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Manuscript/Ec.tex

@@ -162,8 +162,8 @@ The performance of the ground-state gold standard CCSD(T) is also investigated.
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 The present manuscript is organized as follows.
-In Sec \ref{sec:OO-CIPSI}, we provide theoretical details about the CIPSI algorithm and the orbital optimization procedure that we have employed here.
+In Sec.~\ref{sec:OO-CIPSI}, we provide theoretical details about the CIPSI algorithm and the orbital optimization procedure that we have employed here.
-Section.~\ref{sec:compdet} deals with computational details concerning geometries, basis sets, and methods.
+Section \ref{sec:compdet} deals with computational details concerning geometries, basis sets, and methods.
 In Sec.~\ref{sec:res}, we report our reference FCI correlation energies for the five-membered and six-membered cyclic molecules obtained thanks to extrapolated orbital-optimized CIPSI calculations (Sec.~\ref{sec:cipsi_res}).
 These reference correlation energies are then used to benchmark and study the convergence properties of various perturbative and CC methods (Sec.~\ref{sec:mpcc_res}).
 Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
@@ -206,7 +206,7 @@ Most of the technology presented here has been borrowed from complete-active-spa
 Here, we detail our orbital optimization procedure within the CIPSI algorithm and we assume that the variational wave function is normalized, \ie, $\braket*{\Psivar}{\Psivar} = 1$.
 
 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 $\{\kappa_{pq}\}_{1 \le p,q \le \Norb}$.
-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.
+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.
 (For a detailed comparison of coupled, uncoupled, and partially-coupled optimizations within SCI methods, we refer the interested reader to the recent work of Yao and Umrigar. \cite{Yao_2021})
 To do so, we conveniently rewrite the variational energy as
 \begin{equation}
@@ -229,7 +229,7 @@ one can iteratively minimize the variational energy with respect to the paramete
 	\label{eq:kappa_newton}
 	\bk = - \bH^{-1} \cdot \bg,
 \end{equation}
-where $\bg$ and $\bH$ are the orbital gradient and Hessian, respectively, both evaluated at $\bk = \bO$.
+where $\bg$ and $\bH$ are the orbital gradient and Hessian matrices, respectively, both evaluated at $\bk = \bO$.
 Their elements are explicitly given by the following expressions: \cite{Bozkaya_2011,Henderson_2014a}
 \begin{equation}
 \begin{split}
@@ -294,18 +294,19 @@ Because the size of the CI space is much larger than the orbital space, for each
 After each microiteration (\ie, orbital rotation), the one- and two-electron integrals [see Eqs.~\eqref{eq:one} and \eqref{eq:two}] have to be updated. 
 Moreover, the CI matrix must be re-diagonalized and new one- and two-electron density matrices [see Eqs.~\eqref{eq:one_dm} and \eqref{eq:two_dm}] are computed.
 Microiterations are stopped when a stationary point is found, \ie, $\norm{\bg}_\infty < \tau$, where $\tau$ is a user-defined threshold which has been set to $10^{-3}$ a.u.~in the present study, and a new CIPSI selection step is performed.
-Note that a tight convergence is not critical here as a new set of microiterations is performed at each macroiteration and a new production CIPSI run is performed from scratch using the final set of orbitals.
+Note that a tight convergence is not critical here as a new set of microiterations is performed at each macroiteration and a new production CIPSI run is performed from scratch using the final set of orbitals (see Sec.~\ref{sec:compdet}).
+This procedure might sound computationally expensive but one has to realize that the microiterations are usually performed only for relatively compact variational spaces.
+Therefore, the computational bottleneck remains the diagonalization of the CI matrix for very large variational spaces.
 
-This procedure might sound computationally expensive but one has to realize that it is usually performed only for relatively compact variational space 
 %\begin{equation}
 %	\Evar =  \sum_{pq} h_p^q \gamma_p^q + \frac{1}{2} \sum_{pqrs} v_{pq}^{rs} \Gamma_{pq}^{rs},
 %\end{equation}
 
-To enhance the convergence of the microiteration process, we employ a variant of the Newton-Raphson method known as ``trust region''. \cite{Nocedal_1999}
+To enhance the convergence of the microiteration process, we employ an adaptation of the Newton-Raphson method known as ``trust region''. \cite{Nocedal_1999}
 This popular variant defines a region where the quadratic approximation \eqref{eq:EvarTaylor} is an adequate representation of the objective energy function \eqref{eq:Evar_c_k} and it evolves during the optimization process in order to preserve the adequacy via a constraint on the step size preventing it from overstepping, \ie, $\norm{\bk} \leq \Delta$, where $\Delta$ is the trust radius.
 By introducing a Lagrange multiplier $\lambda$ to control the trust-region size, one replaces Eq.~\eqref{eq:kappa_newton} by $\bk = - (\bH + \lambda \bI)^{-1} \cdot \bg$.
 The addition of the level shift $\lambda \geq 0$ removes the negative eigenvalues and ensures the positive definiteness of the Hessian matrix by reducing the step size.
-By choosing the right value of $\lambda$, the step size is constrained into a hypersphere of radius $\Delta$ and is able to evolve from the Newton direction at $\lambda = 0$ to the steepest descent direction as $\lambda$ grows.
+By choosing the right value of $\lambda$, $\norm{\bk}$ is constrained into a hypersphere of radius $\Delta$ and is able to evolve from the Newton direction at $\lambda = 0$ to the steepest descent direction as $\lambda$ grows.
 The evolution of the trust radius during the optimization and the use of a condition to cancel the step when the energy rises ensure the convergence of the algorithm.
 More details can be found in Ref.~\onlinecite{Nocedal_1999}.
 
@@ -328,7 +329,7 @@ Following our usual procedure, \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loo
 Natural orbitals are computed based on this wave function and they are used to perform a new CIPSI run.
 Successive orbital optimizations are then performed, which consist in minimizing the variational CIPSI energy at each iteration up to approximately $2 \times 10^5$ determinants.
 When convergence is achieved in terms of orbital optimization, as our ``production'' run, we perform a new CIPSI calculation from scratch using this set of optimized orbitals.
-Using optimized orbitals has the undeniable advantage to produce, for a given variational energy, more compact CI expansions.
+Using optimized orbitals has the undeniable advantage to produce, for a given variational energy, more compact CI expansions (see Sec.~\ref{sec:res}).
 For the benzene molecule, we also explore the use of localized orbitals (LOs) which are produced with the Boys-Foster localization procedure \cite{Boys_1960} that we apply to the natural orbitals in several orbital windows in order to preserve a strict $\sigma$-$\pi$ separation in the planar systems considered here. \cite{Loos_2020e}
 Because they take advantage of the local character of electron correlation, localized orbitals have been shown to provide faster convergence towards the FCI limit compared to natural orbitals. \cite{Angeli_2003,Angeli_2009,BenAmor_2011,Suaud_2017,Chien_2018,Eriksen_2020,Loos_2020e}
 As we shall see below, employing optimized orbitals has the advantage to produce an even smoother and faster convergence of the SCI energy toward the FCI limit.