diff --git a/Notes/PerturbativeAnalysis.tex b/Notes/PerturbativeAnalysis.tex new file mode 100644 index 0000000..703dc97 --- /dev/null +++ b/Notes/PerturbativeAnalysis.tex @@ -0,0 +1,335 @@ +\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,mleftright} +\usepackage[version=4]{mhchem} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{txfonts} + +\usepackage[ + colorlinks=true, + citecolor=blue, + breaklinks=true + ]{hyperref} +\urlstyle{same} + +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\eg}{\textit{e.g.}} +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\usepackage[normalem]{ulem} +\newcommand{\titou}[1]{\textcolor{red}{#1}} +\newcommand{\trashPFL}[1]{\textcolor{r\ed}{\sout{#1}}} +\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} + +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\QP}{\textsc{quantum package}} +\newcommand{\T}[1]{#1^{\intercal}} + +% coordinates +\newcommand{\br}{\boldsymbol{r}} +\newcommand{\bx}{\boldsymbol{x}} +\newcommand{\dbr}{d\br} +\newcommand{\dbx}{d\bx} + +% methods +\newcommand{\GW}{\text{$GW$}} +\newcommand{\evGW}{ev$GW$} +\newcommand{\qsGW}{qs$GW$} +\newcommand{\GOWO}{$G_0W_0$} +\newcommand{\Hxc}{\text{Hxc}} +\newcommand{\xc}{\text{xc}} +\newcommand{\Ha}{\text{H}} +\newcommand{\co}{\text{c}} +\newcommand{\x}{\text{x}} +\newcommand{\KS}{\text{KS}} +\newcommand{\HF}{\text{HF}} +\newcommand{\RPA}{\text{RPA}} + +% +\newcommand{\Ne}{N} +\newcommand{\Norb}{K} +\newcommand{\Nocc}{O} +\newcommand{\Nvir}{V} + +% operators +\newcommand{\hH}{\Hat{H}} +\newcommand{\hS}{\Hat{S}} +\newcommand{\ani}[1]{\hat{a}_{#1}} +\newcommand{\cre}[1]{\hat{a}_{#1}^\dagger} +\newcommand{\no}[2]{\mleft\{ \hat{a}_{#1}^{#2}\mright\} } + +% energies +\newcommand{\Enuc}{E^\text{nuc}} +\newcommand{\Ec}[1]{E_\text{c}^{#1}} +\newcommand{\EHF}{E^\text{HF}} + +% orbital energies +\newcommand{\eps}{\epsilon} +\newcommand{\reps}{\Tilde{\epsilon}} +\newcommand{\Om}{\Omega} + +% Matrix elements +\newcommand{\SigC}{\Sigma^\text{c}} +\newcommand{\rSigC}{\Tilde{\Sigma}^\text{c}} +\newcommand{\MO}[1]{\phi_{#1}} +\newcommand{\SO}[1]{\psi_{#1}} +\newcommand{\eri}[2]{\braket{#1}{#2}} +\newcommand{\aeri}[2]{\mel{#1}{}{#2}} +\newcommand{\ERI}[2]{(#1|#2)} +\newcommand{\rbra}[1]{(#1|} +\newcommand{\rket}[1]{|#1)} + + +% Matrices +\newcommand{\bO}{\boldsymbol{0}} +\newcommand{\bI}{\boldsymbol{1}} +\newcommand{\bH}{\boldsymbol{H}} +\newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}} +\newcommand{\be}{\boldsymbol{\epsilon}} +\newcommand{\bOm}{\boldsymbol{\Omega}} +\newcommand{\bA}{\boldsymbol{A}} +\newcommand{\bB}{\boldsymbol{B}} +\newcommand{\bC}{\boldsymbol{C}} +\newcommand{\bD}{\boldsymbol{D}} +\newcommand{\bF}{\boldsymbol{F}} +\newcommand{\bU}{\boldsymbol{U}} +\newcommand{\bV}{\boldsymbol{V}} +\newcommand{\bW}{\boldsymbol{W}} +\newcommand{\bX}{\boldsymbol{X}} +\newcommand{\bY}{\boldsymbol{Y}} +\newcommand{\bZ}{\boldsymbol{Z}} +\newcommand{\bc}{\boldsymbol{c}} + +% orbitals, gaps, etc +\newcommand{\IP}{I} +\newcommand{\EA}{A} +\newcommand{\HOMO}{\text{HOMO}} +\newcommand{\LUMO}{\text{LUMO}} +\newcommand{\Eg}{E_\text{g}} +\newcommand{\EgFun}{\Eg^\text{fund}} +\newcommand{\EgOpt}{\Eg^\text{opt}} +\newcommand{\EB}{E_B} + +% shortcuts for greek letters +\newcommand{\si}{\sigma} +\newcommand{\la}{\lambda} + + +\newcommand{\RHH}{R_{\ce{H-H}}} +\newcommand{\ii}{\mathrm{i}} + +% addresses +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} + +\begin{document} + +\title{Perturbative Analysis of the Similarity Renormalisation Group} + +\author{Antoine \surname{Marie}} + \email{amarie@irsamc.ups-tlse.fr} + \affiliation{\LCPQ} + +\author{Pierre-Fran\c{c}ois \surname{Loos}} + \email{loos@irsamc.ups-tlse.fr} + \affiliation{\LCPQ} + +%\begin{abstract} +%Here comes the abstract. +%\bigskip +%\begin{center} +% \boxed{\includegraphics[width=0.5\linewidth]{TOC}} +%\end{center} +%\bigskip +%\end{abstract} + +\maketitle + + +%%%%%%%%%%%%%%%%%%%%%% +\section{Introduction} +%%%%%%%%%%%%%%%%%%%%%% + +The aim of this document is two-fold. +First, we want to re-derive the perturbative analysis of the similarity renormalisation group (SRG) formalism applied to the non-relativistic electronic Hamiltonian. +In a second time, we want to apply the same formalism to the unfolded GW Hamiltonian. +Before jumping into these analysis, we do a brief presentation of the SRG formalism. + +%%%%%%%%%%%%%%%%%%%%%% +\section{The similarity renormalisation group} +%%%%%%%%%%%%%%%%%%%%%% + +The similarity renormalization group aims at continuously transforming an Hamiltonian to a diagonal form, or more often to a block-diagonal form. +Therefore, the transformed Hamiltonian +\begin{equation} + \bH(s) = \bU(s) \, \bH \, \bU^\dag(s) +\end{equation} +depends on a flow parameter $s$. +The resulting Hamiltonian possess up to $N$-body operators with $N$ the number of particle. +\begin{equation} + \bH(s) = E_0(s) + \bF(s) + \bV(s) + \bW(s) + \dots +\end{equation} +In the following, we will truncate every contribution superior to two-body operators. +We can easily derive an evolution equation for this Hamiltonian by taking the derivative of $\bH(s)$. This gives +\begin{equation} + \label{eq:flowEquation} + \dv{\bH(s)}{s} = \comm{\boldsymbol{\eta}(s)}{\bH(s)} +\end{equation} +where $\boldsymbol{\eta}(s)$, the flow generator, is defined as +\begin{equation} +\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s) . +\end{equation} +To solve this equation at a cost inferior to the one of diagonalizing the initial Hamiltonian, one needs to introduce approximation for $\boldsymbol{\eta}(s)$. +Before doing so, we need to define what is the blocks to suppress in order to obtain a block-diagonal Hamiltonian. +Therefore, the Hamiltonian is separated in two parts as +\begin{equation} + \bH(s) = \underbrace{\bH^\text{d}(s)}_{\text{diagonal}} + \underbrace{\bH^\text{od}(s)}_{\text{off-diagonal}}. +\end{equation} +By definition, we have the following condition on $\bH^\text{od}$ +\begin{equation} + \bH^\text{od}(\infty) = \boldsymbol{0}. +\end{equation} + +In this work, we will use Wegner's canonical generator which is defined as +\begin{equation} + \boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)}. +\end{equation} +This generator has the advantage of defining a true renormalisation scheme, \ie the coupling coefficients with the highest energy determinants are removed first. +One of the flaws of this generator is that it generates a stiff set of ODE which is difficult to solve numerically. +However, here we consider analytical perturbative expressions so we will not be affected by this problem. + +%%%%%%%%%%%%%%%%%%%%%% +\section{The electronic Hamiltonian} +%%%%%%%%%%%%%%%%%%%%%% + +In this part, we derive the perturbative