On Some Aspects of Fock-Space Multi-Reference Coupled-Cluster Singles and Doubles Energies and...

CHAPTER 14

ON SOME ASPECTS OF FOCK-SPACE MULTI-REFERENCECOUPLED-CLUSTER SINGLES AND DOUBLES ENERGIESAND OPTICAL PROPERTIES

PRASHANT UDAY MANOHAR1, KODAGENAHALLI R. SHAMASUNDAR2,ARIJIT BAG3, NAYANA VAVAL4, AND SOURAV PAL5

1Department of Chemistry, University of Southern California, Los Angeles, CA 90089-0482, USA, e-mail:[email protected]ät Stuttgart, Institut für Theoretische Chemie, Pfaffenwaldring 55, D-70569 Stuttgart,Germany, e-mail: [email protected] Chemistry Division, National Chemical Laboratory, Pune 411008, India, e-mail:[email protected] Chemistry Division, National Chemical Laboratory, Pune 411008, India, e-mail:[email protected] Chemistry Division, National Chemical Laboratory, Pune 411008, India, e-mail:[email protected]

Abstract: Multi-reference coupled cluster methods are established as accurate and efficient tools fordescribing electronic structure of quasi degenerate states. Recently we have developedmulti-reference coupled cluster linear response approach based on the constrained vari-ation method. The method is very general and can describe challenging problems dueto the multiple-root nature of effective Hamiltonian. Calculation of response propertiesfor the ionized/electron attached or excited state molecules is a challenging task. Withthis formulation it is possible to accurately predict the higher order molecular propertiesof the open shell molecules. In this article we review the response approaches for quasidegenerate cases with emphasis on Fock space multi-reference coupled cluster method.

Keywords: Multireference coupled cluster, Fock space, Optical properties, Linear response

14.1. INTRODUCTION

Spectacular success of the standard single reference coupled-cluster (SRCC) methodin the last few decades in explaining a wide variety of chemical phenomena iswell documented in the quantum chemistry literature [1]. Its ability to treat dy-namical electron correlation accurately for closed shell molecules and satisfy size-extensivity features, makes it an attractive candidate for electronic structure calcula-tions. Another reason for the emergence of SRCC as the state-of-the-art method is

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376 P.U. Manohar et al.

the successful development of efficient analytic energy derivative techniques for thecalculation of molecular properties [2–7]. The response approach in the SRCC con-text was first attempted by Monkhorst [8]. This was originally formulated in anon-stationary framework, and hence, this did not have the simplicities introducedby the generalized Hellmann – Feynman theorem and the (2n+ 1) rule. As a result,evaluation of first-order property in SRCC depended explicitly on first-derivativesof cluster amplitudes with respect to the external perturbation. Thus, to satisfy thenecessary condition for Hellmann – Feynman theorem and get the full use of (2n+1)rule, several attempts have been made to formulate stationary coupled cluster theory[9–12]. All such attempts have ended in more complicated theories. This apparentimpediment was overcome by Bartlett and coworkers [3, 13]. They used the idea ofalgebraic Z-vector method introduced by Handy and Schaefer [14] in the analysis ofanalytic orbital derivatives introduced in the configuration interaction method. Theyhave shown that the requirement of first derivative of cluster amplitudes for suchmode of perturbation may be replaced by a single perturbation independent quantity,known as Z-vector. This and the subsequent developments by Bartlett and coworkerssubstantially facilitated efficient implementation of molecular energy gradients forSRCC, and significantly contributed to its success in quantum chemistry. However,Z-vector type of approach turned out to be a tedious job for higher-order proper-ties such as Hessians, polarizabilities, etc. On the other hand, a conceptually dif-ferent approach, using a constrained variation approach, proposed by Jørgensen andcoworkers, has incorporated all such developments in a single formulation throughthe formation of Lagrangian functional which is easily applicable to higher orderenergy derivative [15–18]. This approach involved recasting of standard SRCC the-ory in a stationary framework by introducing an extra set of de-excitation ampli-tudes. It was shown that this method includes the Z-vector method as a zeroth-orderresult and transparently extends its benefits to higher-order properties. While SRCCmethod has been routinely used for energy and energy derivatives for closed shelllike systems, where a single determinant can be identified to be the most impor-tant part of the wave-function. Description of molecular excited states, open shelland potential energy surfaces in general, demand a description beyond SRCC the-ories. These are the cases where reference state may not be described by a singleSlater determinant. A linear combination of determinants may be more appropriateto describe non-dynamic correlation associated with such states. The use of multi-determinantal reference space to incorporate non-dynamical electron correlation isin general known as multi-reference (MR) theories [19, 20]. There have been severalapproaches developed in last two decades among which the effective Hamiltonianapproaches [21, 22] are dominating. An effective Hamiltonian is diagonalized within a suitably chosen model space to approximately reproduce a part of the spectrumassociated with the exact Hamiltonian.

