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dynker.bib
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@ -1,22 +1,35 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-10-20 11:56:13 +0200
%% Created for Pierre-Francois Loos at 2020-10-20 14:42:08 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Zhang_2004,
Author = {Zhang, Fan and Burke, Kieron},
Date-Added = {2020-10-20 14:41:53 +0200},
Date-Modified = {2020-10-20 14:42:07 +0200},
Doi = {10.1103/PhysRevA.69.052510},
Journal = {Phys. Rev. A},
Pages = {052510},
Title = {Adiabatic connection for near degenerate excited states},
Volume = {69},
Year = {2004},
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.69.052510},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.69.052510}}
@article{Marut_2020,
Author = {C. Marut and B. Senjean and E. Fromager and P. F. Loos},
Date-Added = {2020-10-20 11:41:05 +0200},
Date-Modified = {2020-10-20 11:42:39 +0200},
Date-Modified = {2020-10-20 14:38:50 +0200},
Doi = {10.1039/d0fd00059k},
Journal = {Faraday. Discuss.},
Title = {Weight dependence of local exchange-correlation functionals in ensemble density-functional theory: double excitations in two-electron systems},
Volume = {Advance article},
Year = {2020}}
Year = {advance article},
Bdsk-Url-1 = {https://doi.org/10.1039/d0fd00059k}}
@article{Bottcher_1974,
Author = {C. Bottcher and K. Docken},

53
dynker.tex
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@ -92,8 +92,8 @@ In Eq.~\eqref{eq:kernel},
\end{equation}
is the (spin-resolved) Hartree-exchange-correlation (Hxc) dynamical kernel.
In the case of a spin-independent kernel, we will drop the superscript $\sigma$.
As readily seen from Eq.~\eqref{eq:kernel-Hxc}, only the correlation (c) part of the kernel is frequency dependent \alert{in a wave function context.
However, in a density-functional context, the exchange part of the kernel can be frequency dependent if exact exchange is considered. \cite{Hesselmann_2011,Hellgren_2013}}
As readily seen from Eq.~\eqref{eq:kernel-Hxc}, only the correlation (c) part of the kernel is frequency dependent in a wave function context.
However, in a density-functional context, the exchange part of the kernel can be frequency dependent if exact exchange is considered. \cite{Hesselmann_2011,Hellgren_2013}
In a wave function context, the static Hartree-exchange (Hx) matrix elements read
\begin{equation}
f_{ia,jb}^{\Hx,\sigma} = 2\sigma \ERI{ia}{jb} - \ERI{ib}{ja}
@ -192,8 +192,8 @@ In the next section, we illustrate these concepts and the various tricks that ca
\label{sec:exact}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\alert{Let us consider a two-level quantum system where two opposite-spin electrons occupied the lowest-energy level. \cite{Romaniello_2009b}
In other words, the lowest orbital is doubly occupied and the system has a singlet ground state.}
Let us consider a two-level quantum system where two opposite-spin electrons occupied the lowest-energy level. \cite{Romaniello_2009b}
In other words, the lowest orbital is doubly occupied and the system has a singlet ground state.
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 $\ket{c\bar{c}}$.
@ -232,27 +232,26 @@ with
\end{subequations}
and $\Delta\e{} = \e{c} - \e{v}$.
The energy of the only triplet state is simply $\mel{T}{\hH}{T} = \EHF + \Delta\e{} - \ERI{vv}{cc}$.
\alert{Exact excitation energies are calculated as differences of these total energies.
Note that these energies are exact results within the one-electron space spanned by the basis functions.}
Exact excitation energies are calculated as differences of these total energies.
Note that these energies are exact results within the one-electron space spanned by the basis functions.
For the sake of illustration, we will use the same \alert{molecular systems} throughout this study, and consider the singlet ground state of i) the \ce{H2} molecule ($R_{\ce{H-H}} = 1.4$ bohr) in the STO-3G basis, ii) the \ce{HeH+} molecule ($R_{\ce{He-H}} = 1.4632$ bohr) in the STO-3G basis, and iii) the \ce{He} atom in Pople's 6-31G basis set. \cite{SzaboBook}
\alert{The minimal basis (STO-3G) and double-zeta basis (6-31G) have been chosen to produce two-level systems.
