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Application of the CC(P;Q) Hierarchy of CoupledCluster Methods to the Beryllium Dimer Ilias Magoulas, Nicholas Patrick Bauman, Jun Shen, and Piotr Piecuch J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b10892 • Publication Date (Web): 29 Dec 2017 Downloaded from http://pubs.acs.org on December 31, 2017

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The Journal of Physical Chemistry

Application of the CC(P ;Q) Hierarchy of Coupled-Cluster Methods to the Beryllium Dimer Ilias Magoulas,† Nicholas P. Bauman,† Jun Shen,† and Piotr Piecuch∗,†,‡ †Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA ‡Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA E-mail: [email protected].

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Abstract The performance of coupled-cluster approaches with higher–than–doubly excited clusters,

including the CCSD(T), CCSD(2)T ,

CR-CC(2,3),

CCSD(TQ), and

CR-CC(2,4) corrections to CCSD, the active-space CCSDt, CCSDtq, and CCSDTq methods, and the CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) corrections to CCSDt, CCSDtq, and CCSDTq resulting from the CC(P ;Q) formalism, in reproducing the CCSDT and CCSDTQ potential energy curves and vibrational term values characterizing Be2 in its electronic ground state is assessed. The correlation-consistent augcc-pVnZ and aug-cc-pCVnZ (n = T and Q) basis sets are employed. Among the CCSD-based corrections, the completely renormalized CR-CC(2,3) and CR-CC(2,4) approaches perform the best. The CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) methods, especially CC(t;3) and CC(q;4), outperform other employed approaches in reproducing the CCSDT and CCSDTQ data. Composite schemes combining the all-electron CCSDT calculations extrapolated to the complete basis set limit with the frozen-core CC(q;4) and CCSDTQ computations using the aug-cc-pVTZ basis to account for connected quadruple excitations reproduce the latest experimental vibrational spectrum of Be2 to within 4–5 cm−1 , when the vibrational spacings are examined, with typical errors being below 1–2 cm−1 . The resulting binding energies and equilibrium bond lengths agree with their experimentally derived counterparts to within ∼10 cm−1 and 0.01 ˚ A.

1. Introduction The elusive beryllium dimer has troubled experimentalists and theorists alike for decades. The experimental investigation of the Be2 system is hindered by the low vapor pressure of elemental beryllium, in conjunction with the thermal dissociation of Be2 at temperatures higher than 300K. 1 The situation is further complicated by the fact that elemental beryllium and its oxide are extremely toxic, being responsible for two kinds of lung diseases – acute beryllium pneumonitis and berylliosis. 2–4 It is, therefore, not surprising that it was not until 2


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1975 when the first electronic photoabsorption spectra of Be2 in low-temperature rare gas matrices were recorded by Brom et al. 5 For comparison, the spectra of the nontoxic, heavier group-2 homologs, Mg2 and Ca2 , were already known by 1931. 6 Gas phase spectra of Be2 were finally obtained in 1984 thanks to the use of the laser vaporization of elemental beryllium. 7,8 The vibrational levels of the beryllium dimer in its ground electronic state X 1 Σ+ g are not directly accessible, since, as a homonuclear diatomic molecule, Be2 has no dipole-allowed infrared transitions. Nonetheless, one can probe the X 1 Σ+ g vibrational manifold by exciting the dimer to either the A1 Πu or B 1 Σ+ u excited states and studying the fluorescence spectra. The latest experimental investigation of Be2 using stimulated emission pumping by Merritt et al., 9 which resulted in the rotationally resolved rovibrational spectra, allowed the authors to determine the binding energy De = 929.7(2.0) cm−1 and the equilibrium bond length A. The subsequent re-examination of the spectra reported in ref 9 by Meshkov re = 2.4536 ˚ et al., 10 who used the direct-potential-fit analysis, resulted in the most accurate estimates of De and re to date, namely, De = 934.9(0.4) cm−1 and re = 2.445(5) ˚ A. Contrary to the scarce experimental studies, as summarized above, the theoretical literature on the beryllium dimer spans nearly 90 years and contains more than 120 published investigations, largely because the small size of the beryllium dimer, eight electrons in total, renders it a perfect candidate for various theoretical treatments. By considering the closed-shell nature of the monomers, one might naively anticipate Be2 to be unstable or barely stable with respect to the dissociation into beryllium atoms. For example, in 1929, Herzberg argued, by means of elementary quantum mechanical analysis, that the Be2 dimer should exhibit a very weak interaction similar to the isovalent He2 , meaning a van der Waals minimum of a few wavenumbers at a relatively large internuclear distance. 11 We know today that this is not the case. For a long period of time after Herzberg’s initial analysis, the nature of the Be2 X 1 Σ+ g potential energy curve (PEC) was in a state of flux. Depending on the level of theoretical treatment, the PEC could be purely repulsive or attractive. Some computational studies found a shallow minimum around 5 ˚ A, whereas some others resulted

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in the PECs having a deeper minimum around 2.5 ˚ A, or even possessing two different minima, one at a shorter and another one at a longer Be–Be distances. As already mentioned, the binding in Be2 , which exceeds 900 cm−1 , is much stronger than in the helium dimer, and the corresponding PEC has a single minimum at the relatively short internuclear separation of about 2.4 ˚ A. For an overview of the relevant theoretical literature, the interested reader is referred to refs 12–17 and references therein. Some of the early theoretical work led to erroneous results, since the employed electronic structure treatments were too low. However, the deceptively simple beryllium dimer can also be challenging for the high-level quantum chemistry methods, including some ab initio approaches tested in this work. The main difficulty in an accurate theoretical description of this small, weakly bound diatomic pertains to the near-degeneracy of the 2s and 2p subshells of the Be monomers, combined with unusually large and difficult to balance dynamic correlation effects. Reliable results can only be obtained by using larger basis sets including f and g functions and methods capable of recovering essentially all valence electron correlations. 13,16–31 One often assumes that weakly bound systems and larger dynamic correlation effects can easily be handled by methods based on coupled-cluster (CC) theory, 32–37 but even the CC approaches face challenges when applied to Be2 . For example, the PEC obtained with the CCSD approach, 38,39 even when coupled with very large basis sets, is qualitatively wrong, 22 whereas the widely used CCSD(T) method 40 captures only two thirds of the binding. 13–15,28,41 Even full CCSDT 42,43 is insufficient in this case, 23 prompting the explicit inclusion of connected quadruple excitations. One can incorporate quadruples fully via the CCSDTQ approach 44–47 (which provides an exact, full configuration interaction (CI) description of Be2 if only valence electrons are correlated), but CCSDTQ is generally prohibitively expensive. Thus, the question arises if one can obtain accurate, CCSDTQ-quality, results for Be2 with approximate treatments of connected triples and quadruples in the CC theory. Full CCSDT is not as accurate as CCSDTQ, but it is much more reliable in the case of Be2 than CCSD(T), so one also wonders if it is possible to reproduce the CCSDT PEC of

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the beryllium dimer at a small fraction of the cost of CCSDT calculations using approximate ways of handling connected triply excited clusters which are more robust than CCSD(T). We examine these two questions in the present study, focusing on the ability of methods with up to triply and with up to quadruply excited clusters based on the recently proposed CC(P ;Q) formalism 48–51 to reproduce the ground-state PECs and the corresponding De , re , and vibrational term values of Be2 resulting from the frozen-core and all-electron CCSDT and frozen-core CCSDTQ (= full CI) calculations. We recall that the CC(P ;Q) methodology allows one to merge the completely renormalized (CR) CC methods and their equation-ofmotion (EOM) 52–54 excited-state extensions, which originate from the method of moments of CC (MMCC) equations, 48,55–69 such as CR-CC(2,3), 63–66 CR-CC(2,4), 63,64,70 and CREOMCC(2,3), 65,67,71 with the active-space CC and EOMCC approaches. 47,72–85 Thus, in this work, we report the results of the CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) calculations, which use the CC(P ;Q) formalism to correct energies obtained with the active-space CCSDt, CCSDtq, and CCSDTq approaches for the remaining triples (CC(t;3) and CC(t,q;3)), the remaining triples and quadruples (CC(t,q;3,4)), and the remaining quadruples (CC(q;4)) missing in CCSDt, CCSDtq, and CCSDTq. In order to enrich our discussion and appreciate the advantages offered by the CC(P ;Q)-based CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) schemes compared to other single-reference CC methods, we also report the results obtained with the more established approaches, which correct the CCSD energies for triples or triples and quadruples, including CCSD(T), CCSD(2)T , 86 CR-CC(2,3), CCSD(TQ), 57 and CR-CC(2,4). For CCSDT and approximations to it, including CCSD(T), CCSD(2)T , CR-CC(2,3), CCSDt, and CC(t;3), we use the augmented polarized valence correlationconsistent aug-cc-pVTZ and aug-cc-pVQZ basis sets and their core-valence aug-cc-pCVTZ and aug-cc-pCVQZ counterparts needed in all-electron calculations, developed in ref 87. In the case of CCSDTQ and the various approximate treatments of quadruples or triples and quadruples, including CCSD(TQ), CR-CC(2,4), CCSDtq, CC(t,q;3), CC(t,q;3,4), CCSDTq, and CC(q;4), we rely on the aug-cc-pVTZ basis.

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Although the present study has a predominantly methodological character, where the main goal is to test the recently developed CC(P ;Q)-based CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) approaches against the parent full CCSDT and CCSDTQ data, we also report the results obtained by combining our highest-level frozen-core CC(q;4) and CCSDTQ calculations using the aug-cc-pVTZ basis set with the all-electron, complete basis set (CBS) limit CCSDT data. This allows us to make comparisons with the experimentally derived De , re , and vibrational term values characterizing the beryllium dimer, reported in ref 10, while comparing the CC(q;4) and CCSDTQ methods with each other.

2. Theory and Computational Details 2.1. Theory As explained in the Introduction, the main objective of this work is the examination of the ability of the CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) approaches, which rely on the CC(P ;Q) formalism proposed in refs 48 and 49, to reproduce the parent CCSDT and CCSDTQ results for the PEC and vibrational spectrum of Be2 in the X 1 Σ+ g state. Thus, we begin by summarizing the key elements of the CC(P ;Q) methodology relevant to this study. In the CC(P ; Q) formalism, which can be viewed as a generalization of the biorthogonal MMCC expansions that previously resulted in the CR-CC(mA , mB ) (e.g., CR-CC(2,3) and CR-CC(2,4)) and CR-EOMCC(mA , mB ) approaches 63–67,70 to unconventional truncations in the cluster and excitation operators of CC/EOMCC, such as those characterizing the activespace theories, 85 we start by introducing two subspaces of the N -electron Hilbert space H of interest, referred to as the P and Q spaces, H (P ) and H (Q) , respectively. The first subspace, H (P ) , is spanned by the excited determinants |ΦK = EK |Φ in which we perform the initial CC/EOMCC calculations (EK is the elementary particle-hole excitation operator that generates |ΦK from the reference determinant |Φ). The second subspace, H (Q) ⊆ (H (0) ⊕H (P ) )⊥ , where H (0) is a one-dimensional manifold spanned by |Φ, is spanned by the 6


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excited determinants |ΦK outside the P -space, which are used to construct the noniterative (P )

corrections to the CC/EOMCC energies Eμ

obtained in H (P ) . The main objective of the

CC(P ; Q) methodology is to provide expressions for the noniterative corrections δμ (P ; Q) (P )

to the energies Eμ

resulting from the CC/EOMCC calculations in the P space H (P ) ,

which can describe the many-electron correlation effects due to the Q-space determinants |ΦK ∈ H (Q) . In the following, we focus on the ground-state (μ = 0) CC(P ;Q) equations. The ground-state CC(P ; Q) energy, which results in the CC(t;3), CC(t,q;3), and CC(t,q;3,4) approaches implemented in refs 48–51 and the CC(q;4) method developed for the purpose of this work, is defined as (P +Q)

E0

(P )

= E0

+ δ0 (P ; Q),

(1) (P )

where the noniterative correction δ0 (P ; Q) to the P -space CC energy E0

due to the many-

electron correlation effects captured by the Q-space determinants |ΦK ∈ H (Q) is calculated as 48,49,51 δ0 (P ; Q) =

0,K (P ) M0,K (P ).

