STOCHASTIC APPROACH TO MANY-BODY PROBLEMS

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STOCHASTIC APPROACH TO MANY-BODY PROBLEMS

by

Anna Gribkovskaya

Thesis

for the degree of

Master of Science

Faculty of Mathematics and Natural Sciences University of Oslo

August 2018

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To my friends and family

This work is dedicated to my family. Thank you all for your support.

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Acknowledgements

I would like acknowledge all people who bear with me while I was writing this thesis.

And many thanks to my supervisor Morten H. Jensen for his help and support.

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Introduction

Motivation

The Quantum Monte Carlo methods have been known for decades now, since in 1950 Forsythe and Leibler have presented the method for matrix inversion by stochastic approach [12]. However it was first applied to the physical problem in the 1962 by Kalos to compute the ground state for three- and four-body nuclei [19]. Nowadays the Monte Carlo methods are widely used in computational physics and chemistry.

In this thesis we focus on anab initio algorithm to compute the ground state properties of a quantum system known as Coupled Cluster Quantum Monte Carlo (CC- QMC). This algorithm is based on the Full Configuration Monte Carlo (FCIQMC) developed by Booth and Alavi [5] and was first introduced by Thom [36]. In this thesis we use the free electron gas model to test the method. In the thesis we start with a deterministic coupled cluster method and then move to the stochastic algorithm.

Achievements

The main goal in this thesis was to develop the code for implementation of the CCQMC algorithm. The method is rather new and there are just few articles on the topic. The only implementation of the method we were able to find is done by the same researchers who have been developing the method in the firs place. There is no implementation in C++ so we have to write the program from scratch.

Apart from that we also write a solver for deterministic CCD.

The code was written in collaboration with Andrei Kukharenka. We have been also collaborating on some parts of implementation and discussion of results.

Structure of the thesis

The structure of the report is the following:

• Theory.

1

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2 Contents

– Quantum mechanics. Provide a short review on basic concepts and theorems needed for the methods used in the thesis.

– Quantum Dot. A theoretical description of quantum dots.

– The Hartree-Fock theory. In this chapter we discuss the method and also introduce a concept of Hartree-Fock basis.

– Homogeneous Electron Gas. Here we present another quantum system and discuss its theoretical background.

– Coupled Cluster method. Here the deterministic Coupled Cluster method is presented, together with short review on derivation of the equations needed for the method.

– Monte Carlo Methods in Quantum Physics. Here we present various Monte Carlo methods, such as Variational Monte Carlo, Diffusion Monte Carlo, Full Configuration Monte Carlo and Coupled Cluster Monte Carlo.

• Implementation and Results. Here we present some details regarding how the methods were implemented. Also in this part the results and some comments regarding them are presented.

• Conclusion further research. Here we summarize everything that has been done and discuss the possibilities for the future research on the topic.

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Part I Theory

3

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Chapter 1 Quantum mechanics

In the end of 19th century physics had some unsolved problems that couldn’t be tackled using already developed theories and methods. This had led to a signifi- cantly different theory with a number of essential distinctions from that developed before. This new theory was named quantum mechanics. In the classical (or Newto- nian mechanics) there is a theoretical possibility to obtain a complete knowledge of the system under consideration. In quantum mechanics this is not possible, neither for some particular moment in time nor for all other moments in time.

Let’s introduce some concepts here. The uncertainty principle and the probability interpretation of the wave function. The uncertainty principle (or Heisenberg’s uncertainty principle) puts a limit on the precision of our measure- ment of some particular pairs of physical quantities (e.i. position and momentum).

The probability interpretation of the wave function is a bit harder to explain, because one needs a proper mathematical description of the quantum mechanics to understand this. For now we just say that if we consider one single particle in space the probability of finding it in some certain position is related to the wave function. We will derive the expression for the total wave function of the system later in this thesis. Another substantial difference is associated with so-called principle of complementarity. It was formulated by Niels Bohr, one of the founders of the quantum mechanics. It stands that in order to describe a system we need a pair of certain complementary properties which cannot be observed simultaneously. A good example of such a pair are wave and particle properties of light or electrons.

Classical mechanics considered light as a wave and electron as a particle, but this approach failed to explain the photoelectric effect and the diffraction of electrons on a slit. Moreover, position and momentum of the particle also can be considered as a pair of complementary properties. This makes Bohr’s principle of the complementarity closely connected to Heisenberg’s uncertainty principle. Also one should mention that quantum mechanics allows us to change the number of particles in some particular state. For example, we have a creation operator which increases the number of particles in some given state of the system by one, andannihilation operator which decreases this number by one. The last concept to be mentioned here is the word quantum itself. In physics, ”quantum” determines the smallest

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6 Quantum mechanics Chapter 1

possible difference between two values or minimal amount of quantity involved in an interaction. This concept is associated with the revolutionary supposition made by Max Planck back in 1900. He assumed that electromagnetic energy could be emitted in a form of some discrete quantities, known as quantum. He also introduced a pro- portionality coefficient for a minimal energy difference, so-called Planck’s constant h= 6.626×10⁻³⁴ (kg·m²·s⁻¹). As one can see it has a very small value.

How does it possible to have two theories much different from each other and con- tinue to use both of them? This is perfectly fine even it may seems to be a bit contra- dictory. In science there is a rule that require for any new theory to agree with the previous ones under some conditions. This rule is called a correspondence principle. In case of quantum mechanics these conditions are named classical limit orcorrespondence limit. In particular this means that quantum mechanical description of the system should correspond to those obtained by classical theory for large quantum numbers. Mathematically it can be achieved by requirement h→0, whereh Planck’s constant. This principle allows us to determine whether a specific quantum theory is valid or not.

