Theory

Response functions

A polarizable object that is acted on by an external electric field \(\boldsymbol{E}_\text{ext}\) responds by producing an induced field \(\boldsymbol{E}_\text{induced}\). In general, for linear materials, the response is described by the causal response function \(\boldsymbol{\chi}\), which is specific to the object in question

\[\boldsymbol{E}_\text{induced}(\boldsymbol{r}, t) = \int \boldsymbol{\chi}(\boldsymbol{r} - \boldsymbol{r}', t - t') \boldsymbol{E}_\text{ext}(\boldsymbol{r}', t') \mathrm{d}\boldsymbol{r'}\mathrm{d}t'.\]

It is useful to work in the frequency domain by taking the Fourier transform of all time-dependent quantities. Then, the convolution in the time domain becomes a multiplication

\[\boldsymbol{E}_\text{induced}(\boldsymbol{r}, \omega) = \int \boldsymbol{\chi}(\boldsymbol{r} - \boldsymbol{r}', \omega) \boldsymbol{E}_\text{ext}(\boldsymbol{r}', \omega) \mathrm{d}\boldsymbol{r'}.\]

The electric field can always be expanded in a series of multipoles (Griffiths 1998, Chapter 3)

\[\begin{align} \boldsymbol{E}_\text{induced} = \frac{1}{4\pi\varepsilon_0} \bigg[ \underbrace{ \frac{q\boldsymbol{r}}{r^3} }_\text{monopole} + \underbrace{ \left[\frac{3\boldsymbol{r}\boldsymbol{r}^\text{T}}{r^5} - \frac{1}{r^3}\right]\boldsymbol{d} }_\text{dipole} + \dots \bigg], \end{align}\]

where the quadrupoles and higher-order terms are replaced by the dots. Importantly, retardation effects have been negelected in this expression, so that the field everywhere instantaneously feels a change in the dipole moment. This is a reasonable approximation in a region smaller than the relevant wavelengths. In our notation, \(\boldsymbol{r}^T\boldsymbol{d}\) refers to the dot product \(\boldsymbol{r}\cdot\boldsymbol{d}\) and \(r\) is the norm of \(\boldsymbol{r}\). The charge \(q\) and the dipole moment \(\boldsymbol{d}\)

\[\begin{split}\begin{align} q &= \int \rho(\boldsymbol{r}) \mathrm{d}{\boldsymbol{r}} \\ \boldsymbol{d} &= \int \rho(\boldsymbol{r}) \boldsymbol{r}\mathrm{d}{\boldsymbol{r}} \end{align}\end{split}\]

are both related to the charge distribution \(\rho\) in the object (and the latter is specific to the choice of origin in the coordinate system). Then, the response of the object is characterized by the response of the dipole moment to the external field. If the object is smaller than the relevant wavelengths, it is related to the external field through the polarizability \(\boldsymbol{\alpha}\)

\[\begin{equation} \boldsymbol{d}(\omega) = \boldsymbol{\alpha}(\omega)\boldsymbol{E}_\text{ext}(\omega), \end{equation}\]

The polarizability for a system can be obtained via TDDFT calculations, e.g., using NWChem or gpaw.

The polarizability is related to the optical absorption spectrum. It can be expressed as the oscillator strength function

\[\begin{equation} S(\omega) = \frac{2 \omega}{\pi} \mathrm{Im}\:\alpha(\omega). \end{equation}\]

Dipolar coupling

For an assembly of polarizable units, each with their own polarizability \(\boldsymbol{\alpha}^{(i)}\), the induced fields of the units can be computed self-consistently (Fojt et al. 2021).

Assuming that the units have zero charge and no higher multipole moments than the dipole, the total electric field \(\boldsymbol E_\text{tot}^{(i)}(\omega)\) at the position of each unit must be the external field plus the dipolar field from all other units. This can be written as the tensor equation

