Valence Bond Wave Functions Research Articles

Block-Correlated Coupled Cluster Theory Based on the Generalized Valence Bond Reference for Singlet-Triplet Energy Gaps of Strongly Correlated Systems.

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.

Dynamical Electron Correlation and the Chemical Bond. IV. Covalent Bonds in A2 Molecules (A = N-As and F-Br).

In a series of recent papers, we investigated the effect of dynamical electron correlation on the potential energy curves and spectroscopic constants of several diatomic molecules, including the simple diatomic hydrides (AH) and the more complex diatomic fluorides (AF) and homonuclear diatomic molecules (A2) with A = B-F (AF) or A = C-F (A2), respectively. Our goal was to understand the dependence of the dynamical electron correlation energy, EDEC, on the internuclear distance, R, and quantify how dynamical electron correlation influences the spectroscopic constants (De, Re, and ωe) of these molecules. At large R, we found that the magnitude of EDEC(R) had a simple dependence on R, with EDEC(R) increasing nearly exponentially with decreasing R. However, as R continued to decrease, there were significant variations in EDEC(R). These variations led to differing changes in the predicted spectroscopic constants of the molecules. In many molecules, the changes in EDEC(R) could be correlated with changes in the underlying spin-coupled generalized valence bond wave function, either in the orbitals or the spin-coupling coefficients. In the current paper, we extend these studies to higher main group elements, comparing the effects of EDEC(R) on P2 and As2 versus N2, and on Cl2 and Br2 versus F2. We find that there are significant differences between the effects of dynamical electron correlation on the molecules in the first and subsequent rows of the periodic table.

Quantum Algorithmic Approach to Multiconfigurational Valence Bond Theory: Insights from Interpretable Circuit Design.

Efficient ways to prepare Fermionic ground states on quantum computers are in high demand, and different techniques have been developed over the past few years. Despite having a vast set of methods, it is still unclear which method performs well for which system. In this work, we combine interpretable circuit designs with an effective basis approach in order to optimize a multiconfigurational valence bond wave function. Based on selected model systems, we show how this leads to an explainable performance. We demonstrate that the developed methodology outperforms related methods in terms of the size of the effective basis, as well as individual quantum resources for the involved circuits.

Block-Correlated Coupled Cluster Theory with up to Four-Pair Correlation for Accurate Static Correlation of 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.

Equation-of-Motion Block-Correlated Coupled Cluster Method for Excited Electronic States of Strongly Correlated 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.

Gauge-Theoretic Origin of Rydberg Quantum Spin Liquids.

Recent atomic physics experiments and numerical works have reported complementary signatures of the emergence of a topological quantum spin liquid in models with blockade interactions. However, the specific mechanism stabilizing such a phase remains unclear. Here, we introduce an exact relation between an Ising-Higgs lattice gauge theory on the kagome lattice and blockaded models on Ruby lattices. This relation elucidates the origin of previously observed topological spin liquids by directly linking the latter to a deconfined phase of a solvable gauge theory. By means of exact diagonalization and unbiased quantum MonteCarlo simulations, we show that the deconfined phases extend in a broad region of the parameter space; these states are characterized by a large ground state overlap with resonating valence bond wave functions. These blockaded models include both creation or annihilation and hopping dynamics, and can be experimentally realized with Rydberg-dressed atoms, offering novel and controllable platforms for the engineering and characterization of spin liquid states.

A linear surrogate for optimising functions of an orthogonal matrix with applications in wave function theory

The technique of surrogate optimisation is to use a simpler function to approximate a complex function that is time-consuming to evaluate. We show that the maximum of a special type of surrogate function f ( U ) = Tr ( A U ) , U ∈ O ( n ) is at A T ( A A T ) 1 / 2 , and that there is one and only one local maximum both in S O ( n ) and O ( n ) − S O ( n ) . This function f ( U ) has been found to be useful in various aspects of electronic structure theory, including proving the Carlson-Keller theorem, and localising orbitals. As one other example, we apply it here to optimise the ground state of molecules using the Generalised Valence Bond wavefunction.

