THE JOURNAL OF CHEMICAL PHYSICS 142, 044113 (2015)

A new scheme for perturbative triples correction to (0,1) sector of Fock space multi-reference coupled cluster method: Theory, implementation, and examples Achintya Kumar Dutta,a) Nayana Vaval, and Sourav Pala) Physical Chemistry Division, CSIR-National Chemical Laboratory, Pune 411008, India

(Received 13 October 2014; accepted 7 January 2015; published online 28 January 2015) We propose a new elegant strategy to implement third order triples correction in the light of many-body perturbation theory to the Fock space multi-reference coupled cluster method for the ionization problem. The computational scaling as well as the storage requirement is of key concerns in any many-body calculations. Our proposed approach scales as N6 does not require the storage of triples amplitudes and gives superior agreement over all the previous attempts made. This approach is capable of calculating multiple roots in a single calculation in contrast to the inclusion of perturbative triples in the equation of motion variant of the coupled cluster theory, where each root needs to be computed in a state-specific way and requires both the left and right state vectors together. The performance of the newly implemented scheme is tested by applying to methylene, boron nitride (B2N) anion, nitrogen, water, carbon monoxide, acetylene, formaldehyde, and thymine monomer, a DNA base. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4906233]

I. INTRODUCTION

The study of electron detachment and its induced phenomenon has attracted the attention of generations of theoreticians and experimentalists.1 Among the various theoretical approaches available, the Fock space multi-reference coupled cluster (FSMRCC) method2–6 has emerged as one of the most systematic method for the direct calculation of ionization potential (IP). The FSMRCC is generally used in singles, doubles approximation (FSMRCCSD)3 and is equivalent to corresponding equation of motion coupled cluster (EOMCC) method for principal peaks7 in one valence problem. The FSMRCCSD has a prominent tendency for a systematic overestimation of IP values, as compared to full configuration interaction (FCI) or experiments.8 However, the energy gaps between the states are generally better reproduced than the IP values. The most obvious way to improve the accuracy is to include higher order excitations in FSMRCC. The full inclusion of triples is generally the most straightforward way. Bartlett and co-workers have implemented full triples within the framework of FSMRCC9,10 and EOMCC.11,12 However, the full FSMRCCSDT scales as N8 and has huge storage requirement due to need to store triples amplitudes for both (0,0) and (0,1) sectors. Therefore, full FSMRCCSDT can only be used for purpose of benchmarking in case of very small molecules in a modest basis set. An economical compromise between the accuracy and computational cost can be achieved by inclusion of partial triples based on perturbative orders. The inclusion of perturbative triples correction in FSMRCC goes back to the work of Bartlett and co-workers,8 who have included triples correction complete up to third order in energy for (0,1) sector, a)Electronic addresses: [email protected] and [email protected]

0021-9606/2015/142(4)/044113/11/$30.00

whereas the effective Hamiltonian for the (0,0) sector is kept at the CCSD approximation. The resulting FSMRCCSD+T∗(3) method scales as N6 and did not require the storage of T3(0,1). However, it has been found that FSMRCCSD+T∗(3) method does not give a consistent performance8 and in some cases, it takes the value in wrong direction, i.e., away from the experimental and/or FCI value. Pal and co-workers13 latter corrected for the error by including perturbative triples correction up to fourth order for (0,1) sector and the resulting method leads to IP correction in the proper direction. However, the fourth order triples correction scheme of Pal and coworkers, as described in Ref. 13, scales as non-iterative N7 and requires the storage of triples amplitudes, which significantly restrict its applicability beyond small molecules. On the other hand, the third order triples correction schemes in EOMIP-CCSD by Stanton and co-workers14 (EOMIP-CCSD∗ scheme) as well as Krylov and co-workers15 scale as N6 and do not require the storage of triples amplitudes and yet take the IP correction in the proper direction. In both the schemes of partial triples correction to EOMIP-CCSD, one needs to calculate the left vector, which leads to the fact that triples correction can only be calculated in a state specific way. The partial triples correction schemes in FSMRCC, on the other hand, only require the T amplitudes and correction to multiple states can be computed in a single calculation. The aim of this paper is to implement a new scheme for partial triples correction to the (0,1) sector of FSMRCCSD, which takes the IP values in the correct direction, at the same time, keeping the scaling at N6 and avoiding the storage of triples amplitude. The paper is organized as follows: Sec. II contains the theory, implementation, and computational details of the new scheme for inclusion of perturbative triples in (0,1) sector of FSMRCC. The results and discussion on them are followed in Sec. III. The concluding remarks are presented in Sec. IV.

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II. THEORY AND COMPUTATIONAL DETAILS

For (0,0) sector, 

A. Effective Hamiltonian formulation of FSMRCC

Sˆ1(0,0) =

The FSMRCC method uses a common vacuum, which is generally but not necessarily a closed shell N-electron Hartree-Fock determinant. The holes and the particles are defined with respect to this common vacuum3 and these are further divided into active and inactive holes and particles. A linear combination of suitably chosen active configurations, based on energy criteria, generates the model space. The model space for a (p,h) valence Fock space, having h active hole and p active particle, can be represented as  Ψ(p,h) = (p,h) . Ci(p,h) (1) µ Φi (0)µ

ia  1  ab  † † sij aa ab a j ai . Sˆ2(0,0) = 4 ijab

i

The main aim of any effective Hamiltonian based theory is to extract some selective eigenvalues from the whole eigenvalue spectrum of the original Hamiltonian. To fulfill the objective, one needs to partition the configuration space into the model space and its complementary or orthogonal space. A complete model space (CMS) is obtained when all the possible configurations, produced by distributing the valence electrons among all the active orbitals in all possible ways, are included in the model space. Inclusion of a subset of these configurations results in an incomplete model space (IMS). The projection operator for model space is defined as  Φ(p,h)Φ(p,h) . (p,h) PM = (2) i i i

The projection operator in the orthogonal space, i.e., the virtual space, is defined as Q M = 1 − PM .

(3)

The projection operator PM generates the model space determinants and Q M generates the orthogonal space determinants. The interaction among the model space determinants introduces the non-dynamic correlation. The dynamic correlation arises due to the interactions of the modelspace determinants with the orthogonal space determinants. This interaction is brought in through a valence universal wave operator Ω, which is parameterized in such a way that it can generate the exact wave function by acting on the model space. The Ω has the following form:   ˜ (p, h) Ω = eS . (4) The curly braces indicate normal ordering of the cluster operators and S˜ (p,h) is defined as following: S˜ (p,h) =

p  h 

Sˆ (k,l).