Today we usually say that classical mechanics describes the macroscopic objects and quantum mechanics describes microscopic ones. This arise from the fact that some quantum effects can be observed only for extremely small particles. However this is not enough to describe the difference, because even the observation itself is now different from that in classical physics. For more details on the matter, please refer to [28].

1.1 Quantum theory

In this section we provide a brief description of the main assumptions needed for the quantum theory and some basic notations to be used in the thesis.

As it has been already mentioned quantum theory has been developed though the 20^th century. As any other new theory it is based on some assumptions called the postulates of quantum mechanics (QM). In this thesis we do not aim to provide a detailed description on the topic, however some basic introduction is needed.

Definition 1. Hilbert Space.

LetH be a complex vector space. The inner producthα|βi¹ in that space is defined so that it has the following properties:

1. haα|βi=ahα|βi,a ∈C.

2. hα|bβi=b^∗hα|βi, b ∈C.

3. hα|βi=hβ|αi.

1|αiand|βiare ket-vectors in Dirac notations.

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Section 1.1 Quantum theory 7

4. ||α||² =hα|αi ≥0.

Postulate 1. Every instant state of a system is represented by a vector in Hilbert spaceH .

Comment. This is a very strong demand, because it means that any superposition of the different states is also a state of the system. For example, if|α₁i and|α₂iare vectors describing possible states of the system, then their linear combination|αi is also a state of the same system:

|αi=a₁|α₁i+a₂|α₂i, a₁ and a₂ ∈C.

Definition 2. Operator ˆQ is Hermitian if it satisfies the equation:

Qˆ^†= ˆQ, (1.1)

where ˆQ^† is adjoint of ˆQ.

Postulate 2. Every physical observable is associated with an operator ˆQin a Hilbert space. Action of the operator on state vector results into following eigenvalue equation:

Qˆ|αi=q|αi, (1.2)

where eigenvalues q are the only measurable values associated with the operator.

Eigenvectors determine a complete orthonormal set of vectors for this operator. In addition every operator ˆQ associated with a measurable physical quantity must be a linear Hermitian operator. In particular that means it should posses the following properties:

Qˆ^† = ˆQ, (1.3)

(aQ)ˆ ^† =a^∗Qˆ^†, (1.4)

( ˆQPˆ)^†= ˆP^†Qˆ^†, (1.5) ( ˆQ+ ˆP)^† = ˆQ^†+ ˆP^†, (1.6) wherea∈C, and * denote complex conjugate.

Comment. The eigenvalue equation (1.2) has interesting properties when ˆQ is Her- mitian operator. They are presented in theorems below.

Theorem 1. Set of eigenvalues of any Hermitian operator Qˆ on Hilbert space H is set of real numbers.

Theorem 2. Eigenvectors of any Hermitian operator Qˆ on Hilbert space H that belong to different eigenvalues are orthogonal.

Theorem 3. Set of eigenvectors of any Hermitian operator Qˆ on Hilbert space H can be chosen to be an orthonormal basis for H .

We do not provide proofs for these Theorems. For more detailed description please refer to [3].

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Postulate 3. LetH1 and H2 be the Hilbert spaces corresponding to two systems.

The Hilbert space of joint system is given by tensor product H =H1 ⊗H2. Comment. As it is shown further in this thesis this postulate provides a method to describe many-particle systems.

Postulate 4. The time evolution of the quantum mechanical system is given by the Schr¨odinger equation:

i¯h∂

∂t|Ψi= ˆH|Ψi, (1.7)

where|Ψi=|Ψ(t)i .

In this thesis we do not consider time-evolution of the system and will be focused on the time-independent Schr¨odinger equation or stationary-state equation:

Hˆ|Ψi=E|Ψi, (1.8)

where ˆH in Hamiltonian, |Ψiis state vector of the system and E is the expectation value of the energy. It is the energy spectrum we are interested in, so before solving the Schr¨odinger equation we need to agree on the form of Hamiltonian and also construct the state vector. This is presented in the sections below.

1.1.1 Many-Body problem formulation

As it has been already mentioned all state functions arethe vectors in the Hilbert space. When we deal with a system consisting of many particles we need to define the type of these particles. Bosons are the particles with an integer spin and fermions are the particles with an odd half-integer spin. State vectors of these particles belong to different Hilbert spaces and should be studied independently. In this thesis we consider electrons which are fermions and must obey an exclusion principle, formulated by Wolfgang Pauli in 1925.

Definition 3. The Pauli exclusion principle.

Two or more identical fermions cannot occupy the same quantum state simultaneously in the same system.

Let’s take a closer look at the total wave function and how this principle affect the permutations of the particles withing the system. First we need to mention that the particles are identical and indistinguishable. In quantum mechanics interacting and identical particles are considered indistinguishable, which is different from classical mechanics where all particles are distinguishable. The concept of indistin- guishability requires some discussion about what happens to the wave function if we interchange the particles? At this point a difference between bosons and fermions becomes a significant issue.

In order to make a proper mathematical description and be able to drive properties of wave functions we need to define a new operator for permutation of the particles.

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Section 1.1 Quantum theory 9

Definition 4. Let ˆPij be the operator that interchanges particles i and j.

Pˆ_ij|Ψ(x₁...x_i...x_j...)i=|Ψ(x₁...x_j...x_i...)i Theorem 4. Hermiticity of the permutation operator.

Pˆ_ij is a Hermitian operator in Hilbert space for identical particles.

So that Pˆ_ij⁻¹ = ˆP_ij^†.

Considering this property one may define and solve the eigenvalue equation for the permutation operator.

