Current affiliation: ] Department of Applied Mathematics,
Faculty of Electrical Engineering and Computer Science, VŠB – Technical
University of Ostrava, Czech RepublicCurrent affiliation: ] Chemical Sciences
Division, Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley,
California 94720, USA

# Vibronic dynamics in electron continuum - iterative solvers

Martina Ćosićová  [  Jan Dvořák  [  Martin Čížek<EMAIL_ADDRESS>Charles University, Faculty of Mathematics and Physics, Institute of
Theoretical Physics,
V Holešovičkách 2, 180 00 Prague, Czech Republic

###### Abstract

We present a general two-dimensional model of conical intersection between
metastable states that are vibronically coupled not only directly but also
indirectly through a virtual electron in the autodetachment continuum. This
model is used as a test ground for design and comparison of iterative solvers
for resonance dynamics in low-energy electron-molecule collisions. Two Krylov-
subspace methods with various preconditioning schemes are compared. To
demonstrate the applicability of the proposed methods on even larger models,
we also test the performance of one of the methods on a recent model of
vibrational excitation of CO2 by electron impact based on three vibronically-
coupled discrete states in continuum (Renner-Teller doublet of shape
resonances coupled to sigma virtual state) including four vibrational degrees
of freedom. Two-dimensional electron energy-loss spectra resulting from the
electron-molecule scattering within the models are briefly discussed.

Schrödinger equation, Krylov subspace iterative methods, vibronic coupling.

††preprint: APS/123-QED

##  I Introduction

Despite of a long history of investigations [1, 2, 3], the collisions of low-
energy electrons with molecules still represent a fascinating and challenging
field of study. By low energy we mean here the energy below the electronic-
excitation threshold, i. e. the energy that does not exceed few units of
electron volt. Even these low energies lead to many interesting phenomena like
appearance of sharp structures in cross sections [4, 5] or the possibility to
select dissociation into different anionic fragments by tuning the energy [6,
7]. This topic is both interesting for practical applications [8] and
challenging for the theory even for small polyatomic molecules [9, 10, 11, 12,
13, 14, 15].

In this paper we will focus on the process of the vibrational excitation in
collision of an electron $e^{-}$ with a molecule $M$ initially in a
vibrational state $|v_{i}\rangle$

$e^{-}+M(v_{i})\to M^{-}\to e^{-}+M(v_{f})$ (1)

mediated by one or several metastable anion states $M^{-}$. After the process,
the molecule is left in a final vibrational state $|v_{f}\rangle$. The total
energy $E$ during the collision is conserved

$E=\epsilon_{i}+E_{v_{i}}=\epsilon_{f}+E_{v_{f}},$ (2)

where $\epsilon$ are electron energies and $E_{v}$ energies of vibrational
states of the molecule for the initial and the final states, before and after
the collision. This process is closely related to the process of
photodetachment of an electron $e^{-}$ from a molecular anion $M^{-}$

$\gamma+M^{-}\to(M^{-})^{*}\to e^{-}+M(v_{f}),$ (3)

with initial energy $E$ of the system defined now by the energy of the photon
$\gamma$ shone on the anion to excite it to the state $(M^{-})^{*}$. The
dynamics of both of these processes is driven by potential energy states of
the negative molecular ion and their widths for decay into electronic
continuum channels [16]. The energies of the released electrons are sensitive
to the relative position of the anion and the neutral molecular states and the
selection rules are different than for the radiative transitions [17, 18, 19].

The goal of this paper is to advance the detailed theory of the dynamics of
the electron detachment from anions in such processes. In the development of
the theoretical methods, we keep in mind the description of experiments that
study in detail the energies of released electrons [20, 16, 21], and in
particular, we calculate the two-dimensional electron energy-loss spectrum (2D
EELS) for our model. The 2D electron loss spectroscopy was pioneered by Currel
and Commer [22, 23] and further developed by Allan and collaborators [24]. Up
to now, a dozen of high-resolution spectra for different molecules have been
measured [24, 25, 26, 27, 28, 29, 30, 31, 16] but the detailed understanding
of such spectra for polyatomic molecules is mostly lacking.

In this paper, we present and test a general scheme for solving the nuclear
dynamics of the negative ion formed in the collision of an electron with a
polyatomic molecule. The scheme is tailored for a class of models that are
inspired by the pseudo-Jahn-Teller model of Estrada, Cederbaum and Domcke [32]
with modifications meant to make it a more realistic model of real molecules.
This approach combines a model of vibronic coupling of several anionic states
expanded in low-order polynomials in vibrational coordinates close to
equilibrium geometry of the neutral molecule with projection-operator approach
to include the interaction of the anion discrete states with the electronic
continuum. The model is rather flexible in adding states and vibrational
degrees of freedom and the present scheme has been used to produce the results
in our previous work on CO2 [33, 34, 35]. In these papers we did not explain
the methods and their performance in detail, a gap that is meant to be filled
by this work.

We start Section II by reviewing the projection-operator approach to the
dynamics of vibrational excitation in electron collisions with molecules. We
then proceed by reminding the model of Estrada et al. [32] and propose its
generalization by including the vibronic coupling through the electron
continuum in addition to the direct vibronic coupling present in the original
model. This section is concluded by explaining the representation of the wave-
function components and the Hamiltonian in a basis constructed from neutral
vibrational states. In Section III, we first introduce the used iteration
methods and preconditioning schemes and then we discuss their performance for
the models. The section IV is devoted to a brief description of the obtained
2D spectra for the models and we conclude by summarising the results in
Section V.

##  II Theory

The vibrational and resonance dynamics in electron-molecule collisions has
been studied theoretically for a long time (see, for example, one of the
review papers [2, 36, 37, 38]). The direct brute-force approach is only
tractable for small molecules [39] or for small deformations [40]. Number of
approximate schemes have therefore been developed: Born approximation,
adiabatic-nuclei approximation, zero/effective range or semi-classical
approaches. In the present work we focus on the development of the numerical
schemes for the projection-operator approach based on the existence of an
intermediate anion state (or states) that is responsible for the coupling of
the electronic and vibrational motion. The approach is often used in its
approximate form — the local complex potential approximation, but it is known
to fail in predicting interesting phenomena like Wigner cusps or vibrationally
excited Feshbach resonances. The nonlocal approach is well developed for
diatomic molecules [37, 41] but the attempts to use it for polyatomic
molecules are scarce (see for example [42]). In addition to bringing more
degrees of freedom, the polyatomic molecules also exhibit interesting features
like vibronic coupling of resonances, conical intersections and exceptional
points [43, 44]. Here, we follow the work of of Estrada, Cederbaum and Domcke
[32] (ECD86) and extend it to more general form of model functions and
vibronic coupling. We start by presenting basic formulas resulting from the
projection-operator formalism (see [37] for the comprehensive review of the
approach). Then, we narrow the model to two vibrational degrees of freedom and
two vibronically coupled discrete states.

###  II.1 Nonlocal model for multiple discrete states in continuum

The main idea of the nonlocal-discrete-state in continuum model is the
assumption that the coupling of the electronic and vibrational degrees of
freedom in the electron-molecule collision is mediated by one or a few
discrete states and after their removal from the electronic continuum using
projection-operator formalism of Feshbach [45], the electronic basis
consisting of the discrete states and the orthogonalized continuum is
diabatic. The vibrational excitation or dissociative attachment then proceeds
through capture into the discrete state.

We define the projection operator

$\mathcal{Q}=\sum_{d}|d\rangle\langle d|.$ (4)

as a sum over a set of discrete states $|d\rangle$ and the complementary
operator

$\mathcal{P}=I-\mathcal{Q}$ (5)

projecting on the background continuum. The basis in the background part can
be chosen as the states that solve the background scattering problem

$\mathcal{P}\mathcal{H}_{el}\mathcal{P}|\Phi_{0},\epsilon\mu\rangle=(V_{0}+\epsilon)|\Phi_{0},\epsilon\mu\rangle.$
(6)

Here, $V_{0}(\vec{q}\,)$ is the potential energy surface of the neutral
molecule, i. e. the energy of the ground electronic state $|\Phi_{0}\rangle$
as function of the positions of the nuclei $\vec{q}$. Since we consider only
low-energy electron scattering below threshold for the electronic excitation
of the molecule, the state $\Phi_{0}$ is fixed and we will further omit it
from the notation. The electron continuum states $|\epsilon\mu\rangle$ are
thus uniquely described by the electron energy $\epsilon$ and some other
quantum numbers collectively denoted by $\mu$ (typically angular momentum).
All states $|d\rangle$ and $|\epsilon\mu\rangle$ thus form an orthogonal
basis:

$\displaystyle\langle d|d^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\delta_{dd^{\prime}},$ (7) $\displaystyle\langle
d|\epsilon\mu\rangle$ $\displaystyle=$ $\displaystyle 0,$ (8)
$\displaystyle\langle\epsilon\mu|\epsilon^{\prime}\mu^{\prime}\rangle$
$\displaystyle=$
$\displaystyle\delta_{\mu\mu^{\prime}}\delta(\epsilon-\epsilon^{\prime}).$ (9)

The electronic Hamiltonian in the $\mathcal{Q}$-space is described by a matrix

$\langle
d|\mathcal{H}_{el}|d^{\prime}\rangle=V_{0}\delta_{dd^{\prime}}+U_{dd^{\prime}}$
(10)

where all matrix elements depend on the molecular geometry, i. e. positions of
nuclei $\vec{q}$. The diagonal elements
$V_{d}(\vec{q}\,)=V_{0}(\vec{q}\,)+U_{dd}(\vec{q}\,)$ represent the diabatic
discrete-state potentials and the off-diagonal part
$U_{dd^{\prime}}(\vec{q}\,)$ the direct vibronic coupling among the states.
The coupling between the discrete state $|d\rangle$ and the continuum
$|\epsilon\mu\rangle$ is described by the coupling elements

$\langle
d|\mathcal{H}_{el}|\epsilon\mu\rangle=V_{d\epsilon}^{\mu}(\vec{q}\,).$ (11)

These elements represent the vibronic coupling 111 Note that we consider
$V_{d\epsilon}^{\mu}$ to be real quantities. This is a reasonable assumption
since the coupling can be made real for a single $\vec{q}$ by phase
conventions and the dependence on $\vec{q}$ is supposed to be weak
(diabaticity of the basis). of the discrete state to the continuum and they
also lead to the second order vibronic coupling among the discrete states
mediated by continuum as described below.

