A view on coupled cluster perturbation theory using a bivariational Lagrangian formulation
Kasper Kristensen, Janus Juul Eriksen, Devin A. Matthews, Jeppe Olsen, Poul Jørgensen
AA view on coupled cluster perturbation theory usinga bivariational Lagrangian formulation
Kasper Kristensen, ∗ , † Janus Juul Eriksen, ∗ , † Devin A. Matthews, ‡ Jeppe Olsen, † and Poul Jørgensen † qLEAP Center for Theoretical Chemistry, Department of Chemistry, Aarhus University,Langelandsgade 140, DK-8000 Aarhus C, Denmark, and The Institute for ComputationalEngineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA E-mail: [email protected]; [email protected]
Abstract
We consider two distinct coupled cluster (CC) perturbation series that both expand thedifference between the energies of the CCSD (CC with single and double excitations) andCCSDT (CC with single, double, and triple excitations) models in orders of the Møller-Plessetfluctuation potential. We initially introduce the E-CCSD(T– n ) series, in which the CCSDamplitude equations are satisfied at the expansion point, and compare it to the recently devel-oped CCSD(T– n ) series [J. Chem. Phys. 111444000, 064108 (2014)], in which not only the CCSDamplitude, but also the CCSD multiplier equations are satisfied at the expansion point. Thecomputational scaling is similar for the two series, and both are term-wise size extensive witha formal convergence towards the CCSDT target energy. However, the two series are differ-ent, and the CCSD(T– n ) series is found to exhibit a more rapid convergence up through theseries, which we trace back to the fact that more information at the expansion point is utilized ∗ To whom correspondence should be addressed † Aarhus University ‡ The University of Texas at Austin a r X i v : . [ phy s i c s . c h e m - ph ] J a n han for the E-CCSD(T– n ) series. The present analysis can be generalized to any perturbationexpansion representing the difference between a parent CC model and a higher-level targetCC model. In general, we demonstrate that, whenever the parent parameters depend upon theperturbation operator, a perturbation expansion of the CC energy (where only parent ampli-tudes are used) differs from a perturbation expansion of the CC Lagrangian (where both parentamplitudes and parent multipliers are used). For the latter case, the bivariational Lagrangianformulation becomes more than a convenient mathematical tool, since it facilitates a differentand faster convergent perturbation series than the simpler energy-based expansion. Introduction
Coupled cluster (CC) theory is perhaps the most powerful method for describing dynamicalelectron correlation effects within the realm of modern quantum chemistry. The CC singles anddoubles (CCSD) model, in which the cluster operator is truncated at the level of double excita-tions, is a robust and useful model, but it is well-known that the effects of triple (and higher-level)excitations need to be taken into account in order to obtain highly accurate results that may com-pete with the accuracy of experiments. However, the steep scaling of the CC singles, doubles, andtriples (CCSDT) and CC singles, doubles, triples, and quadruples (CCSDTQ) models limitstheir use to rather modest molecular systems. For this reason, a computationally tractable alter-native to the expensive iterative CCSDT and CCSDTQ models is to develop approximate models,for which the important triples and/or quadruples contributions are determined from a perturbationanalysis, and hence included in a cheap and non-iterative fashion. A plethora of different modelsfor the approximate treatment of triples and/or quadruples effects have been suggested, and we re-fer to Ref. 11 for a recent theoretical overview of approximate non-iterative triples and quadruplesmodels and Refs. 12 and 13 for a numerical comparison of many of these.In the present work, we focus on perturbation theory within a CC framework, where a Møller-Plesset (MP) partitioning of the Hamiltonian is performed, and the energy difference between azeroth-order (parent) CC model and a higher-level (target) CC model is expanded in orders of theperturbation (the fluctuation potential). In particular, we will base the perturbation analysis on abivariational CC Lagrangian obtained by adding to the CC target energy the CC amplitude equa-tions with associated Lagrange multipliers. We note that the linearly parametrized state formallyspanned by the Lagrange multipliers is often referred to as the CC Λ -state, and that this is ingeneral different from the exponentially parametrized CC state. As pointed out by Arponen, extensively exploited in the CC Lagrangian formulation, and recently discussed by Kvaal, the CC energy may be interpreted as a CC functional in both the CC amplitudes and the Λ -stateparameters. We will show that the fastest convergence is obtained when these two sets of stateparameters are treated on an equal footing in the perturbative expansion of the energy difference3etween a parent and a target CC model. Thus, we will distinguish between a perturbation expan-sion of the CC energy, for which only parent amplitudes are used at the expansion point, and aperturbation expansion of the CC Lagrangian, for which both parent amplitudes and parent mul-tipliers are used at the expansion point. At first sight, the bivariational Lagrangian