Closure relation to solve the Ornstein-Zernike equation
In statistical mechanics the hypernetted-chain equation is a closure relation to solve the Ornstein–Zernike equation which relates the direct correlation function to the total correlation function. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function . It is given by:
ln
y
(
r
12
)
=
ln
g
(
r
12
)
+
β
u
(
r
12
)
=
ρ
∫
[
h
(
r
13
)
−
ln
g
(
r
13
)
−
β
u
(
r
13
)
]
h
(
r
23
)
d
r
3
,
{\displaystyle \ln y(r_{12})=\ln g(r_{12})+\beta u(r_{12})=\rho \int \lefth(r_{23})\,d\mathbf {r_{3}} ,\,}
where
ρ
=
N
V
{\displaystyle \rho ={\frac {N}{V}}}
number density of molecules,
h
(
r
)
=
g
(
r
)
−
1
{\displaystyle h(r)=g(r)-1}
g
(
r
)
{\displaystyle g(r)}
radial distribution function ,
u
(
r
)
{\displaystyle u(r)}
β
=
1
k
B
T
{\displaystyle \beta ={\frac {1}{k_{\rm {B}}T}}}
T
{\displaystyle T}
Thermodynamic temperature and
k
B
{\displaystyle k_{\rm {B}}}
Boltzmann constant .
Derivation
The direct correlation function represents the direct correlation between two particles in a system containing N − 2 other particles. It can be represented by
c
(
r
)
=
g
t
o
t
a
l
(
r
)
−
g
i
n
d
i
r
e
c
t
(
r
)
{\displaystyle c(r)=g_{\rm {total}}(r)-g_{\rm {indirect}}(r)\,}
where
g
t
o
t
a
l
(
r
)
=
g
(
r
)
=
exp
[
−
β
w
(
r
)
]
{\displaystyle g_{\rm {total}}(r)=g(r)=\exp}
w
(
r
)
{\displaystyle w(r)}
potential of mean force ) and
g
i
n
d
i
r
e
c
t
(
r
)
{\displaystyle g_{\rm {indirect}}(r)}
u
(
r
)
{\displaystyle u(r)}
g
i
n
d
i
r
e
c
t
(
r
)
=
exp
{
−
β
[
w
(
r
)
−
u
(
r
)
]
}
{\displaystyle g_{\rm {indirect}}(r)=\exp\{-\beta \}}
approximate
c
(
r
)
{\displaystyle c(r)}
c
(
r
)
=
e
−
β
w
(
r
)
−
e
−
β
[
w
(
r
)
−
u
(
r
)
]
.
{\displaystyle c(r)=e^{-\beta w(r)}-e^{-\beta }.\,}
By expanding the indirect part of
g
(
r
)
{\displaystyle g(r)}
y
(
r
)
=
e
β
u
(
r
)
g
(
r
)
(
=
g
i
n
d
i
r
e
c
t
(
r
)
)
{\displaystyle y(r)=e^{\beta u(r)}g(r)(=g_{\rm {indirect}}(r))}
c
(
r
)
{\displaystyle c(r)}
c
(
r
)
=
e
−
β
w
(
r
)
−
1
+
β
[
w
(
r
)
−
u
(
r
)
]
=
g
(
r
)
−
1
−
ln
y
(
r
)
=
f
(
r
)
y
(
r
)
+
[
y
(
r
)
−
1
−
ln
y
(
r
)
]
(
HNC
)
,
{\displaystyle c(r)=e^{-\beta w(r)}-1+\beta \,=g(r)-1-\ln y(r)\,=f(r)y(r)+\,\,({\text{HNC}}),}
with
f
(
r
)
=
e
−
β
u
(
r
)
−
1
{\displaystyle f(r)=e^{-\beta u(r)}-1}
This equation is the essence of the hypernetted chain equation. We can equivalently write
h
(
r
)
−
c
(
r
)
=
g
(
r
)
−
1
−
c
(
r
)
=
ln
y
(
r
)
.
{\displaystyle h(r)-c(r)=g(r)-1-c(r)=\ln y(r).}
If we substitute this result in the Ornstein–Zernike equation
h
(
r
12
)
−
c
(
r
12
)
=
ρ
∫
c
(
r
13
)
h
(
r
23
)
d
r
3
,
{\displaystyle h(r_{12})-c(r_{12})=\rho \int c(r_{13})h(r_{23})d\mathbf {r} _{3},}
one obtains the hypernetted-chain equation :
ln
y
(
r
12
)
=
ln
g
(
r
12
)
+
β
u
(
r
12
)
=
ρ
∫
[
h
(
r
13
)
−
ln
g
(
r
13
)
−
β
u
(
r
13
)
]
h
(
r
23
)
d
r
3
.
{\displaystyle \ln y(r_{12})=\ln g(r_{12})+\beta u(r_{12})=\rho \int \lefth(r_{23})\,d\mathbf {r_{3}} .\,}
See also
Categories :
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↑
is the 
, 
is the 
is the direct interaction between pairs. 
with 
being the 
the 
(with 
the 
is the radial distribution function without the direct interaction between pairs 
. Thus we approximate 
by
we can approximate 
.
