Diagrammatic Coupled Cluster Monte Carlo
Charles J. C. Scott, Roberto Di Remigio, T. Daniel Crawford, Alex J. W. Thom
DDiagrammatic Coupled Cluster Monte Carlo
Charles J. C. Scott, ∗ , † Roberto Di Remigio, ∗ , ‡ , ¶ T. Daniel Crawford, ¶ and AlexJ. W. Thom † † Department of Chemistry, University of Cambridge, Cambridge, UK ‡ Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, Universityof Tromsø - The Arctic University of Norway, N-9037 Tromsø, Norway ¶ Department of Chemistry, Virginia Tech, Blacksburg, Virginia 24061, United States
E-mail: [email protected]; [email protected] a r X i v : . [ phy s i c s . c h e m - ph ] F e b bstract We propose a modified coupled cluster Monte Carlo algorithm that stochasticallysamples connected terms within the truncated Baker–Campbell–Hausdorff expansionof the similarity transformed Hamiltonian by construction of coupled cluster diagramson the fly. Our new approach – diagCCMC – allows propagation to be performed usingonly the connected components of the similarity-transformed Hamiltonian, greatly re-ducing the memory cost associated with the stochastic solution of the coupled clusterequations. We show that for perfectly local, noninteracting systems, diagCCMC is ableto represent the coupled cluster wavefunction with a memory cost that scales linearlywith system size. The favorable memory cost is observed with the only assumption offixed stochastic granularity and is valid for arbitrary levels of coupled cluster theory.Significant reduction in memory cost is also shown to smoothly appear with dissoci-ation of a finite chain of helium atoms. This approach is also shown not to breakdown in the presence of strong correlation through the example of a stretched nitrogenmolecule. Our novel methodology moves the theoretical basis of coupled cluster MonteCarlo closer to deterministic approaches.
Graphical TOC Entry N ! factorial scaling to polynomial, thecomputational cost of CC methods, measured in terms of both required CPU floating-pointoperations and memory, is still an issue. The coupled cluster with single and double substi-tutions (CCSD) and CCSD with perturbative triples correction (CCSD(T)) approximationsprovide a balance between computational cost and accuracy that has led to relatively wideadoption, but are eventually precluded for many large systems.Recent work has made great progress on this issue through application of various ap-proximations, which enable calculations to be performed with reduced memory and compu-tational costs. In particular, various approximations exploiting the locality of electron corre-lation allow calculations with costs asymptotically proportional to measures of system size.These include approaches based on orbital localisation, molecular fragmentation, and decompositions, such as resolution-of-the-identity, Cholesky or singular-value, of thetwo-electron integrals tensors. However, while providing large efficiencies in CCSDcalculations, higher truncation levels will generally exceed available memory resources beforesuch approximations are a reasonable proposition.In this letter we propose and demonstrate a coupled cluster-based projector MonteCarlo (MC) algorithm that enables automatic exploitation of the wavefunction sparsity forarbitrary excitation orders. Our methodology can be particularly beneficial for localisedrepresentations of the wavefunction, but it is not limited by assumptions of locality. Theapproach can fully leverage the sparsity inherent in the CC amplitudes at higher excitationlevels, allowing dramatic reductions in memory costs for higher levels of theory.The CC wavefunction is expressed as an exponential transformation of a reference single-determinant wavefunction | D (cid:105) : | CC (cid:105) = e T | D (cid:105) (1)3here the cluster operator T is given as a sum of second-quantised excitation operators: T = (cid:88) k T k , (2)with the k -th order cluster operators expressed as sums of excitation operators weighted bythe corresponding cluster amplitudes : T k = (cid:88) i ∈ k th replacements t i τ i = 1( k !) (cid:88) a ,a ,...,a k i ,i ,...,i k t i i ...i k a a ...a k τ a a ...a k i i ...i k , (3)in the tensor notation for second quantisation proposed by Kutzelnigg and Mukherjee .Upon truncation of the cluster operator to a certain excitation level l and projection ofthe Schr¨odinger equation onto the corresponding excitation manifold one obtains the linked energy and cluster amplitudes equations: (cid:104) D | ¯ H N | D (cid:105) = E CC (4a)Ω n ( t ) = (cid:104) D n | ¯ H N | D (cid:105) = 0 . (4b)We have introduced the similarity-transformed Hamiltonian, ¯ H N = e − T H N e T , and | D n (cid:105) canbe any state within the projection manifold (up to an l -fold excitation of | D (cid:105) ). These CCequations are manifestly size-extensive order-by-order and term-by-term and furthermoreprovide the basis for the formulation of response theory. CC methods have to be carefully derived order-by-order and their implementation sub-sequently carried out, a process that can be rather time-consuming and error-prone.
