clean up
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@ -67,8 +67,11 @@
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%matrices
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\newcommand{\bO}{\boldsymbol{0}}
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\newcommand{\bI}{\boldsymbol{I}}
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\newcommand{\bA}{\boldsymbol{A}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\bh}{\boldsymbol{h}}
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\newcommand{\bx}{\boldsymbol{x}}
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\newcommand{\bb}{\boldsymbol{b}}
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\newcommand{\bc}{\boldsymbol{c}}
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\renewcommand{\tr}[1]{{#1}^{\dag}}
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@ -395,28 +398,70 @@ diabatization and conical intersections \cite{Kaczmarski_2010}
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\noindent {\textbf{The Concept of dynamical properties.}}
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As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
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Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes.
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To do so, let us consider we want to solve a hard problem given by the Schr{\"o}dinger-like equation of the form $\bH \bc = \omega \bc$.
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If we assume that the Hamiltonian $\bH$ is of size $N \times N$, this \textit{linear} set of equations yields $K$ solutions.
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However, in practice, $K$ can be very large.
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Therefore, it is usually convenient to recast it as
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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}
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To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system.
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In most cases, this can be done by solving a set of linear equations of the form
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\begin{equation}
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\label{eq:lin_sys}
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\bA \bx = \omega \bx,
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\end{equation}
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where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bx$.
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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.
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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
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\begin{equation}
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\label{eq:lin_sys_split}
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\begin{pmatrix}
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\bH_0 & \tr{\bh} \\
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\bh & \bH_1 \\
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\bA_1 & \tr{\bb} \\
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\bb & \bA_2 \\
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\end{pmatrix}
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\begin{pmatrix}
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\bc_0\\
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\bc_1 \\
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\bx_1 \\
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\bx_2 \\
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\end{pmatrix}
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= \omega
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\begin{pmatrix}
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\bc_0 \\
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\bc_1 \\
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\end{pmatrix}
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\bx_1 \\
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\bx_2 \\
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\end{pmatrix},
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\end{equation}
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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.
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Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors.
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This system of equation has exactly the same number of solutions.
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Solving separately each row of the system \eqref{eq:lin_sys_split} yields
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\begin{subequations}
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\begin{gather}
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\label{eq:row1}
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\bA_1 \bx_1 + \tr{\bb} \bx_2 = \omega \bx_1,
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\\
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\label{eq:row2}
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\bx_2 = (\omega \bI - \bA_2)^{-1} \bb \bx_1.
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\end{gather}
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\end{subequations}
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Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator
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\begin{equation}
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\label{eq:non_lin_sys}
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\Tilde{\bA}_1(\omega) \bx_1 = \omega \bx_1
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\end{equation}
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with
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\begin{equation}
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\Tilde{\bA}_1(\omega) = \bA_1 + \tr{\bb} (\omega \bI - \bA_2)^{-1} \bb
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\end{equation}
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which has, by construction, exactly the same solutions than the linear system \eqref{eq:lin_sys} but a smaller dimension.
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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}
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How have we been able to reduce the dimension of the problem while keeping the same number of solutions?
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To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent.
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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.
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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}
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Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension.
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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}.
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Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently.
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One of these approximations is the so-called \textit{static} approximation, which corresponds to fix the frequency to a particular value.
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For example, as commonly done within the Bethe-Salpeter formalism, $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
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In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature.
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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 $K_1$.
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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.
