosen energy range, one for each transition. Both real
and imaginary parts of the pole positions increase with energy, as the
trajectory moves deeper into the complex ${\bf J}$-plane. We note that the
trajectories obtained for each of the three transitions (2), shown in Fig. 4a,
coincide to almost graphical accuracy, indicating that all the transitions are
affected by the same resonance. Like a CE pole Land , a Regge pole affects, in
principle, all elements of an ${\bf S}$-matrix. A degree to which an element
is affected depends on its residue at the pole in question. For many
transitions, the residue is zero, or extremely small, so that only particular
ro-vibrational manifolds (e.g., $v^{\prime}=3$, $j^{\prime}=0,1,2$ in this
study) are affected by the same pole. Finding a serious discrepancy between
the Regge trajectories obtained for the affected transitions, would indicate a
defect of the numerical method used for their evaluation. Finding all three in
good agreement is an indication that the method works correctly.

Figure 4b shows the behaviour of the three residues whose magnitudes, after an
initial rapid increase tend to level off at higher energies. The resonance
contributions to the ICS’s, $\sigma^{res}(E)$, shown in Figs.3b, 3d and 3f,
exhibit modulated Regge oscillations well described by Eq.(10). At lower
energies, they are limited by the magnitude of the corresponding residue,
while at higher energies they vanish as the imaginary part of the pole
position increases [cf. Eq.(10)]. This is a behaviour typical for a Regge
trajectory passing not too close to the real axis JCP2 . We note that Regge
oscillations for different transitions are in phase and, for this reason, they
remain visible in the reactive ICS summed over the rotational quantum numbers
and helicities D2 .

Figure 4:  a) Regge trajectories for the transitions in (2) in the higher
energy region; b) residues for three trajectories shown in a).

Comparison with Fig. 2 suggests that the resonance responsible for the
patterns in Figs. 3a, 3c and 3e is the one we earlier called A. In the next
Section we will provide a more rigorous proof.

##  VI The relation between complex energies and Regge poles

The poles of ${\boldsymbol{S}}$-matrix in the complex ${\boldsymbol{E}}$\- and
${\bf J}$-planes, introduced in Sects. III and IV, are intimately related. An
${\bf S}$-matrix element is a function of two independent variables, $E$ and
$J$, which enters the Schroedinger equation in the combination
$\Lambda=J(J+1)$. Treating $E$ as a real or complex parameter, we recall that
a pole at some ${\boldsymbol{{\Lambda}}}_{n}={\bf J}_{n}({\bf J}_{n}+1)$,
${\bf S}_{\nu^{\prime}\leftarrow\nu}(E,{\bf J}_{n})=\infty$, occurs if the
solution of the Schroedinger equation, regular at the origin, contains only
outgoing waves at infinity REG1 . Thus, ${\boldsymbol{{\Lambda}}}_{n}$ is a
complex eigenvalue of a Sturm-Liouville problem IH1 ; Macek1 . Typically,
there are infinitely many such eigenvalues
${\boldsymbol{{\Lambda}}}_{n}({\boldsymbol{E}})$, $n=1,2,...$, for each value
of $E$, whether real or complex. Overall,
${\boldsymbol{{\Lambda}}}({\boldsymbol{E}})$ can be seen as a single-valued
function defined on a multi-sheet Riemann surface $\mathcal{R}$, whose sheets
may be connected, e.g., at branching points. The $n$-th Regge trajectory,
${\bf J}_{n}(E)$, can then be found by moving along the real $E$-axis on the
$n$-th Riemann sheet, and reading off the values of
${\boldsymbol{{\Lambda}}}(E)$ as we go. Inverting
${\boldsymbol{{\Lambda}}}({\boldsymbol{E}})$ we have a function
${\boldsymbol{\mathcal{E}}}(\boldsymbol{{\Lambda}})$, single-valued on its own
Riemann surface $\mathcal{R^{\prime}}$ which contains the information about
the position of complex energy poles. Although neither
${\boldsymbol{{\Lambda}}}({\boldsymbol{E}})$ nor $\mathcal{R}$ are usually
known, the relation between CE and Regge poles can be explored locally, that
is, for a single isolated trajectory close to the real axis, and in limited
ranges of ${\boldsymbol{E}}$ and ${\boldsymbol{J}}$.

Below we consider this simple case, suspending for a time the subscript $n$
numbering the pole. Suppose we have a segment of the real $\Lambda$-axis,
$\Delta\Lambda$, where, for a CE trajectory
${\boldsymbol{E}}(\Lambda)=E_{1}({\Lambda})+iE_{2}({\Lambda})$, both $E_{1}$
and $E_{2}$ are approximately linear functions of ${\Lambda}$. Since
${\bf\mathcal{E}}({\boldsymbol{{\Lambda}}})$ is analytic, it is,
approximately, a linear function of ${\boldsymbol{{\Lambda}}}$,
${\boldsymbol{E}}({\boldsymbol{\Lambda}})\approx{\bf A}{\Lambda}+{\bf B}$, in
a subset of $\mathcal{R}$ around $\Delta{\Lambda}$, which we will call $w$.
The function ${\bf\mathcal{E}}({\Lambda})$ transforms $w$ into a subset
$w^{\prime}$ of the complex ${\bf E}$-plain, where (we write ${\bf
Z}=Z_{1}+iZ_{2}$)

$\displaystyle\left(\begin{array}[]{rr}E_{1}\\\
E_{2}\end{array}\right)=\left(\begin{array}[]{rr}A_{1}&-A_{2}\\\
A_{2}&A_{1}\end{array}\right)\left(\begin{array}[]{rr}{\Lambda}_{1}\\\
{\Lambda}_{2}\end{array}\right)+\left(\begin{array}[]{rr}B_{1}\\\
B_{2}\end{array}\right).$ (20)

If $E_{2}({\Lambda})$ is sufficiently small, there is also a segment of the
real ${\boldsymbol{E}}$-axis, $\Delta E$, which lies in $w^{\prime}$. The
image of $\Delta E$ in the complex ${\boldsymbol{{\Lambda}}}$-plane can be
obtained by inverting the linear relation (20)

$\displaystyle\left(\begin{array}[]{rr}{\Lambda}_{1}\\\
{\Lambda}_{2}\end{array}\right)=\Delta^{-1}\left(\begin{array}[]{rr}A_{1}&A_{2}\\\
-A_{2}&A_{1}\end{array}\right)\left(\begin{array}[]{rr}E_{1}\\\
E_{2}\end{array}\right)-\Delta^{-1}\left(\begin{array}[]{rr}A_{1}B_{1}+A_{2}B_{2}\\\
A_{1}B_{2}-A_{2}B_{1}\end{array}\right),$ (29)

where $\Delta=A_{1}^{2}+A_{2}^{2}$, and putting $E_{2}=0$. Expressing
${\boldsymbol{J}}$ as a function of ${\boldsymbol{{\Lambda}}}({E})$ yields the
Regge trajectory ${\boldsymbol{J}}(E)$. Alternatively, we may be interested in
how the position of a CE pole ${\bf E}_{n}$ varies with $J$, while $J$ is kept
real, $\mbox{I}mJ=0$. In this case we put ${\Lambda}_{2}=0$ in Eq.(20) and
solve it for ${\boldsymbol{E}}({\Lambda}_{1})$.

