We present a class of correlated wavefunctions generated by exponentials of two-body on-site Hermitian operators that can be evaluated with polynomial computational cost via a Hamiltonian similarity transformation. Wavefunctions of this form have been studied with variational Monte Carlo methods, but we present a formalism to perform non-stochastic calculations. The Hausdorff series generated by these Jastrow factors can be summed exactly without truncation resulting in a set of equations with polynomial computational cost. The correlators include the density-density, collinear spin-spin, spin-density cross terms, and on-site double occupancy operators. The resulting non-Hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with only a small set of correlation terms required for accurate calculations in systems with local interactions. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the one and two-dimensional repulsive Hubbard model where it yields accurate results for large systems with low computational cost. Symmetry projection methods are included to further improve the reference wavefunction and results under strong correlation without sacrificing good quantum numbers resulting in very accurate energies for small systems and producing a better ground state for the calculation of other properties.
We present a class of correlated wavefunctions generated by exponentials of two-body on-site Hermitian operators that can be evaluated with polynomial computational cost via a Hamiltonian similarity transformation. Wavefunctions of this form have been studied with variational Monte Carlo methods, but we present a formalism to perform non-stochastic calculations. The Hausdorff series generated by these Jastrow factors can be summed exactly without truncation resulting in a set of equations with polynomial computational cost. The correlators include the density-density, collinear spin-spin, spin-density cross terms, and on-site double occupancy operators. The resulting non-Hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with only a small set of correlation terms required for accurate calculations in systems with local interactions. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the one and two-dimensional repulsive Hubbard model where it yields accurate results for large systems with low computational cost. Symmetry projection methods are included to further improve the reference wavefunction and results under strong correlation without sacrificing good quantum numbers resulting in very accurate energies for small systems and producing a better ground state for the calculation of other properties.
We present a class of Lie algebraic similarity transformations generated by exponentials of two-body on-site Hermitian operators whose Hausdorff series can be summed exactly without truncation. The correlators are defined over the entire lattice and include the Gutzwiller factor ${n}_{i\ensuremath{\uparrow}}{n}_{i\ensuremath{\downarrow}}$, and two-site products of density $({n}_{i\ensuremath{\uparrow}}+{n}_{i\ensuremath{\downarrow}})$ and spin $({n}_{i\ensuremath{\uparrow}}\ensuremath{-}{n}_{i\ensuremath{\downarrow}})$ operators. The resulting non-Hermitian many-body Hamiltonian can be solved in a biorthogonal mean-field approach with polynomial computational cost. The proposed similarity transformation generates locally weighted orbital transformations of the reference determinant. Although the energy of the model is unbound, projective equations in the spirit of coupled cluster theory lead to well-defined solutions. The theory is tested on the one- and two-dimensional repulsive Hubbard model where it yields accurate results for small and medium sized interaction strengths.
Standard approaches to approximating solutions to the Schroedinger equation rely upon the assumption that there is a single dominant electronic configuration with a small number of low-lying excited states. As such, they are referred to as single reference approaches and are accurate only when the band gap is small in comparison with the inter-electronic repulsion of the valence electrons. Systems for which this is the case are weakly-correlated. If the inter-electronic repulsion larger than the band gap, then a single electronic reference is not an appropriate ansatz for the physical wavefunction. As the strength of the interaction is increased many Slater determinants become near-degenerate, each providing non-negligible contributions. Such systems are strongly-correlated. The goal of the thesis was to employ wavefunctions which are: non-trivial, tractable, size-consistent and extensive, and not given in an active space. We have outlined approaches based on wavefunction forms more suited to describe strong correlation. Built on the Lie algebra su(2), our first trial wavefunction was the Bethe ansatz solution to the reduced BCS Hamiltonian derived by Richardson. This wavefunction is expressed as a product of quasi-particles each parametrized in terms of quasi-momenta. Consistency among the quasi-momenta is ensured by a set of non-linear equations to be solved numerically. This approach is feasible in both variational and projected flavours. We next considered a wavefunction of the same form without requiring it to be an eigenvector of the reduced BCS Hamiltonian. A variational approach in this case is only feasible provided there are two equivalent descriptions in terms of particles and holes. In any case, a bivariational principle and a projected approach are both feasible. Also based on su(2), there is AP1roG (coupled cluster pair doubles). Open-shell singlets and triplets were constructed with the Lie algebras sp(N) and so(4N). The resulting formulae appear to be infeasible unless it is possible to efficiently compute generalizations of determinants to three-dimensional objects. Treating each spatial orbital individually, all four possible occupations close the Lie superalgebra gl(2|2).
