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The Journal of Physical Chemistry Letters

Generalized Valence Bond-Based Block-Correlated Coupled Cluster Theory with up to Five-Pair Correlation for Accurate Static Correlation of Strongly Correlated Systems

Feichao Han, Pengfei Ma, Xiaochuan Ren, Wei Li, Shuhua Li

Published May 29, 2025• 5 citations
Publisher

Abstract

Generalized valence bond-based block-correlated coupled cluster theory with up to five-pair correlation (GVB-BCCC5) has been presented for the first time to describe the electronic structures of strongly correlated systems with singlet ground states. Several efficient techniques have been employed to make GVB-BCCC5 calculations practical for strongly correlated systems. We then apply this method to investigate various systems, including potential energy surfaces of a hydrogen cuboid lattice and a phosphorus cluster (P 4 ); relative energies of two isomers for C 8 and V 2 H 2; and the bond dissociation energy in a metal-oxide compound, FeO 3 . All calculations show that GVB-BCCC5 can provide nearly the exact static correlation energy as the density matrix renormalization group method (based on the same GVB orbitals). This work demonstrates the potential applications of the GVB-BCCC5 method in the accurate description of many strongly correlated systems.

DOI
10.1021/acs.jpclett.5c00990

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Generalized valence bond-based block-correlated coupled cluster theory with up to five-pair correlation (GVB-BCCC5) has been presented for the first time to describe the electronic structures of strongly correlated systems with singlet ground states. Several efficient techniques have been employed to make GVB-BCCC5 calculations practical for strongly correlated systems. We then apply this method to investigate various systems, including potential energy surfaces of a hydrogen cuboid lattice and a phosphorus cluster (P 4 ); relative energies of two isomers for C 8 and V 2 H 2; and the bond dissociation energy in a metal-oxide compound, FeO 3 . All calculations show that GVB-BCCC5 can provide nearly the exact static correlation energy as the density matrix renormalization group method (based on the same GVB orbitals). This work demonstrates the potential applications of the GVB-BCCC5 method in the accurate description of many strongly correlated systems.
A block-correlated coupled cluster method with up to four-pair correlation based on the generalized valence bond wave function (GVB-BCCC4) is first implemented, which offers an alternative method for electronic structure calculations of strongly correlated systems. We developed some techniques to derive a set of compact and cost-effective equations for GVB-BCCC4, which include the definition of n -block ( n = 1–4) Hamiltonian matrices, the combination of excitation operators, and the definition of independent amplitudes. We then applied the GVB-BCCC4 method to investigate several potential energy surfaces of strongly correlated systems with singlet ground states. Our calculations demonstrate that the GVB-BCCC4 method can provide nearly exact static correlation energies as the density matrix renormalization group method (on the basis of the same GVB orbitals). This work highlights the significance of four-pair correlation in quantitative descriptions of static correlation energy for strongly correlated systems.
An optimized implementation of block-correlated coupled cluster theory based on the generalized valence bond wave function (GVB-BCCC) for the singlet ground state of strongly correlated systems is presented. The GVB-BCCC method with two-pair correlation (GVB-BCCC2b) or up to three-pair correlation (GVB-BCCC3b) will be the focus of this work. Three major techniques have been adopted to dramatically accelerate GVB-BCCC2b and GVB-BCCC3b calculations. First, the GVB-BCCC2b and GVB-BCCC3b codes are noticeably optimized by removing redundant calculations. Second, independent amplitudes are identified by constraining excited configurations to be pure singlet states and only independent amplitudes need to be solved. Third, an incremental updating scheme for the amplitudes in solving the GVB-BCCC equations is adopted. With these techniques, accurate GVB-BCCC3b calculations are now accessible for systems with relatively large active spaces (50 electrons in 50 orbitals) and GVB-BCCC2b calculations are affordable for systems with much larger active spaces. We have applied GVB-BCCC methods to investigate three typical kinds of systems: polyacenes, pentaprismane, and [Cu 2 O 2 ] 2+ isomers. For polyacenes, we demonstrate that GVB-BCCC3b can capture more than 94% of the total correlation energy even for 12-acene with 50 π electrons. For the potential energy curve of simultaneously stretching 15 C–C bonds in pentaprismane, our calculations show that the GVB-BCCC3b results are quite close to the results from the density matrix renormalization group (DMRG) over the whole range. For two dinuclear copper oxide isomers, their relative energy predicted by GVB-BCCC3b is also in good accord with the DMRG result. All calculations show that the inclusion of three-pair correlation in GVB-BCCC is critical for accurate descriptions of strongly correlated systems.
