Towards Structural Reconstruction from X-Ray Spectra Anton Vladyka1Christoph J. Sahle2yand Johannes Niskanen1z 1University of Turku Department of Physics and Astronomy 20014 Turun yliopisto Finland

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Towards Structural Reconstruction from X-Ray Spectra

Anton Vladyka,1, ∗Christoph J. Sahle,2, †and Johannes Niskanen1, ‡

1University of Turku, Department of Physics and Astronomy, 20014 Turun yliopisto, Finland

2European Synchrotron Radiation Source, 71 Avenue des Martyrs, 38000 Grenoble, France

(Dated: February 17, 2023)

We report a statistical analysis of Ge K-edge X-ray emission spectra simulated for amorphous

GeO2at elevated pressures. We ﬁnd that employing machine learning approaches we can reliably

predict the statistical moments of the Kβ00 and Kβ2peaks in the spectrum from the Coulomb ma-

trix descriptor with a training set of ∼104samples. Spectral-signiﬁcance-guided dimensionality

reduction techniques allow us to construct an approximate inverse mapping from spectral moments

to pseudo-Coulomb matrices. When applying this to the moments of the ensemble-mean spectrum,

we obtain distances from the active site that match closely to those of the ensemble mean and which

moreover reproduce the pressure-induced coordination change in amorphous GeO2. With this ap-

proach utilizing emulator-based component analysis, we are able to ﬁlter out the artiﬁcially complete

structural information available from simulated snapshots, and quantitatively analyse structural

changes that can be inferred from the changes in the Kβemission spectrum alone.

I. INTRODUCTION

Core-level spectroscopy provides information of struc-

ture of matter at the atomic level, and the constituent

methods are applied from standard material character-

ization to conceptually new experiments at large-scale

facilities such as free-electron lasers. Although refer-

ence data helps, interpretation of core-level spectra is

not always straightforward, especially in the case of soft

condensed or amorphous matter where ensemble statis-

tics plays a drastic role [1–7]. Studies of this statistical

nature, and the implied repeated function evaluations,

could beneﬁt from machine learning (ML), application

of which to core-level spectra has been studied rather

intensively lately [8–15]. In general, when working with

atomic resolution studies have raised the need to engineer

features for both structure [16–20] and spectra [13, 19].

The pressure dependent evolution of the germanium

coordination by oxygen in glassy GeO2has been a long

standing subject of study [21–24]. Besides applications of

amorphous GeO2in technical glasses, the increased sen-

sitivity of a-GeO2to pressure compared to amorphous

SiO2motivates the study of structural changes similar

to those expected to occur in the pressurized analogue

glass a-SiO2but at greatly reduced absolute pressures.

Detailed knowledge of the compaction mechanisms in

these simple glasses will have direct consequences for our

understanding of geological, geochemical, and geophysi-

cal processes involving more complex silicate glasses and

melts.

X-ray emission spectra (XES) of GeO2is an inviting

case for development of spectroscopic analysis for soft

and amorphous condensed matter. First, large spectro-

scopic changes with changing local structure are known

∗anton.vladyka@utu.ﬁ

†christoph.sahle@esrf.fr

‡johannes.niskanen@utu.ﬁ

to exist [24]. Second, simulations are known to repro-

duce the observed ensemble-mean eﬀects well[25]. Third,

XES is local-occupied-orbital derived and a few orbital-

bonding neighbor atoms are expected to be decisive for

the spectrum outcome. This would result in a mini-

mal set of structural parameters needed to predict XES.

Last, owing to the chemical simplicity and simple bond-

ing topology due to non-molecular structure, this system

has promise to be reproduced by ML with the limited

number of data points that the condensed phase allows.

Namely, for such systems the electronic simulation needs

to account for multi-electron eﬀects in numerous interact-

ing atoms – typically on the level of density functional

theory. As a consequence, the number of individual struc-

tural data points for spectroscopy can be expected to be

∼104in an extensive contemporary simulation.

