Seniority-Based Coupled Cluster Theory

Home , Coupled cluster

arXiv:1410.6529v2 [physics.chem-ph] 5 Dec 2014 al.Mc rgeshsbe aei multi-reference in made fail theory, been may cluster has CCSD(T) coupled progress or Much CCSD unfortu- as badly. such single- same, traditional methods which The reference for of treatment CCSD(T). systems, cluster correlated coupled as strongly the for ap- said to be cannot excitations, simply refer nately, triple can excitations we perturbative double one plus which and large, CCSD single or too with (CCSD) not cluster coupled is ply system provided the routine: essentially that accurate is the systems such that of of point description treatment the the to systems, to correlated approach weakly function wave powerful a ceyaogteelns ehdwihte refer orbital they 1-reference doubles cluster which of coupled method product antisymmetric a geminals the lines: as to these along ocvery essential. is systems correlated inexpensive. for strongly techniques computationally cluster or coupled of box developments Continued black means no by odsrb togcreain.Wysol simplifica- a able should be Why to correlations. seems strong CCD, describe unlike preserve to pCCD, which but excitations pairs, re- those electron (CCD) only doubles include cluster to is coupled a stricted surprising like so for looks this pCCD makes correlations that What strong systems. of the variety wide of description reasonable h ope lse C)fml fmethods of family (CC) cluster coupled The n21,Aesadcwresmd upiigdis- surprising a made coworkers and Ayers 2013, In 6–9 A1o)adwihw ilrfrt spair as to refer will we which and (AP1roG) u a os ihlwplnma cln ( scaling polynomial low succes with very so is do (pCCD) doubles can cluster but cal coupled such Pair spac of size. Hilbert scaling tem the a However, in interaction. working configuration while correlations strong scribes ueirfrtedsrpino togycreae system correlated f strongly competitive of description are comparab the that is for results approach superior with cluster methods clus coupled cluster higher-seniority function pair coupled including wave frozen effecti by an This and add are pCCD. lack energy here pCCD severa we DOCI and which here DOCI the tions, What show both We about. comes reproducing frequently orbitals). in molecular pCCD to of atomic from tion obyocpe ofiuainitrcin(OI ihopti with (DOCI) interaction configuration occupied Doubly .INTRODUCTION I. eateto hsc n srnm,Rc nvriy Hous University, Rice Astronomy, and Physics of Department eateto hsc n srnm,Rc nvriy Hous University, Rice Astronomy, and Physics of Department 2 10 eateto hmsr,Rc nvriy oso,T 7700 TX Houston, University, Rice Chemistry, of Department eateto hmsr,Rc nvriy oso,T 7700 TX Houston, University, Rice Chemistry, of Department ob ue u h ehiusare techniques the but sure, be to eateto hmsr,Rc nvriy oso,T 7700 TX Houston, University, Rice Chemistry, of Department 7700 TX Houston, University, Rice Chemistry, of Department pC)poie remarkably a provides (pCCD) eirt-ae ope lse theory cluster coupled Seniority-based hmsM Henderson M. Thomas utv .Scuseria E. Gustavo 1–3 Dtd uy5 2021) 5, July (Dated: rnuzW Bulik W. Ireneusz aa Stein Tamar offer N 5 4 3 irgrigtetoeeto nerltransforma- integral two-electron the disregarding , neato n ria seniority. configuration orbital correlations occupied and doubly how- interaction dynamic discuss this, must beyond accomplish the first To we go provide. ever, of to not does on some wish pCCD perspective we which include some Third, and ap- offer the pCCD to successes. implement wish method’s to we the needs Second, one equations proach. pCCD, the of all description self-contained with a provide to want we able be problems? multi-reference method describe single-reference to fundamentally a of tion ubro nardeetos h dai ipe ev- simple: the is is idea seniority The The spinorbital electrons. ery determinant. unpaired a of of number seniority the of spinorbital, otenme fboe lcrnpairs. related electron broken is of seniority number speaking, the Loosely to contain electron. them between one which only pairs spinorbital of number n otesm pta ria.I htcs,teseniority the case, that In orbital. spatial or- same correspond- the the spinorbitals which two to the ing in are paired pairing, are singlet that bitals to ourselves restrict we nti aucit ese od he hns First, things. three do to seek we manuscript, this In arculdcutrter sbsdo h concept the on based is theory cluster coupled Pair nti ok si u rvoswr ntesubject, the on work previous our in as work, this In I EIRT N OBYOCCUPIED DOUBLY AND SENIORITY II. s. uain ean obntra ihsys- with combinatorial remains culations uhsalrta htnee o full for needed that than smaller much e e mltdswihaeecue from excluded are which amplitudes ter rwal orltdssesadoften and systems correlated weakly or fli erdcn OIenergetically, DOCI reproducing in sful OFGRTO INTERACTION CONFIGURATION ei ott rdtoa closed-shell traditional to cost in le ie riasotnacrtl de- accurately often orbitals mized xmlsilsrtn h success the illustrating examples l φ p ¯ eteteto yai correla- dynamic of treatment ve n h eirt fadtriati the is determinant a of seniority the and , φ n hwhwti success this how show and , p o,T 77005-1892 TX ton, o,T 77005-1892 TX ton, spie ihoeadol n other one only and one with paired is -82and 5-1892 -82and 5-1892 5-1892 5-1892 10 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 operator 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 respect 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 Schrödinger 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). b j jb p pq where fq is an element of the Fock operator and vrs = hφp φq|Vee|φr φsi is a two-electron integral in Dirac no- III. PAIR COUPLED CLUSTER DOUBLES tation. As promised, these equations can be solved in O(N 3) computational cost with the aid of the interme- j jj b In pCCD, we write the wave function as diate yi = b vbb ti . As with traditionalP CC methods, we can define a left- T hand eigenvector hL| of H¯ in CI-like fashion: |Ψi =e |0i (5) hL| = h0|(1 + Z) (13) where |0i is a closed-shell reference determinant and where a † T = ti Pa Pi (6) i † X Z = za Pi Pa. (14) ia X ia