The STO-3G basis for two-center systems (\ce{H2} and \ce{HeH+}) corresponds to one $s$-type gaussian basis function on each center, while the 6-31G basis for the helium atom corresponds to two (contracted) $s$-type gaussian functions with different exponents.}
For the sake of illustration, we will use the same molecular systems throughout this study, and consider the singlet ground state of i) the \ce{H2} molecule ($R_{\ce{H-H}} = 1.4$ bohr) in the STO-3G basis, ii) the \ce{HeH+} molecule ($R_{\ce{He-H}} = 1.4632$ bohr) in the STO-3G basis, and iii) the \ce{He} atom in Pople's 6-31G basis set. \cite{SzaboBook}
The minimal basis (STO-3G) and double-zeta basis (6-31G) have been chosen to produce two-level systems.
The STO-3G basis for two-center systems (\ce{H2} and \ce{HeH+}) corresponds to one $s$-type gaussian basis function on each center, while the 6-31G basis for the helium atom corresponds to two (contracted) $s$-type gaussian functions with different exponents.
These three systems provide prototypical examples of valence, charge-transfer, and Rydberg excitations, respectively, and will be employed to quantity the performance of the various methods considered in the present study for each type of excited states. \cite{Senjean_2015,Romaniello_2009b}
\alert{
In the case of \ce{H2}, the HOMO and LUMO orbitals have $\sigma_g$ and $\sigma_u$ symmetries, respectively.
The electronic configuration of the ground state is $\sigma_g^2$, and the doubly-excited state of configuration $\sigma_u^2$ has an auto-ionising resonance nature. \cite{Bottcher_1974,Barca_2018a,Marut_2020}
The singly-excited states correspond to $\sigma_g \sigma_u$ configurations.
The singly-excited states have $\sigma_g \sigma_u$ configurations.
In He, highly-accurate calculations reveal that the lowest doubly-excited state of configuration $1s^2$ is an auto-ionising resonance state, extremely high in energy and lies in the continuum. \cite{Madden_1963,Burges_1995,Marut_2020}
However, in a minimal basis set such as STO-3G, it is of Rydberg nature as it corresponds to a transition from a relatively compact $s$-type function to a more diffuse orbital of the same symmetry.
In the heteronuclear diatomic \ce{HeH+}, a Mulliken or L\"owdin population analysis associates $1.53$ electrons on the \ce{He} center and $0.47$ electrons on the \ce{H} nucleus for the ground state, \cite{SzaboBook} with an opposite trend for the excited states.
Thus, electronic excitations in \ce{HeH+} correspond to a charge transfer from the \ce{He} nucleus to the proton.}
In the heteronuclear diatomic molecule \ce{HeH+}, a Mulliken or L\"owdin population analysis associates $1.53$ electrons on the \ce{He} center and $0.47$ electrons on the \ce{H} nucleus for the ground state. \cite{SzaboBook}
Thus, electronic excitations in \ce{HeH+} correspond to a charge transfer from the \ce{He} nucleus to the proton.
The numerical values of the various quantities defined above are gathered in Table \ref{tab:params} for each system.
%%% TABLE I %%%
\begin{table*}
\caption{Numerical values \alert{(in eV)} of the valence and conduction orbital energies, $\e{v}$ and $\e{c}$, and two-electron integrals in the orbital basis for various two-level systems.
\caption{Numerical values (in eV) of the valence and conduction orbital energies, $\e{v}$ and $\e{c}$, and two-electron integrals in the orbital basis for various two-level systems.
\label{tab:params}
}
\begin{ruledtabular}
@ -276,10 +275,10 @@ We are going to use these as reference for the remaining of this study.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 to the static kernel a frequency-dependent part by reverse-engineering the exact Hamiltonian: \alert{one single and one double excitations}, assumed to be strongly coupled, are isolated from among the spectrum and added manually to the static kernel.
The very same idea was \alert{taken} further by Huix-Rotllant, Casida and coworkers, \cite{Huix-Rotllant_2011} and tested on a large set of molecules.
More specifically, D-TDDFT adds to the static kernel a frequency-dependent part by reverse-engineering the exact Hamiltonian: one single and one double excitations, assumed to be strongly coupled, are isolated from among the spectrum and added manually to the static kernel.
The very same idea was taken further by Huix-Rotllant, Casida and coworkers, \cite{Huix-Rotllant_2011} and tested on a large set of molecules.
Here, we start instead from a HF reference.
