Orbital-Optimized MP3 and MP2.5 with Density-Fitting and Cholesky

Article pubs.acs.org/JCTC

Orbital-Optimized MP3 and MP2.5 with Density-Fitting and Cholesky Decomposition Approximations Uğur Bozkaya* Department of Chemistry, Hacettepe University, Ankara 06800, Turkey S Supporting Information *

ABSTRACT: Efficient implementations of the orbital-optimized MP3 and MP2.5 methods with the density-fitting (DF-OMP3 and DFOMP2.5) and Cholesky decomposition (CD-OMP3 and CD-OMP2.5) approaches are presented. The DF/CD-OMP3 and DF/CD-OMP2.5 methods are applied to a set of alkanes to compare the computational cost with the conventional orbital-optimized MP3 (OMP3) [Bozkaya J. Chem. Phys. 2011, 135, 224103] and the orbital-optimized MP2.5 (OMP2.5) [Bozkaya and Sherrill J. Chem. Phys. 2014, 141, 204105]. Our results demonstrate that the DF-OMP3 and DF-OMP2.5 methods provide considerably lower computational costs than OMP3 and OMP2.5. Further application results show that the orbital-optimized methods are very helpful for the study of open-shell noncovalent interactions, aromatic bond dissociation energies, and hydrogen transfer reactions. We conclude that the DF-OMP3 and DF-OMP2.5 methods are very promising for molecular systems with challenging electronic structures.

1. INTRODUCTION Orbital-optimized Møller−Plesset (MP) perturbation theory and coupled-cluster (CC) methods have been of great interest in modern computational chemistry.1−30 Orbital optimized post-Hartree−Fock (HF) methods, such as the coupled-cluster doubles (CCD),1,2,6 the linearized coupled-cluster doubles (LCCD),11,15 coupled pair functional theories,31,32 active space and excited state CC methods,3,4 the second-order perturbation theory (MP2),6,10,17,18 the third-order perturbation theory (MP3),7,9,12 and the density-cumulant functional theory (DCFT),30 have been reported in previous studies. Triples excitation corrections for the orbital-optimized CC methods have also been implemented.5,8,28,33,34 In the previous studies it has been demonstrated that the orbital-optimized CC and MP methods are very beneficial for chemical systems with problematic electronic structures, such as symmetry-breaking problems,2,6,7,10−12 transition states,9,15,25 free radicals,9−11,15,25 bond-breaking problems,8,28,35 open-shell noncovalent interactions,13 and direct computation of ionization potentials and electron affinities,14,16 and predictions of the chemical reactivity.36 Tensor decomposition of the electron repulsion integrals (ERIs) has been of considerable interest in modern quantum chemistry. The density fitting (DF) approach is one of the popular approximations.17,18,37−46 With the DF approach, one can express the four-index two-electron integrals (TEIs) in terms of the three-index tensors. Another popular integral approximation is the partial Cholesky decomposition (CD) of the TEI tensor.17,45,47−50 The DF and CD approximations are very helpful to reduce the storage cost of ERIs as well as the © XXXX American Chemical Society

computational time due to the reduced I/O time. In the DF approximation, ERIs of the primary basis set are expanded in terms of a preconstructed auxiliary basis set, whereas in the CD approximation the CD tensors are directly generated from the primary basis set. The DF approximation was employed for orbital-optimized MP2, which is denoted as OO-RI-MP2.25 However, in their study Neese et al.25 employed the DF integrals for the correlation energy, but for the reference energy the conventional integrals were used. In our 2014 study, the orbitaloptimized MP2 method with the DF and CD approaches (DFOMP2 and CD-OMP2) was presented.17 In our studies, the DF and CD approaches were applied to both the reference and the correlation energies.17,18 In this study, the orbital-optimized MP3 and MP2.551 methods (OMP37,9,12 and OMP2.519,20) with the DF and CD approaches are presented, which are denoted as DF-OMP3, CD-OMP3, DF-OMP2.5, and CD-OMP2.5, respectively. The equations reported have been implemented in a new computer code and added to the DFOCC module of the PSI4 package.52 Both restricted and unrestricted formalism are implemented. The computational time of the DF-OMP3, CD-OMP3, DFOMP2.5, and CD-OMP2.5 methods are compared with that of OMP3 and OMP2.5. The DF/CD-OMP3 and DF/CDOMP2.5 methods are applied to open-shell noncovalent interactions, aromatic bond dissociation energies, and hydrogen Received: November 30, 2015

A

DOI: 10.1021/acs.jctc.5b01128 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Article

Journal of Chemical Theory and Computation

where |0⟩ is the reference determinant, Ŵ N is the two-electron component of the normal ordered Hamiltonian operator,55,56 ̂ (2) T̂ (1) 2 and T2 are the first- and second-order cluster double excitation operators, respectively, and the subscript c means only connected diagrams are considered.

transfer reactions, where standard MP3 and MP2.5 have difficulties.

2. DENSITY-FITTING VS CONVENTIONAL INTEGRALS FOR ORBITAL-OPTIMIZED METHODS Before reporting detailed equations for the DF-OMP3 and DFOMP2.5 methods, we would like to highlight the differences between DF-OMP3/DF-OMP2.5 and OMP3/OMP2.5 for energy computations. First of all, the DF-OMP3 and DFOMP2.5 working equations are not obtained just replacing the conventional ERIs in the OMP3 and OMP2.5 expressions with the DF tensors. Because some of terms can be factorized more efficiently using the DF intermediates. Further, with the DF approach the four-index two-particle density matrix (TPDM) can be replaced with the three-index TPDM, which greatly simplifies the orbital optimization procedure as well as greatly reduces the storage cost of TPDM. Moreover, the DF approach enables on the fly computation of the problematic particle− particle ladder (PPL) term of the T2-amplitude equation and the Vabcd term of the conventional TPDM with direct algorithms, which avoids the cumbersome I/O procedures, in the molecular orbital (MO) basis or in the atomic-orbital (AO) basis, for the PPL and Vabcd term.