Ithas long been recognised that the use of normal-ordering,
Wick’s theorem, and theensuing diagrammatic techniques can be leveraged to automate both steps, thoughspin-adaptation can still pose significant challenges. Consider the normal-ordered, elec-4ronic Hamiltonian: H N = F + Φ = (cid:88) pq f qp e pq + 12 (cid:88) pqrs g rspq e pqrs = (cid:88) pq f qp e pq + 14 (cid:88) pqrs ¯ g rspq e pqrs = H − E ref , (5)its similarity transformation admits a Baker–Campbell–Hausdorff (BCH) expansion trun-cating exactly after the four-fold nested commutator. Since all excitation operators arenormal-ordered and commuting, the commutator expansion lets us reduce the Hamiltonian-excitation operator products to only those terms which are connected . Excitation oper-ators will only appear to the right of the Hamiltonian and only terms where each excitationoperator shares at least one index with the Hamiltonian will lead to nonzero terms in theresiduals Ω n ( t ) appearing in eqs. (4):¯ H N = ( H N e T ) c = H N + H N T + 12! H N T T + 13! H N T T T + 14! H N T T T T. (6)Moreover, by virtue of Wick’s theorem, the products of normal-ordered strings appearingin the connected expansion will still be expressed as normal-ordered strings, further simpli-fying the algebra. The requirement of shared indices between the Hamiltonian and clustercoefficients enables the resulting equations to be solved via a series of tensor contractionsbetween multi-index quantities: the sought-after cluster amplitudes and the molecular one-and two-electron integrals. The iterative process required to solve eqs. (4) is highly amenablefor a rapid evaluation on conventional computing architectures, but remains non-trivialto parallelise, especially for higher truncation orders in the CC hierarchy. A proper fac-torisation of intermediates is essential to achieve acceptable time to solution and memoryrequirements.In recent years some of us have been involved in developing a projector MC algorithmto obtain the CC solutions within a stochastic error bar.