† in terms of the pair operators Pa and Pi, where generi- Then the expectation value of H¯ is cally E = h0|(1 + Z) H¯ |0i = h0|(1 + Z)e−T H eT |0i. (15) † † † Pq = cq↑ cq↓ (7) a The equations for the amplitudes ti are just with the singlet pairing we are using. As usual, one can ∂E insert this ansatz into the Schr¨odinger equation to get 0= a (16) ∂zi E = h0|H¯ |0i, (8a) and guarantee by their satisfaction that † ¯ 0= h0|Pi Pa H|0i, (8b) E = h0|H¯ |0i (17)

H¯ for any value of Z; similarly, we obtain the amplitudes where the similarity transformed Hamiltonian is given a by zi from ∂ −T T E H¯ =e H e . (9) 0= a . (18) ∂ti In AP1roG, one instead writes We ﬁnd that the z equations are

17,18 p RHF-based CCD by simply retaining only the pair in terms of one-electron integrals hq and the two-electron aa aa pq amplitudes tii and zii which we have here written as integrals vrs previously defined. The energy gradient is a a simply ti and zi for compactness of notation and to em- then phasize that the pCCD t and z amplitudes are two-index † ∂E(κ) quantities. In practice, one usually finds that Z ∼ T , as = hr γq − hq γr (25) ∂κ h X p r r p we might expect. We note in passing that one can readily pq κ=0 r identify the various channels19,20 of the CCD amplitude + vrs Γqt − vqt Γrs − (p ↔ q) equations in Eqn. 12b, where the ladder terms are found X pt rs rs pt i on the third line, the ring and crossed-ring terms appear rst on the second line, and what we have termed the Brueck- p pq where γq and Γrs are one-body and two-body density ner or mosaic terms appear on the first line. For pCCD, matrices, given by the various ring terms decouple, though our limited nu- merical experience suggests that a pair ring CCD model p −T † T γ = h0|(1 + Z)e c cp e |0i, (26a) is not useful. q X qσ σ σ Like DOCI, pCCD is not invariant to the choice of † pq −T † † T Γrs = h0|(1 + Z)e cr cs ′ cq ′ cpσ e |0i. (26b) which orbitals are used to define the pair operators Pp . X σ σ σ Additionally, pCCD depends on the choice of reference σσ′ determinant |0i. In order to have a well-defined method, We use a Newton-Raphson scheme to minimize the norm we must provide a way of fixing these choices. This can of the orbital gradient, which finds an orbital stationary be accomplished by orbital optimization,21,22 for which point. Having found such a point, we check the eigenval- purpose we introduce the one-body antihermitian opera- ues of the coupled cluster orbital Hessian and, if there is tor a negative eigenvalue, follow the instability until we find † † κ = κpq c cq − c cp (20) a local energy minimum or saddle point (i.e. we look X X pσ σ qσ σ p>q σ for points with zero gradient and non-negative Hessian). The analytic formulae for the density matrices and the which, when exponentiated, creates unitary orbital ro- orbital Hessian are presented in the appendix. As has tations; here, σ indexes spins (i.e. σ =↑, ↓). Note that been previously pointed out, there are multiple solutions in contrast to the typical coupled-cluster orbital opti- to the orbital optimization equations, and because the mization which includes only occupied-virtual mixing, we optimized orbitals are generally local in character if the must allow all orbitals to mix. We have taken κ to be system is strongly correlated,7,10,11 it proves convenient real. to start from the RHF determinant with localized molec- Given the rotation operator, we can simply generalize ular orbitals. We should also point out that convergence the energy to of the pair amplitude and response equations is greatly 23 E(κ)= h0|(1 + Z)e−T e−κ H eκ eT |0i (21) aided by using DIIS. Our Newton-Raphson procedure typically uses the diagonal Hessian and turns on the full and make it stationary with respect to κ, which gives us analytic Hessian only near convergence; this avoids get- ∂E(κ) ting trapped in high energy local minima. 0= (22) It should be noted here that the one-body density ma- ∂κ pq κ=0 trix γ is diagonal in the basis in which we define the −T † † T = h0|(1 + Z)e [H,c cq − c cp ]e |0i pairing. In other words, the molecular orbitals defining X pσ σ qσ σ σ the pCCD T and Z operators are also the natural or- Γ where we work at κ = 0 by transforming the basis in bitals of pCCD. The two-body density matrix is also very sparse and has a kind of semi-diagonal form where which we express the Hamiltonian (i.e. by transforming qq pq qp the one- and two-electron integrals). The commutator only Γpp, Γpq, and Γpq are non-zero. These properties can be evaluated readily: are true both for pCCD and for DOCI (and indeed for any zero-seniority wave function method). Detailed ex- † r † q † [H,c cq ]= h c cq − h c cr (23) pressions for the density matrices can be found in the pσ σ X p rσ σ X r pσ σ r r Appendix. rs † † + v c c ct ′ cq X X pt rσ sσ′ σ σ ′ rst σ IV. PAIR COUPLED CLUSTER AND DOUBLY qt † † OCCUPIED CONFIGURATION INTERACTION − v c ct ′ cs ′ crσ X X rs pσ σ σ rst σ′ where the Hamiltonian is Now that we have given ample detail about pCCD and have introduced DOCI, it will prove useful to compare re- p † 1 pq † † H = h c cq + v c c cs ′ cr sults from the two methods for a variety of small systems X X q pσ σ 2 X X rs pσ qσ′ σ σ pq σ pqrs σσ′ for which the DOCI calculations are feasible. We will (24) compare the energies from the two approaches, and also 5 look at overlaps of the pCCD and DOCI wave functions; −7.92 explicitly, we will compute −7.93 −7.94 ∆E = EpCCD − EDOCI (27) −7.95 to assess the quality of the pCCD energy and −7.96 −7.97 S Z −T T = h0|(1 + )e |DOCIi hDOCI|e |0i (28) −7.98 E (Hartree) to assess the quality of the pCCD wave functions. Note −7.99 that S ≈ 1 when pCCD is close to DOCI; more explicitly, −8.00 FCI −8.01 DOCI we have pCCD −8.02 h0|(1 + Z)e−T eT |0i =1, (29) 1 1.5 2 2.5 3 3.5 4 4.5 5 RLi−H (Å) and inserting the projector |DOCIi hDOCI| should not substantially change this value when pCCD and DOCI 2.5E−06 2.5E−07 roughly coincide. Because pCCD is biorthogonal, we do not have S < 1; indeed, we will frequently see that S is 2.0E−06 2.0E−07 slightly larger than one. We emphasize here that both 1.5E−06 1.5E−07 pCCD and DOCI can be symmetry adapted despite hav- 1.0E−06 1.0E−07 ing individual orbitals which are not symmetry eigen- functions, due to the orbital optimization; indeed, for 5.0E−07 5.0E−08 the examples discussed below pCCD with optimized or- 0.0E+00 0.0E+00 Overlap bitals appears to respect point-group symmetry, though E (Hartree) we have found model Hamiltonians for which this is not −5.0E−07 −5.0E−08 the case. We will always compare DOCI and pCCD −1.0E−06 ∆E −1.0E−07 1 − S with the same orbital set (usually orbitals optimized for −1.5E−06 −1.5E−07 pCCD). Spot checks show that typically orbitals opti- 0 1 2 3 4 5 mized for DOCI are virtually indistinguishable from or- RLi−H (Å) bitals optimized for pCCD. All DOCI and pCCD calculations in this section and indeed throughout the manuscript use in-house pro- FIG. 2. Dissociation of LiH. Top panel: Dissociation energies grams, as do the frozen-pair coupled cluster calculations from FCI, DOCI, and pCCD. Bottom panel: Difference be- discussed in Sec. V; other calculations used the Gaussian tween DOCI and pCCD energies (∆E, defined in Eqn. 27 and −S program package.24 Throughout, we will use Dunning’s measured on the left axis) and in the overlap (1 , measured S cc-pVDZ basis set,25 because we need a sufficiently small on the right axis with defined in Eqn. 28). basis that the DOCI is computationally tractable, though we will use Cartesian rather than spherical d-functions. expect DOCI and pCCD to be very accurate in this case. We start by noting that for H2, as for any two-electron singlet, pCCD with orbital optimization is exact (and is Indeed, Fig. 2 shows that pCCD and DOCI are energeti- equivalent to DOCI). This is just because one can use cally indistinguishable and both are essentially superim- occupied-virtual rotations to make single excitations in posable with FCI (errors are on the order of 0.4 mEH CCSD vanish (in other words, one can do Brueckner cou- throughout the dissociation). Moreover, the DOCI and pled cluster doubles) and then pick a virtual-virtual rota- pCCD wave functions have near unit overlap throughout tion to eliminate the seniority two excitation amplitudes. the dissociation. This is exactly what we would expect One can see this by noting that for a two-electron singlet, for such a problem. we have We next turn our attention to the dissociation of equally spaced hydrogen chains. These serve as impor- 1 ab † † tant prototypes of strongly correlated systems and map T = t c c c1 c1 ; (30) 2 X 1,1 a↑ b↓ ↓ ↑ 26 ab in a loose sense to the Hubbard Hamiltonian. The top panel of Fig. 3 shows the difference between the DOCI the combination of fermionic antisymmetry and spin and pCCD energies per electron pair, while the bottom ab ba symmetry means that t1,1 = t1,1, so we can define a real panel shows the deviation of the overlap S from unity, ab symmetric matrix Mab = t1,1 which can be diagonalized again per electron pair. These results appear to be sat- by a virtual-virtual rotation so that T takes the pCCD urating, though unfortunately the DOCI calculations on form. Numerically, we find that with optimized orbitals, H10 are impracticably expensive with our code. EpCCD = EDOCI = EFCI and S = 1, as we should. We should note that while the equivalence between In Fig. 2 we show results for the dissociation of LiH. DOCI and pCCD has been established for energetically Because LiH is a quasi-two–electron problem, we would optimized orbitals, we see the same general behavior 6