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%%%%%%%%%%%%%%%%%%
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%%% CONCLUSION %%%
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@ -1,13 +1,43 @@
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-04-16 22:33:55 +0200
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%% Created for Pierre-Francois Loos at 2020-04-17 20:05:26 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@article{Gatti_2007,
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Author = {M. Gatti and V. Olevano and L. Reining,and I. V. Tokatly},
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Date-Added = {2020-04-17 10:10:16 +0200},
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Date-Modified = {2020-04-17 10:11:34 +0200},
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Doi = {10.1103/PhysRevLett.99.057401},
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Journal = {Phys. Rev. Lett.},
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Pages = {057401},
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Title = {Transforming Nonlocality into a Frequency Dependence: A Shortcut to Spectroscopy},
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Volume = {99},
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Year = {2007},
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.99.057401}}
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@article{Sangalli_2011,
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Author = {Sangalli, Davide and Romaniello, Pina and Onida, Giovanni and Marini, Andrea},
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Date-Added = {2020-04-17 10:04:29 +0200},
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Date-Modified = {2020-04-17 10:04:29 +0200},
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Doi = {10.1063/1.3518705},
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File = {/Users/loos/Zotero/storage/9S3XW2FJ/Sangalli et al. - 2011 - Double excitations in correlated systems A many--b.pdf},
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Issn = {0021-9606, 1089-7690},
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Journal = {J. Chem. Phys.},
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Language = {en},
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Month = jan,
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Number = {3},
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Pages = {034115},
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Shorttitle = {Double Excitations in Correlated Systems},
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Title = {Double Excitations in Correlated Systems: {{A}} Many\textendash{}Body Approach},
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Volume = {134},
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||||
Year = {2011},
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||||
Bdsk-Url-1 = {https://doi.org/10.1063/1.3518705}}
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@article{Dreuw_2015,
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Author = {Dreuw, Andreas and Wormit, Michael},
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Date-Added = {2020-04-16 22:30:10 +0200},
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@ -185,34 +215,36 @@
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Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevLett.45.290}}
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@article{Perdew_1983,
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title = {Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities},
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year = {1983},
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month = {Nov},
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publisher = {American Physical Society},
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doi = {10.1103/PhysRevLett.51.1884},
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||||
url = {https://link.aps.org/doi/10.1103/PhysRevLett.51.1884}
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||||
}
|
||||
Author = {Perdew, John P. and Levy, Mel},
|
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Doi = {10.1103/PhysRevLett.51.1884},
|
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Issue = {20},
|
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Journal = {Phys. Rev. Lett.},
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Month = {Nov},
|
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Numpages = {0},
|
||||
Pages = {1884--1887},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities},
|
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Url = {https://link.aps.org/doi/10.1103/PhysRevLett.51.1884},
|
||||
Volume = {51},
|
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Year = {1983},
|
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.51.1884},
|
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.51.1884}}
|
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|
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@article{Sham_1983,
|
