Formulation and implementation of a unitary group adapted state universal multi-reference coupled cluster (UGA-SUMRCC) theory: Excited and ionized state energies
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 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.
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.
Some recent advances in the area of multi-reference coupled-cluster theory of the state-universal type are overviewed. An emphasis is placed on the following new developments: (i) the idea of combining the state-universal multi-reference coupled-cluster singles and doubles method (SUMRCCSD) with the multi-reference many-body perturbation theory (MRMBPT), in which cluster amplitudes of the SUMRCCSD formalism that carry only core and virtual orbital indices are replaced by their first-order MRMBPT estimates; and (ii) the idea of combining the recently proposed method of moments of coupled-cluster equations with the SUMRCC formalism. It is demonstrated that the new SUMRCCSD(1) method, obtained by approximating the SUMRCCSD cluster amplitudes carrying only core and virtual orbital indices by their first-order MRMBPT values, provides the results that are comparable to those obtained with the complete SUMRCCSD approach.
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.
Coupled cluster (CC) theory is the “gold standard” of molecular electronic structure theory, yet the full arc of the subject—from its variational origins in Hartree–Fock theory, through the exponential ansatz and its diagrammatic machinery, to its modern reformulation as a system of polynomial equations on an algebraic variety, its multireference generalizations, and its emerging reincarnation as a quantum circuit—is rarely presented as a single connected narrative. This review builds that narrative completely from first principles and carries it through to today’s research frontier. Starting from the electronic Schrödinger equation, we derive Hartree–Fock theory, second quantization, and Wick’s theorem; construct the coupled cluster singles-and-doubles (CCSD) equations via the similarity-transformed Hamiltonian; and show how these same equations reappear, verbatim, as a truncated eigenvalue problem on a projective variety generalizing the Grassmannian, with a computable CC degree controlling the number of solutions. We then develop the perturbative triples correction CCSD(T), the linear-response equationof-motion (EOM-CC) framework for excited, ionized, and electron-attached states, the effective-Hamiltonian formalism underlying multireference coupled cluster theory and its notorious intruder-state problem, and the unitary coupled cluster (UCC) ansatz that restores the variational principle at the cost of an infinite operator expansion—tamed, on a quantum computer, by Trotterization and the variational quantum eigensolver. We close with a synthesis of what is rigorously known about the resulting nonlinear equations, a map of directions deliberately left out of scope, and an assessment of the open question of quantum advantage in this domain. Every claim is stated formally, with a full statement and proof or proof sketch, and cited to its origin; the computational cost of every method is compared explicitly throughout.