Analytic Energy Gradients for the Coupled-Cluster Singles and Doubles with Perturbative Triples Method with the Density-Fitting Approximation

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1 Analytic Energy Gradients for the Coupled-Cluster Singles and Doubles with Perturbative Triples Method with the Density-Fitting Approximation Uğur Bozkaya 1, a) and C. David Sherrill 2 1) Department of Chemistry, Hacettepe University, Ankara 06800, Turkey 2) Center for Computational Molecular Science and Technology, School of Chemistry and Biochemistry, and School of Computational Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA Abstract An efficient implementation of analytic gradients for the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method with the density-fitting (DF) approximation, denoted DF-CCSD(T), is reported. For the molecules considered, the DF approach substantially accelerates conventional CCSD(T) analytic gradients due to the reduced input/output (I/O) time and the acceleration of so-called gradient terms : formation of particle density matrices (PDMs), computation of the generalized Fock-matrix (GFM), solution of the Z-vector equation, formation of the effective PDMs and GFM, back-transformation of the PDMs and GFM, from the molecular orbital (MO) to the atomic orbital (AO) basis, and computation of gradients in the AO basis. For the largest member of the molecular test set considered (C 6 H 14 ) the computational times for analytic gradients (with the cc-pvtz basis set, in serial) are [CCSD(T)] and 49.8 [DF-CCSD(T)] hours, a speedup of more than 2-fold. In the evaluation of gradient terms, the DF approach completely avoids the use of four-index two-electron integrals. Similar to our previous studies on DF- MP2 and DF-CCSD gradients, our formalism employs 2- and 3-index two-particle density matrices (TPDMs) instead of 4-index TPDMs. Errors introduced by the DF approximation are negligible for equilibrium geometries and harmonic vibrational frequencies. a) Author to whom correspondence should be addressed.electronic mail: ugur.bozkaya@hacettepe.edu.tr. 1

2 I. INTRODUCTION It has been demonstrated that coupled-cluster (CC) methods are accurate for the prediction of molecular properties. 1 6 The coupled-cluster singles and doubles (CCSD) method 7 provides quite accurate results for most molecular systems at equilibrium geometries, but nevertheless a triple excitations correction is required to obtain high accuracy The coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method 11,12,15 provides excellent results for a broad range of chemical systems near equilibrium geometries. 13,16 25 Analytic gradient methods are very helpful to locate stationary points on potential energy surfaces, as well as for the computation of one-electron properties Following the Z- vector method of Handy and Schaefer, 31 Adamowicz et al. 32 presented the initial algebraic expressions for analytic gradients of the CCSD method. In 1985, the analytic gradient for the coupled-cluster doubles (CCD) method was implemented by Fitzgerald et al. 33 In a 1987 study, Scheiner et al. 34 implemented the analytic gradient for the closed-shell CCSD method. Later, several CCSD analytic first derivative papers were also published The first implementation of CCSD(T) analytic gradients is reported by Scuseria for the closedshell restricted Hartree-Fock (RHF) reference, 40 later improved by Lee and Rendell. 41 In 1992 and 1993 studies, the CCSD(T) analytic gradients for unrestricted and restricted open-shell Hartree-Fock (UHF and ROHF) references were reported by Watts et al. 22,42 In 2003, Hald et al. 43 reported an integral direct implementation of CCSD(T) analytic gradients. Approximate factorization of the electron repulsion integrals (ERIs) has been of considerable interest in modern quantum chemistry The most common ERI decomposition technique is density fitting (DF) ,58 65 In the DF approximation the four-index ERIs are expanded in terms of three-index tensors. Another popular integral tensor decomposition approach is Cholesky decomposition (CD) ,62,63 The DF and CD approximations are very helpful in reducing the cost of integral transformations and the storage requirements of the ERI tensor. In the DF approximation, integrals of the primary basis set are expanded in terms of a pre-defined iliary basis set, whereas in the CD approach the three-index factors are generated from the primary basis set. Analytic gradients for the density-fitted post-hartree Fock (HF) methods have been reported for second-order Møller Plesset perturbation theory (MP2) with conventional HF reference (RI-MP2), as well as with the DF-HF reference, 60 coupled cluster model 2