The starting point, as withany projector MC method, is the imaginary-time Schr¨odinger equation obtained after a5ick rotation τ ← i t . Repeated application of the approximate linear propagator to a trialwavefunction will yield the ground-state solution: | Ψ( τ + δτ ) (cid:105) = [1 − δτ ( H − S )] | Ψ( τ ) (cid:105) (7)where S is a free parameter that is varied to keep the normalisation of Ψ( τ ) approxi-mately constant. In the CCMC and full configuration interaction quantum Monte Carlo(FCIQMC) approaches, a population of particles in Fock space represents the wavefunctionand evolves according to simple rules of spawning, death, and annihilation. For a CCAnsatz, unit particles may represent nonunit contributions to CC amplitudes by letting theintermediate normalisation condition vary with the population on the reference determi-nant: (cid:104) D | CCMC( τ ) (cid:105) = N ( τ ). A factor of N ( τ ) is removed from the definition of T ( τ )and this determines the granularity of amplitude representation: amplitude values smallerthan N ( τ ) are stochastically rounded during the calculation, vide infra . To avoid confusion,we denote the so-modified cluster operators and amplitudes as T (cid:48) and t (cid:48) n , respectively, so | CCMC (cid:105) = N e T (cid:48) N | D (cid:105) . Thus, in the unlinked formulation first put forward by Thom, thedynamic equation for the amplitudes becomes: t (cid:48) n → t (cid:48) n − δτ (cid:104) D | τ † n [ H − S ] | CCMC (cid:105) , (8)where we have dropped the τ -dependence for clarity. CCMC is fully general with respectto the truncation level in the cluster operator and sidesteps the need to store a full repre-sentation of the wavefunction at any point. CCMC should allow for the effective solution ofthe CC equations with a much reduced memory cost, as previously realised in the FCIQMCmethod. However, while various cases demonstrate memory cost reduction, especiallyin the presence of weak correlation, the corresponding increase in computational cost waslarge even by the standards of projector MC methods and modifications used in relatedapproaches, such as the initiator approximation, proved comparatively ineffective. deterministic community. Franklin et al. have discussed a CCMC algorithm to sampleeqs. (4) using the update step: t (cid:48) n → t (cid:48) n − δτ N (cid:104) D | τ † n ¯ H | D (cid:105) , ( | D n (cid:105) (cid:54) = | D (cid:105) ) (9a) N → N − δτ N (cid:104) D | ¯ H − S | D (cid:105) (9b)The authors however noted that the use of the similarity-transformed Hamiltonian requiredan ad hoc modification: t (cid:48) n → t (cid:48) n − δτ N (cid:104) D | τ † n [ ¯ H − E CC ] | D (cid:105) − δτ ( E CC − S ) t n , (10)to deal with convergence issues with the projected energy prior to the initialisation of pop-ulation control. In addition, due to evaluation of ¯ H via the commutator expansion of thebare Hamiltonian, rather than the sum of connected Hamiltonian-excitation operator prod-ucts (6), some disconnected terms were included. These extraneous terms in the algorithmof Franklin et al. have been observed to correctly cancel out on average, but render unnec-essarily complex the sampling of connected contributions only. Eventually, it is difficult todevelop stochastic counterparts to approximations, such as the CCn hierarchy, proposedwithin deterministic CC theory.We here reconsider the implementation of the linked CCMC algorithm in the light ofthe diagrammatic techniques used in deterministic CC, an approach we name diagrammaticCoupled Cluster Monte Carlo (diagCCMC). The update equation can be easily derived as afinite difference approximation to the exact imaginary-time dynamics of the coupled clusterwavefunction under the assumption of constant intermediate normalisation: t n ( τ + δτ ) = t n ( τ ) − δτ (cid:104) D | τ † n ¯ H N ( τ ) | D (cid:105) . (11)7his has been noted elsewhere, and we will discuss its implications in greater detail in a sub-sequent communication, but for now it will suffice to observe that since this is a projectorMC approach it will eventually converge to the lowest energy solution of the CC equations.The existence of multiple solutions to the nonlinear CC equations is well-documented, and a projector MC approach could result in a different solution to the CC equations thanthe one found via a deterministic procedure, where iteration stabilises upon whichever solu-tion is approached first from a given starting point. In practice a difference is only observedif a highly truncated form of CC has been applied inappropriately to a system, and eventhen only in the