State-specific Multireference Coupled Cluster Theory Research Articles

Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation.

This study examines several techniques to improve the efficiency of the linearized multireference driven similarity renormalization group truncated to one- and two-body operators [MR-LDSRG(2)]. We propose a sequential MR-LDSRG(2) [sq-MR-LDSRG(2)] scheme, in which one-body substitutions are folded exactly into the Hamiltonian. This new approach is combined with density fitting (DF) to reduce the storage cost of two-electron integrals. To further avoid storage of large four-index intermediates, we propose a noninteracting virtual orbital (NIVO) approximation of the Baker-Campbell-Hausdorff series that neglects commutators terms with three and four virtual indices. The NIVO approximation reduces the computational prefactor of the MR-LDSRG(2), bringing it closer to that of coupled cluster with singles and doubles (CCSD). We test the effect of the DF and NIVO approximations on the MR-LDSRG(2) and sq-MR-LDSRG(2) methods by computing properties of eight diatomic molecules. The diatomic constants obtained by DF-sq-MR-LDSRG(2)+NIVO are found to be as accurate as those from the original MR-LDSRG(2) and coupled cluster theory with singles, doubles, and perturbative triples. Finally, we demonstrate that the DF-sq-MR-LDSRG(2)+NIVO scheme can be applied to chemical systems with more than 550 basis functions by computing the automerization energy of cyclobutadiene with a quintuple-ζ basis set. The predicted automerization energy is found to be similar to the value computed with Mukherjee's state-specific multireference coupled cluster theory with singles and doubles.

Relativistic state-specific multireference coupled cluster theory description for bond-breaking energy surfaces.

A four-component (4c) relativistic state specific multireference coupled cluster (4c-SSMRCC) method has been developed and applied to compute the ground state spectroscopic constants of Ag2, Cu2, Au2, and I2. The reference functions used in these calculations are obtained using computationally inexpensive improved virtual orbital-complete active space configuration interaction scheme. Rigorous size-extensivity and insensitivity towards the intruder state problem make our method an interesting choice for the calculation of the dissociation energy surface. To the best of our knowledge, this study is the first implementation of the SSMRCC within the relativistic framework. The overall agreement of our results, employing the smallest model space, with both theoretical and experimental reference values indicates that the 4c-SSMRCC method can be fruitfully used to describe electronic structures and associated properties of systems containing heavy elements. We observe a relativistic bond stabilization for the coinage metal dimers while the I-I bond is weakened by the relativistic effects.

Discontinuities-free complete-active-space state-specific multi-reference coupled cluster theory for describing bond stretching and dissociation.

The earlier proposed multi-reference state-specific coupled-cluster theory with the complete active space reference [CASCC; Lyakh et al., J. Chem. Phys. 122, 024108 (2005)] suffered from a problem of energy discontinuities when the formal reference state was changing in the calculation of the potential energy curve (PEC). A simple remedy to the discontinuity problem is found and is presented in this work. It involves using natural complete active space self-consistent field active orbitals in the complete active space coupled-cluster calculations. The approach gives smooth PECs for different types of dissociation problems, as illustrated in the calculations of the dissociation of the single bond in the hydrogen fluorine molecule and of the symmetric double-bond dissociation in the water molecule.

Unitary group adapted state specific multireference perturbation theory: Formulation and pilot applications.

We present here a comprehensive account of the formulation and pilot applications of the second-order perturbative analogue of the recently proposed unitary group adapted state-specific multireference coupled cluster theory (UGA-SSMRCC), which we call as the UGA-SSMRPT2. We also discuss the essential similarities and differences between the UGA-SSMRPT2 and the allied SA-SSMRPT2. Our theory, like its parent UGA-SSMRCC formalism, is size-extensive. However, because of the noninvariance of the theory with respect to the transformation among the active orbitals, it requires the use of localized orbitals to ensure size-consistency. We have demonstrated the performance of the formalism with a set of pilot applications, exploring (a) the accuracy of the potential energy surface (PES) of a set of small prototypical difficult molecules in their various low-lying states, using natural, pseudocanonical and localized orbitals and compared the respective nonparallelity errors (NPE) and the mean average deviations (MAD) vis-a-vis the full CI results with the same basis; (b) the efficacy of localized active orbitals to ensure and demonstrate manifest size-consistency with respect to fragmentation. We found that natural orbitals lead to the best overall PES, as evidenced by the NPE and MAD values. The MRMP2 results for individual states and of the MCQDPT2 for multiple states displaying avoided curve crossings are uniformly poorer as compared with the UGA-SSMRPT2 results. The striking aspect of the size-consistency check is the complete insensitivity of the sum of fragment energies with given fragment spin-multiplicities, which are obtained as the asymptotic limit of super-molecules with different coupled spins.

