Electron Density in Quantum Theory - Electron Density and Chemical Bonding - page 111

Similar to DKH, the calculation of the electron density causes problems in the
ZORA approach [ 50 ]. The electron density and the current density obtained from

X

N

c i; ZORA ðÞc i; ZORA ðÞ r S ZORA ðÞ¼ (45)

i¼ 0

c X

N

c i; ZORA ðÞa c i; ZORA ðÞ

j S ZORA ðÞ¼

(46)

i¼ 0

are only approximations to the Dirac densities, since the elimination of the small
component causes a picture change and introduces therefore an error. An improved
ZORA electron density (ZORA-4 density) is obtained by a backtransformation of
the small component and the introduction of a scaling factor. Following van Lenthe
and Baerends [ 50 ], a small and a large component density are defined for each
orbital:

y s p i c i; ZORA ðÞ

c 2 s p i c i; ZORA ðrÞ

r i

ðÞ¼ ; (47)

2

ð 2 c 2 V Þ

ðÞ¼c i; ZORA ðÞc i; ZORA ðÞ: r i (48)

The ZORA-4 density, which is normalized to one, is then calculated as:

X

N

r i þ r i

r S ZORA 4 ¼

þ Ð r i d 3 r :

(49)

1

i¼ 0

Inserting the nonrelativistic Schrodinger Hamiltonian into the continuity
equation:

h D C E D r E i (50)

2 m e i X

N
i¼ 1 r

@ Crj C h i h

i Cd ðÞ r r i

Cd ð 3 Þ

2 2

¼ ð Þ ð r r i Þ

i C

d t

yields the nonrelativistic (NR) electron density and current density. In the case of
a wave function approximated by a single Slater determinant, it is given by:

X X

N N

r S NR ðÞ¼ c i

2

ðÞc i ðÞ¼ j c i ðÞ j ; (51)

i ¼ 1 i ¼ 1

2 m e i X

N

:

j S NR ðÞ¼

h

c i Þ c i ðÞrc i ðÞrc i ð ðÞ ðÞ (52)

i¼ 1