13 research outputs found
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(-a<sup>2</sup>t<sup>2</sup>). The
rest of the cases are new and need the other Laplace transformation with
integrand exp(-a<sup>2</sup>t) 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<sub>Ñ</sub>+ H<sub>ne</sub>+ aH<sub>ee</sub>, 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<sub>ee</sub>)
in total electronic energy – apart from virial theorem and the highly detailed
and well-known algorithm for V<sub>ee</sub>, 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<sub>ij</sub><sup>-1</sup>
was overwritten as r<sub>ij</sub><sup>-1</sup> ® ar<sub>ij</sub><sup>-1</sup>, and used “a” as input. The most important finding
beside that the repulsion energy V<sub>ee</sub>(a) is a quasi-linear function
of “a”, is that the extended 1<sup>st</sup> Hohenberg-Kohn theorem (Y<sub>0</sub>(a=1) Û H<sub>ne</sub> Û Y<sub>0</sub>(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