Reduced Density Matrix Cumulants: The Combinatorics of Size-Consistency and Generalized Normal Ordering

Renormalized Internally Contracted Multireference Coupled Cluster with Perturbative Triples

In this work, we combine the many-body formulation of the internally contracted multireference coupled cluster (ic-MRCC) method with Evangelista's multireference formulation of the driven similarity renormalization group (DSRG). The DSRG method can be viewed as a unitary multireference coupled cluster theory, which renormalizes the amplitudes based on a flow equation approach to eliminate numerical instabilities. We extend this approach by demonstrating that the unitary flow equation approach can be adapted for nonunitary transformations, rationalizing the renormalization of ic-MRCC amplitudes. We denote the new approach, the renormalized ic-MRCC (ric-MRCC) method. To achieve high accuracy with a reasonable computational cost, we introduce a new approximation to the Baker-Campbell-Hausdorff expansion. We fully consider the linear commutator while approximating the quadratic commutator, for which we neglect specific contractions involving amplitudes with active indices. Moreover, we introduce approximate perturbative triples to obtain the ric-MRCCSD[T] method. We demonstrate the accuracy of our approaches in comparison to advanced multireference methods for the potential energy curves of H8, F2, H2O, N2, and Cr2. Additionally, we show that ric-MRCCSD and ric-MRCSSD[T] match the accuracy of CCSD(T) for evaluating spectroscopic constants and of full configuration interaction energies for a set of small molecules.

On the notion of strong correlation in electronic structure theory

Strong correlation has been said to have many faces, and appears to have many synonyms of questionable suitability. In this work we aim not to define the term once and for all, but to highlight one possibility that is both rigorously defined and physically transparent, and remains so in reference to molecules and quantum lattice models. We survey both molecular examples - hydrogen systems (Hn, n = 2, 4, 6), Be2, H-He-H, and benzene - and the half-filled Hubbard model over a range of correlation regimes. Various quantities are examined including the extent of spin symmetry breaking in correlated single-reference wave functions, energetic ratios inspired by the Hubbard model and the Virial theorem, and metrics derived from the one- and two-electron reduced density matrices (RDMs). The trace and the square norm of the cumulant of the two-electron reduced density matrix capture what may well be defined as strong correlation. Accordingly, strong correlation is understood as a statistical dependence between two electrons, and is distinct from the concepts of "correlation energy" and more general than entanglement quantities that require a partitioning of a quantum system into distinguishable subspaces. This work enables us to build a bridge between a rigorous and quantifiable regime of strong electron correlation and more familiar chemical concepts such as anti-aromaticity in the context of Baird's rule.

Separation of metric in Wick's theorem

In quantum chemistry, Wick's theorem is an important tool to reduce products of fermionic creation and annihilation operators. It is especially useful in computations employing reference states. The original theorem has been generalized to tackle multiconfigurational wave functions or nonorthogonal orbitals. One particular issue of the resulting structure is that the metric and density matrices are intertwined despite their different origin. Here, an alternative, rather general tensorial formulation of Wick's theorem is proposed. The main difference is the separation of the metric-the coefficients at normal-ordered operators become products of an n-electron density matrix element and the Pfaffian of a matrix formed by orbital overlaps. Different properties of the formalism are discussed, including the use of density cumulants, the particle-hole symmetry, and applications to transition density matrices, i.e., the case of different bra and ket reference states. The metric-separated version of Wick's theorem provides a platform for the derivation of various quantum chemical methods, especially those complicated by non-trivial reference states and nonorthogonality issues.

Automatic derivation of many-body theories based on general Fermi vacua

This paper describes Wick&d, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated references and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. Wick&d employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since Wick&d can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of  Wick&d, we use it to evaluate the single-reference coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multireference perturbation theory.

Cumulants as the variables of density cumulant theory: A path to Hermitian triples

We study the combination of orbital-optimized density cumulant theory and a new parameterization of reduced density matrices in which the variables are the particle-hole cumulant elements. We call this combination OλDCT. We find that this new Ansatz solves problems identified in the previous unitary coupled cluster Ansatz for density cumulant theory: the theory is now free of near-zero denominators between occupied and virtual blocks, can correctly describe the dissociation of H₂, and is rigorously size-extensive. In addition, the new Ansatz has fewer terms than the previous unitary Ansatz, and the optimal orbitals delivered by the exact theory are the natural orbitals. Numerical studies on systems amenable to full configuration interaction show that the amplitudes from the previous ODC-12 method approximate the exact amplitudes predicted by this Ansatz. Studies on equilibrium properties of diatomic molecules show that even with the new Ansatz, it is necessary to include triples to improve the accuracy of the method compared to orbital-optimized linearized coupled cluster doubles. With a simple iterative triples correction, OλDCT outperforms other orbital-optimized methods truncated at comparable levels in the amplitudes, as well as coupled cluster single and doubles with perturbative triples [CCSD(T)]. By adding four more terms to the cumulant parameterization, OλDCT outperforms CCSDT while having the same O(V⁵O³) scaling.

