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Chemical Physics Letters 612 (2014) 209–213

Contents lists available at ScienceDirect

Chemical Physics Letters

jou rn al hom epage: www.elsev ier .com/ locate /cp le t t

xtended coupled cluster method for potential energy surface: decoupled approach

ayali P. Joshi, Nayana Vaval ∗

hysical Chemistry Division, CSIR – National Chemical Laboratory, Pune 411008, India

r t i c l e i n f o

rticle history:eceived 21 May 2014

a b s t r a c t

Extended coupled cluster (ECC) method has been implemented extensively for the calculation of molec-

n final form 7 August 2014vailable online 14 August 2014

ular properties. In this Letter we report the potential energy surface (PES) study using coupled and adecoupled approximation of ECC. HF, N2 and C2 are studied as test systems. N2 and C2 being doubly andtriply bonded, are considered to be interesting systems for PES study. We compare our results with fullCI (FCI) results wherever available. Decoupled approach within ECC framework shows good convergencefor all the molecules.

© 2014 Published by Elsevier B.V.

. Introduction

Study of potential energy surface (PES) provides a key to under-tand various chemical reactions [1], kinetics, dynamics [2–4] andpectroscopic properties. In recent years, there have been a lot oftudies on PES [2–5]. Hartree Fock [6,7] is used as a zeroth orderpproximation for the correlated calculations. Among the corre-ated methods, single reference coupled cluster (SRCC) method8–12] has been accepted as the most accurate method even ints approximate form. SRCC has been extensively used for thealculation of energy [13–15] and energy derivatives [16–21] asell as for potential energy surface calculations. Application of

RCC for the PES calculations at the bond breaking region is onef the most challenging problems. Though SRCC is suitable athe ground state geometry, it fails completely at the dissociationimit in particular when RHF does not dissociate correctly, wherehe non-dynamical correlation dominates [22–24]. To overcomehis problem higher order terms like triples and quadruples arencluded [25] or multi reference methods are used [26,27]. Varioustudies have been performed for the calculation of PES using vari-tional coupled cluster method [28–31]. Being variational it haspper bound in energy and does not collapse like standard SRCCethod. Expectation values coupled cluster (XCC) [32], unitary cou-

led cluster (UCC) [33], extended coupled cluster (ECC) [34–37],

mproved coupled cluster (ICC) [30] and quadratic coupled clusterQCC) [31] are some of the variants of the variational coupled clus-er approach. However, all the variational methods suffer from the

∗ Corresponding author.E-mail addresses: ps.joshi@ncl.res.in (S.P. Joshi), np.vaval@ncl.res.in (N. Vaval).

ttp://dx.doi.org/10.1016/j.cplett.2014.08.016009-2614/© 2014 Published by Elsevier B.V.

problem of non-terminating series and needs to be truncated forpractical application. Various truncation schemes are available inthe literature. Energy functional in variational CC theory includesconnected diagrams, however, when differentiated leads to dis-connected diagrams giving loss of size extensivity. ECC functionaldue to the double linked nature always maintains linked diagramseven after differentiation and therefore the size extensivity is main-tained [37]. However, this has double the number of amplitudesand hence twice the number of equations to be solved. Decoupledapproximation has been proposed to solve this problem. Decoupledapproximation was implemented for electric properties at the equi-librium geometry and is successfully tested for closed shell systems[38].

Last decade has witnessed a wide variety of study on potentialenergy surfaces. The incorrect convergence of RHF based studiesmakes it challenging for any SRCC method to study PESs. Musialand Bartlett [25] have shown that the �CCSD(TQf) approxima-tion, based on �2CCSD(TQf) method, gives improvement in thePESs due to the inclusion of the connected factorized quadruples.Knowles and co-workers [30,31] used different approximations ofthe variational coupled cluster method for the study of potentialenergy surfaces. Piecuch and co-workers [26] used renormalizednon-iterative coupled cluster method for the study of PESs. Cur-rent Letter emphasizes on the study of extended coupled cluster(ECC) method along with promising decoupled approximation forthe potential energy surface. We have studied close shell moleculeslike HF, N2 and C2 in bond breaking region. All the three systems

are very well studied in the literature before, due to the challengesthey possess.

