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SURPRISES: when ab initio meets statistics in extended systems

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2009 Comput. Sci. Disc. 2 015002

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SURPRISES: when ab initio meets statistics inextended systems

Simone Taioli1,2,5, Stefano Simonucci1,3,5 and Maurizio Dapor1,4,6

1 Center for Material and Microsystems, FBK-IRST, Via Sommarive, 18 I-38100,Povo (Trento), Italy2 The European Centre for Theoretical Studies in Nuclear Physics and Related Areas(ECT*), Strada delle Tabarelle 286, I-38100, Villazzano (Trento), Italy3 Department of Physics, University of Camerino, Via Madonna delle Carceri 9, 34012,Camerino, Italy4 Systems Laboratory, Swiss Federal Institute of Technology (ETH), Zurich, SwitzerlandE-mail: taioli@fbk.eu, stefano.simonucci@unicam.it and dapor@fbk.eu

Received 10 April 2009, in final form 27 August 2009Published 9 November 2009Computational Science & Discovery 2 (2009) 015002 (23pp)doi:10.1088/1749-4699/2/1/015002

Abstract. The surface photoelectron and inner shell electron spectroscopy (SURPRISES)program suite performs ab initio calculations of photoionization and non-radiative decayspectra in nanoclusters and solid state systems by using a space-energy similarity procedureto reproduce the band-like part of the spectra. This approach provides an extension of Fanoresonant multichannel scattering theory dealing with the complexity arising from condensedmatter calculations at a computational cost comparable to that of molecules. The bottleneckof electron spectroscopy ab initio calculations in condensed matter is the size of the Hilbertspace where the wavefunctions are expanded and the increase in number of final decaystates in comparison to that of atoms and molecules. In particular, the diagonalization of theinterchannel interaction to take into account the correlation between the double ion and theescaping electron is impracticable when hole delocalization on valence bands and electronicexcitations are included in the model. To overcome this problem SURPRISES uses a ‘space-energy similarity’ approach, which allows the spreading of the Auger probability over thebands without tuning semi-empirical parameters. Furthermore, a completely new feature inthe landscape of ab initio resonant decay processes calculations is represented by includingenergy loss through a statistical approach. Using the calculated lineshape as electron source,a Monte Carlo routine simulates the effect of inelastic losses on the original lineshape. In thisprocess, the computed spectrum can be directly compared with acquired experimental spectra,thus avoiding background subtraction, a procedure not free from uncertainty. The programcan exploit the symmetry of the system under investigation to reduce the calculation scalingand may compute photoemission and Auger decay angular distribution patterns including

5 Corresponding authors for first principles calculations.6 Corresponding author for the energy loss calculation.

Computational Science & Discovery 2 (2009) 015002 www.iop.org/journals/csd© 2009 IOP Publishing Ltd 1749-4699/09/015002+23$30.00

Computational Science & Discovery 2 (2009) 015002 S Taioli et al

energy loss for the electrons emitted in resonance-affected photoionization processes. In thispaper, we present general methods, computational techniques and a number of numericaltests applied to the calculation of Si K–LL and O K–LL Auger spectra from different SiO2

nanoclusters.

Contents

1. Introduction 2

2. Outline of the model 3

3. Program summary, nature and solution of the problem 43.1. Logic formulation of the method and program structure . . . . . . . . . . . . . . . . . . . . . 5

4. Multichannel scattering: SURPRISES 64.1. Scaling performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64.2. The Hamiltonian and its solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.3. Projected potentials and solution of the model Hamiltonian . . . . . . . . . . . . . . . . . . . 8

5. Physical observables 105.1. Auger matrix elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2. Evaluation of spectroscopic observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6. Energy loss 12

7. Algorithms, computational techniques and numerical issues 137.1. Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2. Electronic structure calculations: integrals, HF and CI . . . . . . . . . . . . . . . . . . . . . . 147.3. The interchannel potential and the energy-wavefunction similarity procedure . . . . . . . . . . 147.4. Numerical issues using projected potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.5. Numerical issues using the Monte Carlo scheme . . . . . . . . . . . . . . . . . . . . . . . . . 17

8. The working program 198.1. Test run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Acknowledgments 21

References 21

1. Introduction

The interaction of photon or electron beams with matter plays a crucial role in many areas of theoreticalphysics and astrophysics and its study is essential for our understanding of the Universe and for the discoveryof new chemical, physical and biological processes [1]. Radiation damage of living cells [2], identification ofthe chemical composition and investigation of physical properties of a material at any level of aggregation [3]are just a few examples where the light–matter or electron–matter interaction mechanisms need to be carefullyaddressed. Such scattering processes give crucial information on the dynamics of the excitation processes innanostructures and solids [4–6], providing a unique benchmark for theoretical achievements in the study ofoptical properties, band gaps and quasi-particle spectra where electronic correlation creates states other thansimple Fermi gas [7].

Furthermore, energy redistribution following the excitation can force the system to decay through avariety of radiative, non-radiative and dissociative paths. The study of such decay mechanisms proved tobe particularly important in optical devices, where carrier recombination quenching is the basis for reducedperformances of carbon nanotubes as photoluminescent means [8–10].

But while the knowledge of such mechanisms has progressed greatly for atoms and molecules, largely dueto theoretical advances in treating accurately the correlation in small systems, computational tools able to dealwith the increasing complexity and delocalization issues in core–hole photoemission (PES) and Auger decay

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

from nanoclusters and condensed matter systems are still the subject of intense research [11]. In this work, wefocus on the computational and methodological techniques for calculating PES and non-radiative decay spectrain nanoclusters and solids with a full ab initio treatment of band effects and hole delocalization. Furthermore,we will discuss a unified theoretical and computational framework able to mix quantum mechanical first-principles calculations with a Monte Carlo treatment of the electron energy loss for simulating the interactionof electrons, on their way out of the solid, with plasmons [12] and the surrounding electronic cloud.

2. Outline of the model

In our model the interaction of light and electrons with atoms, bound in clusters or periodical arrays, canproceed either via a resonant state, in which the impinging particle loses its kinetic energy, or via a directionization path. After the collision, different decay paths, each one with different probability dependent onthe initial electron kinetic or photon energy, result in a variety of spectroscopies analyzing separate regions ofthe energy spectrum, such as PES, Auger, secondary emission, energy loss or excitation of collective chargemotions.

If the impinging particle is energetic enough to excite (or emit) an inner-shell electron to an empty(continuum) state, an autoionizing (Auger) metastable state can be created. Often this primary excitation isfollowed by shake-up and shake-off processes. These excited, quasi-bound states undergo a de-excitationprocess by a variety of different paths, such as photoluminescence or secondary electron emission. ResonantPES and Auger transition occur when intermediate metastable states are embedded in the continuum of thenext higher charge state of the system [13, 14]. In this process, the system gets rid of the excess energy byemitting an electron from the outer bands.

The description of the working program suite will be focused on K–VV Auger transitions, appearingwhen the projectile beam causes the ejection of a core level (K) electron. One can observe a non-radiativedecay in which the energetic redistribution, due to the Coulomb interaction, drops out one electron off thevalence band V1 into the unoccupied inner shell, releasing a second electron from an occupied valence bandV2 into the continuum. The energy conservation constrains the value of the kinetic energy of the escapingelectron to

EAuger = (EK − EV1)− EV2 . (1)

Other selection rules can be derived from the conservation of momentum, which, particularly in highlysymmetric solid, can forbid some transitions. In the Auger process above, the final state is a doubly ionizedatom with holes in the V1 and V2 bands. This process is allowed if (EK − EV1)− EV2 > 0. Since the kineticenergy distribution of the Auger electrons depends on the features of the electronic structure, on the natureof bonds and of the surrounding chemical environment, the Auger spectrum is a fingerprint of the systemunder investigation. Such a spectrum is independent on the mechanism of the initial core–hole formation butinfluenced only by the location of the initial and final holes. In core–valence–valence (K–VV) spectra of solids,such as those investigated in this work, the holes are, in principle, delocalized on all the valence bands and thehole–hole repulsion is a function of the dispersion curves. Then, one expects that the final states are created bythe interplay between intra-atomic and inter-atomic transitions.

