Density Functional Theory:
The Classical Hard-Core Gas
Simon Bignold
Supervisor: Christoph Ortner
University of Warwick
October 17, 2012
Simon Bignold Supervisor: Christoph Ortner
Outline
Model
Helmholtz Free Energy
One Particle Density
Introduction to DFT
Ideal Gas
Hard-Core Potential
Further Work
Simon Bignold Supervisor: Christoph Ortner
Model
Simon Bignold Supervisor: Christoph Ortner
Set-up
We have a number of particles N with positions xi ∈ Λ ⊂ Rd and
momentum pi ∈ Rd .
For convenience we define
XN = (x1 , . . . , xN ) ∈ ΛN
PN = (p1 , . . . , pN ) ∈ RdN
and
β=
Simon Bignold Supervisor: Christoph Ortner
1
.
kB T
The Hamiltonian
HΛVN (XN , PN )
X
=
W (xi − xj ) +
V (xi )
i=1
1≤i<j≤N
|
N
X
{z
}
inter-particle interaction U
Simon Bignold Supervisor: Christoph Ortner
|
{z
}
external potential
N
X
pi2
+
2m
|i=1{z }
Kinetic part
Canonical Gibbs Ensemble
β
characterised by γΛ,N
∈ P(Λ, BΛ ) with density
ρβΛ,N (XN , PN ) =
exp[−βHΛVN (XN , PN )]
N!ZΛ (β, N)
with respect to the Lebesgue measure, where BΛ is the Borel σ
-algebra on Λ.
Here ZΛ (β, N) is a normalisation factor known as the Partition
Function.
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Partition Functions
ZΛ (β, N) =
Rd
|
1
×
N!
|
N
p2
exp −β
dp
2m
{z
}
Z
kinetic partition function
Z
N
Y
exp[−βV (xi )]
ΛN i=1
Y
exp[−βW (xi − xj )]dXN .
1≤i<j≤N
{z
ZΛ,con (β,V ) configurational partition function
Simon Bignold Supervisor: Christoph Ortner
}
Average Energy
We define
R
E=
RdN
R
ΛN
HΛVN (XN , PN ) exp[−βHΛVN (XN , PN )]dXN dPN
.
N!ZΛ (β, N)
Simon Bignold Supervisor: Christoph Ortner
Free Energy
Simon Bignold Supervisor: Christoph Ortner
Helmholtz Free Energy
Free energy is minimised at equilibrium if temperature is held
constant.
N
AΛβ [V ] = E − TSΛ .
We can also show
N
AΛβ [V ] = −β −1 ln[ZΛ (β, N)]
#
"N Z
Y
pi2
−1
dpi
= −β ln
exp −β
2m
d
i=0 R
|
{z
}
Kinetic free energy

+ −β −1 ln 
|
1
N!
Z
N
Y

Y
exp[−βV (xi )]
ΛN i=1
exp[−βW (xi − xj )]dXN  .
1≤i<j≤N
N
{z
AΛβ,con [V ] configurational free energy
Simon Bignold Supervisor: Christoph Ortner
}
Free Energy per Particle
For use in the thermodynamic limit (where N → ∞ ). We need
free energy per particle
N
FβΛ [V ] =
N
1 ΛN
A [V ]
N β
Λ
and we can similarly define Fβ,con
[V ].
Simon Bignold Supervisor: Christoph Ortner
One-particle Density
Simon Bignold Supervisor: Christoph Ortner
One-particle Density
Three ways of doing this
• Integrating out N − 1 Variables
R R
N Rd ΛN−1 exp −βHΛVN (XN , PN ) dx2 . . . dxN dPN
(1)
ρΛN (x) =
N!ZΛ (β, N)
• Average over δ-functions
* N
+
X
(1)
δ(x − xi )
ρΛN (x) =
i=0
C
1
=
N!ZΛ (β, N)
Z Z X
N
h
i
δ(x − xi ) exp −βHΛVN (XN , PN ) dXN dPN
×
Rd
ΛN i=0
• Functional Derivative
N
(1)
ρΛN (x) =
Simon Bignold Supervisor: Christoph Ortner
δAΛβ [V ]
δV (x)
.
