James Emil Avery

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207

Program Fullerene: A software package for constructing and analyzing structures of regular fullerenes

Fullerene (Version 4.4) is a general purpose open‐source program that can generate any fullerene isomer, perform topological and graph theoretical analysis, as well as calculate a number of physical and chemical properties. The program creates symmetric planar drawings of the fullerene graph and generates accurate molecular 3D geometries by way of force‐field optimization, serving as a good starting point for further quantum theoretical treatments. It includes a number of fullerene‐to‐fullerene transformations, such as Goldberg–Coxeter transforms, Stone–Wales transforms, Endo–Kroto, Yoshida–Fowler, and Brinkmann–Fowler vertex insertions. The program is written in standard Fortran and C++ and can easily be installed in a Linux or UNIX environment.

Size-Change termination and bound analysis

Despite its simplicity, the size-change termination principle, presen- ted by Lee, Jones and Ben-Amram in [LJB01], is surprisingly strong and is able to show termination for a large class of programs. A significant limitation for its use, however, is the fact that the SCT requires data types to be well-founded, and that all mechanisms used to determine termination must involve decreases in these global, well-founded partial orders. Following is an extension of the size-change principle that allows for non-well founded data types, and a realization of this principle for integer data types. The extended size-change principle is realized through combining abstract interpretation over the domain of convex polyhedra with the use of size-change graphs. In the cases when data types are well founded, the method handles every case that is handled by LJB size-change termination. The method has been implemented in a subject language independent shared library, libesct (available at [Ave05a]), as well as for the ANSI C specializer C−MixII, handling a subset of its internal language Core-C.

The Generalized Sturmian Method for Calculating Spectra of Atoms and Ions

The properties of generalized Sturmian basis sets are reviewed, and functions of this type are used to perform direct configuration interaction calculations on the spectra of atoms and ions. Singlet excited states calculated in this way show good agreement with experimentally measured spectra. When the generalized Sturmian method is applied to atoms, the configurations are constructed from hydrogenlike atomic orbitals with an effective charge which is characteristic of the configuration. Thus, orthonormality between the orbitals of different configurations cannot be assumed, and the generalized Slater–Condon rules must be used. This aspect of the problem is discussed in detail. Finally spectra are calculated in the presence of a strong external electric field. In addition to the expected Stark effect, the calculated spectra exhibit anomalous states. These are shown to be states where one of the electrons is primarily outside the atom or ion, with only a small amplitude inside.

Coulomb Sturmians as a basis for molecular calculations

Almost all modern quantum chemistry programs use Gaussian basis sets even though Gaussians cannot accurately represent the cusp at atomic nuclei, nor can they represent the slow decay of the wave function at large distances. The reason that Gaussians dominate quantum chemistry today is the great mathematical difficulty of evaluating interelectron repulsion integrals when exponential-type orbitals (ETOs) are used. In this paper we show that when many-centre Coulomb Sturmian ETOs are used as a basis, the most important integrals can be evaluated rapidly and accurately by means of the theory of hyperspherical harmonics. For the remaining many-centre integrals, Coulomb Sturmians are shown to have advantages over other ETOs. Pilot calculations are performed on N-electron molecules using the Generalized Sturmian Method.

Structure and Properties of the Nonface-Spiral Fullerenes T -C 380 , D 3 -C 384 , D 3 -C 440 , and D 3 -C 672 and Their Halma and Leapfrog Transforms.

The structure and properties of the three smallest nonface-spiral (NS) fullerenes NS-T-C₃₈₀, NS-D₃-C₃₈₄, NS-D₃-C₄₄₀, and the first isolated pentagon NS-fullerene, NS-D₃-C₆₇₂, are investigated in detail. They are constructed by either a generalized face-spiral algorithm or by vertex insertions followed by a force-field optimization using the recently introduced program Fullerene. The obtained structures were then further optimized at the density functional level of theory and their stability analyzed with reference to Ih-C₆₀. The large number of hexagons results in a higher stability of the NS-fullerenes compared to C60, but, as expected, in a lower stability than most stable isomers. None of the many investigated halma transforms on nonspiral fullerenes, NS-T-C₃₈₀, NS-D₃-C₃₈₄, NS-D₃-C₄₄₀, and NS-D₃-C₆₇₂, admit any spirals, and we conjecture that all halma transforms of NS-fullerenes belong to the class of NS-fullerenes. A similar result was found to not hold for the related leapfrog transformation. We also show that the first known NS-fullerene with isolated pentagons, NS-D₃-C₆₇₂, is a halma transform of D3-C₁₆₈.

Can Coulomb Sturmians Be Used as a Basis for N-Electron Molecular Calculations?

A method is proposed for using isoenergetic configurations formed from many-center Coulomb Sturmians as a basis for calculations on N-electron molecules. Such configurations are solutions to an approximate N-electron Schrodinger equation with a weighted potential, and they are thus closely analogous to the Goscinskian configurations that we have used previously to study atomic spectra. We show that when the method is applied to diatomic molecules, all of the relevant integrals are pure functions of the parameter s = kR, and therefore they can be evaluated once and for all and stored.

Fusion of Parallel Array Operations

We address the problem of fusing array operations based on criteria such as shape compatibility, data reuse, and minimizing communication. We formulate the problem as a partitioning problem (WSP) that is general enough to handle loop fusion, combinator fusion, and other types of fusion analysis. Traditionally, when optimizing for data reuse, the fusion problem has been formulated as a static weighted graph partitioning problem (known as the Weighted Loop Fusion problem). We show that this scheme cannot accurately track data reuse between multiple independent loops, since it overestimates total data reuse of certain cases. Our formulation in terms of partitions allows use of realistic cost functions that can track resource usage accurately. We give correctness proofs, and prove that WSP can maximize data reuse in programs exactly, in contrast to prior work. For the exact optimal solution, which is NP-hard to find, we present a branch-and-bound algorithm together with a polynomial-time preconditioner that reduces the problem size significantly in practice. We further present a polynomialtime greedy approximation that is fast enough to use for JIT-compilation and gives near-optimal results in practice. All algorithms have been implemented in the automatic parallelization platform Bohrium, run on a set of benchmarks, and compared to existing methods from the literature.

Molecular integrals for slater type orbitals using coulomb sturmians

The use of Slater type orbitals in molecular calculations is hindered by the slowness of integral evaluation. In the present paper, we introduce a method for overcoming this problem by expanding STO’s in terms of Coulomb Sturmians, for which the problem of evaluating molecular integrals rapidly has been satisfactorily solved using methods based on the theory of hyperspherical harmonics.

Natural Orbitals from Generalized Sturmian Calculations

The generalized Sturmian method is a direct configuration interaction method for solving the Schrodinger equation of a many-electron system. The configurations in the basis set are solutions to an approximate Schrodinger equation with a weighted potential βνV0(x), the weighting factors βν being chosen in such a way as to make the set of solutions isoenergetic. The method is illustrated by calculation of atomic excited states and used to generate natural orbitals.