SURPRISES deals with such ionization processes and the following decay states in condensed mattersystems where such interplay has an essential role if one is to establish the true nature of hole–hole,hole–electron and electron–electron interaction mechanisms in solids. Many models, described in severalreviews [3], [15–17], have been suggested to tackle the calculation of Auger spectra in solids, a difficulttask due to the high number of energy levels involved and the contribution of additional degrees of freedom,which broaden the lineshapes, such as shake processes and electron–phonon interaction. Among these,successful methods have been developed by Tarantelli and Cederbaum [18], Averbukh and Cederbaum [19],Ohno and Wendin [20] and Cini–Sawatzky–Verdozzi [21–24]. All of these approaches, rooted in Lander’sidea [25] that C–VV spectra in solids should reflect the self-convolution of the valence band densityof states (DOS), recognize the importance of correlation effects in double-hole final states, which breakthe single-particle picture and are responsible for the large discrepancies between computed results and

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

experimental measurements. Such a correlation is taken into account by introducing either a semi-empiricalparameter (Cini–Sawatzky) or using a perturbative scheme (Cederbaum), which corrects the initial ‘atomic-like’ transition matrix elements.

In the Cini–Sawatzky approach, in particular, the ‘band-like’ behavior of the spectra is obtained by tuningan optimal parameter, representing the hole–hole correlation energy, until satisfactory agreement with theexperiment is reached. In this framework, Cini–Sawatzky were able to explain Auger spectra of transitionmetals, such as Zn, where the hole localization effects play a paramount role. While this approach can bevery useful in order to gain insights on such kinds of scattering processes, excluding some of the mostcomplicated effects, the main drawback remains the use of a phenomenological parameter, which may hinderthe predictability of the results. Furthermore, matrix elements and characteristic parameters are taken fromatoms, a fact that can be inappropriate for crystals, where occupation numbers are different from the atomicones.

The striking difference between our approach and those used by all the above cited authors (and furtherextensions of their work) lies in the computation of the many-body Green function. In fact, in our method themany-body Green function is projected onto ‘localized’ states, which are a ‘local mixture’ of atomic states.Therefore, differently from previous works, SURPRISES uses multicentered local projectors that may includesome atomic space points; one can say that the Auger matrix elements computed in this way already include adelocalized character. In other words, the role of the multicentered projectors is somehow similar to that playedby localized Wannier functions in solid state calculations [26], where a Fourier transformed set of plane wavesbrings to a complete basis set centered in some space points selected on the basis of some criteria, such asminimal localization. It is clear that the use of ‘localized’ (not atomic) Auger matrix elements will prove tobe very valuable in the treatment of transitions, such as valence–valence–valence (V–VV), where initial statesare not atomic, but delocalized in nature.

Finally, to have a complete theoretical description of the electron emission mechanisms from nanoclustersor solid targets one has to take into account the post-collisional interactions due to the electronic dynamicalscreening. In this regard, one could move along two independent directions. On the one hand, starting from themeasured spectrum, affected by all kinds of energy losses suffered by the electron on its way out of the solid,one could recover the ‘true’ spectrum by conventional deconvolution procedures.

Whereas many valuable approaches have been used for extracting the ‘true’ non-radiative decay spectrumin condensed matter [27, 28], these procedures are not free from uncertainty, since the characteristics of theexperimental apparatus and different background subtraction procedures can dramatically affect the lineshapesand, therefore, the final decay rates [29]. In this work, we mix the ab initio results with a Monte Carloprocedure, implemented in SURPRISES, in order to use the computed spectrum as electron source for therandom sampling and simulate the effect of inelastic scattering on the original electron energy distribution.In this way, one can avoid background subtraction and superimpose directly the inelastic energy loss to thenon-radiative decay spectrum. The key ingredients for this procedure are the elastic and inelastic scatteringcross sections.

While the Si K–LL and O K–LL Auger spectra from SiO2 clusters are presented as a case study,we believe that our approach and the computational techniques are sufficiently general to be applied tothe interpretation of electron spectroscopies such as PES, Auger (AES) and autoionization in a variety ofnanoclusters and condensed matter systems taking into account, ‘on the fly’, dynamical screening effects.

3. Program summary, nature and solution of the problem

The SURPRISES program suite performs time-independent multichannel scattering simulations forcalculating PES, total and partial decay rates in solid targets (autoionization and Auger spectra) excitedby electron or photon impact. After solving the Lippmann–Schwinger (LS) equation for the continuumorbital in a multicentered projected potential, hole–hole repulsion is treated using a space-energy similarityapproach, which allows accurate treatment of the correlation energy. SURPRISES can calculate emissionangular patterns of photoelectrons and Auger electrons using a modified partial waves analysis, valid in non-rotational symmetric systems [30].

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

Furthermore, it is able to take into account the energy loss of the secondary electrons emitted in non-radiative decay processes from the surface of the system using a Monte Carlo technique.

The geometry and Gaussian basis set are the only ingredients to be provided in the ab initio calculation.The dielectric matrix (or the inelastic mean-free path) to generate the energy loss contribution and elasticmean-free path are needed to generate the energy loss spectrum.

The programming language is FORTRAN 77, 90 and C. The code can be run on any computer capableof compiling Fortran and C, both in serial and parallel environment; SURPRISES is self-contained, having itsown numerical archives, except for numerical libraries performing matrix diagonalizations, such as Lapack,Arpack and Blas. Parallelization has been reached using ‘shared-memory-type’ routines, allowing processorcommunication through variables stored in a shared address space. No MPI or multiprocessor-distributedmemory routines have been attempted at present: some gain, although not enormous for the nature of theproblem under investigation, could be reached by using SCALAPACK, BLACS or any other equivalent parallellibrary in some parts of the suite. Shared memory parallelization has been implemented in the calculation of theCI electronic correlation, of the matrix products, of the intensities, where the Green operator is to be inverted.To specify the performances of our code, the test cases use an INTEL XEON 4 CPU@2.83GHz workstation,mounted with 8 Giga of RAM, meaning that a very high level of accuracy at a very feasible computationalcost can be reached using a desktop machine. The code used in this work is being made generally available viaGNU General Public License.

3.1. Logic formulation of the method and program structure

To treat the resonant collision, main electronic correlation effects and the energy loss, SURPRISES uses acascade of procedures, involving many steps. While PES and resonant Auger spectra can be computed withinour approach, the description of the method will be focused on normal Auger transitions. Electronic correlationis taken into account by configuration interaction (CI), nevertheless any many-body method can be used toobtain the quasi-particle energy levels. The computational steps undertaken are:

(i) Calculation of the ground-state electronic structure: the correlation in the ground state of the systemis included by using Hartree–Fock (HF) and CI with single and double-excited determinants (CISD).

(ii) Calculation of the Auger state electronic structure: the correlation in the initial state of the ionizedtarget is included by using HF and CISD.

(iii) Calculation of the final double hole states electronic structure: the correlation in the final ionic statesof the doubly ionized target is taken into account by HF and CISD expansions.

(iv) Diagonalization of the Hamiltonian including interchannel coupling in a selected open channelssubspace: the correlation in the final states populated by the doubly ionized target and the Auger electronis taken into account by the interchannel coupling. In this procedure a number of transitions, among thosegenerated in the double ion CISD step, are selected, which account for the majority of the total decayprobability. This reduction makes it possible to diagonalize the Hamiltonian, otherwise computationallyvery expensive.

(v) Space-energy similarity procedure: hole delocalization and band-effects are included by splitting eachstate coming out from the previous interchannel interaction in a number of transitions, among thosegenerated in step (ii), close in energy and presenting a maximum space overlap with the selection ofstates in step (iii).

(vi) Auger matrix elements: calculation of the Auger matrix elements on the local multisite basis.

(vii) Cross section: primary ionization is treated at the first level of perturbation theory and Auger spectra arecomputed by using Fano multichannel scattering theory [31].

(viii) Energy loss: superimposition to the theoretical spectrum of the electron energy loss from the solidsurface.

The above logical flow is sketched in figure 1 for clarity.

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

Figure 1. Structure and data flow for the SURPRISES program suite.

4. Multichannel scattering: SURPRISES

4.1. Scaling performances

SURPRISES uses a cluster approach and projected potentials to treat the resonant multichannel scatteringproblem. In principle, the target can be represented by atoms, molecules and solid state matter. Since someaspects of this approach have been described elsewhere for atoms and molecules [32], in this paper we willfocus on relevant methodological and computational procedures implemented for calculating Auger spectra incondensed matter.

