## Date of Award

January 2023

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Chemistry

## First Advisor

Mark Hoffmann

## Abstract

The focal point of this dissertation is unitary transformation containing an important class of operators and their representations. The unitary operators hold a prominent position in quantum chemistry because they can be employed to describe transformations (rotations) between orthonormal bases and leave invariant essential properties of a system (e.g., trace of a matrix and inner product between two vectors). This dissertation investigates computational studies of both energy optimizing and non-optimizing unitary operators. Liberal use of elements of differential geometry is made, which allowed development of second-order (or Hessian) based approaches on a manifold. The study of unitary operators also provided a framework for investigation of application of Symmetry Renormalization Group to questions in quantum chemistry.A set of orthonormal spin orbitals constructs the Fock space. In many circumstances – for instance, during the optimization of an electronic state or in the calculation of an electronic state experiencing an external perturbation – it is required to perform transformation between different sets of orthonormal spin orbitals by the unitary operators. In second quantization language, the unitary transformations can be applied by the exponential of an anti-Hermitian operator, constructed as a linear combination of excitation operators. The stationary point on an energy hypersurface is achieved during the optimization of an electronic state using the unitary transformations when the first variation in energy is zero. For an arbitrary unitary transformation of the spin orbitals, any expectation value is invariant at a minimum stationary point. That is, the spin orbitals which make the total energy stationary are not unique, thus unitary transformations of the spin orbitals can be employed to produce the canonical Hartree-Fock equations. In this concept, producing canonical spin orbitals (or Hamiltonian diagonalization) and localized spin orbitals are considered as redundant orbital rotations. The importance of Localized Molecular Orbitals in correlation treatments beyond mean-field calculation and in the illustration of chemical bonding (and antibonding) can hardly be overstated. However, generation of orthonormal localized occupied MOs is significantly more straightforward than obtaining orthonormal localized virtual MOs. Orthonormal molecular orbitals allow facile use of highly efficient group theoretical methods (e.g., graphical unitary group approach) for calculation of Hamiltonian matrix elements in multireference configuration interaction calculations (such as MRCISD) and in quasidegenerate perturbation treatments, such as the Generalized Van Vleck Perturbation Theory. Moreover, localized MOs can elucidate qualitative understanding of bonding in molecules, in addition to high accuracy quantitative descriptions. We adopt the powers of the fourth moment cost function introduced by Jørgensen and coworkers. Because the fourth moment cost functions are prone to having multiple negative Hessian eigenvalues when starting from easily available canonical (or near canonical) MOs, standard optimization algorithms can fail to obtain the orbitals of the virtual or partially occupied spaces. To overcome this drawback, we applied Trust Region algorithm on orthonormal Riemannian manifold with an approximate retraction from the tangent space built into first and second derivatives of the cost function. Moreover, the Riemannian Trust Region outer iterations were coupled to truncated conjugate gradient inner loops, which avoided any costly solutions of simultaneous linear equations or eigenvector/eigenvalue solutions. Numerical examples are provided on model systems, including the high connectivity H10 set in 1-, 2- and 3-dimensional arrangements, and on a chemically realistic description of cyclobutadiene (c-C4H4), and the propargyl radical (C3H3). The assessment is made in two different approaches: application of the algorithm on (i) the separated occupied and virtual blocks at HF level, and (ii) the active space at MCSCF level of theory. Complementing the theoretical and algorithmic studies of transformations of relevance to quantum chemistry are studies of specific molecular species that have highly nontrivial partitioning of orbitals into doubly occupied, unoccupied and variably occupied. These studies use contemporary best practices for methodology, with consideration of both accuracy and computational resource requirements (i.e. “costs”). In this monograph, finding the equilibrium and transition geometries of cyclobutane-dicarboxylic acids (CBDAs), along with determination of the low-lying electronic states of NdO and NdS are examples of nonredundant orbital rotations, while application of fourth central moment localization on cyclobutadiene and the propargyl radical are illustrations of redundant orbital rotations. For CBDAs, the thermal dissociation mechanism of α-truxillic acid (CBDA-1) and β-truxinic acid (CBDA-4) has been calculated by density functional theory (DFT) and the multireference Møller-Plesset (MRMP) methods to explain experimental findings. Based on uB3LYP/6-31G(d,p) and uMPW1K/6-31G(d,p), the reaction for CBDA-1 takes place through one step and two C-C bonds break at the same time. In contrast this dissociation reaction for CBDA-4 happens through a stepwise mechanism and two C-C bonds break through a transient diradical singlet intermediate. MRMP2/6-311+G(d,p) and uMPW1K/6-311+G(d,p) single point calculations were performed to estimate Gibbs free energy barriers of these dissociation reactions. In MRMP2 technique, four orbitals correlate four electrons (4e,4o) for CBDA-1, and two orbitals for CBDA-4 construct an active space for two electrons (2e,2o). For NdO and NdS molecules, potential energy curves (PECs) were calculated for 21 and 18 low-lying electronic states, respectively. In each case, static electron correlation effects were described by incomplete model space multiconfiguration self-consistent field (MCSCF) wave functions based on an active space that included the most important valence orbitals and dynamic electron correlation was treated by the multireference second-order generalized Van Vleck perturbation theory (GVVPT2). Scalar relativistic contributions were included by the effective core potential (ECP) approach with relevant basis sets. The 21 and 18 electronic states of NdO and NdS were predicted to be in the excitation energy range of ~3.2 eV and ~2.7 eV, respectively. The ground electronic states of NdO and NdS were determined as 15H (6s4f_σ 4f_φ 4f_δ) and 15H (4f_φ 4f_π 4f_π 6s), with spectroscopic constants: bond length R_e=1.780 Å and 2.325 Å, and harmonic frequency ω_e=891 〖cm〗^(-1) and 538 〖cm〗^(-1), respectively.

## Recommended Citation

Sepehri, Aliakbar, "Unitary Transformation And Trust-Region Optimization Of Localized Orbitals Using Riemannian Manifolds" (2023). *Theses and Dissertations*. 5267.

https://commons.und.edu/theses/5267