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minor corrections up to Sec III

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36
dynker.bib
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@ -1,13 +1,24 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-06-26 09:45:06 +0200 %% Created for Pierre-Francois Loos at 2020-07-20 08:52:20 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Lowdin_1963,
Author = {P. L{\"o}wdin},
Date-Added = {2020-07-20 08:49:22 +0200},
Date-Modified = {2020-07-20 08:52:15 +0200},
Doi = {10.1016/0022-2852(63)90151-6},
Journal = {J. Mol. Spectrosc.},
Pages = {12--33},
Title = {Studies in perturbation theory: Part I. An elementary iteration-variation procedure for solving the Schr{\"o}dinger equation by partitioning technique},
Volume = {10},
Year = {1963}}
@article{Petersilka_1996, @article{Petersilka_1996,
Author = {M. Petersilka and U. J. Gossmann and and E. K. U. Gross}, Author = {M. Petersilka and U. J. Gossmann and and E. K. U. Gross},
Date-Added = {2020-06-26 09:43:33 +0200}, Date-Added = {2020-06-26 09:43:33 +0200},
@ -17,7 +28,8 @@
Pages = {1212}, Pages = {1212},
Title = {Excitation Energies From Time-Dependent Density-Functional Theory}, Title = {Excitation Energies From Time-Dependent Density-Functional Theory},
Volume = {76}, Volume = {76},
Year = {1996}} Year = {1996},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.76.1212}}
@article{Nielsen_1980, @article{Nielsen_1980,
Author = {Egon S. Nielsen and Poul Jorgensen}, Author = {Egon S. Nielsen and Poul Jorgensen},
@ -14426,14 +14438,12 @@
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.93.235113}} Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.93.235113}}
@article{Boulanger_2014, @article{Boulanger_2014,
author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier}, Author = {Boulanger, Paul and Jacquemin, Denis and Duchemin, Ivan and Blase, Xavier},
title = {Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe-Salpeter Approach}, Doi = {10.1021/ct401101u},
journal = {J. Chem. Theory Comput.}, Journal = {J. Chem. Theory Comput.},
volume = {10}, Number = {3},
number = {3}, Pages = {1212--1218},
pages = {1212--1218}, Title = {Fast and Accurate Electronic Excitations in Cyanines with the Many-Body Bethe-Salpeter Approach},
year = {2014}, Volume = {10},
doi = {10.1021/ct401101u}, Year = {2014},
} Bdsk-Url-1 = {https://doi.org/10.1021/ct401101u}}

35
dynker.tex
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@ -41,26 +41,26 @@ In particular, using a simple two-level model, we analyze, for each kernel, the
\section{Linear response theory} \section{Linear response theory}
\label{sec:LR} \label{sec:LR}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Linear response theory is a powerful approach that allows to directly access the optical excitations $\omega_s$ of a given electronic system (such as a molecule) and their corresponding oscillator strengths $f_s$ [extracted from their eigenvectors $\T{(\bX_s \bY_s)}$] via the response of the system to a weak electromagnetic field. \cite{Oddershede_1977,Casida_1995,Petersilka_1996} Linear response theory is a powerful approach that allows to directly access the optical excitations $\omega_S$ of a given electronic system (such as a molecule) and their corresponding oscillator strengths $f_s$ [extracted from their eigenvectors $\T{(\bX_S \bY_S)}$] via the response of the system to a weak electromagnetic field. \cite{Oddershede_1977,Casida_1995,Petersilka_1996}
From a practical point of view, these quantities are obtained by solving non-linear, frequency-dependent Casida-like equations in the space of single excitations and de-excitations \cite{Casida_1995} From a practical point of view, these quantities are obtained by solving non-linear, frequency-dependent Casida-like equations in the space of single excitations and de-excitations \cite{Casida_1995}
\begin{equation} \label{eq:LR} \begin{equation} \label{eq:LR}
\begin{pmatrix} \begin{pmatrix}
\bR^{\sigma}(\omega_s) & \bC^{\sigma}(\omega_s) \bR^{\sigma}(\omega_S) & \bC^{\sigma}(\omega_S)
\\ \\
-\bC^{\sigma}(-\omega_s)^* & -\bR^{\sigma}(-\omega_s)^* -\bC^{\sigma}(-\omega_S)^* & -\bR^{\sigma}(-\omega_S)^*
\end{pmatrix} \end{pmatrix}
\cdot \cdot
\begin{pmatrix} \begin{pmatrix}
\bX_s^{\sigma} \bX_S^{\sigma}
\\ \\
\bY_s^{\sigma} \bY_S^{\sigma}
\end{pmatrix} \end{pmatrix}
= =
\omega_s \omega_S
\begin{pmatrix} \begin{pmatrix}
\bX_s^{\sigma} \bX_S^{\sigma}
\\ \\
\bY_s^{\sigma} \bY_S^{\sigma}
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
where the explicit expressions of the resonant and coupling blocks, $\bR^{\sigma}(\omega)$ and $\bC^{\sigma}(\omega)$, depend on the spin manifold ($\sigma =$ $\updw$ for singlets and $\sigma =$ $\upup$ for triplets) and the level of approximation that one employs. where the explicit expressions of the resonant and coupling blocks, $\bR^{\sigma}(\omega)$ and $\bC^{\sigma}(\omega)$, depend on the spin manifold ($\sigma =$ $\updw$ for singlets and $\sigma =$ $\upup$ for triplets) and the level of approximation that one employs.