3 CC2, 69,70 local MP2 (DF-LMP2), 71 complete active space second-order perturbation theory (DF-CASPT2), 72 orbital-optimized MP2 (DF-OMP2), 61 local time-dependent coupled cluster response theory (DF-TD-LCC2), 73 third-order Møller Plesset perturbation theory (DF-MP3), 74 the DF-MP2.5 model, 74 linearized coupled-cluster doubles (DF-LCCD), 74 and recently for the density-fitted CCSD and CCD methods (DF-CCSD and DF-CCD). 64 However, one can not compute analytic gradients with the CD integrals unless one uses an approach like that of Aquilante et al. 75 The computational advantages of the DF approach for analytic gradients are that one can replace the 4-index integrals and 4-index two-particle density matrix (TPDM) with 2- index and 3-index integrals and TPDMs. Thus minimizing input/output (I/O) and storage requirements, and reducing the computational cost of certain terms. Further, the DF approach also facilitates the solution of the Z-vector equation 31 and the computation of the generalized Fock matrix (GFM). 60 Energies for the density-fitted and Cholesky decomposed CCSD(T) methods have been presented in previous studies. 58,65,76 78 In this study, analytic gradient expressions are reported for the density-fitted CCSD(T) method [DF-CCSD(T)], where the density-fitting has been applied to overall energy expression. The equations presented have been coded in C++ language and added to the Dfocc module of the Psi4 package. 79 The computational time of the DF-CCSD(T) gradients is compared with that of the conventional CCSD(T) method (from Cfour). 80 The DF-CCSD(T) method is applied to equilibrium geometries and vibrational frequencies. II. THEORETICAL APPROACH A. Density-Fitting In the DF approximation the atomic-orbital (AO) basis integrals are written as (µν λσ) DF = 3 Q b Q µνb Q λσ, (1)

4 the DF factor b Q µν is defined as b Q µν = (µν P )[J 1/2 ] P Q, (2) P where (µν P ) = χ µ (r 1 )χ ν (r 1 ) 1 r 12 φ P (r 2 ) dr 1 dr 2, (3) and J P Q = φ P (r 1 ) 1 r 12 φ Q (r 2 ) dr 1 dr 2, (4) where {φ P (r)} and {χ µ (r)} are iliary and primary basis functions, respectively. TEIs over molecular-orbital (MO) can be written similarly where b Q pq is a MO basis DF factor. (pq rs) DF = Q b Q pqb Q rs, (5) B. DF-CCSD Energy and Amplitude Equations At first, we employ the conventional notation for the orbital indices: i, j, k, l, m, n for occupied orbitals; a, b, c, d, e, f for virtual orbitals; and p, q, r, s, t, u for general spin orbitals. The DF-CCSD correlation energy can be expressed as follows E = 0 e ˆT Ĥ N e ˆT 0, (6) where ĤN is the normal-ordered Hamiltonian operator, 4,81 0 is the reference determinant, and ˆT is the cluster excitation operator. For the CCSD wave function ˆT = ˆT 1 + ˆT 2, (7) 4

5 where ˆT 1 and ˆT 2 are single- and double-excitation operators, respectively. vir ˆT 1 = t a i â î, (8) i a ˆT 2 = 1 4 vir i,j a,b t ab ij â ˆb ĵî, (9) where â and î are the creation and annihilation operators, and t a i and t ab ij are the single and double excitation amplitudes, respectively. The amplitude equations can be written as Φ a i e ˆT Ĥ N e ˆT 0 = 0, (10) Φ ab ij e ˆT Ĥ N e ˆT 0 = 0, (11) where Φ a i and Φ ab ij are singly- and doubly-excited Slater determinants, respectively. C. DF-CCSD-Λ Energy Functional (Unrelaxed Lagrangian) It is convenient to introduce a Lagrangian 82,83 (DF-CCSD-Λ functional) to obtain a variational energy functional ( L) as follows L = 0 (1 + ˆΛ)e ˆT Ĥe ˆT 0, (12) where Ĥ is the Hamiltonian operator and ˆΛ is the coupled-cluster de-excitation operator. For the CCSD wave function ˆΛ = ˆΛ 1 + ˆΛ 2, (13) where ˆΛ 1 and ˆΛ 2 are the single and double de-excitation operators, respectively. vir ˆΛ 1 = λ i a î â, (14) i a 5

