Compressive sampling in configuration interaction wavefunctions

The problem is considered of generating approximate quantum-mechanical wavefunctions that have as many as possible coefficients held exactly to zero for a given desired accuracy. Two approaches are adopted. In the first, perturbation theory within the Davidson diagonalisation algorithm is used to mask off small coefficients in the wavefunction against a predefined target energy threshold. Second, sparsity is introduced by penalty-function optimisation, with a norm-based compressive-sampling penalty function that decreases with increasing sparsity. The first approach is found to be robust and reliable, whereas the second does not succeed in keeping the wavefunction sparse

A Continuum from Medieval Literary Networks to Modern Counterparts: The Attractions and Operations of Social Networks.

Collaborative doctorate award. The project included three industry placements with Antenna International.While the benefits of analysing social networks within the wider humanities are becoming more accepted, very little work of this kind has been done in medieval studies. This thesis seeks to begin to fill this lacuna by considering the advantages of examining historical moments through the lens of ‘network’. Focusing on the later medieval world (in particular c.1300-1520), but also drawing on parallel evidence from the modern day, it demonstrates how the paradigm of ‘network’ allows a more nuanced reading of, predominantly literary, historical moments, which in turn reveals a deeper understanding of collective social thinking and behaviour. This new methodological approach is threefold, drawing on analytic tools from various disciplines. It blends historical contextual investigation with literary analysis, and frames the results in the sociological and anthropological theories of belonging, exchange, and play. The thesis is structured around four case studies, each of which demonstrates a particular form of network formation, and also shows how far these networks reflect their respective cultural milieus and influences. Three medieval chapters focus on what I term ‘literary networks’, a concept ripe for network analysis thanks to the highly participatory nature of medieval literature, and thus theoretically comparable to modern networks based around information exchange. Across the thesis, instances of formal, informal, and virtual networks are considered from medieval France and England, as well as the twenty-first century West. This combination of interdisciplinary method and structure allows innovative new readings of underappreciated sources, whilst also highlighting a transhistorical continuum of universal appeals to social networks: namely, the satisfaction of the human need to belong, the facilitation of competitive play, and the opportunity to acquire social capital and build reputations. This investigative synthesis between medieval material and more modern network evidence reveals that, while realised through unrecognisably altered technologies and experiencing some resultant disruptions, these fundamental appeals of social network membership, in part, remain constant between the two periods.REACT, AHR

Application of the quasi-variational coupled cluster method to the nonlinear optical properties of model hydrogen systems

We present a pilot application of the recently proposed quasi-variational coupled cluster method to the energies, polarizabilities, and second hyperpolarizabilities of model hydrogen chains. Relative to other single-reference methods of equivalent computational complexity, we demonstrate this method to be highly robust and especially useful when traditional coupled cluster theory fails to perform adequately. In particular, our results indicate it to be a suitable method for the black-box treatment of multiradicals, making it of widespread general interest and applicability

Approximate variational coupled cluster theory

We show that it is possible to construct an accurate approximation to the variational coupled cluster method, limited to double substitutions, from the minimization of a functional that is rigorously extensive, exact for isolated two-electron subsystems and invariant to transformations of the underlying orbital basis. This approximate variational coupled cluster theory is a modification and enhancement of our earlier linked pair functional theory. It is first motivated by the constraint that the inverse square root of the matrix that transforms the cluster amplitudes must exist. Low-order corrections are then included to enhance the accuracy of the approximation of variational coupled cluster, while ensuring that the computational complexity of the method never exceeds that of the standard traditional coupled cluster method. The effects of single excitations are included by energy minimization with respect to the orbitals defining the reference wavefunction. The resulting quantum chemical method is demonstrated to be a robust approach to the calculation of molecular electronic structure and performs well when static correlation effects are strong

Rigorously extensive orbital-invariant renormalized perturbative triples corrections from quasi-variational coupled cluster theory

We show that, by making use of the linked tensor objects inherent to the approach, Orbital-optimised Quasi-Variational Coupled Cluster Theory (OQVCCD) leads naturally to a computationally-trivial, rigorously extensive, and orbital-invariant renormalization of the standard (T) correction for the perturbative inclusion of the effects of connected triple excitations. The resulting prototype method, renormalized perturbative triple OQVCCD (R-OQVCCD(T)), is demonstrated to predict potential energy curves for single bond-breaking processes of significantly higher accuracy than OQVCCD with the standard perturbative triple-excitation correction (OQVCCD(T)) itself, and to be in good numerical correspondence with the existing renormalized (R-CCSD(T)) and completely renormalized (CR-CCSD(T)) coupled-cluster singles doubles triples methods, while continuing to provide descriptions of multiple bond-breaking processes of OQVCCD(T) quality

Breaking multiple covalent bonds with Hartree-Fock-based quantum chemistry: quasi-variational coupled cluster theory with perturbative treatment of triple excitations

We enhance the recently proposed Optimized-orbital Quasi-Variational Coupled Cluster Doubles (OQVCCD) method for the calculation of ground-state molecular electronic structure by augmenting it with the standard perturbative (T) correction for the effects of connected triple excitations. We demonstrate the OQVCCD(T) ansatz to be outstandingly robust and accurate in the description of the breaking of the triple bond in diatomic nitrogen, N2, where traditional CCSD and CCSD(T) completely fail, yet with a computational cost that is nearly the same as that of CCSD(T). This result provides insight into the failure of CCSD(T) and related methods and how it may be overcome

Quasi-variational coupled cluster theory

We extend our previous work on the construction of new approximations of the variational coupled cluster method. By combining several linked pair functional transformations in such a way as to give appropriately balanced infinite-order contributions, in order to approximate 〈〉L well at all orders, we formulate a new quantum chemical method, which we name quasi-variational coupled cluster. We demonstrate this method to be particularly robust in the regime of strong static electron correlation, improving significantly on our earlier approximate variational coupled cluster approach

A linked electron pair functional

A modification of the variational configuration interaction functional in the first-order interacting space for molecular electronic structure is presented. The modified functional is a fully linked expression that by construction is extensive and invariant to transformations of the underlying orbital basis and is exact for an ensemble of separated two-electron subsystems. In addition, an approximation to variational coupled cluster is generated through truncation of the exponential cluster operator. When combined, these methods demonstrate accuracy that exceeds that of the standard coupled-cluster method, in particular in situations where the reference Slater determinant is not a good approximation

Benchmark studies of variational, unitary and extended coupled cluster methods

Comparative benchmark calculations are presented for coupled cluster theory in its standard formulation, as well as variational, extended, and unitary coupled cluster methods. The systems studied include HF, N2, and CN, and with cluster operators that for the first time include up to quadruple excitations. In cases where static correlation effects are weak, the differences between the predictions of molecular properties from each theory are negligible. When, however, static correlation is strong, it is demonstrated that variational coupled cluster theory can be significantly more robust than the traditional ansatz and offers a starting point on which to base single-determinant reference methods that can be used beyond the normal domain of applicability. These conclusions hold at all levels of truncation of the cluster operator, with the variational approach showing significantly smaller errors

Nonuniqueness of algebraic first-order density-matrix functionals

By explicit construction of counterexamples having the same eigenvalue spectrum of one-matrix, but different two-matrix, we show that density-matrix functionals for the electronic energy that are based solely on the eigenvalues of the one-matrix cannot be unique in functional representation of the two-matrix. The one-to-many mapping may be understood either through the number of independent parameters or the contraction relation