### Department of Physics, Chemistry and Biology

### Master’s Thesis

### Vibrationally resolved silicon L-edge spectrum of

### SiCl

_{4}

### in the static exchange approximation

### Johnny Jonsson

LiTH-IFM-EX-08/1909

Department of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden

Master’s Thesis LiTH-IFM-EX-08/1909

### Vibrationally resolved silicon L-edge spectrum of

### SiCl

_{4}

### in the static exchange approximation

### Johnny Jonsson

Adviser: Ulf Ekstr¨om

IFM, Link¨opings Universitet

Patrick Norman

IFM, Link¨opings Universitet

Examiner: Patrick Norman

IFM, Link¨opings Universitet

Avdelning, Institution Division, Department

Computational Physics

Department of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden

Datum Date 2007-02-29 Spr˚ak Language ¤ Svenska/Swedish ¤ Engelska/English ¤ ⊠ Rapporttyp Report category ¤ Licentiatavhandling ¤ Examensarbete ¤ C-uppsats ¤ D-uppsats ¤ ¨Ovrig rapport ¤ ISBN ISRN

Serietitel och serienummer Title of series, numbering

ISSN

URL f¨or elektronisk version

Titel Title

Vibrationellt uppl¨ost spektrum i static exchange approximationen av L-kanten hos kisel i SiCl4

Vibrationally resolved silicon L-edge spectrum of SiCl4 in the static exchange

approximation F¨orfattare Author Johnny Jonsson Sammanfattning Abstract

The X-ray absorption spectrum of silicon in of SiCl4 has been calculated for the

LIII and LII edges. The resulting spectrum has been vibrationally resolved by

considering the symmetric stretch vibrational mode and the results has been com-pared to experiment [4]. One peak from the experiment was found to be missing in the calculated vibrationally resolved spectrum. The other calculated peaks could be matched to the corresponding experimental peaks although significant basis set effects are present. An investigation of one peak beyond the Franck–Condon principle shows it to be a good approximation in the case of the studied system.

Nyckelord Keywords X-ray spectroscopy,vibrational,spectrum ⊠ — LiTH-IFM-EX-08/1909 —

## Abstract

The X-ray absorption spectrum of silicon in of SiCl4 has been calculated for the

LIII and LII edges. The resulting spectrum has been vibrationally resolved by

considering the symmetric stretch vibrational mode and the results has been com-pared to experiment [4]. One peak from the experiment was found to be missing in the calculated vibrationally resolved spectrum. The other calculated peaks could be matched to the corresponding experimental peaks although significant basis set effects are present. An investigation of one peak beyond the Franck–Condon principle shows it to be a good approximation in the case of the studied system.

## Acknowledgements

I would like to thank my supervisor Ulf Ekstr¨om and my examiner Patrick Norman who presented me with this enriching yet challenging work. They never gave up on me.

I am also most grateful to my opponent Andreas Rasimus for his help and advice with this project and elsewhere. Lastly I would like to thank all my friends and family. Without them I would not have made it this far.

## Contents

1 Introduction 1

1.1 Background . . . 1

1.2 Thesis objectives and problem definition . . . 3

1.3 Thesis outline . . . 3

2 Electronic structure theory 5 2.1 Relativistic one-particle theories . . . 5

2.1.1 The Klein–Gordon equation . . . 5

2.1.2 The Dirac equation . . . 6

2.2 Molecular wave functions . . . 7

2.2.1 The molecular Hamiltonian . . . 7

2.2.2 Born–Oppenheimer approximation . . . 8

2.2.3 Orbitals . . . 9

2.2.4 Slater determinants . . . 9

2.2.5 The energy of a Slater determinant . . . 10

2.3 The Hartree–Fock approximation . . . 11

2.3.1 The Hartree–Fock equations . . . 11

2.3.2 Roothaan equations . . . 13

2.3.3 Koopmans theorem . . . 13

2.4 Configuration interaction . . . 13

2.4.1 Electron correlation . . . 13

2.4.2 Excited determinants . . . 14

2.4.3 The configuration interaction wave function . . . 14

2.5 The static exchange approximation . . . 15

2.6 Examples of relativistic effects . . . 16

3 Nuclear motions 19 3.1 Molecular vibrations . . . 19

3.1.1 General vibrational Hamiltonian . . . 19

3.1.2 Vibrations in SiCl4 . . . 20

3.2 Vibronic transitions . . . 21

4 Results and discussion 25 4.1 Computational details . . . 25

4.2 Electronic ground state equilibrium spectrum . . . 26 ix

4.3 Vibrational resolution . . . 27

5 Summary and conclusions 31

Bibliography 33

A Electronic potential curves of SiCl4 35

### Chapter 1

## Introduction

### 1.1

### Background

Spectroscopy is a useful tool when analyzing the structure and composition of known or unknown substances and materials. The applications of spectroscopy include chemical analysis of distant stars to identification of substances in for instance forensic investigations.

The basic concept of spectroscopy is that molecules only absorbs light (or other electromagnetic waves such as X-rays or UV) of certain frequencies (energies). When a photon is absorbed, an electron is excited to a state (orbital) with higher energy. The electron can then return to the ground state and release the energy as a photon. The specific photon energies absorbed and emitted constitute the system’s unique absorption and emission spectra.

There are different types of spectroscopies depending on what is measured. In absorption spectroscopy the intensities and energies at which photons are ab-sorbed are measured, in emission spectroscopy the energy of emitted photons is measured. Other types of spectroscopies include photo-electron spectroscopy and Auger electron spectroscopy. The first of the two is based on the photoelectric effect and the molecule being ionized by the incident radiation followed by mea-surements on the energy of the removed electron. In Auger electron spectroscopy a core electron is excited leaving a hole which is then filled by an electron from a higher energy level. The decrease in energy of the second electron is then released as Auger electrons which are measured upon.

In this thesis we consider X-ray absorption spectroscopy, XAS, in which a
sample is irradiated with X-ray photons and the amount absorbed is measured
along with the energy of the absorbed photons. The high energy of the X-rays
excites core electrons, i.e electrons in the inner shells, which have energies unique
to every element. But when an atom is bound to a molecule the energy levels are
shifted because of interactions with the other atoms in the molecule. The shift
is called chemical shift and is rather small (0-1 eV for Cl, see Fig. 1.1 compared
to the soft X-rays ∼ 102 _{eV to 10}4 _{eV) but sensitive to the environment, i.e}

the other atoms in the molecule. X-ray absorption spectroscopy is thus uniquely 1

sensitive to the local environment of the studied atom and hence provides a means of identifying the chemical surroundings of a probed atom.

Figure 1.1. Four-component Hartree–Fock calculation of the orbital energies of the 2p-shell in chlorine. All calculations were were carried out using the taug-cc-VDZ basis set.

In spectroscopy the ionization energies of individual electrons are known as edges. The edges are named according to the shell from which the ionized electron originates. This thesis deals with the X-ray spectrum below (0-10 eV) the LII

-and LIII-edges, i.e excitations from the 2p-shell (see Fig. 1.2) to states just below

the ionization continuum.

