A finite difference Newton-Raphson solution of the atomic Hartree-Fock problem

Cayford, J.K.; Fimple, W. R.; Unger, Deborah

doi:10.1016/0021-9991(74)90070-9

Cited by 8 publications

(5 citation statements)

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“…The algorithm presented in Ref. [2] is for the special case of one-dimensional submatrices ( M = l), but it can be generalized by considering the elements of the A matrix of the algorithm to be themselves M X M matrices. The GNRI can be applied also to the solution of eqs.…”

Section: M )mentioning

confidence: 99%

“…The Appendix of Ref. [2] gives an algorithm for rapidly solving eq. (44) with the J matrix of the form of eq.…”

Section: M )mentioning

confidence: 99%

“…In the present case it is possible to form a partly block tridiagonal Jacobian matrix which can be handled almost as easily as the original. Let us define a solution vector U, for each value of a with components, As in Ref, [2] the Lagrange multipliers are treated as just another variable with no special considerations.…”

Section: M )mentioning

confidence: 99%

“…For the first SCF iteration all terms containing interactions are set to zero and extra diagonal multiplier terms added where necessary, such that hydrogenic functions (in finite-difference form) are very close to solving the system. From here the system is taken to its final form in a sequence of steps in a procedure called tracking [2].…”

Section: M )mentioning

confidence: 99%

“…Recently a finite-difference Newton-Raphson algorithm, originally developed by Van Dine [l], has proven to be very successful in solving the Hartree Fock (HF) equations for atoms [2] and diatomic molecules [3].…”

Section: Introductionmentioning

confidence: 99%

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A finite‐difference approach to the electron correlation problem

Fimple

Unwin

1976

Int J of Quantum Chemistry

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AbstractsA variant of the transcorrelated method of Boys and Handy employing finite differences is presented. It is based upon the following two properties of the transcorrelated Hamiltonian operator C-'HC: (1) C-'HC possesses an energy eigenvalue spectrum which is identical to that associated with H itself; and (2) if C = T erg/' then C ' H C is free of the singularities of H a t the points where the interelectron separation Y~ is zero. A bivariational principle for approximating the eigenvalues and the left and right eigenfunctions of C -'HC is introduced and the resulting set of coupled integro-differential equations are solved in finite-difference form by means of a coupled self-consistent field, Newton Raphson algorithm. As a preliminary test of the method, a calculation of the ground-state energy of the helium atom is presented.

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Section: M )mentioning

confidence: 99%

“…The Appendix of Ref. [2] gives an algorithm for rapidly solving eq. (44) with the J matrix of the form of eq.…”

Section: M )mentioning

confidence: 99%

Section: M )mentioning

confidence: 99%

Section: M )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A finite‐difference approach to the electron correlation problem

Fimple

Unwin

1976

Int J of Quantum Chemistry

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Numerical electronic structure calculations for atoms. I. Generalized variable transformation and nonrelativistic calculations

Andrae

Hinze

1997

Int. J. Quant. Chem.

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The radial functions P in the nonrelativistic description of the electronic structure of i atoms are solutions of eigenvalue equations. These equations are treated as two-point boundary value problems for bound one-electron states and can be solved to high accuracy with finite difference methods. For these numerical techniques the transformation to a suitable new radial variable is essential. The effects and consequences following from an arbitrary suitable variable transformation are studied in the most general way. Important results following from this general analysis are the extension of the range of application of the standard Numerov scheme and the outline for an algorithm of increased overall efficiency and reliability. Especially, the explicit use of transformed solution functions is shown to be unnecessary. Radial functions can be determined for arbitrary effective potentials resulting from the underlying theoretical description. It is demonstrated that all numerical results can be calculated to within a consistent numerical truncation error of order h 4 in the grid point spacing. The approach can be extended to handle unbound one-electron states and is applicable in other fields, where differential equations of similar character occur, e.g., in the description of the vibration of a diatomic molecule. A similar analysis for relativistic electronic structure calculations for atoms will be given in Part II. i assumed to behave exponentially decaying for r ª ϱ, so that asymptotically a homogeneous differential equation is found. In addition, the effective Ž . potential energy function V r must approach zero i for r ª ϱ. The long-range asymptotic behavior of INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY

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