679 lines
31 KiB
TeX
679 lines
31 KiB
TeX
\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-2}
|
|
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
|
|
\usepackage[version=4]{mhchem}
|
|
|
|
\usepackage[utf8]{inputenc}
|
|
\usepackage[T1]{fontenc}
|
|
\usepackage{txfonts}
|
|
|
|
\usepackage[
|
|
colorlinks=true,
|
|
citecolor=blue,
|
|
breaklinks=true
|
|
]{hyperref}
|
|
\urlstyle{same}
|
|
|
|
\begin{document}
|
|
|
|
\title{A Tale of Three Dynamical Kernels}
|
|
|
|
\author{Pierre-Fran\c{c}ois \surname{Loos}}
|
|
\email{loos@irsamc.ups-tlse.fr}
|
|
\affiliation{\LCPQ}
|
|
|
|
\begin{abstract}
|
|
We discuss the physical properties and accuracy of three distinct dynamical (\ie, frequency-dependent) kernels for the computation of optical excitations within linear response theory:
|
|
i) an \textit{a priori} built kernel inspired by the dressed time-dependent density-functional theory (TDDFT) kernel proposed by Maitra and coworkers [\href{https://doi.org/10.1063/1.1651060}{J.~Chem.~Phys.~120, 5932 (2004)}],
|
|
ii) the dynamical kernel stemming from the Bethe-Salpeter equation (BSE) formalism derived originally by Strinati [\href{https://doi.org/10.1007/BF02725962}{Riv.~Nuovo Cimento 11, 1--86 (1988)}], and
|
|
iii) the second-order BSE kernel derived by Yang and coworkers [\href{https://doi.org/10.1063/1.4824907}{J.~Chem.~Phys.~139, 154109 (2013)}].
|
|
In particular, using a simple two-level model, we analyze, for each kernel, the appearance of spurious excitations, as first evidenced by Romaniello and collaborators [\href{https://doi.org/10.1063/1.3065669}{J.~Chem.~Phys.~130, 044108 (2009)}], due to the approximate nature of the kernels.
|
|
%\\
|
|
%\bigskip
|
|
%\begin{center}
|
|
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
|
|
%\end{center}
|
|
%\bigskip
|
|
\end{abstract}
|
|
|
|
\maketitle
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Linear response theory}
|
|
\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{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{pmatrix}
|
|
\bR(\omega_s) & \bC(\omega_s)
|
|
\\
|
|
-\bC(-\omega_s) & -\bR(-\omega_s)
|
|
\end{pmatrix}
|
|
\begin{pmatrix}
|
|
\bX_s
|
|
\\
|
|
\bY_s
|
|
\end{pmatrix}
|
|
=
|
|
\omega_s
|
|
\begin{pmatrix}
|
|
\bX_s
|
|
\\
|
|
\bY_s
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
where the explicit expressions of the resonant and coupling blocks, $\bR(\omega)$ and $\bC(\omega)$, depend on the level of approximation that one employs.
|
|
Neglecting the coupling block between the resonant and anti-resonants parts, $\bR(\omega)$ and $-\bR(-\omega)$, is known as the Tamm-Dancoff approximation (TDA).
|
|
The non-linear eigenvalue problem defined in Eq.~\eqref{eq:LR} has particle-hole symmetry which means that it is invariant via the transformation $\omega \to -\omega$, and, thanks to its non-linear nature stemming from its frequency dependence, it potentially generates more than just single excitations.
|
|
|
|
In a wave function context, introducing a spatial orbital basis $\lbrace \MO{p} \rbrace$, we assume here that the elements of the matrices defined in Eq.~\eqref{eq:LR} have the following generic form:
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
\begin{split}
|
|
R_{ia,jb}(\omega)
|
|
& = \iint \MO{i}(\br) \MO{a}(\br) \bR(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
|
|
\\
|
|
& = (\e{a} - \e{i}) \delta_{ij} \delta_{ab} + 2 \sigma \ERI{ia}{jb} - \ERI{ib}{ja} + f_{ia,jb}^\sigma(\omega)
|
|
\end{split}
|
|
\\
|
|
\begin{split}
|
|
C_{ia,jb}(\omega)
|
|
& = \iint \MO{i}(\br) \MO{a}(\br) \bC(\omega) \MO{j}(\br') \MO{b}(\br') d\br d\br'
|
|
\\
|
|
& = 2 \sigma \ERI{ia}{bj} - \ERI{ij}{ba} + f_{ia,bj}^\sigma(\omega)
|
|
\end{split}
|
|
\end{gather}
|
|
\end{subequations}
|
|
where $\sigma = 1 $ or $0$ for singlet ($\updw$) and triplet ($\upup$) excited states (respectively), and
|
|
\begin{equation}
|
|
\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}
|
|
are the usual (bare) two-electron integrals.