1.2 Thesis objectives and problem definition 3

### 1.2

### Thesis objectives and problem definition

In an article of Domke et al. [4] a vibrationally resolved experimental L-edge spectrum of SiCl4 is presented. This thesis main objective is to produce a

com-puted equivalent to this spectrum through electronic calculations using the static exchange method and vibrational broadening. The vibrational broadening will be conformed to the Franck–Condon approximation and only the breathing vibration of symmetric stretch of the bonds (see Fig. 1.3) will be considered.

Figure 1.3. The thetrahedron shaped SiCl4

### 1.3

### Thesis outline

In Chapters 2 and 3 there is a brief presentation of the underlying theory along with the methods used in the thesis project. In Chapter 4 the results of the calculations along with analysis are presented. Chapter 5 contains conclusions of this thesis project and future work and possible extensions.

### Chapter 2

## Electronic structure theory

For this chapter my sources has been Refs. [12], [8], [11], [6] and [5].

### 2.1

### Relativistic one-particle theories

The high speed of the inner electrons in atoms and molecules1_{makes the relativistic}

effects far too great to be neglected. The most dominant relativistic feature is the electron spin which through the spin-orbit coupling introduces a split in the energy levels of an atom. This split is of course very important when dealing with absorption spectra because a relativistic approach would accordingly give two peaks in the spectra where the non-relativistic calculations would show only one.

### 2.1.1

### The Klein–Gordon equation

In the relativistic realm the Schr¨odinger equation no longer suffice. We need to incorporate relativistic effects and the easiest way of doing this is simply to make the Hamiltonian comply with Einstein’s special relativity. Starting off with the relativistic dispersion relation for a free particle

E =pp2_{c}2_{+ m}2_{c}4 _{(2.1)}

and making the substitutions p → ˆp = −i~∇ and E → ˆE = i~∂/∂t we get i~∂Ψ

∂t = p

(−i~∇)2_{c}2_{+ m}2_{c}4_{Ψ} _{(2.2)}

If we square both sides to get rid of the square root this becomes
−_{c}12
∂2_{Ψ}
∂t2 = (∇
2_{+}m2c2
~2 )Ψ (2.3)
1

In molecules the inner electrons are mostly confined to single atoms and do not contribute much to the bonds of the molecule thus behaving almost as in the single atoms.

and we get what is known as the Klein–Gordon equation. This is a relativistic correct equation and, in its non-relativistic limit, consistent with the Schr¨odinger equation describing a free spinless particle. Nonetheless it is insufficient for the purposes of quantum chemistry which deals with the motions of electrons which are spin 1/2 particles. Therefore we need an equation which naturally takes the spin into consideration.

### 2.1.2

### The Dirac equation

We have seen above that a straight forward operator substitution in the relativistic dispersion relation leads to a relativistic generalization of the Schr¨odinger equation. But from a relativistic correct theory we would expect a symmetry in the treatment of time and space and not the different orders of the derivatives as in the Klein– Gordon and the Schr¨odinger equations. In order to obtain an equation which is of the first order in time and space variables one can once again start with the relativistic dispersion relation:

E =pp2_{c}2_{+ m}2_{c}4 _{(2.4)}

This time we demand that the right hand side can be written in first order of p
E =pp2_{c}2_{+ m}2_{c}4_{= α · pc + βmc}2 _{(2.5)}

where α and β are constants. Making the usual operator substitutions we arrive at the celebrated Dirac equation

i∂Ψ

∂t = (α · ˆp + βmc

2_{)Ψ} _{(2.6)}

Now, to proceed we can rather easily get expressions for the arbitrary α and β constants by squaring (2.5),

E2_{= (α · pc + βmc}2_{)(α · pc + βmc}2_{)} _{(2.7)}

Comparing with (2.4) three constrains on the α and β emerges:

α2x= α2y= α2z= β2= I (2.8)

©

α, βª= 0 (2.9)

©

αx, αyª=©αy, αzª=©αz, αxª= 0 (2.10)

The only way these equations are valid is if we let α and β be matrices. Also the anti-commutation relations requires α and β to be at least 4-by-4. These constraints do not fully determine α so β and there are different representations of the Dirac equation depending on the choice of these matrices. The standard representation is obtained when the α is constructed out of the anti-commuting Pauli spin matrices σx, σy, σz according to

αx= µ 0 σx σx 0 ¶ , αy= µ 0 σy σy 0 ¶ , αz= µ 0 σz σz 0 ¶ (2.11)

2.2 Molecular wave functions 7 and β is chosen as β = µ I 0 0 −I ¶ (2.12) where I is the 2-by-2 identity matrix.

A consequence of the Dirac equation being a 4x4 matrix equation is of course that its solutions will be vectors. This means that wave functions satisfying the Dirac equation will have four degrees of freedom

Ψ = ΨLα ΨSα ΨLβ ΨSβ (2.13)

two of which corresponds to the two spin states, denoted α and β. Unlike the Schr¨odinger equation the Dirac equation thus naturally includes the electron spin. The other two are called small and large components, where the small components sometimes are referred to as positronic solutions.

### 2.2

### Molecular wave functions

### 2.2.1

### The molecular Hamiltonian

The main problem of quantum chemistry is to solve the equation ˆHΨ = EeΨ,

i.e the Schr¨odinger equation in the non-relativistic case and the Dirac equation in the relativistic case, for a specific molecule and find the wave functions Ψ that are eigenfunctions to the Hamiltonian and their corresponding energies; the eigenvalues Ee.

In atomic units the non-relativistic Hamiltonian for a general molecule is

ˆ
H = −1_{2}
N
X
i=1
∇2i −
1
2MA
M
X
A=1
∇2A−
N
X
i=1
M
X
A=1
ZA
riA +
N
X
i=1
N
X
j>i
1
rij +
M
X
A=1
M
X
B>A
ZAZB
RAB
(2.14)
where N is the number of electrons, M is the number of nuclei, ZAis the atomic

number of nucleus A, rijis the distance between electron i and j, riAis the distance

between electron i and nucleus A, RAB is the distance between nucleus A and

B. The first term corresponds to the kinetic energy of the electrons, the second term to the kinetic energy of the nuclei, the third term is the electromagnetic attraction between the electrons and nuclei, the fourth term corresponds to the electron-electron repulsion and the fifth term the nuclear-nuclear repulsion. A more compact self-explanatory way of writing Eq. (2.14) is

ˆ

H = ˆTe+ ˆTn+ ˆVne+ ˆVee+ ˆVnn (2.15)

Now it becomes apparent that we can separate the Hamiltonian into two dif-ferent parts; a sum of one-electron operators and a sum of two-electron operators.

With this partitioning we can write the Hamiltonian of Eq. (2.14) as ˆ H = N X i=1 ˆ h(i) + N X i=1 N X j>i ˆ g(i, j) (2.16)

This form of the Hamiltonian is also valid in the relativistic case where the one-electron Hamiltonian ˆh(i) is replaced by the Dirac-Hamiltonian which is further discussed in Section 2.6. A similar partitioning will be used in Section 2.2.5 and the following sections to ensure the argument holds also for the relativistic case.

