Department of Physics, Chemistry and Biology
Master’s Thesis
Theoretical investigation of the first-order
hyperpolarizability in the two-photon resonant
region
Mikael Bergstedt
LiTH-IFM-EX-07/1862-SEDepartment of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden
Master’s Thesis LiTH-IFM-EX-07/1862-SE
Theoretical investigation of the first-order
hyperpolarizability in the two-photon resonant
region
Mikael Bergstedt
Adviser: Patrick Norman
The Department of Physics, Chemistry and Biology
Examiner: Patrick Norman
The Department of Physics, Chemistry and Biology
Avdelning, Institution Division, Department Computational Physics
Department of Physics, Chemistry and Biology Link¨opings universitet, SE-581 83 Link¨oping, Sweden
Datum Date 2007-11-05 Spr˚ak Language ¤ Svenska/Swedish ¤ Engelska/English ¤ £ Rapporttyp Report category ¤ Licentiatavhandling ¤ Examensarbete ¤ C-uppsats ¤ D-uppsats ¤ ¨Ovrig rapport ¤
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Titel Title
Teoretisk unders¨okning av andra ordningens susceptibilitet i det tv˚afotonresonanta omr˚adet
Theoretical investigation of the first-order hyperpolarizability in the two-photon resonant region F¨orfattare Author Mikael Bergstedt Sammanfattning Abstract
Time-dependent density functional theory calculations have been carried out to determine the complex first-order hyperpolarizability in the two-photon resonance region of the molecule IDS-Cab. Calculations show that three strongly absorbing states, in the ultraviolet region, are separated to the extent that no significant interference of the imaginary parts of the tensor elements of the first-order hyper-polarizability occurs. Consequently, and in contrast to experimental findings [27], no reduced imaginary parts of the first-order hyperpolarizability in the two-photon resonant region can be seen.
Nyckelord Keywords
two-photon absorption, first-order hyperpolarizability, responce theory, time-dependent density functional theory, Hartree–Fock approximation
£
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-10290
—
LiTH-IFM-EX-07/1862-SE
“Do not let your hearts be troubled. Trust in God; trust
also in me [Jesus].” (John 14:1)
“K¨ann ingen oro. Tro p˚
a Gud, och tro p˚
a mig [Jesus].”
(Joh 14:1)
Abstract
Time-dependent density functional theory calculations have been carried out to determine the complex first-order hyperpolarizability in the two-photon resonance region of the molecule IDS-Cab. Calculations show that three strongly absorbing states, in the ultraviolet region, are separated to the extent that no significant interference of the imaginary parts of the tensor elements of the first-order hyper-polarizability occurs. Consequently, and in contrast to experimental findings [27], no reduced imaginary parts of the first-order hyperpolarizability in the two-photon resonant region can be seen.
Acknowledgements
First of all, I would like to thank my supervisor Patrick Norman for all his help and support during my work, and for the warm relation we have. Second I thank my father Jan- ¨Osten Bergstedt and my mother Anna Bergstedt for all their sup-port. I also thank Brother Ingmund, Brother Frans–Eric, Brother Stefan, and the postulant Staffan in the Franciscan order Third Order Regular for their support and help, and finally I thank all my friends at the Department of Physics, Chem-istry and Biology at Link¨opings University, and my other friends, in the Catholic Church and elsewhere, for splendouring my life during my studies. Thank You!
Contents
1 Introduction 1 1.1 Physical Background . . . 1 1.2 Problem Definition . . . 2 1.3 Background . . . 2 1.3.1 Articles of interest . . . 42 Electronic Structure Theory 7 2.1 Basic Concepts . . . 7
2.1.1 Background . . . 7
2.1.2 Molecular Hamiltonian . . . 8
2.1.3 Born–Oppenheimer Approximation . . . 8
2.1.4 Antisymmetry Principle . . . 9
2.1.5 Spatial and Spin Orbitals . . . 10
2.1.6 Slater Determinants . . . 10
2.2 Hartree–Fock Approximation . . . 11
2.2.1 Hartree–Fock Equations . . . 11
2.3 Restricted Closed-Shell Hartree–Fock Theory . . . 12
2.3.1 Roothaan Equations and Introducing a Basis . . . 13
2.3.2 Density Matrix . . . 14
2.3.3 Fock Matrix . . . 15
2.3.4 Self Consistent Field Procedure . . . 17
2.4 Density Functional Theory . . . 18
2.4.1 First Hohenberg–Kohn Theorem . . . 18
2.4.2 Energy Functionals . . . 18
2.4.3 Second Hohenberg–Kohn Theorem . . . 19
2.4.4 Kohn–Sham Density Functional Theory . . . 19
3 Response Theory 21 3.1 Light Interaction with a Molecule . . . 21
3.2 Off-Resonant Perturbations . . . 22
3.2.1 Time-Dependent Perturbation Theory . . . 23
3.3 Resonant and Near-Resonant Perturbations . . . 26
3.3.1 Rigorous Description in Wave-Function Theory . . . 26
3.3.2 Pragmatic Description in Wave-Function Theory . . . 31 xi
xii Contents
4 Results and Discussion 33
4.1 Computational Details . . . 33
4.2 Molecule . . . 33
4.3 Excited States Investigation . . . 36
4.3.1 Molecular Transitions at the Hartree–Fock Level . . . 36
4.3.2 Non-Degenerate Excited States . . . 37
4.3.3 Molecular Transitions at the Density Functional Theory Level 38 4.3.4 Reduction of Excitation Energies . . . 39
4.4 Basis Set Investigation . . . 40
4.4.1 State X and HF . . . . 40
4.4.2 State X and DFT . . . . 41
4.4.3 State Y Z and HF . . . . 41
4.4.4 State Y Z and DFT . . . . 42
4.4.5 Conclusions . . . 42
4.5 Static Hyperpolarizability Investigation . . . 43
4.6 First-Order Hyperpolarizability . . . 44
4.6.1 Tensor Elements Investigation . . . 44
4.6.2 Hyperpolarizability in Different Directions . . . 46
4.6.3 Comparison Between Theory and Experiment . . . 48
5 Conclusions 49
A Hyperpolarizability Tensor Elements 51
B First-Order Hyperpolarizability 55
Chapter 1
Introduction
This thesis investigate the first-order hyperpolarizability in the two-photon res-onant region. This chapter gives an introduction and background to the basic concepts, the problem definition of this thesis, and an overview of relevant arti-cles.
1.1
Physical Background
The excitation of atomic or molecular systems, by exposing the system to light or other electromagnetic waves, is a fundamental concept in physics. In other words the atom or molecule absorb energy quanta(s), photon(s), and ends up in a higher energy state. The most fundamental law in physics, that the total energy of a system is conserved, is of course valid here. Ordinarily these excitations are a consequence of one photon being absorbed by one molecule, called one-photon absorption, but it is possible, and most important for this thesis, that
two or more photons, that could not excite the molecule by them selves, together
excite the molecule. The former case is called two-photon absorption. Second
harmonic generation (SHG) [12], which the first-order hyperpolarizability describes
and that is an example of a non-linear optical (NLO) effect, is the phenomenon that incoming photons interact with a nonlinear material and this process result in outgoing photons with twice the energy.
