saving work before OO section
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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 2021-06-19 07:02:58 +0200
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%% Created for Pierre-Francois Loos at 2021-07-02 10:54:28 +0200
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%% Saved with string encoding Unicode (UTF-8)
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@ -22,13 +22,15 @@
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\QP}{\textsc{quantum package}}
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\newcommand{\Nel}{n}
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\newcommand{\Norb}{N}
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\newcommand{\Ndet}{N_\text{det}}
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\newcommand{\Norb}{N_\text{orb}}
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\newcommand{\br}{\boldsymbol{r}}
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\newcommand{\bc}{\boldsymbol{c}}
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\newcommand{\bR}{\boldsymbol{R}}
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\newcommand{\bX}{\boldsymbol{X}}
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\newcommand{\bH}{\boldsymbol{H}}
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\newcommand{\hH}{\Hat{H}}
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\newcommand{\hh}{\Hat{h}}
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@ -87,17 +89,17 @@ The nuclei coordinates can then be treated as parameters in the electronic Hamil
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The second central approximation which makes calculations feasible by a computer is the basis set approximation where one introduces a set of pre-defined basis functions to represent the many-electron wave function of the system.
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In most molecular calculations, a set of one-electron, atom-centered gaussian basis functions are introduced to expand the so-called one-electron molecular orbitals which are then used to build the many-electron Slater determinants.
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The third and most relevant approximation in the present context is the ansatz (or form) of the electronic wave function $\Psi$.
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For example, in configuration interaction (CI) methods, the wave function is expanded as a linear combination of Slater determinants, while in (single-reference) coupled-cluster (CC) theory, \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002,Bartlett_2007,Shavitt_2009} a reference Slater determinant $\Psi_0$ [usually taken as the Hartree-Fock (HF) wave function] is multiplied by a wave operator defined as the exponentiated excitation operator $\Hat{T} = \sum_{k=1}^n \Hat{T}_k$ (where $n$ is the number of electrons).
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For example, in configuration interaction (CI) methods, the wave function is expanded as a linear combination of Slater determinants, while in (single-reference) coupled-cluster (CC) theory, \cite{Cizek_1966,Paldus_1972,Crawford_2000,Piecuch_2002,Bartlett_2007,Shavitt_2009} a reference Slater determinant $\Psi_0$ [usually taken as the Hartree-Fock (HF) wave function] is multiplied by a wave operator defined as the exponentiated excitation operator $\Hat{T} = \sum_{k=1}^\Nel \Hat{T}_k$ (where $\Nel$ is the number of electrons).
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The truncation of $\Hat{T}$ allows to define a hierarchy of non-variational and size-extensive methods with improved accuracy:
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CC with singles and doubles (CCSD), \cite{Cizek_1966,Purvis_1982} CC with singles, doubles, and triples (CCSDT), \cite{Noga_1987a,Scuseria_1988} CC with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1992} with corresponding computational scalings of $\order*{N^{6}}$, $\order*{N^{8}}$, and $\order*{N^{10}}$, respectively (where $N$ denotes the number of orbitals).
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CC with singles and doubles (CCSD), \cite{Cizek_1966,Purvis_1982} CC with singles, doubles, and triples (CCSDT), \cite{Noga_1987a,Scuseria_1988} CC with singles, doubles, triples, and quadruples (CCSDTQ), \cite{Oliphant_1991,Kucharski_1992} with corresponding computational scalings of $\order*{\Norb^{6}}$, $\order*{\Norb^{8}}$, and $\order*{\Norb^{10}}$, respectively (where $\Norb$ denotes the number of orbitals).
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Parallel to the ``complete'' CC series presented above, an alternative series of approximate iterative CC models have been developed by the Aarhus group in the context of CC response theory \cite{Christiansen_1998} where one skips the most expensive terms and avoids the storage of the higher-excitation amplitudes: CC2, \cite{Christiansen_1995a} CC3, \cite{Christiansen_1995b,Koch_1997} and CC4 \cite{Kallay_2005,Matthews_2021}
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These iterative methods scale as $\order*{N^{5}}$, $\order*{N^{7}}$, and $\order*{N^{9}}$, respectively, and can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ.
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These iterative methods scale as $\order*{\Norb^{5}}$, $\order*{\Norb^{7}}$, and $\order*{\Norb^{9}}$, respectively, and can be seen as cheaper approximations of CCSD, CCSDT, and CCSDTQ.