The main issue in solid state calculations is represented by the computational effort, exponentiallyincreasing with the number of channels, along with the unfavorable scaling, far from linear, of some quantumchemistry methods, such as CI or multiconfigurational self-consistent field (MCSCF) [33]. In order to reachgood accuracy (∼ 0.1 eV), one needs methods able to recover important parts of the electronic correlationwith linear scaling or small exponent power-law dependence with the number of electrons and size of thebasis set [34]. CI-based approaches become rapidly useless in condensed matter if one adopts a ‘brute force’approach to take into account delocalization effects from the first principles, resulting in a very large size ofthe excited determinants space.

To reduce the number of such determinants in the target, we use two different computational strategies: wefirst use a cluster approach to keep the number of atoms low, calculating the electronic structure of the systemin model clusters of increasing size until we reproduce satisfactorily the band structure of the real solid. In fact,to reduce the computational cost it is important to assess the range of the interactions, particularly in condensedmatter, where one deals with an enormous number of atoms. The cluster approach has been preferred to theperiodic cell model [35], since core-level and Auger spectroscopy probes the local density of states (LDOS)and Auger transitions are affected within a range of only few atomic diameters. Usually, a reduced number ofneighbors is requested to calculate Auger or autoionization spectra in good agreement with experimental datain solids [36]. Nevertheless, this reduction reveals itself not to be satisfactory for a reasonable computationalscalability of the calculation. Then, a new computational procedure has been implemented, which uses a fewlocalized (but not atomic) transitions in the CI procedure and diagonalizes the interchannel Hamiltonian inthis reduced space. Using a ‘space-energy’ similarity procedure, which will be detailed below, each selectedtransition within the previous model space is split in a number of states to recover the band-like part of thespectrum. In practice, the Hamiltonian is ‘block diagonalized at each energy’ close to the selected channels.Such analysis, which is the core of our method, significantly reduces the computational effort to calculate corelevel PES and non-radiative decay in larger systems. With regard to scaling performances, the concern is aboutboth the basis set and the number of open channels: one could have a small, then computationally feasible,basis set but an enormous number of open channels due to the size of the system, or, vice versa, a large numberof basis set with a very limited number of channels. Whereas the basis set scales linearly with the numberof atoms, the channel number scales as the second power. Furthermore, if one is to reproduce accurately thebehavior of the scattering potential, a larger number of functions, approximately scaling linearly with the

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

number of atoms, is needed. The cost of the calculation of the interchannel potential scales cubicly with thenumber of channels, whereas the memory cost with the square. Finally, symmetry may be used to furtherreduce the computational scaling, particularly in the interchannel procedure; in systems where symmetryhalves the spanned functional space, Hamiltonian matrix inversions, such those performed for calculatingthe Green operator, which scale as the cube of the system size, will be eightfold reduced. Symmetry helps inthe calculation of the electronic correlation, reducing the size of the determinants space. In the calculation ofmono- and bi-electronic integrals, symmetry enters only in the generation of the cluster geometry, according tothe one the system belongs to. It is not straightforward to assess generally the problem of the scaling gain withsymmetry, since it is impracticable to know in advance which element of the point group the linear combinationof basis set functions will belong to.

4.2. The Hamiltonian and its solution

The multichannel theory of scattering aims to find positive energy solutions of the many-body Hamiltonian:

(H − E)9−

α,ε = 0, (2)

where

H(1, . . . , N ) =

N∑i=1

[T (i) + V en(i)] +1

2

N∑i 6= j

v(i, j) = H0 +1

2

N∑i 6= j

v(i, j) (3)

and

T (i) = −12∇

2i ; V en(i) =

∑µ

1

|ri − Rµ|, v(i, j) =

1

|ri − r j |. (4)

In equations (3) and (4) H0 is the free Hamiltonian, containing the electron kinetic energy T and theelectron–nuclei interaction V en, and v(i, j) is the electron–electron repulsion. The scattering wavefunction9−α,ε , solution of the many-body Schrödinger equation (2) with ingoing boundary conditions, describes the

motion of a particle with kinetic energy εα = E − Eα, asymptotically not interacting with the scatteringcenter in the state 2α at energy Eα. Assuming that the primary electron carries out enough energy to avoidpost-emission interactions with the remaining core–hole-ionized target, the scattering wavefunction of thedouble-ionized system and the electron ejected in the Auger decay can be written in the long-range limit [37]:

limrN−1→∞

9−

α,ε(1, 2, . . . , N − 1) ∝ |2α(1, 2, . . . , N − 2)〉[|σα(sN−1) ψ

−

εα(rN−1)〉

]+|

∑β

2β(1, 2, . . . , N − 2)〉

[σβ(sN−1)

e−iθβ

(2π)3/2 rN−1

]S(εβ, εα), (5)

where |σαψ−εα

〉 is the escaping electron spin–orbital, θβ the phase shift and S(εβ, εα) the scattering amplitudes,coupling different channels. To give an appropriate representation of the multichannel process, the scatteringwavefunction should include the main correlation effects, those among bound electrons in the final decaystates of the system and between the double ion and the electron in the continuum. The traditional way tosolve equation (2) with ingoing boundary conditions relies on the static exchange approximation (SEA) [38],which splits the scattering process in two steps: the first step is to find the self-consistent solution of the HFequations for the bound states:

F (b)α θαi (r) = εiθαi (r), (6)

F (b)α = T + Ven(r) +N−2∑j=1

[a(b)α j J (α)j (r)− c(b)α j K (α)

j (r)], (7)

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

where F (b)α is the HF operator, calculated as the sum of the kinetic energy (T ), of the electron–nuclei attractionpotential (Ven), of Coulomb ( J (α)j ) and exchange (K (α)

j ) operators weighted by the coefficients aα j and cα j ,depending on the occupation number. The second one is the solution of an HF-like equation at positive energyfor the continuum orbital:

F (c)α |ψ−

αk(r)〉 = εα|ψ−

αk((r))〉, (8)

F (c)α = T + Ven(r) +∑

j

[a(c)α j J (α)j (r)− c(c)α j K (α)

j (r)]

= T + Vα(r), (9)

where the HF potential F (c)α is an effective mean field containing the interaction between the escaping electronand the system. Many calculations on atomic and molecular systems [39–41] showed that the solution of theeffective HF equation (8) introduces larger numerical errors than the one of equation (7). Then, within SEA,SURPRISES aims to find a very accurate solution of the equation (8) that includes appropriate boundaryconditions. In this regard, SURPRISES uses projected potentials and localized basis functions. Extension ofsuch an approach to take into account delocalization, spreading the Auger probability over the bands, and theextrinsic inelastic energy loss will be discussed below.

4.3. Projected potentials and solution of the model Hamiltonian

Since the computer is a discrete, finite memory machine, the eigenstates |ψ−

αk〉 and the interaction potentialVα in equations (8) and (9) have to be represented in a finite-dimensional space. In order to do that, we definea model N -electron Hamiltonian, which is the projection of equation (2) onto an M-dimensional space (G),spanned by L2 square integrable functions {|gl〉; l = 1, . . . ,M}:

H(1, . . . , N ) =

N∑i=1

[T (i) + V enπ (i)] +

1

2

N∑i 6= j

vπ (i, j), (10)

T (i) = −12∇

2i ; V en

π (i) = π(i)V en(i)π(i), (11)

vπ (i, j) = π(i)π( j)v(i, j)π(i)π( j), (12)

π(i) =

M∑l=1

|gl(i)〉〈gl(i)|, (13)

where π is the identity operator in (G). The structure of the scattering wavefunction in the asymptotic region(see equation (5)) suggests that the space (G) used to represent the scattering wavefunction in the interactionregion can be chosen as

9αkα (1, . . . N − 1) =√

N − 1 A[

|2α(1, . . . , N − 2) σα(sN−1)〉 |ψEkα(rN−1)〉

], (14)

where A is the antisymmetrizer under the constraint that continuum orbitals are orthogonal to bound orbitals:

〈ψαk(N − 1)|2β(1, . . . , N − 2)〉 = 0; ∀ α, β, k. (15)