What is the first-order hyperpolarizability, and how does it come up in physics? Light, considered not as energy quanta (photons) but as electromagnetic waves, of course effect the dipole moment µ of a molecule when it is exposed to the light. The effect of a molecule being exposed to light is a small change in the dipole moment of the molecule which can therefore be expressed as a Taylor series according to
µ(t) = µ0+ αE(t) +1 2βE
2(t) +1 6γE
3(t) + · · · (1.1) where µ0 is the permanent electric dipole moment of the molecule. Here α is the linear polarizability, β is the first-order hyperpolarizability and γ is the second-order hyperpolarizability. The first-second-order hyperpolarizability β describes second
2 Introduction harmonic generation, because when the electric field is expressed as
Eα(t) =X ω
Eαωe−iωt (1.2)
where α denotes molecular axis which can be chosen as x-, y- and z-axis, and
Eω
α are the Fourier amplitudes of the electric field, it is obvious that the factor
E2(t) next to β in Eq. (1.1) will double the frequency ω. The time-changing electromagnetic wave will affect the electrons and nuclei in the molecule and make them behave as “small antennas” that will emit electromagnetic radiation. Since the electric field is vectorial, α will be a second-rank tensor, β will be a third-rank tensor, γ will be a fourth-rank tensor and so on. In a Cartesian coordinate system the first-order hyperpolarizability will be denoted βαβγ where the subscripts can adopt x-, y- and z-axis.
The first-order hyperpolarizability is a complex valued quantity because an imaginary damping term is introduced in the expression for β. Without this damping term the real valued β would diverge in the two-photon resonant region and it would be impossible to profit from calculations of β in the two-photon resonant region. When the damping term is introduced the imaginary part of
β is usually dominant in the two-photon resonant region. The articles that are
presented below, in Section 1.3.1, is mostly about the curious effect “real first-order hyperpolarizability in the two-photon resonant region” which is a rare but, according to the articles, real effect.
1.2
Problem Definition
The problem formulation of this thesis is to investigate the first-order hyperpolariz-ability in the two-photon resonant region. This is done by performing high-quality calculations on a certain molecule, and comparing the results with experimental results reported in articles.
1.3
Background
G¨oppert-Mayer [13] first suggested two-photon absorption in 1931, but it was not until 30 years later, in 1961, that Kaiser and Garret [24] showed experimentally that two-photon absorption is a physical reality. The main cause of this delay in time is that the phenomenon is hard to experimentally detect without high intensities of the incident light, as attained with a laser1. Two-photon absorption is still not completely explored.
There is currently an intense effort aimed at developing photonic materials and technologies. Photonic techniques are based on the use of photons for the transport and storage of information, compared to electronic techniques that are based on the
transport of electrons. Photonic techniques exploit nonlinear interactions between
1.3 Background 3 molecules and electromagnetic fields and computational quantum chemistry can therefore contribute by delivering accurate polizabilities and hyperpolizabilities.
Non-linear optics is the study of the interaction between matter and high-intensive laser light. During the passed two decades, and especially in recent years, there has been an intensive research on organic materials for non-linear optical effects, and much effort has been focused on the development of NLO materials. NLO effects of organic molecules arise from electronic motion, and photonic influence on it, in contrast to inorganic materials whose NLO effects are due mainly to the contribution from polar optical lattice vibrations [5, 43, 38]. Advantages of organic NLO materials are low cost and ease of processing, which make them very attractive to the industry.
Currently a large number of research groups are especially interested in these materials, because they produce important effects, such as the so-called
electro-optic effect, which is change of the electro-optical properties (for example refractive index)
of a material because of application of an external electric field that usually varies slowly compared with light, and second-harmonic generation. Based on these effects, important applications in optoelectronic [31], which is the study and ap-plication of electronic devices that interact with light, photonics [31], and optical data storage [10, 8, 14] have been developed. Other applications of NLO materi-als are frequency-doubling devices, optical signal processing, optical interconnects [18], etc.
Though NLO materials have been in use for some time and a wide variety of such materials now exist, there still remains a need for new and better ones. High speed communication would greatly benefit from the development of better electro-optic (NLO) materials. The microscopic first-order hyperpolarizability, β, is an important parameter for the characterization of these materials. Two main techniques for measuring β that are used are electric-field induced second-harmonic
generation (EFISH) and the more recently introduced technique of hyper-Rayleigh scattering (HRS). Certain organic molecules can possess large β-values leading to
efficient nonlinearity. To achieve high second-harmonic generation molecules with high β-values must be incorporated in non-centrosymmetric media such as poled polymers or thin films.
Interesting phenomenon can occur in NLO systems, molecules, with degener-ate excited stdegener-ates and heightened polarizability and/or hyperpolarizabilities, with enhanced imaginary parts, in the resonant regions. Interference of polarizability and/or hyperpolarizabilities, in these systems, give rise to the interesting phe-nomenon, and examples of them are Fano resonance [11], inversion less lasers [21, 46], electromagnetically induced transparency [15, 16] and reduction of the velocity of light [17]. These phenomenon have usually been studied for atomic systems for linear and third-order polarizabilities (α, γ) but not for the first-order hyperpolarizability (β) because even-order susceptibilities vanish in atomic sys-tems. Recently a series of articles have been published that state that β, both amplitude and phase, has been measured for a certain molecule2, and the articles show, theoretically and experimentally, a curios effect, videlicet “real first-order
4 Introduction hyperpolarizability in the two-photon resonant region”. These articles are, among others, presented below.
1.3.1
Articles of interest
Orr and Ward [35] year 1971 present the full sum-over-state (SOS) expression for the first-order hyperpolarizability, which is still used and referred to today.
Recently Meshulam et al. [27, 26, 30] and Berkovic et al. [4] stated that they show theoretically and experimentally how the first-order hyperpolarizability of a three-level molecule3, with quasi-degenerate excited states,4 at a two-photon resonance can obtain a real value with strong resonant enhancement. This is a curios effect. They also state that this phenomenon arises from destructive interference of the usually dominant imaginary parts of the resonant first-order hyperpolarizability. The authors of the articles mean that the molecule can be considered as a two-dimensional molecule with two dominating parts of β, namely
βxxx which correspond to one transition along the long-axis of the molecule, and
βxzz which correspond to one transition along a smaller part of the molecule that is along a line with almost right angle to the long-axis. The authors of the articles mean that βµ, which is the total β projected on the dipole moment vector µ, and the sum βxxx + βxzz, which they say they measure, are virtually identical in magnitude. They also state that βµ is real valued in the two-photon resonant region because the imaginary parts of βxxxand βxzzhave opposite signs and cancel each other. In the Refs. [27] and [4] it is stated that both the absolute value and the phase of βµ is measured and presented, in Ref. [30] only the absolute value of
βµ is presented, and in Ref. [26] only the phase of βµ is presented.
Meshulam et al. [29, 28] state that the first-order hyperpolarizability has been measured for a series of one-dimensional and two-dimensional organic molecules.5 Despite the chemically similar nature of the molecules, a drastic reduction of electric field induced second harmonic β values for the two-dimensional molecules was detected, and based on quantum mechanical calculations the conclusion is that destructive interference of two strong imaginary parts, of βxxx and βxzz, occur. EFISH and hyper Rayleight scattering were used which probes different combinations of the β tensor, so that the two contributions βxxx and βxzz could be deduced.
Berkovic et al. [3] state that the non-resonant two-level model,6first presented by Oudar and Chemla [36], can be extended into the resonant region. The first-order hyperpolarizability is measured, with electric field induced second harmonic generation, on a one-dimensional molecule, and the authors state that the extended 3This means that three states, the ground state and two excited states, are most significant,
and these states are considered in the theory expressions for β in the present article. This is the molecule that I have performed high-performance calculations on.
4The two interesting excited states are degenerate according to the article.
5The effect of varying the molecules donor from dialkylamino, which is origin of the
one-dimensional structure of the molecules, to N-carbazolyl, which is origin of the two-one-dimensional structure of the molecules, has been studied.