|
|
Here, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, and $f^{\sigma}(\omega)$ is the correlation part of the spin-resolved kernel.
|
|
(Note that, usually, only the correlation part of the kernel is frequency dependent.)
|
|
In the case of a spin-independent kernel, we will drop the superscrit $\sigma$.
|
|
Unless otherwise stated, atomic units are used and we assume real quantities throughout this manuscript.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{The concept of dynamical quantities}
|
|
\label{sec:dyn}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%s
|
|
As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
|
|
Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes. \cite{Romaniello_2009b,Sangalli_2011,ReiningBook}
|
|
To do so, let us consider the usual chemical scenario where one wants to get the optical excitations of a given system.
|
|
In most cases, this can be done by solving a set of linear equations of the form
|
|
\begin{equation}
|
|
\label{eq:lin_sys}
|
|
\bA \bc = \omega \bc
|
|
\end{equation}
|
|
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.
|
|
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}
|
|
\label{eq:lin_sys_split}
|
|
\begin{pmatrix}
|
|
\bA_1 & \T{\bb} \\
|
|
\bb & \bA_2 \\
|
|
\end{pmatrix}
|
|
\begin{pmatrix}
|
|
\bc_1 \\
|
|
\bc_2 \\
|
|
\end{pmatrix}
|
|
= \omega
|
|
\begin{pmatrix}
|
|
\bc_1 \\
|
|
\bc_2 \\
|
|
\end{pmatrix}
|
|
\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.
|
|
Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors.
|
|
|
|
Solving separately each row of the system \eqref{eq:lin_sys_split} and assuming that $\omega \bI - \bA_2$ is invertible yields
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
\label{eq:row1}
|
|
\bA_1 \bc_1 + \T{\bb} \bc_2 = \omega \bc_1
|
|
\\
|
|
\label{eq:row2}
|
|
\bc_2 = (\omega \bI - \bA_2)^{-1} \bb \bc_1
|
|
\end{gather}
|
|
\end{subequations}
|
|
Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator
|
|
\begin{equation}
|
|
\label{eq:non_lin_sys}
|
|
\Tilde{\bA}_1(\omega) \bc_1 = \omega \bc_1
|
|
\end{equation}
|
|
with
|
|
\begin{equation}
|
|
\Tilde{\bA}_1(\omega) = \bA_1 + \T{\bb} (\omega \bI - \bA_2)^{-1} \bb
|
|
\end{equation}
|
|
which has, by construction, exactly the same solutions than 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}
|
|
|
|
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.
|
|
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.
|
|
This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Garniron_2018,QP2}
|
|
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}.
|
|
Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently.
|
|
For example, assuming that $\bA_2$ is a diagonal matrix is of common practice (see, for example, Ref.~\onlinecite{Garniron_2018} and references therein).
|
|
|
|
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} $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
|
|
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}
|
|
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.
|
|
All additional solutions associated with higher excitations have been forever lost.
|
|
In the next section, we illustrate these concepts and the various tricks that can be used to recover some of these dynamical effects starting from the static eigenproblem.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Dynamical kernels}
|
|
\label{sec:kernel}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Exact Hamiltonian}
|
|
\label{sec:exact}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Let us consider a two-level quantum system made of two orbitals \cite{Romaniello_2009b} in its singlet ground state (\ie, the lowest orbital is doubly occupied).
|
|
We will label these two orbitals, $\MO{v}$ and $\MO{c}$, as valence ($v$) and conduction ($c$) orbitals with respective one-electron Hartree-Fock (HF) energies $\e{v}$ and $\e{c}$.
|
|
In a more quantum chemical language, these correspond to the HOMO and LUMO orbitals (respectively).
|
|
The ground state $\ket{0}$ has a one-electron configuration $\ket{v\bar{v}}$, while the doubly-excited state $\ket{D}$ has a configuration $\ket{c\bar{c}}$.
|
|
There is then only one single excitation possible which corresponds to the transition $v \to c$ with different spin-flip configurations.