The development of coupled cluster methods incorporating the non-dynamicalelectron correlation is known as multi-reference coupled cluster (MRCC). Thereare two subclasses of MRCC methods, which have been studied extensively. One isthe multi root description via effective Hamiltonian approach [23, 24] and the other

Fock-Space Multi-Reference Coupled-Cluster 377

describes a specific root, known as the state-specific MRCC approach [25]. In multi-root effective Hamiltonian approach, exact energies of the corresponding stronglyinteracting near-degenerate states are obtained as roots of effective Hamiltonian viadiagonalization. This class of methods is further subdivided into Hilbert space andFock space approach depending on the way the dynamical part of electron correlationis described [23, 26–36]. In Hilbert space approach a state-universal wave operatorwhich contains different cluster operators for each of the reference determinants inthe model space [26, 30, 31]. It is based on the concept of multiple vacua and the dif-ferent cluster operators are hole-particle creation operators with respect to differentvacua. This method has been used for studying PES, bond-dissociation, etc. Howeverfor PES, state-selective MRCC method developed by Mukherjee and co-workers[25] has been found to be more suitable from the point of view of circumventingthe important problems of intruder states. The Fock-space (FS) MRCC [26–30, 32–36] in general, is suitable for energy difference calculations i,e ionization, electron-attachment and electronic excitation of molecules and is based on the concept ofcommon-vacuum and a valence universal wave-operator. The model space is com-posed of near degenerate configurations obtained by combinations of electron occu-pancies among what are called active orbitals. The occupancies of the active orbitalsare denoted in terms of number of active particles and active holes with respect to thevacuum. The wave operator in this approach must be able to destroy a set of activeholes and/or particles apart from hole particle excitation. The explicit form of inverseof normal ordered exponential is not known. Therefore, Bloch equation approach isthe only approach that can be followed to obtain the cluster amplitude equations. Ithas been shown that this leads to a connected set of equations for each valence sectorand equations for different Fock space sectors are decoupled from each other. Thisdecoupling of the different Fock space sectors is known as subsystem embeddingcondition (SEC). This emerges naturally as consequence of normal-ordered formof ansatz. Because of these simplifications, the normal-ordered exponential ansatzhas become the standard ansatz for FSMRCC theories and their applications. Theeffective Hamiltonian theory aims at a simultaneous description of a manifold ofstrongly interacting states. A zeroth-order approximation to this manifold is providedby the chosen model space. It is further assumed that this model space is energeticallywell-separated from its orthogonal complement and weakly interacts with it. Thismeans that determinants from orthogonal complement are not dominant in any ofthe targeted states.Thus, the choice of the model space is very important the wrongchoice of model space may lead to intruder state problem. However, the intruder stateproblem may be overcome by the Fock-space (FS) effective Hamiltonian scheme ofMeissner [24] which is very efficient from the computational point of view.

Parallel to this, methods like equations-of-motion (EOM) CC [37–43], cou-pled cluster linear response (CC-LR) [44], symmetry adapted clusters configura-tion interaction (SAC-CI) [45, 46], etc. are also known to handle certain classes ofquasi-degeneracy. Bartlett [38] and co-workers implemented EOMCC method forcalculation of excitation energies for atoms and molecules. Stanton and Bartlett [40]used EOMCC method for excitation energies, transition probabilities and excited


state properties. Nooijen and Bartlett [47] implemented it for electron attached sys-tems. Later, Watts and Bartlett [48] included triples excitation to EOMCC for excita-tion energies. Similarity transformed EOMCC method (STEOMCC) was developedby Nooijen and Bartlett [49] for ionized, electron attached and excited states. Morerecently, spin-flip EOMCC has been introduced by Krylov [50] for description ofexcited states, bond breaking, diradicals and triradicals. For one valence system inFock space i.e. for ionization potential and electron affinity calculation it has beenshown that both EOMCC and FSMRCC are equivalent [49]. However, for excitationenergy calculation, such equivalency disappears. It was shown by Bartlett [51] thatwhile the size-extensivity is maintained in FSMRCC, the same is not true in caseof EOM-CC for higher valence cases. However, recently developed similarity trans-formed (ST) EOM-CC method of Nooijen and co-workers [52, 53] is size-extensive.CC-LR method developed by Jørgensen’s group [54] has also been used extensivelyfor the calculation of excitation energies. The excitation energies obtained by CC-LRare identical to the EOM-CC excitation energies.

EOMCC derivatives for property calculation was also initiated and implementedby Stanton [54] first and then by Stanton and Gauss [41, 55]. Analytic energy deriva-tives for ionized states were described by Stanton and Gauss [41] in EOMCC formal-ism. Subsequently, analytic second derivative for excited state was also introduced byStanton and Gauss [55]. Gradients using STEOM-CC was implemented by Nooijenand coworkers [56] using Lagrange undetermined multiplier, which is similar to themethod followed in the work which is to be reviewed in this article. In STEOM-CC method, two more Z vector like quantities are required to be evaluated for theresponse of the S± coefficients to the perturbation along with the Lagrange multiplierlambda (Λ). The role of Λ is equivalent in both cases.

However, obtaining energy derivatives in the context of multi-root MRCC meth-ods was a challenge. Along the lines of SRCC, the analytic linear response forMRCC method based on Monkhorst’s approach [8] was initiated by Pal [57] longback. The computational developments and implementation of the method was donelater by Ajitha et al. [58–60] for obtaining dipole moment of doublet radicals andlow-lying excited states of molecules. However, the approach was quite unsatisfac-tory, since it required expensive evaluation of cluster amplitude derivatives for everymode of perturbations.

