Also at ]Trento Institute for Fundamental Physics and Applications (TIFPA-
INFN), Trento, Italy &
Peter the Great St. Petersburg Polytechnic University, Russia

Also at ]INFN, Sezione di Perugia, Italy

# The Resonant and Normal Auger spectra of Ozone

Simone Taioli<EMAIL_ADDRESS>European Centre for Theoretical Studies in
Nuclear Physics and Related Areas (ECT*-FBK), Trento, Italy  [  Stefano
Simonucci  School of Science and Technology, University of Camerino, Italy  [

###### Abstract

In this work we outline a general method for calculating Auger spectra in
molecules, which accounts for the underlying symmetry of the system. This
theory starts from Fano’s formulation of the interaction between discrete and
continuum states and generalizes this formalism to deal with the simultaneous
presence of several intermediate quasi-bound states and several non-
interacting decay channels. Our theoretical description is specifically
tailored to resonant autoionization and Auger processes, and includes
explicitly the incoming wave boundary conditions for the continuum states and
an accurate treatment of the Coulomb repulsion. This approach is implemented
and applied to the calculation of the $K-LL$ Auger and autoionization spectra
of ozone, which is a $C_{2v}$ symmetric molecule, whose importance in our
atmosphere to filter out radiation has been widely confirmed. We also show the
effect that the molecular point group, and in particular the localization of
the core-hole in the oxygen atoms related by symmetry operations, has on the
electronic structure of the Auger states and on the spectral lineshape by
comparing our results with experimental data.

††preprint: APS/123-QED

##  I Introduction

Ozone has attracted enormous interest in recent years, due in particular to
its paramount role played in atmospheric chemistry. On the one hand, a
dramatic decrease in ozone concentration within the Earth’s upper atmosphere
has shown a major negative impact on the effective shielding of our planet
from ultraviolet sunlight, whose absence would be fatal to life. [1]. On the
other hand, ozone is known to act as greenhouse gas [2, 3], thus increasing
global warming, as well as to be a harmful pollutant in metropolitan areas
produced by exhaust gases of combustion processes with a considerable human
health hazard potential.

These reports on the importance of ozone in the chemical dynamics of the
atmosphere, as well as in the absorption of virtually all the ultraviolet
light that passes into the stratosphere, fuelled ozone research. However,
while several studies focused on the ground state electronic and vibrational
properties of ozone [4, 5, 6], little to no theoretical and experimental
information about electronic excited state dynamics can be found [7, 8, 9].
This investigation turns out to be crucial e.g. to identify the molecular
species that filter out the highest–energy light, such as $\gamma$, X, and UV-
rays and to determine the amount of radiation reaching the surface of our
Earth.

Moreover, the analysis of the excited states is interesting also in view of
the fact that all of the ozone bands are dissociative [10, 11], meaning that
the molecule falls apart to $O+O_{2}$ upon photon absorption. This
dissociative mechanism drives further helpful or harmful reactions depending
on where in the Earth’s atmosphere they occur [12]. Often in these photo-
excited processes, the system goes through intermediate quasi-bound states,
which have long lifetimes if compared with the collision times. Afterwards,
the excited system undergoes a dissociative path into final channels that are
characterized by the presence of a few asymptotically non-interacting
fragments, the observation of which provides useful information on the
properties of the system under examination. In general, these decay processes
are due to the fact that the resonant excited state produced by the initial
collision with the high-energy photon is embedded in the continua of the final
fragments.

Auger spectroscopy, in particular, is a type of electron spectroscopy based on
scattering processes in which the initial state consists of a projectile,
typically X-ray or UV photons or electrons, that collides with an atomic,
molecular or solid state target, and the final states feature electrons,
photons or heavy particles according to the type of experiment. The analysis
of the energies and the intensity distribution of these secondary electrons
represents the central problem of Auger spectroscopy. Indeed, Auger electrons
are emitted with system-dependent probabilities in several open channels,
characterised by different kinetic energies and identified by different
quantum numbers. Of course the Auger spectral lineshape reflects the symmetry
of the perturbed system.

From the standpoint of computation one notes the existence of an ample body of
techniques developed for atomic systems, whereas for molecules the number is
much more limited [13, 14]. This is mainly due to the difficulties created by
the reduced symmetry of molecular systems, which hinders the use of numerical
techniques for representing the electron as it moves outward through the field
of the ionized molecule. Furthermore, another relevant difficulty of molecules
is represented by the inclusion of the nuclear motion and its coupling with
the electronic degrees of freedom. For example, in ozone, fast excited
dissociation may occur at once with the Auger emission [10], generating an
interplay between the vibrational and electronic degrees of freedom beyond the
Born-Oppenheimer approximation [15, 16, 17], which affects the kinetic energy
of the emitted Auger electrons.

In this work we present first a general theoretical and computational method
for interpreting core-electron spectroscopies on molecules, such as
autoionization and Auger, which allows one to reproduce accurately the
experimental results, by including in principle the vibrational details as
well as the combined effects on the spectral lineshapes due to the intrinsic
features of the transition, the characteristics of the incident radiation and
those of the electron spectrometer. In particular, we discuss the central
feature of our approach, that is the ability to calculate accurate wave
functions for continuum states of polycentric symmetric (or asymmetric)
systems at a computational effort similar to that of standard bound-state
calculations. This method is used for calculating the
$O1s\rightarrow\sigma^{*}$ resonant photoemission and the $K-LL$ Auger spectra
of the ozone molecule. The accuracy of the results is increased progressively
by using different levels of theory, from mean field Hartree-Fock (HF) to
correlated method, such as Configuration Interaction (CI), measuring the
relative importance of the different contributions of the electron-electron
correlation to the experimental spectrum. Furthermore, the effect on the Auger
lineshape of the presence of core-hole localized in the $1s$ orbitals of
symmetry-related oxygen atoms rather than in a delocalized molecular orbital
is discussed.

##  II Theoretical and Computational Methods

###  II.1 Autoionization and Auger decay as resonant multichannel processes

Auger states are excited, quasi-bound states created by inner-shell excitation
or ionization that are embedded in the continuum of the next higher charge
state of the system. They can decay either by electron emission (radiationless
transition) or by radiation emission. The radiationless decay is called
autoionization or Auger transition depending weather the system undergoes
initially a neutral excitation by promoting an electron to an empty orbital or
is ionized by the impinging photons. Often the primary excitation process is
accompanied by the simultaneous ionization (shake-off) or excitation (shake-
up) of a valence electron. The resulting peaks in the spectrum are called
satellite lines. Shake-up and shake-off processes typically account for 10% of
the total spectral intensity and will be neglected in our analysis.

To evaluate the autoionization (or Auger) cross section we rely on the time-
independent approach developed by Fano [18]. This approach interprets the
decay process as due to the interaction between a quasi-bound state, produced
by the primary excitation (ionization), and the continuum states of the single
(double) ionized target.

Let us describe this method in the case of an autoionization process in which
one electron is excited to a discrete (resonant) state $|\Phi>$ that is
degenerate with several continua $\\{|\chi_{\alpha,{\cal E}}^{-}>\\}$ of the
ionized target [19]. We notice that the label $|\alpha>$ is a notation
standing for “good quantum numbers” (angular momentum, spin, etc…) in
spherically symmetric systems, such as atoms, and symmetric molecules. The
total Hamiltonian of the problem, assuming the validity of the Born-
Oppenheimer approximation, is $H=H_{s}+H_{rad}+H_{int}$, where $H_{s}$ is the
many-body Hamiltonian of the isolated system,
$H_{rad}=\sum_{k,\lambda}\hbar\omega
a^{\dagger}_{\lambda}({\bm{k}})a_{\lambda}({\bm{k}})$ is the electromagnetic
radiation Hamiltonian, where ${\bm{k}}$ is the wavenumber of a wave confined
within a box of volume $V=L\times L\times L$, $\lambda$ is the polarization
direction and $\omega=kc$, and finally the interaction operator
$H_{int}=\hat{e}_{\lambda}\cdot\sum_{j}q_{j}\vec{r}_{j}$, where $O_{\lambda}$
is the component of the dipolar operator in the polarization direction
$\hat{e}_{\lambda}$.

The continuum wave function of the system $\\{|\chi_{\alpha,{\cal E}}^{-}>\\}$
is characterized by the incoming wave boundary condition ($-$), which means
that asymptotically represents one electron released, with energy ${\cal E}$,
into a channel specified by the state $\alpha$ of the ionized target at the
energy $E_{\alpha}$. In Fano’s approach the scattering eigenstate
$|\Psi_{\alpha,{\cal E}}^{-}>$ of the Hamiltonian of the isolated system
$H_{s}$ is represented by the following linear combination of discrete and
continuum states:

$|\Psi^{-}_{\alpha,{\cal E}}>=a_{\alpha}({\cal
E})|\Phi>+\sum_{\beta=1}^{N_{c}}\int\limits_{0}^{\infty}|\chi^{-}_{\beta,\tau}>C_{\beta,\alpha}(\tau,{\cal
E})d\tau,$ (1)

with the normalization condition:

$<\Psi^{-}_{\alpha,{\cal E}}|\Psi^{-}_{\beta,{\cal
E}^{\prime}}>=\delta_{\alpha\beta}\delta(E_{\alpha}+{\cal E}-E_{\beta}-{\cal
E}^{\prime}).$ (2)

The states $\\{|\chi^{-}_{\beta,\tau}>\\}$, which appear in Equation (1), are
continuum states obtained through the diagonalization of the many-electron
Hamiltonian matrix of the system $H_{s}$ that is constructed using a set of
interacting continuum states $\\{|\chi_{\beta,\epsilon}\\}>$ as follows:

$<\chi_{\beta,\varepsilon}|H_{s}-E|\chi_{\gamma,\varepsilon^{\prime}}>=(E_{\beta}+\varepsilon-E)\delta(E_{\beta}+\varepsilon-
E_{\gamma}-\varepsilon^{\prime})+V_{\beta\gamma}(\varepsilon,\varepsilon^{\prime};E).$
(3)

A solution of this set of equations is characterized by the following
asymptotic behavior:

$\lim_{r\rightarrow\infty}\chi_{\alpha,\cal
E}^{-}=\sum_{\gamma}{\Omega_{\gamma}\over 2ir}\left(e^{i\theta_{\gamma}({\cal
E},r)}\delta_{\gamma\alpha}-e^{-i\theta_{\gamma}({\cal
E},r)}S^{+}_{\gamma\alpha}\right),$ (4)

where $\\{\Omega_{\gamma}\\}$ are symmetry-adapted wave functions describing
possible states of the ionized target and containing also the angular
coordinates and the spin state of the outgoing electron. The radial phases
$\\{\theta_{\gamma}({\cal E},r)\\}$ depend on the nature of the long-range
interaction inside each channel and $S^{+}_{\gamma\alpha}$ is the scattering
amplitude into channel $\gamma$. The coefficients of Equation (1) are obtained
by solving the following equations