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Coupled-cluster methods have been particularly successful at computing accurately various properties for small- and medium-sized molecules.
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\cite{Kallay_2003,Kallay_2004a,Gauss_2006,Kallay_2006,Gauss_2009}
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A similar systematic truncation strategy can be applied to CI methods leading to the well-established family of methods known as CISD, CISDT, CISDTQ, \ldots~where one systematically increases the maximum excitation degree of the determinants taken into account.
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Except for full CI (FCI) where all determinants from the Hilbert space (\ie, with excitation degree up to $N$) are considered, truncated CI methods are variational but lack size-consistency.
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Except for full CI (FCI) where all determinants from the Hilbert space (\ie, with excitation degree up to $\Nel$) are considered, truncated CI methods are variational but lack size-consistency.
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The non-variationality of truncated CC methods being less of an issue than the size-inconsistency of the truncated CI methods, the formers have naturally overshadowed the latters in the electronic structure landscape.
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However, a different strategy has recently made a come back in the context of CI methods. \cite{Bender_1969,Whitten_1969,Huron_1973}
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Indeed, selected CI (SCI) methods, \cite{Booth_2009,Giner_2013,Evangelista_2014,Giner_2015,Holmes_2016,Tubman_2016,Liu_2016,Ohtsuka_2017,Zimmerman_2017,Coe_2018,Garniron_2018} where one iteratively selects the energetically relevant determinants from the FCI space (usually) based on a perturbative criterion, has been recently shown to be highly successful in order to produce reference energies for ground and excited states in small- and medium-size molecules \cite{Holmes_2017,Li_2018,Li_2020,Loos_2018a,Chien_2018,Loos_2019,Loos_2020b,Loos_2020c,Loos_2020e,Garniron_2019,Eriksen_2020,Yao_2020,Veril_2021,Loos_2021} thanks to efficient deterministic, stochastic or hybrid algorithms well suited for massive parallelization.
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@ -110,9 +112,9 @@ Note that, very recently, several groups \cite{Aroeira_2021,Lee_2021,Magoulas_20
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A rather different strategy in order to reach the holy grail FCI limit is to resort to M{\o}ller-Plesset (MP) perturbation theory, \cite{Moller_1934}
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which popularity originates from its black-box nature, size-extensivity, and relatively low computational scaling, making it easily applied to a broad range of molecular systems.
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Again, at least in theory, one can obtain the exact energy of the system by ramping up the degree of the perturbative series. \cite{Marie_2021}
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The second-order M{\o}ller-Plesset (MP2) method \cite{Moller_1934} [which scales as $\order*{N^{5}}$] has been broadly adopted in quantum chemistry for several decades, and is now included in the increasingly popular double-hybrid functionals \cite{Grimme_2006} alongside exact HF exchange.
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The second-order M{\o}ller-Plesset (MP2) method \cite{Moller_1934} [which scales as $\order*{\Norb^{5}}$] has been broadly adopted in quantum chemistry for several decades, and is now included in the increasingly popular double-hybrid functionals \cite{Grimme_2006} alongside exact HF exchange.
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Its higher-order variants [MP3, \cite{Pople_1976}
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MP4, \cite{Krishnan_1980} MP5, \cite{Kucharski_1989} and MP6 \cite{He_1996a,He_1996b} which scales as $\order*{N^{6}}$, $\order*{N^{7}}$, $\order*{N^{8}}$, and $\order*{N^{9}}$ respectively] have been investigated much more scarcely.
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MP4, \cite{Krishnan_1980} MP5, \cite{Kucharski_1989} and MP6 \cite{He_1996a,He_1996b} which scales as $\order*{\Norb^{6}}$, $\order*{\Norb^{7}}$, $\order*{\Norb^{8}}$, and $\order*{\Norb^{9}}$ respectively] have been investigated much more scarcely.
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However, it is now widely recognised that the series of MP approximations might show erratic, slow, or divergent behavior that limit its applicability and systematic improvability. \cite{Laidig_1985,Knowles_1985,Handy_1985,Gill_1986,Laidig_1987,Nobes_1987,Gill_1988,Gill_1988a,Lepetit_1988,Malrieu_2003}
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Again, MP perturbation theory and CC methods can be coupled.
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The CCSD(T) method, \cite{Raghavachari_1989} known as the gold-standard of quantum chemistry for weakly correlated systems, is probably the most iconic example of such coupling.
