j
J b I a ðrÞ ¼ À
l 0
4p
dbc
ðr c À R Jc Þ
j r À R J j
3
J
m Ia
d ðrÞ;
ð7:57Þ
(a tensor defined all over the molecular domain, measured in T
2 J
−1 m
−3 in the SI
system), yield fundamental complementary information on the nuclear coupling
phenomenon, transmission paths and electron-nuclear interaction. The j
IJ function
can be integrated to obtain the local magnetic field
h ^
B
n
J b
i ¼ ÀK
J b I a m I a ;
and the coupling constant, Eq. (7.54). These expressions can also be regarded as
generalized forms of the integral BS law from classical electrodynamics [46].
7.3 Singularities, Stagnation Lines and Stagnation Graph
of a Current Density Field
The current density of an n-electron quantum mechanical system may form 3n −
2-dimensional vortices in 3-dimensional configuration space [78–84]. If the system
admits a description in terms of density matrices, natural orbitals are obtained by
diagonalizing c
ð0Þ
ð rÞ, Eq. (7.9). Then the total current density vector, J
B or J
m I , can
be analyzed in terms of distinct contributions from each orbital.
The topology of the J
B vector field deserves a careful and detailed investigation.
Its most interesting features are observed in the proximity of an SP at which the
modulus jJ
B
j vanishes. An SP is classified in terms of topological index
3
i [85, 86],
and of a (rank, signature) label [15, 55–57, 87–90]. A continuous, open or closed,
path of SPs is referred to as stagnation line (SL), consisting of either vortex points
(index i ¼ þ 1), or saddle points (index i ¼ À1).
In the vicinity of a stagnation point at r 0 , the fields J
B
ðrÞ and J
m I ðrÞ can be
described by a truncated Taylor series expansion about r 0 , e.g.,
J
B
c r
ð Þ ¼ r a À r 0a
ð
Þ r a J
B
c
h
i
r¼r 0
þ
1
2
r a À r 0a
ð
Þ r b À r 0b
À
Á r a r b J
B
c
h
i
r¼r 0
þ Á Á Á
ð7:58Þ
3
The topological index i counts the number of times that the current density vector J
B rotates
completely while one walks counterclockwise around a circle of radius , so small that J
B has no
zeroes inside except the SP at its centre. The topological index i of a saddle (vortex) line is −1
(+1). Both SPs have ðr; sÞ ¼ ð2; 0Þ.
7 Topology of Quantum Mechanical Current Density …
165
J b I a ðrÞ ¼ À
l 0
4p
dbc
ðr c À R Jc Þ
j r À R J j
3
J
m Ia
d ðrÞ;
ð7:57Þ
(a tensor defined all over the molecular domain, measured in T
2 J
−1 m
−3 in the SI
system), yield fundamental complementary information on the nuclear coupling
phenomenon, transmission paths and electron-nuclear interaction. The j
IJ function
can be integrated to obtain the local magnetic field
h ^
B
n
J b
i ¼ ÀK
J b I a m I a ;
and the coupling constant, Eq. (7.54). These expressions can also be regarded as
generalized forms of the integral BS law from classical electrodynamics [46].
7.3 Singularities, Stagnation Lines and Stagnation Graph
of a Current Density Field
The current density of an n-electron quantum mechanical system may form 3n −
2-dimensional vortices in 3-dimensional configuration space [78–84]. If the system
admits a description in terms of density matrices, natural orbitals are obtained by
diagonalizing c
ð0Þ
ð rÞ, Eq. (7.9). Then the total current density vector, J
B or J
m I , can
be analyzed in terms of distinct contributions from each orbital.
The topology of the J
B vector field deserves a careful and detailed investigation.
Its most interesting features are observed in the proximity of an SP at which the
modulus jJ
B
j vanishes. An SP is classified in terms of topological index
3
i [85, 86],
and of a (rank, signature) label [15, 55–57, 87–90]. A continuous, open or closed,
path of SPs is referred to as stagnation line (SL), consisting of either vortex points
(index i ¼ þ 1), or saddle points (index i ¼ À1).
In the vicinity of a stagnation point at r 0 , the fields J
B
ðrÞ and J
m I ðrÞ can be
described by a truncated Taylor series expansion about r 0 , e.g.,
J
B
c r
ð Þ ¼ r a À r 0a
ð
Þ r a J
B
c
h
i
r¼r 0
þ
1
2
r a À r 0a
ð
Þ r b À r 0b
À
Á r a r b J
B
c
h
i
r¼r 0
þ Á Á Á
ð7:58Þ
3
The topological index i counts the number of times that the current density vector J
B rotates
completely while one walks counterclockwise around a circle of radius , so small that J
B has no
zeroes inside except the SP at its centre. The topological index i of a saddle (vortex) line is −1
(+1). Both SPs have ðr; sÞ ¼ ð2; 0Þ.
7 Topology of Quantum Mechanical Current Density …
165