### 2.2.2

### Born–Oppenheimer approximation

Due to the great difference in mass (the proton mass is almost 2000 times that of the electron) the electrons in a molecule moves a lot faster than the nuclei. From the electrons’ point of view the nuclei are very heavy and almost at rest at fixed positions. Thus the electrons can be said to move in the static potential of the nuclei.

For the nuclei, on the other hand, the electrons move so fast that they will only appear to be in the average positions. The electronic energy constitutes a potential energy for the nuclei which can thus be said to move in the average field of the electrons. So the many-body Hamiltonian in Eq. (2.14) can be modified into two parts.

The electronic part of the Hamiltonian (2.15) becomes ˆ

He= ˆTe+ ˆVne( ¯R) + ˆVee (2.17)

and is called the electronic Hamiltonian. This yields the electronic Schr¨odinger equation:

ˆ

He( ¯R)ψe(¯ri; ¯R) = Ee( ¯R)Ψe(¯ri; ¯R) (2.18)

Since the electronic Hamiltonian depends on the nuclear coordinates so will the energy eigenvalues Eeand eigenfunctions Ψe.

In addition to the nuclear-nuclear repulsion, the nuclear energy is also affected by the average electron energy Ee( ¯R) = hΨe| ˆHe|Ψei:

ˆ

Hn= ˆTn+ ˆVnn( ¯R) + Ee( ¯R) (2.19)

So evaluating Eq. (2.18) for different arrangements of the nuclei yields a potential surface ˆVn( ¯R) = ˆVnn+ Ee in which the Schr¨odinger equation of the nuclei

ˆ

HnΨn= EtotΨn (2.20)

can be solved. Since the molecule vibrating or rotating is mainly a movement of the nuclei, solving this equation makes it possible to calculate the vibrational and rotational movement of the molecule. The vibrational dynamics of the molecule will be further explored in Section 3.1

Finally the total wave function can be formed as

2.2 Molecular wave functions 9

### 2.2.3

### Orbitals

An orbital is a wave function for a single electron. If the electron is bound to a molecule, its wave function is referred to as a molecular orbital that takes the form

χ(¯r) =
µ
ψα_{(¯}_{r)}
ψβ_{(¯}_{r)}
¶
(2.22)
where each component represents a spin direction. This way we get a wave function
which in the general case lets the spin depend on the spatial distribution ψ of the
electron so that the spin of the electron varies in space. Opposite of this we can
put the spatial components equal to get a description of an electron which has the
same spin everywhere in space. This is the case in the non-relativistic realm where
we can choose χ(¯r) as
χ(¯r) =
µ
ψα_{(¯}_{r)}
0
¶
(2.23)
or
χ(¯r) =
µ
0
ψβ_{(¯}_{r)}
¶
(2.24)
The spin functions are also assumed to be orthogonal and since the spatial orbitals
are orthonormal so are the spin orbitals formed from them.

### 2.2.4

### Slater determinants

In order to obtain the total wave function for all electrons in a molecule we need to construct a many-electron wave function Ψ(x1, . . . , xN) out of the one-electron

orbitals χi(x). There are many ways to combine the orbitals into a many-electron

wave function. One way is to simply take the product of all orbitals to form a Hartree-product:

Ψ(¯r1, . . . , ¯rN) = χ1(¯r1)χ2(¯r2) . . . χN(¯rN) (2.25)

But in order to fulfill the Pauli exclusion principle the wave function must be antisymmetric according to

Ψ(¯r1, . . . , ¯ra, . . . , ¯rb. . . , ¯rN) = −Ψ(¯r1, . . . , ¯rb, . . . , ¯ra. . . , ¯rN) (2.26)

This is clearly not fulfilled by the Hartree-product.

We need this antisymmetry to only leave the wave function unchanged2_{when}

switching two electrons in the system (it should not matter which electron is in which orbital since the electrons are indistinguishable). Another consequence of the antisymmetry is that two electrons cannot occupy the same orbital since a = b in the equation above forces the wave function to be identically zero.

A form of antisymmetric many-electron wave function that fulfills the Pauli principle is the Slater determinant. The Slater determinant is a linear combina-tion of all possible permutacombina-tions of the orbitals in Hartree-products. A Slater determinant of two particles and orbitals is shown in Eq. (2.27).

2

Ψ(¯r1, ¯r2) =√1

2(χ1(¯r1)χ2(¯r2) − χ1(¯r2)χ2(¯r1)) (2.27) In the general case of N electrons and N orbitals the Slater determinant can be written as Ψ(¯r1, . . . , ¯rN) = 1 √ N ! χ1(¯r1) χ2(¯r1) · · · χN(¯r1) χ1(¯r2) χ2(¯r2) · · · χN(¯r2) .. . ... · · · ... χ1(¯rN) χ2(¯rN) · · · χN(¯rN) (2.28)

A common way of writing the Slater determinant is to put the Hartree product from which it was generated in a ket:

Ψ(¯r1, . . . , ¯rN) = |χ1, χ2, . . . , χNi (2.29)

### 2.2.5

### The energy of a Slater determinant

Since the Slater determinant is a widely used type of wave function the calculation of its energy is central to many of the methods of quantum chemistry.

What we want to calculate is the expectation value of the electronic Hamilto-nian of Eq. (2.17)

E = hΨ| ˆHe|Ψi (2.30)

If we write the electronic Hamiltonian of Eq. (2.17) in full detail ˆ He= − 1 2 N X i=1 ∇2i − N X i=1 M X A=1 ZA riA + N X i=1 N X j>i 1 rij (2.31)

it can easily be split into two parts. The first part
−1_{2}
N
X
i=1
∇2i −
N
X
i=1
M
X
A=1
ZA
riA =
N
X
i=1
¡
−1_{2}∇2i −
M
X
A=1
ZA
riA
¢
=
N
X
i=1
ˆ
h(i) (2.32)

which is the kinetic energy of single electrons and their Coulomb attraction to the nuclei, is a sum of one-electron operators ˆh(i) acting only on electron i. The second part of Eq. (2.17) is the sum of the Coulomb repulsion between every pair of electrons and hence is a sum of two-electron operators, ˆg(i, j).

With this splitting we can write the electronic Hamiltonian as ˆ He= N X i=1 ˆ h(i) + N X i=1 N X j>i ˆ g(i, j) (2.33)

Inserting this into Eq. (2.35) yields hΨ| ˆHe|Ψi = N X i=1 hΨ|ˆh(i)|Ψi + N X i=1 N X j>i hΨ|ˆg(i, j)|Ψi (2.34)

2.3 The Hartree–Fock approximation 11

At this point we have left to expand the Slater determinants in full detail and transform Eq. (2.34) into an equation of spin orbitals. For the one-electron part this is straight forward. The two-electron part, on the other hand, requires more consideration of the determinantal properties of the Slater determinant to be manageable. The details will be left out and here only the result will be presented

hΨ| ˆHe|Ψi = N X i=1 hχi|ˆh(i)|χii + N X i=1 N X j>i ³ hχiχj|ˆg(1, 2)|χiχji − hχiχj|ˆg1, 2|χjχii ´ (2.35) We can write this more compact by writing the first term as a matrix element

hii= hχi|ˆh|χii (2.36)

and introduce the Coulomb and exchange integrals Jij= hχiχi|ˆg(1, 2)|χjχji = Z |χi(¯r1)|2ˆg(1, 2)|χj(¯r2)|2d¯r1d¯r2 (2.37) and Kij= hχiχj|ˆg(1, 2)|χjχii = Z χ∗i(¯r1)χj(¯r1)ˆg(1, 2)χ∗j(¯r2)χi(¯r2)d¯r1d¯r2 (2.38)

The Coulomb integral can be interpreted as the classical electrostatic repulsion between two charge distributions |χi|2 and |χj|2 while the exchange integral

rep-resents spin correlation effects. Then Eq. (2.35) becomes hΨ| ˆHe|Ψi = N X i=1 hii+ N X i=1 N X j>i (Jij− Kij) (2.39)

For a full derivation of Eq. (2.39) see Szabo & Ostlund [12] or Jensen [8].

