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Integrals with Applications to Chemistry
～ Approach from YZ Plain over Hyperbolic Prolate Spheroidal Coordinate ～
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.
SP^{3} Hybride Orbital of Methane molecular is shown as right figure.
Each HCH bond angle is same 109.5°.
In first, we understand that Methane molecular has 4fold degeneracy energy level.
The dot line in the figure show on YZ plain.


On the other hand, 4fold 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 3fold rotation axias.
It's a rotarion spectrum.
This 3fold rotation axias is shown in the right figure.
Three energy state is degeneracy by 3fold rotarion, and another energy state is the CH bond on the 3fold 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 4fold degeneracy state splits to 3fold (rotation) degeneracy state and another state.


The right figure is ammonia molecular.
Ammonia molecular has also SP^{3} 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.
HCH bond angle should be 109.5° because of SP^{3} 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 SP^{2} 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.


The right figure is Acetylene molecular.
Acetylene is the most simple molecular of SP Hybride Orbitals.
By approaching from YZ plain, π orbital pop out to x direction that is vertical to YZ plain.
Furthermore another π orbital on YZ plain is vertical to Z axis.
We can also identify visually the stereo configuration.


* Optical Isommer
Left Figure :
The anticlockwise of inner molecular rotation ①→②→③ of left figure amplifies the anticlockwise fraction of the meansuring wave. And it also attenuates the clockwise fraction of the meansuring wave.
The meansuring light, which transmitted (R)2Chlorobutane solution, is rotated to the anticlockwise.
Right Figure :
The clockwise of inner molecular rotation ①→②→③ of right figure amplifies the clockwise fraction of the meansuring wave. And it also attenuates the anticlockwise fraction of the meansuring wave.
The meansuring light, which transmitted (S)2Chlorobutane solution, is rotated to the clockwise.
The difference of optical isommer R and S2Chlorobutane can be meansured only by this ORD (Optical Rotatory Dispersionan) and CD (Circular Dichroism) optical analysis.

Only in the case of SP^{3} 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.

Hybride Orbitals
SP^{3} Hybride Orbital : CH_{4} 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 CH bond is same for Methane.

SP^{2} Hybride Orbital : C_{2}H_{4} 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.

SP Hybride Orbital : C_{2}H_{2} Acetylene
The center of σ bond also sets on origin for individual integrals of Acetylene.
And the rotation axias sets along σ bond.
Furthermore for individual integrals of π bond of Acetylene, the rotation axias also sets along σ bond.
χ_{2px} and χ_{2py} are used as π bond here.

SP^{3}d, SP^{3}d^{2}, and SP^{3}d^{3} Hybride Orbitals
SP^{3}d, SP^{3}d^{2}, and SP^{3}d^{3} Hybride Orbitals, shown below, exist in nature.
PCl_{5}, shown left edge, is SP^{3}d Hybride Orbital and has one C3 axis (Z axis).
ICl_{7}, shown right edge, is SP^{3}d^{3} Hybride Orbital and has one C5 axis (Z axis).
On the other hand, SCl_{6}, shown the center of above figure, is SP^{3}d^{2} Hybride Orbital and has three C4 axis (X, Y, Z axis).
This SP^{3}d^{2} is regular octahedron.
But if four bonds are fixed on XY plain, shown as right figure, this symmetrical regular octahedron break out.
And they split 4fold degeneracy energy level (four bonds on XY plain) and 2fold degeneracy energy level (two bonds along Z axis).



SP^{3}d^{4} and SP^{3}d^{5} Hybride Orbitals
SP^{3}d^{4} and SP^{3}d^{5} Hybride Orbitals was guessed theoretically.
SP^{3}d^{5} Hybride Orbital is shown below.




Atomic Orbitals from hydrogen atom
K shell

L shell

M shell

N shell




Laplacian
～ Approach from YZ Plain over Hyperbolic Prolate Spheroidal Coordinate ～
For the sphere, typically, the procedure of the conversion of Laplacian from Cartesian(XYZ) to Polar Coordinates systems is via Cylindrical Coordinates. For the spheroid, the same procedure is useable too. And the hyperbolic function is useable too.
Here, for a spheroid, the conversion of Laplacian from Cartesian to hyperbolic Prolate Spheroidal Coordinates systems was carried out via hyperbolic Elliptic Cylindrical Coordinates.
At first, the Laplacian of hyperbolic Elliptic Cylindrical Coordinates was calculated.
And next(finally), the Laplacian of hyperbolic Prolate Spheroidal Coordinates was calculated.
1.1 Hyperbolic Elliptic Cylindrical Coordinates (μ_{1}, ν_{1}, y_{1})
The coordinates hyperbolic function cosh_{ }μ_{1} and cosine function cos_{ }ν_{1} are given by the equations below.
The range of the variable μ_{1} is from 0 to ∞ and the range of the variable ν_{1} is from 0 to 2π. Right figure shows XZ plane over Cartesian coordinates, in which electron 1 is point P_{1} (x_{1}, y_{1}, z_{1}), nuclear A is point A (0, 0, a), and nuclear B is point B (0, 0, a) over Cartesian coordinates. r_{AB} 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 r_{A1} and r_{B1} is the distance of nuclear A and B, respectively, from electron 1.
From the Pitagorass theorem of the rightangled triangle A z’_{1}P_{1} and B z’_{1}P_{1}, z_{1} and x_{1} appear below.