expression for the SRG applied to the non-relativistic electronic Hamiltonian +\begin{equation} + \label{eq:hamiltonianSecondQuant} + \hH = \sum_{pq} f_{pq} \cre{p}\ani{q} + \frac{1}{4} \sum_{pqrs} \aeri{pq}{rs} \cre{p}\cre{q}\ani{r}\ani{s} +\end{equation} +which can also be written in normal order wrt a reference determinant as +\begin{equation} + \label{eq:hamiltonianNormalOrder} + \hH = E_0 + \sum_{pq} + f_p^q\no{q}{p} + \frac{1}{4} \sum_{pqrs}v_{pq}^{rs}\no{rs}{pq}. +\end{equation} + +In this case, we want to decouple the reference determinant from every singly and doubly excited determinants. +Hence, we define the off-diagonal Hamiltonian as +\begin{equation} + \label{eq:hamiltonianOffDiagonal} + \hH^{\text{od}}(s) = \sum_{ia} + f_i^a(s)\no{a}{i} + \frac{1}{4} \sum_{ijab}v(s)_{ij}^{ab}\no{ab}{ij}. +\end{equation} +Note that each coefficients depend on $s$. + +The perturbative parameter $\la$ is such that +\begin{equation} + \bH(0) = E_0(0) + F(0) + \la V(0) +\end{equation} +In addition, we know the following initial conditions. +We use the HF basis set of the reference such that $F^{\text{od}}(0) = 0$ and $F^{\mathrm{d}}(0)=\delta_{pq}\epsilon_p$ +Therefore, we have +\begin{align} + \bH^\text{d}(0)&=E_0(0) + F^{\mathrm{d}}(0) + \la V^{\mathrm{d}}(0) & \bH^\text{od}(0)&= \la V^{\mathrm{od}}(0) +\end{align} +Now, we want to compute the terms at each order of the following development +\begin{equation} + \label{eq:hamiltonianPTExpansion} + \bH(s) = \bH^{(0)}(s) + \bH^{(1)}(s) + \bH^{(2)}(s) + \bH^{(3)}(s) + \dots +\end{equation} +by integrating Eq.~\eqref{eq:flowEquation}. +First, we start by showing that the zeroth order Hamiltonian is independant of $s$ and therefore equal to $\bH^{(0)}(0)$ +\begin{align} + \bH(\delta s) &= \bH(0) + \delta s \dv{\bH(s)}{s}\bigg|_{s=0} + O(\delta s^2) \\ + \dv{\bH(s)}{s}\bigg|_{s=0} &= [ \boldsymbol{\eta}(0), \bH(0)] \\ +\end{align} +However, we have seen that $\bH^\text{od}(0)$ is of order 1. +Hence, $\boldsymbol{\eta}(0)$ does not have a zero order contribution. +Which gives us the following equality +\begin{equation} + \bH^{(0)}(\delta s) = \bH^{(0)}(0) +\end{equation} +meaning that the zeroth order Hamiltonian is independant of $s$. + +The right-hand side of \eqref{eq:flowEquation} has no zeroth order, so we want to compute its first order term. +To do so, we first need to compute the first order term of $\boldsymbol{\eta}(s)$. +We have seen that $\bH^\text{od}(0)$ has no zeroth order contribution so we have +\begin{equation} + \boldsymbol{\eta}^{(1)}(s) = \comm{\bH^{\text{d},(0)}(0)}{\bH^{\text{od},(1)}(s)} +\end{equation} +According to the appendix the one-body part of $\boldsymbol{\eta}^{(1)}(s) $ has four contributions. However, two of them involve the two-body part of $\bH^{\text{d},(0)}(0)$ which is equal to zero. +In addition, the term $\left[A_{1}, B_{2}\right]_{p}^{q}$ involves the coefficients $A_i^a = f_i^a(0) = 0$. +So finally we only have one term +\begin{align} + \eta_a^{i,(1)}(s) &= \comm{\bH_1^{\text{d},(0)}(0)}{\bH_1^{\text{od},(1)}(s)} \\ + &= \sum_r \left( f_a^r(0) f_r^{i,(1)}(s) - f_r^i(0) f_a^{r,(1)}(s) \right) \\ + &= (\epsilon_a - \epsilon_i)f_a^{i,(1)}(s) +\end{align} +Now turning to the two-body part of $\boldsymbol{\eta}^{(1)}(s) $ and once again two terms are zero because the two-body part of $\bH^{\text{d},(0)}(0)$ is equal