(1)

T2̂ =

(2)

T2̂ =

(μν|λσ )DF =

∑

occ vir i,j

†

(8)

a,b

tab(2) ij

1 4

occ vir

∑ ∑ tijab⟨ij∥ab⟩DF i,j

(10)

where ⟨ij∥ab⟩DF is the antisymmetrized ERI within the DF approximation, EMP3 and Eref are the MP3 and reference energies, respectively. The amplitude equations can be written as7,53,54 d (1) ⟨Φijab|(fN̂ T2̂ + ŴN )c |0⟩ = 0

(11)

d (2) (1) ⟨Φijab|(fN̂ T2̂ + ŴN T2̂ )c |0⟩ = 0

(12)

where fdN̂ is the diagonal part of the normal ordered Fock operator, ⟨Φab ij | is a doubly excited Slater determinant. The MP3 amplitude equations can be written more explicitly as follows

Naux

vir

tijab(1)Dijab

(2)

P

+ P −̂ (ab) ∑ tijae(1)fbe

= ⟨ij ab⟩DF

e≠b

where

occ

(μν|P) =

∫ ∫ χμ (r1)χν (r1) r1 φP(r2) dr1 dr2 12

ab(1) − P −̂ (ij) ∑ tim fmj

(13)

m≠j

(3)

and JPQ =

(9)

a,b

tijab = tijab(1) + tijab(2)

(1)

∑ (μν|P)[J−1/2 ]PQ

∑ ∑ tijab(2)a†̂ b ̂ j ̂i ̂

(7)

a,b

EMP3 = Eref +

In the CD approach, the CD vectors bQμν are obtained directly from the CD procedure, and Q is a Cholesky vector. In the DF approach, the DF factors bQμν are defined as Q bμν =

1 4

i,j

†

where and are the first- and second-order double excitation amplitudes, respectively, and â† and i ̂ are the creation and annihilation operators, respectively. We can write the MP3 energy more explicitly as follows

Q Q bμν bλσ

Q

∑ ∑ tijab(1)a†̂ b ̂ j ̂i ̂

tab(1) ij

3. THEORETICAL APPROACH 3.1. Integral Approximations. In the DF and CD approaches the atomic-orbital (AO) basis integrals are written as Naux

occ vir

1 4

vir

∫ ∫ φP(r1) r1 φQ (r2) dr1 dr2 12

(4)

+

where χμ(r) and φP(r) are the primary and auxiliary basis functions, respectively. Similarly, the molecular-orbital (MO) basis integrals can be written as

+

Naux

(pq|rs)DF =

∑ bpqQbrsQ

ΔEMP3 =

+

(2) ⟨0|(ŴN T2̂ )c |0⟩

m≠j

1 2

ab(1) ⟨mn∥ij⟩DF ∑ ∑ tmn

1 2

∑ ∑ tijef (1)⟨ab∥ef ⟩DF

m vir e

n vir f

ae(1) + P −̂ (ij)P −̂ (ab) ∑ ∑ tim ⟨mb∥ej⟩DF

where bQpq is a MO basis DF/CD factor. 3.2. DF-MP3-Λ Energy Functional and Amplitude Equations. The conventional notation is used for the different type of orbitals: 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 MP3 correlation energy can be written as follows7,53,54 (1) ⟨0|(ŴN T2̂ )c |0⟩

e≠b occ occ

occ vir

(5)

Q

occ

ab(2) tijab(2)Dijab = P −̂ (ab) ∑ tijae(2)fbe − P −̂ (ij) ∑ tim fmj

m

where f pq is the Fock matrix,

Dab ij

e

(14)

and P̂ −(pq) are given by

Dijab = fii + f jj − faa − fbb

(15)

P −̂ (pq) = 1 − P(̂ pq)

(16)

where P̂(pq) is the permutation operator.

(6) B


Article

Journal of Chemical Theory and Computation To obtain a variational energy functional (Ẽ MP3) it is convenient57,58 to introduce a Lagrangian (DF-MP3-Λ functional) as follows7

† ̂ where eK is the orbital rotation operator, p ̂ ̃ , p ̂ ,̃ and |p̃⟩ are the transformed creation, annihilation operators, and a transformed spin−orbital, respectively. The operator K̂ can be written as

(1) (2) ̃ ΔEMP3 = ⟨0|(ŴN T2̂ )c |0⟩ + ⟨0|(ŴN T2̂ )c |0⟩ (1) d ⟨0|{Λ̂ 2 (fN̂

+

(1) T2̂

K̂ =

p,q

+ ŴN )c }c |0⟩

(1) d (2) (1) + ⟨0|{Λ̂ 2 (fN̂ T2̂ + ŴN T2̂ )c }c |0⟩ (2) d (1) + ⟨0|{Λ̂ 2 (fN̂ T2̂ + ŴN )c }c |0⟩

(18)

(2) † (2) Λ̂ 2 = T2̂

(19)

C(κ) = C(0)e K

κ

(2) κ (1) κ + ⟨0|{Λ̂ 2 (fN̂ T2̂ + Ŵ N )c }c |0⟩

κ

2

(2) † (2) Λ̂ 2 = T2̂

(25)

̂

̂

̂

̂

p ̂ ̃ = e K pê −K ̂

| p ̃ ⟩ = e K | p⟩

(1)

1 κ (2) ⟨0|(Ŵ N T2̂ )c |0⟩ 2

(1) κ (2) 1 κ (1) ⟨0|{Λ̂ 2 (fN̂ T2̂ + Ŵ N T2̂ )c }c |0⟩ 2 (2) κ (1) 1 κ + ⟨0|{Λ̂ 2 (fN̂ T2̂ + Ŵ N )c }c |0⟩ (33) 2 κ κ κ κ ̂ where Ĥ , fN, Ŵ N, and Ĥ N are the transformed operators