Relating CE and CAM pole trajectories can be useful if, for example, the two
trajectories have been obtained independently, and the consistency of the
calculations needs to be checked. It may also be useful if there are many
trajectories of each kind, and one needs to know which CAM trajectory
corresponds to a particular CE one. Finally, the physical mechanism of a
resonance, e.g., capture in the van der Waals well of the exit channel, is
most easily established for CE poles D1 , D2 . The resonance contributions to
observable cross sections, on the other hand, are best described with the help
of CAM trajectories CPC2 . Again, the ability to relate the two types of
trajectories can be helpful.

As an example, we check whether the Regge trajectory shown in Fig.4a for the
$(3,0,0)\leftarrow(0,0,0)$ transition is indeed the CAM counterpart of the CE
trajectory in Fig.2, which we have earlier labelled A. We do it by first
evaluating the CE trajectory for the pole A in the higher energy region
numerically. We then fit the pole position by a linear function of
$\Lambda=J(J+1)$, thus obtaining the values for coefficients $A_{1,2}$ and
$B_{1,2}$ in Eq.(20). These values are inserted into Eq.(29), and the
approximate Regge trajectory for the pole A is drawn. This trajectory is then
compared with the exact Regge trajectory obtained independently, by Padé
reconstruction of the ${\bf S}$-matrix element in the energy range of
interest. A good agreement between the two CAM trajectories shown in Fig. 5
demonstrates the validity of our analysis.

It is instructive to revert the discussion to more physical terms, in the
simple case when the imaginary part of the pole position in the CE plane does
not depend on the angular momentum and is sufficiently small to put $A_{2}=0$
in Eq.(20) FH2 . Now, on the real $J$-axis for $J>>1$ we rewrite (20) as

$\displaystyle E_{1}\approx A_{1}J^{2}+B_{1}\equiv J^{2}/2I+E_{0}$ (30)
$\displaystyle E_{2}\approx B_{2}\equiv-1/\tau,$

where $\tau=1/E_{2}$ is the lifetime of the resonance, which, we assumed, is
the same whether the complex rotates or not. The first of Eqs.(30) is known as
the ’$J$-shifting approximation’ JSH  and is widely used in the discussion of
resonance phenomena D1 ; D2 . In the present context, it means that the energy
of the rotating complex consists of two independent parts: its internal
binding energy $E_{0}$ and the rotational energy of a rigid symmetric top,
$J^{2}/2I$. This makes $I=1/2A_{1}$ the moment of inertia of the metastable
triatomic. On the real $E$-axis, equations (29) now read

$\displaystyle\Lambda_{1}\approx J_{1}^{2}=2I(E-E_{0})$ (31)
$\displaystyle\Lambda_{2}\approx 2J_{1}J_{2}=2I/\tau.$

The first equation, which is just the inverse of the first of the Eqs.(30),
uses the energy conservation to select the angular momentum at which the
complex would be formed for a given energy $E$. The second equation tells us
how far the complex would rotate during its lifetime. The typical angle of
rotation is given by the life-angle $\varphi_{life}$, which is, in this case,
the product of the angular velocity of the complex, $\omega\equiv J_{1}/I$,
and the lifetime of the resonance $\tau$, $\varphi_{life}\equiv
1/J_{2}=\omega\tau$.

Figure 5:  a) Real part of the position of the pole A in the complex energy
plane vs. $\Lambda=J(J+1)$ evaluated by Padé reconstruction of the ${\bf
S}$-matrix element for real integer values of $J$, $J=17,...,27$ (dots), and
its linear fit corresponding to Eq.(20) (solid); b) same as (a), but for the
imaginary part of the pole’s position; c) real part of
${\boldsymbol{{\Lambda}}}(E)$ evaluated for the pole A by Padé reconstruction
of the ${\bf S}$-matrix element for $78.54\quad meV<E<197.54\quad meV$
(solid), and as predicted by Eq.(29) (dashed); d) same as (c), but for the
imaginary part.

In the above analysis we considered a subset of $\mathcal{R}$ well removed
from branching singularities, and small enough to justify using the Taylor
expansion of ${\boldsymbol{E}}(\Lambda)$ truncated after linear terms, so that
the condition defining ${\boldsymbol{E}}({\boldsymbol{{\Lambda}}})$ or
${\boldsymbol{{\Lambda}}}({\boldsymbol{E}})$ can be written as

$\displaystyle{\boldsymbol{F}}({\boldsymbol{E}},{\boldsymbol{{\Lambda}}})\equiv{\bf
a}{\boldsymbol{E}}+{\bf b}{\boldsymbol{{\Lambda}}}+{\bf c}=0,$ (32)

where ${\bf a}$,${\bf b}$ and ${\bf c}$ are complex valued constants. The
approximation (32) would break down, for example if the energies of two
resonances are aligned for certain values of the centrifugal potential. This
is the case for example of the F+H2 reaction FH3 ; DARIO04  where the
alignment of the resonances A and B gives specific effects in the energy
dependences of the reaction probabilities TCA11  and on the angular
distributions FH5  also observed experimentally Sci06 . If this happens, a
branching point of the Riemann surface moves close to the real axis, and can
no longer be ignored. There the function ${\bf
F}({\boldsymbol{E}},{\boldsymbol{{\Lambda}}})$ in (32) must be approximated by
a polynomial quadratic in both ${\boldsymbol{E}}$ and
${\boldsymbol{{\Lambda}}}$, as discussed in details in Refs. FH3 ; ICS3 ; JCP2
. Finally, in the unlikely case of three or more resonances aligned for
certain value of $J$, ${\bf F}$ must be approximated by a polynomial of the
third or higher order, so that a larger number of Riemann sheets will be taken
into account.