The Letter is organized in the following manner. In Section 2 webriefly discuss the extended coupled cluster (ECC) method along

2 Physic

wiS

2

td

E

∣∣⟨

Hp

T

∑

dtk

t

E

Hvft[g

flot�s

e

10 S.P. Joshi, N. Vaval / Chemical

ith the decoupled approximation. Results and discussion on thems done in Section 3. Conclusions on the results are presented inection 4.

. Theory

We briefly discuss the extended coupled cluster (ECC) func-ional, that we have used for PES study. The ECC functional usesifferent ket and bra vectors.

=⟨

� ′|H|�⟩

(1)

�⟩

= eT∣∣�

⟩(2)

�′∣∣ =⟨

�∣∣ e˙e−T (3)

ere∑

is a hole particle de-excitation operator while T is holearticle excitation operator.

They can be defined as

=∑

i

tiC†i

(4)

=∑

i

�iCi (5)

Here � ′ and � are bi-orthogonal to each other and they areifferently parameterized. Arponen [34] and Bishop et al. [35] firstime proposed the functional for calculating energy and is popularlynown as ECC.

After performing double similarity transformation the func-ional is written in the following form:

=⟨

˚0|e˙(HeT )L|˚0⟩

DL(6)

DL implies that the right operator must be connected to theamiltonian H. The subscript DL (double linked) signifies the leftector � is either connected to the Hamiltonian H or to two dif-erent T operators. Thus, DL conforms that the series is naturallyerminating and thus gives size extensive energy and properties33]. Double linking also ensures that we have only connected dia-rams.

The form of the functional used in this Letter is as follows:

E =⟨

˚0

∣∣ V t2 + �2V + �1V t1 + �2V t2 + �1V t2 + �2V t1 + �1Ft1

+�2Ft2 + 12

�1�1V + 12

V t1t1+12

�1V t1t1+ 12

�2V t1t1+ 12

�1�1V t1

+�1�2V t2 + �1V t2t1 + �2V t2t1 + 12

�2V t2t2 + 12!

�2V t2t21

+ 13!

�2V t31 + 1

4!�2V t4

1 + 13!

�31 V t2 + 1

2!�2V t1t1

∣∣˚0⟩

(7)

It can be seen that, the contribution of the right operator is takenull within CCSD approximation; also the terms we use are doubleinked within CCSD model. The initial implementation was basedn the cubic truncation of the left and right vector. Thus, the extraerms compared to the cubic approximation are �2Vt2t2

1 , �2Vt31 ,

2Vt41 , �1Vt2. However, we have not included terms like �2

2 Vt22 and

ome other terms like this.The amplitude equations are obtained by differentiating energy

xpression with respect to cluster amplitudes i.e.

∂E

∂t(0)i

= 0 i = 1, 2, . . . (8a)

∂E

∂�(0)i

= 0 i = 1, 2, . . . (8b)

s Letters 612 (2014) 209–213

Differentiation with respect to t amplitudes gives equation for� amplitudes, while differentiation with respect to � amplitudesgives equation for t amplitudes. The superscript (0) denotes thatthe amplitudes are unperturbed.

Equations for t(0)2 and t(0)

1 amplitudes are given below

∂E

∂�(0)2

=⟨

˚pqab

∣∣ V + V t(0)1 + (F + V)t(0)

2 + 12!

V t(0)1 t(0)

1 + V t(0)1 t(0)

2

+ 12!

V t(0)2 t(0)

2 + �(0)1 V t(0)

2 + 12!

V t(0)2 (t(0)

1 )2 + 1

3!V(t(0)

1 )3

+ 14!