In SRCC context, this problem was solved by incorporation of Z-vector technique[4] or equivalently, the constrained variational approach (CVA) [7], which was basedon the method of Lagrange undetermined multipliers. The CVA method in FSMRCCframework was studied by Szalay [61], independently. However, this method wasapplicable for complete model spaces (CMS) only. Similar developments for bothFock and Hilbert space MRCC [62, 63] for ionic, electron attached and excited stateswas pursued by Pal and co-workers in recent years [62]. The CVA formulation ofShamasundar et al. [62]. is applicable for a general incomplete model space (IMS)cases and simplifies to Szalay’s formulation in the case of CMS. This single rootmethod provides a cost-effective tool to obtain higher order energy derivatives with


the knowledge of lower order cluster amplitude derivatives and a set of perturbationindependent vectors, the Lagrange multipliers.

The computational developments of FSMRCC using CVA were recently startedby Manohar et al. [64, 65] and Bag et al. [66] using singles and doubles (SD)truncation of the wave operator. The implementation for analytic dipole momentsand polarizabilities of doublet radicals has been done by Manohar et al. [64] and Baget al. [66, 67]. Implementation of CVA-FSMRCC method for the energy derivativesof the singlet and the triplet excited states have been done by Bag et al. [66]. In thisreview article, we present an overview of the FSMRCC method for energy deriva-tives. In particular, we review the Lagrangian based formulation using CVA whichis the most efficient implementation of the method. We review the basic theory anddiscuss some of the results obtained using the state of the art CVA based analyticderivative formulation.

14.2. FOCK-SPACE MULTI-REFERENCE COUPLED-CLUSTERMETHOD

We now present the developments so far in FSMRCC and related methods. As men-tioned earlier, FSMRCC is based on the concept of a common vacuum. We choosean N-electron RHF configuration as vacuum, which defines the holes and particles.In the Fock-space, the model space determinants contain h-holes and p-particles dis-tributed within a set of what are termed as active holes and active particles, usuallyaround the Fermi level. We denote the above p-active particle, h-active hole modelspace determinant by

{

Φ(p,h)i

}

. Thus, the model space of a (p, h) valence Fock-spacecan be written as

∣

∣Ψ(p,h)(0)μ 〉 =

∑

i

C(p,h)iμ

∣

∣Φ(p,h)i 〉 (14-1)

The dynamical electron correlation arises due to comparatively weak interactions ofthe model-space configurations with the virtual space configurations. This interac-tion is brought in through a universal wave operator Ω which is parameterized suchthat the states generated by its action on the reference function satisfy Schrödingerequation. To generate the exact states for the (p, h) valence system, the wave operatormust generate all valid excitations from the model space. The correlated μth wavefunction in the MRCC formalism is written as

∣

∣Ψ (p,h)μ 〉 = Ω∣

∣Ψ(p,h)(0)μ 〉 (14-2)

where

Ω = {

eT(p,h)}(14-3)

The brace-bracket indicates normal ordering of the cluster-operators. In the Fock-space approach T (p,h) amplitudes contain the lower valence amplitudes and thus givethe additional flexibility in the theory.


T (p,h) =p∑

k=0

h∑

l=0

T (k,l) (14-4)

The super scripted bracket in the right hand side of the above expression indicatesthat the cluster operator T is capable of destroying exactly k active particles and l ac-tive holes, in addition to creation of holes and particles. The T (p,h) operator subsumesall such lower T (k,l) operators.

The Schrödinger equation for the manifold of quasi-degenerate states can bewritten as

H∣

∣Ψ(p,h)i 〉 = Ei

∣

∣Ψ(p,h)i 〉

which leads to

HΩ(∑

i

C(p,h)iμ Φ

(p,h)i

) = EμΩ(∑

i

C(p,h)iμ Φ

(p,h)i

)

(14-5)

The projection operator for model space is defined as

P(p,h) =∑

i

∣

∣Φ(p,h)i 〉〈Φ(p,h)

i

∣

∣ (14-6)

The orthogonal component of the model space, i.e. the virtual space is defined as

Q = 1− P (14-7)

The effective Hamiltonian for (p, h) valence system can be defined such that

∑

j

(

H(p,h)eff

)

ijCjμ = EμCiμ (14-8)

(

H(p,h)eff

)

ij = 〈Φ(k,l)i

∣

∣Ω−1HΩ∣

∣Φ(k,l)j 〉

which can be written as

H(p,h)eff = P(p,h)Ω−1HΩP(p,h) (14-9)

The form the inverse ofΩ , in general may not be well defined. Hence, above defini-tion is seldom used to obtain the effective Hamiltonian. Instead, the Bloch – Lindgrenapproach is commonly used to define the effective Hamiltonian. The Bloch equationis just a modified form of Schrödinger equation.

HΩP = ΩHeffP (14-10)


The Bloch-Lindgren approach not only eliminates the requirement of Ω−1, but alsoprovides an important criterion the effective Hamiltonian must fulfill. The effectiveHamiltonian is, in general, non-hermitian. Mainly two approaches are used to obtainΩ and the effective Hamiltonian. One of them, known as Bloch projection approach,involves left projection of above equation by P and Q.

P(k,l)(HΩ −ΩH(k,l)eff

)

P(k,l) = 0

Q(k,l)(HΩ −ΩH(k,l)eff

)

P(k,l) = 0;

∀ k = 0, . . . , p;

l = 0, . . . , h (14-11)

The normalization condition is specified indirectly through parameterization ofΩ . Incase of complete model spaces (CMS), the intermediate normalization is commonlyemployed.

The diagonalization of the effective Hamiltonian within the P space gives theenergies of the corresponding states and the left and the right eigenvectors.