|
|
As usual, this can produce a singlet singly-excited state $\ket{S} = (\ket{v\bar{c}} + \ket{c\bar{v}})/\sqrt{2}$, and a triplet singly-excited state $\ket{T} = (\ket{v\bar{c}} - \ket{c\bar{v}})/\sqrt{2}$. \cite{SzaboBook}
|
|
|
|
For the singlet manifold, the exact Hamiltonian in the basis of the (spin-adapted) configuration state functions reads
|
|
\begin{equation} \label{eq:H-exact}
|
|
\bH^{\updw} =
|
|
\begin{pmatrix}
|
|
\mel{0}{\hH}{0} & \mel{0}{\hH}{S} & \mel{0}{\hH}{D} \\
|
|
\mel{S}{\hH}{0} & \mel{S}{\hH}{S} & \mel{S}{\hH}{D} \\
|
|
\mel{D}{\hH}{0} & \mel{D}{\hH}{S} & \mel{D}{\hH}{D} \\
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
with
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\mel{0}{\hH}{0} & = 2\e{v} - \ERI{vv}{vv} = \EHF
|
|
\\
|
|
\mel{S}{\hH - \EHF}{S} & = \Delta\e{} + \ERI{vc}{cv} - \ERI{vv}{cc}
|
|
\\
|
|
\begin{split}
|
|
\mel{D}{\hH - \EHF}{D}
|
|
& = 2\Delta\e{} + \ERI{vv}{vv} + \ERI{cc}{cc}
|
|
\\
|
|
& + 2\ERI{vc}{cv} - 4\ERI{vv}{cc}
|
|
\end{split}
|
|
\\
|
|
\mel{0}{\hH}{S} & = 0
|
|
\\
|
|
\mel{S}{\hH}{D} & = \sqrt{2}[\ERI{vc}{cc} - \ERI{cv}{vv}]
|
|
\\
|
|
\mel{0}{\hH}{D} & = \ERI{vc}{cv}
|
|
\end{align}
|
|
\end{subequations}
|
|
and $\Delta\e{} = \e{c} - \e{v}$.
|
|
The energy of the only triplet state is simply $\mel{T}{\hH}{T} = \EHF + \Delta\e{} - \ERI{vv}{cc}$.
|
|
|
|
For the sake of illustration, we will use the same numerical example throughout this study, and consider the singlet ground state of the \ce{He} atom in Pople's 6-31G basis set.
|
|
This system contains two orbitals and the numerical values of the various quantities defined above are
|
|
\begin{subequations}
|
|
\begin{align}
|
|
\e{v} & = -0.914\,127
|
|
&
|
|
\e{c} & = + 1.399\,859
|
|
\\
|
|
\ERI{vv}{vv} & = 1.026\,907
|
|
&
|
|
\ERI{cc}{cc} & = 0.766\,363
|
|
\\
|
|
\ERI{vv}{cc} & = 0.858\,133
|
|
&
|
|
\ERI{vc}{cv} & = 0.227\,670
|
|
\\
|
|
\ERI{vv}{vc} & = 0.316\,490
|
|
&
|
|
\ERI{vc}{cc} & = 0.255\,554
|
|
\end{align}
|
|
\end{subequations}
|
|
This yields the following exact singlet and triplet excitation energies
|
|
\begin{align} \label{sec:exact}
|
|
\omega_{1}^{\updw} & = 1.92145
|
|
&
|
|
\omega_{3}^{\updw} & = 3.47880
|
|
&
|
|
\omega_{1}^{\upup} & = 1.47085
|
|
\end{align}
|
|
that we are going to use a reference for the remaining of this study.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Maitra's dynamical kernel}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
The kernel proposed by Maitra and coworkers \cite{Maitra_2004,Cave_2004} in the context of dressed TDDFT (D-TDDFT) corresponds to an \textit{ad hoc} many-body theory correction to TDDFT.
|
|
More specifically, D-TDDFT adds manually to the static kernel a frequency-dependent part by reverse-engineering the exact Hamiltonian \eqref{eq:H-exact}.
|
|
The very same idea was taking further by Huix-Rotllant, Casida and coworkers. \cite{Huix-Rotllant_2011}
|
|
For the singlet states, we have
|
|
\begin{equation} \label{eq:f-Maitra}
|
|
f_M^{\updw}(\omega) = \frac{\abs*{\mel{S}{\hH}{D}}^2}{\omega - (\mel{D}{\hH}{D} - \mel{0}{\hH}{0}) }
|
|
\end{equation}
|
|
while $f_M^{\upup}(\omega) = 0$.