$<\Phi|H_{s}-E|\Psi^{-}_{\alpha,{\cal
E}}>=<\chi^{-}_{\beta,\varepsilon}|H_{s}-E|\Psi^{-}_{\alpha,{\cal E}}>=0,$ (5)

which explicitly read:

$(E_{\Phi}-E)a_{\alpha}({\cal
E})+\sum_{\beta=1}^{N_{c}}\int\limits_{0}^{\infty}M_{\beta}^{-}(\tau,E)C_{\beta\alpha}(\tau,{\cal
E})d\tau=0;$ (6) $a_{\alpha}({\cal
E})M_{\beta}^{-}(\varepsilon,E)^{*}+(\varepsilon+E_{\beta}-E)C_{\beta\alpha}(\varepsilon,{\cal
E})=0,$ (7)

where

$M_{\beta}^{-}(\varepsilon,E)=<\Phi|H_{s}-E|\chi_{\beta,\varepsilon}^{-}>;~{}~{}~{}~{}(E_{\Phi}-E)=<\Phi|H_{s}-E|\Phi>~{}.$
(8)

By moving into the complex plane to avoid singularities that appear in the
coefficients $\\{a_{\alpha},C_{\beta\alpha}\\}$ of Equations (6),(7), one gets
the following expression

$|\Psi^{-}_{\alpha,{\cal E}}>=|\chi^{-}_{\alpha,{\cal
E}}>+{M^{-}_{\alpha}({\cal E},E)\over E-E_{r}-i{\Gamma\over
2}}\left[|\Phi>+\lim_{\nu\rightarrow
0}\sum_{\beta}\int\limits_{0}^{\infty}{|\chi^{-}_{\beta,\tau}>M^{-}_{\beta}(\tau,E)^{*}\over
E-E_{\beta}-\tau-i\nu}d\tau\right],$ (9)

with $\Gamma$ and $E_{r}$ defined as follows

$\Gamma=\sum_{\beta}\Gamma_{\beta}=2\pi\sum_{\beta}|M^{-}_{\beta}({\cal
E}_{r},E)|^{2};\qquad{\cal E}_{r}=E_{r}-E_{\alpha};$ (10)
$E_{r}=E_{\Phi}+\Delta;\qquad\Delta=\sum_{\beta}{\cal
P}\int\limits_{0}^{\infty}{|M^{-}_{\beta}(\tau,E)|^{2}\over
E-E_{\beta}-\tau}d\tau~{}.$ (11)

Finally, knowing the stationary states $\\{|\Psi^{-}_{\alpha,{\cal E}}>\\}$
for each possible combination of the quantum numbers $\alpha$ and ${\cal E}$,
one can obtain the cross section for the autoionization process in which
photon absorption of appropriate energy promotes an electron to a discrete
resonant state $|\Phi>$, and afterward the ionized target decays non-
radiatively into channel $|\chi^{-}_{\alpha,{\cal E}}>$. Indeed, according to
the general theory of scattering [20, 21] the autoionization cross section is
proportional to the square element of the transition matrix. Using Equations
(8)-(11) one obtains:

$\displaystyle{\bf T}^{+}_{fi}({\cal E})=<\chi^{-}_{\alpha,{\cal
E}};N-1,\omega|\hat{H}_{int}|o;N,\omega>+$
$\displaystyle+{<\chi^{-}_{\alpha,{\cal
E}}|\hat{H}_{s}-E|\Phi><\Phi^{-};N-1,\omega|H_{int}|o;N,\omega>\over{\cal
E}-{\cal E}_{r}+i{\Gamma\over 2}}$ , (12)

connecting the initial state $|\Psi^{+}_{i}>=|o>|N,\omega>$, which is the
tensorial product of the ground state $|o>$ of the system and of the
$|N,\omega>$ state of the radiation with $N$ $\omega$-frequency photons, to
the final state $|\Psi^{-}_{f}>=|\Psi_{\alpha,\cal E}^{-}>|N-1,\omega>$, which
is the tensorial product of the continuum state $|\Psi_{\alpha,\cal E}^{-}>$
of the system and of a state of the radiation in which one photon of frequency
$\omega$ has been absorbed. In Equation (12) the following definition has been
used:

$\displaystyle|\Phi^{-}>$ $\displaystyle=$
$\displaystyle|\Phi>+\lim_{\nu\rightarrow
0}\sum_{\beta=1}^{N_{c}}\int\limits_{0}^{\infty}{|\chi^{-}_{\beta,\tau}>M^{-}_{\beta}(\tau,E)^{*}\over
E-E_{\beta}-\tau-i\nu}d\tau;$ $\displaystyle{\cal E}$ $\displaystyle=$
$\displaystyle\omega-(E_{\alpha}-E_{o}).$ (13)

By neglecting the first term in the right hand side of Equation (12), which
represents the contribution to the cross section of direct photoemission, the
cross section of the resonant autoionization process can be approximated as
follows:

${d\sigma_{fi}({\cal E})\over d{\cal E}}={4\pi^{2}\omega\over
Nc}|T^{+}_{fi}({\cal E})|^{2}=\sigma(\omega){1\over
2\pi}{\Gamma_{\alpha}\over({\cal E}-{\cal E}_{r})^{2}+\Gamma^{2}/4},$ (14)

where $\sigma(\omega)\propto|<\Phi;N-1,\omega|H_{int}|o;N,\omega>|^{2}$ is the
cross section of the excitation process $|o>\rightarrow|\Phi>$. One can
observe that, as long as $\Gamma_{\alpha}$, which represents the decay rate
into channel $|\alpha>$ and $\Gamma$, which represents the total decay rate,
are slowly energy dependent in the range of interest, the cross section per
unit energy defined by Equation (14) has a Lorentzian profile. The energy
shift $\Delta$, defined in Equation (11) usually represents a small correction
to $E_{\Phi}$, so typically can be neglected.

We notice that similar results can be obtained also for the Auger decay where,
at odds with the resonant autoionzation process, the intermediate state is
constituted by a quasi-bound state of the ionized target plus one electron
emitted upon the initial photoionization, while the final state is
characterized by the presence of two electrons in the continuum. In our model,
which includes explicitly the incoming wave boundary conditions for the
continuum states, the Auger decay is described as a two-step process in which
no interaction occurs between the primary photo-emitted and Auger electrons.

The ${\bf T}$ matrix of the process can be, therefore, written as follows:

$T^{+\prime}_{\alpha i}=<\chi^{-}_{\alpha,{\cal E}_{1}{\cal
E}_{2}};N-1,\omega|\hat{H}_{int}|o;N,\omega>+\int\limits_{0}^{\infty}{<\chi^{-}_{\alpha,{\cal
E}_{1}{\cal
E}_{2}}|\hat{H}_{s}-E|\Phi_{\tau}><\Phi_{\tau};N-1,\omega|H_{int}|o;N,\omega>d\tau\over{\cal
E}_{1}+{\cal E}_{2}-{\cal E}_{r}-\tau+i\Gamma/2},$ (15)

where ${\cal E}_{1}$ and ${\cal E}_{2}$ are the energies of the outgoing
electrons related to the other characteristic energies of the problem by the
following relationship: ${\cal E}_{1}+{\cal E}_{2}=\omega-(E_{\alpha}-E_{o})$
($E_{o}$ is the ground state energy of the system). The intermediate state
$|\Phi_{\tau}>$ is a continuum state that, as long as
$\omega>>E_{\alpha}-E_{r}$, can be approximated by an antisymmetrized product,
$\hat{A}\\{|\Phi>|\tau>\\}$, where $|\tau>$ represents the state of the
primary electron. In a similar way, the final state can be represented by
$\hat{A}\\{|\chi^{-}_{\beta,{\cal E}_{2}}>|{\cal E}_{1}>\\}$, where $|{\cal
E}_{1}>$ is the state of the primary electron.

Neglecting the first terms in the right-hand side of Equation (15), that
represents the probability amplitude of the double direct ionization process,
one gets

$\displaystyle{d\sigma_{\alpha}({\cal E})\over d{\cal
E}}=\sigma(\overline{\cal E}){1\over 2\pi}{\Gamma_{\alpha}\over({\cal E}-{\cal
E}_{r})^{2}+\Gamma^{2}/4};$ $\displaystyle\overline{\cal
E}=\omega-(E_{r}-E_{o}),$ (16)

where $\sigma(\overline{\cal E})$ is the photoionization cross section of the
primary process. This result, again, displays a Lorentzian behaviour and
indicates that $\Gamma_{\alpha}$ is the rate of the non-radiative decay
process from the ion intermediate state $|\Phi>$ into the channel $|\alpha>$,
and $\Gamma$ is the total Auger decay rate.

The cross sections (14), (16) represent the starting point for constructing
the “theoretical” spectrum to be compared with the experimental one. They give
the contributions to the spectrum due to the intrinsic features of the target,
while the finite resolution of the electron spectrometer, the specific
characteristics of the incident photon beam, and the broadening of the Auger
lineshapes due to vibrational modes can be taken into account by performing a
convolution with a Gaussian function [14].

###  II.2 The projected potential approach and the many-body Hamiltonian

Basically, to assess the cross sections (14), (16) we are left with the
problems of calculating the intermediate resonant state and the continuum
wavefunction. In particular, the problem of finding the eigensolutions of the
Hamiltonian is separated with respect to finding the bound and continuum
orbitals. This means to neglect the effects that the continuum orbital has on
the bound orbitals, which is obtained without taking into account the presence
of the outgoing electron. This is the so-called ”Static Exchange
Approximation”. In general, the interacting decay channels
$\\{\tilde{\chi}_{\alpha\vec{k}}\\}$ are represented by the tensor product

$\chi_{\alpha\vec{k}}(1,..N-1,N)=\hat{\cal
A}[\Theta_{\alpha}(1,...,N-1)\varphi_{\alpha\vec{k}}(N)]$ (17)

between a set of functions $\varphi_{\alpha\vec{k}}(1)$ that describes the
spin-orbital of the unbound electron and $\Theta_{\alpha}(1,...,N-1)$ that is
the determinant representing the bound state of the remaining $N-1$ system
($N-2$ for Auger processes), and $\hat{\cal A}$ is the antisymmetrizer that
includes also the normalization constant.