formulationmight seem like an unnecessary complication, since, for the target model, the Lagrangian formallyequals the energy. The purpose of this work is, however, to highlight and exemplify that not onlyis the Lagrangian formulation a convenient mathematical tool that may simplify the derivation ofvarious perturbation expansions; in many cases, the Lagrangian formulation will actually lead todifferent perturbation series than the corresponding energy formulation.To exemplify this difference, we consider two perturbation series which both expand the differ-ence between the energies of the CCSD and CCSDT models. We initially introduce the E-CCSD(T– n )series, in which the CCSD amplitude equations are satisfied at the expansion point, and next com-pare it to the recently developed CCSD(T– n ) series, in which not only the CCSD amplitudeequations are satisfied at the expansion point, but also the CCSD multiplier equations. Despitedepending on the fluctuation potential to infinite order in the space of all single and double ex-citations, the CCSD amplitudes are formally considered as zeroth-order parameters in both theE-CCSD(T– n ) and CCSD(T– n ) series, since the CCSD model represents the expansion point.Similarly, the CCSD multipliers, which too depend on the fluctuation potential, are consideredas zeroth-order parameters for the CCSD(T– n ) series, but not so for the E-CCSD(T– n ) series. Thepath from the CCSD expansion point towards the CCSDT target energy, as defined by a perturba-tion expansion, is thus different within the E-CCSD(T– n ) and CCSD(T– n ) series, and, as will beshown in the present work, the CCSD(T– n ) series is the more rapidly converging of the two, sincemore information is utilized at the expansion point. We finally reiterate that the lowest-order con-tribution of the CCSD(T– n ) series (that of the CCSD(T–2) model) is identical to the triples-onlypart of the CCSD(2) model of Gwaltney and Head-Gordon and the second-order model ofthe CC(2)PT( m ) series of Hirata et al. , the CCSD(T–3) model is identical to the triples-onlypart of the third-order CC(2)PT( m ) model, while for fourth and higher orders, the CCSD(T– n ) and4C(2)PT( n ) series are different. The E-CCSD(T– n ) series, however, differs from the aforemen-tioned perturbation series for all corrections.The present study is outlined as follows. In Section 2, we derive the E-CCSD(T– n ) seriesand compare it to the CCSD(T– n ) series in order to illustrate the importance of treating parentamplitudes and multipliers on an equal footing. In Section 3, we present numerical results for theE-CCSD(T– n ) and CCSD(T– n ) energies, while some concluding remarks are given in Section 4. In this section, we consider two perturbation series that expand the difference between the CCSDand CCSDT energies in orders of the perturbation. In Section 2.1, we develop a new energy-basedperturbation series denoted the E-CCSD(T– n ) series, which can be formulated in terms of CCamplitudes without the need for invoking a CC Lagrangian. Next, in Section 2.2, we develop acommon bivariational framework, in which we recast the E-CCSD(T– n ) and CCSD(T– n ) series.Finally, we present a theoretical comparison between the two series in Section 2.3 in order toexemplify how the CC Lagrangian framework may lead to a perturbation series that is inherentlydifferent from that which arise from the corresponding energy formulation. In this work, we use a MP partitioning of the Hamiltonianˆ H = ˆ f + ˆ Φ (2.1.1)where ˆ f is the Fock operator and ˆ Φ is the fluctuation potential. We consider first the CCSD model,which we choose as the common reference point for the perturbation expansions to follow. The5CSD energy, E CCSD , and associated amplitude equations may be written as E CCSD = (cid:104) HF | ˆ H ∗ ˆ T | HF (cid:105) (2.1.2)and 0 = (cid:104) µ i | ˆ H ∗ ˆ T | HF (cid:105) ( i = , ) (2.1.3)where (cid:104) µ | and (cid:104) µ | represent a singly and a doubly excited state with respect to the HF determi-nant, | HF (cid:105) , and the (non-Hermitian) CCSD similarity-transformed Hamiltonian is given byˆ H ∗ ˆ T = e − ∗ ˆ T ˆ He ∗ ˆ T , ∗ ˆ T = ∗ ˆ T + ∗ ˆ T (2.1.4)with ∗ ˆ T and ∗ ˆ T being the CCSD singles and doubles cluster operators. Throughout the paper, wewill use asterisks to denote CCSD quantities and generally use the generic notation ˆ T i = ∑ µ i t µ i ˆ τ µ i for a cluster operator at excitation level i , where ˆ τ µ i is an excitation operator and t µ i is the associatedamplitude.We now parametrize the difference between the CCSD and CCSDT energy in terms of correc-tion amplitudes, δ t µ i , which represent the difference between the CCSD and CCSDT amplitudes.The correction amplitudes are expanded in orders of the fluctuation potential, and the CCSDTamplitudes, t µ i , may thus be written as t µ i = t ( ) µ i + δ t ( ) µ i + δ t ( ) µ i + . . . ( i = , , ) (2.1.5)where t ( ) µ i = ∗ t µ i for i = , t ( ) µ =