+

κ ̂ ̂ Ĥ = e−K Hê K

(34)

κ ̂ d ̂ fN̂ = e−K fN̂ e K

(35)

κ ̂ ̂ Ŵ N = e−K WN̂ e K

(36)

κ ̂ ̂ HN̂ = e−K HN̂ e K

(37)

where Ĥ is the Hamiltonian operator. The first and second derivatives of Ẽ (κ) with respect to κ (at κ = 0) can be written as follows

3.4. Parametrization of the DF-OMP3 and DF-OMP2.5 Wave Functions. For the parametrization of DF-OMP3 and DF-OMP2.5 wave functions, we will follow our previous studies.6−15 The orbital variations can be achieved by means of an unitary operator59−62 †

(32)

(1) κ (1) κ + ⟨0|{Λ̂ 2 (fN̂ T2̂ + Ŵ N )c }c |0⟩

N c c

(24)

κ

E(̃ κ) = ⟨0|Ĥ |0⟩ + ⟨0|(Ŵ N T2̂ )c |0⟩ +

(21)

a,b

† (1) T2̂

p ̂ ̃ = e K p†̂ e−K

(2)

DF-OMP2.5

(1) d (2) (1) 1 + ⟨0|{Λ̂ 2 (fN̂ T2̂ + ŴN T2̂ )c }c |0⟩ 2 (2) d (1) 1 + ⟨0|{Λ̂ 2 (fN̂ T2̂ + ŴN )c }c |0⟩ (23) 2 As in the case of MP3, the de-excitation operators are as follows:

=

κ

(1) κ (2) κ (1) + ⟨0|{Λ̂ 2 (fN̂ T2̂ + Ŵ N T2̂ )c }c |0⟩

as different from MP3, the combined amplitudes are defined by 1 tijab = tijab(1) + tijab(2) (22) 2 19 The DF-MP2.5-Λ functional is given by (1) (2) 1 ̃ ΔEMP2.5 = ⟨0|(ŴN T2̂ )c |0⟩ + ⟨0|(ŴN T2̂ )c |0⟩ 2 (1) d (1) + ⟨0|{Λ̂ (f ̂ T̂ + Ŵ ) } |0⟩

(1) Λ̂ 2

(1)

(1) κ (1) κ + ⟨0|{Λ̂ 2 (fN̂ T2̂ + Ŵ N )c }c |0⟩

occ vir

N

κ

E(̃ κ) = ⟨0|Ĥ |0⟩ + ⟨0|(Ŵ N T2̂ )c |0⟩ + ⟨0|(Ŵ N T2̂ )c |0⟩

∑ ∑ tijab⟨ij∥ab⟩DF

2

(31)

where C(0) and C(κ) are the initial and transformed MO coefficient matrices, respectively. For DF-OMP3 and DF-OMP2.5, the following variational energy functionals (Lagrangians) can be written7,19as follows DF-OMP3

It should be noted that ΔẼ MP3 is stationary with respect to excitation and de-excitation amplitudes. 3.3. DF-MP2.5-Λ Energy Functional and Amplitude Equations. The MP2.5 correlation energy can be written as follows19 (1) (2) 1 ΔEMP2.5 = ⟨0|(ŴN T2̂ )c |0⟩ + ⟨0|(ŴN T2̂ )c |0⟩ (20) 2 we can write the MP2.5 energy more explicitly as follows

i,j

(30)

where κpq are the orbital rotation parameters. The transformed molecular orbital (MO) coefficients may be written as

(17)

(1) † (1) Λ̂ 2 = T2̂

1 4

(29)

p>q

K = Skew(κ)

̂ (2) where Λ̂(1) 2 and Λ2 are the first- and second-order double deexcitation operators. It is trivial to show that7

EMP2.5 = Eref +

∑ K pqp†̂ q ̂ = ∑ κpq(p†̂ q ̂ − q†̂ p̂)

wpq =

∂E ̃ ∂κpq

A pq , rs =

(26)

(38)

κ= 0

∂ 2E ̃ ∂κpq∂κrs

(39)

κ= 0

Then Ẽ (κ) can be expanded up to second-order as

(27)

(2)

(0)

E ̃ (κ ) = E ̃

(28) C

+ κ†w +

1 † κ Aκ 2

(40)


Article

Journal of Chemical Theory and Computation where κ is the orbital rotation vector, w is the orbital gradient vector, and A is the orbital Hessian matrix. Hence, minimizing the Lagrangian with respect to κ we obtain the following equation −1

occ

The unique nonzero blocks of the separable TPDMs can be written as occ

This final equation is identical to the Newton−Raphson step. Hence, the orbitals are transformed until the convergence achieved. 3.5. Response Density Matrices. The one-particle density matrix (OPDM) can be decomposed as follows

corr Q ΓQij (sep) = δij(γQ + γQ̃ ) + γijcorrJQ − P+(ij) ∑ γjm bim m

= δij

ΓQab(sep) = γabcorrJQ

(43)

= −.ij

occ

γQ =

vir

γQ̃ =

(45)

occ vir

1 2

.ae = −

ef in(1) λef + ∑ ∑ tmn n

e,f

1 2

occ vir

1 2

ef mn(1) λaf − ∑ ∑ tmn m,n

f

1 2

(47)

occ vir ef (1) mn(2) λaf ∑ ∑ tmn m,n

f

.ae = −

(60)

ΓQia (corr) = TiaQ + yiaQ

(61) vir

Q ΓQab(corr) = −2V ab +

occ vir

∑∑ n

1 2

ef in(1) tmn λef

e,f

1 + 4

occ vir ef mn(1) λaf − ∑ ∑ tmn m,n

f

= 4∑

∑∑ n

1 4

DF-OMP2.5 (49)