H(p,h)eff C(p,h) = C(p,h)E

C(p,h)H(p,h)eff = EC(p,h) (14-12)

C(p,h)C(p,h) = C(p,h)C(p,h) = 1 (14-13)

Because of normal ordering, the contractions amongst different cluster operatorswithin the exponential are avoided, leading to partial hierarchical decoupling ofcluster equations. This is commonly referred to as sub-system embedding condi-tion (SEC). The lower valence cluster equations are completely decoupled fromthe higher valence cluster equations because of SEC. Hence, the Bloch equationsare solved progressively from the lowest valence (0, 0) sector upwards up to (p, h)valence sector.

14.3. FSMRCC LINEAR RESPONSE

14.3.1. The Explicit Differentiation Method

The LR in FSMRCC framework was initiated by Pal and then implemented byPal and co-workers for dipole moments of doublet radicals and excited states ofclosed-shell molecules. The method is non-variational and involves explicit differen-tiation of Bloch equation with respect to uniform external field. In presence of time-independent uniform external field, the parameters Υ = {

H(p,h)eff , C(p,h), C(p,h), E,Ω

}

become perturbation dependent and can be expanded in Taylor series of g.

Υ (g) = Υ (0) + gΥ (1) + 1

2!g2Υ (2) + 1

3!g3Υ (3) + . . . (14-14)


The differentiation of the Bloch equations following left projections by model spaceand virtual space configurations with respect to g yields the equations for wave-function derivatives and the derivative effective Hamiltonian.

P(k,l)(H(1)Ω (0) + H(0)Ω (1) −Ω (1)H(k,l)(0)

eff −Ω (0)H(k,l)(1)

eff

)

P(k,l) = 0 (14-15)

Q(k,l)(H(1)Ω (0) + H(0)Ω (1) −Ω (1)H(k,l)(0)

eff −Ω (0)H(k,l)(1)

eff

)

P(k,l) = 0 (14-16)

;∀k = 0, . . . , p;l = 0, . . . , h

The equations are linear in the perturbation dependent quantities. It is interestingto note that the homogeneous parts of the Ω derivative equations are identical tothe linear homogeneous part of the undifferentiated cluster equations. The SECtransparently holds at every order. The method provides multiple roots of derivativeeffective-Hamiltonian which can be obtained simultaneously by solving followingequations.

∑

i

{(

H(1)eff

)

jiC(0)iμ +

(

H(0)eff

)

jiC(1)iμ

} = E(1)μ C(0)

jμ + E(0)μ C(1)

jμ (14-17)

However, due to its non-variational nature, the method does not obey the generalized

Hellmann – Feynman theorem for energy derivatives. Therefore, the evaluation ofnth order energy derivatives demands the knowledge of cluster amplitudes and theirderivatives up to nth order.

14.3.2. The Z-vector Method

The explicit differentiation method described in the last subsection, although straightforward, is cumbersome, particularly for higher order derivatives. Before discussingz-vector method for multi-reference problem we will briefly state formalism for sin-gle reference first. The response approach for SRCC was formulated by Monkhorst[8] to enable analytic computation of such properties. The SRCC was originally for-mulated in a non-stationary framework, and due to this it did not have the simplicitiesintroduced by the generalized Hellmann – Feynman theorem and the (2n+ 1) -rule.As a result, the expression for a first-order property in SRCC depended explicitlyon first- derivatives of cluster amplitudes with respect to the external perturbation,which means it is necessary calculate these cluster amplitude derivatives for allmodes of perturbation. Bartlett and coworkers [3, 13] introduced what is known asa z-vector method For first-order properties, this problem was overcome by Bartlettand coworkers using the idea of algebraic Z-vector method based on Dalgarno’sinterchange theorem [68]. This substantially facilitated efficient implementation ofmolecular energy gradients for SRCC, and significantly contributed to its successin quantum chemistry. In lines of Z-vector formalism in SRCC, Pal and co-workersproposed Z-vector technique for FSMRCC response [58–60]. Expanding Ω and its


derivatives in terms of T (p,h) and absorbing the lower valence T operators in the

transformed Hamiltonian H (H = [H exp (T (i,j))]c), the equations for H(p,h)(1)

eff and

T (p,h)(1)can be simplified and rearranged. The H(p,h)(1)

eff can thus be rewritten as.

P(p,h)(H(1) + H(1)T (p,h)(0) − T (p,h)(0)H(1) + H(0)T (p,h)(1) − T (p,h)(1)

H(0))P(p,h)

= P(p,h)(H(p,h)(1)

eff

)

P(p,h) (14-18)

Similarly, from Eq. (14-15) we arrive to

Q(p,h)(H(0)T (k,l)(1) − T (k,l)(1)H(0) − T (k,l)(1)

H(p,h)(0)

eff − T (k,l)(0)H(p,h)(1)

eff

)

P(p,h)

= Q(p,h)(H(1) + H(1)T (k,l)(0) − T (k,l)(0)H(1))P(p,h) (14-19)

Applying the resolution of identity it follows

〈Q(p,h)|H(0)|Q(p,h)〉〈Q(p,h)|T (p,h)(1) |P(p,h)〉−〈Q(p,h)|T (p,h)(1) |P(p,h)〉〈P(p,h)|H(0)|P(p,h)〉−〈Q(p,h)|T (p,h)(1) |P(p,h)〉〈P(p,h)|H(p,h)(0)

eff |P(p,h)〉= 〈Q(p,h)|H(1) + H(1)T (k,l)(0) − T (k,l)(0)