### 2.3

### The Hartree–Fock approximation

The simplest way of approximating an N -electron wave function is to use a single Slater determinant. This is due to the fact that since an arbitrary N -electron wave function can be expanded in a complete basis set a single determinant is very limited basis expansion. So for a more complete and general description one would rather use a series of determinants of an infinite basis set made out of orbitals (See Section 2.4 Configuration Interaction). But in the Hartree–Fock approximation we will settle for a single determinant.

### 2.3.1

### The Hartree–Fock equations

The main goal of the Hartree–Fock method is to find the single determinant which best approximates the ground state wave function. We will consider as the best

approximation the determinant which minimizes the energy and thus corresponds to the lowest possible ground state energy. Since the energy depends on the determinant Ψ,

E[Ψ] = hΨ| ˆHe|Ψi (2.40)

it also depends on the spin orbitals the determinant is made out of.

This way we can simply vary the orbitals of Eq. (2.39) under the constraint that they are orthogonal, until a minimum is reached yielding the sought after set of spin orbitals.

To do this formally we rewrite the Coulomb and exchange integrals of Eq. (2.39) in terms of Coulomb and exchange operators:

ˆ Jj|χii = h Z |χj(¯r2)|2g(1, 2)d¯ˆ r2 i χi(¯r1) = hχj|ˆg(1, 2)|χji|χii (2.41) and ˆ Kj|χii = " Z χ∗j(¯r2)ˆg(1, 2)χi(¯r2)d¯r2 # χi(¯r1) = hχj|ˆg(1, 2)|χii|χji (2.42) yielding EHF= N X i=1 hχi|ˆh|χii + N X i=1 N X j>i ³ hχi| ˆJj|χii − hχi| ˆKj|χii ´ (2.43) Using Lagrange multipliers and the variational principle one can show that in order to minimize Eq. (2.43) the orbitals must fulfill the eigenvalue equation

" ˆ h + N X j6=i ( ˆJj− ˆKj) # χi(¯r1) = εiχi(¯r1) (2.44)

By noting that ˆJj|χji = ˆKj|χji the right-hand sum can be written to include the

term where j = i: _{"}
ˆ
h +
N
X
j
( ˆJj− ˆKj)
#
χi(¯r1) = εiχi(¯r1) (2.45)

These are the Hartree–Fock equations. Another way of writing them is to introduce the Fock-operator

ˆ

f = ˆh +X

j

( ˆJj− ˆKj) (2.46)

The first term of the Fock operator, ˆh, is the one particle Hamiltonian. The second term is the Hartree–Fock potential an average electric potential.

With the Fock operator the Hartree–Fock equations becomes more compact ˆ

f χi(¯r) = εiχi(¯r) (2.47)

Note that the Fock operator depends on the orbitals it acts on making Eq. (2.47) non-linear. But they can nevertheless be solved using iterative methods. It is also worth noting that the sum of the orbital energies of Eq. (2.47) is not the total energy EHF of Eq. (2.43)

2.4 Configuration interaction 13

### 2.3.2

### Roothaan equations

In order to solve the Hartree–Fock equations numerically (Eq. (2.47)) for other systems than single atoms we need further approximations. One way is to expand the molecular spin orbitals of the Slater determinant in a limited basis set of M basis functions called atomic orbitals thus forming a linear combination of atomic orbitals, LCAO. χi(¯r1) = M X j=1 Cijφj(¯r1) (2.48)

When inserting the expansion into Eq. (2.47) we get ˆ f M X j=1 Cijφj(¯r1) = εi M X j=1 Cijφj(¯r1) (2.49)

which is more or less an equation for the LCAO expansion coefficients. Multiplying by φℓ(1)∗ and integrating over ¯r1 we get

M X j=1 Cij Z φ∗ℓ(¯r1) ˆf φj(¯r1)d¯r1= εi M X j=1 Cij Z φ∗ℓ(¯r1)φj(¯r1)d¯r1 (2.50)

thus transforming the HF-equations into a single matrix equation

FC= SCε (2.51)

where F is the Fock matrix with elements Fℓj =

R φ∗

ℓ(¯r1) ˆf φj(¯r1)d¯r1, C is the

matrix of Cij, S is the overlap matrix with elements Sℓj =

R φ∗

ℓ(¯r1)φj(¯r1)d¯r1and

εis a diagonal matrix with ε_{i} on the diagonal.

### 2.3.3

### Koopmans theorem

When the Roothan equations [Eq. (2.51)] are solved we end up with a set of M spin orbitals of which the N lowest in energy are combined as a Slater determinant forming the HF ground state. These N spin orbitals are called occupied whereas the M − N remaining orbitals are called virtual orbitals. The orbitals are not the only result of solving the Hartree–Fock since there are also a set of eigenvalues εi

corresponding to each orbitals. These eigenvalues can be interpreted as ionization energies for the populated orbitals and electron affinities for the virtual.

### 2.4

### Configuration interaction

### 2.4.1

### Electron correlation

According to the variational principle the Hartree–Fock energy (EHF) is greater

than the exact (E). The difference in energy,

is called the correlation energy. It is so called because the Hartree–Fock wave function is uncorrelated i.e it allows for electrons with different spins to overlap in space when in reality the Coulomb repulsion would keep the electrons further apart. The movement of a single electron in the system is thus in reality correlated to the other electrons’ movements.

### 2.4.2

### Excited determinants

As mentioned in Section 2.3 the exact N -electron wave function Φ can be ex-panded in a series of N -electron Slater determinants. A straight forward choice of expansion determinants is all the possible determinants formed from the set of M orbitals we get when solving the Roothaan equations. These determi-nants can be constructed by taking the Hartree–Fock ground state determinant Ψ0 = |χ1, χ2, . . . , χa, χb, . . . , χNi replacing a spin orbital χa with a virtual spin

orbital χr. This corresponds to exciting the electron occupying χa to χr.

Deter-minants formed this way are called excited deterDeter-minants and are denoted |Ψr

ai = |χ1, χ2, . . . , χr, χb, . . . , χNi (2.53)

where we have used the HF-ground state as reference state, that is we have taken the determinant Ψ0 = |χ1, χ2, . . . , χa, χb, . . . , χNi and replaced the spin orbital

χa with the virtual spin orbital χr thus exciting the electron in χa to χr. This is

called a singly excited determinant.