Analytic Energy Gradients for the Driven Similarity Renormalization Group Multireference Second-Order Perturbation Theory

We derive analytic energy gradients of the driven similarity renormalization group (DSRG) multireference second-order perturbation theory (MRPT2) using the method of Lagrange multipliers. In the Lagrangian, we impose constraints for a complete-active-space self-consistent-field reference wave function and the semicanonical orthonormal molecular orbitals. Solving for the associated Lagrange multipliers is found to share the same asymptotic scaling of a single DSRG-MRPT2 energy computation. A pilot implementation of the DSRG-MRPT2 analytic gradients is used to optimize the geometry of the singlet and triplet states of p-benzyne. The equilibrium bond lengths and angles are similar to those computed via other MRPT2s and Mukherjee's multireference coupled cluster theory. An approximate DSRG-MRPT2 method that neglects the contributions of the three-body density cumulant is found to introduce negligible errors in the geometry of p-benzyne, lending itself to a promising low-cost approach for molecular geometry optimizations using large active spaces.

Dressing of Vertices by Cumulants in Multi-Reference Coupled Cluster

A new scheme is introduced in Multi-Reference (MR) Coupled Cluster (CC) based on the MR Generalized Normal Ordering (MRGNO) and the corresponding MR Generalized Wick Theorem (MRGWT) of Kutzelnigg and Mukherjee. The key element is the identification of a structure in MRGWT generated terms, facilitated by Goldstone diagram techniques. This allows for bundling the many terms of the MRGWT expansion and introduces a hierarchy in the equations that can be harnessed in devising approximations. One- and two-particle interaction vertices are found to be uniformly substituted for their counterpart dressed by density cumulants. This allows for a straightforward rewriting of the ordinary energy expression of CC with interaction dressed (id) one- and two-particle terms and reveals the presence of three- and higher-rank dressed interaction vertices too. Cumulants appearing out of dressed interaction vertices contribute to the amplitude equations and can be interpreted to have an amplitude dressing role. Dressing of one- and two-particle interaction vertices is the most straightforward to implement and does not hinder computational feasibility, provided that the reference function involves strictly limited active space sizes. The Generalized Valence Bond wave function, underlying pilot numerical tests, fulfills this criterion. Results on multiple bond breaking scenarios point to the need of stepping beyond one- and two-particle id. An extremely simple version of incorporating amplitude dressing in addition to one- and two-particle id is seen to cure the potential energy curves remarkably, stimulating further investigations along this line.

Challenges for variational reduced-density-matrix theory with three-particle N-representability conditions

The direct variational optimization of the two-electron reduced density matrix (2RDM) can provide a reference-independent description of the electronic structure of many-electron systems that naturally capture strong or nondynamic correlation effects. Such variational 2RDM approaches can often provide a highly accurate description of strong electron correlation, provided that the 2RDMs satisfy at least partial three-particle N-representability conditions (e.g., the T2 condition). However, recent benchmark calculations on hydrogen clusters [N. H. Stair and F. A. Evangelista, J. Chem. Phys. 153, 104108 (2020)] suggest that even the T2 condition leads to unacceptably inaccurate results in the case of two- and three-dimensional clusters. We demonstrate that these failures persist under the application of full three-particle N-representability conditions (3POS). A variety of correlation metrics are explored in order to identify regimes under which 3POS calculations become unreliable, and we find that the relative squared magnitudes of the cumulant three- and two-particle reduced density matrices correlate reasonably well with the energy error in these systems. However, calculations on other molecular systems reveal that this metric is not a universal indicator for the reliability of the reduced-density-matrix theory with 3POS conditions.

Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics

We report a spin-free formulation of the multireference (MR) driven similarity renormalization group (DSRG) based on the ensemble normal ordering of Mukherjee and Kutzelnigg [J. Chem. Phys. 107, 432 (1997)]. This ensemble averages over all microstates of a given total spin quantum number, and therefore, it is invariant with respect to SU(2) transformations. As such, all equations may be reformulated in terms of spin-free quantities and they closely resemble those of spin-adapted closed-shell coupled cluster (CC) theory. The current implementation is used to assess the accuracy of various truncated MR-DSRG methods (perturbation theory up to third order and iterative methods with single and double excitations) in computing the constants of 33 first-row diatomic molecules. The accuracy trends for these first-row diatomics are consistent with our previous benchmark on a small subset of closed-shell diatomic molecules. We then present the first MR-DSRG application on transition-metal complexes by computing the spin splittings of the [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ molecules. A focal point analysis (FPA) shows that third-order perturbative corrections are essential to achieve reasonably converged energetics. The FPA based on the linearized MR-DSRG theory with one- and two-body operators and up to a quintuple-ζ basis set predicts the spin splittings of [Fe(H₂O)₆]²⁺ and [Fe(NH₃)₆]²⁺ to be -35.7 and -17.1 kcal mol^-1, respectively, showing good agreement with the results of local CC theory with singles, doubles, and perturbative triples.

Assessing the orbital-optimized unitary Ansatz for density cumulant theory

The previously proposed Ansatz for density cumulant theory that combines orbital-optimization and a parameterization of the 2-electron reduced density matrix cumulant in terms of unitary coupled cluster amplitudes (OUDCT) is carefully examined. Formally, we elucidate the relationship between OUDCT and orbital-optimized unitary coupled cluster theory and show the existence of near-zero denominators in the stationarity conditions for both the exact and some approximate OUDCT methods. We implement methods of the OUDCT Ansatz restricted to double excitations for numerical study, up to the fifth commutator in the Baker-Campbell-Hausdorff expansion. We find that methods derived from the Ansatz beyond the previously known ODC-12 method tend to be less accurate for equilibrium properties and less reliable when attempting to describe H₂ dissociation. New developments are needed to formulate more accurate density cumulant theory variants.