6The level model is the simplified full sum-over-state model presented by [35]. The
1.3 Background 5 theoretically two-level model predicts both the amplitude and the phase of the measurements. It is also shown that both on-resonance and off-resonance EFISH yield the same hyperpolarizability extrapolated to the zero frequency limit.
Chapter 2
Electronic Structure Theory
In writing this chapter, I have mainly used the following sources: Szabo and Ostlund [42], Jensen [22], Bransden and Joachain [7] and Parr and Yang [37].
2.1
Basic Concepts
2.1.1
Background
Atoms and molecules can be considered as negatively charged electrons surround-ing positively charged nuclei. The Coulomb force, which always exist between two charged particles, is fundamental when one considers atoms and molecules, and the Coulomb potential between two particles with charges qiand qj, and with the distance rij between them, is in atomic units given by
Vij = V (rij) =qiqj
rij
(2.1) In classical mechanics the equation that describe the dynamics of the system is
Newton’s second law which reads
−dV dr = m
d2r
dt2 (2.2)
where m is the particle mass and r is the position vector. The electron is though a very light and small elementary particle which move very fast so a classical mechanical treatment is impossible. Instead one has to use a quantum mechanical treatment. The electrons display both wave and particle properties so they must be described in terms of a wave function, Φ. The quantum mechanical equation that correspond to Newton’s second law is the time-dependent Schr¨odinger equation:
ˆ
H|Φi = i~∂
∂t|Φi (2.3)
If the Hamilton operator ˆH is independent of time the time dependence of the wave
function |Φi can be separated out as a simple phase factor according to Eq. (2.5) 7
8 Electronic Structure Theory below:
ˆ
H(r, t) = ˆH(r) (2.4)
|Φ(r, t)i = |Φ(r)ie−iEt/~ (2.5) ˆ
H(r)|Φ(r)i = E|Φ(r)i (2.6) where Eq. (2.6) is the time-independent Schr¨odinger equation.
2.1.2
Molecular Hamiltonian
The main interest, in this chapter, is to find approximate solutions to the non-relativistic time-independent Schr¨odinger equation
ˆ
H|Φi = E|Φi (2.7)
where ˆH is the Hamiltonian operator for a system of electrons and nuclei with
position vectors ri and RA, respectively. The distance between the ith electron and the Ath nucleus is riA= |ri− RA|, the distance between the ith electron and the jth electron is rij = |ri− rj|, and the distance between the Ath nucleus and the Bth nucleus is RAB = |RA− RB|. The Hamiltonian for N electrons and M nuclei, in atomic units, is
ˆ H = − N X i=1 1 2∇ 2 i− M X A=1 1 2MA∇ 2 A− 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 (2.8) In the above equation MAis the ratio of the mass of nucleus A to the mass of an electron, and ZA is the atomic number of nucleus A. The Laplacian operators ∇2 i and ∇2
A involve differentiation with respect to the coordinates of the ith electron and the Ath nucleus. The first term in Eq. (2.8) is the operator for the kinetic energy of the electrons. The second term is the operator for the kinetic energy of the nuclei. The third term represents the Coulomb attraction between the electrons and the nuclei. The fourth term represents the Coulomb repulsion between the electrons, and the fifth term represents the Coulomb repulsion between the nuclei.
2.1.3
Born–Oppenheimer Approximation
The Born–Oppenheimer approximation is central to quantum chemistry, and it states that, since nuclei are much heavier then electrons, they move much more slowly, and the Schr¨odinger equation can therefore, to a good approximation, be separated into one part that describes the electronic wave function, |Ψeleci, for a fixed set of nuclei, and one part that describes the nuclear wave function, |Φnuci. When calculating the nuclear wave function the energy from the electronic wave function plays the role of a potential energy, i.e., the nuclei move on potential energy surfaces which are solutions to the electronic Schr¨odinger equation.
The Born–Oppenheimer Approximation is expressed in equations according to Eqs. (2.9)–(2.13) below: ˆ H = − N X i=1 1 2∇ 2 i− M X A=1 1 2MA ∇2 A− 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 (2.9)
2.1 Basic Concepts 9 ˆ Helec= − N X i=1 1 2∇ 2 i − 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 (2.10) ˆ
Helec|Ψeleci = Eelec|Ψeleci (2.11)
|Φi = |Φnuci|Ψeleci (2.12) ³ − M X A=1 1 2MA ∇2 A+ Eelec ´
|Φnuci = Etot|Φnuci (2.13)
The last term of Eq. (2.10) can be considered constant. Any constant added to an operator only adds to the operator eigenvalues and has no effect on the operator eigenfunctions. We now define the electronic Hamiltonian as
ˆ Helec = − N X i=1 1 2∇ 2 i − N X i=1 M X A=1 ZA riA + N X i=1 N X j>i 1 rij (2.14)
The total energy for fixed nuclei must also include the constant nuclear repulsion.
Etot= Eelec+ M X A=1 M X B>A ZAZB RAB (2.15) Eq. (2.11), (2.14), and (2.15) constitute the electronic problem, which will be studied in this chapter. From now on we drop the subscript “elec” and only consider the electronic Hamiltonian and electronic wave functions.
2.1.4
Antisymmetry Principle
The electronic Hamiltonian in Eq. (2.14) depends only on the spatial coordinates of the electrons. To completely describe an electron, it is necessary, however, to specify its spin. In the context of our non-relativistic theory we introduce two spin functions α(ω) and β(ω) which correspond to spin up and spin down respectively. These are functions of an unspecified spin variable ω. An electron is described by the three spatial coordinates r and by one spin coordinate ω. We denote these four coordinates collectively by x = {r, ω}. The wave function for an N -electron system is then a function of x1, x2, . . . , xN which we write Ψ(x1, x2, . . . , xN). Be-cause the Hamiltonian operator makes no reference to spin, simply making the wave function depend on spin (in the way just described) does not lead to doubly occupied orbitals. A satisfactory theory can be obtained if we make the following additional requirement on a wave function: A many electron wave function must
be antisymmetric with respect to the interchange of the coordinates (both space and spin) of any two electrons,
Ψ(x1, . . . , xi, . . . , xj, . . . , xN) = −Ψ(x1, . . . , xj, . . . , xi, . . . , xN) (2.16) This is called the antisymmetry principle and it goes hand in hand with the Pauli
10 Electronic Structure Theory says that only one fermion (read electron) can occupy a given quantum state. From Eq. (2.16) one can easily see that if two electrons occupy the same coordinates xi then the many electron wave function will be zero, i.e. it is impossible for two electrons to occupy the same coordinates.
The exact wave function must satisfy both the Schr¨odinger equation and the antisymmetry principle.
2.1.5
Spatial and Spin Orbitals
An orbital is a wave function of a single particle, an electron, per definition. There are both spatial orbitals and spin orbitals. We write a spatial orbital as ψi(r) and
|ψi(r)|2dr is the probability of finding an electron in the small volume element dr centered at r.
The state of an electron is, however, also specified by its spin which can be spin up, α(ω), or spin down, β(ω). From every spatial orbital ψ(r) one can form two spin orbitals, representing spin up and spin down respectively, according to
χ(r) = ψ(r)α(ω) or ψ(r)β(ω) (2.17)
Restricted spin orbitals have the same spatial function for spin up and spin down,
as in Eq. (2.17). Unrestricted spin orbitals have different spatial functions for spin up and spin down according to
χ(r) = ψα(r)α(ω) or ψβ(r)β(ω) (2.18)
2.1.6
Slater Determinants
A powerful way to construct a wave function is to make use of Slater determinants:
Ψ(x1, x2, . . . , xN) =√1 N ! ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ χi(x1) χj(x1) . . . χk(x1) χi(x2) χj(x2) . . . χk(x2) .. . ... ... χi(xN) χj(xN) . . . χk(xN) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ (2.19)
This wave function represents N electrons in N spin orbitals. The rows de-scribe the coordinates, and the columns dede-scribe the spin orbitals. Interchanging coordinates of two electrons is the same as changing to rows in the Slater determi-nant which result in change of sign (the antisymmetry principle). If two electrons have the same spin orbitals two columns will be identical and the Slater deter-minant will be zero (Pauli exclusion principle). The probability for two electrons with identical spins to be at the same point in space is zero and corresponds to two equal rows in the Slater determinants.
