Calculating Conductivity from Molecular Dynamics Simulation
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Methods to calculate the conductivity of a species from molecular dynamics simulation
Green-Kubo relations
The velocity autocorrelation function, \(C_V(t)\), is
$$C_V(t)=\frac{1}{N}\sum_{i=1}^{N}(\vec{v_i}(0)\cdot\vec{v_i}(t))=<\vec{v_i}(0)\cdot\vec{v_i}(t)>$$
where \(N\) is the total number of particles (atoms/molecules in the selection) and \(\vec{v_i}\) is a vector storing the three components of the velocity (\(v_x\), \(v_y\), and \(v_z\)) for the \(i\)-th particle.
\(\vec{v_i}(0)=\vec{v_i}(t=t_0)\) and \(\vec{v_i}(t)=\vec{v_i}(t=t_0+n \Delta t)\), where \(n\) is the timestep and \(\Delta t\) is the timestep size
Given this function decays to zero at long time, the diffusion constant \(D\) may be found from the integral of \(C_V(T)\) as
$$D=\frac{1}{3}\int_{t=0}^{t=\infty}<\vec{v_i}(0)\cdot\vec{v_i}(t)>dt$$
Electrical conductivity, \(\sigma\), is calculated using the normalized autocorrelation function of the total current \(J(t)\) as
$$J(t)=<(\sum_{i}\vec{v}_{i+}(t)-\sum_{j}\vec{v}_{j-}(t))>\times<(\sum_{i}\vec{v}_{i+}(0)-\sum_{j}\vec{v}_{j-}(0))>$$
where \(\vec{v}_{i+}\) and \(\vec{v}_{j-}\) are the velocity vectors for the cations and anions in the system, respectively
Conductivity is then calculated as
$$\sigma=\frac{e^2}{3Vk_BT}\int_{0}^{\infty}J(t)dt$$
Nernst-Einstein
Compute the mean-square displacement (\(MSD\)) of atoms or molecules from a set of initial positions
$$MSD\equiv<(\vec{x}(t)-\vec{x_0})^2>=\frac{1}{N}\sum_{i=1}^{N} |\vec{x}^{(i)}(t)-\vec{x}^{(i)}(0)|^2$$
where \(N\) is the total number of particles (atoms/molecules) in the selection, vectors \(\vec{x}^{i}(t)\) and \(\vec{x}^{(i)}(0)\) are the position of the \(i\)-th particle at time \(t\) and the reference position of the \(i\)-th particle.
The command will calculate a diffusion constant, \(D\), from \(MSD\) according to the Einstein relation
$$MSD=2Dt$$
where \(t\) is the simulation time used to calculate the \(MSD\)
Calculating \(D\) for cations and anions in the system (\(D_+\) and \(D_-\), respectively), the ionic conductivity can be calculated by the Nernst-Einstein equation
$$\sigma=\frac{e^2}{Vk_BT}(N_+z_+D_++N_-z_-D_-)$$
where \(e\) and \(k_B\) are the unit charge and Boltzmann constant; \(V\), \(T\), \(N\pm\), and \(z\pm\) are the volume, temperature, number of each ionic species, and charge of the ionic species in the system.