(5)

(6)

For (0,1) sector, Sˆ1(0,1) =



 s um au† a m ,

mu   ˆS (0,1) = 1 s ua au† aa† a j ai , 2 2 ijau ij

(7)

where u is the active hole. For (1,0) sector,   Sˆ1(1,0) = s ex ae† a x , ex  1  ab  † † Sˆ2(1,0) = s xi aa ab ai a x , 2 ixab

(8)

where x is the active particle. For (1,1) sector, Sˆ1(1,1) =



Sˆ2(1,1) =



 s ux au† a x ,

ux

 † † s au xi aa au a i a x .

(9)

aiux

Here, it should be noted that a and i levels in S2(1,1) in Eq. (9) cannot both refer to active orbitals at the same time. The Schrödinger equation for the manifold of quasidegenerate states can be written as   Hˆ Ψi(p,h) = Ei Ψi(p,h) . (10) The correlated µth wave function in MRCC formalism can be written as Ψ(p,h) = Ω Ψ(p,h) . (11) (0)µ µ From Eqs. (1) and (10), we get   ⌢ (p,h)+ (p,h)+ H Ω * Ci(p,h) = E µ Ω * Ci(p,h) . (12) µ Φi µ Φi , i , i The effective Hamiltonian for (p,h) valence system can be defined as ( ) (p,h) Hˆ eff C j µ = E µCi µ , (13) ij

j

where (

(p,h) Hˆ eff

) ij

  Ω−1 HΩ ˆ Φ(p,h) . = Φ(p,h) i j

(14)

Equation (14) can be written as

k=0 l=0

The cluster operator S˜ (k,l) can destroy exactly k active particles and l active holes, in addition to creation of holes and particles. The S˜ (p,h) subsumes all lower sector Fock space S˜ (k,l) operators. The S˜ (0,0) is equivalent to standard singlereference coupled cluster amplitudes (T (0,0)). They have the following form in the normal ordered second quantized notation.

 s ai aa† ai ,

(p,h) −1 ˆ (p,h) Hˆ eff = PM Ω HΩPM . −1

(15)

However, Ω may not be well defined in all the cases. Therefore, the above definition of effective Hamiltonian cannot be used in a general way. On the other hand, the Bloch-Lindgren approach eliminates the requirement of Ω−1 for the solution of effective Hamiltonian, and therefore, the FSMRCC equations are generally solved using this approach.

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The Bloch equation is a modified form of the Schrödinger equation, ˆ ˆ HΩP M = Ω Heff PM .

(16)

The Bloch projection approach for solution of the FSMRCC equations involves left projection of Eq. (16) with PM and Q M operators and it results in  (k,l) (k,l)  ˆ PM HΩ − Ω Hˆ eff PM = 0, (17)   ˆ − Ω Hˆ eff P(k,l) = 0 ,∀k = 0,...,p; l = 0,...,h. Q(k,l) HΩ (18) M M To generate the sufficiency condition, an additional constraint is imposed through parameterization of Ω. It is generally introduced, in case of CMS, by imposing the intermediate normalization condition PM ΩPM = PM . However, in case of incomplete model space, the situation is slightly different. It has been shown by Mukherjee16 that in case of incomplete model space, the valence universality of the wave operator is sufficient to guarantee linked-cluster theorem. However, one needs to relax the intermediate normalization for that. Pal et al.2 latter shown that the intermediate normalization can be used for a special case of quasi-complete model space in (1,1) sector, without any loss of generality. Now, the equations for Ω and Heff are coupled with each other through Eqs. (17) and (18). Therefore, Heff cannot be expressed explicitly in terms of Ω. However, by imposing intermediate normalization, Heff can directly be written as a function of Ω. In that case, Eq. (17) gets simplified to (p,h) ˆ (p,h) (p,h) (p,h) (p,h) ˆ Heff PM . = PM HΩPM PM

(19)

After solving the equations for Ω and Heff , the diagonalization of the effective Hamiltonian within the P space gives the energies of the corresponding states and the left and right eigen vectors, (p,h) (p,h) C = C (p,h) E, Hˆ eff

(20)

(p,h) = E C˜ (p,h), C˜ (p,h) Hˆ eff

(21)

˜ (p,h) (p,h)

C

C

=C

(p,h) ˜ (p,h)

C

= 1.

(22)

The contractions amongst the cluster operators of different sectors of Fock space are avoided due to the normal ordering, which leads to partial hierarchical decoupling of cluster equations. Thus, once the amplitudes of a particular sector of Fock space are solved, they appear as parameters in the equations for the higher sector of Fock space. This phenomenon is commonly referred to as sub-system embedding condition (SEC). Therefore, the lower valence cluster equations are decoupled from the higher valence cluster equations, because of the SEC, which makes their solution less complicated. Hence, the Bloch equations are solved progressively from the lowest valence (0,0) sector upwards up to (p,h) valence sector. B. Perturbative analysis of the MRCC equations

Coupled cluster method has inherent relationship with many-body perturbation theory and exponential structure of FSMRCC provides a transparent framework for perturbative

order analysis of the underlying equations.17 But before going into that, let us note that in this implementation of perturbative triples correction in FSMRCC, the correction is included only in the (0,1) sector of FSMRCC. The (0,0) sector has been kept at singles and doubles truncation, i.e., the similarity transformed Hamiltonian H given by the following equation is constructed using only the T1 and T2 amplitudes: H = e−T

(0,0)

HeT

(0,0)

.

