Thermodynamic Integration (TI) is one of the most widespread methods for calculating free energy differences. By taking the derivative of the free energy difference with respect to [math]\displaystyle{ \lambda }[/math], this equation can be easily derived. It is one of the few methods that require calculation of [math]\displaystyle{ \frac{\partial U(\lambda,\vec{q})}{\partial\lambda} }[/math], however, is still one of the easiest free energy calculation to work with. The need to calculate the derivative is also one of its limitations as many simulation packages will not calculate this natively, and can cause problems when numerically evaluating it at [math]\displaystyle{ r = 0 }[/math]. Despite this, it is still one of the most accurate methods if used correctly and it is recommended that those new to the field because of its simplicity and ease of use.

Derivation

Starting with the identity of free energy

[math]\displaystyle{ \displaystyle A = -\beta^{-1} \ln Q }[/math]

then taking the derivative with respect to [math]\displaystyle{ \lambda }[/math] yields

[math]\displaystyle{ \displaystyle dA/d\lambda = -\beta^{-1}\frac{d}{d\lambda} \ln \int e^{-\beta U(\lambda,\vec{q})}d\vec{q} = -\beta^{-1} \frac{\frac{d}{d\lambda}\int e^{-\beta U(\lambda,\vec{q})}d\vec{q}}{Q} }[/math]

which can then be written as

[math]\displaystyle{ \displaystyle dA/d\lambda = -\beta^{-1}\frac{-\beta\int \frac{dU(\lambda,\vec{q})}{d\lambda} e^{-\beta U(\lambda,\vec{q})}d\vec{q}}{Q} = \left\langle \frac{dU(\lambda,\vec{q})}{d\lambda}\right\rangle_\lambda }[/math].

Finally, one can do integration over the whole range of [math]\displaystyle{ \lambda }[/math] to get the final TI equation

[math]\displaystyle{ \displaystyle \Delta A = \int_0^1 \left\langle \frac{dU(\lambda,\vec{q})}{d\lambda}\right\rangle_\lambda d\lambda }[/math].

Estimating Free Energies with TI

The above derivation makes it rather simple to estimate free energies from TI as there is no iterative solution needed, and only nearby states are needed to calculate the derivative. Since there is often a finite and discrete number of [math]\displaystyle{ \lambda }[/math] states, numeric integration schemes are frequently required. All numeric integration schemes have the form

[math]\displaystyle{ \displaystyle \Delta A \approx \sum_{k=1}^K w_k \left\langle \frac{dU(\lambda,\vec{q})}{d\lambda}\right\rangle_k }[/math]

where the weights, [math]\displaystyle{ w_k }[/math] will depend on which numeric integration style is chosen. It is recommenced to those starting in free energy calculations to integrate with the trapezoid rule as it is very straightforward and maximizes flexibility in choice of [math]\displaystyle{ \lambda }[/math]. Under the trapezoid rule, even lambda spacing weights are [math]\displaystyle{ \displaystyle w_1 = w_k = 1/[2(K-1)] }[/math] and [math]\displaystyle{ w_{k \ne 1,K} = 1/(K-1) }[/math].

Variance of TI

The variance of TI is one of the simplest, but has one catch experimenters should be aware of. Because the information from calculating [math]\displaystyle{ \left\langle \frac{dU}{d\lambda} \right\rangle }[/math] requires information from neighboring states, the variance at each state does not add independently.

Since each of the averages is generated from different simulations, the variance for TI is

[math]\displaystyle{ \mathrm{var}\left(\Delta A\right) = \sum_{k=1}^{K}w_k^2 \mathrm{var}\left(\frac{dU}{d\lambda}\right)_k }[/math].

