The Chemistry Maths Book, Second Edition

10.6 Other coordinate systems 311

EXAMPLE 10.14The hydrogen molecule

In the method of ‘linear combinations of atomic orbitals’ (LCAO) for the ground

state of the hydrogen molecule, the simplest form of the occupied molecular orbital is

ψ 1 = 1 C(1s

A

1 + 11 s

B

), where (in atomic units) and

are normalized 1 sorbitals centred on the protons at Aand B(see Figure 10.10). C is a

constant chosen to normalize the orbital :. Thus

The integrals S

AA

and S

BB

are the normalization integrals for the 1 sorbital, as in

Example 10.8, and equal unity. Then. To evaluate the overlap integral

we use confocal elliptic coordinates (ξ, 1 η, 1 φ). Then, ifR 1 = 12 ais the bond length,

and the volume element is

The volume integral over all space is then

=−

==−

+

=

−

=

R

eddd

R

3

0

2

1

1

1

2

0

2

8 π

ππ

ZZ Z ZZ

φη ξ

ξ

φ

ξξηφ

∞

ηηξ

ξ

ηξηφ

=−

+

=

−





















1

1

1

2

Z

∞

eddd

R

Sed

R

e

R

R

AB

==

−

==−

+

=

−

1

8

3

0

2

1

1

1

2

ππ

π

ZZZZ

ξ

φη ξ

ξ

ξ

v

∞

( −−ηξηφ

2

)ddd

dv

R

=−ddd

3

22

8

()ξη ξηφ

ξη=

• 
  

=

rr −

R

rr

R

AB AB

,

Sed

rr

AB

AB

=

−+

1

π

Z

()

v

CS=+121 ()

AB

=++













CS S S

2

2

AA BB AB

1

11

2

2

2

22

=+











 =++

−− −−

ZZ

ππ

π

eed

C

ee

rr rr

AB AB

v eed

−+rr

()

()

AB

v

Zψ

2

dv= 1

11 se

r

B

B

=

−

11 se ()π

r

A

A

=

−

()π

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Figure 10.10