On the one hand, since both the intermediate and the final ionic states are
bound states of the molecule, one can calculate their energies using standard
quantum mechanical techniques proposed for bound-state calculations, such as
the single Slater determinant HF or many configuration wave functions methods,
such as CI or MC-SCF, or Green’s function based techniques [14]. In
particular, the solutions of the secular problem for the $(N-1)$-electron
system ($N-2$ for Auger processes) are obtained inside the space spanned by
the orbitals taken from a set of $n$ bound orbitals $|\theta_{j}>$. Both the
HF single-determinant wavefunction and the CI treatment of symmetry-adapted
orbitals will be used to obtain the bound state eigenfunctions in Equation
(17) as:

$<\Theta_{\alpha}|\hat{H}_{s}^{N-1}|\Theta_{\beta}>=E_{\alpha}\delta_{\alpha\beta}.$
(18)

This solution is searched within the Hilbert space spanned by a basis set that
we use to expand the orbital wavefunctions. Typically our basis set is built
with Hermite Gaussian functions (HGF)

$g(\mathbf{r})=g(u,v,w;a,\mathbf{R};\mathbf{r})=N{\partial^{u+v+w}\over\partial
X^{u}\partial Y^{v}\partial
Z^{w}}({2\alpha\over\pi})^{3/4}exp[-\alpha(\mathbf{r}-\mathbf{R})^{2}],$

where $\mathbf{R}\equiv(X,Y,Z)$ gives the position where $g$ is centered,
$\alpha$ is a coefficient determining the HGF width, the order of derivation
($u,v,w$) determines the symmetry type, and $N$ is a normalization factor

$N=[\alpha^{l}(2u-1)!!(2v-1)!!(2w-1)!!]^{-1/2}{,}\qquad l=u+v+w,$

Using a mixed basis set of HGFs of every order and centrature we evaluate the
mono and bi-electronic integrals of the many-body Hamiltonian (18).

On the other hand, to construct the continuum orbital of Equation (17) in the
effective field of the bound orbitals we developed a method that is capable to
include also the interchannel coupling among various
$\\{\chi_{\alpha\vec{k}}\\}$ and, furthermore, the interaction among continuum
and discrete states. In order to include these effects we use the projected
potential approach. Within the projected potential framework, a model
Hamiltonian is defined in which both the monoelectronic and the bielectronic
part of the potential are represented in terms of $L^{2}$-functions. The use
of this model Hamiltonian allows us to include also the interchannel coupling
and to obtain directly the correct, non-interacting decay channels
$\\{|\chi^{-}_{\alpha\vec{k}}>\\}$ defined in the previous section (see
Equation (4)).

To present this approach we consider the case of a $N$-electron system with
one electron in the continuum. We define a projector
$\hat{\pi}(i)=\sum_{l=1}^{m}~{}|g_{l}(i)><g_{l}(i)|$ into a $m$-dimensional
space $({\cal G})$ of $L^{2}({\cal R}^{3})-$functions, spanned by the
orthonormal set $\\{|g_{l}>~{};~{}l=1,..,m\\}$. We project both the electron-
nuclei attraction potential
$\hat{V}^{en}(i)=-\sum_{\mu}{Z_{\mu}\over|\vec{r}_{i}-\vec{R}_{\mu}|}$, and
the electron-electron Coulomb repulsion
$\hat{v}(i,j)={1\over|\vec{r}_{i}-\vec{r}_{j}|}$ operators, obtaining the
following model electronic Hamiltonian

$\hat{H}_{s}(1,..,N)=\sum^{N}_{i=1}[\hat{T}(i)+\hat{V}_{en}^{\pi}(i)]+{1\over
2}\sum^{N}_{i\neq j}\hat{v}^{\pi}(i,j),$ (19) $\hat{T}(i)=-{1\over
2}\nabla^{2}_{i}~{}~{}~{};~{}~{}~{}\hat{V}_{en}^{\pi}(i)=\hat{\pi}(i)\hat{V}_{en}(i)\hat{\pi}(i),$
(20)
$\hat{v}^{\pi}(i,j)=\hat{\pi}(i)\hat{\pi}(j)\hat{v}(i,j)\hat{\pi}(i)\hat{\pi}(j).$
(21)

By projecting out only the potential terms of the Hamiltonian we come up with
the solution to the issues of i) representing accurately the orbital inside
the scattering region, which is the important volume where the matrix elements
that couple bound and continuum states (see Equation (8)) have to be
appropriately calculated; ii) recovering the continuum part of the spectrum by
means of the unprojected kinetic term. The region where the potential is
projected can of course be incremented by including e.g. more diffuse HGFs so
to approximate more accurately the Coulomb potential tail. Outside this
region, where only the long-range part of the potential survives, one can
represent the continuum orbital as a linear combination of eigenfunctions of
the long-range potential, which are typically known analytically (e.g. Coulomb
wave functions).

In order to construct the product (17) and find the continuum eigenfunctions
$\varphi_{\alpha\bf k}$ of the Hamiltonian (19) with positive eigenvalues, we
notice that the Hamiltonian matrix diagonalization within the entire
functional space $\cal G$ is equivalent to solve the following projected
Lippmann-Schwinger (LS) equation, that includes also the appropriate boundary
condition:

$\varphi_{\alpha{\bf k}}(\bm{r})=e^{i\bf k\cdot\bf
r}+\hat{G}_{0}^{-}(\varepsilon)\hat{T}_{\alpha}(\varepsilon)e^{i\bf k\cdot\bf
r},$ (22)

where
$G_{0}^{-}(\epsilon=\frac{k^{2}}{2})=\frac{1}{\epsilon-\frac{\hat{k}^{2}}{2}-i\epsilon}$
is the free-particle Green’s function and $\hat{T}_{\alpha}$ is the transition
operator defined by the equation:

$\hat{T}_{\alpha}=\hat{V}^{\pi}_{\alpha}+\hat{V}^{\pi}_{\alpha}\hat{G}_{0}^{-}(\varepsilon)\hat{T}_{\alpha}.$
(23)

In Equation (23) $\hat{V}^{\pi}_{\alpha}(\bm{r})$ is the approximate
representation of the Coulomb operator (see Equation (20)), including both
electron-electron and electron-nuclei interactions, projected by a finite set
of $L^{2}$ functions. The elements of this basis set are chosen to minimize
the difference
$(\hat{V}_{\alpha}-\hat{V}^{\pi}_{\alpha})|\varphi_{\alpha\vec{k}}>$ inside
the scattering volume. We notice that in Equation (22) the correction to the
free-wave state to obtain the scattering wavefunction is represented by a
linear combination of functions of the type $G_{0}^{-}|\alpha\rangle g_{j}$.

By using the previous definitions one gets the following expression for the
matrix element that couples two interacting channels:

$<\chi_{\alpha\vec{k}}|\hat{H}^{N}|\chi_{\beta\vec{p}}>=(2\pi)^{3}\delta(\vec{k}-\vec{p})\delta_{\alpha\beta}({k^{2}\over
2}+E_{\alpha})+<\varphi_{\alpha\vec{k}}|\hat{V}^{en}_{\pi}\delta_{\alpha\beta}+\hat{W}_{\pi}^{\alpha\beta}|\varphi_{\beta\vec{p}}>,$
(24)

where

$\hat{W}_{\pi}^{\alpha\beta}(1)=\sum_{j=2}^{N}<\Theta_{\alpha}(2,.j.,N)|\hat{v}_{\pi}(1,j)(\hat{1}-\hat{\cal
P}_{1,j})|\Theta_{\beta}(2,.j.,N)>,$ (25)

and $\hat{\cal P}_{1,j}$ is the operator that interchanges the $(1,j)$
variables.

If now we assume that the bound-state problem has been solved, i.e. the
eigenvectors $(|\Theta_{1}>,|\Theta_{2}>,..,|\Theta_{M}>)$ of $\hat{H}^{N-1}$
($\hat{H}^{N-2}$ for the Auger process) have been found inside the space of
the Slater determinants built up using the $\\{\theta_{j}\\}$ orbitals, we can
look at the matrix elements defined in Equation (24) as the representation of
an effective one-particle Hamiltonian over a set of basis vectors
$\\{|\varphi_{\alpha\vec{k}}>\\}$. This Hamiltonian, indeed, is that of a
particle, with internal degrees of freedom, that moves in an effective
potential depending on the internal states $\\{|\alpha>\\}$ of the particle
itself. We notice that the basis vectors $\\{|\varphi_{\alpha\vec{k}}>\\}$ are
labelled by two indices, one continuous $(\vec{k})$ and one discrete
$(\alpha)$, and that they shall satisfy orthonormality constraints both with
respect to $\vec{k}$ and to $\alpha$. Finally, the bound orbitals
$\Theta_{\alpha}(1,...,N-1)$ and continuum wavefunctions
$\varphi_{\alpha\vec{k}}(\vec{r})$ can be made mutually orthogonal.

##  III Results and discussion

###  III.1 The ozone molecular geometry and its electronic structure

Ozone ground state has a trigonal planar bent molecular geometry belonging to
the $C_{2v}$ symmetry group (similar to the water molecule), whereby the
central oxygen atom is in a $sp^{2}$-hybridized configuration. To optimize the
ozone atomic coordinates we started from experimental oxygen positions and
accommodated the molecule in a cell with side of 10 Å. The ozone geometry was
relaxed below $10^{-3}$ Ry/Å for the interatomic forces via first-principles
density functional theory (DFT) calculations as implemented in the Quantum
Espresso code suite [22], using a PBE-GGA functional [23]. We have used the
Troullier-Martins (TM) norm-conserving pseudopotentials tabulated in the
Quantum Espresso web page. Including the $\Gamma$ point only to sample the
Brillouin zone and using a kinetic energy cut-off of 130 Ry, the self-
consistent DFT convergence is reached within the energy error of $10^{-5}$.
Upon optimization, the O–O bond length turns out to be 1.273 Å, while the
O–O–O angle is 117.16∘, which well compare with experimental data [24] (1.272
Å and 116.78∘, respectively). The ozone ground state geometry is shown in
Figure 1. The molecular bonds in ozone can be represented as a resonance
between two contributing structures, each with a single bond on one side and
double bond on the other, where the terminal OT atoms are more electron rich
than the central OC atom. The 3 $sp^{2}$ hybrid orbitals form a net of 2 O–O
$\sigma$ bonds and 5 lone pairs (two on each terminal OT and one on the
central OC). The remaining 4 valence electrons are distributed among the 3
unhybridized $2p_{z}$ orbitals on each oxygen atom, forming one lowest energy
$\pi$ bond, one highest energy $\pi^{*}$ antibonding and one intermediate
energy $\pi$ nonbonding orbitals, respectively.

Figure 1: Ozone geometry in the ground state.

###  III.2 The Auger spectrum of ozone

Electronic structure calculations have been carried out initially at mean-
field HF level for the ground state, a set of highly-excited core-hole
intermediate states and the double-ionized molecule. We remind that the HF
method is based on a set of non-linear equations, minimized to deliver the
best variational wavefuntions and energy. Thus, they are sensitive to the
kick-off orbitals. In this regard, we will show below the effect of localizing
the inner hole on atomic or molecular symmetry orbitals.

The ground state of the ozone molecule is described by the following
configuration
${}^{1}A_{1}:~{}1a_{1}^{2}\,1b_{1}^{2}\,2a_{1}^{2}\,3a_{1}^{2}\,2b_{1}^{2}\,4a_{1}^{2}\,5a_{1}^{2}\,3b_{1}^{2}\,1b_{2}^{2}\,4b_{1}^{2}\,6a_{1}^{2}\,1a_{2}^{2}$.