\[\begin{split}\begin{align} \underbrace{ \begin{bmatrix} \boldsymbol E_\text{tot}^{(1)}(\omega) \\ \boldsymbol E_\text{tot}^{(2)}(\omega) \\ \vdots \\ \boldsymbol E_\text{tot}^{(N)}(\omega) \\ \end{bmatrix} }_{\textstyle\underline{\boldsymbol E}_\text{tot}(\omega)} &= \underbrace{ \begin{bmatrix} \boldsymbol E_\text{ext}^{(1)}(\omega) \\ \boldsymbol E_\text{ext}^{(2)}(\omega) \\ \vdots \\ \boldsymbol E_\text{ext}^{(N)}(\omega) \\ \end{bmatrix} }_{\textstyle \underline{\boldsymbol E}_\text{ext}(\omega)} + \underbrace{ \begin{bmatrix} 0 & \boldsymbol T_{12} & \ldots & \boldsymbol T_{1N} \\ \boldsymbol T_{21} & 0 & & \boldsymbol T_{2N} \\ \vdots & & \ddots & \vdots \\ \boldsymbol T_{N1} & \boldsymbol T_{N2} & \ldots & 0 \end{bmatrix} }_{\textstyle \underline{\boldsymbol T}} \underbrace{ \begin{bmatrix} \boldsymbol d^{(1)}(\omega) \\ \boldsymbol d^{(2)}(\omega) \\ \vdots \\ \boldsymbol d^{(N)}(\omega) \\ \end{bmatrix} }_{\textstyle\underline{\boldsymbol d}(\omega)} , \end{align}\end{split}\]

where the dipole-dipole coupling is

\[\begin{align} \boldsymbol{T}_{ij} = \frac{3\boldsymbol{r}^{(ij)}(\boldsymbol{r}^{(ij)})^\text{T}}{\left|r^{(ij)}\right|^5} - \frac{1}{\left|r^{(ij)}\right|^3} \end{align}\]

and \(\boldsymbol{r}^{(ij)}\) is the vector from unit i to j. Each unit responds with the formation of a dipole \(\boldsymbol{d}^{(i)}(\omega) = \boldsymbol\alpha_0^{(i)}(\omega) \boldsymbol{E}_\text{tot}^{(i)}(\omega)\) to the total electric field at the position of the unit. The total electic field is substituted for

\[\begin{align} \underline{\boldsymbol d}(\omega) &= \left[\underline{\boldsymbol I} + \underline{\boldsymbol\alpha}_0(\omega) \underline{\boldsymbol T} \right]^{-1} \underline{\boldsymbol \alpha}_0(\omega) \underline{\boldsymbol E}_\text{ext}(\omega), \end{align}\]

where \(\underline{\boldsymbol{\alpha}}_0(\omega)\) is the tensor with \(\boldsymbol\alpha_0^{(i)}(\omega)\) on the diagonal and \(\underline{\boldsymbol I}\) is the identity tensor. The proportionality tensor is interpreted as a reducible unit-wise polarizability

\[\begin{split}\begin{align} \underline{\boldsymbol{\alpha}}(\omega) &= \left[\underline{\boldsymbol I} + \underline{\boldsymbol\alpha}_0(\omega) \underline{\boldsymbol T} \right]^{-1} \underline{\boldsymbol \alpha}_0(\omega) \\ &= \left[\underline{\boldsymbol\alpha}_0(\omega)^{-1} + \underline{\boldsymbol T}\right]^{-1}, \end{align}\end{split}\]

where the last line is an explicit expression for this quantity. In practice, it is faster to solve a linear system of equations than to invert a matrix. The inversion is avoided by writing the first line as

\[\begin{align} \left[\underline{\boldsymbol I} + \underline{\boldsymbol\alpha}_0(\omega) \underline{\boldsymbol T} \right]\underline{\boldsymbol{\alpha}}(\omega) &= \underline{\boldsymbol \alpha}_0(\omega) \end{align}\]

and solving for \(\underline{\boldsymbol{\alpha}}(\omega)\).

The total dipole is then the sum of the individual dipoles. Assuming that the external electric field is constant, the effective polarizability of the assembly is

\[\begin{align} \boldsymbol\alpha(\omega) = \sum_{i}^{N} \sum_{j}^{N} [\underline{\boldsymbol \alpha}]_{ij}(\omega). \end{align}\]

The summation over columns can be safely carried out before solving the linear system of equations, reducing the matrix size further.

\[\begin{align} \left[\underline{\boldsymbol I} + \underline{\boldsymbol\alpha}_0(\omega) \underline{\boldsymbol T} \right]\sum_{j}^{N} [\underline{\boldsymbol{\alpha}}]_{ij}(\omega) &= \sum_{j}^{N} [\underline{\boldsymbol \alpha}_0]_{ij}(\omega) \end{align}\]

References

• 4. 10. Griffiths, Introduction to Electrodynamics, Upper Saddle River, NJ, Pearson (1998)

• 10. Fojt, T. P. Rossi, T. J. Antosiewicz, M. Kuisma, P. Erhart, Dipolar coupling of nanoparticle-molecule assemblies: An efficient approach for studying strong coupling, J. Chem. Phys 154, 094109 (2021)