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Manuscript/BSE_JPCL.tex
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@ -67,8 +67,11 @@
%matrices %matrices
\newcommand{\bO}{\boldsymbol{0}} \newcommand{\bO}{\boldsymbol{0}}
\newcommand{\bI}{\boldsymbol{I}}
\newcommand{\bA}{\boldsymbol{A}}
\newcommand{\bH}{\boldsymbol{H}} \newcommand{\bH}{\boldsymbol{H}}
\newcommand{\bh}{\boldsymbol{h}} \newcommand{\bx}{\boldsymbol{x}}
\newcommand{\bb}{\boldsymbol{b}}
\newcommand{\bc}{\boldsymbol{c}} \newcommand{\bc}{\boldsymbol{c}}
\renewcommand{\tr}[1]{{#1}^{\dag}} \renewcommand{\tr}[1]{{#1}^{\dag}}
@ -395,28 +398,70 @@ diabatization and conical intersections \cite{Kaczmarski_2010}
\noindent {\textbf{The Concept of dynamical properties.}} \noindent {\textbf{The Concept of dynamical properties.}}
As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness. 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. Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes. \cite{Romaniello_2009,Sangalli_2011,ReiningBook}
To do so, let us consider we want to solve a hard problem given by the Schr{\"o}dinger-like equation of the form $\bH \bc = \omega \bc$. To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system.
If we assume that the Hamiltonian $\bH$ is of size $N \times N$, this \textit{linear} set of equations yields $K$ solutions. In most cases, this can be done by solving a set of linear equations of the form
However, in practice, $K$ can be very large.
Therefore, it is usually convenient to recast it as
\begin{equation} \begin{equation}
\label{eq:lin_sys}
\bA \bx = \omega \bx,
\end{equation}
where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bx$.
If we assume that the operator $\bA$ has a matrix representation of size $K \times K$, this \textit{linear} set of equations yields $K$ excitation energies.
However, in practice, $K$ might be very large, 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} \begin{pmatrix}
\bH_0 & \tr{\bh} \\ \bA_1 & \tr{\bb} \\
\bh & \bH_1 \\ \bb & \bA_2 \\
\end{pmatrix} \end{pmatrix}
\begin{pmatrix} \begin{pmatrix}
\bc_0\\ \bx_1 \\
\bc_1 \\ \bx_2 \\
\end{pmatrix} \end{pmatrix}
= \omega = \omega
\begin{pmatrix} \begin{pmatrix}
\bc_0 \\ \bx_1 \\
\bc_1 \\ \bx_2 \\
\end{pmatrix} \end{pmatrix},
\end{equation} \end{equation}
where the blocks $\bA_1$ and $\bA_2$, of sizes $K_1 \times K_1$ and $K_2 \times K_2$ (with $K_1 + K_2 = K$), 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 system of equation has exactly the same number of solutions. Solving separately each row of the system \eqref{eq:lin_sys_split} yields
\begin{subequations}
\begin{gather}
\label{eq:row1}
\bA_1 \bx_1 + \tr{\bb} \bx_2 = \omega \bx_1,
\\
\label{eq:row2}
\bx_2 = (\omega \bI - \bA_2)^{-1} \bb \bx_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) \bx_1 = \omega \bx_1
\end{equation}
with
\begin{equation}
\Tilde{\bA}_1(\omega) = \bA_1 + \tr{\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 basis of single excitations 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{Romaniello_2009,Sangalli_2011,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{Gatti_2007,Garniron_2018}
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 analogue 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.
One of these approximations is the so-called \textit{static} approximation, which corresponds to fix the frequency to a particular value.
For example, as commonly done within the Bethe-Salpeter formalism, $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature.
This approximation comes 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 $K$ to $K_1$.
Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%
%%% CONCLUSION %%% %%% CONCLUSION %%%

482
Manuscript/xbbib.bib
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@ -1,13 +1,43 @@
%% 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 2020-04-16 22:33:55 +0200 %% Created for Pierre-Francois Loos at 2020-04-17 20:05:26 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Gatti_2007,
Author = {M. Gatti and V. Olevano and L. Reining,and I. V. Tokatly},
Date-Added = {2020-04-17 10:10:16 +0200},
Date-Modified = {2020-04-17 10:11:34 +0200},
Doi = {10.1103/PhysRevLett.99.057401},
Journal = {Phys. Rev. Lett.},
Pages = {057401},
Title = {Transforming Nonlocality into a Frequency Dependence: A Shortcut to Spectroscopy},
Volume = {99},
Year = {2007},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.99.057401}}
@article{Sangalli_2011,
Author = {Sangalli, Davide and Romaniello, Pina and Onida, Giovanni and Marini, Andrea},
Date-Added = {2020-04-17 10:04:29 +0200},
Date-Modified = {2020-04-17 10:04:29 +0200},
Doi = {10.1063/1.3518705},
File = {/Users/loos/Zotero/storage/9S3XW2FJ/Sangalli et al. - 2011 - Double excitations in correlated systems A many--b.pdf},
Issn = {0021-9606, 1089-7690},
Journal = {J. Chem. Phys.},
Language = {en},
Month = jan,
Number = {3},
Pages = {034115},
Shorttitle = {Double Excitations in Correlated Systems},
Title = {Double Excitations in Correlated Systems: {{A}} Many\textendash{}Body Approach},
Volume = {134},
Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3518705}}
@article{Dreuw_2015, @article{Dreuw_2015,
Author = {Dreuw, Andreas and Wormit, Michael}, Author = {Dreuw, Andreas and Wormit, Michael},
Date-Added = {2020-04-16 22:30:10 +0200}, Date-Added = {2020-04-16 22:30:10 +0200},
@ -185,34 +215,36 @@
Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevLett.45.290}} Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevLett.45.290}}
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month = {Nov}, Title = {Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities},
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} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.51.1884},
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month = {Nov}, Title = {Density-Functional Theory of the Energy Gap},
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} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.51.1888},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.51.1888}}
@article{Hybertsen_1986, @article{Hybertsen_1986,
Author = {Hybertsen, Mark S. and Louie, Steven G.}, Author = {Hybertsen, Mark S. and Louie, Steven G.},
@ -278,93 +310,100 @@
Bdsk-Url-2 = {https://doi.org/10.1088/0034-4885/61/3/002}} Bdsk-Url-2 = {https://doi.org/10.1088/0034-4885/61/3/002}}
@article{Northrup_1991, @article{Northrup_1991,
title = {Many-body Calculation of the Surface-State Energies for Si(111)2\ifmmode\times\else\texttimes\fi{}1}, Author = {Northrup, John E. and Hybertsen, Mark S. and Louie, Steven G.},
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year = {1991}, Publisher = {American Physical Society},
month = {Jan}, Title = {Many-body Calculation of the Surface-State Energies for Si(111)2\ifmmode\times\else\texttimes\fi{}1},
publisher = {American Physical Society}, Url = {https://link.aps.org/doi/10.1103/PhysRevLett.66.500},
doi = {10.1103/PhysRevLett.66.500}, Volume = {66},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.66.500} Year = {1991},
} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.66.500},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.66.500}}
