# 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

## Related Questions

### Berry’s curvature and magnetic moment in TMDCs

2  Asked on August 19, 2021 by carmen-gonzlez

### Analog computing in matter modeling today: Any applications?

1  Asked on August 19, 2021 by ksousa

### What are the pitfalls for new users of DFT?

4  Asked on August 19, 2021

### Calculating diffusion coefficient from Mean Squared Displacement

2  Asked on August 19, 2021

### What are the types of MCSCF?

3  Asked on August 19, 2021

### What are the types of SCF?

5  Asked on August 19, 2021

### Which schedulers are compatible with a virtual machine?

1  Asked on August 19, 2021

### Is the number of possible Bravais lattices a mathematical fact?

1  Asked on August 19, 2021 by camps

### What are the great unsolved questions in Matter Modeling?

2  Asked on August 19, 2021

### Quantum Dot properties using VASP

2  Asked on August 19, 2021 by suseel-rahul

### Tools for high-throughput DFT studies?

5  Asked on August 19, 2021

### What are the types of Quantum Monte Carlo?

3  Asked on August 19, 2021 by nike-dattani

### Dealing with symmetry of ordered primitive cell during DFT structure relaxation

1  Asked on August 19, 2021 by doublekx

### Deep Neural Networks: Are they able to provide insights for the many-electron problem or DFT?

1  Asked on February 20, 2021

### Temperature effect on elastic constant using VASP

1  Asked on February 15, 2021 by niraja-moharana

### Can we consider SOC in FM materials using QE?

2  Asked on February 12, 2021 by atom

### Why linear response is absent in a non-centrosymmetric system with time reversal symmetry?

0  Asked on February 12, 2021 by walber97

### Formulate the Model Quantum Spin Hamiltonian for low dimensional (1D or 2D ) magnetic materials

0  Asked on January 26, 2021 by chumbak

### What methods are available for excited state calculations in solids?

2  Asked on January 6, 2021 by profm

### How to set vacuum space for slabs?

1  Asked on December 25, 2020 by barix