This wave function does not specify which electron is in which orbital, the electrons are regarded as indistinguishable.
2.2 Hartree–Fock Approximation 11
2.2
Hartree–Fock Approximation
Since the birth of quantum mechanics, attempts have been made to find approx-imate solutions for the Schr¨odinger equation for many-electron systems. The Hartree–Fock approximation is central to solving such problems and it plays a central role in quantum chemistry.
The basic idea of the Hartree–Fock approximation is the following: accord-ing to the variational principle the wave function Ψ that minimize the energy
E = hΨ| ˆH|Ψi is the best wave function in a given subspace of the Hilbert space
that describe the many-electron system. The wave function |Ψi is a Slater deter-minant, the same as in Eq. (2.19), and in the Hartree–Fock approximation one minimize E with respect to the choice of spin orbitals χi in a systematic way, which leads to an equation which is called the Hartree–Fock equation. In the Hartree–Fock approximation one replace the complicated many-electron problem by a one-electron problem in which electron-electron repulsion is treated in an averaged way. The Hartree–Fock equation is nonlinear and must be solved itera-tively. This is done according to a method that is called the self-consistent field approach.
2.2.1
Hartree–Fock Equations
The equation that define the best spin orbitals, which gives the best Slater deter-minant wave function and that consequently minimizes the electronic energy E, is the Hartree–Fock integro-differential equation
ˆh(1)χa(1) +X b6=a h Z dx2|χb(2)|2r−112 i χa(1) −X b6=a h Z dx2χ∗b(2)χa(2)r12−1 i χb(1) = ²aχa(1) (2.20)
where the number arguments is short for x1 and x2, and ˆh(1) = −1 2∇ 2 i − X A ZA riA (2.21)
Eq. (2.20) is derived by minimizing the energy expression E = hΨ| ˆH|Ψi, where
the wave function |Ψi is a single Slater determinant, by systematically varying the spin orbitals {χa} of which the single Slater determinant wave function consists.
The two terms in Eq. (2.20) that contains summation over spin orbitals is the electron-electron interaction. Without these terms Eq. (2.20) would simply be a one electron Schr¨odinger equation for a single electron in the field of the nuclei:
ˆh(1)χa(1) = ²aχa(1) (2.22)
According to Eq. (2.20) electron one in spin orbital χaexperience a one-electron Coulomb potential as due to the other N −1 electrons
vcoul a (1) = X b6=a Z dx2|χb(2)|2r12−1 (2.23)
12 Electronic Structure Theory It is convenient to define a Coulomb operator ˆJb(1) from
ˆ
Jb(1)χa(1) =h Z dx2χ∗b(2)r12−1χb(2) i
χa(1) (2.24)
and an exchange operator from ˆ
Kb(1)χa(1) =h Z dx2χ∗b(2)r12−1χa(2) i
χb(1) (2.25)
Observe that χa has “exchanged” place with χb compared with Eq. (2.24). The exchange operator can not be understood with classical physics, and is based on the antisymmetry principle.
The Hartree–Fock equation is then h ˆh(1) +X b ˆ Jb(1) −X b ˆ
Kb(1)iχa(1) = ²aχa(1) (2.26)
where we have used the fact that the self-interaction energies is zero in the Hartree– Fock approximation, i.e.
[ ˆJb(1) − ˆKb(1)]χa(1) = 0 (2.27) Alternatively we may write
f |χai = ²a|χai (2.28) where ˆ f (1) = ˆh(1) +X b [ ˆJb(1) − ˆKb(1)] (2.29)
and this is the most common form of the Hartree–Fock equations, and it introduces the Fock operator f .
2.3
Restricted Closed-Shell Hartree–Fock Theory
So far we have considered the Hartree–Fock approximation for a general set of spin orbitals χa. Below we will explore actual calculations of the Hartree–Fock wave functions, and we will have to specify the spin orbitals more exact, and we will introduce what we call a basis.
In this section we will calculate only restricted Hartree–Fock wave functions, and consider closed-shell systems. Thus our molecular Slater determinant wave function consist of an even number of electrons, N, and the spin orbitals in the wave function are restricted, i.e. there are N/2 spatial orbitals, all doubly occupied with one electron with spin up and one electron with spin down. In practice this restricts our discussion to the optimization of ground states.
Accordingly, our spin orbitals are restricted, as in Eq. (2.17), and because of this we can work at the Hartree–Fock equation, (2.28), and get the closed-shell
spatial Hartree–Fock equation
ˆ
2.3 Restricted Closed-Shell Hartree–Fock Theory 13
2.3.1
Roothaan Equations and Introducing a Basis
To solve Eq. (2.30) numerically, which Roothaan showed how to do [39], one introduces a set of K known basis functions {φµ(r)|µ = 1, 2, . . . , K} and expands the unknown molecular (electron) orbitals as a linear expansion according to
ψi= K X µ=1
Cµiφµ i = 1, 2, . . . , K (2.31)
The problem, of calculating the Hartree–Fock molecular orbitals, is now turned into determining the coefficients Cµi. We will do this with matrix calculations. We now insert Eq. (2.31) in Eq. (2.30) and get
ˆ f (1)X ν Cνiφν(1) = ²i X ν Cνiφν(1) (2.32) By multiplying with φ∗
µ(1) to the left and integrating gives us the matrix equation X ν Cνi Z dr1φ∗µ(1) ˆf (1)φν(1) = ²i X ν Cνi Z dr1φ∗µ(1)φν(1) (2.33)
We define the overlap matrix S which has the elements
Sµν = Z
dr1φ∗µ(1)φν(1) (2.34)
and is a K × K Hermitian matrix with diagonal elements equal to unity, because the basis functions {φµ} are assumed to be normalized, and off-diagonal elements
which are numbers less then one in magnitude, because the basis functions are not in general orthogonal. In the calculation the basis functions are assumed to be linearly independent. If the determinant of S is close to zero there is a redundancy in the basis set.
We also define the Fock matrix F which has the elements
Fµν = Z
dr1φ∗µ(1) ˆf (1)φν(1) (2.35) which also is a K × K Hermitian matrix.
We can now write the integrated Hartree–Fock equation as X ν FµνCνi= ²i X ν SµνCνi i = 1, 2, . . . , K (2.36)
or simpler as a matrix equation
FC = SC² (2.37)
where C is a K × K square matrix
C = C11 C12 . . . C1K C21 C22 . . . C2K .. . ... ... CK1 CK2 . . . CKK (2.38)
14 Electronic Structure Theory and ² is a diagonal matrix with the orbital energies ²i as diagonal elements
² = ²1 0 . . . 0 0 ²2 0 . . . . .. 0 . . . 0 ²K (2.39)
Eq. (2.36) and (2.37) are the Roothaan equations.
The problem of determining the Hartree–Fock molecular orbitals {ψi} involves
solving the matrix equation FC = SC², and to do that we use the density matrix.