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year = {1983},
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month = {Nov},
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publisher = {American Physical Society},
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doi = {10.1103/PhysRevLett.51.1888},
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|
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}
|
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Author = {Sham, L. J. and Schl\"uter, M.},
|
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Doi = {10.1103/PhysRevLett.51.1888},
|
||||
Issue = {20},
|
||||
Journal = {Phys. Rev. Lett.},
|
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Month = {Nov},
|
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Numpages = {0},
|
||||
Pages = {1888--1891},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Density-Functional Theory of the Energy Gap},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevLett.51.1888},
|
||||
Volume = {51},
|
||||
Year = {1983},
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.51.1888},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.51.1888}}
|
||||
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@article{Hybertsen_1986,
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Author = {Hybertsen, Mark S. and Louie, Steven G.},
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@ -278,93 +310,100 @@
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Bdsk-Url-2 = {https://doi.org/10.1088/0034-4885/61/3/002}}
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@article{Northrup_1991,
|
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title = {Many-body Calculation of the Surface-State Energies for Si(111)2\ifmmode\times\else\texttimes\fi{}1},
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doi = {10.1103/PhysRevLett.66.500},
|
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url = {https://link.aps.org/doi/10.1103/PhysRevLett.66.500}
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}
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Author = {Northrup, John E. and Hybertsen, Mark S. and Louie, Steven G.},
|
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Doi = {10.1103/PhysRevLett.66.500},
|
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Issue = {4},
|
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Journal = {Phys. Rev. Lett.},
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Month = {Jan},
|
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Numpages = {0},
|
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Pages = {500--503},
|
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Publisher = {American Physical Society},
|
||||
Title = {Many-body Calculation of the Surface-State Energies for Si(111)2\ifmmode\times\else\texttimes\fi{}1},
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||||
Url = {https://link.aps.org/doi/10.1103/PhysRevLett.66.500},
|
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Volume = {66},
|
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Year = {1991},
|
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Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.66.500},
|
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Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.66.500}}
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|
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@article{Blase_1994,
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title = {Self-Energy Effects on the Surface-State Energies of H-Si(111)1\ifmmode\times\else\texttimes\fi{}1},
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|
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url = {https://link.aps.org/doi/10.1103/PhysRevB.49.4973}
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}
|
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Author = {Blase, X. and Zhu, Xuejun and Louie, Steven G.},
|
||||
Doi = {10.1103/PhysRevB.49.4973},
|
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Issue = {7},
|
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Journal = {Phys. Rev. B},
|
||||
Month = {Feb},
|
||||
Numpages = {0},
|
||||
Pages = {4973--4980},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Self-Energy Effects on the Surface-State Energies of H-Si(111)1\ifmmode\times\else\texttimes\fi{}1},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.49.4973},
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||||
Volume = {49},
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||||
Year = {1994},
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||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.49.4973},
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||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.49.4973}}
|
||||
|
||||
@article{Blase_1995,
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title = {Quasiparticle Band Structure of Bulk Hexagonal Boron Nitride and Related Systems},
|
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author = {Blase, X. and Rubio, Angel and Louie, Steven G. and Cohen, Marvin L.},