|
|
The expression \eqref{eq:f-Maitra} can be easily obtained by folding the double excitation onto the single excitation starting from the Hamiltonian \eqref{eq:H-exact}, as explained in Sec.~\ref{sec:dyn}.
|
|
It is clear that one must know \textit{a priori} the structure of the Hamiltonian to construct such dynamical kernel, and this obviously hampers its applicability to realistic photochemical systems where it is sometimes hard to get a clear picture of the interplay between excited states. \cite{Boggio-Pasqua_2007}
|
|
|
|
For the two-level model, the non-linear equations defined in Eq.~\eqref{eq:LR} provides the following effective Hamiltonian
|
|
\begin{equation} \label{eq:H-M}
|
|
\bH_{M}(\omega) =
|
|
\begin{pmatrix}
|
|
R_M(\omega) & C_M(\omega)
|
|
\\
|
|
-C_M(-\omega) & -R_M(-\omega)
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
with
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
R_M(\omega) = \Delta\e{} + 2 \sigma \ERI{vc}{vc} - \ERI{vc}{vc} + f_M^{\sigma}(\omega)
|
|
\\
|
|
C_M(\omega) = 2 \sigma \ERI{vc}{cv} - \ERI{vv}{cc} + f_M^{\sigma}(\omega)
|
|
\end{gather}
|
|
\end{subequations}
|
|
which provides the following excitation energies when diagonalized:
|
|
\begin{align} \label{sec:M}
|
|
\omega_{1}^{\updw} & = 1.89314
|
|
&
|
|
\omega_{3}^{\updw} & = 3.44865
|
|
&
|
|
\omega_{1}^{\upup} & = 1.43794
|
|
\end{align}
|
|
Although not particularly accurate, this kernel provides exactly the right number of solutions (2 singlets and 1 triplet).
|
|
Its accuracy could be certainly improved in a DFT context.
|
|
However, this is not the point of the present investigation.
|
|
Because $f_M^{\upup}(\omega) = 0$, the triplet excitation energy is equivalent to the TDHF excitation energy.
|
|
In the static approximation where $f_M^{\updw}(\omega) = 0$, the singlet excitations are also TDHF excitation energies.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Dynamical BSE kernel}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Within MBPT, one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
|
|
Assuming that the dynamically-screened Coulomb potential has been calculated at the random-phase approximation (RPA) level and within the Tamm-Dancoff approximation (TDA), the expression of the $\GW$ quasiparticle energy is
|
|
\begin{equation}
|
|
\e{p}^{\GW} = \e{p} + Z_{p}^{\GW} \SigGW{p}(\e{p})
|
|
\end{equation}
|
|
where $p = v$ or $c$,
|
|
\begin{equation}
|
|
\label{eq:SigC}
|
|
\SigGW{p}(\omega) = \frac{2 \ERI{pv}{vc}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{pc}{cv}^2}{\omega - \e{c} - \Omega}
|
|
\end{equation}
|
|
are the correlation parts of the self-energy associated with wither the valence of conduction orbitals,
|
|
\begin{equation}
|
|
Z_{p}^{\GW} = \qty( 1 - \left. \pdv{\SigGW{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
|
|
\end{equation}
|
|
is the renormalization factor.
|
|
In Eq.~\eqref{eq:SigC}, $\Omega = \Delta\eGW{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\eGW{} = \eGW{c} - \eGW{v}$.
|
|
|
|
One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, which reads
|
|
\begin{equation} \label{eq:HBSE}
|
|
\bH^{\dBSE}(\omega) =
|
|
\begin{pmatrix}
|
|
R(\omega) & C(\omega)
|
|
\\
|
|
-C(-\omega) & -R(-\omega)
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
with
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
R(\omega) = \Delta\eGW{} + 2 \sigma \ERI{vc}{cv} - W_R(\omega)
|
|
\\
|
|
C(\omega) = 2 \sigma \ERI{vc}{cv} - W_C(\omega)
|
|
\end{gather}
|
|
\end{subequations}
|
|
($\sigma = 1$ for singlets and $\sigma = 0$ for triplets) and
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
W_R(\omega) = \ERI{vv}{cc} + \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\eGW{}}
|
|
\\
|
|
W_C(\omega) = \ERI{vc}{cv} + \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega}
|
|
\end{gather}
|
|
\end{subequations}
|
|
are the elements of the dynamically-screened Coulomb potential for the resonant and coupling blocks of the dBSE Hamiltonian.
