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fix few typos

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Pierre-Francois Loos 2021-07-20 16:28:51 +02:00
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16
Manuscript/Ec.bib
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@ -1,13 +1,27 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-07-19 13:49:57 +0200
%% Created for Pierre-Francois Loos at 2021-07-20 16:08:44 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Bozkaya_2011,
author = {Bozkaya,U{\u g}ur and Turney,Justin M. and Yamaguchi,Yukio and Schaefer,Henry F. and Sherrill,C. David},
date-added = {2021-07-20 16:08:28 +0200},
date-modified = {2021-07-20 16:08:40 +0200},
doi = {10.1063/1.3631129},
journal = {J. Chem. Phys.},
number = {10},
pages = {104103},
title = {Quadratically convergent algorithm for orbital optimization in the orbital-optimized coupled-cluster doubles method and in orbital-optimized second-order M{\o}ller-Plesset perturbation theory},
url = {https://doi.org/10.1063/1.3631129},
volume = {135},
year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1063/1.3631129}}
@book{Nocedal_1999,
address = {New York, NY, USA},
author = {Nocedal, Jorge and Wright, Stephen J.},

6
Manuscript/Ec.tex
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@ -222,7 +222,7 @@ Here, we detail our orbital optimization procedure within the CIPSI algorithm an
As stated in Sec.~\ref{sec:intro}, $\Evar$ depends on both the CI coefficients $\{ c_I \}_{1 \le I \le \Ndet}$ [see Eq.~\eqref{eq:Psivar}] but also on the orbital rotation parameters $\{\kappa_{pq}\}_{1 \le p,q \le \Norb}$.
Here, motivated by cost saving arguments, we have chosen to optimize separately the CI and orbital coefficients by alternatively diagonalizing the CI matrix after each selection step and then rotating the orbitals until the variational energy for a given number of determinants is minimal.
To do so, we conveniently rewrite the variational energy as
To do so, we conveniently rewrite the variational energy as
\begin{equation}
\label{eq:Evar_c_k}
\Evar(\bc,\bk) = \mel{\Psivar}{e^{\hk} \hH e^{-\hk}}{\Psivar},
@ -231,7 +231,7 @@ where $\bc$ gathers the CI coefficients, $\bk$ the orbital rotation parameters,
\begin{equation}
\hk = \sum_{p < q} \sum_{\sigma} \kappa_{pq} \qty(\cre{p\sigma} \ani{q\sigma} - \cre{q\sigma} \ani{p\sigma})
\end{equation}
is a real-valued one-electron antisymmetric operator, which creates an orthogonal transformation of the orbital coefficients when exponentiated, $\ani{p\sigma}$ ($\cre{p\sigma}$) being the second quantization annihilation (creation) operator which annihilates (creates) a spin-$\sigma$ electron in the (real-valued) spatial orbital $\MO{p}(\br)$.
is a real-valued one-electron antisymmetric operator, which creates an orthogonal transformation of the orbital coefficients when exponentiated, $\ani{p\sigma}$ ($\cre{p\sigma}$) being the second quantization annihilation (creation) operator which annihilates (creates) a spin-$\sigma$ electron in the (real-valued) spatial orbital $\MO{p}(\br)$. \cite{Helgaker_2013}
Applying the Newton-Raphson method by Taylor-expanding the variational energy to second order around $\bk = \bO$, \ie,
\begin{equation}
@ -243,7 +243,7 @@ one can iteratively minimize the variational energy with respect to the paramete
\bk = - \bH^{-1} \cdot \bg,
\end{equation}
where $\bg$ and $\bH$ are the orbital gradient and Hessian, respectively, both evaluated at $\bk = \bO$.
Their elements are explicitly given by the following expressions: \cite{Henderson_2014a}
Their elements are explicitly given by the following expressions: \cite{Bozkaya_2011,Henderson_2014a}
\begin{equation}
\begin{split}
g_{pq}