6 ˆΛ 2 = 1 4 vir i,j a,b λ ij ab î ĵ ˆbâ, (15) where λ i a and λ ij ab are the single and double de-excitation amplitudes, respectively. The DF-CCSD T - and Λ-amplitude equations 35,37,38,84,85 are obtained using variational properties of L. 0 (1 + ˆΛ) [ ] e ˆT Ĥe ˆT E Φ a i = 0, (16) 0 (1 + ˆΛ) [ ] e ˆT Ĥe ˆT E Φ ab ij = 0, (17) where E is the DF-CCSD total energy. In our previous study 64 all explicit equations for DF-CCSD T and Λ amplitudes, and particle density matrices (PDMs) were provided in the spin-orbital formalism. Hence, we will not repeat those equations. However, spin-free equations (for T and Λ amplitudes, and PDMs) for a closed-shell RHF reference based DF-CCSD method are reported in the Supporting Information. D. Triples Energy Correction for DF-CCSD 1. Spin-Orbital Equations Following the prescription of Bartlett and co-workers, 3,5,6,86 the perturbative triples corrections for the DF-CCSD method 65 can be obtained from the following general equation E [4] T = 0 ( ˆT 1 + ˆT [2] 2 )ŴN ˆT ˆT ˆf 2 N o [2] ˆT 3 0, (18) where ˆf o N is the off-diagonal part of the normal-ordered Fock operator, and ŴN is the twoelectron component of Ĥ N. With the canonical HF orbitals the second term of Eq.(18) vanishes. The second-order triples excitation operator is defined as follows where t(c) abc ijk ˆT [2] 3 = 1 36 vir ijk abc are the connected triple amplitudes. 6 t(c) abc ijk â ˆb ĉ ˆkĵî, (19)

7 It is convenient to express the (T) correction as follows 22,23 E (T ) = 1 36 vir ijk abc [t(c) abc ijk + t(d) abc ijk] D abc ijk t(c) abc ijk, (20) where t(d) abc ijk are the disconnected triple amplitudes. t(c)abc ijk Dijk abc t(c) abc ijk = P (ij/k)p (ab/c) vir e occ P (ij/k)p (ab/c) m and t(d)abc ijk t ae ij bc ek DF are defined as t ab im mc jk DF, (21) D abc ijk t(d) abc ijk = P (ij/k)p (ab/c) t c k ab ij DF + P (ij/k)p (ab/c) t ab ij f kc, (22) where D abc ijk = f ii + f jj + f kk f aa f bb f cc. (23) The permutation operator is defined as P (pq/r) = 1 P(pr) P(qr). (24) where P(pr) acts to permute indices p and r. Then, we may define the combined triple amplitudes as follows t abc ijk = t(c) abc ijk + t(d) abc ijk. (25) Finally, the (T) correction can be written as E (T ) = 1 36 vir ijk Further, we may also define the combined ˆT 3 operator as ˆT 3 = 1 36 abc vir ijk abc 7 t abc ijk D abc ijk t(c) abc ijk. (26) t abc ijk â ˆb ĉ ˆkĵî. (27)

8 2. Closed-Shell Spin-Free Equations The closed-shell spin-free equation for the triples corrections can be written as follows, 87,88 where E (T ) = 1 3 vir ijk abc ( )( ) 4Wijk abc + Wijk bca + Wijk cab Vijk abc Vijk cba Dijk, abc (28) Wijk abc = Pijk abc ( vir e t ce kj(ia be) DF m t ab im(mj kc) DF ) / D abc ijk, (29) V abc ijk = W abc ijk + ( t a i (jb kc) DF + t b j(ia kc) DF + t c k(ia jb) DF ) / D abc ijk, (30) and P abc ijk X abc ijk = X abc ijk + X acb ikj + X bac jik + X bca jki + X cab kij + X cba kji. (31) E. Triples Lagrangian For the triples correction we introduce the following unrelaxed Lagrangian, L (T ) = 0 ( ˆT 1 + ˆT [2] 2 )ŴN ˆT ˆT ˆf 2 N o [2] ˆT ˆΛ 3 ( ˆf d N ˆT [2] 3 + ŴN ˆT 2 ) 0, (32) where ˆΛ 3 = 1 36 vir ijk abc λ ijk abc î ĵ ˆk ĉˆbâ. (33) Since the Lagrangian is stationary with respect to triples amplitudes, it can readily be shown that λ ijk abc = tabc ijk. (34) 8

9 Therefore, we may re-write the Lagrangian as follows L (T ) = 0 ( ˆT 1 + ˆT [2] 2 )ŴN ˆT ˆT ˆf 2 N o [2] ˆT ˆT ( 3 ˆf d [2] N ˆT 3 + ŴN ˆT ) 2 0. (35) F. Triples Contributions to Λ-Amplitudes 1. Spin-Orbital Equations Differentiating the Lagrangian with respect to singles and doubles amplitudes we obtain contributions of L (T ) to Λ-amplitudes, which are denoted as (T ) λ i a and (T ) λ ij ab. Hence, contributions of L (T ) to Λ 1 -amplitudes can be written as (T ) λ i a = 1 4 vir jk and their contributions to the overall amplitudes are bc t(c) abc ijk jk bc DF, (36) λ i a = DF CCSD λ i a + (T ) λ i a/d a i. (37) Similarly, contributions of L (T ) to Λ 2 -amplitudes can be written as where (T ) λ ij ab = 1 2 P vir (ab) M aef ijk bk ef DF e,f m,n k 1 2 P vir (ij) Mimn mn jc abc DF k c c vir + f kc t(c) abc ijk, (38) M abc ijk = 2t(c) abc ijk + t(d) abc ijk, (39) and their contributions to the overall amplitudes are λ ij ab = DF CCSD λ ij ab + (T ) λ ij ab /Dab ij. (40) 9