|
|
It can be easily shown that solving the equation
|
|
\begin{equation}
|
|
\det[\bH^{\dBSE}(\omega) - \omega \bI] = 0
|
|
\end{equation}
|
|
yields 6 solutions (per spin manifold): 3 pairs of frequencies opposite in sign, which corresponds to the 3 resonant states and the 3 anti-resonant states.
|
|
As mentioned in Ref.~\cite{Romaniello_2009b}, spurious solutions appears due to the approximate nature of the dBSE kernel.
|
|
Indeed, diagonalizing the exact Hamiltonian would produce two singlet solutions corresponding to the singly- and doubly-excited states, while there is only one triplet state (see discussion earlier in the section).
|
|
Therefore, there is one spurious solution for the singlet manifold and two spurious solution for the triplet manifold.
|
|
|
|
Within the static approximation, the BSE Hamiltonian is
|
|
\begin{equation}
|
|
\bH^{\BSE} =
|
|
\begin{pmatrix}
|
|
R^{\stat} & C^{\stat}
|
|
\\
|
|
-C^{\stat} & -R^{\stat}
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
with
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
R^{\stat} = \Delta\eGW{} + 2 \sigma \ERI{vc}{vc} - W_R(\omega = \Delta\eGW{})
|
|
\\
|
|
C^{\stat} = 2 \sigma \ERI{vc}{vc} - W_C(\omega = 0)
|
|
\end{gather}
|
|
\end{subequations}
|
|
In the static approximation, only one pair of solutions (per spin manifold) is obtained by diagonalizing $\bH^{\BSE}$.
|
|
There are, like in the dynamical case, opposite in sign.
|
|
Therefore, the static BSE Hamiltonian does not produce spurious excitations but misses the (singlet) double excitation.
|
|
|
|
Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant part of the BSE Hamiltonian, \ie, $C(\omega) = 0$, allows to remove some of these spurious excitations.
|
|
In this case, the excitation energies are obtained by solving the simple equation $R(\omega) - \omega = 0$, which yields two solutions for each spin manifold.
|
|
There is thus only one spurious excitation in the triplet manifold, the two solutions of the singlet manifold corresponding to the single and double excitations.
|
|
|
|
Another way to access dynamical effects while staying in the static framework is to use perturbation theory.
|
|
To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that
|
|
\begin{equation}
|
|
\bH^{\dBSE}(\omega) = \underbrace{\bH^{\BSE}}_{\bH^{(0)}} + \underbrace{\qty[ \bH^{\dBSE}(\omega) - \bH^{\BSE} ]}_{\bH^{(1)}}
|
|
\end{equation}
|
|
Thanks to (renormalized) first-order perturbation theory, one gets
|
|
\begin{equation}
|
|
\omega_{1,\sigma}^{\BSE1} = \omega_{1,\sigma}^{\BSE} + Z_{1} \T{\bV} \cdot \qty[ \bH^{\dBSE}(\omega = \omega_{1,\sigma}^{\BSE}) - \bH^{\BSE} ] \cdot \bV
|
|
\end{equation}
|
|
where
|
|
\begin{equation}
|
|
\bV =
|
|
\begin{pmatrix}
|
|
X \\ Y
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
are the eigenvectors of $\bH^{\BSE}$, and
|
|
\begin{equation}
|
|
Z_{1} = \qty{ 1 - \T{\bV} \cdot \left. \pdv{\bH^{\dBSE}(\omega)}{\omega} \right|_{\omega = \omega_{1,\sigma}^{\BSE}} \cdot \bV }^{-1}
|
|
\end{equation}
|
|
This corresponds to a dynamical correction to the static excitations, and the TDA can be applied to the dynamical correction, a scheme we label as dTDA in the following.
|
|
|
|
|
|
which yields
|
|
\begin{align}
|
|
\Omega & = 2.769\,327
|
|
&
|
|
\eGW{v} & = -0.863\,700
|
|
&
|
|
\eGW{c} & = +1.373\,640
|
|
\end{align}
|
|
|
|
%%% FIGURE 1 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{dBSE}
|
|
\caption{
|
|
$\det[\bH^{\dBSE}(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (red) and triplet (blue) manifolds.
