The Hamiltonian of the two-particle system in real space with coordinates
with index 1
for the proton (mass M) and index 2 for the electron (mass m) can be written as:
2
2
2
p1
p2
Ze
H = -------- + ------- – --------------2M 2m 4 0
with
2
hb
– -------2M
2
1
hb
– ------2m
2
Ze
+ --------------- = E
4 0
2
where the Laplacian with index 1 operates in the subspace of the electron:
2
1
=
2
1
2
+
2
1
2
+
2
1
and the wave function depends on the six coordinates of the two particles.
The coordinates of the centre-of-mass are:
M 1+m 2
X = -------------------------x = 2– 1
M+m
M 1+m 2
Y = ---------------------------y = 2– 1
M+m
M 1+m 2
Z = -------------------------z = 2– 1
M+m
and:
2
2
2
= x +y +z = r
Insertion of the new parameters in the differential equation yields:
2
2
1
M
= ---------------M+m
2
2
X
2
2
2
M
– 2 ---------------M+m X x
+
2
2
x
2
and it follows:
1
----M
1
1
+ ---m
2
1
= ---------------M+m
2
X
2
+
Y
2
2
+
Z
1
+ ---
2
x
2
With the reduced mass:
mM
= ---------------m+M
This equation can be separated:
=
( x , y, z ) ( X , Y , Z )
E = E+E
Giving two equations, one for the centre-of-mass motion:
2
hb
– -----------------------2(m + M )
( X, Y , Z ) = E
( X, Y , Z )
2
+
y
2
2
+
z
2
and for the relative electron motion:
2
hb
– -----2
2
Ze
( x, y, z ) – -------------- = E ( X , Y , Z )
4 0r
Thus the atomic units can be adopted to take into account the motion of the nucleus.
taken as the new atomic unit of mass.
can be