H(1)|P(p,h)〉 (14-20)

Substituting Eq. (14-18) in the above equation, it follows

〈Q(p,h)|H(0)|Q(p,h)〉〈Q(p,h)|T (p,h)(1) |P(p,h)〉−〈Q(p,h)|T (p,h)(1) |P(p,h)〉〈P(p,h)|H(0)|P(p,h)〉−〈Q(p,h)|T (p,h)(1) |P(p,h)〉〈P(p,h)|H(p,h)(0)

eff |P(p,h)〉= 〈Q(p,h)|(H(0) + H(1)T (p,h)(0) − T (p,h)(0)

H(1)|P(p,h)〉+〈Q(p,h)|T (p,h)(0) |P(p,h)〉〈P(p,h)|(H(0) + H(1)T (p,h)(0) |P(p,h)〉 (14-21)

From the above equation, one can see that T (p,h)(1)in the left hand side appears in

the right of H in the Q-space and to the left of Heff in P-space. Since, in general,Heff has a matrix structure, the factorization of the T (p,h)(1)

can be effected only upona diagonal assumption of Heff. In the limit of single reference due to single-rootof the model-space, this approximation becomes exact. In case of degenerate statesand states with different symmetries also, the diagonal approximation is appropriate.Thus, under the diagonal approximation, the equations for T (p,h)(1)

can be written as


(〈Q(p,h)|(H(0)|Q(p,h)〉 − 〈P(p,h)|(H(0)|P(p,h)〉−〈P(p,h)|H(p,h)(0)

eff |P(p,h)〉 + 〈Q(p,h)∣

∣H(1)T (p,h)(0)

−T (p,h)(0)H(1)|P(p,h)〉)〈Q(p,h)|T (p,h)(1) |P(p,h)〉

= 〈Q(p,h)|(H(1) + H(1)T (p,h)(0) − T (p,h)(0)H(1))

+T (p,h)(0)(H(1) + H(1)T (p,h)(0)

)|P(p,h)〉 (14-22)

Rearranging the above equation, it follows

〈Q(p,h)|T (p,h)(1) |P(p,h)〉 = 〈Q(p,h)|(H(1) + [H(1), T (p,h)(0)])+ T (p,h)(0)

(H(1)

+H(1)T (p,h)(0))|P(p,h)〉 ∗ A(p,h)(−1)

(14-23)

where,

A(p,h) = 〈Q(p,h)|H(0)|Q(p,h)〉 − 〈P(p,h)|H(0)|P(p,h)〉−〈P(p,h)|H(p,h)(0)

eff |P(p,h)〉 − 〈Q(p,h)|T (p,h)(0)H(0)|P(p,h)〉 (14-24)

Ajitha and Pal proposed a perturbation independent set of amplitudes ζ such thatfor every Q(p,h)T (p,h)P(p,h) sector amplitude, there is a corresponding P(p,h)ζ (p,h)Q(p,h)

amplitude. Defining the ζ vectors as

〈P(p,h)|ζ (p,h)|Q(p,h)〉 = 〈P(p,h)|(H(1) + [H(1), T (p,h)(0)])+ T (p,h)(0)

(H(1)

+H(1)T (p,h)(0))|Q(p,h)〉 ∗ A(p,h)−1

; (14-25)

the derivative effective Hamiltonian equation can be written in terms of these ζvectors as

〈P(p,h)|H(p,h)(0)

eff |P(p,h)〉 = 〈P(p,h)|(H(1) + H(1)T (p,h)(0))|P(p,h)〉

〈Q(p,h)|ζ (p,h)|Q(p,h)〉〈Q(p,h)|H(0)|P(p,h)〉(14-26)

This equation is a consequence of Dalgarno’s interchange theorem. We observethat the derivative effective Hamiltonian equation is independent of the derivativeT-amplitudes of the (p, h)-sector of the Fock space.

14.3.3. The Constrained-Variation Method

In the lines of SRCC, the constrained-variational approach (CVA) for energy deriva-tives was introduced in FSMRCC context by Szalay and was applicable for com-plete model spaces (CMS). Later, Pal and co-workers independently formulated the


CVA-FSMRCC for general incomplete model spaces (IMS) and showed that thefunctional simplifies to the one proposed by Szalay if applied for CMS and quasi-complete model space (QMS). The CVA-FSMRCC method of Pal an co-workersprovides response of a specific root of the multiple roots of FSMRCC. One has toproject a single desired state (root of effective Hamiltonian) for doing constrainedvariation. In FSMRCC context, the energy of a specific state of the (p, h) FS sector isgiven by

Eμ =∑

ij

C(p,h)μi (Heff)

(p,h)ij C(p,h)

jμ (14-27)

We construct the Lagrangian to minimize the energy expression given above, withthe constraint that the MRCC equations (Eqs. 14-11 and 14-13) are satisfied for thestate μ.

