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@ -328,8 +328,8 @@ Although the FCI energy has the enjoyable property of being independent of the s
In the present study, we investigate, in particular, the convergence behavior of the CIPSI energy for two sets of orbitals: natural orbitals (NOs) and optimized orbitals (OOs). In the present study, we investigate, in particular, the convergence behavior of the CIPSI energy for two sets of orbitals: natural orbitals (NOs) and optimized orbitals (OOs).
Following our usual procedure, \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c,Loos_2020e} we perform first a preliminary SCI calculation using HF orbitals in order to generate a SCI wave function with at least $10^7$ determinants. Following our usual procedure, \cite{Scemama_2018,Scemama_2018b,Scemama_2019,Loos_2018a,Loos_2019,Loos_2020a,Loos_2020b,Loos_2020c,Loos_2020e} we perform first a preliminary SCI calculation using HF orbitals in order to generate a SCI wave function with at least $10^7$ determinants.
Natural orbitals are computed based on this wave function and they are used to perform a new CIPSI run. 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. Successive orbital optimizations are then performed, which consist in minimizing the variational CIPSI energy at each macroiteration 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. 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 (see Sec.~\ref{sec:res}). 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} 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} 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}