10 In the paper of Watts, Gauss, and Bartlett 22 there is t abc ijk However, this appears to be a typographical error. in Eq.(38) instead of t(c)abc ijk. 2. Closed-Shell Spin-Free Equations At first, we would like to note that the spin-free equations for Λ-amplitudes and PDMs were reported by Scuseria 40 and by Lee and Rendell 41 in terms of so-called Z-amplitudes. However, their definition of Z 1 - and Z 2 -amplitudes and PDMs are different than ours. For example, the relation between Z 2 -amplitudes and Λ 2 -amplitudes may be written as Z ab ij = 4λ ij ab 2λji ab. Therefore, instead of using their formulas, we spin-adapted our spin-orbital equations following the prescription of Scuseria and Schaefer 89 for spin-adaptation of triples amplitudes, which is not as trivial as double amplitudes. Hence, we obtain the following formula for triples contributions to Λ 1 -amplitudes, (T ) λ i a = 1 2 vir jk (jb kc) bc ( 4Wijk abc + Wijk bca + Wijk cab ) 3Wijk cba 2Wijk acb Wijk bac. (41) Similarly, our spin-free equation for Λ 2 -amplitudes, where (T ) λ ij ab = P vir +(ia, jb) e,f m,n k c ( 2M aef ijk M fea ijk ) M afe ijk (kf be) DF vir ( ) P + (ia, jb) 2Mimn abc Mimn cba Mimn acb (mj nc) DF vir + k c ( f kc 2Wijk abc Wijk cba Wijk acb ), (42) M abc ijk = W abc ijk + V abc ijk. (43) 10

11 G. Triples Contributions to PDMs 1. Spin-Orbital Equations Using our definitions of one- and two-particle density matrices (OPDM and TPDM) 60,64 we can write the following general equation for triples corrections for OPDM diagonal elements γ p (corr) = 1 2 P +(pq) 0 ˆT [2] 3 {ˆp ˆp} ˆT 3 0. (44) More explicitly, we have the following equations, γ (corr) i = 1 12 vir jk abc t(c) abc ijk t abc ijk, (45) γ (corr) a = 1 12 vir ijk bc t(c) abc ijk t abc ijk. (46) At this stage, we would like to note that in our previous studies 60,64 we employed the noncanonical perturbed orbitals (NCPO) to obtain our non-canonical gradient (NCG) equations. However, we may also use the canonical-perturbed orbitals (CPO) to obtain a strictly canonical gradient (SCG) formalism. In the SCG we need to compute only diagonal blocks of the unrelaxed OPDM, which is scales as O(N 5 ) for DF-CCSD. However, in SCG we need to a more complex Z-vector equation compared with NCG. Hence, there is no advantage of SCG over NCG in the case of DF-CCSD. However, in the case of DF-CCSD(T), the SCG approach is very helpful since the scaling of full OPDM is O(N 7 ). Thus, with the SCG approach, one needs only diagonal OPDMs, which are scale as O(N 6 ), instead of the full OPDM, which is scales as O(N 7 ). The 3-index TPDM can be written as follows, Γ Q(corr) im = 1 2 P vir +(im) M ijam b Q ja, (47) j a 11

12 Γ Q(corr) ia = 1 2 occ j M jiam b Q jm + 1 vir M ijab b Q jb vir vir M iabd b Q db 2, (48) m j b b d Γ Q(corr) ab = 1 2 P vir +(ab) M icba b Q ic, (49) i c where M ijam = 1 2 vir k bc M abc ijk t bc mk, (50) vir M ijab = t(c) abc ijk t c k, (51) k c M iabd = 1 2 vir jk c M abc ijk t dc jk. (52) Finally, we note that there is no difference between the NCG and SCG for TPDMs. 2. Closed-Shell Spin-Free Equations Spin-free equations for the OPDM can be written as γ (corr) i vir = jk abc ( ) 4Wijk abc + Wjki abc + Wkij abc Wkji abc 2Wikj abc Wjik abc Vijk abc, (53) γ (corr) a = vir ijk Similarly, TPDMs can be written as bc ( ) 4Wijk abc + Wijk cab + Wijk bca 3Wijk cba 2Wijk acb Wijk bac Vijk abc. (54) Γ Q(corr) im vir = P + (im) M ijam b Q ja, (55) j a 12

