# Is there a way to obtain solution-phase dielectric constants?

Matter Modeling Asked by nathanielng on November 18, 2021

I wish to calculate solution-phase dielectric constants (required for a Monte-Carlo model) for CoCl$$_2$$ and TaS$$_2$$ dissolved in DMF.

Is it possible to estimate these constants from the solid-state values, or to obtain them from first-principles / molecular dynamics calculations?

It's possible to estimate solution-phase dielectric constant from a molecular dynamics simulation using this formula:

$$epsilon_{r} = 1 + frac{4pi}{3Vk_{B}T}(langle mathbf{P}^{2} rangle - langle mathbf{P} rangle^{2})$$

Where $$V$$ is the volume, $$k_{B}$$ is Boltzmann's constant, $$T$$ is temperature, and $$P$$ is the dipole moment defined as: $$mathbf{P} = sum_{i} vec{mu}_{i}$$ the summation of molecular dipole moments.

In the absence of any external electric field (which I assume is the case here), from electrostatics, you have:

$$mathbf{P}(mathbf{r}) = chi int_{Omega} mathbf{T}(mathbf{r}-mathbf{r}^{'})cdot mathbf{P}(mathbf{r}^{'}) d^{3} mathbf{r}^{'}$$

$$chi$$ is the susceptibility, which is unknown here. Also, $$mathbf{T}$$ is the dipole-dipole tensor defined as:

$$T_{ij} = frac{partial^{2}}{partial x_{i}partial x_{j}}(-ln(r))$$

Now if you replace the integral with a summation and replace $$mathbf{P}(mathbf{r})$$ with the discretized dipole moment at molecular locations shown as $$mathsf{P}$$ and the discretized dipole-dipole tensor (matrix $$mathsf{T}$$), you have:

$$mathsf{P} = chi mathsf{T} cdot mathsf{P}$$

or:

$$mathsf{T} cdot mathsf{P} = frac{1}{chi} mathsf{P}$$

This is an eigenvalue problem where you know dipole-dipole tensor $$mathsf{T}$$, while the eigenvector (dipole moment $$mathsf{P}$$) and eigenvalue (susceptibility $$chi$$) are unknowns. You can solve this eigenvalue problem for your system and then you'll get the dielectric constant by estimating the fluctuation of dipole moment.

Answered by Mithridates the Great on November 18, 2021

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