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@ -132,8 +134,7 @@ The performance of the ground-state gold standard CCSD(T) is also investigated.
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Five-membered rings (top) and six-membered rings (bottom) considered in this study.
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\label{fig:mol}}
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\end{figure*}
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%%% FIG 1 %%%
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%%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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@ -142,33 +143,31 @@ The geometries of the twelve systems considered in the present study have been a
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Note that, for the sake of consistency, the geometry of benzene considered here is different from one of Ref.~\onlinecite{Loos_2020e} which has been computed at a lower level of theory [MP2/6-31G(d)]. \cite{Schreiber_2008}
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The MP2, MP3, MP4, CC2, CC3, CC4, CCSD, CCSDT, and CCSDTQ calculations have been performed with CFOUR, \cite{Matthews_2020} while the CCSD(T) and MP5 calculations have been computed with Gaussian 09. \cite{g09}
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The CIPSI calculations have been performed with {\QP}. \cite{Garniron_2019}
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\titou{The particularity of the current implementation is that the selection step and the PT2 correction are computed \textit{simultaneously} via a hybrid semistochastic algorithm \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
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Moreover, a renormalized version of the PT2 correction (dubbed rPT2 below) has been recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher-order of perturbation. \cite{Garniron_2019}
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We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the rPT2 correction and the CIPSI algorithm.}
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In the current implementation, the selection step and the PT2 correction are computed simultaneously via a hybrid semistochastic algorithm. \cite{Garniron_2017,Garniron_2019} (which explains the statistical error associated with the PT2 correction in the following).
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Here, we employ the renormalized version of the PT2 correction which has been recently implemented and tested for a more efficient extrapolation to the FCI limit thanks to a partial resummation of the higher-order of perturbation. \cite{Garniron_2019}
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We refer the interested reader to Ref.~\onlinecite{Garniron_2019} where one can find all the details regarding the implementation of the PT2 correction and the CIPSI algorithm.
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For all these calculations, we consider Dunning's correlation-consistent double-$\zeta$ basis (cc-pVDZ).
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Although the FCI energy has the enjoyable property of being independent of the set of one-electron orbitals used to construct the many-electron Slater determinants, as a truncated CI method, the convergence properties of CIPSI strongly dependent on this orbital choice.
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In the present study, we investigate the convergence behavior of the CIPSI energy for four distinct sets: natural orbitals (NOs), localized orbitals (LOs), and optimized orbitals (OOs).
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In the present study, we investigate the convergence behavior of the CIPSI energy for two sets of orbitals in particular: natural orbitals (NOs) and optimized orbitals (OOs).
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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.
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Then, natural orbitals (NOs) are computed based on this wave function, and subsequently localized orbitals.
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The Boys-Foster localization procedure \cite{Boys_1960} that we apply to the natural orbitals is performed in several orbital windows: \titou{i) core, ii) valence $\sigma$, iii) valence $\pi$, iv) valence $\pi^*$, v) valence $\sigma^*$, vi) the higher-lying $\sigma$ orbitals, and vii) the higher-lying $\pi$ orbitals.}
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Like Pipek-Mezey, \cite{Pipek_1989} this choice of orbital windows allows to preserve a strict $\sigma$-$\pi$ separation in planar systems like the ones considered here.
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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}
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Using these localized orbitals as starting point, we also perform successive orbital optimizations, which consist in minimizing the variational CIPSI energy at each iteration up to approximately $2 \times 10^5$ determinants.
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Natural orbitals are computed based on this wave function and they are used to perform a new CIPSI run.
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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.
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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.
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As we shall see below, employing optimized orbitals has the advantage to produce a smoother and faster convergence of the SCI energy toward the FCI limit.
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In some cases, we also explore the use of localized orbitals (LOs) which are produced with the help of 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.
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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}
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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.
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Note that, unlike excited-state calculations where it is important to enforce that the wave functions are eigenfunctions of the $\Hat{S}^2$ spin operator, \cite{Chilkuri_2021} the present wave functions do not fulfil this property as we aim for the lowest possible energy of a singlet state.
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We have found that $\expval*{\Hat{S}^2}$ is, nonetheless, very close to zero for each system.
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%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{CIPSI with optimized orbitals}
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%%%%%%%%%%%%%%%%%%%%%%%%%
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Here, we provide key details about the CIPSI method.