Pˆ_ij|Ψi=_ij|Ψi, (1.9)

Pˆ_ijPˆ_ij|Ψi=²_ij|Ψi, (1.10)

²_ij = 1→_ij =±1. (1.11)

Definition 5. Symmetricity of wave function.

Ifij = 1, |Ψi considered to be symmetric. In this case it corresponds to bosons.

If _ij = −1, |Ψi considered to be antisymmetric. In this case it corresponds to fermions.

The permutation operator is used in Chapter 3 for the construction of total many fermion wave function and in Chapter 5 for the so-called wave function amplitudes.

The Hamiltonian of many-body system

As it has been already mentioned, the Hamiltonian is a Hermitian operator. It can be expressed as follows:

Hˆ = ˆT + ˆV , (1.12)

where ˆT is the kinetic energy operator and ˆV is the potential energy operator.

Here we assume the electrons are confined by a pure isotropic harmonic oscillator (H.O.) potential. Also in this thesis we consider closed shell systems. It means that all possible single-particle states below a certain level are occupied. Such level often called a Fermi level of the system. In particular this assumption means that addition or removal of one electron to such system requires more energy than same the action in a system with non-occupied lowest levels. Using natural units (¯h =c= e =m_e = 1) one can write Hamiltonian of a such system in Cartesian coordinates as

Hˆ =

N

X

i=1

−1

2∇²_i +1 2ω²r_i²

+

N

X

i<j

1 rij

, (1.13)

N here is number of electrons, ω is oscillator frequency and rij distance between two electrons. The first sum here corresponds to the harmonic oscillator and second sum corresponds to the interaction part. The Hamiltonian can be rewritten as

Hˆ = ˆH₀+ ˆH_I. (1.14)

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More detailed the Hamiltonian can be written as Hˆ =

N

X

i=1

ˆh₀(i) +

N

X

i<j

ˆ

w(i, j), (1.15)

here ˆh0(i) represents the kinetic energy of the particle, possibly an external potential and the ˆw(i, j) term represents the potential energy of the Coulomb interaction between two particles.

1.1.2 Second quantization

The second quantization is a framework that allows us to write long and cumbersome expressions, such as Slater Determinants and many-body Hamiltonians, in a compact way. This is achived by the usage of so-called creation and annihilation operators.

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Definition 6. Creation operator.

We define creation operator as follows:

c^†_i|−i=|ii, (1.16)

where|−i is a true vacuum state.

Creation operator acting on an arbitrary state of some system results into the following expression:

c^†_i |p1p2. . . pNi=|ip1p2. . . pNi (1.17) Definition 7. Annihilation operator is defined as hermitian adjoint to creation operator.

c_i|ii=|−i (1.18)

Some important results following from the definition of the operators:

1. c_i|−i= 0 (no particles).

2. c^†_i|p₁p₂. . . p_Ni= 0 ifi=p_i (particle already exists in state vector).

3. c_i|p₁p₂. . . p_Ni= 0 if i6=p_i (particle does not exist in state vector).

1.2 Operator representation in second quantized form

The Hamiltonian now considered in form (1.15). Omitting the summations and presenting operators in more generic way, it can be written as:

Hˆ = ˆH₀+ ˆW (1.19)

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Section 1.3 Normal ordering and Wick’s theorem 11

where ˆH0 is the so-called one-body term and ˆW is the two-body term. In a second quantized form the one-body term can be written as:

Hˆ0 =

N

X

i

h(i) =ˆ

N

X

pq

hp|hˆ|qic^†_pcq =X

pq

hpqc^†_pcq, (1.20) where

hpq = Z ∞

−∞

φp(x)^∗ˆhφq(x)dx. (1.21) Similarly, the two-body term can be expressed as follows:

Wˆ =

N

X

i<j

ˆ

w(i, j) = 1 2

N

X

pqrs

w^pq_rsc^†_pc^†_qc_sc_r, (1.22) where

w_rs^pq =hpq|wˆ|rsi= Z

dx₁ Z

dx₂φ_p(x₁)^∗φ_p(x₂)^∗ˆhφ_r(x₁)φ_s(x₂). (1.23) As soon as we study fermions it’s more convenient to write the two-body term in an antisymmetric form:

Wˆ = 1 4

X

pqrs

hpq|wˆ|rsi_ASc^†_pc^†_qc_sc_r, (1.24) where

hpq|wˆ|rsi_AS ≡ hpq|wˆ|rsi − hpq|wˆ|sri (1.25) From here on we just use antisymmetric form, so subscriptAS can be omitted.

The Hamiltonian in second quantized form can be then written as:

Hˆ =

N

X

pq

hp|ˆh|qic^†_pc_q+ 1 4

X

pqrs

hpq|wˆ|rsi_ASc^†_pc^†_qc_sc_r (1.26)

1.3 Normal ordering and Wick’s theorem

As it has been already mention we need second quantization to write long expressions in a compact way. However, we also need rules to deal with this expression written in a second quantized form to compute for example matrix elements of Hamiltonain matrix. After writing the Hamiltonian in a second quantized form we are able use for this purpose the following anti-commutator relations:

{cp, cq}= 0 (1.27)

{c^†_p, c^†_q}= 0 (1.28)

{c_p, c^†_q}=δ_pq (1.29)

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Equation (1.29) is a fundamental anti-commutator relation. At the same time when the number of particles growing larger this might become too hard to compute even after all simplification have been done so far. There is an easier way to compute matrix elements. To present it we have to introduce some concepts first.

Definition 8. Vacuum expectation value.

For some arbitrary operator ˆO written as string of operators C₁. . . C_N, such that C_i ∈ {c^†_p} ∪ {c_p} is defined as follows: h−|O|−iˆ =h−|C₁C₂. . . C_N|−i.