The static problem \alert{(\ie, the frequency-independent Hamiltonian)} corresponds then to the time-dependent HF (TDHF) Hamiltonian, while in the TDA, it reduces to configuration interaction with singles (CIS). \cite{Dreuw_2005}
The static problem (\ie, the frequency-independent Hamiltonian) corresponds then to the time-dependent HF (TDHF) Hamiltonian, while in the TDA, it reduces to configuration interaction with singles (CIS). \cite{Dreuw_2005}
For the two-level model, the reverse-engineering process of the exact Hamiltonian \eqref{eq:H-exact} yields
\begin{equation} \label{eq:f-Maitra}
@ -345,10 +344,10 @@ Because, there is nothing to dress for the triplet state, the TDHF and D-TDHF tr
%%% %%% %%% %%%
Although not particularly accurate for the single excitations, Maitra's dynamical kernel allows to access the double excitation with good accuracy and provides exactly the right number of solutions (two singlets and one triplet).
Note that this correlation kernel is known to work best in the weak correlation regime (which is the case here) \alert{in the situation where one single and one double excitations are energetically close and well separated from the others,} \cite{Maitra_2004,Loos_2019,Loos_2020d} but it is not intended to explore strongly correlated systems. \cite{Carrascal_2018}
Note that this correlation kernel is known to work best in the weak correlation regime (which is the case here) in the situation where one single and one double excitations are energetically close and well separated from the others, \cite{Maitra_2004,Loos_2019,Loos_2020d} but it is not intended to explore strongly correlated systems. \cite{Carrascal_2018}
Its accuracy for the single excitations could be certainly improved in a density-functional theory context.
However, this is not the point of the present investigation.
\alert{In Ref.~\onlinecite{Huix-Rotllant_2011}, the authors observed that the best results are obtained using a hybrid kernel for the static part.}
In Ref.~\onlinecite{Huix-Rotllant_2011}, the authors observed that the best results are obtained using a hybrid kernel for the static part.
Table \ref{tab:Maitra} also reports the slightly improved (thanks to error compensation) CIS and D-CIS excitation energies.
In particular, single excitations are greatly improved without altering the accuracy of the double excitation.
@ -356,13 +355,13 @@ Graphically, the curves obtained for CIS and D-CIS are extremely similar to the
In the case of \ce{H2} in a minimal basis, because $\mel{S}{\hH}{D} = 0$, \cite{SzaboBook} there is no dynamical correction for both singlets and triplets, and one cannot access the double excitation with Maitra's kernel.
It would be, of course, a different story in a larger basis set where the coupling between singles and doubles would be non-zero.
\alert{The fact that $\mel{S}{\hH}{D} = 0$ for \ce{H2} in a minimal basis is the direct consequence of the lack of orbital relaxation in the excited states, which is itself due to the fact that the molecular orbitals in that case are unambiguously defined by symmetry.}
The fact that $\mel{S}{\hH}{D} = 0$ for \ce{H2} in a minimal basis is the direct consequence of the lack of orbital relaxation in the excited states, which is itself due to the fact that the molecular orbitals in that case are unambiguously defined by symmetry.
%%% FIGURE 1 %%%
\begin{figure}
\includegraphics[width=\linewidth]{fig1}
\caption{
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in \alert{eV}) for both the singlet (gray and black) and triplet (orange) manifolds of \ce{HeH+}.
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange) manifolds of \ce{HeH+}.
The static TDHF Hamiltonian (dashed) and dynamic D-TDHF Hamiltonian (solid) are considered.
\label{fig:Maitra}
}
@ -496,7 +495,7 @@ Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dyn
\begin{figure}
\includegraphics[width=\linewidth]{fig2}
\caption{
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ \alert{(in eV)} for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}.
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}.
The static BSE Hamiltonian (dashed) and dynamic dBSE Hamiltonian (solid) are considered.
\label{fig:dBSE}
}
@ -505,7 +504,7 @@ Let us highlight the fact that, unlike in Ref.~\onlinecite{Loos_2020e} where dyn
In the static approximation, only one solution per spin manifold is obtained by diagonalizing $\bH_{\BSE}^{\sigma}$ (see Fig.~\ref{fig:dBSE} and Table \ref{tab:BSE}).
Therefore, the static BSE Hamiltonian misses the (singlet) double excitation (as it should), and it shows that the physical single excitation stemming from the dBSE Hamiltonian is indeed the lowest in energy for each spin manifold, \ie, $\omega_1^{\dBSE,\updw}$ and $\omega_1^{\dBSE,\upup}$.