@ -68,7 +68,7 @@ Neglecting the coupling block [\ie, $\bC^{\sigma}(\omega) = 0$] between the reso
In the absence of symmetry breaking, \cite{Dreuw_2005} the non-linear eigenvalue problem defined in Eq.~\eqref{eq:LR} has particle-hole symmetry which means that it is invariant via the transformation $\omega \to -\omega$. In the absence of symmetry breaking, \cite{Dreuw_2005} the non-linear eigenvalue problem defined in Eq.~\eqref{eq:LR} has particle-hole symmetry which means that it is invariant via the transformation $\omega \to -\omega$.
Therefore, without loss of generality, we will restrict our analysis to positive frequencies. Therefore, without loss of generality, we will restrict our analysis to positive frequencies.
In the one-electron basis of (real) spatial orbitals $\lbrace \MO{p} \rbrace$, we will assume that the elements of the matrices defined in Eq.~\eqref{eq:LR} have the following generic forms: \cite{Dreuw_2005} In the one-electron basis of (real) spatial orbitals $\lbrace \MO{p}(\br) \rbrace$, we will assume that the elements of the matrices defined in Eq.~\eqref{eq:LR} have the following generic forms: \cite{Dreuw_2005}
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
R_{ia,jb}^{\sigma}(\omega) = (\e{a} - \e{i}) \delta_{ij} \delta_{ab} + f_{ia,jb}^{\Hxc,\sigma}(\omega) R_{ia,jb}^{\sigma}(\omega) = (\e{a} - \e{i}) \delta_{ij} \delta_{ab} + f_{ia,jb}^{\Hxc,\sigma}(\omega)
@ -76,7 +76,7 @@ In the one-electron basis of (real) spatial orbitals $\lbrace \MO{p} \rbrace$, w
C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\Hxc,\sigma}(\omega) C_{ia,jb}^{\sigma}(\omega) = f_{ia,bj}^{\Hxc,\sigma}(\omega)
\end{gather} \end{gather}
\end{subequations} \end{subequations}
where $\delta_{pq}$ is the Kronecker delta, $\e{p}$ is the one-electron energy associated with $\MO{p}$, and where $\delta_{pq}$ is the Kronecker delta, $\e{p}$ is the one-electron (or quasiparticle) energy associated with $\MO{p}(\br)$, and
\begin{equation} \label{eq:kernel} \begin{equation} \label{eq:kernel}
f_{ia,jb}^{\Hxc,\sigma}(\omega) f_{ia,jb}^{\Hxc,\sigma}(\omega)
= \iint \MO{i}(\br) \MO{a}(\br) f^{\Hxc,\sigma}(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br' = \iint \MO{i}(\br) \MO{a}(\br) f^{\Hxc,\sigma}(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
@ -97,7 +97,7 @@ where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively)
\ERI{ia}{jb} = \iint \MO{i}(\br) \MO{a}(\br) \frac{1}{\abs{\br - \br'}} \MO{j}(\br') \MO{b}(\br') d\br d\br' \ERI{ia}{jb} = \iint \MO{i}(\br) \MO{a}(\br) \frac{1}{\abs{\br - \br'}} \MO{j}(\br') \MO{b}(\br') d\br d\br'
\end{equation} \end{equation}
are the usual two-electron integrals. are the usual two-electron integrals.
The central point here is that, thanks to its non-linear nature stemming from their frequency dependence, a dynamical kernel potentially generates more than just single excitations. The launchpad of the present study is that, thanks to its non-linear nature stemming from its frequency dependence, a dynamical kernel potentially generates more than just single excitations.
Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript. Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -110,10 +110,10 @@ To do so, let us consider the usual chemical scenario where one wants to get the
In most cases, this can be done by solving a set of linear equations of the form In most cases, this can be done by solving a set of linear equations of the form
\begin{equation} \begin{equation}
\label{eq:lin_sys} \label{eq:lin_sys}
\bA \cdot \bc = \omega \, \bc \bA \cdot \bc = \omega_S \, \bc
\end{equation} \end{equation}
where $\omega$ is one of the optical excitation energies of interest and $\bc$ its transition vector . where $\omega$ is one of the optical excitation energies of interest and $\bc$ its transition vector .