We can of course continue to replace one occupied orbital more with virtual orbital to get a doubly excited determinant.

|Ψrs

abi = |χ1, χ2, . . . , χr, χs, . . . , χNi (2.54)

Replacing three occupied orbitals with virtual we will get triply excited
determi-nants and so on. With this procedure a total of¡M_{N}¢N -electron Slater determinants
can be formed from the M spin orbitals obtained with the Roothaan equations.

### 2.4.3

### The configuration interaction wave function

Using these determinants as a basis for the total electronic wave function we can form the Configuration Interaction or CI wave function Φ as

|Φi = c0|Ψ0i+ X a,r cra|Ψrai+ X a<b r<s crsab|Ψrsabi+ X a<b<c r<s<t crstabc|Ψrstabci+ X a<b<c<d r<s<t<u

crstuabcd|Ψrstuabcdi+. . .

(2.55) The optimization procedure of the CI wave function is essentially the same as for the Hartree–Fock except now we diagonalize the Hamiltonian in the basis of excited determinants.

Of course one can truncate the expansion in Eq. (2.55) to include terms up to singly excited determinants or up to doubly excited determinants or. The wave functions constructed this way are called CI-singles (CIS), CI-doubles (CID), CI-singles-doubles (CISD) and so forth depending on how many excitations are included in the expansion of Eq. (2.55).

2.5 The static exchange approximation 15

### 2.5

### The static exchange approximation

Figure 2.1. Theoretical L-edge spectrum for HCl in the static exchange approximation. Calculations made with the taug-cc-CVTZ basis set and the experimental equilibrium geometry [10]. The inset shows a spectrum calculated with the HF-ground state as reference.

When a photon is absorbed by a molecule in the state |Ψ0i it ends up in an

excited state |ΨKi (the energy difference of the states being the energy of the

photon). The probability of the molecule absorbing the photon is proportional to the square modulus of the transition moment

Mα0→K= hΨ0|ˆµα|ΨKi (2.56)

where µα is the dipole moment operator of the molecule along the α-axis.

Calcu-lating the transition moments for many different ΨK therefore, in principle, gives

us the Ψ0 absorption spectrum.

The transition moment can easily be evaluated using the HF-optimized ground
state and a CI-singles wave function,P_{r,a}cr

aΨra, as excited final state.

Now, when a core electron, as any of the six 2p-electrons, is excited the valence electrons will be screened by one less electron and thus experience an increased attraction to the nuclei. This makes the electron cloud contract spatially and thus lowers the orbital energies. To incorporate the relaxation energy, we can construct the CIS by forming the excited determinants from an optimized ionized state Ψion

in which a core electron has been completely removed from the system. A com-parison between calculations on HCl with use of either the Hartree–Fock ground state or the ionic state can be seen in Fig. 2.1 where the relaxation energy appears roughly as the displacement to higher energy of the upper spectrum compared to

the lower. Changing the reference in the excited state of Eq. (2.56), we get Mα0→K= hΨ0|ˆµα|ΨionCISi (2.57)

which is a fairly easy way of calculating the transition dipole moments and includ-ing the relaxation energy.

Another concern is how to include excitations from the whole 2p-shell when constructing the excited state. We are dealing with excitations from any of the six electrons of the 2p-shell, so the straight forward way of calculating the spec-trum would be to form and optimize two separate CIS wave functions, |Ψ2p1/2i

and |Ψ2p3/2i, each involving only excitations from either the 2p3/2 or 2p1/2 shell.

After calculating the dipole moments with the two states, one simply adds the respective spectrum to obtain the total spectrum of 2p-shell absorption. However, this treatment does not take the channel interaction into account, i.e the overlap of excitations from different orbitals occurring when the orbitals are close in energy. Since this is the case in 2p-shell we need to consider another method.

A more rigorous way of treating the many excitations, and the method that is used in STEX, is to optimize the excited state as a single CIS wave function

|ΨionCISi = |Ψ2p1/2i + |Ψ2p3/2i = X a,r cra|Ψrai + X b,s csb|Ψsbi (2.58)

incorporating holes in either of the six 2p-orbitals instead of treating 2p3/2 and

2p1/2 shells separately. This method allows for the excitations to originate in both

shells and thus be a mixture of excitations from both shells. The hole contribution to each excited state is given by the constants cs

b and csa in Eq. (2.58).

Instead of the transition dipole moments, spectra are often presented in terms of oscillator strengths which are dimensionless quantities proportional to the tran-sition moment. The oscillator strength is defined as

fI→K= 2 3(EK− EI) X α∈{x,y,z} ¯ ¯MI→K α ¯ ¯2 (2.59)

and will be used through out the thesis when a spectrum is presented.

As an example, STEX-calculations on HCl are presented in Fig. 2.1 and de-tails are found in Table 2.1. The oscillator strengths obtained through the static exchange calculations are convoluted using a Lorentzian band profile to produce the naturally broadened peaks. Notable features of Fig. 2.1 include the spin-orbit splits of certain peaks. The

### 2.6

### Examples of relativistic effects

One of the most important relativistic features in L-edge spectroscopy is the spin-orbit coupling. This effect arises from the electron spin coupling with the magnetic field generated from the nucleus moving around the electron. Whether the spin is aligned with the magnetic field from the nucleus or not will lower or raise the electron’s energy. In a spectrum this enters as peaks appearing in pairs split by

2.6 Examples of relativistic effects 17

Table 2.1. Excitation energies (eV) and oscillator strengths (×10−4_{) for the dominating}

transitions below the chlorine LII-edge calculated in the STEX approximation with the

aug-cc-CVTZ basis set. The fourth and eighth column contains the hole contribution in percent.

State ∆E f p3/2 State ∆E f p1/2 ∆ESO

A 201.89 36.12 99.7 A’ 203.51 22.01 98.6 1.62 B 202.06 8.11 99.1 B’ 203.64 2.66 97.5 1.58 C 204.25 13.25 99.8 C’ 205.92 6.01 99.6 1.67 D 204.33 60.31 98.6 D’ 205.97 32.82 99.3 1.64 E 205.05 3.35 100 E’ 206.75 4.05 39.7 1.70 F 205.06 5.94 99.9 F’ 206.75 7.35 74.2 1.69

the energy difference of the spin-orbit coupling. In Fig. 2.2 the relativistic shifts
in the orbital energy levels in 2p-shell of Cl− _{are shown and the magnitude of the}

split is about 2 eV in the 2p-shell. The spin-orbit split can be compared to the large energy splits of almost 70 eV and 180 eV between the 2p and the 2s- and 3s-shells respectively. The STEX approach described above is Fig. 2.2 also shows a rather large change in energy of the s-orbitals due to relativistic contraction.

Figure 2.2. Non-relativistic (left) and relativistic (right) calculations of the electronic energy levels in Cl−

### Chapter 3

## Nuclear motions

In this chapter my main sources has been Ref. [2] and Ref. [7].

