High-Level Coupled-Cluster Energetics by Monte Carlo Sampling and Moment Expansions: Further Details and Comparisons
aa r X i v : . [ phy s i c s . c h e m - ph ] F e b High-Level Coupled-Cluster Energetics by Monte Carlo Sampling andMoment Expansions: Further Details and Comparisons
J. Emiliano Deustua, Jun Shen, and Piotr Piecuch
1, 2, a) Department of Chemistry, Michigan State University, East Lansing, Michigan 48824,USA Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824,USA (Dated: 23 February 2021)
We recently proposed a novel approach to converging electronic energies equivalent to high-level coupled-cluster (CC) computations by combining the deterministic CC( P ; Q ) formalism with the stochastic configu-ration interaction (CI) and CC Quantum Monte Carlo (QMC) propagations. This article extends our initialstudy [J. E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. , 223003 (2017)], which focused onrecovering the energies obtained with the CC method with singles, doubles, and triples (CCSDT) using theinformation extracted from full CI QMC and CCSDT-MC, to the CIQMC approaches truncated at triplesand quadruples. It also reports our first semi-stochastic CC( P ; Q ) calculations aimed at converging the en-ergies that correspond to the CC method with singles, doubles, triples, and quadruples (CCSDTQ). Theability of the semi-stochastic CC( P ; Q ) formalism to recover the CCSDT and CCSDTQ energies, even whenelectronic quasi-degeneracies and triply and quadruply excited clusters become substantial, is illustrated bya few numerical examples, including the F–F bond breaking in F , the automerization of cyclobutadiene, andthe double dissociation of the water molecule. I. INTRODUCTION
One of the main goals of quantum chemistry is to pro-vide an accurate and systematically improvable descrip-tion of many-electron correlation effects needed to deter-mine molecular potential energy and property surfacesand understand chemical reactivity and various typesof spectroscopy. In searching for the best solutions inthis area, the size extensive methods based on the expo-nential wave function ansatz of coupled-cluster (CC)theory, | Ψ i = e T | Φ i , (1)where T = N X n =1 T n (2)is the cluster operator, T n is the n -body component of T , N is the number of correlated electrons, and | Φ i is thereference determinant, and their extensions to excited,open-shell, and multi-reference states are among thetop contenders. In this study, we focus on the higher-rank members of the single-reference CC hierarchy be-yond the basic CC singles and doubles (CCSD) level,where T is truncated at T , especially on the CCapproach with singles, doubles, and triples (CCSDT),where T is truncated at T , and the CC approachwith singles, doubles, triples, and quadruples (CCS-DTQ), where T is truncated at T . This is moti-vated by the fact that in great many cases relevant to a) Corresponding author; e-mail: [email protected]. chemistry, including molecular properties at equilibriumgeometries, multi-reference situations involving smallernumbers of strongly correlated electrons, as in the caseof bond breaking and formation in the course of chemi-cal reactions, noncovalent interactions, and photochem-istry, the single-reference CCSD, CCSDT, CCSDTQ, etc.methods and their equation-of-motion (EOM) andlinear response extensions rapidly converge to theexact, full configuration interaction (FCI) limit, allowingone to incorporate the relevant many-electron correlationeffects in a conceptually straightforward manner throughparticle-hole excitations from a single Slater determinantdefining the Fermi vacuum without loss of accuracy asthe system becomes larger characterizing truncated CImethods. The convergence of the single-reference CCSD,CCSDT, CCSDTQ, etc. hierarchy toward FCI in situ-ations other than larger numbers of strongly entangledelectrons is fast, but costs of the post-CCSD computa-tions needed to achieve a quantitative description, whichare determined by the iterative n o n u steps in the CCSDTcase and the iterative n o n u steps in the case of CCSDTQ,where n o ( n u ) is the number of occupied (unoccupied)correlated orbitals, are usually prohibitively expensive.This is why part of the CC method development efforthas been devoted to finding approximate ways of incorpo-rating higher–than–two-body components of the clusteroperator T , i.e., T n components with n >
2, and theanalogous higher-order components of the EOMCC ex-citation, electron-attachment, and electron-detachmentoperators, which could reduce enormous computationalcosts of the CCSDT, CCSDTQ, and similar schemes,while eliminating failures of the CCSD[T], CCSD(T), CCSDT-1,
CC3, and other perturbative CC ap-proaches (cf. Ref. 10 for a review) that fail when bondbreaking, biradicals, and other typical multi-reference sit-uations in chemistry are examined.
In fact, theanalogous effort has been taking place in other areasof many-body theory, such as studies of nuclear mat-ter, where a systematic, computationally efficient, androbust incorporation of higher-order many-particle cor-relation effects is every bit as important as in the caseof electronic structure theory and where the quantum-chemistry-inspired CC and EOMCC methods, thanks, inpart, to our group’s involvement, have become quitepopular (see, e.g., Ref. 56 and references therein). Whilesubstantial progress in the above area, reviewed, for ex-ample, in Refs. 10, 48, and 57, has already been made,the search for the optimum solution that would allow usto obtain the results of the full CCSDT, full CCSDTQ,or similar quality at the fraction of the cost and with-out having to rely on perturbative concepts or user- andsystem-dependent ideas, such as the idea of active or-bitals to select higher–than–two-body components of thecluster and EOMCC excitation operators, continues.In order to address this situation, we have started ex-ploring a radically new way of converging accurate elec-tronic energetics equivalent to those obtained with thehigh-level CC approaches of the full CCSDT, full CCS-DTQ, and similar types, at the small fraction of thecomputational cost and preserving the black-box charac-ter of conventional single-reference methods, even whenhigher–than–two-body components of the cluster and ex-citation operators characterizing potential energy sur-faces along bond stretching coordinates become large. The key idea of the approach suggested in Ref. 58,which we have recently extended to excited states, is a merger of the deterministic formalism, abbreviatedas CC( P ; Q ), which enables one to correct en-ergies obtained with conventional as well as unconven-tional truncations in the cluster and EOMCC excita-tion operators for any category of many-electron corre-lation effects of interest, with the stochastic FCI Quan-tum Monte Carlo (FCIQMC) and CC Monte Carlo(CCMC) methods (cf. Refs. 72–74 for alterna-tive ways of combining FCIQMC with the deterministicCC framework). As shown in Refs. 58 and 60, wherewe reported preliminary calculations aimed at recover-ing full CCSDT and EOMCCSDT energetics, theresulting semi-stochastic CC( P ; Q ) methodology, usingthe FCIQMC and CCSDT-MC approaches to identifythe leading determinants or cluster amplitudes in thewave function and the a posteriori CC( P ; Q ) correctionsto capture the remaining correlations, rapidly convergesto the target energetics based on the information ex-tracted from the early stages of FCIQMC or CCSDT-MC propagations. If confirmed through additional testsand comparisons involving various QMC and CC levels,the merger of the deterministic CC( P ; Q ) and stochas-tic CIQMC and CCMC ideas, originally proposed in Ref.58, may substantially impact accurate quantum calcula-tions for many-electron and other many-fermion systems,opening interesting new possibilities in this area. The present study is our next step in the develop-ment and examination of the semi-stochastic CC( P ; Q )methodology. In this work, we extend our initial study, which focused on recovering the full CCSDT energeticsbased on the information extracted from the FCIQMCand CCSDT-MC propagations, to the CIQMC methodstruncated at triples (CISDT-MC) or triples and quadru-ples (CISDTQ-MC), which may offer significant savingsin the computational effort compared to FCIQMC andwhich are formally compatible with the CCSDT andCCSDTQ excitation manifolds we would like to cap-ture. We also report our initial results of the semi-stochastic CC( P ; Q ) calculations aimed at convergingthe full CCSDTQ energetics. The ability of the semi-stochastic CC( P ; Q ) approaches to recover the CCSDTand CCSDTQ energies based on the truncated CISDT-MC and CISDTQ-MC propagations, even when elec-tronic quasi-degeneracies and T and T clusters becomesubstantial, is illustrated using the challenging cases ofthe F–F bond breaking in F , the automerization of cy-clobutadiene, and the double dissociation of the watermolecule as examples. II. THEORY AND ALGORITHMIC DETAILS
As pointed out in the Introduction, the semi-stochasticCC( P ; Q ) approach proposed in Ref. 58 is based oncombining the deterministic CC( P ; Q ) framework, devel-oped mainly in Refs. 57, 61, and 63, with the CIQMCand CCMC ideas that were originally laid down in Refs.64, 65, and 68. Thus, we divide this section into twosubsections. In Section II A, we summarize the key ele-ments of the deterministic CC( P ; Q ) formalism, focusingon the ground-state problem relevant to the calculationsreported in this study. Section II B provides informa-tion about the semi-stochastic CC( P ; Q ) methods devel-oped and tested in this work, which aim at convergingthe CCSDT and CCSDTQ energies with the help of theFCIQMC, CISDT-MC, and CISDTQ-MC propagations. A. Basic Elements of the Ground-State CC( P ; Q )Formalism The CC( P ; Q ) formalism has emerged out ofour interest in generalizing the biorthogonal mo-ment energy expansions, which in the past re-sulted in the completely renormalized (CR) CCand EOMCC approaches, including CR-CC(2,3), CR-EOMCC(2,3), δ -CR-EOMCC(2,3), and theirhigher-order extensions, such that one can cor-rect the CC/EOMCC energies obtained with unconven-tional truncations in the cluster and EOMCC excitationoperators, in addition to the conventional ones at a givenmany-body rank, for essentially any category of many-electron correlation effects of interest. The CC( P ; Q )framework is general, i.e., it applies to ground as well asexcited states, but since this work deals with the calcu-lations that aim at recovering the ground-state CCSDTand CCSDTQ energetics, in the description below we fo-cus on the ground-state CC( P ; Q ) theory.According to the formal CC( P ; Q ) prescription, theground-state energy of a N -electron system is determinedin two steps. In the initial, iterative, CC( P ) step, wesolve the CC equations in the subspace H ( P ) of the N -electron Hilbert space H . We assume that subspace H ( P ) , which we also call the P space, is spanned by theexcited determinants | Φ K i = E K | Φ i that together withthe reference determinant | Φ i provide the leading contri-butions to the target ground state | Ψ i ( E K designates theusual elementary particle-hole excitation operator gener-ating | Φ K i from | Φ i ). In other words, we approximatethe cluster operator T in Eq. (1) by T ( P ) = X | Φ K i∈ H ( P ) t K E K (3)and solve the usual system of CC equations, M K ( P ) = 0 , | Φ K i ∈ H ( P ) , (4)where M K ( P ) = h Φ K | ¯ H ( P ) | Φ i (5)are the generalized moments of the P -space CCequations and¯ H ( P ) = e − T ( P ) He T ( P ) = ( He T ( P ) ) C (6)is the relevant similarity-transformed Hamiltonian, forthe cluster amplitudes t K (subscript C in Eq. (6) desig-nates the connected operator product). Once the clusteroperator T ( P ) and the ground-state energy E ( P ) = h Φ | ¯ H ( P ) | Φ i (7)that corresponds to it are determined, we proceed tothe second step of CC( P ; Q ) considerations, which is thecalculation of the noniterative correction δ ( P ; Q ) to theCC( P ) energy E ( P ) that accounts for the many-electroncorrelation effects captured by another subspace of the N -electron Hilbert space H , designated as H ( Q ) andcalled the Q space, which satisfies the condition H ( Q ) ⊆ ( H (0) ⊕ H ( P ) ) ⊥ , where H (0) is a one-dimensional sub-space of H spanned by the reference determinant | Φ i .The formula for the δ ( P ; Q ) correction is δ ( P ; Q ) = X | Φ K i∈ H ( Q ) rank( | Φ K i ) ≤ min( N ( P )0 , Ξ ( Q ) ) ℓ K ( P ) M K ( P ) , (8)where integer N ( P ) defines the highest many-bodyrank of the excited determinants | Φ K i relative to | Φ i (rank( | Φ K i )) for which moments M K ( P ), Eq. (5), arestill non-zero and Ξ ( Q ) is the highest many-body rank of the excited determinant(s) | Φ K i included in H ( Q ) .In practical CC( P ; Q ) calculations, including those dis-cussed in Section III, the ℓ K ( P ) coefficients entering Eq.