Electron Density in Quantum Theory - Electron Density and Chemical Bonding - page 110

a single Slater determinant (SD), the electron density and the current density are
then given as:

X X

N N

c i

2 2

r S DCB ðÞ¼ c i þ c i ðÞc i ðÞ¼ ðÞ ðÞ ; (39)

i¼ 1 i¼ 1

c X

N

c i

SD

j

DCB ðÞ¼

ðÞa c i ðÞ: (40)

i¼ 1

In contrast to the DCB Hamiltonian, the electron density in the DKH framework
is not obtained as the sum of squared DKH orbitals. The latter deviates from the
electron density, especially in the region around the nucleus. At larger distances
from the nucleus, these errors decrease [ 64 , 65 ]. It is possible to relate the square of
the Dirac 4-spinors to the square of the two-component DKH spinor by the
introduction of a position-dependent error Dr i ( r ):

2

2

c i; 4comp ðÞ ¼ c i ; DKH ðÞ þ Dr i ðÞ: (41)

Considering the evaluation of expectation values over operators with a trans-
formed wave function, one must not forget to transform the operator, too. The
position-dependent error Dr ( ) in the calculation of the electron density vanishes

when the transformed (picture-change corrected) density operator

r

r r is used [ 66 ].
The electron density obtained in this way is then equal to the Dirac electron density.
The continuity equation is obtained in the same way as for the Dirac-Coulomb-
Breit Hamiltonian, but with the transformed density operator and Hamiltonian
applied. Then, the continuity equation reads:

UC UC

D E D E

@
@ t UC Ur r U y

ÞU ð 1 Þ y UC U ð 1 Þ a 1 d ð 3 Þ ¼r cN ð r r 1 : (42)

The electron density and the current density are therefore obtained as:

C

D E

X

N

ÞU y

C Ud ð 3 Þ

r S DKH ðÞ¼ ð r r i ; (43)

i ¼ 1

C

D E

c X

N

ÞU ðiÞ y

C U ðiÞ a i d ð 3 Þ

SD
DKH

j ðÞ¼ ð r r i : (44)

i ¼ 1

In the case of more than one electron, the unitary transformation is given as the
direct product of N unitary transformations

i

¼ 1 U ðiÞ :

U ¼