##  VII The near-threshold range: resonances B, C, D, and E

The lower energy region, $58.54$ $meV\leq E\leq 78.54$ $meV$, is rich in
resonances, and the resonance patterns in Fig.1a are more diverse (cf. Fig.
1b). We will analyse each of the transitions in (2) in detail after a brief
remark concerning the types of Regge trajectories we are likely to encounter.

Based on an analysis of a simple potential scattering model, in Ref. PLA  it
was shown that the behaviour of a Regge trajectory in the complex
${\boldsymbol{J}}$-plane depends on whether at $J=0$ it correlates with a
metastable state at a complex energy ${\boldsymbol{E}}_{0}=E_{1}+iE_{2}$,
$E_{1}>0$ (trajectory of type I), or with a bound state at a negative energy
$E_{1}<0$ (trajectory of type II). The difference between two kinds of
trajectories can be understood as follows. For a sharp resonance, finding a
Regge pole at an energy $E>0$ amounts to looking for a value of $J$, such that
the centrifugal potential (CP) proportional to $J(J+1)$ would align the real
part of the energy of the metastable state with the chosen $E$. Starting in a
bound state with a negative energy, for any $E>0$ we require a positive CP
capable of lifting the state to the desired level. Thus, for a trajectory of
type $(II)$ the real part of ${\boldsymbol{J}}$ increases with $E$, while
$\mbox{I}m{\boldsymbol{J}}$, at first small, increases as the state is being
forced out of the well. The situation is different if we start from a
metastable state. For energies smaller than $E_{1}$, the state needs to be
lowered, and that requires a negative CP or, equivalently, an imaginary
${\boldsymbol{J}}$. Thus, a type $(I)$ trajectory, for $E<E_{1}$ must first
have a considerable decreasing imaginary part, while its real part should
increase for $E>E_{1}$, where the state will, again, have to be lifted.

This over-simplistic picture is, however, useful, and can be adapted to multi-
channel scattering, currently of interest to us. Let us assume that a
particular transition becomes energetically possible for $E$ above its
threshold energy $E_{thr}$. For $J=0$, some of the many metastable states of
the system will lie below, and some will lie above $E_{thr}$ (in terms of real
parts of their energies). Those lying below will have to be ’lifted’, so that
their Regge trajectories will belong to the type $(II)$ and exhibit a
behaviour similar to that shown in Fig. 4a. Those above $E_{thr}$ will need to
be ’lowered’ initially, and their imaginary parts, will first have to decrease
before, possibly, picking up later.

Figure 6:  a) Real parts of Regge trajectories B, C, $\mbox{D}1$, $\mbox{D}2$,
$\mbox{E}1$ and $\mbox{E}2$ evaluated by Padé reconstruction of the ${\bf
S}$-matrix elements for the transitions listed in (2). Trajectory B is
obtained from the $(3,0,0)\leftarrow(0,0,0)$ transition in the whole energy
range. For other trajectories, the vertical dashed lines mark the energy
ranges where the part of a trajectory was found in a particular transition
$(3,j^{\prime},\Omega^{\prime})\leftarrow(0,0,0)$. Two parts of the
$\mbox{D}2$ trajectory are joined by a dashed line where it does not appear in
the Padé reconstruction. b) the imaginary parts of the trajectories shown in
a).

The Regge trajectories obtained by Padé reconstruction of the ${\bf S}$-matrix
elements for the three transitions in (2) in the lower energy range are shown
in Fig.6. As one would expect (cf. Fig. 2a), for the HF($v^{\prime}$=3)
product the Regge trajectories corresponding to resonances E and D behave as
trajectories of type $(I)$, whereas those corresponding to the resonances B
and C are of type $(II)$. We also note that the resonances D and E are in
reality multiplets, of which only one member was found in the CE Padé
reconstruction of the $(3,0,0)\leftarrow(0,0,0)$ matrix element shown in Fig.
2. The physical origin of the splitting of these resonances has been
investigated in depth in Ref. JCP05  for the case of the F+H2 reaction, where
an inelastic adiabatic model attributes metastable states to bound levels
supported by the adiabatic curves. In the space-fixed (SF) representation, one
can use three quantum numbers, initial $\\{v,j,l\\}$ or final
$\\{v^{\prime},j^{\prime},l^{\prime}\\}$, in order to identify each component
of the multiplet. The last of the quantum numbers, $l$ ($l^{\prime}$) , labels
the orbital angular momentum, and can take integer values from from $|J-j|$ to
$J+j$, allowed by inversion parity conservation. Alternatively, in the body-
frame (BF), a component of the multiplet can be labeled by
$\\{v^{\prime},j^{\prime},\Omega^{\prime}\\}$ quantum numbers, where
$\Omega^{\prime}$ stands for the projection of the hindered rotor angular
momentum $j^{\prime}$ along the principal axis of the BF, which correlates
adiabatically with the bending motion levels of the triatomic complex. In
particular, a simple hyper-spherical model JCP05  applied to the resonance D,
relates the amount of the splitting to the magnitudes of the Coriolis forces.
In the inelastic adiabatic model, the D resonance is trapped in the adiabatic
curve of $j^{\prime}$=1, and it is a doublet, whereas the E resonance has
three different components.

In general, our Padé technique is able to find trajectories whose imaginary
parts are not too large, and whose residues are not too small. The real parts
of the poles positions are usually determined more accurately than their
imaginary parts. (For someone watching a boat from land, it is also easier to
evaluate its displacement to the right or left of the observer than its
distance from the shore). Because the Coriolis effects are small for the
system we study, the higher components of the multiplets (correlating with
higher $\Omega^{\prime}$ states) should be less pronounced in transitions from
the ground rotational state, $j=0$. With smaller residues, we expect these
higher component trajectories to be also harder to find with the help of our
Padé technique.

In Fig. 6 we can distinguish six different trajectories. The trajectory
corresponding to the resonance B is found in the $(3,0,0)\leftarrow(0,0,0)$
and $(3,1,1)\leftarrow(0,0,0)$ transitions in the range $58.54$ to $68.64$
$meV$. The Regge trajectory C is found only in the $(3,0,0)\leftarrow(0,0,0)$
transition in the energy range $58.54$ to $61.94$ $meV$, where two other
transitions are still closed. Both B and C trajectories belong to the type
$(II)$, as expected.