Compact and accurate ab initio valence bond wave functions for electron transfer: The classic but challenging covalent-ionic interaction in LiF

This paper combines the valence bond block diabatization approach with the idea of orbital breathing. With highly compact wave functions, the breathing orbital valence bond (BOVB) method is applied to investigate several atomic and molecular properties, including the electron affinity of F, the adiabatic and diabatic potential energy curves and the dipole moment curves of the two lowest-lying 1Σ+ states, the electronic coupling curve and the crossing distance of the two diabatic states, and the spectroscopic constants of the ground states for LiF. The configuration selection scheme proposed in this work is quite general, requiring only the selection of several de-excitation and excitation orbitals in a sense like the restricted active space self-consistent field method. Practically, this is also the first time that BOVB results are extrapolated to complete basis set limit. Armed with the chemical intuition provided by valence bond theory, the classic but challenging covalent-ionic interaction in the title molecule is not only conceptually interpreted but also accurately computed.

Efficient Implementation of Block-Correlated Coupled Cluster Theory Based on the Generalized Valence Bond Reference 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 [Cu2O2]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.

Dynamical electron correlation and the chemical bond. I. Covalent bonds in AH and AF (A = B-F).

Dynamical electron correlation has a major impact on the computed values of molecular properties and the energetics of molecular processes. This study focused on the effect of dynamical electron correlation on the spectroscopic constants (Re, ωe, De), and potential energy curves, ΔE(R), of the covalently bound AH and AF molecules, A = B-F. The changes in the spectroscopic constants (ΔRe, Δωe, ΔDe) caused by dynamical correlation are erratic and, at times, even surprising. These changes can be understood based on the dependence of the dynamical electron correlation energies of the AH and AF molecules as a function of the bond distance, i.e., ΔEDEC(R). At large R, the magnitude of ΔEDEC(R) increases nearly exponentially with decreasing R, but this increase slows as R continues to decrease and, in many cases, even reverses at very short R. The changes in ΔEDEC(R) in the region around Re were as unexpected as they were surprising, e.g., distinct minima and maxima were found in the curves of ΔEDEC(R) for the most polar molecules. The variations in ΔEDEC(R) for R ≲ Re are directly correlated with major changes in the electronic structure of the molecules as revealed by a detailed analysis of the spin-coupled generalized valence bond wave function. The results reported here indicate that we have much to learn about the nature of dynamical electron correlation and its effect on chemical bonds and molecular properties and processes.

Electronic densities and valence bond wave functions.

Valence bond (VB) wave functions are studied from the density point of view. The density is plotted as a difference with the quasi-state built on the same orbitals. The densities of the components of the VB wave function are also shown. The breathing orbital effect leads to small modifications of the density. It is shown that while the densities of ionic and covalent components are the same, their coupling ends-up in modifications of the electronic density.

Dispersion Interactions between Molecules in and out of Equilibrium Geometry: Visualization and Analysis.

The London dispersion interactions between systems undergoing bond breaking, twisting, or compression are not well studied due to the scarcity and the high computational cost of methods being able to describe both the dynamic correlation and the multireference character of the system. Recently developed methods based on the Generalized Valence Bond wave function, such as EERPA-GVB and SAPT(GVB) (SAPT = symmetry-adapted perturbation theory), allow one to accurately compute and analyze noncovalent interactions between multireference systems. Here, we augment this analysis by introducing a local indicator for dispersion interactions inspired by Mata and Wuttke’s Dispersion Interaction Density [J. Comput. Chem.2017, 38, 15−2327761924] applied on top of an EERPA-GVB computation. Using a few model systems, we show what insights into the nature and evolution of the dispersion interaction during bond breaking and twisting such an approach is able to offer. The new indicator can be used at a minimal cost additional to an EERPA-GVB computation and can be complemented by an energy decomposition employing the SAPT(GVB) method. We explain the physics behind the initial increase, followed by a decrease in the interaction of linear molecules upon bond stretching. Namely, the elongation of covalent bonds leads to the enhancement of attractive dispersion interactions. For even larger bond lengths, this effect is canceled by the increase of the repulsive exchange forces resulting in a suppression of the interaction and finally leading to repulsion between monomers.

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.

The Valence-Bond (VB) Model and Its Intimate Relationship to the Symmetric or Permutation Group.

VB and molecular orbital (MO) models are normally distinguished by the fact the first looks at molecules as a collection of atoms held together by chemical bonds while the latter adopts the view that each molecule should be regarded as an independent entity built up of electrons and nuclei and characterized by its molecular structure. Nevertheless, there is a much more fundamental difference between these two models which is only revealed when the symmetries of the many-electron Hamiltonian are fully taken into account: while the VB and MO wave functions exhibit the point-group symmetry, whenever present in the many-electron Hamiltonian, only VB wave functions exhibit the permutation symmetry, which is always present in the many-electron Hamiltonian. Practically all the conflicts among the practitioners of the two models can be traced down to the lack of permutation symmetry in the MO wave functions. Moreover, when examined from the permutation group perspective, it becomes clear that the concepts introduced by Pauling to deal with molecules can be equally applied to the study of the atomic structure. In other words, as strange as it may sound, VB can be extended to the study of atoms and, therefore, is a much more general model than MO.

