BioPHYSICAL chemistry

272 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Operators (Table 9.1)

r→r

To solve this equation, we make some assumptions. The interactions

between the two electrons, and the two nuclei, are assumed to be neg-

ligible compared to the interactions between the nuclei and electrons: this

eliminates two of the six terms for the potential. With these assumptions,

Schrödinger’s equation for the hydrogen molecule can be written as:

(13.4)

The wavefunction describing the two electrons is simplified as the product

of wavefunctions for each electron ψ(r 1 ,r 2 ) =ψ(r 1 )ψ(r 2 ): the validity of

this assumption will be addressed later. The equation can be rewritten

by realizing that the derivative operators are specific to either electron 1

or 2, and so for each derivative operator part of the wavefunction can

be considered to be a constant, yielding:

=Eψ(r 1 ,r 2 ) (13.5)

This expression can be simplified by dividing the entire equation by

ψ(r 1 )ψ(r 2 ) and grouping together the terms that depend upon r 1 and r 2 :

(13.6)

These two equations can be separated by setting each of the individual

parts equal to E 1 and E 2 , which together add up to E. In this case, the asso-

ciation of the wavefunctions to the nuclei A and B are shown explicitly

by the subscript.

−∇−−

()

()

Z^2

2

2

2

2

2

02

2

(^20)

1

4

1

4

1

mr

r

e

r

e

ψ

ψ

πε A πε rr

E

B2

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

=

−∇−−

Z^2

1

1

2

1

2

01

2

(^20)

1

4

1

4

1

mr

r

e

r

e

ψ r

ψ

() πε πε

()

A BB1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

−∇+∇−

Z^2

21

2

112

2

2

2

24 m 0

rrrr

e

[()()()()]ψψψψ

πε

11111

rrrrrr^12

A1 A2 B1 B2

+++ (, )

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ψ

−∇+∇ − +

Z^2

1

2

2

2

12

2

(^240)

11

m

rr

e

rr

()(,)ψ

πε A1 A 2 2B1B2

++ (, ) (,

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

=

11

rrψψrr^12 E rr^12 ))

p

i

→∇

Z