ΓQij (corr) =

occ vir ef (1) mn(2) λaf ∑ ∑ tmn m,n

f

(50)

(63)

1 Q y 2 ia

Q ΓQab(corr) = −V ab +

1 2

(64) vir

∑ Vaebf befQ e,f

(65)

The intermediates that appear in eqs 60−65 are defined as (51)

occ vir

TiaQ =

where Γprqs is the four-index TPDM. The explicit equations of Γprqs for MP3 and MP2.5 were given in our previous studies.7,19 TPDM can be decomposed as in the case of OPDM as follows

∑ ∑ tijabbQjb j

b

(66)

occ

(52)

V ijQ =

where and are the reference and correlation contributions to TPDM, and ΓQ(sep) is the separable part of pq TPDM. The reference TPDM is given by

V ijQ ′ =

Q ∑ Vimjnbmn m,n

ΓQ(corr) pq

ΓQij (ref) = δijJQ − bijQ

1 Q V ij − V ijQ ′ 2

ΓQia (corr) = TiaQ +

ΓprqsbrsQ

ΓQpq = ΓQpq(ref) + ΓQpq(corr) + ΓQpq(sep)

(62)

e,f

ef (1) in(2) tmn λef

e,f

r ,s

ΓQ(ref) pq

∑ Vaebf befQ

occ vir

Note that in eqs 47−50 the definition of the combined amplitudes (tab ij ) are different. With the DF approximation we can replace the four-index TPDM with the three-index TPDM, which is defined as follows

ΓQpq

ΓQij (corr) = V ijQ − 2V ijQ ′

(48)

DF-OMP2.5 1 .mi = 2

(59)

Finally, the unique nonzero blocks of the correlation TPDM can be expressed as follows: DF-OMP3

ef (1) in(2) λef ∑ ∑ tmn e,f

∑ γefcorrbefQ e,f

occ vir n

(58)

m,n

where . intermediates are defined as follows: DF-OMP3 .mi =

Q ∑ γmncorrbmn

(44)

= −.ab

(57)

where bQpq is again computed in the DF-REF basis, and intermediates are given by

where δij denotes the Kronecker delta. The unique nonzero blocks of the correlation OPDM can be written as

γabcorr

(56)

(42)

corr where γref pq and γpq are the reference and correlation parts of OPDM, respectively. The reference OPDM is given by

γijcorr

(55)

vir

ΓQia (sep) = −∑ γeacorrbieQ e

ref corr γpq = γpq + γpq

γijref

(54)

m

(41)

κ = −A w

Q ∑ bmm

JQ =

(67)

vir

∑ Viejf befQ e,f

(53)

(68)

occ Q V ab =

bQij

where is computed using the reference auxiliary (DF-REF) basis, the JK-FIT basis set,42 and JQ is defined as

Q ∑ Vmanbbmn m,n

D

(69) DOI: 10.1021/acs.jctc.5b01128 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

Article

Journal of Chemical Theory and Computation occ vir Q yiaQ = 2 ∑ ∑ Viemabme m

In our direct algorithms we assume that there is enough memory to store Wabef/Vabcd type tensors for a fixed virtual index a, dynamic arrays with the size of V3 (where V denotes virtual orbitals). Further, whenever four-index integrals including two or less virtual indices are necessary, they are built on the fly from the DF integral tensors. Moreover, in the present formulation, three-index intermediates are defined where the DF approach enables a favorable factorization. Finally, we would like to note that for non-DF methods, where the conventional four-index integrals are used, computations were performed with a standard disk-based method as other post-HF methods implemented in the PSI4 package.

(70)

e

bQpq

where is computed using the correlation auxiliary (DF-MP) basis, the resolution of the identity (RI) basis set,63 and the four-index intermediates are defined as Vijkl =

1 2

vir

e

(71)

f

occ occ

1 2

Vabcd =

Viajb =

vir

∑ ∑ tijef (1)λefkl(1) cd(1) mn(1) λab ∑ ∑ tmn m

n

(72)

occ uocc

1 2

∑ ∑ timbe(1)λaejm(1) m

e

4. RESULTS AND DISCUSSION The computational cost of the OMP2.5, DF-OMP2.5, CDOMP2.5, OMP3, DF-OMP3, and CD-OMP3 methods were compared using a set of alkanes. For energy computations of alkanes Dunning’s correlation-consistent polarized valence triple-ζ basis set (cc-pVTZ) was used.65,66 For the cc-pVTZ primary basis set, cc-pVTZ-JKFIT42 and cc-pVTZ-RI63 auxiliary basis set pairs were used for reference and correlation energies, respectively. Further, a set of open-shell complexes13 was considered to investigate the performance of DF-OMP2.5 and DF-OMP3 for open-shell noncovalent interactions. For the open-shell complexes, the aug-cc-pVQZ basis set65,66 was used. The aug-cc-pVQZ-JKFIT42 and aug-cc-pVQZ-RI63 auxiliary basis sets were employed as fitting basis sets. Moreover, aromatic bond dissociation energies18 (ABDEs) and hydrogen transfer reaction energies (HTRE)10,19,67 were considered to assess the performance of DF-OMP2.5 and DF-OMP3. For the ABDE and HTRE sets the cc-pVTZ primary basis set, and the cc-pVTZ-JKFIT and cc-pVTZ-RI63 auxiliary basis set were employed. In the CD-OMP3 and CD-OMP2.5 computations, a CD threshold of 10−4 was employed. For open-shell species, the unrestricted orbitals were employed, and no mixing of spin up and spin down orbitals was allowed. All electrons were correlated in all computations. Noncovalent interaction energies were counterpoise corrected.68 All computations were performed with the PSI4 program package,52 which includes our DFOCC module. 4.1. Efficiency of DF-OMP3 and DF-OMP2.5. In order to assess the efficiency of the DF-OMP3 and DF-OMP2.5 methods, we consider a set of alkanes. For the alkanes set considered, the computational time for the OMP2.5, DFOMP2.5, CD-OMP2.5, OMP3, DF-OMP3, and CD-OMP3 methods are presented graphically in Figure 1. The DF-OMP3 and CD-OMP3 methods substantially reduce the computational cost compared to OMP3, there are 26- (DF-OMP3) and 18-fold (CD-OMP3) reductions in the computational time compared to OMP3 for the largest member (C8H18) of the alkanes set. Similarly, the DF-OMP2.5 and CD-OMP2.5 methods significantly reduce the wall-time compared to OMP2.5, there are 20- (DF-OMP2.5) and 14-fold (CDOMP2.5) reductions in the wall-time compared to OMP2.5, for the largest member of the alkanes set. Further, the cost of the DF-OMP3 and DF-OMP2.5 methods are also lower than that of CD-OMP3 and CD-OMP2.5, by 1.5 times on the average. The difference between the computational cost of DF-OMP3/ DF-OMP2.5 and CD-OMP3/CD-OMP2.5 is arising from the difference between number of auxiliary basis functions and Cholesky vectors. For the alkanes set considered, the number of