@article{Blase_1994, @article{Blase_1994,
title = {Self-Energy Effects on the Surface-State Energies of H-Si(111)1\ifmmode\times\else\texttimes\fi{}1}, Author = {Blase, X. and Zhu, Xuejun and Louie, Steven G.},
author = {Blase, X. and Zhu, Xuejun and Louie, Steven G.}, Doi = {10.1103/PhysRevB.49.4973},
journal = {Phys. Rev. B}, Issue = {7},
volume = {49}, Journal = {Phys. Rev. B},
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pages = {4973--4980}, Numpages = {0},
numpages = {0}, Pages = {4973--4980},
year = {1994}, Publisher = {American Physical Society},
month = {Feb}, Title = {Self-Energy Effects on the Surface-State Energies of H-Si(111)1\ifmmode\times\else\texttimes\fi{}1},
publisher = {American Physical Society}, Url = {https://link.aps.org/doi/10.1103/PhysRevB.49.4973},
doi = {10.1103/PhysRevB.49.4973}, Volume = {49},
url = {https://link.aps.org/doi/10.1103/PhysRevB.49.4973} Year = {1994},
} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.49.4973},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.49.4973}}
@article{Blase_1995, @article{Blase_1995,
title = {Quasiparticle Band Structure of Bulk Hexagonal Boron Nitride and Related Systems}, Author = {Blase, X. and Rubio, Angel and Louie, Steven G. and Cohen, Marvin L.},
author = {Blase, X. and Rubio, Angel and Louie, Steven G. and Cohen, Marvin L.}, Doi = {10.1103/PhysRevB.51.6868},
journal = {Phys. Rev. B}, Issue = {11},
volume = {51}, Journal = {Phys. Rev. B},
issue = {11}, Month = {Mar},
pages = {6868--6875}, Numpages = {0},
numpages = {0}, Pages = {6868--6875},
year = {1995}, Publisher = {American Physical Society},
month = {Mar}, Title = {Quasiparticle Band Structure of Bulk Hexagonal Boron Nitride and Related Systems},
publisher = {American Physical Society}, Url = {https://link.aps.org/doi/10.1103/PhysRevB.51.6868},
doi = {10.1103/PhysRevB.51.6868}, Volume = {51},
url = {https://link.aps.org/doi/10.1103/PhysRevB.51.6868} Year = {1995},
} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.51.6868},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.51.6868}}
@article{Rohlfing_1995, @article{Rohlfing_1995,
title = {Efficient Scheme for GW Quasiparticle Band-Structure Calculations with Aapplications to Bulk Si and to the Si(001)-(2\ifmmode\times\else\texttimes\fi{}1) Surface}, Author = {Rohlfing, Michael and Kr{\"u}ger, Peter and Pollmann, Johannes},
author = {Rohlfing, Michael and Kr{\"u}ger, Peter and Pollmann, Johannes}, Doi = {10.1103/PhysRevB.52.1905},
journal = {Phys. Rev. B}, Issue = {3},
volume = {52}, Journal = {Phys. Rev. B},
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pages = {1905--1917}, Numpages = {0},
numpages = {0}, Pages = {1905--1917},
year = {1995}, Publisher = {American Physical Society},
month = {Jul}, Title = {Efficient Scheme for GW Quasiparticle Band-Structure Calculations with Aapplications to Bulk Si and to the Si(001)-(2\ifmmode\times\else\texttimes\fi{}1) Surface},
publisher = {American Physical Society}, Url = {https://link.aps.org/doi/10.1103/PhysRevB.52.1905},
doi = {10.1103/PhysRevB.52.1905}, Volume = {52},
url = {https://link.aps.org/doi/10.1103/PhysRevB.52.1905} Year = {1995},
} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.52.1905},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.52.1905}}
@article{Verdozzi_1995, @article{Verdozzi_1995,
title = {Evaluation of $\mathit{GW}$ Approximations for the Self-Energy of a Hubbard Cluster}, Author = {Verdozzi, C. and Godby, R. W. and Holloway, S.},
author = {Verdozzi, C. and Godby, R. W. and Holloway, S.}, Doi = {10.1103/PhysRevLett.74.2327},