(8) are calculated as ℓ K ( P ) = h Φ | (1 + Λ ( P ) ) ¯ H ( P ) | Φ K i /D K ( P ) , (9)where is the unit operator,Λ ( P ) = X | Φ K i∈ H ( P ) λ K ( E K ) † (10)is the hole-particle deexcitation operator defining the brastate h ˜Ψ ( P ) | = h Φ | (1 + Λ ( P ) ) e − T ( P ) corresponding to theCC( P ) ket state | Ψ ( P ) i = e T ( P ) | Φ i , and D K ( P ) = E ( P ) − h Φ K | ¯ H ( P ) | Φ K i . (11)One determines Λ ( P ) , or the amplitudes λ K that defineit, by solving the linear system of equations representingthe left eigenstate CC problem in the P space, i.e., h Φ | ( + Λ ( P ) ) ¯ H ( P ) | Φ K i = E ( P ) λ K , | Φ K i ∈ H ( P ) , (12)where E ( P ) is the previously determined CC( P ) energy.Once the noniterative correction δ ( P ; Q ) is determined,the CC( P ; Q ) energy is obtained as E ( P + Q ) = E ( P ) + δ ( P ; Q ) . (13)In practice, we often distinguish between the completeversion of the CC( P ; Q ) theory, designated, followingRefs. 58 and 62, as CC( P ; Q ) EN , which uses the Epstein–Nesbet-like denominator D K ( P ), Eq. (11), in calculat-ing the ℓ K ( P ) amplitudes, and the approximate versionof CC( P ; Q ), abbreviated as CC( P ; Q ) MP , which relies onthe Møller–Plesset form of D K ( P ) obtained by replacing¯ H ( P ) in Eq. (11) by the bare Fock operator (cf., e.g.,Refs. 58, 62, and 63). Both of these variants of theCC( P ; Q ) formalism are considered in this study.We must now come up with the appropriate choicesof the P and Q spaces entering the CC( P ; Q ) consider-ations that would allow us to match the quality of thehigh-level CC computations of the CCSDT, CCSDTQ,and similar type at the small fraction of the cost. Asis often the case in the CC work, one could start fromthe conventional choices, where the P space H ( P ) isspanned by all excited | Φ a ...a n i ...i n i determinants with theexcitation rank n ≤ m A , where i , i , . . . ( a , a , . . . ) des-ignate the spin-orbitals occupied (unoccupied) in | Φ i ,and the Q space H ( Q ) by those with m A < n ≤ m B ,where m B ≤ N . In that case, one ends up with thewell-stablished CR-CC( m A , m B ) hierarchy, including the aforementioned CR-CC(2,3) approxima-tion, where m A = 2 and m B = 3, and the relatedCCSD(2) T88 (cf., also, Refs. 89–91), CCSD(T) Λ , Λ-CCSD(T), and similar schemes that al-low one to correct the CCSD energies for triples. TheCR-CC(2,3) method is useful, improving, for example,poor performance of CCSD(T) in covalent bond breakingsituations and in certain classes of nonco-valent interactions without a substantial increase ofthe computational effort, but neither CR-CC(2,3) nor itsCCSD(2) T , CCSD(T) Λ , and Λ-CCSD(T) counterparts(which are all approximations to CR-CC(2,3)) are freefrom drawbacks. One of the main problems with CR-CC(2,3), CCSD(2) T , Λ-CCSD(T), and other nonitera-tive corrections to CCSD is the fact that, in analogyto CCSD(T), they decouple the higher-order T n compo-nents with n > m A , such as T or T and T , from theirlower-order n ≤ m A (e.g., T and T ) counterparts. Thiscan result in substantial errors, for example when theactivation energies and chemical reaction profiles involv-ing rearrangements of π bonds and singlet–triplet gapsin certain classes of biradical species are examined. The automerization of cyclobutadiene, which is one ofthe benchmark examples in Section III, provides an il-lustration of the challenges the noniterative correctionsto CCSD, including CCSD(2) T and CR-CC(2,3), facewhen the coupling of the lower-order T and T andhigher-order T clusters becomes significant (see Ref. 61for further analysis and additional remarks). One canaddress problems of this type by using active orbitalsto incorporate the dominant higher–than–doubly exciteddeterminants, in addition to all singles and doubles, inthe P space, as in the successful CC(t;3), CC(t,q;3),and CC(t,q;3,4) hierarchy, which uses theCC( P ; Q ) framework to correct the results of the active-space CCSDt or CCSDtq calcu-lations for the remaining T or T and T correlationsthat were not captured via active orbitals, but the result-ing methods are no longer computational black boxes.The semi-stochastic CC( P ; Q ) methodology, introducedin Ref. 58, extended to excited states in Ref. 60, andfurther developed in this work, which takes advantage ofthe FCIQMC or truncated CIQMC/CCMC propagationsthat can identify the leading higher–than–doubly-exciteddeterminants for the inclusion in the P space, while us-ing the noniterative δ ( P ; Q ) corrections to capture theremaining correlations of interest, offers an automatedway of performing CC( P ; Q ) computations without anyreference to the user- and system-dependent active or-bitals. The semi-stochastic CC( P ; Q ) methods developedand tested in this study are discussed next. B. Semi-stochastic CC( P ; Q ) Approaches Using FCIQMCand its Truncated CISDT-MC and CISDTQ-MCCounterparts In our original examination of the semi-stochasticCC( P ; Q ) framework and its recent extension to ex-cited states, where we focused on converging the fullCCSDT and EOMCCSDT energetics, we demonstratedthat the FCIQMC and CCSDT-MC approaches are capa-ble of generating meaningful P spaces for the subsequentCC( P )/EOMCC( P ) iterations, which precede the deter- mination of the δ ( P ; Q ) moment corrections, already inthe early stages of the respective QMC propagations.The main objective of this work is to explore if the sameremains true when FCIQMC is replaced by its less ex-pensive truncated CISDT-MC and CISDTQ-MC coun-terparts, in which spawning beyond the triply excited(CISDT-MC) or quadruply excited (CISDTQ-MC) de-terminants is disallowed, and if one can use the CIQMC-driven CC( P ; Q ) calculations to converge the higher-levelCCSDTQ energetics with similar efficiency.The key steps of the semi-stochastic CC( P ; Q ) algo-rithm exploited in this study, which allows us to convergethe CCSDT and CCSDTQ energetics using the P spacesextracted from the FCIQMC and truncated CISDT-MCand CISDTQ-MC propagations, are as follows:1. Initiate a CIQMC run appropriate for the CCmethod of interest by placing a certain number ofwalkers on the reference state | Φ i , which in all ofthe calculations reported in this article is the re-stricted Hartree-Fock (RHF) determinant. Amongthe CIQMC schemes that can provide meaningful P spaces for the CC( P ; Q ) calculations targetingthe CCSDT energetics are the FCIQMC approachused in our earlier work and the CISDT-MCand CISDTQ-MC methods examined in the presentstudy. If the objective is to converge the CCS-DTQ energetics, one can use FCIQMC or CISDTQ-MC, which are the two choices pursued in thepresent work, but not CISDT-MC, which ignoresquadruply excited determinants. As in our earliersemi-stochastic CC( P )/EOMCC( P ) and CC( P ; Q )work, all of the calculations reported in thisarticle adopt the initiator CIQMC ( i -CIQMC) al-gorithm, originally proposed in Ref. 65, based oninteger walker numbers.2. After a certain number of CIQMC time steps, calledMC iterations, i.e., after some QMC propagationtime τ , extract a list of higher–than–doubly ex-cited determinants relevant to the CC theory of in-terest to construct the P space for executing theCC( P ) calculations. If one is interested in tar-geting the CCSDT-level energetics, the P spaceused in the CC( P ) iterations consists of all singlyand doubly excited determinants and a subset oftriply excited determinants identified by the un-derlying FCIQMC, CISDT-MC, or CISDTQ-MCpropagation, where each triply excited determinantin the subset is populated by at least n P positiveor negative walkers. In analogy to our previousCC( P )/EOMCC( P ) and CC( P ; Q ) studies, allof the CC( P ) and CC( P ; Q ) computations carriedout in this work use n P = 1. If the goal is to con-verge the CCSDTQ energetics, the P space for theCC( P ) computations is defined as all singly anddoubly excited determinants and a subset of triplyand quadruply excited determinants identified bythe underlying FCIQMC or CISDTQ-MC propaga-tion, where, again, each triply and quadruply ex-cited determinant in the subset is populated by aminimum of n P positive or negative walkers.3. Solve the CC( P ) and left-eigenstate CC( P ) equa-tions, Eqs. (4) and (12), respectively, where E ( P ) is given by Eq. (7), for the cluster operator T ( P ) and the deexcitation operator Λ ( P ) in the P spacedetermined in step 2. If the objective is to convergethe CCSDT-level energetics, we define T ( P ) = T + T + T (MC)3 and Λ ( P ) = Λ + Λ + Λ (MC)3 , where T (MC)3 and Λ (MC)3 are the three-body components of T ( P ) and Λ ( P ) , respectively, defined using the list oftriples identified by the FCIQMC, CISDT-MC, orCISDTQ-MC propagation at time τ , as describedin step 2. If one is targeting the CCSDTQ-levelenergetics, T ( P ) = T + T + T (MC)3 + T (MC)4 andΛ ( P ) = Λ + Λ + Λ (MC)3 + Λ (MC)4 , where T (MC)3 andΛ (MC)3 are the three-body and T (MC)4 and Λ (MC)4 four-body components of T ( P ) and Λ ( P ) , respec-tively, defined using the lists of triples and quadru-ples identified by the FCIQMC or CISDTQ-MCpropagation at time τ .4. Use the CC( P ; Q ) correction δ ( P ; Q ), Eq. (8), tocorrect the energy E ( P ) obtained in step 3 for theremaining correlation effects of interest, meaningthose correlations that were not captured by theCC( P ) calculations performed at the time τ the listof higher–than–doubly excited determinants enter-ing the relevant P space was created. If the ob-jective is to converge the CCSDT-level energetics,the Q space entering the definition of δ ( P ; Q ) con-sists of those triply excited determinants that inthe FCIQMC, CISDT-MC, or CISDTQ-MC prop-agation at time τ are populated by less than n P positive or negative walkers (in this study, where n P = 1, the triply excited determinants that werenot captured by the FCIQMC, CISDT-MC, orCISDTQ-MC propagation at time τ ). If the goalis to recover the CCSDTQ-level energetics, the Q space used to calculate δ ( P ; Q ) consists of the triplyand quadruply excited determinants that in theFCIQMC or CISDTQ-MC propagation at time τ are populated by less than n P positive or negativewalkers.5. Check the convergence of the CC( P ; Q ) energy E ( P + Q ) , Eq. (13), obtained in step 4, by repeatingsteps 2–4 at some later CIQMC propagation time τ ′ > τ . If the resulting energy E ( P + Q ) no longerchanges within a given convergence threshold, theCC( P ; Q ) calculation can be stopped. As pointedout in Refs. 58–60, one can also stop it once thefraction (fractions) of higher–than–doubly exciteddeterminants captured by the CIQMC propagationrelevant to the target CC theory level, included inthe P space, is (are) sufficiently large to obtain the desired accuracy. This is further discussed in Sec-tion III, where the numerical results obtained inthis study are presented.The above semi-stochastic CC( P ; Q ) algorithm, al-lowing us to recover the CCSDT and CCSDTQ ener-getics using the P spaces identified with the help ofFCIQMC or truncated CISDT-MC and CISDTQ-MCpropagations, has been implemented by modifying ourpreviously developed standalone deterministic CC( P ; Q )codes, which rely on the RHF, restricted open-shell Hartree-Fock, and integral routines in the GAMESSpackage, such that they could handle the stochas-tically determined lists of triples and quadruples, andby interfacing the resulting program with the i -CIQMCroutines available in the HANDE software. Asin our earlier semi-stochastic CC( P )/EOMCC( P ) andCC( P ; Q ) work, we rely on the original form of theinitiator CIQMC ( i -CIQMC) algorithm proposed in Ref.65, where only those determinants that acquire walkerpopulation exceeding a preset value n a are allowed to at-tempt spawning new walkers onto empty determinants,but one could consider interfacing our CC( P ; Q ) frame-work with the improved ways of converging CIQMC, suchas the adaptive-shift method developed in Refs. 67 and114. We will consider such an interface in the future.In the case of the semi-stochastic CC( P ; Q ) codesaimed at converging the CCSDT energetics, which wehave extended in the present study by allowing them towork with the CISDT-MC and CISDTQ-MC approaches,in addition to the previously examined FCIQMC and CCSDT-MC options, we follow the algorithmsummarized in steps 1–5 without any alterations. Inparticular, all of the quantities entering Eq. (8) forthe noniterative correction δ ( P ; Q ) are treated in thepresent study fully. This is an improvement comparedto our original semi-stochastic CC( P ; Q ) computationsutilizing FCIQMC and CCSDT-MC, reported in Ref.58, where we adopted an approximation in which thethree-body component Λ (MC)3 of the deexcitation opera-tor Λ ( P ) used to determine amplitudes ℓ K ( P ) enteringEq. (8) was neglected. In analogy to this work, thesimilarity-transformed Hamiltonian ¯ H ( P ) , defining mo-ments M K ( P ) and entering the linear system defined byEq. (12), which is used to determine Λ ( P ) , was treatedin Ref. 58 fully, i.e., ¯ H ( P ) employed in the CC( P ; Q )calculations aimed at recovering the CCSDT energeticswas defined as ( He T + T + T (MC)3 ) C , so that the one- andtwo-body components of Λ ( P ) employed in Ref. 58 wereproperly relaxed in the presence of the three-body com-ponent T (MC)3 of the cluster operator T ( P ) obtained in thepreceding CC( P ) calculations, but Λ (MC)3 was neglected.Although all of our numerical tests to date indicate thatthis approximation has a small effect on the results ofthe semi-stochastic CC( P ; Q ) calculations utilizing fulland truncated CIQMC and no effect on our main conclu-sions, we no longer use it in this work. In other words,all of the calculations reported in the present study relyon the complete