(73)

Then, the energy of the DF-MP3-Λ and DF-MP2.5-Λ functionals can be expressed in terms of PDMs as follows Ẽ =

∑ γpqhpq + p,q

1 2

Naux

∑ ∑ ΓQpqbpqQ Q

(74)

p,q

3.6. Generalized-Fock and Orbital Gradient. The orbital gradient can be written in terms of GFM as6,7,11,19 wpq = 2(Fpq − Fqp)

(75)

where Fpq is the generalized-Fock matrix. It is possible to separate GFM into reference, correlation, and separable parts:17,18,46 ref corr sep Fpq = Fpq + F pq + F pq

(76)

More explicitly, we can write ref Fpq =

Naux

∑ hpr γrqref + ∑ ∑ ΓQqr(ref)bprQ r

corr F pq =

Q Naux

∑ hpr γrqcorr + ∑ ∑ ΓQqr(corr)bprQ r

sep F pq =

Q

r

(78)

Naux

∑ ∑ ΓQqr(sep)bprQ Q

Fref pq

(77)

r

r

Fsep pq

(79)

Fcorr pq

for and the DF-REF basis integrals, while for the DF-MP basis integrals are employed. 3.7. Orbital Optimization Procedure. The DF-OMP3 and DF-OMP2.5 wave functions are defined by a set of orbital rotation parameters κ, and the first- and second-order double excitation amplitudes t(1) and t(2) 2 2 . For the computational (2) efficiency, t(1) , t , and κ are simultaneously optimized, as 2 2 previous studies.2,6,7,10−12 Orbital rotation parameters are obtained from eq 41 using the approximate Hessians described in our previous study.17 The direct inversion in the iterative subspace (DIIS) extrapolation technique64 is implemented for orbital rotation parameters.11 3.8. Implementation Specific Notes. In our implementation no four-index tensors with three- and four-virtuals are stored in the core memory or on the disk. For example, one needs to build the Wabef intermediate for the T2 amplitude equation and the Vabcd intermediate for the 3-index TPDM. These tensors are built on the fly, and their contributions are directly added to the corresponding amplitudes and densities. E


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Figure 1. Wall-time (min) on the logarithmic scale for computations of single-point energies for the CnH2n+2 (n = 1−8) set from the OMP2.5, DF-OMP2.5, CD-OMP2.5, OMP3, DF-OMP3, and CDOMP3 (CD threshold is 10−4) methods with the cc-pVTZ basis set. All computations were performed with a 10−8 energy convergence tolerance on a single node (6 cores) Intel(R) Core(TM) i7-4930K CPU at 3.40 GHz computer (memory ∼64 GB).

Figure 2. Mean absolute errors (kcal mol−1) in noncovalent interaction energies for the open-shell complexes from the MP2.5, OMP2.5, DF-OMP2.5, CD-OMP2.5, MP3, OMP3, DF-OMP3, CDOMP3, and CCSD methods with respect to CCSD(T) (aug-cc-pVQZ basis set was employed).

The MAE values are 0.70 (MP2.5), 0.12 (OMP2.5), 0.14 (DF-OMP2.5), 0.14 (CD-OMP2.5), 0.65 (MP3), 0.25 (OMP3), 0.27 (DF-OMP3), 0.27 (CD-OMP3), and 0.38 (CCSD) kcal mol−1. Hence, performances of the orbitaloptimized methods are substantially better than their standard variants. The OMP2.5 and DF/CD-OMP2.5 methods provide 6 and 5 times lower MAE values compared with MP2.5, respectively. Similarly, the OMP3 and DF/CD-OMP3 methods provide ∼2.5 times lower MAE values compared with MP3. The MAE values of DF/CD-OMP3 and DF/CD-OMP2.5 are only 0.02 kcal mol−1 higher than that of OMP3 and OMP2.5. Hence, the DF and CD approximations introduce negligible errors compared to conventional methods, while they substantially reduce the computational cost. Further, it is also

Cholesky vectors is higher, by 1.7−1.8 times, than that of auxiliary basis functions. 4.2. Open-Shell Noncovalent Interactions. In order to assess the accuracy of the DF-OMP3 and DF-OMP2.5 methods, we first consider open-shell noncovalent interactions. For the considered test set,13 noncovalent interaction energies (kcal mol−1) at the aug-cc-pVQZ MP2.5, OMP2.5, DFOMP2.5, CD-OMP2.5, MP3, OMP3, DF-OMP3, CD-OMP3, coupled-cluster singles and doubles (CCSD),69 and coupledcluster singles and doubles with perturbative triples [CCSD(T)]70,71 levels are reported in Table 1. The corresponding mean absolute errors (MAEs) with respect to CCSD(T)/augcc-pVQZ are presented graphically in Figure 2.