2.3.2
Density Matrix
The probability of finding an electron, described by the spatial function ψa(r), in the small volume element dr at position r is |ψa(r)|2dr. The probability dis-tribution function is |ψa(r)|2. The total charge density, or the total probability distribution function, for a closed-shell molecule described by a Slater determinant wave function where every spatial orbital ψa is occupied with two electrons is
ρ(r) = 2
N/2 X a
|ψa(r)|2 (2.40)
and ρ(r)dr is the probability of finding an electron (any electron) in dr at r. If we now insert the molecular orbital expansion (2.31) in Eq. (2.40) for the total charge density we get:
ρ(r) = 2 N/2 X a ψ∗ a(r)ψa(r) = 2 N/2 X a X ν C∗ νaφ∗ν(r) X µ Cµaφµ(r) =X µν h 2 N/2 X a C∗ νaCµa i φ∗ ν(r)φµ(r) =X µν Pνµφ∗ν(r)φµ(r) (2.41)
This defines the density matrix P which has elements
Pνµ= 2 N/2 X a C∗ νaCµa (2.42)
2.3 Restricted Closed-Shell Hartree–Fock Theory 15
2.3.3
Fock Matrix
It can be shown that the elements in the Fock matrix, for restricted closed-shell Hartree–Fock, is given by Fµν= Z dr1φ∗µ(1) ˆf (1)φν(1) = Hcore µν + N/2 X a h 2(µν|aa) − (µa|aν)i (2.43)
where we have defined the core-Hamiltonian matrix
Hcore µν =
Z
dr1φ∗µ(1)ˆh(1)φν(1) (2.44)
and the two electron integral (µν|λσ) =
Z
dr1dr2φ∗µ(1)φν(1)r12−1φ∗λ(2)φσ(2) (2.45)
The major difficulty in Hartree–Fock calculations is to evaluate and manipulate these two-electron integrals because of their large number. The core-Hamiltonian, with its kinetic energy part and its nuclear attraction part, needs only to be evaluated once because it remains constant during the iterative Hartree–Fock cal-culations.
We insert Eq. (2.31) in Eq. (2.43) and get, which is a bit complicated to see,
Fµν = Hµνcore+ N/2 X a X λσ C∗ λaCσa[2(µν|λσ) − (µσ|λν)] = Hcore µν + X λσ Pλσ[(µν|λσ) −1 2(µσ|λν)] = Hcore µν + Gµν (2.46)
and this defines the matrix G which is the two-electron part of the Fock matrix. The Fock matrix depends on the density matrix according to Eq. (2.46), i.e. F = F(P) or equivalently for the expansion coefficients F = F(C) and thus the Roothaan equations are nonlinear, F(C)C = SC², and must be solved iteratively. If the basis set {φa} would be orthogonal then S would be the unity matrix and
F(C)C = C², and we could find the eigenvectors C and the eigenvalues ² by diagonalizing F. Though, our basis set is not orthogonal and that is why we need to work at it, orthogonalized it, so that the Roothaans equations become on the usual matrix eigenvalue form.
Given a basis set {φa}, it is possible to find a transformation matrix X so that
the transformed basis set
φ0µ= X
ν
16 Electronic Structure Theory forms an orthonormal set. Every φ0µ is a linear combination of {φν}. The trans-formed basis set {φ0µ} fulfills
Z drφ0∗ µ(r)φ 0 ν(r) = δµν (2.48) so it is orthonormal.
We insert Eq. (2.47) into Eq. (2.48) and we keep in mind Eq. (2.34). This gives us an equation that can be written in matrix notation according to
X†SX = 1 (2.49)
where X†
µν = Xνµ∗ and 1 is the unit matrix.
Thus, a transformation matrix X that orthogonalizes the basis set {φa} must fulfill Eq. (2.49). Two usual choices of the transformation matrix X are the
sym-metric orthogonalization matrix and the canonical orthogonalization matrix. Both
have advantages and disadvantages. S is Hermitian, and because of this it can be diagonalized by a unitary matrix U accordingly to
U†SU = s (2.50)
where s is a diagonal matrix with the eigenvalues of S as diagonal elements. In symmetric orthogonalization one choose the transformation matrix to be the inverse square root of S
X ≡ S−1/2= Us−1/2U† (2.51)
This transformation matrix fulfills Eq. (2.49) because if S is Hermitian so is S−1/2, so (S−1/2)† = S−1/2, and
S−1/2SS−1/2 = S−1/2S1/2= S0= 1 (2.52) But be aware that if we have linear dependence or near linear dependence some of the eigenvalues will approach zero and in Eq. (2.51) we will divide with a very small number which give numerical precision problems.
In canonical orthogonalization one choose the transformation matrix to be
X ≡ Us−1/2 (2.53)
which also fulfills Eq. (2.49) according to
X†SX = (Us−1/2)†SUs−1/2= s−1/2U†SUs−1/2= s−1/2ss−1/2= 1 (2.54) The last part that remains now in the work of solving the Roothaan equations, so that we can determine the spatial orbitals and calculate physical expectation values, is the following:
We introduce a new coefficient matrix C0 which is related to the old coefficient matrix C by
2.3 Restricted Closed-Shell Hartree–Fock Theory 17 and here we have assumed that X has an inverse which is the case if there is no linear dependencies. We now substitute C = XC0 into the Roothaan equation and get
FXC0 = SXC0² (2.56)
and we multiplicate with X† from the left and get
(X†FX)C0 = (X†SX)C0 ² (2.57)
but according to Eq. (2.49) we now have (X†FX)C0 = C0 ² and we define F0 = X†FX we get the eigenvalue equation
F0C0 = C0² (2.58)
which can be solved by diagonalizing F0 for C0 and ², and C can be calculated from Eq. (2.55).
Now when we can calculate C we can determine the spatial orbitals {ψi} from
Eq. (2.31), and the spatial Hartree–Fock Eq. (2.30) is thereby solved.
2.3.4
Self Consistent Field Procedure
The actual computation of restricted closed-shell Hartree–Fock wave functions for molecules can be done with the self-consistent-field procedure which is described below.
1. Specify a molecule (a set of coordinates {RA} , atomic numbers {ZA}, and number of electrons N ) and a basis set {φµ}.
2. Calculate all required molecular integrals, Sµν, Hcore
µν and (µν|λσ).
3. Diagonalize the overlap matrix S and obtain a transformation matrix X from either (2.51) or (2.53).
4. Obtain a guess at the density matrix P.
5. Calculate the matrix G of Eq. (2.46) from the density matrix P and the two-electron integrals (µν|λσ).
6. Add G to the core-Hamiltonian to obtain the Fock matrix F = Hcore+ G. 7. Calculate the transformed Fock matrix F0 = X†FX.
8. Diagonalize F0 to get C0 and ². 9. Calculate C = XC0.
10. Form a new density matrix P from C using (2.42).
11. Determine whether the procedure has converged, i.e., determine whether the new density matrix of step (10) is the same as the previous density matrix within a specified criterion. If the procedure has not converged, return to step (5) with the new density matrix.
18 Electronic Structure Theory 12. If the procedure has converged, then use the resultant solution, represented
by C, P, F, etc., to calculate expectation values and other quantities of interest.
I have collected the procedure from [42].
2.4
Density Functional Theory
2.4.1
First Hohenberg–Kohn Theorem
The basis for Density Functional Theory is that it is a one-to-one correspondence between the electron density ρ and the energy. According to the first Hohenberg–
Kohn theorem [20], the ground-state electronic energy is determined completely
by the electron density. One realize the importance of this when one compare the electron density approach with the wave function approach. The electron density,
ρ(r1) = N Z
|Ψ(x1, x2, . . . , xN)|2dω1dx2. . . dxN (2.59)
where Ψ(x1, x2, . . . , xN) is the ground state wave function describing the N-electron system, only depend on three (spatial) coordinates, while the wave function Ψ de-pends on 4N coordinates. The complexibility of the wave function increases with the number of electrons while the electron density has the same number of variables independently of the size of the system.