� =∑

ij

C(p,h)μi (Heff)

(p,h)ij C(p,h)

jμ +p∑

k=0

h∑

l=0

{

P(k,l)Λ(k,l)P(k,l)P(k,l)[HΩ −ΩH(k,l)eff

]

P(k,l)

+P(k,l)Λ(k,l)Q(k,l)Q(k,l)[HΩ −ΩH(k,l)eff

]

P(k,l)}+ Eμ[∑

ij

C(p,h)μi C(p,h)

jμ − 1]

(14-28)

The Λs in the above equation are the undetermined Lagrange multipliers obtainedby applying stationarity condition on the Lagrangian � with respect to the clusteramplitudes of the corresponding sectors. The stationarity condition on � with re-spect to the Λ vectors yields the MRCC equations for the cluster amplitude. Obvi-ously, the cluster-amplitudes are completely decoupled from the Λ vectors. The Λvectors follow the partial sector wise decoupling exactly in the reverse SEC, i.e.,the Λ vectors of the highest sector are totally decoupled from the lowest ones.The lowest sector Λ vectors are coupled with the Λs of all the higher sectorsthrough the inhomogeneous part of the linear equations. The eigenvectors and ef-fective Hamiltonian can be obtained by applying stationarity condition on the �. Incase of CMS and IMS, since the effective Hamiltonian can be explicitly defined interms of the cluster amplitudes, the CVA method simplifies to the one proposed bySzalay.

14.3.3.1. First Order Properties Using CVA-FSMRCC

For simplicity, we consider the ionization problem. In FSMRCC context, the model-space configurations belong to (0,1) sector of the FS, with the unionized RHFreference as vacuum. One can easily extend the algebra for general (p, h) valencecase. The configurations of the ionized states are given by


|Ψ (0,1)μ 〉 =

∑

i

Ciμ|Φ(0,1)i 〉 (14-29)

The dynamical electron correlation effects are brought in through a universal waveoperator Ω .

Ω = {eT(0,1)} (14-30)

T (0,1) = T (0,1) + T (0,0) (14-31)

The wave operatorΩ is parameterized such that the states generated by its action onthe reference function satisfy Bloch – Lindgren equation for effective Hamiltoniangiven by Eq. (14-10). From the earlier definition (14-28), the Lagrangian can bedefined as

� =∑

ij

C(0,1)μi (Heff)

(0,1)ij C(0,1)

jμ

+P(0,1)Λ(0,1)P(0,1)P(0,1)[HΩ −ΩH(0,1)eff

]

P(0,1)

+P(0,1)Λ(0,1)Q(0,1)Q(0,1)[HΩ −ΩH(0,1)eff

]

P(0,1)

+P(0,0)Λ(0,0)P(0,0)P(0,0)HΩP(0,0)

+P(0,0)Λ(0,0)Q(0,0)Q(0,0)HΩP(0,0)

−Eμ

⎛

⎝

∑

ij

C(0,1)μi C(0,1)

jμ − 1

⎞

⎠ (14-32)

Being the CMS, the effective Hamiltonian has explicit expression in terms of thecluster amplitudes due to which, the closed part of the Lagrangian vanishes and theabove equation reduces to

� =∑

ij

C(0,1)μi (Heff)

(0,1)ij C(0,1)

jμ

+P(0,1)Λ(0,1)Q(0,1)Q(0,1)[HΩ −ΩH(0,1)eff

]

P(0,1)

+P(0,0)Λ(0,0)Q(0,0)Q(0,0)HΩP(0,0)

−Eμ

⎛

⎝

∑

ij

C(0,1)μi C(0,1)

jμ − 1

⎞

⎠ (14-33)

In presence of external field, the Lagrangian and the parameters Υ ={Heff, C, C, E,Ω ,Λ} become perturbation dependent. These can be expanded in Tay-lor series.


Υ (g) = Υ (0) + gΥ (1) + 1

2!g2Υ (2) + 1

3!g3Υ (3) + . . . (14-34)

The Lagrangian defined in Eq. (14-33) can be differentiated with respect to the fieldg to obtain the Lagrangians at every order. The zeroth order and the first order La-grangians can therefore, be written as

�(0) =(

C(0,1)(0)H(0,1)(0)eff C(0,1)(0)

)

μμ

+P(0,1)Λ(0,1)(0)[H(0)Ω (0) −Ω (0)H(0,1)(0)eff

]

P(0,1)

+P(0,0)Λ(0,0)(0)[H(0)Ω (0)]P(0,0)

−Eμ

⎛

⎝

∑

ij

C(0,1)(0)μi C(0,1)(0)

jμ − 1

⎞

⎠ (14-35)

�(1) =(

C(0,1)(1)H(0,1)(0)eff C(0,1)(0)

)

μμ+(

C(0,1)(0)H(0,1)(1)eff C(0,1)(0)

)

μμ

+(

C(0,1)(0)H(0,1)eff

(0)C(0,1)(1)

)

μμ

+P(0,1)Λ(0,1)(1)[H(0)Ω (0) −Ω (0)H(0,1)(0)eff

]

P(0,1)

+P(0,1)Λ(0,1)(0)[H(1)Ω (0) + H(0)Ω (1) −Ω (1)H(0,1)(0)eff −Ω (0)H(0,1)(1)

eff

]

P(0,1)

+P(0,0)Λ(0,0)(1)H(0)Ω (0)P(0,0) + P(0,0)Λ(0,0)(0)H(1)Ω (0) + H(0)Ω (1)P(0,0)

−E(0)μ

∑

ij

(

C(0,1)(0)μi C(0,1)(1)

jμ + C(0,1)(1)μi C(0,1)(0)

jμ

)

−E(1)μ

⎛

⎝

∑

ij

C(0,1)(0)μi C(0,1)(0)

jμ − 1

⎞

⎠ (14-36)

The Eqs. (14-35) and (14-36) give the energy and the first order energy derivativefor the state μ. Because of stationarity of Lagrangian with respect to Λ and Ω , theabove expressions are further simplified. The energy derivatives follow (2n+ 1) rulewith respect to the Ω amplitudes and (2n + 2) rule with respect to Λ amplitudes.There is a (2n+ 1) rule for the eigenvectors C(0,1) and C(0,1) for evaluation of energyderivatives. With these, the expressions for Lagrangians given in Eqs. (14-35) and(14-36) simplify. We denote this simplified Lagrangian as �opt.