(2)

(Q)

|ΦK ∈H (P ) rank(|ΦK )≤min(N0 ,Ξ(Q) )

The M0,K (P ) quantities entering eq 2, given by ¯ (P ) |Φ, M0,K (P ) = ΦK |H

(3)

¯ (P ) = e−T (P ) HeT (P ) H

(4)

where

and T (P ) =

tK EK ,

(5)

|ΦK ∈H (P )

are the generalized moments 55,56 of the P -space CC equations corresponding to projections of these equations on the |ΦK determinants from the Q space H (Q) . In other words, we 7



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first solve the CC equations in the P space H (P ) , i.e., M0,K (P ) = 0, where |ΦK ∈ H (P ) , (P )

to obtain the cluster amplitudes tK defining T (P ) and the corresponding energy E0

= (P )

¯ (P ) |Φ, and then proceed toward the determination of the δ0 (P ; Q) correction to E0 Φ|H

using moments M0,K (P ), eq 3, obtained by projecting the left-hand side of the connected¯ (P ) |Φ, on the Q-space determinants cluster form of the electronic Schrödinger equation, H (P )

|ΦK ∈ H (Q) . The N0

integer in eq 2 is the highest many-body rank of the excited

determinants |ΦK relative to |Φ (rank(|ΦK )), for which the generalized moments M0,K (P ) of the P -space CC equations are still nonzero, and Ξ(Q) is the highest many-body rank of the excited determinant(s) |ΦK included in H (Q) . The issue remains how to determine the 0,K (P ) amplitudes that multiply moments M0,K (P ) in eq 2. As in our earlier work on the CC(P ; Q) approaches, 48–51 we adopt the quasi-perturbative formula reminiscent of the CR-CC(mA , mB ) methods, namely, ¯ (P ) |ΦK /D0,K (P ), 0,K (P ) = Φ|(1 + Λ(P ) )H

(6)

where the denominator D0,K (P ) is given by the Epstein–Nesbet-like expression (P )

D0,K (P ) = E0

¯ (P ) |ΦK , − ΦK |H

(7)

which can be approximated further, if desired (see below). The Λ(P ) operator in eq 6 is the hole-particle deexcitation operator, given by Λ(P ) =

|ΦK ∈H

λK (EK )† ,

(8)

(P )

(P ) ) (P ) (P ) −T (P ) ˜ (P which defines the CC bra state Ψ )e matching the |Ψ0 = eT |Φ 0 | = Φ|(1 + Λ

ket obtained in the calculations in the P space H (P ) . One determines Λ(P ) by solving (P ) ¯ open |ΦK = 0, where |ΦK ∈ H (P ) , the left CC equations in H (P ) , i.e., Φ|(1 + Λ(P ) )H (P ) ¯ open ¯ (P ) −E0(P ) 1, and 1 is the unit operator. To enforce strict invariance of the δ0 (P ; Q) H =H

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(P )

(P +Q)

correction to E0 , or of the total energy E0

, with respect to rotations among degenerate

orbitals (relevant to non-Abelian groups), while using the Epstein–Nesbet partitioning of ¯ (P ) in the Q-space block of it that leads to eqs 6 and 7, the above expressions for the H 0,K (P ) amplitudes must be replaced by linear systems of equations involving small blocks ¯ (P ) that couple the degenerate determinants |ΦK ∈ H (Q) (see of the matrix representing H ref 48 for the details). The CC(P ;Q) formalism, as summarized above, is very flexible, allowing us to formulate novel approaches, such as the CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) hierarchy tested in the present study, while encompassing various other conventional and unconventional CC approximations within a single, unifying mathematical framework. This is reflected in Table 1, which displays the key elements of the CC methods employed in this work, all summarized using the language of the CC(P ;Q) theory, along with the corresponding CPU time scalings, where no , nu , No , and Nu are the numbers of correlated occupied, unoccupied, active occupied, and active unoccupied orbitals, respectively. For example, the groundstate CC(P ; Q) formalism contains the previously formulated biorthogonal CR-CC(mA , mB ) approximations, including CR-CC(2,3) and its CCSD(2)T counterpart and CR-CC(2,4) as special cases. Indeed, we obtain the CR-CC(mA , mB ) scheme when the P space H (P ) ...an in the above considerations is spanned by all |Φai11...i determinants with the excitation n

rank n ≤ mA and the Q space H (Q) by those with mA < n ≤ mB , where mB ≤ N . As usual, i1 , i2 , . . . or i, j, . . . and a1 , a2 , . . . or a, b, . . . are the occupied and unoccupied ...an ...an spin-orbitals, respectively, in the reference determinant |Φ, |Φai11...i = Eia11...i |Φ, and n n ...an = nκ=1 aaκ aiκ , where ap and ap are the creation and annihilation operators associated Eia11...i n

with spin-orbital p. The conventional CCSD(T) and CCSD(TQ) approaches can be expressed using the language of the CC(P ;Q) formalism as well, since CCSD(T) is an approximation to CCSD(2)T and CR-CC(2,3), whereas the CCSD(TQ) scheme adopted in this work, which is equivalent to the CCSD(TQ),b method of ref 57, is an approximation to CR-CC(2,4). The relationships between the CCSD(T) approach and the CCSD(2)T and CR-CC(2,3) methods

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have been discussed in refs 63, 64, and 66. For the relationship between CCSD(TQ) and CR-CC(2,4), we refer the reader to ref 88 and references therein. However, in designing the CC(P ; Q) energy expansions given by eqs 1 and 2, where moments M0,K (P ) and amplitudes 0,K (P ) for |ΦK ∈ H (Q) are defined by eqs 3 and 6, respectively, one has the flexibility of choosing other, less conventional P and Q spaces that go beyond the reconstruction of the existing CCSD(T), CCSD(TQ), CCSD(2)T , CR-CC(2,3), and CR-CC(2,4) (in general, CR-CC(mA , mB )) approaches, while retaining the relatively low computational costs when compared to the full CCSDT and CCSDTQ methods. This is exactly what is done in the CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) approaches benchmarked in this study. As shown in Table 1, we obtain the CC(t;3) energy by solving the CCSDt equations in the P space H (P ) = G (SD) ⊕ G (t) and correcting the resulting CCSDt energy for the triples outside the active set using the δ0 (P ; Q) correction defined in the Q space H (Q) = G (T) G (t) , (t) = span |ΦAbc where G (SD) = span |Φai , |Φab ij i < j, a < b , G ijK i < j < K, A < b < c , and G (T) = span |Φabc ijk i < j < k, a < b < c . Here and elsewhere in this article, we adopt the notation in which bold capital-case indices represent active spin-orbitals (I, J, . . . for the active occupied and A, B, . . . for the active unoccupied ones). We continue using lower-case italics indices, if the active/inactive character of a given spin-orbital is not specified. In the CC(t,q;3,4) approach, we obtain the corresponding energy by solving the CCSDtq equations in the P space H (P ) = G (SD) ⊕G (t) ⊕G (q) and correcting the resulting CCSDtq energy for the subsets of triples and quadruples in the subspace H (Q) = (G (T) G (t) )⊕(G (Q) G (q) ), where G (t) is defined as in the CC(t;3) case, G (q) = span |ΦABcd ijKL i < j < K < L, A < B < c < i < j < k < l, a < b < c < d . The CC(t,q;3) scheme is an d , and G (Q) = span |Φabcd ijkl approximation to CC(t,q;3,4), in which the contributions due to quadruples from the G (Q)

G (q) subspace, present in CC(t,q;3,4), are simply ignored. Finally, we obtain the CC(q;4) energy by solving the CCSDTq equations in the P space H (P ) = G (SD) ⊕ G (T) ⊕ G˜(q) and correcting the resulting CCSDTq energy for the subsets of quadruply excited determinants i < j < k < L, A < b < c < d . In in H (Q) = G (Q) G˜(q) , where G˜(q) = span |ΦAbcd ijkL

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defining the above subspaces G (t) , G (q) , and G˜(q) , we have followed a general recipe how to design the active-space CC methods at any level of truncation in the cluster operator laid down in refs 74 and 85. As one can see in Table 1, in the triples corrections δ0 (P ; Q) entering the CR-CC(2,3), CC(t;3), and CC(t,q;3) energy expressions and in the triples contributions to δ0 (P ; Q) defining the CR-CC(2,4) and CC(t,q;3,4) approaches adopted in the present study, we use the Epstein–Nesbet-like denominators D0,K (P ) defined by eq 7. We do it, since the replacement of the Epstein–Nesbet D0,K (P ) denominators by their Møller–Plesset counterparts, which ¯ (P ) on can formally be obtained by approximating the similarity-transformed Hamiltonian H the right-hand side of eq 7 by the bare Fock operator, assuming that a canonical Hartree– Fock basis set is employed at the same time, does not lead to any savings in the CPU timings associated with the triples corrections, which all scale as n3o n4u independent of the form of D0,K (P ), while worsening the results (see refs 48–51 and 63–67 for additional details and examples). The latter problem can be seen in this paper too, when we compare the CRCC(2,3) results for Be2 obtained with the Epstein–Nesbet form of the D0,K (P ) denominator with the results obtained using CCSD(2)T , which relies on the Møller–Plesset-like expression for D0,K (P ), while being otherwise identical to CR-CC(2,3). The situation changes when the quadruples corrections δ0 (P ; Q) entering the CC(q;4) energy expression and the quadruples contributions to δ0 (P ; Q) defining the CR-CC(2,4) and CC(t,q;3,4) approaches are examined. In this case, we use the Møller–Plesset form of the relevant D0,K (P ) denominators, since this allows us to reduce the n4o n5u steps associated with the use of the Epstein–Nesbet-like expressions for D0,K (P ) to a considerably more manageable n2o n5u level while avoiding formal issues related to the coupling of triples and quadruples in the CR-CC(2,4) and CC(t,q;3,4) corrections, which disappear when the quadruples contributions to these corrections use the Møller–Plesset form of D0,K (P ) (see refs 48, 51, and 88 for further information). To summarize, in reporting the CC(t;3), CC(t,q;3), and CC(t,q;3,4) results, we focus on the calculations that use the better Epstein–Nesbet-like D0,K (P ) denominators in defining

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the respective triples corrections, since this improves the results without increasing the computational costs. The Møller–Plesset form of the D0,K (P ) denominators is used only in the quadruples corrections of CC(t,q;3,4) and CC(q;4) to avoid the more expensive n4o n5u scaling. A comparison of the PECs and the De , re , and vibrational term values of Be2 resulting from the CC(t;3), CC(t,q;3), and CC(t,q;3,4) calculations based on the Epstein–Nesbet-like denominators D0,K (P ) in the triples corrections with their counterparts using the Møller– Plesset form of the D0,K (P ) denominators in the triples as well as quadruples corrections can be found in the Supporting Information. This comparison confirms our earlier observations, reported in refs 50 and 51, that the replacement of the Epstein–Nesbet form of the D0,K (P ) denominators in the triples corrections by their Møller–Plesset-like counterparts worsens the CC(t;3), CC(t,q;3), and CC(t,q;3,4) PECs and their spectroscopic characteristics, validating our decision to focus on the former form when discussing the CC(t;3), CC(t,q;3), and CC(t,q;3,4) results for the beryllium dimer in Section 3.