In this work, we focus on Ge KβXES calculations

of amorphous GeO2at elevated pressures. Our previ-

ous work on the water molecule indicated that predicting

spectral features is easier than predicting structural fea-

tures [14]. In the condensed phase, where the structural

features to be predicted are more numerous, the task is

arguably even more complicated. As a solution to this

dilemma, we build a procedure on spectrum prediction

for structures, dimensionality reduction and iterative op-

timization algorithms. This approach is possible because

the evaluation of an ML model requires much less compu-

tational resources than the corresponding quantum me-

chanical calculation does. We predict statistical moments

of XES lines from a Coulomb matrix[16] that describes

the local atomic structure around the site of characteris-

tic X-ray emission. Next, we study obtainable structural

information for the occurring spectral changes in the

pressure progression of the XES by emulator-based com-

ponent analysis (ECA)[15]. Last, we investigate an ap-

proximate solution to the spectrum-to-structure inverse

problem by ﬁrst transforming it into an optimization task

in the dimension-reduced ECA space, followed by expan-

sion to the full multi-dimensional Coulomb matrix. A

dedicated evaluation data set allows for assessment of

arXiv:2210.13909v2 [cond-mat.dis-nn] 16 Feb 2023

performance of the approach in each of the aforemen-

tioned tasks.

II. METHODS

After ionisation from a Ge 1s orbital, the electronic

system is left in a highly excited state, which decays by

either Auger decay or by emission of a photon, accompa-

nied by a transition of an electron from a higher-energy

orbital. For germanium, transitions from 3p to 1s or-

bital give rise to so called Kβemission spectra. Since the

Ge 3p orbitals constitute valence orbitals, Ge KβXES

is highly sensitive to chemical bonding of the active Ge

site.

We study data of amorphous GeO2from statistical

spectral simulations over a range of 11 pressure values

from 0 GPa to 120 GPa. These XES spectra, simulated

using the OCEAN code [26, 27] (version 2.5.2), are based

on real-space conﬁgurations from ab initio molecular dy-

namics simulations reported earlier by Du et al. [28]. We

used the Quantum ESPRESSO program package (version

5.0) [29, 30] for sampling the ground state wave func-

tions and electron densities at the gamma point with a

plane wave cutoﬀ of 100 Ry (see Ref. 25 for more details).

Transition matrix elements are then calculated using the

Haydock recursion method [31] as implemented in the

OCEAN code using a Lorentzian width of 1.0 eV for the

continued fraction. At each pressure point, Ge KβXES

spectra of 18 structurally uncorrelated AIMD simulation

snapshots containing 72 GeO2formula units were cal-

culated for each Ge atom. For 5 pressure points only

17 out of 18 snapshots yielded spectra in a ﬁnite time

frame due to convergence issues, resulting in 13896 XES

spectra. The spectra of individual Ge sites were aligned

and normalized for each pressure to yield a constant Kβ5

line peak position and intensity in its ensemble average

spectrum.

Even though extensive from a statistical simulation

viewpoint, the available dataset is still rather limited for

sophisticated ML algorithms. In this case, using a de-

scriptive numerics allows for condensing structural and

spectral information to a few parameters, resulting in an

improvement of ML performance. We apply descriptors

to both the spectrum and the atomic structure of the

system (see below).

A. Spectral-line descriptor

The XES spectrum is given as a function presenting

intensity against photon energy in eV (I=I(E)). For

a distinguishable peak in the spectrum, we use raw mo-

ments deﬁned as follows:

M1=RI(E)EdE

RI(E)dE,(1)

Mn=RI(E)(E−M1)ndE

RI(E)dE,for n= 2,3,4.(2)

Corresponding spectral descriptors used in the model are

spectrum peak position mean µ=M1(eV), standard

deviation σ=√M2(eV), skewness sk = M3/σ3and

excess kurtosis ex = M4/σ4−3. These descriptors are

referred to as “spectral moments”, and are presented as

a vector mthroughout the manuscript. In this work, 4

moments were calculated for both Kβ00 and Kβ2peaks,

which resulted in 8 descriptors per spectrum.