The Regge trajectory corresponding to the CE trajectory D is found in
$(3,0,0)\leftarrow(0,0,0)$ and $(3,1,1)\leftarrow(0,0,0)$ transitions in the
ranges $58.54$ to $63.14$ and $63.14$ to $67.94$ $meV$, respectively. This is
a trajectory of type $(I)$, whose imaginary part shows a characteristic
minimum at about $68.2$ $meV$. As expected, it appears to be a part of a
doublet, so we label it $\mbox{D}1$. The other member of the doublet, labelled
$\mbox{D}2$, is much weaker. We find the evidence of this in the transition
$(3,0,0)\leftarrow(0,0,0)$ between $61.44$ and $63.14$ $meV$, and in the
transition $(3,1,1)\leftarrow(0,0,0)$ between $64.94$ and $67.14$ $meV$.
Although we have been unable to see it for $63.13$ $meV<E<64.94$ $meV$, there
is little doubt that the two curves in Figs. 6a and 6b belong to the same
Regge trajectory.

As in the case of the D resonance, we find two similar Regge trajectories in
the energy range $68.44$ and $78.14$ $meV$, henceforth called $\mbox{E}1$ and
$\mbox{E}2$. Trajectory $\mbox{E}1$ is the Regge counterpart of the CE
trajectory E in Fig. 2 and is seen in all three transitions, contributing
various parts of the curve in Figs. 6a and 6b. Like $\mbox{D}1$, it is a
trajectory of type $(II)$. We see the trajectory $\mbox{E}2$ in the
$(3,2,2)\leftarrow(0,0,0)$ transition for $E$ between $72.94$ and $77.54$
$meV$. It is weaker than $\mbox{E}1$ and its imaginary part can be determined
less accurately.

With the list of relevant Regge trajectories complete, we proceed to identify
the trajectories responsible for various patterns in the ICS’s in Fig.1.

###  VII.1 The $(3,0,0)\leftarrow(0,0,0)$ transition

This transition, which opens at a collision energy of $58.19$ $meV$, and
covers all the lower energy range, is affected by most of the Regge
trajectories shown in Fig. 6. Among the patterns, whose origin we need to
explain, there are sharp peaks at $E\approx 59$, $61$ and $63.2$ $meV$,
smoother humps at $E\approx 59.8$, and $65.5$ $meV$, and a jugged structure
between $69.5$ and $74$ $meV$ (see Fig.7a ). There is also a $W$-shaped
feature at $E\approx 62.15$ $meV$. The results of our analysis are shown in
Figs.7 b, c, and d.

Trajectory B contributes to the ICS the second and the third sharp peaks,
together with the hump at $65.5$ $meV$, as is shown in Fig.7c. All three
features arise as the trajectory B passes near a real integer value
$\mbox{R}e{\boldsymbol{J}}_{A}=9$, $10$ and $11$. As
$\mbox{I}m{\boldsymbol{J}}_{A}$ increases, the features become less
pronounced. For comparison we have plotted by a dashed line the Breit-Wigner
special case ($\mbox{R}e[\boldsymbol{\gamma}_{n}^{K}]=0$) of the Fano
approximation (7) including only the $K=9$, $10$ and $11$ terms. As expected,
the approximation correctly describes the peak with the smallest
$\mbox{I}m{\bf J}_{A}$, and becomes much less accurate at higher energies.
Trajectory C is similar to the trajectory B, and is responsible for the first
peak and the first hump at $\mbox{R}e\boldsymbol{J}_{C}=5$ and $6$,
respectively (see Fig. 7b). Figure Fig. 7d shows that trajectory $\mbox{E}1$
is responsible for the saw-like structure between $69.5$ and $74$ $meV$, where
its imaginary part is at minimum (cf. Fig. 6b). The maxima of
$\sigma^{res}_{E1}(E)$ coincide with the energies at which
$\mbox{R}e{\boldsymbol{J}}_{E1}=3,4...8$. The structure is also reasonably
well described by the Fano formula (7), shown in Fig. 7d by a dashed line.

Subtracting from $\sigma_{(3,0,0)\leftarrow(0,0,0)}(E)$ resonance
contributions from trajectories B,C, and $\mbox{E}1$ does not yet give us a
smooth structureless background (see Fig. 7a). There is still the $W$-shaped
structure at $61.15$ $meV$. Another such structure, previously hidden by the
large peak at $62.15$ $meV$, has appeared at $E\approx 63$ $meV$, together
with additional dips at $60.5$, $61$ and, $61.5$ $meV$. All these features are
the resonance contributions from weak resonances $\mbox{D}1$ and $\mbox{D}2$,
whose trajectories pass close to the real axis (see Fig. 6b), and which only
contribute when $\mbox{R}e{\boldsymbol{J}}_{D1,D2}$ is very close to an
integer value $K$. Since the energy grid of $0.1$ $meV$ used in these
calculations is relatively coarse, the contributions appears at only one
energy in the vicinity of $E^{K}$, producing there a $V$-shaped dip. Two such
dips close to each other yield the $W$-like patterns seen in Fig. 7a. To show
that this is the case, in Fig. 8 we increased the energy scale and plotted the
resonance contributions from the trajectories $\mbox{D}1$ and $\mbox{D}2$,
which affect $\sigma_{(3,0,0)\leftarrow(0,0,0)}(E)$ at the energies $E^{K}$
where $\mbox{R}e{\boldsymbol{J}}_{D1}=2,3,4,5,6$ and
$\mbox{R}e{\boldsymbol{J}}_{D2}=3,4$. Subtracting all resonance contributions
from the exact ICS now gives us a reasonably smooth background, and this
concludes out analysis of the $(3,0,0)\leftarrow(0,0,0)$ transition.

Figure 7:  a) Integral cross sections $\sigma_{\nu^{\prime}\leftarrow\nu}(E)$,
$\nu=(0,0,0)$, $\nu^{\prime}=(3,0,0)$, in the lower energy range (solid). Also
shown by a thick solid line is
$\sigma_{\nu^{\prime}\leftarrow\nu}-\sigma_{B}^{res}-\sigma_{C}^{res}-\sigma_{E1}^{res}$;
b) resonance contribution from the trajectory C, $\sigma_{C}^{res}(E)$. The
maxima of $\sigma_{C}^{res}(E)$ appear at $E^{K}$ such that $\mbox{R}e{\bf
J}_{C}(E^{K})\approx K$, $K=5,6$; c) same as b) but for the trajectory B. The
dashed line corresponds to the Fano approximation (7); d) same as b) but for
the trajectory $\mbox{E}1$. Figure 8: a) Integral cross sections
$\sigma_{\nu^{\prime}\leftarrow\nu}-\sigma_{B}^{res}-\sigma_{C}^{res}-\sigma_{E1}^{res}$
(solid) and
$\sigma_{\nu^{\prime}\leftarrow\nu}-\sigma_{B}^{res}-\sigma_{C}^{res}-\sigma_{E1}^{res}-\sigma_{D1}^{res}-\sigma_{D2}^{res}$
(dashed) for the same transition of Fig. 7. ; b) resonance contributions from
the trajectories $\mbox{D}1$ and $\mbox{D}2$, $\sigma_{D1,D2}^{res}(E)$. The
maxima of $\sigma_{D1,D2}^{res}(E)$ appear at $E^{K}$ such that $\mbox{R}e{\bf
J}_{C}(E^{K})\approx K$.