Erratum: “Unitary group adapted state-specific multi-reference coupled cluster theory: Formulation and pilot numerical applications” [J. Chem. Phys. 137, 024105 (2012)

Erratum: “Unitary group adapted state-specific multi-reference coupled cluster theory: Formulation and pilot numerical applications” [J. Chem. Phys. 137, 024105 (2012)

Erratum: “Inactive excitations in Mukherjee's state-specific multireference coupled cluster theory treated with internal contraction: Development and applications” [J. Chem. Phys. 136, 164104 (2012)

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Linear-response theory for Mukherjee's multireference coupled-cluster method: Static and dynamic polarizabilities

The formalism of response theory is applied to derive expressions for static and dynamic polarizabilities within the state-specific multireference coupled-cluster theory suggested by Mukherjee and co-workers (Mk-MRCC) [J. Chem. Phys. 110, 6171 (1998)]. We show that the redundancy problem inherent to Mk-MRCC theory gives rise to spurious poles in the Mk-MRCC response functions, which hampers the reliable calculation of dynamic polarizabilities. Furthermore, we demonstrate that in the case of a symmetry-breaking perturbation a working response theory is obtained only if certain internal excitations are included in the responses of the cluster amplitudes. Exemplary calculations within the singles and doubles approximation (Mk-MRCCSD) are carried out on aryne compounds to illustrate the impact of a multireference ansatz on the polarizability.

Unitary group adapted state-specific multi-reference coupled cluster theory: Formulation and pilot numerical applications

We present the formulation and the implementation of a spin-free state-specific multi-reference coupled cluster (SSMRCC) theory, realized via the unitary group adapted (UGA) approach, using a multi-exponential type of cluster expansion of the wave-operator Ω. The cluster operators are defined in terms of spin-free unitary generators, and normal ordered exponential parametrization is utilized for cluster expansion instead of pure exponentials. Our Ansatz for Ω is a natural spin-free extension of the spinorbital based Jeziorski-Monkhorst (JM) Ansatz. The normal ordered cluster Ansatz for Ω results in a terminating series of the direct term of the MRCC equations, and it uses ordinary Wick algebra to generate the working equations in a straightforward manner. We call our formulation as UGA-SSMRCC theory. Just as in the case of the spinorbital based SSMRCC theory, there are redundancies in the cluster operators, which are exploited to ensure size-extensivity and avoidance of intruders via suitable sufficiency conditions. Although there already exists in the literature a spin-free JM-like Ansatz, introduced by Datta and Mukherjee, its structure is considerably more complex than ours. The UGA-SSMRCC offers an easier access to spin-free MRCC formulation as compared to the Datta-Mukherjee Ansatz, which at the same time provides with quite accurate description of electron correlation. We will demonstrate the efficacy of the UGA-SSMRCC formulation with a set of numerical results. For non-singlet cases, there is pronounced M(s) dependence of the energy for the spinorbital based SSMRCC results. Although M(s) = 1 results are closer to full configuration interaction (FCI), the extent of spin-contamination is more. In most of the cases, our UGA-SSMRCC results are closer to FCI than the spinorbital M(s) = 0 results.

