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Notes/Commands.tex
Executable file
137
Notes/Commands.tex
Executable file
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%============================================================%
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%%% NEWCOMMANDS %%%
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% ============================================================%
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%%% This latex gather all the latex commands that I use to facilitate the writing of electronic structure manuscript, report...
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%%% You can import these commands by adding the following line in you .text
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%%% \input{commands}
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%%% Latin %%%
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\newcommand{\ie}{\textit{i.e.}~}
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\newcommand{\eg}{\textit{e.g.}~}
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\newcommand{\etal}{\textit{et al.}~}
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%%% Operators %%%
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\newcommand{\hH}{\Hat{H}} % Hamiltonian operator
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\newcommand{\HN}{\Hat{\mathnormal{H}}_{\text{N}}} % Normal ordered Hamiltonian
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\newcommand{\Hsim}{\hat{\bar{H}}} % Similarity transformed Hamiltonian
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\newcommand{\hC}{\Hat{C}} % CI operator
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\newcommand{\hT}{\Hat{T}} % Cluster operator
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\newcommand{\T}[1]{\Hat{\mathnormal{T}}_{#1}} % Cluster operator of a given excitation number
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\newcommand{\hsig}{\Hat{\sigma}} % Unitary cluster operator
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\newcommand{\hK}{\Hat{K}} % Anti-hermitian orbital rotation operator
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\newcommand{\hS}{\Hat{S}} % Anti-hermitian CI coefficients rotation operator
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\newcommand{\hP}[1]{\Hat{\mathnormal{P}}_{#1}} % Permutation operators
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\newcommand{\hE}{\Hat{E}} % Spin averaged single excitation operator
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\newcommand{\cre}[1]{a_{#1}^\dagger} % Creation operator
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\newcommand{\ani}[1]{a_{#1}} % Annihilation operator
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\newcommand{\bcre}[1]{b_{#1}^\dagger} % Boson creation operator
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\newcommand{\bani}[1]{b_{#1}} % Boson annihilation operator
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\newcommand{\no}[2]{\mleft\{ \hat{a}_{#1}^{#2}\mright\} }
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%%% Matrices %%%
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\newcommand{\bA}{\boldsymbol{A}}
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\newcommand{\bB}{\boldsymbol{B}}
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\newcommand{\bC}[2]{\boldsymbol{C}_{#1}^{#2}}
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\newcommand{\bD}{\boldsymbol{D}}
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\newcommand{\bE}{\boldsymbol{E}}
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\newcommand{\bF}[2]{\boldsymbol{F}_{#1}^{#2}}
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\newcommand{\bG}{\boldsymbol{G}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\bJ}{\boldsymbol{J}}
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\newcommand{\bK}{\boldsymbol{K}}
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\newcommand{\bL}{\boldsymbol{L}}
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\newcommand{\bM}{\boldsymbol{M}}
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\newcommand{\bN}{\boldsymbol{N}}
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\newcommand{\bP}{\boldsymbol{P}}
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\newcommand{\bQ}{\boldsymbol{Q}}
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\newcommand{\bR}{\boldsymbol{R}}
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\newcommand{\bS}{\boldsymbol{S}}
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\newcommand{\bT}{\boldsymbol{T}}
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\newcommand{\bU}{\boldsymbol{U}}
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\newcommand{\bV}[2]{\boldsymbol{V}_{#1}^{#2}}
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\newcommand{\bW}{\boldsymbol{W}}
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\newcommand{\bX}{\boldsymbol{X}}
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\newcommand{\bY}{\boldsymbol{Y}}
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\newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
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\newcommand{\bO}{\boldsymbol{0}}
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\newcommand{\bI}{\boldsymbol{1}}
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\newcommand{\bc}{\boldsymbol{c}}
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\newcommand{\br}{\boldsymbol{r}}
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\newcommand{\bx}{\boldsymbol{x}}
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\newcommand{\bSig}{\boldsymbol{\Sigma}}
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\newcommand{\bSigC}{\boldsymbol{\Sigma}^{\text{c}}}
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\newcommand{\be}{\boldsymbol{\epsilon}}