2.0E−05 1.0E−02 H4 0.0E+00 H6 H −2.0E−05 8 1.0E−03 −4.0E−05 −6.0E−05

−8.0E−05 E (Hartree) ∆

E/N (Hartree) 1.0E−04 ∆ −1.0E−04 H4 −1.2E−04 H6 H8 −1.4E−04 1.0E−05 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5

RH−H (Å) RH−H (Å)

2.0E−05 1.0E−01 H 0.0E+00 4 H6 −2.0E−05 1.0E−02 H8 −4.0E−05 1.0E−03 −6.0E−05

−8.0E−05 1 − S

(1−S)/N 1.0E−04 −1.0E−04 −1.2E−04 H4 1.0E−05 −1.4E−04 H6 H8 −1.6E−04 1.0E−06 0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5

RH−H (Å) RH−H (Å)

FIG. 3. Dissociation of equally spaced hydrogen chains. Top FIG. 4. Dissociation of equally spaced hydrogen chains in the panel: Differences between DOCI and pCCD energies (∆E, canonical RHF basis rather than the pCCD-optimized basis defined in Eqn. 27) per electron pair. Bottom panel: devia- used elsewhere. Top panel: Differences between DOCI and tions in the overlap (1 − S, with S defined in Eqn. 28) per pCCD energies (∆E, defined in Eqn. 27). Bottom panel: electron pair. deviations in the overlap (1 − S, with S defined in Eqn. 28). when DOCI and pCCD pair canonical RHF orbitals in- stricted Hartree-Fock (UHF) result, despite being closed- stead, though not to the same degree. That is, even shell wave functions, though as we shall see later, this is pairing canonical RHF orbitals rather than optimized somewhat fortuitous. These methods miss a significant orbitals, pCCD and DOCI give energies that agree to amount of the correlation compared to UHF-based CCSD within a few milliHartree, with the agreement predictably and CCSD(T); the dynamic correlation, then, is clearly degrading as the systems become more strongly corre- not well described. lated. We can see this in hydrogen chains in Fig. 4. Similar conclusions can be reached from examining the Strangely, the agreement between DOCI and pCCD ap- dissociation of N2. As Fig. 1 reveals, DOCI does not pears to improve as we move from H4 to H6 to H8 when give all the strong correlation needed to dissociate the using canonical RHF orbitals, while in the optimized or- triple bond in N2 correctly, but does offer substantial bital case we see the opposite behavior. We should em- improvements over RHF. We see similar results in Fig. phasize that the deviations in the energy and overlap in 6. In these calculations, we froze the nitrogen 1s core Fig. 4 are not shown per electron pair. orbitals after the orbital optimization and compare the Our next example is the symmetric double dissocia- frozen-core DOCI to the frozen-core pCCD. We also note tion of H2O, as shown in Fig. 5. Again, pCCD and that our procedure of repeatedly following instabilities in DOCI provide nearly identical energies throughout the the pCCD orbital Hessian led to an unphysical reference dissociation process, and the overlaps of the pCCD and determinant for which the pCCD broke down; we have DOCI wave functions are large. The coincidence of DOCI thus used a stationary point rather than a minimum of and pCCD, in other words, is true not just for one pair of the pCCD energy functional to define the reference. Our strongly correlated electrons, but for two pairs as well. At results reiterate that pCCD and DOCI get most but not dissociation, DOCI and pCCD give essentially the unre- all of the strong correlation in N2, and fail to account for 7

−75.50 −108.00 RHF −75.60 UHF −108.20 DOCI −75.70 pCCD −108.40 UCCSD RCCSD −75.80 −108.60 −75.90 −108.80 −76.00 E (Hartree) UHF E (Hartree) −109.00 −76.10 DOCI pCCD −76.20 UCCSD −109.20 UCCSD(T) −76.30 −109.40 0.5 1 1.5 2 2.5 3 1 1.5 2 2.5 3

RO−H (Å) RN−N (Å)

1.0E−04 1.0E−04 3.0E−04 3.0E−04 ∆E 5.0E−05 1 − S 5.0E−05 0.0E+00 0.0E+00 0.0E+00 0.0E+00 −3.0E−04 −3.0E−04 −5.0E−05 −5.0E−05 −6.0E−04 −6.0E−04 −1.0E−04 −1.0E−04 −9.0E−04 −9.0E−04 −1.5E−04 −1.5E−04 Overlap Overlap E (Hartree) E (Hartree) −1.2E−03 −1.2E−03 −2.0E−04 −2.0E−04 −2.5E−04 −2.5E−04 −1.5E−03 ∆E −1.5E−03 1 − S −3.0E−04 −3.0E−04 −1.8E−03 −1.8E−03 0.5 1 1.5 2 2.5 3 1 1.5 2 2.5 3