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numpages = {0},
|
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year = {1995},
|
||||
month = {Mar},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevB.51.6868},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevB.51.6868}
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||||
}
|
||||
Author = {Blase, X. and Rubio, Angel and Louie, Steven G. and Cohen, Marvin L.},
|
||||
Doi = {10.1103/PhysRevB.51.6868},
|
||||
Issue = {11},
|
||||
Journal = {Phys. Rev. B},
|
||||
Month = {Mar},
|
||||
Numpages = {0},
|
||||
Pages = {6868--6875},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Quasiparticle Band Structure of Bulk Hexagonal Boron Nitride and Related Systems},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.51.6868},
|
||||
Volume = {51},
|
||||
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,
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||||
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},
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author = {Rohlfing, Michael and Kr{\"u}ger, Peter and Pollmann, Johannes},
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volume = {52},
|
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|
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|
||||
numpages = {0},
|
||||
year = {1995},
|
||||
month = {Jul},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevB.52.1905},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevB.52.1905}
|
||||
}
|
||||
Author = {Rohlfing, Michael and Kr{\"u}ger, Peter and Pollmann, Johannes},
|
||||
Doi = {10.1103/PhysRevB.52.1905},
|
||||
Issue = {3},
|
||||
Journal = {Phys. Rev. B},
|
||||
Month = {Jul},
|
||||
Numpages = {0},
|
||||
Pages = {1905--1917},
|
||||
Publisher = {American Physical Society},
|
||||
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},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevB.52.1905},
|
||||
Volume = {52},
|
||||
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,
|
||||
title = {Evaluation of $\mathit{GW}$ Approximations for the Self-Energy of a Hubbard Cluster},
|
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|
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numpages = {0},
|
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year = {1995},
|
||||
month = {Mar},
|
||||
publisher = {American Physical Society},
|
||||
doi = {10.1103/PhysRevLett.74.2327},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevLett.74.2327}
|
||||
}
|
||||
Author = {Verdozzi, C. and Godby, R. W. and Holloway, S.},
|
||||
Doi = {10.1103/PhysRevLett.74.2327},
|
||||
Issue = {12},
|
||||
Journal = {Phys. Rev. Lett.},
|
||||
Month = {Mar},
|
||||
Numpages = {0},
|
||||
Pages = {2327--2330},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Evaluation of $\mathit{GW}$ Approximations for the Self-Energy of a Hubbard Cluster},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevLett.74.2327},
|
||||
Volume = {74},
|
||||
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,
|
||||
title = {Inelastic Lifetimes of Hot Electrons in Real Metals},
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numpages = {0},
|
||||
year = {1999},
|
||||
month = {Sep},
|
||||
publisher = {American Physical Society},
|
||||
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|
||||
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|
||||
}
|
||||
Author = {Campillo, I. and Pitarke, J. M. and Rubio, A. and Zarate, E. and Echenique, P. M.},
|
||||
Doi = {10.1103/PhysRevLett.83.2230},
|
||||
Issue = {11},
|
||||
Journal = {Phys. Rev. Lett.},
|
||||
Month = {Sep},
|
||||
Numpages = {0},
|
||||
Pages = {2230--2233},
|
||||
Publisher = {American Physical Society},
|
||||
Title = {Inelastic Lifetimes of Hot Electrons in Real Metals},
|
||||
Url = {https://link.aps.org/doi/10.1103/PhysRevLett.83.2230},
|
||||
Volume = {83},
|
||||
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,
|
||||
Author = {Onida, Giovanni and Reining, Lucia and Rubio, Angel},
|
||||
@ -708,13 +747,14 @@
|
||||
|
||||
@article{Dreuw_2004,
|
||||
Author = {Dreuw, Andreas and Head-Gordon, Martin},
|
||||
Date-Modified = {2020-04-17 20:05:23 +0200},
|
||||
Doi = {10.1021/ja039556n},
|
||||
Eprint = {https://doi.org/10.1021/ja039556n},
|
||||
Journal = {Journal of the American Chemical Society},
|
||||
Note = {PMID: 15038755},
|
||||
Number = {12},
|
||||
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},
|
||||
Volume = {126},
|
||||
Year = {2004},
|
||||
@ -1248,143 +1288,125 @@
|
||||
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.81.115433},
|
||||
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.81.115433}}
|
||||
|
||||
|
||||
@article{Rangel_2016,
|
||||
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},
|
||||
journal = {J. Chem. Theory Comput. },
|
||||
volume = {12},
|
||||
number = {6},
|
||||
pages = {2834-2842},
|
||||
year = {2016},
|
||||
doi = {10.1021/acs.jctc.6b00163},
|
||||
note ={PMID: 27123935},
|
||||
URL = { https://doi.org/10.1021/acs.jctc.6b00163},
|
||||
eprint = { https://doi.org/10.1021/acs.jctc.6b00163}
|
||||
}
|
||||
|
||||
