# Free energy estimates logged in the GROMACS .log file in 2D alchemical metadynamics
In well-tempered metadynamics, $$V(\vec{s}, t)=\sum_{k\tau < t}W(k\tau)\exp\left( -\sum_{i=1}^{d}\frac{(s_i-s_i(q(k\tau)))^{2}}{2 \sigma_{i}^{2}}\right)$$ where $$W(k\tau)=W_{0}\exp\left( -\frac{V(\vec{s}(q(k\tau)), k\tau)}{k_{B}\Delta T}\right)$$.
In our case, $W_0=1.00050 \; \text{k}_{B}\text{T}$.
- At step 500 (1.0 ps), $\vec{s}(t)=(\theta(t), \lambda(t))=(-3.030568, 0)$. No Gaussian has been added before this point, so $V(\vec{s}(q(k\tau)), k\tau)=0$. That is, $W(t=1\;\text{ps})=W_0 \exp(-0/k_{B}T)=W_0=1.00050 \; \text{k}_{B}\text{T}$. The first Gaussian added at this point has the following functional form: $$G_1(\theta, \lambda)=V({\theta, \lambda, t=1\;\text{ps}})=W(t=1\;\text{ps})\exp(-\frac{(\theta - \theta(t=1\;\text{ps}))^{2}}{2\sigma_{\theta}^{2}})\exp(-\frac{(\lambda - \lambda(t=1\;\text{ps}))^{2}}{2\sigma_{\lambda}^{2}})$$ Or $$G_1(\theta, \lambda)=V({\theta, \lambda, t=1\;\text{ps}})=1.00050 \exp(- \frac{(\theta + 3.030568)^{2}}{2 \times 0.5^{2}})\exp(-\frac{\lambda^{2}}{2 \times 0.0001 ^{2}})$$
- At step 510 (1.02 ps), $\vec{s}(t)=(\theta(t), \lambda(t))=(-2.881022, 0)$. The value of free energy estimated documented in the log file the value of another point on the same free energy landscape (constructed by only the deposition of the first Gaussian), which is as follows: $$G_1(\theta=-2.881022, \lambda=0)=1.00050 \exp(-\frac{(-2.881022 + 3.030568)^{2}}{2\times 0.5^{2}})\exp(-\frac{0^{2}}{2 \times 0.0001^{2}})=0.95674$$