With this structure, wavefunction (14) will have the correct asymptotic behavior, representing a double-ionizedsystem in the state 2α at energy Eα and a free electron with kinetic energy k2

α/2 = E − Eα, where E is thetotal energy. Furthermore, in this way, the functional space can be naturally written as the tensorial product oftwo orthogonal Hilbert spaces {|αl, g j 〉 = |αl〉 ⊗ |g j 〉, l = 1, . . . ,m, j = 1, . . . , n}: the former spannedby the basis set {|αl〉; l = 1, . . . ,m}, approximating the eigenstates {|2α〉} of the double ion, built withthe eigenfunctions of equation (7); the latter spanned by the basis functions {|g j 〉; j = 1, . . . , 〉, n}, whichapproximate the scattering state of the escaping electron and are eigenstates of the kinetic energy operator T

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

at a given energy εα. Since both the continuum and bound wavefunctions are expansions within the Hilbertspace G, two different strategies can be used to satisfy the orthogonality constraint (see equation (15)). One isdirect orthogonalization, the other uses projection of the kinetic operator onto the orthogonal complementof the bound states. The latter procedure defines an effective potential, which has the same structure ofthe Phillips–Kleinman potential [42] used in condensed matter calculations for building valence orbitalsorthogonal to the core states. In general, we find that the Phillips–Kleinman technique leads to numericalinstabilities whenever the eigenvalues of the Gaussian functions overlap matrix are smaller than 10−4.

Assuming to have solved the bound states problem (see equation (7)), we replace the equation (8) withthe following LS equation:

|ψ−

αk〉 = |φk〉 + G−

0 (εα)Vα|ψ−

αk〉, (16)

where

G−

0 (εα) = limν→0

[εα − iν − H0]−1; H0 = −

12∇

2 (17)

is the free single-particle Green’s function at energy εα = E − Eα and Vα is defined in equation (9). The LSequation (16) is a Dyson-like equation with the unperturbed term |φk〉 given by a plane wave and representsthe formal solution to the scattering problem (2) inside the Hilbert space (G). In fact, a simple substitution ofequation (16) to (2) shows that |ψ−

αk〉 is an eigenstate of the Hamiltonian H if |φk〉 is eigenstate of the kineticenergy operator T .

Vα is the screened Coulomb potential representing the interaction of the escaping electron with theremaining system in the state |2α〉. This interaction is long range, with a tail approximately representedby a monocenter Coulomb potential plus higher order corrections, such dipole terms, that depend on thetype of system under investigation. To treat the long-range behavior of the screened Coulomb potential,SURPRISES splits the scattering problem into two separate regions: a problem for large distances, whereanalytical hydrogenic-like solutions are available, and a more challenging problem for small distances, wherethe interaction of the electron with the remaining system is not negligible. In our approach, we do not solvethe asymptotic scattering problem using an ‘R-matrix like’ procedure. In this method, according to an ideadeveloped by Wigner for nuclear interactions, the orbital obtained through numerical solution of the internalregion problem can be matched with the analytically known asymptotic expression of the external regionand its derivative, once the best matching point has been determined. SURPRISES instead utilizes a ‘twoHeaviside potentials’ approach, which resembles only ideally to the Wigner one: the first is a bi-electronicpotential numerically computed in the interaction region to treat accurately the scattering problem and assumedzero outside this space; the other one, asymptotic, is assumed zero within the scattering region and mono-electronic in the outer region. Differently from the R-matrix approach, which uses directly the configurationspace in the standard meaning of space volume, SURPRISES working space is a functional Hilbert space.Therefore, a functional separation in bound and scattering space maps the separation in the real ordinaryspace. Furthermore, in this way, the internal region, fixed ‘a priori’ in the R-matrix approach, can be enlarged‘on the fly’ to describe better the scattering potential increasing the basis set, while the asymptotic potentialcan be treated more accurately by simply using mono-electronic integrals, computationally less expensive. Theprojection onto the Hilbert space (G) gives an explicit matrix expression for the screened Coulomb potential:

VPα =

m∑k,l

n∑i, j

|αk〉|gi 〉Vαk ,αl

i j 〈g j |〈αl |, (18)

where

V αk ,αli j = 〈gi |

[V enπ δαkαl + W αβ

π

]|g j 〉 (19)

and

W(1, . . . , N − 1| E) =

N−1∑j=2

〈2α(2, . j., N − 1)|vπ (1, j)(1 − P1, j )|2β(2, . j., N )〉. (20)

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

In equation (20), P1, j is the operator that interchanges the (1, j) variables. The choice of the functional spaceG, where the Coulomb potential is projected, proves crucial to obtain good accuracy, involving the cut-off of theinteraction region. The idea implemented in SURPRISES is to project the Coulomb operator onto a functionalspace, large enough to reproduce correctly the behavior of the true potential in the interaction region in sucha way as to minimize the difference between the projected and the true potential acting on the continuumwavefunction

(V Pα − Vα)|ψ−

αk〉 ' 0. (21)

The satisfaction of this criterion allows one to obtain good accuracy continuum orbitals and spectroscopicmatrix elements in the region where the interaction is important. Finally, defining the transition operator withinG as

Tα = VPα + VP

αG−

0 Tα (22)

the LS equation (16) projected onto the model space can be written as

ψ−

αk(r) = φk(r) + G−

0 (εα)Tα(E)φk(r) = φk(r) +∑

l j

〈gl |Tα|g j 〉〈g j |φk〉G−

0 (εα)|gl〉. (23)

Therefore, the general expression of the continuum wavefunction is obtained by applying the free-particleGreen function to the elements of the chosen Hilbert space. Furthermore, since the free-particle Green functioncan be written as

G±

0 (εα) = limε→0

[E − Eα ± iε − H0]−1=

PE − Eα − H0

∓ iπδ(E − Eα − H0), (24)

the continuum orbital contains a linear combination of elements

δ(E − H0)|g j 〉. (25)

In other words, equation (25) states that the continuum wavefunction for a given channel |2α〉 isproportional to the continuum DOS at energy E . Such an expression, used by SURPRISES to calculate thescattering wavefunction, can be calculated analytically if Hermite Gaussian functions (HGFs) are chosen tospan G.

5. Physical observables

5.1. Auger matrix elements

After calculating the scattering wavefunction one has to extract the observables of interest. Following the time-independent Fano approach [31], we describe the decay process as due to the interaction between a discrete,quasi-bound state, produced in the initial ionization or excitation process and the continuum of decay states.

For an Auger process, the resonant core–hole intermediate state |8〉 of the system is degenerate withseveral continua {|χ−

β,εβ〉}, asymptotically describing the electron released with kinetic energy εβ into the

double-ionized target channel |β〉 at energy Eβ . Therefore, the scattering eigenstates 9−α,εα

in equation (2) arerepresented by a linear combination of degenerate states:

|9−

α,εα〉 = aα(εα)|8〉 +

Nc∑β=1

∫∞

0|χ−

β,τ 〉Cβ,α(τ, ετ )dτ, (26)

where the non-interacting continuum states {|χ−

β,εβ〉}, are obtained by solving the LS equation (23) with

the diagonalization of the interchannel projected potential (see equation (19)). Once the coefficientsaα(εα),Cβ,α(τ, ετ ) in equation (26) have been obtained by projecting equation (2) into both the decay

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

channels and the intermediate state, the final expression of the scattering wavefunction is [14]

|9−

α,εα〉 = |χ−

α,εα〉 +

M−

jα(εα, E)

E − Er − i02

|8〉 + limν→0

∑β

∫∞

0

|χ−

β,τβ〉M∗−

β (τβ, E)

E − Eβ − τβ − iνdτβ

= |χ−

α,εα〉 +

M−

jα(εα, E)

E − Er − i02|8−

〉, (27)

where Er = 〈8|H |8〉 +∑

β

∫ |M−

β |2

E−Eβ−τβ, and the total decay rate is given by

0 =

∑β

0β = 2π∑β

|M−

β (εβ, E)|2. (28)

In equation (27), the discrete–continuum coupling matrix elements are given by

M−

β (εβ, E) = 〈8|H − E |χ−

β,εβ〉. (29)