Table 1. Open-Shell Noncovalent Interaction Energies (kcal mol−1) at the aug-cc-pVQZ MP2.5, OMP2.5, DF-OMP2.5, CDOMP2.5, MP3, OMP3, DF-OMP3, CD-OMP3, CCSD, and CCSD(T) Levels, Mean Absolute Errors (Δmae), Standard Deviation of Errors (Δstd), and Maximum Absolute Errors (Δmax)

a

complex

MP2.5

OMP2.5

DF-OMP2.5

CD-OMP2.5

MP3

OMP3

DF-OMP3

CD-OMP3

CCSD

CCSD(T)

HOH···CH3 NH···NHa Li···Lib H2O···HNH2+ H2···Li H2O···F FH···BH2 He···Li He···OH HF···CO+ H2O···Cl H2O···Br H2O···Li FH···NH2 NC···Ne He···NH Δmae Δstd Δmax

−1.58 −1.03 −0.14 −25.36 −0.02 0.35 −3.98 0.00 −0.01 −33.89 −2.47 −2.53 −11.67 −10.15 −0.06 −0.02 0.70 1.46 4.01

−1.68 −1.02 −0.17 −25.34 −0.02 −3.34 −4.10 0.00 −0.02 −29.74 −3.37 −3.14 −12.07 −10.22 −0.08 −0.02 0.12 0.22 0.79

−1.85 −1.01 −0.21 −25.37 −0.02 −3.33 −4.19 0.00 0.26 −29.75 −3.36 −3.14 −12.62 −10.21 −0.06 −0.02 0.14 0.24 0.75

−1.84 −1.01 −0.25 −25.37 −0.02 −3.33 −4.23 0.01 0.26 −29.75 −3.35 −3.14 −12.59 −10.21 −0.06 −0.03 0.14 0.23 0.71

−1.54 −1.02 −0.34 −25.43 −0.02 0.63 −3.93 0.00 −0.02 −32.56 −2.19 −2.23 −11.97 −10.06 −0.06 −0.03 0.65 1.29 4.05

−1.61 −1.01 −0.38 −25.43 −0.02 −1.81 −4.00 0.00 −0.03 −30.11 −2.78 −2.61 −12.36 −10.09 −0.05 −0.03 0.25 0.45 1.62

−1.73 −1.00 −0.42 −25.49 −0.02 −1.81 −4.08 0.01 0.23 −30.12 −3.45 −2.98 −12.87 −10.10 0.17 −0.03 0.27 0.48 1.62

−1.73 −1.00 −0.48 −25.49 −0.02 −1.80 −4.08 0.01 0.23 −30.12 −3.45 −2.98 −12.86 −10.09 0.19 −0.03 0.27 0.48 1.62

−1.45 −1.00 −0.94 −24.80 −0.02 −1.90 −3.88 0.00 −0.02 −28.37 −2.39 −2.33 −12.16 −9.81 −0.06 −0.03 0.38 0.52 1.52

−1.70 −1.01 −0.96 −25.11 −0.02 −3.42 −4.14 0.00 −0.04 −29.88 −3.26 −3.12 −12.32 −10.12 −0.06 −0.03

The lowest quintet state of the dimer is considered. bThe lowest triplet state of the dimer is considered. F


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Table 2. Aromatic Bond Dissociation Energies (kcal mol−1) at the cc-pVTZ MP2.5, OMP2.5, DF-OMP2.5, CD-OMP2.5, MP3, OMP3, DF-OMP3, CD-OMP3, CCSD, and CCSD(T) Levels, Mean Absolute Errors (Δmae), Standard Deviation of Errors (Δstd), and Maximum Absolute Errors (Δmax) radical

MP2.5

OMP2.5

DF-OMP2.5

CD-OMP2.5

MP3

OMP3

DF-OMP3

CD-OMP3

CCSD

CCSD(T)

phenyl 2-methylbenzyl 3-methylbenzyl 4-methylbenzyl 2,3-dimethylphenyl 3,4-dimethylphenyl 2,4-dimethylphenyl 2,5-dimethylphenyl 2,6-dimethylphenyl 3,5-dimethylphenyl benzyl 2-methylphenyl 3-methylphenyl 4-methylphenyl 2-aminophenyl 3-aminophenyl anilino 3-hydroxyphenyl phenoxy Δmae Δstd Δmax

139.4 115.0 114.8 114.8 139.6 140.1 140.1 139.6 139.5 139.4 114.9 139.5 139.5 140.2 139.1 137.2 117.8 138.1 115.7 17.7 1.0 19.5

118.8 96.6 96.5 96.3 118.6 119.0 119.0 118.4 118.6 118.6 96.7 118.6 118.6 119.1 119.4 118.6 98.7 118.8 95.4 2.3 0.7 2.8

118.8 96.6 96.5 95.3 118.6 119.0 119.0 118.4 118.6 118.6 96.7 118.6 118.7 119.2 119.4 118.6 98.7 118.8 95.4 2.4 0.7 3.0

118.8 96.6 96.5 95.3 118.6 119.0 119.0 118.4 118.5 118.5 96.7 118.6 118.6 119.1 119.4 118.5 98.7 118.8 95.4 2.4 0.7 3.0

135.1 109.4 109.3 109.3 134.8 135.3 135.3 134.7 134.7 134.7 109.6 134.9 134.9 135.6 134.8 133.2 111.4 134.0 109.2 12.7 1.1 13.7

119.1 96.0 96.0 95.8 118.9 119.3 119.3 118.7 118.8 118.9 96.2 118.9 119.0 119.4 119.7 118.8 97.4 119.1 94.6 2.4 0.3 2.5

119.1 96.0 96.0 95.9 118.9 119.3 119.3 118.7 118.8 118.9 96.2 118.9 119.0 119.4 119.7 118.8 97.4 119.1 94.6 2.3 0.3 2.5

