calculate_relative_permittivity¶

mdhelper.analysis.electrostatics.calculate_relative_permittivity(M: ndarray[float], temperature: float, volume: float, *, reduced: bool = False) → float[source]¶

Calculates the relative permittivity (or static dielectric constant) \(\varepsilon_\mathrm{r}\) of a medium using the instantaneous dipole moments \(\mathbf{M}(t)\).

The dipole moment fluctuation formula [1] relates the relative permittivity to the dipole moment via

\[\varepsilon_\mathrm{r}=1+\frac{\overline{\langle\mathbf{M}^2\rangle -\langle\mathbf{M}\rangle^2}}{3\varepsilon_0 Vk_\mathrm{B}T}\]

where the angular brackets \(\langle\,\cdot\,\rangle\) denote the ensemble average, the overline signifies the spatial average, \(\varepsilon_0\) is the vacuum permittivity, \(k_\mathrm{B}\) is the Boltzmann constant, and \(T\) is the system temperature.

Note

If residues (molecules) in your system have net charges, the dipole moments must be made position-independent by subtracting the product of the net charge and the center of mass or geometry.

Parameters:

Marray-like

Instantaneous dipole moments over \(N_t\) frames.

Shape: \((N_t,\,3)\).

Reference unit: \(\mathrm{e\cdotÅ}\).

temperaturefloat

System temperature \(T\).

Note

If reduced=True, temperature should be equal to the energy scale. When the Lennard-Jones potential is used, it generally means that \(T^* = 1\), or temperature=1.

Reference unit: \(\mathrm{K}\).

volumefloat

System volume \(V\).

Reference unit: \(\mathrm{Å^3}\).

reducedbool, keyword-only, default: False

Specifies whether the data is in reduced units.

Returns:

dielectricfloat: Relative permittivity (or static dielectric constant).

References

[1]

Neumann, M. Dipole Moment Fluctuation Formulas in Computer Simulations of Polar Systems. Molecular Physics 1983, 50 (4), 841–858. https://doi.org/10.1080/00268978300102721.