representations of ¯ H ( P ) and Λ ( P ) whenconstructing moments M K ( P ) and amplitudes ℓ K ( P ) en-tering Eq. (8). This means that ¯ H ( P ) and Λ ( P ) used todetermine the CC( P ; Q ) correction δ ( P ; Q ) in the calcu-lations aimed at the CCSDT energetics are defined as( He T + T + T (MC)3 ) C and Λ + Λ + Λ (MC)3 , respectively.We have, however, introduced an approximation in thesemi-stochastic CC( P ; Q ) routines that are used to con-verge the CCSDTQ-level energetics. Given the pilot na-ture of these routines, the noniterative correction δ ( P ; Q )that they produce corrects the E ( P ) energy, which is ob-tained in this case by solving the CC( P ) equations inthe space of all singles and doubles and subsets of triplesand quadruples captured by FCIQMC or CISDTQ-MC,for the remaining triples not included in the P space,but the quadruples contributions to δ ( P ; Q ) are ignored.This approximation is acceptable, since in the τ = ∞ limit, where the P space contains all triples and quadru-ples, i.e., the corresponding Q space is empty, the uncor-rected CC( P ) and partially or fully corrected CC( P ; Q )calculations recover the CCSDTQ energetics. All of ourtests to date, including those discussed in Section III,indicate that the convergence of the CC( P ; Q ) compu-tations, in which the quadruples component of δ ( P ; Q )is ignored, toward CCSDTQ is rapid, even when the T effects become significant, so the above approximationdoes not seem to have a major effect on the convergencerate, but we will implement the full correction δ ( P ; Q )due to the missing triples as well as quadruples in the fu-ture to examine if one can accelerate convergence towardCCSDTQ even further.As explained in Refs. 58 and 60 (cf., also, Ref. 59), thesemi-stochastic CC( P ; Q ) approaches of the type summa-rized above offer a number of advantages. Among themare substantial savings in the computational effort com-pared to the parent high-level CC theories they targetand a systematic behavior of the resulting E ( P + Q ) ener-gies as τ approaches ∞ . The latter feature is a directconsequence of the fact that if we follow the definitionsof the P and Q spaces introduced in steps 2 and 4 above,the initial, τ = 0, CC( P ; Q ) energies are identical tothose obtained with CR-CC(2,3) or CR-CC(2,4), whichare approximations to CCSDT and CCSDTQ, respec-tively, that account for some T (CR-CC(2,3)) or T and T (CR-CC(2,4)) correlations. In the τ = ∞ limit, theCC( P ; Q ) energies E ( P + Q ) become equivalent to their re-spective high-level CC parents, which account for the T n components with n >
2, such as T or T and T , fully, sothat the QMC propagation time τ becomes a parameterconnecting CR-CC(2,3) with CCSDT and CR-CC(2,4)with CCSDTQ. In the case of our current implementa-tion of the semi-stochastic CC( P ; Q ) approach aimed atconverging the CCSDTQ energetics, where the quadru-ples contributions to correction δ ( P ; Q ) are ignored, theinitial, τ = 0, CC( P ; Q ) energy is equivalent to that ob-tained with the CR-CC(2,3) approach, i.e., the QMCpropagation time τ connects CR-CC(2,3) with CCSDTQ.When τ approaches ∞ , the uncorrected CC( P ) energies E ( P ) converge to their CCSDT and CCSDTQ parentsas well, but the convergence toward CCSDT and CCS-DTQ is in this case slower, since the CC( P ) energies at τ = 0 are equivalent to those of CCSD, which has noinformation about the T n components with n >
2, and,as shown in our earlier work, and as clearly demon-strated in the present study, the CC( P ; Q ) corrections δ ( P ; Q ) greatly accelerate the convergence toward thetarget CC energetics. The above relationships betweenthe semi-stochastic CC( P ) and CC( P ; Q ) approaches andthe deterministic CCSD, CR-CC(2,3)/CR-CC(2,4), andCCSDT/CCSDTQ theories are also helpful when debug-ging the CC( P ) and CC( P ; Q ) codes.As far as the savings in the computational effort of-fered by the semi-stochastic CC( P ; Q ) methods, whencompared to their high-level CC parents, such as CCSDTor CCSDTQ, are concerned, they were already discussedin Refs. 58 and 60, so here we focus on the informa-tion relevant to the calculations discussed in Section III.There are three main factors that contribute to these sav-ings. First, the computational times associated with theearly stages of the CIQMC walker propagations, whichare sufficient to recover the parent CCSDT or CCSDTQenergetics to within small fractions of a millihartree whenthe semi-stochastic CC( P ; Q ) framework is employed, arevery short compared to the converged CIQMC runs.They are already short when one uses FCIQMC, andthey are even shorter when one replaces FCIQMC bythe CISDT-MC and CISDTQ-MC truncations.Second, the CC( P ) calculations using small fractionsof higher–than–doubly excited determinants, which ishow the P spaces used in these calculations look likewhen the early stages of the CIQMC walker propaga-tions are considered, are much faster than the parent CCcomputations. For example, when the most expensive h Φ abcijk | [ H, T ] | Φ i or h Φ abcijk | [ ¯ H (2) , T ] | Φ i contributions tothe CCSDT equations, where ¯ H (2) = e − T − T He T + T ,are isolated and implemented using programming meth-ods similar to those exploited in selected CI algorithms(rather than the usual diagrammatic techniques that as-sume continuous excitation manifolds labeled by all oc-cupied and all unoccupied orbitals), one can acceleratetheir determination by a factor of up to ( D/d ) , where D is the number of all triples and d is the number oftriples included in the P space, captured with the helpof CIQMC propagations. Other contributions to theCCSDT equations that involve T or the projections onthe triply excited determinants, such as h Φ abij | [ H, T ] | Φ i and h Φ abcijk | [ H, T ] | Φ i , may offer additional speedups, onthe order of ( D/d ). Our current CC( P ) codes are still inthe pilot stages, but the speedups on the order of ( D/d ) inthe determination of the most expensive h Φ abcijk | [ H, T ] | Φ i (or h Φ abcijk | [ ¯ H (2) , T ] | Φ i ) terms are attainable. Similar re-marks apply to the CC( P )/CC( P ; Q ) calculations aimedat converging the CCSDTQ energetics, where one canconsiderably speed up the determination of the most ex-pensive h Φ abcdijkl | [ H, T ] | Φ i or h Φ abcdijkl | [ ¯ H (2) , T ] | Φ i contri-butions and other terms containing the T and T clus-ters and the projections on the triply and quadruply ex-cited determinants. It should also be noted that theCC( P ) calculations do not require storing the entire T and T vectors. The T (MC)3 and T (MC)4 operators usemuch smaller numbers of amplitudes than their full T and T counterparts.Third, the computation of the noniterative correction δ ( P ; Q ) is much less expensive than a single iteration ofthe target CC calculation. In the case of the CC( P ; Q )calculations aimed at converging the CCSDT energet-ics, the computational time required to determine thecorresponding correction δ ( P ; Q ) scales no worse than ∼ n o n u , which is much less than the n o n u scaling ofeach iteration of CCSDT. In the case of the CC( P ; Q )approach aimed at CCSDTQ, the computational time re-quired to determine correction δ ( P ; Q ) scales as ∼ n o n u in the case of the contributions due to the remain-ing triples and is on the order of n o n u in the case ofthe quadruples part of δ ( P ; Q ), when the more com-plete CC( P ; Q ) EN approach is used, or n o n u , when theCC( P ; Q ) MP form of δ ( P ; Q ) is employed. This is allmuch less than the n o n u scaling of every CCSDTQ itera-tion. As mentioned above, in our current implementationof the semi-stochastic CC( P ; Q ) approach aimed at con-verging the CCSDTQ energetics, the quadruples contri-bution to correction δ ( P ; Q ) is neglected, so the computa-tional time required to obtain δ ( P ; Q ) scales as ∼ n o n u ,at worst, which points to the usefulness of such an ap-proximation, especially that the convergence of the re-sulting CC( P ; Q ) energies toward CCSDTQ is, as shownin Section III, very fast. III. NUMERICAL EXAMPLES
In order to demonstrate the benefits offered by thesemi-stochastic CC( P ; Q ) framework, especially the newCC( P ; Q ) approaches implemented in this work that re-place FCIQMC by the less expensive CISDT-MC andCISDTQ-MC propagations, we applied the FCIQMC-,CISDT-MC-, and CISDTQ-MC-driven CC( P ; Q ) meth-ods aimed at converging the CCSDT and CCSDTQ en-ergetics to a few molecular problems, for which the par-ent full CCSDT and CCSDTQ results had previouslybeen determined or were not too difficult to be recal-culated. Thus, we carried out an extensive series ofthe CISDT-MC- and CISDTQ-MC-driven CC( P ; Q ) cal-culations, along with the analogous computations usingFCIQMC, which was utilized in our earlier study, toexamine the ability of the semi-stochastic CC( P ; Q ) ap-proaches using various types of CIQMC to recover theCCSDT energetics for the F–F bond dissociation in thefluorine molecule (Section III A) and the automerizationof cyclobutadiene (Section III B). In order to illustratethe performance of the FCIQMC- and CISDTQ-MC-driven CC( P ; Q ) methods in calculations aimed at con-verging the CCSDTQ energetics, we considered the sym- metric stretching of the O–H bonds in the water molecule(Section III C). We chose bond breaking in F , which isaccurately described by full CCSDT, sincewe examined the same system in our original FCIQMC-and CCSDT-MC-driven CC( P ; Q ) work and in thepreceding deterministic CC( P ; Q )-based CC(t;3) calcula-tions reported in Ref. 57. Our choice of the automeriza-tion of cyclobutadiene, which is accurately described byCCSDT as well, was motivated by similar reasons.We studied this problem, where all noniterative triplescorrections to CCSD, including CCSD(T), Λ-CCSD(T),CCSD(2) T , and CR-CC(2,3) fail, using the de-terministic CC(t;3) approach exploiting the CC( P ; Q )ideas in Ref. 61, and we studied it again using the semi-stochastic CC( P ; Q ) framework utilizing FCIQMC andCCSDT-MC in Ref. 58. We would like to explore nowwhat the effect of replacing FCIQMC propagations bytheir less expensive CISDT-MC and CISDTQ-MC coun-terparts on the convergence of the CC( P ; Q ) energies to-ward CCSDT is. We would also like to learn if the incor-poration of the previously neglected three-body com-ponent of the deexcitation operator Λ ( P ) , which is usedto construct amplitudes ℓ K ( P ) entering Eq. (8), helpsthe accuracy of the resulting semi-stochastic CC( P ; Q )energies. We studied the C v -symmetric double disso-ciation of H O, since by simultaneously stretching bothO–H bonds by factors exceeding 2, one ends up with acatastrophic failure of CCSDT.
One needs an ac-curate description of the T and T clusters to obtain amore reliable description of the water potential energysurface in that region. Following our earlier semi-stochastic and deterministicCC( P ; Q ) work, which also provides the parentCCSDT and CCSDTQ energetics, we used thecc-pVDZ, cc-pVTZ, and aug-cc-pVTZ basis setsfor F and the cc-pVDZ bases for cyclobutadiene and wa-ter. For consistency with Refs. 57, 58, and 61, in all ofthe post-RHF computations for the F–F bond breakingin F and the automerization of cyclobutadiene, the coreelectrons corresponding to the 1s shells of the fluorineand carbon atoms were kept frozen. As in Refs. 63 and118, which provide the reference CCSDTQ data and, inthe case of Ref. 118, the geometries of the equilibriumand stretched water molecule used in our semi-stochasticCC( P ; Q ) calculations aimed at converging the CCSDTQenergetics, we correlated all electrons. Each of the rele-vant i -FCIQMC (all systems), i -CISDT-MC (F and cy-clobutadiene), and i -CISDTQ-MC (all systems) runs wasinitiated by placing 100 walkers on the RHF referencedeterminant and we set the initiator parameter n a at 3.All of the i -FCIQMC, i -CISDT-MC, and i -CISDTQ-MCpropagations used the time step δτ of 0.0001 a.u. A. Bond Breaking in F We begin our discussion of the semi-stochasticCC( P ; Q ) calculations carried out in this study with theF–F bond dissociation in the fluorine molecule, as de-scribed by the cc-pVDZ basis set using the Cartesiancomponents of d orbitals (see Table I and Figs. 1–3). In analogy to Ref. 58, where our initial FCIQMC-and CCSDT-MC-based CC( P ; Q ) results for F were pre-sented, we considered the equilibrium geometry R e =2 . R , including R = 1 . R e ,2 R e , and 5 R e , which are characterized by the increas-ingly large nondynamical correlations. The increasinglyimportant role of nondynamical correlation effects as theF–F bond is stretched is reflected in the magnitude of T contributions, defined by forming the difference ofthe CCSDT and CCSD energies, which grows, in abso-lute value, from 9.485 millihartree at R = R e to 32.424,45.638, and 49.816 millihartree at R = 1 . R e , 2 R e , and5 R e , respectively, when the cc-pVDZ basis set is em-ployed. The T effects in the R = 2 R e − R e region are solarge that they exceed the depth of the CCSDT potentialwell, estimated at about 44 millihartree when the differ-ence between the CCSDT energies at R = 5 R e , whereF is essentially dissociated, and R = R e is considered.They grow with R so fast that the popular perturbativeCCSD(T) correction to CCSD fails at larger F–F separa-tions, producing the − . − . − .
