Chemistry Reference
In-Depth Information
with
t
n
(
I
)and
t
e
(
i
) denoting the kinetic energy operators for nuclei
I
and electrons
i
.
v
nn
(
I
,
J
) is the repulsion energy operator between nuclei
I
and
J
,
v
ee
(
i
,
j
) the electron-
electron repulsion operator of electrons
i
and
j
,and
v
ne
(
I
,
J
) describes the attraction of
electron
i
and nucleus
I
. The explicit form of the expressions for the operators
is defined through the basic physics on which an approximation to the exact
Hamiltonian relies. When one aims at an adequate description of all elements in
the periodic table, including heavy metal atoms like actinides, the Dirac-Coulomb-
Breit (DCB) Hamiltonian is the most suitable choice for chemistry. It is deduced from
the Einsteinian relativity principle using classical electromagnetic fields [
22
]. There
may exist extreme cases, where quantum electrodynamical (QED) corrections play a
role [i.e., where a quantization of the electromagnetic field (photons) is necessary],
but we consider these cases to be unimportant for general chemistry.
For the DCB Hamiltonian, we first introduce the one-electron Dirac Hamiltonian
h
D
(
i
) for an electron in an external potential. It contains all one-electron operators
and the Coulomb interaction between the single electron and the nuclei (in Gaussian
units; used throughout):
X
M
Z
I
e
2
r
i
R
I
e
c
A
m
e
c
2
h
D
ð
i
Þ¼
c
a
i
p
i
þ
þ b
i
ð
1
Þ
ef
j
:
(19)
j
I
¼
1
and the
scalar potential
f
, are introduced via minimal coupling which ensures Lorentz
covariance of the one-electron Dirac equation of motion. For an isolated atom,
molecule, or crystal, we have
External electromagnetic fields, represented by the vector potential
A
0. In the more general case of a
system containing
M
nuclei and
N
electrons, the DCB Hamiltonian includes the
one-electron Dirac Hamiltonian as:
A ¼
0 and
f ¼
"
#
X
X
X
M
N
M
I
Z
I
e
2
r
i
R
I
p
e
c
A
m
e
c
2
H
DCB
¼
2
m
I
þ
c
a
i
p
i
þ
þ b
i
ð
Þ
ef
1
j
j
I¼
1
i¼
1
I¼
1
X
X
X
X
M
M
N
N
Z
I
Z
J
e
2
R
I
R
J
þ
j
þ
gi
ðÞ
;
j
ð
20
Þ
j
I
¼
1
J
¼
I
þ
1
i
¼
1
j
¼
i
þ
1
with
I
and
i
being the particle indices,
R
I
and
r
i
the coordinates,
Z
I
e
and
e
the
p
i
the momenta, and
m
I
and
m
e
the masses of the nuclei and
the electrons, respectively. The speed of light is denoted as
c
.
p
I
and
charges,
a
matrix for the
i
-th electron, where
a ¼
(
a
x
,
a
y
,
a
z
) is a three-dimensional vector of
four-by-four matrices:
a
i
is a Dirac
0
1
0
1
0
1
00
00
s
x
00
00
s
y
00
00
s
z
@
A
; a
y
¼
@
A
; a
z
¼
@
A
;
a
x
¼
00
00
00
00
00
00
s
x
s
y
s
z
(21)