If we assume that the operator $\bA$ has a matrix representation of size $N \times N$, this \textit{linear} set of equations yields $N$ excitation energies. If we assume that the matrix $\bA$ is diagonalizable and of size $N \times N$, the \textit{linear} set of equations \eqref{eq:lin_sys} yields $N$ excitation energies.
However, in practice, $N$ might be (very) large (\eg, equal to the total number of single and double excitations generated from a reference Slater determinant), and it might therefore be practically useful to recast this system as two smaller coupled systems, such that However, in practice, $N$ might be (very) large (\eg, equal to the total number of single and double excitations generated from a reference Slater determinant), and it might therefore be practically useful to recast this system as two smaller coupled systems, such that
\begin{equation} \begin{equation}
\label{eq:lin_sys_split} \label{eq:lin_sys_split}
@ -133,9 +133,9 @@ However, in practice, $N$ might be (very) large (\eg, equal to the total number
\end{pmatrix} \end{pmatrix}
\end{equation} \end{equation}
where the blocks $\bA_1$ and $\bA_2$, of sizes $N_1 \times N_1$ and $N_2 \times N_2$ (with $N_1 + N_2 = N$), can be associated with, for example, the single and double excitations of the system. where the blocks $\bA_1$ and $\bA_2$, of sizes $N_1 \times N_1$ and $N_2 \times N_2$ (with $N_1 + N_2 = N$), can be associated with, for example, the single and double excitations of the system.
Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors. This decomposition technique is often called L\"owdin partitioning in the literature. \cite{Lowdin_1963}
Solving separately each row of the system \eqref{eq:lin_sys_split} and assuming that $\omega \bI - \bA_2$ is invertible, it follows that Solving separately each row of the system \eqref{eq:lin_sys_split} and assuming that $\omega \bI - \bA_2$ is invertible, we get
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
\label{eq:row1} \label{eq:row1}
@ -156,10 +156,11 @@ with
\end{equation} \end{equation}
which has, by construction, exactly the same solutions as the linear system \eqref{eq:lin_sys} but a smaller dimension. which has, by construction, exactly the same solutions as the linear system \eqref{eq:lin_sys} but a smaller dimension.
For example, an operator $\Tilde{\bA}_1(\omega)$ built in the single-excitation basis can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2}. \cite{ReiningBook} For example, an operator $\Tilde{\bA}_1(\omega)$ built in the single-excitation basis can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2}. \cite{ReiningBook}
Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies.
How have we been able to reduce the dimension of the problem while keeping the same number of solutions? How have we been able to reduce the dimension of the problem while keeping the same number of solutions?
To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent. To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent.
In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension. In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension [see Eq.~\eqref{eq:non_lin_sys}].
This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Sottile_2003,Garniron_2018,QP2} This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Sottile_2003,Garniron_2018,QP2}
Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension. Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension.
However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analog given by Eq.~\eqref{eq:lin_sys}. However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analog given by Eq.~\eqref{eq:lin_sys}.
@ -169,7 +170,7 @@ For example, assuming that $\bA_2$ is a diagonal matrix is of common practice (s
Another of these approximations is the so-called \textit{static} approximation, where one sets the frequency to a particular value. Another of these approximations is the so-called \textit{static} approximation, where one sets the frequency to a particular value.
For example, as commonly done within the Bethe-Salpeter equation (BSE) formalism of many-body perturbation theory (MBPT), \cite{Strinati_1988} $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$. For example, as commonly done within the Bethe-Salpeter equation (BSE) formalism of many-body perturbation theory (MBPT), \cite{Strinati_1988} $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature. In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature.
A similar example in the context of time-dependent density-functional theory (TDDFT) \cite{Runge_1984} is provided by the ubiquitous adiabatic approximation, \cite{Tozer_2000} which neglects all memory effects by making static the exchange-correlation (xc) kernel (\ie, frequency-independent). \cite{Maitra_2016} A similar example in the context of time-dependent density-functional theory (TDDFT) \cite{Runge_1984} is provided by the ubiquitous adiabatic approximation, \cite{Tozer_2000} which neglects all memory effects by making static the exchange-correlation (xc) kernel (\ie, frequency independent). \cite{Maitra_2016}
These approximations come with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $N$ to $N_1$. These approximations come with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $N$ to $N_1$.
Coming back to our example, in the static (or adiabatic) approximation, the operator $\Tilde{\bA}_1$ built in the single-excitation basis cannot provide double excitations anymore, and the $N_1$ excitation energies are associated with single excitations. Coming back to our example, in the static (or adiabatic) approximation, the operator $\Tilde{\bA}_1$ built in the single-excitation basis cannot provide double excitations anymore, and the $N_1$ excitation energies are associated with single excitations.
All additional solutions associated with higher excitations have been forever lost. All additional solutions associated with higher excitations have been forever lost.