(23)

For the rest of the manuscript, T (0,0) has been used to denote (0,0) sector amplitudes and S (0,1) to denote (0,1) sector amplitudes to avoid any confusion. If all the vacuum diagrams of H, i.e., Eref are dropped (0,1) while constructing Heff , then we are left with Heff , the eigenvalues of which gives (Etarget − Eref ), i.e., desired IP values. This (H − Eref ) can be expressed in terms of one body operator F, two body operator V , and three body operator W . Here, it should be noted that the hole-hole F and V contribute minimum at zeroth order and first order in perturbation, respectively. On the other hand, W contributes at the second order. Now, let us analyze the effective Hamiltonian for (0,1) sector within singles, doubles, and triples approximation, ( (0,1) Heff = P(0,1) F + F S1(0,1) +V S2(0,1) ) + F S2(0,1) +V S3(0,1) P(0,1). (24) The first four terms in the above equation are also present in the CCSD approximation. So, the only additional term arises due to inclusion of triples comes from the contraction of S3(0,1) (0,1) with V . Diagram 1 in Figure 1 depicts the Heff diagram (0,1) coming from V S3 . Now, the only other contribution of S3(0,1) (0,1) to Heff can come through the contraction with W . The type of W , which has five lines pointing downward, required to the (0,1) contract with S3(0,1) to generate Heff , cannot be formed by contraction between V and T (0,0) and therefore need not to be considered. Now, the determining equation for S3(0,1) is given by ( W +V S2(0,1) +W S1(0,1) + F S3(0,1) +V S3(0,1) Q(0,1) 3 ) (0,1) + W S3(0,1) +W S2(0,1) − S3(0,1) Heff P(0,1) = 0. (25) It is well known that S2(0,1) and S1(0,1) contribute at least at the first and second orders in perturbation. The minimal order contribution to S3(0,1) comes in the second order and taking contribution only up to second order, Eq. (25) is reduced to ( ) (0,1) Q3(0,1) W + F S3(0,1) +V S2(0,1) − S3(0,1) Heff P(0,1) = 0. (26) Solving Eq. (26) for S3(0,1) with converged S1(0,1) and S2(0,1) within CCSD approximation and taking the contribution (0,1) of S3(0,1) into Heff according to Eq. (24) leads to ∗ FSMRCCSD+T (3) scheme of Bartlett and co-workers.8 The resulting IP values are correct at least up to third order in perturbation. The problem with the above method is that it fails to take the IP values towards a proper direction, i.e., towards the full CI or experimental values for most of the cases. Pal and co-workers13 later corrected the problem by

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FIG. 1. The diagrammatic representation of FSMRCCSD+T(3)3rd scheme in skeleton form. (0,1) (a) Diagram 1 for Heff . (b) Diagrams 2-5 for (0,1)

determination of S3

(c) Diagrams 6-7 for contribution

(0,1) (0,1) of S3 into S2 . (d) Diagram 8 for contribution of (0,1) (0,1) S3 and S1 . The double line denotes the H vertex

and curly lines denote the bare V integral vertex, and the double arrow denotes the active line.

taking up to third order contribution in Eq. (25), ( W + F S3(0,1) +V S2(0,1) +V S3(0,1) +W S2(0,1) Q(0,1) 3 ) (0,1) − S3(0,1) Heff P(0,1) = 0.

(27)

(0,1) Therefore, the Heff is correct at least up to fourth order in perturbation and it takes the IP correction in a proper direction. However, the desired accuracy is gained at the expense of increased computational cost. In Eq. (27), the first term (W ) contains terms up to third order in the perturbation, which includes the contribution coming from contraction of V and T3(0,0). The calculation of the lowest order of contribution to T3(0,0) scales as non-iterative N7 and it becomes the bottleneck of the method. Moreover, the fourth-order triples correction scheme, as described in Ref. 13, requires the storage of S3(0,1). The need to store the five index S3(0,1) amplitudes restricts its application beyond small molecules.18 We are going to call this method as FSMRCCSD+T(3)4th for rest of the manuscript. Another problem with the FSMRCCSD+T(3)4th scheme of Pal and co-workers13 is that it is actually not complete up to fourth order in perturbation. To make it complete up to fourth order, one needs to include the lowest order effect of T3(0,0) into T2(0,0), which is third order in perturbation and will contribute to Heff in fourth order. This term was neglected in the fourth order triples scheme of Pal and co-workers,13 which essentially means that in FSMRCCSD+T(3)4th scheme, the target state is more correlated than the reference state and should result in systematic underestimation of IP values. The third order triples correction to EOMIP-CCSD method, as proposed by both Stanton and co-workers14 and Krylov and co-workers,15 takes the IP correction towards the direction of FCI or experiment, scales as N6, and also does not require storage of R3. In both the approaches, the similarity transformed Hamiltonian is diagonalized over a space spanned by 1h and 2h1p particle operators and correction due to R3 is calculated non-iteratively. Now, in FSMRCC, the effective Hamiltonian is diagonalized over 1h space only.3 The effect

of higher excitations (2h1p, 3h2p, and so on) is introduced through S (0,1) amplitudes. Therefore, for a proper inclusion of the effect S3(0,1), one needs to consider the effect of S3(0,1) amplitudes coming through the S1(0,1) and S2(0,1) amplitudes, i.e., one should consider the effect of S3(0,1) into S1(0,1) and S2(0,1) amplitudes. This was the part missing in FSMRCCSD+T∗(3) scheme of Bartlett and co-workers8 and presumably, the source of error in that particular approximation.

C. The FSMRCCSD+T(3)3rd

In the present scheme, the minimal order contribution to S3(0,1) has been determined using Eq. (26) and the effect of S3(0,1) into S1(0,1) and S2(0,1) has been considered using the following equations, which was missing in the FSMRCCSD+T∗(3) scheme:8 ( Q1(0,1) F + F S1(0,1) +V S2(0,1) + F S2(0,1) +V S3(0,1) ) (0,1) − S1(0,1) Heff P(0,1) = 0, (28) ( Q(0,1) V + F S2(0,1) +V S2(0,1) +W S2(0,1) +V S3(0,1) 2 ) (0,1) − S2(0,1) Heff P(0,1) = 0. (29) This leads to a proper inclusion of the effect of partial triples into Heff . In this scheme, both reference and target state energies are complete at least up to third order in energy with some selective suitable higher order terms introduced by the infinite order summation in the iterative process. In this new third order triples scheme for (0,1) sector of FSMRCC, which we are going to denote as FSMRCCSD+T(3)3rd for rest of the manuscript, first FSMRCC amplitude equations are solved in singles and doubles approximation. The triples contribution is calculated from converged S1(0,1) and S2(0,1) amplitudes using Eq. (26). (0,1) Then, we proceed to calculate the part of the Heff , which involves the contribution of S3(0,1). The contribution of S(0,1) to 3

