Thermodynamic integration is a method used to compare the difference in free energy between two given states (e.g., A and B) whose potential energies
U
A
and
U
B
have different dependences on the spatial coordinates. Because the free energy of a system is not simply a function of the phase space coordinates of the system, but is instead a function of the Boltzmann-weighted integral over phase space (i.e. partition function), the free energy difference between two states cannot be calculated directly. In thermodynamic integration, the free energy difference is calculated by defining a thermodynamic path between the states and integrating over ensemble-averaged enthalpy changes along the path. Such paths can either be real chemical processes or alchemical processes. An example alchemical process is the Kirkwood's coupling parameter method.
Consider two systems, A and B, with potential energies
U
A
and
U
B
. The potential energy in either system can be calculated as an ensemble average over configurations sampled from a molecular dynamics or Monte Carlo simulation with proper Boltzmann weighting. Now consider a new potential energy function defined as:
U
(
λ
)
=
U
A
+
λ
(
U
B
−
U
A
)
Here,
λ
is defined as a coupling parameter with a value between 0 and 1, and thus the potential energy as a function of
λ
varies from the energy of system A for
λ
=
0
and system B for
λ
=
1
. In the canonical ensemble, the partition function of the system can be written as:
Q
(
N
,
V
,
T
,
λ
)
=
∑
s
exp
[
−
U
s
(
λ
)
/
k
B
T
]
In this notation,
U
s
(
λ
)
is the potential energy of state
s
in the ensemble with potential energy function
U
(
λ
)
as defined above. The free energy of this system is defined as:
F
(
N
,
V
,
T
,
λ
)
=
−
k
B
T
ln
Q
(
N
,
V
,
T
,
λ
)
,
If we take the derivative of F with respect to λ, we will get that it equals the ensemble average of the derivative of potential energy with respect to λ.
Δ
F
(
A
→
B
)
=
∫
0
1
∂
F
(
λ
)
∂
λ
d
λ
=
−
∫
0
1
k
B
T
Q
∂
Q
∂
λ
d
λ
=
∫
0
1
k
B
T
Q
∑
s
1
k
B
T
exp
[
−
U
s
(
λ
)
/
k
B
T
]
∂
U
s
(
λ
)
∂
λ
d
λ
=
∫
0
1
⟨
∂
U
(
λ
)
∂
λ
⟩
λ
d
λ
=
∫
0
1
⟨
U
B
(
λ
)
−
U
A
(
λ
)
⟩
λ
d
λ
The change in free energy between states A and B can thus be computed from the integral of the ensemble averaged derivatives of potential energy over the coupling parameter
λ
. In practice, this is performed by defining a potential energy function
U
(
λ
)
, sampling the ensemble of equilibrium configurations at a series of
λ
values, calculating the ensemble-averaged derivative of
U
(
λ
)
with respect to
λ
at each
λ
value, and finally computing the integral over the ensemble-averaged derivatives.
Umbrella sampling is a related free energy method. It adds a bias to the potential energy. In the limit of an infinite strong bias it is equivalent to thermodynamic integration.
