Quantum Chemistry by Patel K

Author:Patel, K [Patel, K]
Language: eng
Format: epub
Publisher: Unknown
Published: 2015-07-18T16:00:00+00:00

-ve

V ( r )e 2 r r

+ve

H

H2 2 2m Helium atom : (z = 2)

r 1

r2 V

V (r )

2e 2 r

1

A r attraction of e 1 by 12 B the nucleus

2e 2

r

2

attraction of e 2 by the

nucleus

e 2

r

12

repulsing between e 1 and e 2

2

H 2H m 1 2 2 2 V

Carbon atom : (z = 6)

There are six e outside the nucleus of charge +6e. The total P.E. (v ) comprises six

attractive terms of the type 6 e 2 , one for each electron and fifteen interelectronic

repulsive r i

2 e

terms of the type r ij , one for each distinct repelling pair of electrons. Thus :

V

6e 2 6e 2 6e 2 6e 2 6e 2 6e 2

r r r r r r 1 2 3 4 56

2 e 2 e 2 e 2 e 2 e

r r repulsion of

12 r 13 14 r 15 r 16

e 1 by all other e ' s e 2 e 2 e 2 e 2

r r r r repulsion of

23 24 25 26

2 e 2 e 2 e

r r r repulsion of

34 35 36

2 e 2 e

r r repulsion of

45 46 e 2

r repulsion of

56

H 2 6 6 16 1

e 2 by all other e ' s

e 3 by all other e ' s

e 4 by all other e ' s e 5 by all other e ' s

H i 2 6 e 2 e 2 r

2m i 1 i 1 r i i j ij

In general for an atom with atomic number Z,

2 z z 1 e 2 1 H H z

i 2 Z e 2 r

2m i 1 i 1 r i i j ij

Exercise : State explicitly the potential function and hence write the Hamiltonian operators for (i) Lithium atom (z = 3) (ii) Oxygen atom (z = 8) and (iii) Hydrogen molecule, H2 .

THIRD POSTULATES OF QUANTUM MECHANICS

SCHRODINGER EQUATION:

We have said earlier that is a rich source of information about the state and have described how operators can serve to extract information contained in . So far we have said nothing about the source of this powerful quantity: How to get this ? This question is answered in Postulate 3 which states:

The possible state functions of a system are the solution of the differential equation.

H i H

t

This equation, known as the time-dependent Schrodinger equation, is the fundamental equation of motion in quantum mechanics. It describes how develops with time. It must be mentioned here that, although it is easy to write the Schrodinger equation for any system, it is very difficult to solve it exactly for most of the systems. In fact it has been solved exactly for only the one-electron systems (hydrogen atom and hydrogen-like ions). For all other systems. only the approximate solutions are obtained using standard approximation methods such as the variation principle and the perturbation methods.

Let us practice writing Schrodinger equation using simple cases.

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