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S1(0,1) and S2(0,1) amplitudes is also calculated using Eqs. (28) and (29) and the FSMRCCSD equations are re-iterated to get convergence using the S1(0,1) and S2(0,1) amplitudes updated with (0,1) the effect of S(0,1) and S2(0,1), 3 . With this new converged S1 (0,1) remaining parts of Heff are constructed using Eq. (24). The (0,1) diagrams for the S3(0,1) contribution into Heff , the determining (0,1) (0,1) equations for S3 and contribution of S3 to S1(0,1) and S2(0,1) amplitudes are presented in the skeletal form in Figure 1 and the explicit programmable expressions in the spin adapted form are presented in Appendix. The contribution of S3(0,1) to S1(0,1) and S2(0,1), i.e., V S3(0,1) kind of term needs to be calculated only once and can be stored to be reused at each step of the FSMRCCSD iterations. Thus, the need to save the S3(0,1) amplitudes is removed and it significantly reduces the storage requirement. The determining equation for S3(0,1) scales as non-iterative N6. Although, some special attention needs to be paid towards V S2(0,1) terms coming from diagram 5 (Expression (A8) in Appendix). The terms should be implemented as following: first, S2(0,1) amplitudes are contracted with the bare V integrals to form an intermediate, which in turn contracted with the T2(0,0) amplitudes to calculate the contribution to S3(0,1). This approach helps to maintain the scaling as N6 rather than using the traditional V type intermediates, which increase the (0,1) scaling to N7 for this case. The contribution of S3(0,1) to Heff and S1(0,1) and S2(0,1) scale as N6 and need to be calculated only once. Therefore, the new FSMRCCSD+T(3)3rd scheme for the (0,1) sector scales as non-iterative N6 and does not require storage of triples. However, the situation is more favorable than what the above statement indicates. The most expensive terms, which occur due to our third order triples correction scheme to (0,1) sector of Fock space, scales as nh3np3 and nh4np2, where np and nh represent the number of unoccupied and occupied orbitals, respectively, in the Hartree-Fock reference state. The reference state CCSD calculation involves an iterative step that scales with nh2np4. Now, for typical applications in a reasonable basis set, np is much greater than nh. Therefore, the total cost of the calculation for (0,1) sector in FSMRCCSD+T(3)3rd scheme should be less than a single iteration of the reference state CCSD equations. The scaling is still dominated by reference state (0,0) sector CCSD calculations and the FSMRCCSD+T(3)3rd scheme overall scales as iterative N6.

D. Computational details

All the FSMRCC calculations are done using our in-house coupled cluster codes in a hierarchy of Dunning’s correlation consistent cc-pVXZ(X = D,T,Q) basis set.19 The one and two electron integrals converged Hartree-Fock coefficients and eigenvalues are taken from Gamess US package.20 All the EOMCC calculations are performed using CFOUR.21 Experimental geometries are used in all the cases, except the case of thymine, where DFT-B3LYP/6-311G∗∗ + + optimized geometry has been used. For the case of methylene and B2N anion, the geometry is taken from the work of Krylov and co-workers.15 All the used geometries, Hartree-Fock energy, and reference state CCSD energies ((0,0) sector) are provided in the supplementary material.22 The IP values at the complete basis-set (CBS) limit22–25 are obtained by extrapolating the computed IP values, using following formula, used by Kamiya and Hirata26 in their benchmark study, 2

E (n) = E (∞) + η 1e−(n−1) + η 2e−(n−1) ,

where E (∞) is the energy at the CBS limit, and n = 2, 3, 4 correspond to the cc-pVDZ, cc-pVTZ, and cc-pVQZ basis sets, respectively. E (n) are the corresponding energies, and η 1 and η 2 are parameters that are used to fit the energies. III. RESULTS AND DISCUSSIONS A. Comparison with FCI 1. Methylene

In Table I, we compare the first three vertical IP values of methylene calculated using different partial triples corrected variants of FSMRCC, with that of FCI, in 6-31G basis set. The absolute error in FSMRCCSD varies from 0.040 eV (12A1) to 0.802 eV (2 2A1 state), i.e., 0.42%-3.57%. The absolute error in Bartlett’s8 FSMRCCSD+T∗(3) scheme varies from 0.039 eV to 0.866 eV. The IP correction for the 12B1 and 2 2A1 states of methylene in FSMRCCSD+T∗(3) scheme goes away from the FCI. However, for 12A1 state, it goes in correct direction. The error in our new FSMRCCSD + (T)3rd shows significant improvement over FSMRCCSD, with absolute error bar in the range of 0.032-0.532 eV, i.e., 0.22%-2.45%. The EOMIP-CCSD∗ scheme also shows a similar absolute error

TABLE I. Vertical ionization energies (eV) of methylene (1A1), 6-31G basis. 12A1

(30)

12B1

22A1

Method

Total energy

IP

Total energy

IP

Total energy

IP

FSMRCCSD FSMRCCSD+T(3)∗ FSMRCCSD+T(3)3rd FSMRCCSD+T(3)4th EOMIP-CCSD∗ EOMIP-CCSDT EOMIP-CCSDTQ EOMIP-CCSDTQP FCI limit

−38.589 36 −38.586 44 −38.589 35 −38.590 25 −38.589 42 −38.589 89 −38.589 96 −38.589 97 −38.581 00

9.593 9.672 9.593 9.568 9.591 9.632 9.633 9.633 9.633

−38.406 92 −38.406 32 −38.408 87 −38.410 83 −38.408 78 −38.411 80 −38.412 10 −38.412 11 −38.412 11

14.557 14.574 14.504 14.451 14.507 14.478 14.473 14.473 14.472

−38.115 56 −38.113 19 −38.125 46 −38.131 35 −38.127 56 −38.144 84 −38.147 04 −38.147 09 −38.147 10

22.486 22.550 22.216 22.056 22.158 21.742 21.685 21.684 21.684

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TABLE II. Total energy (Hartree) of the ground state and the relative energy (eV) of first excited state of B2N in cc-pVDZ basis set. 2Σ

Method

u

−104.009 39 −104.008 36 −104.017 43 −104.020 95 −104.018 40 −104.033 57 −104.034 82