V(t(0)1 )

4 ∣∣˚0⟩

= 0

(9a)

∂E

∂�(0)1

=⟨

˚pa

∣∣�(0)1 V + (F + V)t(0)

1 + V t(0)2 + 1

2!V t(0)

1 t(0)1

+V t(0)1 t(0)

2 + �(0)1 V t(0)

1 + �(0)2 V t(0)

2 + 12!

(�(0)1 )

(2)V t(0)

2

∣∣˚0⟩

= 0

(9b)

In similar way, equations for �(0)2 and �(0)

1 are given as

∂E

∂t(0)2

=⟨

˚0

∣∣ V + �(0)1 V + �(0)

2 (F + V) + 12!

�(0)1 �(0)

1 V

+�(0)1 �(0)

2 V + �(0)1 V t(0)

1 + �(0)2 V t(0)

1 + �(0)2 V t(0)

2

+ 12!

�(0)2 V(t(0)

1 )2 + 1

3!(�(0)

1 )3V

∣∣˚pqab

⟩= 0

(10a)

∂E

∂t(0)1

=⟨

˚0

∣∣ V t(0)1 + �(0)

1 (F + V) + �(0)2 V + 1

2!�(0)

1 �(0)1 V

+V t(0)1 t(0)

2 + �(0)1 V t(0)

1 + �(0)2 V t(0)

2 + �(0)2 V t(0)

1

+�(0)1 V t(0)

2 + �(0)2 V t(0)

2 t(0)1 + 1

2!�(0)

2 V(t(0)1 )

(2)

+ 13!

�(0)2 V(t(0)

1 )(3) ∣∣˚p

a

⟩= 0

(10b)

For cluster amplitude calculations, we solve equations (9a) and(9b) along with (10a) and (10b). These equations are solved itera-tively to get the accurate energy. It can be seen that ECC methodcontains double the number of terms and hence amplitude equa-tions compared to the SRCC approximation. This at times givesproblem in convergence. Decoupled approximation is used as analternative to eliminate this problem. The discussion on it is donein next section.

2.1. Decoupled scheme

The ECC functional behaves very well at equilibrium andstretched geometries around equilibrium. However, near bondbreaking region ECCSD possess problem of convergence. To over-come this problem we have used a decoupled scheme for the studyof PES.

Current decoupled scheme separates the left and right clusteroperators. Eq. (8b), in detail Eqs. (9a) and (9b) give the ampli-tudes within singles and doubles approximations and they arevery similar to non-variational coupled cluster (NVCC) equation.Thus, initially we determine the t amplitudes eliminating � ampli-tudes. Once t amplitudes are known, we treat them as constant.We then solve the equation for � amplitudes i.e. Eq. (8a) and Eqs.(10a) and (10b) i.e. detail expression. Then we come back to thet amplitude equation. This time we consider all the terms includ-ing � amplitudes, however, � amplitudes are treated as constant.Again, we go back to the � amplitude equation. Thus, in the decoup-

led approximation we have both the amplitudes but we treat theother amplitude as a constant and hence every time only half thenumbers of cluster amplitudes are included. Using these left andright cluster amplitudes we evaluate the potential energy surface

S.P. Joshi, N. Vaval / Chemical Physics Letters 612 (2014) 209–213 211

rve fo

(ttrtmtt

3

aoatble(dtbfFsNhRG

3

w[w

Figure 1. Potential energy cu

PES) for singly and multiply bonded systems. The number of clus-er amplitudes are reduced, in the decoupled scheme as comparedo ECC, though the numbers of equations are same. At the equilib-ium geometry, coupled and decoupled scheme take almost sameime. However, at stretched geometries decoupled scheme takes

ore computer time compared to coupled scheme. Despite thishe decoupled scheme is preferred due to convergence problemhat we face for the coupled scheme.