To illustrate ow this is different from independent adding, consider the trapezoid rule example. Taking into account the correct equation the variance would be

[math]\displaystyle{ \mathrm{var}\left(\Delta A_{1,K}\right) = \frac{1}{4}\mathrm{var}\left(\frac{dU}{d\lambda}\right)_1 + \mathrm{var}\left(\frac{dU}{d\lambda}\right)_2 + \cdots + \mathrm{var}\left(\frac{dU}{d\lambda}\right)_{K-1} +\frac{1}{4}\mathrm{var}\left(\frac{dU}{d\lambda}\right)_K }[/math].

Compare this to the incorrect method shown below.

[math]\displaystyle{ \mathrm{var}\left(\Delta A_{i,i+1}\right) = \frac{1}{4}\mathrm{var}\left(\frac{dU}{d\lambda}\right)_i + \frac{1}{4}\mathrm{var}\left(\frac{dU}{d\lambda}\right)_{i+1} }[/math]

[math]\displaystyle{ \begin{alignat}{2} \mathrm{var}\left(\Delta A_{1,K}\right) &=\sum_{i=1}^{K-1}\mathrm{var}\left(\Delta A_{i,i+1}\right) \\ &= \frac{1}{4}\mathrm{var}\left(\frac{dU}{d\lambda}\right)_1 + \frac{1}{2}\mathrm{var}\left(\frac{dU}{d\lambda}\right)_2 + \cdots + \frac{1}{2}\mathrm{var}\left(\frac{dU}{d\lambda}\right)_{K-1} +\frac{1}{4}\mathrm{var}\left(\frac{dU}{d\lambda}\right)_K \\ \end{alignat} }[/math]

The second set of equations is clearly quite different from the correct first set. To get the error, each individual average should be multiplied by [math]\displaystyle{ \sqrt{2\tau} }[/math] to correct for the correlation time at each state; error is then [math]\displaystyle{ \sqrt{\mathrm{var}\left(\Delta A_{i,K}\right)} }[/math].

Problems with TI

Although TI is one of the simplest methods, it also suffers from some of the more difficult challenges that must be overcome. For instance, if the curvature of [math]\displaystyle{ \frac{dU}{d\lambda} }[/math] is large, the bias introduced by discrete [math]\displaystyle{ \lambda }[/math] states becomes significant. This said, when using TI, it is of the utmost importance that researchers verify that sufficient states are present such that the free energy is independent of number of states.

Infinite [math]\displaystyle{ dU/d\lambda }[/math]

One of the largest problems with TI is evaluation of [math]\displaystyle{ \frac{dU}{d\lambda} }[/math] at [math]\displaystyle{ r = 0 }[/math] when standard Lennard-Jones potentials are used. The simplest thermodynamic pathway one can chose is the linear pathway of

[math]\displaystyle{ U(\lambda,\vec{q}) = (1-\lambda)U_0(\vec{q}) + \lambda U_1(\vec{q}) + U_{unaffected}(\vec{q}) }[/math]

which is acceptable for changes in parameters, but not for annihilating or decoupling a site because of the [math]\displaystyle{ r^{-12} }[/math] term in the Lennard-Jones potentials in [math]\displaystyle{ U_i }[/math]. The linear transformation will always have an infinite potential at [math]\displaystyle{ r=0 }[/math] leading to numeric instabilities for evaluating [math]\displaystyle{ \frac{dU}{d\lambda} }[/math] in TI. Although there are ways to get around this, they do not converge very will with any function of [math]\displaystyle{ \lambda }[/math] taking the form shown above. However, if one were to use a soft core potential, this problem can be mostly avoided.

Modifying Simulations

Because most other free energy methods do not need to evaluate [math]\displaystyle{ \frac{dU}{d\lambda} }[/math], they do not natively support evaluating this in the code. If the thermodynamic path is constructed with a linear transformation, then the derivative can be evaluated in post-processing knowing the energy of the system. However, if the transformation is done with soft core potentials, then the derivative will need to be evaluated in code, and it will often be necessary for researchers to modify the code in-house as many simulation packages do not evaluate this quantity.

Thermodynamic Integration

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