0. We emphasize that, since we have chosen to expandthe CCSDT amplitudes around the CCSD reference point, the CCSD amplitudes, { ∗ t µ , ∗ t µ } , arezeroth-order by definition. The { δ t µ , δ t µ } amplitudes thus represent corrections to the CCSDamplitudes, while { δ t µ } are the CCSDT triples amplitudes.6he CCSDT cluster operator may now be written as ˆ T = ∗ ˆ T + δ ˆ T , where δ ˆ T contains the cor-rection amplitudes, and the CCSDT energy may be obtained by projecting the CCSDT Schrödingerequation, e − δ ˆ T ˆ H ∗ ˆ T e δ ˆ T | HF (cid:105) = E CCSDT | HF (cid:105) , against (cid:104) HF | E CCSDT = (cid:104) HF | e − δ ˆ T ˆ H ∗ ˆ T e δ ˆ T | HF (cid:105) = E CCSD + (cid:104) HF | [ ˆ Φ ∗ ˆ T , δ ˆ T + δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] | HF (cid:105) (2.1.6)where we have carried out a Baker-Campbell-Hausdorff (BCH) expansion, while the CCSDT am-plitude equations are obtained by projection against the combined excitation manifold of all single,double, and triple excitations out of the HF reference0 = (cid:104) µ i | e − δ ˆ T ˆ H ∗ ˆ T e δ ˆ T | HF (cid:105) ( i = , , ) . (2.1.7)The order analysis of Eq. (2.1.7) is identical to the one performed in Ref. 11 (orders are countedin ˆ Φ ), and the resulting amplitudes are thus the same. Compactly, these are given by δ t ( n ) µ i = − ε − µ i (cid:18) (cid:104) µ i | [ ˆ Φ ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) ( n ) ( i = , ) (2.1.8a) δ t ( n ) µ = − ε − µ (cid:18) (cid:104) µ | ˆ Φ ∗ ˆ T + [ ˆ Φ ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) ( n ) (2.1.8b)where ε µ i is the orbital energy difference between the virtual and occupied spin-orbitals of exci-tation µ i , and the right-hand sides of the equations contain all terms of order n , i.e., the sum ofthe orders of all δ T operators plus one (for the fluctuation potential) equals n . For example, thefirst-order singles and doubles corrections are zero, δ t ( ) µ = δ t ( ) µ =
0, while the first-order triplescorrections are given as δ t ( ) µ = − ε − µ (cid:104) µ | ˆ Φ ∗ ˆ T | HF (cid:105) . By collecting terms in orders of the fluctuation7otential, the CCSDT energy in Eq. (2.1.6) may now be expanded as E CCSDT = E CCSD + ∞ ∑ n = E ( n ) (2.1.9a) E ( n ) = (cid:104) HF | [ ˆ Φ ∗ ˆ T , δ ˆ T ( n − ) + δ ˆ T ( n − ) ] | HF (cid:105) + n − ∑ m = (cid:104) HF | [[ ˆ Φ ∗ ˆ T , δ ˆ T ( m ) ] , δ ˆ T ( n − m − ) ] | HF (cid:105) (2.1.9b)where we have used the fact that the first-order singles and doubles amplitudes vanish to restrict thesummations. We denote the perturbation series defined by Eq. (2.1.9) as the E-CCSD(T–n) series to emphasize that it is based on a perturbation expansion of the CCSDT energy around the CCSD energy point, at which the CCSD amplitude equations are satisfied. We note that this notation isnot to be confused with ECC, which is usually used as an acronym for extended coupled clustertheory in the literature
From Eq. (2.1.9), it follows that the first non-vanishing energy correction to the E-CCSD(T– n )series is of third order. The two lowest-order corrections are given in Eq. (A.0.4a) and Eq. (A.0.4b)of Appendix A. The E-CCSD(T– n ) series is evidently different from the CCSD(T– n ) series de-veloped in Ref. 11, which starts at second order. Both series, however, describe the differencebetween the CCSD and CCSDT energies using a MP partitioning of the Hamiltonian, and the cor-rection amplitudes are identical. The only apparent difference is that the CCSDT energy is thecentral quantity for the E-CCSD(T– n ) series, while the CCSDT Lagrangian forms the basis for theCCSD(T– n ) series. In Section 2.2, we develop a general Lagrangian framework to enable a directcomparison of the E-CCSD(T– n ) and CCSD(T– n ) series in Section 2.3.8 .2 A general bivariational Lagrangian framework The CCSDT Lagrangian may be obtained by adding to the CCSDT energy the amplitude equationsin Eq. (2.1.7) with associated (undetermined) multipliers L CCSDT ( t ( ) , ¯t ( ) , δ t , δ ¯t ) = E CCSDT + ∑ j = ∑ ν j ( ¯ t ( ) ν j + δ ¯ t ν j ) (cid:104) ν j | e − δ ˆ T ˆ H T ( ) e δ ˆ T | HF (cid:105) (2.2.1)where we have chosen the following parametrization of the CCSDT multipliers in analogy withEq. (2.1.5) ¯ t µ i = ¯ t ( ) µ i + δ ¯ t ( ) µ i + δ ¯ t ( ) µ i + . . . ( i = , , ) . (2.2.2)If the expansion of the CCSDT multipliers in Eq. (2.2.2) is left untruncated, these will equalthe parameters of the linearly parametrized CCSDT Λ -state. The notation for the Lagrangian, L CCSDT ( t ( ) , ¯t ( ) , δ t , δ ¯t ) , highlights that the amplitudes and multipliers at the expansion point are t ( ) and ¯t ( ) , respectively, while δ t and δ ¯t represent the correction amplitudes and correction mul-tipliers, respectively. By setting t ( ) = ¯t ( ) = , we arrive at an MP-like perturbation expansion,albeit one that has the CCSDT energy as the target energy instead of the full configuration inter-action (FCI) energy. In this work, however, we focus on the case for which the CCSD amplitudesare used as zeroth-order amplitudes ( t ( ) µ = ∗ t µ , t ( ) µ = ∗ t µ , t ( ) µ = n ) seriesmay be recovered by choosing ¯ t ( ) µ i = n ) series corresponds to usingCCSD multipliers as zeroth-order multipliers (¯ t ( ) µ = ∗ ¯ t µ , ¯ t ( ) µ = ∗ ¯ t µ , ¯ t ( ) µ = L CCSD = (cid:104) HF | ˆ H ∗ ˆ T | HF (cid:105) + ∑ j = ∑ ν j ∗ ¯ t ν j (cid:104) ν j | ˆ H ∗ ˆ T | HF (cid:105) (2.2.3)9o be stationary with respect to variations in the CCSD amplitudes0 = ∂ L CCSD ∂ ∗ t µ i = (cid:104) HF | [ ˆ Φ ∗ ˆ T , ˆ τ µ i ] | HF (cid:105) + ∑ j = ∑ ν j ∗ ¯ t ν j (cid:104) ν j | [ ˆ H ∗ ˆ T , ˆ τ µ i ] | HF (cid:105) ( i = , ) . (2.2.4)In the following, we let ¯ t ( ) µ i represent a general zeroth-order multiplier to treat the two se-ries on an equal