### 3.1

### Molecular vibrations

A molecule vibrating, as seen in the Born–Oppenheimer approximation, is mainly if not only the movement of the nuclei. This means, as mentioned earlier in Section 2.2.2, that we can construct a potential energy surface (PES) for the nuclei to move on by calculating the electronic energy E at different nuclear geometries.

For a diatomic molecule such as HCl the problem is of course one dimensional;
the stretch of the bond is the only degree of freedom for the nuclei1_{.}

Many complex vibration modes of larger molecules can often be reduced to a system of linear coupled one-dimensional oscillators which greatly simplifies the treatment of such systems. But there are also cases of one-dimensional vibration modes in large molecules. In the case of the SiCl4 molecule, we will consider the

symmetric stretch of all bonds with the bond angles held constant. Although the movement is carried out by four chlorine atoms, because of symmetry this is a one-dimensional breathing vibration.

When the potential surface has been calculated, the Schr¨odinger equation is solved in the potential and vibrational wave functions are obtained.

### 3.1.1

### General vibrational Hamiltonian

In a general form the Hamiltonian for the nuclei can be written ˆ H = n X i=1 ˆ p2 2mi + V (ˆq1, ˆq2, . . . , ˆqn) (3.1)

or rewritten on matrix form: ˆ H = 1

2Q˙

†_{M ˙}_{Q + V (Q)} _{(3.2)}

1

Of course there is a rotational degree of freedom but the energies connected to the molecule rotating is so much smaller than for the vibrations that we can neglect them.

where Q is a vector of the Cartesian coordinates qi, M is the mass matrix; the

diagonal matrix with mi on the diagonal and we have used the relation pi= miqi.

Here we can choose a coordinate system suitable to the molecule using a unitary
coordinate transformation. But by doing so the kinetic energy term often becomes
unnecessary complex involving cross terms. This can however be avoided by the
introduction of mass weighted coordinates eQ = M−1/2_{Q. Making this substitution}

of variables the Hamiltonian (3.2) transforms into H = 1

2Q˙e

†

˙e

Q + V (M1/2Q)e (3.3)

Now we can take any unitary transformation of the mass weighted coordinates as new coordinates

q = U eQ (3.4)

without complicating the kinetic energy term: H = 1

2˙q

†_{U}†_{U ˙q + V (M}1/2_{U}†_{q) =} 1

2˙q

†_{˙q + V (M}1/2_{U}†_{q)} _{(3.5)}

where in the last step we have used the property U−1 _{= U}† _{of unitary }

transfor-mations.

### 3.1.2

### Vibrations in SiCl

4Considering the SiCl4molecule we can introduce a coordinate system centered at

the silicon atom. The coordinate vector describing the geometry of the molecule can then be written

Q = (x1 y1 z1 x2 y2 . . .)T (3.6)

where we have let the index 1 designate the silicon atom and index 2-5 the four chlorine atoms. The symmetric stretch of SiCl4is a one-dimensional movement of

the nuclei and can be described as ¯

Q(s) = Q0+ s∆Q (3.7)

where s is the coordinate of the stretch, i.e the displacement along the unit vector ∆Q from the equilibrium geometry Q0. The vector ∆Q for the tetrahedron shaped

SiCl4 is

∆Q =¡ 0 0 0 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 ¢T (3.8)

The effective mass of the vibration can be computed if we consider the classical kinetic energy of the nuclei

T =1 2

X

i

miQ˙¯2i (3.9)

where qiis according to Eq. (3.6) and miis the corresponding mass. Using Eq. (3.7)

3.2 Vibronic transitions 21
T =1
2
15
X
i=1
mi˙s2∆qi2=
1
2˙s
2_{· 12 · m}
Cl (3.10)

Now s is of course the displacement of the chlorine atoms in the Cartesian co-ordinate system. Since we are dealing with symmetric stretch, it would be more intuitive if the value of s represented the bond length during the vibration. This is easily done by scaling

sbond= s

√

3 (3.11)

This turns Eq. (3.10) into T =1 2˙s 2 · 12mCl=1 2˙s 2 bond· 4mCl (3.12)

Which is exactly what we would expect the kinetic energy of four moving chlorine atoms to be. Now we can identify the effective mass of the vibration as

M = 4mCl (3.13)

Finally if we insert Eq. (3.7) into the Hamiltonian of Eq. (3.2), we get
H = 1
2( ˙s∆Q)
†_{M ˙s∆Q + V (Q}
0+ s∆Q) = 1
2˙s
2_{∆Q}†_{M ∆Q + V (Q}
0+ s∆Q) (3.14)

where the scalar ∆Q†_{M ∆Q}† _{can be identified as the mass in Eq. (3.13) and a}

change to mass-weighted coordinates ˆs can be made: H = 1

2˙ˆs + V (Q0/2 √

mCl+ ˆs∆Q) (3.15)

This way we can easily solve the Schr¨odinger equation for the nuclear movement in different potentials.

### 3.2

### Vibronic transitions

In a molecule, the electrons move, as established in Sec. 2.2.2, much faster than the nuclei. An electronic transition is therefore almost instantaneous from the nuclear point of view and the electronic transition thus occurs for a fixed nuclear geometry. The change in electron density accompanying the transition alters the potential surface experienced by the nuclei which readjust their positions to a new equilibrium. This approximation with instantaneous electronic transition and slowly adjusting nuclei is called the Franck–Condon principle and it is illustrated in Fig. 3.1.

We can calculate the absorption spectrum of vibronic transitions from a given
state |K, ki, i.e the molecule is in vibrational state k of the electronic state K (see
Fig. 3.1). Then the probability that it is found in the state |K′_{, k}′_{i, after an }

exci-tation has occurred, is proportional to the square modulus of the transition dipole moment (Eq. (2.56). To simplify the calculations of the transition probabilities we

Figure 3.1. Illustration of a vibronic transition between states |K, ki and |K′_{, k}′_{i }

ac-cording to the Franck–Condon principle.

first notice that the dipole operator can be split up in an electronic and a nuclear part, ˆ µ = −eX i ¯ ri+ e X A ZAR¯A= ˆµe+ ˆµn (3.16)

and since the state also is separable in a nuclear and electronic part (Born– Oppenheimer approximation, cf. Eq. (2.21)),

|K, ki = |Ki|ki (3.17)

where the electronic state |Ki depends parametrically on the nuclear coordinates and hence on |ki the nuclear state, we can expand the transition moment as

MK,k→K′,k′ = hK, k|ˆµe+ ˆµn|K′, k′i

= hk|hK|ˆµe_{|K}′_{i|k}′_{i + hK|hk|ˆ}_{µ}n_{|k}′_{i|K}′_{i} _{(3.18)}

The electronic dipole moment, on the other hand, does depend on the nuclear
co-ordinates. This is because the electronic state, in the Born–Oppenheimer
approx-imation, |Ki depends parametrically on the nuclear coordinates (see Eq. (2.21)).
The electronic dipole moment µe_{( ¯}_{R) = hK|ˆ}_{µ}e_{|K}′_{i in Eq. (3.18) therefore is a}

function of the nuclear coordinates ¯R. But if we make use of the Franck–Condon
principle and consider the electronic dipole moment for a fixed geometry µe_{( ¯}_{R}

0),

we can lift it outside the integration of the nuclear coordinates and form MK,k→K′,k′ = hK|ˆµe|K′ihk|k′i + hK|K′i

| {z }

=0

hk|ˆµn|k′i

= µe( ¯R0)hk|k′i (3.19)

where we in the last term has used that the nuclear dipole moment does not depend on the electronic coordinates and that the electronic states are orthogonal. The

3.2 Vibronic transitions 23

transition moment, and ultimately the transition probability, is thus proportional
to the square modulus of the overlap hk|k′_{i of the vibrational wave functions so}

the greater the overlap the more likely the transition is to occur. This is called the Franck–Condon principle and the square modulus of overlap integrals

S(k, k′_{) = |hk|k}′_{i|}2 _{(3.20)}

are called Franck–Condon factors.