13 Γ Q(corr) ia = occ j occ M jiam b Q jm + vir vir M ijab b Q jb + vir M iabd b Q bd, (56) m j b b d Γ Q(corr) ab vir = P + (ab) M icab b Q ic, (57) i c where vir ( ) M ijam = 2Mijk abc + 2Mijk cab + 2Mijk bca Mijk cba Mijk acb 4Mijk bac t bc mk, (58) k bc vir ( ) M iabd = 4Mijk abc + Mijk cab + Mijk bca 2Mijk cba 2Mijk acb 2Mijk bac t dc jk, (59) jk c vir M ijab = k c t c k ( ) 4Wijk abc + Wijk cab + Wijk bca 2Wijk cba 2Wijk acb 2Wijk bac. (60) H. DF-CCSD(T) Analytic Gradients For the DF-CCSD(T) wave function the following Lagrangian can be employed (in the spin-orbital formalism) L = 0 (1 + ˆΛ)e ˆT Ĥe ˆT 0 + pq z pq f pq, (61) where the Z-vector 31 satisfies z pq = z qp, (62) z pp = 0. (63) More explicitly L = 0 (1 + ˆΛ)e ˆT Ĥe ˆT ai z ai f ai + ij z ij f ij + ab z ab f ab. (64) 13

14 The solution of the Z-vector equations will be presented in the next section. The gradient of energy can be written as follows 82,83,90 96 de dx = L, (65) x x=x0 x=x0 when we insert the explicit expressions for integral derivatives into above equation, the gradient equation can be cast into the following form 60 de dx = p,q γ eff pq h x pq p,q F eff pq S x pq + Q pq Γ Q(eff) pq (Q pq) x P,Q Γ eff P Q Jx P Q, (66) where γ eff pq, Γ eff P Q Q(eff), Γ pq, and Fpq eff GFM. The unrelaxed OPDM 25,97 is are the relaxed OPDM, 2- and 3-index TPDM, and γ pq = 1 2 P +(pq) 0 (1 + ˆΛ) e ˆT ˆp ˆq e ˆT 0, (67) and the unrelaxed 2- and 3-index TPDMs are defined by 60,64 Γ Q pq = 1 2 ˆP + (pq) r,s Γ Q pq = P 0 (1 + ˆΛ)e ˆT ˆp ˆr ŝˆq e ˆT 0 b Q rs, (68) Γ P pq [J 1/2 ] P Q, (69) Γ P Q = 1 c P 2 pq Γ Q pq = 1 Γ P 2 pqc Q pq, (70) c Q pq = p,q P p,q b P pq [J 1/2 ] P Q. (71) Explicit equations for the unrelaxed GFM is provided in our previous studies ,64 In SCG we need to consider only diagonal elements of the unrelaxed OPDM. Hence, with the following definition γ (corr) pq = δ pq γ (corr) p, (72) we can use our previously reported formulas for the GFM and the separable part of the TPDM. 64 In SCG, Z-vector contributions to the OPDM and the separable part of the TPDM can 14

15 be written as follows γ eff pq = γ pq + z pq, (73) Γ Q(eff) ij = Γ Q ij + z ij J Q ( ) + δ ij ZQ + Z Q + Z Q ( Z Q ij + ZQ ji 1 2 ) Z Q ij, (74) Γ Q(eff) ai = Γ Q(eff) ia = Γ Q ia + z aij Q Z Q ai Z Q ai, (75) Γ Q(eff) ab = Γ Q ab + z abj Q. (76) Similarly, the Z-vector contributions to the GFM can be written as follows F eff ia = F ia + ε i z ai, (77) F eff ab = F ab + ε a z ab, (78) F eff ij = F ij + ε i z ij Q Q Q b Q im ZQ mj m b Q ij Z Q 1 2 b Q ij Z Q Q Q Q b Q ij Z Q Q l vir d vir b Q ie ZQ ej e b Q il ZQ lj b Q id ZQ dj, (79) 15