footing. Equations for the correction amplitudes are determined by requiring L CCSDT ( ∗ t , ¯t ( ) , δ t , δ ¯t ) to be stationary with respect to variations in the correction multipliers ∂ L CCSDT ( ∗ t , ¯t ( ) , δ t , δ ¯t ) ∂ δ ¯ t µ i = ( i = , , ) (2.2.5)which reproduces the CCSDT amplitude equations in Eq. (2.1.7). It follows that the equationsfor the correction amplitudes are independent of the choice of zeroth-order multipliers, and thecorrection amplitudes for the E-CCSD(T– n ) and CCSD(T– n ) series are therefore identical to allorders. Equations for the CCSDT multipliers are obtained by requiring L CCSDT ( ∗ t , ¯t ( ) , δ t , δ ¯t ) tobe stationary with respect to variations in the amplitudes ∂ L CCSDT ( ∗ t , ¯t ( ) , δ t , δ ¯t ) ∂ δ t µ i = ( i = , , ) . (2.2.6)Unlike for the correction amplitudes in Eq. (2.1.7), the precise form of the multiplier equationswill depend upon the choice of zeroth-order multipliers, and the correction multipliers for theE-CCSD(T– n ) and CCSD(T– n ) series will therefore be different. To keep this distinction clear,we will refer to the correction multipliers associated with the choices ¯ t ( ) = and ¯ t ( ) = ∗ ¯t as δ ¯t E and δ ¯t L , respectively, while the generic notation δ ¯t may refer to either of the series. We notethat at infinite order, the multipliers—as defined within either the E-CCSD(T– n ) or CCSD(T– n )series—are identical (assuming that both expansions converge), i.e. ∞ ∑ n = δ ¯ t E ( n ) = ∗ ¯ t + ∞ ∑ n = δ ¯ t L ( n ) = ¯ t (2.2.7)10here ¯ t is the final set of converged CCSDT multipliers ( Λ -state parameters).To simplify the comparison of the two series, it proves convenient to expand the Lagrangianin Eq. (2.2.1) in a form that emphasizes the dependence on the CCSD multiplier equation inEq. (2.2.4) L CCSDT ( ∗ t , ¯t ( ) , δ t , δ ¯t ) = E CCSD + ∑ i = ∑ µ i δ t µ i (cid:18) (cid:104) HF | [ ˆ Φ ∗ ˆ T , ˆ τ µ i ] | HF (cid:105) + ∑ j = ∑ ν j ¯ t ( ) ν j (cid:104) ν j | [ ˆ H ∗ ˆ T , ˆ τ µ i ] | HF (cid:105) (cid:19) + (cid:104) HF | [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] | HF (cid:105) + ∑ j = ∑ ν j ¯ t ( ) ν j (cid:18) (cid:104) ν j | [ ˆ Φ ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) + ∑ j = ∑ ν j δ ¯ t ν j (cid:18) (cid:104) ν j | [ ˆ H ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) + ∑ ν δ ¯ t ν (cid:18) (cid:104) ν | ˆ Φ ∗ ˆ T + [ ˆ H ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) (2.2.8)where we have set t ( ) = ∗ t and used the CCSD amplitude equations in Eq. (2.1.3).By choosing ¯t ( ) = in Eq. (2.2.8), we arrive at the Lagrangian L CCSDT ( ∗ t , , δ t , δ ¯t E ) , whichreads L CCSDT ( ∗ t , , δ t , δ ¯t E ) = E CCSD + (cid:104) HF | [ ˆ Φ ∗ ˆ T , δ ˆ T + δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] | HF (cid:105) + ∑ j = ∑ ν j δ ¯ t E ν j (cid:18) (cid:104) ν j | [ ˆ H ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) + ∑ ν δ ¯ t E ν (cid:18) (cid:104) ν | ˆ Φ ∗ ˆ T + [ ˆ H ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) . (2.2.9)Since the δ ¯t E multipliers multiply the CCSDT amplitude equations in Eq. (2.1.7), they may be11liminated from Eq. (2.2.9), and it follows that L CCSDT ( ∗ t , , δ t , δ ¯t E ) equals the CCSDT energy inEq. (2.1.6) L CCSDT ( ∗ t , , δ t , δ ¯t E ) = E CCSDT . (2.2.10)By evaluating L CCSDT ( ∗ t , , δ t , δ ¯t E ) to different orders in the fluctuation potential, we thus arriveat the E-CCSD(T– n ) series in Eq. (2.1.9), for which the energy corrections may be expressed exclu-sively in terms of correction amplitudes. The evaluation of the E-CCSD(T– n ) energy correctionsusing Eq. (2.1.9) corresponds to using the n + n determine theenergy to order n +
1. Alternatively, by exploiting that L CCSDT ( ∗ t , , δ t , δ ¯t E ) is variational in theamplitudes as well as the multipliers, it may be evaluated to different orders in the fluctuationpotential by using the 2 n + n + for the amplitudes and multipliers (i.e., theamplitudes/multipliers to order n determine the Lagrangian to order 2 n + n + n + n + n ) series to the CCSD(T– n ) series (cf. Appendix A).By setting ¯t ( ) = ∗ ¯t in Eq. (2.2.8), the resulting Lagrangian L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) becomes L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) = E CCSD + (cid:104) HF | [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] | HF (cid:105) + ∑ j = ∑ ν j ∗ ¯ t ν j (cid:18) (cid:104) ν j | [ ˆ Φ ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) + ∑ j = ∑ ν j δ ¯ t L ν j (cid:18) (cid:104) ν j | [ ˆ H ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) + ∑ ν δ ¯ t L ν (cid:18) (cid:104) ν | ˆ Φ ∗ ˆ T + [ ˆ H ∗ ˆ T , δ ˆ T ] + [[ ˆ Φ ∗ ˆ T , δ ˆ T ] , δ ˆ T ] + . . . | HF (cid:105) (cid:19) (2.2.11)where we have used the CCSD multiplier equations in Eq. (2.2.4). An expansion of L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) in orders of the fluctuation potential defines the recently proposed CCSD(T– n ) series. The E-CCSD(T– n )12eries begins at third order (see Eq. (2.1.9)), while the CCSD(T– n ) series starts already at secondorder. The series are evidently different, and in Section 2.3 we perform an explicit comparison ofthem. n ) and CCSD(T– n ) series In this section, we compare the E-CCSD(T– n ) and CCSD(T– n ) series—first, we discuss the originof the difference between the series from a formal point