Author = {Rangel, Tonatiuh and Hamed, Samia M. and Bruneval, Fabien and Neaton, Jeffrey B.},
|
||||
Doi = {10.1021/acs.jctc.6b00163},
|
||||
Eprint = {https://doi.org/10.1021/acs.jctc.6b00163},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 27123935},
|
||||
Number = {6},
|
||||
Pages = {2834-2842},
|
||||
Title = {Evaluating the GW Approximation with CCSD(T) for Charged Excitations Across the Oligoacenes},
|
||||
Url = {https://doi.org/10.1021/acs.jctc.6b00163},
|
||||
Volume = {12},
|
||||
Year = {2016},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00163}}
|
||||
|
||||
@article{Bruneval_2013,
|
||||
author = {Bruneval, Fabien and Marques, Miguel A. L.},
|
||||
title = {Benchmarking the Starting Points of the GW Approximation for Molecules},
|
||||
journal = {J. Chem. Theory Comput. },
|
||||
volume = {9},
|
||||
number = {1},
|
||||
pages = {324-329},
|
||||
year = {2013},
|
||||
doi = {10.1021/ct300835h},
|
||||
note ={PMID: 26589035},
|
||||
URL = { https://doi.org/10.1021/ct300835h},
|
||||
eprint = { https://doi.org/10.1021/ct300835h}
|
||||
}
|
||||
|
||||
Author = {Bruneval, Fabien and Marques, Miguel A. L.},
|
||||
Doi = {10.1021/ct300835h},
|
||||
Eprint = {https://doi.org/10.1021/ct300835h},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 26589035},
|
||||
Number = {1},
|
||||
Pages = {324-329},
|
||||
Title = {Benchmarking the Starting Points of the GW Approximation for Molecules},
|
||||
Url = {https://doi.org/10.1021/ct300835h},
|
||||
Volume = {9},
|
||||
Year = {2013},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/ct300835h}}
|
||||
|
||||
@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},
|
||||
title = {Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules III: A Benchmark of GW Methods},
|
||||
journal = {J. Chem. Theory Comput. },
|
||||
volume = {12},
|
||||
number = {2},
|
||||
pages = {615-626},
|
||||
year = {2016},
|
||||
doi = {10.1021/acs.jctc.5b00871},
|
||||
note ={PMID: 26731609},
|
||||
URL = { https://doi.org/10.1021/acs.jctc.5b00871},
|
||||
eprint = { https://doi.org/10.1021/acs.jctc.5b00871}
|
||||
}
|
||||
|
||||
|
||||
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},
|
||||
Doi = {10.1021/acs.jctc.5b00871},
|
||||
Eprint = {https://doi.org/10.1021/acs.jctc.5b00871},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 26731609},
|
||||
Number = {2},
|
||||
Pages = {615-626},
|
||||
Title = {Accurate Ionization Potentials and Electron Affinities of Acceptor Molecules III: A Benchmark of GW Methods},
|
||||
Url = {https://doi.org/10.1021/acs.jctc.5b00871},
|
||||
Volume = {12},
|
||||
Year = {2016},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b00871}}
|
||||
|
||||
@article{Kaplan_2016,
|
||||
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},
|
||||
journal = {J. Chem. Theory Comput. },
|
||||
volume = {12},
|
||||
number = {6},
|
||||
pages = {2528-2541},
|
||||
year = {2016},
|
||||
doi = {10.1021/acs.jctc.5b01238},
|
||||
note ={PMID: 27168352},
|
||||
URL = { https://doi.org/10.1021/acs.jctc.5b01238},
|
||||
eprint = { https://doi.org/10.1021/acs.jctc.5b01238}
|
||||
}
|
||||
|
||||
Author = {Kaplan, F. and Harding, M. E. and Seiler, C. and Weigend, F. and Evers, F. and van Setten, M. J.},
|
||||
Doi = {10.1021/acs.jctc.5b01238},
|
||||
Eprint = {https://doi.org/10.1021/acs.jctc.5b01238},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 27168352},
|
||||
Number = {6},
|
||||
Pages = {2528-2541},
|
||||
Title = {Quasi-Particle Self-Consistent GW for Molecules},
|
||||
Url = {https://doi.org/10.1021/acs.jctc.5b01238},
|
||||
Volume = {12},
|
||||
Year = {2016},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b01238}}
|
||||
|
||||
@article{Caruso_2016,
|
||||
author = {Caruso, Fabio and Dauth, Matthias and van Setten, Michiel J. and Rinke, Patrick},
|
||||
title = {Benchmark of GW Approaches for the GW100 Test Set},
|
||||
journal = {Journal of Chemical Theory and Computation},
|
||||
volume = {12},
|
||||
number = {10},
|
||||
pages = {5076-5087},
|
||||
year = {2016},
|
||||
doi = {10.1021/acs.jctc.6b00774},
|
||||
note ={PMID: 27631585},
|
||||
URL = { https://doi.org/10.1021/acs.jctc.6b00774},
|
||||
eprint = { https://doi.org/10.1021/acs.jctc.6b00774}
|
||||
}
|
||||
|
||||
Author = {Caruso, Fabio and Dauth, Matthias and van Setten, Michiel J. and Rinke, Patrick},
|
||||
Doi = {10.1021/acs.jctc.6b00774},
|
||||
Eprint = {https://doi.org/10.1021/acs.jctc.6b00774},
|
||||
Journal = {Journal of Chemical Theory and Computation},
|
||||
Note = {PMID: 27631585},
|
||||
Number = {10},
|
||||
Pages = {5076-5087},
|
||||
Title = {Benchmark of GW Approaches for the GW100 Test Set},
|
||||
Url = {https://doi.org/10.1021/acs.jctc.6b00774},
|
||||
Volume = {12},
|
||||
Year = {2016},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00774}}
|
||||
|
||||
@article{Korbel_2014,
|
||||
author = {K{\"{o}}rbel, Sabine and Boulanger, Paul and Duchemin, Ivan and Blase, Xavier and Marques, Miguel A. L. and Botti, Silvana},
|
||||
title = {Benchmark Many-Body GW and Bethe–Salpeter Calculations for Small Transition Metal Molecules},
|
||||
journal = {J. Chem. Theory Comput. },
|
||||
volume = {10},
|
||||