This can be further verified by switching off gradually the electron-electron interaction as one would do in the adiabatic connection formalism. \cite{Langreth_1975,Gunnarsson_1976}
This can be further verified by switching off gradually the electron-electron interaction as one would do in the adiabatic connection formalism. \cite{Langreth_1975,Gunnarsson_1976,Zhang_2004}
Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant parts of the dBSE Hamiltonian \eqref{eq:HBSE}, does not change the situation in terms of spurious solutions: there is still one spurious excitation in the triplet manifold ($\omega_{2}^{\BSE,\upup}$), and the two solutions for the singlet manifold which corresponds to the single and double excitations.
However, it does increase significantly the static excitations while the magnitude of the dynamical corrections is not altered by the TDA.
@ -687,14 +686,14 @@ In the case of BSE2, the perturbative partitioning (pBSE2) is simply
\begin{figure}
\includegraphics[width=\linewidth]{fig3}
\caption{
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ \alert{(in eV)} for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}.
$\det[\bH(\omega) - \omega \bI]$ as a function of $\omega$ (in hartree) for both the singlet (gray and black) and triplet (orange and red) manifolds of \ce{HeH+}.
The static BSE2 Hamiltonian (dashed) and dynamic dBSE2 Hamiltonian (solid) are considered.
\label{fig:BSE2}
}
\end{figure}
%%% %%% %%% %%%
As compared to dBSE, dBSE2 produces much larger dynamical corrections to the static excitation energies, \alert{$\omega_1^{\updw}$ and $\omega_1^{\upup}$}, (see values in square brackets in Table \ref{tab:BSE2}) probably due to the poorer quality of its static reference (TDHF or CIS).
As compared to dBSE, dBSE2 produces much larger dynamical corrections to the static excitation energies, $\omega_1^{\updw}$ and $\omega_1^{\upup}$, (see values in square brackets in Table \ref{tab:BSE2}) probably due to the poorer quality of its static reference (TDHF or CIS).
Similarly to what has been observed in Sec.~\ref{sec:Maitra}, the TDA vertical excitations are slightly more accurate due to error compensations.
Note also that the perturbative treatment is a remarkably good approximation to the dynamical scheme for single excitations (except for \ce{H2}, see below), especially in the TDA.
This justifies the use of the perturbative treatment in Refs.~\onlinecite{Zhang_2013,Rebolini_2016}.
@ -711,14 +710,14 @@ However, one cannot access the double excitation.
The take-home message of the present paper is that dynamical kernels have much more to give that one would think.
In more scientific terms, dynamical kernels can provide, thanks to their frequency-dependent nature, additional excitations that can be associated to higher-order excitations (such as the infamous double excitations), an unappreciated feature of dynamical quantities.
However, they sometimes give too much, and generate spurious excitations, \ie, excitation which does not correspond to any physical excited state.
The appearance of these \alert{fictitious} excitations is due to the approximate nature of the dynamical kernel.
The appearance of these fictitious excitations is due to the approximate nature of the dynamical kernel.
Moreover, because of the non-linear character of the linear response problem when one employs a dynamical kernel, it is computationally more involved to access these extra excitations.
Using a simple two-model system, we have explored the physics of three dynamical kernels: i) a kernel based on the dressed TDDFT method introduced by Maitra and coworkers, \cite{Maitra_2004} ii) the dynamical kernel from the BSE formalism derived by Strinati in his hallmark 1988 paper, \cite{Strinati_1988} as well as the second-order BSE kernel derived by Zhang \textit{et al.}, \cite{Zhang_2013} and Rebolini and Toulouse. \cite{Rebolini_2016,Rebolini_PhD}
Prototypical examples of valence, charge-transfer, and Rydberg excited states have been considered.
From these, we have observed that, overall, the dynamical correction usually improves the static excitation energies, and that, although one can access double excitations, the accuracy of the BSE and BSE2 kernels for double excitations is rather average.
If one has no interest in double excitations, a perturbative treatment is an excellent alternative to a non-linear resolution of the dynamical equations.
\alert{Although it would be interesting to study the performance of such kernels in the case of stretched bonds, the appearance of singlet and triplet instabilities makes such type of invesgations particularly difficult.}
Although it would be interesting to study the performance of such kernels in the case of stretched bonds, the appearance of singlet and triplet instabilities makes such type of investigations particularly difficult.
We hope that the present contribution will foster new developments around dynamical kernels for optical excitations, in particular to access double excitations in molecular systems.