16 F eff ai = F ai + z ai ε a Q b Q amz Q mi m Q b Q ai Z Q Q vir b Q aez Q ei e ( w ai + w ai), (80) where ε p are orbital energies and intermediates are defined as J Q = b Q mm, (81) m Z Q = 2 z ai b Q ai, (82) ai Z Q ai = m Z Q ij = 2 a z am b Q mi, (83) z ai b Q aj, (84) Z Q = kl z kl b Q kl, (85) Z Q ij = 2 k z ik b Q kj, (86) Z Q = cd z cd b Q cd, (87) Z Q ai = b z ab b Q bi, (88) w ai = 2 Q b Q ai Z Q Q l b Q al ZQ li, (89) w ai = 2 Q b Q ai Z Q 2 Q vir d b Q ad ZQ di. (90) We would like to note that for the Z-vector contributions to the PDMs and GFM, the b Q pq integrals should employ the same iliary basis set (e.g., the JK-FIT basis set) as 16

17 density-fitted reference function, since these contributions originate from the Fock matrix derivatives. The gradient is evaluated by back-transforming the relaxed PDMs and the relaxed GFM into the AO basis and contracting them against AO integral derivatives, as usual F µν = pq γ µν = pq Γ Q µν = pq C µp C νq F eff pq, (91) C µp C νq γ eff pq, (92) C µp C νq ΓQ(eff) pq, (93) where F µν, γ µν, and Γ Q µν are the AO basis GFM, OPDM, and 3-index TPDM, respectively. The 2-index TPDM can be written as Γ P Q = 1 2 c Q µν = p,q P c P pq Γ Q(eff) pq = 1 2 c P µνγ Q µν, (94) µν b P µν [J 1/2 ] P Q, (95) hence, the final gradient equation in the AO basis can be written as follows 60 de dx = µν γ µν h x µν µν F µν S x µν + Q Γ Q µν(q µν) x µν P,Q Γ P Q J x P Q. (96) I. SCG Z-Vector Equations Before presenting Z-vector equations, let us define the orbital gradient as follows, w pq = 2 ( F pq F qp ). (97) We obtain a set of linear equations for Z-vectors by differentiating the Lagrangian in Eq.(61) with respect to orbital rotation parameters. We solve the Z-vector equations in three steps. At first, we obtain the OO-block Z-vector as follows w ij z ij = 2 ( ), (98) ε i ε j 17

18 Then, the VV-block Z-vector as w ab z ab = 2 ( ), (99) ε a ε b Finally, the VO-block Z-vector, w eff ai + bj A ai,bj z bj = 0, (100) where A is the HF MO Hessian, and w eff ai = w ai + w ai + w ai. (101) A detailed discussion for the iterative solution of Eq.(100) is reported in our previous study. 60 Further, singularities in Eq.(98) and Eq.(99), in the case of orbital degeneracies, are avoided applying the method suggested by Lee and Rendell. 41 III. RESULTS AND DISCUSSION Results from the CCSD(T) and DF-CCSD(T) methods were obtained for a set of alkanes for comparison of the computational cost for single-point analytic gradient computations, using geometries from our recent study. 59 For the alkanes Dunning s correlationconsistent polarized valence triple-ζ basis set (cc-pvtz) was employed with the frozen core approximation. 101,102 For the cc-pvtz primary basis sets, cc-pvtz-jkfit 49 and cc-pvtz- RI 103 iliary basis sets were employed for reference and correlation energies, respectively. Additionally, the CCSD(T) and DF-CCSD(T) methods were applied to a set of small molecules 104 for comparison of geometries and vibrational frequencies. These computations used Dunning s correlation-consistent polarized valence quadruple-ζ basis set (cc-pvqz) with the frozen core approximation. 101,102 For the cc-pvqz primary basis sets, cc-pvqz- JKFIT 49 and cc-pvqz-ri 103 iliary basis sets were employed for reference and correlation energies, respectively. The DF-CCSD(T) computations were performed with our present code, which is available in the development version of the Psi4 program package, 79 while the conventional CCSD(T) computations were carried out with the Cfour1.0 program. 80 Energies were converged tightly to 10 8 hartree. We verified our DF-CCSD(T) analytic 18