2.2. Computational Details All of the CC calculations for Be2 performed in this study were based on the restricted Hartree–Fock (RHF) reference functions. The CCSD, CCSD(T), CCSD(TQ), CCSD(2)T , and CR-CC(2,3) computations were performed using codes developed by our group, 56,57,63,64,89 which are available in the GAMESS package 90,91 as standard options. The CC(t;3), CC(t,q;3), and CC(t,q;3,4) calculations, along with the underlying CCSDt and CCSDtq and parent CCSDT and CCSDTQ computations, were performed using our in-house package interfaced with the integral, Hartree–Fock, and integral transformation routines in GAMESS, developed in refs 48–51, although in the case of CCSDTQ, where we froze core electrons, we also used the GAMESS determinantal full CI code, since CCSDTQ and full CI are equivalent when only four electrons are correlated and the efficient implementation of full CI in GAMESS can take advantage of the spatial symmetry, which our existing CCSDTQ routines cannot yet do. The CR-CC(2,4) calculations used our standalone codes, interfaced with GAMESS as 12


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well. The CCSDTq and CC(q;4) computations were performed using the pilot codes developed for the purpose of this study by making suitable modifications in the active-space CC and CC(P ;Q) routines described in ref 51. Because of their pilot character, they are not as efficient as theoretically possible, especially when compared with the more mature CCSDt, CC(t;3), and CCSDT routines, but they do produce correct results, so that we can examine the accuracy of the CCSDTq and CC(q;4) methods, when applied to Be2 . All of the active-space CC approaches and the various CC(P ;Q) corrections employed in this work used two active occupied and six active unoccupied orbitals corresponding to the 2s and 2p valence shells of the beryllium atoms, i.e., the No and Nu values that enter the formulas for CPU time scalings in Table 1 were set at 2 and 6, respectively. In the case of the frozen-core CC calculations, we used the aug-cc-pVTZ (all methods) and aug-cc-pVQZ (all methods up to full CCSDT) basis sets, abbreviated in this work as ATZ and AQZ, respectively, developed in ref 87 and taken from the EMSL Basis Set Library. 92,93 In the all-electron calculations, which we performed for all methods including up to triply excited clusters, we used the core-valence aug-cc-pCVTZ (ACTZ) and aug-cc-pCVQZ (ACQZ) bases, 87 taken from the EMSL Basis Set Library as well. For the composite schemes discussed in Section 3.3, in which our highest-level frozen-core CC(q;4)/ATZ and CCSDTQ/ATZ calculations are combined with the all-electron CCSDT energetics extrapolated to the CBS limit, we also used the aug-cc-pCV5Z basis, abbreviated in this article as AC5Z, which we needed to determine the converged RHF energies. We took this basis set, which was originally developed in ref 87, from the Peterson group’s website. 94 Finally, the vibrational term values corresponding to the ground-state PECs of Be2 resulting from the various CC calculations reported in this work were determined by numerically integrating the radial Schrödinger equation from 2.1 to 12.0 ˚ A using the Numerov-Cooley algorithm 95 found in the Level 8.2 code. 96 For each method, we used the following internuclear separations r to determine the relevant electronic energies: 2.1, 2.2, 2.3, 2.4, 2.454, 2.5, 2.55, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 3.8, 4.0, 4.4, 4.8, 5.2, 5.6, 6.0, 7.0, 8.0, and 12.0 ˚ A. We also used Level 8.2 to determine the De and re values.

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Although our existing CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) routines, especially the newest quadruples corrections of CC(t,q;3,4) and CC(q;4), need more work to improve their efficiency, some of them, especially the CC(t;3) code and its underlying CCSDt and parent CCSDT counterparts, are efficient enough to demonstrate the computational benefits of employing the CC(P ;Q) methodology. This is illustrated in Table 2, where we show the computational (CPU) timings characterizing the various CC approaches approximating CCSDT, including CCSD(T), CR-CC(2,3), CCSDt, and CC(t;3), along with CCSD and CCSDT, in the all-electron calculations for the beryllium dimer at r = 2.454 ˚ A using the largest basis set employed in this work, namely, ACQZ, consisting of 218 orbitals. To make our analysis meaningful, in reporting the timings for CCSD, CCSD(T), and CR-CC(2,3), we used the spin-integrated spin-orbital codes, which can work with closed- and open-shell reference determinants and which were implemented in exactly the same way as CCSDt, CC(t;3), and CCSDT, i.e., with the help of our home-grown automated derivation and implementation software discussed, for example, in ref 51. The faster CCSD, CCSD(T), and CR-CC(2,3) routines, available in GAMESS as standard options, were not used for this purpose, since they were developed in a completely different manner (using manybody diagrammatics and manual code optimization). All of the timings shown in Table 2 correspond to single-core runs without taking advantage of the spatial D∞h symmetry of Be2 . The CPU times associated with the RHF and integral transformation steps are ignored. As demonstrated in Table 2, the computational benefits of using the CC(t;3) methodology as a substitute for full CCSDT are enormous, especially when we realize that the CC(t;3) PECs and the corresponding De , re , and vibrational term values are practically as accurate as their CCSDT counterparts (see Section 3.1). In the case of the all-electron calculations for Be2 employing the ACQZ basis set, the CC(t;3) approach reduces the CPU time required by the parent CCSDT calculation by a factor of more than 20, with only minimal loss of accuracy in the description of the PEC and its spectroscopic characteristics. The computational cost of improving the CCSDt energy by the noniterative CC(t;3) correction is only about 2 %

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of the CPU time required by the CCSDt iterations, which is an important observation too, since, as shown in the next section, CC(t;3) is a lot more accurate. The CC(t;3) calculation considered in Table 2 is 14–18 times longer than its CR-CC(2,3) and CCSD(T) counterparts, but we must keep in mind that CC(t;3) is also considerably more accurate, particularly when the De and vibrational term values are examined (see Section 3.1). Furthermore, the CC(t;3) results remain virtually identical to their full CCSDT counterparts in situations involving covalent bond breaking, where CCSD(T) fails. 48–51 The CR-CC(2,3) method improves the performance of CCSD(T) in such cases, but CC(t;3) is substantially more effective in this regard, 48–51 while being more accurate than CR-CC(2,3) for the weakly bound Be2 system. It is, therefore, useful to invest in the CC(t;3) methodology, since it improves the CCSDbased CCSD(T) and CR-CC(2,3) results for covalent and, as shown in this work, non-covalent interactions, while being a lot less expensive than full CCSDT. Work is underway to improve the performance of our CC(t;3) codes by further optimizing the underlying CCSDt routines. Although the main focus of this study is CC(t;3) and its CC(t,q;3), CC(t,q;3,4), and CC(q;4) extensions to quadruples, it is worth noticing that the CR-CC(2,3) approach is only slightly more expensive than its less robust CCSD(T) counterpart. Indeed, as shown in Table 2, the CPU time required by the all-electron CR-CC(2,3) calculations for the beryllium dimer at r = 2.454 ˚ A, as described by the ACQZ basis set, is only 32 % longer than that required by CCSD(T). At the same time, as demonstrated in Section 3.1, the quality of the PECs and the De and vibrational term values obtained with CR-CC(2,3) is substantially higher than the quality of the corresponding CCSD(T) results. This shows once again that it is generally better to use the biorthogonal MMCC expansions, such as those developed in refs 48 and 63–65, in deriving the noniterative corrections to CC energies than the traditional perturbation theory considerations. The excellent performance of the CC(t;3) approach and its extensions to connected quadruples, especially CC(q;4), discussed in the next section, combined with the savings in the computational effort compared to CCSDT and CCSDTQ, summarized in Table 1 and illustrated by the timings in Table 2, reinforces this point.

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3. Results and Discussion The results of our calculations for the beryllium dimer are summarized in Tables 3–13 and Figures 1 and 2. We divide our discussion into three parts. First, in Section 3.1, we compare the ground-state PECs and the corresponding De , re , and vibrational term values obtained with CCSDT with the results of calculations using the various approximate treatments of triples, including the CCSD(T), CCSD(2)T , and CR-CC(2,3) corrections to CCSD and the CC(t;3) correction to CCSDt. Next, in Section 3.2, we examine the ability of the various approximations to CCSDTQ, including the CCSD(TQ) and CR-CC(2,4) corrections to CCSD, the CC(t,q;3) and CC(t,q;3,4) corrections to CCSDtq, and the CC(q;4) correction to CCSDTq, to reproduce the ground-state PEC and the De , re , and vibrational term values obtained with CCSDTQ. Finally, in Section 3.3, we compare the De and re values and the vibrational spectra obtained using two composite schemes, in which our highest-level frozen-core CC(q;4) and CCSDTQ calculations are combined with the all-electron CCSDT energetics extrapolated to the CBS limit, with experiment and with each another.

3.1. Methods Incorporating Connected Triple Excitations The ground-state PECs of Be2 resulting from the approximate treatments of triples, including the CCSD-based CCSD(T), CCSD(2)T , and CR-CC(2,3) approaches, the active-space CCSDt method, and the CC(t;3) correction to CCSDt, using the ATZ and AQZ basis sets in the frozen-core calculations and the ACTZ and ACQZ bases when correlating all electrons, and their full CCSDT counterparts can be found in Tables 3–6 (electronic energies at selected internuclear separations r) and Figure 1. The corresponding De , re , and vibrational term values are reported in Tables 7–10. As shown in Tables 3–10 and Figure 1, the CR-CC(2,3) method outperforms the other employed triples corrections to CCSD. Indeed, for the four basis sets used in this work, the CR-CC(2,3) calculations recover 86–88 % of the corresponding CCSDT De values, as opposed