FSMRCCSD FSMRCCSD+T(3)∗ FSMRCCSD+T(3)3rd FSMRCCSD+T(3)4th EOMIP-CCSD∗ EOMIP-CCSDT EOMIP-CCSDTQ

2Σ g

0.737 0.752 0.725 0.725 0.715 0.700 0.695

it should be noted that the FSMRCCSD+T(3)4th partial triples scheme gives a better description of the final state energy of both ground and first excited states of BNB than both FSMRCCSD+T(3)3rd and EOMIP-CCSD∗ methods. This makes the fourth order triples scheme more suitable for final state property calculation than the third order triples correction scheme to both FSMRCC and EOMCC, however, at the expense of higher computational cost. B. Comparison with experiment 1. N2

bar in the range of 0.035-0.474 eV. The FSMRCCSD+(T)4th scheme of Pal and co-workers13 shows slightly smaller absolute error bar of 0.021-0.372 eV, i.e., 0.15%-1.72%. In the full EOMIP-CCSDT method, the error bars are reduced to 0.001-0.058 eV, i.e., 0.1%-0.27%. The maximum error in EOMIP-CCSDTQ method reduces to 0.001 and it almost approached full CI level. 2. B2N anion

The vibronic coupling between the ground (2Σu) and the low lying excited states (2Σg) of B2N and the resulting “artificial symmetry breaking” make it a complicated system to be studied in standard single reference methods.27 However, EOMIP-CC based methods are known to give very good performance for the BNB radical and Krylov and co-workers15 have used it as a test case to benchmark their partial triples correction scheme to EOMIP-CC. Table II presents the vertical gaps between the lowest two states in different variants of FSMRCC methods in cc-pVDZ basis set. The absolute error in FSMRCCSD is 0.042 eV, as compared to EOMIP-CCSDTQ method. The latter provides a very good approximation to FCI result. The FSMRCCSD+T∗(3) scheme increases the error to 0.057 eV. Both FSMRCCSD+T(3)3rd and FSMRCCSD+T(3)4th schemes reduce the absolute error to 0.030 eV. On the other hand, the EOMIP-CCSD∗ method gives an absolute error of 0.020 eV. The full EOMIPCCSDT method shows a negligible error 0.005 eV. Here,

Table III presents the vertical IP values of first three ionized states of N2 in different truncation level of FSMRCC and compare them with experimental values. It can be seen that the IP values for all the three states in FSMRCCSD method are overestimated compared to its full triples corrected version, in all the three basis sets. The FSMRCCSD+T∗(3) scheme takes the IP correction in the direction of EOMIP-CCSDT value only for the 1πu state. For rest of the two states, the IP correction moves away from that in EOMIP-CCSDT method. The new FSMRCCSD+T(3)3rd scheme takes the IP correction towards the direction of EOMIP-CCSDT for all the three states, however, the values are slightly underestimated as compared to the EOMIP-CCSDT value. The EOMIP-CCSD∗ scheme gives a similar performance. The underestimation of IP values is more prominent in FSMRCCSD+T(3)4th method, which is expected from the imbalance arising due to missing fourth order term, as discussed in the Subsection II B. In CBS limit, the FSMRCCSD method shows an error bar of 0.23 eV-0.49 eV, as compared to the experimental results. On the other hand, the IP values in FSMRCCSD+T(3)3rd scheme are within 0.10 eV of experimental value. The EOMIP-CCSD∗ method shows a similar accuracy range and predicts IP values within 0.13 eV of experimental results. Here, it should be noted that the improvement from FSMRCCSD is more prominent in case of inner valence one than that in outer valence ionization. The FSMRCCSD+T(3)4th scheme leads to more underestimation of IP values than that in the

TABLE III. Ionization potential of N2 (in eV).

Basis

State

FSMRCCSD

FSMRCCSD+T(3)∗

FSMRCCSD+T(3)3rd

FSMRCCSD+T(3)4th

EOMIP-CCSD∗

EOMIPCCSDT

cc-pVDZ

3σg 1πu 2σu

15.22 16.97 18.49

15.24 16.68 18.74

15.02 16.54 18.37

14.89 16.38 18.24

15.02 16.45 18.35

15.10 16.64 18.36

cc-pVTZ

3σg 1πu 2σu

15.61 17.24 18.84

15.64 16.96 19.09

15.34 16.76 18.62

15.21 16.63 18.50

15.33 16.66 18.61

15.45 16.91 18.66

cc-pVQZ

3σg 1πu 2σu

15.74 17.35 18.95

15.75 17.06 19.19

15.43 16.83 18.71

15.31 16.70 18.56

15.43 16.75 18.69

15.56 17.01 18.76

CBS

3σg 1πu 2σu

15.81 17.42 19.01

15.81 17.12 19.25

15.48 16.87 18.76

15.37 16.74 18.59

15.49 16.80 18.74

15.62 17.07 18.81

Expt.32

15.58 16.93 18.75

044113-7

Dutta, Vaval, and Pal

J. Chem. Phys. 142, 044113 (2015)

TABLE IV. Ionization potential of H2O.

Basis

State

FSMRCCSD

FSMRCCSD+T(3)∗

FSMRCCSD+T(3)3rd

FSMRCCSD+T(3)4th

EOMIP-CCSD∗

EOMIPCCSDT

cc-pVDZ

1b2 3a1 1b1

11.80 14.13 18.47

12.16 14.47 18.66

11.90 14.22 18.49

11.86 14.18 18.45

11.91 14.23 18.50

11.93 14.26 18.53

cc-pVTZ

1b2 3a1 1b1

12.42 14.65 18.85

12.77 14.91 19.06

12.39 14.63 18.79

12.32 14.56 18.73

12.39 14.62 18.79

12.47 14.70 18.87

cc-pVQZ

1b2 3a1 1b1

12.62 14.82 19.00

12.96 15.16 19.23

12.54 14.76 18.91

12.47 14.70 18.84

12.55 14.76 18.92

12.64 14.85 19.00

CBS

1b2 3a1 1b1

12.73 14.92 19.11

13.07 15.31 19.33

12.62 14.83 18.98

12.56 14.78 18.91

12.64 14.84 19.00

12.74 14.94 19.08

Expt.32

12.62 14.73 18.55

FSMRCCSD+T(3)3rd scheme, and in CBS limit, the former shows deviation as high as 0.33 eV from experimental values. The FSMRCCSD+T∗(3) scheme shows some improvement over FSMRCCSD for the 1πu states. However, it increases the deviation from experiment for the 3σg and 2σu states and for these two states, it shows higher error bar even than the FSMRCCSD method.

error is slightly higher (0.44 eV) for the 1b1 state. However, even the full EOMIP-CCSDT method gives similar error bar for the 1b1 state. The EOMIP-CCSD∗ method gives identical performance as that of the FSMRCCSD+T(3)3rd method. The fourth order triples scheme, on the other hand, leads to slightly underestimated results, though the values are very close to that in FSMRCCSD+T(3)3rd method.