The intermediate quasi-bound states populated by the primary ionization are
the following:

  * •

${}^{2}A_{1}:~{}1a_{1}^{1}\,1b_{1}^{2}\,2a_{1}^{2}\,3a_{1}^{2}\,2b_{1}^{2}\,4a_{1}^{2}\,5a_{1}^{2}\,3b_{1}^{2}\,1b_{2}^{2}\,4b_{1}^{2}\,6a_{1}^{2}\,1a_{2}^{2}$;

  * •

${}^{2}B_{1}:~{}1a_{1}^{2}\,1b_{1}^{1}\,2a_{1}^{2}\,3a_{1}^{2}\,2b_{1}^{2}\,4a_{1}^{2}\,5a_{1}^{2}\,3b_{1}^{2}\,1b_{2}^{2}\,4b_{1}^{2}\,6a_{1}^{2}\,1a_{2}^{2}$;

  * •

${}^{2}A_{1}:~{}1a_{1}^{2}\,1b_{1}^{2}\,2a_{1}^{1}\,3a_{1}^{2}\,2b_{1}^{2}\,4a_{1}^{2}\,5a_{1}^{2}\,3b_{1}^{2}\,1b_{2}^{2}\,4b_{1}^{2}\,6a_{1}^{2}\,1a_{2}^{2}$.

The final states of the doubly ionized target are characterized by two holes
distributed in all the possible ways among the 3-6$a_{1}$, 2-4$b_{1}$,
1$a_{2}$, and 1$b_{2}$ orbitals, which results in 81 different possible final
channels. We will limit the discussion to the electronic spectrum, using only
the electronic Hamiltonian and taking into account the effects of the nuclear
motion in a simplified manner by adding a broadening of the Auger lines only
at the end, in order to compare the theoretical and experimental spectra. A
further line broadening will be included to take into account the finite
resolution of the spectrometer used in electron spectroscopy measurements.
Furthermore, the two-step and static exchange approximations are used and only
the decay process from the intermediate quasi-bound state is analyzed. In the
case of the Auger decay, we further assume that the primary electron is fast
enough to avoid any appreciable interaction with the ionized ozone molecule.

| $1a_{1}$ | $1b_{1}$ | $2a_{1}$ | $3a_{1}$ | $2b_{1}$ | $4a_{1}$
---|---|---|---|---|---|---
$\epsilon_{HF}$ non rel. | -20.9179 | -20.7070 | -20.7070 | -1.7453 | -1.4281 | -1.0948
$\epsilon_{HF}$ rel. | -20.9124 | -20.7015 | -20.7015 | -1.7460 | -1.4286 | -1.0957
| $5a_{1}$ | $3b_{1}$ | $1b_{2}$ | $4b_{1}$ | $6a_{1}$ | $1a_{2}$
$\epsilon_{HF}$ non rel. | -0.8301 | -0.7983 | -0.7785 | -0.5640 | -0.5531 | -0.4870
$\epsilon_{HF}$ rel. | -0.8296 | -0.7979 | -0.7778 | -0.5633 | -0.552 | -0.4865
$E_{HF}$ non rel. | -224.356 |  |  |  |  |
$E_{HF}$ rel. | -224.408 |  |  |  |  |
Table 1: Relativistic (rel.) vs. non relativistic (non rel.) HF total
($E_{HF}$) and orbital energies ($\epsilon_{HF}$), calculated for the ozone
ground state. Values are reported in a.u.

The basis functions used in the expansion of bound and continuum orbitals is
taken from the aug–cc–pVQZ basis set with $(12s,6p,3d,2f,1g)$ HGF contracted
to $(5s,4p,3d,2f,1g)$ centered on each oxygen nucleus. To increase the
accuracy and rely on a larger number of diffuse functions to represent the
continuum orbital, in our simulations we completely decontracted the
aug–cc–pVQZ HGF basis set to $(13s,7p,4d)$ HGF without the $g$-symmetry
Gaussian to limit the computational cost. Indeed, this decontraction procedure
will add significantly to the computational cost of the calculation of the
Auger spectra. The HGF basis set has been decontracted in order to have
eigenvectors of an average HF operator at energies near to those of the Auger
electron in the various decay channels. Using this HGF basis set, we
calculated the total and orbital energies of the neutral ground state of ozone
by numerical solution of the HF equations with and without the inclusion of
relativistic effects [25], where in the latter simulation the electronic
repulsion takes into account also the Gaunt term [26]. These results are
reported in Table (1) in a.u. Analysing these values we conclude that
relativistic effects are negligible and a non-relativistic approximation can
be adopted to simulate the Auger lineshape of ozone.

The final states produced by the decay process are represented by the wave
functions of Equation (17), where $|\Theta_{\alpha}\rangle$ is the wave
function for the state $|\alpha\rangle$ representing the final single and
double ionized target, respectively, while $\varphi_{\alpha\kappa}$ is the
spin-orbital describing the Auger electron. We remind that the use of
$\chi_{\alpha\vec{k}}$ instead of the well behaving wave function
$\chi_{\alpha\vec{k}}^{-}$ that diagonalizes the Hamiltonian (3) is equivalent
to disregard the coupling among the decay channels. This approximation is in
general too drastic [13, 14] and will be abandoned in our calculations.
Intermediate and final bound states of the single and double charged molecule
are represented by the following linear combinations of Slater determinants:

$\displaystyle|\Phi(1,\dots,N-1)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(}N-1)!}\sum_{i}{a_{i}|\varphi_{1}^{i}(1),\dots,\varphi_{N-1}^{i}(N-1)|},$
(26) $\displaystyle|\Theta_{\alpha}(1,\dots,N-1)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{(}N-2)!}\sum_{j}{b_{\alpha
j}|\theta_{1,\alpha}^{j}(1),\dots,\theta_{N-2,\alpha}^{j}(N-2)|}.$ (27)

The coefficients $a_{i}$ and $b_{\alpha,j}$ are determined variationally by
solving the secular problem with respect to the standard electronic
Hamiltonian. The bound orbitals $\\{\varphi\\}$ and $\\{\theta\\}$ are
obtained by solving separate HF equations for the various states of interest
and therefore the resulting orbitals for a given state are not orthogonal to
those of a different state of charge.

The matrix elements (8) that couple intermediate and final states are
calculated between Slater determinants built up in terms of orbitals belonging
to mutually non-orthogonal sets as follows:

$\displaystyle
M_{\alpha,\vec{k}}(E)=<\Phi|\hat{H}-E|\chi_{\alpha,\vec{k}}>=\sum_{i}\sum_{j}a_{i}b_{\alpha,j}\left\\{\sum_{l=1}^{N-1}\left(\sum_{m=2}^{N-1}<\varphi^{i}_{l}|\hat{h}|\vartheta^{j}_{m,\alpha}>|S_{\alpha}^{ij}(l;m)|\right.\right.$
$\displaystyle+\left.<\varphi^{i}_{l}|\hat{h}|\eta_{\alpha,\vec{k}}>|S_{\alpha}^{ij}(l;1)|\right)+\sum_{l=2}^{N-1}\sum_{r=1}^{l-1}\left[\sum_{m=3}^{N-1}\sum_{s=2}^{m-1}\left(<\varphi^{i}_{l},\varphi^{i}_{r}|\hat{g}|\vartheta^{j}_{m,\alpha},\vartheta^{j}_{s,\alpha}>\right.\right.$
$\displaystyle\left.-<\varphi^{i}_{l},\varphi^{i}_{r}|\hat{g}|\vartheta^{j}_{s,\alpha},\vartheta^{j}_{m,\alpha}>\right)|S_{\alpha}^{ij}(l,r;m,s)|+\sum_{m=2}^{N-1}\left(<\varphi^{i}_{l},\varphi^{i}_{r}|\hat{g}|\vartheta^{j}_{m,\alpha},\eta_{\alpha,\vec{k}}>\right.$
$\displaystyle-<\varphi^{i}_{l},\varphi^{i}_{r}|\hat{g}|\eta_{\alpha,\vec{k}}\vartheta^{j}_{m,\alpha}>)|S_{\alpha}^{ij}(l,r;m,1)|]-E|S^{ij}_{\alpha}|\\},$
(28)

where $S_{\alpha}^{ij}$ is the following overlap matrix

$\displaystyle
S^{ij}_{\alpha}=\begin{pmatrix}<\varphi^{i}_{1}|\vartheta^{j}_{1,\alpha}>&\ldots&<\varphi^{i}_{1}|\vartheta^{j}_{N-2,\alpha}>&<\varphi^{i}_{1}|\eta_{\alpha,\vec{k}}>\cr\vdots&&\vdots&\vdots\cr<\varphi^{i}_{N-1}|\vartheta^{j}_{1,\alpha}>&\ldots&<\varphi^{i}_{N-1}|\vartheta^{j}_{N-2,\alpha}>&<\varphi^{i}_{N-1}|\eta_{\alpha,\vec{k}}>\end{pmatrix}$
(29)

and $|S^{ij}_{\alpha}(l;m)|$ is the determinant of the minor obtained by
taking away row $l$ and column $m$ and $|S^{ij}_{\alpha}(l,r;m,s)|$ that of
the minor obtained by taking away rows $(l,r)$ and columns $(m,s)$. In order
to calculate these matrix elements, which give the relative decay rates onto
the various channels, one has to evaluate integrals between $L^{2}$-functions,
used to represent both the bound and the continuum orbitals. Analytical
expressions of these integrals using HGF of any order and centre as basis
functions for the bound orbitals can be derived [14], which decrease the
computational cost.

We stress that, besides a truncated multi-configuration expansion, the only
approximation in our bound state calculations is represented by the decoupling
of the electronic and nuclear motion, without including the effects of the
latter from first-principles. Therefore, both the intermediate and the final
states are purely electronic states obtained at the equilibrium geometry of
the molecule.

Auger state | $E_{1}$ | $E_{2}$ | $E_{3}$
---|---|---|---
$1a_{1}^{-1}$ | -203.705 (562.0) | -204.168(549.4) | -204.268(546.6)
$1b_{1}^{-1}$ | -203.853(557.9) | -204.373(543.8) | -204.482(540.8)
$2a_{1}^{-1}$ | -203.854(557.9) | -204.373(543.8) | -204.482(540.8)
Table 2: Auger state energies in a.u. for different core-hole configurations.
$E_{1}$ is the single-configuration HF energy; $E_{2}$ is calculated by using
single-excitation CI (CIS) with an active space up to the 27th excited state
with respect to the relevant configuration. In $E_{3}$ active space was
increased to include up to 63 molecular orbitals. Ionization energies (in eV)
at the same level of theory are also reported in parenthesis for each Auger
state.

