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Structure factor (SQ) is a characterization quantity which is frequently calculated in molecular simulations. It is easily to code the calculation program according to the definition. In most molecular simulations, periodic boundary condition is adopted therefore $$S(\mathbf{q}):=\mathcal{FT}\lbrace\langle\rho(\mathbf{r})\rho(\mathbf{0})\rangle\rbrace(\mathbf{q})$$ is actually a circular convolution, where FFT can give a dramatic boost in contrast to calculate directly. The steps using FFT are: Calculate $\rho(\mathbf{r})$, which is a summation of Dirac-Delta function that can be estimated as a hisotgram; Calculate $\hat{\rho}(\mathbf{q})$ by FFT; $S(\mathbf{q})=|\hat{\rho}(\mathbf{q})|^2$; Calculate mean over modulus: $S(q)=\int S(\mathbf{q})\delta(|\mathbf{q}|-q)\mathrm{d}\mathbf{q}/\int \delta(|\mathbf{q}|-q)\mathrm{d}\mathbf{q}$ Efficiency If the simulation box is divided into $N$ (in 3D systems, $N=N_xN_yN_z$ for example) bins, the FFT gives $O(N\log(N))$ complexity and step

Formulae of umbrella sampling method can be found on wikipedia. In umbrella sampling, a reaction coordinate $\xi$ is pre-defined from the atomic coordinates, a bias potential is added to the atoms of interest to keep $\xi$ of the system at a specific window $\xi_w$. The bias form is usually a harmonic potential: $$u^b_w(\xi)=0.5k_w(\xi-\xi_w)^2$$ Therefore, the energy of biased system $A^b_w = A^{ub}_w + u^b_w(\xi)$. The superscript $ub$ is short for "un-biased". In the simulations, we can sample the reaction coordinate in each window and evaluate their distribution $P^b(\xi)$, since the free energy $A=-k_BT\ln(P)$, we have: $$A_w^{ub}(\xi) = -k_BT\ln(P^b_w(\xi))-u^b_w(\xi)-F_w$$ with $F_w$ is a reference free energy of each window and remains an unknown constant. One method to derive $F_w$ is WHAM, in year 2005, Kästner et al.  (The Journal of Chemical Physics 123, no. 14 (October 8, 2005): 144104.) have proposed a new method whose idea is to take derivative of $A^u_w$ w