Inactive excitations in Mukherjee's state-specific multireference coupled cluster theory treated with internal contraction: Development and applications

One generic difficulty of most state-specific many-body formalisms using the Jeziorski-Monkhorst ansatz: ψ = Σ(μ)exp(T(μ))|φ(μ)>c(μ) for the wave-operators is the large number of redundant cluster amplitudes. The number of cluster amplitudes up to a given rank is many more in number compared to the dimension of the Hilbert Space spanned by the virtual functions of up to the same rank of excitations. At the same time, all inactive excitations--though linearly independent--are far too numerous. It is well known from the success of the contracted multi-reference configuration interaction (MRCI(SD)) that, at least for the inactive double excitations, their model space dependence (μ-dependence) is weak. Considerable simplifications can thus be obtained by using a partially internally contracted description, which uses the physically appealing approximation of taking the inactive excitations T(i) to be independent of the model space labels (μ-independent). We propose and implement in this paper such a formalism with internal contractions for inactive excitations (ICI) within Mukherjee's state-specific multi-reference coupled cluster theory (SS-MRCC) framework (referred to from now on as the ICI-SS-MRCC). To the extent the μ-independence of T(i) is valid, we expect the ICI-SS-MRCC to retain the conceptual advantages of size-extensivity yet using a drastically reduced number of cluster amplitudes without sacrificing accuracy. Moreover, greater coupling is achieved between the virtual functions reached by inactive excitations as a result of the internal contraction while retaining the original coupling term for the μ-dependent excitations akin to the parent theory. Another major advantage of the ICI-SS-MRCC, unlike the other analogous internally contracted theories, such as IC-MRCISD, CASPT2, or MRMP2, is that it can use relaxed coefficients for the model functions. However, at the same time it employs projection manifolds for the virtuals obtained from inactive n hole-n particle (nh-np) excitations on the entire reference function containing relaxed model space coefficients. The performance of the method has been assessed by applying it to compute the potential energy surfaces of the prototypical H(4); to the torsional potential energy barrier for the cis-trans isomerism in C(2)H(4) as well as that of N(2)H(2), automerization of cyclobutadiene, single point energy calculation of CH(2), SiH(2), and comparing them against the SS-MRCC results, benchmark full CI results, wherever available and those from the allied MR formalisms. Our findings are very much reminiscent of the experience gained from the IC-MRCISD method.

Superior performance of Mukherjee’s state-specific multi-reference coupled-cluster theory at the singles and doubles truncation scheme with localized active orbitals

The state-specific multi-reference coupled-cluster (SS-MRCC) theory of Mukherjee et al., in its singles and doubles truncation scheme (SS-MRCCSD), misses important couplings between the virtual functions reached by single and double excitations from different model functions. Since the SS-MRCC theory is not invariant with respect to the transformations among the active orbitals, the results are dependent on the active orbitals chosen. We demonstrate in this paper with results for potential energy curves for several example molecules involving single and multiple bond dissociation that the performance of SS-MRCCSD is significantly improved if localized active orbitals are used. The improvement is remarkable both in terms of the non-parallelity error and the magnitude of correlation energy recovered vis-a-vis the full configuration interaction results with the same basis set. The results bolster our claim that SS-MRCCSD with localized orbitals is an accurate general theory for potential energy surfaces.

Ab initio wavefunctions in bioinorganic chemistry: More than a succès d’estime?

Ab initio wavefunctions in bioinorganic chemistry: More than a succès d’estime?

The spin-free analogue of Mukherjee's state-specific multireference coupled cluster theory

