Chemistry Reference
In-Depth Information
The Hohenberg-Kohn theorems are the foundation for the development of
density functional theory (DFT) [
17
] and its extensions, e.g., time-dependent
DFT, conceptual DFT, current DFT, and subsystem DFT. These methods are
based on the electron density rather than on the wave function as central
quantity for the calculation of molecular properties and chemical descriptors.
All contributions to the total energy are represented by functionals which
depend on the electron density only such that the total energy and in principle
even the wave function
C
(
,
t
)] are given as functionals of
the density. Early attempts to define such density functionals were made by
Thomas [
18
] and by Fermi [
19
] presenting the first kinetic energy density
functionals in 1927/1928. Molecular properties and chemical descriptors are
then defined as derivatives of the total-energy density functional
E
[
r
]with
respect to external perturbations.
In practice, the electron density is usually calculated from a wave function (even
within DFT
r
1
,
...
,
r
N
,
t
)
ΒΌ C
[
r
(
r
Kohn-Sham DFT). For this, one has to choose a suitable approxi-
mate Hamiltonian operator and an ansatz for the wave function. In order to arrive at
a consistent theory that overcomes all pitfalls and covers all interactions and effects
important for the chemistry of the whole periodic table, including heavy atoms, one
must apply a theory which is based on the Dirac equation [
20
,
21
]. A comprehen-
sive description of matter is therefore solely given by the Dirac-Coulomb-Breit or
the Dirac-Coulomb Hamiltonian [
22
] which we use as the standard reference when
we investigate the performance or accuracy of any approximate relativistic method
(the electron-electron interaction is usually approximated by the instantaneous
Coulomb interaction). The most important approximate Hamiltonian operators
will be discussed in detail in the theory section of this work.
Different types of many-electron functions are known as approximations to the
exact wave function and are built from one-electron functions, i.e., from orbitals
c
i
(
!
). Such an independent-particle model in which the wave function can be
assembled from an antisymmetrized product of
N
one-electron functions entirely
neglects the correlated motion of the electrons and causes therefore errors in the
description of systems containing more than one electron. It is therefore important
to carry out a systematic analysis of the method-inherent approximations to ensure
that a sufficiently high accuracy for electron densities obtained from quantum
chemical calculations can be guaranteed (correlation effects on the electron density
will be discussed in Sect.
4.2
of this work).
This work is organized as follows. In Sect.
2
, we demonstrate how equations for
the electron density are derived from fundamental principles to analyze in the next
section how these equations depend on the choice of the Hamiltonian and the ansatz
for the wave function. Then we proceed in Sect.
4
with the analysis of approximate
electron densities obtained from different choices for the Hamiltonian with the
focus on relativistic effects. In Sect.
5
, we regard the electron density at the position
of the nucleus, which is prone to errors for most of the approximate Hamiltonians.
Section
6
deals with the electron density in the context of conceptual DFT and
atoms in molecules in combination with relativistic electronic structure methods.
r