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to 73–81 % obtained with CCSD(T) and the even worse 58–65 % obtained with CCSD(2)T . This is a consequence of the more accurate description of the X 1 Σ+ g state of Be2 by the CRCC(2,3) approach, when compared to CCSD(T) and CCSD(2)T , which we can illustrate by examining the mean unsigned error (MUE) and non-parallelity error (NPE) values relative to CCSDT characterizing the corresponding PECs. As shown in Table 3, when the ATZ basis set is employed and the core electrons of Be2 are frozen in post-RHF calculations, the MUE and NPE values relative to CCSDT characterizing the CCSD(T) and CCSD(2)T potentials are 0.299 and 0.829 millihartree, respectively, in the former case and 0.502 and 1.609 millihartree, respectively, in the case of the latter method. This should be compared to the noticeably lower MUE and NPE values characterizing the corresponding CR-CC(2,3) calculations, which are 0.149 and 0.458 millihartree, respectively. The CR-CC(2,3) approach offers similar improvements in the CCSD(T) and CCSD(2)T electronic energies, when other basis sets are employed and when all electrons are correlated. For example, when we replace the valence ATZ basis set by its core-valence ACTZ counterpart and correlate all electrons (Table 4), the CR-CC(2,3) method reduces the MUE and NPE values relative to CCSDT characterizing the CCSD(T) and CCSD(2)T PECs from 0.327 and 0.785 millihartree in the former case and 0.620 and 1.626 millihartree in the case of the latter method to 0.238 and 0.514 millihartree, respectively. The improvements in the CCSD(T) energies offered by the CR-CC(2,3) approach appear to be less significant when we turn to the ACQZ basis (Table 6), but this is misleading, since the actual shape of the all-electron CR-CC(2,3)/ACQZ potential is much closer to that obtained in the corresponding full CCSDT calculations than the shape of the CCSD(T)/ACQZ PEC. This will become clearer when we start discussing the resulting vibrational term values. The fact that the shapes of the ground-state PECs of Be2 obtained with the CR-CC(2,3) method are closer to the shapes of the corresponding CCSDT potentials than those resulting from the CCSD(T) and CCSD(2)T calculations can be seen in multiple ways. For example, as shown in Figure 1, each CCSD(T) PEC has an unphysical shoulder around r = 3.4 ˚ A, which becomes even more pronounced when the

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CCSD(2)T method is used instead. This shoulder is much less pronounced when we use the CR-CC(2,3) approach, in perfect agreement with the corresponding full CCSDT curves. The superiority of the CR-CC(2,3) PECs over those obtained with CCSD(T) and CCSD(2)T is clearly reflected in the corresponding vibrational term values G(v). Although CCSD(T) recovers the same number of vibrational levels as CR-CC(2,3) (compared to the parent CCSDT spectra, both methods miss one vibrational state), the CR-CC(2,3) vibrational term values agree with their CCSDT counterparts more accurately than those obtained with CCSD(T). Indeed, as shown in Tables 7–10, in the majority of cases, the CR-CC(2,3) approach reduces the deviations from CCSDT characterizing the CCSD(T) G(v) values by factors of 1.5–2.5 or so. The situation is even better, in favor of CR-CC(2,3), when the CR-CC(2,3) vibrational term values are compared with those obtained with CCSD(2)T . In this case, the CR-CC(2,3) approach reduces the errors in the G(v) values resulting from the CCSD(2)T calculations relative to CCSDT by factors that often are as large as 2.5–5. Thus, in spite of the fact that the underlying CCSD potentials display very shallow minima in the long-range r ≈ 4.4–4.5 ˚ A region, which is a totally wrong description, the triples correction of CR-CC(2,3) is powerful enough to produce PECs that are of generally good quality, when compared to CCSDT, while being substantially better than those produced by the CCSD(T) and CCSD(2)T methods, independent of the number of electrons correlated in the calculations or the size of the basis set. The CR-CC(2,3) approach constitutes a significant improvement over the other noniterative triples corrections to the CCSD energies, but the results of the CR-CC(2,3) calculations for the ground-state PEC of Be2 are still not as good as desired, when compared to the parent CCSDT data. For example, as shown in Table 10, the CR-CC(2,3)/ACQZ zero-point vibrational energy, G(v = 0), differs from the corresponding CCSDT/ACQZ value by 7.3 cm−1 or 6 %. This is a substantial improvement over CCSD(2)T , but not over CCSD(T). For the last, v = 9, vibrational level supported by the CR-CC(2,3)/ACQZ potential, the difference between the CR-CC(2,3)/ACQZ and CCSDT/ACQZ G(v) values of 105.5 cm−1 or 14 % is

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considerably smaller than the 146.3 cm−1 or 19 % difference obtained with CCSD(T), and the CCSD(2)T potential does not even support the v = 9 bound state, but the 14 % error relative to CCSDT characterizing the CR-CC(2,3)/ACQZ G(v = 9) value is rather large. Furthermore, all CCSDT PECs, including that resulting from the largest all-electron CCSDT calculations performed in this study using the ACQZ basis set, support 11 vibrational states, as opposed to 10 states supported by the CR-CC(2,3) PECs. This is a consequence of the fact that the CR-CC(2,3) potentials, although considerably better than those provided by the CCSD(T) and CCSD(2)T methods, are too shallow, when compared to their full CCSDT counterparts, by 77–110 cm−1 or 12–14 %, when the corresponding binding energies De are analyzed. Clearly, it would be useful to improve these results while retaining the simplicity of the idea of noniterative triples corrections. Let us, therefore, examine the CC(t;3) calculations, in which instead of correcting the CCSD energies for all triples, as is done in CCSD(T), CCSD(2)T , and CR-CC(2,3), we correct the CCSDt energies for the subset of triples outside the G (t) subspace defined in Section 2.1, missing in CCSDt. First of all, it is interesting to note that unlike CCSD, which neglects the connected triply excited (T3 ) clusters altogether, the CCSDt approach that includes some of them through the use of active orbitals tries to capture the right minimum in the r ≈ 2.5 ˚ A region (see Figure 1). Although this alone is not sufficient to produce the correct PEC, since the remaining triple excitations outside the G (t) subspace are important for eliminating the secondary minima on the CCSDt PECs at larger internuclear separations, which are remnants of the analogous minima characterizing the corresponding CCSD potentials, it is quite clear that the CCSDt methodology improves the description of the ground electronic state of the beryllium dimer at shorter internuclear separations compared to CCSD. Obviously, the existence of two minima on each CCSDt PEC calculated in this work, one in the shorter range, which is physically meaningful, although much too shallow, and another, unphysical one, in the region of larger r values, makes the CCSDt PECs useless for the vibrational analysis, but we can expect that the CCSDt calculations improve

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the quality of the singly and doubly excited clusters, T1 and T2 , respectively, compared to CCSD, since in CCSDt they are determined in the presence of the dominant T3 contributions captured using active orbitals. The question then emerges how to improve the CCSDt description of the ground-state PEC of the beryllium dimer without going all the way to full CCSDT. We could try to improve the CCSDt results by increasing the numbers of active orbitals used to select the dominant triply excited cluster amplitudes, including, for example, the RHF orbitals correlating with the 3s and 3p or even higher shells of the beryllium atoms in the active space, but in this study we choose the less expensive and, in our view, more interesting alternative, which is based on adding the noniterative CC(t;3) corrections due to the triples missing in CCSDt to the CCSDt energies. When the active-space CCSDt energies are corrected for the triples that have not been captured by CCSDt using the CC(t;3) methodology, we see a dramatic improvement in the description of the X 1 Σ+ g PEC of Be2 by the underlying CCSDt and CCSD-based CR-CC(2,3) approaches. Indeed, as shown in Figure 1, the CC(t;3) method performs as good as CRCC(2,3) near the shoulder around r = 3.4 ˚ A, improving the relatively poor description of this region by CCSD(T) and CCSD(2)T , while further deepening the minimum and bringing the resulting PECs to an excellent agreement with their CCSDT counterparts. This is reflected in the tiny MUE and NPE values relative to CCSDT characterizing the CC(t;3) potentials (see Tables 3–6). The MUE values relative to CCSDT obtained in the CC(t;3) calculations range from 0.079 millihartree, when the ATZ basis set is employed, to 0.185 millihartree in the all-electron ACQZ case. The corresponding NPE values range from 0.165 millihartree, obtained in the all-electron CC(t;3) calculations using the ACTZ basis, to 0.216 millihartree when the AQZ basis set is utilized. These very small differences between the CC(t;3) and CCSDT PECs should be compared to the noticeably larger MUE and NPE values relative to CCSDT, of 0.149–0.294 and 0.458–0.631, millihartree, respectively, obtained in the CRCC(2,3) calculations, not to mention the much worse MUEs and NPEs obtained with CCSDt, which can be as large as 2.184 millihartree for the MUEs and 3.125 millihartree for the NPEs.

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As a result of all this, the binding energies De , the equilibrium bond lengths re , and the vibrational term values G(v) resulting from the CC(t;3) calculations are substantially better, when compared to their CCSDT parents, than their CR-CC(2,3) analogs. Indeed, as shown in Tables 7–10, the CC(t;3) calculations reduce the 77–110 cm−1 and 0.011–0.020 ˚ A errors relative to CCSDT in the De and re values obtained with CR-CC(2,3) to the much smaller A errors, respectively. Similarly impressive improvements are 37–47 cm−1 and 0.000–0.003 ˚ observed when the vibrational term values are examined. For example, when all electrons are correlated and the largest basis set employed in this study, namely, ACQZ, is considered, the CC(t;3) approach reduces the 7–106 cm−1 or 6–14 % errors relative to CCSDT obtained in the CR-CC(2,3) calculations for the v = 0–9 vibrational states supported by the CRCC(2,3) potential to 2–44 cm−1 or 2–6 %. Furthermore, with an exception of the ACTZ basis set, the ground-state PECs of Be2 obtained in the CC(t;3) calculations support the same number of vibrational levels as the full CCSDT potentials (eleven), which no other approximate treatment of triples examined in this study was able to do. In particular, when the all-electron calculations using the ACQZ basis set are examined, the highest, v = 10, vibrational state obtained with the CC(t;3) approach, which the corresponding full CCSDT calculation places at about 784 cm−1 , is within 45 cm−1 from the CCSDT value. All of the above discussion demonstrates that CC(t;3) is a robust method, capable of producing the results of nearly full CCSDT quality at a small fraction of the computational cost, even when the challenging case of the beryllium dimer is considered. The triples correction of CC(t;3), derived with the help of the CC(P ;Q) formalism, clearly benefits from the use of the singly and doubly excited clusters relaxed in the presence of the dominant T3 contributions captured by the CCSDt approach, enabling one to improve the CR-CC(2,3) calculations that employ the nonrelaxed T1 and T2 amplitudes, obtained in the absence of information about T3 with CCSD. Furthermore, by including the leading effects due to connected triply excited clusters, in addition to singles and doubles, in the P space and by using the CC(P ;Q) correction δ0 (P ; Q) to describe the remaining rather than all

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triples, we increase the likelihood of producing PECs that are much closer to their CCSDT counterparts than those obtained using the CR-CC(2,3) corrections to CCSD. Our next task, which is the focus on the next subsection, is to examine how effective the CC(P ;Q)-inspired CC(t,q;3) and CC(t,q;3,4) corrections to CCSDtq and the CC(q;4) correction to CCSDTq are in reproducing the ground-state PEC and the De , re , and vibrational term values of Be2 obtained with the full CCSDTQ approach, especially when compared to the conventional CCSD(TQ) and completely renormalized CR-CC(2,4) calculations.