In this paper, we develop a rigorously spin-adapted version of Mukherjee's state-specific multireference coupled cluster theory (SS-MRCC, also known as Mk-MRCC) [U. S. Mahapatra, B. Datta, and D. Mukherjee, J. Chem. Phys. 110, 6171 (1999)] for reference spaces comprising open-shell configurations. The principal features of our approach are as follows: (1) The wave operator Ω is written as Ω = ∑(μ)Ω(μ)|φ(μ)>c(μ), where {φ(μ)} is the set of configuration state functions spanning a complete active space. (2) In contrast to the Jeziorski-Monkhorst Ansatz in spin-orbital basis, we write Ω(μ) as a power series expansion of cluster operators R(μ) defined in terms of spin-free unitary generators. (3) The operators R(μ) are either closed-shell-like n hole-n particle excitations (denoted as T(μ)) or they involve valence (active) destruction operators (denoted as S(μ)); these latter type of operators can have active-active scatterings, which can also carry the same active orbital labels (such S(μ)'s are called to have spectator excitations). (4) To simulate multiple excitations involving powers of cluster operators, we allow the S(μ)'s carrying the same active orbital labels to contract among themselves. (5) We exclude S(μ)'s with direct spectator scatterings. (6) Most crucially, the factors associated with contracted composites are chosen as the inverse of the number of ways the S(μ)'s can be joined among one another leading to the same excitation. The factors introduced in (6) have been called the automorphic factors by us. One principal thrust of this paper is to show that the use of the automorphic factors imparts a remarkable simplicity to the final amplitude equations: the equations consist of terms that are at most quartic in cluster amplitudes, barring only a few. In close analogy to the Mk-MRCC theory, the inherent linear dependence of the cluster amplitudes leading to redundancy is resolved by invoking sufficiency conditions, which are exact spin-free analogues of the spin-orbital based Mk-MRCC theory. This leads to manifest size-extensivity and an intruder-free formulation. Our formalism provides a relaxed description of the nondynamical correlation in presence of dynamical correlation. Pilot numerical applications to doublet systems, e.g., potential energy surfaces for the first two excited (2)A' states of asymmetric H(2)S(+) ion and the ground (2)Σ(+)state of BeH radical are presented to assess the viability of our formalism over an wide range of nuclear geometries and the manifest avoidance of intruder state problem.

State-specific multireference coupled-cluster theory of molecular electronic excited states

A state-specific coupled cluster (CC) theory with the CAS (complete active space) reference (CASCC) and based on an approach that employs single reference determinant (so-called “formal reference” determinant) in the expression for the wave function has been developed to study electronic excited states with different spatial and spin symmetries. The formal reference determinant is used to generate the reference wave function in the form of a linear combination of the CAS determinants contracted to configurations with the spin and spatial symmetries of the target state. Such properly symmetrized multideterminantal reference provides the zero-order description of the state under consideration. To that reference an exponential CC operator is applied to describe the dynamic electron correlation effects in the CASCC wave function of the considered state. All necessary equations for the CASCC energy and configurational amplitudes are generated using an automated computer-based method which constructs all necessary coupled cluster diagrams for any arbitrary level of the electronic excitation.

Inclusion of selected higher excitations involving active orbitals in the state-specific multireference coupled-cluster theory

The parent state-specific multireference coupled-cluster (SS-MRCC) theory proposed by Mukherjee et al. [J. Chem. Phys. 110, 6171 (1999)], though rigorously size-extensive and also size-consistent with localized orbitals, has some deficiencies in the minimal truncation scheme, viz. at the singles and doubles (SD) level (SS-MRCCSD). SS-MRCCSD does not involve the direct coupling of all the model functions with a given virtual function belonging to the uncontracted multiconfiguration CISD space. It also does not involve, even in the linear power of a cluster operator T(μ), the direct coupling of the virtual functions χ(l(μ)), which are up to doubly excited with respect to a model function φ(μ) to the other virtual functions of the MRCISD space which can be generated by triple and quadruple excitations from φ(μ). We argue that inclusion of a selection of triples and quadruples involving at most two inactive orbital excitations from every φ(μ) would ameliorate the shortcoming of the incomplete coupling of the triply and quadruply excited virtual functions which can couple with the singly and doubly excited ones. This extended ansatz for our SS-MRCC theory, to be called SS-MRCCSDtq by us, would still miss the direct coupling of the manifold of the model functions {φ(λ),λ ≠ μ} to singly and doubly excited virtual functions. However, this effect is expected to be less significant than the lack of the more complete virtual space couplings, these functions being many more numerous, suggesting the new methods to be significantly improved schemes. Excellent results on the potential energy surfaces of small molecules involving single, double, and triple bond dissociation bear out our expectations fully.