119.0 96.0 96.0 95.8 118.8 119.3 119.2 118.7 118.8 118.9 96.2 118.9 118.9 119.4 119.7 118.8 97.4 119.0 94.6 2.4 0.3 2.5

120.8 96.7 96.7 96.5 120.5 120.9 121.0 120.3 120.5 120.5 96.9 120.6 120.6 121.1 121.2 120.3 97.4 120.6 94.1 1.1 0.5 2.1

121.5 98.5 98.5 98.3 121.3 121.7 121.8 121.2 121.3 121.3 98.6 121.4 121.4 121.9 122.0 121.0 99.2 121.4 96.2

noteworthy that the performance of DF-OMP2.5 and DFOMP3 are better than that of CCSD. 4.3. Aromatic Bond Dissociation Energies. As the second step of our assessment, we assess the performance of DF-OMP3 and DF-OMP2.5 for aromatic bond dissociation energies,18 where the standard methods dramatically fail due to the high spin-contamination problem. The following general reaction is employed in the evaluation of ABDE for an aromatic radical. Ar − H → Ar· + ·H

For the considered set, the ABDE values (kcal mol−1) at the ccpVTZ MP2.5, OMP2.5, DF-OMP2.5, CD-OMP2.5, MP3, OMP3, DF-OMP3, CD-OMP3, CCSD, and CCSD(T) levels are reported in Table 2. The MAE values with respect to CCSD(T)/cc-pVTZ are presented graphically in Figure 3. The MAE values are 17.7 (MP2.5), 2.3 (OMP2.5), 2.4 (DFOMP2.5), 2.4 (CD-OMP2.5), 12.7 (MP3), 2.4 (OMP3), 2.3 (DF-OMP3), 2.4 (CD-OMP3), and 1.1 (CCSD) kcal mol−1, indicating a reduction in MP3 and MP2.5 errors by more than factors of 5 and 7 when optimized orbitals are employed, respectively. Failures of the MP3 and MP2.5 methods can be attributed to spin-contamination in the reference unrestricted HF (UHF) wave function as well as the lack of orbital relaxation effects. As it is shown in our previous study,18 for aromatic radicals ⟨S2⟩ values for the reference UHF wave function is significantly contaminated; hence, UHF wave functions have considerably larger ⟨S2⟩ values than the theoretical value of 0.75. Our results demonstrate that if the reference wave function suffers from spin-contamination, then standard MP methods dramatically fail, as previously discussed.9,15,20,25 On the other hand, orbital-optimized MP methods are resistant to spin-contamination in the reference wave function. In our previous study,18 it was demonstrated that OMP2 reference wave functions include quite negligible

Figure 3. Mean absolute errors (kcal mol−1) in aromatic bond dissociation energies for the MP2.5, OMP2.5, DF-OMP2.5, CDOMP2.5, MP3, OMP3, DF-OMP3, CD-OMP3, and CCSD methods with respect to CCSD(T) (cc-pVTZ basis set was employed).

spin-contamination for aromatic radicals. Hence, it was concluded that the contamination of the UHF reference is arising from the lack of correlation effects in the UHF wave function. Unfortunately, we have not yet implemented the ⟨S2⟩ value for OMP3 and OMP2.5, which would provide us a more detailed analysis. The CCSD method is not as sensitive to spincontamination in the reference wave function as MP2, MP2.5, and MP3, since the cluster single excitation operator partly compensates the orbital relaxation effects.69,72 4.4. Hydrogen Transfer Reactions. Finally we turn our attention to the energies of hydrogen transfer reactions. The test set considered20 is shown in Table 3. For hydrogen transfer reactions, the reaction energies (kcal mol−1) from the cc-pVTZ MP2.5, OMP2.5, DF-OMP2.5, CD-OMP2.5, MP3, OMP3, DF-OMP3, CD-OMP3, CCSD, and CCSD(T) levels are G


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Table 3. Reaction Energies (kcal mol−1) of the Hydrogen Transfer Reactions at the cc-pVTZ MP2.5, OMP2.5, DF-OMP2.5, CD-OMP2.5, MP3, OMP3, DF-OMP3, CD-OMP3, CCSD, and CCSD(T) Levels, Mean Absolute Errors (Δmae), Standard Deviation of Errors (Δstd), and Maximum Absolute Errors (Δmax) reaction

MP2.5

OMP2.5

DFOMP2.5

CDOMP2.5

MP3

OMP3

DFOMP3

CDOMP3

CCSD

CCSD(T)

CH3 + H2 → CH4 + H C2H + H2 → C2H2 + H C2H3 + H2 → C2H4 + H C(CH3)3 + H2 → HC(CH3)3 + H C6H5 + H2 → C6H6 + H C2H + C2H4 → C2H2 + C2H3 C(CH3)3 + C2H4 → HC(CH3)3 + C2H3 C6H5 + C2H4 → C6H6 + C2H3 C2H + HC(CH3)3 → C2H2 + C(CH3)3 C6H5 + HC(CH3)3 → C6H6 + C(CH3)3 C2H + C6H6 → C2H2 + C6H5 C2H + CH4 → C2H2 + CH3 C2H3 + CH4 → C2H4 + CH3 C(CH3)3 + CH4 → HC(CH3)3 + CH3 C6H5 + CH4 → C6H6 + CH3 Δmae Δstd Δmax

− 6.1 −48.9 −19.4 0.7 −35.6 −29.5 20.1 −16.2 −49.6 −36.3 −13.3 −42.8 −13.4 6.8 −29.5 10.8 9.1 22.6

− 6.0 −35.1 −12.7 1.5 −12.8 −22.4 14.2 −0.1 −36.6 −14.3 −22.3 −29.1 −6.7 7.5 −6.8 1.5 1.9 3.0

−6.0 −35.1 −12.7 2.2 −12.8 −22.4 14.9 −0.1 −37.3 −15.1 −22.3 −29.1 −6.7 8.2 −6.8 1.5 1.8 3.0