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\newcommand{\bOm}{\boldsymbol{\Omega}}
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\newcommand{\bEta}[1]{\boldsymbol{\eta}^{(#1)}(s)}
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\newcommand{\bHd}[1]{\bH_\text{d}^{(#1)}}
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\newcommand{\bHod}[1]{\bH_\text{od}^{(#1)}}
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\newcommand{\FC}[1]{F_{#1}^{\text{C}}} % Core Fock matrix
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\newcommand{\FA}[1]{F_{#1}^{\text{A}}} % Active Fock matrix
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%%% Wave functions %%%
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\newcommand{\MO}[1]{\phi_{#1}}
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\newcommand{\SO}[1]{\psi_{#1}}
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%%% Matrix and tensor elements %%%
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\newcommand{\oa}{O_{\alpha}}
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\newcommand{\ob}{O_{\beta}}
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\newcommand{\eri}[2]{\braket{#1}{#2}} % Electron repulsion integral physician notation
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\newcommand{\ceri}[2]{\mleft(#1|#2\mright)} % Electron repulsion integral chemist notation
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\newcommand{\aeri}[2]{\mel{#1}{}{#2}} % Double bar integral
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\newcommand{\kron}[1]{\delta_{#1}} % Kronecker delta
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\newcommand{\cbra}[1]{(#1|} % Chemist bra
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\newcommand{\cket}[1]{|#1)} % Chemist ket
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%%% Mathematics %%%
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\newcommand{\ii}{\mathrm{i}}
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%%% Text acronyms and abbreviations %%%
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\newcommand{\FCI}{\text{FCI}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\MCSCF}{\text{MCSCF}}
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\newcommand{\occ}{\text{occ}}
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\newcommand{\vir}{\text{vir}}
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\newcommand{\HOMO}{\text{HOMO}}
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\newcommand{\LUMO}{\text{LUMO}}
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\newcommand{\nuc}{\text{nuc}}
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\newcommand{\KS}{\text{KS}}
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\newcommand{\RPA}{\text{RPA}}
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\newcommand{\phRPA}{\text{ph-RPA}}
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\newcommand{\ppRPA}{\text{pp-RPA}}
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\newcommand{\h}{\text{1h}}
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\newcommand{\p}{\text{1p}}
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\newcommand{\hp}{\text{1h+1p}}
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\newcommand{\hhp}{\text{2h1p}}
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\newcommand{\pph}{\text{2p1h}}
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\newcommand{\dRPA}{\text{dRPA}}
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\newcommand{\RPAx}{\text{RPAx}}
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\newcommand{\QP}{\textsc{quantum package}}
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\newcommand{\Hxc}{\text{Hxc}}
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\newcommand{\xc}{\text{xc}}
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\newcommand{\x}{\text{x}}
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\newcommand{\GW}{\text{GW}}
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\newcommand{\GF}{\text{GF(2)}}
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\newcommand{\GT}{\text{$GT$}}
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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%%% Notations %%%
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\newcommand{\Ne}{N}
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\newcommand{\Norb}{K}
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\newcommand{\Nocc}{O}
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\newcommand{\Nvir}{V}
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@ -688,8 +688,8 @@ Which finally gives
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The expression for the GF(2) case is
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\begin{align}
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\epsilon_r^{(2)}(s) &= \sum _{ija} \frac{\aeri{ra}{ij}^2}{\epsilon_r ^{(0)}- \Delta_{ij}^a} \left(1 - e^{-2(\epsilon_r^{(0)} - \Delta_{ij}^a)^2s}\right) \notag \\
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&+ \sum _{iab} \frac{\aeri{ri}{ab}^2}{\epsilon_r^{(0)} - \Delta_{i}^{ab}} \left(1 - e^{-2(\epsilon_r^{(0)} - \Delta_{i}^{ab})^2s}\right) \notag
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\epsilon_r^{(2)}(s) &= \frac{1}{2} \sum _{ija} \frac{\aeri{ra}{ij}^2}{\epsilon_r ^{(0)}- \Delta_{ij}^a} \left(1 - e^{-2(\epsilon_r^{(0)} - \Delta_{ij}^a)^2s}\right) \notag \\
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&+ \frac{1}{2} \sum _{iab} \frac{\aeri{ri}{ab}^2}{\epsilon_r^{(0)} - \Delta_{i}^{ab}} \left(1 - e^{-2(\epsilon_r^{(0)} - \Delta_{i}^{ab})^2s}\right) \notag
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\end{align}
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\end{itemize}
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