A block-correlated coupled cluster (BCCC) method based on the generalized valence bond (GVB) wave function (GVB-BCCC in short) is proposed and implemented at the ab initio level, which represents an attractive multireference electronic structure method for strongly correlated systems. The GVB-BCCC method is demonstrated to provide satisfactory descriptions for multiple bond breaking in small molecules, although the GVB reference function is qualitatively wrong for the studied processes. For a challenging prototype of strongly correlated systems, tridecane with all 12 single C–C bonds at various distances, our calculations have shown that the GVB-BCCC2b method can provide highly comparable results as the density matrix renormalization group method for potential energy surfaces along simultaneous dissociation of all C–C bonds.
A block-correlated coupled cluster (BCCC) method based on the triplet generalized valence bond (GVB) wave function (GVB-BCCC) has been implemented for the first time. By introducing several techniques, we have developed a practical and efficient GVB-BCCC code. The GVB-BCCC3 method (with up to three-pair correlation) can be used to deal with strongly correlated (SC) systems with triplet or singlet ground states, allowing singlet-triplet (S-T) energy gaps in the active space of SC systems computationally available. For selected SC systems, our calculations show that GVB-BCCC3 can always provide correct ground-state spin multiplicity as the complete active space configuration interaction (CASCI) or density matrix renormalization group (DMRG). Furthermore, we found that the S-T energy gaps from GVB-BCCC3 are quite consistent with CASCI or DMRG results. This work demonstrates that GVB-BCCC3 is a promising theoretical tool for describing S-T energy gaps within the active space of SC systems with large active spaces.
Strongly correlated (SC) systems present significant challenges for classical quantum chemistry methods. Quantum computing, particularly the variational quantum eigensolver (VQE), offers a promising framework to address these challenges by inherently supporting exponentially large configuration spaces. However, its application to SC systems remains limited due to the single-reference nature of the widely used ansatzes such as unitary coupled cluster (UCC). To address this challenge, we propose the generalized valence bond-based unitary block correlated coupled cluster (GVB-UBCCC) method. This novel ansatz incorporates the multiconfigurational nature of generalized valence bond (GVB) and the accuracy of block correlated coupled cluster (BCCC) methods, making it well-suited for SC systems. We have implemented the GVB-UBCCC method with up to two-block correlation (GVB-UBCCC2) and applied it to investigate ground-state energies for several SC systems, including H 4, the water dimer, N 2 H 2, and S 6, at most described by 24 qubits. Our approach demonstrates that for these systems, GVB-UBCCC2 can achieve more accurate ground-state energies than UCCSD in most cases while requiring only O ( N 2 ) quantum gates and parameters, as opposed to the O ( N 4 ) scaling of UCCSD. The results highlight the effectiveness and potential advantages of GVB-UBCCC in SC systems.
An equation-of-motion block-correlated coupled cluster method based on the generalized valence bond wave function (EOM-GVB-BCCC) is proposed to describe low-lying excited states for strongly correlated systems. The EOM-GVB-BCCC2b method with up to two-pair correlation has been implemented and tested for a few strongly correlated systems. For a water hexamer with stretched O-H bonds, which is beyond the capability of the CASSCF method, EOM-GVB-BCCC2b provides very close results as the density matrix renormalization group (DMRG). For four conjugated diradical species with triplet ground states, we found that their vertical S-T gaps from EOM-GVB-BCCC2b are also quite consistent with the DMRG results. This new method is expected to be a promising theoretical tool for describing the low-lying excited states of strongly correlated systems with large active spaces.