348 mil-lihartree errors relative to CCSDT at R = 1 . R e , 2 R e ,and 5 R e , respectively, misrepresenting the physics of T correlations in the stretched F molecule.The triples corrections to CCSD that rely on thebiorthogonal moment expansions of the CC( P ; Q ) type,including CR-CC(2,3), work much better than CCSD(T).This is especially true when the most complete variant ofthe CR-CC(2,3) approach using the Epstein–Nesbet formof the D K ( P ) denominator in determining the ℓ K ( P )amplitudes that enter the corresponding triples correc-tion to CCSD, abbreviated sometimes as CR-CC(2,3),Dor CR-CC(2,3) D62,63,78,80,83 and represented in Table Iby the τ = 0 CC( P ; Q ) EN results, is considered. Indeed,the CR-CC(2,3) D calculations reduce large errors in theCCSD(T) energies at R = 1 . R e , 2 R e , and 5 R e to 1.735,1.862, and 1.613 millihartree, respectively, improving theCCSD(2) T or the equivalent CR-CC(2,3),Aor CR-CC(2,3) A calculations, which adopt the Møller–Plesset D K ( P ) denominators, at the same time (see the τ = 0 CC( P ; Q ) MP values in Table I)). The CR-CC(2,3) D approach eliminates the failure of CCSD(T) at stretchednuclear geometries, while being more effective in cap-turing the physics of T correlations than CCSD(2) T ,but the only way to obtain further improvements to-ward CCSDT is by incorporating at least some triplesin the iterative part of the calculations, relaxing the T and T amplitudes, which in CCSD(T), CCSD(2) T ,and CR-CC(2,3) are fixed at their CCSD values, in thepresence of the leading T contributions, and correct-ing the resulting energies for the remaining T effectsaccordingly. One can do this deterministically by turn-ing to the previously mentioned CC(t;3) method, which uses the CC( P ; Q ) formalism to correct the energies ob-tained in the active-space CCSDt calculations for theremaining T correlation effects that the CCSDt ap-proach did not capture, or by the approximation toCC(t;3) that replaces the CC( P ; Q ) triples correction toCCSDt by its perturbative CCSD(T) analog, abbrevi-ated as CCSD(T)-h. Alternatively, one can resortto the semi-stochastic CC( P ; Q ) framework advocated inthis work, in which the same goal is accomplished byusing full or truncated CIQMC propagations to identifythe leading triply excited determinants for the inclusionin the underlying P space without having to use activeorbitals.The semi-stochastic CC( P ; Q ) results and the under-lying CC( P ) energies shown in Table I and Figs. 1–3confirm the above expectations. Indeed, with only about30–40 % of the triples in the P space, captured after therelatively short FCIQMC, CISDT-MC, and CISDTQ-MC runs at R = R e and 1 . R e , and even less than that( ∼ R = 2 R e and 5 R e geometries areconsidered, the errors in the uncorrected CC( P ) energiesrelative to their CCSDT parents are already on the orderof 1 millihartree or smaller. This is a massive error reduc-tion compared to the initial, τ = 0, CC( P ), i.e., CCSDenergy values, especially at the larger F–F separations,where the differences between the CCSD and CCSDTenergies are as high as 45.638 millihartree at R = 2 R e or 49.816 millihartree at R = 5 R e . The CC( P ; Q ) correc-tions based on Eq. (8) accelerate the convergence towardCCSDT even further, allowing one to reach the submil-lihartree accuracy levels relative to the parent CCSDTenergetics almost instantaneously, out of the early stagesof the FCIQMC, CISDT-MC, and CISDTQ-MC propa-gations, when no more than 10 % of all triples are in-cluded in the corresponding P spaces. The CC( P ; Q ) EN correction, which adopts the Epstein–Nesbet form of the D K ( P ) denominator in determining the ℓ K ( P ) ampli-tudes entering Eq. (8), is particularly effective in thisregard. With less than 10 % triples in the stochasti-cally determined P spaces, captured after 20,000 or fewer δτ = 0 . P ; Q ) EN energies and their CCSDT parentsare on the order of 0.1 millihartree, being usually evensmaller. This is not only true at the equilibrium geome-try, but also at the larger values of R , including R = 5 R e ,where the F–F bond in F is already de facto broken.When we perform somewhat longer FCIQMC, CISDT-MC, and CISDTQ-MC propagations, allowing them tocapture about 40–50 % of the triples in the P space, when R = R e and 1 . R e , and 20–30 % when R ≥ R e , theCC( P ; Q ) EN calculations recover the CCSDT energeticsto within 10 or so microhartree. The CC( P ; Q ) MP correc-tion, in which the Epstein–Nesbet D K ( P ) denominator,Eq. (11), in the definition of ℓ K ( P ) amplitudes enteringEq. (8) is replaced by its simplified Møller–Plesset form,is not as accurate as CC( P ; Q ) EN , but it still acceleratesthe convergence of the underlying CC( P ) energies, allow-ing one to recover the parent CCSDT energies to within ∼ R = R e and 1 . R e )or 15–20 % ( R = 2 R e , and 5 R e ) of the triples are cap-tured by the FCIQMC, CISDT-MC, and CISDTQ-MCpropagations.The results shown in Table I and Figs. 1–3 demon-strate that it is practically irrelevant whether one usesFCIQMC or one of its less expensive truncated forms,such as CISDT-MC and CISDTQ-MC examined in thisstudy, to identify the leading triply excited determinantsfor the inclusion in the P space used in the CC( P ; Q )and the underlying CC( P ) calculations. Clearly, as τ ap-proaches ∞ , the FCIQMC, CISDT-MC, and CISDTQ-MC propagations converge to completely different limits(FCI in the case of FCIQMC, CISDT-MC in the caseof CISDT-MC, and CISDTQ in the case of CISDTQ-MC), but this has virtually no impact on the conver-gence patterns observed in our semi-stochastic CC( P )and CC( P ; Q ) calculations. This is a consequence of thefact that the uncorrected CC( P ) and corrected CC( P ; Q )computations are capable of recovering the parent high-level CC energetics, such as those corresponding to fullCCSDT discussed in this subsection, based on the in-formation extracted from the early stages of the corre-sponding CIQMC runs. In particular, if we are targetingCCSDT, all we need from the CIQMC calculations isa meaningful list of the leading triply excited determi-nants, which any CIQMC calculation that is allowed tosample the triples subspace of the Hilbert space, eventhe crude CISDT-MC approach, can provide. One cansee, for example, in Table I that the fractions of triplescaptured by the FCIQMC, CISDT-MC, and CISDTQ-MC runs at the various numbers of MC iterations (var-ious propagation times τ ) are very similar. Detailed in-spection of the corresponding lists of triply excited de-terminants shows that while the numbers of walkers onthe individual determinants may substantially differ, thelists of triples identified by the FCIQMC, CISDT-MC,and CISDTQ-MC propagations, especially the more im-portant ones that result in larger T (MC)3 amplitudes inthe subsequent deterministic CC( P ) steps, are not muchdifferent. Once the lists of the leading triples are iden-tified, we turn to the CC( P ) computations, correctingthem for the remaining triples not captured by CIQMC,and this makes the semi-stochastic CC( P ) and CC( P ; Q )calculations rather insensitive to the type of the CIQMCapproach used to construct these lists.All of the above observations regarding the abilityof the semi-stochastic CC( P ; Q ) calculations using theFCIQMC, CISDT-MC, and CISDTQ-MC propagationsto rapidly converge the full CCSDT energetics remaintrue when the cc-pVDZ basis set is replaced by its largercc-pVTZ and aug-cc-pVTZ counterparts (both using thespherical components of d and f functions). This is il-lustrated in Table II, where we examine the stretchedF molecule, in which the F–F distance R is set at 2 R e .We chose R = 2 R e , since, in analogy to the previously discussed cc-pVDZ basis set, the T effects at this ge-ometry, obtained by calculating differences of the re-spective CCSDT and CCSD energies, which are − . − .
036 millihartree, when the aug-cc-pVTZ ba-sis is used, are not only very large, but also larger, inabsolute value, than the corresponding CCSDT disso-ciation energies (differences between the CCSDT ener-gies at R = 5 R e , where the F–F bond is broken, and R = R e obtained with the cc-pVTZ and aug-cc-pVTZbasis sets are about 57 and 60 millihartree, respectively).We also chose it, since the R = 2 R e stretch of theF–F bond length is large enough for the conventionalCCSD(T) approach to fail in a major way when the cc-pVTZ and aug-cc-pVTZ basis sets are employed, result-ing in the − .
354 and − .
209 millihartree errors rela-tive to CCSDT, respectively. The CCSD(2) T correctionto CCSD or the equivalent CR-CC(2,3) A approximation,represented in Table II by the τ = 0 CC( P ; Q ) MP results,helps, but large differences between the CCSD(2) T andCCSDT energies, of 9.211 millihartree in the cc-pVTZcase and 9.808 millihartree when the aug-cc-pVTZ basisset is employed, remain. The CR-CC(2,3) D approach,represented in Table II by the τ = 0 CC( P ; Q ) EN data,is more effective than other triples corrections to CCSD,reducing the large errors relative to CCSDT observed inthe CCSD(T) and CCSD(2) T calculations to 4.254 (cc-pVTZ) and 5.595 (aug-cc-pVTZ) millihartree, but noneof the above results are as good as the energies result-ing from the semi-stochastic CC( P ; Q ) calculations usingFCIQMC, CISDT-MC, and CISDTQ-MC.Indeed, as shown in Table II, we observe a rapid er-ror reduction relative to the parent CCSDT data oncewe start migrating the triply excited determinants iden-tified during the FCIQMC, CISDT-MC, and CISDTQ-MC propagations into the underlying P space. Withabout 20–30 % (cc-pVTZ) or 30–40 % (aug-cc-pVTZ)of the triples in the P space, the 62.819 and 65.036 mil-lihartree errors resulting from the initial CCSD ( τ = 0CC( P )) computations decrease to a 1–2 millihartree levelwhen the CC( P ) method is employed. The CC( P ; Q )corrections due to the remaining triples not captured byFCIQMC, CISDT-MC, and CISDTQ-MC accelerate theconvergence toward CCSDT even further, with the semi-stochastic CC( P ; Q ) EN approach being particularly effi-cient in this regard. With only 2–4 % of the triples inthe stochastically determined P spaces, captured after20,000–30,000 δτ = 0 . P ; Q ) EN calcula-tions recover the full CCSDT energetics corresponding tothe cc-pVTZ and aug-cc-pVTZ basis sets to within 0.1–0.2 millihartree. After 50,000 (cc-pVTZ) or 60,000 (aug-cc-pVTZ) MC iterations, where the FCIQMC, CISDT-MC, and CISDTQ-MC runs are still far from conver-gence, capturing only about 20–30 % (cc-pVTZ) or 30–40 % (aug-cc-pVTZ) of the triples, the errors in theCC( P ; Q ) EN energies relative to CCSDT reduce to a 100microhartree level. In analogy to the cc-pVDZ basisset, the CC( P ; Q ) MP correction is less accurate than itsCC( P ; Q ) EN counterpart when the cc-pVTZ and aug-cc-pVTZ basis sets are employed, recovering the CCSDT en-ergetics to within 0.1–0.2 millihartree after 50,000 ratherthan 20,000–30,000 MC iterations, i.e., after about 20–30 % rather than 2–4 % of the triples are captured bythe CIQMC propagations, but the overall error reduc-tion compared to the underlying CC( P ) calculations orthe various noniterative triples corrections to CCSD isstill impressive.Similarly to the cc-pVDZ basis set, the semi-stochasticCC( P ; Q ) calculations using larger cc-pVTZ and aug-cc-pVTZ bases are rather insensitive to the type of theCIQMC approach used to identify the leading triplesfor the inclusion in the P space. Based on the re-sults in Table II, one might try to argue that the en-ergies obtained with the uncorrected CC( P ) approachusing the CISDT-MC propagations are characterized byslower convergence compared to their CISDTQ-MC- andFCIQMC-driven counterparts, but this would be mis-leading, since CISDT-MC captures the leading triples ata somewhat slower rate, while being less expensive thanCISDTQ-MC and FCIQMC at the same time. For ex-ample, the CISDT-MC-driven CC( P ) computations forF at R = 2 R e using the cc-pVTZ basis set need 60,000 δτ = 0 . ∼ P ) approach using CISDTQ-MC and FCIQMCreaches the same accuracy level sooner, after 50,000 MCiterations. One should keep in mind, however, that ittakes 60,000 MC time steps for the CISDT-MC propa-gation to capture about 30 % of the triples, needed toreach a ∼ P ) calculations, and the analogous CISDTQ-MC andFCIQMC runs capture a similar fraction of the triples af-ter 50,000 time steps. Ultimately, one needs to rememberthat all CIQMC-driven CC( P ) computations consideredin this subsection converge to CCSDT as τ → ∞ , inde-pendent of the type of the CIQMC approach used to de-fine the underlying P spaces, as long as the CIQMC prop-agation is allowed to spawn walkers on the triply exciteddeterminants. Perhaps more importantly, the CC( P ; Q )corrections to the CC( P ) energies make the convergencetoward CCSDT not only much faster, but also less de-pendent on the type of the CIQMC approach used in thecalculations, since they take care of the triples that werenot captured by the respective QMC propagations.Before discussing our next molecular example, it isworth pointing out that the FCIQMC-driven CC( P ; Q )calculations reported in Tables I and II and Fig. 1, inwhich, as explained in Section II B, we used completerepresentations of ¯ H ( P ) and Λ ( P ) in determining correc-tions δ ( P ; Q ), approach the parent CCSDT energetics ofthe stretched F system in the early stages of the un-derlying FCIQMC propagations faster than the analo-gous calculations reported in Ref. 58, where the three-body component of Λ ( P ) was neglected. For example, the CC( P ; Q ) energies of F at R = 2 R e using the aug-cc-pVTZ basis set obtained in this work after 10,000,20,000, and 30,000 δτ = 0 . P ; Q ) energies shown in Tables Iand II and Figs. 1–3, determined by treating the deexci-tation operator Λ ( P ) in Eq. (9) fully, i.e., by defining Λ ( P ) as Λ + Λ + Λ (MC)3 , with their FCIQMC- and CCSDT-MC-based counterparts obtained in Ref. 58, where Λ ( P ) was approximated by Λ + Λ , we can conclude that aslong as Λ (MC)3 is not neglected one can replace FCIQMCby CISDTQ-MC or, even, CISDT-MC and still improvethe rate of convergence of the CC( P ; Q ) energies towardCCSDT in the early stages of the QMC propagationscompared to that reported in Ref. 58.The above observations, combined with the superiorperformance of the CC( P ; Q ) EN approach compared to itsCC( P ; Q ) MP counterpart, suggest that a complete treat-ment of correction δ ( P ; Q ), as dictated by Eqs. (8), (9),and (11), is more important, especially when one is inter-ested in accelerating convergence of the semi-stochasticCC( P ; Q ) calculations for stretched or more multirefer-ence molecules in the early stages of the QMC propa-gations, than the actual type of the underlying CIQMCapproach. It is interesting to examine if the same remainstrue when other molecular examples, including those dis-cussed in the next two subsections, are considered. B. Automerization of Cyclobutadiene
Our next example is the challenging and frequentlystudied automerization of cyclobutadiene(see Fig. 4). In this case, in order to obtain reliableenergetics using computational means, especially the ac-tivation energy, one has to provide an accurate and well-balanced description of the nondegenerate closed-shell re-actant (or the equivalent product) species, in which themany-electron correlation effects have a predominantlydynamical character, and the quasi-degenerate, birad-icaloid transition state characterized by substantial non-dynamical correlations. Experiment suggests that theactivation energy for the automerization of cyclobuta-diene is somewhere between 1.6 and 10 kcal/mol.