Potential energy surface studies via a single root multireference coupled cluster theory

We have employed complete active space based single root multireference coupled cluster method (the resulting method is referred to by the acronym sr-MRCC) to compute the potential energy surfaces (PESs) of some well studied "protypical model" systems for which a highly accurate and reliable database is available for comparison. As that of state-specific theory, the sr-MRCC approach focuses and correlates one state while using a multiconfigurational reference and thus it naturally avoids intruder states. The present method is structurally different from the well known state specific multireference coupled cluster (SS-MRCC) method introduced by Mahapatra et al. [Mol. Phys. 94, 157 (1998)]. As that of the SS-MRCC theory, the present method is also based on the Jeziorski-Monkhorst ansatz where a different exponential cluster operator exp(T(mu)) acts on its corresponding model function phi(mu). The final cluster finding equations contain coupling between the cluster operators for all the mu, which are mainly responsible to prove the extensivity of both the cluster amplitudes and the energy. The present sr-MRCC theory is size-extensive and size-consistent when localized orbitals are used. The systems considered here exhibit varying degrees of degeneracy at different regions of PES. The treatment of these systems via traditional effective Hamiltonian based methods suffers from divergence problems in the iterative solution of the CC equations (the issue termed as "intruder state"). The sr-MRCC results lie closer to the ones obtained by the SS-MRCC method for these systems. To judge the efficacy of the present method, we have compared our results with other previously published theoretical estimations, which clearly indicate that the present method is reliable in studying the dissociation PES of states plagued by electronic degeneracy as well as notorious intruder effects. The highly satisfactory performance of the sr-MRCC method, vis-a-vis the other sophisticated methods, in describing the lowest and the first excited singlet states of BeH(2) at points of high degeneracy is noticeable.

Insights into the orbital invariance problem in state-specific multireference coupled cluster theory

In this communication we report the results of our studies on the orbital invariance properties of the state-specific multireference coupled cluster approach suggested by Mukherjee and co-workers (Mk-MRCC). In particular, we have gathered numerical evidence to show that even when the linear excitation manifold is modified in order to span the same space for each reference, the resulting method is not orbital invariant. In order to test this conjecture we have proposed a new truncation scheme (Mk-MRCCSDtq) which, in addition to full single and double excitations, contains partial triple and quadruple excitations. For a reference space generated by all possible combinations of two electrons in two orbitals, the linear excitation manifold of Mk-MRCCSDtq spans the same set for each reference determinant. Mk-MRCCSDtq is found to lack energy invariance for rotations among active molecular orbitals but it is less sensitive to orbital rotations than the conventional scheme which includes only singles and doubles (Mk-MRCCSD). Nevertheless, Mk-MRCCSDtq is a very accurate method, superior with respect to multireference configuration interaction approaches, and competitive with the active-space coupled cluster method and the MRexpT ansatz.

Perturbative triples corrections in state-specific multireference coupled cluster theory

We formulated and implemented a perturbative triples correction for the state-specific multireference coupled cluster approach with singles and doubles suggested by Mukherjee and co-workers, Mk-MRCCSD [Mol. Phys. 94, 157 (1998)]. Our derivation of the energy correction [Mk-MRCCSD(T)] is based on a constrained search for stationary points of the Mk-MRCC energy functional together with a perturbative expansion with respect to the appearing triples cluster operator. The Lambda-Mk-MRCCSD(T) approach derived in this way consists in (1) a correction to the off-diagonal matrix elements of the effective Hamiltonian which is unique to coupled cluster methods based on the Jeziorski-Monkhorst ansatz, and (2) an asymmetric energy correction to the diagonal elements of the effective Hamiltonian. The Mk-MRCCSD(T) correction is obtained from the Lambda-Mk-MRCCSD(T) method by approximating the singles and doubles Lagrange multipliers with the corresponding cluster amplitudes. We investigate the performance of the Mk-MRCCSD(T) method by applying it to the potential energy curve of the BeH(2) model and F(2) and the geometry and harmonic vibrational frequencies of ozone. Computation of the energy difference between the mono- and bicyclic forms of the 2,6-pyridyne diradical illustrates the potential of Mk-MRCCSD(T) as a tool for the study of realistic chemical problems requiring multireference zeroth-order wave functions.