In Table 2 we report the HF energy $E_{1}$, calculated by assuming a single
determinant wavefunction with a core-hole in the Auger states $1a_{1}^{-1}$,
$1b_{1}^{-1}$ and $2a_{1}^{-1}$, respectively. Moreover, we also report in
Table 2 the energies calculated using CIS from the reference configuration for
an active space of 27 and 63 orbitals, labelled $E_{2}$ and $E_{3}$,
respectively. Both $E_{1}$ and $E_{2}$ were calculated preserving the
molecular symmetry, whereby the hole is not localized on a particular oxygen
atom rather “delocalized” on a molecular symmetry orbital. Furthermore, in
Table 2 we report also the ionization energies (IE) necessary to extract inner
electrons. We notice that IEs from $1b_{1}$ and $2a_{1}$ orbitals are almost
identical, being these molecular orbitals almost localized on the two C2v
symmetry-related oxygen atoms. While HF orbital optimization was achieved
starting from a hole initially delocalized on the $1a_{1},1b_{1},2a_{1}$ inner
shells, a partial relocalization of the holes is achieved by adding dynamic
correlation via CIS.

However, the most difficult part is the treatment of the continuum
wavefunction. Experience tells us that a major source of error is due to poor
representation of the continuum orbital, which is typically expanded using
plane-waves. Our approach overcome this problem by the use of the model
Hamiltonian with projected potential defined in Equation (19). The
construction of its scattering stationary states inside the space spanned by
the orthonormal wave functions $\\{\tilde{\chi}_{\alpha\vec{k}}\\}$, defined
in Equation (17), is obtained by the numerical solution of the LS equation
(22). The analytic expressions of the elements of the matrix representative of
the transition operator (23) using HGF can be obtained from the knowledge of
those relative to $V_{\alpha}^{\pi}$ and $\hat{G}_{0}$. To improve the
theoretical reproduction of the experimental Auger spectra, we finally
diagonalize the interchannel Hamiltonian (24), which includes the coupling
among the various decay channels, each one described by a wave function that
takes into account, at CIS level of theory, the intrachannel correlation
effects. The diagonalization of the interchannel coupling between interacting
channels redistributes the decay probability among independent channels.

In Figures (2), (4), and (5) we plot the $K-LL$ Auger spectra of ozone
calculated with the decoupled channel approach in which the intermediate
states are characterized by single inner-shell vacancies in the
$K=1a_{1},1b_{1},2a_{1}$ molecular orbitals and the final states by two
vacancies in all possible valence orbitals. In particular, the green line
represents 80 principal decay channels using HF level of theory, while black,
red and blue lines represent simulations with 20, 70, and 140 final states,
respectively, using CIS level of theory. In these calculations the active
space consists of 27 molecular orbitals. In Table (3) we report the total
$K-LL$ Auger decay rates from the $1a_{1},1b_{1},2a_{1}$ core-hole
intermediate states as a function of the number of final states included in
the calculations. The calculated transition energies, partial and total decay
rates of the $K-LL$ Auger processes are reported in Tables (4) ($K=1a_{1}$),
(5) ($K=1b_{1}$), (6) ($K=2a_{1}$), respectively, along with the final state
orbital occupations for the 20 brightest final states. We notice that the
$K-LL$ Auger transition spectra in Figures (4) and (5) are almost identical,
and also the total Auger probability does not differ significantly, being
2.9127 $\times 10^{-3}$ a.u. and 2.9485 $\times 10^{-3}$ a.u. (see Table (3),
third column), respectively. This result is obtained despite the symmetry of
the intermediate state is different, as the initial hole created upon
photoionization belongs to different irriducible representation of the
$C_{2v}$ symmetry group. This is interpreted as the effect of the inclusion of
the single excitations from the reference state in the description of the
orbital wavefunctions, which results in the quasi-relocalization of the core-
hole into one of the two symmetry-equivalent oxygen atoms despite the HF
orbitals were optimized starting from a hole “delocalized” in each of the
molecular orbitals. The gain in energy of this hole relocalization via multi-
configuration interaction is about 17 eV. At variance, the total $K-LL$ Auger
probability for an initial hole in the deepest $K=1a_{1}$ molecular orbital is
lower independently of the number of final channels used (see Table (3)). We
notice that the use of CIS to treat the orbital wavefunctions has large impact
on the accuracy of the Auger decay rate and peak position, showing a large
blueshift with respect to the HF values in excess of 10 (see Figure (2)) to 20
eV (see Figures (4) and (5)).

The Auger lines, peaked at the transition energies calculated as by Equation
(10), were broadened by Lorentzian functions, whose widths were obtained from
Equation (10). Finally, to take into account the finite resolution of the
experimental set-up and the vibrational broadening, we convoluted the
theoretical lineshape with a Gaussian having full width at half maximum (FWHM)
of $0.1$ eV.

|  | # of channels |
---|---|---|---
Auger state | 20 | 70 | 140
$1a_{1}^{-1}$ | $1.4321$ | $2.1208$ | $2.6241$
$1b_{1}^{-1}$ | $1.9824$ | $2.4548$ | $2.9127$
$2a_{1}^{-1}$ | $1.9957$ | $2.4644$ | $2.9485$
Table 3: Total $K-LL$ Auger decay rates of the three different intermediate
states $1a_{1},1b_{1},2a_{1}$ using CIS with an active space that includes up
to 27 molecular orbitals for 20, 70, and 140 final channels. Data are reported
in $\times 10^{3}$ a.u.

We observe that by increasing the number of non interacting final channels,
from 20 (black lines) to 70 (red lines) and 140 (blue lines), the partial
probability of Auger decay of course redistributes among different channels,
and also the total probability changes significatively, moving e.g. from
1.4321 $\times 10^{-3}$ a.u. to 2.6241 $\times 10^{-3}$ a.u. for the
$K=1a^{-1}$ Auger state (see Table (3)).

Finally, in Figure (3) we compare our ab-initio calculations with the
experimental data of Ref. [10] recorded at a photon energy of 536.7 eV,
finding an overall good agreement. We stress that the experimental data refer
to a spectator transition $O1s\rightarrow\sigma^{*}$, which is a process
similar (not exactly the same) to that one we considered here. Auger
lineshapes were broadened by convolution with a 0.8 eV Gaussian function.
These large observed linewidths have been attributed both to the dissociative
character of the final states and to nuclear vibrations.

Figure 2: $K-LL$ Auger spectrum of ozone, where $K=1a_{1}$. Green line: HF
level of theory with 80 independent final states, obtained by diagonalization
of the multichannel Hamiltonian (24). Black, red, and blue lines represent the
spectral lineshape obtained with an active space of 27 molecular orbitals and
a number of 20, 70, 140 independent final channels, respectively.  Figure 3:
Comparison between the $K-LL$ ($K=1a_{1}$) Auger experimental spectrum of
ozone (black line) [10] and our first-principles simulation (green line)
obtained with an active space of 27 molecular orbitals and 140 independent
final channels. Our lineshapes were convoluted via 0.8 eV Gaussian function to
achieve the experimental broadening.  Figure 4: $K-LL$ Auger spectrum of
ozone, where $K=1b_{1}$. Green line: HF level of theory with 80 independent
final states, obtained by diagonalization of the multichannel Hamiltonian
(24). Black, red, and blue lines represent the spectral lineshape obtained
with an active space of 27 molecular orbitals and a number of 20, 70, 140
independent final channels, respectively.  Figure 5: $K-LL$ Auger spectrum of
ozone, where $K=2a_{1}$. Green line: HF level of theory with 80 independent
final states, obtained by diagonalization of the multichannel Hamiltonian
(24). Black, red, and blue lines represent the spectral lineshape obtained
with an active space of 27 molecular orbitals and a number of 20, 70, 140
independent final channels, respectively.

###  III.3 The autoionization spectrum of ozone

In this section we deal with the resonant autoionization spectrum of ozone
following the photo-electronic excitation $O1s\rightarrow\sigma^{*}$. These
results have been obtained at the ground state geometry, disregarding the
nuclear motion.

The transition energy of this process is 538.39 eV for the two $O_{T}$ atoms
related by $C_{2v}$ symmetry operations and 543.73 eV if the excitation
involves the central oxygen atom $O_{C}$. Indeed, while molecular orbitals
were obtained by carrying out CI on HF optimized symmetry orbitals, which are
thus delocalized all over ozone, the core-hole almost relocalizes in $O_{T}$
and $O_{C}$ upon CI.

In Figure (6) we plot the behavior of both the resonant and direct
contributions to the spectrum as a function of the autoionized electron
kinetic energy. Plotted lineshapes have been obtained after convolution with
Lorentzian functions, whose width is provided by our ab-initio calculations
(see Equation (10)), and by a further convolution with 1 eV FWHM Gaussian
profile to take into account the nuclear broadening and the dissociative
nature of the excited states of ozone. In particular, red and blue lines
represent CIS calculations of the resonant autoionization spectra of ozone, in
which a core-hole is created upon excitation from the molecular symmetry
orbital that is almost localized in one of the two oxygen atoms ($O_{T}$)
related by symmetry operations. In green we plot the spectral lineshape for a
core-hole created in the symmetry orbital almost localized in the central
oxygen atom ($O_{C}$), whose atomic orbital of $1s$ character (corresponding
to the $1a^{-1}$ molecular symmetry orbital) is lower in energy than the other
two $1s$ oxygen orbitals ($1b^{-1},2a^{-1}$). In these simulations 140 final
states have been included.

Figure 6: $O1s\rightarrow\sigma^{*}$ autoionization spectrum of ozone. Red,
blue lines: CIS lineshapes with 140 final states obtained upon diagonalization
of the multichannel Hamiltonian (24). Core-hole belongs to molecular symmetry
orbitals, which are optimized using HF and extend all over the ozone molecule.
However, upon CI the hole relocalizes in one of the two oxygen atoms ($O_{T}$)
related by symmetry operations. Green line: spectral lineshape for a core-hole
relocalized after the CIS procedure in the central oxygen atom $O_{C}$.

###  III.4 Core-hole orbital symmetry and localization

In Auger experiments core-electrons are extracted from a localized inner
orbital of one particular oxygen. In our previous calculations core-hole
localization was obtained by including single excitations in HF molecular
symmetry orbitals. This means that at HF level molecular orbitals were
optimized by assuming that the core-hole was “delocalized” over the ozone
molecule. Core-hole localization was achieved basically by adding part of the
dynamical correlation neglected by HF via CIS procedure.

However, it is interesting (and actually more representative of what actually
happens in experiments) to show the $K-LL$ Auger spectrum obtained by
optimizing the molecular orbitals starting from an atomic core-hole completely
localized in one of the two $O_{T}$s, equivalent by $C_{2v}$ symmetry
operation (thus using atomic orbitals in the HF optimization to break the
orbital symmetry). Indeed the holes, created upon primary ionization in
localized atomic sites, favour intra-atomic more than inter-atomic
transitions. This difference in the HF kick-off orbitals reflects into the
final spectrum plotted in Figure (7) (black line), obtained at the same level
of theory as in previous calculations (red line). In particular, the spectral
intensity of the highest energy peaks is quite modified by core-hole
localization.