An Investigation into the Approximations Used in Wave Packet Molecular Dynamics for the Study of Warm Dense Matter

Wave packet molecular dynamics (WPMD) has recently received a lot of attention as a computationally fast tool with which to study dynamical processes in warm dense matter beyond the Born–Oppenheimer approximation. These techniques, typically, employ many approximations to achieve computational efficiency while implementing semi-empirical scaling parameters to retain accuracy. We investigated three of the main approximations ubiquitous to WPMD: a restricted basis set, approximations to exchange, and the lack of correlation. We examined each of these approximations in regard to atomic and molecular hydrogen in addition to a dense hydrogen plasma. We found that the biggest improvement to WPMD comes from combining a two-Gaussian basis with a semi-empirical correction based on the valence-bond wave function. A single parameter scales this correction to match experimental pressures of dense hydrogen. Ultimately, we found that semi-empirical scaling parameters are necessary to correct for the main approximations in WPMD. However, reducing the scaling parameters for more ab-initio terms gives more accurate results and displays the underlying physics more readily.

Resonating valence bond realization of spin-1 non-Abelian chiral spin liquid on the torus

We propose resonating valence bond wave functions for a spin-1 system on the torus that realize a non-Abelian chiral spin liquid. The wave functions take the form of infinite-dimensional matrix product states constructed from conformal blocks of the $\mathrm{SO}{(3)}_{1}$ Wess-Zumino-Witten model. This means that they are lattice analogs of the bosonic Moore-Read state introduced in fractional quantum Hall systems. The topological order of this system is revealed by explicit construction of threefold-degenerate ground states and analytical computation of the modular $S$ and $T$ matrices.

Quantum trimer models and topological SU(3) spin liquids on the kagome lattice

We construct and study quantum trimer models and resonating SU(3)-singlet models on the kagome lattice, which generalize quantum dimer models and the Resonating Valence Bond wavefunctions to a trimer and SU(3) setting. We demonstrate that these models carry a Z_3 symmetry which originates in the structure of trimers and the SU(3) representation theory, and which becomes the only symmetry under renormalization. Based on this, we construct simple and exact parent Hamiltonians for the model which exhibit a topological 9-fold degenerate ground space. A combination of analytical reasoning and numerical analysis reveals that the quantum order ultimately displayed by the model depends on the relative weight assigned to different types of trimers -- it can display either Z_3 topological order or form a symmetry-broken trimer crystal, and in addition possesses a point with an enhanced U(1) symmetry and critical behavior. Our results accordingly hold for the SU(3) model, where the two natural choices for trimer weights give rise to either a topological spin liquid or a system with symmetry-broken order, respectively. Our work thus demonstrates the suitability of resonating trimer and SU(3)-singlet ansatzes to model SU(3) topological spin liquids on the kagome lattice.

Calculation of dispersion interactions with the geminal-based ring Coupled Cluster Doubles method

The performance of the recently developed multi-reference extension of ring coupled cluster doubles is investigated for dispersion energy calculations, applied to the generalized valence bond wave function. The leading-order contribution to the dispersion energy is shown to have the correct asymptotic behaviour. Illustrative calculations on noble gas dimers are presented.

Describing Strong Correlation with Block-Correlated Coupled Cluster Theory.

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.

Three-center two-electron bonds in the boranes B2H6 and B3H8− from the quantum interference perspective

In this work the nature of the three-center two-electron (3c2e) chemical bonds in the diborane (6) molecule and the octahydrotriborate anion (B3H8−) is investigated by the Generalized Product Function Energy Partitioning (GPF-EP) method using modern valence bond wave functions. The Interference Energy Analysis shows that 3c2e bonds have the same features of a 2c2e bond: interference dominates the energetic balance for the formation of the bond and the kinetic energy drop is responsible for its stabilization. Also, 3c2e are generally way more energetic than a 2c2e analogous one, because interference can occur in three different ways (multiple interference effect), rather than just in one as for the two-center bonds. Furthermore, with the GPF-EP method one could establish a more rigorous way for discussing the idea of a “supported” bond in the diborane (6) molecule.