Chemistry Reference
In-Depth Information
Table 2 Values of
r
(
r
)in
e
˚
3
and
L
(
r
)in
e
˚
5
at the
bond critical point BCP1 of
titanium tetrachloride
BCP1
r
(
r
)
L
(
r
)
Supermolecular
0.65
1.73
Embedding
0.62
1.98
Difference
0.03
0.25
deficiencies of the embedding density has its origin in the complex formation,
where electron density is shifted from the Cl
fragment toward the titanium atom.
These changes are not fully recovered in the region of the border between both
subsystems. The largest changes are observed between both subsystems directly
next to the TiCl
3
fragment on the axis perpendicular to the bonding axis. The
difference density between the KS-DFT reference calculation of the full system and
the embedding density is shown in Fig.
8e
.
Considering the topology of the electron density of titanium tetrachloride, four
BCPs are found of which two are located in the plane that is depicted in Fig.
8
.
BCP1 is located directly next to the border between the subsystems. The two BCPs
in the plane are located at the same positions in both the supermolecular density
and the embedding density. The values of the electron density and the negative
Laplacian at the BCP1, taken from Table
2
, differ only by ~0.03 and 0.25 e
˚
5
,
respectively. In contrast to the result for ammonia borane, the negative Laplacian
has the correct sign at both BCPs.
8 Conclusion
In this work, we reviewed important issues related to the calculation of the electron
density in quantum theory. We discussed the definition of the quantities “electron
density” and “current density” as defined by a continuity equation depending on the
many-electron Hamiltonian and wave function. In practice, both types of densities
are calculated from (quasi)relativistic Hamiltonians and wave function approxima-
tions. For this reason, we discussed the most important Hamiltonians - namely the
relativistic Dirac-Coulomb-Breit reference, the Dirac-Coulomb, the Douglas-
Kroll-Hess, the ZORA and the nonrelativistic Schrodinger Hamiltonians - and
wave functions.
In an analysis of approximate electron densities, the most important observations
to make regarding the accuracy of approximate Hamiltonians are that in most cases
scalar-relativistic variants of the approximate Hamiltonian operators as ZORA and
DKH are sufficient to obtain a reasonable description of the electron density. The
differences to two-component Hamiltonians including spin-orbit effects are negli-
gible as long as one considers quantities that do not primarily feature spin-orbit
effects. These approximate Hamiltonian operators are in general reliable but may
produce densities deviating from the fully relativistic reference at the position of the
nucleus. For the innermost atomic shells, ZORA and picture-change-affected DKH
