Hydrogen Molecule - A computer study

Abstract

The purpose of this project was to obtain the energy at ground state of a Hydrogen molecule by computer. This was achieved by solving numerically by computer programs the time-independent Schrodinger equation for a Hydrogen molecule.

This included the building-up of the equation, mathematical manipulation on the equation for purposes of separation of variables and appropriate substitution, formation of matrix eigenvalue system by application of finite difference method and design of algorithm and programs for solving numerically the matrix eigenvalue system to obtain the eigenvalue with the minimum absolute magnitude - the required energy.

The time-independent Schrodinger equation in both polar and cylindrical co-ordinate were considered in this project. This project first dealt with less complicated case having similar nature and proceeded step by step to achieve the final objective. Therefore this project totally studied four cases and they were in the following order :

the simple Hydrogen atom (H),
the Hydrogen molecule ion (H2+),
the Helium atom (He) and finally
the Hydrogen molecule (H2).

Iteration method was employed for solving the matrix eigenvalue system and programs were written in Pascal language to implement the iteration method algorithm together with the technique of performing matrix shift.

The Borland Turbo Pascal version 7.0 (for DOS) compiler was used for programming. The MS Excel version 5.0 was used for graph plotting. Desktop PC computers with configuration of DX4-100 and Pentium-75 having 16 Megabytes memory were used to execute the programs.


All the numerical results produced by programs were expressed in Atomic Units for the purpose of comparison with accepted and known data. The project had also considered the accuracy of numerical storage in computer memory with reference to the compiler used and the accuracy of numerical computation as a result of very large number of iteration.

The value of "pi", correct to 40 decimal places, was used as a reference value.

The matrix formed as a result of application of finite difference method was very large and this posed a practical problem as to both the length of computation time by programs and the computer memory available. The order of magnitude involved in the length of computation time was one of hours and even days. The memory available had restricted the maximum size of the column vector in the iteration method and therefore the maximum number of divisions in the finite difference method. These two factors had eventually restricted the algorithms available for the implementation of the iteration method. The best was then to have all matrix operations performed row by row. The maximum size of the column vector achieved in this project was one of 15876, a grid size of 128 points by 128 points for two variables and only 13 points for four variables, as in the case of Hydrogen molecule.

The constraint in memory was actually a problem with the compiler, it only addressed the conventional 640 Kilobytes memory in every computer system irrespective of any extended or expanded memory available for the computer system.

The project had also explored the possibility and practicality of using the technique of RAM-disk to circumvent this memory pitfall. All programs written in this project could be modified easily to use the RAM-disk technique.



This project had achieved the following :-

(a) had calculated the ground state energy, the energy for five excited states of Hydrogen atom and had used the same set of programs, by changing the electrostatic charge Z, calculated the ground state energy for He+ ion, Li+2 ion and Be+3 ion, the results were quite good;

(b) had calculated the binding energy and inter-nucleus distance of the Hydrogen molecule ion, the result was rather good too;

(c) had calculated the ground state energy of Helium atom and had tried to do the excited state which could only give an idea of order of magnitude, however, had used the same set of programs to estimate the ground state energy for Li+1 ion and Be+2 ion, and with the result obtained in (a) above, also gave a fairly good estimation to the ionization energy of Helium atom and these two ions;

(d) had calculated the ground state energy and inter-nucleus distance of Hydrogen molecule, but the result could only be described as satisfactorily good;

(e) had studied the algebraic solution of tridiagonal matrix in the Hydrogen atom case.