scholarly journals Quasi-Linear buildup of Coulomb integrals via the coupling strength parameter in the non-relativistic electronic schrödinger equation

2020 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Schrödinger Equation ◽  
Coupling Strength ◽  
Schrodinger Equation ◽  
Strength Parameter ◽  
Coulomb Integrals

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 Solving the Non-Relativistic Electronic Schrödinger Equation with Manipulating the Coupling Strength Parameter over the Electron-Electron Coulomb Integrals

2019 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Coupling Strength ◽  
Correlation Effect ◽  
Electronic Energy ◽  
Basis Set ◽  
Strength Parameter ◽  
Repulsion Energy ◽  
Consistent Field ◽  
Self Consistent Field ◽  
Moller Plesset ◽  
Coulomb Integrals

The non-relativistic electronic Hamiltonian, H(a)= Hkin+Hne+aHee, extended with coupling strength parameter (a), allows to switch the electron-electron repulsion energy off and on. First, the easier a=0 case is solved and the solution of real (physical) a=1 case is generated thereafter from it to calculate the total electronic energy (Etotal electr,K) mainly for ground state (K=0). This strategy is worked out with utilizing generalized Moller-Plesset (MP), square of Hamiltonian (H2) and Configuration interactions (CI) devices. Applying standard eigensolver for Hamiltonian matrices (one or two times) buys off the needs of self-consistent field (SCF) convergence in this algorithm, along with providing the correction for basis set error and correlation effect. (SCF convergence is typically performed in the standard HF-SCF/basis/a=1 routine in today practice.)

scholarly journals Solving the Non-Relativistic Electronic Schrödinger Equation with Manipulating the Coupling Strength Parameter over the Electron-Electron Coulomb Integrals

2019 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Coupling Strength ◽  
Correlation Effect ◽  
Electronic Energy ◽  
Basis Set ◽  
Strength Parameter ◽  
Repulsion Energy ◽  
Consistent Field ◽  
Self Consistent Field ◽  
Moller Plesset ◽  
Coulomb Integrals

The non-relativistic electronic Hamiltonian, H(a)= Hkin+Hne+aHee, extended with coupling strength parameter (a), allows to switch the electron-electron repulsion energy off and on. First, the easier a=0 case is solved and the solution of real (physical) a=1 case is generated thereafter from it to calculate the total electronic energy (Etotal electr,K) mainly for ground state (K=0). This strategy is worked out with utilizing generalized Moller-Plesset (MP), square of Hamiltonian (H2) and Configuration interactions (CI) devices. Applying standard eigensolver for Hamiltonian matrices (one or two times) buys off the needs of self-consistent field (SCF) convergence in this algorithm, along with providing the correction for basis set error and correlation effect. (SCF convergence is typically performed in the standard HF-SCF/basis/a=1 routine in today practice.)

scholarly journals Quasi-Linear Buildup of Coulomb Integrals via the Coupling Strength Parameter in the Non-Relativistic Electronic Schrödinger Equation

2019 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Ground State ◽  
Coupling Strength ◽  
Virial Theorem ◽  
Slater Determinant ◽  
Electronic Energy ◽  
Strength Parameter ◽  
Total Electronic Energy ◽  
Repulsion Energy ◽  
Physical Case ◽  
Coulomb Integrals

The non-relativistic electronic Hamiltonian, Hkin+Hne+aHee, is linear in coupling strength parameter (a), but its eigenvalues (electronic energies) have only quasi-linear dependence on it. Detailed analysis is given on the participation of electron-electron repulsion energy (Vee) in total electronic energy (Etotal electr,k) in addition to the wellknown virial theorem and standard algorithm for vee(a=1)=Vee calculated during the standard- and post HF-SCF routines. Using a particular modification in the SCF part of the Gaussian package, we have analyzed the ground state solutions via the parameter “a”. Technically, with a single line in the SCF algorithm, operator was changed as 1/rij-> a/rij with input “a”. The most important findings are, 1, vee(a) is quasi-linear function of “a”, 2, the extension of 1st Hohenberg-Kohn theorem (PSI0(a=1) <=> Hne <=> Y0(a=0)) and its consequences in relation to “a”. The latter allows an algebraic transfer from the simpler solution of case a=0 (where the single Slater determinant Y0 is the accurate form) to the physical case a=1. Moreover, we have generalized the emblematic Hund’s rule, virial-, Hohenberg-Kohn- and Koopmans theorems in relation to the coupling strength parameter.

scholarly journals Quasi-Linear Buildup of Coulomb Integrals via the Coupling Strength Parameter in the Non-Relativistic Electronic Schrödinger Equation

2019 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Ground State ◽  
Coupling Strength ◽  
Virial Theorem ◽  
Slater Determinant ◽  
Electronic Energy ◽  
Strength Parameter ◽  
Total Electronic Energy ◽  
Repulsion Energy ◽  
Physical Case ◽  
Coulomb Integrals

The non-relativistic electronic Hamiltonian, Hkin+Hne+aHee, is linear in coupling strength parameter (a), but its eigenvalues (electronic energies) have only quasi-linear dependence on it. Detailed analysis is given on the participation of electron-electron repulsion energy (Vee) in total electronic energy (Etotal electr,k) in addition to the wellknown virial theorem and standard algorithm for vee(a=1)=Vee calculated during the standard- and post HF-SCF routines. Using a particular modification in the SCF part of the Gaussian package, we have analyzed the ground state solutions via the parameter “a”. Technically, with a single line in the SCF algorithm, operator was changed as 1/rij-> a/rij with input “a”. The most important findings are, 1, vee(a) is quasi-linear function of “a”, 2, the extension of 1st Hohenberg-Kohn theorem (PSI0(a=1) <=> Hne <=> Y0(a=0)) and its consequences in relation to “a”. The latter allows an algebraic transfer from the simpler solution of case a=0 (where the single Slater determinant Y0 is the accurate form) to the physical case a=1. Moreover, we have generalized the emblematic Hund’s rule, virial-, Hohenberg-Kohn- and Koopmans theorems in relation to the coupling strength parameter.

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