###  VII.2 The $(3,1,1)\leftarrow(0,0,0)$ transition

This transition opens at $E=62.72$ $meV$ and the corresponding ICS cannot,
therefore, be affected by the resonance C (see Fig. 6). We see that it is also
not influenced by the resonance B, whose residues are very small. The two
peaks and a hump in Fig. 9a come from the resonance $\mbox{D}1$ (see Fig. 9b)
as its Regge trajectory passes near integer values $K=6$, $7$ and $8$. The
plot of $\sigma_{\nu^{\prime}\leftarrow\nu}-\sigma_{D1}^{res}$ retains small
slow oscillations which can be attributed to the resonance $\mbox{D}2$, but we
will not follow the matter further. At slightly higher energies, the saw-like
structure at $69$ $meV$ $<E<$ $73$ $meV$ comes from the resonance $\mbox{E}1$,
and is similar to the pattern seen in the ICS of the
$(3,0,0)\leftarrow(0,0,0)$ transition (cf. Fig. 7d).

One difference is that after subtracting $\sigma_{E1}^{res}$ from the exact
ICS, the resulting curve still contains a cusp at $\approx 68.7$ $meV$, as
well as some minor oscillations (see Fig. 9c). To understand the nature of the
these features, we also plot in Fig. 9c the integral term in Eq.(6) (shown by
a dot-dashed line). The dot-dashed curve also contains the cusp, which must,
therefore, be attributed to the background term $\sigma^{back}$ in Eq.(6). In
Ref. PLA , we used a simple one-channel model to argue that $\sigma^{res}$ in
Eq.(6) does not contain the entire resonance contribution, so that a part of
it remains included in $\sigma^{back}$. The cusp in Fig.9a is likely to be
such a feature, produced in $\sigma^{back}$ by the resonance $\mbox{D}1$.

Figure 9:  a) Integral cross section $\sigma_{\nu^{\prime}\leftarrow\nu}(E)$,
$\nu=(0,0,0)$, $\nu^{\prime}=(3,1,1)$, in the lower energy range (solid). Also
shown by a dashed line is
$\sigma_{\nu^{\prime}\leftarrow\nu}-\sigma_{D1}^{res}$; b) the resonance
contribution from the trajectory $\mbox{D}1$, $\sigma_{D1}^{res}(E)$; c) the
same cross sections in a different energy range. Also shown are
$\sigma_{\nu^{\prime}\leftarrow\nu}-\sigma_{E1}^{res}$ (dashed), and the
integral in Eq.(6) (dot-dashed); d) the resonance contribution from the
trajectory $\mbox{E}1$, $\sigma_{E1}^{res}(E)$.

###  VII.3 The $(3,2,2)\leftarrow(0,0,0)$ transition

This transition opens at a higher energy, $E=71.77$ $meV$, and can only be
affected by the resonances $\mbox{E}1$ and $\mbox{E}2$. The resonance
$\mbox{E}1$ contributes to the ICS two Lorentzian peaks, the first of which is
accurately described by the formula (7) (see Fig. 10b). The peaks occur as
$\mbox{R}eJ_{E1}$ approaches the values $K=8$ and $9$, respectively. The
contribution from the trajectory $\mbox{E}2$ is about five times smaller, and
consists of a Fano-like shape at $E\approx 72.05$ $meV$ and a Lorentzian peak
at $E\approx 74.25$ $meV$ (see Fig. 10c). The smooth background cross section
$\sigma^{back}$ is shown in Fig.10a.

Figure 10:  a) Integral cross sections
$\sigma_{\nu^{\prime}\leftarrow\nu}(E)$, $\nu=(0,0,0)$,
$\nu^{\prime}=(3,2,2)$, in the lower energy range (solid). Also shown by a
dashed line is
$\sigma_{\nu^{\prime}\leftarrow\nu}-\sigma_{E1}^{res}-\sigma_{E2}^{res}$; b)
resonance contribution of the trajectory $\mbox{E}1$, $\sigma_{E1}^{res}(E)$.
The dashed line corresponds to the Fano approximation (7); c) same as b) but
for the trajectory $\mbox{E}2$.

##  VIII Conclusion and discussion

In summary, there are several resonances which affect the behaviour of the
integral cross sections of the title reaction in the energy range $58.54$ to
$197.54$ $meV$ D2 . Some have been identified earlier in D1  as one transition
state resonance (A) and several Feshbach resonances arising from the capture
in metastable states of the van der Waals well in the exit channel (B,C, D and
E). In this work we present a quantitative CAM analysis of the effects
produced on state-to-state ICS of the title reaction. As the input, we use the
scattering matrix elements evaluated with the FXZ potential energy surface
(PES) PES4 . This recent ab initio PES is known to accurately reproduce many
resonance features experimentally observed for the F+HD reaction EXP6 ; EXP7 ;
EXP8 ; EXP9 .
Positions and widths of the metastable states affecting the state-to-sate
integral cross sections, obtained by Padé reconstruction of the $S$-matrix
element in the complex energy plane, are shown in Fig. 2 for different values
of the total angular momentum $J$. The behaviour of the complex energy poles
of the resonances A,B,C and D are in good agreement with those obtained by
$Q$-matrix analysis in D1 , notwithstanding significant differences in the
exit Van der Waals well of the PES’s employed in D1  and the present work (for
a comparison, see Fig. 1 of Ref.D2 ). There is also a resonance E at somewhat
higher energies, not discussed by the authors of D2 . The similarity of the
resonance poles structures obtained for different PES’s, and the high energy
resolution achieved in the state of the art molecular beam experiments EXP6 ;
EXP7 ; EXP8 ; EXP9 , suggest that the resonance effects, studied in this work,
should be accessible to experimental observation. Such an experiment would
provide a highly sensitive test of the accuracy of quantum chemistry
calculations.

