1. Theory

Hybrid particle–field simulations switch out the ordinary particle–particle Lennard-Jones interactions with interactions between particles and a slowly varying density field. In this way, the most expensive part of normal molecular dynamics simulations is circumvented. Hybrid particle–field densities are defined as

$$\varphi (\mathbf{r})=\sum _{i=1}^{N}P(\mathbf{r}-{\mathbf{r}}_i),$$

where $\mathbf{r}$ is a spatial coordinate, $P$ is a window function used to distribute particle number densities onto a computational grid, and ${\mathbf{r}}_i$ is the position of particle $i$ (of total $N$). By default, HyMD uses a Cloud-In-Cell (CIC) window function. A Hamiltonian form for a system of $N$ particles in $M$ molecules is

$$\mathcal{H}(\{\mathbf{r}\})=\sum _{m=1}^M{H}_0(\{\mathbf{r},\mathbf{v}{\}}_m)+W[\varphi (\mathbf{r})]$$

with $H_0$ being a standard intramolecular Hamiltonian form (see Intramolecular bonds) including kinetic terms, while $W$ is a density dependent interaction energy functional.

In the Hamiltonian hPF-MD formalism [Bore and Cascella, 2020], the density field is filtered using a grid-independent filtering function $H$,

$$\tilde{\varphi }(\mathbf{r})=\int \mathrm{d}\mathbf{x}\varphi (\mathbf{x})H(\mathbf{r}-\mathbf{x}).$$

The filter smooths the density, ensuring that $\tilde{\varphi }$ and $W[\tilde{\varphi }([\varphi ])]$ both converge as the grid size is reduced.

1.1. External potential

The external potential acting on a particle is defined as the functional derivative of $W$ with respect to $\varphi$. In the filtered formalism, the potential takes the form

$$\begin{array}{rl}V(\mathbf{r})& =\int \mathrm{d}\mathbf{y}\frac{\delta w}{\delta \varphi (\mathbf{r})}\\ & =\int \mathrm{d}\mathbf{y}\frac{\delta w}{\delta \tilde{\varphi }(\mathbf{y})}\frac{\delta \tilde{\varphi }(\mathbf{y})}{\delta \varphi (\mathbf{r})},\end{array}$$

under the assumption of a local form of the interaction energy functional, $W[\tilde{\varphi }]=\int \mathrm{d}\mathbf{r}w[\tilde{\varphi }(\mathbf{r})]$. Note that

$$\frac{\delta \tilde{\varphi }(\mathbf{y})}{\delta \varphi (\mathbf{r})}=H(\mathbf{y}-\mathbf{r}).$$

1.2. Force interpolation

The forces on particle $i$ are obtained by differentiation of the external potential,

$${\mathbf{F}}_i=-\int \mathrm{d}\mathbf{r}\mathrm{\nabla }V(\mathbf{r})P(\mathbf{r}-{\mathbf{r}}_i).$$

1.3. Reciprocal space calculations

The field operations in HyMD are discretised and performed on a grid in reciprocal space using (discrete) fast Fourier transform algorithms. After interpolating the density ${\varphi }_{ijk}$ with CIC, we apply the filtering and obtain the discrete version of the external potential by

$${\tilde{\varphi }}_{ijk}={\mathrm{FFT}}^{-1}[\mathrm{FFT}(\varphi )\mathrm{FFT}(H)]$$

and

$$V_{ijk}={\mathrm{FFT}}^{-1}[\mathrm{FFT}\left(\frac{\delta w(\tilde{\varphi })}{\delta \tilde{\varphi }}\right)\mathrm{FFT}(H)].$$

The forces are obtained by differentiation of $V$ in Fourier space as

$$\mathrm{\nabla }{V}_{ijk}={\mathrm{FFT}}^{-1}[i\mathbf{k}\mathrm{FFT}\left(\frac{\delta w(\tilde{\varphi })}{\delta \tilde{\varphi }}\right)\mathrm{FFT}(H)].$$

1.4. Filter

By default, the filter used in HyMD is a simple Gaussian of the form

$$\begin{array}{rl}H(x)& =\frac{1}{\sqrt{2\pi }\sigma }\exp \left[\frac{-x^2}{2{\sigma }^2}\right]\\ \hat{H}(k)& =\exp \left[\frac{-{\sigma }^2{k}^2}{2}\right].\end{array}$$

For more details about the filtering, see Filtering.

1.5. Hamiltonian form

The default form of the interaction energy functional in HyMD is

$$W=\frac{1}{2{\varphi }_0}\int \mathrm{d}\mathbf{r}\sum _{\text{i},\text{j}}{\tilde{\chi }}_{\text{i}-\text{j}}{\tilde{\varphi }}_{\text{i}}(\mathbf{r}){\tilde{\varphi }}_{\text{j}}(\mathbf{r})+\frac{1}{2\kappa }{(\sum _{\text{k}}{\tilde{\varphi }}_{\text{k}}(\mathbf{r})-{\varphi }_0)}^2.$$

See Interaction energy functionals for details.