2. H2O

3. CO

The vertical ionization energies for first three states of water have been presented in Table IV. The FSMRCCSD method gives very good agreement with the full EOMIPCCSDT method. All the partial triples corrected version of FSMRCC underestimates the IP values compared to full triples corrected versions, except the FSMRCCSD+T∗(3) method, which severely overestimates the IP values for all the three states. The IP values for all the three states in FSMRCCSD+T(3)3rd method shows improved agreement with experimental values than that in FSMRCCSD method. In CBS limit, the FSMRCCSD+T(3)3rd method predicts IP value within 0.1 eV of experimental result for 1b2 and 3a1 states. The

Table V presents the vertical ionization energies for first three states of carbon monoxide. The FSMRCCSD method overestimates the IP values for the 5σ and 4σ states, and underestimates for the 1π state, as compared to EOMIPCCSDT method. The FSMRCCSD+T(3)3rd method leads to red shift of the IP values for all the three states as compared to FSMRCCSD method. The FSMRCCSD+T(3)4th method and EOMIP-CCSD∗ scheme also lead to similar trend. In CBS limit, the FSMRCCSD+T(3)3rd method predicts very accurate vertical IP values, which are within 0.13 eV of experimental values, whereas, the FSMRCCSD method shows a maximum error of 0.31 eV. The EOMIP-CCSD∗ method

TABLE V. Ionization potential of CO.

State

FSMRCCSD

FSMRCCSD+T(3)∗

FSMRCCSD+T(3)3rd

FSMRCCSD+T(3)4th

EOMIP-CCSD∗

EOMIPCCSDT

cc-pVDZ

5σ 1π 4σ

13.85 16.64 19.46

13.65 16.78 20.06

13.47 16.53 19.39

13.37 16.44 19.30

13.44 16.55 19.38

13.59 16.71 19.34

cc-pVTZ

5σ 1π 4σ

14.17 16.94 19.74

14.00 17.07 20.32

13.76 16.74 19.54

13.66 16.63 19.44

13.72 16.75 19.54

13.90 16.98 19.55

cc-pVQZ

5σ 1π 4σ

14.27 17.05 19.86

14.10 17.16 20.42

13.84 16.81 19.62

13.75 16.70 19.52

13.81 16.82 19.63

14.00 17.07 19.65

CBS

5σ 1π 4σ

14.33 17.11 19.93

14.16 17.21 20.48

13.88 16.85 19.67

13.80 16.74 19.60

13.86 16.86 19.69

14.06 17.12 19.71

Basis

Expt.32

14.01 16.91 19.72

044113-8

Dutta, Vaval, and Pal

J. Chem. Phys. 142, 044113 (2015)

TABLE VI. Ionization potential of C2H2.

Basis

State

FSMRCCSD

FSMRCCSD+T(3)∗

FSMRCCSD+T(3)3rd

FSMRCCSD+T(3)4th

EOMIP-CCSD∗

EOMIPCCSDT

cc-pVDZ

2Π u 2Σ + g 2Σ + u

11.36 17.02 18.88

11.23 17.10 19.08

11.10 16.84 18.74

10.98 16.73 18.65

11.08 16.82 18.75

11.23 16.89 18.78

cc-pVTZ

2Π u 2Σ + g 2Σ + u

11.56 17.22 19.10

11.42 17.24 19.31

11.24 16.97 18.91

11.13 16.86 18.81

11.22 16.98 18.89

11.43 17.09 18.96

cc-pVQZ

2Π u 2Σ + g 2Σ + u

11.64 17.31 19.18

11.48 17.36 19.38

11.28 17.03 18.96

11.17 16.90 18.86

11.27 17.04 18.94

11.50 17.17 19.02

CBS

2Π u 2Σ + g 2Σ + u

11.69 17.36 19.23

11.51 17.44 19.42

11.30 17.07 18.99

11.19 16.92 18.89

11.30 17.08 18.97

11.54 17.22 19.05

gives nearly identical performance. However, the fourth order triples scheme in FSMRCC leads to slightly higher error bar in IP values and maximum deviation from experiments is 0.21 eV observed for 5σ state. The FSMRCCSD+T∗(3) method, on the other hand, leads to disastrous performance and shows error as high as 0.76 eV in CBS limit. 4. C2H2

The vertical IP values for the first three states of acetylene are presented in Table VI. The FSMRCCSD method overestimates the IP values as compared to the full EOMIPCCSDT method. The FSMRCCSD+T∗(3) method takes the IP correction in the wrong direction, i.e., away from the EOMIP-CCSDT results for the 2Σ+g and 2Σ+u states; however, the IP values go in the correct direction for the 2Πu state. The FSMRCCSD+T(3)3rd, FSMRCCSD+T(3)4th, and the EOMIP-CCSD∗ method take IP values in the correct direction for all the three states. However, the FSMRCCSD+T(3)4th method slightly underestimates the IP values as compared to other two variants.

Expt.33

11.49 16.70 18.70

In the CBS limit, the FSMRCCSD method significantly overestimates as compared to the experimental values, specially the deviation for the state 2Σ+g state is as high as 0.66 eV. The error gets reduced to 0.37 eV in FSMRCCSD+T(3)3rd, and it also shows improvement for the other two states also. The EOMIP-CCSD∗ method gives identical results. The FSMRCCSD+T(3)4th method slightly underestimates the IP values; however, the results are in similar range as that of FSMRCCSD+T(3)3rd method. On the other hand, the FSMRCCSD+T∗(3) method increases the error from that in FSMRCCSD, except for the 2Πu state, where it gives good agreement with experiment. 5. H2CO

Table VII presents the vertical IP values for the first three states of formaldehyde. The FSMRCCSD method generally overestimates the IP values, as compared to the full EOMIP-CCSDT method, except the 2b2 state in ccpVDZ basis set. The FSMRCCSD+T∗(3) method takes the IP correction in the wrong direction, i.e., away from the

TABLE VII. Ionization potential of H2CO.

