# 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?

• Are low-frequency DC-limit dielectric constants of interest, or the full spectrum of the refractive index? Jan 24 at 8:35
• These are for low frequency DC-limit dielectric constants Jan 24 at 9:30

## 1 Answer

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.

• Does this method capture the contribution from the polarizability of the electron cloud of the molecules? It seems that, e.g. if you estimate the dielectric constant of a rare gas liquid using this method, you will get exactly 1, which is clearly unphysical Jan 21 at 11:58
• @wzkchem5 That would depend on the forcefield being used, would it not? Jan 21 at 12:11
• @B.Kelly But no force fields, even polarizable force fields, will predict a non-zero dipole moment for a rare gas liquid. The point is that a system can display a non-unity dielectric constant even if it does not have a fluctuating dipole moment. A dielectric constant can arise from electronic polarizability alone. Jan 21 at 12:20
• So your issue is with classical force-fields and not the maths? I would expect a drude oscillator model to get the dipole moment of a rare gas liquid. It might not match experiment, but that is a different issue Jan 21 at 13:07
• For completeness, this would need (a) references, and (b) a discussion of ergodicity. In many systems, including solutions, it can take a long time for the fluctuations to become ergodic, which has to do with the rates of interconversion between internal states of the system. For solutions, this can include jammed rotations or changes in solvation-shell structure around the solute. Jan 24 at 8:41