However, the knowledge of complex energies is not enough to quantify the
effects which the resonances produce in the state-to-state integral cross
sections. For this purpose, we require the Regge trajectories corresponding to
the resonances A,B,C, D and E, which are shown in Figs. 4 and 6. Our analysis
demonstrates that the resonances D and E are multiplets, confirming the
splitting phenomenon of bending excited metastable states studied in JCP05
for the F+H2 reaction. The Regge trajectories exhibit two different types of
behaviour PLA . While their real parts tend to always increase with energy,
the imaginary parts may increase (resonances A, B and C), or first decrease
with the energy (resonances $\mbox{D}1$ and $\mbox{E}1$), depending on whether
the position of the corresponding metastable state at $J=0$,
$\mbox{R}e{\boldsymbol{E}}_{X}(J=0)$, $X=\mbox{A},\mbox{B},\mbox{C}...$ lies
above or below the energy threshold at which the channel opens. The
explanation is similar to the one given in PLA : lowering the state to an
energy $E<\mbox{R}e{\boldsymbol{E}}_{X}(J=0)$ requires a negative centrifugal
potential and, therefore, a large $\mbox{I}m{\boldsymbol{J}}$ (cf. Figs.2 and
6) . Raising the state to an energy $E>\mbox{R}e{\boldsymbol{E}}_{X}(J=0)$
requires a positive centrifugal potential. However, a large centrifugal
barrier also alters the overall potential, which can no longer support long-
lived metastable states, whose lifetimes and life angles must, therefore,
decrease. As a result, the Regge trajectories move deeper into the complex
${\boldsymbol{J}}$-plane, and cease to affect the ICS’s as energy increases.
Accordingly, resonance effects remain predominantly a low-energy phenomenon.

It is worth mentioning the close relation which exists between complex
energies and Regge poles, so that the knowledge of a trajectory of one type
may allow reconstruction of its other counterpart. We considered the simplest
case of this relation in Sect. V, and used it to relate the CE and the Regge
trajectories of the resonance A in the higher energy region (see Fig.5).

Having identified the relevant Regge trajectories, we followed them to
evaluate the contributions they make to the ICS of the chosen transition.
These contributions may take various forms depending on the proximity of the
trajectory to the real ${\boldsymbol{J}}$-axis, and the magnitude of its
residue. In the higher energy range a single trajectory corresponding to the
resonance A is responsible for modulated sinusoidal oscillations [see Eq.(10)]
for all transitions considered (see Fig.3). The oscillations for different
transitions are in phase, and the resonance pattern survives summation over
helicities and rotational quantum numbers D2 . This allows these oscillations
to be visible in recent molecular beams experiments EXP7 . In the lower energy
range, for the $(3,0,0)\leftarrow(0,0,0)$ transition, we assign the first two,
and the subsequent three peaks, to the trajectories C and B, respectively. The
first peaks of the sequences have nearly Lorentzian shapes and are well
described by the Fano-like formula (7). A similar sequence of peaks is
contributed to the ICS of the $(3,1,1)\leftarrow(0,0,0)$ transition by the
trajectory $\mbox{D}1$. The contribution of the trajectory $\mbox{E}1$ to
$\sigma_{(3,0,0)\leftarrow(0,0,0)}(E)$ consists of seven asymmetric Fano
shapes, which are produced while its imaginary part is small. A similar
pattern is produced by it in $\sigma_{(3,1,1)\leftarrow(0,0,0)}(E)$, where the
Fano shapes are inverted, with initial gradual rise followed by a sharp fall.
The trajectory $\mbox{E}1$ also affects the $(3,2,2)\leftarrow(0,0,0)$
transition, although in a different manner. There it produces two nearly
Lorentzian peaks, followed by a low broad hump. The $(3,2,2)\leftarrow(0,0,0)$
ICS is also affected by a much weaker $\mbox{E}2$ resonance, with
$\sigma^{res}_{(3,2,2)\leftarrow(0,0,0)}(E)$ consisting of an asymmetric Fano
shape followed by a Lorentzian peak.

Finally, the accuracy of a Padé reconstruction depends on the accuracy of the
input data and on the numerical format (in this work, exponential with four
significant digits) they are stored in. In general, the quality of the input
data was found sufficient to accurately reproduce a Regge trajectory, with its
different parts consistently reproduced in the analysis of different
transitions. Determination of weak (small residue) trajectories passing close
to the real axis proved to be most difficult. Such are the trajectories
$\mbox{D}1$ and $\mbox{D}2$, which make a notable contribution only at
$E\approx E^{K}$ when ${\boldsymbol{J}}_{X}$ is close to an integer $K$, where
$\mbox{I}m{\boldsymbol{J}}_{X}$ can be determined with a good accuracy. For
energies between $E^{K}$, $\mbox{I}m{\boldsymbol{J}}_{X}$ found by Padé
reconstruction fluctuates. Fortunately, this is not important, since there the
contribution of such a trajectory vanishes. Thus, we can connect
$\mbox{I}m{\boldsymbol{J}}_{X}(E^{K})$ by a smooth line, as was done for the
resonances $\mbox{D}1$ and $\mbox{D}2$ in Fig. 6).

To conclude, we would like to emphasise that the present CAM approach is
suitable for analysing state-to-state ICS for transitions with non-zero
initial and final helicities $\Omega$ and $\Omega^{\prime}$. This makes it a
convenient tool for investigating resonance effects in a broad class of
reactions, e.g., those involving ions and molecules, where recent time
independent close coupling studies He12 ; DDF14  have found a large number of
non-zero helicity states contributing to the integral cross sections.

##  IX Acknowledgements:

DS acknowledges support of the Basque Government (Grant No. IT-472-10), and
the Ministry of Science and Innovation of Spain (Grant No.
FIS2009-12773-C02-01). EA is grateful for support through BCAM Severo Ochoa
Excellence Accreditation SEV-2013-0323 and the MINECO project
MTM2013-46553-C3-1-P. DDF wants to give a particular acknowledgement to the
Basque Foundation for Science for an Ikerbasque Visiting Fellowship grant in
the University of the Basque Country of Leioa (Bilbao), during which most of
the work was done. DDF also thanks the Italian MIUR for financial support
through the PRIN 2010/2011 grant N. 2010ERFKXL. The SGI/IZO-SGIker UPV/EHU and
I2BASQUE are acknowledged for providing computational resources. Quantum
reactive scattering calculations ware performed under the HPC-EUROPA2 project
(project number:228398) with support of the European Commission - Capacities
Area- Research. Some of the computational time was also supplied by CINECA
(Bologna) under ISCRA project N. HP10CJX3D4. In addition, the authors are
grateful to Prof. J.N.L. Connor and Prof. V. Aquilanti for useful discussions.