of view, and next, we compare the twolowest-order multipliers and energy corrections for the two series.As shown in Section 2.2, both series can be derived from Eq. (2.2.8) with different choices ofparent multipliers, resulting in the Lagrangians, L CCSDT ( ∗ t , , δ t , δ ¯t E ) and L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) ,for the E-CCSD(T– n ) and CCSD(T– n ) series, respectively. The L CCSDT ( ∗ t , , δ t , δ ¯t E ) Lagrangianreduces to the standard CCSDT energy expression, because no zeroth-order multipliers enter theLagrangian and because the correction multipliers, δ ¯ t , are multiplied by the CCSDT amplitudeequations (cf. Eq. (2.2.9) and Eq. (2.2.10)). However, the L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) Lagrangiancannot be subject to a similar reduction, since certain terms in the CCSDT amplitude equationswere cancelled when the CCSD multiplier equations were used to manipulate Eq. (2.2.8) to arriveat Eq. (2.2.11). Consequently, the CCSD multipliers cannot be removed from Eq. (2.2.11), and L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) can therefore not be reduced to the standard CCSDT energy expression.In the same way as the CCSD amplitudes formally depend on the fluctuation potential to infi-nite order in the space of all single and double excitations, so do the CCSD multipliers (or CCSD Λ -state parameters). Thus, since the CCSD multipliers are not counted as zeroth-order parametersin the E-CCSD(T– n ) series (unlike in the CCSD(T– n ) series), orders are necessarily counted differ-ently in the perturbative expansions of L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) and L CCSDT ( ∗ t , , δ t , δ ¯t E ) ; however,we may compare the two series by comparing their leading-order, next-to-leading-order, etc., cor-rections to one another, which we will do numerically in Section 3.On that note, we have theoretically compared the lowest- and next-to-lowest-order correc-tions of the two series in Appendix A. In summary, we find that for the L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) Λ -state in the space of all single anddouble excitations is used (the CCSD Λ -state), and the lowest-order multiplier correction thereforeoccurs in the triples space (cf. Eq. (A.0.8)). For the L CCSDT ( ∗ t , , δ t , δ ¯t E ) Lagrangian, on theother hand, there is no representation of the CCSDT Λ -state at zeroth order, and the leading-ordercontributions to this state (in the form of the first-order multipliers, δ ¯ t E ( ) µ i ) hence occur within thesingles and doubles space, while triples multipliers first enter the E-CCSD(T– n ) series at the fol-lowing (second) order (cf. Eq. (A.0.3)). In fact, we find that the first- and second-order singles anddoubles multipliers, { δ ¯ t E ( ) µ i , δ ¯ t E ( ) µ i } for i = ,
2, are nothing but the two lowest-order contributionsto the CCSD Λ -state parameters, if the CCSD multiplier equations in Eq. (2.2.4) are solved pertur-batively (cf. Eq. (A.0.12)). Furthermore, the lowest-order E-CCSD(T– n ) and CCSD(T– n ) energycorrections are found to have the same structural form, however, they are expressed in terms ofdifferent sets of multipliers (cf. Eq. (A.0.7) and Eq. (A.0.11)). Thus, the E-CCSD(T– n ) series maybe viewed as attempting to compensate for the poor (non-existing) guess at the CCSDT Λ -state bymimicking the CCSD(T– n ) series as closely as possible within a perturbational framework. Forboth series, a perturbative solution of the CCSDT Λ -state is embedded into the energy corrections,and the E-CCSD(T– n ) series is thus trailing behind the CCSD(T– n ) series from the onset of theperturbation expansion. Again, this motivates the claim that it is advantageous to consider theCC energy as the stationary point of an energy functional in both the CC and Λ -state parame-ters, and hence that perturbative expansions are optimally carried out whenever the two states aretreated on an equal footing. For higher orders in the perturbation, a direct comparisonof the E-CCSD(T– n ) and CCSD(T– n ) series is more intricate, but we note that they will differ toall orders. Only in the infinite-order limit are the two bound to agree ( L CCSDT ( ∗ t , , δ t , δ ¯t E ) and L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) both equal the CCSDT energy), but this is a natural consequence of the factthat the CC energy may be described in terms of fully converged CC amplitudes alone.In conclusion, the L CCSDT ( ∗ t , , δ t , δ ¯t E ) and L CCSDT ( ∗ t , ∗ ¯t , δ t , δ ¯t L ) Lagrangians have the sameparent energy (CCSD) and the same target energy (CCSDT), and all contributions of both se-14ies will be trivially term-wise size extensive to all orders, as they are all expressed in terms of(linked) commutator expressions. However, the paths between these two CC energies, as definedby a perturbation expansion, are obviously very different for the two series. Conceptually, theexpansion point for the E-CCSD(T– n ) series is formally the CCSD energy (only the CCSD am-plitude equations are satisfied), while the expansion point for the CCSD(T– n ) series is the CCSDLagrangian (the CCSD amplitude and
