Thermodynamic integration

Thermodynamic integration is used to calculate the difference in the Helmholtz energy function, $A$, between two states. The path must be continuous and reversible, i.e., the system must evolve through a succession of equilibrium states (Ref. 1 Eq. 3.5)

$\mathrm{\Delta }A=A(\lambda )-A({\lambda }_0)=\underset{{\lambda }_0}{\overset{\lambda }{\int }}{⟨\frac{\partial U(\mathbf{r},\lambda )}{\partial \lambda }⟩}_{\lambda }\mathrm{d}\lambda$

Isothermal integration

At constant temperature (Ref. 2 Eq. 5):

$\frac{A({\rho }_2,T)}{N{k}_{B}T}=\frac{A({\rho }_1,T)}{N{k}_{B}T}+\underset{{\rho }_1}{\overset{{\rho }_2}{\int }}\frac{p(\rho )}{k_{B}T{\rho }^2}\mathrm{d}\rho$

Isobaric integration

At constant pressure (Ref. 2 Eq. 6):

$\frac{G(T_2,p)}{N{k}_B{T}_2}=\frac{G(T_1,p)}{N{k}_B{T}_1}-\underset{T_1}{\overset{T_2}{\int }}\frac{H(T)}{N{k}_B{T}^2}\mathrm{d}T$

where $G$ is the Gibbs energy function and $H$ is the enthalpy.

Isochoric integration

At constant volume (Ref. 2 Eq. 7):

$\frac{A(T_2,V)}{N{k}_B{T}_2}=\frac{A(T_1,V)}{N{k}_B{T}_1}-\underset{T_1}{\overset{T_2}{\int }}\frac{U(T)}{N{k}_B{T}^2}\mathrm{d}T$

where $U$ is the internal energy.

See also

• Gibbs-Duhem integration

References

1. J. A. Barker and D. Henderson "What is "liquid"? Understanding the states of matter ", Reviews of Modern Physics 48 pp. 587 - 671 (1976)
2. C. Vega, E. Sanz, J. L. F. Abascal and E. G. Noya "Determination of phase diagrams via computer simulation: methodology and applications to water, electrolytes and proteins", Journal of Physics: Condensed Matter 20 153101 (2008) (section 4)

Related reading

• Enrique de Miguel "Estimating errors in free energy calculations from thermodynamic integration using fitted data", Journal of Chemical Physics 129 214112 (2008)