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Seniority-Based Coupled Cluster Theory

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Seniority-Based Coupled Cluster Theory Seniority-based coupled cluster theory Thomas M. Henderson Department of Chemistry, Rice University, Houston, TX 77005-1892 and Department of Physics and Astronomy, Rice University, Houston, TX 77005-1892 Ireneusz W. Bulik Department of Chemistry, Rice University, Houston, TX 77005-1892 Tamar Stein Department of Chemistry, Rice University, Houston, TX 77005-1892 Gustavo E. Scuseria Department of Chemistry, Rice University, Houston, TX 77005-1892 and Department of Physics and Astronomy, Rice University, Houston, TX 77005-1892 (Dated: July 5, 2021) Doubly occupied configuration interaction (DOCI) with optimized orbitals often accurately de- scribes strong correlations while working in a Hilbert space much smaller than that needed for full configuration interaction. However, the scaling of such calculations remains combinatorial with sys- tem size. Pair coupled cluster doubles (pCCD) is very successful in reproducing DOCI energetically, but can do so with low polynomial scaling (N 3, disregarding the two-electron integral transforma- tion from atomic to molecular orbitals). We show here several examples illustrating the success of pCCD in reproducing both the DOCI energy and wave function, and show how this success frequently comes about. What DOCI and pCCD lack are an effective treatment of dynamic correla- tions, which we here add by including higher-seniority cluster amplitudes which are excluded from pCCD. This frozen pair coupled cluster approach is comparable in cost to traditional closed-shell coupled cluster methods with results that are competitive for weakly correlated systems and often superior for the description of strongly correlated systems. I. INTRODUCTION tion of a fundamentally single-reference method be able to describe multi-reference problems? The coupled cluster (CC) family of methods1–3 offer In this manuscript, we seek to do three things. First, a powerful wave function approach to the description of we want to provide a self-contained description of pCCD, weakly correlated systems, to the point that the accurate with all the equations one needs to implement the ap- treatment of such systems is essentially routine: provided proach. Second, we wish to offer some perspective on that the system is not too large, one can simply ap- the method’s successes. Third, we wish to go beyond ply coupled cluster with single and double excitations4 pCCD and include some of the dynamic correlations (CCSD) or CCSD plus perturbative triple excitations,5 which pCCD does not provide. To accomplish this, how- which we refer to as CCSD(T). The same, unfortu- ever, we first must discuss doubly occupied configuration nately, cannot be said for the coupled cluster treatment of interaction and orbital seniority. strongly correlated systems, for which traditional single- reference methods such as CCSD or CCSD(T) may fail badly. Much progress has been made in multi-reference coupled cluster theory,2 to be sure, but the techniques are II. SENIORITY AND DOUBLY OCCUPIED by no means black box or computationally inexpensive. CONFIGURATION INTERACTION arXiv:1410.6529v2 [physics.chem-ph] 5 Dec 2014 Continued developments of coupled cluster techniques for strongly correlated systems is essential. Pair coupled cluster theory is based on the concept In 2013, Ayers and coworkers made a surprising dis- of the seniority of a determinant. The seniority is the ocvery along these lines: a method which they refer number of unpaired electrons. The idea is simple: ev- to as the antisymmetric product of 1-reference orbital ery spinorbital φp is paired with one and only one other geminals6–9 (AP1roG) and which we will refer to as pair spinorbital, φp¯, and the seniority of a determinant is the coupled cluster doubles10 (pCCD) provides a remarkably number of spinorbital pairs which between them contain reasonable description of the strong correlations for a only one electron. Loosely speaking, seniority is related wide variety of systems. What makes this so surprising is to the number of broken electron pairs. that pCCD looks like coupled cluster doubles (CCD) re- In this work, as in our previous work on the subject,10 stricted to include only those excitations which preserve we restrict ourselves to singlet pairing, in which the or- electron pairs, but pCCD, unlike CCD, seems to be able bitals that are paired are the two spinorbitals correspond- to describe strong correlations. Why should a simplifica- ing to the same spatial orbital. In that case, the seniority 2 operator is just −3.9 Ω= N − 2 D (1) −4 where N is the number operator −4.1 † † N = c cp + c cp = np + np (2) −4.2 X p↑ ↑ p↓ ↓ X ↑ ↓ p p RHF E (Hartree) −4.3 Ω=0 and D is a double-occupancy operator Ω=0,2 Ω=0,2,4 −4.4 Ω † † =0,2,4,6,8=FCI D = c c cp cp = np np . (3) X p↑ p↓ ↓ ↑ X ↑ ↓ p p −4.5 0.5 1 1.5 2 2.5 3 3.5 Throughout this work, we will use indices i, j, k, l for R (Å) occupied spatial orbitals, a, b, c, d for virtual spatial −108.6 orbitals, and p, q, r, s for general spatial orbitals. It is important to notice that seniority depends on −108.7 which orbitals we use to define the double-occupancy op- erator D, because a unitary transformation which mixes −108.8 the orbitals leaves N invariant but changes the form of D. If we define seniority with respect to the molecular or- −108.9 bitals of the restricted Hartree-Fock (RHF) determinant E (Hartree) −109 RHF |RHFi, then we see that the RHF determinant is a se- Ω=0 niority eigenfunction and has seniority zero. If we define Ω=0,2 −109.1 Ω=0,2,4 seniority with respect to a different basis, this need not Ω=0,2,4,6,8=FCI be true. It is also important to note that seniority is not −109.2 a symmetry of the molecular Hamiltonian – [H, Ω] 6= 0 0.5 1 1.5 2 2.5 3 – which means that the exact wave function is not an R (Å) eigenfunction of Ω. The utility of the seniority concept comes from using FIG. 1. Top panel: Dissociation of the equally-spaced H8 it as an alternative to organize Hilbert space.11 Conven- chain. Bottom panel: Dissociation of N2. Both calculations tionally, we describe determinants in terms of their exci- are done in the cc-pvdz basis set and restrict the CI problem tation level, which we can extract from the particle-hole to a minimal active space. We emphasize that curves are obtained with an RHF wave function. Results taken from number operator Ref. 11. 2 Nph = na + na + 2 − ni − ni . (4) X ↑ ↓ X ↑ ↓ a i zero seniority sector of Hilbert space, which we refer to as As with seniority, the excitation level is neither orbitally doubly occupied configuration interaction (DOCI).11–16 invariant (because defining particles and holes with re- Because DOCI is not invariant to the orbitals with re- spect to a different Fermi vacuum changes the excita- spect to which seniority is defined, we optimize this tion level) nor a symmetry of the Hamiltonian, but it choice energetically. This is analagous to optimizing the nevertheless provides a valuable framework within which identity of the reference determinant in an excitation- we can organize Hilbert space and solve the Schr¨odinger truncated CI calculation, or with optimizing the orbitals equation in a subspace. The exact wave function is gen- in CAS-SCF, though DOCI is generally size consistent. erally a linear combination of determinants of all possi- As we and others have shown, DOCI with orbital opti- ble excitation levels, and similarly it is generally a linear mization provides a valuable tool for the description of combination of determinants of all possible seniorities. strong correlations. This can be shown in Fig. 1, which The success of single-reference coupled cluster theory for shows that DOCI gives the correct limit in the dissocia- weakly correlated systems is grounded on the fact that tion of the equally spaced H8 chain and gives most of the the coupled cluster expansion in terms of particle-hole ex- strong correlation in N2 as well. Note that these plots citations out of the Hartree-Fock determinant converges are generated using a minimal active space to remove, to rapidly toward full configuration interaction (FCI). The the degree possible, dynamic correlation at dissociation. ground state of weakly correlated systems, then, is char- The chief drawback of DOCI is that of computational acterized by having a low number of particle-holes. cost: the number of determinants with Ω = 0 is just We posit that the ground state of strongly correlated the square root of the number of all determinants with a systems is characterized by having a low seniority num- given particle number, so the cost of DOCI is the square ber in a suitable one-electron basis. One can test this by root of the cost of full CI. Worse yet, it is more difficult defining configuration interaction (CI) restricted to the to use symmetry to eliminate determinants from DOCI 3 than it is to eliminate determinants from FCI. For exam- Explicitly, the pCCD energy and amplitudes are given ple, every DOCI determinant is a spin singlet with our by singlet pairing scheme, so we cannot use spin symmetry E = h0|H|0i + ta vii (12a) to reduce the number of determinants to be included. In X i aa practice, DOCI calculations on systems with more than ia a few dozen electrons are prohibitively expensive. 0= vaa +2 f a − f i − vjj ta − vii tb ta (12b) ii a i X aa j X bb i i This is where pCCD enters the picture: pCCD gen- j b erally provides results which for the molecular Hamilto- ia ia ii a a nian are nearly indistinguishable from those of DOCI, − 2 2 via − vai − vaa ti ti but whereas the computational cost of DOCI scales com- + vaa tb + vjj ta + vjj ta tb binatorially with system size, the cost of pCCD scales as X bb i X ii j X bb j i O(N 3).
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