�(0)opt =

(

C(0,1)(0)H(0,1)(0)eff C(0,1)(0)

)

μμ(14-37)


�(1)opt =

(

C(0,1)(0)H(0,1)(1)effΩ (0) C(0,1)(0)

)

μμ

+P(0,1)Λ(0,1)(0)[H(1)Ω (0) −Ω (0)H(0,1)(1)effΩ (0)

]

P(0,1)

+P(0,0)Λ(0,0)(0)H(1)Ω (0)P(0,0) (14-38)

The subscript Ω (0) indicates that the derivative effective Hamiltonian does not con-tain any term formed from derivatives of the cluster amplitudes. The first order prop-erties can thus be obtained simply with the knowledge of Ω and Λ amplitudes only.

14.3.3.2. Second Order Properties Using CVA-FSMRCC

Now, we differentiate the Lagrangian (Eq. 14-33) twice with respect to the externalfield.

�(2) =(

C(0,1)(2)H(0,1)(0)eff C(0,1)(0)

)

μμ+(

C(0,1)(1)H(0,1)(1)eff C(0,1)(0)

)

μμ

+2(

C(0,1)(1)H(0,1)(0)eff C(0,1)(1)

)

μμ+(

C(0,1)(0)H(0,1)(2)eff C(0,1)(0)

)

μμ

+(

C(0,1)(0)H(0,1)(1)eff C(0,1)(1)

)

μμ+(

C(0,1)(0)H(0,1)eff

(0)C(0,1)(2)

)

μμ

+P(0,1)Λ(0,1)(2)[H(0)Ω (0) −Ω (0)H(0,1)(0)eff

]

P(0,1)

+P(0,1)Λ(0,1)(1)[H(1)Ω (0) + H(0)Ω (1) −Ω (0)H(0,1)(1)eff −Ω (1)H(0,1)(0)

eff

]

P(0,1)

+P(0,1)Λ(0,1)(0)[H(2)Ω (0) + H(1)Ω (1) + H(0)Ω (2)]P(0,1)

−P(0,1)Λ(0,1)(0)[Ω (0)H(0,1)(2)eff +Ω (1)H(0,1)(1)

eff +Ω (2)H(0,1)(0)eff

]

P(0,1)

+P(0,0)Λ(0,0)(2)[H(0)Ω (0)]P(0,0) + P(0,0)Λ(0,0)(1)[H(1)Ω (0) + H(0)Ω (1)]P(0,0)

+P(0,0)Λ(0,0)(0)[H(2)Ω (0) + H(1)Ω (1) + H(0)Ω (2)]P(0,0)

−E(0)μ

∑

ij

(

C(0,1)(2)μi C(0,1)(0)

jμ + 2C(0,1)(1)μi C(0,1)(1)

jμ + C(0,1)(0)μi C(0,1)(2)

jμ

)

−E(1)μ

∑

ij

(

C(0,1)(1)μi C(0,1)(0)

jμ + C(0,1)(0)μi C(0,1)(1)

jμ

)

−E(2)μ

∑

ij

(

C(0,1)(0)μi C(0,1)(0)

jμ − 1)

(14-39)

TheΛ(1) containing terms in the above expressions vanish as they are the products ofthe Λ(1) with the derivative MRCC equations (Eqs. 14-15 and 14-16). Similarly, theΛ(2) containing terms are the MRCC equations multiplied to the �(2) equations andhence, vanish. Analogously, the C(2) and C(2) terms also vanish. With these cancella-tions, the energy second derivative equation reduces to the form which contains onlyundifferentiatedΛ vectors and up to first derivatives of cluster amplitudes and eigen-


vectors. single-root nature of the CVA method, one has to obtain the Λ amplitudesseparately for every state unlike the non-variational response of FSMRCC structure.However, the expensive evaluation of wave-function derivatives for each mode ofperturbation is avoided in CVA-FSMRCC. This feature becomes more prominentwhile obtaining higher order properties like polarizability. Also, the single-root fea-ture makes CVA more attractive for the cases like curve-crossing studies of excitedstates, etc. than the non-variational response method.