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Here, we provide key details about the CIPSI method. \cite{Huron_1973,Garniron_2019}
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Note that we focus on the ground state but the present discussion can be easily extended to excited states. \cite{Scemama_2019,Veril_2021}
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At each iteration $k$, the total CIPSI energy $E_\text{CIPSI}^{(k)}$ is defined as the sum of the variational energy
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At the $k$th iteration, the total CIPSI energy $E_\text{CIPSI}^{(k)}$ is defined as the sum of the variational energy
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\begin{equation}
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E_\text{var}^{(k)} = \frac{\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\Psi_\text{var}^{(k)}}}{\braket*{\Psi_\text{var}^{(k)}}{\Psi_\text{var}^{(k)}}}
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\end{equation}
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@ -178,21 +177,27 @@ and a second-order perturbative correction
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= \sum_{\alpha \in \mathcal{A}_k} e_{\alpha}
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= \sum_{\alpha \in \mathcal{A}_k} \frac{\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\alpha}}{E_\text{var}^{(k)} - \mel*{\alpha}{\Hat{H}}{\alpha}}
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\end{equation}
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where $\Psi_\text{var}^{(k)} = \sum_{I \in \mathcal{I}_k} c_I^{(k)} \ket*{I}$ is the variational wave function, $\mathcal{I}_k$ is the set of internal determinants $\ket*{I}$ and $\mathcal{A}_k$ is the set of external determinants $\ket*{\alpha}$ which do not belong to the variational space but are linked to it via a nonzero matrix element, \ie, $\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\alpha} \neq 0$.
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where $\hH$ is the (non-relativistic) electronic Hamiltonian,
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\begin{equation}
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\label{eq:Psivar}
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\Psi_\text{var}^{(k)} = \sum_{I \in \mathcal{I}_k} c_I^{(k)} \ket*{I}
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\end{equation}
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is the variational wave function, $\mathcal{I}_k$ is the set of internal determinants $\ket*{I}$ and $\mathcal{A}_k$ is the set of external determinants $\ket*{\alpha}$ which do not belong to the variational space but are linked to it via a nonzero matrix element, \ie, $\mel*{\Psi_\text{var}^{(k)}}{\Hat{H}}{\alpha} \neq 0$.
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The sets $\mathcal{I}_k$ and $\mathcal{A}_k$ define, at the $k$th iteration, the internal and external spaces, respectively.
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In practice, $E_\text{var}^{(k)}$ is computed by diagonalizing the CI matrix in the reference space and the magnitude of $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit.
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In practice, $E_\text{var}^{(k)}$ is computed by diagonalizing the $\Ndet^{(k)} \times \Ndet^{(k)}$ CI matrix $\bH$ with elements $H_{IJ} = \mel{I}{\hH}{J}$ via Davidson's algorithm \cite{Davidson_1975} and the magnitude of $E_\text{PT2}$ provides a qualitative idea of the ``distance'' to the FCI limit. \cite{Garniron_2018}
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We then linearly extrapolate, using large variational space, the CIPSI energy to $E_\text{PT2} = 0$ (which effectively corresponds to the FCI limit).
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Further details concerning the extrapolation procedure are provided below (see Sec.~\ref{sec:res}).
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Orbital optimization techniques at the SCI level are theoretically straightforward, but practically challenging. \cite{Yao_2020,Yao_2021}
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Here, we detail our orbital optimization procedure within the CIPSI algorithm and we assume that the variational wave function is normalized, \ie, $\braket*{\Psi_\text{var}}{\Psi_\text{var}} = 1$.
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Then, the variational energy can be written as
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\begin{equation}
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E_\text{var}(\bc,\bX) = \mel{\Psi_\text{var}}{e^{\hX} \hH e^{-\hX}}{\Psi_\text{var}},
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\end{equation}
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where $\bc$ gathers the CI coefficients, $\bX$ the orbital rotation parameters and $\hX$ is a one-electron anti-hermitian operator, which creates a rotation matrix when exponentiated, \ie, $\bR = e^{\bX}$.
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From a more general point of view, the variational energy $E_\text{var}$ depends on both the coefficient $\{ c_I \}_{1 \le I \le \Ndet^{(k)}}$ [see Eq.~\eqref{eq:Psivar}] but also on the orbital rotation parameter $\{X_{pq}\}_{1 \le p,q \le \Norb}$.
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Orbital optimization techniques at the SCI level are theoretically straightforward, but practically challenging.