worst cases.The second term on the right-hand side is the contribution to the CC vector functionΩ n ( t ) resulting from the projection upon the determinant | D n (cid:105) and is representable as a finitesum of enumerable diagrams. Thus, at each iteration, we wish to randomly select n a diagramsfrom (cid:104) D | τ † n ¯ H N | D (cid:105) . Each of these will be in the form of an excitation operator, τ i , andcorresponding weight, w i , selected with some known, normalised probability, p diagram , suchthat we expect to select any given contributing diagram p diagram × n a times at each iteration.As by construction (cid:104) D | τ † j τ i | D (cid:105) = δ ij , a selected term can be found to contribute to theupdate of a single coefficient with no additional sign considerations. Rather than explicitlyintroduce a particulate representation of the coefficients, as in FCIQMC and previous CCMCapproaches, we stochastically round all coefficients t n with magnitude below some strictlypositive granularity parameter ∆. If | t n | < ∆, then | t n | is rediscretised to either ∆ (withprobability (cid:12)(cid:12) t n ∆ (cid:12)(cid:12) ) or 0 (with probability 1 − (cid:12)(cid:12) t n ∆ (cid:12)(cid:12) ). This can be shown to be equivalent toa representation with unit particles and constant intermediate normalisation .We perform diagram selection by reading off terms from right-to-left in (cid:104) D | τ † n ¯ H N | D (cid:105) :1. Select a random cluster of excitation operators with probability p select utilising theeven selection scheme restricted to clusters of at most 4 excitation operators. Thiscorresponds to simultaneously selecting a term in the BCH expansion (6) and theexcitation level of each excitation operator in the commutator.8. Select one of the 13 possible H N vertices with some probability p hvertex .3. Select the contraction pattern of the chosen cluster and Hamiltonian vertex. Thisidentifies a specific Kucharski–Bartlett sign sequence for the diagram we areconsidering and which excitation operators are associated with which term within thesign sequence with probability p contract .4. Select which indices of each excitation operator will be contracted with the Hamiltonianvertex. Having selected the contraction pattern this is a matter of simple combinatorics,with a given set of indices selected with probability p internal .5. Select the external indices of the Hamiltonian vertex with probability p external .6. Evaluate the index of resulting projection determinant in the update step, i.e. (cid:104) D | τ † n ,and the diagrammatic amplitude including all parity factors.This obtains a single specific diagram with probability: p diagram = p select p hver p cont p int p ext , (12)where the obvious abbreviations have been used to refer to each of the previously statedprobabilities. These are conditional probabilities, as the various events leading to the com-puted p diagram are not independent. This procedure to select diagrams can be visualised asgraphically building the diagram bottom-up, see Figure 1.To evaluate the contribution of a selected diagram to our propagation, we slightly modifythe standard rules of diagrammatic interpretation. Instead of summing over all indices, andthus having to correct for any potential double counting, our algorithm selects a specific diagram along with a specific set of indices for all lines.To ensure proper normalisation of our sampling probability, we require there be only asingle way to select diagrams related by: • The antipermutation of antisymmetrised Goldstone vertex indices.9igure 1: Graphical depiction of the diagrammatic CCMC algorithm. This example showsthe steps involved in the generation of one of the possible diagrams contributing to the T equations. • The antipermutation of cluster operator particle or hole indices. • The commutation of cluster operators.All these modifications can be viewed as replacing sums (cid:80) ij with (cid:80) i>j + δ ij . In the first twocases summation runs over equivalent indices and the i = j term must be zero, while in thethird case summation runs over excitation operators and the i = j term corresponds to adiagram with additional symmetry that as such must be treated more carefully to ensureunique selection of a Kucharski–Bartlett sign sequence. Specifically, we do not requirean additional factor of for: • Each pair of equivalent internal or external lines. • Two cluster operators of the same rank but with different specific indices, providedthey have a well-determined ordering on selection.10dditionally, to include the effect of permutation