Accurately predicting the effects of correlations caused by strong interactions between electrons represents one of the key unsolved challenges in quantum chemistry. This thesis details theoretical developments addressing two aspects of this challenge. First, advances in Canonical Transformation (CT) theory address the task of describing weak interactions between the chemically active valence electrons and the chemically inert core electrons when the valence electrons are strongly correlated. Second, new methodologies for Jastrow factor wavefunctions are explored in an attempt to describe strong interactions amongst the valence electrons. In its original form, CT theory suffered from poorly understood numerical instabilities in strongly interacting systems and poor accuracy relative to coupled cluster theory in weakly interacting systems. Here the numeric issues are identified as consequences of CT theory's central approximation and shown to be similar to the intruder states of many-body perturbation theory. It is demonstrated that these instabilities can be avoided by placing particular restrictions on the theory's degrees of freedom. In weakly interacting systems, we show that using a quadratic, rather than linear, commutator approximation improves accuracy, and that the accuracy in any system can be improved by making use of the system's 3-body reduced density matrix. These improvements are demonstrated in a variety of systems, including N2 , H2 O, NiO, and free base porphin. Jastrow factor wavefunctions have traditionally been handled using quan- tum Monte Carlo techniques such as variational Monte Carlo (VMC). Here we explore how VMC can be made to accommodate the unconventionally large numbers of variational parameters present in many-body Jastrow factors and show that the stochastic reconfiguration algorithm can be thus improved by employing the method of conjugate gradients to solve its central linear equation. In addition to extending the reach of VMC, we have also developed nonstochastic methods for working with Jastrow factor wavefunctions that do not rely on random sampling. These methods, similar in design to those used with the coupled cluster wavefunction, produce energies very close to those of VMC in applications to the Heisenberg, spinless Hubbard, and full Hubbard model Hamiltonians.
We present a similarity transformation theory based on a polynomial form of a particle-hole pair excitation operator. In the weakly correlated limit, this polynomial becomes an exponential, leading to coupled cluster doubles. In the opposite strongly correlated limit, the polynomial becomes an extended Bessel expansion and yields the projected BCS wave function. In between, we interpolate using a single parameter. The effective Hamiltonian is non-Hermitian and this polynomial similarity transformation theory follows the philosophy of traditional coupled cluster, left projecting the transformed Hamiltonian onto subspaces of the Hilbert space in which the wave function variance is forced to be zero. Similarly, the interpolation parameter is obtained through minimizing the next residual in the projective hierarchy. We rationalize and demonstrate how and why coupled cluster doubles is ill suited to the strongly correlated limit, whereas the Bessel expansion remains well behaved. The model provides accurate wave functions with energy errors that in its best variant are smaller than 1% across all interaction strengths. The numerical cost is polynomial in system size and the theory can be straightforwardly applied to any realistic Hamiltonian.
The goal of electronic structure theory is to solve the time-independent Schrodinger equation with a method that scales polynomially with the system size. The problem has been solved in the weakly-correlated regime, where coupled cluster theory (CC) with singles, doubles and perturbative triples (CCSD(T)) can achieve chemical accuracy. However, in the strongly-correlated regime, where the two-body interaction dominates, there are still no methods that can achieve a comparable accomplishment. Strong correlation always comes hand in hand with spontaneous symmetry breaking in mean-field theory. For example, coupled cluster theory faces a symmetry dilemma in the strongly correlated regime, where it either completely fails to describe the system if symmetry adapted, or has to artificially break certain symmetries in order to produce reasonable energies. It is plausible that the problem of strong correlation can be solved by curing the symptoms, i.e., fixing the symmetries. Symmetry projection has been carried out at the mean-field level, but it was unclear until this work how to incorporate correlation on top of it because of the language barrier between symmetry projection and coupled cluster theory. This thesis is devoted to combining CC and symmetry projection, focusing on spin and number symmetry. Different flavours of theories are obtained depending on whether a restricted or unrestricted reference is used. On the restricted side, the spin projected unrestricted Hartree Fock (SUHF) state can be rewritten as a polynomial of particle-hole excitations, acting on the restricted reference. Thus, this formalism opens up many ways to combine SUHF with CC. One the unrestricted side, the concept of disentangled cluster is introduced to describe the CC wavefunction transformed by symmetry rotation. A scheme basing on differential equations is devised to approximate these disentangled clusters, which scales polynomially and is systematically improvable. By combining symmetry projection and coupled cluster theory, the symmetry dilemma can be solved to a good extent.
The past several years have seen renewed interest in the use of symmetry-projected Hartree-Fock for the description of strong correlations. Unfortunately, these symmetry-projected mean-field methods do not adequately account for dynamic correlation. Presumably, this shortcoming could be addressed if one could combine symmetry-projected Hartree-Fock with a many-body method such as coupled-cluster theory, but this is by no means straightforward because the two techniques are formulated in very different ways. However, we have recently shown that the singlet S2-projected unrestricted Hartree-Fock wave function can in fact be written in a coupled-cluster-like wave function. That is, the spin-projected unrestricted Hartree-Fock wave function can be written as a polynomial of a double-excitation operator acting on some closed-shell reference determinant. Here, we extend this result and show that the spin-projected generalized Hartree-Fock wave function (which has both S2 and Sz projection) is likewise a polynomial of low-order excitation operators acting on a closed-shell determinant and provide a closed-form expression for the resulting polynomial coefficients. The spin projection of the generalized Hartree-Fock wave function introduces connected triple and quadruple excitations which are absent when spin-projecting an unrestricted Hartree-Fock determinant. We include a few preliminary applications of the combination of this spin-projected Hartree-Fock and coupled-cluster theory to the Hubbard Hamiltonian and comment on generalizations of the methodology. Results here are not for production level, but a similarity-transformed theory that combines the two offers the promise of being accurate for both weak and strong correlation, and may offer significant improvements in the intermediate correlation regime where neither projected Hartree-Fock nor coupled cluster is particularly accurate.