|
|
\label{fig:dBSE}
|
|
}
|
|
\end{figure}
|
|
%%% %%% %%% %%%
|
|
|
|
Figure \ref{fig:dBSE} shows the three resonant solutions (for the singlet and triplet spin manifold) of the dynamical BSE Hamiltonian $\bH(\omega)$ defined in Eq.~\eqref{eq:HBSE}, the curve being invariant with respect to the transformation $\omega \to - \omega$ (electron-hole symmetry).
|
|
Numerically, we find
|
|
\begin{align}
|
|
\omega_{1,\updw}^{\dBSE} & = 1.90527
|
|
&
|
|
\omega_{2,\updw}^{\dBSE} & = 2.78377
|
|
&
|
|
\omega_{3,\updw}^{\dBSE} & = 4.90134
|
|
\end{align}
|
|
for the singlet states, and
|
|
\begin{align}
|
|
\omega_{1,\upup}^{\dBSE} & = 1.46636
|
|
&
|
|
\omega_{2,\upup}^{\dBSE} & = 2.76178
|
|
&
|
|
\omega_{3,\upup}^{\dBSE} & = 4.91545
|
|
\end{align}
|
|
for the triplet states.
|
|
it is interesting to mention that, around $\omega = \omega_1^{\sigma}$ ($\sigma =$ $\updw$ or $\upup$), the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the two other solutions, $\omega_2^{\sigma}$ and $\omega_3^{\sigma}$, stem from poles and consequently the slope is very large around these frequency values.
|
|
|
|
Diagonalizing the static BSE Hamiltonian yields the following singlet and triplet excitation energies:
|
|
\begin{align}
|
|
\omega_{1,\updw}^{\BSE} & = 1.92778
|
|
&
|
|
\omega_{1,\upup}^{\BSE} & = 1.48821
|
|
\end{align}
|
|
which shows that the physical single excitation stemming from the dynamical BSE Hamiltonian is the lowest one for each spin manifold, \ie, $\omega_1^{\updw}$ and $\omega_1^{\upup}$.
|
|
|
|
%%% FIGURE 2 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{dBSE-TDA}
|
|
\caption{
|
|
$\det[\bH^{\TDAdBSE}(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (red) and triplet (blue) manifolds within the TDA.
|
|
\label{fig:dBSE-TDA}
|
|
}
|
|
\end{figure}
|
|
%%% %%% %%% %%%
|
|
|
|
Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
|
|
As one can see, the spurious solution $\omega_2^{\sigma}$ has disappeared, and two pairs of solutions remain for each spin manifold.
|
|
Numerically, we have
|
|
\begin{align}
|
|
\omega_{1,\updw}^{\TDAdBSE} & = 1.94005
|
|
&
|
|
\omega_{3,\updw}^{\TDAdBSE} & = 4.90117
|
|
\end{align}
|
|
for the singlet states, and
|
|
\begin{align}
|
|
\omega_{1,\upup}^{\TDAdBSE} & = 1.47070
|
|
&
|
|
\omega_{3,\upup}^{\TDAdBSE} & = 4.91517
|
|
\end{align}
|
|
while the static values are
|
|
\begin{align}
|
|
\omega_{1,\updw}^{\TDABSE} & = 1.95137
|
|
&
|
|
\omega_{1,\upup}^{\TDABSE} & = 1.49603
|
|
\end{align}
|
|
|
|
All these numerical results are gathered in Table \ref{tab:BSE}.
|
|
This evidences that BSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree.
|
|
A
|
|
|
|
The perturbatively-corrected values are also reported, which shows that this scheme is very efficient at reproducing the dynamical value.
|
|
Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE1, it is quite close to the exact excitation energy.
|
|
|
|
%%% TABLE I %%%
|
|
\begin{table*}
|
|
\caption{BSE singlet and triplet excitation energies at various levels of theory.
|
|
\label{tab:BSE}
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{|c|ccccccc|c|}
|
|
Singlets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
|
|
\hline
|
|
$\omega_1$ & 1.92778 & 1.90022 & 1.91554 & 1.90527 & 1.95137 & 1.94004 & 1.94005 & 1.92145 \\
|
|
$\omega_2$ & & & & 2.78377 & & & & \\
|
|
$\omega_3$ & & & & 4.90134 & & & 4.90117 & 3.47880 \\
|
|
\hline
|
|
Triplets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
|
|
\hline
|
|
$\omega_1$ & 1.48821 & 1.46860 & 1.46260 & 1.46636 & 1.49603 & 1.47070 & 1.47070 & 1.47085 \\
|
|
$\omega_2$ & & & & 2.76178 & & & & \\
|
|
$\omega_3$ & & & & 4.91545 & & & 4.91517 & \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
%%% %%% %%% %%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Second-order BSE kernel}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
Here, we follow a different strategy and compute the dynamical second-order BSE kernel as illustrated by Yang and collaborators \cite{Zhang_2013}, and Rebolini and Toulouse \cite{Rebolini_2016}.
