Generalization of Brillouin theorem for the non-relativistic electronic Schr\"odinger equation in relation to coupling strength parameter, and its consequences in single determinant basis sets for configuration interactions

The Brillouin theorem has been generalized for the extended non-relativistic electronic Hamiltonian (Hkin+ Hne+ aHee) 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 rij-1 was overwritten as 1/rij to a/rij, 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.Comment: 27 pages, 2 figure

On the statistical distribution of prime numbers, a view from where the distribution of prime numbers is not erratic

Currently there is no known efficient formula for primes. Besides that, prime numbers have great importance in e.g., information technology such as public-key cryptography, and their position and possible or impossible functional generation among the natural numbers is an ancient dilemma. The properties of the functions 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers. The characterization of odd primes vs. odd non-primes can be done with 2ab+a+b among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection, as well as with the Fermat and Euler numbers as examples.Comment: 27 pages, 4 tables, 3 figure

Quasi-linear buildup of Coulomb integrals via the coupling strength parameter in the non-relativistic electronic Schrodinger equation

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 well-known 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)HneY0(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 rule, virial-, Hohenberg-Kohn- and Koopmans theorems in relation to the coupling strength parameter.Comment: 9 pages, 2 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:1709.0735

Solving the non-relativistic electronic Schrodinger equation with manipulating the coupling strength parameter over the electron-electron 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.)Comment: 9 pages, 2 tables. arXiv admin note: text overlap with arXiv:1709.0735

Excited electronic potential-energy surfaces and transition moments for the H3 system

Four electronic states of H3 have been studied using a multiple-reference double-excitation configuration-interaction method with an extensive basis set of 75 Gaussian-type atomic orbitals. A total of 1340 ab initio points were calculated over a wide range of H3 molecular geometries. These four states include the ground state and the Rydberg 2s A12′ and 2pz A22″ states, as well as the state that in equilateral triangular geometry is related to the ground state by a conical intersection. Electric-dipole transition moments were also obtained between these states. The results show that the atomic and diatomic energetic asymptotes are accurately described. The barriers, wells, and energy differences also show good agreement compared to literature values, where available. The potential energies of the ground state and the 2pz A22″ Rydberg state display smooth and regular behavior and were fitted over the whole molecular geometries using a rotated Morse curve-cubic spline approach. The other two potential-energy surfaces reveal more complicated behaviors, such as avoided crossings, and will require a different fitting procedure to obtain global fitting. Finally, dynamical implications of these potential surfaces and electric-dipole transition moments are discussed

Analytic evaluation of Coulomb integrals for one, two and three-electron distance operators

<p> The state of the art for integral evaluation is that analytical solutions to integrals are far more useful than numerical solutions. We evaluate certain integrals analytically that are necessary in some approaches in quantum chemistry. In the title, where R stands for nucleus-electron and r for electron-electron distances, the (n,m)=(0,0) case is trivial, the (n,m)=(1,0) and (0,1) cases are well known, fundamental milestone in integration and widely used in computation chemistry, as well as based on Laplace transformation with integrand exp(-a²t²). The rest of the cases are new and need the other Laplace transformation with integrand exp(-a²t) also, as well as the necessity of a two dimensional version of Boys function comes up in case. These analytic expressions (up to Gaussian function integrand) are useful for manipulation with higher moments of inter-electronic distances, for example in correlation calculations. </p

Quasi-linear dependence of Coulomb forces on coupling strength parameter in the non-relativistic electronic Schrödinger equation and its consequences in Hund’s rule, Mølle -Plesset perturbation- , virial - , Hohenberg-Kohn - and Koopmans theorem

<p> The extended non-relativistic electronic Hamiltonian, H_Ñ+ H_ne+ aH_ee, is linear in coupling strength parameter (a), but its eigenvalues (interpreted as electronic energies) have only quasi-linear dependence on “a”. No detailed analysis has yet been published on the ratio or participation of electron-electron repulsion energy (V_ee) in total electronic energy – apart from virial theorem and the highly detailed and well-known algorithm for V_ee, which is calculated during the standard HF-SCF and post-HF-SCF routines. Using a particular modification of the SCF part in the Gaussian package we have analyzed the ground state solutions via the parameter “a”. Technically, this modification was essentially a modification of a single line in an SCF algorithm, wherein the operator r_ij^-1 was overwritten as r_ij^-1 ® ar_ij^-1, and used “a” as input. The most important finding beside that the repulsion energy V_ee(a) is a quasi-linear function of “a”, is that the extended 1^st Hohenberg-Kohn theorem (Y₀(a=1) Û H_ne Û 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 is the accurate form) to the realistic wanted case a=1. Moreover, we have generalized the emblematic theorems in the title in relation to the coupling strength parameter. </p