Charles Weatherford | Florida A&M University (original) (raw)

Papers by Charles Weatherford

International Journal of Quantum Chemistry, 2004

ABSTRACT The strategy of using expansions in spherical harmonics of displaced Slater-type (exp(−α... more ABSTRACT The strategy of using expansions in spherical harmonics of displaced Slater-type (exp(−αx)) orbitals to evaluate multicenter molecular integrals is applied to three-centered nuclear attraction integrals (potential due to the product of separated Slater-type orbitals). Closed formulae are produced for nine terms of an infinite series for interior region, (inside − ipot) r2 < a, and 14 terms for Exterior Region, (outside − epot) r2 > a. The value a = 2 is chosen for the interior region and r = 14 for the exterior region. Our focus is to obtain a closed formula for the Slater formula. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Molecular Physics, 2006

The integral bottleneck in evaluating molecular energies arises from the two-electron contributio... more The integral bottleneck in evaluating molecular energies arises from the two-electron contributions. These are difficult and time-consuming to evaluate, especially over exponential type orbitals, used here to ensure the correct behaviour of atomic orbitals. In this work, it is ...

The computational theory for calculation of the solution of the time-dependent Schrödinger equati... more The computational theory for calculation of the solution of the time-dependent Schrödinger equation for two electrons [C.A. Weatherford, Computational Chemistry: Reviews of Current Trends, Vol. 5, ed. J. Leszczynski, World Scientific 2000] is reviewed and adapted to the case of the one-dimensional helium atom. This results in a new computational time-dependent exchange/correlation theory. A solution algorithm which discretizes space using a spectral discrete variable basis of synthetic cartesian polynomials, and discretizes time using a spectral element discrete variable basis of Chebyshev polynomials, is presented. Supported by NSF CREST grant HRD-9707076, and by NASA grant NAG5-10148.

Technology & Innovation, 2012

ABSTRACT Fusion research is a key element in the nation's long-term energy supply strateg... more ABSTRACT Fusion research is a key element in the nation's long-term energy supply strategy. Florida A&M University's Center for Plasma Science and Technology (FAMU CePaST) is building a spheromak fusion device, where fusion energy research is on the forefront of clean energy technology. The Spheromak Turbulent Physics Experiment (STPX) is in a class of experiments used to investigate the physics principles of self-organized plasmas. This is relevant for the development of an economically feasible, controlled fusion reactor as a source of electrical power. The project involves collaboration between FAMU CePaST, West Virginia University, and Auburn University. This project will determine, using theory, experiment, and simulation, the essential elements required for full kinetic modeling of an entire spheromak plasma using ab initio magnetohydrodynamic (MHD) with direct modifications from new turbulence physics. High time resolution measurements of electron temperatures, ion temperatures, and magnetic field fluctuations will be used to study turbulence during magnetic reconnection, anomalous ion heating, dusty plasmas, and equilibrium modification. This article assesses the clean energy technology of the spheromak device, the plasma regimes that a spheromak will provide, and discusses its role in understanding turbulence, an essential part of fusion.

Computational Chemistry: Reviews of Current Trends, 2000

International Journal of Quantum Chemistry, 1999

A form of classical electrodynamic field exists which gives exact agreement with the operator fie... more A form of classical electrodynamic field exists which gives exact agreement with the operator field of quantum electrodynamics (QED) for the Lamb shift of a harmonically bound point electron. Here it is pointed out that this form of classical theory with its physically acceptable interpretation is the result of an unconventional resolution of a mathematically ambiguous term in classical field theory. Finally a quantum-classical correspondence principle is shown to exist in the sense that the classical field and expectation value of the QED operator field are identical if retardation is neglected in the latter.

International Journal of Quantum Chemistry, 1999

A form of classical electrodynamic field exists which gives exact agreement with the operator fie... more A form of classical electrodynamic field exists which gives exact agreement with the operator field of quantum electrodynamics (QED) for the Lamb shift of a harmonically bound point electron. Here it is pointed out that this form of classical theory with its physically acceptable interpretation is the result of an unconventional resolution of a mathematically ambiguous term in classical field theory. Finally a quantum-classical correspondence principle is shown to exist in the sense that the classical field and expectation value of the QED operator field are identical if retardation is neglected in the latter.