Figure 7: $K-LL$ Auger spectrum of ozone obtained from CIS calculations after
convolution with Lorentzian functions having the theoretical widths and with 1
eV FWHM Gaussian functions to reproduce the combined effect of the finite
resolution of the spectrometer and of the nuclear vibration. Black line: HF
orbitals optimized starting from an atomic core-hole localized in on of the
two $O_{T}$ oxygen centers. Red line: HF orbitals are optimized with the
initial core-hole “delocalized” in the molecular symmetry orbitals.
Calculations were carried out with an active space of 27 molecular orbitals
and 140 independent final channels, including the interchannel coupling (24).

##  IV Conclusions

In this work a first-principles method for the calculation and interpretation
of core-electron spectra of ozone has been described. This method is based on
Fano’s theory of decay, due to the interaction between discrete and continuum
states. We discussed this theoretical framework in the general case of several
continua interacting with several quasi-bound states, including explicitly the
boundary conditions appropriate to the study of Auger emission.

To achieve a relevant reduction in the computational efforts required by the
application of the theory, our approach uses a model Hamiltonian with a
projected potential represented in terms of $L^{2}$-functions. The spectral
properties of this projected Hamiltonian are such that the subspace of the
scattering eigenstates significant for the physical problem is finite and
isomorphous to the subspace of the functions used for representing the
potential energy operator. Furthermore, the many-body problem can be reduced
to an effective single-particle problem, in which the scattering states can be
obtained from the resolution of a projected Lippmann-Schwinger equation with
the proper boundary condition.

Wave functions of the bound and continuum orbitals have been expanded onto a
$L^{2}$ Gaussian basis set, which includes decontracted diffuse functions,
therefore reducing the Schrödinger equation to a matrix equation. The
calculation of mono and bielectronic integrals (and of the transition matrix
elements between discrete and continuum states with non-orthonormal orbitals)
is carried out numerically on the basis of analytical expressions.
Furthermore, to assess accurately the partial decay rates into different
channels we introduced the effects of interactions among final decay states
along with the many-body interaction within the remaining ionic system.

Finally, the analytic expression of the autoionization and Auger cross
sections has been applied to predict the ozone radiationless decay lineshape
as functions of the kinetic energy of the emitted electron. These Auger
lineshapes have been broadened with a Lorentzian profile, whose width is a
direct result of our ab-initio calculation. We included also “a posteriori”
the specific features of the incident radiation, of the electron spectrometer,
and of the effects of nuclear motion in the framework of the Born-Oppenheimer
approximation by convolution with a Gaussian profile, finding a good
comparison with available experimental data.

The effect of electronic correlation and the influence of core-hole
localization on the Auger peak energy positions and intensities have been also
analyzed, finding different spectral properties when the HF molecular orbitals
are optimized starting from a core-hole localized in a specific oxygen atom
that favours intra-atomic transitions. While localization breaks the symmetry
of the molecular orbitals, which is afterwards recovered by multi-
configuration expansion of the wavefunctions, both intensity and energy peak
differ than those obtained by delocalizing the inner-hole on the two symmetry-
equivalent oxygen centers.

Our method is applicable to molecular systems owing to the polycentric nature
of the basis functions and to the fact that all the integrals involved in the
calculation of the Auger matrix elements are analytical, cost-effective and
easily programmable. Moreover, it is possible to take into account the effects
of interaction among the final channels without resorting to numerical
integration on the energy as foreseen by Fano’s formulae. Finally, the
possibility of obtaining continuum wave functions makes the method useful also
for studying other scattering problems, such as photoionization, internal
conversion, electron-atom and electron-molecule scattering and so on.

## References

  * McElroy and Fogal [2008] McElroy, C.; Fogal, P.  Ozone: From discovery to protection.  Atmosphere-Ocean 2008, 46, 1–13.
  * Shindell et al. [2005] Shindell, D.T.; Faluvegi, G.; Bell, N.; Schmidt, G.A.  An emissions-based view of climate forcing by methane and tropospheric ozone.  Geophysical Research Letters 2005, 32.
  * Fuglestvedt et al. [1999] Fuglestvedt, J.S.; Berntsen, T.K.; Isaksen, I.S.; Mao, H.; Liang, X.Z.; Wang, W.C.  Climatic forcing of nitrogen oxides through changes in tropospheric ozone and methane; global 3D model studies.  Atmospheric Environment 1999, 33, 961–977.
  * Ju-Xiang et al. [2007] Ju-Xiang, S.; Zheng-He, Z.; Duo-Hui, H.; Jun, W.; Xin-Lu, C.; Xiang-Dong, Y.  Potential energy surfaces of ozone in the ground state.  Chinese Physics 2007, 16, 2650–2655.
  * Banichevich et al. [1993] Banichevich, A.; Peyerimhoff, S.D.; Grein, F.  Potential energy surfaces of ozone in its ground state and in the lowest-lying eight excited states.  Chemical Physics 1993, 178, 155–188.
  * Tyuterev et al. [2013] Tyuterev, V.G.; Kochanov, R.V.; Tashkun, S.A.; Holka, F.; Szalay, P.G.  New analytical model for the ozone electronic ground state potential surface and accurate ab initio vibrational predictions at high energy range.  The Journal of Chemical Physics 2013, 139, 134307.
  * Lopaev et al. [2008] Lopaev, D.V.; Malykhin, E.M.; Namiot, V.A.  UV absorption of vibrationally excited ozone.  Journal of Physics B: Atomic, Molecular and Optical Physics 2008, 41, 085104.
  * Allan et al. [1996] Allan, M.; Mason, N.J.; Davies, J.A.  Study of electronically excited states of ozone by electron‐energy‐loss spectroscopy.  The Journal of Chemical Physics 1996, 105, 5665–5671.
  * Chien et al. [2018] Chien, A.D.; Holmes, A.A.; Otten, M.; Umrigar, C.J.; Sharma, S.; Zimmerman, P.M.  Excited States of Methylene, Polyenes, and Ozone from Heat-Bath Configuration Interaction.  The Journal of Physical Chemistry A 2018, 122, 2714–2722.
  * Rosenqvist et al. [2001] Rosenqvist, L.; Wiesner, K.; Naves de Brito, A.; Bässler, M.; Feifel, R.; Hjelte, I.; Miron, C.; Wang, H.; Piancastelli, M.N.; Svensson, S.; Björneholm, O.; Sorensen, S.L.  Femtosecond dissociation of ozone studied by the Auger Doppler effect.  The Journal of Chemical Physics 2001, 115, 3614–3620.
  * Ndengué et al. [2016] Ndengué, S.; Dawes, R.; Wang, X.G.; Carrington, T.; Sun, Z.; Guo, H.  Calculated vibrational states of ozone up to dissociation.  The Journal of Chemical Physics 2016, 144, 074302.
  * McConnell and Jin [2008] McConnell, J.C.; Jin, J.J.  Stratospheric ozone chemistry.  Atmosphere-Ocean 2008, 46, 69–92.
  * Taioli et al. [2009] Taioli, S.; Simonucci, S.; Dapor, M.  SURPRISES: when ab initio meets statistics in extended systems.  Computational Science & Discovery 2009, 2, 015002.
  * Taioli et al. [2010] Taioli, S.; Simonucci, S.; Calliari, L.; Dapor, M.  Electron spectroscopies and inelastic processes in nanoclusters and solids: Theory and experiment.  Physics reports 2010, 493, 237–319.
  * Verucchi et al. [2012] Verucchi, R.; Aversa, L.; Nardi, M.V.; Taioli, S.; a Beccara, S.; Alfè, D.; Nasi, L.; Rossi, F.; Salviati, G.; Iannotta, S.  Epitaxy of nanocrystalline silicon carbide on Si (111) at room temperature.  Journal of the American Chemical Society 2012, 134, 17400–17403.
  * Tatti et al. [2016] Tatti, R.; Aversa, L.; Verucchi, R.; Cavaliere, E.; Garberoglio, G.; Pugno, N.M.; Speranza, G.; Taioli, S.  Synthesis of single layer graphene on Cu (111) by C 60 supersonic molecular beam epitaxy.  RSC advances 2016, 6, 37982–37993.
  * Segatta et al. [2017] Segatta, F.; Cupellini, L.; Jurinovich, S.; Mukamel, S.; Dapor, M.; Taioli, S.; Garavelli, M.; Mennucci, B.  A quantum chemical interpretation of two-dimensional electronic spectroscopy of Light-Harvesting complexes.  Journal of the American Chemical Society 2017, 139, 7558.
  * Fano [1961] Fano, U.  Effects of Configuration Interaction on Intensities and Phase Shifts.  Phys. Rev. 1961, 124, 1866–1878.
  * Åberg and Howat [1982] Åberg, T.; Howat, G.  Theory of the Auger Effect.  Handbuch der Physik 1982, 6, 469–619.
  * Newton [1982] Newton, R.G.  Scattering Theory of Waves and Particles; 1982.
  * Taylor [2000] Taylor, J.R.  Scattering Theory; 2000.
  * Giannozzi et al. [2009] Giannozzi, P.; Baroni, S.; Bonini, N.; Calandra, M.; Car, R.; Cavazzoni, C.; Ceresoli, D.; Chiarotti, G.L.; Cococcioni, M.; Dabo, I.; Corso, A.D.; de Gironcoli, S.; Fabris, S.; Fratesi, G.; Gebauer, R.; Gerstmann, U.; Gougoussis, C.; Kokalj, A.; Lazzeri, M.; Martin-Samos, L.; Marzari, N.; Mauri, F.; Mazzarello, R.; Paolini, S.; Pasquarello, A.; Paulatto, L.; Sbraccia, C.; Scandolo, S.; Sclauzero, G.; Seitsonen, A.P.; Smogunov, A.; Umari, P.; Wentzcovitch, R.M.  QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials.  Journal of Physics: Condensed Matter 2009, 21, 395502.
  * Perdew et al. [1996] Perdew, J.P.; Burke, K.; Ernzerhof, M.  Generalized Gradient Approximation Made Simple.  Physical Review Letters 1996, 77, 3865–3868.
  * Tanaka and Morino [1970] Tanaka, T.; Morino, Y.  Coriolis interaction and anharmonic potential function of ozone from the microwave spectra in the excited vibrational states.  Journal of Molecular Spectroscopy 1970, 33, 538–551.
  * Morresi et al. [2018] Morresi, T.; Taioli, S.; Simonucci, S.  Nuclear Beta Decay: Relativistic Theory and Ab Initio Simulations of Electroweak Decay Spectra in Medium-Heavy Nuclei and of Atomic and Molecular Electronic Structure (Adv. Theory Simul. 11/2018).  Advanced Theory and Simulations 2018, 1, 1870030.
  * Reiher and Wolf [2009] Reiher, M.; Wolf, A.  Relativistic Quantum Chemistry: The Fundamental Theory of Molecular Science; Wiley-VCH Verlag GmbH & Co. KGaA, 2009.