2.4.2
Energy Functionals
A function produces a number from a set of variables, while a functional produces a number from a function, which in turn depends on variables. The wave function Ψ and the electron density ρ are functions, while the energy depending on the wave function or on the electron density is a functional. A function is denoted
f (x) and a functional is denoted F [f ].
The total energy of a N -electron system can be expressed as a functional
E[ρ] = hΨ|H|Ψi = T [ρ] + Vne[ρ] + Vee[ρ] (2.60) where T [ρ] is the true kinetic energy of the system, and Vee[ρ] is the true electron-electron interaction, and Vne[ρ] is the true potential describing the interaction between the nuclei and the electrons.
The Hohenberg–Kohn functional, FHK, is defined as
FHK[ρ] ≡ T [ρ] + Vee[ρ] (2.61)
and the external potential, v(r), relate to Vne according to
Vne[ρ] = Z
2.4 Density Functional Theory 19
2.4.3
Second Hohenberg–Kohn Theorem
The second Hohenberg–Kohn theorem [37] states that, for the exact (true) ground state energy E0 and the energy functional E[ρ], the equation
E0≤ E[ρ] (2.63)
is valid, and there is equality if and only if ρ is equal to the exact (true) ground state electron density.
Thus E0= min ρ {E[ρ]} = minρ n FHK[ρ] + Z v(r)ρ(r)dro (2.64)
Applying the variational principle with the constraint Z
ρ(r)dr = N (2.65)
one can use a Lagrange multiplier µ according to
δ n E[ρ] − µ h Z ρ(r)dr − N io = 0 (2.66)
which can be rewritten as an Euler–Lagrange equation according to
δFHK
δρ + v(r) = µ (2.67)
where µ is the chemical potential [37]. One can solve Eq. (2.67) and get the electron density, but this is hard to do because FHK[ρ] is unknown. The solution to this problem is called the Kohn–Sham method.
2.4.4
Kohn–Sham Density Functional Theory
The main idea in the Kohn–Sham method [25] is to consider noninteracting par-ticles and strive to derive the same electron density as the interacting parpar-ticles have. In a noninteracting system Vee[ρ] will simply be zero, and T [ρ] will be the known functional of the kinetic energy of the noninteracting system
Ts[ρ] = hΨ0| −1 2∇ 2|Ψ 0i = N X i=1 hψi| −1 2∇ 2|ψ ii (2.68)
where Ψ0 is the wave function, of a many particle system, that minimizes the kinetic energy, and the |ψii is one-electron orbitals which is called the Kohn–Sham
orbitals. Accordingly FHK[ρ] is just Ts[ρ] and Eq. (2.67) becomes
δTs
δρ + vs(r) = µ (2.69)
where vs(r) is an external potential and the link between the interacting and noninteracting systems.
20 Electronic Structure Theory We now again look at the exact functionals:
Vee[ρ] = J[ρ] + nonclassical term (2.70) where J[ρ] is the Coulomb energy of the noninteracting system.
Consider
FHK[ρ] = Ts[ρ] + J[ρ] + Exc[ρ] (2.71) where the exchange-correlation energy is
Exc[ρ] = (T [ρ] − Ts[ρ]) + (Vee[ρ] − J[ρ]) (2.72) Inserting Eq. (2.71) in Eq. (2.67) gives
δTs δρ + δJ δρ + δExc δρ + v(r) = µ (2.73)
and comparing Eq. (2.73) with Eq. (2.69) gives
vs(r) = δJ δρ + δExc δρ + v(r) = Z ρ(r0) |r − r0 |dr 0 +δExc[ρ] δρ + v(r) (2.74)
The Schr¨odinger equation for noninteracting particles with the external poten-tial vs(r) is simply: ³
−1
2∇
2+ vs(r)´|ψ
ii = ²i|ψii (2.75) and the electron density is given by
ρ(r) =
N X i=1
|ψi|2 (2.76)
Eq. (2.74) – (2.76) are the well-known Kohn–Sham equations, and they can be solved by applying a self-consistent field method according to the following: one begins with a guessed ρ(r) and calculates the external potential from Eq. (2.74), and then one calculate the Kohn–Sham orbitals with Eq. (2.75) and a new electron density by using Eq. (2.76). The total energy can be computed, by using Eq. (2.71), from
E[ρ] =
Z
ρ(r)v(r)dr + FHK[ρ] (2.77) A “problem” is that there is no explicit expression for the exact exchange-correlation functional Exc[ρ]. Much work has been put into, and much has been written about, estimating this exchange-correlation functional. There are various approximations proposed. For further reading about this, see Parr and Yang [37] and Jensen [22].
Salek et al. [41] present density-functional theory for linear and nonlinear response functions.
Chapter 3
Response Theory
In writing this chapter, I have mainly used the following sources: Norman and Ruud [34], Boyd [6], Norman at al. [32] and Norman at al. [33].
3.1
Light Interaction with a Molecule
In physics one can consider light in two different ways, as particles (photons) or as electromagnetic waves. In the first way to consider light, as photons, one can understand its interaction with a molecule as scattering. A photon comes in, hits the molecule that for a short moment τ lands in a higher virtual energy level Ei and then transmits a photon with the same energy, as the incoming one, and ends in its ground state |ψ0i. If the molecules first excited state is |ψ1i, with belonging energy E1 = Ei + ∆E, then τ is short enough not to violate the uncertainty principle
τ ∆E v ~ (3.1)
The molecule can also absorb two (or more) photons at the same time and transmit a photon with the double energy (frequency). This is known as second-harmonic generation and is an example of a nonlinear optical process.
If the frequency of the incoming photons increase so that the energy gap ∆E violates Eq. (3.1), then one can not consider the event as instantaneous scattering because the molecule will absorb photons and get in a excited state |ψ1i.
How can one understand lights interaction with a molecule when light is con-sidered as an electromagnetic wave? The light we are considering here is visible or infrared light so it has a wave-length of a few hundred nanometers, while the molecule is just a few nanometer large. The time-changing electromagnetic wave will affect the electrons and the nuclei in the molecule, and their movement, and make them behave as “small antennas” that will transmit electromagnetic radi-ation of the same frequency as the incoming wave. The dipole moment of the molecule becomes
µ(t) = µ0+ αE(t) (3.2)
22 Response Theory where α is the polarizability, and µ0 is the permanent electric dipole moment of the molecule.
But how can nonlinear effects, as second-harmonic generation, be explained when light is being considered as a plane wave? The answer is to generalize Eq. (3.2) as a Taylor series according to
µ(t) = µ0+ αE(t) + 1 2βE 2(t) +1 6γE 3(t) + · · · (3.3)
This can be done because the electric dipole moment of the molecule will alter, when it is exposed to a small perturbation, i.e., when it is exposed to the time-dependent electric field. It is obvious that the electric dipole moment can have the double frequency, or the triple frequency, of the incoming light because of the square, and cubic, dependencies. The physical quantity β is the first-order hyperpolarizability, and γ is the second-order hyperpolarizability.
Since the electric field is vectorial, α will be a second-rank tensor, β will be a third-rank tensor, γ will be a fourth-rank tensor and so on.
One can express the electric field as
Eα(t) =X ω
Eω
αe−iωt (3.4)
where α denotes molecular axis which can be chosen as x-, y- and z-axis, and
Eω
α are the Fourier amplitudes of the electric field. The summation includes both positive and negative frequencies, and because E(t) is real Eω= [E−ω]∗. Generally
E(t) consists of a static component and of one or more time-oscillating components.