CCSD multiplier equations are satisfied). For the energy-based E-CCSD(T– n ) series, the Lagrangian is thus merely a mathematical tool that allows for cor-rection energies to be obtained using amplitude and multiplier corrections that satisfy the 2 n + n + n ) series is deeply rooted within a bivariationalLagrangian formulation and has no energy-based analogue. The performance of the models of the CCSD(T– n ) series has recently been theoretically as wellas numerically compared to a variety of alternative triples models for two sets of closed-shell and open-shell species, and the formal convergence of the series (through sixth order in theperturbation) has been confirmed. Furthermore, the performance of the higher-order CCSD(T– n )models with respect to the target CCSDT model was found to be essentially independent of the HFreference used, and thus, independent of the spin of the ground state. In this section, we assess thenumerical performance of the E-CCSD(T– n ) models (once again measured against results obtainedwith the target CCSDT model) in order to compare the rate of convergence throughout the seriesto that of the CCSD(T– n ) series.We here use the two test sets previously used in Refs. 12, 13, 31, and 32: (i)
17 closed-shellmolecules, all optimized at the all-electron CCSD(T)/cc-pCVQZ level of theory, and (ii)
18 open-shell atoms and radicals, all optimized at the all-electron CCSD(T)/cc-pVQZ level of theory. For aspecification of the members of the closed- and open-shell test sets as well as tabularized moleculargeometries, cf. Refs. 6 and the papers describing the HEAT thermochemical model, respec-15ively. All of the closed-shell calculations are based on a restricted HF (RHF) reference, whileunrestricted HF (UHF) as well as restricted open-shell HF (ROHF) trial functions have been usedfor the open-shell calculations. The correlation-consistent cc-pVTZ basis set is used throughoutfor all of the reported valence-electron (frozen-core) results, and the Aquarius program has beenused for all of the calculations.In Figure 1, we consider the performance of the five lowest-order models of the E-CCSD(T– n )and CCSD(T– n ) hierarchies. Mean recoveries (in %) of the triples correlation energy, E T = E CCSDT − E CCSD , are presented in Figure 1a, Figure 1c, and Figure 1e, while mean deviationsfrom E T (in kcal/mol) are presented in Figure 1b, Figure 1d, and Figure 1f. In all cases, we reportstatistical error measures generated from the individual results, cf. the supplementary material. As noted in Section 2.3, the E-CCSD(T– n ) and CCSD(T– n ) series start at third and second order, re-spectively, but we may group these together like E-CCSD(T–3)/CCSD(T–2), E-CCSD(T–4)/CCSD(T–3),etc.The results in Figure 1 show that the CCSD(T– n ) models in general yield smaller mean andstandard errors than their E-CCSD(T– n ) counterparts, and the CCSD(T– n ) series furthermore ex-hibits a more stable convergence than the E-CCSD(T– n ) series. In other words, the rate of conver-gence is improved in the CCSD(T– n ) series over the E-CCSD(T– n ) series. For most of the consid-ered molecules, the E-CCSD(T– n ) corrections are negative/positive for uneven/even orders, whichleads to the oscillatory convergence behavior observed for the E-CCSD(T– n ) series in Figure 1.Some oscillatory behavior is also observed for the CCSD(T– n ) series, but this is much less promi-nent, and primarily observed beyond fourth order. The superior stability of the CCSD(T– n ) seriescompared to the E-CCSD(T– n ) series is also manifested in the smaller standard deviations for theformer (in Figure 1 represented in terms of standard errors of the mean). Some of the molecules,however, differ considerably from the mean trends in Figure 1. For example, methylene (CH )and ozone (O ) are notoriously difficult cases due to significant multireference character, and,for both, we observe a rather slow convergence throughout either of the series. While for O , theconvergence towards the CCSDT limit is oscillatory, for CH , the convergence is stabile, yet slow,16f. Table S4 of the supplementary material for the E-CCSD(T– n ) results. Similar, but significantlyless pronounced problems, are observed for the CCSD(T– n ) series. Finally, we note how theresults obtained using the two open-shell references (UHF and ROHF) are similar, and also thatthe general behavior for the mean deviations are similar to the RHF results.From an application point of view, the CCSD(T– n ) series is practically converged onto theCCSDT limit at the CCSD(T–4) model (robust for closed- and open-shell systems), while two addi-tional corrections (two additional orders in the perturbation) are needed in the E-CCSD(T– n ) seriesin order to match these results (the E-CCSD(T–7) model). However, even for the E-CCSD(T–7)model, the standard deviations (for both recoveries and deviations) are larger than for the CCSD(T–4)model. If results more accurate than those provided by the CCSD(T–4)/E-CCSD(T–7) modelsare desired, it is in general necessary to also account for the effects of quadruple excitations, asthe quadruples energy contribution may easily exceed the difference between the CCSDT andCCSD(T–4) energies. In such cases, the recently proposed CCSDT(Q– n ) models may of-fer attractive alternatives to the iterative CCSDTQ model. These models are theoretically on parwith the CCSD(T– n ) models, but expand the CCSDTQ–CCSDT energy difference, rather than theCCSDT–CCSD difference, in orders of the MP fluctuation potential.In conclusion, a number of