operators ˆ P for inequivalent external lineswe must permute the hole and particle indices of a resulting excitation operator to a uniqueantisymmetrised ordering for storage. This ensures proper cancellation between all equiva-lent orderings, which could otherwise differ due to the stochastic sampling. Eventually, theamplitude of the contribution of the selected diagram, w diagram , is given as the product of thecluster amplitude, w clus = (cid:81) i t i , and Hamiltonian element, w hamil , with appropriately deter-mined parity ( − σ . The overall contribution of a single selected diagram to the coefficient t n determined by the open lines of the diagram will be: w diagram p diagram = ( − σ w clus w hamil p select p hver p cont p int p ext , (13)wherever possible we aspire to have p diagram ∝ | w diagram | . We will now demonstrate the ability of diagCCMC to recover energies at high levels ofCC theory on the nitrogen molecule in a stretched geometry ( r NN = 3 . a ). It has previouslybeen shown that connected contributions up to hextuples are vital to obtaining high accuracyfor this system. Correlation energies for a range of basis sets and truncation levels arereported in Table 1, showing agreement within error bars with deterministic results andthe existing literature in all but the most extreme cases, where convergence to a differentsolution is observed as noted previously.We then turn our attention to test systems of beryllium and neon atoms at a variety oftruncation levels. Extending these systems by introducing noninteracting replicas illustratesthe behaviour of our approach in the presence of locality in comparison to previous Fock-space stochastic methods, namely the original unlinked CCMC (hereafter simply referred toas CCMC) and FCIQMC.To allow reasonable comparison between diagCCMC, CCMC, and FCIQMC all calcula-tions were performed with: • Granularity parameter ∆ equal to 10 − . This is the threshold for the stochastic round-11ng of the cluster amplitudes. • δτ and n attempts such that, on each iteration, a spawning event may have maximumsize of 3 × − .For Coupled Cluster Monte Carlo (CCMC) and FCIQMC this corresponds to a stable cal-culation with reference population of N = 10 and a timestep such that no spawning eventproduces more than three particles. CCMC and FCIQMC calculations were performed withthe HANDE-QMC code using the default, uniform excitation generators. For CCMC,we adopted the even selection scheme of Scott and Thom . The molecular integrals weregenerated in FCIDUMP format using the Q-Chem and Psi4 quantum chemistry pro-gram packages, see the Supporting Information for more details. Table 1: Correlation energy for different levels of theory and basis sets for N with r NN = 3 . a . Molecular integrals were generated in FCIDUMP format withthe Psi4 program package. The STO-3G and 6-31G results were computedusing MRCC.
The canonical restricted Hartree–Fock orbitals were used, giv-ing E ref = − .
937 562 E h and − .
360 046 E h in the STO-3G and 6-31G bases,respectively. STO-3G 6-31GSD CC − .
589 163 − .
491 480diagCCMC − . a − . − .
589 923 − .
533 600diagCCMC − . a − . − .
523 049diagCCMC − . b SDTQ5 CC − .
523 036diagCCMC − . b SDTQ56 CC − .
527 863diagCCMC − . b a In these cases the stochastic, imaginary-time propagation was found to initially convergeto the conventional CC solution, before relaxing to another, lower energy solution.
Value not computed due to computational constraints.We report the correlation energies obtained for an isolated Be atom and the noninter-acting replicas systems in Table 2. We compare CC results up to and including quadrupleexcitations with FCIQMC. For these systems CCSDTQ is equivalent to FCI, thus providing12 good sanity check for the diagCCMC approach. In addition, results at each level of theoryare expected to agree within statistical errors due to the size-consistency of all consideredapproaches, as is observed.
Table 2: Correlation energy for different levels of theory using 1, 2 and 4 Bereplicas in a cc-pVDZ basis set. Note that for these systems CCSDTQ is equiv-alent to FCI. Molecular integrals were generated in FCIDUMP format with theQ-Chem program package.
The canonical Hartree–Fock orbitals for a single-atom calculation were used, and no spin symmetry breaking was observed, giving E ref = − .
572 341 E h . n replicas − .
045 032(2) − .
090 07(2) − . − .
045 00(5) − .
090 11(7) − . − .
045 067(2) − .
090 12(2) − .
180 34(7)diagCCMC − .
045 12(4) − . − . − .
045 070(2) − .
090 15(2) − .
180 44(9)diagCCMC − .
045 04(4) − . − . − .
045 072 1(7) − .
090 151(5) − .