Introduction to Density Functional Theory
Simon Bignold Supervisor: Christoph Ortner
Introduction to DFT
We want to express free energy as the sum of a functional of
one-particle density only and another term.
(1)
V (x) is a conjugate variable to the one-particle density ρΛN (x).
Since free energy is a functional of the external potential, we can
use a Legendre transform to re-write the free energy.
h
i Z
N
(1)
(1)
AΛβ [V ] = FHK ρΛN (x) + V (x)ρΛN (x)dx
Λ
FHK is known as the Hohnberg-Kohn functional.
Simon Bignold Supervisor: Christoph Ortner
Ideal Gas
Simon Bignold Supervisor: Christoph Ortner
Ideal Gas
In this case the internal potential is zero
1
ZΛ (β, N) =
N!
N Z
Z
exp[−βV (x)]dx
Rd
Λ
N
−βp12
exp
dp1
.
2m
We introduce the notation
zβ (dxi ) = dxi exp[−βV (xi )]
and
N
Z
z(Λ) =
Λ
Simon Bignold Supervisor: Christoph Ortner
N
.
zβ (dx)
Ideal Gas: Free Energy
We can re-write the partition function as
1
ZΛ (β, N) =
N!
z(Λ)
λd
N
λ=
β
2πm
1
2
which allows us to write
N
AΛβ [V ] = β −1 (ln[N!] + Nd ln[λ] − N ln[z(Λ)]) .
Simon Bignold Supervisor: Christoph Ortner
Density Functional Form
We seek to re-write the free energy in a density functional form.
Using the functional derivative of the free energy we can find the
one-particle density
(1)
ρΛN (x) =
N exp[−βV (x)]
.
z(Λ)
Re-arranging we can find an expression for the external potential
" (1)
#
ρ
(x)z(Λ)
N
V (x) = −β −1 ln Λ
.
N
Simon Bignold Supervisor: Christoph Ortner
DFT Free energy
We therefore find
Z
Z
h
i
z(Λ)
(1)
(1)
(1)
.
ρΛN (x)V (x)dx = −β −1 ρΛN (x) ln ρΛN (x) dx−β −1 N ln
N
Λ
Λ
We recall a generalisation of Stirling’s approximation
√
2πN
N
e
N
N
√
1
1
N
exp
exp
≤ N! ≤ 2πN
.
12N + 1
e
12N
Using this and that
energy as
N
AΛβ [V ]
R
(1)
Λ ρΛN (x)dx
Z
= N we can re-write the free
(1)
(1)
=β
ρΛN (x) ln[ρΛN (x)] + d ln λ − 1 dx
Λ
Z
(1)
+ ρΛN (x)V (x)dx + O(ln N).
−1
Λ
Simon Bignold Supervisor: Christoph Ortner
Hard-Core Gas
Simon Bignold Supervisor: Christoph Ortner
Hard-Core Potential
(
Wa (xi − xj ) = 0 kxi − xj k > a
Wa (xi − xj ) = ∞ kxi − xj k ≤ a
Given that a < acp where acp is the close-packing density.
Then our internal potential is
X
U=
1≤i<j≤N
Simon Bignold Supervisor: Christoph Ortner
Wa (xi − xj ).
1D Free Energy
Here we are in one dimension Λ = L and we can write
ZΛ,con (β, V ) = Z(L)
Z L
Z L
1
...
=
N! 0
0
×
N
Y
Y
exp [−βWa (xi − xj )]
1≤i<j≤N
exp[−βV (xi )]dx1 . . . dxN .
i=1
Since we are in one dimension we can order the particles so that
there positions lie between 0 and L
0 < x1 ≤ x2 · · · ≤ xN < L.
Simon Bignold Supervisor: Christoph Ortner
Convolution form
We can use two changes of variables
yja = xj − (j − 1)a
λaN = l a − yNa
l a = L − (N − 1)a
a
λai = yi+1
− yia
and that the potential is translation invariant
V (xj ) = V (yja ).
