commit 8df8f336cb80cbfad62b6fc72a63a4744ca51958 Author: Pierre-Francois Loos Date: Sun May 17 22:24:50 2020 +0200 Initial Overleaf Import diff --git a/main.tex b/main.tex new file mode 100644 index 0000000..6d245e0 --- /dev/null +++ b/main.tex @@ -0,0 +1,132 @@ +\documentclass[journal=jctcce,manuscript=article]{achemso} + +%%\usepackage{natbib} +%%\usepackage{notoccite} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{graphicx}% Include figure files:q + +\usepackage{dcolumn}% Align table columns on decimal point +\usepackage{bm}% bold math +\usepackage{longtable} +\usepackage[usenames, dvipsnames]{color} +\usepackage{subfigure} +\usepackage{multirow} +\usepackage{dsfont} +\usepackage{ulem} +\usepackage{xcolor} + +\DeclareMathOperator*{\argmax}{argmax} +\DeclareMathOperator*{\argmin}{argmin} + + +\newcommand\vari{{\varepsilon}_i} +\newcommand\vara{{\varepsilon}_a} +\newcommand\varj{{\varepsilon}_j} +\newcommand\varb{{\varepsilon}_b} +\newcommand\varn{{\varepsilon}_n} +\newcommand\varm{{\varepsilon}_m} +\newcommand\Oms{{\Omega}_s} +\newcommand\hOms{\frac{{\Omega}_s}{2}} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +\newcommand{\correction}[1]{\textcolor{blue}{#1}} +%\definecolor{myblue}{rgb}{0.0, 0.0, 1.0} + +\author{Pierre-Fran\c{c}ois Loos} + \email{loos@irsamc.ups-tlse.fr} + \affiliation[LCPQ, Toulouse]{Laboratoire de Chimie et Physique Quantiques, Universit\'e de Toulouse, CNRS, UPS, France} + +\author{Xavier Blase} + \email{xavier.blase@neel.cnrs.fr} + \affiliation[NEEL, Grenoble]{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France} + + + +\title{ Dynamical BSE } + +\date{\today} +\begin{document} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{abstract} + + \end{abstract} + + %%%%%%%%%%%%%%%%%%%%%%%%%%%% + + \section{Theory} + +The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')$ reads, dropping the (space/spin)-variables: +\begin{align*} +[iL_0]( \omega_1 ) = \frac{ 1 }{ 2\pi } \int d \omega \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2} ) e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} } + \end{align*} +with $\tau_{34} = t_3 - t_4$ and +$t^{34} = (t_3 + t_4)/2$. Plugging now the 1-body Green's function Lehman representation, e.g. +$$ +G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn}(\varepsilon_n - \mu) } +$$ +and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component +\begin{align*} +\int dx_1 dx_{1'} \; & \phi_a^*(x_1) \phi_i(x_{1'}) L_0(x_1,3;x_{1'},4; \Oms) = e^{i \Oms t^{34} } \times \\ + & \frac{ \phi_a^*(x_3) \phi_i(x_4) } { \Oms - ( \vara - \vari ) + i \eta } + \Big( \theta( \tau ) e^{i ( \vari + \hOms) \tau } + + \theta( - \tau ) e^{i (\vara - \hOms \tau } \Big) +\end{align*} +with $\tau = \tau_{34}$. +We further obtain the spectral representation of +$\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle$ +expanding the field operators over a complete orbital basis creation/destruction operators: +\begin{align*} +\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) & | N,s \rangle = - \Big( e^{ -i \Omega_s t^{34} } \Big) \sum_{mn} \phi_m(x_3) \phi_n^*(x_4) \langle N | {\hat a}_n^{\dagger} {\hat a}_m | N,s \rangle \times \nonumber \\ +\times & \Big( \theta( \tau ) e^{- i ( \varepsilon_m - \hOms ) \tau } + + \theta( -\tau ) e^{ - i ( \varepsilon_n + \hOms) \tau } \Big) +\end{align*} +with $\tau = \tau_{34}$ and where the $ \lbrace \varepsilon_{n/m} \rbrace$ are proper addition/removal energies such that e.g. +$$ +e^{ i H \tau } {\hat a}_m^{\dagger} | N \rangle = e^{ i (E_0^N + \varepsilon_m ) \tau } {\hat a} _m^{\dagger} | N \rangle +$$ +Selecting (n,m)=(j,b) yields the largest components +$A_{jb}^{s} = \langle N | {\hat a}_j^{\dagger} {\hat a}_b | N,s \rangle $, while (n,m)=(b,j) yields much weaker +$B_{jb}^{s} = \langle N | {\hat a}_b^{\dagger} {\hat a}_j | N,s \rangle $ contributions. We used chemist notations with (i,j) indexing occupied orbitals and (a,b) virtual ones. Neglecting the $B_{jb}^{s}$ leads to the Tamm Dancoff approximation (TDA). Obtaining similarly the spectral representation of $ \langle N | T {\hat \psi}(1) {\hat \psi}^{\dagger}(1') | N,s \rangle$ ($t_{1'} = t_1^{+}$) projected onto $\phi_a^*(x_1) \phi_i(x_{1'})$, +one obtains after a few tedious manipulations (see Supplemental Information) the dynamical Bethe-Salpeter equation (DBSE) : + \begin{align} + ( \varepsilon_a - \varepsilon_i - \Omega_s ) A_{ia}^{s} + &+ \sum_{jb} \Big( v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) \Big) A_{jb}^{s} \\ + &+ \sum_{bj} \Big( v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) \Big) B_{jb}^{s} + = 0 +\end{align} + with an effective dynamically screened Coulomb potential (see Pina eq. 24): + \begin{align} + \widetilde{W}_{ij,ab}(\Oms) &= { i \over 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) \times \\ + \hskip 1cm &\times \left[ \frac{1}{ (\Oms - \omega) - ( \varb - \vari ) +i \eta } + \frac{1}{ (\Oms + \omega) - ( \vara - \varj ) + i\eta } \right] \nonumber +\end{align} +In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level: +\begin{align*} +W_{ij,ab}(\omega) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\ + & \times \Big( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big) +\end{align*} +so that +\begin{align} + \widetilde{W}_{ij,ab}( \Oms ) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\ + & \times \left[ \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta } + \frac{ 1}{ \Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta } + \right] \nonumber +\end{align} +with e.g. $ \Omega_{ib}^{s} = \Oms - ( \varepsilon_b - \varepsilon_i) $. \textcolor{red}{Due to excitonic effects, the lowest BSE ${\Omega}_1$ excitation energy stands lower than the lowest $\Omega_m^{RPA}$ excitation energy, so that +e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and cannot diverge. Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that +$$ +\left[ \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta } + \frac{ 1}{ \Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta } + \right] + < + \Big( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big) < 0 +$$ +in the limit $(\omega \rightarrow 0)$ of the standard adiabatic BSE . WELL, do we know the sign of +$[ij|m] [ab|m]$ ?? } + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\end{document}