The canonical transformation (CT) theory has been developed as a multireference electronic structure method to compute high-level dynamic correlation on top of a large active space reference treated with the ab initio density matrix renormalization group method. This article describes a parallelized algorithm and implementation of the CT theory to handle large computational demands of the CT calculation, which has the same scaling as the coupled cluster singles and doubles theory. To stabilize the iterative solution of the CT method, a modification to the CT amplitude equation is introduced with the inclusion of a level shift parameter. The level-shifted condition has been found to effectively remove a type of intruder state that arises in the linear equations of CT and to address the discontinuity problems in the potential energy curves observed in the previous CT studies.
The canonical transformation (CT) theory has been developed as a multireference electronic structure method to compute high-level dynamic correlation on top of a large active space reference treated with the ab initio density matrix renormalization group method. This article describes a parallelized algorithm and implementation of the CT theory to handle large computational demands of the CT calculation, which has the same scaling as the coupled cluster singles and doubles theory. To stabilize the iterative solution of the CT method, a modification to the CT amplitude equation is introduced with the inclusion of a level shift parameter. The level-shifted condition has been found to effectively remove a type of intruder state that arises in the linear equations of CT and to address the discontinuity problems in the potential energy curves observed in the previous CT studies.
We describe the joint application of the density matrix renormalization group and canonical transformation theory to multireference quantum chemistry. The density matrix renormalization group provides the ability to describe static correlation in large active spaces, while the canonical transformation theory provides a high-order description of the dynamic correlation effects. We demonstrate the joint theory in two benchmark systems designed to test the dynamic and static correlation capabilities of the methods, namely, (i) total correlation energies in long polyenes and (ii) the isomerization curve of the [Cu(2)O(2)](2+) core. The largest complete active spaces and atomic orbital basis sets treated by the joint DMRG-CT theory in these systems correspond to a (24e,24o) active space and 268 atomic orbitals in the polyenes and a (28e,32o) active space and 278 atomic orbitals in [Cu(2)O(2)](2+).
Canonical transformation (CT) theory describes dynamic correlation in multireference systems with large active spaces. Here we discuss CT theory's intruder state problem and why our previous approach of overlap matrix truncation becomes infeasible for sufficiently large active spaces. We propose the use of strongly and weakly contracted excitation operators as alternatives for dealing with intruder states in CT theory. The performance of these operators is evaluated for the H(2)O, N(2), and NiO molecules, with comparisons made to complete active space second order perturbation theory and Davidson-corrected multireference configuration interaction theory. Finally, using a combination of strongly contracted CT theory and orbital-optimized density matrix renormalization group theory, we evaluate the singlet-triplet gap of free base porphin using an active space containing all 24 out-of-plane 2p orbitals. Modeling dynamic correlation with an active space of this size is currently only possible using CT theory.
This article presents an efficient and parallelized implementation of the density matrix renormalization group (DMRG) algorithm for quantum chemistry calculations. The DMRG method as a large-scale multireference electronic structure model is by nature particularly efficient for one-dimensionally correlated systems, while the present development is oriented toward applications for polynuclear transition metal compounds, in which the macroscopic one-dimensional structure of electron correlation is absent. A straightforward extension of the DMRG algorithm is proposed with further improvements and aggressive optimizations to allow its application with large multireference active space, which is often demanded for metal compound calculations. Special efficiency is achieved by making better use of sparsity and symmetry in the operator and wave function representations. By accomplishing computationally intensive DMRG calculations, the authors have found that a large number of renormalized basis states are required to represent high entanglement of the electron correlation for metal compound applications, and it is crucial to adopt auxiliary perturbative correction to the projected density matrix during the DMRG sweep optimization in order to attain proper convergence to the solution. Potential energy curve calculations for the Cr(2) molecule near the known equilibrium precisely predicted the full configuration interaction energies with a correlation space of 24 electrons in 30 orbitals [denoted by (24e,30o)]. The energies are demonstrated to be accurate to 0.6mE(h) (the error from the extrapolated best value) when as many as 10,000 renormalized basis states are employed for the left and right DMRG block representations. The relative energy curves for [Cu(2)O(2)](2+) along the isomerization coordinate were obtained from DMRG and other correlated calculations, for which a fairly large orbital space (32e,62o) is modeled as a full correlation space. The DMRG prediction nearly overlaps with the energy curve from the coupled cluster with singles, doubles, and perturbative triple [CCSD(T)] calculations, while the multireference complete active space self-consistent field (CASSCF) calculations with the small reference configuration (8e,8o) are found to overestimate the biradical character of the electronic state of [Cu(2)O(2)](2+) according to the one-electron density matrix analysis.
In this thesis, we will describe several extensions of the Density Matrix Renormalization Group (DMRG) and Canonical Transformation Theory (CT). In the first part we describe a new way to solve for excited states in DMRG. To overcome the limitations of the traditional state-averaging approaches in excited state calculations, where one solves for all states between the ground state and excited state of interest, we have investigated a number of new excited state algorithms. Building on the work of van der Vorst and Sleijpen (SIAM J. Matrix Anal. Appl., 17, 401 (1996)), we have implemented Harmonic Davidson and State-Averaged Harmonic Davidson algorithms within the context of DMRG. We have assessed their accuracy and stability of convergence in complete active space DMRG calculations on the low-lying excited states in the acenes ranging from naphthalene to pentacene. We find that both algorithms offer increased accuracy over the traditional State-Averaged Davidson approach, and in particular, the State-Averaged Harmonic Davidson algorithm offers an optimal combination of accuracy and stability in convergence. In the second part, we propose an analytic response theory for DMRG whereby response properties correspond to analytic derivatives of DMRG observables with respect to the applied perturbations. Both static and frequencydependent response theories are formulated and implemented. We evaluate our pilot implementation by calculating static and frequency-dependent polar- isabilities of short oligo-di-acetylenes. The analytic response theory is competitive with dynamical DMRG methods and yields significantly improved accuracies when using a small number of density matrix renormalisation group states. Strengths and weaknesses of the analytic approach are discussed. In the third part, we describe how to calculate density matrices in CT theory. Density matrices are useful not only for computing observables but also for characterizing the nature of individual states. We demonstrate this with a preliminary application of the CT theory to understand the low-lying excited states of oligo-phenylvinylenes. We finish by presenting the theory and equations for calculating the response (e.g. to an external field) in the CT and DMRG-CT theories.
We propose the use of parameter-free preentanglers as initial states for quantum algorithms. We apply this idea to the electronic structure problem, combining a quantized version of the Canonical Transformation by Yanai and Chan [J. Chem. Phys. 124, 194106 (2006)] with the Complete Active Space Density Matrix Renormalization Group. This new ansatz allows to shift the computational burden between the quantum and the classical processor. In the vicinity of multi-reference points in the potential energy surfaces of H2O, N2, BeH2 and the P4 system, we find this strategy to require significantly less parameters than corresponding generalized unitary coupled cluster circuits. We propose a new algorithm to prepare Matrix Product States based on the Linear Combination of Unitaries and compare it to the Sequential Unitary Algorithm proposed by Ran in [Phys. Rev. A 101, 032310 (2020)].
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.