Christopher W. Murray

Quadrature schemes for integrals of density functional theory

The evaluation of integrals which arise in density functional theory, as applied to molecules, as discussed. Beckes scheme for reducing them to a sum of integrals over atom based polyhedra is used. Within each of these regions, quadratures for the spherical polar coordinates are examined; in particular we compare a Euler-Maclaurin based scheme with Gauss schemes. Upon specific investigations we find that the Euler-Maclaurin scheme is favoured for radial quadrature and Gauss-Legendre quadrature is preferred for theta. We investigate the number of quadrature points required for a given accuracy, and we demonstrate our favoured approach by calculations on a variety of molecules.

The harmonic frequencies of benzene

Abstract We report calculations for the harmonic frequencies of C6H6 and C6D6. Our most sophisticated quantum chemistry values are obtained with the MP2 method and a TZ2P+f basis set (288 basis functions), which are the largest such calculations reported on benzene to date. Using the SCF density, we also calculate the frequencies using the exchange and correlation expressions of density functional theory. We compare our calculated harmonic frequencies with those deduced from experiment by Goodman, Ozkabak and Thakur. The density functional frequencies appear to be more reliable predictions than the MP2 frequencies and they are obtained at significantly less cost.

Perturbation theory for open shell systems

Abstract Two new forms of perturbation theory are presented that are applicable to open shell systems. They are based entirely on the restricted open shell Hartree-Fock formalism and can be used on the lowest energy of a given symmetry at that geometry. The methods are compared with other approaches and some test cases are presented.

Kohn—Sham bond lengths and frequencies calculated with accurate quadrature and large basis sets

Abstract We report calculations of bond lengths and frequencies using Kohn—Sham theory, defined as replacing the exchange term in the Hathree—Fock self-consistent field procedure by potentials of density functional theory. Several functionals are tested including the local density approximation to the exchange energy. Beckes non-local correction to the exchange, the Vosko—Wilk—Nusair functional for correlation with Perdews non-local correction to the correlation energy and the Lee—Yang—Parr correlation functional. High accuracy quadrature is used, which enables the gradient of the energy to be calculated straightforwardly. The results are compared to Hartree—Fock theory and to hydrid DFT methods based on the Hartree—Fock density. On average, bond lengths from the hybrid method are much better than SCF bond lengths, and often better than those from second-order Moller—Plesset theory. The Kohn—Sham bond lengths are rather long, but improve as the basis set is increased, and for large basis sets bond lengths, dipole moments and frequencies appear on to be a significant improvement over SCF theory.

The determination of hyperpolarisabilities using density functional theory

Abstract The theory for the calculation of the static polarisability α and hyperpolarisability β using density functional theory is presented. In particular the computational implementation of the coupled-perturbed Kohn—Sham equations is discussed. Calculations on CH 2 O and CH 3 CN are reported using large basis sets and accurate quadrature using the local density approximation (S-VWN). The results suggest that DFT is a promising method for the determination of these properties. Finally it is argued that the theory of Vignale, Rasolt and Geldart can be used as a basis for the determination of frequency-dependent polarisabilities.

Improved algorithms for the lowest few eigenvalues and associated eigenvectors of large matrices

Abstract Some modifications of Davidsons eigenvalue algorithm are discussed and their performances on a number of test cases are assessed. They are found to offer improvements over the original algorithm. A method for solving the equations stemming from the quasi degenerate variational perturbation theory is presented. Solutions can be obtained as simply as with the eigenvalue algorithm for both ground and excited states.

A study of O3, S3, CH2, and Be2 using Kohn–Sham theory with accurate quadrature and large basis sets

Density functional calculations with accurate quadrature and large basis sets are reported on ozone, thiozone, methylene, and the beryllium dimer. The Kohn–Sham self‐consistent procedure is used throughout and nonlocal gradient corrected functionals (B‐LYP, B‐P) are used. The systems and properties chosen are notoriously difficult problems for accurate computation, nevertheless, the density functional methods perform well. It is found, in line with calculations on more straightforward problems, that the functionals involving gradient corrections are more reliable than local density functionals. The B‐LYP prediction is within 5 kcal/mol of the most accurate ab initio data on the C2v–D3h energy gap in ozone and thiozone. The B‐LYP functional performs reasonably well on the vibrational frequencies of ozone whilst the B‐P functional gives very accurate results for the structural parameters of ozone. B‐LYP is used to estimate the singlet–triplet splitting in methylene with the triplet treated in both a restric...

Comparison and assessment of different forms of open shell perturbation theory

Existing methods for the calculation of the correlation energy of open shell atoms and molecules using Mo/ller–Plesset perturbation theory are discussed with special emphasis on recent advances in methods based on the restricted Hartree–Fock wave function. The convergence of the Mo/ller–Plesset series is examined to high order for a number of small systems and it is shown that some of the methods offer significant advantage over traditional unrestricted Mo/ller–Plesset theory. The optimized bond length and vibrational frequencies of CN, NO, and O2 are also compared for second and fourth order Mo/ller–Plesset theory, and again it is found that the same methods give much better results than the corresponding unrestricted results. Further supporting calculations are presented on the electron affinity of CN and the dissociation energy of HCN. The new methods involve relatively minor adaptation of existing unrestricted Mo/ller–Plesset (UMP) codes, and in the light of their superior performance are recommended ...

Structures and vibrational frequencies of FOOF and FONO using density functional theory

Abstract Calculations of the equilibrium structure and vibrational frequencies of FOOF using the local density approximation are in good agreement with experimental results. However using a theoretically more accurate gradient corrected (non-local) density functional produces a worse structure. Three isomers of FONO are also studied. The geometry of C 2v isomer FNO 2 is predicted accurately by the local density approximation, with gradient corrected functions again giving a poorer structure, but better vibrational frequencies. The structure of the trans-isomer of FONO is in agreement with recent coupled cluster studies, however calculations on cis-FONO disagree with the coupled cluster results, but may be in better agreement with the experimental geometry.

Theoretical calculation of the photoelectron spectrum of ethylene

Abstract A theoretical study of the carbon 2s region of the photoelectron spectrum of ethylene is reported. The calculations are much larger than those previously performed and are successful in qualitatively describing the spectrum.