This way, we parameterized the matrix elements given by Eqns. (6), (10), (11)
of the Hamiltonian $\mathcal{H}_{el}$ for the electron scattering from the
molecule for each fixed position of the nuclei $\vec{q}$ by functions
$V_{0}(\vec{q}\,)$, $U_{dd^{\prime}}(\vec{q}\,)$ and
$V_{d\epsilon}^{\mu}(\vec{q}\,)$. To describe the electron scattering from the
molecule including the vibronic dynamics, we start from the definition of the
vibrational states $|v\rangle$ of the target neutral molecule:

$H_{0}|v\rangle=(T_{N}+V_{0})|v\rangle=E_{v}|v\rangle,$ (12)

where $T_{N}$ is the kinetic-energy operator for the nuclei and $v$ is a set
of quantum numbers that uniquely determine the vibrational states with energy
$E_{v}$. It can be shown (see for example [37]) that the vibronic motion of
the anion is described by the effective Hamiltonian

$H_{ef}=H_{0}+U+F,$ (13)

which is the matrix in the indices $d$, $d^{\prime}$ and the operator in the
space of vibrational degrees of freedom. In the equation above, $H_{0}$ is the
Hamiltonian operator for the vibrations of the molecule multiplied by unity
matrix $\delta_{dd^{\prime}}$ in the discrete state indices, $U$ is the matrix
with the elements $U_{dd^{\prime}}$ defined above and the operator $F$
describes the dynamical coupling of the discrete-state space to the electronic
continuum

$F_{dd^{\prime}}(E-H_{0})=\\\
\sum_{\mu}\int_{0}^{\infty}V_{d\epsilon}^{\mu}(\vec{q}\,)[E-H_{0}-\epsilon+i\eta]^{-1}V_{d^{\prime}\epsilon}^{\mu}(\vec{q}\,^{\prime})\mathrm{d}\epsilon,$
(14)

where $\eta$ is a positive infinitesimal. This operator is a matrix in the
discrete-state indices and nonlocal operator in the nuclear coordinate
$\vec{q}$.

The discrete-state contribution to the $T$-matrix for vibrational excitation
by electron scattering in a continuum state $|\epsilon_{i}\mu_{i}\rangle$ from
the initial vibrational state $v_{i}$ to final state $v_{f}$ and leaving in
continuum state $|\epsilon_{f}\mu_{f}\rangle$ is given by [32]

$T_{\mu_{f}v_{f}\leftarrow\mu_{i}v_{i}}=\sum_{dd^{\prime}}\langle
v_{f}|V_{d\epsilon_{f}}^{\mu_{f}}[E-H_{ef}]_{dd^{\prime}}^{-1}V_{d^{\prime}\epsilon_{i}}^{\mu_{i}}|v_{i}\rangle$
(15)

and is closely related to the integral cross section for the vibrational
excitation event

$\sigma_{v_{f}\leftarrow
v_{i}}=\frac{2\pi^{3}}{\epsilon_{i}}\sum_{\mu_{i}\mu_{f}}\left|T_{\mu_{f}v_{f}\leftarrow\mu_{i}v_{i}}\right|^{2}.$
(16)

Finally, to simulate the full 2D electron energy-loss spectra, we have to
collect vibrational excitation cross sections for all accessible final states

$S(\epsilon_{i},\Delta\epsilon)=\sum_{v_{f}}\sigma_{v_{f}\leftarrow
v_{i}}(\epsilon_{i})\rho(\Delta\epsilon-\Delta\epsilon_{v_{f}}),$ (17)

where $\rho(\epsilon)$ is the resolution function of the spectrometer
(simulated here with a Gaussian function with full width at half maximum
equaled to 10 meV, which is comparable to the values in the experiment [24]).
The energy loss
$\Delta\epsilon_{v_{f}}=E_{v_{f}}-E_{v_{i}}=\epsilon_{i}-\epsilon_{f}$ in each
term in Eq. (17) is fixed by the energy conservation. The function
$S(\epsilon_{i},\Delta\epsilon)$ gives the full experimental information in
the electron energy-loss spectroscopy except for the angular resolution that
can also be included [35] but it is not of the interest in the present paper.

###  II.2 Pseudo Jahn-Teller model of Estrada et al.

Figure 1: Sections through the model potential energy surfaces in the $q_{g}=0$ (left) and $q_{u}=0$ (right) planes for the ECD86 model. Blue-shaded areas give the position and the width of the fixed nuclei electronic resonance.  Figure 2: Perspective view of the potential energy manifold for the ECD86 model. The width (inverse lifetime) is marked by the color scale.  Table 1: The values of the parameters describing the pseudo Jahn-Teller model of Estrada et al. [32].  Parameter | Value | Parameter | Value
---|---|---|---
$\omega_{g}$ | $0.258$ | $\omega_{u}$ | $0.091$
$E_{1}$ | $2.45$ | $E_{2}$ | $2.85$
$\kappa_{1}$ | $-0.212$ | $\kappa_{2}$ | $0.254$
$\lambda$ | $0.318$ |  |
$a_{1}$ | $0.086$ | $a_{2}$ | $0.186$
$b_{1}$ | $0.833$ | $b_{2}$ | $0.375$
$l_{1}$ | $2$ | $l_{2}$ | $1$

Model by Estrada, Cederbaum and Domcke [32] assumes a molecule with an Abelian
group of symmetry. They consider two discrete states $d=1,2$ that transform
according to different irreducible representations of the symmetry group and
are coupled vibronically through a nontotally symmetric vibrational mode
$q_{u}$. They also consider excitation of another, totally symmetric mode
$q_{g}$, so that the geometry of the molecule within the model is describe by
a vector $\vec{q}=(q_{g},q_{u})$. The symmetry then dictates the structure of
the matrix $U$:

$U=\left(\begin{array}[]{cc}E_{1}+\kappa_{1}q_{g}&\lambda q_{u}\\\ \lambda
q_{u}&E_{2}+\kappa_{2}q_{g}\end{array}\right).$ (18)

This is a completely general form of the dependence of the matrix $U$ on
coordinates when the terms are restricted up to the first order in $\vec{q}$
and the symmetry requirements are taken into account. Similarly, we can expand
the matrix of the discrete-state-continuum coupling. For simplicity, we
consider only two partial waves $|\epsilon\mu\rangle$ for $\mu=e,o$
representing one even and one odd linear combination of partial waves coupled
to the discrete-state space. In principle, we could consider more partial
waves but they could be decoupled from the problem by a unitary
transformation, grouping thus partial waves into effective channels with
number of channels not exceeding the dimension of the $\mathcal{Q}$-space
[46]. Estrada et al [32] considered the coupling matrix $V_{d\epsilon}^{\mu}$
independent of the nuclear coordinates (we are going to lift this restriction
in the next section). The symmetry selection rules then forbid the coupling
$V_{d\epsilon}^{\mu}$ to the different symmetry of the discrete state $d$ and
the partial wave $\mu$. We therefore assume that the discrete state $d=1$ has
even symmetry as the $\mu=e$ partial wave and $d=2$ has the symmetry of the
odd partial wave $\mu=o$. Using Eq. (14), we then see that only the diagonal
matrix elements $F_{11}(\epsilon)$ and $F_{22}(\epsilon)$ of the level-shift
operator are nonzero. They can be generated from their imaginary parts
(widths)

$\displaystyle\Gamma_{11}(\epsilon)$ $\displaystyle=$
$\displaystyle-2\,\mathrm{Im}F_{11}=2\pi|V_{1\epsilon}^{e}|^{2},$
$\displaystyle\Gamma_{22}(\epsilon)$ $\displaystyle=$
$\displaystyle-2\,\mathrm{Im}F_{22}=2\pi|V_{2\epsilon}^{o}|^{2}$

by means of the integral transform $\mathcal{F}[\Gamma_{ii}(\epsilon)/2\pi]$
defined as (compare Eq. (14))

$\mathcal{F}[f(\epsilon)]=\int\frac{f(x)}{\epsilon-x+i\eta}{\rm d}x.$ (19)

This transform can be worked out analytically for the assumed form of the
widths

$\Gamma_{dd}(\epsilon)=a_{d}\epsilon^{l_{d}+1/2}\exp(-b_{d}\epsilon)$ (20)

(see [47]). To complete the model description we must give the vibrational
Hamiltonian $H_{0}$ of the neutral molecule. The model assumes simply harmonic
vibrations

$H_{0}=T_{N}+V_{0}=-{\textstyle\frac{1}{2}}\omega_{g}\frac{\partial^{2}}{\partial
q_{g}^{2}}-{\textstyle\frac{1}{2}}\omega_{u}\frac{\partial^{2}}{\partial
q_{u}^{2}}+{\textstyle\frac{1}{2}}\omega_{g}q_{g}^{2}+{\textstyle\frac{1}{2}}\omega_{u}q_{u}^{2}.$
(21)

The vibrational eigenstates $|v\rangle$ satisfying Eq. (12) with this harmonic
Hamiltonian can be numbered by two quantum numbers $\nu=(n_{g},n_{u})$ and the
vibrational energies are given by standard harmonic oscillator formula
$E_{\nu}=\omega_{g}(n_{g}+{\textstyle\frac{1}{2}})+\omega_{u}(n_{u}+{\textstyle\frac{1}{2}})$.