If we insert Eq. (3.4) to Eq. (3.3), we get
µα(t) = µ0 α+ X ω ααβ(−ω; ω)Eω βe−iωt +1 2 X ω1,ω2 βαβγ(−ωσ; ω1, ω2)Eβω1Eγω2e−iωσt +1 6 X ω1,ω2,ω3 γαβγδ(−ωσ; ω1, ω2, ω3)Eβω1Eγω2Eδω3e−iωσt+ . . . (3.5)
where we have used the Einstein summation convention, which is used here and elsewhere, and the Greek indices runs over x, y and z. The frequencies ωσdenotes the sum of optical frequencies: for terms involving β then ωσ = (ω1+ ω2), and for terms involving γ then ωσ= (ω1+ ω2+ ω3).
3.2
Off-Resonant Perturbations
In this section we consider light interaction with a molecule that could be consid-ered as a system that is perturbed by a small time-dependent perturbation and that is why we now will study time-dependent perturbation theory.
3.2 Off-Resonant Perturbations 23
3.2.1
Time-Dependent Perturbation Theory
The time evolution of the state vector is given by the time-dependent Schr¨odinger equation
i~∂
∂t|ψi = ˆH|ψi (3.6)
where the Hamiltonian is independent or dependent. When the time-dependence in the Hamiltonian is a small perturbation one can express the solution to the time-dependent Schr¨odinger equation in terms of the eigenstates of the unperturbed system. We separate the Hamiltonian in one time-independent part
ˆ
H0, which is the molecular Hamiltonian for the unperturbed system, and in one time-dependent part ˆV (t), which is a small perturbation, according to
ˆ
H = ˆH0+ ˆV (t) (3.7) In our case, with the electric dipole approximation, the interaction between the molecule and the electric field is given by the operator
ˆ
V (t) = −ˆµαEα(t) (3.8) where we have used Einstein summation, and α = {x, y, z}. That the dipole moment is an operator and the electric field is an amplitude corresponds to the quantum mechanical treatment of the molecule and the classical treatment of the external field.
The wave-function |ψii is denoted |ii, in the following of this chapter, because
of simplicity. We assume that the solutions to the eigenvalue equation ˆ
H0|ni = En|ni (3.9) are known, where |ni is the exact eigenstates and Enis the corresponding energies. Since the set {|ni} is complete we can write the state vector, in any point in time, as
|ψ(t)i =X
n
cn(t)|ni (3.10)
where cn(−∞) = δ0n which guarantee that the molecule is in its ground state |0i at the beginning of time. Our aim is to decide the probabilities for the molecule to be in another state at a later point in time, i.e. to decide |cn(t)|2 for all n. If the system where unperturbed then cn(t) would simply be given by
cn(t) = cn(0)e−iEnt/~ (3.11)
but since we have a small perturbation we write
cn(t) = dn(t)e−iEnt/~ (3.12)
and the state vector is then given by
|ψ(t)i =X
n
24 Response Theory The coefficients dn can be written as
dn(t) = d(0)
n + d(1)n (t) + d(2)n (t) + . . . (3.14) where the superscript in brackets is indicating the order of the perturbation. If we write the state vector as
|ψ(t)i =X
j,n
d(j)
n (t)e−iEnt/~|ni (3.15)
then · i~∂ ∂t − ˆH ¸ |ψ(t)i = 0 ⇐⇒ X j · i~∂d (j) m ∂t − X n eiωmntd(j) n hm| ˆV (t)|ni ¸ = 0 (3.16)
where we have multiplied from the left with the bra vector hm|(eiEmt/~) and where
ωmn=Em− En
~ (3.17)
Now we sort out terms of equal order in Eq. (3.16), which gives
i~∂d (N ) m ∂t = X n eiωmntd(N −1) n hm| ˆV (t)|ni (3.18)
which can be solved recursively since d(0)n = δn0 is known. If we insert the expres-sion for ˆV from Eq. (3.8) in Eq. (3.18) and perform the time integration, after
some simplifying, we get
d(1)m(t) = 1 ~ X ω1 hm|ˆµα|0iEαω1 ωm0− ω1 e i(ωm0−ω1)t (3.19)
and if we insert this result in Eq. (3.18) and perform the time integration we get
d(2)m(t) = 1 ~2 X ω1ω2 X n hm|ˆµα|nihn|ˆµβ|0iEαω1Eβω2 (ωm0− ω1− ω2)(ωn0− ω2)e i(ωm0−ω1−ω2)t (3.20)
We now write the state vector as
|ψ(t)i = |ψ(0)(t)i + |ψ(1)(t)i + |ψ(2)(t)i + · · · (3.21) where
|ψ(N )(t)i =X n
d(N )n e−iEnt/~|ni (3.22) The expectation value of the electric dipole moment operator is of special inter-est to us because we can attain the polarizability ααβ and the hyperpolarizability
βαβγ from it.
3.2 Off-Resonant Perturbations 25 where the superscript in brackets is indicating the order of the perturbation ˆV .
This gives
hˆµi(0)= h0|ˆµ|0i (3.24)
hˆµi(1)= hψ(0)|ˆµ|ψ(1)i + hψ(1)|ˆµ|ψ(0)i (3.25)
hˆµi(2) = hψ(0)|ˆµ|ψ(2)i + hψ(1)|ˆµ|ψ(1)i + hψ(2)|ˆµ|ψ(0)i (3.26) and so on.
The zeroth-order correction to the wave function will simply be given by
|ψ(0)i = e−iE0t/~|0i and the first-order correction to the wave function |ψ(1)i will be given by Eq. (3.19) and Eq. (3.22) so we are now able to derive an expression for the first-order polarization from Eq. (3.25) . After some simplifications, it is given by hˆµαi(1) = hψ(0)|ˆµα|ψ(1)i + hψ(1)|ˆµα|ψ(0)i =X ω1 1 ~ X n · h0|ˆµα|nihn|ˆµβ|0i ωn0− ω1 +h0|ˆµβ|nihn|ˆµα|0i ωn0+ ω1 ¸ Eω1 β e−iω1t (3.27) If we compare Eq. (3.27) with Eq. (3.5) we see that the linear polarizability is given by ααβ(−ω; ω) = 1 ~ X n · h0|ˆµα|nihn|ˆµβ|0i ωn0− ω + h0|ˆµβ|nihn|ˆµα|0i ωn0+ ω ¸ (3.28) In a similar way as we derived the linear polarizability above we can derive the first hyperpolarizability. The second-order correction to the wave function |ψ(2)i is given by Eq. (3.20) and Eq. (3.22), and with this knowledge we can express, after some simplifications, the second-order polarization as
hˆµαi(2) = hψ(0)|ˆµα|ψ(2)i + hψ(1)|ˆµα|ψ(1)i + hψ(2)|ˆµα|ψ(0)i = 1 2 X ω1ω2 1 ~2 X P1,2 X np ·
h0|ˆµα|nihn|ˆµβ|pihp|ˆµγ|0i (ωn0− ω1− ω2)(ωp0− ω2)
h0|ˆµβ|nihn|ˆµα|pihp|ˆµγ|0i (ωn0+ ω1)(ωp0− ω2)
h0|ˆµγ|pihp|ˆµβ|nihn|ˆµα|0i (ωn0+ ω1+ ω2+)(ωp0+ ω2)
¸
Eω1
β Eγω2e−i(ω1+ω2)t (3.29) where the operator P1,2 is permuting the pairs (ˆµβ, ω1) and (ˆµγ, ω2). By a direct comparison between Eq. (3.29) and Eq. (3.5) one can easily see that the first-order hyperpolarization is given by βαβγ(−ωσ; ω1, ω2) = 1 ~2 X P1,2 X np ·
h0|ˆµα|nihn|ˆµβ|pihp|ˆµγ|0i (ωn0− ωσ)(ωp0− ω2)
h0|ˆµβ|nihn|ˆµα|pihp|ˆµγ|0i (ωn0+ ω1)(ωp0− ω2)
h0|ˆµγ|pihp|ˆµβ|nihn|ˆµα|0i (ωn0+ ωσ)(ωp0+ ω2)
¸
26 Response Theory
3.3
Resonant and Near-Resonant Perturbations
As can be seen in Eq. (3.28) the linear polarizability ααβ(−ω; ω) diverges in the resonant region, videlicet when ω approaches the excitation frequency ωn0 for all excitation-energies ~ωn0. As can be seen in Eq. (3.30) the first-order hyperpolar-izability βαβγ(−ωσ; ω1, ω2) also diverges in the resonant region, videlicet when
ω1, ω2 and, most importantly for this thesis, ω1+ ω2 approaches ωn0 for all excitation-energies ~ωn0. We have used standard time-dependent perturbation theory, to achieve Eq. (3.28) and Eq. (3.30), and this theory is only applicable in the non-resonant region and is not valid in the resonant region. To solve this problem, and get expressions for ααβ(−ω; ω) and βαβγ(−ωσ; ω1, ω2) in the reso-nant and near-resoreso-nant region, two strategies are possible, namely to turn to a few-states-model or to include damping terms in the expressions for ααβ(−ω; ω) and βαβγ(−ωσ; ω1, ω2). The strategy to include damping is described below, in a rigorous and in a pragmatic approach.