180 36(6)In order to assess the computational performance of diagCCMC we compare two measuresof efficiency: • n attempts /δτ , that is, the number of stochastic samples performed per unit imaginarytime. This metric is a measure of the minimum CPU cost, provided that the length ofpropagation in imaginary time is roughly constant between approaches, or equivalentlya roughly constant inefficiency between the approaches. • n states , that is, the number of occupied excitation operators. This metric is a measureof the minimum memory cost. For a deterministic calculation this would amount tothe Hilbert space size for the selected truncation level.The promise of stochastic methods is to greatly reduce the cost of high-level correlatedcalculations by naturally exploiting the wavefunction sparsity. Figure 2 reports the ratioof n states per replica and the size of the Hilbert space for an isolated atom at the given13runcation level. For an isolated Be atom, the reduction in memory footprint is clearlyevident: all methods compared require significantly less than the full size of the Hilbertspace (ratio <
1) to successfully achieve convergence and recover the deterministic results.Unsurprisingly and correctly, diagCCMC requires the same amount of storage as its unlinkedcounterparts. Notice also that the ratio decreases in going from CCSD to CCSDTQ showinghow stochastic methods single out the important portions of the Hilbert space. For perfectlylocal systems, such as the noninteracting 2- and 4-atom replicas, one also expects the numberof states per replica to roughly stay constant. This expectation stems from the linked diagramtheorem and is met by the diagCCMC approach where at each iteration only connecteddiagrams are sampled. The same is, quite emphatically, not true for either FCIQMC orCCMC: the number of states per replica approaches and surpasses the size of the single-atom Hilbert space.In Figure 3 we can see that diagCCMC outperforms each of the corresponding CCMCapproaches also when estimating the CPU cost of the calculations on the Be systems hereconsidered. It is particularly striking to note the order of magnitude difference betweenthe diagrammatic and unlinked approaches at the CCSD level of theory even for this tinysystem.The same observation also holds true for higher orders of CC theory, as can clearly be seenfrom Figure 4 where we plot the n states metric for an isolated Ne atom and its corresponding2 and 4 noninteracting replicas system. Table 3 reports the correlation energies per replicafor a systems of noninteracting Ne atoms. diagCCMC affords calculations practically atconstant memory cost per replica in contrast with CCMC for which the increasing costexceeded available computational resources for the higher order excitations.Finally, we studied the dissociation of a chain of 5 helium atoms as an example of in-teracting system. The diagrammatic algorithm shows favourable CPU and memory cost fornoninteracting systems, further suggesting that it might also straightforwardly leverage lo-calisation in the orbital space to achieve reduced cost for calculations on interacting systems.14igure 2: The ratio of states per-replica and corresponding reduced Hilbert space size for 1,2 and 4 Be replicas in a cc-pVDZ basis set at various levels of theory. The n states metric is ameasure of the memory cost of the calculation. For a single Be atom the Hilbert space sizesare 121, 529, and 1093 states for CCSD, CCSDT, and CCSDTQ, respectively, and the cor-responding reduced Hilbert space multiplies these values by the number of Be atoms. Notethat for these systems CCSDTQ is equivalent to FCI. Solid, dotted and dash-dotted linesare used for diagCCMC, CCMC and FCIQMC results, respectively. Molecular integralswere generated in FCIDUMP format with the Q-Chem program package. The canoni-cal Hartree–Fock orbitals for a single-atom calculation were used, and no spin symmetrybreaking was observed.As a preliminary test for this conjecture, Figure 5 shows the memory cost for the dissociationcurve of an interacting chain of five helium atoms. We localised the occupied and virtualorbital sets with the Foster–Boys and the Pipek–Mezey criteria, respectively. We