Chemistry Reference
In-Depth Information
a positively charged “hole” (Dirac hole theory). With this interpretation, Dirac
predicted the existence of an antiparticle, which contains the same mass as the
electron but carries the opposite charge. In practice, the positronic states are not
occupied in a quantum chemical calculation. The positronic states lead to the
consequence that the Hamiltonian is no longer bounded from below because any
electron can in principle take an infinitely large negative energy, which may result
in a variational collapse during the optimization of the wave function. Another
related pathology of the Dirac formalism is known as the Brown-Ravenhall [
30
]
disease, or continuum dissolution. In fully relativistic calculations, these patholo-
gies are circumvented by technical tricks that can be formalized in terms of
iteratively optimized projection operators.
However, the positronic solutions are not relevant to chemistry, and their
calculation is therefore an unnecessary burden. We now discuss two possibilities
to avoid the calculation of these positronic states by decoupling the large and the
small components of the spinor. Eliminating two components from the spinor yields
efficient two-component methods. Either they can be projected out, e.g., using the
generalized Douglas-Kroll-Hess (DKH) unitary transformation technique [
31
-
33
]
or they are eliminated in the so-called regular approximations [
34
]. Recently,
efficient four-component formulations of the unitary block-diagonalization in one
shot hold promises for future routine applications [
35
-
40
]. However, low-order
regular approximations and DKH schemes are sufficiently accurate, efficient, and
well embedded in quantum chemistry program packages tailored for routine quan-
tum chemical calculations (like, e.g., MOLCAS [
41
] or ADF [
42
,
43
]).
The DKH transformation technique is based on an idea by Douglas and Kroll
[
44
] which was later rediscovered and turned into a practical method by Hess [
31
].
The DKH formalism has then further been developed by Reiher and Wolf [
32
,
33
,
45
-
49
]. The general unitary DKH transformation block-diagonalizes the Dirac
Hamiltonian:
0
1
00
00
h
þ
@
A
;
Uh
D
U
y
¼
h
bd
¼
(26)
00
00
h
resulting in two decoupled two-component matrix operators
h
+
and
h
for the
electronic and the positronic eigenstates, respectively. The unitary transformation
is constructed as a product of infinitely many unitary transformations
U
¼
U
1
...
U
3
U
2
U
1
U
0
each Taylor expanded in terms of an antihermitian operator. In
practical applications,
the expansion of the unitary transformation is usually
truncated.
Also the wave function
c ¼
Uc
must be transformed in the same way. If
expectation values of hermitian operators are calculated in the DKH picture, one
must take care that the operators are transformed, too. The neglect of these
transformations leads to the so-called picture change error (PCE) [
50
-
52
]. As a
consequence, the electron density in the DKH picture is not obtained by simply