## References

  * (1) Cold Molecules: Theory, Experiments, Applications. Ed. by R.V. Krems, W.C. Stwalley, B. Friedrich. CRC Press, Boca, USA, 2009.
  * (2) S. Cavalli, V. Aquilanti, K.C. Mundim, D. De Fazio, J. Phys. Chem. A 118 (2014) 6632.
  * (3) V. Aquilanti, K.C. Mundim, M. Elango, S. Kleijn, T. Kasai. Chem. Phys. Lett., 498, (2010) 209.
  * (4) V. Aquilanti, K.C. Mundim, S. Cavalli, D. De Fazio, A. Aguilar, J.M Lucas, . Chem. Phys. , 398, (2012) 186
  * (5) J.N.L. Connor, Chem. Soc. Rev. 5 (1976) 125.
  * (6) J.N.L. Connor, D. Farrelly, D.C. Mackay, J. Chem. Phys. 74 (1981) 3278.
  * (7) K.-E. Thylwe, J.N.L. Connor, J. Phys. A 18 (1985) 2957.
  * (8) J.N.L. Connor, D.C. Mackay, K.-E. Thylwe, J. Chem. Phys. 85 (1986) 6368.
  * (9) D. Bessis, A. Haffad, A.Z. Msezane, Phys. Rev. A. 49 (1994) 3366.
  * (10) J.N.L. Connor, K.-E. Thylwe, J. Chem. Phys. 86 (1987) 188.
  * (11) P. McCabe, J.N.L. Connor, K.-E. Thylwe, J. Chem. Phys. 98 (1993) 2947.
  * (12) D. Sokolovski, J.N.L. Connor, G.C. Schatz, Chem. Phys. Lett. 238 (1995) 127.
  * (13) D. Sokolovski, J.N.L. Connor, G.C. Schatz, J. Chem. Phys. 103 (1995) 5979.
  * (14) D. Sokolovski, J.F. Castillo, C. Tully, Chem. Phys. Lett. 313 (1999) 225.
  * (15) D. Sokolovski, J.F. Castillo, Phys. Chem. Chem. Phys. 2 (2000) 507.
  * (16) F.J. Aoiz, L. Bañares, J.F. Castillo, D. Sokolovski, J. Chem. Phys. 117 (2002) 2546.
  * (17) D. Sokolovski, Chem. Phys. Lett. 370 (2003) 805.
  * (18) D. Sokolovski, S.K. Sen, V. Aquilanti, S. Cavalli, D. De Fazio, J. Chem. Phys. 126 (2007) 084305.
  * (19) D. Sokolovski, D. De Fazio, S. Cavalli, V. Aquilanti, Phys. Chem. Chem. Phys. 9 (2007) 5664.
  * (20) D. Sokolovski, Phys. Scr. 78 (2008) 058118.
  * (21) J.N.L. Connor, J. Chem. Phys. 138 (2013) 124310.
  * (22) J. N. L. Connor, J. Chem. Soc. Faraday Trans. 305 (1990) 1627.
  * (23) P.G. Burke, C. Tate, Comput. Phys. Commun. 1 (1969) 97.
  * (24) D. Sokolovski, Phys. Rev. A. 62 (2000) 024702.
  * (25) D. Sokolovski, A.Z. Msezane, Z. Felfli, S.Yu. Ovchinnikov, J.H. Macek, Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 261 (2007) 133.
  * (26) D. Sokolovski, E. Akhmatskaya, Phys. Lett. A 375 (2011) 3062.
Please note that there is a typo in Eq.(5), which should read  $\displaystyle
S(k,\lambda)=\Delta^{(2)}(E,\lambda)/\Delta^{(1)}(E,{\lambda})\equiv\quad\quad\quad\quad\quad\quad$
$\displaystyle-\frac{H^{(2)}_{\lambda}(kr)\\{[\ln
H^{(2)}_{{\lambda}}(kr)]^{\prime}-[\ln\phi_{well}(r,k,{\lambda})]^{\prime}-2\Omega\\}}{H^{(1)}_{\lambda}(kr)\\{[\ln
H^{(1)}_{{\lambda}}(kr)]^{\prime}-[\ln\phi_{well}(r,k,{\lambda})]^{\prime}-2\Omega\\}}{\biggr{\rvert}}_{r=R},$
with $\phi(r)=A(k,{\lambda})\sqrt{r}\phi_{well}(r)$ and
$q\equiv(k^{2}+2V)^{1/2}$.

