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.
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.
The traditional state universal multi-reference coupled cluster (SUMRCC) theory uses the Jeziorski-Monkhorst (JM) based Ansatz of the wave operator: Ω = Σ(μ)Ω(μ)|φ(μ)><φ(μ)|, where Ω(μ) = exp(T(μ)) is the cluster representation of the component of Ω inducing virtual excitations from the model function φ(μ). In the first formulations, φ(μ)s were chosen to be single determinants and T(μ)s were defined in terms of spinorbitals. This leads to spin-contamination for the non-singlet cases. In this paper, we propose and implement an explicitly spin-free realization of the SUMRCC theory. This method uses spin-free unitary generators in defining the cluster operators, {T(μ)}, which even at singles-doubles truncation, generates non-commuting cluster operators. We propose the use of normal-ordered exponential parameterization for Ω:Σ(μ){exp(T(μ))}|φ(μ)><φ(μ)|, where {} denotes the normal ordering with respect to a common closed shell vacuum which makes the "direct term" of the SUMRCC equations terminate at the quartic power. We choose our model functions {φ(μ)} as unitary group adapted (UGA) Gel'fand states which is why we call our theory UGA-SUMRCC. In the spirit of the original SUMRCC, we choose exactly the right number of linearly independent cluster operators in {T(μ)} such that no redundancies in the virtual functions {χ(μ)(l)} are involved. Using example applications for electron detached/attached and h-p excited states relative to a closed shell ground state we discuss how to choose the most compact and non-redundant cluster operators. Although there exists a more elaborate spin-adapted JM-like ansatz of Datta and Mukherjee (known as combinatoric open-shell CC (COS-CC), its working equations are more complex. Results are compared with those from COS-CC, equation of motion coupled cluster methods, restricted open-shell Hartree-Fock coupled cluster, and full configuration interaction. We observe that our results are more accurate with respect to most other theories as a result of the use of the cluster expansion structure for our wave operator. Our results are comparable to those from the more involved COS-CC, indicating that our theory captures the most important aspects of physics with a considerably simpler scheme.
The Unitary Group Adapted State Universal Multireference Coupled Cluster (UGA-SUMRCC) theory, recently developed by us (J. Chem. Phys.2012, 137, 074104), contains exactly the right number of linearly independent cluster operators. This avoids any redundancy of the excitation manifold in a way exactly paralleling the traditional spin-orbital based SUMRCC. The choice of the linearly independent cluster operators inducing the same change of orbital occupancy becomes increasingly cumbersome if we go over to the cases of active CSFs with more than two active quasiparticles. In the present development, we explore several aspects of the UGA-SUMRCC theory: (a) The first is a variant where we have deliberately incorporated redundancy of the cluster amplitudes to simplify the working equations and have shown that it can serve as a very good approximation to the parent UGA-SUMRCC theory for states with more than two valence occupancies. This in turn suggests that it could be a useful avenue to pursue for arbitrary mh-np situation since the working equations assume simpler algebraic structure in such cases. (b) The analyses of the aspects of size extensivity are known to involve greater complexity if they involve various reduced density matrices (RDMs), since the RDMs are not size-extensive quantities. We have presented the proof for UGA-SUMRCC starting from equations containing h-p RDMs via a decomposition involving products of size-extensive cumulants and argue that it has relevance for general cases beyond the h-p model spaces. (c) A useful extension of UGA-SUMRCC lies in formulating the theory for direct calculations of energy differences of spectroscopic interest such as excitation energies, ionization potentials, and electron affinities relative to a closed shell ground state, thus providing attractive alternatives to other allied methods such as SAC-CI, CC-LRT, EOM-CC, STEOM-CC, or ADC. This extension, called UGA-based Quasi-Fock MRCC by us, also leads to exact cancellation of common correlation terms between the initial and final states. Taking a cue from the hierarchical development in Fock-space theories but keeping in mind the advantages of a state-universal (equivalently called a valence specific) theory, our formulation proposes a spin-adapted, accurate, and compact scheme for studying such energy differences. Our results demonstrate superior performance of the method as compared to EOM-CC.