Since the basis set used by SURPRISES is multicentered, these Auger matrix elements are projected onto‘localized’ states, which may include many atoms and, then, have an intrinsic delocalized character. Thisis a fundamental difference, which will be essential in the treatment of transitions where Auger states aredelocalized, between our approach and those using atomic Auger matrix elements. Finally, the scatteringwavefunctions are orthonormalized according to

〈9−

α,εα|9−

β,εβ〉 = δαβδ(Eα + εα − Eβ − εβ). (30)

5.2. Evaluation of spectroscopic observables

The Auger matrix elements, defined in equation (29), play a fundamental role in the calculation ofautoionization and Auger cross sections. Within the frozen phonon approximation, for a photon beam (buta similar expression would be valid, mutatis mutandis, for an electron beam) with frequency ω polarized alongthe λ-direction, first-order perturbation theory and equation (27) give for the Auger cross section:

∂σ0→α

∂Ek∂ Ep(Ek, Ep;ω, λ) =

(4π2ω

c

)|〈0|Oλ|9

−

αEk Ep〉|

2δ

(E0 + hω −

(Eα +

k2 + p2

2

))

' δ(E0 + hω − E)

(4π2ω

c

) ∣∣〈0|Oλ|8−ηp〉

〈8|H − E |χ−

αEka〉

E − Er − i02

∣∣2, (31)

where |9−

αEk Ep〉 ' |9−

αEk〉 ⊗ |ηp〉, assuming that the escaping photoelectron (|ηp〉) is not interacting with the

system. In equation (31), the small direct double-ionization term and the interference between resonant anddirect contributions have been neglected. Defining the partial decay probability as

0α = 2π |M−

α (εβ, E)|2, (32)

one can write

∂σ0→α

∂Ek∂ Ep(Ek, Ep;ω, λ) =

(2πω

c

)|〈0|Oλ|8

−ηp〉|2

E − Er − i020αδ

(E0 + hω −

(Eα +

k2 + p2

2

)). (33)

In practice, to obtain the Auger or autoionizing decay probability, one needs to calculate the Auger matrixelements.

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

6. Energy loss

In order to gain new insights into the nature of the emission process from a solid it is essential to take intoaccount the energy loss suffered by the electrons escaping from the solid surface. This kinetic energy loss maybe due to single or multiple inelastic scattering and to the interaction with collective excitations of the system.In this respect, the typical procedure to compare the computed and experimental spectra is to reconstruct the‘true’ lineshape by conventional deconvolution procedures [27, 28]. This approach is not free from uncertainty,since acquired spectra need to be corrected for the energy dependence of the analyzer transmission function.

In order to avoid these deconvolution procedures, we proceed, alternatively, along the direction ofsimulating the effect of inelastic scattering on the original electron energy distribution. The basic ideaimplemented in SURPRISES is to use the ab initio emission probability distribution as the initial distributionpattern for electrons undergoing inelastic energy losses. After adding the inelastic energy loss to the ab initiospectrum, one can compare directly theoretical and acquired experimental spectra. In this way, along witha full ab initio treatment of the hole–hole correlation (which possibly includes intrinsic energy loss due toshake processes), extrinsic energy losses due to the interaction of the escaping electron with the surroundingelectronic clouds and collective excitations of the system are taken into account.

A possible approach to account for the extrinsic loss is the Monte Carlo method [44]. Energy loss peaksmay appear after the Monte Carlo treatment of the Auger electrons energy distribution, giving importantinformation on the underlying electronic structure.

In this section, we summarize the details of the Monte Carlo procedure implemented inSURPRISES [45–47]. Here we assume that the theoretical lineshape is provided as input energy distributionand that the energy is lost in discrete collisions. The step-length χ is a random variable defined in the interval[0, ∞) and assumed to follow a Poisson-type law:

pχ (x) =1

λexp

(−

x

λ

), (34)

where pχ (x) is the probability density and λ is the expected value of χ . λ rules the step length in the MonteCarlo procedure

1s = −λ ln(µ1), (35)

where µ1 is a random number uniformly distributed in the range [0, 1] and can be interpreted as the electronmean free path

λ(E) =1

N [σinel(E) + σel(E)]. (36)

In equation (36), N is the density of scattering centers, σel(E) and σinel(E) are the elastic and inelasticscattering cross sections. In SURPRISES σel(E) and σinel(E) are assumed to be user provided inputs. Insection 7.5 we will propose a way to calculate them, using the relativistic partial wave expansion method [48]and the Ritchie theory, respectively [49]. In a Monte Carlo run, before each collision, a random number µ2

uniformly distributed in the range [0, 1] is generated and compared with the probability of inelastic scatteringqinel = σinel/(σinel+σel).Whenµ2 is smaller or equal to the probability of inelastic scattering, then the collisionis inelastic; otherwise, it is elastic. If the collision is elastic, θ , the polar scattering angle, is selected so thatthe integrated scattering probability in the range [0, θ] is equal to the random number µ3 uniformly distributedwithin [0, 1]:

µ3 =1

σel

∫ θ

0

dσel

d�2π sinϑdϑ, (37)

where � is the solid angle of scattering and dσel/d� is the differential elastic scattering cross section. If thecollision is inelastic, the energy loss ω of an electron, impinging with kinetic energy E , is computed via arandom number µ4, uniformly distributed in the range [0, 1]:

µ4 =1

σinel

∫ W

0

dσinel

dωdω, (38)

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

where dσinel/dω is the differential inelastic scattering cross section. Therefore, using random sampling andequations (37) and (38), SURPRISES can calculate both the polar scattering angle for the elastic collisionsand the energy loss and, therefore, the energy loss spectrum.

7. Algorithms, computational techniques and numerical issues

7.1. Wavefunctions

Following the flow chart in figure 1, the first issue to face in order to minimize the run time and the memory at agiven accuracy is the efficient representation of both bound orbitals and continuum wavefunctions. While manymethods have been proposed to solve efficiently the self-consistent equation for the bound states (equation (7)),the calculation of the scattering wavefunction from first principles in solids is still an open problem. As fromequation (25), the continuum wavefunction relies on a representation of the free-particle Green’s function (orof the DOS) on the basis set elements. Since analytical expressions can be obtained for such a representationusing Gaussian functions, SURPRISES adopts contracted symmetry adapted HGF, centered on the nuclei, asbasis set in the calculation of both the bounds and continuum states. HGFs are defined by [50–52]

g(r) = g(u, v, w;α,R; r) = N∂u+v+w

∂Xu ∂Yv ∂Zw

(2α

π

)3/4

exp[−α(r − R)2] (39)

where R ≡ (X,Y,Z) gives the center position of g and N is the normalization factor

N = [αl(2u − 1)!!(2v − 1)!!(2w − 1)!!]−1/2, l = u + v + w. (40)

HGF basis sets have many properties able to reduce the computational cost [33]: firstly, one-body and two-bodyintegrals can be analytically calculated and, since the dimensionality of such integrals is halved with HGF,the number of bi-electronic integrals scales as K 4/8, where K is the number of contracted basis functions;secondly, integrals involving HGF with angular momenta higher than 0, such as p-, d-, f-type Gaussians, can becalculated differentiating terms containing s-type Gaussians only. The multiplication of Gaussian functions bysymmetry adapted Hermite polynomials reproduces the oscillating behavior of the continuum wavefunctionsinside the scattering region.

Gaussian functions characterized by the same value of u, v, w and nuclear coordinate R are groupedtogether in a unique ‘pseudocenter’; different Gaussians belonging to the same pseudocenter give linear-dependent expressions for equation (25) reducing the elements of the space in the scattering calculation to thenumber of pseudocenters. Nevertheless, due to the large number of atoms and open channels in comparisonto molecules, the size of the Hilbert space where the wavefunctions are expanded increases rapidly fornanoclusters and solids. An increasing number of Gaussians may lead to convergence to non-physical statesdue to basis set superposition errors and linear dependence problems [53], particularly in the calculation ofmetastable states. In order to keep the number of Gaussians low, we split the problem in two steps: we firstcalculate the bound orbitals in a restricted Gaussian basis set; we then introduce an auxiliary basis set ofmodified Gaussian functions to compute the matrix elements of the scattering potential (equation (18)).