International Journal of Quantum Chemistry, 1984

International Journal of Quantum Chemistry, 2009

International Journal of Quantum Chemistry, 2009

ABSTRACT In this article, I consider the use of the Kohn variational principle in obtaining the r... more ABSTRACT In this article, I consider the use of the Kohn variational principle in obtaining the reaction matrix, which determines the positive energy solution to the partial wave Schrödinger equation in coordinate space. The formulation considered is appropriate for electron scattering from an atom or molecule. In its usual form, the Kohn variational principle provides the relationship between the exact reaction matrix, independent of first-order errors in the trial wavefunction, and the trial reaction matrix. This relationship is usually called the Kato identity. It is obtained using the second form of Green's theorem for a sphere of infinite radius. I show how a Kohn-like variational principle, using Green's theorem for a sphere of arbitrary radius, provides an expression for the exact, coordinate space, integral equation amplitudes in terms of the trial amplitudes. The results are again exact to second order in the trial wavefunction. Applications of the present results to electron-molecule frame transformations and algebraic scattering theory are then suggested.

In a recent article [C.A. Weatherford, E. Red, A. Wynn III, International Journal of Quantum Chem... more In a recent article [C.A. Weatherford, E. Red, A. Wynn III, International Journal of Quantum Chemistry 90, 1289-1294 (2002)], an algorithm was described whereby a synthetic weighted polynomial basis may be constructed which is adapted (designed) to a particular potential. It was applied therein to the Schroedinger equation with a coulomb potential in one dimension (-1/|x| ). A weighted polynomial basis with weight function w(x)=exp(-a|x|) was employed. It was observed that this potential had no even parity solutions - only odd parity solutions. The question arises as to the relationship of the solutions (eigenfunctions and eigenvalues) for this hard coulomb potential to the solutions for the soft coulomb potential (-1/ √x^2+b^2^1/2 ). In particular, since the soft coulomb potential is clearly expected to possess both even and odd parity solutions, how do these solutions behave as b->0 and thus what happens to the even solutions. This problem is deceptively difficult none of the standard basis sets produce a variational minimum as a function of 'a' for nonzero 'b'. This is apparently why this problem has never been done before. A new orthonormal basis was designed with weight function w(x)=exp(-a√x^2+b^2) which did produce a variational minimum for variable a and arbitrary fixed 'b'. The present paper describes these solutions and clearly indicates how they behave as b->0 .

The evaluation of a matrix exponential function is a classic problem of computational linear alge... more The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review20 4], none of which produce a definitive algorithm which is broadly applicable and sufficiently accurate, as well as being reasonably fast. Herein, we employ a method which evaulates a matrix exponential as the solution to a first-order initial value problem in a pseudo-time variable. The new aspect of the present implementation of this method is to use finite elements in the pseudo-time variable. [Weatherford C A, Red E, and Wynn A 2002 Journal of Molecular Structure592 47] Then using an expansion in a properly chosen pseudo-time basis, we are able to make accurate calculations of the exponential of any given matrix as the solution to a set of simultaneous equations, even when the matrix is singular or rectangular.

The implementation of the Dirac representation is facilitated by the finite element space-time al... more The implementation of the Dirac representation is facilitated by the finite element space-time algorithm.[1] Multicenter integral computations are also facilitated by this same algorithm. The present work is the first application of this original algorithm to the computation of bound states of atoms and molecules. The Dirac representation is employed such that H0 is the sum of the one-electron operators while the residual H1 is the sum of the two-electron operators. Soft-Coulomb geminals are then used as the basis for the time-dependent calculation of a superposition of the bound-states. The eigenstates and eigenvalues are then extracted by filter-diagonalization. An addition theorem is given for the soft-coulomb geminals and the geminals are translated again using the space-time algorithm, so that multicenter integrals may be computed. Several small atoms and molecules are considered as an illustration of the method. [1]D.H. Gebremedhin, C.A. Weatherford, X. Zhang, A. Wynn III, and G. Tanaka, ``Evaluation of the matrix exponential function using finite elements in time,'' arXiv:0811.2612v1 [math-ph] 17 Nov 2008.

