
2. Eight Atomic Integrals Over Hyperbolic Prolate Spheroidal Coordinates For The Lowest State's Energy Of Hydrogen Molecular
Simple Hydrogen Molecular is two electron and two nuclear. The grand state energy of the Hydrogen Molecular is calculated with Slater function 1s electronic orbital. As the approximation, Valence Bond orbital method (VB method or HL method, W. Heitler and F. London) or Simple LCAO Molecular Orbital Method (MO method) are used, and the integrals are calculated with HamiltonJacobi method over hyperbolic Confocal Elliptic coordinates (hyperbolic Prolate Spheroidal coordinate).
The Hamiltonian of Hydrogen Molecular electronic orbital is below.
∇_{1}^{2} and ∇_{2}^{2} are the Laplacians of electron 1 and electron 2, respectively.
r_{AB} is the distance of nuclear A from nuclear B, and is constant according to Born Oppenheimer approximation in the electronic orbital of Schrödinger equation. The nuclear motions involving the vibration and the rotation is not discussed here.
The wave function involving the spin function must be antisymmetric under the exchange of electron 1 and electron 2. Therefore the lowest energy level electronic wave function, with the singlet spin function, of VB method and the simplest LCAO MO method are as below, respectively.
The wave function of VB method
The wave function of the simplest LCAO MO method
S is overlap integral as below.
The lowest energy of VB method and the simplest LCAO MO method are E_{VB +} and E_{MO +} respectively.
E_{VB +} for VB method is expressed as below.
Below spin function integral is 0, because of the spin function orthogonality.
And below spin function integral is 1, because of the spin function normalization.
The lowest energy of VB method
Therefore E_{VB+} for VB method is expressed as below.
The lowest energy of the simplest LCAO MO method
From the same procedure, E_{MO+} for the simplest LCAO MO method is expressed as below.
The Hydrogen nonperturbation energy E_{H} and the Normalization equation are as below.
As a result, eight integrals remain in E_{VB +} and E_{MO +}.
Preparation
Before eight integrals calculation, Electronic Orbital, Coordinates Patterns, and MyLibrary are mentioned in this preparation.
○Electronic Orbital

