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arXiv:2103.12381 · Journal of Chemical Theory and Computation

Assessing the Accuracy of Tailored Coupled Cluster Methods Corrected by Electronic Wave Functions of Polynomial Cost

Aleksandra Leszczyk, Mihály Máté, Örs Legeza, Katharina Bogusławski

Published Dec 29, 2021• 42 citations
arXivHTMLPDF

Abstract

Tailored coupled cluster theory represents a computationally inexpensive way to describe static and dynamical electron correlation effects. In this work, we scrutinize the performance of various coupled cluster methods tailored by electronic wave functions of polynomial cost. Specifically, we focus on frozen-pair coupled cluster (fpCC) methods, which are tailored by pair-coupled cluster doubles (pCCD), and coupled cluster theory tailored by matrix product state wave functions optimized by the density matrix renormalization group (DMRG) algorithm. As test system, we selected a set of various small- and medium-sized molecules containing diatomics (N 2, F 2, C 2, CN +, CO, BN, BO +, and Cr 2 ) and molecules (ammonia, ethylene, cyclobutadiene, benzene, hydrogen chains, rings, and cuboids) for which the conventional single-reference coupled cluster singles and doubles (CCSD) method is not able to produce accurate results for spectroscopic constants, potential energy surfaces, and barrier heights. Most importantly, DMRG-tailored and pCCD-tailored approaches yield similar errors in spectroscopic constants and potential energy surfaces compared to accurate theoretical and/or experimental reference data. Although fpCC methods provide a reliable description for the dissociation pathway of molecules featuring single and quadruple bonds, they fail in the description of triple or hextuple bond-breaking processes or avoided crossing regions.

