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1s-1s 1s-2s 1s-2pz 1s-2pxy 2s-2pz 2s-2pxy
2s-2s 2pz-2pz 2pz-2pxy 2px-2px 2py-2py 2px-2py

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. 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. It's a rotarion spectrum. 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.

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 anti-clockwise of inner molecular rotation ①→②→③ of left figure amplifies the anti-clockwise fraction of the meansuring wave. And it also attenuates the clockwise fraction of the meansuring wave. The meansuring light, which transmitted (R)-2-Chlorobutane solution, is rotated to the anti-clockwise.

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 anti-clockwise fraction of the meansuring wave. The meansuring light, which transmitted (S)-2-Chlorobutane solution, is rotated to the clockwise.
The difference of optical isommer R- and S-2-Chlorobutane can be meansured only by this ORD (Optical Rotatory Dispersionan) and CD (Circular Dichroism) optical analysis.

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.

Hybride Orbitals

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. χ2px is used as π bond here.


SP Hybride Orbital : C2H2 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. χ2px and χ2py are used as π bond here.



SP3d, SP3d2, and SP3d3 Hybride Orbitals

SP3d, SP3d2, and SP3d3 Hybride Orbitals, shown below, exist in nature. PCl5, shown left edge, is SP3d Hybride Orbital and has one C3 axis (Z axis). ICl7, shown right edge, is SP3d3 Hybride Orbital and has one C5 axis (Z axis).

On the other hand, SCl6, shown the center of above figure, is SP3d2 Hybride Orbital and has three C4 axis (X, Y, Z axis). This SP3d2 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 4-fold degeneracy energy level (four bonds on XY plain) and 2-fold degeneracy energy level (two bonds along Z axis).

SP3d4 and SP3d5 Hybride Orbitals

SP3d4 and SP3d5 Hybride Orbitals was guessed theoretically. SP3d5 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 (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.

XZ plane of hyperbolic Elliptic Cylindrical Coordinates

Below equations show the correlations between cosh μ1 and cos ν1, over hyperbolic Elliptic Cylindrical coordinates, and rA1 and rB1, 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.
hyperbolic Elliptic Cylindrical Coordinates

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).

Laplacian over hyperbolic  Elliptic Cylindrical Coordinates

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 y1ρ1. From the Pitagorass theorem of the right-angled triangle A z’1P1 and B z’1P1, z1 and ρ1 appear also below.

XZ plane of hyperbolic Prolate Spheroidal Coordinates

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 x1 and y1 are (x1,0,0) and (0,y1,0), respectively.

The Hyperbolic Prolate Spheroidal Coordinates is below.
hyperbolic Prolate Spheroidal Coordinates

The ranges of the variables are below.

XY plane of hyperbolic Prolate Spheroidal Coordinates

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.
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 Born-Oppehimer 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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