number = {9},
|
||||
pages = {3934-3943},
|
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year = {2014},
|
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doi = {10.1021/ct5003658},
|
||||
note ={PMID: 26588537},
|
||||
URL = { https://doi.org/10.1021/ct5003658},
|
||||
eprint = { https://doi.org/10.1021/ct5003658}
|
||||
}
|
||||
|
||||
Author = {K{\"{o}}rbel, Sabine and Boulanger, Paul and Duchemin, Ivan and Blase, Xavier and Marques, Miguel A. L. and Botti, Silvana},
|
||||
Doi = {10.1021/ct5003658},
|
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Eprint = {https://doi.org/10.1021/ct5003658},
|
||||
Journal = {J. Chem. Theory Comput.},
|
||||
Note = {PMID: 26588537},
|
||||
Number = {9},
|
||||
Pages = {3934-3943},
|
||||
Title = {Benchmark Many-Body GW and Bethe--Salpeter Calculations for Small Transition Metal Molecules},
|
||||
Url = {https://doi.org/10.1021/ct5003658},
|
||||
Volume = {10},
|
||||
Year = {2014},
|
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Bdsk-Url-1 = {https://doi.org/10.1021/ct5003658}}
|
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|
||||
@article{Blase_2011b,
|
||||
author = {Blase,X. and Attaccalite,C. },
|
||||
title = {Charge-Transfer Excitations in Molecular Donor-Acceptor Complexes within the Many-Body Bethe-Salpeter Approach},
|
||||
journal = {Appl. Phys. Lett. },
|
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volume = {99},
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|
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eprint = { https://doi.org/10.1063/1.3655352}
|
||||
}
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||||
|
||||
|
||||
@article{Duchemin_2012,
|
||||
title = {Short-Range to Long-Range Charge-Transfer Excitations in the Zincbacteriochlorin-Bacteriochlorin Complex: A Bethe-Salpeter Study},
|
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author = {Duchemin, I. and Deutsch, T. and Blase, X.},
|
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journal = {Phys. Rev. Lett.},
|
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volume = {109},
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month = {Oct},
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doi = {10.1103/PhysRevLett.109.167801},
|
||||
url = {https://link.aps.org/doi/10.1103/PhysRevLett.109.167801}
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||||
}
|
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|
||||
|
||||
Author = {Blase,X. and Attaccalite,C.},
|
||||
Doi = {10.1063/1.3655352},
|
||||
Eprint = {https://doi.org/10.1063/1.3655352},
|
||||
Journal = {Appl. Phys. Lett.},
|
||||
Number = {17},
|
||||
Pages = {171909},
|
||||
Title = {Charge-Transfer Excitations in Molecular Donor-Acceptor Complexes within the Many-Body Bethe-Salpeter Approach},
|
||||
Url = {https://doi.org/10.1063/1.3655352},
|
||||
Volume = {99},
|
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Year = {2011},
|
||||
Bdsk-Url-1 = {https://doi.org/10.1063/1.3655352}}
|
||||
|
||||
@article{Stein_2009,
|
||||
author = {Stein, Tamar and Kronik, Leeor and Baer, Roi},
|
||||
title = {Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory},
|
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journal = {J. Am. Chem. Soc.},
|
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volume = {131},
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pages = {2818-2820},
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eprint = { https://doi.org/10.1021/ja8087482}
|
||||
}
|
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|
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|
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|
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Author = {Stein, Tamar and Kronik, Leeor and Baer, Roi},
|
||||
Doi = {10.1021/ja8087482},
|
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Eprint = {https://doi.org/10.1021/ja8087482},
|
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Journal = {J. Am. Chem. Soc.},
|
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Note = {PMID: 19239266},
|
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Number = {8},
|
||||
Pages = {2818-2820},
|
||||
Title = {Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory},
|
||||
Url = {https://doi.org/10.1021/ja8087482},
|
||||
Volume = {131},
|
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Year = {2009},
|
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Bdsk-Url-1 = {https://doi.org/10.1021/ja8087482}}
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|
||||
@article{Garniron_2018,
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Author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos},
|
||||
Date-Added = {2018-11-29 14:23:11 +0100},
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Date-Modified = {2018-11-29 14:23:11 +0100},
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|
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Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}}
|
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|
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Reference in New Issue
Block a user