Principles - Methods of Molecular Quantum Mechanics - page 15

(e) The hydrogen molecule H 2
This is a diatomic two-electron molecular system. The Born-
Oppenheimer Hamiltonian will be

<

1
r 12 þ

1
R

H ¼ h 1 þ h 2 þ

! !

1
2 r

1
r A1

1
r B1

1
2 r

1
r A2

1
r B2

1
r 12 þ

1
R

2 2

: ¼

1

þ

2

þ

¼ h A1 þ h B2 þ V

ð 1 : 71 Þ

where h A1 and h B2 are the one-electron Hamiltonians (1.68) for
atoms A and B (with Z ¼ 1) and

1
r B1

1
r A2 þ

1
r 12 þ

1
R

V ¼ ð 1 : 72 Þ

is the interatomic potential between A and B.

1.3.2 State Function and Average Value of Observables

We assume there is a state function (or wavefunction, in general complex)
Y

(x,t) that describes in a probabilistic way the dynamical state of a
microscopic system. In coordinate space, Y is a regular function of

coordinate x and time t such that

Yð x ; t ÞY ð x ; t Þ dx

¼ probability at time t of finding at dx the system in state

ð 1 : 73 Þ

Y

Y

provided is normalized to 1:

ð dx Y

ð x ; t ÞYð x ; t Þ¼ 1 ð 1 : 74 Þ

where integration covers the whole space.
The average value of any physical observable 8 A described by the
Hermitian operator A is obtained from

Ð dx Y

ð dx A Yð x ; t ÞY

ð x ; t Þ A Yð x ; t Þ

ð x ; t Þ

Ð dx Y Ð dx Y

h A i¼

ð x ; t ÞYð x ; t Þ ¼

ð 1 : 75 Þ

ð x ; t ÞYð x ; t Þ

8 Its expectation value, that can be observed by experiment.