  * (27) D. Sokolovski, Z. Felfli, A.Z. Msezane, Phys. Lett. A 376 (2012) 733.
  * (28) J.H. Macek, P.S. Krstić, S.Yu. Ovchinnikov, Phys. Rev. Lett. 93 (2004) 183203.
  * (29) S.Yu. Ovchinnikov, P.S. Krstić, J.H. Macek, Phys. Rev. A. 74 (2006) 042706.
  * (30) H.P. Mulholland, Proc. Cambridge Philos. Soc. 24 (1928) 280.
  * (31) D. Sokolovski, D. De Fazio, S. Cavalli, V. Aquilanti, J. Chem. Phys. 126 (2007) 12110.
  * (32) D. Sokolovski, Z. Felfli, S.Yu. Ovchinnikov, J.H. Macek, A.Z. Msezane, Phys. Rev. A 76 (2007) 012705.
  * (33) Z. Felfli, A.Z. Msezane, D. Sokolovski, J. Phys. B. 45 (2012) 045201.
  * (34) G.A. Baker Jr., The Essentials of Padé Approximations, Academic, New York, 1975.
  * (35) R. Piessens, E. de Doncker-Kapenga, C.W. Überhuber, D. Kahaner, QUADPACK, A Subroutine Package for Automatic Integration, Springer-Verlag, 1983.
  * (36) E. de Doncker-Kapenga, ACM SIGNUM Newsl. 13 (1978) 12.
  * (37) P. Wynn, Math. Tables Aids Comput. 10 (1956) 91.
  * (38) D. Sokolovski, A.Z. Msezane, Phys. Rev. A. 70 (2004) 032710.
  * (39) D. Vrinceanu, A.Z. Msezane, D. Bessis, J.N.L. Connor, D. Sokolovski, Chem. Phys. Lett. 324 (2000) 311.
  * (40) D. Sokolovski, S. Sen, Semiclassical and Other Methods for Understanding Molecular Collisions and Chemical Reactions, Collaborative Comput. Project on Molecular Quantum Dynamics (CCP6), Daresbury, UK, 2005, pp. 104.
  * (41) D. Sokolovski, E. Akhmatskaya, S.K. Sen, Comput. Phys. Commun. 182 (2011) 448.
  * (42) E. Akhmatskaya, D. Sokolovski, C. Echeverría-Arrondo, Comp. Phys. Comm. 185 (2014) 2127.
  * (43) D. Sokolosvki, E. Akhmatskaya, in Lecture Notes in Computer Science (LNCS) ICCSA 2014, Part I, 8579, pp. 522–537, B. Murgante et al. (Eds.), Springer International Publishing Switzerland (2014)
  * (44) D.M. Neumark, et al., J. Chem. Phys. 82 (1985) 3067.
  * (45) R.T. Skodje, D. Skouteris, D.E. Manolopoulos, S.-H. Lee, F. Dong, K. Liu, J. Chem. Phys. 112 (2000) 4536; Phys. Rev. Lett. 85 (2000) 1206.
  * (46) F. Dong, S.-H. Lee, K. Liu, J. Chem. Phys. 113 (2000) 3633; ibid 124 (2006) 224312.
  * (47) S.-H. Lee, F. Dong, K. Liu, J. Chem. Phys. 116 (2002) 7839; ibid 125 (2006) 133106; Faraday Discuss. 127 (2004) 49.
  * (48) W.W. Harper, S.A. Nizkorodov, D.J. Nesbitt, J. Chem. Phys. 116 (2002) 5622.
  * (49) Z. Ren, et al., Proc. Natl. Acad. Sci. U. S. A. 105 (2008) 12662.
  * (50) W. Dong, C. Xiao, T. Wang, D. Dai, X. Yang, D.H. Zhang, Science 327 (2010) 1501.
  * (51) T. Wang, et al., Science 342 (2013) 1499.
  * (52) T. Wang, et al., J. Phys. Chem. Lett. 5 (2014) 3049.
  * (53) K. Stark, H.-J. Werner, J. Chem. Phys. 104 (1996) 6515.
  * (54) V. Aquilanti, S. Cavalli, F. Pirani, A. Volpi, D. Cappelletti, J. Phys. Chem. 105 (2001) 2401.
  * (55) V. Aquilanti, S. Cavalli, D. De Fazio, A. Volpi A., Int. Jour. Quantum Chem. 85 (2001) 368.
  * (56) V. Aquilanti, R. Candori, D. Cappelletti, E. Luzzatti, F. Pirani, Chem. Phys. 145 (1990) 293.
  * (57) V. Aquilanti, S. Cavalli, D. De Fazio, A. Volpi, A. Aguilar, J.M. Lucas, Chem. Phys. 308 (2005) 237.
  * (58) D. De Fazio, V. Aquilanti, S. Cavalli, A. Aguilar, J. M. Lucas, J. Chem. Phys. 125 (2006) 133109.
  * (59) M. Tizniti, S.D. Le Picard, F. Lique, C. Berteloite, A. Canosa, M.H. Alexander, I.R. Sims, Nat. Chem., 6, (2014)141.
  * (60) D. De Fazio, V. Aquilanti, S. Cavalli, A. Aguilar, J. M. Lucas, J. Chem. Phys. 129 (2008) 064303.
  * (61) D. Skouteris, D. De Fazio, S. Cavalli, V. Aquilanti, J. Phys. Chem. A 113 (2009) 14807.
  * (62) D. De Fazio, S. Cavalli, V. Aquilanti, A.A. Buchachenko, T.V. Tscherbul, J. Phys. Chem. A 111 (2007) 12538.
  * (63) B. Fu, X. Xu, D.H. Zhang, J. Chem. Phys. 129 (2008) 011103.
  * (64) D. De Fazio, J. M. Lucas, V. Aquilanti, S. Cavalli, Phys. Chem. Chem. Phys. 13 (2011) 8571.
  * (65) M.B. Krasilnikov, et. al., J. Chem. Phys. 138 (2013) 244302.
  * (66) J. Chen, Z. Sun and D.H. Zhang J Chem Phys 142 (2015) 024303.
  * (67) D. De Fazio, M. de Castro, A. Aguado, V. Aquilanti, S. Cavalli, J. Chem. Phys. 137 (2012) 244306.
  * (68) D. De Fazio, Phys. Chem. Chem. Phys. 16 (2014) 11662.
  * (69) F. T. Smith, Phys. Rev. 118 (1960) 349; ibid. 119 (1960) 2098 (erratum); J. Chem. Phys. 36 (1962) 248.
  * (70) V. Aquilanti, S. Cavalli, A. Simoni, A. Aguilar, J.M. Lucas, D. De Fazio, J. Chem. Phys. 121 (2004) 11675.
  * (71) V. Aquilanti, S. Cavalli, D. De Fazio, A. Simoni, T. Tscherbul, J. Chem. Phys. 123 (2005) 054314.
  * (72) L. D. Landau, E. M. Lifshitz, Quantum Mechanics, 3rd ed., Pergamon, Oxford, 1977.
  * (73) J. F. Castillo, D. E. Manolopoulos, K. Stark, H.-J. Werner, J. Chem. Phys. 104 (1996) 6531.
  * (74) S. Cavalli, D. De Fazio, Theor. Chem. Acc. 129 (2011) 141.
  * (75) M. Qui, et al., Science 311 (2006) 144.
  * (76) U. Fano, Phys. Rev. 124, 1866 (1961).
  * (77) A. Miroshnichenko, S. Flach, Y.S. Kivshar, Rev. Mod. Phys., 82, 2258 (2010).
  * (78) Thus, for a resonance at $E=E^{K}$ Fano’s formula reads FANO2  ($q$=const) $\sigma^{res}=\frac{[(E-E^{K}]+\Gamma q/2]^{2}}{(E-E^{K})^{2}+\Gamma^{2}/4}$, while Eq.(7) gives $\sigma^{res}_{K}=\frac{Re[\gamma^{K}](E-E^{K})+Im[\gamma^{K}]\Gamma/2}{(E-E^{K})^{2}+\Gamma^{2}/4}$.
  * (79) D. H. Zhang, J. Z. H. Zhang, J. Chem. Phys. 110, (1999), 7622.