14.4. IMPLEMENTATIONS AND RESULTS AND DISCUSSIONS

In the initial implementation of the FSMRCC response for the first order propertiesAjitha et al. used the formulation which involved explicit differentiation method asdiscussed in section III A. of the cluster amplitudes. Using this approach dipolemoments of the OH, OOH, HCOO, CH , SiH and NO radicals [58–60]. However,this is not the efficient way of the calculation of molecular properties in particularhigher order properties. The Z-vector method described in Section 14.3.2 was quitecomplicated in structure and has not been implemented yet for practical applications.On the other hand, Pal and co-workers have implemented most efficient approach inthis article. CVA-FSMRCCSD has been implemented for the first and second orderproperties of the one valence radicals as well and N electron excited states [64–67].Bag et al. have also implemented the third order property i.e. first hyperpolarizabil-ity for the one valence problem [69]. These implementations are presented in thereferences [64–67]. Taking the advantage of SEC, first the (0, 0) sector amplitudesare calculated. Before solving for the valence sector amplitudes, the (HeT(0,0)

)c, in-termediates are calculated and stored these are denoted as H . The closed part ofH, i.e. Hcl is the ground state energy. The open parts of H can be further classifiedinto one body, two body, three body parts and so on. Under the singles and doubles(SD) approximation, only up to three body parts of H contribute to the T (0,1), T (1,0)

and T (1,1) as well as to the Λ(0,1), Λ(1,0) and Λ(1,1) equations. Because of the largedimensions, the three body parts of the H cannot be stored in the hard disk. Theseare, therefore, evaluated in the code as and when required. The one and two-bodyparts of H are calculated only once and are stored. The equations for Λ amplitudesare linear simultaneous equations. However, unlike the T amplitudes Λ amplitudesfollow reverse decoupling. Thus, first equations for the Λ amplitudes of the highestsector are solved, followed by the solution of the lower sectors of the Λ. The Jacobiiterative procedure has been used for the obtaining Λ amplitudes and the clusteramplitudes. The first derivatives of cluster amplitudes are obtained in the similarway using differentiated FSMRCC equations for cluster amplitude derivatives andderivative effective Hamiltonian. Taking the advantage of SEC, the derivative H areobtained and stored in the similar way. The terms arising from ∂ΛΩHeff

∂Ωin which,

there is Λ−Ω contraction, have been merged together, since, the differentiation ofΩ contained in Heff is structurally identical for these terms. The Λ − Ω contrac-tions for these terms is programmed and stored in an array of dimensions of the


effective Hamiltonian, for computational simplicity. To validate the implementationof Lagrange based code for the (1,1) sector Pal and co-workers [66] implementedthe non-relaxed finite field calculations for the first and second order properties.Each Fock space sector starting from (0,0) each sector results were checked withthe analytic code till the desired accuracy.

14.4.1. Ionized and Electron Attached States

As mentioned earlier the lagrangian based constrained response approach was im-plemented for the ionized, electron attached and excited state first and second orderproperties [64–67]. For the one valence problem, the OH, OOH, HCOO, NO, NO2,CH, and SiH radicals were studied. If the RHF of the closed shell anion is consideredas a vacuum, the model space configuration of the radical belongs to the (0,1)/(1,0)sector of the Fock space. For the Hydroxy radical, results of the CVA FSMRCCmethod were compared with the EOMCC non-relaxed as well as Full CI resultsavailable. CC-pVDZ basis used was used for the calculation. The RHF of the ofhydroxide was chosen as a vacuum and removal of the electron from HOO leadsto degenerate doublet 2Π state of the radical. The dipole moment result (0.634 au)was in good agreement with the FCI (0.663 au) as well as EOMCC (0.639 au) [64].However, for polarizabilities no other results we available for the same. Manoharet al. compared analytic results with the non-relaxed finite field calculations. Ana-lytic polarizability diagonal value along the inter-nuclear axis using CVA FSMRCCmethod yields 6.61 au which is in good agreement with the finite field results 6.0 au.Thus, it can be concluded that the effect of relaxation is marginal. The model spaceconfigurations of a radical belonging to this class corresponds to the (1,0) sectorof the Fock space with the RHF of the corresponding cation as a vacuum. Twosystems chosen in this category are CH and SiH radicals. For the dipole momentand polarizabilities of the CH and SiH, Sadlej basis optimized for the properties wasused. For both the radicals CVAFSMRCC results were compared with finite field aswell as UGA-CCSD results[70]. The dipole moment for CH radical was found tobe 0.543 au using analytic FSMRCC, where as finite field FSMRCC was 0.520 authe UGA-CCSD gives 0.535 au. For SiH dipole moment was very small. The valueusing analytic FSMRCC was 0.063 au and UGA-CCSD (0.037 au) [70].

14.4.2. Excited State Dipole Moment and Polarizabilities

The excited state properties of CH+, H2O and Ozone have been computed usinga reference space of one hole one particle excited determinants with respect to therestricted Hartree Fock of the ground state of the molecules as vacuum. The codefor (1,1) FS sector was validated using non-relaxed finite field calculations for CH+molecule [67]. Basis set convergence of the properties was also studied for the CH+molecule. It was observed that both singlet and triplet state dipole moments increasefrom DZ to TZ basis, DZP to TZP basis and so on, but the trend is just opposite


for polarizabilities. It was observed that singlet dipole moments decrease with theaddition of polarization functions, where as triplet dipole moments increase withthe addition of polarization functions. However, polarizabilities of both singlet andtriplet states increase with the addition of polarization functions. The dipole mo-ments and polarizabilities of both the states virtually converged after the additionof a second polarization function. The adiabatic excited state dipole moment andpolarizabilities for the water and ozone were also studied by the method [66].

ACKNOWLEDGMENTS

One of us (SP) acknowledges partial financial assistance from J. C. Bose Fellowshipgrand of DST and Shanti Swarup Bhatnagar (SSB) prize grant of CSIR towards thiswork. Authors also acknowledge Center of Excellence at N.C.L Pune.

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