このエントリーをはてなブックマークに追加 follow us in feedly BACK<< >>NEXT

1. Laplacian over Hyperbolic Prolate Spheroidal Coordinates

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.
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).

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 x1ρ1. 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.


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.
Laplacian over hyperbolic Prolate Spheroidal Coordinates



このエントリーをはてなブックマークに追加 follow us in feedly
Home Top BACK<< >>NEXT