The numerical values of $\omega_{i}$ and the parameters defining direct
coupling matrix $U$ and discrete-state-continuum matrix $V_{e}$ for the model
studied in [32] and used here for testing are given in Table 1. Note that
several variants of the model were studied in [32]. Here we study only the
most complex form of the model with the values of parameters as given in the
table. To visualize the character of the model, we show the one dimensional
sections through the model potentials in Fig. 1. The functions shown are
$V_{0}(\vec{q})$, $V_{d}(\vec{q})=V_{0}(\vec{q})+U_{dd}(\vec{q})$ and the
local complex potential $V_{loc}(\vec{q})$. The last function is defined as
the position $V_{\mathrm{loc}}=E_{R}-\frac{i}{2}\Gamma_{R}$ of the pole of the
fixed nuclei S matrix, which has to be located iteratively [34]. The
perspective view of the local complex potential
$\operatorname{Re}V_{\mathrm{loc}}$ colored by values of
$-2\operatorname{Im}V_{\mathrm{loc}}$ is also shown in Fig. 2.

###  II.3 Generalized model with vibronic coupling with continuum states

Figure 3:  Sections through the model potentials in the $q_{g}=0$ (left) and $q_{u}=0$ (right) planes for the new model.  Figure 4:  Perspective view of the potential energy manifold for the new model, colored by the width of the resonance.  Table 2: The values of the parameters describing the generalization of the model.  Parameter | Value | Parameter | Value
---|---|---|---
$a_{1}^{e}$ | $0.07$ | $a_{2}^{e}$ | $0.1$
$b_{1}^{e}$ | $0.25$ | $b_{2}^{e}$ | $0.5$
$l_{1}^{e}$ | $0$ | $l_{2}^{e}$ | $0$
$a_{1}^{o}$ | $0.186$ | $a_{2}^{o}$ | $0.15$
$b_{1}^{o}$ | $0.375$ | $b_{2}^{o}$ | $0.8$
$l_{1}^{o}$ | $1$ | $l_{2}^{o}$ | $1$
$\lambda_{1}$ | $0.2$ | $\lambda_{2}$ | $0.1$

The vibronic model above assumes the most simple structure of the discrete-
state-continuum coupling matrix $V_{\epsilon}=\\{V_{d\epsilon}^{\mu}\\}$ with
row index $d$ and column index $\mu$:

$V_{\epsilon}=\left(\begin{array}[]{cc}V_{1\epsilon}^{e}(\vec{q}\,)&V_{1\epsilon}^{o}(\vec{q}\,)\\\
V_{2\epsilon}^{e}(\vec{q}\,)&V_{2\epsilon}^{o}(\vec{q}\,)\end{array}\right)=\left(\begin{array}[]{cc}\sqrt{\Gamma_{11}/2\pi}&0\\\
0&\sqrt{\Gamma_{22}/2\pi}\end{array}\right).$

Here, we would like to go one step beyond the approximation of the coupling
matrix by constant terms and we expand the matrix to the first order in the
normal vibrational coordinates. This generalization is very useful in the
description of the interaction of resonances through the electronic continuum
that is switched off in the equilibrium geometry, but becomes nonzero with
deformation as, for example, in pyrrole molecule [48] . This feature was also
important ingredient of the model for CO2 molecule [33, 34]. We will further
assume that the dependence of $V_{d\epsilon}(\vec{q}\,)$ on the electron
energy $\epsilon$ and the normal coordinates is separable. Taking into account
the symmetry of the system, we get

$V_{\epsilon}=\left(\begin{array}[]{cc}f_{1}^{e}(\epsilon)[1+\lambda_{1}q_{g}]&f_{1}^{o}(\epsilon)q_{u}\\\
f_{2}^{e}(\epsilon)q_{u}&f_{2}^{o}(\epsilon)[1+\lambda_{2}q_{g}]\end{array}\right),$

where the terms that couple a discrete state to the partial wave of different
symmetry must be odd functions of $q_{u}$. We see that half of the total
number of 12 terms (up to first order in $\vec{q}$) in the coupling matrix are
zero due to the symmetry. For the purposes of the testing of the numerical
methods, we choose the same form of the energy dependence as in the original
model:

$2\pi|f_{d}^{\mu}|^{2}=a_{d}^{\mu}\epsilon^{l_{d}^{\mu}+1/2}\exp(-b_{d}^{\mu}\epsilon)$
(22)

with the values of the parameters given in Table 2.

Using formula (14), we see that the structure of the nonlocal level-shift
operator $F(E-H_{0})$ is much richer:

$\displaystyle F_{11}$ $\displaystyle=$
$\displaystyle(1+\lambda_{1}q_{g})\mathcal{F}[f_{1}^{e}f_{1}^{e}](1+\lambda_{1}q_{g})+q_{u}\mathcal{F}[f_{1}^{o}f_{1}^{o}]q_{u},$
$\displaystyle F_{22}$ $\displaystyle=$ $\displaystyle
q_{u}\mathcal{F}[f_{2}^{e}f_{2}^{e}]q_{u}+(1+\lambda_{2}q_{g})\mathcal{F}[f_{2}^{o}f_{2}^{o}](1+\lambda_{2}q_{g}),$
$\displaystyle F_{12}$ $\displaystyle=$
$\displaystyle(1+\lambda_{1}q_{g})\mathcal{F}[f_{1}^{e}f_{2}^{e}]q_{u}+q_{u}\mathcal{F}[f_{1}^{o}f_{2}^{o}](1+\lambda_{2}q_{g}),$
$\displaystyle F_{21}$ $\displaystyle=$ $\displaystyle
q_{u}\mathcal{F}[f_{1}^{e}f_{2}^{e}](1+\lambda_{1}q_{g})+(1+\lambda_{2}q_{g})\mathcal{F}[f_{1}^{o}f_{2}^{o}]q_{u},$
(23)

where we used the integral transform (19) again. The ordering of the terms
that depend on $q_{i}$ with respect to $\mathcal{F}(\epsilon)$ must be kept
because we substitute the operator $\epsilon=E-H_{0}$, which does not commute
with the normal coordinates.

The potentials for the new generalized model are visualised in Figs. 3 and 4.
Note that the structure of such conical intersections in continuum have been
investigated in [44, 43]. In accordance with their findings the potential
manifolds shown in Figs. 2 and 4 do not intersect in a single point like
regular conical intersections but in a line segment bounded by two exceptional
points. The form of our model as given by Eq. (23) is more general than the
expansion investigated in [44, 43] because they studied a linear coordinate
expansion of the width function $\Gamma$ whereas we prescribe the linear
expansion of the coupling matrix $V_{\epsilon}$, which is more natural for the
subsequent treatment of the dynamics. In our case, the linear form of the
coupling matrix produces also quadratic terms in widths in Eq. (23). When the
quadratic terms are omitted, we recover the form used in [44, 43]. However, we
can not omit these terms in the dynamics since it would distort the unitarity
of the S matrix.

###  II.4 Numerical representation of the dynamics

For the numerical solution of the dynamics, we expand wave-function components
in the harmonic oscillator basis $|\nu\rangle=|n_{g},n_{u}\rangle$ associated
with the model Hamiltonian of the neutral molecule (21). We first rewrite Eqs.
(15), (16) for the cross section as

$\sigma_{v_{f}\leftarrow
v_{i}}=\frac{2\pi^{3}}{\epsilon_{i}}\sum_{\mu_{i}\mu_{f}}\left|\langle\Phi_{v_{f}}^{\mu_{f}}|\Psi_{v_{i}}^{\mu_{i}}\rangle\right|^{2},$
(24)

where we defined auxiliary wave-functions $|\Phi_{v}^{\mu}\rangle$ with
components

$\Phi^{v\mu}_{d,n_{g},n_{u}}=\langle
n_{g},n_{u}|V_{d\epsilon}^{\mu}|v\rangle,$ (25)

and with energy according to the conservation law (2), i. e.
$\epsilon=E-E_{v}$. The anion wave function $|\Psi_{v_{i}}^{\mu_{i}}\rangle$
satisfies

$(E-H_{ef})|\Psi_{v_{i}}^{\mu_{i}}\rangle=|\Phi_{v_{i}}^{\mu_{i}}\rangle.$
(26)

In the harmonic oscillator basis, this equation represents a system of linear
equations for unknown components of the discrete-state wave function

$\Psi_{\alpha}\equiv\Psi_{d,n_{g},n_{u}}=\langle d|\langle
n_{g},n_{u}|\Psi_{v_{i}}^{\mu_{i}}\rangle,$ (27)

where we introduced a compound index $\alpha\equiv(d,n_{g},n_{u})$. Using this
notation the matrix of this system $A_{\alpha,\alpha^{\prime}}$ reads

$A_{\alpha,\alpha^{\prime}}=\langle
n_{g},n_{u}|(E-H_{ef})_{dd^{\prime}}|n_{g}^{\prime},n_{u}^{\prime}\rangle$
(28)

and the scalar product in Eq. (24) can be written as the sum
$\sum_{\alpha}\equiv\sum_{d}\sum_{n_{g}}\sum_{n_{u}}$ over the components of
$\Phi^{v_{f}\mu_{f}}_{\alpha}$ and $\Psi_{\alpha}$. We cut of the basis in
each dimension keeping the states $|n_{g}\rangle$ for $n_{g}=0,1,...,N_{g}-1$
and $|n_{u}\rangle$ for $n_{u}=0,1,...,N_{u}-1$. The states $\Psi_{\alpha}$
are thus represented by $N=2N_{g}N_{u}$ component vectors and $A$ is a
$N\times N$ matrix.

