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arXiv:2005.10132 · The Journal of Chemical Physics

Connected three-body terms in single-reference unitary many-body theories: Iterative and perturbative approximations

Chenyang Li, Francesco A. Evangelista

Published Jun 18, 2020• 12 citations
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Abstract

This work introduces various approaches to include connected three-body terms in unitary many-body theories, focusing on the driven similarity renormalization group (DSRG). Starting from the least approximate method-the linearized DSRG truncated to one-, two-, and three-body operators [LDSRG(3)]-we develop several approximate LDSRG(3) models with reduced computational cost. Through a perturbative analysis, we motivate a family of iterative LDSRG(3)-n and -n' (n = 1, 2, 3, 4) methods that contain a subset of the LDSRG(3) diagrams. Among these variants, the LDSRG(3)-2 scheme has the same computational complexity of coupled cluster theory with singles, doubles, and triples (CCSDT), but it outperforms CCSDT in the accuracy of the predicted correlation energies. We also propose and implement two perturbative triples corrections based on the linearized DSRG truncated to one- and two-body operators augmented with recursive semi-quadratic commutators [qDSRG(2)]. The resulting qDSRG(2)+(T) approach matches the accuracy of the "gold-standard" coupled cluster theory with singles, doubles, and perturbative triples model on the energetics of twenty-eight closed-shell atoms and small molecules.

DOI
10.1063/5.0008353

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This work introduces various approaches to include connected three-body terms in unitary many-body theories, focusing on the driven similarity renormalization group (DSRG). Starting from the least approximate method-the linearized DSRG truncated to one-, two-, and three-body operators [LDSRG(3)]-we develop several approximate LDSRG(3) models with reduced computational cost. Through a perturbative analysis, we motivate a family of iterative LDSRG(3)-n and -n' (n = 1, 2, 3, 4) methods that contain a subset of the LDSRG(3) diagrams. Among these variants, the LDSRG(3)-2 scheme has the same computational complexity of coupled cluster theory with singles, doubles, and triples (CCSDT), but it outperforms CCSDT in the accuracy of the predicted correlation energies. We also propose and implement two perturbative triples corrections based on the linearized DSRG truncated to one- and two-body operators augmented with recursive semi-quadratic commutators [qDSRG(2)]. The resulting qDSRG(2)+(T) approach matches the accuracy of the "gold-standard" coupled cluster theory with singles, doubles, and perturbative triples model on the energetics of twenty-eight closed-shell atoms and small molecules.
Applications of the similarity renormalization group (SRG) approach [F. Wegner, Ann. Phys. 506, 77 (1994) and S. D. Głazek and K. G. Wilson, Phys. Rev. D 49, 4214 (1994)] to the formulation of useful many-body theories of electron correlation are considered. In addition to presenting a production-level implementation of the SRG based on a single-reference formalism, a novel integral version of the SRG is reported, in which the flow of the Hamiltonian is driven by a source operator. It is shown that this driven SRG (DSRG) produces a Hamiltonian flow that is analogous to that of the SRG. Compared to the SRG, which requires propagating a set of ordinary differential equations, the DSRG is computationally advantageous since it consists of a set of polynomial equations. The equilibrium distances, harmonic vibrational frequencies, and vibrational anharmonicities of a series of diatomic molecules computed with the SRG and DSRG approximated with one- and two-body normal ordered operators are in good agreement with benchmark values from coupled cluster with singles, doubles, and perturbative triples. Particularly surprising results are found when the SRG and DSRG methods are applied to C2 and F2. In the former case, both methods fail to converge, while in the latter case an unbound potential energy curve is obtained. A modified commutator approximation is shown to correct these problems in the case of the DSRG method.
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.
We present a non-iterative triples correction to the relativistic quadratic unitary coupled cluster singles and doubles (qUCCSD) method, denoted as qUCCSD[T]. The method builds upon the Hermitian structure of the similarity-transformed Hamiltonian in the unitary coupled cluster method and can be derived by using perturbational truncation to the unitary coupled cluster energy functional. Relativistic effects are incorporated using the exact two-component atomic mean-field Hamiltonian, and the computational cost is further reduced through the frozen natural spinor and the Cholesky decomposition approximations. Benchmark results demonstrate that qUCCSD[T] outperforms previously proposed triples corrections to the unitary coupled cluster method and yields excellent agreement with experimental data and Full CI results. In addition, the method shows high accuracy in computing the bond dissociation enthalpies, molecular geometries, vibrational frequencies, ionization potentials, and electron affinities of heavy-element-containing systems. The new qUCCSD[T] method is competitive to popular CCSD(T) even on a classical computer.
Over the past decade the in-medium similarity renormalization group (IMSRG) approach has proven to be a powerful and versatile ab initio many-body method for studying medium-mass nuclei. So far, the IMSRG was limited to the approximation in which only up to two-body operators are incorporated in the renormalization group flow, referred to as the IMSRG(2). In this work, we extend the IMSRG(2) approach to fully include three-body operators yielding the IMSRG(3) approximation. We use a perturbative scaling analysis to estimate the importance of individual terms in this approximation and introduce truncations that aim to approximate the IMSRG(3) at a lower computational cost. The IMSRG(3) is systematically benchmarked for different nuclear Hamiltonians for 4He^{4}\mathrm{He}4He and 16O^{16}\mathrm{O}16O in small model spaces. The IMSRG(3) systematically improves over the IMSRG(2) relative to exact results. Approximate IMSRG(3) truncations constructed based on computational cost are able to reproduce much of the systematic improvement offered by the full IMSRG(3). We also find that the approximate IMSRG(3) truncations behave consistently with expectations from our perturbative analysis, indicating that this strategy may also be used to systematically approximate the IMSRG(3).
We propose a non-iterative, post-hoc correction to the unitary coupled cluster theory with the single, double, and triple excitations (UCCSDT) Ansatz, which considers the leading-order effects of neglected quadruple excitations. We present two ways to derive this correction, henceforth referred to as [Q-6], which leads to an improvement in the correlation energy shown to be truncated to sixth-order in many-body perturbation theory. A comparison between the UCC-based [Q-6] correction proposed in this work and analogous, "platinum standard" quadruple corrections proposed in conventional coupled cluster theory recognizes that [Q-6] is distinct from prior corrections since it is constructed entirely from internally connected components. Although trotterized (t) and full operator variants of UCCSDT exhibit errors in scans of small molecule potential energy surfaces that routinely exceed 1.6 mH, we find that t/UCCSDT[Q-6] is, nevertheless, able to achieve chemical accuracy as measured by the mean unsigned error.
Fourth-order many-body corrections to matrix elements for atoms with one valence electron are derived. The obtained diagrams are classified using coupled-cluster-inspired separation into contributions from n-particle excitations from the lowest-order wave function. The complete set of fourth-order diagrams involves only connected single, double, and triple excitations and disconnected quadruple excitations. Approximately half of the fourth-order diagrams are not accounted for by the popular coupled-cluster method truncated at single and double excitations (CCSD). Explicit formulas are tabulated for the entire set of fourth-order diagrams missed by the CCSD method and its linearized version, i.e., contributions from connected triple and disconnected quadruple excitations. A partial summation scheme of the derived fourth-order contributions to all orders of perturbation theory is proposed.