Chemistry Reference
In-Depth Information
The key point to handle these two equations was formulated by Blum and Torruela
(1972) in a series of now landmark papers by using a rotational invariant expansion
that would allow a separation between the radial and the angular dependence, the
latter being handled through the Wigner elements
R
µν
()
Ω
, which are a generaliza-
tion of the spherical harmonics to all three Euler angles (Gray and Gubbins 1984).
This procedure creates an infinite series of “projections”
ar
ij;
mn
µν
()
of the various cor-
relation functions
a
(1,2), both in
r
-space and Fourier space, for an expansion that is
adapted to the rotationally invariant isotropic phase. The number integer of indexes
(
m
,
n
,
l
;μ,ν) depends on the symmetry of the phase. Without getting into such unnec-
essary complications, since we will be mainly considering macroscopic homoge-
neous and isotropic phase, the generic expansion reads:
∑
µν
()
()
()
mnl
mnl
a
1,2
=ar
ϕ
1,2
(7.38)
µν
where the “invariants” are defined as
mn l
µνλ
mnl
mnl
m
n l
() () ()
ϕ
(, )
12
=
ϕ
(
ΩΩΩ
,
,
r
/
r
)
=
R
Ω
1
R
Ω
Rr
r
/
(7.39)
µν
µν
12
µµ
′
νν
′
2
λ
′
0
µνλ
′′′
′′ ′
where the matrix element is a Racah 3-
j
symbol (Gray and Gubbins 1984), and
λ
= -λ, a notation that we shall use throughout. The form of the invariant is entirely
dictated by the symmetry of the phase. Note that the invariant does not carry the
species pair indexing, since it depends only on the angles. This notation allows one
to rewrite both exact Equation 7.35 and Equation 7.37 solely in terms of the projec-
tions
ar
ij;
mn
µν
()
, hence discarding all the angular dependence. The MOZ Equation 7.35
can be rewritten into the convenient matrix form, if one takes care of expressing
Equation 7.35 in the intermolecular frame, where the vector
k
is aligned with the
local z-axis. Since the mixture is isotropic and homogeneous, this operation does not
alter the symmetry contained in the expansion of the type given in Equation 7.38, but
it takes now a new form in the molecular frame (Perera and Patey 1988),
=
∑
ˆ
mn
χ
h
(, )
12
h
( )
kR
m
(
ΩΩΩ
2
)
)
R
n
(
(7.4 0)
ij
µχ
1
ν
χ
ij
;
µν
mn
χµν
where the orientation dependence on
k
has been lifted. The new expansion coeffi-
cients
hk
ij
mn
;
χ
are related to those in the lab fixed frame by a simple linear relation
(Fries and Patey 1985),
()
µν
m
n
l
∑
ˆ
ˆ
mn
χ
mnl
hk
()
=
hk
()
(7.41)
ij
;
µν
ij
;
µν
χ
χ
0
l