19 gradients code with respect to numerical gradients (with the 5-point formulas). A. Efficiency of DF-CCSD(T) Analytic Gradients We consider a set of alkanes 59 to assess the efficiency of the DF-CCSD(T) analytic gradients. The total computational (wall) times for single-point frozen-core analytic gradient computations using the CCSD(T) and DF-CCSD(T) methods are presented in Figure 1. These computations were performed on a single core of an Intel(R) Xeon(R) CPU E GHz computer, (although we perform computations with just a single core for a clearer comparison between programs, our code has been partially parallelized, through basic linear algebra subprogram (BLAS) routines, to use multiple cores). For each software 64 GB memory is provided. Further, in CCSD(T) computations (with Cfour) the AO-basis algorithm is chosen for the particle-particle ladder (PPL) term with the ABCD- TYPE=AOBASIS option. For the largest member of the alkanes set, C 6 H 14, the number of active occupied orbitals, virtual orbitals, and iliary basis functions are O = 19, V = 351, and N = 906. As shown in Figure 1, the DF-CCSD(T) method significantly reduces the total computational cost compared to conventional CCSD(T); there is more than 2-fold reduction in wall time compared to CCSD(T) for the largest member of the alkanes test set. Since we do not have detailed timings for different parts of the CCSD(T) gradient code in Cfour, we cannot compare results for different parts of the CCSD(T) gradient computations. However, our recent study 65 demonstrates that our DF-CCSD energy code is significantly faster (about 5-fold in the case of C 8 H 18 ) than the conventional CCSD code of Cfour. Further, another recent study 64 shows that for DF-CCSD [hence for DF-CCSD(T)] the computation of gradient related terms, 105 which was the most expensive part for CCSD gradients of the Molpro program, 106,107 is substantially accelerated (more than 8-fold for C 10 H 22 ) with the help of DF approximations. Hence, due to dramatically reduced cost for gradient terms and significantly reduced I/O time, the DF approach enables substantially accelerated analytic gradient codes for CC methods. Next, we analyze the cost of different parts of the DF-CCSD(T) analytic gradient computations. Figure 2 shows the total computational time broken down into four major components: (1) the total time to converge the DF-CCSD energy and amplitudes, (2) the total 19

20 time for (T) correction and its contributions to the Λ equations and PDMs (3) the total time to converge the Λ equations, and (4) the cost of the remaining gradient terms. 105 For the largest member of the alkanes test set, C 6 H 14, 3.4% of the time is spent in the DF-CCSD energy and amplitude equations, 91.0 % is spent in (T) correction and its contributions to the Λ equations and PDMs, 4.9% is spent in the Λ equations, and 0.7% is spent on the remaining gradient terms. For the remaining molecules, % of the time is spent in the (T) part. Clearly the (T) correction part is the most cost expensive part of the DF-CCSD(T) gradient, as expected, for large molecules. Finally, we illustrate the scaling with respect to the number of computer cores for the DF-CCSD(T) gradients for the alkane test set in Figure 3. In our previous study, 64 it has been demonstrated that the DF-CCSD code scales fairly well for the large molecules. Hence, in this study we illustrate the scaling of the (T) part. However, we note that our present (T) gradient code is not fully parallelized, yet. It is partially threaded through BLAS routines only. As shown in Figure 3, our (T) gradient algorithm is partially parallelized. B. Accuracy of DF-CCSD(T) Analytic Gradients To assess the accuracy of the DF-CCSD(T) method, we consider equilibrium geometries and harmonic vibrational frequencies. In order to assess the performance of DF-CCSD(T) for bond lengths, we consider a set of small molecules employed in previous studies. 104,108 Table I reports the bond lengths of the molecules considered in increasing order. Errors for DF-CCSD(T) with respect to conventional CCSD(T) are depicted in Figure 4. The mean absolute error ( mae ), the standard deviation of errors ( std ), and the maximum absolute error ( max ) with respect to CCSD(T) are , , and Å, respectively. Further, we compare the performance of DF-CCSD(T) and CCSD(T) for bond lengths with respect to the experimental values. 108 mae values for DF-CCSD(T) and CCSD(T) are and Å, respectively, as depicted in Figure 5. Hence, the performance of the CCSD(T) and DF-CCSD(T) methods is essentially the same, as in the case of CCSD/DF-CCSD 64 and MP2/DF-MP2. 60 Overall, the error introduced by the DF approximation is quite negligible. Next, we consider harmonic vibrational frequencies of the molecules shown in Table I. The mae, std, and max values are 0.7, 1.0, and 3.6 cm -1, respectively. Thus, for harmonic vibrational frequencies results of CCSD(T) and DF-CCSD(T) are essentially the same, sim- 20