RO−H (Å) RN−N (Å)

FIG. 5. Symmetric double dissociation of H2O. Top panel: FIG. 6. Dissociation of N2. Top panel: Dissociation energies Dissociation energies from DOCI and pCCD, as well as from from DOCI and pCCD, as well as from RHF, UHF, and RHF- unrestricted Hartree-Fock (UHF) and CCSD and CCSD(T) and UHF-based CCSD. Bottom panel: Errors in the energy based thereon. Bottom panel: Errors in the energy (∆E, (∆E, defined in Eqn. 27 and measured on the left axis) and in defined in Eqn. 27 and measured on the left axis) and in the the overlap (1 − S, measured on the right axis with S defined overlap (1 − S, measured on the right axis with S defined in in Eqn. 28). Eqn. 28).

retrieve about 36% of the correlation energy even after TABLE I. Energies and overlaps in the neon atom. Here, orbital optimization, with optimized orbitals very close E Ref denotes the energy of the reference determinant. We to the canonical RHF molecular orbitals. The bulk of show results for both the optimized determinant for pCCD the correlations must then involve determinants of higher and for the canonical RHF determinant as a reference. seniority. In order to remedy this deficiency, we turn Optimized Canonical to what we call frozen-pair coupled cluster,10 as we will E Ref -128.488 823 -128.488 866 describe shortly. E DOCI -128.559 677 -128.546 705 First, however, it may be instructive to take a closer EpCCD -128.559 674 -128.546 701 look at the T -amplitudes of pCCD and the CI coefficients ECCSD -128.683 931 -128.683 958 − − 1 − S 1.43 × 10 7 1.16 × 10 7 of DOCI, to understand why the two methods coincide so neatly. Often, what we find, as in the examples above, is that the pCCD T -amplitudes are such that each occupied orbital is strongly correlated with at most one virtual the dynamic correlation effectively. Nonetheless, even for a orbital, so that each row of the matrix ti has at most one this triple bond we see that DOCI and pCCD have close large entry, while most of the amplitudes are small. The agreement. DOCI vector follows this same basic structure, which is One can see that DOCI and pCCD do not describe unsurprising since the DOCI and pCCD wave functions dynamic correlation particularly well by considering the are essentially the same. In these cases, pCCD and DOCI neon atom, as seen in Tab. I. While DOCI and pCCD are similar to a kind of perfect pairing wave function.27–29 are in excellent agreement with one another, they only For example, for the stretched H2O case, the pCCD and 8

DOCI wave functions are qualitatively −75.50 2 4 2 2 2 |Ψi ≈ |O O OH − α OH ⋆ i (31) −75.60 1s lp σ σ where α approaches 1 at dissociation and where O1s,Olp, −75.70 OHσ, and OHσ⋆ respectively denote the oxygen 1s or- −75.80 bital, oxygen lone-pair orbitals, OH bonding orbitals, and −75.90 OH antibonding orbitals. In the case of stretched H2O, it is the small deviations from this perfect pairing struc- −76.00 UHF ture which cause the energy to be close to the UHF limit. E (Hartree) pCCD −76.10 UCCSD(T) That is, the only wave function amplitudes larger than fpCCD ∼ 0.05 correspond to excitations from an OH bonding −76.20 fpCCSD fpCCSDT orbital into its antibonding orbital, but correlating the −76.30 bonding orbitals alone yields an energy somewhat above 0.5 1 1.5 2 2.5 3 the sum of restricted open-shell Hartree-Fock atomic en- RO−H (Å) ergies. Thus, we might not expect pCCD to describe strong correlations beyond those accessible with the perfect pairing structure, even though we must emphasize FIG. 7. Frozen pair symmetric double dissociation of H2O. that the pCCD wave function is not inherently limited to this form. Indeed, it is important to note that we have found cases −75.85 in the repulsive Hubbard Hamiltonian26 for which neither −75.90 pCCD nor DOCI adopt a perfect pairing structure, yet the two methods still agree closely. We also note that −75.95 30 for the attractive pairing Hamiltonian or the attractive −76.00 Hubbard Hamiltonian (results not shown), one can ﬁnd instances in which pCCD does not resemble DOCI. In −76.05 these cases, the DOCI coeﬃcients and the pCCD ampli- −76.10 E (Hartree) tudes are dense and neither DOCI nor pCCD displays −76.15 CCSD a perfect pairing structure. While pCCD and DOCI in- FCI clude a perfect pairing wave function as a special case, −76.20 fpCCD CCSDT they are more general methods. The fact that pCCD −76.25 closely resembles DOCI seems a key feature of fermionic 1 1.5 2 2.5 3 repulsive Hamiltonians like the molecular one. Re