Using the definition above, matrix elements can be obtained by the following expression:

hΦ|Hˆ0|Φi=X

pq

hp|ˆh|qi h−|cN. . . c1c^†_pcqc^†₁. . . c^†_N|−i, (1.30) and

hΦ|Wˆ |Φi= 1 4

X

pqrs

hpq|wˆ|rsi h−|c_N. . . c₁c^†_pc^†_qc_sc_rc^†₁. . . c^†_N|−i. (1.31) As one can see from (1.30) and (1.31) matrix elements are written in form of vacuum expectation value. After this we introduce Wick’s Theorem, which allows us to compute these values using normal ordered operators.

Definition 9. Normal ordering.

Let ¯C = C₁. . . C_n be an arbitrary operator string consisting of creation and annihilation operators. Let σ ∈ S_n be a permutation, that results in all the creation operators in the string ¯C be on the left side and all the annihilation operators to the right side. Normal ordered string denoted using the braces as {C₁. . . C_n}. Normal ordering is defined as:

{C₁. . . C_n} ≡(−1)^|σ|[creation operators]×[annihilation operators] (1.32) One should remember that normal order is not a unique sequence of operators, since it is possible to arrange them in different ways.

Another important concept we need to mention before we can go to Wick’s theorem is contraction between operators.

Definition 10. Contraction.

Contraction between two operators is a difference between their current order and a normal order:

XY =XY − {XY}. (1.33)

For creation and annihilation operators one may write four different possible contractions:

c_pc_q=c_pc_q− {c_pc_q}= 0, (1.34) c^†_pc^†_q=c^†_pc^†_q− {c^†_pc^†_q}= 0, (1.35) c^†_pc_q=c^†_pc_q− {c^†_pc_q}= 0, (1.36) c_pc^†_q=c_pc^†_q− {c_pc^†_q}=δ_pq. (1.37)

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Section 1.3 Normal ordering and Wick’s theorem 13

As one can see the only possible non zero contraction is the last one as it correspond to the anti-commutator relation (1.29) above.

Below, the contraction inside a normal ordered string is defined.

Definition 11. Contraction inside the operator string.

Let ¯C = C₁. . . C_n be an arbitrary operator string consisting of creation and annihilation operators. Let (C_q, C_p) be a pair of operators and σ be any possible permutation that places C_q to the first place in the string and C_p to the second.

{C₁. . . C_q. . . C_p. . . C_n} ≡(−1)^|σ|{C_qC_pC_σ(3). . . C_σ(n)}. (1.38) For an arbitrarym contractions inside one string we have:

m contractions

z }| {

{C₁. . . C_n}= (−1)^|σ|{C_p₁C_q₁. . . C_p_mC_q_mC_σ(2m+1). . . C_σ(n)}. (1.39) Now we can finally state Wick’s theorem.

Theorem 5. Wick’s theorem.

Any operator string can that contains creation and annihilation operators can be also written as sum of a normal ordered product of these operators and all possible contractions inside this normal ordered product.

Let C¯ =C₁. . . C_n be an arbitrary operator string consisting of creation and annihilation operators.

C₁. . . C_n ={C₁. . . C_n}+ X

all single contractins

one contraction

z }| {

{C₁. . . C_n}+ (1.40)

X

all double contractins

two contraction

z }| {

{C₁. . . C_n}+· · ·+ X

all ⁿ₂ contractins

n

2 contraction

z }| {

{C₁. . . C_n}. (1.41)

Outcomes from Wick’s theorem:

1.

h−| {C₁. . . C_n} |−i= 0.

2.

h−|C₁. . . C_n|−i= 0,∀ odd n.

3.

h−|C₁. . . C_n|−i=X

n 2

all contraction

z }| {

{C₁. . . C_n},∀ even n.

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For the derivation of the coupled cluster equations we need to consider a product of normal-ordered strings. To do this efficiently we also state a generalized Wick’s theorem.

Theorem 6. Generalized Wick’s theorem.

The generalized Wick’s theorem extends the ordinary Wick’s theorem for the case of multiple products of normal ordered strings. In this case the only valid contractions are those between the different strings.

Let’s consider a set of operator strings. C₁¹...C_i¹, C₁²...C_j² and C₁ⁿ...C_kⁿ. Here n is total number of strings. Then if we need to evaluate the following product of a set of normal-ordered strings:

{C₁¹. . . C_i¹}{C₁². . . C_j²}...{C₁ⁿ. . . C_kⁿ}={C₁¹. . . C_i¹|C₁². . . C_j²|...|C₁ⁿ. . . C_kⁿ}+

X

all single contractins

one contraction between strings

z }| {

{C₁¹. . . C_kⁿ}={C₁¹. . . C_i¹|C₁². . . C_j²|...|C₁ⁿ. . . C_kⁿ}+

X

all double contractins

two contractions between strings

z }| {

{C₁¹. . . C_kⁿ}={C₁¹. . . C_i¹|C₁². . . C_j²|...|C₁ⁿ. . . C_kⁿ}+

· · ·+ X

all ⁿ₂ contractins

n

2 contractions between strings

z }| {

{C₁¹. . . C_kⁿ}={C₁¹. . . C_i¹|C₁². . . C_j²|...|C₁ⁿ. . . C_kⁿ}, (1.42) where contractions between strings mean we are only considering contractions of the a following type:

{C¹₁. . . C_i¹|C²₁. . . C_j²|...|C₁ⁿ. . . C_kⁿ}, (1.43) {C¹₁. . . C_i¹|C₁². . . C_j²|...|C₁ⁿ. . . C_kⁿ}, (1.44) {C¹₁. . . C¹_i|C₁²...C_j²|...|C₁ⁿ. . . C_kⁿ}, (1.45) and so on.