3.2. Methods with Connected Triple and Quadruple Excitations The ground-state PECs of the beryllium dimer obtained with the approximate treatments of connected triply and quadruply excited clusters, including the CCSD-based CCSD(TQ) and CR-CC(2,4) methods, the active-space CCSDtq approach, and the CC(t,q;3) and CC(t,q;3,4) corrections to CCSDtq, along with the results of the active-space CCSDTq and CCSDTqbased CC(q;4) calculations, which approximate the connected quadruples while treating the triples fully, and the parent electronic energies at the various values of r obtained with CCSDTQ can be found in Table 11 and Figure 2. The corresponding De , re , and vibrational term values are shown in Table 12. We used only one basis set, namely, ATZ, and correlated valence electrons, so that the CCSDTQ results are equivalent to the exact, full CI, data. We begin our discussion with the triples and quadruples corrections to CCSD, represented in this work by the conventional CCSD(TQ) and completely renormalized CR-CC(2,4) methods. In analogy to CR-CC(2,3) vs CCSD(T), the CR-CC(2,4) calculations improve the description of the X 1 Σ+ g PEC of Be2 provided by CCSD(TQ). Indeed, as shown in Tables 11 and 12 and Figure 2, the CR-CC(2,4) potential is deeper than that provided by the CCSD(TQ) approach by about 104 cm−1 , when the ATZ basis set is employed, recovering 83 % of the corresponding CCSDTQ binding energy and eliminating the unphysical shoulder in the region of intermediate internuclear distances seen in the CCSD(TQ) calculations. There is no such shoulder when the full CCSDTQ method is used and the CR-CC(2,4) PEC does 22


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not have it either. As a result, the overall shape of the CR-CC(2,4) PEC of Be2 is in good agreement with the parent CCSDTQ potential, reducing the MUE and NPE values relative to CCSDTQ characterizing the CCSD(TQ) calculations using the ATZ basis set from 0.392 and 1.145 millihartree to 0.226 and 0.744 millihartree, respectively. Similar improvements are observed when we compare the CCSD(TQ) and CR-CC(2,4) vibrational term values G(v) with their CCSDTQ counterparts. Indeed, as shown in Table 12, the CR-CC(2,4) approach reduces the relatively large deviations from CCSDTQ characterizing the CCSD(TQ) G(v) values corresponding to the v = 0–9 levels supported by the CCSD(TQ) potential, which range from ∼13 cm−1 for v = 0 to more than 219 cm−1 for v = 9, by a factor of about 2. Furthermore, unlike the CCSD(TQ) PEC, the CR-CC(2,4) potential supports the same number of vibrational levels as that obtained in the full CCSDTQ calculations (eleven). We can conclude this part of our discussion by stating that the CR-CC(2,4) method improves the CCSD(TQ) results for the beryllium dimer in a substantial manner, but it is also quite clear from Tables 11 and 12 and Figure 2 that the ground-state PEC of Be2 obtained with CR-CC(2,4) is not as accurate as one would like it to be. For example, the CRCC(2,4) zero-point energy, G(v = 0), differs from the corresponding CCSDTQ value by about 7 cm−1 or 6 % and the error for the last, v = 10, vibrational level relative to CCSDTQ is 121 cm−1 or 17 %. In analogy to the previously discussed CCSDt calculations, the active-space CCSDtq approach tries to capture the correct minimum in the r ≈ 2.5 ˚ A region, but because CCSDtq is missing the higher-order dynamic correlation effects beyond the small subsets of triples and quadruples identified using active orbitals one ends up with two minima, one in the shorter range, which is physically meaningful, and another one in the region of larger r values, similar to the unphysical minimum observed in the CCSD calculations (see Figure 2). In spite of this, the performance of the CC(t,q;3) and CC(t,q;3,4) approaches, which correct the CCSDtq energies for the missing triples (CC(t,q;3)) or missing triples and quadruples (CC(t,q;3,4)) outside the G (t) and G (q) subspaces defined in Section 2.1, is very good. Both CC(P ;Q) methods outperform the CCSD-based CR-CC(2,4) approach, especially around the

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equilibrium geometry, since they use the higher-quality T1 and T2 amplitudes obtained with CCSDtq, which are relaxed in the presence of the dominant connected triply and quadruply excited contributions captured by CCSDtq. As shown in Table 11, when the ATZ basis set is employed, the CC(t,q;3) and CC(t,q;3,4) methods reduce the MUE values relative to the CCSDTQ potential obtained in the underlying CCSDtq and CCSD-based CR-CC(2,4) calculations from 1.187 millihartree in the former case and 0.226 millihartree in the case of the latter approach to 0.176 and 0.135 millihartree, respectively. Similar error reductions are observed when the NPE values relative to the CCSDTQ PEC are examined. In this case, the CC(t,q;3) and CC(t,q;3,4) corrections to CCSDtq reduce the 0.744 and 3.284 millihartree NPE values relative to CCSDTQ resulting from the CR-CC(2,4) and CCSDtq calculations to 0.478 and 0.332 millihartree, respectively. This substantially improved description of the X 1 Σ+ g PEC of the beryllium dimer offered by the CC(t,q;3) and CC(t,q;3,4) schemes is reflected in the resulting De , re , and vibrational term values shown in Table 12. Indeed, the CC(t,q;3,4) and CCSDTQ zero-point energies differ by less than 3 cm−1 or 2 %, as opposed to 7 cm−1 or 6 % observed in the CR-CC(2,4) calculations, when the ATZ basis set is employed. For the last, v = 10, vibrational level supported by the CCSDTQ/ATZ potential, the difference between the CC(t,q;3,4) and CCSDTQ results is about 70 cm−1 or 10 %, compared to the much worse 121 cm−1 or 17 % obtained with CR-CC(2,4). The A, respectively, are considerably CC(t,q;3,4)/ATZ De and re values of 647.0 cm−1 and 2.501 ˚ A resulting from the corresponding CCSDTQ calculations closer to 718.6 cm−1 and 2.498 ˚ than the De and re values of 595.9 cm−1 and 2.518 ˚ A obtained with CR-CC(2,4)/ATZ. Although the CC(t,q;3) approach, which corrects the CCSDtq energies for the triples outside the active set, but not for the missing quadruples, gives a somewhat worse description of the vibrational term values of Be2 than the more complete CC(t,q;3,4) method, it is still more accurate than CCSD(TQ), CR-CC(2,4), and CCSDtq (see Tables 11 and 12 and Figure 2). The performance of the CC(t,q;3) and CC(t,q;3,4) approaches is certainly encouraging, especially when we realize the huge savings in the computer effort offered by these methods

24


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compared to full CCSDTQ (see Table 1), but they still underbind Be2 relative to CCSDTQ by about 94 and 72 cm−1 , respectively, when the ATZ basis set is employed in the calculations, while producing similar errors in the description of the high-lying vibrational states near the dissociation threshold. This can be addressed by turning to the higher-level CC(P ;Q)-based methodology, abbreviated as CC(q;4), which corrects the CCSDTq energies for the subset of quadruples missing in CCSDTq. By treating the connected triples fully, the CCSDTq approach alone is already very accurate, improving the CC(t,q;3) and CC(t,q;3,4) energies at all internuclear separations (see Table 11 and Figure 2). The CC(q;4) correction to CCSDTq improves the high-accuracy CCSDTq energies even further, so that it is hard to tell the differences between the CC(q;4) and CCSDTQ PECs. Indeed, as shown in Table 12, the CC(q;4) scheme captures 97 % of the CCSDTQ binding energy, having the virtually identical equilibrium bond length, while reducing the already very small MUE and NPE values relative to CCSDTQ resulting from the CCSDTq calculations, of 0.062 and 0.144 millihartree, respectively, to the even smaller 0.039 millihartree for MUE and 0.090 millihartree for NPE, when the ATZ basis set is employed (see Table 11). As a result, the entire vibrational manifold resulting from the CC(q;4) calculations is very close to that obtained with full CCSDTQ. As can be seen in Table 12, the CC(q;4) and CCSDTQ zero-point energies differ by a mere 0.4 cm−1 . For the last, v = 10, level supported by the CC(q;4)/ATZ and CCSDTQ/ATZ potentials, the difference between the CC(q;4) and CCSDTQ data is only about 19 cm−1 or 3 %, which is a substantial improvement over 31 cm−1 obtained with the underlying CCSDTq calculations, not to mention 121 and 70 cm−1 obtained with the CR-CC(2,4) and CC(t,q;3,4) corrections to CCSD and CCSDtq, respectively. It is quite clear from the results in Tables 11 and 12 and Figure 2 that we should be able to use the CC(q;4) approach as a highly reliable substitute for the more expensive full CCSDTQ method, when examining the ground-state PEC and the corresponding spectroscopic characteristics of the beryllium dimer. An illustration of this is provided in the next subsection, where we discuss the results of composite calculations using the CC(q;4) and

25



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CCSDTQ calculations to capture the connected quadruples on top of the converged CCSDT information, which are compared with each other and with the experimentally derived data.

3.3. Comparison with Experiment In order to provide final insights into the performance of the CC(P ;Q) approaches, especially the most accurate CC(P ;Q) approximation examined in this work, CC(q;4), which accounts for the triply and quadruply excited clusters, we devised two composite schemes, referred to as schemes A and B, in which the ground-state electronic energies are calculated as (A)

E0

(RHF/AC5Z)

= E0

(CCSDT/CBS)

+ ΔE0

(CC(q;4)/ATZ)

+ (ΔE0

(CCSDT/ATZ)

− ΔE0

)

(9)

and (B)

E0

(RHF/AC5Z)

= E0

(CCSDT/CBS)

+ ΔE0

(CCSDTQ/ATZ)

+ (ΔE0

(CCSDT/ATZ)

− ΔE0

),

(10)

respectively. The first term on the right-hand side of each of the above equations designates the RHF reference energy obtained using the AC5Z basis set. Due to the very fast (exponential) convergence of the Hartree–Fock energies with respect to the basis set, we can treat the RHF/AC5Z energies as equivalent to the RHF/CBS values (we verified this by inspecting the differences between the RHF/AC5Z and RHF/ACQZ energies, which do not exceed 99 microhartree; more importantly in the context of the determination of the De , re , and G(v) values, the NPE characterizing the RHF/ACQZ PEC of Be2 relative to its RHF/AC5Z counterpart is only 12 microhartree, which is less than 3 cm−1 ). The second term on the right-hand sides of eqs 9 and 10 is the CBS limit of the CCSDT correlation energy extrapolated from the all-electron CCSDT/ACTZ and CCSDT/ACQZ calculations using the well-known two-point formula of refs 97 and 98, which in our case reads (CCSDT/CBS)

ΔE0

(CCSDT/ACnZ)

=

n3 ΔE0

(CCSDT/AC(n−1)Z)

− (n − 1)3 ΔE0 n3 − (n − 1)3

26


,

(11)

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where n and (n − 1) are the cardinal numbers of the ACnZ and AC(n − 1)Z basis sets, respectively. Since our extrapolation of the CBS limit of the CCSDT correlation energy is based on the results obtained with the ACTZ and ACQZ basis sets, n in eq 11 must be set at 4. The two composite schemes considered here differ in the way the effects due to connected quadruply excited (T4 ) clusters are handled. In scheme A, defined by eq 9, we estimate the T4 effects by forming the difference between the frozen-core CC(q;4)/ATZ and CCSDT/ATZ energies. In scheme B, defined by eq 10, which is a parent scheme for judging the performance of scheme A, we replace the CC(q;4)/ATZ energy by its full CCSDTQ/ATZ (in our case, full CI/ATZ) counterpart. Our goal is to compare the De , re , and vibrational term values that characterize the ground-state PEC of Be2 calculated using the CC(q;4)-based scheme A with those resulting from the CCSDTQ-based scheme B and the latest experimentally derived data reported by Meshkov et al. 10 This is done in Table 13. As shown in Table 13, the agreement between the vibrational term values obtained with schemes A and B is excellent, especially if we take into account savings in the computational time offered by the CC(q;4) approach compared to CCSDTQ (cf. Table 1). It is, in fact, hard to distinguish between the results obtained with scheme A and those calculated using the parent scheme B, when the energy spacings between the successive vibrational levels, ΔGv+1/2 ≡ G(v+1)−G(v), are examined. Indeed, the differences between the ΔGv+1/2 values obtained with schemes A and B are 1.8 cm−1 on average, not exceeding 3.7 cm−1 , when all of the vibrational levels, including those near the dissociation threshold, are taken into account. The differences between the vibrational term values G(v) resulting from schemes A and B are larger than in the case of the corresponding energy spacings ΔGv+1/2 when v ≥ 3, but they are still in generally good agreement with one another, ranging from less than 1 cm−1 for v = 0 and 1 and less than 10 cm−1 for v < 5 to less than 20 cm−1 , when the entire vibrational spectrum is considered. Both extrapolation schemes give identical equilibrium bond lengths to within 0.001 ˚ A and the difference between the binding energies resulting from schemes A and B is less than 20 cm−1 or 2%, so the CC(q;4)-based scheme A can be