3.3.1
Rigorous Description in Wave-Function Theory
A rigorous approach for attaining expressions for the polarizability and the first-order hyperpolarizability is to use the density matrix formulation of quantum mechanics. In quantum chemistry, though, we work with the wave function, and not with the density matrix, so in the following of this section an equation of motion that takes relaxation into account is presented in wave-function theory, and the resulting expressions for ααβ(−ω; ω) and βαβγ(−ωσ; ω1, ω2) are non-divergent in the resonant and off-resonant regions.
An alternative way of describing the wave-function is to use unitary operators and express the wave-function as
|ψ(t)i = ei ˆP (t)|0i (3.31) where clearly hψ(t)|ψ(t)i = 1 for all Hermitian operators ˆP (t). To describe the
time-evolution from the ground-state |0i to an arbitary ket vector we do not need to consider all Hermitian operators, as will be seen below, but can restrict ourselves to the Hermitian operators which have the following form
ˆ
P (t) =X
n>0
[Pn(t)|nih0| + P∗
n(t)|0ihn|] (3.32)
Clearly the following expressions are valid ˆ P (t)|0i =X n>0 Pn|ni (3.33) ˆ P (t)2|0i = |0iX n>0 |Pn|2 (3.34)
To realize that Eq. (3.31) is equivalent with Eq. (3.10), apart from a phase factor, the former equation is expressed according to
|ψ(t)i = ei ˆP (t)|0i = |0i cos α + iX
n>0
Pn|nisin α
3.3 Resonant and Near-Resonant Perturbations 27 where
α =sX
n>0
|Pn|2 (3.36)
A complete parametrization can accordingly be written as
|ψ(t)i = ei ˆP (t)|0ieiφ(t) (3.37) where φ(t) is real, but in this thesis we are interested in quantum mechanical averages so φ(t) is not needed.
We will now determine the time dependence of the amplitudes Pn. The equa-tion
∂
∂thψ(t)| ˆΩnm|ψ(t)i =
1
i~hψ(t)|[ ˆΩnm, ˆH]|ψ(t)i (3.38)
is valid for the general state-transfer operator ˆΩnm = |nihm|, and if ˆρ(t) = |ψ(t)ihψ(t)| is introduced Eq. (3.38) can be written as
∂
∂thm|ˆρ|ni =
1
i~hm|[ ˆH, ˆρ]|ni (3.39)
This is the Liouville equation, and it is equivalent to the time-dependent Schr¨odinger equation (Eq. (3.6)), and from it, it is possible to attain the already expressed expressions for the linear polarizability, Eq.(3.28), and the first-order hyperpolar-izability, Eq.(3.30), that are valid in the off-resonant region. With the Liouville equation it is possible to include effects, like relaxation of the excited states and intermolecular collisions, that are not easily included in the Hamiltonian. These interactions are included, without a detailed knowledge about them, by introduc-ing phenomenological dampintroduc-ing accordintroduc-ing to
∂ ∂thm|ˆρ|ni = 1 i~hm|[ ˆH, ˆρ]|ni − γmn(hm|ˆρ|ni − ρ eq mn) (3.40)
where hm|ˆρ|ni relaxes to its equilibrium value ρeq
mn at rate γmn, which is a decay rate. Since there is no decay out of the ground state γ00= 0. For ˆΩnmthe damped equation of motion is ∂ ∂thψ(t)| ˆΩnm|ψ(t)i = 1 i~hψ(t)|[ ˆΩnm, ˆH]|ψ(t)i − γmn[hψ(t)| ˆΩnm|ψ(t)i −hψeq(t)| ˆΩnm|ψeq(t)i] (3.41)
Time-Dependent Perturbation Theory
Perturbation theory will now be used to determine Pn, and to solve the damped equation of motion. The complex amplitudes Pn can be expressed as
28 Response Theory where the superscripts indicates the order of perturbation. The time-dependent wave function can be expressed according to
|ψ(t)i = ei ˆP (t)|0i = |0i + i ˆP (t)|0i −1 2P (t)ˆ 2|0i − i 6P (t)ˆ 3|0i + · · · (3.43) or as |ψ(t)i = |ψ(0)i + |ψ(1)i + |ψ(2)i + · · · (3.44) where |ψ(0)i = |0i (3.45) |ψ(1)i = iX n>0 P(1) n |ni (3.46) |ψ(2)i = iX n>0 P(2) n |ni − 1 2 X n>0 |P(1) n |2|0i (3.47)
To decide the amplitudes Pn(1) Eq. (3.41) is applied to the state-transfer operators ˆ
Ωn0 and ˆΩ0n= ˆΩ†n0 which gives
∂ ∂t[h0| ˆΩn0|ψ (1)i + hψ(1)| ˆΩn0|0i] = 1 i~[h0|[ ˆΩn0, ˆH0]|ψ (1)i + hψ(1)|[ ˆΩn0, ˆH 0]|0i + h0|[ ˆΩn0, ˆV (t)]|0i] −γn0[h0| ˆΩn0|ψ(1)i + hψ(1)| ˆΩn0|0i] (3.48) If we insert Eq. (3.46), evaluate the commutators, and perform an overall complex conjugation we get ∂ ∂tP (1) n + Pn(1)(iωn+ γn0) = − 1 ~hn| ˆV (t)|0i (3.49)
which can be solved, with the initial condition Pn(1) = 0 for t = −∞, according to
P(1) n = e−(iωn+γn0)t t Z −∞ −1 ~hn| ˆV (t 0)|0ie−(iωn+γn0)t0dt0 = −1 ~ ∞ Z −∞ hn| ˆVω|0ie−iωe²t iωn− iω + γn0+ ²dω (3.50)
which gives the first-order correction to the wave function according to
|ψ(1)i = 1 ~ ∞ Z −∞ X n>0 hn| ˆVω|0ie−iωe²t
