mbd_dipole Module

Form either a real or a complex dipole matrix.

The real-typed version is equivalent to $\mathbf q=0$.

$$\boldsymbol{\mathcal A}:=[\mathbf a_1\mathbf a_2\mathbf a_3],\qquad\boldsymbol{\mathcal B}:=[\mathbf b_1\mathbf b_2\mathbf b_3] \\ \boldsymbol{\mathcal B}=2\pi(\boldsymbol{\mathcal A}^{-1})^\mathrm T,\qquad \partial\boldsymbol{\mathcal B}=-\big((\partial\boldsymbol{\mathcal A})\boldsymbol{\mathcal A}^{-1}\big)^\mathrm T\boldsymbol{\mathcal B} \\ \mathbf R_\mathbf n=\boldsymbol{\mathcal A}\mathbf n,\qquad\partial\mathbf R_\mathbf n=(\partial\boldsymbol{\mathcal A})\mathbf n, \\ \mathbf G_\mathbf m=\boldsymbol{\mathcal B}\mathbf m,\qquad \partial\mathbf G_\mathbf m=-\big((\partial\boldsymbol{\mathcal A})\boldsymbol{\mathcal A}^{-1}\big)^\mathrm T\mathbf G_\mathbf m, \\ \frac{\partial G_{\mathbf ma}}{\partial A_{bc}}=-\mathcal A^{-1}_{ca}G_{\mathbf mb}$$

$$ \begin{gathered} \mathbf T_{ij}(\mathbf q)=\sum_{\mathbf n}\mathbf T(\mathbf R_{\mathbf nij})\mathrm e^{-\mathrm i\mathbf q\cdot\mathbf R_{\mathbf nij}},\quad\mathbf R_{\mathbf nij}=\mathbf R_j+\mathbf R_\mathbf n-\mathbf R_i \ \frac{\mathrm d\mathbf R_{\mathbf nij}}{\mathrm d\mathbf R_k}=(\delta_{jk}-\delta_{ik})\mathbf I \ \mathbf{T}{ij}(\mathbf{q})\approx\mathbf{T}^\text{Ew}(\mathbf{q}) =\sum_\mathbf n^{|\mathbf R_{\mathbf nij}|

$$T_{ab}(\mathbf r)=\frac{\partial^2}{\partial r_a\partial r_b}\frac1r= \frac{-3r_ar_b+r^2\delta_{ab}}{r^5},\qquad \frac{\partial T_{ab}(\mathbf r)}{\partial r_c}=-3\left( \frac{r_a\delta_{bc}+r_b\delta_{ca}+r_c\delta_{ab}}{r^5}- \frac{5r_ar_br_c}{r^7} \right)$$

Arguments

Return Value real(kind=dp), (3,3)

$$\begin{aligned} B(R,\gamma) &=\operatorname{erfc}(\gamma R) +\frac{2\gamma R}{\sqrt\pi}\mathrm e^{-(\gamma R)^2} \\ \partial B(R,\gamma) &=-\frac4{\sqrt\pi}(\gamma R)^2\mathrm e^{-(\gamma R)^2} (R\partial\gamma+\gamma\partial R) \end{aligned}$$

Arguments

Return Value real(kind=dp)

$$\begin{aligned} C(r,\gamma) &=3\operatorname{erfc}(\gamma R) +\frac{2\gamma R}{\sqrt\pi}(3+2(\gamma R)^2)\mathrm e^{-(\gamma R)^2} \\ \partial C(R,\gamma) &=-\frac8{\sqrt\pi}(\gamma R)^4\mathrm e^{-(\gamma R)^2} (R\partial\gamma+\gamma\partial R) \end{aligned}$$

Arguments

Return Value real(kind=dp)

$$T_{ab}^\text{erfc}(\mathbf r,\gamma) =-3\frac{r_ar_b}{r^5}C(r,\gamma)+\frac{\delta_{ab}}{r^3}B(r,\gamma)$$

Arguments

Return Value real(kind=dp), (3,3)

$$\begin{aligned} T^\text{GG}_{ab}(\mathbf r,\sigma)&= \frac{\partial^2}{\partial r_a\partial r_b}\frac{\operatorname{erf}(\zeta)}r= \big(\operatorname{erf}(\zeta)-\Theta(\zeta)\big)T_{ab}(\mathbf r)+ 2\zeta^2\Theta(\zeta)\frac{r_ar_b}{r^5} \\ \Theta(\zeta)&=\frac{2\zeta}{\sqrt\pi}\exp(-\zeta^2),\qquad \zeta=\frac r\sigma \\ \frac{\mathrm d T_{ab}^\text{GG}(\mathbf r,\sigma)}{\mathrm dr_c}&= 2\zeta\Theta(\zeta)\left(T_{ab}(\mathbf r)+(3-2\zeta^2)\frac{r_ar_b}{r^5}\right) \frac{\mathrm d\zeta}{\mathrm dr_c} \\ &+\big(\operatorname{erf}(\zeta)-\Theta(\zeta)\big) \frac{\partial T_{ab}(\mathbf r)}{\partial r_c}- 2\zeta^2\Theta(\zeta)\left( \frac13\frac{\partial T_{ab}(\mathbf r)}{\partial r_c}+ \frac{r_c\delta_{ab}}{r^5} \right) \\ \qquad\frac{\mathrm d\zeta}{\mathrm dr_c}&= \frac{r_c}{r\sigma}-\frac r{\sigma^2}\frac{\mathrm d\sigma}{\mathrm dr_c} \end{aligned}$$

Arguments

Return Value real(kind=dp), (3,3)