DOI
10.1021/acs.jctc.1c00284

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Tailored coupled cluster theory represents a computationally inexpensive way to describe static and dynamical electron correlation effects. In this work, we scrutinize the performance of various coupled cluster methods tailored by electronic wave functions of polynomial cost. Specifically, we focus on frozen-pair coupled cluster (fpCC) methods, which are tailored by pair-coupled cluster doubles (pCCD), and coupled cluster theory tailored by matrix product state wave functions optimized by the density matrix renormalization group (DMRG) algorithm. As test system, we selected a set of various small- and medium-sized molecules containing diatomics (N 2, F 2, C 2, CN +, CO, BN, BO +, and Cr 2 ) and molecules (ammonia, ethylene, cyclobutadiene, benzene, hydrogen chains, rings, and cuboids) for which the conventional single-reference coupled cluster singles and doubles (CCSD) method is not able to produce accurate results for spectroscopic constants, potential energy surfaces, and barrier heights. Most importantly, DMRG-tailored and pCCD-tailored approaches yield similar errors in spectroscopic constants and potential energy surfaces compared to accurate theoretical and/or experimental reference data. Although fpCC methods provide a reliable description for the dissociation pathway of molecules featuring single and quadruple bonds, they fail in the description of triple or hextuple bond-breaking processes or avoided crossing regions.
Wave functions based on electron-pair states provide inexpensive and reliable models to describe quantum many-body problems containing strongly correlated electrons, given that broken-pair states have been appropriately accounted for by, for instance, a posteriori corrections. In this article, we analyze the performance of electron-pair methods in predicting orbital-based correlation spectra. We focus on the (orbital-optimized) pair-coupled cluster doubles (pCCD) ansatz with a linearized coupled-cluster (LCC) correction. Specifically, we scrutinize how orbital-based entanglement and correlation measures can be determined from a pCCD-tailored CC wave function. Furthermore, we employ the single-orbital entropy, the orbital-pair mutual information, and the eigenvalue spectra of the two-orbital reduced density matrices to benchmark the performance of the LCC correction for the one-dimensional Hubbard model with the periodic boundary condition as well as the N2 and F2 molecules against density matrix renormalization group reference calculations. Our study indicates that pCCD-LCC accurately reproduces the orbital-pair correlation patterns in the weak correlation limit and for molecules close to their equilibrium structure. Hence, we can conclude that pCCD-LCC predicts reliable wave functions in this regime.
The central problem in the modern electronic structure theory is the calculation of cor- relation energy, possibly by an approach that would account for both static and dynamic correlation in an efficient, balanced and accurate way. In this thesis, I present a collection of methods that combine the effective treatment of dynamic correlation by the coupled cluster theory with density matrix renormalization group, a well-established technique for calculations of strongly correlated systems. The connection between them is achieved via the tailored coupled clusters (TCC) ansatz, which conveniently does not impose any ad- ditional computational costs. After the successful initial assessment, we developed more efficient implementations of these methods by employing the local approaches based on pair natural orbitals. This way, we extended the range of possible applications to larger systems with thousands of basis functions. To assess the accuracy of TCC as well as its local counterparts, we performed a variety of benchmark calculations ranging from small, yet challenging systems such as the nitrogen molecule or tetramethyleneethane diradical, to larger molecules like oxo-Mn(Salen) or Fe(II)-porphyrin model. 1
In the past decade, the quantum chemical version of the density matrix renormalization group (DMRG) method has established itself as the method of choice for calculations of strongly correlated molecular systems. Despite its favorable scaling, it is in practice not suitable for computations of dynamic correlation. We present a novel method for accurate “post-DMRG” treatment of dynamic correlation based on the tailored coupled cluster (CC) theory in which the DMRG method is responsible for the proper description of nondynamic correlation, whereas dynamic correlation is incorporated through the framework of the CC theory. We illustrate the potential of this method on prominent multireference systems, in particular, N 2 and Cr 2 molecules and also oxo-Mn(Salen), for which we have performed the first post-DMRG computations in order to shed light on the energy ordering of the lowest spin states.
Perhaps the most important approximations to the electronic structure problem in quantum chemistry are those based on coupled cluster and density functional theories. Coupled cluster theory has been called the ``gold standard'' of quantum chemistry due to the high accuracy that it achieves for weakly correlated systems. Kohn--Sham density functionals based on semilocal approximations are, without a doubt, the most widely used methods in chemistry and material science because of their high accuracy/cost ratio. The root of the success of coupled cluster and density functionals is their ability to efficiently describe the dynamic part of the electron correlation. However, both traditional coupled cluster and density functional approximations may fail catastrophically when substantial static correlation is present. This severely limits the applicability of these methods to a plethora of important chemical and physical problems such as, e.g., the description of bond breaking, transition states, transition metal-, lanthanide- and actinide-containing compounds, and superconductivity. In an attempt to tackle this problem, nonstandard (single-reference) coupled cluster-based techniques that aim to describe static correlation have been recently developed: pair coupled cluster doubles (pCCD) and singlet-paired coupled cluster doubles (CCD0). The ability to describe static correlation in pCCD and CCD0 comes, however, at the expense of important amounts of dynamic correlation so that the high accuracy of standard coupled cluster becomes unattainable. Thus, the reliable and efficient description of static and dynamic correlation in a simultaneous manner remains an open problem for quantum chemistry and many-body theory in general. In this thesis, different ways to combine pCCD and CCD0 with density functionals in order to describe static and dynamic correlation simultaneously (and efficiently) are explored. The combination of wavefunction and density functional methods has a long history in quantum chemistry (practical implementations have appeared in the literature since the 1970s). However, this kind of techniques have not achieved widespread use due to problems such as double counting of correlation and the symmetry dilemma---the fact that wavefunction methods respect the symmetries of Hamiltonian, while modern functionals are designed to work with broken symmetry densities. Here, particular mathematical features of pCCD and CCD0 are exploited to avoid these problems in an efficient manner. The two resulting families of approximations, denoted as pCCD+DFT and CCD0+DFT, are shown to be able to describe static and dynamic correlation in standard benchmark calculations. Furthermore, it is also shown that CCD0+DFT lends itself to combination with correlation from the direct random phase approximation (dRPA). Inclusion of dRPA in the long-range via the technique of range-separation allows for the description of dispersion correlation, the remaining part of the correlation. Thus, when combined with the dRPA, CCD0+DFT can account for all three-types of electron correlation that are necessary to accurately describe molecular systems. Lastly, applications of CCD0+DFT to actinide chemistry are considered in this work. The accuracy of CCD0+DFT for predicting equilibrium geometries and vibrational frequencies of actinide molecules and ions is assessed and compared to that of well-established quantum chemical methods. For this purpose, the f0f^0f0 actinyl series (UO22+_2^{2+}22+​, NpO23+_ 2^{3+}23+​, PuO24+_2^{4+}24+​), the isoelectronic NUN, and Thorium (ThO, ThO2+^{2+}2+) and Nobelium (NoO, NoO2_22​) oxides are studied. It is shown that the CCD0+DFT description of these species agrees with available experimental data and is comparable with the results given by the highest-level calculations that are possible for such heavy compounds while being, at least, an order of magnitude lower in computational cost.
Doubly occupied configuration interaction (DOCI) with optimized orbitals often accurately describes strong correlations while working in a Hilbert space much smaller than that needed for full configuration interaction. However, the scaling of such calculations remains combinatorial with system size. Pair coupled cluster doubles (pCCD) is very successful in reproducing DOCI energetically, but can do so with low polynomial scaling (N(3), disregarding the two-electron integral transformation from atomic to molecular orbitals). We show here several examples illustrating the success of pCCD in reproducing both the DOCI energy and wave function and show how this success frequently comes about. What DOCI and pCCD lack are an effective treatment of dynamic correlations, which we here add by including higher-seniority cluster amplitudes which are excluded from pCCD. This frozen pair coupled cluster approach is comparable in cost to traditional closed-shell coupled cluster methods with results that are competitive for weakly correlated systems and often superior for the description of strongly correlated systems.
The works included in this thesis aim to extend the applications of pair-coupled cluster doubles (pCCD) theory. pCCD produces reliable results for strongly correlated chemical systems and performs better than other traditional quantum chemical methods. Still, achieving greater accuracy requires post-pCCD corrections, which steeply increase computational cost, limiting its widespread application. Quantum embedding approaches, which reserve accurate but high-cost methods for the active site of chemical change and approximated methods for other regions of a chemical system, can be used to circumvent this problem. In this context, two new pCCD-based embedding formulations have been implemented and tested. Both the pCCD-in-DFT and pCCD-in-pCCD methods produce reliable results across a wide range of intermolecular interactions. In particular, the latter method shows significantly better performance, as it does not depend on approximated density functionals. Additionally, estimation of molecular properties is also pertinent for practical applications with pCCD. As the traditional response-theory-based formulation for pCCD analyzed here did not meet the required level of accuracy, an alternate approach based on the expectation-value method was implemented for pCCD and post-pCCD methods. This approach produces excellent agreement with the references at a much lower computational cost, as we bypass the need to solve the response equations. Thus, the pCCD-in-pCCD embedding method and the expectation-value-pCCD approach provide routes for large-scale simulations of challenging, complex systems.