Basis

State

FSMRCCSD

FSMRCCSD+T(3)∗

FSMRCCSD+T(3)3rd

FSMRCCSD+T(3)4th

EOMIP-CCSD∗

EOMIPCCSDT

cc-pVDZ

2b2 1b1 5a1

10.34 14.29 15.73

10.63 14.40 16.16

10.33 14.15 15.72

10.29 14.06 15.60

10.33 14.14 15.66

10.40 14.27 15.72

cc-pVTZ

2b2 1b1 5a1

10.76 14.58 16.06

11.02 14.67 16.47

10.68 14.35 15.94

10.58 14.24 15.80

10.65 14.34 15.88

10.75 14.53 15.98

cc-pVQZ

2b2 1b1 5a1

10.91 14.70 16.20

11.20 14.75 16.59

10.77 14.42 16.03

10.70 14.32 15.89

10.75 14.42 15.97

10.87 14.63 16.08

CBS

2b2 1b1 5a1

11.00 14.77 16.28

11.31 14.79 16.66

10.82 14.46 16.08

10.77 14.37 15.94

10.81 14.47 16.02

10.94 14.69 16.14

Expt.32

10.88 14.50 16.00

044113-9

Dutta, Vaval, and Pal

J. Chem. Phys. 142, 044113 (2015)

TABLE VIII. Maximum absolute, minimum absolute, average absolute and root mean square deviation of calculated IP (in eV) from experimental results in extrapolated CBS limit.

Method FSMRCCSD FSMRCCSD+T(3)∗ FSMRCCSD+T(3)3rd FSMRCCSD+T(3)4th EOMIP-CCSD∗ EOMIP-CCSDT

Maximum absolute deviation

Minimum absolute deviation

Average absolute deviation

RMS deviation

0.66 0.78 0.43 0.36 0.45 0.53

0.11 0.20 0.01 0.06 0.02 0.01

0.31 0.45 0.13 0.17 0.13 0.18

0.12 0.26 0.03 0.4 0.03 0.06

EOMIP-CCSDT results for all the three states. On the other hand, the FSMRCCSD+T(3)3rd, FSMRCCSD+T(3)4th, and EOMIP-CCSD∗, all of the three methods, take the IP correction in the proper direction for all the three states. In the CBS limit, the FSMRCCSD+T(3)3rd method shows excellent agreement with experiment and predicts IP values, which are within 0.08 eV of experimental results. The FSMRCCSD+T(3)4th and EOMIP-CCSD∗ methods give result in the similar range. The FSMRCCSD method, on the other hand, shows an error bar of 0.12-0.28 eV. The FSMRCCSD+T∗(3) method leads to drastic deterioration in the quality of the result, with error as high as 0.66 eV, in case of 5a1 state.

C. Error analysis

Table VIII presents an analysis of the trends for the errors in different truncation schemes of FSMRCC and EOMCC. It can be seen than FSMRCCSD+T∗(3) method shows the largest error with maximum absolute deviation of 0.78 eV, which is even higher than the FSMRCCSD method. The FSMRCCSD+T(3)3rd method shows a maximum deviation of 0.43, which is nearly identical with that in EOMIP-CCSD∗ method. Here, it should be noted that the maximum deviation from experiment is observed for 1b1 state of water and the error introduced in the FSMRCCSD+T(3)3rd method is comparable to that in full EOMIP-CCSDT method. The FSMRCCSD+T(3)3rd method shows average absolute deviation of 0.13 eV, which is again identical with that in EOMIP-CCSD∗ method, and slightly better than that in the FSMRCCSD+T(3)4th scheme. The FSMRCCSD+T∗(3)

method, on the other hand, shows an average absolute deviation of 0.45 eV, which is even higher than that in the FSMRCCSD method. The RMS deviation follows the same trend as that of average absolute deviation. Therefore, it can be said that the FSMRCCSD+T(3)3rd method can correct for the high error observed in the IP values calculated in FSMRCCSD+T∗(3) method and the former gives slightly better performance that FSMRCCSD+T(3)4th scheme of Pal and co-workers13 in less computational cost.

D. Vertical IP of thymine

The accurate determination of ionization and ionization induced structural changes of DNA nucleobases are extremely important for a proper understanding of the effects of radiation damage on genetic materials. Krylov and co-workers28–31 have used EOMIP-CCSD method for accurate determination of IP values of base, base pairs, and tetramer, with great success. To show the applicability of FSMRCCSD+T(3)3rd method, we have calculated the ionization energy of first six states of thymine in cc-pVDZ and cc-pVTZ basis set. Table IX shows that in both the basis sets, the FSMRCCSD+T(3)3rd method leads to lowering of IP values from that in FSMRCCSD method. The IP values in FSMRCCSD+T(3)3rd/cc-pVTZ level of theory are systematically underestimated as compared to the experimental results (except the 3 2A′′ state) and the values are within 0.27 eV of experimental range. On the other hand, the FSMRCCSD method systematically overestimates the IP values as compared to the experimental results, and it shows error as high as 0.55 eV, for the 3 2A′ state.

TABLE IX. Vertical ionization energies of thymine (in eV). Molecule cc-pVDZ basis set FSMRCCSD FSMRCCSD+T(3)3rd cc-pVTZ basis set FSMRCCSD FSMRCCSD+T(3)3rd Experimental34 Experimental35 Experimental36 Experimental37

1 2A′′

1 2A′

2 2A′′

2 2 A′

3 2A′′

3 2 A′

8.80 8.48

9.72 9.53

10.10 9.83

10.63 10.43

12.35 12.04

13.59 13.35

9.17 8.79 9.19 9.20 9.18 9.02

10.18 9.89 10.14 10.05 10.03 9.95

10.55 10.20 10.45 10.44 10.39 10.40

11.07 10.77 10.89 10.88 10.82 10.80

12.70 12.33 12.27 12.30 12.27 12.10

13.86 13.58 13.31 ... ... ...