27



viewed as a reliable substitute for the more expensive scheme B based on CCSDTQ. The above agreement between the results obtained with schemes A and B translates into a similarly good agreement between the De , re , G(v), and ΔGv+1/2 values obtained with both schemes and the latest experimentally derived data reported by Meshkov and co-workers in ref 10. In particular, as shown in Table 13, the De and re values resulting from the calculations using the CC(q;4)-based scheme A agree with the results obtained in ref 10 to within 7 cm−1 and 0.011 ˚ A, respectively, being comparable to scheme B in this regard, A error which gives a 13 cm−1 error relative to experiment for the binding energy and a 0.011 ˚ relative to the experimentally derived equilibrium bond length. The vibrational term values resulting from scheme A match their experimental counterparts to within ∼10 cm−1 , with scheme B performing in a similar manner. The agreement with experiment becomes even better when we examine the vibrational energy spacings ΔGv+1/2 resulting from schemes A and B. Indeed, both schemes reproduce the experimental ΔGv+1/2 values to within ∼4–5 cm−1 , with typical errors being below 1–2 cm−1 . Interestingly, in agreement with the prediction made in ref 27 and consistent with the most recent analysis of the experimental data obtained by Merritt et al., 9 which was carried out by Meshkov and co-workers, 10 both of our composite schemes, including, in particular, the CC(q;4)-based scheme A, predict the existence of the elusive v = 11 bound state slightly below the dissociation threshold. Although the total vibrational term values of this high-lying state resulting from schemes A and B, which are 927.7 and 947.2 cm−1 , respectively, differ from the G(v = 11) energy reported in ref 10 by 6.7 and 12.8 cm−1 , the agreement between our calculations, as far as the location of the v = 11 state relative to the asymptotes of the corresponding potentials is concerned, and ref 10 is quite remarkable. Indeed, Meshkov and co-workers locate the elusive v = 11 level 0.5 cm−1 below the dissociation threshold. Scheme A and its parent scheme B locate it 0.3 and 0.6 cm−1 below the corresponding potential asymptotes, respectively (see Table 13). The ability of both composite schemes examined in this subsection, especially the less

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expensive CC(q;4)-based scheme A, to closely reproduce the experimental vibrational spectrum of the challenging beryllium dimer to within a few wavenumbers, with the maximum errors in the calculated ΔGv+1/2 values being about 4–5 cm−1 , is very promising from the point of view of future applications of the CC(P ;Q) methodology. However, in evaluating the results reported in Table 13, we should keep in mind that we neglected the post-Born– Oppenheimer effects and the effects of relativity in our calculations. Although, according to ref 16, the former effects in the beryllium dimer are very small, contributing only about 0.1 cm−1 to its binding energy, the effects of relativity are more significant, changing De by ∼4–5 cm−1 . 13,15,16,28 Encouraged by the results obtained in the present study, we will return to the examination of the De , re , and vibrational term values of Be2 using the higher-level CC(P ;Q) methods, such as CC(q;4), which is almost as accurate as the full CCSDTQ approach, in the future work, further improving the quality of our calculations by using basis sets that are even larger than those employed in this work and correcting the resulting potentials and various spectroscopic properties for the post-Born–Oppenheimer and relativistic effects.

4. Conclusions We have examined the performance of several CC approaches with the connected triply as well triply and quadruply excited clusters, including the CCSD(T), CCSD(2)T , CR-CC(2,3), CCSD(TQ), and CR-CC(2,4) corrections to CCSD, the active-space CCSDt, CCSDtq, and CCSDTq methods, and the recently developed CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) corrections to CCSDt, CCSDtq, and CCSDTq that originate from the CC(P ;Q) formalism, in describing the ground-state PEC and the De , re , and vibrational term values of the challenging beryllium dimer. Our focus has been on comparing the ability of the various approximate treatments of triples (the CCSD(T), CCSD(2)T , CR-CC(2,3), CCSDt, and CC(t;3) approaches), triples and quadruples (the CCSD(TQ), CR-CC(2,4), CCSDtq, CC(t,q;3), and CC(t,q;3,4) schemes), and quadruples (the CCSDTq and CC(q;4) methods) to reproduce

29



the corresponding frozen-core and all-electron full CCSDT and frozen-core CCSDTQ (in this work, full CI) data. In generating the relevant PECs, we have used the augmented correlation-consistent basis sets of the triple and quadruple zeta quality. We have demonstrated that among methods that correct the CCSD energies for the effects of triples or triples and quadruples, the CR-CC(2,3) and CR-CC(2,4) approaches are most accurate, reproducing the parent CCSDT and CCSDTQ PECs and spectroscopic data more faithfully than the popular approximations of the CCSD(T) and CCSD(TQ) type. We have seen improvements in the CCSD(T) and CCSD(TQ) results provided by the CR-CC(2,3) and CR-CC(2,4) methods before, especially when examining potential energy surfaces involving covalent bond breaking, but much less has been known about the advantages offered by the completely renormalized CC approximations in applications involving challenging weakly bound systems, such as Be2 . This study is our first detailed examination of this particular theory aspect and we plan to continue exploring it even further in the future. The most important finding of the present work is the observation that the CC(P ;Q)based CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) methods, which correct the energies obtained in the CCSDt (the CC(t;3) approach), CCSDtq (the CC(t,q;3) and CC(t,q;3,4) approximations), and CCSDTq (the CC(q;4) scheme) calculations for the subsets of triples, triples and quadruples, or quadruples that have not been captured by the respective activespace CC computations, especially the CC(t;3) and CC(q;4) approaches, outperform the other tested methods in reproducing the corresponding full CCSDT and CCSDTQ PECs and De , re , and vibrational term values. Again, we have seen this before, when examining covalent bond dissociations, chemical reaction pathways involving significant bond rearrangements, and other cases involving stronger nondynamic correlation effects, but this is the first study that demonstrates the usefulness of the CC(P ;Q)-based approximations combining the active-space and completely renormalized CC ideas in the application involving a weakly bound system with the large and hard to balance dynamic correlation effects. As emphasized throughout the present paper, the CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4)

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methods benefit from the relaxation of the singly and doubly excited (CC(t;3), CC(t,q;3), and CC(t,q;3,4)) or singly, doubly, and triply excited (CC(q;4)) clusters in the presence of the dominant triply (CC(t;3)), triply and quadruply (CC(t,q;3) and CC(t,q;3,4)), and quadruply (CC(q;4)) excited cluster components captured in the underlying active-space CC calculations. This should be contrasted with the CR-CC(2,3) and CR-CC(2,4) approaches, and their CCSD(T), CCSD(2)T , and CCSD(TQ) counterparts that use the nonrelaxed T1 and T2 amplitudes obtained in the CCSD calculations. The results reported in this work clearly demonstrate that one is much better off by including the leading effects due to higher– than–two-body clusters, in addition to singles and doubles, in the P space and by using the CC(P ;Q) corrections δ0 (P ; Q) to capture the remaining rather than all triples or triples and quadruples. Although the present study has been predominantly methodological, testing the recently developed CC(P ;Q)-based CC(t;3), CC(t,q;3), CC(t,q;3,4), and CC(q;4) approaches against the parent full CCSDT and CCSDTQ data, we have also reported the results obtained by combining our highest-level frozen-core CC(q;4) and CCSDTQ calculations using the augmented correlation-consistent basis set of the triple zeta quality with the all-electron, CBS-limit CCSDT data. This has allowed us to make comparisons with the experimentally derived De , re , and vibrational term values, and vibrational level spacings characterizing the beryllium dimer, reported in ref 10, while comparing the CC(q;4) and CCSDTQ methods with each other at the same time. We have demonstrated that the CC(q;4) approach can be viewed as a highly reliable substitute for the more expensive CCSDTQ method and that the composite scheme based on combining the all-electron, CBS-limit CCSDT energetics with the frozen-core CC(q;4) calculations to capture the effects of connected quadruply excited clusters works as well as its CCSDTQ-based analog. We have shown that such a composite scheme reproduces the experimentally derived binding energy and equilibrium bond length of Be2 , obtained in ref 10, to within a few cm−1 and ∼ 0.01 ˚ A, describing the entire vibrational spectrum of the beryllium dimer, including states near the dissociation

31



threshold, to within ∼ 4 cm−1 , when the vibrational level spacings are examined, with typical errors relative to experiment being less than 1–2 cm−1 . In agreement with the prediction made in ref 27 and the latest analysis of the experimental data reported in ref 10, our composite scheme using the all-electron, CBS-limit CCSDT energies corrected for the effects of connected quadruples using the frozen-core CC(q;4) approach and its CCSDTQ-based counterpart predict the existence of the elusive v = 11 bound state of Be2 slightly below the dissociation threshold. This reinforces our conclusions regarding the usefulness of the CC(P ;Q) formalism in designing accurate approximations to the high-level methods of CC theory, such as CCSDTQ.

Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.XXXXXX. Tables comparing the PECs and the De , re , and vibrational term values of the Be2 dimer obtained in the CC(t;3), CC(t,q;3), and CC(t,q;3,4) calculations based on the Epstein– Nesbet-like denominators D0,K (P ) in the respective triples corrections with their counterparts using the Møller–Plesset form of the D0,K (P ) denominators in the corrections due to triples as well as quadruples, along with the underlying CCSDt and CCSDtq and parent CCSDT and CCSDTQ data.

Acknowledgement This work has been supported by the Chemical Sciences, Geosciences and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy (Grant No. DE-FG02-01ER15228).