The most accurate single- and multi-reference calcu-lations performed to date, reviewed, for example, inRefs. 61, 117, and 140, imply that the purely elec-tronic value of the energy barrier falls into the 6–10kcal/mol range. In particular, as pointed out in Ref.61 (cf., also, Ref. 116), one can obtain a reliable de-scription of the activation energy using the full CCSDTapproach. Given this information and the methodolog-1ical nature of the present study, in which we had toperform a large number of semi-stochastic CC( P ) andCC( P ; Q ) calculations, exploring three different types ofthe CIQMC method, including FCIQMC, CISDT-MC,and CISDTQ-MC, and probing many values of the QMCpropagation time τ , in a discussion below we focus onconverging the CCSDT energetics obtained using thespherical cc-pVDZ basis set. As shown in Ref. 61and Table III, the CCSDT/cc-pVDZ activation energycharacterizing the automerization of cyclobutadiene, as-suming the reactant/product and transition-state geome-tries obtained with the multireference average-quadraticCC (MR-AQCC) approach in Ref. 134, whichwe adopt in the CC( P ) and CC( P ; Q ) calculations re-ported in this work as well, is 7.627 kcal/mol, in rea-sonable agreement with the most accurate ab initio re-sults reported to date. The results of our semi-stochasticCC( P ) and CC( P ; Q ) calculations, aimed at recover-ing the CCSDT/cc-pVDZ energetics of the reactant andtransition-state species and the corresponding activationenergy using the FCIQMC, CISDT-MC, and CISDTQ-MC propagations to identify the leading triply exciteddeterminants for constructing the underlying P spaces,are summarized in Table III and Fig. 5.As already mentioned, all of the noniterative triplescorrections to CCSD, including CCSD(T), Λ-CCSD(T),CCSD(2) T , and CR-CC(2,3), perform very poorly in thiscase, producing activation barriers in a 16–17 kcal/molrange when the cc-pVDZ basis set is considered, in-stead of ∼ τ = 0 CC( P ) barrier in Table III), but theimprovements offered by the noniterative triples correc-tions to CCSD are far from sufficient. This, in particu-lar, applies to the CCSD(2) T = CR-CC(2 , A and CR-CC(2,3) D approaches, represented in Table III by the τ = 0 CC( P ; Q ) MP and CC( P ; Q ) EN data, respectively,where errors in the resulting activation energies relativeto CCSDT are 9.611 kcal/mol (126 %) in the formercase and 8.653 kcal/mol (113 %) in the case of the lat-ter method. As explained in Ref. 61, the poor perfor-mance of the noniterative triples corrections to CCSD indescribing the automerization of cyclobutadiene is a con-sequence of neglecting the coupling between the T clus-ters and their lower-order T and T counterparts, whichis accounted for in CCSDT, but ignored in methods suchas CCSD(T), Λ-CCSD(T), CCSD(2) T , and CR-CC(2,3).This coupling is particularly large at the transition-stategeometry, where the magnitude of T contributions, de-fined as the absolute value of the difference between theCCSDT and CCSD energies, is nearly 48 millihartree,when the cc-pVDZ basis set is employed, and where er-rors in the CCSD(T), Λ-CCSD(T), CCSD(2) T , and CR-CC(2,3) energies relative to CCSDT range from about 14 to 20 millihartree, as opposed to ∼ , if we want to cap-ture the coupling of the T , T , and T clusters withouthaving to solve full CCSDT equations, while preservingthe idea of noniterative triples corrections to energies ob-tained in lower-order CC calculations, we must solve forthe T and T amplitudes, which in the CCSD(T), Λ-CCSD(T), CCSD(2) T , and CR-CC(2,3) approaches areobtained with CCSD, in the presence of the dominant T components by incorporating some triples in the itera-tive CC steps, and then correct the resulting energies forthe remaining T effects neglected in the CC iterations.Again, this can be done deterministically by solving theactive-space CCSDt equations, in which the dominant T amplitudes are selected using active orbitals, and cor-recting the CCSDt energies for the remaining T corre-lations using the CC( P ; Q ) corrections δ ( P ; Q ), as in theCC(t;3) calculations reported in Ref. 61, or by turning tothe semi-stochastic form of the CC( P ; Q ) formalism pur-sued in this study, which eliminates the need for definingactive orbitals, when identifying the leading triples, byresorting to CIQMC propagations. Interestingly, usingthe CCSD(T)-type correction to CCSDt, as in the afore-mentioned CCSD(T)-h approach, in the calculations forthe automerization of cyclobutadiene worsens the activa-tion energies obtained with CCSDt, moving them awayfrom their parent CCSDT values. This underlines thesignificance of treating corrections δ ( P ; Q ) due to the cor-relation effects outside the underlying P spaces as com-pletely as possible, following Eqs. (8), (9), and (11),avoiding drastic approximations in these equations thatlead to the triples corrections of CCSD(T).As shown in Table III and Fig. 5, the semi-stochasticCC( P ; Q ) calculations using FCIQMC, CISDT-MC, andCISDTQ-MC are remarkably efficient in capturing thedesired T correlation effects. Independent of the typeof the CIQMC approach, they allow us to convergethe CCSDT values of the transition-state and activa-tion energies, which are poorly described by the nonit-erative triples corrections to CCSD, to within 1–2 milli-hartree or 1–2 kcal/mol out of the early stages of theCIQMC propagations, while further improving an ac-curate description of the reactant by methods such asCR-CC(2,3) D . Similarly to F , the performance of theCC( P ; Q ) EN approach, which uses the Epstein–Nesbetform of the D K ( P ) denominator in calculating the ℓ K ( P )amplitudes entering Eq. (8), is particularly impressive.With just 5–6 % of the triples in the stochastically de-termined P spaces, captured by the FCIQMC, CISDT-MC, and CISDTQ-MC propagations after 30,000 δτ =0 . P ; Q ) EN approach reduces the initial 0.848 mil-lihartree, 14.636 millihartree, and 8.653 kcal/mol errorsin the reactant, transition-state, and activation energiesrelative to CCSDT obtained in the τ = 0 CC( P ; Q ) EN or CR-CC(2,3) D calculations by factors of 2–4, to 0.228–0.279 millihartree, 3.648–6.651 millihartree, and 2.146–23.999 kcal/mol, respectively. After the additional 10,000MC time steps, which result in capturing 12–16 % ofthe triples in the underlying P spaces, errors in theCC( P ; Q ) EN reactant, transition-state, and activation en-ergies relative to their CCSDT values become 0.080–0.164 millihartree, 1.556–3.367 millihartree, and 0.919–2.011 kcal/mol, respectively. These are remarkable im-provements compared to the initial CR-CC(2,3) D val-ues, especially if we realize that the early stages of theFCIQMC, CISDT-MC, and CISDTQ-MC calculations,such as 30,000–40,000 δτ = 0 . P ) calculations and significant reductionsin the T amplitude storage requirements. After 50,000MC iterations, where, as shown in Fig. 5, the FCIQMC,CISDT-MC, and CISDTQ-MC runs are still far from con-vergence, capturing about 20–30 % of the triples, i.e., stillrelatively small fractions of all triply excited determi-nants, the CC( P ; Q ) EN calculations recover the CCSDTvalues of the reactant, transition-state, and activationenergies to within 22–57 microhartree, 0.243–0.602 mil-lihartree, and 0.138–0.343 kcal/mol, respectively, whichis a massive error reduction compared to CR-CC(2,3) D and other noniterative triples corrections to CCSD. Asin the case of bond breaking in F , the CC( P ; Q ) MP cor-rection, which uses the Møller–Plesset D K ( P ) denomi-nator in Eq. (9) instead of its more elaborate Epstein–Nesbet form given by Eq. (11), is less accurate thanits CC( P ; Q ) EN counterpart, but its ability to acceler-ate convergence of the underlying CC( P ) energies andimproving the results obtained with CR-CC(2,3) andother triples corrections to CCSD is still quite impres-sive. For example, with about 20–30 % of the triplescaptured by the FCIQMC, CISDT-MC, and CISDTQ-MC propagations after 50,000 MC iterations, the differ-ences between the CC( P ; Q ) MP reactant, transition-state,and activation energies and their CCSDT counterparts,of 0.877–1.235 millihartree, 1.488–2.238 millihartree, and0.361–0.629 kcal/mol, are much smaller than the analo-gous errors relative to CCSDT resulting from the cor-responding CC( P ) calculations, which are 6.895–9.202millihartree, 9.727–12.495 millihartree, and 1.601–2.067kcal/mol, respectively, although they are not as small asthe aforementioned 22–57 microhartree, 0.243–0.602 mil-lihartree, and 0.138–0.343 kcal/mol errors obtained usingthe CC( P ; Q ) EN correction.In analogy to the fluorine molecule, the semi-stochasticCC( P ; Q ) calculations aimed at converging the CCSDTresults for the automerization of cyclobutadiene are gen-erally insensitive to the type of the CIQMC approachused to identify the leading triples for the inclusion in theunderlying P spaces. It is sufficient to resort to the leastexpensive forms of the CIQMC propagations capable ofcapturing the triples, such as CISDT-MC or CISDTQ-MC, to obtain the fast convergence of the CC( P ; Q ) re-actant, transition-state, and activation energies towardtheir CCSDT parents observed in Table III and Fig. 5. Treating the CC( P ; Q ) correction δ ( P ; Q ) fully, follow-ing Eqs. (8), (9), and (11), is, however, important. Wehave already discussed the benefits of using the Epstein–Nesbet form of the D K ( P ) denominator, Eq. (11), indetermining the ℓ K ( P ) amplitudes entering Eq. (8). Acomplete treatment of the deexcitation operator Λ ( P ) inEq. (9), which in the case of the triples corrections to theCC( P ) energies considered here means representing it asΛ + Λ + Λ (MC)3 , is important too. One can consideran approximation in which the three-body componentΛ (MC)3 is neglected, which is what we did in Ref. 58,but it is generally better, especially in the earlier stagesof the CIQMC propagations, to keep all of the relevantmany-body components of Λ ( P ) in calculating the ℓ K ( P )amplitudes that enter the CC( P ; Q ) correction δ ( P ; Q ).This can be illustrated by comparing the results of theFCIQMC-driven CC( P ; Q ) computations shown in TableIII, where we used a complete representation of Λ ( P ) ,in which the three-body component Λ (MC)3 was included,with the analogous results reported in Ref. 58, whereΛ (MC)3 was neglected. For example, the differences be-tween the CC( P ; Q ) EN reactant, transition-state, and ac-tivation energies and their CCSDT counterparts obtainedin this work after 40,000 δτ = 0 . τ becomes longer, differentways of handling the Λ ( P ) operator or different ways ofdefining the D K ( P ) denominator in Eq. (9) become lessimportant, but if we are interested in accurately approxi-mating the parent CC energetics in the early stages of theunderlying CIQMC propagations, treating these quanti-ties fully is essential.As shown in this subsection and Section III A, usingcomplete representations of the Λ ( P ) and ¯ H ( P ) opera-tors and the Epstein–Nesbet-type denominators D K ( P )in determining corrections δ ( P ; Q ) benefits the semi-stochastic CC( P ; Q ) calculations aimed at converging theCCSDT energetics. In Section III C, which is the finalpart of our discussion of the numerical results obtained inthis work, we investigate if similar applies to the CIQMC-driven CC( P ; Q ) computations targeting CCSDTQ. C. Double Dissociation of H O Our last example, which illustrates the ability of thesemi-stochastic CC( P ) and CC( P ; Q ) approaches to con-verge the CCSDTQ energetics, is the C v -symmetric cutof the ground-state potential energy surface of the watermolecule, in which both O–H bonds are simultaneouslystretched without changing the ∠ (H–O–H) angle, result-ing in large T and T contributions. Following Ref. 118,and consistent with our earlier deterministic CC( P ; Q )3study, where we also obtained the reference CCSDTQenergies, we used the spherical cc-pVDZ basis set, corre-lated all electrons, and considered four stretches of the O–H bonds, including R O - H = 1 . R e , 2 R e , 2 . R e , and 3 R e ,in addition to the equilibrium geometry, R O - H = R e . Weused the same geometries, which the reader can find inRef. 118, in the semi-stochastic CC( P ) and CC( P ; Q )calculations for H O carried out in this work, summa-rized in Table IV and Fig. 6. The authors of Ref. 118obtained the CCSDTQ energies too, but we rely on ourown CCSDTQ data, published in Ref. 63 and recalcu-lated in this study, since Ref. 118 does not provide theCCSDTQ results for R O - H = 2 . R e and 3 R e and theCCSDTQ energies for R O - H = 1 . R e and 2 R e reportedin Ref. 118 are in slight disagreement with the correctlyconverged values.Up to twice the equilibrium O–H bond lengths, theCCSDT approach provides an accurate description ofthe electronic energies of water, resulting in the 0.493,1.423, and − .
405 millihartree signed errors relativeto FCI at R O - H = R e , 1 . R e , and 2 R e , respectively,when the cc-pVDZ basis set is employed, but when R O - H > R e , CCSDT completely fails, and theCCSD(T), CCSD(2) T or CR-CC(2,3) A (in Table IV, τ = 0 CC( P ; Q ) MP ), CR-CC(2,3) D (in Table IV, τ = 0CC( P ; Q ) EN ), CCSDt, and CC(t;3) approximations toCCSDT, which were examined in Refs. 63, 75, 88, and118, fail with it (CCSD(T) fails already at R O - H = 2 R e ).In particular, the difference between the CCSDT and FCIenergies obtained with the cc-pVDZ basis set at R O - H =2 . R e is − .
752 millihartree. At R O - H = 3 R e , the situa-tion becomes even more dramatic, with the CCSDT/cc-pVDZ energy falling 40.126 millihartree below its FCIcounterpart. One needs to incorporate T clustersto reduce these massive errors in the R O - H > R e re-gion, and in order to do it in a reliable manner onehas to use full CCSDTQ or one of the robust approx-imations to it, such as the CCSDtq, CC(t,q;3), andCC(t,q;3,4) methods tested in Ref. 63. The conventional T plus T corrections to CCSD, such as CCSD(TQ f ), or their CCSD(2) TQ88,90 and CR-CC(2,4) coun-terparts examined in Refs. 63 and 88 do not suffice. TheCCSDT(2) Q quadruples correction to CCSDT is notrobust enough either. When the cc-pVDZ basis set is employed, the differ-ences between the CCSDTQ and FCI energies at R O - H = R e , 1 . R e , 2 R e , 2 . R e , and 3 R e are 0.019, 0.121, 0.030, − . − .
