Biomedical Engineering Reference
In-Depth Information
∂
2
∂
x
i
+
n
n
−
1
n
h
2
8π
2
m
e
∂
2
∂
y
i
+
∂
2
∂
z
i
Ze
2
4πε
0
r
i
e
2
4πε
0
r
ij
H
=
−
−
+
(14.1)
i
=
1
i
=
1
j
=
i
+
1
14.1 One- and Two-Electron Operators
In order to stop the notation becoming unwieldy, I will group these terms as follows. Each
term in the first bracket refers to the coordinates of the individual electrons, and gives the
kinetic energy of each electron together with its attraction to the nucleus. I will call such
terms
one-electron operators
and authors normally write them as
∂
2
∂
x
i
+
h
2
8π
2
m
e
∂
2
∂
y
i
+
∂
2
∂
z
i
Ze
2
4πε
0
r
i
h
(1)
(
r
i
)
=−
−
(14.2)
Each term in the second double summation gives the Coulomb repulsion of a pair of
electrons; I will refer to them as
two-electron operators
and write
g
r
i
,
r
j
=
e
2
4πε
0
r
ij
ˆ
(14.3)
The Hamiltonian is then, in our compact notation
n
n
−
1
n
g
r
i
,
r
j
H
h
(1)
(
r
i
)
=
+
1
ˆ
(14.4)
i
=
1
i
=
1
j
=
i
+
and we wish to investigate the solutions of the Schrödinger equation
H
Ψ (
r
1
,
r
2
, ...,
r
n
)
=
εΨ (
r
1
,
r
2
, ...,
r
n
)
(14.5)
14.2 Many-Body Problem
The many-electron atom is an example of a so-called
many-body problem
. These are not
unique to quantum theory; a familiar example is planetary motion. Newton's equations
of motion can be solved exactly for the motion of any one of the planets round the sun
individually, but the planets also attract each other. During the eighteenth and nineteenth
centuries a great deal of effort was expended trying to find an exact solution to planetary
motion, but all efforts failed and it is generally accepted that exact solutions do not exist,
even for just three bodies. Astronomers are lucky in the sense that the gravitational force
depends on the product of twomasses and themass of the sun ismuch greater than themasses
of the individual planets. Ingenious techniques were developed to treat the interplanetary
attractions as small perturbations, and the planetary problem can be solved numerically to
any accuracy required.
To continue with the planetary motion analogy, chemists are less lucky in one sense
because the electrostatic force depends on the product of two charges and in the case of an
electron and a proton these forces are roughly equal in magnitude. It will therefore come