. Results and discussion

We present the study of potential energy surfaces for HF, C2nd N2 molecules using ECC functional. Here we report the resultsbtained for these systems with ECCSD method, where equationsre solved in a coupled as well as in a decoupled manner. All thehree systems are very well studied for the PES. We have chosen theasis sets such that we can compare our results with the existing

iterature results. We compare our results with FCI method wher-ver possible. Hydrogen fluoride molecule has been studied in DZDunning) [39] basis set. C2 molecule has gained a vital importanceue to its role in combustion reaction and also due to its mul-ireference character at equilibrium geometry. Hence the systemecomes an interesting candidate for the study. We study the PESor C2 in 6-31 G* [40,41] basis set and compare our results withCI results reported in the literature. N2 being strongly correlatedystem proves to be a good example to study PES. We have studied2 in DZP [42] basis set and for comparison the CCSD (T) methodas been used. To start with the calculations we have consideredHF determinant. CCSD (T) as well as FCI results are obtained usingAMESS [43] package. Results for PES are shown in Figures 1–3.

.1. Hydrogen fluoride

PES for hydrogen fluoride molecule has been studied veryell with various methods and is documented in literature

25,30,31,44]. All electrons are correlated in our calculations ande compare our results with FCI values. Energies are calculated for

r HF molecule reference FCI.

bond lengths ranging from 1.2996 (0.75Re) to 6.238 (3.6Re) Bohr.The equilibrium (Re) bond length is 1.7328 Bohr. Figure 1 repre-sents the potential energy curve for HF. The potential energy curvefor HF can be divided into two parts. Part one consists from 0.75Re

to 1.5Re where the RHF configuration is a good approximation forthe exact wave function. The electronic configuration at the equi-librium geometry is

(1�)2(2�)2(3�)2(1�)2(2�)2|(4�)0. . .

Around equilibrium only dynamic correlation is dominant. Aswe go towards stretched geometry static correlation becomes cru-cial. At 1.5Re onwards the RHF electronic configuration changes to

(1�)2(2�)2(1�)2(2�)2(3�)2|(4�)0

The configuration which involves bi-excitation (3�)2 → (4�)0

becomes equally important as the ground state configuration. Thismakes it difficult for the convergence of the ECC equation in partic-ular when solved in a coupled manner. As a result we are not able toconverge the ECC equation beyond 2.75Re. At 3.6Re FCI expansioncoefficient for the above bi-excitation is 0.6163 compared to theground state coefficient of 0.6474. This indicates the importance ofstatic correlation at 3.6Re.

From the curves in Figure 1 we can see that with the decoup-led approach curve performs very well and shows the sametrend like FCI i.e. reference curve. The difference between FCI andECCSD decoupled approach is very small i.e. 0.005 a.u. From 3 Bohronwards coupled ECCSD starts to differ with the FCI. Our resultsobtained using decoupled approach represents the curve is veryclose to the FCI and thus this scheme is significantly better thancoupled ECCSD in terms of faster the convergence.

3.2. Nitrogen

The electronic configuration of N2 molecule is given as

(1�)2(2�)2(3�)2(4�)2(5�)2(1�)2(2�)2|(6�)0

212 S.P. Joshi, N. Vaval / Chemical Physics Letters 612 (2014) 209–213

ergy c

t

pNNElar

Figure 2. Potential en

We have studied N2 molecule in DZP (Dunning) basis. However,here are no FCI results are available in this basis, in the literature.

Since the system is strongly correlated, the static correlationlays dominant role starting from equilibrium geometry itself. Also2 being triply bonded is an interesting system for the PES study.2 is studied by Piecuch and co-workers in STO-3G [29] basis using

CCSD method. Although, in their study functional is not doubleinked and also many higher order terms like �2

2vt22 , �1�2vt2

2 , etc.re included, which are missing in our study. All electrons are cor-elated in our study. PES is calculated for bond lengths ranging from

Figure 3. Potential energy c

urve for N2 molecule.

1.8 to 4.25 Bohr. The equilibrium bond length is taken as 2.07 Bohr.It is known that for N2 higher excited amplitudes i.e. T3 and T4 playan important role [25,29].

