Applications to Chemistry

The molecular, which has the chiral center, is known as the optical isomer. The absolute configuration by RS method is unique in order to recognize the optical isomer. This RS method visually decides the stereo configuration. In this context, we would like to introduce "from YZ plain approaching". Because it is simple by reason of being similar. Nevertheless R type and S type is same energy level for Molecular Orbital method. In this "Application to Chemistry", we will approach from YZ plain only by reason of being visually similar. SP³ Hybride Orbital of Methane molecular is shown as right figure. Each H-C-H bond angle is same 109.5°. In first, we understand that Methane molecular has 4-fold degeneracy energy level. The dot line in the figure show on YZ plain.

On the other hand, 4-fold degeneracy energy state and a respective energy state can be meansured by the Electronic Spectrum of Methane Spectroscopy. This respective energy state origins from the four kinds of 3-fold rotation axias. This 3-fold rotation axias is shown in the right figure. Three energy state is degeneracy by 3-fold rotarion, and another energy state is the C-H bond on the 3-fold rotation axis. When Metane molecular is exposed under the electromagnetic field by meansuring with the electromagnetic wave and is transferred into the rotation spectrum state, the above 4-fold degeneracy state splits to 3-fold (rotation) degeneracy state and another state.

The right figure is ammonia molecular. Ammonia molecular has also SP³ Hybride Orbital. Atomic number 7, Nitrogen N, has five valence electrons without 1s orbital two electrons. Three valence electrons are used in the bond with hydrogen atom. The remained two electrons is as "Lone Pair" electron on Nitrogen atom. H-C-H bond angle should be 109.5° because of SP³ Hybride Orbital. But this angle is 107.5° with "Lone Pair" electron repulsion as a cause. Many trying has attempted for dividing between s Orbital and p Orbital.

The right figure is Ethylene molecular. Ethylene is the most simple molecular of SP² Hybride Orbitals. By approaching from YZ plain, π orbital pop out to x direction that is vertical to YZ plain. We can also identify visually the stereo configuration.

Optical Isommer

Only in the case of SP³ Hybride Orbital, there is a lot of exceptional phenomena mentioned above. But, here, we will show the basical Integrals appearing in VB method and LCAO MO method. In this first page, we list the basical Hybride Orbitals, Atomic Orbitals from hydrogen atom, Laplacian over Hyperbolic Prolate Sheroidal coordinates, and the several Coordinate Patterns used in one electron Integrals and two electrons Integrals.

SP³ Hybride Orbital : CH₄ Methane

On the other hand for individual integrals of Methane, the center of the bond sets on origin. And the rotation axias sets along σ bond. Four C-H bond is same for Methane.

SP² Hybride Orbital : C₂H₄ Ethylene

On the other hand for individual integrals of σ bond of Ethylene, the center of the bond sets on origin. And the rotation axias sets along σ bond. Furthermore for individual integrals of π bond of Ethylene, the rotation axias also sets along σ bond. χ_2px is used as π bond here.

K shell

L shell

M shell

N shell

1.1 Hyperbolic Elliptic Cylindrical Coordinates (μ₁, ν₁, y₁)

The coordinates hyperbolic function cosh μ1 and cosine function cos ν1 are given by the equations below. The range of the variable μ1 is from 0 to ∞ and the range of the variable ν1 is from 0 to 2π. Right figure shows XZ plane over Cartesian coordinates, in which electron 1 is point P1 (x1, y1, z1), nuclear A is point A (0, 0, -a), and nuclear B is point B (0, 0, a) over Cartesian coordinates. rAB or 2a is the distance of nuclear A from nuclear B, and is constant in accordance with Born Oppenheimer approximation in the electronic orbital of Schrödinger equation. Therefore nuclear A and B are fixed on point A and B, respectively. In the other side, electron 1 is variable on XZ plane. And rA1 and rB1 is the distance of nuclear A and B, respectively, from electron 1. From the Pitagorass theorem of the right-angled triangle A z’1P1 and B z’1P1, z1 and x1 appear below.

Below equations show the correlations between cosh μ₁ and cos ν₁, over hyperbolic Elliptic Cylindrical coordinates, and r_A1 and r_B1, over Cartesian coordinates.


The formulas of the hyperbolic function and the trigonometric function
Hyperbolic function

The hyperbolic functions are definited as below, but these formulas are not used here.

Trigonometric function


Ellipsoid on XZ plane is moved along Y axias. The hyperbolic Elliptic Cylindrical Coordinates is below.


The ranges of the variables are below.