B. Structural descriptors

As the structural descriptor we use the Coulomb ma-

trix [16], the elements of which are deﬁned as

Cij =(0.5Z2.4

iif i=j,

ZiZj

Rij if i6=j, (3)

where Ziis the atomic number of the i-th atom, and

Rij is the distance between the i-th and j-th atoms. In

this work, we arrange the atoms by the distance from

the active site in ascending order, and grouped Ge atoms

ﬁrst, followed by the O atoms. Since in this approach

the order of atoms is the same for a given number of

Ge and O atoms, the diagonal elements of the Coulomb

matrix are identical for each structure. Therefore, owing

to symmetry of the matrix, only the upper triangle of

the Coulomb matrix is used (see Fig. 1) as vector pas

an input data. From an optimization search, we deduced

the optimal number of the atoms used for the Coulomb

matrix calculation to be the 10 closest Ge and 7 closest

O atoms, which leads to 153-dimensional feature vectors

(see details in SI, Fig. S1).

The deﬁnition of the Coulomb matrix implies that it

can be inverted to a distance matrix containing inter-

atomic distances by

Rij =ZiZj

Cij

with i6=j, (4)

where Rij is the distance between atoms iand j. This

conversion is possible as the Ziof the chemical elements

in each matrix element Cij is known. Furthermore, apart

from the handedness of the coordinate system, the atomic

geometry can be reconstructed from the distance matrix

R, and therefore, from the Coulomb matrix C(see Sup-

plementary Information for algorithm).

To check the performance of the Coulomb matrix de-

scriptor against a many-body-tensor-representation[32]

spirited descriptor, we used snapshot-wise evaluated ra-

dial distribution functions (RDF) from the active site.

Although similar predictive power was obtained via the

RDF, its performance in the later steps of the analysis

(spectral coverage of decomposition) was inferior to that

of the Coulomb matrix.

C. Algorithms

The structural and spectral data are presented as

feature-wise standardized matrices ˜

Pand ˜

M, respec-

tively (individual data points ˜

pand ˜

moccupy rows in

these matrices). The analysis algorithms aim at discover-

ing the correlations between the two data sets, for which

we ﬁrst applied the emulator-based component analysis

(ECA) method as described in Ref. 15. This algorithm

relies on a machine-learning based emulator for spectral

features at a vector of structural descriptors, that may

be previously unseen to it. The algorithm uses projec-

tion of structural data on a subspace so that projected

data maximize the generalized covered spectral variance

(R2score) when a prediction is made using the emulator.

Here, ECA is applied to standardized Coulomb matrix

parameters ˜

pand the corresponding standardized spec-

tral moments ˜

m. The decomposition algorithm results

in orthonormal standardized-structural-parameter-space

vectors ˜

v1,˜

v2, . . . so that spectral moments ˜

memu =

Semu(˜

p(k))for projections

p(k)=

i=1

vi(˜

vi·˜

| {z }

=:ti

,(5)

predicted using trained neural network Semu, cover most

of spectral variance of the respective set of points ˜

pat

the given rank k. Scores tiare coordinates of the approx-

imate point ˜

p(k)in the k-dimensional subspace.

The ECA method requires an emulator capable of pre-

dicting spectral moments for new structural data points.

As an emulator, a trained multilayer perceptron (MLP)

with 2 hidden layers and 64 neurons in each layer was

used. We dedicated 80% of data for training, and 20% for

evaluation of the prediction. Overall, all conﬁgurations

of MLPs with 2–3 hidden layers and 64 or 128 neurons

were evaluated on the training dataset (∼11000 spectra)

using mean squared error as a training metric.

For comparison, we used partial least squares ﬁtting

based on singular value decomposition (PLSSVD) [33] as

applied to the X-ray spectroscopic problem in Ref. 15.

The PLSSVD algorithm relies on projections of spectral

and structural feature vectors on latent variables between

which a linear ﬁt is made. The method results in an

approximation of the data up to rank k

M≈˜

i=1

U(i)ciV(i)T,(6)

where U(i)is the i-th (column) basis vector of the struc-

tural descriptors and V(i)is the i-th (column) basis vec-

tor of the spectral descriptor space. The coeﬃcient ci

FIG. 1. The principle of spectral moment prediction for a Ge

KβXES peak for amorphous GeO2. A Coulomb matrix is

generated from a structure, and its upper triangle is fed as

input for MLP, which is trained to predict spectral moments

of the line of interest.