For the solution of Eq. (26) in the original model, Estrada et al. [32]
devised a specially tailored method based on the block-tridiagonal structure
of the matrix $A$. Our aim in this paper is to develop a more general method
capable of solving a larger class of models and test it both on the original
model and on our generalization. The matrix $A$ is large but sparse. From the
character of the problem, it is also complex symmetric, but not Hermitian. The
structure of the matrix $A$ depends on the order of the basis vectors as
illustrated in Fig. 5 for the generalized model.

Figure 5:  Structure of the matrix $A$ of the generalized model depending on
the ordering of the basis, where the order is given by three letters $d$, $g$,
and $u$ representing the discrete states and vibratonal modes $q_{g}$ and
$q_{u}$. The first index changes the fastest. The size of the matrix elements
is represented by color in a logarithmic scale. The block-diagonal parts used
for preconditioning are shown by the squares and for some orderings, the
blocks corresponding to the discrete state are also indicated.

##  III Krylov-subspace iteration methods

The Krylov-subspace iteration methods are well suited for solving Eq. (26).
The main idea of all of the Krylov-subspace methods is that they solve a
linear system

$Ax=b$

iteratively producing a sequence $x_{0},x_{1},x_{2},\ldots,x_{n}$ of
approximations of the solution vector $x$ in the Krylov-subspace, i. e. in the
space

$\mathcal{K}_{n}\left(A;r_{0}\right)=span\left\\{r_{0},Ar_{0},A^{2}r_{0},...,A^{n-1}r_{0}\right\\}.$

where $r_{0}=b-Ax_{0}$ denotes the initial residual vector. The methods differ
by the definition of the ”ideal” approximation of the solution $x_{n}$ within
the Krylov subspace and usually proceed by application of simple recursive
formulas. To produce the Krylov subspace we only need to implement the matrix
multiplication $Aw$ for an arbitrary vector $w$. The choice of the harmonic
basis is very convenient for implementation of this matrix multiplication.
From Eqs. (13), (18), (23), we can see that the multiplication by the matrix
$A$ can be decomposed to successive multiplications by energy-dependent
diagonal factors, for example,

$\displaystyle[(E-H_{0})w]_{\alpha}$ $\displaystyle=$
$\displaystyle(E-E_{n_{g},n_{u}})w_{d,n_{g},n_{u}},$
$\displaystyle{[\mathcal{F}(E-H_{0})w]}_{\alpha}$ $\displaystyle=$
$\displaystyle\mathcal{F}(E-E_{n_{g},n_{u}})w_{d,n_{g},n_{u}}.$

and by operators of the coordinates $q_{g}$ and $q_{u}$

$\displaystyle[q_{g}w]_{\alpha}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{\sqrt{2}}}\left(w_{d,n_{g}{-}1,n_{u}}\sqrt{n_{g}}+w_{d,n_{g}{+}1,n_{u}}\sqrt{n_{g}+1}\right)$
$\displaystyle{[q_{u}w]}_{\alpha}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{\sqrt{2}}}\left(w_{d,n_{g},n_{u}{-}1}\sqrt{n_{u}}+w_{d,n_{g},n_{u}{+}1}\sqrt{n_{u}+1}\right).$

These operations can be implemented very efficiently. Note that all energy
factors and square roots can be precalculated and stored before starting the
iteration process. Compared to a matrix-vector multiplication, which requires
$O(N^{2})$ operations for full matrices, the above procedure requires only
$O(N)$ operations (with the operation count being approximately three times
larger for the generalized model due to more complicated structure of the
operator $F$). The efficiency of the method of solution of Eq. (26) is then
given by the rate of convergence of the sequence $x_{n}$ to the solution,
which is judged by monitoring the size of the norm of the residuum
$\left\|r_{n}\right\|=\left\|b-Ax_{n}\right\|$. In the following tests we stop
the iterations when the value $\left\|r_{n}\right\|<10^{-6}\left\|b\right\|$
is reached.

###  III.1 Methods of interest

Saad and Schultz [49] developed the generalized minimal residual (GMRES)
method, one of the most widely used Krylov-subspace methods. This method
constructs an orthonormal basis $v_{1},v_{2},\ldots,v_{n}$ of the $n$-th
Krylov subspace $\mathcal{K}_{n}\left(A;r_{0}\right)$ using the Arnoldi
algorithm that can be written in the matrix form as

$AV_{n}=V_{n+1}\tilde{H}_{n},\quad V_{n}=[v_{1}|v_{2}|\cdots|v_{n}]$

, where $\tilde{H}_{n}\in\mathbb{C}^{(n+1)\times n}$ is an upper Hessenberg
matrix. The approximation $x_{n}$ of the solution in each step is given by the
condition that the residual vector $r_{n}=b-Ax_{n}$ satisfies the optimality
property:

$\left\|r_{n}\right\|=\min\limits_{x_{n}\in
x_{0}+\mathcal{K}_{n}\left(A;r_{0}\right)}\left\|b-Ax_{n}\right\|.$ (29)

If we write $x_{n}=x_{0}+V_{n}y_{n}$, this condition leads to the $(n+1)\times
n$ least-square problem for $y_{n}$:

$y_{n}=\operatorname*{arg\,min}\limits_{y\in\mathbb{C}^{n}}\left\|\left\|r_{0}\right\|e_{1}-\tilde{H}_{n}y\right\|,$

which has to be solved in every iteration.

An advantage of the GMRES method is that it can be used for any matrix $A$
with no other special properties required than the regularity. On the other
hand, each new basis vector at every iteration step has to be orthogonalized
to all previous vectors. Thus, the size of the matrix $V_{n}$ grows during the
iteration process and if the method does not converge quickly, the storage
space and the time needed for each step grows significantly. Note that the
matrix A of our system of equations is complex symmetric (i. e. not
Hermitian). For this reason, the Conjugate gradient method (which, unlike the
GMRES method, uses short-term recurrences to construct the basis of the Krylov
subspace, making it much less computationally demanding) can not be used to
solve it. It is known [50] that for non-normal matrices it is not possible to
define an ‘optimal’ iterative process (i. e. a process that minimizes the
residual or certain norm of the error over the Krylov subspace) that
constructs the basis of Krylov subspace using short-term recurrences [51]. For
complex symmetric matrices van der Vorst and Mellisen [52] presented an
alternative way to define an iterative process based on three-term recurrences
and derived the conjugate orthogonal conjugate gradient (COCG) method. Setting
$v_{0}=r_{0}=b-Ax_{0}$, the basis of the Krylov subspace is constructed using
the recursive formula

$v_{n+1}=Av_{n}+\alpha_{n}v_{n}+\beta_{n}v_{n-1},$

where $\alpha_{n}$ and $\beta_{n}$ follow from the conditions $\left\langle
v_{n+1}|v_{n}\right\rangle_{S}=0$ and
$\left\langle{v}_{n+1}|v_{n-1}\right\rangle_{S}=0$. These conditions are
analogous to those that define the Conjugate gradient method, in which,
however, we have replaced the standard scalar product with the symmetrized
bilinear form

$\left\langle a|b\right\rangle_{S}=\left\langle a^{*}|b\right\rangle,\quad
a,b\in\mathbb{C}^{N}.$

Note that the complex conjugation in the left argument cancels the complex
conjugation in the standard definition of the scalar product. The bilinear
form $\left\langle\cdot|\cdot\right\rangle_{S}$ is not therefore positive
definite but it preserves the symmetry of the matrix $A$. This process ensures
that vectors $v_{0},v_{1},\ldots,v_{n}$ satisfy the conjugate orthogonality
property (vectors $a,b\in\mathbb{C}^{N}$ are conjugate orthogonal if
$\left\langle{a}|b\right\rangle_{S}=0$). The whole iterative process is thus
analogous to the conjugate gradient method but the convergence after $N$ steps
($N$ being the dimension of the matrix $A$) is not guaranteed in exact
arithmetic. In addition, the iterations do not have to converge at all since
the symmetrized product can be zero (or very small) even for nonzero vectors.
In practice, however, the convergence is usually achieved. Moreover,
(especially with proper preconditioning) it is often rather fast.

###  III.2 Preconditioning

In this paper, by preconditioning we understand the transformation of the
original linear system $Ax=b$ into an equivalent problem (see, for example,
[53])

$M^{-1}AM^{-T}y=M^{-1}b,\qquad\mbox{for~{}}y=M^{T}x,$ (30)

with a regular matrix $M$. This way, the preconditioning preserves the
symmetry of the matrix $A$. As a rule of thumb, a good preconditioner $M$ is
represented by some fast invertible approximation of the original matrix $A$,
but there are no exact guidelines for choosing the ideal matrix $M$ ensuring a
fast convergence for the transformed problem. There is a large variety of
preconditioners known in the literature but usually they are proposed for
specific problems with matrices $A$ having some special properties. We tested
some of the preconditiononing options but we found that the most simple
methods work best for the present problem [54, 55]. In the following, we
discuss different possibilities of block-diagonal preconditioning. For this
choice we define the matrix $M$ as a block diagonal section of the original
matrix $A$. Depending on the structure of the blocks, the individual blocks
can either be inverted directly to construct $M^{-1}$ or a banded structure of
the blocks can be used. The exact structure of the block depends on the
ordering of the basis (see Fig. 5). We thus define three different
preconditioners $M^{dg}$, $M^{gu}$, $M^{du}$ with small diagonal blocks of the
sizes $N_{u}\times N_{u}$, $2\times 2$ and $N_{g}\times N_{g}$, respectively,
and three preconditioners $M^{d}$, $M^{g}$, $M^{u}$ with large blocks of sizes
$N_{g}N_{u}\times N_{g}N_{u}$, $2N_{u}\times 2N_{u}$ and $2N_{g}\times
2N_{g}$. To be more specific, in the $(dug)$ ordering of the basis functions,
the small-block preconditioning matrix $M^{ug}$ has $N_{u}N_{g}$ blocks
$M^{(n_{u},n_{g})}$ of the size $2\times 2$ with the matrix elements