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Here, we detail our orbital optimization procedure with the CIPSI algorithm.
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From a more general point of view, the variational energy $E_\text{var}^{(k)}$ depends on both the coefficient $\{ c_I \}_{1 \le I \le \Ndet^{(k)}}$ but also on the orbital rotation parameter $\{X_{pq}\}_{1 \le \mu p,q \le \Norb}$.
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%such that the $p$th orbital is
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%\begin{equation}
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% \phi_p(\br) = \sum_{\mu} C_{\mu p} \chi_{\mu}(\br)
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%\end{equation}
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%where $\chi_{\mu}(\br)$ is a basis function.
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The diagonalization of the CI matrix ensures that
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For a given set of orbitals, The diagonalization of the CI matrix ensures that
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\begin{equation}
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\pdv{E_\text{var}(\bc,\bX)}{c_I} = 0,
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\end{equation}
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@ -200,16 +205,10 @@ but, a priori, we have
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\begin{equation}
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\pdv{E_\text{var}(\bc,\bX)}{X_{pq}} \neq 0,
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\end{equation}
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Although one could use a second order method to minimize the corresponding energy, one has to realize that the size of the CI space is much larger than orbital space.
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It is therefore more appropriate to perform a minimization of the variational energy with respect to the orbital rotation parameters and then compute the new CI coefficients by re-diagonalizing the CI matrix.
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Here, we assume that the variational wave function is normalized, \ie, $\braket*{\Psi_\text{var}}{\Psi_\text{var}} = 1$.
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Then, the previous equation can be rewritten,
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\begin{equation}
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E(\bc,\bX) = \mel{\Psi_\text{var}}{e^{\hX} \hH e^{-\hX}}{\Psi_\text{var}},
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\end{equation}
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where $\bc$ gathers the CI coefficients, $\bX$ the orbital rotation parameters and $\hX$ is a one-electron anti-hermitian operator, which creates a rotation matrix when exponentiated, \ie, $\bR = e^{\bX}$.
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The energy $E$ depends on the determinants and their number, thus the orbital optimization will not be the same for two different sets of determinants.
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To understand the concept of orbital rotation, we look at this operator $\bX$ in more details,
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\begin{equation}
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\hX = \sum_{p > q} \sum_{\sigma} X_{pq} (\hat{a}_{p \sigma}^{\dagger} \hat{a}_{q \sigma} - \hat{a}_{q \sigma}^{\dagger} \hat{a}_{p \sigma}),
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@ -262,18 +261,33 @@ And the Hessian of the energy with respect to the orbital rotation, $H_{pq,rs}$,
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with $\delta_{ij}$ is the Kronecker delta, $\mathcal{P}_{pq}$ the permutation operator $\mathcal{P}_{pq} = 1
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- (p \leftrightarrow q)$ and where $(p \leftrightarrow q)$ applied to an equation returns the same equation with the indices $p$ and $q$ swapped.
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Trust region method and Newton's method are very similar, they use a quadratical approximation of a real function, i.e.,
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Taylor expansion truncated at the second order. In the Newton method, the step is given by the minimizer of the
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quadratical approximation, contrary to the trust region method which gives the minimizer of the quadratical approximation
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in the trust region. The trust region defines a region where the quadratical approximation is a adequate representation
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of the real function and it evolves during the optimization process in order to preserve the adequacy. The constraint
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for the step size is the following, $\norm{\bm{X}_{k+1}} \leq \Delta_k$ with $\Delta_k$ the trust radius. By putting the
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constraint with a Lagrange multiplier $\lambda$ and derivating the Lagrangian, the solution is
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$\bm{X}_{k+1} = - (\bm{H_k} + \lambda \bm{I})^{-1} \cdot \bm{g}_k$.
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The addition of a constant $\lambda \geq 0$ on the diagonal of the hessian removes the negative eigenvalues and
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reduce the size of the step since the calculation uses its inverse. By choosing the right $\lambda$ the step size is contraint
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into a hypersphere of radius $\Delta_k$. In addition, the evolution of $\Delta_k$ during the optimization and the use of
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a condition to cancel a step ensure the convergence of the algorithm.
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More details could be found in Numerical Optimization, Nocedal \& Wright (1999)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results and discussion}
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\label{sec:res}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{table*}
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\caption{Total energy $E$ (in \Eh) and correlation energy $\Ec$ (in \mEh) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set.