1.4 Normal-Ordered Electronic Hamiltonian

In section 1.2 we have presented the operator representation in the second quantized form. Equation (1.26) that provides the second quantized form of the electronic Hamiltonian can be rewritten using Wick’s theorem as the normal ordered operator string. This is a very convenient approach for the derivation of the coupled cluster equations that are provided in Chapter 5.

Let’s start with the one-electron part given by equation (1.20):

Hˆ₀ =X

pq

hp|ˆh|qi {c^†_pc_q}+X

i

hi|h|iiˆ =X

pq

h_pq{c^†_pc_q}+X

i

h_ii (1.46)

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Section 1.5 Normal-Ordered Electronic Hamiltonian 15

The second term in the Hamiltonian equation is the two-body part given by (1.24).

One can rewrite it using Wick’s theorem as follows:

c^†_pc^†_qc_sc_r ={c^†_pc^†_qc_sc_r}+{c^†_pc^†_qc_sc_r}+{c^†_pc^†_qc_sc_r}+

{c^†_pc^†_qc_sc_r}+{c^†_pc^†_qc_sc_r}+{c^†_pc^†_qc_sc_r}+{c^†_pc^†_qc_sc_r} (1.47) Remembering that the contraction is non-zero only for the operator acting on hole state to the left, we may rewrite (1.47) as:

{c^†_pc^†_qc_sc_r} −δp∈iδ_ps{c^†_qc_r}+δq∈iδ_qs{c^†_pc_r}+δp∈iδ_pr{c^†_qc_s} −

δq∈iδqr{c^†_pcs} −δp∈iδpsδq∈jδqr+δp∈iδprδq∈jδqs (1.48) whereq ∈j (the indexj belongs to an occupied state) andδq∈j (equalityq=j must hold). After this we may rewrite the two-body term in the Hamiltonian and obtain:

1 4

X

pqrs

hpq|rsi {c^†_pc^†_qc_sc_r} −1 4

X

qri

hiq|rii {c^†_qc_r}+ 1 4

X

pri

hpi|rii {c^†_pc_r}+ +1

4 X

qsi

hiq|isi {c^†_qc_s} − 1 4

X

psi

hpi|isi {c^†_pc_s} − 1 4

X

ij

hij|iji+1 4

X

ij

hij|jii

= 1 4

X

pqrs

hpq|rsi {c^†_pc^†_qc_sc_r}+X

pri

hpi|rii {c^†_pc_r}+1 2

X

ij

hij|iji. (1.49) And finally the Hamiltonian (1.26) can be rewritten as follows:

Hˆ =X

pq

h_pq{c^†_pc_q}+X

i

h_ii+ 1 4

X

pqrs

hpq|rsi {c^†_pc^†_qc_sc_r}+X

pri

hpi|rii {c^†_pc_r}+ 1

2 X

ij

hij|iji=X

pq

f_pq{c^†_pc_q}+1 4

X

pqrs

hpq|rsi {c^†_pc^†_qc_sc_r}+hΦ₀|H|Φ₀i,(1.50) with

F_N=X

pq

f_pq{c^†_pc_q}=X

pq

h_pq{c^†_pc_q}+X

i

hpi|qii {c^†_pc_q}

. (1.51)

F_N is the normal-ordered Fock operator. It is discussed in more detailed manner in Chapter 3.

After this the normal-ordered Hamiltonian can be written as:

Hˆ_N = ˆH− hΦ₀|H|Φ₀i (1.52)

One may say that the normal-ordered form of the operator is obtained by subtracting the reference expectation value of this operator from the operator itself. In this case Hˆ_N may be referred to as correlation operator.

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1.5 Particle-Hole representation

Definition 12. Fermi vacuum.

Let|Φ₀ibe n-electron reference determinant constructed from the true vacuum|−i.

It can be written as a string of creation operators acting on a true vacuum:

|Φ₀i=c^†_ic^†_j..|−i.

Such reference determinant is often called a ”Fermi vacuum”.

The state |Φ₀i is composed by occupied orbitals. However they are chosen from a set of single-particle functions that contains also other functions. This additional functions are called virtual orbitals. There is a convention regarding labeling this occupied and virtual orbitals. For occupied orbitals we use name ”hole states” and for virtual orbitals we use ”particle states”. Hole states are labeled with lettersi,j,k and particle states are labeled with lettersa,b,c.

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Chapter 2 Quantum Dot

2.1 Introduction to Quantum Dots

The mathematical description of quantum dots (QD) is presented below, but in the most general sense one may say that QDs are man-made devices that are small enough to posses quantum properties, such as energy shell structure, tunneling effect and etc. Most commonly such devices are fabricated using semiconductors and their size vary from few nanometers to hundreds of nanometers (one nanometer or nm is equal to 1×10⁻⁹m). In literature one may find name ”artificial atoms” when referring to such semiconductor nanostructures. This name reflect the fact that QDs and atoms share many similar properties, however this is not completely legit name, though QD are larger then atoms. For atoms size is usually measured in picometres (one picometre or pm is equal to 1×10⁻¹²m). Normal size of an atom vary form 53 pm for hydrogen atom (this quantity is also known as Bohr radius), to 273 pm for cesium (which is considered to be one of the largest atoms). As one can mention even the small QDs correspond 10 atoms in diameter. Apart from this QDs are very similar to the atoms. The name Quantum Dot reflect the fact that we have a structure that is small enough to have quantum properties and also that this structure is spatially localized. The properties of QDs lie between those of individual discrete atoms or molecules and bulk semiconductors. This fact make such particles matter of great interest both for science and industry.

In this part we provide a theoretical description of two-dimensional quantum dots.

