1. Laplacian over Hyperbolic Prolate Spheroidal Coordinates

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.

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.
x₁ and z₁ both of them are the functions of μ₁ and ν₁.


∂μ₁/∂x₁ , ∂ν₁/∂x₁ , ∂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 x1 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 x₁ and z₁.




∂/∂x₁ is calculated.
∂μ₁/∂x₁ and ∂ν₁/∂x₁ are calculated, individually.

Introducing ∂μ₁/∂x₁ and ∂ν₁/∂x₁, we obtain ∂/∂x₁ below.


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

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



∂²/∂x₁² and ∂²/∂z₁² are calculated.
We obtain ∂²/∂x₁² as below.

We obtain ∂²/∂z₁² as below.



∂²/∂x₁² + ∂²/∂z₁² is calculated,
in which sinh²μ₁=(cosh²μ₁–1) and sin²ν₁ =(1-cos²ν₁) are used for cosh²μ₁ sin²ν₁ + sinh²μ₁cos²ν₁.

We obtain ∂²/∂x₁² + ∂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).

Coordinates are below over hyperbolic Prolate Spheroidal coordinates.


x₁ of ∂²/∂x₁² + ∂²/∂z₁² over hyperbolic Elliptic Cylindrical coordinates is converted to ρ₁ over hyperbolic Prolate Spheroidal coordinates.


The same procedure, x₁ of ∂/∂x₁ 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.