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\caption{Total energy $E$ (in \Eh) and correlation energy $\Delta E$ (in \mEh) for the frozen-core ground state of five-membered rings in the cc-pVDZ basis set.
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\label{tab:Tab5-VDZ}}
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\begin{ruledtabular}
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\begin{tabular}{lcccccccccc}
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& \mc{2}{c}{Cyclopentadiene} & \mc{2}{c}{Furan} & \mc{2}{c}{Imidazole} & \mc{2}{c}{Pyrrole} & \mc{2}{c}{Thiophene} \\
|
||||
\cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9} \cline{10-11}
|
||||
Method & $E$& $\Ec$ & $E$ & $\Ec$ & $E$ & $\Ec$ & $E$ & $\Ec$ & $E$ & $\Ec$ \\
|
||||
Method & $E$& $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ \\
|
||||
\hline
|
||||
HF & $-192.8083$ & & $-228.6433$ & & $-224.8354$ & & $-208.8286$ & & -551.3210 & \\
|
||||
\hline
|
||||
@ -300,14 +314,14 @@ with $\delta_{ij}$ is the Kronecker delta, $\mathcal{P}_{pq}$ the permutation op
|
||||
|
||||
\begin{squeezetable}
|
||||
\begin{table*}
|
||||
\caption{Total energy $E$ (in \Eh) and correlation energy $\Ec$ (in \mEh) for the frozen-core ground state of six-membered rings in the cc-pVDZ basis set.
|
||||
\caption{Total energy $E$ (in \Eh) and correlation energy $\Delta E$ (in \mEh) for the frozen-core ground state of six-membered rings in the cc-pVDZ basis set.
|
||||
\label{tab:Tab6-VDZ}}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{lcccccccccccccc}
|
||||
& \mc{2}{c}{Benzene} & \mc{2}{c}{Pyrazine} & \mc{2}{c}{Pyridazine} & \mc{2}{c}{Pyridine} & \mc{2}{c}{Pyrimidine} & \mc{2}{c}{Tetrazine} & \mc{2}{c}{Triazine} \\
|
||||
\cline{2-3} \cline{4-5} \cline{6-7} \cline{8-9} \cline{10-11} \cline{12-13} \cline{14-15}
|
||||
Method & $E$ & $\Ec$ & $E$ & $\Ec$ & $E$ & $\Ec$ & $E$ & $\Ec$
|
||||
& $E$ & $\Ec$ & $E$ & $\Ec$ & $E$ & $\Ec$ \\
|
||||
Method & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$
|
||||
& $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ \\
|
||||
\hline
|
||||
HF & $-230.7222$ & & $-262.7030$ & & $-262.6699$ & & $-246.7152$ & & $-262.7137$ & & $-294.6157$ & & $-278.7173$ \\
|
||||
\hline
|
||||
@ -326,47 +340,52 @@ with $\delta_{ij}$ is the Kronecker delta, $\mathcal{P}_{pq}$ the permutation op
|
||||
\hline
|
||||
CCSD(T) & $-231.5798$ & $-857.5$ & $-263.6024$ & $-899.4$ & $-263.5740$ & $-904.1$ & $-247.5929$ & $-877.7$ & $-263.6099$ & $-896.2$ & $-295.5680$ & $-952.2$ & $-279.6305$ & $-913.1$ \\
|
||||
\hline
|
||||
CIPSI & & $-863.0$ & & & & $-908.8$ & & $-883.4$ & & $-900.4$ & & $-957.3$ & & $-918.5$\\
|
||||
CIPSI & & $-863.0$ & & $-904.6$ & & $-908.8$ & & $-883.4$ & & $-900.4$ & & $-957.3$ & & $-918.5$\\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table*}
|
||||
\end{squeezetable}
|
||||
|
||||
\begin{figure*}
|
||||
% \includegraphics[width=0.15\textwidth]{Cyclopentadiene_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Furan_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Imidazole_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Pyrrole_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Thiophene_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Benzene_vs_Ndet}
|
||||
% \\
|
||||
% \includegraphics[width=0.15\textwidth]{Pyrazine_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Pyridazine_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Pyridine_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Pyrimidine_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Tetrazine_vs_Ndet}
|
||||
% \includegraphics[width=0.15\textwidth]{Triazine_vs_Ndet}
|
||||
\caption{$E_\text{var}$ and $E_\text{var} + E_\text{PT2}$ as functions of the number of determinants $\Ndet$ in the variational space for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
|
||||
The CCSDTQ correlation energy is represented as a thick black line.