similarities exist between the E-CCSD(T– n ) and CCSD(T– n ) se-ries, but both the magnitude of the (individual) errors as well as the oscillatory convergence patternare significantly reduced in the CCSD(T– n ) series, as compared to the E-CCSD(T– n ) series. Thisimprovement is in line with the theoretical analysis in Section 2.3. More information about theexpansion point is used in the CCSD(T– n ) series, where the CCSD amplitudes and CCSD mul-tipliers are both built into the perturbative corrections, to yield a faster and more balanced rateof convergence than that observed for the E-CCSD(T– n ) series, in which only the CCSD ampli-tudes are used to construct the energy corrections. Based on the results for the five lowest-ordermodels in Figure 1, the E-CCSD(T– n ) and CCSD(T– n ) series both appear to converge for all ofthe considered molecules, although slowly for some of the notoriously difficult cases. However,a formal convergence analysis is required to firmly establish whether the series indeed converge17 T-2)E-(T-3) (T-3)E-(T-4) (T-4)E-(T-5) (T-5)E-(T-6) (T-6)E-(T-7)CCSD(T-n) / E-CCSD(T-n) model859095100 R e c o v e r y ( i n % ) RHF-CCSD(T-n)RHF-E-CCSD(T-n) (a) RHF recoveries (T-2)E-(T-3) (T-3)E-(T-4) (T-4)E-(T-5) (T-5)E-(T-6) (T-6)E-(T-7)CCSD(T-n) / E-CCSD(T-n) model0.00.51.01.52.02.5 D e v i a t i o n ( i n k c a l / m o l ) RHF-CCSD(T-n)RHF-E-CCSD(T-n) (b) RHF deviations (T-2)E-(T-3) (T-3)E-(T-4) (T-4)E-(T-5) (T-5)E-(T-6) (T-6)E-(T-7)CCSD(T-n) / E-CCSD(T-n) model859095100 R e c o v e r y ( i n % ) UHF-CCSD(T-n)UHF-E-CCSD(T-n) (c) UHF recoveries (T-2)E-(T-3) (T-3)E-(T-4) (T-4)E-(T-5) (T-5)E-(T-6) (T-6)E-(T-7)CCSD(T-n) / E-CCSD(T-n) model0.00.51.01.52.02.5 D e v i a t i o n ( i n k c a l / m o l ) UHF-CCSD(T-n)UHF-E-CCSD(T-n) (d) UHF deviations (T-2)E-(T-3) (T-3)E-(T-4) (T-4)E-(T-5) (T-5)E-(T-6) (T-6)E-(T-7)CCSD(T-n) / E-CCSD(T-n) model859095100 R e c o v e r y ( i n % ) ROHF-CCSD(T-n)ROHF-E-CCSD(T-n) (e) ROHF recoveries (T-2)E-(T-3) (T-3)E-(T-4) (T-4)E-(T-5) (T-5)E-(T-6) (T-6)E-(T-7)CCSD(T-n) / E-CCSD(T-n) model0.00.51.01.52.02.5 D e v i a t i o n ( i n k c a l / m o l ) ROHF-CCSD(T-n)ROHF-E-CCSD(T-n) (f) ROHF deviations
Figure 1: Mean recoveries of (in %, Figure 1a, Figure 1c, and Figure 1e) and mean deviations from(in kcal/mol, Figure 1b, Figure 1d, and Figure 1f) the triples energy E T for the CCSD(T– n ) andE-CCSD(T– n ) series using RHF, UHF, and ROHF references. The error bars show the standarderror of the mean. 18i.e., establish the radius of convergence for the series). This will be the subject of a forthcomingpaper. We have developed the E-CCSD(T– n ) perturbation series and compared it to the recently pro-posed CCSD(T– n ) series in order to gain new insights into the importance of treating amplitudesand multipliers (parameters of the Λ -state) on an equal footing whenever perturbation expansionsare developed within CC theory. Both series represent a perturbation expansion of the differencebetween the CCSD and CCSDT energies, and they share the same common set of correction am-plitudes. The E-CCSD(T– n ) series formally describes an expansion around the CCSD energy point(CCSD amplitude equations are satisfied), while the CCSD(T– n ) series may be viewed as an ex-pansion around the CCSD Lagrangian point (CCSD amplitude equations and CCSD multiplierequations are satisfied). The two series are therefore different, and the CCSD(T– n ) series is foundto converge more rapidly towards the CCSDT target energy, since all available information at theCCSD expansion point is utilized.The presented analysis may be generalized to any perturbation expansion representing the dif-ference between a parent CC model and a higher-level target CC model. For developments of CCperturbation expansions, we thus generally advocate the use a bivariational Lagrangian CC formu-lation to ensure an optimal rate of convergence in terms of term-wise size extensive correctionstowards the target energy. For example, two perturbation series formulated around the CCSDTenergy (E-CCSDT(Q– n )) and CCSDT Lagrangian (CCSDT(Q– n )) expansion points, respectively,to describe an expansion towards the CCSDTQ target energy in order the Møller-Plesset fluctua-tion potential, are also bound to exhibit different rates of convergence, following a similar line ofarguments.In quantum chemistry, a Lagrangian energy functional has traditionally been viewed merely19s a convenient mathematical tool for deriving perturbative expansions, however, one that wouldgive rise to expansions that are identical to those based on the standard energy. The present workhighlights how this equivalence between energy- and Lagrangian-based perturbation theory onlyholds whenever the zeroth-order parameters do not depend on the perturbing operator, as is, forexample, the case for standard MP perturbation theory where the zeroth-order parameters vanish.Thus, when the zeroth-order parameters are independent of the perturbing operator, a Lagrangianformulation is merely of mathematical convenience, but, for perturbation-dependent zeroth-orderparameters (e.g., like those of the right- (CC) and left-hand ( Λ ) eigenstates of a non-Hermitian CCsimilarity-transformed Hamiltonian), a bivariational Lagrangian formulation is in general expectedto lead to a faster and more stable convergence than a corresponding energy formulation. This is animportant point to keep in mind for future developments and applications involving perturbationexpansions. Acknowledgments