FIG. 2. Raw XES spectra. Colored curves depict the mean

spectra for each pressure, black curve shows the global mean

spectrum. Dark and light shaded areas indicate ±σfrom the

mean spectrum and max/min range, respectively. Vertical

dashed lines mark the intervals of the two studied peaks, Kβ00

and Kβ2.

is obtained by a ﬁt to the scores (˜

PU(i),˜

MV(i)). The

orthonormal basis vectors are obtained from a singular

value decomposition of the covariance matrix of the data

cov( ˜

P,˜

M) = ˜

PT˜

M; ordering along descending magni-

tude of the singular values λiis applied. For analysis

using the PLSSVD algorithm, the same evaluation data

set as for ECA was used.

III. RESULTS

The ensemble-averaged Ge KβXES of GeO2shows

a transition induced by pressure, as seen in Figure 2.

Even though a smooth progression of the spectrum as a

function of pressure is observed, the underlying statistical

variation in the condensed-phase XES is large [3, 5, 25].

This is manifested by gray shading in Figure 2 that shows

FIG. 3. (a–c) Sample XES spectra at diﬀerent pressures. For each spectrum the inset shows the corresponding 3D structure,

where the active Ge site is yellow, other Ge sites are pink and O sites are red. The symbols are used to indicate the data point

in the panels below. (d–k) Results of the MLP training: predicted spectral moments of the evaluation data set for the Kβ00

(d–g) and Kβ2peak (h–k). The color of each point indicates the corresponding pressure for the structure, and colored crosses

in every panel depict the mean values for each pressure subset (known: moments of known mean spectrum, predicted: mean

of predicted moments). Positions of the sample spectra from panels (a–c) are marked with black markers. Number in every

panel shows the Pearson’s correlation coeﬃcient rbetween known and predicted data.

the minimum-maximum variation of intensity in the data

set. We proceed with our analysis for the two lines with

clear pressure dependence: the Kβ00 and the Kβ2.

Figure 3a–c presents structures and spectra of three

individual snapshots at pressures of 0, 30, and 120 GPa,

respectively. Figures 3d–k in turn show the prediction

and training performance of the chosen MLP for these

descriptors. In the ﬁgure, perfect match between known

and predicted data lie on the diagonal dashed line. Fur-

thermore, the positions of the three illustrated spectra of

Figure 3a–c, as well as the mean moment values for each

pressure point against moment values of the known mean

spectrum are indicated by crosses.

The spectra and their statistical moments show a clear

trend as a function of pressure. Moreover, the overall

quality of the prediction performance yields Pearson cor-

relation coeﬃcients above 0.94. The pressure-induced

progression in the spectra is transferred into spectral mo-

ments, for which the ML task proved to be easier than

predicting spectra as vectors of channel-wise-listed in-

tensity values (see Fig. S2). Analogously with simple

intensity prediction, spectral moments of an ensemble-

averaged spectrum can be estimated by the mean of pre-

dicted moments to a good accuracy (see crosses in Figure

3d–k). However, this is an approximate ﬁnding instead

of a mathematical equality.

For the evaluation data set, some 77% of generalized

covered spectral variance (R2score) can be explained by

only a single ECA component ˜

v1(83% with two compo-

nents {˜

v1,˜

v2}). These components represent individu-

ally standardized elements of a Coulomb matrix unrolled

to 153-dimensional vectors (for {˜

v1,˜

v2}rolled back to the

standardized Coulomb matrix diﬀerences, see Fig. S3)

For PLSSVD, corresponding spectral variance coverages

were 73% and 77% for one and two components, respec-

tively. The added contribution of the second component

indicates a rapid drop of improvement in higher ranks.

Before entering the inverse problem, it is instructive

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TowardsStructuralReconstructionfromX-RaySpectraAntonVladyka,1,ChristophJ.Sahle,2,yandJohannesNiskanen1,z1UniversityofTurku,DepartmentofPhysicsandAstronomy,20014Turunyliopisto,Finland2EuropeanSynchrotronRadiationSource,71AvenuedesMartyrs,38000Grenoble,France(Dated:February17,2023)Wereportastatistica...

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Towards Structural Reconstruction from X-Ray Spectra Anton Vladyka1Christoph J. Sahle2yand Johannes Niskanen1z 1University of Turku Department of Physics and Astronomy 20014 Turun yliopisto Finland

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