$M^{(n_{u},n_{g})}_{d,d^{\prime}}=A_{(d,n_{g},n_{u}),(d^{\prime},n_{g},n_{u})}$
(31)

and the large-block preconditioning matrix $M^{g}$ has $N_{g}$ blocks
$M^{(n_{g})}$ of the size $2N_{u}\times 2N_{u}$ with the matrix elements

$M^{(n_{g})}_{dn_{u},d^{\prime}n_{u}^{\prime}}=A_{(d,n_{g},n_{u}),(d^{\prime},n_{g},n_{u}^{\prime})}.$
(32)

To apply the preconditioning we need to act with $M^{-1}$ on a vector $w$ with
components $w_{d,n_{g},n_{u}}$. This can be done block by block; for example,
to apply the preconditioner $M^{u}$ we invert each block $M^{(n_{u})}$ using
the $LL^{T}$ (or $LDL^{T}$) decomposition and act on the section of
$w_{d,n_{g},n_{u}}$-vector for this fixed $n_{u}$

$[{M^{(n_{u})}}^{-1}w]_{d,n_{g},n_{u}}=\sum_{d^{\prime},n_{g}^{\prime}}[{M^{(n_{u})}}^{-1}]_{dn_{g},d^{\prime}n_{g}^{\prime}}w_{d^{\prime},n_{g}^{\prime},n_{u}}.$
(33)

This is repeated for each $n_{u}$. Note that the inverting of the
preconditioning matrices $M^{(n_{u})}$ should be done just once and stored
before starting iterations. Moreover, in practice the $L$ (and $D$) matrix is
stored in memory instead of ${M^{(n_{u})}}^{-1}$.

Figure 6:  Convergence of GMRES method for the model of Estrada et al. [32]
for various preconditioners. Number of iterations needed for each electron
energy (top) and convergence of residuum for $\epsilon_{i}=$2 eV and 4 eV
(bottom two panels). The top part shows results for both right-hand sides
$\mu=e,~{}o$ of the linear system; bottom two parts display just $\mu=e$.

Figure 7:  Convergence of COCG method for the model of Estrada et al. [32],
see also caption of Fig. 6 for details.

###  III.3 Numerical testing

We applied the above described methods GMRES and COCG to solve the system (26)
with the matrix (28) for the original ECD86 model and for our generalization
of the model. The performance of each method for different preconditioning is
discussed separately for the two models in the next two paragraphs. The last
paragraph also discusses the performance of the COCG method for a realistic
model that describes inelastic electron scattering from the CO2 molecule.

#### Perfomance of the methods for ECD86 model

is shown in Fig. 6 and Fig. 7. Each of the figures is devoted to one of the
methods comparing different preconditioning schemes. The top graph summarises
the number of iterations needed for convergence for all energies and bottom
two graphs demonstrate the decrease of the residuum norm for two selected
energies $\epsilon_{i}=$2eV and 4eV. The different preconditioning methods are
shown with different colors. The curves of the same color correspond to the
two different right-hand sides $\mu_{i}=o,e$ in Eq. (26).

Let us first focus on graphs at the top of Fig. 6 showing the performance of
the GMRES method. The method converges rather well (less then 700 iterations)
even without any preconditioning. The convergence is extremely fast below 2 eV
(several dozens of iterations) but gets slower above this energy with maximum
around 4eV. This is related to the spectrum of the anion. The electron with
energy below 2 eV does not have enough energy to populate vibrational states
of the anionic potential. The process of the electron scattering is therefore
almost elastic, which means that the wave function is not much perturbed with
respect to the initial state used for starting the iterations. Above this
energy, the dynamics is much richer, which is reflected in the increased
number of iterations needed to reach the converged wave function. For the most
of the energies in the range of interest, the preconditioning reduces the
number of iterations considerably. The least efficient preconditioning
matrices $M^{dg}$, $M^{gu}$ (overlapping curves in Fig. 6) include only
diagonal portion of the matrix $A$ and are therefore numerically very cheap to
implement. The preconditioner $M^{du}$ includes also terms proportional to
coupling constants $\kappa_{1}$, $\kappa_{2}$. For the ECD86 model, there are
no terms in the matrix $A$ added by increasing the size of preconditioner to
$M^{d}$ and $M^{u}$, the convergence curves thus overlap for these three
preconditioners. The best results are obtained with the preconditioning matrix
$M^{g}$ which has blocks of size $2N_{u}\times 2N_{u}$ and includes terms
proportional to $\lambda$ in Eq. (18). The convergence of residuum norm in the
lower part of Fig. 6 shows a difference in behavior of different
preconditioned methods. While the best method with $M^{g}$ preconditioner
converges exponentionally for all energies, there is kind of plateau in the
other methods and the iterations without preconditioning even became more
efficient at high energies.

The behavior of the COCG method (Fig. 7( is different in several aspects. The
overall number of iterations is approximately three time larger (for
unpreconditioned iterations) but we have to keep in mind that the COCG method
is much simpler with computational demands constant over the course of the
iterations. For GMRES, the computational demands for one iteration grows
quadraticaly with the number of iterations. The COCG method does not have the
minimization property (29). This is reflected in the shape of the convergence
curves (two bottom graphs in Fig. 7). Unlike in similar curves for GMRES, here
the residuum can locally grow although in general it finally converges to
zero. The efficiency of the different preconditioning schemes is similar like
in GMRES, although for higher energies only the $M^{g}$ preconditioner is
useful.

#### Performance of the methods for generalized model.

Figure 8:  Convergence of GMRES method for the generalized model, see also
caption of Fig. 6 for details.

Figure 9:  Convergence of COCG method for the generalized model, see also
caption of Fig. 6 for details.

The generalized model has a more complicated structure (23) of the level-shift
operator $F(E)$, which is reflected in a more complicated structure of the
matrix $A$, see Fig. 5. Surprisingly, the iteration methods converge faster
with this matrix. There are no clear criteria relating rigorously the
structure of the matrix to the speed of convergence. We believe that the
faster convergence here may be related to the fact that operator $F$ in
generalized model increases diagonal elements of the matrix $A$. Apart from a
little bit faster convergence, the graphs in Fig. 8 for the GMRES method in
the new model look qualitatively similar as for ECD86 model. The norm of the
residuum is monotonously decreasing for all methods and the preconditioner
$M^{g}$ is again the most efficient. The individual preconditioners now lead
to different convergence rates, because all choices of the diagonal blocks are
distinct for the richer structure of $A$. The exception is the equivalence of
$M^{du}$ and $M^{u}$ preconditioning (the green and yellow lines are
overlapping in the graphs). This can be nicely understood from the structure
of the matrix $A$ depicted in Fig. 5. We see that the large and small black
diagonal boxes in the bottom right matrix differ by a blank area of zero
matrix elements.

The faster convergence for the new model is even more apparent for the COCG
method in Fig. 9. Now all preconditioning schemes except for $M^{dg}$ and
$M^{gu}$ are faster than direct iterations.

To conclude the numerical experiment section, we would like to add a few notes
on the implementation. Even without utilizing the structure of the matrix $A$,
we have got by one order of magnitude faster calculation of the spectra
utilizing the Krylov-subspace iteration methods as compared to a direct
solver. Optimizing the matrix-vector multiplication using the structure of the
matrix $A$ explained at the beginning of Sec. III leads to the another order
of magnitude speed up. From the previous examples, we see that the proper
choice of preconditioning leads to the decrease of number of iterations needed
for convergence by another one order of magnitude for both models and both
methods.

#### Performance for model of $e^{-}$+CO2.

In the final part of this section, we discuss our earlier work [33, 34, 35] on
the electron collisions with the carbon dioxide (CO2) molecule in the context
of the present paper. The vibronic coupling model for the $e+\mathrm{CO}_{2}$
system [34] follows the general approach presented here in Sec. II.3, however,
the model is more complex. We considered the nuclear motion within the full
four-dimensional vibrational space in combination with three electronic states
(${}^{2}\Sigma_{g}^{+}$ virtual state and two components of ${}^{2}\Pi_{u}$
shape resonance), which are coupled upon bending of the molecule. The
Hamiltonian is thus a $3\times 3$ matrix in the electronic space and we did
not restrict its elements only to the first order in the normal coordinates
(some of the elements were expanded up to the fourth order). Additionally, the
three discrete states were coupled to four electron partial waves. The
vibrational dynamics is described analogically to the scheme given in Sec.
II.4 but there are four vibrational indices instead of two. The vibrational
basis was constructed from products of eigenfunctions of 1D harmonic
oscillators for symmetric and stretching modes and eigenfunctions of 2D
harmonic oscillator expressed in polar coordinates for the two-dimensional
bending mode.

Using the COCG method without any preconditioning, the number of iterations
needed to reach the convergence with the stopping criterion of $10^{-3}$
(sufficient to obtain converged cross sections) rapidly grows with the
electron energy, see Fig. 10. For energies above 3 eV, even $2\times 10^{5}$
iterations were insufficient to reach the convergence, therefore, a suitable
preconditioning is essential.

The slow rate of convergence or no convergence at all is caused by the
coupling of the discrete states through the bending mode. The stretching modes
do not affect the convergence much since we found that the COCG method
converges badly even for the case where we did not consider the stretching
modes.222We can easily freeze a vibrational mode by considering only the
ground state as the basis within this mode and setting all relevant model
parameters to zero. Thus, taking a block-diagonal preconditioner where blocks
contain the discrete states and the two-dimensional bending was a natural
choice. Such a preconditioner is analogous to the preconditioner $M^{g}$ that
performs the best for the ECD86 model and its generalization. In the case of
CO2, around 200 iterations were sufficient to reach the convergence for
initial electron energy of 3 eV, see Fig. 10.