*

##  Appendix A The Auger spectrum of ozone

Table 4: Auger spectrum of ozone with a primary hole localized in the $1a_{1}^{-1}$ molecular orbital. First column: double occupancy of the final transition states. Second: Auger transition energy ($E_{\alpha}$, eV). Third column: partial Auger probabilities ($\Gamma_{\alpha}$, units: a.u. $\times 10^{-3}$) according to Equation (10). The theory used is CIS with an active space of 27 orbitals.  Trans. state occ. | Trans. En. | Trans. Prob.
---|---|---
2.00 — 2.00 — 1.97 — 1.59 — 1.42 — 1.64 — 0.09 — 1.71 | 495.17 | 0.1100
0.39 — 2.00 — 1.94 — 1.65 — 1.60 — 0.13 — 1.86 |  |
2.00 — 2.00 — 1.58 — 1.63 — 1.77 — 1.86 — 0.37 — 1.69 | 467.14 | 0.1818
0.15 — 2.00 — 1.58 — 1.67 — 1.84 — 0.25 — 1.61 |  |
2.00 — 2.00 — 2.00 — 1.95 — 1.63 — 1.53 — 0.19 — 1.38 | 502.3 | 0.0879
0.34 — 2.00 — 1.99 — 1.89 — 1.59 — 0.17 — 1.34 |  |
2.00 — 2.00 — 2.00 — 1.96 — 1.70 — 1.62 — 0.14 — 1.39 | 506.25 | 0.0767
0.54 — 2.00 — 1.99 — 1.90 — 1.68 — 0.05 — 1.01 |  |
2.00 — 2.00 — 1.99 — 1.94 — 1.80 — 1.53 — 0.34 — 1.21 | 500.02 | 0.0804
0.21 — 2.00 — 1.99 — 1.89 — 1.46 — 0.24 — 1.40 |  |
2.00 — 2.00 — 2.00 — 1.96 — 1.94 — 1.78 — 0.06 — 1.50 | 506.065 | 0.0697
0.55 — 2.00 — 1.98 — 1.31 — 1.91 — 0.03 — 0.98 |  |
2.00 — 2.00 — 1.99 — 1.93 — 1.74 — 1.54 — 0.14 — 1.44 | 506.03 | 0.0761
0.53 — 2.00 — 1.99 — 1.88 — 1.65 — 0.06 — 1.12 |  |
2.00 — 2.00 — 2.00 — 1.88 — 1.67 — 1.49 — 0.21 — 1.90 | 503.195 | 0.0596
0.18 — 2.00 — 1.98 — 1.33 — 1.70 — 0.12 — 1.55 |  |
2.00 — 2.00 — 1.69 — 1.67 — 1.45 — 1.83 — 0.19 — 1.79 | 476.33 | 0.0352
0.44 — 2.00 — 1.61 — 1.77 — 1.73 — 0.15 — 1.67 |  |
2.00 — 2.00 — 1.99 — 1.90 — 1.47 — 1.35 — 0.17 — 1.81 | 502.25 | 0.0693
0.26 — 2.00 — 1.97 — 1.73 — 1.50 — 0.27 — 1.58 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.36 — 1.22 — 0.03 — 1.95 | 508.36 | 0.0705
0.30 — 2.00 — 1.99 — 1.82 — 1.59 — 0.03 — 1.74 |  |
2.00 — 2.00 — 2.00 — 1.96 — 1.46 — 1.52 — 0.10 — 1.97 | 504.16 | 0.0518
0.54 — 2.00 — 1.99 — 1.67 — 1.21 — 0.11 — 1.47 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.94 — 1.13 — 0.02 — 1.54 | 512.03 | 0.0553
0.59 — 2.00 — 2.00 — 1.98 — 1.93 — 0.01 — 0.89 |  |
2.00 — 2.00 — 1.34 — 1.59 — 1.78 — 1.86 — 0.30 — 1.87 | 453.39 | 0.1166
0.29 — 2.00 — 1.41 — 1.76 — 1.72 — 0.27 — 1.81 |  |
2.00 — 2.00 — 1.99 — 1.73 — 1.67 — 1.59 — 0.09 — 1.77 | 503.62 | 0.0466
0.44 — 2.00 — 1.97 — 1.68 — 1.25 — 0.05 — 1.77 |  |
2.00 — 2.00 — 1.99 — 1.87 — 1.80 — 1.67 — 0.28 — 1.42 | 496.73 | 0.0512
0.28 — 2.00 — 1.99 — 1.63 — 1.50 — 0.26 — 1.31 |  |
2.00 — 2.00 — 2.00 — 1.95 — 1.95 — 1.49 — 0.49 — 1.83 | 502.45 | 0.0416
0.16 — 2.00 — 1.99 — 1.35 — 1.84 — 0.06 — 0.90 |  |
2.00 — 2.00 — 1.99 — 1.82 — 1.81 — 1.67 — 0.21 — 1.61 | 500.80 | 0.0432
0.23 — 2.00 — 1.99 — 1.60 — 1.59 — 0.27 — 1.21 |  |
Total Transition Probability ($\Gamma$) |  $1.4321\times 10^{-3}$ A.U. |
Table 5: Auger spectrum of ozone with a primary hole localized in the $1b_{1}^{-1}$ molecular orbital. First column: double occupancy of the final transition states. Second: Auger transition energy ($E_{\alpha}$, eV). Third column: partial Auger probabilities ($\Gamma_{\alpha}$, units: a.u. $\times 10^{-3}$) according to Equation (10). The theory used is CIS with an active space of 27 orbitals.  Trans. state occ. | Trans. En. | Trans. Prob.
---|---|---
2.00 — 2.00 — 2.00 — 1.99 — 1.98 — 1.92 — 0.04 — 1.93 | 510.751 | 0.2255
0.03 — 2.00 — 1.99 — 1.98 — 1.87 — 0.03 — 0.24 |  |
2.00 — 2.00 — 2.00 — 2.00 — 1.97 — 1.90 — 0.03 — 1.86 | 510.17 | 0.1713
0.08 — 2.00 — 1.99 — 1.98 — 1.10 — 0.02 — 1.07 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.46 — 1.58 — 0.05 — 1.89 | 502.71 | 0.1755
0.38 — 2.00 — 1.98 — 1.80 — 1.19 — 0.03 — 1.67 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.36 — 1.22 — 0.03 — 1.95 | 502.79 | 0.1453
0.30 — 2.00 — 1.99 — 1.82 — 1.59 — 0.03 — 1.74 |  |
2.00 — 2.00 — 2.00 — 1.97 — 1.49 — 1.94 — 0.03 — 1.78 | 504.44 | 0.1286
0.33 — 2.00 — 1.99 — 1.98 — 1.56 — 0.02 — 0.90 |  |
2.00 — 2.00 — 2.00 — 1.99 — 1.97 — 1.09 — 0.02 — 1.90 | 510.93 | 0.1439
0.07 — 2.00 — 2.00 — 1.98 — 1.94 — 0.02 — 1.03 |  |
2.00 — 2.00 — 2.00 — 1.97 — 1.84 — 1.29 — 0.02 — 1.95 | 508.79 | 0.1289
0.18 — 2.00 — 1.99 — 1.96 — 1.09 — 0.02 — 1.69 |  |
2.00 — 2.00 — 2.00 — 1.97 — 1.84 — 1.03 — 0.02 — 1.98 | 509.47 | 0.1359
0.19 — 2.00 — 2.00 — 1.97 — 1.16 — 0.01 — 1.82 |  |
2.00 — 2.00 — 2.00 — 1.99 — 1.98 — 1.81 — 0.04 — 1.10 | 504.35 | 0.1162
0.17 — 2.00 — 1.99 — 1.88 — 1.93 — 0.04 — 1.09 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.91 — 1.67 — 0.07 — 1.62 | 503.07 | 0.0798
0.33 — 2.00 — 1.99 — 1.52 — 1.82 — 0.03 — 1.07 |  |
2.00 — 2.00 — 2.00 — 1.92 — 1.87 — 1.69 — 0.02 — 1.92 | 503.99 | 0.0706
0.25 — 2.00 — 1.98 — 1.32 — 1.20 — 0.01 — 1.83 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.94 — 1.67 — 0.02 — 1.68 | 504.29 | 0.0608
0.29 — 2.00 — 1.98 — 1.52 — 1.86 — 0.01 — 1.06 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.94 — 1.13 — 0.02 — 1.54 | 506.46 | 0.0554
0.59 — 2.00 — 2.00 — 1.98 — 1.93 — 0.01 — 0.89 |  |
2.00 — 2.00 — 2.00 — 1.94 — 1.79 — 1.10 — 0.01 — 1.81 | 504.77 | 0.0777
0.47 — 2.00 — 1.99 — 1.59 — 1.65 — 0.01 — 1.64 |  |
2.00 — 2.00 — 1.91 — 1.62 — 1.55 — 1.65 — 0.29 — 1.83 | 472.09 | 0.0856
0.32 — 2.00 — 1.65 — 1.50 — 1.65 — 0.22 — 1.80 |  |
2.00 — 2.00 — 1.99 — 1.84 — 1.77 — 1.16 — 0.42 — 1.95 | 497.19 | 0.0339
0.28 — 2.00 — 1.96 — 1.59 — 1.23 — 0.04 — 1.76 |  |
2.00 — 2.00 — 1.73 — 1.70 — 1.70 — 1.57 — 0.34 — 1.91 | 467.51 | 0.0790
0.15 — 2.00 — 1.62 — 1.44 — 1.62 — 0.30 — 1.90 |  |
2.00 — 2.00 — 1.99 — 1.60 1.88 — 1.54 — 0.25 — 1.67 | 499.45 | 0.0032
0.18 — 2.00 — 1.99 — 1.83 — 1.87 — 0.05 — 1.15 |  |
2.00 — 2.00 — 1.68 — 1.70 — 1.68 — 1.68 — 0.19 — 1.68 | 468.17 | 0.0622
0.51 — 2.00 — 1.70 — 1.70 — 1.53 — 0.18 — 1.78 |  |
2.00 — 2.00 — 1.99 — 1.88 — 1.86 — 1.81 — 0.22 — 1.54 | 491.23 | 0.0032
0.37 — 2.00 — 1.73 — 1.62 — 1.74 — 0.14 — 1.09 |  |
Total Transition Probability ($\Gamma$) |  $1.9824\times 10^{-3}$ A.U. |
Table 6: Auger spectrum of ozone with a primary hole localized in the $2a_{1}^{-1}$ molecular orbital. First column: double occupancy of the final transition states. Second: Auger transition energy ($E_{\alpha}$, eV). Third column: partial Auger probabilities ($\Gamma_{\alpha}$, units: a.u. $\times 10^{-3}$) according to Equation (10). Last row: total decay rate. The theory used is CIS with an active space of 27 orbitals.  Trans. state occ. | Trans. En. | Trans. Prob.
---|---|---
2.00 — 2.00 — 2.00 — 1.99 — 1.98 — 1.92 — 0.04 — 1.93 | 510.07 | 0.2515
0.03 — 2.00 — 1.99 — 1.98 — 1.87 — 0.03 — 0.24 |  |
2.00 — 2.00 — 2.00 — 2.00 — 1.97 — 1.90 — 0.03 — 1.86 | 510.17 | 0.1693
— 0.08 — 2.00 — 1.99 — 1.98 — 1.10 — 0.02 — 1.07 |  |
2.00 — 2.00 2.00 — 1.98 — 1.46 — 1.58 — 0.05 — 1.89 | 502.71 | 0.1708
0.38 — 2.00 — 1.98 — 1.80 — 1.19 — 0.03 — 1.67 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.36 — 1.22 — 0.03 — 1.95 | 502.79 | 0.1478
0.30 — 2.00 — 1.99 — 1.82 — 1.59 — 0.03 — 1.74 |  |
2.00 — 2.00 — 2.00 — 1.97 — 1.49 — 1.94 — 0.03 — 1.78 | 504.44 | 0.1290
0.33 — 2.00 — 1.99 — 1.98 — 1.56 — 0.02 — 0.90 |  |
2.00 — 2.00 — 2.00 — 1.97 — 1.84 — 1.29 — 0.02 — 1.95 | 508.79 | 0.1401
0.18 — 2.00 — 1.99 — 1.96 — 1.09 — 0.02 — 1.69 |  |
2.00 — 2.00 2.00 — 1.99 — 1.97 — 1.09 — 0.02 — 1.90 | 510.92 | 0.1443
0.07 — 2.00 — 2.00 — 1.98 — 1.94 — 0.02 — 1.03 |  |
2.00 — 2.00 2.00 — 1.97 — 1.84 — 1.03 — 0.02 — 1.98 | 509.46 | 0.1225
0.19 — 2.00 — 2.00 — 1.97 — 1.16 — 0.01 — 1.82 |  |
2.00 — 2.00 — 2.00 — 1.99 — 1.98 — 1.81 — 0.04 — 1.10 | 504.35 | 0.1091
0.17 — 2.00 — 1.99 — 1.88 — 1.93 — 0.04 — 1.09 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.91 — 1.67 — 0.07 — 1.62 | 503.06 | 0.0797
0.33 — 2.00 — 1.99 — 1.52 — 1.82 — 0.03 — 1.07 |  |
2.00 — 2.00 — 2.00 — 1.92 — 1.87 — 1.69 — 0.02 — 1.92 | 503.99 | 0.0808
0.25 2.00 — 1.98 — 1.32 — 1.20 — 0.01 — 1.83 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.94 — 1.13 — 0.02 — 1.54 | 506.45 | 0.0552
0.59 — 2.00 — 2.00 — 1.98 — 1.93 — 0.01 — 0.89 |  |
2.00 — 2.00 — 2.00 — 1.98 — 1.94 — 1.67 — 0.02 — 1.68 | 504.28 | 0.0610
0.29 — 2.00 — 1.98 — 1.52 — 1.86 — 0.01 — 1.06 |  |
2.00 — 2.00 — 2.00 — 1.94 — 1.79 — 1.10 — 0.01 — 1.81 | 504.77 | 0.0663
0.47 — 2.00 — 1.99 — 1.59 — 1.65 — 0.01 — 1.64 |  |
2.00 — 2.00 — 1.91 — 1.62 — 1.55 — 1.65 — 0.29 — 1.83 | 472.09 | 0.0878
0.32 — 2.00 — 1.65 — 1.50 — 1.65 — 0.22 — 1.80 |  |
2.00 — 2.00 — 1.99 — 1.84 — 1.77 — 1.16 — 0.42 — 1.95 | 497.19 | 0.030
0.28 — 2.00 — 1.96 — 1.59 — 1.23 — 0.04 — 1.76 |  |
2.00 — 2.00 — 1.99 — 1.60 — 1.88 — 1.54 — 0.25 — 1.67 | 499.45 | 0.0033
0.18 — 2.00 — 1.99 — 1.83 — 1.87 — 0.05 — 1.15 |  |
2.00 — 2.00 — 1.73 — 1.70 — 1.70 — 1.57 — 0.34 — 1.91 | 467.51 | 0.0799
0.15 — 2.00 — 1.62 — 1.44 — 1.62 — 0.30 — 1.90 |  |
2.00 — 2.00 — 1.68 — 1.70 — 1.68 — 1.68 — 0.19 — 1.68 | 468.17 | 0.0637
0.51 — 2.00 — 1.70 — 1.70 — 1.53 — 0.18 — 1.78 |  |
2.00 — 2.00 — 1.99 — 1.88 — 1.86 — 1.81 — 0.22 — 1.54 | 491.23 | 0.0032
0.37 — 2.00 — 1.73 — 1.62 — 1.74 — 0.14 — 1.09 |  |
Total Transition Probability ($\Gamma$) |  $1.9957\times 10^{-3}$ A.U. |