com-pare the n states metric with the memory cost at the dissociation limit for a deterministic anda diagCCMC CCSD calculation. The former (dotted line) is the maximum memory costfor performing CCSD calculations on the isolated atoms: below it, the cost is comparableto that for a wavefunction with excitations localised to each He atom. The onset of suchbehaviour is evident from Figure 5, which also shows the recovery of the noninteracting limitat large separations.In conclusion, we have described a stochastic realisation of linked CC theory that fully15igure 3: Number of stochastic samples performed ( n attempts ) per unit imaginary time perreplica for 1, 2 and 4 Be replicas in a cc-pVDZ basis set at various levels of theory. Assumingthat the length of propagation in imaginary time is roughly constant between approaches thismetric is a measure of the CPU cost of the calculation. Note that for these systems CCSDTQis equivalent to FCI. Solid, dotted and dash-dotted lines are used for diagCCMC, CCMCand FCIQMC results, respectively. Molecular integrals were generated in FCIDUMP formatwith the Q-Chem program package. The canonical Hartree–Fock orbitals for a single-atomcalculation were used, and no spin symmetry breaking was observed.exploits the connectedness of the similarity-transformed Hamiltonian, as exemplified in thediagrammatic expansion of the CC equations. Our stochastic diagrammatic implementa-tion avoids the computational and memory cost issues associated with deterministic andunlinked stochastic approaches, by generating diagrams on-the-fly and accumulating thecorresponding amplitudes. Finally, we have shown how the stochastic and deterministicimplementations can be rationalised within the same framework. This bridges the existinggap between the two strategies: by clearing possible misunderstandings on how and why stochastic methods work and enabling future cross-fertilisation.16 able 3: Correlation energy for different levels of theory using 1, 2 and 4Ne replicas in a cc-pVDZ basis set. Molecular integrals were generated inFCIDUMP format with the Psi4 program package and exact CC results ob-tained using MRCC.
The canonical Hartree–Fock orbitals for a single-atomcalculation were used and no spin symmetry breaking was observed, giving E ref = − .
488 776 E h . n replicas − .
190 865(3) − .
381 72(3) − . − .
190 94(5) − . − . − .
190 861 − .
381 723 b − .
763 446 b SDT CCMC − .
191 951(4) − .
383 89(4) − . − .
191 85(10) − . − . − .
191 945 − .
383 891 b − .
767 781 b SDTQ CCMC − .
192 092(4) − .
384 18(6) a diagCCMC − . − . − . − .
192 095 − .
384 191 b − .
768 382 b SDTQ5 CCMC − .
192 103(4) − .
384 36(9) a diagCCMC − . − . − . − .
192 106 − .
384 212 b − .
768 424 b SDTQ56 CCMC − .
192 119(5) a a diagCCMC − . − . − . − .
192 106 − .
384 211 b − .
768 422 b FCI − .
192 106(5) a aa
Value not computed due to computational constraints. b Value obtained as multiple of single atom result for comparison.
Acknowledgement
C.J.C.S. is grateful to the Sims Fund for a studentship and A.J.W.T. to the Royal Societyfor a University Research Fellowship under Grant Nos. UF110161 and UF160398. Bothare grateful for support under ARCHER Leadership Project grant e507. R.D.R. acknowl-edges partial support by the Research Council of Norway through its Centres of Excellencescheme, project number 262695 and through its Mobility Grant scheme, project number261873. R.D.R. is also grateful to the Norwegian Supercomputer Program through a grantfor computer time (Grant No. NN4654K). T.D.C. was supported by grants CHE-1465149and ACI-1450169 from the U.S. National Science Foundation.17 upporting Information Available
The following files are available free of charge.Additional data related to this publication, including a copy of the diagCCMC code, rawand analysed data files and analysis scripts, is available at the University of Cambridge datarepository https://doi.org/10.17863/CAM.34952 and https://doi.org/10.17863/CAM.36097 .We used the goldstone L A TEX package, available on GitHub https://github.com/avcopan/styfiles , to draw the coupled cluster diagrams. We used matplotlib for all theplots in the paper.
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