044113-10

Dutta, Vaval, and Pal

J. Chem. Phys. 142, 044113 (2015)

IV. CONCLUSIONS

where

The present work describes theory, implementation, and performance of a new scheme for triples correction to (0,1) sector of Fock space multi-reference coupled cluster method. The scheme scales as N6 and avoids the storage of triples amplitudes and still able to take the IP correction towards the proper direction. The present scheme avoids the problem of accuracy in FSMRCCSD+T∗(3) scheme8 and high computational cost in FSMRCCSD+T(3)4th scheme.13 The new FSMRCCSD+T(3)3rd scheme gives similar performance to that of EOMIP-CCSD∗ method,14 with the additional advantage of its capacity to calculate IP correction to all the states in a single calculation. The partial triples correction scheme in EOMIP-CC requires calculation of left vector,14,15 which compels the correction to be calculated in a state specific manner. The FSMRCCSD+T(3)3rd scheme gives better agreement with experiment than the previous FSMRCCSD+T(3)4th scheme of Pal and co-workers,13 which systematically underestimates the IP values due to the partially neglected reference state correction terms. The FSMRCCSD+T(3)3rd scheme can also be applied beyond small molecules and our calculation on thymine gives good agreement with experimental values. Similar partial triples correction scheme can also be persuaded for other sectors of FSMRCC. However, the partial triples correction to (1,0) and (1,1) sectors requires the triples correction to the reference state ((0,0) sector) also, which can be and is indeed neglected in present implementation to (0,1) sector. Work is currently underway to implement a balanced partial triples correction scheme to (1,0) and higher sectors of FSMRCC and will be reported in some future publication.

(A3) ′ ′ D ij kba l = Fi + Fa + Fb − F j − Fk − Fl.

(A4)

1. W terms from diagram 2   ′ AD(1)ij kab Vbi ′l c t ca Vj ai ′c t lbc l = jk + k c

− −

c

 m 

Vi ′m j k t lba m+



Vk i ′ac t cb jl +



c

ac Vj bi ′c t kl

c

Vi ′m jl t kabm .

m

(A5) 2. W terms from diagram 3   ′ db db Vmi ′c dt ca Vmi ′c dt ac AD(2)ij kab l =2 mk t jl − mk t jl mcd

− −

 mcd 

ab Vmi ′dc t dc j k t ml −

mcd 

+



V

bc t da j mt l k +



ab Vmi ′c dt dc jl t k m

mcd db ac Vmi ′c dt ml tk j +

mcd

+2

da cb Vmi ′c dt mk t jl

mcd mi ′c d

mcd



ca Vmi ′c dt db j mt l k

mcd



da Vmi ′c dt cb ml t j k −

mcd



da Vmi ′c dt bc ml t j k .

mcd

(A6) V S2(0,1)

terms from diagram 4    ′ ′ ′ V baci S ijlc + V bacl S ij kc − V amkl S ij mb AD(3) j k l =

3.

i ′ab

c

− ACKNOWLEDGMENTS

c



′

ia V bml j Smk −

m

The authors acknowledge the grant from CSIR XII five year plan project on Multi-scale Simulations of Material (MSM) and facilities of the Centre of Excellence in Scientific Computing at NCL. A.K.D. thanks the Council of Scientific and Industrial Research (CSIR) for a Senior Research Fellowship. S.P. acknowledges the DST J.C. Bose Fellowship project and CSIR SSB grant towards completion of the work.

′

′

′

′

′

i ab i ab i ab i ab AD ij kab l = AD(1) j k l + AD(2) j k l + AD(3) j k l + AD(4) j kl ,

m



′

V k l mb S ij ma −

m



′

ib V k j am Sml .

m

(A7) 4.

V S2(0,1)

terms from diagram 5  ( ′ ) ′ i ′c ad AD(4)ij kab Vbmdc 2S ij mc − Sm l = j tkl mcd

+ +



( ′ ) i ′c ad Vl c nm 2S ij mc − Sm j tk n

mnc  ( ′ ) i ′c bd Vamdc 2S ij mc − Sm j tl k mcd

+ APPENDIX: PROGRAMMABLE EXPRESSIONS (0,1) Heff

Expression for the additional term in due to from diagram 1,   ′ ′ (0,1) ′ ′ (i , j ) = −2 V klab S ij ′ab V klab Ski jab Heff ′l kl + 2 −

′ V klab Sli jab ′k +



klab

′

V klba S ij ′ab kl .

) ( ′ i ′c ba Vk c nm 2S ij mc − Sm j tl m

mnc

S3(0,1)

+



′

Vl c mn S ij mc t kabn −



(A1) −

mcd ′

ba Vmck n S ij mc t ln +

mcd 

i ′c ad Vbmdc Sml tk j +

+

′

D ij kab l



′

i b ac Vncl m Smn tk j +

mnc

′

′

,



′

i c ab Vmc j n Sml tk n

(A2)

+

 mnc

mcd 

′

i b ac Vmc j n Smn t kl

mnc

mcd

Expression for S3(0,1), S ij kab l =

′

ad Vbmdc S ij mc t kl

mnc mnc   i ′c bd i ′c bd − Vamdc S j mt k l − Vamdc Smk tl j

klab

AD ij kab l



mnc

−

klab

klab







′

i a bc Vmc j n Smn tl k

mnc i ′a bc Vnck m Smn tl j .

(A8)

044113-11

Dutta, Vaval, and Pal

J. Chem. Phys. 142, 044113 (2015)

Effect of S3(0,1) into S2(0,1) diagram 7,    ′ ′ ′ ′ S ij ka = 2 V abcl S ij kab V abcl S ijlab V abcl Sli jba l − k − k bcl

12M.

bcl

bcl    ′ i ′ab i ′ba −2 V ml k b S j ml + V ml k b S j ml + V ml k b Sli jba m ml b ml b ml b    ′ba i ′ab i ′ba −2 V ml j b Smk V ml j b Slimk + V ml j b Smk l+ l. ml b ml b ml b

(A9) Effect of

S3(0,1)

Effect of

S3(0,1)

S2(0,1)

into diagram 8,    ′ ′ ′ ′ S ij ka = 2 F l b S ij kab F l b S ijlab F l b Slikab l − k − j . (A10) lb

′

lb

in to

S ij = −2

S1(0,1) 

lb

diagram 9, ′

V klab S ij kab l +2



′

V klab Ski jlab

klab  klab ′ i ′ab − V klab Sl j k + V klba S ij kab l . klab

klab

The superscript ( ′ ) denotes the active labeling, S denotes (0,1) sector amplitudes, and T denotes (0,0) sector amplitudes. The V denote the bare integral, and F and V denotes the standard one body and two body intermediates of similarly transformed Hamiltonian within CCSD approximation, the exact programmable expression can be found in Ref. 38.

1R. A. Marcus and N. Sutin, Biochim. Biophys. Acta 811(3), 265-322 (1985). 2S. Pal, M. Rittby, R. J. Bartlett, D. Sinha, and D. Mukherjee, J. Chem. Phys.

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