## Free energy surface Somethings we have to consider contributions other than total energy and potentials, like entropy. Then we may consider free energy surface. **Helmholtz Free energy** (min in equilibrium at NVT constant, aka [[Ensemble#Canonical ensemble]]): $A = -k_B T \ln \left[ C \int \cdots \int e^{-\beta U} \mathrm{d}q \right]$ **Gibbs Free energy**: $G = -k_B T \ln \left[ C \int \cdots \int e^{-\beta (PV + U)} \mathrm{d}q \right]$ If calculate the difference of $A$, then $\Delta A = -\frac{1}{\beta} \ln \left[ \frac{\int_{\Gamma_1} \exp(-\beta \cdot U) \, \mathrm{d}q}{\int_{\Gamma_2} \exp(-\beta \cdot U) \, \mathrm{d}q} \right]$ Here $\beta$ is still $\frac{1}{kT}$, $\Gamma_{1}$, $\Gamma_{2}$ describe certain states of the system. - the energy difference can tell the relative likelihood of finding a system in a given state. - Often these states can be distinguished by a single number, an "order parameter", and even the reaction pathway. In this case, we say it's a reaction coordinate. Comparison between total & free energy: - Total Energy: small system, ordered system. -> [[Potential energy surface|PES]] - Free Energy: large system, disordered system. -> After integration, one gets [[Free energy surface]]. >[!Notice] >The integration is because entropy and free energy are not average of phase states coordinates, but volumetric properties. Therefore, integration is required. Then, how to compute free energy difference? There are typically two ways: 1. Calculate directly (for small $\Delta H$) $F_i = -kT \ln Q_i, \quad F_0 = -kT \ln Q_0$ $F_i - F_0 = -kT \ln \frac{Q_i}{Q_0} = -\frac{1}{\beta} \ln \left( \frac{\int e^{-\beta H} \, \mathrm{d}\mathbf{r} \, \mathrm{d}\mathbf{p}}{\int e^{-\beta H_0} \, \mathrm{d}\mathbf{r} \, \mathrm{d}\mathbf{p}} \right) = -\frac{1}{\beta} \ln \frac{\int e^{-\beta H} \, \mathrm{d}\mathbf{r} \, \mathrm{d}\mathbf{p}}{\int e^{-\beta H_0} \, \mathrm{d}\mathbf{r} \, \mathrm{d}\mathbf{p}}=-\frac{1}{\beta} \ln \langle e^{-\beta \Delta H (\mathbf{r}^{N}, \mathbf{p}^{N})}\rangle $ Here the bracket means average. This method only works for small enthalpy change. 2. Build transition path For large $\Delta H$, one should write $H_{1} = H_{0}+\lambda (H_{1}-H_{0})$ and build a path for transition. This would require simulation. Then, $\Delta A_{a \to b} = \int_{\lambda_a}^{\lambda_b} \left\langle \frac{\partial H(\mathbf{x}, \mathbf{p}; \lambda)}{\partial \lambda} \right\rangle \, \mathrm{d}\lambda$ >[!Info] >See [[Potential energy surface]] and [[Evaluating FES]] for more.