journal = {Phys. Rev. Lett.}, Issue = {12},
volume = {74}, Journal = {Phys. Rev. Lett.},
issue = {12}, Month = {Mar},
pages = {2327--2330}, Numpages = {0},
numpages = {0}, Pages = {2327--2330},
year = {1995}, Publisher = {American Physical Society},
month = {Mar}, Title = {Evaluation of $\mathit{GW}$ Approximations for the Self-Energy of a Hubbard Cluster},
publisher = {American Physical Society}, Url = {https://link.aps.org/doi/10.1103/PhysRevLett.74.2327},
doi = {10.1103/PhysRevLett.74.2327}, Volume = {74},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.74.2327} Year = {1995},
} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.74.2327},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.74.2327}}
@article{Campillo_1999, @article{Campillo_1999,
title = {Inelastic Lifetimes of Hot Electrons in Real Metals}, Author = {Campillo, I. and Pitarke, J. M. and Rubio, A. and Zarate, E. and Echenique, P. M.},
author = {Campillo, I. and Pitarke, J. M. and Rubio, A. and Zarate, E. and Echenique, P. M.}, Doi = {10.1103/PhysRevLett.83.2230},
journal = {Phys. Rev. Lett.}, Issue = {11},
volume = {83}, Journal = {Phys. Rev. Lett.},
issue = {11}, Month = {Sep},
pages = {2230--2233}, Numpages = {0},
numpages = {0}, Pages = {2230--2233},
year = {1999}, Publisher = {American Physical Society},
month = {Sep}, Title = {Inelastic Lifetimes of Hot Electrons in Real Metals},
publisher = {American Physical Society}, Url = {https://link.aps.org/doi/10.1103/PhysRevLett.83.2230},
doi = {10.1103/PhysRevLett.83.2230}, Volume = {83},
url = {https://link.aps.org/doi/10.1103/PhysRevLett.83.2230} Year = {1999},
} Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.83.2230},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.83.2230}}
@article{Onida_2002, @article{Onida_2002,
Author = {Onida, Giovanni and Reining, Lucia and Rubio, Angel}, Author = {Onida, Giovanni and Reining, Lucia and Rubio, Angel},
@ -708,13 +747,14 @@
@article{Dreuw_2004, @article{Dreuw_2004,
Author = {Dreuw, Andreas and Head-Gordon, Martin}, Author = {Dreuw, Andreas and Head-Gordon, Martin},
Date-Modified = {2020-04-17 20:05:23 +0200},
Doi = {10.1021/ja039556n}, Doi = {10.1021/ja039556n},
Eprint = {https://doi.org/10.1021/ja039556n}, Eprint = {https://doi.org/10.1021/ja039556n},
Journal = {Journal of the American Chemical Society}, Journal = {Journal of the American Chemical Society},
Note = {PMID: 15038755}, Note = {PMID: 15038755},
Number = {12}, Number = {12},
Pages = {4007-4016}, Pages = {4007-4016},
Title = {Failure of Time-Dependent Density Functional Theory for Long-Range Charge-Transfer Excited States:  The Zincbacteriochlorin−Bacteriochlorin and Bacteriochlorophyll−Spheroidene Complexes}, Title = {Failure of Time-Dependent Density Functional Theory for Long-Range Charge-Transfer Excited States: The Zincbacteriochlorin-Bacteriochlorin and Bacteriochlorophyll-Spheroidene Complexes},
Url = {https://doi.org/10.1021/ja039556n}, Url = {https://doi.org/10.1021/ja039556n},
Volume = {126}, Volume = {126},
Year = {2004}, Year = {2004},
@ -1248,143 +1288,125 @@
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.81.115433}, Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.81.115433},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.81.115433}} Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.81.115433}}
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author = {Rangel, Tonatiuh and Hamed, Samia M. and Bruneval, Fabien and Neaton, Jeffrey B.}, Author = {Rangel, Tonatiuh and Hamed, Samia M. and Bruneval, Fabien and Neaton, Jeffrey B.},