Molecular Physics, 2005

A new technique is presented for the solution of Poisson's equati... more A new technique is presented for the solution of Poisson's equation in spherical coordinates. The method employs an expansion of the solution in a new set of functions defined herein for the first time, called ‘spectral forms’. The spectral forms have spherical harmonics as their angular part, but use a new set of radial functions that automatically statisfy the boundary

International Journal of Quantum Chemistry, 2004

International Journal of Quantum Chemistry, 2004

ABSTRACT A technique is given for solving the time-dependent Schrödinger equation for electron+po... more ABSTRACT A technique is given for solving the time-dependent Schrödinger equation for electron+potential scattering at low energies (k2 ≤ 1.0 a.u.). This is accomplished by avoiding wavepackets, with the inevitable wavepacket spreading, and solving a time-integral form of the Schrödinger equation directly. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

International Journal of Quantum Chemistry, 2008

To develop a viable detection system for 2,4-dinitrotoluene (DNT), its unimolecular decomposition... more To develop a viable detection system for 2,4-dinitrotoluene (DNT), its unimolecular decomposition mechanisms were studied by using the density functional theory. Three decomposition mechanisms were investigated: (1) homolytic cleavage; (2) migration of one of the O atoms in the NO2 groups to its neighboring C atoms; (3) migration of one of the H atoms in the methyl group to a

Journal of Molecular Structure: THEOCHEM, 2002

Progress in Theoretical Chemistry and Physics, 2011

ABSTRACT A new compact two-range addition theorem for Coulomb Sturmians is presented. This theore... more ABSTRACT A new compact two-range addition theorem for Coulomb Sturmians is presented. This theorem has been derived by breaking up the exponential-type orbitals into convenient elementary functions: the Yukawa potential (e − αr ∕ r) and “evenly-loaded solid harmonics,” (r2ν+lYml(r^) for which translation formulas are available. The resulting two-range translation formula for the exponential orbital is presented and used to construct a new addition theorem for the Coulomb Sturmians.

International Journal of Quantum Chemistry, 2004

ABSTRACT The strategy of using expansions in spherical harmonics of displaced Slater-type (exp(−α... more ABSTRACT The strategy of using expansions in spherical harmonics of displaced Slater-type (exp(−αx)) orbitals to evaluate multicenter molecular integrals is applied to three-centered nuclear attraction integrals (potential due to the product of separated Slater-type orbitals). Closed formulae are produced for nine terms of an infinite series for interior region, (inside − ipot) r2 < a, and 14 terms for Exterior Region, (outside − epot) r2 > a. The value a = 2 is chosen for the interior region and r = 14 for the exterior region. Our focus is to obtain a closed formula for the Slater formula. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Molecular Physics, 2006

The integral bottleneck in evaluating molecular energies arises from the two-electron contributio... more The integral bottleneck in evaluating molecular energies arises from the two-electron contributions. These are difficult and time-consuming to evaluate, especially over exponential type orbitals, used here to ensure the correct behaviour of atomic orbitals. In this work, it is ...

The computational theory for calculation of the solution of the time-dependent Schrödinger equati... more The computational theory for calculation of the solution of the time-dependent Schrödinger equation for two electrons [C.A. Weatherford, Computational Chemistry: Reviews of Current Trends, Vol. 5, ed. J. Leszczynski, World Scientific 2000] is reviewed and adapted to the case of the one-dimensional helium atom. This results in a new computational time-dependent exchange/correlation theory. A solution algorithm which discretizes space using a spectral discrete variable basis of synthetic cartesian polynomials, and discretizes time using a spectral element discrete variable basis of Chebyshev polynomials, is presented. Supported by NSF CREST grant HRD-9707076, and by NASA grant NAG5-10148.

