scholarly journals Generalization of Brillouin theorem for the non-relativistic electronic Schrödinger equation in relation to coupling strength parameter, and its consequences in single determinant basis sets for configuration interactions

2017 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Schrödinger Equation ◽  
Coupling Strength ◽  
Schrodinger Equation ◽  
Basis Set ◽  
Basis Sets ◽  
Single Line ◽  
Strength Parameter ◽  
Physical Case ◽  
Ci Calculations

<p> The Brillouin theorem has been generalized for the extended non-relativistic electronic Hamiltonian (H<sub>Ñ</sub>+ H<sub>ne</sub>+ aH<sub>ee</sub>) in relation to coupling strength parameter (a), as well as for the configuration interactions (CI) formalism in this respect. For a computation support, we have made a particular modification of the SCF part in the Gaussian package: essentially a single line was changed in an SCF algorithm, wherein the operator r<sub>ij</sub><sup>-1</sup> was overwritten as r<sub>ij</sub><sup>-1</sup> ® ar<sub>ij</sub><sup>-1</sup>, and “a” was used as input. The case a=0 generates an orto-normalized set of Slater determinants which can be used as a basis set for CI calculations for the interesting physical case a=1, removing the known restriction by Brillouin theorem with this trick. The latter opens a door from the theoretically interesting subject of this work toward practice. </p>

scholarly journals Generalization of Brillouin theorem for the non-relativistic electronic Schrödinger equation in relation to coupling strength parameter, and its consequences in single determinant basis sets for configuration interactions

2017 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Schrödinger Equation ◽  
Coupling Strength ◽  
Schrodinger Equation ◽  
Basis Set ◽  
Basis Sets ◽  
Single Line ◽  
Strength Parameter ◽  
Physical Case ◽  
Ci Calculations

<p> The Brillouin theorem has been generalized for the extended non-relativistic electronic Hamiltonian (H<sub>Ñ</sub>+ H<sub>ne</sub>+ aH<sub>ee</sub>) in relation to coupling strength parameter (a), as well as for the configuration interactions (CI) formalism in this respect. For a computation support, we have made a particular modification of the SCF part in the Gaussian package: essentially a single line was changed in an SCF algorithm, wherein the operator r<sub>ij</sub><sup>-1</sup> was overwritten as r<sub>ij</sub><sup>-1</sup> ® ar<sub>ij</sub><sup>-1</sup>, and “a” was used as input. The case a=0 generates an orto-normalized set of Slater determinants which can be used as a basis set for CI calculations for the interesting physical case a=1, removing the known restriction by Brillouin theorem with this trick. The latter opens a door from the theoretically interesting subject of this work toward practice. </p>

scholarly journals Solution of the Schrödinger torsion equation in the basis set of Mathieu functions: verification by numerical experiment

2021 ◽  
Vol 2052 (1) ◽  
pp. 012004
Author(s):  
A N Belov ◽  
V V Turovtsev ◽  
Yu A Fedina ◽  
Yu D Orlov
Keyword(s):  
Schrödinger Equation ◽  
Numerical Experiment ◽  
Numerical Solution ◽  
Schrodinger Equation ◽  
Periodic Potential ◽  
Plane Waves ◽  
Basis Set ◽  
Basis Sets ◽  
Mathieu Functions ◽  
Computational Stability

Abstract The efficiency of the algorithm for the numerical solution of the Schrödinger torsion equation in the basis of Mathieu functions has been considered. The computational stability of the proposed algorithm is shown. The energies of torsion transitions determined in the basis sets of plane waves and Mathieu functions have been compared with the results of spectroscopy. A conclusion about the applicability of the algorithm using the basis set of Mathieu functions to the solution of the Schrödinger equation with a periodic potential has been derived.

LOCAL DENSITY CALCULATIONS FOR LARGE SYSTEMS USING MULTIPLE SCATTERING THEORY

1995 ◽  
Vol 02 (01) ◽  
pp. 71-79
Author(s):  
D.M.C. NICHOLSON ◽  
G.M. STOCKS ◽  
Y. WANG ◽  
W.A. SHELTON ◽  
Z. SZOTEK ◽  
...  
Keyword(s):  
Schrödinger Equation ◽  
Multiple Scattering ◽  
Scattering Theory ◽  
Schrodinger Equation ◽  
Local Density ◽  
Basis Set ◽  
Scattering Method ◽  
Multiple Scattering Theory ◽  
Large Systems ◽  
Multiple Scattering Method

The accuracy of energy differences calculated from first principles within the local density approximation (LDA) has been demonstrated for a large number of systems. Armed with these energy differences researchers are addressing questions of phase stability and structural relaxation. However, these techniques are very computationally intensive and are therefore not being used for the simulation of large complex systems. Many of the methods for solving the Kohn-Sham equations of the LDA rely on basis set methods for solution of the Schrodinger equation. An alternative approach is multiple scattering theory (MST). We feel that the locally exact solutions of the Schrodinger equation which are at the heart of the multiple scattering method give the method an efficiency which cannot be ignored in the search for methods with which to attack large systems. Furthermore, the analytic properties of the Green function which is determined directly in MST result in computational shortcuts.

scholarly journals Simplistic calculation of the tunneling splittings for proton transfer in malonaldehyde and formic acid dimer using the one-dimensional Schrödinger equation

2021 ◽  
Author(s):  
Denis S. Tikhonov
Keyword(s):  
Schrödinger Equation ◽  
Proton Transfer ◽  
Formic Acid ◽  
Schrodinger Equation ◽  
Zero Point ◽  
Basis Set ◽  
One Dimensional ◽  
Hartree Fock ◽  
Tunneling Splittings ◽  
Complete Basis Set

Abstract In this manuscript we present an approach for computing tunneling splittings for large amplitude motions. The core of the approach is a solution of an effective one-dimensional Schrödinger equation with an effective mass and an effective potential energy surface composed of electronic and harmonic zero-point vibrational energies of small amplitude motions in the molecule. The method has been shown to work in cases of three model motions: nitrogen inversion in ammonia, single proton transfer in malonaldehyde, and double proton transfer in the formic acid dimer. In the current work we also investigate the performance of different DFT and post-Hartree-Fock methods for prediction of the proton transfer tunneling splittings, quality of the effective Schrödinger equation parameters upon the isotopic substitution, and possibility of a complete basis set (CBS) extrapolation for the resulting tunneling splittings.

scholarly journals A simplistic computational procedure for tunneling splittings caused by proton transfer

2021 ◽  
Author(s):  
Denis S. Tikhonov
Keyword(s):  
Schrödinger Equation ◽  
Proton Transfer ◽  
Schrodinger Equation ◽  
Zero Point ◽  
Computational Procedure ◽  
Basis Set ◽  
Hartree Fock ◽  
Tunneling Splittings ◽  
Complete Basis Set ◽  
Large Amplitude Motions

AbstractIn this manuscript, we present an approach for computing tunneling splittings for large amplitude motions. The core of the approach is a solution of an effective one-dimensional Schrödinger equation with an effective mass and an effective potential energy surface composed of electronic and harmonic zero-point vibrational energies of small amplitude motions in the molecule. The method has been shown to work in cases of three model motions: nitrogen inversion in ammonia, single proton transfer in malonaldehyde, and double proton transfer in the formic acid dimer. In the current work, we also investigate the performance of different DFT and post-Hartree–Fock methods for prediction of the proton transfer tunneling splittings, quality of the effective Schrödinger equation parameters upon the isotopic substitution, and possibility of a complete basis set (CBS) extrapolation for the resulting tunneling splittings.

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