However it’s worth consider first what are quantum dots and why are they so interesting. In literature QDs are sometimes called artificial atoms. This comes from the fact that QD share many of their properties with real atoms despite being artificially created. The most commonly QDs are composed by using elements from periodic table of groups II-VI, III-V and IV-VI. For example, GaAs, InAs, ZnS, CdSe and etc.

Today we have many types of QDs, with a large field of application. It is a growing research area. History of quantum dots traces back to 1980, when they were first discovered in glass crystals [11]. However this discovery doesn’t result in immediate blow up of the research on the topic. It took quite a time before Murray et al.

17

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18 Quantum Dot Chapter 2

[27] managed to make a colloidal QD. Since then the interest to QDs is constantly growing. Today QDs have a large field of applications from medicine to quantum computing. QDs are used in lasers, for solid state lighting, for solar cells and also for biological and medical applications [38].

2.1.1 Mathematical description of Quantum Dots

Before presenting equations for quantum dots we should make some basic assumptions. The main approximation considers the form of Hamiltonian of the system.

• Electrons are confined by Harmonic Oscillator potential V(r)_HO = ^mω₂²^r².

• Electrons interact via two-body Coulomb potential V(r_ij) = _r¹

ij.

• The Hamiltonian is considered to be two-dimensional.

• The HO potential is spherically symmetric, with parabolic quantum dot.

• External magnetic field is not present.

The mathematical description of Quantum Dots has been provided in many other master thesis, for example in [23], so we are not aiming to derive all the equations and present a detailed description here. We have already presented the electronic Hamiltonian of in Section (1.2). Fro QD the one-body operator corresponds to kinetic energy and external potential and a two-body operator corresponds to Coulomb interaction between two particles:

Hˆ₀ = ˆT + ˆV_HO → 1

2r²− ∇²_r

2 , (2.1)

Vˆ =X

i<j

1

r_ij, (2.2)

here r is position of the particle, r_ij is distance between particles. In the previous Chapter we mentioned single-particle wave functions. These functions can be obtained using the one-body part of the Hamiltonian and are well-known functions corresponding to so-called quantum Harmonic oscillator. For a more detailed infor- mation please refer to [30]. The single particle wave functions in polar coordinates can be then written as:

φ_SP(r, θ) =

2n!

(n+|m|)!

¹₂ 1

2πe^imθr^|m|L^|m|_n (r²)e^−r

2

2 , (2.3)

here r=|r|,L_n are the Laguerre polynomials, n and m are magnetic and principal quantum numbers respectively. The single particle energy can be presented as :

(i) = 1 +|m_i|+ 2n_i, (2.4)

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Section 2.1 Introduction to Quantum Dots 19

the single particle energy is measured in units ¯hω. In Fig. 2.1 the shell structure of QD is presented. The two-body matrix elements are computed using the algorithm presented in the article [2]. The details for this calculations are provided in Appendix B.

Table 2.1: Quantum numbers for the single-particle basis using a harmonic oscillator in two dimensions.

Shell number (n, m) Energy Degeneracy N

7 (0,−6) (1,−4) (2,−2) (3,0) (2, 2) (1, 4) (0, 6) 7¯hω 14 56 6 (0,−5) (1,−3) (2,−1) (2, 1) (1, 3) (0, 5) 6¯hω 12 42

5 (0,−4) (1,−2) (2,0) (1, 2) (0, 4) 5¯hω 10 30

4 (0,−3) (1,−1) (1,1) (0, 3) 4¯hω 8 20

3 (0,−2) (1,0) (0, 2) 3¯hω 6 12

2 (0,−1) (0, 1) 2¯hω 4 6

1 (0,0) ¯hω 2 2

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Chapter 3 The Hatree-Fock theory

Here we present a brief overview of the Hatree-Fock (HF) theory. It is a well-known and defined way to study systems of large number of particles. Hartree-Fock method is the simplest and nevertheless rather efficient methods. Initially introduced by Hartree and then improved by Fock it is one of the most popularab initio methods in quantum chemistry. It is easy to implement, but has some disadvantages, for example it fails to provide high accuracy. However, more precise methods are often built on the HF results. This makes HF theory a good starting point for anyone who wants to tackle a many-body problem. Methods that are using HF as an input are usually referred to as post-Hartree-Fock. Among them are Configuration Interaction (CI) and Coupled Cluster (CC) theories.

The main idea of HF theory is to approximate the unknown wave function with a single Slater Determinant constructed using a single-particle wave functions representing the occupied states in a system under consideration.

3.1 Introduction to HF

Here we start with the time independent Schr¨odinger equation for the ground state:

Hˆ |Φ₀i=E₀|Φ₀i (3.1)

In order to derive the HF equations we approximate the ground state wave function with a single SD:

|Φ₀i=

N

Y

i=1

c^†_i |−i=|φ₀, . . . φ_Ni (3.2) Hereφ₀, . . . φ_N are single-particle wave functions.

This method uses an approximation to the exact many-body wave function by a Slater determinant of N orthonormal single-particle wave functions. In this case

21

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22 The Hatree-Fock theory Chapter 3

the approximate wave function of the system is given by:

Φ(x₁, x₂, . . . , x_N, α, β, . . . , σ) = 1

√N!

ψ_α(x₁) ψ_α(x₂) . . . ψ_α(x_N) ψ_β(x₁) ψ_β(x₂) . . . ψ_β(x_N)

. . . . . . . . ψ_σ(x₁) ψ_σ(x₂) . . . ψ_σ(x_N)

, (3.3)

In this equationψ(x_i) stands for the single electron wave function, x_i stand for the coordinates and spin values of a particleiandα, β, , σare quantum numbers needed to describe remaining quantum numbers. However this expression can be simplified by introducing a new operator ˆA. This operator is given by

Aˆ= 1 N!