Figure 2 gives the PES calculated with coupled and decoupledECC method along with coupled cluster singles and doubles (CCSD)and coupled cluster singles and doubles using partial triples CCSD

(T). There has been a wide study over N2 molecule using CC approxi-mations. We have reported the curve using CCSD (T) for comparisonaccompanied by CCSD. From Figure 2 we can see that the differencebetween the CCSD and CCSD (T) is about 0.05 a.u. The difference

urve for C2 molecule.

Physic

ieftact

3

(

emmhorgo(tac

IfdqimWa

4

feWtadrstbct

mto

[[[[[[[[[[[[[[[[[[[[[[[[

[[

[[[[[[[

S.P. Joshi, N. Vaval / Chemical

ncreases rapidly after 3 Bohr showing the importance of higherxcitation amplitudes i.e. T3. Inclusion of T4 and higher excitationsurthermore with a variational approach may help in improvinghe results towards FCI. However, CCSD as well as CCSD (T) fall offfter 3.5 Bohr. The strong triple bond breaking might be compli-ated due to effect of static correlation. Decoupled scheme makeshe evaluation of static correlation feasible.

.3. Carbon dimer

The electronic configuration of C2 is given as

1�)2(2�)2(3�)2(4�)2(5�)2(1�)2|(6�)0

C2 is an interesting system to study because of the near degen-racy of 3�u state with the ground state, along with a strongultireference character. C2 has been tested by various correlatedethods due to its quasi degenerate nature [25,27,30,31,45,46]. We

ave chosen 6-31 G* basis set for our study due to the availabilityf results in the literature for comparison [45,46]. The equilib-ium bond length is 2.377 Bohr (1.243 A), taken from experimentaleometry. The bond is stretched until 5.67 Bohr (3 A). We compareur results with FCI [45] and completely renormalized [CR-CCSDTQ)] coupled cluster method with triple and quadruples excita-ion studied by Sherrill and Piecuch [46]. CR-CCSD (TQ) methodlso overestimates energy compared to full CI, however, it is lowerompared to ECCSD.

Figure 3 gives all the details of curves using different methods.t can be seen that full ECCSD and decoupled ECCSD methods per-orm similar for C2 molecule. Compared to FCI, both ECCSD andecoupled ECCSD are on the higher side as expected. However,ualitative trend is obtained correctly. Thus, inclusion of the non-

terative triples and quadruples should improve the results for ECCethod too. However, they are computationally very expensive.ithin CCSD approximation ECC being variational does not fall off

nd gives qualitatively correct behaviour.

. Conclusion

For efficient and accurate calculation of potential energy sur-aces of HF, N2 and C2 molecules we have used ECC method. ECCquations were solved in a coupled as well as decoupled manner.e present results using both the methods. We have compared

he results with FCI wherever available. Though both couplednd decoupled ECC methods contain same number of terms, theecoupled scheme proves a better option near bond breakingegion. Since coupled ECC contains 2n number of equations to beolved at stretched geometries it is difficult to converge. Howeverhe decoupled scheme takes advantage of considering half the num-er of equation at a time, thereby reducing the complications inomputational calculations. For all the molecules studied in Letter,he decoupled scheme behaves very well.

For N2 molecule we have compared our results with CCSD (T)ethod. Decoupled scheme provides better results for all the sys-

ems at the dissociation limit. The decoupled ECC method takes caref static correlation to some extent. However, in cases where static

[[[[

s Letters 612 (2014) 209–213 213

correlation dominates even in the ground state it may fail to takecare of it i.e. Cr2. In such cases quadratic terms along with the higherbody excitation term like triples, quadruples may help. However,use of multireference based methods will be more effective.

Acknowledgments

The authors acknowledge the facilities of the Centre of Excel-lence in Scientific Computing at NCL. NV would like to thank thegrant CSIR 12th five year project on MSM, for financial support. SJwould like to thank Council of Scientific and Industrial Research(CSIR) for Senior Research Fellowship and Mr. Turvasu Senguptafor his help.

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