733 millihartree, respectively, whichis a huge improvement over CCSDT. One might arguethe need for the inclusion of T n clusters with n > R O - H = 2 . R e and 3 R e , or try to obtain further im-provements in describing the R O - H > R e region by re-placing the RHF reference determinants used throughoutthis work by their unrestricted counterparts, but studiesof this kind are outside the scope of this article. Thegoal of the calculations for the water molecule discussedin this subsection is to explore the potential offered bythe semi-stochastic CC( P ) and CC( P ; Q ) approaches, es- pecially the CC( P ; Q ) corrections to the CC( P ) energiescalculated with the help of the FCIQMC and CISDTQ-MC propagations, in converging the CCSDTQ energeticsobtained with the spin- and symmetry-adapted RHF ref-erences.As shown in Table IV and Fig. 6, the semi-stochasticCC( P ; Q ) calculations using FCIQMC and CISDTQ-MCare extremely efficient in capturing the combined effectsof T and T correlations, even in the most challenging R O - H > R e region, where the T contributions, whichhave to overcome the massive failures of the CCSDTapproach, are very large and difficult to balance withtheir T counterparts. They accurately reproduce theparent CCSDTQ energetics already in the early stagesof the FCIQMC and CISDTQ-MC propagations, greatlyaccelerating convergence of the underlying CC( P ) calcu-lations, in spite of the fact that in our current implemen-tation of the semi-stochastic CC( P ; Q ) routines aimedat CCSDTQ the noniterative correction δ ( P ; Q ) correctsthe energy obtained by solving the CC( P ) equations inthe space of all singles and doubles and subsets of triplesand quadruples captured by FCIQMC or CISDTQ-MCfor the triples outside the stochastically determined P space, but not for the quadruples missed by CIQMC.Similarly to the previously discussed CC( P ; Q ) calcu-lations aimed at CCSDT, the CC( P ; Q ) approach tar-geting CCSDTQ that adopts the CC( P ; Q ) EN correc-tion is generally most effective, although the resultsof the CC( P ; Q ) MP calculations, in which the Epstein–Nesbet denominator D K ( P ) in Eq. (9) is replaced by itsMøller–Plesset form, are as accurate as their CC( P ; Q ) EN counterparts in the quasi-degenerate R O - H > R e re-gion. Indeed, when we look at the results in TableIV corresponding to R O - H = 2 . R e and 3 R e , wherethe T effects, estimated by forming the differences ofthe CCSDTQ and CCSDT energies, exceed 22 and 35millihartree, respectively, and where the differencesbetween the CCSDT and CCSD energies, which mea-sure the magnitude of T contributions, are about − −
51 millihartree, respectively, the FCIQMC-and CISDTQ-MC-based CC( P ; Q ) EN computations re-duce the large − .
739 ( R O - H = 2 . R e ) and − . R O - H = 3 R e ) millihartree errors relative to CCSDTQobtained in the initial CR-CC(2,3) D ( τ = 0 CC( P ; Q ) EN )calculations to fractions of a millihartree after only 20,000 δτ = 0 . P spaces. The FCIQMC- and CISDTQ-MC-drivenCC( P ; Q ) MP calculations using the same QMC propaga-tion time τ are similarly effective though. They reducethe large − .
469 and − .
302 millihartree errors rela-tive to CCSDTQ resulting from the initial CCSD(2) T orCR-CC(2,3) A ( τ = 0 CC( P ; Q ) MP ) computations to asubmillihartree level too.The situation changes in the R O - H = R e − R e re-gion, where the T effects are much smaller than thoseoriginating from the T clusters. In this case, the con-4vergence of the energies obtained in the semi-stochasticCC( P ; Q ) MP calculations toward CCSDTQ is slower thanthat obtained with the CC( P ; Q ) EN approach, i.e., ourearlier conclusion, drawn from the calculations discussedin Sections III A and III B and Ref. 58, that the use ofthe CC( P ; Q ) EN corrections to the semi-stochastic CC( P )energies is generally most effective still stands. This be-comes particularly clear when we compare the results ofthe FCIQMC- and CISDTQ-MC-driven CC( P ; Q ) MP andCC( P ; Q ) EN calculations at R O - H = R e and 1 . R e . Forexample, it takes only 40,000 δτ = 0 . P space, for the CC( P ; Q ) EN ap-proach to reach a 0.1 millihartree accuracy level relativeto CCSDTQ at R O - H = R e . The CC( P ; Q ) MP calcu-lations reach the same accuracy level after 100,000 MCtime steps that capture about 35 % of the triples and 10% of the quadruples. When the R O - H = 1 . R e geom-etry is considered, the CC( P ; Q ) EN calculations reach a0.1 millihartree accuracy level relative to CCSDTQ af-ter 60,000–70,000 MC iterations that capture about 30% of the triples and 6–9 % of the quadruples, i.e., inthe relatively early stages of the FCIQMC and CISDTQ-MC propagations. The CC( P ; Q ) MP calculations reach asimilar accuracy level 20,000–30,000 MC iterations later,after capturing about 40 % of the triples and more than10 % of the quadruples. It is certainly reassuring that theCC( P ; Q ) EN calculations using FCIQMC and CISDTQ-MC to identify the leading triply and quadruply exciteddeterminants for the inclusion in the underlying P spacesare capable of reproducing the CCSDTQ energies of thewater molecule over a wide range of geometries along the C v -symmetric cut of the ground-state potential energysurface considered in Table IV and Fig. 6 to within ∼ R O - H = R e ) or30 % ( R O - H > R e ) of the triples and 2 % ( R O - H = R e )or about 10 % ( R O - H > R e ) of the quadruples. Havingsaid this, it is interesting to observe that both types ofthe CC( P ; Q ) corrections tested in this study, abbrevi-ated as CC( P ; Q ) MP and CC( P ; Q ) EN , perform equallywell when R O - H > R e , i.e., when the T and T effectsare both very large. We observed a similar behavior inRef. 63, when examining the relative performance of theCC( P ; Q )-based CC(t,q;3) A and CC(t,q;3) D correctionsto CCSDtq using the double dissociation of water as oneof the examples. This should not be surprising, sincethe CC(t,q;3) A and CC(t,q;3) D methods investigated inRef. 63 can be regarded as the deterministic counterpartsof the semi-stochastic CC( P ; Q ) MP and CC( P ; Q ) EN ap-proaches targeting the CCSDTQ energetics implementedin this work.As in the case of the CC( P ; Q ) calculations targetingCCSDT, discussed in Sections III A and III B, the ob-served fast convergence of the semi-stochastic CC( P ; Q )calculations aimed at recovering the CCSDTQ energet-ics does not seem to be affected by the type of theCIQMC approach used to identify the leading triply and quadruply excited determinants. This should facili-tate future applications of the semi-stochastic CC( P ; Q )methodology, including cases of stronger electronic quasi-degeneracies characterized by large T and T contribu-tions, helping us to converge the CCSDTQ-level energet-ics at the small fraction of the deterministic CCSDTQeffort by taking advantage of the least expensive formsCIQMC capable of capturing triples and quadruples, rep-resented in this study by CISDTQ-MC. IV. CONCLUSIONS
We have recently started exploring a novel way of ob-taining accurate electronic energetics equivalent to high-level CC calculations, at the small fraction of the com-putational effort and preserving the black-box characterof conventional single-reference computations, by merg-ing the deterministic CC( P ; Q ) formalism, originally pro-posed in Refs. 57 and 61, along with the underly-ing CC( P )/EOMCC( P ) framework, with the stochas-tic CIQMC and CCMC approaches. Whencombined with the FCIQMC and CCSDT-MC wavefunction sampling, used to identify the leading triplyexcited determinants or cluster/excitation amplitudes,and correcting the CC( P ) and EOMCC( P ) ener-gies for the remaining triples not captured by FCIQMCor CCSDT-MC, the resulting semi-stochastic CC( P ; Q )methodology and its excited-state extension turnedout to be very promising, allowing us to converge theCCSDT and EOMCCSDT energetics out of the earlystages of the underlying QMC propagations.This study can be regarded as the next key step inthe development and exploration of the semi-stochasticCC( P ; Q ) approaches, in which we have extended our ini-tial work, focusing on recovering the CCSDT energeticsand relying on FCIQMC and CCSDT-MC, to more effi-cient ways of identifying the leading higher–than–doublyexcited determinants for the inclusion in the underly-ing P spaces. We have accomplished this goal by re-placing FCIQMC by its less expensive CISDT-MC andCISDTQ-MC counterparts. We have also developed andtested the initial variant of the semi-stochastic CC( P ; Q )method aimed at converging the CCSDTQ energetics, inwhich the results of CC( P ) calculations in the subspacesspanned by singles, doubles, and subsets of triples andquadruples identified by FCIQMC or CISDTQ-MC arecorrected for the remaining triples outside the stochasti-cally determined P spaces. By comparing the FCIQMC-driven CC( P ; Q ) calculations targeting CCSDT, carriedout in this work, in which the noniterative corrections δ ( P ; Q ) to the CC( P ) energies have been treated fully,as required by Eqs. (8), (9), and (11), with the analo-gous computations reported in Ref. 58, where the samecorrections were treated in a somewhat simplified man-ner by neglecting the three-body component of the de-excitation operator Λ ( P ) used to construct amplitudes ℓ K ( P ) entering Eq. (8), we have examined the sig-5nificance of the full vs approximate treatment of thesecorrections for the accuracy of the resulting CC( P ; Q )energies. Other important issues, such as the benefitsof using the Epstein–Nesbet form of the denominators D K ( P ) that enter the definition of corrections δ ( P ; Q ),resulting in the CC( P ; Q ) EN variant of CC( P ; Q ), as com-pared to their Møller–Plesset counterparts defining theCC( P ; Q ) MP corrections, have been investigated as well.The ability of the semi-stochastic CC( P ; Q ) approachesto converge the CCSDT and CCSDTQ energies, basedon the truncated CISDT-MC and CISDTQ-MC propa-gations, and their FCIQMC counterparts in which thenoniterative corrections δ ( P ; Q ) have been treated fully,has been illustrated using a few molecular examples, forwhich the deterministic CCSDT and CCSDTQ calcula-tions that provide the reference data are feasible andwhich require a high-level CC treatment to obtain areliable description. Thus, we have reported the re-sults of the semi-stochastic CC( P ; Q ) calculations usingCISDT-MC, CISDTQ-MC, and FCIQMC aimed at con-verging the CCSDT energetics for the F–F bond breakingin F and the automerization of cyclobutadiene, whichrequire an accurate treatment of T clusters account-ing for the relaxation of T and T amplitudes in thepresence of large T contributions, and the CISDTQ-MC- and FCIQMC-driven CC( P ; Q ) computations forthe C v -symmetric stretching of the O–H bonds in thewater molecule targeting CCSDTQ, where the T and T clusters become large and difficult to balance.The numerical results reported in this article clearlyshow that the semi-stochastic CC( P ; Q ) calculations arecapable of accurately reproducing the parent CCSDTand CCSDTQ energetics, even when electronic quasi-degeneracies and higher–than–two-body components ofthe cluster operator become large, out of the early stagesof the corresponding CIQMC propagations, acceleratingconvergence of the underlying CC( P ) computations atthe same time. The convergence of the CC( P ; Q ) ener-gies toward their CCSDT and CCSDTQ parents does notseem to be affected by the type of the CIQMC approachused to identify the leading triply or triply and quadruplyexcited determinants. In the case of the CC( P ; Q ) cal-culations targeting the CCSDT energetics, one can useFCIQMC or one of its less expensive truncated forms,such as CISDTQ-MC, or even the crude CISDT-MC ap-proach, with virtually no impact on the systematic con-vergence pattern toward CCSDT as the propagation time τ approaches ∞ . Similarly, one can replace FCIQMCby CISDTQ-MC without any significant effect on theconvergence of the semi-stochastic CC( P ; Q ) calculationstoward CCSDTQ. Our calculations also suggest that acomplete treatment of the CC( P ; Q ) corrections δ ( P ; Q ),as defined by Eqs. (8), (9), and (11), including the useof the CC( P ; Q ) EN approach, as opposed to its more ap-proximate CC( P ; Q ) MP version, is more important thanthe actual type of the CIQMC approach used to deter-mine the relevant P spaces, especially when one is inter-ested in accelerating convergence of the semi-stochastic CC( P ; Q ) calculations in the early stages of the QMCpropagations. We have demonstrated that independentof the type of the CIQMC approach used to identify theleading triply or triply and quadruply excited determi-nants for the inclusion in the relevant P spaces and inde-pendent of the magnitude of T and T effects, the semi-stochastic CC( P ; Q ) calculations allow us to reach sub-millihartree accuracy levels relative to the parent CCSDTand CCSDTQ energetics with small fractions of higher–than–doubly excited determinants captured in the earlystages of the corresponding CIQMC runs.By relaxing T and T clusters in the presence oftheir T or T and T counterparts defined using theexcitation lists provided by full or truncated CIQMC,the semi-stochastic CC( P ; Q ) computations are capableof considerably improving accuracy of the more estab-lished noniterative corrections to CCSD without mak-ing the calculations a lot more expensive. In thissense, the semi-stochastic CC( P ; Q ) methodology usingCIQMC is very similar to the deterministic CC(t;3),CC(t,q;3), and CC(t,q;3,4) hierarchy developed andtested in Refs. 57, 61–63, 82, and 100, which uses theCC( P ; Q ) corrections to correct the results of the active-space CCSDt or CCSDtq calculations for the remain-ing T or T and T correlations that were not cap-tured via active orbitals. There is, however, one ma-jor advantage of the semi-stochastic CC( P ; Q ) frame-work over the CC(t;3), CC(t,q;3), and CC(t,q;3,4) ap-proaches, namely, the use of FCIQMC or truncatedCIQMC propagations, which can efficiently identify theleading higher–than–doubly-excited determinants for theinclusion in the relevant P spaces, combined with the δ ( P ; Q ) corrections to capture the remaining correla-tions of interest, offers an automated way of perform-ing accurate CC( P ; Q ) computations without any refer-ence to the user- and system-dependent active orbitals.The analogies between the active-space CCSDt (for ex-cited states, EOMCCSDt ) and semi-stochasticCC( P )/EOMCC( P ) approaches, on which the deter-ministic CC(t;3) (in the case of CCSDt/EOMCCSDt)and CIQMC-driven (in the case of semi-stochasticCC( P )/EOMCC( P )) CC( P ; Q ) approaches are based,have been investigated in Ref. 60.The findings presented in this article are encourag-ing from the point of view of future applications of thesemi-stochastic CC( P ; Q ) methodology using CIQMC, in-cluding challenging cases of stronger electronic quasi-degeneracies characterized by large T or T and T con-tributions that other approximations to CCSDT or CCS-DTQ may struggle with, but the story is not over yet. Wecertainly need to improve the efficiency of our CC( P ; Q )codes, especially the underlying CC( P ) routines, to ob-tain full benefits offered by the semi-stochastic CC( P ; Q )approaches, discussed in Section II B. This is especiallytrue in the case of our current CC( P ; Q ) codes aimedat converging the CCSDTQ energetics, which have alargely pilot character. In this case, we also need toexamine if one can further improve the convergence of6the FCIQMC- or CISDTQ-MC-driven CC( P ; Q ) calcu-lations aimed at CCSDTQ by correcting the underlyingCC( P ) energies for both the missing triples and quadru-ples not captured by CIQMC at a given time τ , not justfor the missing triples, as has been done in this work.It would also be useful to examine if one can extendthe semi-stochastic CC( P ) and CC( P ; Q ) approaches tothe higher CC theory levels, beyond CCSDTQ exam-ined in this work and beyond EOMCCSDT explored inRefs. 59 and 60, and investigate if our observations re-garding the utility of the truncated CIQMC methods,such as CISDT-MC and CISDTQ-MC, remain true inthe excited-state and open-shell CC( P ; Q ) calculations.In this study, we have adopted the original form of the i -CIQMC algorithm proposed in Ref. 65, but it would beinteresting to examine if one could obtain additional ben-efits by interfacing our semi-stochastic CC( P ; Q ) methodswith the improved ways of converging CIQMC, such asthe adaptive-shift approach developed Refs. 67 and 114.All of the above ideas are presently pursued in our group,and the results will be reported as soon as they becomeavailable. Last, but not least, we have recently inter-faced our CC( P ) and CC( P ; Q ) routines with some ofthe modern versions of the selected CI approaches, whichdate back to the late 1960s and early 1970s andwhich have recently regained significant attention. Our initial numerical results, which we hope to reportin a separate publication, indicate that selected CImethods can be as effective in generating meaningful P spaces for the CC( P ) calculations, which precede the de-termination of the δ ( P ; Q ) moment corrections, as thestochastic CIQMC propagations advocated in this andour earlier studies. ACKNOWLEDGMENTS
This work has been supported by the Chemical Sci-ences, Geosciences and Biosciences Division, Office of Ba-sic Energy Sciences, Office of Science, U.S. Departmentof Energy (Grant No. DE-FG02-01ER15228 to P.P), theNational Science Foundation (Grant No. CHE-1763371to P.P.), and Phase I and II Software Fellowships awardedto J.E.D. by the Molecular Sciences Software Institutefunded by the National Science Foundation grant ACI-1547580. P.P. thanks Professors Ali Alavi, George H.Booth, and Alex J. W. Thom for useful discussions.