X

p

(−)^pP ,ˆ (3.4)

here the sum goes over all possible permutations of two particles and p stands for the number of permutations.

We also need to introduce the so-called Hartree-Fock wave function, which is given by the product of all possible single-particle wave functions

Φ_H(x₁, x₂, . . . , x_N, α, β, . . . , ν) = ψ_α(x₁)ψ_β(x₂). . . ψ_ν(x_N). (3.5) Using these definitions, the Slater determinant can be rewritten as

Φ(x₁, x₂, . . . , x_N, α, β, . . . , ν) = 1

√ N!

X

P

(−)^PP ψˆ _α(x₁)ψ_β(x₂). . . ψ_ν(x_N) =√

N! ˆAΦ_H, (3.6) Using a Slater determinant and assuming the Hamiltonian is given on the form of (1.13), we obtain the functional E[Φ] for the energy. According to the variational principle

E[Φ]≥E₀. (3.7)

hereE₀denotes the exact ground state energy. There are two main strategies we may use now in order to obtain the ground state energy. In order to find the minimum of the energy functional we may either vary a Slater determinant, or we may expand the single-particle functions in some known basis and then vary the coefficients of the expansions. In this thesis we use the second method. For those who want more detailed insight in the theory presented above in this chapter, please take a look at chapter 15 in [26].

3.2 Derivation of a Hartree-Fock equations

In order to derive the Hartree-Fock equations we need to chose an orthogonal basis to be used for the expansion. As soon as we have H.O. potential that confine particles

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Section 3.2 Derivation of a Hartree-Fock equations 23

in the system it’s reasonable to choose H.O. functions as a basis functions for the expansion as well. In this case we can be sure the functions are orthogonal by nature, that is

ψ_p =X

λ

C_pλφ_λ., (3.8)

whereφ_λ is our new basis functions andC_pλrepresent expansion coefficients. This is a very important property for us, as we will vary the coefficients in order to minimize the energy. The sum in (3.8) goes to infinity, however we will use a truncation to a certain value.

Using these definitions we may write the energy functional as E[Φ] =

N

X

µ=1

hµ|ˆh0|µi+1 2

N

X

µ=1 N

X

ν=1

hµν|ˆv|µνiAS. (3.9) Hereµandνare basis functions we use in the Slater determinant Φ defined in (3.6).

After expansion in a new basis the functional for the energy turns into:

E[Ψ] =

N

X

i=1 Z

X

αβ

C_iα^∗ C_iβhα|h|βi+1 2

N

X

ij=1 Z

X

αβγδ

C_iα^∗ C_jβ^∗ C_iγC_jδhαβ|ˆv|γδi_AS. (3.10) Here Ψ is a new Slater determinant andα, β, γ, δ correspond to new basis functions and Z is energy cut-off representing the total possible number of states in a chosen basis. The coefficientsC are expansion coefficients from (3.8). They form a unitary matrix that performs a transformation to a new basis and also preserve orthogonality of the basis functions. This allows us to use the Lagrange multiplier method to find the local minimum of the energy functional. We use orthogonality requirement as a constraint and set up a Lagrange functional, were all multipliers have to be in units of energy as soon as matrix C contains only some numbers. Those energies are often called Hartree-Fock single particle energies. They are different from the single-particle energies corresponding to the basis functions because we chose the basis to be H.O. functions without any perturbations. However as it discussed below this new single-particle energies follow the same degeneracy pattern as a pure H.O. energies for the system (below the Fermi level). After applying the Lagrange multipliers method we obtain the following expression

Z

X

β

C_iβhα|h|βi+

N

X

j=1 Z

X

βγδ

C_jβ^∗ C_jδC_iγhαβ|ˆv|γδi_AS =^HF_i C_iα. (3.11)

here ^HF_i are new single-particle energies. Now we define Hartree-Fock matrix as h^HF_αβ =hα|h|βi+

N

X

j=1

X

γδ

C_jγ^∗ C_jδhαγ|ˆv|βδi_AS, (3.12)

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24 The Hatree-Fock theory Chapter 3

once we obtain a matrix eigenvalue problem:

ˆh^HFC =^HFC. (3.13)

To simplify computations in our program we use so called density matrix. It is defined as

ρ_γδ =X

i≤F

hγ|iihi|δi=X

i≤F

C_iγC_iδ^∗. (3.14)

In this case Hamiltonian can be rewritten as ˆh^HF_αβ =_αδ_α,β+X

γδ

ρ_γδhαγ|ˆv|βδi_AS. (3.15)

3.3 Hartree-Fock basis

After we have implemented the Hartree-Fock algorithm, we have compute the ground state energy and also have a coefficient matrixC. This coefficient allow us to perform a transformation to a Hartree-Fock basis. This can be done as follows:

hab|cdi=X

ijkl

C_i^aC_j^bC_k^cC_l^dhij|kli. (3.16) Naive implementation of the formula above scales asN⁸. However one may mention that there are only one common index for each coefficient and the TBME so the transformation can be dome by performing intermediate computations such as:

haj|kli=X

i

C_i^ahij|kli, (3.17)

hab|kli=X

j

C_j^bhaj|kli, (3.18)

hab|cli=X

k

C_k^chab|kli, (3.19)

hab|cdi=X

l

C_l^dhab|cli. (3.20)

Such transformations scale asN⁵.

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Chapter 4 Homogeneous Electron Gas

The Homogeneous Electron Gas (HEG) or, as it sometimes called, the free electron gas is a very useful model in condensed matter physics as it allows to study the many-fermion system without additional complication caused by lattice symmetry.

This makes it useful for studies of metals, i.e modeling the properties of valence electrons.