Below equations show the correlations between cosh μ_{1} and cos_{ }ν_{1}, 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.
∇_{1}^{2} is below over XYZ coordinates.
Laplacian over hyperbolic Elliptic Cylindrical coordinates appears with the same procedure of the Cylindrical coordinates.
y_{1} and z_{1} both of them are the functions of μ_{1} and ν_{1}.
∂μ_{1}/∂y_{1} , ∂ν_{1}/∂y_{1} , ∂μ_{1}/∂z_{1} , and ∂ν_{1}/∂z_{1} are calculated.
Differential Formula
Hyperbolic function
Archyperbolic function
Cosine and sine function
Arccosine function
Differential calculus examples
And μ_{1} and ν_{1} are appeared by y_{1} and z_{1}.
∂/∂y_{1} is calculated.
∂μ_{1}/∂y_{1} and ∂ν_{1}/∂y_{1} are calculated, individually.
Introducing ∂μ_{1}/∂y_{1} and ∂ν_{1}/∂y_{1}, we obtain ∂/∂y_{1} below.
∂/∂z_{1} is calculated.
∂μ_{1}/∂z_{1} and ∂ν_{1}/∂z_{1} are calculated, individually.
Introducing ∂μ_{1}/∂z_{1} and ∂ν_{1}/∂z_{1}, we obtain ∂/∂z_{1} below.
∂^{2}/∂y_{1}^{2} and ∂^{2}/∂z_{1}^{2} are calculated.
We obtain ∂^{2}/∂y_{1}^{2} as below.
We obtain ∂^{2}/∂z_{1}^{2} as below.
∂^{2}/∂y_{1}^{2} + ∂^{2}/∂z_{1}^{2} is calculated,
in which sinh^{2}μ_{1}=(cosh^{2}μ_{1}–1) and sin^{2}ν_{1} =(1cos^{2}ν_{1}) are used for cosh^{2}μ_{1} sin^{2}ν_{1} + sinh^{2}μ_{1}cos^{2}ν_{1}.
We obtain ∂^{2}/∂y_{1}^{2} + ∂^{2}/∂z_{1}^{2} below.
And we obtain Laplacian ∇_{1}^{2} over hyperbolic Elliptic Cylindrical coordinates,
in which cosh^{2}μ_{1} =( sinh^{2}μ_{1}+1) and cos^{2}ν_{1}=(1sin^{2}ν_{1}) are used for (cosh^{2}μ_{1}cos^{2}ν_{1}).

1.3 Hyperbolic Prolate Spheroidal Coordinates
The coordinates hyperbolic function cosh μ_{1} and cosine function cos_{ }ν_{1} are also given by the equations below.
The range of the variable μ_{1} is from 0 to ∞ and the range of the variable ν_{1} 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 y_{1}→ρ_{1}.
From the Pitagorass theorem of the rightangled triangle A z’_{1}P_{1} and B z’_{1}P_{1}, z_{1} and ρ_{1} appear also below.


In the next step, the plane involving triangle A B P_{1} 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 P_{1} to X axis, and the point of intersection is a point x’_{1}. Further we draw a perpendicular line from P_{1} to Y axis , and the point of intersection is a point y’_{1}. The point x’_{1} and y’_{1} are (x_{1},0,0) and (0,y_{1},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.
y_{1} of ∂^{2}/∂y_{1}^{2} + ∂^{2}/∂z_{1}^{2} over hyperbolic Elliptic Cylindrical coordinates is converted to ρ_{1} over hyperbolic Prolate Spheroidal coordinates.
The same procedure, y_{1} of ∂/∂y_{1} over hyperbolic Elliptic Cylindrical coordinates coordinates is converted to ρ_{1} over hyperbolic Prolate Spheroidal coordinates.
∂^{2}/∂x_{1}^{2} + ∂^{2}/∂y_{1}^{2} over hyperbolic Prolate Spheroidal coordinates is the same over Cylindrical coordinates.
∂^{2}/∂x_{1}^{2} + ∂^{2}/∂y_{1}^{2} and ∂^{2}/∂z_{1}^{2} is as below.
Introducing ∂/∂ρ_{1}, ∂^{2}/∂x_{1}^{2} + ∂^{2}/∂y_{1}^{2} + ∂^{2}/∂z_{1}^{2} becomes as below.
Introducing ρ_{1}= a_{ }sinh μ_{1} sin_{ }ν_{1}, ∂^{2}/∂x_{1}^{2} + ∂^{2}/∂y_{1}^{2} + ∂^{2}/∂z_{1}^{2} becomes as below..
we obtain Laplacian over hyperbolic Prolate Spheroidal coordinates.




Coordinates Patterns
"Original" coordinates fix the Nuclear A and B on the rotation Z axis.
And the length between Nuclear A and B is fixed 'a' according with the BornOppehimer approximation.
Original1
Original1 is for Electron 1.
Original2
Original2 is for Electron 2.

"A" Coordinates patterns fix Nuclear A and one Electron on the rotation Z axis.
And the length between Nuclear A and one Electron is fixed.
There is no Nuclear B.
A1
A1 pattern is for Nuclear A and Electron 1. And Electron 2 is fixed.
A2
A2 pattern is for Nuclear A and Electron 2. And Electron 1 is fixed.

"B" Coordinates patterns fix Nuclear B and one Electron on the rotation Z axis.
And the length between Nuclear B and one Electron is fixed.
There is no Nuclear A.
B1
B1 pattern is for Nuclear B and Electron 1. And Electron 2 is fixed.
B2
B2 pattern is for Nuclear B and Electron 2. And Electron 1 is fixed.




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