Another way of viewing this is to say that when an electronic transition occurs between two states (see Fig. 3.1), the vibrational wave function of the first state is placed in the potential surface of the other state. In the first state the wave function is an eigenstate to the vibrational Hamiltonian of that state whereas in the other state it generally is not. It is on the other hand possible to describe the transferred wave function as a superposition of the eigenfunctions of the Hamilto-nian of the excited electronic state’s potential surface. The overlaps then enters as the expansion coefficients of the wave packet. An electronic excitation thus in general puts the molecule in a superposition of vibrational states, i.e the electronic transition is vibrationally broadened.

In Fig. 3.2 the Franck–Condon overlaps h0|ki between the ground state and the 50 lowest vibrational energy levels of each electronic excited state of SiCl4 is

presented. The most notable feature of Fig. 3.2 is that B, C, and D consist almost entirely of transitions from vibrational ground state to ground state. This has an obvious explanation in that the corresponding potential curves (Fig. A.1) each have their equilibrium very close to the ground states equilibrium. The other ones are more spread out and thus consist of several vibrational transitions.

Figure 3.2. The Franck–Condon factors corresponding to transitions from the ground state to the 50 lowest vibrational energy levels of the potential curves of the symmetric stretch mode of SiCl4in Fig. A.1.

### Chapter 4

## Results and discussion

### 4.1

### Computational details

The ground state potential curve for the symmetric stretch of SiCl4 has been

calculated with Gaussian03 [1] using an equilibrium bond length obtained from a DFT geometry optimization with the B3LYP [3] functional and the aug-cc-pVDZ [13] basis set. All relativistic STEX calculations were carried out in a locally modified version of the Dirac program [9]. These calculations also utilize the geometry optimized in Gaussian03 for the equilibrium geometry. For the chlorine atoms the basis aug-cc-pVDZ is used in the Dirac calculations. The silicon is described using an uncontracted basis set of size [18s12p5d6f ].

The sequence of oscillator strengths obtained from the STEX calculations are convoluted using a normalized Lorentzian according to

L(x, γ) = 1
π
"
γ
x2_{+ γ}2
#
(4.1)

where γ is a parameter determining the half-width at half-maximum (HWHM). Each oscillator strength is thus widened to an area equal to its height. In Fig. 4.1 a γ of 0.075 eV was used to convolute the spectrum. In Fig. 4.3, the five peaks (1,2,3,1’ and 3’) were convoluted with a wider Lorentzian (γ=0.075 eV) than the higher energy peaks (γ=0.025 eV) to correct for the shorter lifetime of the excited vibrational states with low energy. The orbitals plotted in Fig. 4.2 were calculated in the equivalent core approximation, i.e the orbitals were calculated for a molecule where we have switched the silicon atom to a phosphorus ion creating the molecule PCl+4. The idea behind this approximation is that the outermost electrons are

delocalized and experience the core hole mainly as an increase in the nuclear charge. So the virtual orbitals calculated with a (Z + 1) charged nucleus would resemble the diffuse orbitals in a core-excited state. The equivalent-core orbitals were calculated with Hartree–Fock in Gaussian03 using the basis set aug-cc-pVTZ.

### 4.2

### Electronic ground state equilibrium spectrum

Figure 4.1. Theoretical silicon L-edge spectrum for SiCl4 in the static exchange

ap-proximation. For details regarding basis set, geometry and convolution, see Section 4.1.

Table 4.1. Excitation energies (eV) and oscillator strengths (×10−4_{) for the dominant}

transitions below the silicon L-edge of SiCl4 in the STEX approximation. The fourth

and eighth column contains the hole contribution in percent. For computational details see Section 4.1.

State ∆E f p3/2 State ∆E f p1/2 ∆ESO

A 106.10 69.33 91.8 A’ 106.72 113.5 91.6 0.62 B 107.73 34.43 97.3 B’ 108.34 0.0 68 0.61 C 107.89 39.80 99.7 C’ 108.43 53.04 96.7 0.54 D 108.35 5.44 99.9 D’ 108.99 2.02 100 0.64 E 108.91 1.55 99.9 E’ 109.56 1.33 100 0.65 F 109.43 1.75 100 F’ 110.07 1.07 100 0.64 G 109.50 0.05 100 H 109.77 0.25 100 I 110.26 2.22 100 110.90 1.19 100 0.64 J 110.35 0.03 100 110.84 5.00 99.8 111.49 4.50 100 0.65 110.96 2.59 99.7 111.59 0.05 100 110.97 13.98 99.8 111.62 8.58 100 0.65

The L-edge spectrum of Si in SiCl4 has been calculated and is presented in

Fig. 4.1. The 22 most dominant transitions are presented in detail in Table 4.1. In Fig. 4.1, the notation A(A′), . . . , F (F′) for p3/2(p1/2) transitions is used. The

spin-orbit splittings are marked along with the corresponding energy differences. The ionization thresholds were calculated from the averaged ∆SCF value, and assuming

4.3 Vibrational resolution 27

by a 0.65 eV spin-orbit split, to be 110.70 eV and 111.35 eV for 2p3/2 and 2p1/2,

respectively. In Fig. 4.1 the energy of (B′), the p1/2transitions to the final state of

(B), is marked although its oscillator strength is close to zero. Also worth noting is
the relatively low energy difference between the transitions (C) and (C′) is 0.54 eV
compared to the spin-orbit split of ≈ 0.65 eV of the other transitions. In Table 4.1
the transitions to the states A,B,D,F,G and H corresponds to electronic transitions
to single orbitals whereas the other final states are mixtures of several orbitals.
To visualize the orbitals, equivalent core1 _{calculations were performed and the}

resulting orbitals corresponding to the A,B and D final states are presented in
Fig. 4.2. Only the three orbitals lowest in energy are presented since the calculation
becomes unreliable at higher energies due to the incompleteness of the basis set.
In the plot the σ∗ _{character of the A orbital is visible.}

(a) Virtual orbital corre-sponding to transition A in Fig. 4.1 (b) Virtual orbital corresponding to transition B in Fig. 4.1

(c) Virtual orbital correspond-ing to transition D in Fig. 4.1

Figure 4.2. Virtual molecular orbitals of SiCl4calculated in the equivalent core

approx-imation. The plotted orbitals corresponds to the electronic transitions involving single final states. Surfaces are plotted with contour values of 0.1 (a and b) and 0.03 (c).