Figure 10:  Number of iterations needed to solve the Schrödinger equation for
the $e+\mathrm{CO}_{2}$ system using the COCG method without and with
preconditioning. In the latter case, the curve is multiplied by a factor of
10.

Figure 11:  2D electron energy-loss spectrum for ECD86 model (top) and its
decomposition to the gerade and ungerade symmetries (bottom). The intensity of
spectrum is shown in a logarithmic scale.

##  IV Discussion of resulting spectra for test models

It is not purpose of this paper to study in detail the calculated spectra and
their interpretation. This will require a detailed analysis of the final-state
distribution and shape of the individual components of the wave function in
the coordinate representation and its relation to the shape of potentials and
also study of the dependence of the results on the model parameters. It is
quite voluminous work that deserves a separate paper. We would also like to
identify specific molecules that can be treated with the model of the current
setup or a proper generalization. We already published the generalization of
the model [34] needed to describe the resulting spectra for the CO2 molecule
[33] and performed the detailed analysis [35] including the final-state
distribution, the wave functions and decomposition of spectra due to
contribution of components of different symmetry.

In the following, we just show and briefly describe the 2D spectra for the
ECD86 model (which were not subject of their original paper) and for our new
generalization of the model. We also separate the contribution of the two
right hand sides in Eq. (26) corresponding to the gerade and ungerade
symmetry.

###  IV.1 2D spectrum for ECD86 model

The calculated 2D spectrum for the ECD86 model is shown in Fig. 11. The
intensity given by Eq. (17) is plotted as a function of both energy loss
$\Delta\epsilon$ and initial electron energy $\epsilon_{i}$ in a color
logarithmic scale. It is fully converged result, i. e. it is independent of
the method used to calculate it. Interestingly enough, the spectrum is
qualitatively quite similar to the 2D spectrum for the CO2 molecule [23, 33].
The bulk of the spectrum is located at energies of the incident electron
between 2-4 eV. This is a consequence of the shape of the anion potential
manifold (see Figs. 1 and 2) and its location relative to the potential of the
neutral molecule. The understanding of the detailed shape is not trivial. For
small electron energy losses, the spectrum is discretized by vibrational
frequencies whose ratio is approximately 3:1. But since this ratio is not
exact, the spectrum becomes quasi-continuous for energies above 1 eV. At the
same time, we see that there is some selection mechanism that singles out
narrower structures close to the diagonal threshold line. There are also
diagonal rays appearing in the structure of the spectrum (better apparent in
the decomposition of the spectrum according to symmetries). Both of these
features were present in the case of CO2, where we performed the detailed
analysis [35].

Figure 12:  The same data as in Fig. 11 but for the new model.

###  IV.2 2D spectrum for new model

We proposed the new model above to consistently introduce the vibronic
coupling in the level-shift operator in the ECD86 model and to test the
iteration schemes to solve the dynamics in this model. The choice of the model
parameters was guided by our experience with the diatomic molecules, but apart
from that the choice is completely random. To our surprise, the resulting
spectrum has a quite interesting intricate structure, which is furthermore
similar to experimental data for some molecules, like benzene and its
derivatives [56]. Particularly, we are speaking about the wedge-shaped
structure with the tip touching the vertical axis at the incident electron
energy of 2.2 eV, limited by this energy from below and limited approximately
with the diagonal line corresponding to the electron energy loss
$\Delta\epsilon=\epsilon_{i}+2$ eV from above. The origin of this structure is
not clear and since it is quite common in experimental data, we will dedicate
the future study to this phenomenon. It indicates some selection mechanism in
the dynamics that forces the system to skip through a region with small energy
losses to large losses.

##  V Conclusions

We derived a generalization of the model of conical intersection in electronic
continuum proposed originally by Estrada et al. [32] by including terms linear
in the vibrational coordinates also in the term that couples the two discrete
states of the original model to two partial waves of the electronic continuum.
The generalization thus produces quadratic terms in the nonlocal level-shift
operator $F(E)$ that describes the dynamics of the vibrational excitation of
the molecule by collision with an electron.

We also implemented two Krylov-subspace iteration methods GMRES and COCG for
solving the dynamics and calculation of 2D electron energy-loss spectra, and
we tested the two methods on the original ECD86 model and our generalization.
The Krylov-subspace methods are ideally suited for this kind of models because
the multiplication by model functions expanded in polynomials of vibrational
coordinates can be implemented very efficiently in the oscillator basis. For
both models we observed a very good convergence of both methods even without
preconditioning. The slower convergence of the COCG method is compensated by
the simplicity of its implementation. The computational demands of GMRES also
grow in the course of iteration procedure. The preconditioning by block-
diagonal matrix works well only for some choice of the blocks (depends on the
ordering of the basis).

Out of all tested preconditioners, the $M^{g}$ preconditioner proved to be the
most efficient for both the two-dimensional models and our earlier realistic
model of the CO2 molecule. This seems to be a natural results since we exactly
invert the blocks that involve the discrete-state space and the coupling mode.
On the other hand, the most time-consuming preconditioner $M^{d}$ works rather
badly taken into account that we invert the blocks corresponding to the full
vibrational space. Thus, the $M^{g}$ preconditioner is the preconditioner of
choice for more complicated models, as the one for CO2, where the
unpreconditioned iterations are expected not to converge for all energies of
interest.

We believe that the methods tested here can be used for more complicated
molecules to get better understanding of the 2D electron energy-loss
spectroscopy. We plan a more extensive parameter study to obtain a deeper
understanding of the results. The proposed method is conceptually simple and
can further be generalized in a straightforward way to include more anion
states, more vibrational degrees of freedom and higher order polynomial
functions. More challenging generalization will be needed to include also
dissociative channels and anharmonicity in the neutral molecule.

###### Acknowledgements.

We gratefully acknowledge the financial support provided by the Czech Science
Foundation Project No. 19-20524S and by the Charles University Grant Agency,
Project No. 552120.