To re-write the configurational partition function
Z la
Z l a −λa
N
Z(L) =
dλaN exp [−βV (l a − λaN )]
dλaN−1
0
0
× exp −βV l a − λaN − λaN−1 . . .
"
!#
N
X
a
a
× exp −βV l −
λi
.
i=1
Simon Bignold Supervisor: Christoph Ortner
Z
l a−
N
X
i=2
0
λN
i
dλa1
Laplace Transform
Since Z(l a ) when V = 0 is a convolution we write the Laplace
transform
Z ∞
exp[−sl a ]Z(l a )dl a
Z (s) =
0
= s −(N+1)
since
Z
0
∞
1
exp[−sλa ]dλa = .
s
Thus using the inverse transform we find
I
1
a
Z(l ) =
exp[sl a ]s −(N+1) ds
2πi C
where C is the Bromwich contour.
Simon Bignold Supervisor: Christoph Ortner
The Saddle-point
We write the integrand of our integral as the exponential of g (s).
g (s) = l a s − (N + 1) ln s
this is clearly complex differentiable and has a minimum on the
real-axis so has a saddle-point.
Using the saddle-point condition g 0 (s0 ) = 0 we have at the
saddle-point
N +1
s0 =
.
L − a(N − 1)
We also find the second derivative
g 00 (s0 ) =
Simon Bignold Supervisor: Christoph Ortner
(N + 1)
.
s02
Saddle-point Approximation
We now expand g (s) around the saddle-point and alter the contour
to give an approximation of the configurational partition function
I
1
a
Z(l ) =
exp[g (s)]ds
2πi C
00
Z x0 +i∞
1
g (s0 )(s − s0 )2
exp[g (s0 )]
ds
≈
exp
2πi
2
x0 −i∞
00
Z ∞
1
g (s0 )y 2
≈
exp[g (s0 )]
exp
dy s − s0 = iy
2π
2
−∞
exp[g (s0 )]
1
≈p
.
1+O
00
N
2πg (s0 )
Thus we can see
N
AΛβ,con [V ]
≈ −β
−1
1
1
00
.
g (s0 ) − ln[2πg (s0 )] + ln 1 + O
2
N
Simon Bignold Supervisor: Christoph Ortner
The Zero Hard-Core Limit
Using the values of g (s0 ) and g 00 (s0 ) we find
1
N,L
−1
Aa ≈ −β
N + 1 − N ln[N + 1] + N ln[L − a(N − 1)] − ln[2π]
2
1
1
− ln[N + 1] + ln 1 + O
.
2
N
The zero hard-core limit is
N +1
1
N,L
−1
lim Aa ≈ −β
N + 1 − N ln
− ln[2π]
a→0
L
2
1
1
− ln[N + 1] + ln 1 + O
.
2
N
Simon Bignold Supervisor: Christoph Ortner
The Ideal Gas Case
From above we have an ideal gas expression for the configurational
energy and using Stirling’s approximation we have
1
1
1
N
N,L
−1
− ln[2π] − ln N + O
A0 = −β
N − N ln
L
2
2
N
the free energy per particle in this case is
N
1
1
1
F0N,L = −β −1 1 − ln
ln[2π] −
ln N + O
−
.
L
2N
2N
N2
Simon Bignold Supervisor: Christoph Ortner
Comparison
We take the thermodynamic limit in both cases and note that
ρ = N/L.
For the ideal gas
F0th = lim F0N,L
N→∞
L→∞
= −β −1 (1 − ln[ρ])
whereas in the thermodynamic and zero hard-core limit of the
hard-core case we have
lim lim FaN,L = lim
N→∞
L→∞
a→0
N→∞
L→∞
lima→0 AN,L
a
N
= −β −1 (1 − ln[ρ])
Simon Bignold Supervisor: Christoph Ortner
Further Work
Simon Bignold Supervisor: Christoph Ortner
Further Work
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Grand Canonical Ensemble
Quantum Problems
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Acknowledgements
I would like to thank Christoph Ortner and Stefan Adams for all
their help and advice.
Simon Bignold Supervisor: Christoph Ortner