core}}_{\text{distillation}}+\eta_{\text{4K}}P^{\text{tot}}_{\text{4K}}t^{\text{hardware}}_{tot}+\eta_{\text{MXC}}P^{\text{tot}}_{\text{MXC}}t^{\text{hardware}}_{tot}$.

###  A.2 Widgetization via Subcircuit Dependency Graph

Figure 16: Schematic of a decomposed circuit using Rigetti’s circuit
decomposition tool. Veritces (nodes) represent moments. Square nodes indicate
anchor vertices and colors indicate widgets. For example moments at all blue
veritices comprise a single widget. The letter at each vertex indicates the
order in which the moment was analyzed.

As described in I.1.1, our resource estimation tooling supports widgetization
of `Cirq` circuits. The primary reason for such a widgetization tool is that
fully unrolling a large circuit, for a utility scale application can place
severe pressure on memory and compute resources. Moreover, for the purposes of
resource estimation using RRE, it is unnecessary and inefficient to revisit,
re-examine, and recompile subcircuits that have already been prepared for
consumption by RRE.

In order to widgetize without unrolling, we perform the following steps:

  1. 1.

Construct a subcircuit dependency graph

  2. 2.

Identify anchor vertices

  3. 3.

Fully decompose subcircuits belonging to anchor vertices

  4. 4.

Concatenate in time to form widgets

Each of these steps will be described below.

###  A.3 Constructing the Subcircuit Dependency Graph

The first step in widgetization without full unrolling is to construct a
directed graph representing subcircuit dependency. In this graph, vertices
represent time slices of the circuit. A directed edge connects a parent vertex
to a child vertex if the child vertex represents a time slice that was created
by decomposing the time slice represented by the parent. The creation of this
graph is accomplished by recursively decomposing time slices, on level at a
time, and stopping when a time slice cannot be decomposed (called a leaf
vertex). When a leaf vertex is reached, one returns to the parent vertex and
decomposes the next time slice in the temporal sequence. If no time slices
remain to decompose, move up to the subsequent parent vertex, and so forth.

In addition to constructing the topology of the subcircuit dependencies, we
keep track of the depth of each time slice. By definition, leaf vertices have
depth 1. These depths are aggregated up the graph, where a parent vertex has a
depth equal to the sum of the depths of its children. This has the effect of
allowing one to know how many time slices a fully decomposed circuit would be
if fully decomposing a circuit from any vertex.

Crucially, during graph construction, a running cache of time slices is
maintained, and if a time slice is encountered which has already been
decomposed, it is not decomposed again.

###  A.4 Identify anchor vertices

Once the subcircuit dependency graph is complete, "anchor vertices" based on a
user defined "maximum depth", can be identified. Anchor vertices are defined
to be vertices for which both of the following are true:

  * •

The concatenation in time of fully decomposed time slices belonging to anchor
vertices will reconstruct the entire, fully decomposed circuit, and

  * •

The depth of an anchor vertex is less than or equal to the maximum depth, but
the depth of its parent vertex is greater than the maximum depth.

###  A.5 Decompose subcircuits belonging to anchor vertices

For each anchor vertex we fully decompose the subcircuit which belongs to it.

###  A.6 Concatenate in time

Beginning with the first time slice of the first anchor vertex, concatenate
the next time slice in temporal order until the number of time slices reaches
the maximum depth. This completes a "widget." Begin a new widget starting with
the next time slice, concatenate the next time slice, and so on. This
procedure concludes when the last time slice of the last anchor vertex is
reached.

The reason for carrying out this concatenation is to ensure widgets are "large
enough" which makes processing in RRE more efficient while still being small
enough to be performant.

###  A.7 Example

Assume the user has set a maximum depth of 3, and consider the circuit in
Figure 16. We can consider the starting circuit to be a single, monolith time
slice, with gates acting on $n$ qubits. This circuit corresponds to the Full
vertex in the figure and is considered as Layer 0 of the circuit.

We can then decompose Full into $N_{1}=4$ time slices, represented by vertices
A, G, H, and K. These four time slices comprise Layer 1. Then, beginning with
the first time slice (vertex A), we treat this time slice as the new circuit,
and repeat the process, by decomposing it into time slices represented by
vertices B, C and D. We then attempt to decompose the time slice represented
by vertex B but note that B cannot be decomposed further. Thus, vertex B is
thus a leaf vertex with depth 1. We then attempt to decompose the time slice
at vertex C, but again C is a leaf vertex. Finally we move on to the time
slice at vertex D, which can be decomposed into single depth time slices
represented by leaf vertices E and F. With all of the children of A fully
decomposed, we then move to the time slice at vertex G and so on. Note that
the depth values for each vertex (in the parenthesis) are the sums of the
depths of their children and exactly match the number of leaf vertices below
it.

With the subcircit dependency graph created, we can now identify vertices B,
C, D, G, H, and K, and anchor vertices. After identifying the anchor vertices,
we can then fully decompose these vertices and start creating widgets.

To begin teh process of widget creation, we start with a widget containing
zero time slices, and add to it the first time slice of the first anchor
vertex, which is the time slice at vertex B. We then add the next time slice
(at vertex C), and then the time slice at vertex E. We now have a complete
widget of depth 3. We begin again with an empty widget and add the time slices
at vertices F, then G, and so on. Eventually we have:

  * •

Widget 1: B, C, E

  * •

Widget 2: F,G,J

  * •

Widget 3: L, M

Notice that executing the widgets in order, reconstructs the execution of the
original circuit, and each widget has maximum depth less than or equal to
three. The circuit is now widgetized and are ready for use in RRE.