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

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:

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

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.