We present two efficient and intruder-free methods for treating dynamic correlation on top of general multiconfiguration reference wave functions — including such as obtained by the density matrix renormalization group (DMRG) with large active spaces. The new methods are the second order variant of the recently proposed multireference linearized coupled cluster method (MRLCC) [Sharma, S.; Alavi, A. J. Chem. Phys. 2015, 143, 102815] and of N-electron valence perturbation theory (NEVPT2), with expected accuracies similar to MRCI+Q and (at least) CASPT2, respectively. Great efficiency gains are realized by representing the first order wave function with a combination of internal contraction (IC) and matrix product state perturbation theory (MPSPT). With this combination, only third order reduced density matrices (RDMs) are required. Thus, we obviate the need for calculating (or estimating) RDMs of fourth or higher order; these had so far posed a severe bottleneck for dynamic correlation treatments involving the large active spaces accessible to DMRG. Using several benchmark systems, including first and second row containing small molecules, Cr 2, pentacene, and oxo-Mn(Salen), we show that active spaces containing at least 30 orbitals can be treated using this method. On a single node, MRLCC2 and NEVPT2 calculations can be performed with over 550 and 1100 virtual orbitals, respectively. We also critically examine the errors incurred due to the three sources of errors introduced in the present implementation – calculating second order instead of third order energy corrections, use of internal contraction, and approximations made in the reference wave function due to DMRG.
The purpose of this paper is to introduce a quasiparticle-based multi-reference coupled-cluster (MRCC) approach. The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference (MR) basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. On the basis of these quasiparticles, a generalization of the normal-ordered operator products for the MR case can be introduced as an alternative to the approach of Mukherjee and Kutzelnigg [Recent Prog. Many-Body Theor. 4, 127 (1995); Mukherjee and Kutzelnigg, J. Chem. Phys. 107, 432 (1997)]. Based on the new normal ordering any quasiparticle-based theory can be formulated using the well-known diagram techniques. Beyond the general quasiparticle framework we also present a possible realization of the unitary transformation. The suggested transformation has an exponential form where the parameters, holding exclusively active indices, are defined in a form similar to the wave operator of the unitary coupled-cluster approach. The definition of our quasiparticle-based MRCC approach strictly follows the form of the single-reference coupled-cluster method and retains several of its beneficial properties. Test results for small systems are presented using a pilot implementation of the new approach and compared to those obtained by other MR methods.
Despite tremendous theoretical efforts, corroborated by state-of-the-art computational techniques toward the development of a genuine multireference coupled cluster (MRCC) theory, the search seems to be far from complete. This is due to the fact that a host of non-trivial problems are still present in the process of developing such approaches, and it now appears that it is really an arduous task to retain the simplistic beauty of a typical single-reference approach while formulating a multireference (MR) theory. The chief sources of difficulty in setting up MR theories are (a) preservation of size extensivity (connectedness) of the formalism, (b) the perfectly determined nature of the problem, that is, having an equal number of parameters and equations, (c) conservation of symmetry, (d) avoidance of intruder states, and (e) correlation to a solution of the projected Schrödinger equation.
The recent development of the density matrix renormalization group (DMRG) method in multireference quantum chemistry makes it practical to evaluate static correlation in a large active space, while dynamic correlation provides a critical correction to the DMRG reference for strong-correlated systems and is usually obtained using multireference perturbation (MRPT) or configuration interaction (MRCI) methods with internal contraction (ic) approximation. These methods can use an active space scalable to relatively larger size references than has previously been possible. However, they are still hardly applicable to systems with an active space larger than 30 orbitals and/or a large basis set because of high computation and storage costs of high-order reduced density matrices (RDMs) and the crucial dependence of the MRCI Hamiltonian dimension on the number of virtual orbitals. In this work, we propose a new effective implementation of DMRG-MRCI, in which we use reconstructed CASCI-type configurations from DMRG wave function via the entropy-driving genetic algorithm (EDGA) [Luo et al. J. Chem. Theory Comput. 2017, 13, 4699−4710.] and integrate it with MRCI by an external contraction (ec) scheme. This bypasses the bottleneck of computing high-order RDMs in traditional DMRG dynamic correlation methods with ic approximation, and the number of MRCI configurations is not dependent on the number of virtual orbitals. Therefore, the DMRG-ec-MRCI method is promising for dealing with a larger active space than 30 orbitals and large basis sets. We demonstrate the capability of our DMRG-ec-MRCI method in several benchmark applications, including the evaluation of the potential energy curve of Cr 2, single-triplet gaps of higher n-acene molecules, and the energy of the Eu-BTBP(NO 3 ) 3 complex.