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H (Q) spaceb {∅} G (T) G (T) {∅} (T) G G (t) {∅} G (T) ⊕ G (Q) {∅} (T) G G (t) (T) (G G (t) ) ⊕ (G (Q) G (q) ) {∅} (Q) G

G˜(q) {∅}

H (P ) spaceb

G (SD) G (SD) G (SD) (SD) G ⊕ G (t) G (SD) ⊕ G (t) G (SD) ⊕ G (T) G (SD) G (SD) ⊕ G (t) ⊕ G (q) G (SD) ⊕ G (t) ⊕ G (q) G (SD) ⊕ G (t) ⊕ G (q) G (SD) ⊕ G (T) ⊕ G˜(q) G (SD) ⊕ G (T) ⊕ G˜(q) G (SD) ⊕ G (T) ⊕ G (Q) — MP EN — EN — EN(T),MP(Q) — EN EN(T),MP(Q) — MP —

D0,K (P ) typec

CPU time scaling Iterative Noniterative n2o n4u — 2 4 no nu n3o n4u n2o n4u n3o n4u 2 4 N o N u no nu — 2 4 N o N u no nu n3o n4u n3o n5u — n2o n4u n3o n4u + n2o n5u No2 Nu2 n2o n4u — 2 2 2 4 N o N u no nu n3o n4u 2 2 2 4 3 4 N o N u no nu no nu + n2o n5u No Nu n3o n5u — 3 5 N o N u no nu n2o n5u n4o n6u —

In describing the CPU time scalings, we focus on the most expensive computational steps characterizing a given calculation. The conventional CCSD(T) and CCSD(TQ) approaches are not included, since CCSD(T) is an approximation to CCSD(2)T and CR-CC(2,3), and CCSD(TQ), which is equivalent to the CCSD(TQ),b method of ref 57, is an approximation to CR-CC(2,4). The relationships between the CCSD(T) approach and the CCSD(2)T and CR-CC(2,3) methods have been discussed in refs 63, 64, and 66. For the relationship between CCSD(TQ) and CR-CC(2,4), see ref 88 and references therein. b The P and Q spaces are defined in terms of subsets of Slater determinants in the many-electron Hilbert space spanning the G (SD) , G (t) , G (T) , G (q) , G˜(q) , and G (Q) subspaces introduced in Section 2.1. c EN stands for the Epstein–Nesbet-like form of the D0,K (P ) denominator, eq 7. MP stands for the Møller–Plesset-type approximation to eq 7, obtained by replacing the similarity-transformed Hamiltonian (P ) ¯ H on the right-hand side of eq 7 by the bare Fock operator, assuming a canonical Hartree–Fock basis. If a given δ0 (P ; Q) expression describes the correction due to triples and quadruples and if different forms of the denominator D0,K (P ) are used in the triples and quadruples parts, the relevant EN and MP symbols are accompanied by the additional letters, T for the triples and Q for the quadruples contributions. The D0,K (P ) denominator types listed in this table apply to the methods discussed in the main text. The CC(t;3), CC(t,q;3), and CC(t,q;3,4) approaches using the Møller–Plesset D0,K (P ) denominators in defining the triples corrections to CCSDt or CCSDtq are examined in the Supporting Information.

a

CCSD CCSD(2)T CR-CC(2,3) CCSDt CC(t;3) CCSDT CR-CC(2,4) CCSDtq CC(t,q;3) CC(t,q;3,4) CCSDTq CC(q;4) CCSDTQ

Method

Table 1: The Key Elements of the Various CC Methodologies Employed in this Work Summarized Using the Language of the CC(P;Q) Formalism, Including the P Spaces H (P ) Adopted in the Iterative CC Calculations and the Q Spaces H (Q) and D0,K (P ) Denominators Defining the Appropriate Noniterative δ0 (P ; Q) Corrections (If Any), Along with the Corresponding CPU Time Scalings.a

The Journal of Physical Chemistry Page 44 of 60

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Table 2: Computational Timings (in CPU Minutes) for the All-Electron CC Calculations with up to Triply Excited Clusters Using the ACQZ Basis Set for A. a the Be2 Dimer at 2.454 ˚ Method CCSD CCSD(T) CR-CC(2,3) CCSDtb CC(t;3)b CCSDT

Iterative Steps 171 171 223 3,090 3,144 64,790

CPU time Noniterative Steps – 4 8 – 8 –

a

Total 171 175 231 3,090 3,152 64,790

The reported timings correspond to single-core runs on the PowerEdge R910 server from Dell equipped with Intel Xeon X7560 2.26GHz processor boards. All of the calculations were performed using the spin-integrated spin-orbital codes interfaced with the integral, Hartree–Fock, and integral transformation routines in GAMESS, developed in refs 48–51. The convergence threshold for the CCSD, CCSDt, and CCSDT iterations was 10−7 hartree and the C1 symmetry was employed throughout. b The number of active occupied orbitals, No , and the number of active unoccupied orbitals, Nu , used in the CCSDt and CC(t;3) calculations were 2 and 6, respectively. This should be compared to the total numbers of occupied (no ) and unoccupied (nu ) orbitals corresponding to the all-electron calculation for Be2 using the ACQZ basis set, which are 4 and 214, respectively.

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Table 3: Electronic Energies of the Be2 Dimer at Selected Internuclear Separations r (in ˚ A) Obtained in the Various Frozen-Core CC Calculations with up to Triply Excited Clusters Using the ATZ Basis Set.a r 2.2 2.454 2.6 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 8.0 MUEb NPEc a

CCSD 5.990 4.805 4.126 3.275 1.975 1.154 0.664 0.382 0.222 0.132 0.080 0.050 0.007 1.759 5.983

CCSD(T) 0.830 0.803 0.737 0.619 0.388 0.223 0.125 0.070 0.040 0.023 0.014 0.009 0.001 0.299 0.829 (CCSDT)

CCSD(2)T 1.612 1.326 1.160 0.948 0.603 0.365 0.216 0.127 0.075 0.046 0.028 0.018 0.003 0.502 1.609

CR-CC(2,3) 0.460 0.372 0.325 0.274 0.190 0.120 0.074 0.046 0.030 0.020 0.013 0.009 0.002 0.149 0.458 (CCSDT)

CCSDt 2.974 2.767 2.556 2.207 1.488 0.918 0.539 0.311 0.180 0.106 0.064 0.039 0.006 1.089 2.968

CC(t;3) 0.165 0.175 0.173 0.162 0.126 0.083 0.052 0.033 0.022 0.015 0.010 0.007 0.001 0.079 0.174

CCSDT -0.236772 -0.239768 -0.239744 -0.239199 -0.238315 -0.237965 -0.237754 -0.237555 -0.237375 -0.237233 -0.237131 -0.237060 -0.236942 — —

The CCSDT energies E0 are reported as (E0 + 29.0) hartree, whereas all of the remaining energies are errors relative to CCSDT in millihartree. b Mean unsigned error relative to CCSDT. c Non-parallelity error relative to CCSDT.

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Table 4: Electronic Energies of the Be2 Dimer at Selected Internuclear Separations r (in ˚ A) Obtained in the Various All-Electron CC Calculations with up to Triply Excited Clusters Using the ACTZ Basis Set.a r 2.2 2.454 2.6 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 8.0 MUEb NPEc a

CCSD 7.049 5.773 5.061 4.183 2.871 2.057 1.577 1.302 1.147 1.059 1.009 0.980 0.939 2.693 6.111

CCSD(T) 0.829 0.809 0.746 0.633 0.412 0.255 0.161 0.109 0.080 0.064 0.056 0.051 0.043 0.327 0.785 (CCSDT)

CCSD(2)T 1.750 1.445 1.273 1.055 0.709 0.475 0.330 0.244 0.194 0.166 0.149 0.139 0.124 0.620 1.626

CR-CC(2,3) 0.596 0.487 0.430 0.377 0.275 0.195 0.148 0.121 0.106 0.097 0.091 0.087 0.082 0.238 0.514 (CCSDT)

CCSDt 3.586 3.414 3.225 2.904 2.234 1.699 1.344 1.129 1.005 0.934 0.894 0.870 0.839 1.852 2.747

CC(t;3) 0.224 0.238 0.237 0.235 0.193 0.146 0.116 0.100 0.090 0.083 0.079 0.076 0.073 0.145 0.165

CCSDT -0.325118 -0.327846 -0.327736 -0.327121 -0.326182 -0.325815 -0.325600 -0.325404 -0.325232 -0.325094 -0.324994 -0.324924 -0.324806 — —


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Table 5: Electronic Energies of the Be2 Dimer at Selected Internuclear Separations r (in ˚ A) Obtained in the Various Frozen-Core CC Calculations with up to Triply Excited Clusters Using the AQZ Basis Set.a r 2.2 2.454 2.6 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 8.0 MUEb NPEc a

CCSD 6.165 4.936 4.235 3.360 2.024 1.183 0.683 0.393 0.229 0.136 0.082 0.052 0.008 1.807 6.158

CCSD(T) 0.714 0.719 0.667 0.565 0.356 0.205 0.115 0.064 0.037 0.022 0.013 0.008 0.001 0.268 0.718 (CCSDT)

CCSD(2)T 1.538 1.266 1.109 0.908 0.580 0.353 0.210 0.124 0.074 0.045 0.028 0.018 0.003 0.481 1.535

CR-CC(2,3) 0.534 0.456 0.390 0.319 0.200 0.123 0.073 0.046 0.030 0.020 0.013 0.009 0.002 0.170 0.532 (CCSDT)

CCSDt 3.131 2.917 2.697 2.331 1.575 0.973 0.573 0.332 0.193 0.113 0.068 0.042 0.006 1.150 3.125

CC(t;3) 0.191 0.217 0.209 0.192 0.135 0.087 0.053 0.034 0.023 0.015 0.010 0.007 0.001 0.090 0.216

CCSDT -0.238467 -0.241203 -0.241074 -0.240419 -0.239408 -0.239009 -0.238778 -0.238566 -0.238379 -0.238234 -0.238130 -0.238060 -0.237942 — —


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Table 6: Electronic Energies of the Be2 Dimer at Selected Internuclear Separations r (in ˚ A) Obtained in the Various All-Electron CC Calculations with up to Triply Excited Clusters Using the ACQZ Basis Set.a r 2.2 2.454 2.6 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 8.0 MUEb NPEc a

CCSD 7.554 6.212 5.470 4.560 3.209 2.381 1.895 1.616 1.458 1.368 1.318 1.288 1.246 3.044 6.308

CCSD(T) 0.742 0.751 0.700 0.599 0.396 0.251 0.164 0.116 0.090 0.075 0.067 0.062 0.056 0.313 0.695 (CCSDT)

CCSD(2)T 1.751 1.448 1.278 1.064 0.728 0.503 0.363 0.280 0.232 0.204 0.188 0.178 0.164 0.645 1.587

CR-CC(2,3) 0.741 0.625 0.540 0.450 0.312 0.228 0.177 0.149 0.134 0.125 0.119 0.115 0.110 0.294 0.631 (CCSDT)

CCSDt 3.959 3.787 3.594 3.265 2.581 2.032 1.666 1.443 1.314 1.240 1.197 1.173 1.140 2.184 2.819

CC(t;3) 0.289 0.314 0.304 0.286 0.226 0.178 0.145 0.127 0.117 0.110 0.106 0.103 0.100 0.185 0.214

CCSDT -0.332351 -0.334719 -0.334466 -0.333712 -0.332632 -0.332221 -0.331989 -0.331779 -0.331594 -0.331451 -0.331349 -0.331280 -0.331164 — —


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Table 7: Vibrational Energies G(v) (in cm-1 ), Dissociation Energies D e (in A) for Be2 Obtained in the Various cm-1 ), and Equilibrium Bond Lengths r e (in ˚ Frozen-Core CC Calculations with up to Triply Excited Clusters Using the ATZ Basis Set.a vb 0 1 2 3 4 5 6 7 8 9 10 De re

CCSD -91.3 -241.9 -341.4 -400.1 —c —c —c —c —c —c —c 57.1 4.473

CCSD(T) -9.5 -37.6 -82.9 -108.8 -121.2 -131.4 -140.9 -150.3 -159.5 -166.1 —c 470.7 2.534

CCSD(2)T -17.7 -69.8 -144.7 -176.1 -194.6 -210.9 -226.7 -242.6 -256.8 —c —c 370.3 2.558