‘Good quality’ basis sets for bound orbital calculations can be found in the literature for atomsand molecules [54, 55] but they are usually not transferable to solid state calculations, since valenceelectron orbitals substantially differ from free-atomic orbitals. Furthermore, an adequate representationof the continuum orbital, comparable in accuracy to the solution of the multiconfiguration procedure inbound orbitals, forces us to enlarge the basis set in the scattering wavefunction calculation. This enlargedbasis set is used both in the expansion of the continuum orbital and in the projection of the interchannelpotential.

In practice, the basis set choice is a trade-off between accuracy and computational cost and it is built byenlarging a variationally optimized basis set for bound orbitals, until variational stability of the Auger decayrate with respect to changes in the basis set is reached. This recipe is, from our experience, the best way todescribe the wavefunction in the energy region where the electrons are emitted avoiding numerical instabilitiesdue to a small overlap between the basis set elements.

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

7.2. Electronic structure calculations: integrals, HF and CI

The following steps of the flow chart in figure 1 are electronic structure calculations on ground, taken asreference state for the system, intermediate core–hole and final Auger states. Exploitation of symmetry,implemented in SURPRISES, can help to keep the cost of computation low, particularly in the interchannelcoupling and its use is always advisable. SURPRISES is able to classify systems up to D2h symmetry pointgroup [56]. A reduction of the number of symmetry operations, due to symmetry breaking defects andaccording to any D2h subgroup, can also be used.

The same computational approach is used in SURPRISES for calculating the electronic structure of thesystem in different excitation conditions. First, overlap integrals among Gaussians are calculated to assesslinear dependency in the chosen basis set, eliminating the functions leading to small eigenvalues of the metrics.Fortran 90 modules then calculate nuclear attraction, mono- and bi-electronic (Coulomb and exchange)integrals among spin–orbitals, in terms of multicentered Gaussians:

〈g j (1)|T |gk(2)〉 =

∫d31d32g∗

j (1)d2

dr2gk(2), (41)

〈g j (1)gl(2)|e2

r12|gk(3)gm(4)〉 =

∫d31d32d33d34g∗

j (1)g∗

l (2)e2

r12gk(3)gm(4) (42)

and Green function matrix elements in the complex plane

〈g j (1)|1

E − H0 + iε|gk(2)〉, (43)

where 1, . . . , 4 are the position vectors and j, . . . ,m refers to different types of Gaussians. The HF calculationtakes the variationally optimized atomic wavefunctions of each of the neutral components in the system as astarting point for the excited state electronic structure calculation. This choice ensures the convergence to thecorrect electronic total energy, since starting from random wavefunctions or eigenvectors of the kinetic energyoperator compromises the numerical stability of the self-consistent procedure.

In the case of both neutral and ionic electronic excitation, orbital relaxation due to a core–hole or finaldouble holes has a large effect on the electronic structure and the use of relaxed orbitals may help theconvergence of the HF equations system. From the solution of the HF equations one obtains the completeset of orbitals corresponding to the variationally lowest energy value.

As a last step in the electronic structure calculations, correlation has to be included in order to treataccurately electron–electron interaction and electronic transitions. In this regard, using all the orbitals of theHF complete solution, SURPRISES performs a linear transformation to the HF basis set and then writes thewavefunctions as an expansion on this set (CI). Since the convergence is very slow, one may select a limitednumber of determinants or excitations, usually corresponding to single or double excitations from the HFground state in a given excitation subspace (CASSCF), using some constraints already in use in SURPRISES.In this way, SURPRISES finds upper bounds to the exact energies of both ground and excited states. We willsee in the case study of SiO2 nanoclusters that the use of this technique consistently reduces the discrepanciesbetween computed and experimental results.

7.3. The interchannel potential and the energy-wavefunction similarity procedure

In SURPRISES the electronic correlation is taken into account first, by performing CI calculations on theintermediate core–hole and double-ionized final states as explained in the previous section; second, by mixingthe open channels via the interchannel interaction Vα defined in equation (19) to take into account thecorrelation between the double ion and the escaping electron.

The computational cost to assess the interchannel coupling is impracticable, when the double ionstates {|2α〉} are calculated by including hole delocalization on all the valence bands and the electronicexcitations. While it is straightforward to generate the perturbated states starting from an HF ground state,the diagonalization of the Hamiltonian on these states, becoming increasingly expensive with the dimension

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

of such a Hilbert space, represents the bottleneck of the scattering calculation. The diagonalization of theHamiltonian (see equation (10)) mixes the interacting open channels, defined in equation (14), and distributesamong them the decay probability to obtain steady state solutions for the scattering problem.

One possibility to perform such a diagonalization is to search for an optimal algorithm able to diagonalizethe large-scale Hamiltonian in the rapidly increasing multichannel space. Such an approach is valuable if oneneeds to find all the eigenvectors in a very large matrix but, in condensed matter, the needed computer powermay be so high that it could not be directly performed as previously done for molecules [57]. In fact, the numberof energy levels n scales linearly with the number of atoms N in the cluster, whereas the computational costto take into account hole delocalization scales as (M = O(n2m+2)), where m is the excitation level. It is clearthat the computational scaling is impractical even for small clusters.

Unfortunately, interchannel interaction cannot be neglected, because its absence is the main source oferrors and of large discrepancies between computed and measured spectra in scattering experiments [57]. Thedevelopment of a new theoretical and computational method implemented in SURPRISES, which uses a fewlocalized orbitals in the CI calculation for the double ions and for the interchannel coupling, makes it possibleto perform accurate resonant scattering calculations in solids with limited computer power. The main idea is toavoid the direct diagonalization of the Hamiltonian on all the active space by selecting a number of channels,which represent, after the interchannel procedure, the brightest transitions or the highest peaks in the spectraand account for the majority of the total decay probability. The dimension of the excited determinants spacein this initial selection step is the major approximation in our procedure. In fact, such a space has to be chosenlarge enough to reproduce well the main transition energies and decay probabilities, but, because of the largecomputational cost, must be a trade-off between computational feasibility and numerical accuracy. This trade-off results in a lack of correlation energy in both the intermediate and final states. In principle, enlarging theexcited determinantal space at this stage would recover an additional part of the correlation energy.

After choosing the channels subsequently to the interchannel potential diagonalization, we select anumber of transitions (which we will call ‘tight-binding’ states) among those generated by the extendedCI calculation of the double ions, close in energy and presenting a maximum overlap with each of thesechannels. In other words, each channel coming out from the first interchannel calculation has been splitin a number of states, similar in terms of energy and wavefunction. These transitions obtained through themaximum overlap analysis recover an important part of the correlation energy and mimic the effect of holedelocalization (or ‘band effects’) and electronic excitation on the localized states. Once obtained the double ionfinal states, in principle, the localized (not atomic, in the sense specified above) Auger matrix elements shouldbe updated with the new electronic continuum wavefunction to take into account the discrete–continuumcoupling. This correcting procedure is not implemented in SURPRISES since this does not dramaticallyinfluence the lineshape. Therefore, Auger matrix elements are kept fixed in the space-energy similarity stepsand their square modulus enters as a multiplicative factor in the lineshape analysis.

Summarizing, our approach reduces initially the number of double ion final states in the lineshape analysisat a level of a CI molecular calculation, allowing a correct cancellation of the errors between the core–holeintermediate state and the decay states, and then recovers the hole–hole correlation via the space-energysimilarity procedure.

7.4. Numerical issues using projected potentials

To test the reliability of the projected potential approach we compare, for the well-studied case of the hydrogenatom, the solution obtained using our model with that obtained by analytical solution of the Schrödingerequation in a Coulomb field at positive energy [58]:[

d2

dρ2+ k2 +

2

ρ−

l(l + 1)

ρ2

](ρRl(ρ)) = 0. (44)

Taking the s-wave scattering (l = 0), the radial part of the hydrogenic wavefunction with E > 0 can be writtenin terms of the confluent hypergeometric function [59]

Rk(ρ) = exp ±ikρF(

1 ∓1

ik, 2,∓2ikρ

). (45)

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

0 50 100r (a.u.)

0

0.05

0.1

0.15

0.2

Wav

efun

ctio

n

a) E=0.000001 a.u.

0 50 100r (a.u.)

0

0.05

0.1

0.15

0.2

0.25

Wav

efun

ctio

n

b) E=0.00001 a.u.

0 50 100r (a.u.)