DATA AVAILABILITY
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CC( P ; Q ) ENa
R/R e MC iterations FCI b CIQ c CIT d FCI b CIQ c CIT d FCI b CIQ c CIT d FCI b CIQ c CIT d . e . f − . g .
692 5 .
692 5 .
229 0 .
760 0 .
760 0 . − . − . − . .
548 3 .
804 3 .
962 0 .
444 0 .
473 0 . − . − . − . .
290 2 .
498 2 .
769 0 .
284 0 .
301 0 . − . − . − . .
791 1 .
523 1 .
765 0 .
212 0 .
184 0 . − . − . − . .
933 0 .
940 1 .
151 0 .
113 0 .
115 0 . − . − . − . .
536 0 .
498 0 .
698 0 .
064 0 .
058 0 . − . − . − . .
383 0 .
308 0 .
410 0 .
044 0 .
036 0 . − . − . − . .
177 0 .
164 0 .
224 0 .
020 0 .
018 0 . − . − . − . .
044 0 .
050 0 .
073 0 .
005 0 .
006 0 .
008 0 . − . − . .
013 0 .
010 0 .
024 0 .
001 0 .
001 0 .
003 0 .
000 0 .
000 0 . ∞ − . h — —1.5 0 0 32 . e . f . g .
312 14 .
220 15 .
874 2 .
198 1 .
980 2 .
115 0 .
351 0 .
321 0 . .
589 3 .
572 5 .
564 0 .
629 0 .
428 0 . − . − .
000 0 . .
728 2 .
391 2 .
206 0 .
323 0 .
285 0 . − .
002 0 .
020 0 . .
065 0 .
706 1 .
387 0 .
142 0 .
084 0 .
171 0 .
020 0 .
009 0 . .
482 0 .
459 0 .
687 0 .
062 0 .
055 0 .
087 0 .
009 0 .
006 0 . .
273 0 .
219 0 .
336 0 .
029 0 .
027 0 .
041 0 .
001 0 .
000 0 . .
128 0 .
106 0 .
231 0 .
013 0 .
011 0 .
028 0 .
000 0 .
000 0 . .
064 0 .
048 0 .
102 0 .
006 0 .
004 0 .
010 0 . − . − . .
012 0 .
009 0 .
026 0 .
001 0 .
001 0 .
003 0 .
000 0 .
000 0 . .
001 0 .
002 0 .
005 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . ∞ − . h — —2.0 0 0 45 . e . f . g .
199 17 .
779 12 .
687 0 .
998 1 .
886 1 . − .
063 0 . − . .
127 2 .
529 3 .
672 0 .
328 0 .
245 0 . − .
014 0 . − . .
802 1 .
172 1 .
393 0 .
081 0 .
115 0 .
128 0 .
008 0 .
011 0 . .
456 0 .
499 0 .
627 0 .
040 0 .
047 0 . − .
001 0 .
000 0 . .
216 0 .
215 0 .
305 0 .
018 0 .
019 0 . − .
001 0 . − . .
083 0 .
112 0 .
160 0 .
007 0 .
010 0 . − . − . − . .
037 0 .
048 0 .
074 0 .
003 0 .
004 0 .
006 0 . − . − . .
013 0 .
019 0 .
034 0 .
001 0 .
002 0 .
003 0 .
000 0 .
000 0 . .
001 0 .
002 0 .
007 0 .
000 0 .
000 0 .
001 0 .
000 0 .
000 0 . .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . ∞ − . h — —5.0 0 0 49 . e . f . g .
887 13 .
326 9 .
776 0 .
455 0 .
672 0 . − .
005 0 .
059 0 . .
968 2 .
535 1 .
315 0 .
152 0 .
165 0 .
102 0 .
040 0 .
026 0 . .
529 0 .
752 1 .
042 0 .
041 0 .
056 0 .
081 0 .
001 0 .
006 0 . .
295 0 .
351 0 .
346 0 .
022 0 .
024 0 .
025 0 . − . − . .
116 0 .
147 0 .
166 0 .
008 0 .
011 0 . − .
001 0 . − . .
047 0 .
059 0 .
070 0 .
003 0 .
004 0 . − .
001 0 . − . .
016 0 .
020 0 .
030 0 .
001 0 .
001 0 .
002 0 .
000 0 .
000 0 . .
006 0 .
006 0 .
014 0 .
000 0 .
000 0 .
001 0 .
000 0 .
000 0 . .
000 0 .
000 0 .
001 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 .
000 0 . ∞ − . h — — a Unless otherwise stated, all energies are reported as errors relative to CCSDT in millihartree. b FCI stands for i -FCIQMC. c CIQ stands for i -CISDTQ-MC. d CIT stands for i -CISDT-MC. e Equivalent to CCSD. f Equivalent to the CCSD energy corrected for the effects of T clusters using the CCSD(2) T approach of Ref. 88, which is equivalentto the approximate form of the completely renormalized CR-CC(2,3) approach of Refs. 75 and 76, abbreviated sometimes asCR-CC(2,3),A or CR-CC(2,3) A . Equivalent to the CCSD energy corrected for the effects of T clusters using the most complete variant of the completely renormalizedCR-CC(2,3) approach of Refs. 75 and 76, abbreviated sometimes as CR-CC(2,3),D or CR-CC(2,3) D . Total CCSDT energy in hartree. TABLE II. Convergence of the CC( P ), CC( P ; Q ) MP , and CC( P ; Q ) EN energies toward CCSDT, where the P spaces consistedof all singles and doubles and subsets of triples identified during the i -FCIQMC, i -CISDTQ-MC, or i -CISDT-MC propagationswith δτ = 0 . Q spaces consisted of the triples not captured by the corresponding QMCsimulations, for the F molecule in which the F–F distance R was set at twice the equilibrium bond length, using the cc-pVTZand aug-cc-pVTZ basis sets, abbreviated as VTZ and AVTZ, respectively. The i -FCIQMC, i -CISDTQ-MC, and i -CISDT-MCcalculations preceding the CC( P ) and CC( P ; Q ) steps were initiated by placing 100 walkers on the RHF determinant and the n a parameter of the initiator algorithm was set at 3. In all post-RHF calculations, the lowest two core orbitals were kept frozenand the spherical components of d and f orbitals were employed throughout. % of triples CC( P ) a CC( P ; Q ) MPa
CC( P ; Q ) ENa
Basis set MC iterations FCI b CIQ c CIT d FCI b CIQ c CIT d FCI b CIQ c CIT d FCI b CIQ c CIT d VTZ 0 0 62 . e . f . g .
714 31 .
973 31 .
571 2 .
738 3 .
104 2 .
636 0 .
728 0 .
896 0 . .
179 14 .
687 20 .
194 0 .
824 1 .
097 1 .
487 0 .
071 0 .
151 0 . .
787 6 .
031 9 .
294 0 .
400 0 .
425 0 .
617 0 .
028 0 .
030 0 . .
406 2 .
574 4 .
203 0 .
160 0 .
171 0 .
284 0 .
002 0 .
001 0 . .
193 1 .
237 2 .
177 0 .
076 0 .
078 0 . − . − . − . .
490 0 .
489 1 .
144 0 .
029 0 .
029 0 . − . − . − . .
178 0 .
171 0 .
576 0 .
011 0 .
010 0 . − . − . − . .
045 0 .
054 0 .
309 0 .
003 0 .
003 0 .
020 0 .
000 0 . − . .
002 0 .
003 0 .
130 0 .
000 0 .
000 0 .
009 0 .
000 0 .
000 0 . ∞ − . h — —AVTZ 0 0 65 . e . f . g .
316 38 .
874 42 .
801 3 .
641 4 .
144 4 .
851 1 .
594 1 .
786 2 . .
190 20 .
799 26 .
557 1 .
276 1 .
656 2 .
288 0 .
382 0 .
512 0 . .
065 9 .
272 13 .
279 0 .
549 0 .
623 0 .
928 0 .
138 0 .
138 0 . .
408 4 .
677 7 .
477 0 .
291 0 .
307 0 .
499 0 .
057 0 .
062 0 . .
208 2 .
425 3 .
951 0 .
136 0 .
150 0 .
244 0 .
016 0 .
019 0 . .
021 1 .
137 2 .
052 0 .
058 0 .
070 0 .
124 0 .
002 0 .
005 0 . .
385 0 .
455 0 .
385 0 .
021 0 .
025 0 .
059 0 .
000 0 .
000 0 . .
125 0 .
154 0 .
125 0 .
007 0 .
008 0 .
026 0 .
000 0 .
000 0 . .
007 0 .
009 0 .
007 0 .
000 0 .
001 0 .
004 0 .
000 0 .
000 0 . ∞ − . h — — a Unless otherwise stated, all energies are reported as errors relative to CCSDT in millihartree. b FCI stands for i -FCIQMC. c CIQ stands for i -CISDTQ-MC. d CIT stands for i -CISDT-MC. e Equivalent to CCSD. f Equivalent to the CCSD energy corrected for the effects of T clusters using the CCSD(2) T approach of Ref. 88, which is equivalentto the approximate form of the completely renormalized CR-CC(2,3) approach of Refs. 75 and 76, abbreviated sometimes asCR-CC(2,3),A or CR-CC(2,3) A . Equivalent to the CCSD energy corrected for the effects of T clusters using the most complete variant of the completely renormalizedCR-CC(2,3) approach of Refs. 75 and 76, abbreviated sometimes as CR-CC(2,3),D or CR-CC(2,3) D . Total CCSDT energy in hartree. TABLE III. Convergence of the CC( P ), CC( P ; Q ) MP , and CC( P ; Q ) EN energies toward CCSDT, where the P spaces consistedof all singles and doubles and subsets of triples identified during the i -FCIQMC, i -CISDTQ-MC, or i -CISDT-MC propagationswith δτ = 0 . Q spaces consisted of the triples not captured by the correspondingQMC simulations, for the reactant (R) and transition state (TS) structures defining the automerization of cyclobutadiene, asdescribed by the cc-pVDZ basis set, optimized in the MR-AQCC calculations reported in Ref. 134, and for the correspondingactivation barrier. The i -FCIQMC, i -CISDTQ-MC, and i -CISDT-MC calculations preceding the CC( P ) and CC( P ; Q ) stepswere initiated by placing 100 walkers on the RHF determinant and the n a parameter of the initiator algorithm was set at 3.In all post-RHF calculations, the lowest four core orbitals were kept frozen and the spherical components of d orbitals wereemployed throughout. % of triples CC( P ) a CC( P ;3) MPa
CC( P ;3) ENa
Species MC iterations FCI b CIQ c CIT d FCI b CIQ c CIT d FCI b CIQ c CIT d FCI b CIQ c CIT d R 0 0 26 . e . f . g .
758 25 .
985 25 .
484 4 .
437 4 .
535 4 .
324 0 .
696 0 .
763 0 . .
532 22 .
513 22 .
462 3 .
684 3 .
621 3 .
612 0 .
496 0 .
418 0 . .
369 17 .
857 18 .
880 2 .
599 2 .
676 2 .
889 0 .
230 0 .
228 0 . .
845 12 .
034 13 .
834 1 .
635 1 .
649 2 .
007 0 .
092 0 .
080 0 . .
895 7 .
176 9 .
202 0 .
877 0 .
913 1 .
235 0 .
022 0 .
023 0 . .
273 3 .
524 5 .
205 0 .
386 0 .
417 0 .
645 0 .
001 0 .
000 0 . .
321 1 .
498 2 .
594 0 .
146 0 .
170 0 . − . − . − . .
512 0 .
563 1 .
181 0 .
056 0 .
060 0 . − . − . − . ∞ − . h — —TS 0 0 47 . e . f . g .
875 46 .
427 45 .
777 18 .
899 19 .
135 18 .
037 13 .
680 13 .