V. FROZEN PAIR COUPLED CLUSTER ◦ FIG. 8. Symmetric double dissociation of H2O at 110 bond angle, with frozen pair coupled cluster and traditional coupled The basic idea of frozen pair coupled cluster is very cluster methods. FCI and CCSD data taken from Ref. 33. All results use closed-shell (restricted) wave functions. simple. One could imagine decomposing the T2 double- (0) excitation operator into a pair part T2 and a non-pair part T˜2; one would then solve the pCCD equations for the pair amplitudes and then solve the usual CCD equations excitations in the cluster operator. What we wish to without allowing the pair amplitudes to change. Note do here is to briefly consider frozen pair coupled clus- that the non-pair operator T˜ creates seniority non-zero ter with single-, double-, and triple-excitation amplitudes 2 31,32 determinants, which we rely upon to provide the dynamic (fpCCSDT). correlation which pCCD lacks; T˜2 on a seniority zero de- In Fig. 7, we show the symmetric double dissoci- terminant returns a linear combination of determinants ation of H2O, this time with the frozen pair approxi- with seniorities two and four. Note also that the Fock op- mation. The effect of single excitations is in this case erator for orbital-optimized pCCD is in general neither small (fpCCD and fpCCSD give similar results) and diagonal nor in the semicanonical form which diagonal- fpCCSD gives results fairly similar to the UHF-based izes the occupied-occupied and virtual-virtual blocks, so CCSD and CCSD(T) curves. Adding full triple excita- the full non-canonical form of the amplitude equations tions in fpCCSDT gives larger correlation at dissociation must be used. This is not a concern for pCCD, where only and probably overcorrelates somewhat. the diagonal elements of the (generally non-diagonal) For comparison purposes, we show results from FCI Fock operator contribute to the amplitude equations. and RHF-based CCSD and CCSDT in Fig. 8. These cal- What we have described above we would call frozen culations fix the H-O-H bond angle at 110◦ rather than pair CCD (fpCCD). One could of course extend this ba- at the 104.474◦ used in our other calculations, and use sic idea to include single excitations and triple or higher spherical d functions; the CCSD, CCSDT, and FCI data 9

TABLE II. Energies in the neon atom. Here, ERef denotes −108.00 RHF the energy of the reference determinant. −108.20 UHF pCCD Method Energy UCCSD ERef -128.488 823 −108.40 RCCSD EpCCD -128.559 674 fpCCSD −108.60 EfpCCD -128.687 585 E CCD -128.683 851 −108.80 EfpCCSD -128.687 619 E (Hartree) ECCSD -128.683 931 −109.00 EfpCCSDT -128.688 497 ECCSDT -128.685 089 −109.20 −109.40 1 1.5 2 2.5 3 R (Å) are taken from Ref. 33. We see that as one stretches N−N the bond, CCSD and CCSDT go through a maximum and turn over; for larger bond lengths, we would expect FIG. 9. Dissociation of N2 with various coupled cluster meth- CCSD and CCSDT to overcorrelate more. In contrast, ods. fpCCD is coincidentally very close to FCI, and while fpCCSD and fpCCSDT overcorrelate somewhat more, they provide sensibly-shaped dissociation curves without VI. CONCLUSIONS requiring symmetry breaking. Table II shows fpCCD and fpCCSD results for the neon While traditional coupled cluster theory is highly suc- atom. While pCCD undercorrelates significantly com- cessful for the description of weakly correlated systems, pared to CCSD, making the frozen pair approximation it generally fails to describe strong correlation. Para- (0) doxically, by simply eliminating the vast bulk of the yields results that differ from those without freezing T2 by about 4 milliHartree. As with the double dissocia- cluster operator, one can form pair coupled cluster dou- tion of H2O, frozen pair coupled cluster overcorrelates bles, which accurately reproduces DOCI, and to the ex- slightly. tent that DOCI can describe strong correlations, so too can pCCD. Moreover, pCCD accomplishes this task with As a final example, we consider fpCCSD for the dis- mean-field computational scaling for the coupled clus- sociation of N2, as seen in Fig. 9. As should by now ter part. Not only does pCCD reproduce the DOCI en- be familiar, fpCCSD gives a reasonable accounting for ergy, it also reproduces the DOCI wave function. The dynamic correlation but overcorrelates somewhat. Both DOCI wave function, in other words, is essentially factor- fpCCSD and RHF-based CCSD break down for large izable into the pCCD form. Loosely, this can be accom- bond lengths, and have an artificial bump in the dis- plished because, upon orbital optimization, the pCCD sociation curve; while fpCCSD does not eliminate this and DOCI wave functions studied in this work adopt a unphysical effect, it at least mitigates it somewhat. perfect–pairing-like structure. Our results show that frozen pair coupled cluster While pCCD can describe strong correlations, it is should be understood as an easy way to incorporate the much less successful at modeling dynamic correlation, reasonable pCCD description of strong correlation while which apparently requires the breaking of electron pairs retaining much of the ability of traditional coupled clus- to obtain higher seniority determinants when we define ter to also describe dynamic correlation. However, while pairs in terms of the spatial orbitals in a particle-hole easy to implement and conceptually simple, it is also im- representation. Using pCCD to obtain the zero-seniority portant to note that a frozen pair full coupled cluster ap- part of the cluster operator and then solving the tradi- proach would give the wrong answer. In other words, in tional coupled cluster equations for the rest of the am- the exact theory one must clearly allow the zero-seniority plitudes yields frozen-pair coupled cluster, which seems T2 amplitudes to relax from their pCCD values. In prac- to be able to describe both weakly and strongly corre- tice, fpCCSD should allow for a reasonable description lated systems with reasonable accuracy and with a com- of both strongly and weakly correlated systems at essen- putational cost not much different from that of standard tially the cost of a CCSD calculation, without breaking coupled cluster methods. spin symmetry, although fpCCSD would be expected to Of course pCCD is not a panacea and there are occa- break down somewhat for cases where pCCD is unable sions when pCCD fails to account for the strong corre- to capture all the strong correlations, as is the case with lation present in the DOCI wave function, although we N2. For two-electron singlets, fpCCD and fpCCSD are have not seen such a case for the molecular Hamiltonian. both exact, because as we have previously noted, pCCD Likewise, it is possible that the DOCI form is too re- is already FCI, which implies that T1 and the non-zero stricted to allow for a complete description of the strong seniority parts of T2 vanish. correlations present, as appears to happen in the disso- 10 ciation of N2, for example. In such cases, the frozen-pair with coupled-cluster approach would be of less utility. We speculate that it may be possible to include these strong † † κ = κpq c cq − c cp (A2) X X pσ σ qσ σ correlations by generalizing the pairing structure to non- p>q σ singlet pairing, so that the pairs included in pCCD and DOCI are not just the two electrons in the same spatial where the orbital rotation is given by the unitary trans- orbital. Regardless, we hope that pCCD and its frozen formation exp(κ). At every step of the Newton-Raphson pair extensions will be useful tools for the description of scheme, we solve for κ, build exp(κ) which rotates to a both weakly and strongly correlated systems without the new orbital basis, transform the integrals, and begin a need for symmetry breaking or higher excitation opera- new iteration. tors. We have already seen that the gradient is simply