|
|
|
|
First, let us compute the second-order quasiparticle energies, which reads
|
|
\begin{equation}
|
|
\eGF{p} = \e{p} + Z_{p}^{\GF} \SigGF{p}(\e{p})
|
|
\end{equation}
|
|
where the second-order self-energy is
|
|
\begin{equation}
|
|
\label{eq:SigGF}
|
|
\SigGF{p}(\omega) = \frac{2 \ERI{pv}{vc}^2}{\omega - \e{v} + \e{c} - \e{v}} + \frac{2 \ERI{pc}{cv}^2}{\omega - \e{c} - (\e{c} - \e{v})}
|
|
\end{equation}
|
|
and
|
|
\begin{equation}
|
|
Z_{p}^{\GF} = \qty( 1 - \left. \pdv{\SigGF{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
|
|
\end{equation}
|
|
This expression can be easily obtained in the present case by the substitution $\Omega \to \e{c} - \e{v}$ which transforms the $GW$ self-energy into its GF2 analog.
|
|
|
|
The static Hamiltonian for this theory is just the usual RPAx (or TDHF) Hamiltonian, \ie,
|
|
\begin{equation}
|
|
\bH^{\RPAx} =
|
|
\begin{pmatrix}
|
|
A^{\stat} & B^{\stat}
|
|
\\
|
|
-B^{\stat} & -A^{\stat}
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
with
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
A^{\stat} = \Delta\eGF{} + 2 \sigma \ERI{vc}{vc} - \ERI{vv}{cc}
|
|
\\
|
|
B^{\stat} = 2 \sigma \ERI{vc}{vc} - \ERI{vc}{cv}
|
|
\end{gather}
|
|
\end{subequations}
|
|
The dynamical part of the kernel for BSE2 (that we will call dRPAx for notational consistency) is a bit ugly but it simplifies greatly in the case of the present model to yield
|
|
\begin{equation}
|
|
\bH^{\dRPAx} = \bH^{\RPAx} +
|
|
\begin{pmatrix}
|
|
A(\omega) & B
|
|
\\
|
|
-B & -A(-\omega)
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
with
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
A^{\updw}(\omega) = - \frac{4 \ERI{cv}{vv} \ERI{vc}{cc} - \ERI{vc}{cc}^2 - \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}}
|
|
\\
|
|
B^{\updw} = - \frac{4 \ERI{vc}{cv}^2 - \ERI{cc}{cc} \ERI{vc}{cv} - \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
|
|
\end{gather}
|
|
\end{subequations}
|
|
and
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
A^{\upup}(\omega) = - \frac{ \ERI{vc}{cc}^2 + \ERI{cv}{vv}^2 }{\omega - 2 \Delta\eGF{}}
|
|
\\
|
|
B^{\upup} = - \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
|
|
\end{gather}
|
|
\end{subequations}
|
|
Note that the coupling blocks $B$ are frequency independent, as they should.
|
|
This has an important consequence as this lack of frequency dependence removes one of the spurious pole.
|
|
The singlet manifold has then the right number of excitations.
|
|
However, one spurious triplet excitation remains (see Fig.~\ref{fig:dBSE2}).
|
|
Numerical results for the two-level model are reported in Table \ref{tab:RPAx} with the usual approximations and perturbative treatments.
|
|
In the case of dRPAx, the perturbative partitioning is simply
|
|
\begin{equation}
|
|
\bH^{\dRPAx}(\omega) = \underbrace{\bH^{\RPAx}}_{\bH^{(0)}} + \underbrace{\qty[ \bH^{\dRPAx}(\omega) - \bH^{\RPAx} ]}_{\bH^{(1)}}
|
|
\end{equation}
|
|
This might not be the smartest way of decomposing the Hamiltonian though but it seems to work fine.
|
|
|
|
%%% TABLE II %%%
|
|
\begin{table*}
|
|
\caption{RPAx singlet and triplet excitation energies at various levels of theory.