0

0.1

0.2

0.3

0.4

0.5

Wav

efun

ctio

n

c) E=0.01 a.u.

0 50 100r (a.u.)

0

0.1

0.2

0.3

0.4

0.5

Wav

efun

ctio

n

d) E=1.0 a.u.

Figure 2. Comparison of the exact hydrogenic wavefunction (continuous line) with the approximatesolution of the model (dashed line) for different values of the energy ((a) 0.000001 au, (b) 0.00001 au,(c) 0.01 au, (d) 1.0 au) as a function of the radial coordinate r and using s-type 50 tempered Gaussians.

In SURPRISES the solution of the scattering problem is obtained from equation (16) in the model space G,spanned by a set of Gaussian functions. Since the hydrogenic Hamiltonian is

H = T + V, (46)

where

T =p2

2m, V = −

1

r(47)

the LS equation (16) equivalent to (46) is given by

|ψ〉 = |φ〉 + G+(E)Vπ |φ〉, (48)

where Vπ = 5V5 and 5 is the projector in G. In practice, one has to calculate the matrix elements of theresolvent (see equation (25)):

|φ j (E)〉 = δ(E − T )|h j 〉. (49)

In order to have the same normalization factor and asymptotic behavior on both the exact and approximatewavefunctions we look for scattering solutions normalized under the condition

〈φ j (E′)|φl(E)〉 = δ jlδ(E − E ′). (50)

In figure 2, we compare the exact (continuous line) with the model (dashed line) solutions of the hydrogenicproblem for different values of the energy, as a function of the radial coordinate r, using a basis set made of

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

0 50 100r (a.u.)

–0.2

0

0.2

0.4

Wav

efun

ctio

n

exact406080

0 50 100r (a.u.)

–0.2

0

0.2

0.4

Wav

efun

ctio

n

exact100120

Figure 3. Comparison of the exact hydrogenic wavefunction (continuous black line) with theapproximate solution of the model at energy 0.01 au for different numbers of Gaussians: on the left-hand side, 40 (dashed blue line), 60 (dot-dashed red line), 80 (dotted green line); on the right-handside, 100 (dotted red line), 120 (dashed green line), logarithmic tempered Gaussians basis set withinthe exponent range [0.0001 : 1000000].

50 s-type tempered Gaussians with exponents in the range [0.001 : 10000]. One can see that the two curvesare in very good agreement in all the range of considered r values, with some exception at very small energyand large coordinate values. The former behavior is because at threshold the electron kinetic energy is of thesame order of magnitude of the Coulomb potential and this causes huge oscillations of the wave vector insidethe scattering region, difficult to reproduce with a reduced basis set. In fact, this behavior disappears enlargingthe basis set to reproduce such oscillations as one can see in figure 3.

On the other hand, the mismatch of the two curves at large values of the coordinate is due to the differentphase shift of the exact and approximate solution in such a region. The basis set can reproduce correctlythe wavefunction within a limited region of space, where the interaction is important. Whereas the exactsolution keeps oscillating with a phase shift proportional to the logarithm of the coordinate, the approximatesolution has a constant phase shift in the outer region. This problem could be solved by increasing the basisset, the only constraint being represented by the overcompleteness or linear dependence of the Gaussianfunctions.

7.5. Numerical issues using the Monte Carlo scheme

The accuracy of the Monte Carlo procedure to assess the energy loss of escaping electrons strongly dependson the values of the elastic and inelastic scattering cross sections used in the simulation (see equations (37)and (38)). In SURPRISES the differential elastic scattering cross section is obtained by solving eitherequation (44) or the Dirac equation in a radial potential to account for spin effects. These equations describethe scattering from a central potential and are integrable numerically for any angular momenta. From Diracscattering theory, the differential elastic scattering cross section can be expressed in terms of the form factoras

dσel

d�=| f (ϑ) |2 + |g(ϑ) |2, (51)

where f (ϑ) and g(ϑ) are the direct and spin-flip scattering amplitudes (see [60]). These quantities can beconnected to the phase shifts through the partial wave expansion of the scattering wavefunction and representthe probability amplitude of an electron to be elastically scattered in a given direction ϑ . Once the differentialcross section has been obtained, the total elastic scattering cross-section is numerically calculated from:

σel(E) = 2π∫ π

0

dσel(ϑ, E)

d�sinϑ dϑ. (52)

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

Figure 4. Plot of the elastic (continuous line) and inelastic (dashed line) scattering cross sections (Å2)as a function of the electron kinetic energy (eV).

On the other hand, assessment of the inelastic cross section implies the calculation of the energy lossfunction [45]:

σinel(E) =me2

2π h2 N E

∫ ωmax

0Im

[−1

ε(0, ω)

]S

(ωE

)dω, (53)

where ε(ω) is the long-wavelength limit of the dielectric function, ω is the energy loss (ωmax the maximumenergy transfer) and, according to Ashley [61], the function S is given by

S(x) = (1 − x) ln4

x−

7

4x + x3/2

−33

32x2. (54)

Calculation of σinel(E) is a difficult task due to the dependence of the dielectric matrix from the energy ofthe traveling electron (dynamic dielectric constant), even assuming the electron momentum transfer to benegligible. In metals, Drude theory for the free electron gas can be used to assess the frequency dependence ofthe dielectric response on the external electromagnetic field. In semiconductors or insulators, one can refer toavailable experimental data or ab initio calculations.

In figure 4, we report the elastic and inelastic scattering cross sections as a function of the impingingelectron kinetic energy for silicon dioxide calculated using equations (52) and (53). The accuracy of thecalculation presented in figure 4 is estimated as about 5–6% of the absolute values, in light of an accuracyof about 1–2% in the differential elastic cross section for scattering angles higher than 5◦. The totalinelastic scattering cross section, obtained by interpolating experimental energy loss functions at differentfrequencies [62], is in good agreement with those presented by other authors [63, 64], with an accuracyincreasing with the electron kinetic energies. Nevertheless, the disagreement at low energy does not concernthe Monte Carlo accuracy, since electrons with energy lower than 50 eV or leaving the solid are not taken intoaccount in the procedure. However, the calculation of the elastic and inelastic cross sections is performed,SURPRISES assumes that data are provided as input.

A third quantity affecting the numerical accuracy of the Monte Carlo procedure is the number of electronpaths used which, in order to obtain statistically significant results, has been set higher than 106.

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

Figure 5. Sketch of the Si5O4H12 (left) and Si2O7H6 (right) nanocluster optimized structures. Oxygenis in red, silicon in light blue and hydrogen in green.

8. The working program

8.1. Test run

The test cases, which ensure that there is a test run for each module, is based on a calculation of Si K–LL(test 1) and O K–LL (test 2) Auger decay from solid SiO2, such as:

e− + Si5O4H12 −→ e−

ph + Si5O4H−∗

12 −→ e−

Au + Si5O4H−−

12 test 1

e− + Si2O7H6 −→ e−

ph + Si2O7H−∗

6 −→ e−

Au + Si2O7H−−

6 test 2.

Since an exhaustive description of the O K–LL spectrum from Si2O7H6 has been given elsewhere [65], here wewill focus on the Si K–LL Auger transition from Si5O4H12. Within the Born–Oppenheimer approximation, thetest uses the optimized structure of Si5O4H12 (Si2O7H6 in test 2) with a central silicon (oxygen in test 2) atomobtained by minimizing the free energy of the electrons at each nuclei position with a smooth Fermi–Diracsmearing, relaxing the atomic bond length until the forces are less then 0.01 eV Å−1 per atom. The resultingatomic configurations used in the tests are plotted in figure 5. Stars of neighbors, centered on the central silicon(oxygen in test 2) atom, are found at a distance of 3.061 a0 (3.103 a0) for oxygen atoms (silicon in test 2) and5.763 a0 (4.953 a0 in test 2) for silicon atoms (oxygen), in good agreement with previous geometry calculationsof similar SiO2 clusters [66]. Calculations of the bound states have been carried out using a basis set of HGFswith (s, p, d)-type character centered on the nuclei. The basis set has been obtained by variationally optimizingthe exponents of standard 6-31G∗ basis sets for oxygen and silicon. In more detail we scan a finite range ofvalues above and below the proposed 6-31G

∗

ones, taking the values which lower the total energy. An auxiliarybasis set, more dense and enlarged with diffuse functions, has been used to represent properly the projectedpotential V β,α in equation (19), until variational stability of the Auger decay rate with respect to changes inthe basis set is reached. This criterion allows an accurate description of the continuum orbitals around theAuger decay energies. The scattering calculations highlight the fundamental role of d-symmetry functions inthe expansion of the projected potential. The resulting basis set is made up of [9 s(= 4sSi + 3sO + 2sH) + 6p(=3pSi + 2pO + 1pH)+ 3d(= 1dSi + 2dO)] contracted HGFs, for a total of 270 (215 in test 2) contracted Gaussians.The exponents αs, αp, αd are in the range (αSi

s = 0.001–1.0, αSip = 0.004–1.2, αSi

d = 1.0) for the (s, p, d)-typeGaussians on silicon, (αO

s = 0.002–0.5, αOp = 0.07–1.0, αO

d = 1.0) for the (s, p, d)-type Gaussians on oxygen,(αH

s = 0.03–1.0, αHp = 1.0) for the (s, p)-type Gaussians on hydrogen.