842 12 . .
577 37 .
689 39 .
655 14 .
220 12 .
522 13 .
774 9 .
452 7 .
793 8 . .
836 28 .
405 33 .
111 9 .
660 7 .
404 10 .
798 5 .
785 3 .
648 6 . .
976 19 .
811 23 .
797 4 .
046 4 .
313 6 .
457 1 .
556 1 .
661 3 . .
795 9 .
727 12 .
495 1 .
634 1 .
488 2 .
238 0 .
309 0 .
243 0 . .
936 4 .
136 6 .
217 0 .
501 0 .
525 0 .
886 0 .
026 0 .
025 0 . .
491 1 .
488 2 .
841 0 .
173 0 .
168 0 .
363 0 .
003 0 .
001 0 . .
525 0 .
591 1 .
260 0 .
058 0 .
065 0 .
148 0 .
000 0 .
000 0 . ∞ − . h — —Barrier 0 0/0 13 . e . f . g .
624 12 .
828 12 .
734 9 .
075 9 .
162 8 .
605 8 .
148 8 .
208 7 . .
696 9 .
523 10 .
789 6 .
612 5 .
586 6 .
377 5 .
620 4 .
628 5 . .
450 6 .
619 8 .
931 4 .
431 2 .
967 4 .
963 3 .
487 2 .
146 3 . .
475 4 .
881 6 .
252 1 .
513 1 .
672 2 .
793 0 .
919 0 .
992 2 . .
820 1 .
601 2 .
067 0 .
475 0 .
361 0 .
629 0 .
181 0 .
138 0 . .
416 0 .
384 0 .
635 0 .
073 0 .
068 0 .
151 0 .
016 0 .
016 0 . . − .
006 0 .
155 0 . − .
001 0 .
038 0 .
003 0 .
002 0 . .
008 0 .
018 0 .
050 0 .
001 0 .
003 0 .
011 0 .
001 0 .
001 0 . ∞ . i — — a Unless otherwise stated, all energies are reported as errors relative to CCSDT, in millihartree for the reactant and transition state andin kcal/mol for the activation barrier. b FCI stands for i -FCIQMC. c CIQ stands for i -CISDTQ-MC. d CIT stands for i -CISDT-MC. e Equivalent to CCSD. f Equivalent to the CCSD energy corrected for the effects of T clusters using the CCSD(2) T approach of Ref. 88, which is equivalentto the approximate form of the completely renormalized CR-CC(2,3) approach of Refs. 75 and 76, abbreviated sometimes asCR-CC(2,3),A or CR-CC(2,3) A . Equivalent to the CCSD energy corrected for the effects of T clusters using the most complete variant of the completely renormalizedCR-CC(2,3) approach of Refs. 75 and 76, abbreviated sometimes as CR-CC(2,3),D or CR-CC(2,3) D . Total CCSDT energy in hartree. i The CCSDT activation barrier in kcal/mol. TABLE IV. Convergence of the CC( P ), CC( P ; Q ) MP , and CC( P ; Q ) EN energies toward CCSDTQ, where the P spaces consistedof all singles and doubles and subsets of triples and quadruples identified during the i -FCIQMC or i -CISDTQ-MC propagationswith δτ = 0 . Q spaces consisted of the triples not captured by the corresponding QMCsimulations, for the equilibrium and four displaced geometries of the H O molecule, as described by the cc-pVDZ basis set,taken from Ref. 118. The i -FCIQMC and i -CISDTQ-MC calculations preceding the CC( P ) and CC( P ; Q ) steps were initiatedby placing 100 walkers on the RHF determinant and the n a parameter of the initiator algorithm was set at 3. All electronswere correlated and the spherical components of d orbitals were employed throughout. % of triples/quadruples CC( P ) b CC( P ; Q ) MPb
CC( P ; Q ) ENb
R/R e a MC iterations FCI c CIQ d FCI c CIQ d FCI c CIQ d FCI c CIQ d . e . f . g / / .
291 3 .
291 0 .
718 0 .
718 0 .
220 0 . / / .
874 2 .
874 0 .
633 0 .
629 0 .
205 0 . / / .
637 2 .
637 0 .
544 0 .
600 0 .
143 0 . / / .
052 2 .
052 0 .
441 0 .
471 0 .
142 0 . / / .
910 1 .
910 0 .
390 0 .
358 0 .
105 0 . / / .
481 1 .
481 0 .
304 0 .
323 0 .
087 0 . / / .
238 1 .
238 0 .
245 0 .
249 0 .
065 0 . / / .
956 0 .
956 0 .
207 0 .
216 0 .
073 0 . /
10 35 /
10 0 .
586 0 .
586 0 .
127 0 .
143 0 .
048 0 . ∞ − . h — —1.5 0 0/0 9 . e . f . g / / .
612 6 .
545 1 .
393 1 .
501 0 .
290 0 . / / .
068 4 .
168 0 .
898 0 .
799 0 .
236 0 . / / .
000 3 .
032 0 .
613 0 .
698 0 .
144 0 . / / .
878 2 .
207 0 .
481 0 .
503 0 .
231 0 . / / .
465 1 .
507 0 .
377 0 .
366 0 .
185 0 . / / .
993 0 .
959 0 .
254 0 .
270 0 .
133 0 . / / .
786 0 .
706 0 .
229 0 .
206 0 .
133 0 . /
10 38 /
11 0 .
552 0 .
548 0 .
186 0 .
156 0 .
130 0 . /
17 48 /
18 0 .
259 0 .
263 0 .
086 0 .
086 0 .
061 0 . ∞ − . h — —2.0 0 0/0 22 . e . f − . g / / .
766 11 .
803 1 .
966 2 . − .
044 0 . / / .
172 4 .
937 1 .
129 1 .
295 0 .
567 0 . / / .
132 3 .
788 0 .
708 0 .
683 0 .
323 0 . / / .
728 1 .
966 0 .
603 0 .
668 0 .
436 0 . / / .
123 1 .
120 0 .
421 0 .
509 0 .
324 0 . / / .
794 0 .
719 0 .
305 0 .
221 0 .
246 0 . / / .
429 0 .
427 0 .
129 0 .
144 0 .
094 0 . /
11 35 /
11 0 .
327 0 .
293 0 .
106 0 .
103 0 .
079 0 . /
18 47 /
18 0 .
107 0 .
102 0 .
036 0 .
026 0 .
029 0 . ∞ − . h — —2.5 0 0/0 22 . e − . f − . g / / . − . − . − . − . − . / / .
254 7 .
207 0 .
010 0 . − . − . / / .
278 2 .
109 0 .
513 0 .
988 0 .
298 0 . / / .
021 1 .
170 0 .
304 0 .
542 0 .
220 0 . / / .
459 0 .
585 0 .
264 0 .
287 0 .
254 0 . / / .
340 0 .
424 0 .
105 0 .
222 0 .
096 0 . /
12 29 / .
133 0 .
411 0 .
059 0 .
020 0 . − . /
16 36 /
13 0 .
088 0 .
155 0 .
014 0 .
052 0 .
011 0 . /
28 49 /
22 0 .
020 0 .
027 0 .
013 0 .
020 0 .
012 0 . ∞ − . h — —3.0 0 0/0 15 . e − . f − . g / / .
165 12 . − . − . − . − . / / .
282 2 . − . − . − . − . / / .
616 3 .
019 0 .
544 0 .
357 0 .
414 0 . / / .
969 0 .
830 0 .
267 0 .
378 0 .
199 0 . / / .
523 0 .
400 0 .
251 0 .
196 0 .
231 0 . / / .
185 0 .
237 0 .
097 0 .
093 0 .
090 0 . /
12 28 /
10 0 .
082 0 .
128 0 .
039 0 .
076 0 .
036 0 . /
16 34 /
14 0 .
030 0 .
050 0 .
022 0 .
030 0 .
021 0 . /
28 48 /
24 0 .
005 0 .
012 0 .
005 0 .
008 0 .
005 0 . ∞ − . h — — a The equilibrium geometry, R O - H = R e , and the geometries that represent a simultaneous stretching of both O–H bonds by factors of1.5, 2.0, 2.5, and 3.0 without changing the ∠ (H–O–H) angle were taken from Ref. 118. b Unless otherwise stated, all energies are reported as errors relative to CCSDTQ in millihartree. c FCI stands for i -FCIQMC. d CIQ stands for i -CISDTQ-MC. e Equivalent to CCSD. f Equivalent to the CCSD energy corrected for the effects of T clusters using the CCSD(2) T approach of Ref. 88, which is equivalentto the approximate form of the completely renormalized CR-CC(2,3) approach of Refs. 75 and 76, abbreviated sometimes asCR-CC(2,3),A or CR-CC(2,3) A . Equivalent to the CCSD energy corrected for the effects of T clusters using the most complete variant of the completely renormalizedCR-CC(2,3) approach of Refs. 75 and 76, abbreviated sometimes as CR-CC(2,3),D or CR-CC(2,3) D . Total CCSDTQ energy in hartree. −505101520 E rr o r r e l . o CC S D T ( m E h ) (a) (b) I era ions (×10 ) −505101520 E rr o r r e l . o CC S D T ( m E h ) (c) I era ions (×10 ) (d) )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s FIG. 1. Convergence of the CC( P ) (red filled circles and dashed lines) and CC( P ; Q ) EN (black open squares and solid lines)energies toward CCSDT for the F /cc-pVDZ molecule in which the F–F distance R was set at (a) R e , (b) 1 . R e , (c) 2 R e , and(d) 5 R e , where R e = 2 . P spaces consisted of all singles and doubles and subsetsof triples identified during the i -FCIQMC propagations with δτ = 0 . Q spaces consisted of the triples not captured by i -FCIQMC. All energies are errorsrelative to CCSDT in millihartree and the insets show the percentages of triples captured during the i -FCIQMC propagations. −505101520 E rr o r r e l . o CC S D T ( m E h ) (a) (b) I era ions (×10 ) −505101520 E rr o r r e l . o CC S D T ( m E h ) (c) I era ions (×10 ) (d) )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s FIG. 2. Same as Fig. 1 except that the subsets of triples included in the CC( P ) calculations are now identified by the i -CISDTQ-MC simulations and the corresponding Q spaces consist of the triples not captured by i -CISDTQ-MC. E rr o r r e l . t o CC S D T ( m E h ) (a) (b) Iterations (×10 ) E rr o r r e l . t o CC S D T ( m E h ) (c) Iterations (×10 ) (d) )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s FIG. 3. Same as Fig. 1 except that the subsets of triples included in the CC( P ) calculations are now identified by the i -CISDT-MC simulations and the corresponding Q spaces consist of the triples not captured by i -CISDT-MC.FIG. 4. The key molecular structures defining the automerization of cyclobutadiene. The leftmost and rightmost structuresrepresent the degenerate reactant/product minima, whereas the structure in the center corresponds the transition state. E rr o r r e l . t o CC S D T ( m E h ) (a) (b) (c) Iterations (×10 ) E rr o r r e l . t o CC S D T ( m E h ) (d) Iterations (×10 ) (e) Iterations (×10 ) (f) )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s )020406080100 % o f T r i p l e s FIG. 5. Convergence of the CC( P ) (red filled circles and dashed lines) and CC( P ; Q ) EN (black open squares and solidlines) energies toward CCSDT for the reactant [panels (a)–(c)] and transition state [panels (d)–(f)] structures defining theautomerization of cyclobutadiene, as described by the cc-pVDZ basis set. The relevant i -CIQMC runs (all using δτ = 0 . P spaces employed in the CC( P ) steps consisted of all singles and doubles and subsets of triplesidentified during the i -FCIQMC propagations; the Q spaces needed to define the corresponding δ ( P ; Q ) corrections consistedof the triples that were not captured by i -FCIQMC. Panels (b) and (e) correspond to the calculations in which the P spacesemployed in the CC( P ) steps consisted of all singles and doubles and subsets of triples identified during the i -CISDTQ-MCpropagations; in this case, the Q spaces needed to define the δ ( P ; Q ) corrections consisted of the triples that were not capturedby i -CISDTQ-MC. Panels (c) and (f) correspond to the calculations in which the P spaces employed in the CC( P ) stepsconsisted of all singles and doubles and subsets of triples identified during the i -CISDT-MC propagations; in this case, the Q spaces needed to define the δ ( P ; Q ) corrections consisted of the triples that were not captured by i -CISDT-MC. All reportedenergies are errors relative to CCSDT in millihartree. The insets show the percentages of triples captured during the relevant i -CIQMC propagations. E rr o r r e l . t o CC S D T Q ( m E h ) (a) (b) Iteration (×10 ) −5051015 E rr o r r e l . t o CC S D T Q ( m E h ) (c) Iteration (×10 ) −10010203040 (d) )020406080100 % o f T r i p l e a n d Q u a d )020406080100 % o f T r i p l e a n d Q u a d )020406080100 % o f T r i p l e a n d Q u a d )020406080100 % o f T r i p l e a n d Q u a d FIG. 6. Convergence of the CC( P ) (red filled circles and dashed lines) and CC( P ; Q ) EN (black open squares and solidlines) energies toward CCSDTQ for the water molecule, as described by the cc-pVDZ basis set. The relevant i -CIQMCruns (all using δτ = 0 . P spaces employed in the CC( P ) steps consisted of all singles anddoubles and subsets of triples and quadruples identified during the i -FCIQMC propagations; the Q spaces needed to define thecorresponding δ ( P ; Q ) corrections consisted of the triples that were not captured by i -FCIQMC. Panels (c) and (d) correspondto the calculations in which the P spaces employed in the CC( P ) steps consisted of all singles and doubles and subsets of triplesand quadruples identified during the i -CISDTQ-MC propagations; in this case, the Q spaces needed to define the corresponding δ ( P ; Q ) corrections consisted of the triples that were not captured by i -CISDTQ-MC. Panels (a) and (c) correspond to theequilibrium geometry. Panels (b) and (d) correspond to the geometry in which both O–H bonds in water are simultaneouslystretched by a factor of 3 without changing the ∠ (H–O–H) angle. All reported energies are errors relative to CCSDTQ inmillihartree. The insets show the percentages of triples (blue line) and quadruples (purple line) captured during the relevant ii