## References

  * Bardsley and Mandl [1968] J. N. Bardsley and F. Mandl, Resonant scattering of electrons by molecules, Rep. Prog. Phys. 31, 471 (1968).
  * Lane [1980] N. F. Lane, The theory of electron-molecule collisions, Rev. Mod. Phys. 52, 29 (1980).
  * Allan [1989] M. Allan, Study of triplet-states and short-lived negative-ions by means of electron-impact spectroscopy, J. Electron Spectr. Rel. Phenom. 48, 219 (1989).
  * Schulz [1973] G. J. Schulz, Resonances in electron impact on diatomic molecules, Rev. Mod. Phys. 45, 423 (1973).
  * Hotop  _et al._ [2003] H. Hotop, M. W. Ruf, M. Allan, and I. I. Fabrikant, Resonance and threshold phenomena in low-energy electron collisions with molecules and clusters, Adv. In At. Mol. Opt. Phys. 49, 85 (2003).
  * Ptasińska  _et al._ [2005] S. Ptasińska, S. Denifl, B. Mróz, M. Probst, V. Grill, E. Illenberger, P. Scheier, and T. D. Märk, Bond selective dissociative electron attachment to thymine, J. Chem. Phys. 123, 10.1063/1.2035592 (2005).
  * Ibanescu and Allan [2009] B. C. Ibanescu and M. Allan, Selective cleavage of the C–O bonds in alcohols and asymmetric ethers by dissociative electron attachment, Phys. Chem. Chem. Phys. 11, 7640 (2009).
  * Fabrikant  _et al._ [2017] I. I. Fabrikant, S. Eden, N. J. Mason, and J. Fedor, Recent progress in dissociative electron attachment: From diatomics to biomolecules, Adv. In At. Mol. Opt. Phys. 66, 545 (2017).
  * Rescigno  _et al._ [2002] T. N. Rescigno, W. A. Isaacs, A. E. Orel, H.-D. Meyer, and C. W. McCurdy, Theoretical study of resonant vibrational excitation of CO2 by electron impact, Phys. Rev. A 65, 032716 (2002).
  * McCurdy  _et al._ [2003] C. W. McCurdy, W. A. Isaacs, H.-D. Meyer, and T. N. Rescigno, Resonant vibrational excitation of ${\mathrm{CO}}_{2}$ by electron impact: Nuclear dynamics on the coupled components of the ${}^{2}{\Pi}_{u}$ resonance, Phys. Rev. A 67, 042708 (2003).
  * Rescigno  _et al._ [2006] T. N. Rescigno, C. S. Trevisan, and A. E. Orel, Dynamics of low-energy electron attachment to formic acid, Phys. Rev. Lett 96, 213201 (2006).
  * Gallup  _et al._ [2009] G. A. Gallup, P. D. Burrow, and I. I. Fabrikant, Electron-induced bond breaking at low energies in HCOOH and glycine: The role of very short-lived $\sigma^{*}$ anion states, Phys. Rev. A 79, 042701 (2009).
  * Čurík  _et al._ [2008] R. Čurík, P. Čárský, and M. Allan, Vibrational excitation of methane by slow electrons revisited: theoretical and experimental study, J. Phys. B: At., Mol. Opt. Phys. 41, 115203 (2008).
  * Čurík  _et al._ [2014] R. Čurík, I. Paidarová, M. Allan, and P. Čárský, Joint experimental and theoretical study on vibrational excitation cross sections for electron collisions with diacetylene, J. Phys. Chem. A 118, 9734 (2014).
  * Nag  _et al._ [2020] P. Nag, R. Čurík, M. Tarana, M. Polášek, M. Ehara, T. Sommerfeld, and J. Fedor, Resonant states in cyanogen NCCN, Phys. Chem. Chem. Phys. 22, 23141 (2020).
  * Anstöter  _et al._ [2020] C. S. Anstöter, G. Mensa-Bonsu, P. Nag, M. Ranković, R. Kumar T. P., A. N. Boichenko, A. V. Bochenkova, J. Fedor, and J. R. R. Verlet, Mode-specific vibrational autodetachment following excitation of electronic resonances by electrons and photons, Phys. Rev. Lett. 124, 203401 (2020).
  * Gallup [1986] G. A. Gallup, Selection rules for vibrational energy loss by resonant electron impact in polyatomic molecules, Phys. Rev. A 34, 2746 (1986).
  * Gallup [1993] G. A. Gallup, Symmetry selection rules for vibrational excitation by resonant electron impact and a unified treatment of vibronic coupling between resonances and to the continuum: A complete symmetry analysis of vibrational excitation in benzene, J. Chem. Phys. 99, 827 (1993).
  * Čurík  _et al._ [2015] R. Čurík, P. Čárský, and M. Allan, Electron-impact vibrational excitation of cyclopropane, J. Chem. Phys. 142, 144312 (2015).
  * Marion  _et al._ [2023] R. Marion, M. Čížek, and X. Urbain, Autodetachment spectroscopy of metastable D${}_{2}^{-}$ and HD-, Phys. Rev. A 107, 052808 (2023).
  * Ranković  _et al._ [2022] M. Ranković, P. Nag, C. S. Anstöter, G. Mensa-Bonsu, R. Kumar T. P., J. R. R. Verlet, and J. Fedor, Resonances in nitrobenzene probed by the electron attachment to neutral and by the photodetachment from anion, J. Chem. Phys. 157, 064302 (2022).
  * Reddish  _et al._ [1988] T. Reddish, F. Currell, and J. Comer, Studies of the 2 eV shape resonance in N2 using a two-dimensional scanning technique, J. Phys. E: Sci. Instrum. 21, 203 (1988).
  * Currell and Comer [1995] F. Currell and J. Comer, Phys. Rev. Lett. 74, 1319 (1995).
  * Regeta and Allan [2013] K. Regeta and M. Allan, Autodetachment dynamics of acrylonitrile anion revealed by two-dimensional electron impact spectra, Phys. Rev. Lett. 110, 203201 (2013).
  * Regeta and Allan [2015] K. Regeta and M. Allan, Two-dimensional spectra of electron collisions with acrylonitrille and methacrylonitrile reveal nuclear dynamics, J. Chem. Phys. 142, 184307 (2015).
  * Allan  _et al._ [2016] M. Allan, K. Regeta, J. D. Gorfinkiel, Z. Mašín, S. Grimme, and C. Bannwarth, Recent research directions in Fribourg: nuclear dynamics in resonances revealed by 2-dimensional EEL spectra, electron collisions with ionic liquids and electronic excitation of pyrimidine, Eur. Phys. J. D 70, 123 (2016).
  * Allan  _et al._ [2018] M. Allan, M. Lacko, P. Papp, Š. Matejčík, M. Zlatar, I. I. Fabrikant, J. Kočišek, and J. Fedor, Dissociative electron attachment and electronic excitation in Fe(CO)5, Phys. Chem. Chem. Phys. 20, 11692 (2018).
  * Ranković  _et al._ [2018] M. Ranković, P. Nag, M. Zawadzki, L. Ballauf, J. Žabka, M. Polášek, J. Kočišek, and J. Fedor, Electron collisions with cyanoacetylene HC3N: Vibrational excitation and dissociative electron attachment, Phys. Rev. A 98, 052708 (2018).
  * Ranković  _et al._ [2020] M. Ranković, R. Kumar T P, P. Nag, J. Kočišek, and J. Fedor, Temporary anions of the dielectric gas C3F7CN and their decay channels, J. Chem. Phys. 152, 244304 (2020).
  * Med  _et al._ [2020] J. Med, Š. Sršeň, P. Slavíček, A. Domaracka, S. Indrajith, P. Rousseau, M. Fárník, J. Fedor, and J. Kočišek, Vibrationally mediated stabilization of electrons in nonpolar matter, J. Phys. Chem. Lett. 11, 2482 (2020).
  * Kumar T. P.  _et al._ [2020] R. Kumar T. P., J. Kočišek, K. Bravaya, and J. Fedor, Electron-induced vibrational excitation and dissociative electron attachment in methyl formate, Phys. Chem. Chem. Phys. 22, 518 (2020).
  * Estrada  _et al._ [1986] H. Estrada, L. S. Cederbaum, and W. Domcke, Vibronic coupling of short‐lived electronic states, J. Chem. Phys. 84, 152 (1986).
  * Dvořák  _et al._ [2022a] J. Dvořák, M. Ranković, K. Houfek, P. Nag, R. Čurík, J. Fedor, and M. Čížek, Vibronic coupling through the continuum in the $e+{\mathrm{CO}}_{2}$ system, Phys. Rev. Lett. 129, 013401 (2022a).
  * Dvořák  _et al._ [2022b] J. Dvořák, K. Houfek, and M. Čížek, Vibrational excitation in the $e+{\mathrm{CO}}_{2}$ system: Nonlocal model of $\mathrm{\Sigma}\mathrm{\Pi}$ vibronic coupling through the continuum, Phys. Rev. A 105, 062821 (2022b).
  * Dvořák  _et al._ [2022c] J. Dvořák, M. Ranković, K. Houfek, P. Nag, R. Čurík, J. Fedor, and M. Čížek, Vibrational excitation in the $e+{\mathrm{CO}}_{2}$ system: Analysis of two-dimensional energy-loss spectrum, Phys. Rev. A 106, 062807 (2022c).
  * Gianturco and Jain [1986] F. A. Gianturco and A. Jain, The theory of electron scattering from polyatomic molecules, Phys. Rep. 143, 347 (1986).
  * Domcke [1991] W. Domcke, Theory of resonance and threshold effects in electron-molecule collisions: The projection-operator approach, Phys. Rep. 208, 97 (1991).
  * Čížek and Houfek [2012] M. Čížek and K. Houfek, Nonlocal Theory of Resonance Electron-Molecule Scattering, in  _Low-energy Electron Scattering from Molecules, Biomolecules and Surfaces_, edited by P. Čársky and R. Čurík (CRC Press, 2012) Chap. 4, pp. 91–125.
  * Scarlett  _et al._ [2021] L. H. Scarlett, I. Bray, and V. D. Fursa, Electronic and vibrational close-coupling method for resonant electron-molecule scattering, Phys. Rev. Lett. 127, 223401 (2021).
  * Čurík and Čárský [2003] R. Čurík and P. Čárský, Vibrationally inelastic electron scattering on polyatomic molecules by the discrete momentum representation (DMR) method, J. Phys. B 36, 2165 (2003).
  * Kossoski and Mario [2020] F. Kossoski and B. Mario, Nonadiabatic dynamics in multidimensional complex potential energy surfaces, Chem. Sci. 11, 9827 (2020).
  * Ambalampitiya and Fabrikant [2020] H. B. Ambalampitiya and I. I. Fabrikant, Nonlocal complex potential theory of dissociative electron attachment: Inclusion of two vibrational modes, Phys. Rev. A 102, 022802 (2020).
  * Feuerbacher  _et al._ [2004] S. Feuerbacher, T. Sommerfeld, and L. S. Cederbaum, Intersections of potential energy surfaces of short-lived states: The complex analogue of conical intersections, J. Chem. Phys. 120, 3201 (2004).
  * Feuerbacher and Cederbaum [2004] S. Feuerbacher and L. S. Cederbaum, Jahn-Teller effect for short-lived states: Study of the complex potential energy surfaces, J. Chem. Phys. 121, 5 (2004).
  * Feshbach [1962] H. Feshbach, A unified theory of nuclear reactions II, Ann. Phys. 19, 287 (1962).
  * Mies [1968] F. H. Mies, Configuration interaction theory. Effect of overlapping resonances, Phys. Rev. 175, 164 (1968).
  * Berman  _et al._ [1983] M. Berman, H. Estrada, L. S. Cederbaum, and W. Domcke, Nuclear dynamics in resonant electron-molecule scattering beyond the local approximation: The 2.3-eV shape resonance in $N_{2}$, Phys. Rev. A 28, 1363 (1983).
  * Ragesh Kumar  _et al._ [2022] T. P. Ragesh Kumar, P. Nag, M. Ranković, T. F. M. Luxford, J. Kočišek, Z. Mašín, and J. Fedor, Distant symmetry control in electron-induced bond cleavage, J. Phys. Chem. Lett. 13, 11136 (2022).
  * Saad and Schultz [1986] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. on Scientific & Statistical Computing 7, 856 (1986).
  * Faber and Manteuffel [1984] V. Faber and T. Manteuffel, Necessary and sufficient conditions for the existence of a conjugate gradient method, SIAM J. on Num. Analysis 21, 352 (1984).
  * Liesen and Strakoš [2013] J. Liesen and Z. Strakoš,  _Krylov subspace methods : principles and analysis_ (Oxford University Press, 2013).
  * van der Vorst and Melissen [1990] H. A. van der Vorst and J. Melissen, A Petrov-Galerkin type method for solving Ax=b, where A is symmetric complex, IEEE Transactions on Magnetics 26, 706 (1990).
  * Saad [2003] Y. Saad,  _Iterative methods for sparse linear systems_ (Society for Industrial and Applied Mathematics, 2003).
  * Šarmanová [2020] M. Šarmanová,  _Iterative calculation of vibrational dynamics in electron scattering from molecule_ , Bcl thesis, Charles University, Prague, http://hdl.handle.net/20.500.11956/121270 (2020).
  * Šarmanová [2022] M. Šarmanová,  _Mathematical modeling of vibrational dynamics in electron scattering from molecule_ , Ms thesis, Charles University, Prague, http://hdl.handle.net/20.500.11956/175327 (2022).
  * Allan [2019] M. Allan, Two-dimensional electron-energy loss spectra reveal nuclear dynamics of negative ion resonances. (2019), Talk, Telluride, https://homeweb.unifr.ch/allanm/pub/ma/Lectures.html (unpublished).