|
|
\label{tab:RPAx}
|
|
}
|
|
\begin{ruledtabular}
|
|
\begin{tabular}{|c|ccccccc|c|}
|
|
Singlets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
|
|
\hline
|
|
$\omega_1$ & 1.84903 & 1.90940 & 1.90950 & 1.90362 & 1.86299 & 1.92356 & 1.92359 & 1.92145 \\
|
|
$\omega_2$ & & & & & & & & \\
|
|
$\omega_3$ & & & & 4.47124 & & & 4.47097 & 3.47880 \\
|
|
\hline
|
|
Triplets & RPAx & RPAx1 & RPAx1(dTDA) & dRPAx & RPAx(TDA) & RPAx1(TDA) & dRPAx(TDA) & Exact \\
|
|
\hline
|
|
$\omega_1$ & 1.38912 & 1.44285 & 1.44304 & 1.42564 & 1.40765 & 1.46154 & 1.46155 & 1.47085 \\
|
|
$\omega_2$ & & & & & & & & \\
|
|
$\omega_3$ & & & & 4.47797 & & & 4.47767 & \\
|
|
\end{tabular}
|
|
\end{ruledtabular}
|
|
\end{table*}
|
|
%%% %%% %%% %%%
|
|
|
|
%%% FIGURE 3 %%%
|
|
\begin{figure}
|
|
\includegraphics[width=\linewidth]{dBSE2}
|
|
\caption{
|
|
$\det[\bH^{\dRPAx}(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (red) and triplet (blue) manifolds.
|
|
\label{fig:dBSE2}
|
|
}
|
|
\end{figure}
|
|
%%% %%% %%% %%%
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Sangalli's kernel}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\titou{This section is experimental...}
|
|
In Ref.~\cite{Sangalli_2011}, Sangalli proposed a dynamical kernel (based on the second RPA) without (he claims) spurious excitations thanks to the design of a number-conserving approach which correctly describes particle indistinguishability and Pauli exclusion principle.
|
|
We will first start by writing down explicitly this kernel as it is given in obscure physicist notations.
|
|
|
|
The dynamical BSE Hamiltonian with Sangalli's kernel is
|
|
\begin{equation}
|
|
\bH^\text{NC}(\omega) =
|
|
\begin{pmatrix}
|
|
H(\omega) & K(\omega)
|
|
\\
|
|
-K(-\omega) & -H(-\omega)
|
|
\end{pmatrix}
|
|
\end{equation}
|
|
with
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
H_{ia,jb}(\omega) = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \Xi_{ia,jb} (\omega)
|
|
\\
|
|
K_{ia,jb}(\omega) = \Xi_{ia,bj} (\omega)
|
|
\end{gather}
|
|
\end{subequations}
|
|
and
|
|
\begin{subequations}
|
|
\begin{gather}
|
|
\Xi_{ia,jb} (\omega) = \sum_{m \neq n} \frac{ C_{ia,mn} C_{jb,mn} }{\omega - ( \omega_{m} + \omega_{n})}
|
|
\\
|
|
C_{ia,mn} = \frac{1}{2} \sum_{jb,kc} \qty{ \qty[ \ERI{ij}{kc} \delta_{ab} + \ERI{kc}{ab} \delta_{ij} ] \qty[ R_{m,jc} R_{n,kb}
|
|
+ R_{m,kb} R_{n,jc} ] }
|
|
\end{gather}
|
|
\end{subequations}
|
|
where $R_{m,ia}$ are the elements of the RPA eigenvectors.
|
|
|
|
For the two-level model, Sangalli's kernel reads
|
|
\begin{align}
|
|
H(\omega) & = \Delta\eGW{} + \Xi_H (\omega)
|
|
\\
|
|
K(\omega) & = \Xi_K (\omega)
|
|
\end{align}
|
|
|
|
\begin{gather}
|
|
\Xi_H (\omega) = 2 \frac{ [\ERI{vv}{vc} + \ERI{vc}{cc}]^2 }{\omega - 2\omega_1}
|
|
\\
|
|
\Xi_K (\omega) = 0
|
|
\end{gather}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Take-home messages}
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
What have we learnt here?
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\acknowledgements{
|
|
%%%%%%%%%%%%%%%%%%%%%%%%
|
|
The author thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
|
|
|
|
|
|
% BIBLIOGRAPHY
|
|
\bibliography{../BSEdyn}
|
|
|
|
\end{document}
|