We adopted a larger basis set on the central silicon (oxygen in test 2) with (4s+4p+3d) Gaussians toproperly represent hole localization effects.

In table 1, we have reported the energies, referred to the vacuum level, and the relative decay rates of themost intense transitions, all with double holes localized on the central silicon (oxygen in test 2) atom, obtainedafter switching on the interchannel coupling. The computed spectra in figure 6 are generated by convoluting

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

Table 1. Kinetic energies (Ekin) in eV, referred to the vacuum level, and probabilities (0α) in arbitraryunits, normalized to the maximum peak, for the O K–LL (left) and Si K–LL (right) Auger localizedstates of the SiO2 nanoclusters under investigations according to the double holes configurations(O K–LL, Si K–LL columns) in the central silicon (oxygen in test 2) atom and final total spin S2

(0 = singlet, 1 = triplet).

O K–LL S2 Ekin 0α Si K–LL S2 Ekin 0α

2s–2s (0) 458.75 0.570 2s–2s (0) 1499.99 0.3352s–2p (0) 473.4 0.511 2s–2p (0) 1544.66 0.8602s–2p (0) 477.41 0.653 2s–2p (1) 1563.35 0.2262s–2p (0) 477.99 0.624 2p–2p (0) 1598.19 0.3622s–2p (1) 481.64 0.156 2p–2p (0) 1603.72 0.9982s–2p (1) 484.96 0.182 2p–2p (0) 1603.73 12s–2p (1) 485.73 0.190 2p–2p (0) 1661.11 0.0512p–2p (0) 493.94 0.670 2p–2p (0) 1714.15 0.0392p–2p (0) 497.89 0.801 2p–2p (0) 1716.26 0.4042p–2p (0) 498.74 0.8622p–2p (0) 500.28 0.8292p–2p (0) 501.98 0.9752p–2p (0) 502.45 1.0

Figure 6. Left: Si K–LL Auger spectrum in Si5O4H12. Comparison between the quantum mechanicalfirst-principles calculation (continuous line) and the Monte Carlo results for different SiO2 layerthicknesses increasing from the bottom to the top: 5 nm (dashed line), 10 nm (spaced dotted line), 15 nm(dot-dashed line), 20 nm (small-dashed line), 25 nm (dotted line). Right: O K–LL Auger spectrum inSi2O7H6. Comparison between the quantum mechanical theoretical data (continuous line), the MonteCarlo results (dashed line) and the experimental data from Taioli et al [65] (dotted line).

the theoretical transition rates with a Gaussian (6g = 1 eV) to take into account both the bandwidth of thex-ray source and the finite resolving power of the electron spectrometer and a Lorentzian with 6l = 0.1 eV,which is the computed total Auger decay rate. Both spectra are normalized to a common height of the mainpeak, while no energy shift has been applied.

Auger transitions are distributed over three different regions in the Si K–LL (O K–LL in test 2) spectrumin the left part (right in test 2) of figure 6. In the Si K–LL Auger spectrum, from right to left, features between(1600–1612) eV ((494–510) eV in test 2) are due to 2p–2p hole configurations in the final states (K-L23L23

transitions); the most intense peaks are singlet transitions around (1608–1610) eV (maximum at 1609.13 eV),close to the experimental peak (1609.08 eV) ((501–503) eV, with the maximum at 502.76 eV in test 2, closeto the experimental peak (502.58 eV)). Between (1565–1572) eV a second group of features is found, wherethe hole configuration in the final states is 2s–2p (K-L1L23 transitions) ((475–492) eV in test 2); the mostintense peaks are singlet transitions around (1568–1570) eV (maximum at 1568.96 eV) ((480–482) eV, with themaximum at 481.1 eV in test 2, close to the experimental peak (481.38 eV)). Finally, a third group of features

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Computational Science & Discovery 2 (2009) 015002 S Taioli et al

is observed between (1546–1554) eV ((460–470) eV in test 2), where the hole configuration in the final statesis 2s–2s (K-L1L1 transitions); the most intense peaks are singlet transitions around (1549–1551) eV (maximumat 1550.02 eV), lower than the experimental peak (1550.91 eV) ((463–465) eV (maximum at 463.86 eV) intest 2, lower than the experimental peak (465.25 eV)). In general the most intense peaks are singlet transitionswith both holes on the central silicon (oxygen in test 2) atom, whereas the shoulders are triplet transitions,unfavored by selection rules.

Good agreement in both transition energies and decay rates is obtained by comparing our calculatedSi K–LL (O K–LL in test 2) Auger spectrum to experimental results by Baba et al [67], and Kashiwakuraet al [68] (van Riessen et al [69], Ramaker et al [70, 71] and Taioli et al [65] in test 2). The s to p contributionis slightly overestimated, due to the finite size of the determinant space; in principle a larger size wouldrecover part of the correlation energy, paying the price of a computationally very expensive procedure. Inthe Monte Carlo simulation, Auger electron generation is simulated assuming a variable depth distribution,whose thickness was increased from 50 to 250 Å until saturation of the low-energy background is reached (intest 2 the thickness is set to a constant value of 40 Å, according to experiments by van Riessen et al [69]).

To obtain good accuracy in the results, the number of random walks is 108. A plot of the QMMCcalculation is given in figure 6, where the original theoretical spectrum is also shown for reference. One can seethat QMMC enhances and broadens the Auger probability increasingly upon decreasing the electron kineticenergy. While in the O K–LL spectrum the large broadening of the K–L1L2,3 peak after Monte Carlo treatment,due to the main plasmon of SiO2, is hidden from the K–L23L23 and K–L1L23 peaks, such a plasmon peak isclearly visible at a distance of about 23 eV below the elastic peak in the Si K–LL Auger spectrum, in theleft side of figure 6. Plasmon energy losses are found in the low energy parts of the spectra, at 1527 eV andat 460 eV, respectively, for Si K–LL and O K–LL spectra. We found that increasing the number of atoms tosimulate a bulk situation enlarges the number of available transitions to the final states to a point where thecomputational cost is enormous, while no appreciable change in the spectroscopic observables is obtained [65].Shake processes should be included in the lineshape analysis of the ab initio calculations to include intrinsicenergy losses.

8.2. Summary

We have presented a detailed account of a first-principles method, computational techniques and a numberof numerical checks to assess the accuracy of a mixed quantum mechanical and Monte Carlo approach forcalculating photoionization, autoionization and Auger spectra in condensed matter. This approach reduces theincreasing computational cost to calculate decay spectra in solids through a space-energy similarity procedure,which allows one to simulate the delocalization with a performance comparable with that of molecules.

In this method, both the initial state core–hole and the final state valence holes are first localized on thecentral atom and multisite interactions enter naturally in the model Hamiltonian as a perturbation.

Furthermore, we demonstrated the capability of such approach in reproducing the Auger spectral lineshape from different SiO2 nanoclusters, taking into account both hole–hole correlation from first-principles andpost-collisional effects due to inelastic scattering and collective charge motion with a Monte Carlo approach.

Acknowledgments

We thank Dr L Calliari for helpful discussions and for carefully reading this manuscript. This work was fundedby the Autonomous Province of Trento, through the FBK-Institute for Scientific and Technological Research.

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