Triple excitations in state-specific multireference coupled cluster theory: Application of Mk-MRCCSDT and Mk-MRCCSDT-n methods to model systems

We report the first implementation with correct scaling of the Mukherjee multireference coupled cluster method with singles, doubles, and approximate iterative triples (Mk-MRCCSDT-n, n=1a,1b,2,3) as well as full triples (Mk-MRCCSDT). These methods were applied to the classic H4, P4, BeH(2), and H8 model systems to assess the ability of the Mk-MRCCSDT-n schemes to accurately account for triple excitations. In all model systems the inclusion of triples via the various Mk-MRCCSDT-n approaches greatly reduces the nonparallelism error (NPE) and the mean nonparallelism derivative diagnostics for the potential energy curves, recovering between 59% and 73% of the full triples effect on average. The most complete triples approximation, Mk-MRCCSDT-3, exhibits the best average performance, reducing the mean NPE to below 0.6 mE(h), compared to 1.4 mE(h) for Mk-MRCCSD. Both linear and quadratic truncations of the Mk-MRCC triples coupling terms are viable simplifications producing no significant errors. If the off-diagonal parts of the occupied-occupied and virtual-virtual blocks of the Fock matrices are ignored, the storage of the triples amplitudes is no longer required for the Mk-MRCCSDT-n methods introduced here. This proves to be an effective approximation that gives results almost indistinguishable from those derived from full consideration of the Fock matrices.

Development and pilot molecular applications of the uncoupled state-specific MRCC (UC-SS-MRCC) theory

In this paper, we propose and apply an uncoupled approximation to the rigorous, size-extensive state-specific multireference coupled cluster theory (SS-MRCC), developed earlier by Mukherjee and coworkers [U.S. Mahapatra, B. Datta, D. Mukherjee, J. Chem. Phys. 110 (1999) 6171]. Both the parent formulation and the uncoupled approximant use the Jeziorski–Monkhorst Ansatz: Ω = ∑ μ exp ( T μ ) | ϕ μ 〉 〈 ϕ μ | involving a different cluster operator exp( T μ ) acting on its corresponding model function ϕ μ . The approximant presented in this paper builds on a preliminary formulation presented by us recently [THEOCHEM 79 (2006) 771]. The working equations of the SS-MRCC, following a set of physically motivated sufficiency conditions, are characterized by certain projection amplitudes connecting each ϕ μ to each virtual function χ l, which is equated to zero. These equations have two types of terms: a ‘direct’ term where for each ϕ μ only powers of T μ appear, and a ‘coupling’ term where T ν ( ν ≠ μ) also figures. In the uncoupled SS-MRCC approximant (UC-SS-MRCC) we explore the possibility of avoiding the coupling term by approximating the coupling matrix element 〈 χ l∣exp(− T μ )exp( T ν )∣ ϕ μ 〉 by 〈 χ l | exp ( - T μ ) exp ( T μ ′ ( ν ) ) | ϕ μ 〉 , where T μ ′ ( ν ) s form the subset of the components of T μ operators which has the same excitation structure of those operators T ν that give nontrivial contributions by their actions on ϕ μ . Absence of the coupling terms leads to considerable simplifications of the working equations. Pilot molecular applications of the UC-SS-MRCC theory to the singlet states of CH 2 and SiH 2, which possess strongly multi-reference character and to the potential energy surface of the ground state of Li 2 indicate the extreme closeness of the computed energies with those from the rigorous but more involved parent SS-MRCC theory. This indicates the potentiality and the efficacy of the UC-SS-MRCC approximation.

Potential energy surface of the electron excited states in the state-specific multi-reference coupled cluster theory. Hydrogen fluoride dissociation

The recently proposed state-specific multi-reference coupled cluster (SSMRCC) theory with the complete active space (CAS) reference has been tested in calculations of the potential energy surfaces of electronically-excited states. The algorithm for the method is derived using the computer-based automated approach for generating the coupled cluster diagrams and the energy and amplitude equations. The active space for the calculation of the target state is constructed using natural orbital expansion obtained from the configuration interaction method with single and double excitations (CISD) one-particle density matrix. The spin–orbitals for the CASCC calculation are obtained using the complete active space self consistent field (CASSCF) method. The numerical example shown concerns several electronically excited states with different spatial and spin symmetries of the hydrogen fluoride molecule. The calculations are performed at several internuclear distances and compared with the results obtained using other methods such as the full configuration interaction (FCI) method, the equation-of-motion (EOM) method, and the multi-reference perturbation theory (MRPT). The comparison shows that the CASCC results are better than the EOM and MRPT results.