−6.0 −35.1 −12.7 2.2 −12.8 −22.4 14.9 −0.1 −37.3 −15.0 −22.3 −29.1 −6.7 8.2 −6.8 1.5 1.8 3.0

−4.3 −45.3 −17.2 2.8 −28.9 −28.1 19.9 −11.8 −48.0 −31.7 −16.3 −40.9 −12.8 7.1 −24.6 8.2 7.1 16.0

−4.2 −33.8 −11.2 3.4 −11.2 −22.6 14.6 −0.1 −37.2 −14.7 −22.5 −29.6 −7.0 7.6 −7.0 1.3 1.6 3.3

−4.2 −33.8 −11.2 3.3 −11.2 −22.6 14.4 −0.1 −37.0 −14.5 −22.5 −29.6 −7.0 7.5 −7.0 1.3 1.6 3.3

−4.2 −33.8 −11.2 3.3 −11.2 −22.6 14.4 0.0 −37.0 −14.5 −22.5 −29.6 −7.0 7.5 −7.0 1.3 1.6 3.3

−2.7 −31.5 −9.8 4.6 −12.0 −21.8 14.4 −2.2 −36.1 −16.6 −19.6 −28.9 −7.1 7.3 −9.3 0.4 0.5 1.0

−3.7 −32.2 −10.4 3.7 −13.0 −21.8 14.1 −2.6 −35.9 −16.6 −19.3 −28.5 −6.7 7.3 −9.3

OMP3 and DF/CD-OMP2.5 methods has been compared with that of OMP37 and OMP2.5.19 Our results demonstrate that the DF/CD-OMP3 and DF/CD-OMP2.5 methods provide significantly lower computational costs than OMP3 and OMP2.5. For the largest member of the alkanes set considered (C8H18), there are 26- (DF-OMP3) and 18-fold (CD-OMP3) reductions in the computational time compared to OMP3. Similarly, there are 20- (DF-OMP2.5) and 14-fold (CDOMP2.5) reductions in the wall-time compared to OMP2.5. Further, the DF/CD-OMP3 and DF/CD-OMP2.5 methods have been applied to open-shell noncovalent interactions, aromatic bond dissociation energies, and reaction energies of hydrogen transfer reactions, where standard MP3 and MP2.5 have difficulties. For open-shell noncovalent interactions, the DF-OMP2.5 (MAE = 0.14 kcal mol−1) and DF-OMP3 (0.27 kcal mol−1) methods outperform MP2.5 (0.70 kcal mol−1), MP3 (0.65 kcal mol−1), and CCSD (0.38 kcal mol−1) and provide quite accurate results. For aromatic bond dissociation energies, the DF-OMP2.5 (2.4 kcal mol−1), DF-OMP3 (2.3 kcal mol−1), and CCSD (1.1 kcal mol−1) methods provide accurate ABDEs, while the results of MP2.5 (17.7 kcal mol−1) and MP3 (12.7 kcal mol−1) are dramatically in errors. Failures of canonical MP methods can be attributed to spincontamination in the reference UHF wave functions. For hydrogen transfer reactions, the DF-OMP2.5 (1.5 kcal mol−1), DF-OMP3 (1.3 kcal mol−1), and CCSD (0.4 kcal mol−1) methods again provide accurate reaction energies, while the results of MP2.5 (10.8 kcal mol−1) and MP3 (8.2 kcal mol−1) are dramatically in errors. Further, the DF and CD approaches introduce negligible errors compared to OMP3 and OMP2.5, by 0.01−0.09 kcal mol−1. Moreover, mean absolute total energy differences between the DF/CD and conventional methods are 0.110−1.041 (DF-OMP2.5), 0.261−1.099 (CD-OMP2.5), 0.458−1.467 (DF-OMP3), and 0.687−1.394 (CD-OMP2.5) mhartree. Our results demonstrate that the DF-OMP2.5 and DFOMP3 methods are very helpful for the molecular systems with the challenging electronic structures, such as open-shell

reported in Table 3. The MAE values with respect to CCSD(T)/cc-pVTZ are presented graphically in Figure 4.

Figure 4. Mean absolute errors (kcal mol−1) in hydrogen transfer reaction energies for the MP2.5, OMP2.5, DF-OMP2.5, CD-OMP2.5, MP3, OMP3, DF-OMP3, CD-OMP3, and CCSD methods with respect to CCSD(T) (cc-pVTZ basis set was employed).

The MAE values are 10.8 (MP2.5), 1.5 (OMP2.5), 1.5 (DFOMP2.5), 1.5 (CD-OMP2.5), 8.2 (MP3), 1.3 (OMP3), 1.3 (DF-OMP3), 1.3 (CD-OMP3), and 0.4 (CCSD) kcal mol−1, indicating a reduction in MP3 and MP2.5 errors by more than factors of 6 and 7 when optimized orbitals are employed, respectively. Dramatic failures of standard MP approaches are again attributed to the spin contamination in the UHF reference wave function.

5. CONCLUSIONS Efficient implementations of the orbital-optimized MP3 and MP2.5 methods with the density-fitting and Cholesky decomposition approximations have been presented, which are denoted as DF-OMP3, CD-OMP3, DF-OMP2.5, and CDOMP2.5, respectively. The computational cost of the DF/CDH


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noncovalent interaction complexes and free radicals. Hence, considering the computational efficiency and the demonstrated accuracy of the DF-OMP3 and DF-OMP2.5 methods, we conclude that the DF-OMP3 and DF-OMP2.5 methods emerge as very useful tools for computational quantum chemistry.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.5b01128. Absolute energies (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the Scientific and Technȯ AK-113Z203). logical Research Council of Turkey (TÜ BIT U.B. also acknowledges support from the Turkish Academy of Sciences, Outstanding Young Scientist Award (TÜ BA-GEBIṖ 2015).

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