Below are the main assumptions to be made for this model:

• We assume to have a certain number of electrons N_ein a cubic box of a certain length L. Volume of the cube is Ω =L².

• No external forces are present, except those provided by background ions.

The density of the background charge is constant and given by N/Ω, here N is number of ions.

• The System is neutral and ions are stationary.

The HEG model allows us to solve the Hartree-Fock equations for system of many interacting particles in the analytical form. Additionally it also allows us to get the total energy and Hamiltonian matrix elements in the basis for Hartree-Fock. This make the model one of the best options to implement a so called post Hartree-Fock methods, for example, coupled cluster (CC), full configuration interaction (FCI) and Monte Carlo methods for many-body problems. These properties make the HEG a perfect system to test the many-body solvers before using it for other systems.

A theoretical description of the HEG in this chapter is based on the lecture notes of S.Kvaal [29].

4.1 Hamiltonian for Homogeneous Electron Gas

The Hamiltonian for the HEG is given by:

Hˆ = ˆH_el+ ˆH_b+ ˆHel−b, (4.1) 25

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26 Homogeneous Electron Gas Chapter 4

with ˆHel being the electronic part given by:

Hˆel=

N

X

i=1

p²_i 2m +e²

2 X

i6=j

e^−µ|rⁱ^−r^j^|

|r_i−r_j|, (4.2)

and ˆH_b being the operator corresponding to the background charge from the ions, given by:

Hˆ_b = e² 2

Z Z

drdr⁰n(r)n(r⁰)e^−µ|r−r⁰^|

|r−r⁰| , (4.3)

and ˆH_el−b being the operator corresponding to interactions between electrons and the positive background charge, given by:

Hˆel−b =−e² 2

N

X

i=1

Z

drn(r)e^−µ|r−xⁱ^|

|r−x_i| , (4.4)

hereµis a convergence factor, n(r) is background charge density. In the thermody- namical limit µ→0.

The single-particle wave functions are given py plane wave:

ψ_kσ(r) = 1

√Ωexp (ikr)ξ_σ, (4.5)

here kis a wave number and ξσ denotes spin (up and down):

ξ_σ=+1/2 = 1

0

ξ_σ=−1/2 = 0

1

. (4.6)

The periodic boundary conditions are assumed, so that wave numbers are only allowed to have some certain values:

k_i = 2πn_i

L i=x, y, z n_i = 0,±1,±2, . . . (4.7) The single-particle energy is then given by:

εnx,ny,nz = ¯h² 2m

2π L

2

n²_x+n²_y+n²_z

. (4.8)

The antisymmetrized matrix elements are given by:

hk_pm_s_pk_qm_s_q|˜v|k_rm_s_rk_sm_s_si_AS (5)

= 4π

L³δ_k_p_+k_q_,k_r_+k_s

δ_m_sp_m_srδ_m_sq_m_ss 1−δ_k_p_k_r 1

|k_r−k_p|²

−δ_m_sp_m_ssδ_m_sq_m_sr 1−δ_k_p_k_s 1

|k_s−k_p|²

, (4.9)

here δ_k_p_k_r and δ_k_p_k_s are Kronecker delta functions.

Table 4.1 presents the shell structure for the HEG in three dimensions.

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Section 4.1 Hamiltonian for Homogeneous Electron Gas 27

n²_x+n²_y+n²_z nx ny nz N↑↓

0 0 0 0 2

1 -1 0 0

1 1 0 0

1 0 -1 0

1 0 1 0

1 0 0 -1

1 0 0 1 14

2 -1 -1 0

2 -1 1 0

2 1 -1 0

2 1 1 0

2 -1 0 -1

2 -1 0 1

2 1 0 -1

2 1 0 1

2 0 -1 -1

2 0 -1 1

2 0 1 -1

2 0 1 1 38

Table 4.1: Single-particle state energies for HEG in atomic units. N↑↓stands for the total number of spin-orbitals.

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Chapter 5 Coupled Cluster method

Coupled cluster (CC) method for quantum chemistry was first introduced by J. Cizek in the late 1960s [7]. A few years later he published a new article on the topic in collaboration with and J. Paldus [8]. CC is anab initio numerical method widely used for approximate solution of the electronic Schr¨odinger equation because it is both reliable and computationally affordable. This section is based on a very detailed and comprehensive overview of the method provided by Crawford and Schaefer in [10].

The coupled cluster method is based on the same basic concepts that are underlying many other many-body methods, such as many-body perturbation theory and full configuration interaction. The main critical difference is the use of the ”exponential ansatz” of the wave function which is discussed below.

In this chapter we discuss some critical ideas for the CC method, such as cluster expansion of the wave function , exponential ansatz, Campbell-Baker-Hausdorff (BCH) expansion, second quantization and particle-hole formalism, normal-ordering and correlation operator in application for the CC method. Some of these ideas we have already mentioned in our previous chapters and some are completely new. The CC method can be described in two different ways using algebraic and diagrammatic formalisms. Both are correct and provide the same results, but the diagrammatic one is way faster to apply. However, for the understanding of the method and its origin we need to begin with algebraic form of the equations.

5.1 Cluster functions and Exponential Anzats

As we have already discussed above, the Slater Determinant can be used to describe a wave function of the electrons. In Dirac notations it can be written as follows:

Φ₀ =|φ_i(x₁)φ_j(x₂)...φ_l(x_n)i (5.1) hereφi(x1) is a one-electron wave function that describes the motion of each electron separately, the x₁ is a vector of coordinates, both spatial and spin.

Keeping in mind that we are working with fermions, the electronic wave function 29

STOCHASTIC APPROACH TO MANY-BODY PROBLEMS