ACKNOWLEDGMENTS ∂E(κ) † = Ppq h[H,c cq ]i (A3) ∂κ X pσ σ pq κ=0 σ This work was supported by the National Science Foundation (CHE-1110884). GES is a Welch Founda- where P is a permutation operator P = 1 − (p ↔ q) tion chair (C-0036). T.S. is an awardee of the Weizmann pq pq and the notation for the expectation value means Institute of Science – National Postdoctoral Award Pro- gram for Advancing Women in Science. We would like −T T to thank Carlos Jim´enez-Hoyos for helpful discussion. hOi = h0|(1 + Z)e O e |0i. (A4) Similarly, the Hessian is

Appendix A: Density Matrices and Orbital Hessian ∂2E(κ) Hpq,rs = (A5) ∂κ ∂κ pq rs κ=0 For completeness, we include here expressions for the 1 † † pCCD density matrices and orbital rotation Hessian; to- = Ppq Prs h[[H,c cq ],c cs ]i 2 X pσ σ rη η gether with the orbital rotation gradient of Eqn. 25, these σ,η provide everything needed for the Newton-Raphson algo- 1 † † + Ppq Prs h[[H,c cs ],c cq ]i rithm we use for orbital optimization. 2 X rη η pσ σ Recall that the energy is written as σ,η

E(κ)= h0|(1 + Z)e−T e−κ H eκ eT |0i (A1) where η is another spin index. We obtain

1 u s s u u q q u s q q s Hpq,rs = Ppq Prs δqr h γ + h γ + δps (h γ + h γ ) − h γ + h γ (A6) n2 X p u u p r u u r p r r p u 1 + δ vuv Γst + vst Γuv + δ vqt Γuv + vuv Γqt 2 X qr pt uv uv pt ps uv rt rt uv tuv + vuv Γqs + vqs Γuv − vst Γqu + vts Γqu + vqu Γst + vqu Γts . X pr uv uv pr X pu rt pu tr rt pu tr pu o uv tu

The one-particle density matrix we have deﬁned as also its natural orbital basis. We then have

p −T † T γ = h0|(1 + Z)e c cp e |0i. (A7) q X qσ σ j j σ γi =2 1 − xi δij , (A8a) b b γa =2 xa δab, (A8b) i a Because T and Z both preserve the seniority of the wave γa = γi =0, (A8c) function, and the reference |0i has seniority zero, it is immediately clear that the one-particle density matrix is diagonal in the basis in which we have deﬁned the pairing; the optimized orbital basis for pCCD, in other words, is where δpq is the Kronecker delta and where we have de- 11

bb b ﬁned Γaa =2 xa, (A13)

j a j xi = ti za, (A9a) X Γij =4 1 − xi − xj +2 δ 3 xi − 1 , (A14) a ij i j ij i xb = tb zi . (A9b) a X i a i Γia =Γai =4 xa − ta zi , (A15) ia ai a i a Recall that i and a are respectively occupied and virtual orbital indices. Γab =2 δ xa, (A16) Similar considerations show that the two-particle den- ab ab a sity matrix is also sparse in the natural orbital basis. The non-zero elements of the two-particle density matrix are qp 1 pq Γpq = − Γpq. (A17) q=6 p 2 jj j i Γ =2 x + δij 1 − 2 x , (A10) ii h i ii We have deﬁned the additional intermediate

xa = tb ta zj . (A18) i X i j b Γaa =2 ta + xa − 2 ta xa + xi − ta zi , (A11) jb ii i i i a i i a Note that the sparsity of the one- and two-particle density matrices allows one to considerably reduce the cost ii i Γaa =2 za, (A12) of evaluating the Hessian.