K. K., J. J. E., and P. J. acknowledge support from the European Research Council under theEuropean Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No.291371. J. O. acknowledges support from the Danish Council for Independent Research, DFF-4181-00537, and D. A. M. acknowledges support from the US National Science Foundation (NSF)under grant number ACI-1148125/1340293. 20
Explicit lowest-order E-CCSD(T– n ) and CCSD(T– n ) correc-tion energies In the present appendix, we compare the lowest- and next-to-lowest-order corrections of the E-CCSD(T– n )and CCSD(T– n ) series. For this, we need a closed-form expression for the CCSDT multipliers (i.e.,the CCSDT Λ -state parameters), which, from Eq. (2.2.6), reads¯ t µ i = − ε − µ i (cid:0) (cid:104) HF | [ Φ ˆ T , ˆ τ µ i ] | HF (cid:105) + ∑ j = ∑ ν j ¯ t ν j (cid:104) ν j | [ Φ ˆ T , ˆ τ µ i ] | HF (cid:105) (cid:1) (A.0.1)where we will again partition the CCSDT cluster operator, ˆ T , as ˆ T = ∗ ˆ T + δ ˆ T . Eq. (A.0.1) maynow be expanded in orders of the fluctuation potential (cf. Eq. (2.2.2))¯ t = ¯ t ( ) + δ ¯ t ( ) + δ ¯ t ( ) + . . . (A.0.2)If ¯ t ( ) = n )series, and the two lowest-order corrections are given by δ ¯ t E ( ) µ i = − ε − µ i (cid:104) HF | [ ˆ Φ ∗ ˆ T , ˆ τ µ i ] | HF (cid:105) (A.0.3a) δ ¯ t E ( ) µ i = − ε − µ i ∑ j = ∑ ν j δ ¯ t E ( ) ν j (cid:104) ν j | [ ˆ Φ ∗ ˆ T , ˆ τ µ i ] | HF (cid:105) . (A.0.3b)It follows that the E-CCSD(T– n ) series has non-vanishing first-order multipliers only in the singlesand doubles space ( δ ¯ t E ( ) µ = i = , , n ) series may be evalu-ated using the n + E ( ) = ∑ j = (cid:104) HF | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) j ] | HF (cid:105) (A.0.4a) E ( ) = ∑ j = (cid:104) HF | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) j ] | HF (cid:105) . (A.0.4b)21lternatively, using the Lagrangian in Eq. (2.2.9) and the 2 n + n + for the ampli-tudes/multipliers, E ( ) and E ( ) may be written as E ( ) = ∑ i = ∑ µ i δ ¯ t E ( ) µ i (cid:104) µ i | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) ] | HF (cid:105) (A.0.5a) E ( ) = ∑ k = (cid:8) ∑ i = ∑ µ i δ ¯ t E ( ) µ i (cid:104) µ i | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) k ] | HF (cid:105) (cid:9) . (A.0.5b)The expressions in Eq. (A.0.4) and Eq. (A.0.5) are of course equivalent as may be verified byexplicit comparison. Finally, the E ( ) energy in Eq. (A.0.5b) may be further recast by expandingthe second-order correction amplitudes, δ t ( ) , given in Eq. (2.1.8) E ( ) = − ∑ k = ∑ ν k (cid:8) ε − ν k ∑ i = ∑ µ i δ ¯ t E ( ) µ i (cid:104) µ i | [ ˆ Φ ∗ ˆ T , ˆ τ ν k ] | HF (cid:105)(cid:104) ν k | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) ] | HF (cid:105) (cid:9) = ∑ k = ∑ ν k δ ¯ t E ( ) ν k (cid:104) ν k | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) ] | HF (cid:105) . (A.0.6)By taking the sum of the third- and fourth-order E-CCSD(T– n ) energies in Eq. (A.0.4a) andEq. (A.0.6), respectively, we can write E ( ) + E ( ) = ∑ i = ∑ µ i (cid:0) δ ¯ t E ( ) µ i + δ ¯ t E ( ) µ i (cid:1) (cid:104) µ i | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) ] | HF (cid:105) + ∑ µ δ ¯ t E ( ) µ (cid:104) µ | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) ] | HF (cid:105) . (A.0.7)To evaluate the two lowest-order energy corrections of the CCSD(T– n ) series (¯ t ( ) = ∗ ¯t ), we onlyneed to consider the first-order multipliers, which read δ ¯ t L ( ) µ = δ ¯ t L ( ) µ = δ ¯ t L ( ) µ = − ε − µ ∑ j = ∗ ¯ t ν j (cid:104) ν j | [ ˆ Φ ∗ ˆ T , ˆ τ µ ] | HF (cid:105) . (A.0.8b)By applying the 2 n + n + n )22eries are given as L ( ) = ∑ i = ∑ µ i ∗ ¯ t µ i (cid:104) µ i | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) ] | HF (cid:105) (A.0.9)and L ( ) = ∑ µ δ ¯ t L ( ) µ (cid:104) µ | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) ] | HF (cid:105) (A.0.10)the sum of which becomes L ( ) + L ( ) = ∑ i = ∑ µ i ∗ ¯ t µ i (cid:104) µ i | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) ] | HF (cid:105) + ∑ µ δ ¯ t L ( ) µ (cid:104) µ | [ ˆ Φ ∗ ˆ T , δ ˆ T ( ) ] | HF (cid:105) . (A.0.11)We may now compare the energy sums in Eq. (A.0.7) and Eq. (A.0.11). Since the two leading-order multiplier corrections of the E-CCSD(T– n ) series are independent of the triple excitationsin the CCSDT ansatz, these will equal the two lowest-order contributions to the CCSD Λ -stateparameters, i.e. ∗ ¯ t µ i = δ ¯ t E ( ) µ i + δ ¯ t E ( ) µ i + O ( ) ( i = , ) (A.0.12)where O ( ) denotes terms of third and higher orders in the fluctuation potential. For this reason,the first term on the right-hand side of Eq. (A.0.7) may be viewed as mimicking the first term onthe right-hand side of Eq. (A.0.11), and the same applies for the second term on the right-handside of the two equations, by noticing that the ¯ t E ( ) multipliers of Eq. (A.0.3b) are similar to the¯ t L ( ) multipliers of Eq. (A.0.8b), with the notable exception that ¯ t E ( ) → ∗ ¯ t in moving from theE-CCSD(T– n ) to the CCSD(T– n ) series. 23 otes and References (1) ˇCíˇcek, J. J. Chem. 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ParallelDistrib. , 74, 3176.(38) See supplementary material at [AIP URL] for individual recoveries and deviations. IndividualCCSD(T– nn