title = {Evaluating the GW Approximation with CCSD(T) for Charged Excitations Across the Oligoacenes}, Doi = {10.1021/acs.jctc.6b00163},
journal = {J. Chem. Theory Comput. }, Eprint = {https://doi.org/10.1021/acs.jctc.6b00163},
volume = {12}, Journal = {J. Chem. Theory Comput.},
number = {6}, Note = {PMID: 27123935},
pages = {2834-2842}, Number = {6},
year = {2016}, Pages = {2834-2842},
doi = {10.1021/acs.jctc.6b00163}, Title = {Evaluating the GW Approximation with CCSD(T) for Charged Excitations Across the Oligoacenes},
note ={PMID: 27123935}, Url = {https://doi.org/10.1021/acs.jctc.6b00163},
URL = { https://doi.org/10.1021/acs.jctc.6b00163}, Volume = {12},
eprint = { https://doi.org/10.1021/acs.jctc.6b00163} Year = {2016},
} Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00163}}
@article{Bruneval_2013, @article{Bruneval_2013,
author = {Bruneval, Fabien and Marques, Miguel A. L.}, Author = {Bruneval, Fabien and Marques, Miguel A. L.},
title = {Benchmarking the Starting Points of the GW Approximation for Molecules}, Doi = {10.1021/ct300835h},
journal = {J. Chem. Theory Comput. }, Eprint = {https://doi.org/10.1021/ct300835h},
volume = {9}, Journal = {J. Chem. Theory Comput.},
number = {1}, Note = {PMID: 26589035},
pages = {324-329}, Number = {1},
year = {2013}, Pages = {324-329},
doi = {10.1021/ct300835h}, Title = {Benchmarking the Starting Points of the GW Approximation for Molecules},
note ={PMID: 26589035}, Url = {https://doi.org/10.1021/ct300835h},
URL = { https://doi.org/10.1021/ct300835h}, Volume = {9},
eprint = { https://doi.org/10.1021/ct300835h} Year = {2013},
} Bdsk-Url-1 = {https://doi.org/10.1021/ct300835h}}
@article{Knight_2016, @article{Knight_2016,
author = {Knight, Joseph W. and Wang, Xiaopeng and Gallandi, Lukas and Dolgounitcheva, Olga and Ren, Xinguo and Ortiz, J. Vincent and Rinke, Patrick and Körzdörfer, Thomas and Marom, Noa}, Author = {Knight, Joseph W. and Wang, Xiaopeng and Gallandi, Lukas and Dolgounitcheva, Olga and Ren, Xinguo and Ortiz, J. Vincent and Rinke, Patrick and K{\"o}rzd{\"o}rfer, Thomas and Marom, Noa},
title = {Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules III: A Benchmark of GW Methods}, Doi = {10.1021/acs.jctc.5b00871},
journal = {J. Chem. Theory Comput. }, Eprint = {https://doi.org/10.1021/acs.jctc.5b00871},
volume = {12}, Journal = {J. Chem. Theory Comput.},
number = {2}, Note = {PMID: 26731609},
pages = {615-626}, Number = {2},
year = {2016}, Pages = {615-626},
doi = {10.1021/acs.jctc.5b00871}, Title = {Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules III: A Benchmark of GW Methods},
note ={PMID: 26731609}, Url = {https://doi.org/10.1021/acs.jctc.5b00871},
URL = { https://doi.org/10.1021/acs.jctc.5b00871}, Volume = {12},
eprint = { https://doi.org/10.1021/acs.jctc.5b00871} Year = {2016},
} Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b00871}}
@article{Kaplan_2016, @article{Kaplan_2016,
author = {Kaplan, F. and Harding, M. E. and Seiler, C. and Weigend, F. and Evers, F. and van Setten, M. J.}, Author = {Kaplan, F. and Harding, M. E. and Seiler, C. and Weigend, F. and Evers, F. and van Setten, M. J.},
title = {Quasi-Particle Self-Consistent GW for Molecules}, Doi = {10.1021/acs.jctc.5b01238},
journal = {J. Chem. Theory Comput. }, Eprint = {https://doi.org/10.1021/acs.jctc.5b01238},
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Date-Added = {2018-11-29 14:23:11 +0100},
Date-Modified = {2018-11-29 14:23:11 +0100},
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