### 4.3

### Vibrational resolution

In Ref. [4] it is concluded that since their experimental spectrum show only a reg-ularly spaced vibrational splitting only a single vibration frequency, corresponding to the symmetric stretch mode, is present. This means that we only need to con-sider the one-dimensional potential energy of the stretch mode in our calculations. A potential energy curve for symmetric stretch was obtained by calculating the electronic energy on a evenly spaced grid of 13 points between R0 − 0.3 ˚A to

R0+ 0.9 ˚A where R0= 2.0651 ˚A is the DFT optimized equilibrium geometry. In

this potential we can then solve the nuclear Schr¨odinger equation, Eq. (2.20), for the vibrational states and eigenvalues.

1

In this thesis we consider only transitions from the electronic and vibrational ground state to the states accounted for in Table 4.1. To obtain the electronic energy levels, STEX spectra for the 13 transitions of Table 4.1 were calculated using the same grid of geometries as for the ground state. Tracing each tran-sition through the collection of spectra we obtained the potential energy curves shown in Fig. A.1. Calculated properties and details are found in Table A.1. The Franck–Condon factors were calculated according to Sec. 3.2 and are presented in Fig. 3.2. Together with the dipole moments received from the STEX calculations, the transition moments and ultimately the oscillator strengths were calculated for the vibronic transitions. The resulting vibrationally broadened spectrum is presented in Fig. 4.3 together with the experimental spectrum from Ref. [4].

Figure 4.3. Vibrationally resolved silicon L-edge spectrum of SiCl4: (a) Calculated

spectrum within the Franck–Condon approximation. The lines has been convoluted with a Lorentzian kernel with HWHM of 0.1 eV for the first five peaks and 0.05 eV for the rest of the spectrum. The calculations were carried out using the same basis set as for the spectrum of Fig. 4.1. (b) Experimental spectrum from Ref. [4].

4.3 Vibrational resolution 29

The peaks have been numbered according to the experimental spectrum and the notation with primes, as described above, is used. A noticeable difference with the experimental spectrum is the contraction of the calculated spectrum with the first peaks appearing approximately 2 eV higher in energy. This is due to the use of the ion as reference state in the STEX approximation. Hence we get a good description of the ionization thresholds. But when modeling the excited neutral atom with the optimized orbitals of the ion, the description is not as accurate and as a result the transition energy will be higher than for the true system.

Comparing with the experimental spectrum of Fig. 4.3, the separate peaks (1), (1′), (2), (3) and (C′) clearly corresponds to the peaks (1), (1′), (2), (3) and (3′) respectively whereas the peak (2′) of Fig. 4.3 does not have a counterpart in the calculated spectra. This can be traced back to the low oscillator strength of transition B’ in the electronic spectrum which would have given rise to a peak 2′ had it been of larger magnitude.

The first peak showing the vibrational splitting, peak 4 in Fig. 4.3, was also calculated beyond the Franck–Condon approximation, i.e the electronic dipole moment µ( ¯R) (see Eq. (3.19)), is no longer held constant but evaluated for different nuclear geometries. The resulting peak is presented in Fig. 4.4 together with comparison with an enhanced image from Ref. [4]. Since the peaks lie very close, there is no doubt that the Franck–Condon approximation is a good approximation in this case.

Figure 4.4.(a) Peak 4 vibrationally resolved beyond the Franck–Condon approximation (solid curve in the middle) plotted together with the vibrational widening of the two electronic transitions of which it consists. The peak in the Franck–Condon approximation (upper dotted curve) is added for reference. All peaks convoluted using a Lorentzian with HWHM of 0.075 eV. (b) Peak 4 from Ref. [4].

### Chapter 5

## Summary and conclusions

The calculated spectrum is well describing the first two peaks, (1) and (1′), and peaks (3) and (3′), it does not however show any peak corresponding to (2′). Higher up in the spectrum the vibrationally interesting part is also described quite well and peaks (4) and (4′) show a satisfactory correspondence to the experimental spectrum. The absence of peak 2′ can be traced back to the electronic spectrum where the B’ transition would have given rise to such a peak, but is found to have an oscillator strength very close to zero. This was also the case when extending the basis set so it is not likely a basis set effect. A calculation of peak (4) outside the Franck–Condon approximation shows that the Franck–Condon approximation is good and certainly motivated in the case of SiCl4. The basis set investigation

presented in Appendix B shows a strong basis set dependency in the higher energies of the spectrum, and in order to address this region a more rigorous basis set investigation is called for.

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### Appendix A

## Electronic potential curves

## of SiCl

_{4}

Table A.1. The harmonic frequency (cm−1_{) in the equilibrium point (pm) of each}

potential curve in Fig. A.1 including the ground state (X in the table).

State ∆E ω ∆R0 State ∆E ω ∆R0

X 0.0 228.6 0.0 A 105.53 183.5 13.51 A’ 106.15 184.9 13.61 B 107.71 185.3 2.79 C 107.87 202.6 2.43 C’ 108.40 201.8 2.67 D 108.24 225.8 -5.04 D’ 108.83 244.3 -6.13 E 108.72 250.8 -6.48 E’ 109.37 242.1 -6.56 F 109.26 239.9 -6.21 F’ 109.90 251.5 -6.16 G 109.32 244.4 -6.43 H 109.59 250.1 -6.38 I 110.10 258.3 -5.78 J 110.16 249.4 -6.57 35

Figure A.1. The energy levels corresponding to the 15 largest electronic oscillator
strengths f0→K_{(Q) before the ionization threshold in the spectrum of Si in SiCl}

4. The

### Appendix B

## Basis set effects

The silicon L-edge spectrum of SiCl4 was also calculated with a bigger basis set

to explore the basis set effects. Resulting spectrum is presented in Fig. B.1 and details in Table B.1. Notable features are the contraction of the spectrum, i.e. the transitions (A) and (A’) are shifted up in energy along with the lowering of the ionization thresholds, compared to Fig. 4.1. Another feature is that the splitting up of the E transition. This is due to the better description of the final states in the larger basis set. The B’ transition is again found to have an oscillator strength close to zero, so it is not remedied by the change to a larger basis set.

Table B.1. Excitation energies (eV) and oscillator strengths (×10−4_{) for SiCl}
4 in the

HF-equilibrium geometry calculated in an extended basis set (see Fig. 4.1).

No. E f p3/2(%) No. E f p1/2 (%) ∆E

A 106.35 80.51 92.2 A’ 106.97 132.35 91.1 0.62 B 107.58 36.07 97.3 B’ 108.17 0 95.4 0.59 C 107.72 52.38 99.9 C’ 108.26 66.23 95.3 0.54 D 107.84 8.91 99.8 D’ 108.48 3.74 0.64 E 108.37 6.12 98.2 108.38 2.10 100 E’ 109.02 2.10 100 0.64 37

Figure B.1. Theoretical L-edge spectrum for SiCl4 in the static exchange

approxima-tion. Calculations are carried out with a basis extended with g-type functions. The lower spectrum is the same as Fig. 4.1; for details see Sec. 4.2.

### P˚

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