Frank E. Harris

Bio: Frank E. Harris is an academic researcher from University of Florida. The author has contributed to research in topics: Wave function & Hartree–Fock method. The author has an hindex of 43, co-authored 284 publications receiving 7022 citations. Previous affiliations of Frank E. Harris include Harvard University & Uppsala University.

Molecular Orbital Theory

Abstract: Publisher Summary This chapter deals with the formal problem of reducing molecular orbital calculations to expressions involving one- and two-electron integrals over the spatial coordinates, with coefficients determined by the group theoretical properties of the spin functions and the electronic permutations. This problem is encountered, for example, when one undertakes to write the expectation value of the Hamiltonian for a given anti-symmetrized spin-orbital product, and in that particular case, the answer is well-known. The focus is on wave functions, which are constructed to be eigenfunctions of the spin, and shall consider the reduction of expressions not only for the energy and other spin-free one- and two-electron operators, but also for general one- and two-electron spin-dependent operators, such as the spin density or the Fermi contact interaction. It has been shown as how a spin-projected single-determinantal wave function based on different spatial orbitals for different spins can be related to the matrix representation method, and it is shown, how to calculate expectation values of both spin-free and spin-dependent operators.

Mathematical methods for physicists : a comprehensive guide

Abstract: Now in its 7th edition, "Mathematical Methods for Physicists" continues to provide all the mathematical methods that aspiring scientists and engineers are likely to encounter as students and beginning researchers. This bestselling text provides mathematical relations and their proofs essential to the study of physics and related fields. While retaining the key features of the 6th edition, the new edition provides a more careful balance of explanation, theory, and examples. Taking a problem-solving-skills approach to incorporating theorems with applications, the book's improved focus will help students succeed throughout their academic careers and well into their professions. Some notable enhancements include more refined and focused content in important topics, improved organization, updated notations, extensive explanations and intuitive exercise sets, a wider range of problem solutions, improvement in the placement, and a wider range of difficulty of exercises. This book is the revised and updated version of the leading text in mathematical physics. It focuses on problem-solving skills and active learning, offering numerous chapter problems. It includes clearly identified definitions, theorems, and proofs promote clarity and understanding. New to this edition: improved modular chapters; new up-to-date examples; and, more intuitive explanations.

More Special Functions

Abstract: This chapter surveys a number of sets of special functions of importance in physics. Where appropriate, the survey includes generating functions, Rodrigues formulas, the relevant differential equation, orthogonality conditions, and applications. The discussion covers Hermite functions (with applications to the quantum harmonic oscillator and to molecular vibrations), Laguerre functions (with application to the hydrogen atom), Chebyshev polynomials (with application to numerical analysis), hypergeometric and confluent hypergeometric functions, the dilogarithm (with application to electronic structure computations), and elliptic integrals.

Sturm-Liouville Theory

Abstract: The chapter starts by considering ODEs subject to boundary conditions, and then considers the conditions under which the operator defining the ODE, together with the boundary conditions is Hermitian. This analysis, Sturm-Liouville theory, includes techniques for making an operator self-adjoint by multiplying its ODE by a weight factor and making an appropriate definition of the scalar product. The use of specific properties of ODEs to aid in the solution of boundary-value problems is illustrated for the Legendre and Hermite operators, and a two-region problem is used to illustrate the effect of matching conditions. The variation method (common in quantum mechanics) is described and illustrated.

Ab Initio Calculations on 62 Low‐Lying States of the O2 Molecule

Abstract: Ab initio calculations have been made on the 62 low‐lying states of molecular O2 which result from the combination of O atoms in 3P, 1D, and 1S atomic states. The calculations are done at nine different internuclear separations, and potential‐energy curves are presented for all states. Twelve bound states were found: the lowest seven have been observed; two others have been predicted before; three are new. The state ordering agrees with experiment except for the c 1Σu− state. Possible reasons for this discrepancy are discussed. The remaining errors in the bound‐state energy separations are rationalized. Data possibly bearing on the unobserved bound states are cited. Repulsive‐state curves are used to discuss predissociation in the Schumann–Runge bands and to illustrate avoided‐crossing phenomena.

Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen

Abstract: In the past, basis sets for use in correlated molecular calculations have largely been taken from single configuration calculations. Recently, Almlof, Taylor, and co‐workers have found that basis sets of natural orbitals derived from correlated atomic calculations (ANOs) provide an excellent description of molecular correlation effects. We report here a careful study of correlation effects in the oxygen atom, establishing that compact sets of primitive Gaussian functions effectively and efficiently describe correlation effects i f the exponents of the functions are optimized in atomic correlated calculations, although the primitive (s p) functions for describing correlation effects can be taken from atomic Hartree–Fock calculations i f the appropriate primitive set is used. Test calculations on oxygen‐containing molecules indicate that these primitive basis sets describe molecular correlation effects as well as the ANO sets of Almlof and Taylor. Guided by the calculations on oxygen, basis sets for use in correlated atomic and molecular calculations were developed for all of the first row atoms from boron through neon and for hydrogen. As in the oxygen atom calculations, it was found that the incremental energy lowerings due to the addition of correlating functions fall into distinct groups. This leads to the concept of c o r r e l a t i o n c o n s i s t e n t b a s i s s e t s, i.e., sets which include all functions in a given group as well as all functions in any higher groups. Correlation consistent sets are given for all of the atoms considered. The most accurate sets determined in this way, [5s4p3d2f1g], consistently yield 99% of the correlation energy obtained with the corresponding ANO sets, even though the latter contains 50% more primitive functions and twice as many primitive polarization functions. It is estimated that this set yields 94%–97% of the total (HF+1+2) correlation energy for the atoms neon through boron.

Self-interaction correction to density-functional approximations for many-electron systems

Abstract: The exact density functional for the ground-state energy is strictly self-interaction-free (i.e., orbitals demonstrably do not self-interact), but many approximations to it, including the local-spin-density (LSD) approximation for exchange and correlation, are not. We present two related methods for the self-interaction correction (SIC) of any density functional for the energy; correction of the self-consistent one-electron potenial follows naturally from the variational principle. Both methods are sanctioned by the Hohenberg-Kohn theorem. Although the first method introduces an orbital-dependent single-particle potential, the second involves a local potential as in the Kohn-Sham scheme. We apply the first method to LSD and show that it properly conserves the number content of the exchange-correlation hole, while substantially improving the description of its shape. We apply this method to a number of physical problems, where the uncorrected LSD approach produces systematic errors. We find systematic improvements, qualitative as well as quantitative, from this simple correction. Benefits of SIC in atomic calculations include (i) improved values for the total energy and for the separate exchange and correlation pieces of it, (ii) accurate binding energies of negative ions, which are wrongly unstable in LSD, (iii) more accurate electron densities, (iv) orbital eigenvalues that closely approximate physical removal energies, including relaxation, and (v) correct longrange behavior of the potential and density. It appears that SIC can also remedy the LSD underestimate of the band gaps in insulators (as shown by numerical calculations for the rare-gas solids and CuCl), and the LSD overestimate of the cohesive energies of transition metals. The LSD spin splitting in atomic Ni and $s\ensuremath{-}d$ interconfigurational energies of transition elements are almost unchanged by SIC. We also discuss the admissibility of fractional occupation numbers, and present a parametrization of the electron-gas correlation energy at any density, based on the recent results of Ceperley and Alder.

Quantum mechanical continuum solvation models.

Abstract: 6.2.2. Definition of Effective Properties 3064 6.3. Response Properties to Magnetic Fields 3066 6.3.1. Nuclear Shielding 3066 6.3.2. Indirect Spin−Spin Coupling 3067 6.3.3. EPR Parameters 3068 6.4. Properties of Chiral Systems 3069 6.4.1. Electronic Circular Dichroism (ECD) 3069 6.4.2. Optical Rotation (OR) 3069 6.4.3. VCD and VROA 3070 7. Continuum and Discrete Models 3071 7.1. Continuum Methods within MD and MC Simulations 3072

Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions

Abstract: The calculation of accurate electron affinities (EAs) of atomic or molecular species is one of the most challenging tasks in quantum chemistry. We describe a reliable procedure for calculating the electron affinity of an atom and present results for hydrogen, boron, carbon, oxygen, and fluorine (hydrogen is included for completeness). This procedure involves the use of the recently proposed correlation‐consistent basis sets augmented with functions to describe the more diffuse character of the atomic anion coupled with a straightforward, uniform expansion of the reference space for multireference singles and doubles configuration‐interaction (MRSD‐CI) calculations. Comparison with previous results and with corresponding full CI calculations are given. The most accurate EAs obtained from the MRSD‐CI calculations are (with experimental values in parentheses) hydrogen 0.740 eV (0.754), boron 0.258 (0.277), carbon 1.245 (1.263), oxygen 1.384 (1.461), and fluorine 3.337 (3.401). The EAs obtained from the MR‐SD...