|
||||
\includegraphics[width=0.24\textwidth]{Cyclopentadiene_EvsNdet}
|
||||
\includegraphics[width=0.24\textwidth]{Furan_EvsNdet}
|
||||
\includegraphics[width=0.24\textwidth]{Imidazole_EvsNdet}
|
||||
\includegraphics[width=0.24\textwidth]{Pyrrole_EvsNdet}
|
||||
\\
|
||||
\includegraphics[width=0.24\textwidth]{Thiophene_EvsNdet}
|
||||
\includegraphics[width=0.24\textwidth]{Benzene_EvsNdet}
|
||||
\includegraphics[width=0.24\textwidth]{Pyrazine_EvsNdet}
|
||||
\includegraphics[width=0.24\textwidth]{Pyridazine_EvsNdet}
|
||||
\\
|
||||
\includegraphics[width=0.24\textwidth]{Pyridine_EvsNdet}
|
||||
\includegraphics[width=0.24\textwidth]{Pyrimidine_EvsNdet}
|
||||
\includegraphics[width=0.24\textwidth]{Tetrazine_EvsNdet}
|
||||
\includegraphics[width=0.24\textwidth]{Triazine_EvsNdet}
|
||||
\caption{$\Delta E_\text{var}$ (solid) and $\Delta E_\text{var} + E_\text{PT2}$ (dashed) as functions of the number of determinants $\Ndet$ in the variational space for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
|
||||
Two sets of orbitals are considered: natural orbitals (NOs, in red) and optimized orbitals (OOs, in blue).
|
||||
The CCSDTQ correlation energy is represented as a thick black line.
|
||||
\label{fig:vsNdet}}
|
||||
\end{figure*}
|
||||
|
||||
\begin{figure*}
|
||||
% \includegraphics[width=0.15\textwidth]{Cyclopentadiene_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Furan_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Imidazole_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Pyrrole_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Thiophene_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Benzene_vs_EPT2}
|
||||
% \\
|
||||
% \includegraphics[width=0.15\textwidth]{Pyrazine_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Pyridazine_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Pyridine_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Pyrimidine_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Tetrazine_vs_EPT2}
|
||||
% \includegraphics[width=0.15\textwidth]{Triazine_vs_EPT2}
|
||||
\caption{$E_\text{var}$ as a function of $E_\text{PT2}$ for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
|
||||
The CCSDTQ correlation energy is represented as a thick black line.
|
||||
\includegraphics[width=0.24\textwidth]{Cyclopentadiene_EvsPT2}
|
||||
\includegraphics[width=0.24\textwidth]{Furan_EvsPT2}
|
||||
\includegraphics[width=0.24\textwidth]{Imidazole_EvsPT2}
|
||||
\includegraphics[width=0.24\textwidth]{Pyrrole_EvsPT2}
|
||||
\\
|
||||
\includegraphics[width=0.24\textwidth]{Thiophene_EvsPT2}
|
||||
\includegraphics[width=0.24\textwidth]{Benzene_EvsPT2}
|
||||
\includegraphics[width=0.24\textwidth]{Pyrazine_EvsPT2}
|
||||
\includegraphics[width=0.24\textwidth]{Pyridazine_EvsPT2}
|
||||
\\
|
||||
\includegraphics[width=0.24\textwidth]{Pyridine_EvsPT2}
|
||||
\includegraphics[width=0.24\textwidth]{Pyrimidine_EvsPT2}
|
||||
\includegraphics[width=0.24\textwidth]{Tetrazine_EvsPT2}
|
||||
\includegraphics[width=0.24\textwidth]{Triazine_EvsPT2}
|
||||
\caption{$\Delta E_\text{var}$ as a function of $E_\text{PT2}$ for the twelve cyclic molecules represented in Fig.~\ref{fig:mol}.
|
||||
Two sets of orbitals are considered: natural orbitals (NOs, in red) and optimized orbitals (OOs, in blue).
|
||||
The four-point linear fit using the four largest variational wave functions for each set is depicted as a dashed black line.
|
||||
The CCSDTQ correlation energy is also represented as a thick black line.
|
||||
\label{fig:vsNdet}}
|
||||
\end{figure*}
|
||||
|
||||
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Reference in New Issue
Block a user