to zero. +So we have +\begin{align} + \eta_{ab}^{ij,(1)}(s) &= \comm{\bH_1^{\text{d},(0)}(0)}{\bH_2^{\text{od},(1)}(s)} \\ + &= \sum_t [P(ab) f_a^t(0) v_{tb}^{ij,(1)}(s) - P(ij) f_t^i(0) v_{ab}^{tj,(1)}(s) ] \\ + &= \sum_t [ P(ab) \epsilon_a \delta_{at}v_{tb}^{ij,(1)}(s) - P(ij) \epsilon_i \delta_{it} v_{ab}^{tj,(1)}(s) ] \\ + &= P(ab) \epsilon_a v_{ab}^{ij,(1)}(s) - P(ij) \epsilon_i v_{ab}^{ij,(1)}(s) \\ + &= \left( \epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j \right) v_{ab}^{ij,(1)} +\end{align} + +WIP... + +%%%%%%%%%%%%%%%%%%%%%% +\section{The unfolded GW Hamiltonian} +%%%%%%%%%%%%%%%%%%%%%% + +\appendix + +%%%%%%%%%%%%%%%%%%%%%% +\section{Matrix elements of $C=[A, B]_{1,2}$} +%%%%%%%%%%%%%%%%%%%%%% + +An operator $A$ containing at most two-body terms may be written in normal ordered form with respect to the reference $\Phi$ as + +$$ +A=A_{0}+A_{1}+A_{2}, +$$ + +where $A_{0}$ is a scalar, and + +$$ +\begin{gathered} +A_{1}=\sum_{p q} A_{p}^{q}\left\{\hat{a}_{q}^{p}\right\}, \\ +A_{2}=\frac{1}{4} \sum_{p q r s} A_{p q}^{r s}\left\{\hat{a}_{r s}^{p q}\right\}, +\end{gathered} +$$ + +with the second quantization operator written compactly as $\hat{a}_{q}^{p}=\hat{a}_{p}^{\dagger} \hat{a}_{q}$ and $\hat{a}_{r s}^{p q}=\hat{a}_{p}^{\dagger} \hat{a}_{q}^{\dagger} \hat{a}_{s} \hat{a}_{r}$. The commutator $C=[A, B]_{1,2}$ contains contributions from the following terms: + +$$ +\begin{gathered} +C_{0}=\left\langle\Phi\left|\left[A_{1}, B_{1}\right]\right| \Phi\right\rangle+\left\langle\Phi\left|\left[A_{2}, B_{2}\right]\right| \Phi\right\rangle, \\ +C_{p}^{q}=\left[A_{1}, B_{1}\right]_{p}^{q}+\left[A_{1}, B_{2}\right]_{p}^{q}-\left[B_{1}, A_{2}\right]_{p}^{q}+\left[A_{2}, B_{2}\right]_{p}^{q}, \\ +C_{p q}^{r s}=\left[A_{1}, B_{2}\right]_{p q}^{r s}-\left[B_{1}, A_{2}\right]_{p q}^{r s}+\left[A_{2}, B_{2}\right]_{p q}^{r s}, +\end{gathered} +$$ + +where the unique contributions to the matrix elements are + +$$ +\begin{gathered} +\left\langle\Phi\left|\left[A_{1}, B_{1}\right]\right| \Phi\right\rangle=\sum_{p} \sum_{i}\left(A_{i}^{p} B_{p}^{i}-B_{i}^{p} A_{p}^{i}\right), \\ +\left\langle\Phi\left|\left[A_{2}, B_{2}\right]\right| \Phi\right\rangle=\frac{1}{4} \sum_{i j} \sum_{a b}\left(A_{i j}^{a b} B_{a b}^{i j}-B_{i j}^{a b} A_{a b}^{i j}\right), \\ +{\left[A_{1}, B_{1}\right]_{p}^{q}=\sum_{r}\left(A_{p}^{r} B_{r}^{q}-B_{p}^{r} A_{r}^{q}\right),} \\ +{\left[A_{1}, B_{2}\right]_{p}^{q}=\sum_{i} \sum_{a} A_{i}^{a} B_{p a}^{q i}-A_{a}^{i} B_{p i}^{q a},} \\ +{\left[A_{2}, B_{2}\right]_{p}^{q}=\frac{1}{2} \sum_{i j} \sum_{a}\left(A_{a p}^{i j} B_{i j}^{a q}-A_{i j}^{a q} B_{a p}^{i j}\right)} \\ ++\frac{1}{2} \sum_{i} \sum_{a b}\left(A_{i p}^{a b} B_{a b}^{i q}-A_{a b}^{i q} B_{i p}^{a b}\right), \\ +{\left[A_{1}, B_{2}\right]_{p q}^{r s}=\sum_{t}\left[P(p q) A_{p}^{t} B_{t q}^{r s}-P(r s) A_{t}^{r} B_{p q}^{t s}\right],} \\ +{\left[A_{2}, B_{2}\right]_{p q}^{r s}=\frac{1}{2} \sum_{a b}\left(A_{p q}^{a b} B_{a b}^{r s}-A_{a b}^{r s} B_{p q}^{a b}\right)} \\ +-\frac{1}{2} \sum_{i j}\left(A_{p q}^{i j} B_{i j}^{r s}-A_{i j}^{r s} B_{p q}^{i j}\right) \\ ++\sum_{i} \sum_{a} P(p q) P(r s)\left[A_{p i}^{r a} B_{q a}^{s i}-A_{p a}^{r i} B_{q i}^{s a}\right] . +\end{gathered} +$$ + +In these equations $P(r s)$ is the antisymmetric permutation operator. + +\end{document} \ No newline at end of file