1.2 Laplacian over hyperbolic Elliptic Cylindrical Coordinates

Coordinates is below over hyperbolic Elliptic Cylindrical coordinates.




∇₁² is below over XYZ coordinates.


Laplacian over hyperbolic Elliptic Cylindrical coordinates appears with the same procedure of the Cylindrical coordinates.
y₁ and z₁ both of them are the functions of μ₁ and ν₁.


∂μ₁/∂y₁ , ∂ν₁/∂y₁ , ∂μ₁/∂z₁ , and ∂ν₁/∂z₁ are calculated.


Differential Formula

Hyperbolic function


Arc-hyperbolic function


Cosine and sine function


Arc-cosine function


Differential calculus examples

μ1 and ν1 are appeared by y1 and z1. Initially, rA1 and rB1 are appeared by x1 and z1. cosh μ1 and cos ν1 are appeared by x1 and z1.

And μ₁ and ν₁ are appeared by y₁ and z₁.




∂/∂y₁ is calculated.
∂μ₁/∂y₁ and ∂ν₁/∂y₁ are calculated, individually.

Introducing ∂μ₁/∂y₁ and ∂ν₁/∂y₁, we obtain ∂/∂y₁ below.


∂/∂z₁ is calculated.
∂μ₁/∂z₁ and ∂ν₁/∂z₁ are calculated, individually.

Introducing ∂μ₁/∂z₁ and ∂ν₁/∂z₁, we obtain ∂/∂z₁ below.



∂²/∂y₁² and ∂²/∂z₁² are calculated.
We obtain ∂²/∂y₁² as below.

We obtain ∂²/∂z₁² as below.



∂²/∂y₁² + ∂²/∂z₁² is calculated,
in which sinh²μ₁=(cosh²μ₁–1) and sin²ν₁ =(1-cos²ν₁) are used for cosh²μ₁ sin²ν₁ + sinh²μ₁cos²ν₁.

We obtain ∂²/∂y₁² + ∂²/∂z₁² below.


And we obtain Laplacian ∇₁² over hyperbolic Elliptic Cylindrical coordinates,
in which cosh²μ₁ =( sinh²μ₁+1) and cos²ν₁=(1-sin²ν₁) are used for (cosh²μ₁-cos²ν₁).

1.3 Hyperbolic Prolate Spheroidal Coordinates

The coordinates hyperbolic function cosh μ1 and cosine function cos ν1 are also given by the equations below. The range of the variable μ1 is from 0 to ∞ and the range of the variable ν1 is from 0 to π. Right figure shows XZ plane (1=0) over Cartesian coordinates, the coordinates is same with Hyperbolic Elliptic Cylindrical Coordinates except y1→ρ1. From the Pitagorass theorem of the right-angled triangle A z’1P1 and B z’1P1, z1 and ρ1 appear also below.

In the next step, the plane involving triangle A B P1 is rotated on Z axis, and the rotation angle is 1. The range of the variable 1 is from 0 to 2π. Right figure shows XY plane over Cartesian coordinates. We draw a perpendicular line from P1 to X axis, and the point of intersection is a point x’1. Further we draw a perpendicular line from P1 to Y axis , and the point of intersection is a point y’1. The point x’1 and y’1 are (x1,0,0) and (0,y1,0), respectively. The Hyperbolic Prolate Spheroidal Coordinates is below. The ranges of the variables are below.

1.4 Laplacian over hyperbolic Prolate Spheroidal Coordinates

Coordinates are below over hyperbolic Prolate Spheroidal coordinates.


y₁ of ∂²/∂y₁² + ∂²/∂z₁² over hyperbolic Elliptic Cylindrical coordinates is converted to ρ₁ over hyperbolic Prolate Spheroidal coordinates.


The same procedure, y₁ of ∂/∂y₁ over hyperbolic Elliptic Cylindrical coordinates coordinates is converted to ρ₁ over hyperbolic Prolate Spheroidal coordinates.


∂²/∂x₁² + ∂²/∂y₁² over hyperbolic Prolate Spheroidal coordinates is the same over Cylindrical coordinates.


∂²/∂x₁² + ∂²/∂y₁² and ∂²/∂z₁² is as below.


Introducing ∂/∂ρ₁, ∂²/∂x₁² + ∂²/∂y₁² + ∂²/∂z₁² becomes as below.


Introducing ρ₁= a sinh μ₁ sin ν₁, ∂²/∂x₁² + ∂²/∂y₁² + ∂²/∂z₁² becomes as below..


we obtain Laplacian over hyperbolic Prolate Spheroidal coordinates.

Original1

Original2

A1

A2

B1

B2