Technology & Innovation, 2012

ABSTRACT Fusion research is a key element in the nation's long-term energy supply strateg... more ABSTRACT Fusion research is a key element in the nation's long-term energy supply strategy. Florida A&M University's Center for Plasma Science and Technology (FAMU CePaST) is building a spheromak fusion device, where fusion energy research is on the forefront of clean energy technology. The Spheromak Turbulent Physics Experiment (STPX) is in a class of experiments used to investigate the physics principles of self-organized plasmas. This is relevant for the development of an economically feasible, controlled fusion reactor as a source of electrical power. The project involves collaboration between FAMU CePaST, West Virginia University, and Auburn University. This project will determine, using theory, experiment, and simulation, the essential elements required for full kinetic modeling of an entire spheromak plasma using ab initio magnetohydrodynamic (MHD) with direct modifications from new turbulence physics. High time resolution measurements of electron temperatures, ion temperatures, and magnetic field fluctuations will be used to study turbulence during magnetic reconnection, anomalous ion heating, dusty plasmas, and equilibrium modification. This article assesses the clean energy technology of the spheromak device, the plasma regimes that a spheromak will provide, and discusses its role in understanding turbulence, an essential part of fusion.

Computational Chemistry: Reviews of Current Trends, 2000

International Journal of Quantum Chemistry, 1999

A form of classical electrodynamic field exists which gives exact agreement with the operator fie... more A form of classical electrodynamic field exists which gives exact agreement with the operator field of quantum electrodynamics (QED) for the Lamb shift of a harmonically bound point electron. Here it is pointed out that this form of classical theory with its physically acceptable interpretation is the result of an unconventional resolution of a mathematically ambiguous term in classical field theory. Finally a quantum-classical correspondence principle is shown to exist in the sense that the classical field and expectation value of the QED operator field are identical if retardation is neglected in the latter.

International Journal of Quantum Chemistry, 1999

A form of classical electrodynamic field exists which gives exact agreement with the operator fie... more A form of classical electrodynamic field exists which gives exact agreement with the operator field of quantum electrodynamics (QED) for the Lamb shift of a harmonically bound point electron. Here it is pointed out that this form of classical theory with its physically acceptable interpretation is the result of an unconventional resolution of a mathematically ambiguous term in classical field theory. Finally a quantum-classical correspondence principle is shown to exist in the sense that the classical field and expectation value of the QED operator field are identical if retardation is neglected in the latter.

International Journal of Quantum Chemistry, 1984

International Journal of Quantum Chemistry, 2009

International Journal of Quantum Chemistry, 2009

ABSTRACT In this article, I consider the use of the Kohn variational principle in obtaining the r... more ABSTRACT In this article, I consider the use of the Kohn variational principle in obtaining the reaction matrix, which determines the positive energy solution to the partial wave Schrödinger equation in coordinate space. The formulation considered is appropriate for electron scattering from an atom or molecule. In its usual form, the Kohn variational principle provides the relationship between the exact reaction matrix, independent of first-order errors in the trial wavefunction, and the trial reaction matrix. This relationship is usually called the Kato identity. It is obtained using the second form of Green's theorem for a sphere of infinite radius. I show how a Kohn-like variational principle, using Green's theorem for a sphere of arbitrary radius, provides an expression for the exact, coordinate space, integral equation amplitudes in terms of the trial amplitudes. The results are again exact to second order in the trial wavefunction. Applications of the present results to electron-molecule frame transformations and algebraic scattering theory are then suggested.

In a recent article [C.A. Weatherford, E. Red, A. Wynn III, International Journal of Quantum Chem... more In a recent article [C.A. Weatherford, E. Red, A. Wynn III, International Journal of Quantum Chemistry 90, 1289-1294 (2002)], an algorithm was described whereby a synthetic weighted polynomial basis may be constructed which is adapted (designed) to a particular potential. It was applied therein to the Schroedinger equation with a coulomb potential in one dimension (-1/|x| ). A weighted polynomial basis with weight function w(x)=exp(-a|x|) was employed. It was observed that this potential had no even parity solutions - only odd parity solutions. The question arises as to the relationship of the solutions (eigenfunctions and eigenvalues) for this hard coulomb potential to the solutions for the soft coulomb potential (-1/ √x^2+b^2^1/2 ). In particular, since the soft coulomb potential is clearly expected to possess both even and odd parity solutions, how do these solutions behave as b->0 and thus what happens to the even solutions. This problem is deceptively difficult none of the standard basis sets produce a variational minimum as a function of 'a' for nonzero 'b'. This is apparently why this problem has never been done before. A new orthonormal basis was designed with weight function w(x)=exp(-a√x^2+b^2) which did produce a variational minimum for variable a and arbitrary fixed 'b'. The present paper describes these solutions and clearly indicates how they behave as b->0 .