1 J. Paldus and X. Z. Li, Adv. Chem. Phys. 110, 1 (1999). (2003). 2 R. J. Bartlett and M. Musial, Rev. Mod. Phys. 79, 291 17 G. E. Scuseria, A. C. Scheiner, T. J. Lee, J. E. Rice, and (2007). H. F. Schaefer III, J. Chem. Phys. 86, 2881 (1987). 3 I. Shavitt and R. J. Bartlett, Many-Body Methods in 18 G. E. Scuseria, C. L. Janssen, and H. F. Schaefer III, J. Chemistry and Physics (Cambridge University Press, New Chem. Phys. 89, 7382 (1988). York, 2009). 19 G. E. Scuseria, T. M. Henderson, and D. C. Sorensen, J. 4 G. D. Purvis and R. J. Bartlett, J. Chem. Phys. 76, 1910 Chem. Phys. 129, 231101 (2008). (1982). 20 G. E. Scuseria, T. M. Henderson, and I. W. Bulik, J. 5 K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head- Chem. Phys. 139, 104113 (2013). Gordon, Chem. Phys. Lett. 157, 479 (1989). 21 G. E. Scuseria and H. F. Schaefer III, Chem. Phys. Lett. 6 P. A. Limacher, P. W. Ayers, P. A. Johnson, S. de 142, 354 (1987). Baerdemacker, D. van Neck, and P. Bultinck, J. Chem. 22 U. Bozkaya, J. M. Turney, Y. Yamaguchi, H. F. Schaefer Theory Comput. 9, 1394 (2013). III, and C. D. Sherrill, J. Chem. Phys. 135, 104103 (2011). 7 P. A. Limacher, T. D. Kim, P. W. Ayers, P. A. Johnson, 23 G. E. Scuseria, T. J. Lee, and H. F. Schaefer III, Chem. S. de Baerdemacker, D. van Neck, and P. Bultinck, Mol. Phys. Lett. 130, 236 (1986). Phys. 112, 853 (2014). 24 M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, 8 P. Tecmer, K. Boguslawski, P. A. Johnson, P. A. Limacher, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, M. Chan, T. Verstraelen, and P. W. Ayers, J. Phys. Chem. B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Cari- A (2014), published online; DOI 10.1021/jp502127v. cato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, 9 K. Boguslawski, P. Tecmer, P. W. Ayers, P. Bultinck, G. Zheng, J. L. Sonnenberg, W. Liang, M. Hada, M. Ehara, S. de Baerdemacker, and D. van Neck, Phys. Rev. B 89, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Naka- 201106(R) (2014). jima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. 10 T. Stein, T. M. Henderson, and G. E. Scuseria, J. Chem. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, Phys. 140, 214113 (2014). J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, 11 L. Bytautas, T. M. Henderson, C. A. Jim´enez-Hoyos, J. K. T. Keith, R. Kobayashi, J. Normand, K. Raghavachari, Ellis, and G. E. Scuseria, J. Chem. Phys. 135, 044119 A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, (2011). M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. 12 T. L. Allen and H. Shull, J. Phys. Chem. 66, 2281 (1962). Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, 13 D. W. Smith and S. J. Fogel, J. Chem. Phys. 43, S91 R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, (1965). R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, 14 F. Weinhold and E. Bright Wilson Jr., J. Chem. Phys. 46, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, 2752 (1967). J. J. Dannenberg, S. Dapprich, P. V. Parandekar, N. J. 15 M. Couty and M. B. Hall, J. Phys. Chem. A 101, 6936 Mayhall, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. (1997). Ortiz, J. Cioslowski, , and D. J. Fox, “Gaussian Devel- 16 C. Kollmar and B. A. Heß, J. Chem. Phys. 119, 4655 opment Version, Revision H.21,” (2010), Gaussian, Inc., 12

Wallingford CT. 9190 (2002). 25 T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989). 30 T. M. Henderson, G. E. Scuseria, J. Dukelsky, A. Signo- 26 J. Hubbard, Proc. Roy. Soc. London 276, 238 (1963). racci, and T. Duguet, Phys. Rev. C 89, 054305 (2014). 27 A. C. Hurley, J. Lennard-Jones, and J. A. Pople, Proc. 31 J. Noga and R. J. Bartlett, J. Chem. Phys. 86, 7041 (1987). Roy. Soc. London, Ser. A 220, 446 (1953). 32 G. E. Scuseria and H. F. Schaefer III, Chem. Phys. Lett. 28 W. A. Goddard and L. A. Harding, Annu. Rev. Phys. 152, 382 (1988). Chem. 29, 363 (1978). 33 J. Olsen, P. Jørgensen, H. Koch, A. Balkova, and R. J. 29 T. Van Voorhis and M. Head-Gordon, J. Chem. Phys. 117, Bartlett, J. Chem. Phys. 104, 8007 (1996).