21 ilar to the CCSD/DF-CCSD case, 64 and the error introduced by the DF approach is again negligible. IV. CONCLUSIONS In this research, an efficient implementation of analytic gradients for the CCSD(T) method with the density-fitting (DF) approximation, which is denoted as DF-CCSD(T), has been reported. The computational time of the DF-CCSD(T) single point analytic gradients have been compared with that of the conventional CCSD(T) (from the Cfour1.0 package). The DF-CCSD(T) method significantly reduces the computational cost compared to CCSD(T), with a more than 2-fold reduction for the largest member of alkane test set considered, C 6 H 14 in a cc-pvtz basis set. Due to the dramatically reduced cost for so-called gradient terms 105 and significantly reduced I/O time, the DF approach enables substantially accelerated analytic gradients for CC methods. Since four-dimensional electron repulsion integral and TPDM tensors can be avoided with the help of the DF approach, which leads to scaling reduction for certain terms and reduced I/O and storage requirements, the DF-CCSD(T) method is substantially more efficient than CCSD(T), especially for gradient terms. DF-CCSD(T) has been applied for a variety of molecules to assess its accuracy for equilibrium bond lengths and harmonic vibrational frequencies.the DF approximation introduces negligible errors compared to CCSD(T): equilibrium bond lenghts have a mean absolute error of Å (max error of Å), and harmonic vibrational frequencies exhibit mean absolute errors of 0.7 cm 1 for the cases considered. V. SUPPLEMENTARY MATERIAL Spin-free equations (for T and Λ amplitudes, and PDMs) for a closed-shell RHF reference based DF-CCSD method were reported in the Supplementary Material. ACKNOWLEDGMENTS This research was supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK-114Z786), the European Cooperation in Science and Technology 21

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28 TABLE I. Bond lengths of molecules considered in order of increasing values (in Å). Molecule Bond 1 HF R HF 2 H 2 O R OH 3 H 2 O 2 R OH 4 HOF R OH 5 HNC R NH 6 NH 3 R NH 7 N 2 H 2 R NH 8 C 2 H 2 R CH 9 HNO R NH 10 HCN R CH 11 C 2 H 4 R CH 12 CH 4 R CH 13 N 2 R NN 14 CH 2 O R CH 15 CO R CO 16 HCN R CN 17 CO 2 R CO 18 HNC R CN 19 C 2 H 2 R CC 20 CH 2 O R CO 21 HNO R NO 22 N 2 H 2 R NN 23 C 2 H 4 R CC 24 HOF R OF 25 H 2 O 2 R OO 26 BH R BH 28

29 6000 CCSD(T) DF-CCSD(T) Wall-Time (min) Number of carbons FIG. 1. Total wall-time (in min) for computations of single-point frozen-core analytic gradients for the C n H 2n+2 (n=1 6) set from the DF-CCSD(T) and CCSD(T) (from Cfour) methods with the cc-pvtz basis set. For the largest member, the number of basis functions is 376. All computations were performed with an 10 8 energy convergence tolerance on a single node (1 core) Intel(R) Xeon(R) CPU E GHz computer (memory 64 GB). 29

30 Wall-Time (min) DF-CCSD(T) Grad Total Time DF-CCSD Total Time (T) with its grad terms Λ-Amplitudes Grad Terms Number of carbons FIG. 2. Total wall-time (in min) and its major components, such as DF-CCSD, (T)-correction, Λ-ampitudes, and gradient terms, 105 for computations of single-point frozen-core analytic gradients for the C n H 2n+2 (n=1 6) set from the DF-CCSD(T) method with the cc-pvtz basis set. For the largest member, the number of basis functions is 376. All computations were performed with an 10 8 energy convergence tolerance on a single node (1 core) Intel(R) Xeon(R) CPU E GHz computer (memory 64 GB). 30

31 Cores 4-Cores 6-Cores Ratio of Wall Time C 4 H 10 C 5 H 12 C 6 H 14 FIG. 3. Ratio of total wall-time for single-point frozen-core analytic gradient computations with 2, 4, and 6 cores with respect to the serial mode for the C 4 H 10, C 5 H 12, and C 6 H 14 molecules from the (T) part of the DF-CCSD(T) method with the cc-pvtz basis set. All computations were performed with an 10 8 energy convergence tolerance on a single node (1 core) Intel(R) Xeon(R) CPU E GHz computer (memory 64 GB). 31

32 r e (Angstrom) Set entry FIG. 4. Errors in bond lengths of molecules considered (Table I) for the DF-CCSD(T) method with respect to CCSD(T) (cc-pvqz basis set was employed). 32

33 MAE for r e (Angstrom) CCSD(T) DF-CCSD(T) FIG. 5. Mean absolute errors in bond lengths of molecules considered (Table I) for the CCSD(T) and DF-CCSD(T) methods with respect to experiment. 33

34 6000 CCSD(T) DF-CCSD(T) Wall-Time (min) Number of carbons

35 Wall-Time (min) DF-CCSD(T) Grad Total Time DF-CCSD Total Time (T) with its grad terms Λ-Amplitudes Grad Terms Number of carbons

36 Cores 4-Cores 6-Cores Ratio of Wall Time C 4 H 10 C 5 H 12 C 6 H 14

37 r e (Angstrom) Set entry

38 MAE for r e (Angstrom) CCSD(T) DF-CCSD(T)