CR-CC(2,3) -3.6 -13.6 -30.0 -43.7 -50.1 -55.2 -60.0 -64.9 -69.8 -73.9 —c 564.2 2.527

CC(t;3) -0.8 -3.5 -10.1 -17.8 -21.9 -25.0 -27.9 -30.9 -33.9 -36.6 -37.6 602.4 2.516

CCSDT 104.9 277.4 390.8 455.6 504.3 545.8 580.4 607.4 626.1 636.3 639.9 640.8 2.516

The CCSDT vibrational energies are total G(v) values, whereas all of the remaining vibrational term values are errors relative to CCSDT. b Vibrational quantum number. c PEC is too shallow to support this vibrational level. a

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Table 8: Vibrational Energies G(v) (in cm-1 ), Dissociation Energies D e (in cm-1 ), A) for Be2 Obtained in the Various Alland Equilibrium Bond Lengths r e (in ˚ Electron CC Calculations with up to Triply Excited Clusters Using the ACTZ Basis Set.a vb 0 1 2 3 4 5 6 7 8 9 10 De re

CCSD -96.4 -257.7 -368.7 -434.7 —c —c —c —c —c —c —c 55.7 4.480

CCSD(T) -8.8 -32.9 -73.3 -103.8 -117.1 -127.2 -136.3 -145.0 -153.7 -160.9 —c 514.6 2.506

CCSD(2)T -17.1 -64.9 -139.5 -178.3 -198.3 -214.6 -230.0 -245.2 -259.8 —c —c 405.0 2.537

CR-CC(2,3) -4.1 -14.6 -33.0 -50.1 -57.5 -63.0 -68.0 -72.8 -77.7 -81.9 —c 594.4 2.509

CC(t;3) -0.3 -1.8 -8.0 -16.6 -21.0 -24.1 -26.8 -29.5 -32.2 -34.5 —c 642.9 2.495

CCSDT 109.6 292.5 417.4 489.3 539.8 582.0 617.0 644.5 664.0 675.0 678.7 679.4 2.495


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Table 9: Vibrational Energies G(v) (in cm-1 ), Dissociation Energies D e (in A) for Be2 Obtained in the Various cm-1 ), and Equilibrium Bond Lengths r e (in ˚ Frozen-Core CC Calculations with up to Triply Excited Clusters Using the AQZ Basis Set.a vb 0 1 2 3 4 5 6 7 8 9 10 De re

CCSD -98.9 -267.2 -388.1 -462.7 —c —c —c —c —c —c —c 55.7 4.455

CCSD(T) -7.4 -27.6 -61.4 -93.4 -106.9 -116.8 -125.7 -134.3 -142.7 -150.2 —c 571.5 2.499

CCSD(2)T -14.8 -54.7 -119.9 -164.7 -185.0 -201.5 -217.0 -232.3 -247.2 -258.9 —c 462.0 2.528

CR-CC(2,3) -5.3 -18.4 -37.1 -55.6 -63.9 -69.7 -75.1 -80.3 -85.7 -90.7 —c 632.1 2.507

CC(t;3) -1.5 -5.3 -12.7 -22.3 -27.7 -31.3 -34.5 -37.7 -40.9 -44.2 -46.0 679.9 2.495

CCSDT 112.5 302.6 437.3 517.6 571.9 617.4 655.6 685.9 707.8 720.9 725.7 726.7 2.492


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Table 10: Vibrational Energies G(v) (in cm-1 ), Dissociation Energies D e (in A) for Be2 Obtained in the Varicm-1 ), and Equilibrium Bond Lengths r e (in ˚ ous All-Electron CC Calculations with up to Triply Excited Clusters Using the ACQZ Basis Set.a vb 0 1 2 3 4 5 6 7 8 9 10 De re

CCSD -105.2 -287.4 -423.3 -508.9 —c —c —c —c —c —c —c 57.2 4.427

CCSD(T) -6.5 -24.1 -53.3 -88.2 -103.7 -113.7 -122.5 -130.8 -138.9 -146.3 —c 633.6 2.473

CCSD(2)T -14.7 -53.0 -115.6 -170.3 -192.8 -209.6 -225.2 -240.5 -255.4 —c —c 510.8 2.500

CR-CC(2,3) -7.3 -22.5 -43.8 -66.7 -77.3 -83.8 -89.4 -95.0 -100.4 -105.5 —c 675.8 2.486

CC(t;3) -2.1 -5.6 -12.4 -22.3 -28.6 -32.2 -35.4 -38.4 -41.4 -44.2 -45.4 739.0 2.469

CCSDT 119.0 323.4 473.5 565.2 622.9 670.0 709.4 741.1 764.2 778.6 784.4 785.7 2.466


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Table 11: Electronic Energies of the Be2 Dimer at Selected Internuclear Separations r (in ˚ A) Obtained in the Various Frozen-Core CC Calculations with Quadruply Excited Clusters Using the ATZ Basis Set.a r 2.2 2.454 2.6 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 8.0 MUEb NPEc

CCSD(TQ) CR-CC(2,4) CCSDtq CC(t,q;3) CC(t,q;3,4) CCSDTq CC(q;4) CCSDTQ 1.147 0.746 3.290 0.480 0.334 0.140 0.084 -0.237316 1.063 0.591 3.033 0.441 0.331 0.144 0.090 -0.240151 0.964 0.512 2.786 0.402 0.313 0.139 0.090 -0.240052 0.800 0.414 2.388 0.343 0.277 0.125 0.084 -0.239424 0.493 0.265 1.596 0.233 0.192 0.090 0.061 -0.238436 0.282 0.161 0.984 0.149 0.120 0.060 0.038 -0.238036 0.157 0.096 0.581 0.093 0.073 0.039 0.023 -0.237798 0.087 0.059 0.337 0.059 0.045 0.025 0.014 -0.237582 0.050 0.037 0.197 0.038 0.029 0.016 0.008 -0.237393 0.029 0.024 0.116 0.025 0.019 0.010 0.005 -0.237244 0.017 0.016 0.070 0.017 0.012 0.007 0.003 -0.237138 0.011 0.010 0.044 0.011 0.008 0.004 0.002 -0.237065 0.002 0.002 0.006 0.002 0.001 0.001 0.000 -0.236943 0.392 0.226 1.187 0.176 0.135 0.062 0.039 — 1.145 0.744 3.284 0.478 0.332 0.144 0.090 — (CCSDTQ)

The CCSDTQ energies E0 , which are equivalent in this case to the energies obtained in the full (CCSDTQ) + 29.0) hartree, whereas all of the remaining energies are CI calculations, are reported as (E0 errors relative to CCSDTQ in millihartree. b Mean unsigned error relative to CCSDTQ. c Non-parallelity error relative to CCSDTQ.

a

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Table 12: Vibrational Energies G(v) (in cm-1 ), Dissociation Energies D e (in A) for Be2 Obtained in the Various cm-1 ), and Equilibrium Bond Lengths r e (in ˚ Frozen-Core CC Calculations with Quadruply Excited Clusters Using the ATZ Basis Set.a v b CCSD(TQ) CR-CC(2,4) CC(t,q;3) CC(t,q;3,4) CCSDTq CC(q;4) CCSDTQ 0 -12.9 -6.9 -4.5 -2.6 -0.9 -0.4 110.8 1 -47.8 -23.9 -15.6 -9.7 -3.7 -1.7 297.1 2 -104.0 -49.4 -32.8 -22.3 -8.6 -4.7 427.5 3 -142.8 -72.4 -50.7 -36.8 -14.5 -8.9 505.8 4 -160.3 -82.9 -59.7 -44.3 -18.0 -11.4 560.2 5 -173.8 -90.5 -66.2 -49.6 -20.6 -13.1 605.9 6 -186.1 -97.5 -72.1 -54.2 -22.8 -14.5 644.2 7 -198.1 -104.3 -78.0 -58.8 -25.0 -15.9 674.8 8 -209.9 -111.2 -83.9 -63.4 -27.2 -17.3 697.2 9 -219.4 -117.7 -89.5 -67.8 -29.4 -18.6 710.9 -120.8 -92.4 -70.3 -30.7 -19.4 716.6 10 —c De 492.3 595.9 624.5 647.0 687.2 698.8 718.6 2.520 2.518 2.507 2.501 2.499 2.495 2.498 re a

The CCSDTQ vibrational energies, which are equivalent in this case to the energies obtained in the full CI calculations, are total G(v) values, whereas all of the remaining vibrational term values are errors relative to CCSDTQ. b Vibrational quantum number. c PEC is too shallow to support this vibrational level.

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Table 13: Vibrational Energies G(v) (in cm-1 ), Energy Spacings Between Successive Vibrational Levels ΔGv+1/2 ≡ G(v + 1) − G(v) (in cm-1 ), Dissociation EnA) of the Be2 Dimer, ergies D e (in cm-1 ), and Equilibrium Bond Lengths r e (in ˚ as Obtained with the Two Composite Schemes Discussed in the Text. va 0 1 2 3 4 5 6 7 8 9 10 11 De re a

2.8 7.2 10.4 10.3 6.6 3.6 1.6 0.1 -1.6 -3.5 -5.3 -6.7

Scheme Ab G(v)e ΔGv+1/2 e (-0.2) 4.4 (-0.7) (-0.9) 3.2 (-1.6) (-2.5) -0.1 (-3.2) (-5.6) -3.7 (-3.7) (-9.4) -3.0 (-2.2) (-11.6) -1.9 (-1.6) (-13.2) -1.6 (-1.5) (-14.7) -1.7 (-1.4) (-16.1) -1.8 (-1.4) (-17.5) -1.8 (-1.3) (-18.8) -1.4 (-0.7) (-19.5) — — 928.0 2.434

Scheme Bc G(v)f ΔGv+1/2 f 3.0 5.1 8.1 4.8 12.9 3.1 16.0 0.0 15.9 -0.8 15.1 -0.3 14.8 -0.1 14.7 -0.3 14.4 -0.5 14.0 -0.5 13.5 -0.7 12.8 — 947.8 2.434

Expt.d G(v) ΔGv+1/2 126.7 222.9 349.7 174.5 524.2 121.0 645.2 76.7 721.8 56.7 778.5 47.2 825.8 38.9 864.7 30.5 895.2 21.7 916.9 12.7 929.6 4.8 934.4 — 934.9 2.445

Vibrational quantum number. b Composite scheme A defined by eq 9. c Composite scheme B defined by eq 10. d Experimentally derived values taken from ref 10. e Errors relative to experiment and, in parentheses, relative to scheme B. f Errors relative to experiment.

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(a)

(b)

(c)

(d)

Figure 1: The ground-state PECs of Be2 resulting from the CCSD and various CC calculations with up to triply excited clusters using the (a) ATZ, (b) ACTZ, (c) AQZ, and (d) ACQZ basis sets. All PECs have been aligned such that the corresponding electronic energies at the internuclear separation r = 12 ˚ A are identical and set at 0 hartree. 57



Figure 2: The ground-state PECs of Be2 resulting from the CCSD and various CC calculations with quadruply excited clusters using the ATZ basis set. All PECs have been aligned such that the corresponding electronic energies at the internuclear separation r = 12 ˚ A are identical and set at 0 hartree.

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Graphical TOC Entry

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TOC Graphic 44x27mm (300 x 300 DPI)


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Application of the CC(P;Q) Hierarchy of Coupled ... - ACS Publications

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