Chemistry Reference
In-Depth Information
which are built from the Pauli spin matrices
s ¼
(
s
x
,
s
y
,
s
z
):
; s
y
¼
; s
z
¼
01
10
0
i
10
0
s
x
¼
(22)
i
0
1
b
i
¼
and
1) is a diagonal matrix. There are different levels of
approximation for the electron-electron interaction. The Coulomb-Breit operator
reads [
22
]:
diag(1,1,
1,
"
#
a
j
e
2
r
i
r
j
a
i
2
a
i
a
j
r
i
r
j
r
i
r
j
gi
ðÞ¼
;
j
1
:
(23)
2
2
r
i
r
j
It is often convenient to employ to Gaunt operator [
28
,
29
] to approximate the
Breit operator. A rigorous approximation to the DCB Hamiltonian is obtained in the
nonrelativistic limit by substituting the Coulomb-Breit operator by the instanta-
neous Coulomb interaction (which we already adopted for the electron-nucleus
interaction). The resulting DC Hamiltonian neglects retardation effects of the
electron-electron interaction which arise from the transmission of the interaction
due to the finite value of the speed of light.
Since the
a
i
and
b
i
parameters in the one-electron Dirac Hamiltonian have
a4
4 structure, it immediately follows that the one-electron functions have
four components. Owing to the 2
2 superstructure of the Hamiltonian [see
(
21
)], they are grouped into two 2-spinors, which are for historical reasons denoted
as large (L) and small (S) component:
0
1
c
i
c
i
c
i
c
i
!
:
@
A
c
i
c
i
c
i
¼
¼
(24)
Since the Dirac Hamiltonian possesses a 2
2 superstructure, any other opera-
tor
O
can also be expressed as:
O
LL
O
LS
O
¼
:
(25)
O
SL
O
SS
The Dirac equation for a single electron yields two sets of solutions, namely
positive (electronic) and negative (positronic) energy eigenstates. The electronic
eigenstates describe either freely moving electrons or electrons that are bound by an
external potential, whereas the positronic states lead to conceptual and practical
difficulties. Dirac interpreted all the positronic states to be occupied by electrons
such that the excitation of an electron from this “sea of electrons” produces