* Mogavi et al. [2019] Reza Hadi Mogavi, Sujit Gujar, Xiaojuan Ma, and Pan Hui.  Hrcr: Hidden markov-based reinforcement to reduce churn in question answering forums.  In _PRICAI 2019: Trends in Artificial Intelligence: 16th Pacific Rim International Conference on Artificial Intelligence, Cuvu, Yanuca Island, Fiji, August 26–30, 2019, Proceedings, Part I 16_ , pages 364–376. Springer, 2019.
  * Fu et al. [2021] Yiyang Fu, Benjun Shen, Yuting Chen, and Linpeng Huang.  Tdmatcher: A topic-based approach to task-developer matching with predictive intelligence for recommendation.  _Applied Soft Computing_ , 110:107720, 2021.
  * Abhinav et al. [2023] Kumar Abhinav, Gurpriya Kaur Bhatia, Alpana Dubey, Sakshi Jain, and Nitish Bhardwaj.  Crowdassist: A multidimensional decision support system for crowd workers.  _Journal of Software: Evolution and Process_ , 35(6):e2404, 2023.
  * Ford et al. [2022] Denae Ford, Nischal Shrestha, and Thomas Zimmermann.  Reboc: Recommending bespoke open source software projects to contributors.  In _2022 IEEE Symposium on Visual Languages and Human-Centric Computing (VL/HCC)_ , pages 1–5. IEEE, 2022.
  * Ge et al. [2022] Xiuting Ge, Shengcheng Yu, Chunrong Fang, Qi Zhu, and Zhihong Zhao.  Leveraging android automated testing to assist crowdsourced testing.  _IEEE Transactions on Software Engineering_ , 49(4):2318–2336, 2022.
  * Jiang et al. [2017] Jyun-Yu Jiang, Pu-Jen Cheng, and Wei Wang.  Open source repository recommendation in social coding.  In _Proceedings of the 40th International ACM SIGIR Conference on Research and Development in Information Retrieval_ , pages 1173–1176, 2017.
  * Huang et al. [2021] Yuekai Huang, Junjie Wang, Song Wang, Zhe Liu, Dandan Wang, and Qing Wang.  Characterizing and predicting good first issues.  In _Proceedings of the 15th ACM/IEEE international symposium on Empirical Software Engineering and Measurement (ESEM)_ , pages 1–12, 2021.
  * Ren et al. [2023] Hao Ren, Mingliang Ma, Xiaowei Zhang, Yulu Cao, and Changhai Nie.  Effective recommendation of cross-project correlated issues based on issue metrics.  In _Proceedings of the 14th Asia-Pacific Symposium on Internetware_ , pages 1–1, 2023.
  * Xiao et al. [2022] Wenxin Xiao, Hao He, Weiwei Xu, Xin Tan, Jinhao Dong, and Minghui Zhou.  Recommending good first issues in github oss projects.  In _Proceedings of the 44th International Conference on Software Engineering_ , pages 1830–1842, 2022.
  * Wang and Sarma [2011] Jianguo Wang and Anita Sarma.  Which bug should i fix: helping new developers onboard a new project.  In _Proceedings of the 4th International Workshop on Cooperative and Human Aspects of Software Engineering_ , pages 76–79, 2011.
  * Fu [2020] Chaogang Fu.  User intimacy model for question recommendation in community question answering.  _Knowledge-Based Systems_ , 188:104844, 2020.
  * Karim et al. [2018] Muhammad Rezaul Karim, Ye Yang, David Messinger, and Guenther Ruhe.  Learn or earn?-intelligent task recommendation for competitive crowdsourced software development.  2018\.
  * YUEN Man-Ching et al. [2015] Connie YUEN Man-Ching, Irwin King, and LEUNG Kwong Sak.  Probabilistic matrix factorization with active learning for quality assurance in crowdsourcing systems.  2015\.
  * Leban [2013] Gregor Leban.  Semantic tools for improving software development in open source communities.  In _ISWC (Posters & Demos)_, pages 97–100, 2013.
  * Dai et al. [2023] Jie Dai, Qingshan Li, Hui Xue, Zhao Luo, Yinglin Wang, and Siyuan Zhan.  Graph collaborative filtering-based bug triaging.  _Journal of Systems and Software_ , 200:111667, 2023.
  * Wu et al. [2022] Hongrun Wu, Yutao Ma, Zhenglong Xiang, Chen Yang, and Keqing He.  A spatial–temporal graph neural network framework for automated software bug triaging.  _Knowledge-Based Systems_ , 241:108308, 2022.
  * He et al. [2022] Hao He, Haonan Su, Wenxin Xiao, Runzhi He, and Minghui Zhou.  Gfi-bot: Automated good first issue recommendation on github.  In _Proceedings of the 30th ACM Joint European Software Engineering Conference and Symposium on the Foundations of Software Engineering_ , pages 1751–1755, 2022.
  * Kashiwa [2019] Yutaro Kashiwa.  Raptor: Release-aware and prioritized bug-fixing task assignment optimization.  In _2019 IEEE International Conference on Software Maintenance and Evolution (ICSME)_ , pages 629–633. IEEE, 2019.
  * Yu et al. [2019] Dunhui Yu, Yi Wang, and Zhuang Zhou.  Software crowdsourcing task allocation algorithm based on dynamic utility.  _IEEE Access_ , 7:33094–33106, 2019.
  * Zhang et al. [2014] Lingxiao Zhang, Yanzhen Zou, Bing Xie, and Zixiao Zhu.  Recommending relevant projects via user behaviour: an exploratory study on github.  In _Proceedings of the 1st International Workshop on Crowd-based Software Development Methods and Technologies_ , pages 25–30, 2014.
  * Yadav et al. [2022] Asmita Yadav, Mohammed Baljon, Shailendra Mishra, Sandeep Kumar Singh, Sharad Saxena, and Sunil Kumar Sharma.  Developer load balancing bug triage: Developed load balance.  _Expert Systems_ , page e13006, 2022.
  * Fu [2019] Chaogang Fu.  Tracking user-role evolution via topic modeling in community question answering.  _Information Processing & Management_, 56(6):102075, 2019.
  * Zhang et al. [2017] Yun Zhang, David Lo, Pavneet Singh Kochhar, Xin Xia, Quanlai Li, and Jianling Sun.  Detecting similar repositories on github.  In _2017 IEEE 24th International Conference on Software Analysis, Evolution and Reengineering (SANER)_ , pages 13–23. IEEE, 2017.
  * Guendouz et al. [2015] Mohamed Guendouz, Abdelmalek Amine, and Reda Mohamed Hamou.  Recommending relevant open source projects on github using a collaborative-filtering technique.  _International Journal of Open Source Software and Processes (IJOSSP)_ , 6(1):1–16, 2015.
  * Koskela et al. [2018] Miika Koskela, Inka Simola, and Kostas Stefanidis.  Open source software recommendations using github.  In _Digital Libraries for Open Knowledge: 22nd International Conference on Theory and Practice of Digital Libraries, TPDL 2018, Porto, Portugal, September 10–13, 2018, Proceedings 22_ , pages 279–285. Springer, 2018.
  * Liu et al. [2018] Chao Liu, Dan Yang, Xiaohong Zhang, Baishakhi Ray, and Md Masudur Rahman.  Recommending github projects for developer onboarding.  _IEEE Access_ , 6:52082–52094, 2018.
  * Zhou et al. [2021] Yuqi Zhou, Jiawei Wu, and Yanchun Sun.  Ghtrec: a personalized service to recommend github trending repositories for developers.  In _2021 IEEE International conference on web services (ICWS)_ , pages 314–323. IEEE, 2021.
  * Kim et al. [2021] JaeWon Kim, JeongA Wi, and YoungBin Kim.  Sequential recommendations on github repository.  _Applied Sciences_ , 11(4):1585, 2021.
  * Nguyen et al. [2018] Phuong T Nguyen, Juri Di Rocco, Riccardo Rubei, and Davide Di Ruscio.  Crosssim: Exploiting mutual relationships to detect similar oss projects.  In _2018 44th Euromicro conference on software engineering and advanced applications (SEAA)_ , pages 388–395. IEEE, 2018.
  * Xiao et al. [2023] Wenxin Xiao, Jingyue Li, Hao He, Ruiqiao Qiu, and Minghui Zhou.  Personalized first issue recommender for newcomers in open source projects.  In _2023 38th IEEE/ACM International Conference on Automated Software Engineering (ASE)_ , pages 800–812. IEEE, 2023.
  * Yang et al. [2022] Peng Yang, Chao Chang, and Yong Tang.  Crowdsourced testing task assignment based on knowledge graphs.  In _2022 IEEE 22nd International Conference on Software Quality, Reliability and Security (QRS)_ , pages 663–671. IEEE, 2022.
  * Fu et al. [2023] Zixuan Fu, Chenghua Wang, and Jiajie Xu.  Graph contextualized self-attention network for software service sequential recommendation.  _Future Generation Computer Systems_ , 149:509–517, 2023.
  * Abbasi et al. [2021] Soohan Abbasi, Zulfiqar Ali, Rajesh Kumar, Madiha Shaikh, Komal Maheshwari, Paras Lal, and Sagar Kumar.  Relevant projectrecommendation system using developer behavior and project features.  _International Journal_ , 10(5), 2021.