The evaluation of a matrix exponential function is a classic problem of computational linear alge... more The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review20 4], none of which produce a definitive algorithm which is broadly applicable and sufficiently accurate, as well as being reasonably fast. Herein, we employ a method which evaulates a matrix exponential as the solution to a first-order initial value problem in a pseudo-time variable. The new aspect of the present implementation of this method is to use finite elements in the pseudo-time variable. [Weatherford C A, Red E, and Wynn A 2002 Journal of Molecular Structure592 47] Then using an expansion in a properly chosen pseudo-time basis, we are able to make accurate calculations of the exponential of any given matrix as the solution to a set of simultaneous equations, even when the matrix is singular or rectangular.

The implementation of the Dirac representation is facilitated by the finite element space-time al... more The implementation of the Dirac representation is facilitated by the finite element space-time algorithm.[1] Multicenter integral computations are also facilitated by this same algorithm. The present work is the first application of this original algorithm to the computation of bound states of atoms and molecules. The Dirac representation is employed such that H0 is the sum of the one-electron operators while the residual H1 is the sum of the two-electron operators. Soft-Coulomb geminals are then used as the basis for the time-dependent calculation of a superposition of the bound-states. The eigenstates and eigenvalues are then extracted by filter-diagonalization. An addition theorem is given for the soft-coulomb geminals and the geminals are translated again using the space-time algorithm, so that multicenter integrals may be computed. Several small atoms and molecules are considered as an illustration of the method. [1]D.H. Gebremedhin, C.A. Weatherford, X. Zhang, A. Wynn III, and G. Tanaka, ``Evaluation of the matrix exponential function using finite elements in time,'' arXiv:0811.2612v1 [math-ph] 17 Nov 2008.

Molecular Physics, 2005

A new technique is presented for the solution of Poisson's equati... more A new technique is presented for the solution of Poisson's equation in spherical coordinates. The method employs an expansion of the solution in a new set of functions defined herein for the first time, called ‘spectral forms’. The spectral forms have spherical harmonics as their angular part, but use a new set of radial functions that automatically statisfy the boundary

International Journal of Quantum Chemistry, 2004

International Journal of Quantum Chemistry, 2004

ABSTRACT A technique is given for solving the time-dependent Schrödinger equation for electron+po... more ABSTRACT A technique is given for solving the time-dependent Schrödinger equation for electron+potential scattering at low energies (k2 ≤ 1.0 a.u.). This is accomplished by avoiding wavepackets, with the inevitable wavepacket spreading, and solving a time-integral form of the Schrödinger equation directly. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

International Journal of Quantum Chemistry, 2008

To develop a viable detection system for 2,4-dinitrotoluene (DNT), its unimolecular decomposition... more To develop a viable detection system for 2,4-dinitrotoluene (DNT), its unimolecular decomposition mechanisms were studied by using the density functional theory. Three decomposition mechanisms were investigated: (1) homolytic cleavage; (2) migration of one of the O atoms in the NO2 groups to its neighboring C atoms; (3) migration of one of the H atoms in the methyl group to a

Journal of Molecular Structure: THEOCHEM, 2002

Progress in Theoretical Chemistry and Physics, 2011

ABSTRACT A new compact two-range addition theorem for Coulomb Sturmians is presented. This theore... more ABSTRACT A new compact two-range addition theorem for Coulomb Sturmians is presented. This theorem has been derived by breaking up the exponential-type orbitals into convenient elementary functions: the Yukawa potential (e − αr ∕ r) and “evenly-loaded solid harmonics,” (r2ν+lYml(r^) for which translation formulas are available. The resulting two-range translation formula for the exponential orbital is presented and used to construct a new addition theorem for the Coulomb Sturmians.