mbd_dipole Module

Form either a real or a complex dipole matrix.

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

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

$$\begin{gathered} \mathbf T_{ij}(\mathbf q)=\sum_{\mathbf n}\mathbf T(\mathbf R_{\mathbf nij})\mathrm e^{-\mathrm i\mathbf q\cdot\mathbf R_{\mathbf nij}},\quad\mathbf R_{\mathbf nij}=\mathbf R_j+\mathbf R_\mathbf n-\mathbf R_i \\ \frac{\mathrm d\mathbf R_{\mathbf nij}}{\mathrm d\mathbf R_k}=(\delta_{jk}-\delta_{ik})\mathbf I \\ \mathbf{T}_{ij}(\mathbf{q})\approx\mathbf{T}^\text{Ew}_{ij}(\mathbf{q}) =\sum_\mathbf n^{|\mathbf R_{\mathbf nij}|<R_\text c}\mathbf T^\text{erfc}(\mathbf R_{\mathbf nij};\gamma)\mathrm e^{-\mathrm i\mathbf q\cdot\mathbf R_{\mathbf nij}} +\frac{4\pi}{V_\text{uc}}\sum_{\mathbf m}^{0<|\mathbf k_\mathbf m|<k_\text c}\mathbf{\hat k}_\mathbf m\otimes\mathbf{\hat k}_\mathbf m\,\mathrm e^{-\frac{k_\mathbf m^2}{4\gamma^2}-\mathrm i\mathbf G_\mathbf m\cdot\mathbf R_{ij}} \\ -\frac{4\gamma^3}{3\sqrt\pi}\delta_{ij}\mathbf I +\delta(\mathbf q)\frac{4 \pi}{3 V_\text{uc}}\mathbf I,\qquad \mathbf k_\mathbf m=\mathbf G_\mathbf m+\mathbf q \end{gathered}$$

$$\partial\left(\frac{4\pi}{V_\text{uc}}\right)=-\frac{4\pi}{V_\text{uc}}\frac{\partial V_\text{uc}}{V_\text{uc}},\qquad\frac{\partial V_\text{uc}}{\partial\boldsymbol{\mathcal A}}=V_\text{uc}(\boldsymbol{\mathcal A}^{-1})^\mathrm T \\ \partial(k^2)=2\mathbf k\cdot\partial\mathbf k \\ \mathbf{\hat k}\otimes\partial\mathbf{\hat k}=\frac{\mathbf k\otimes\partial\mathbf k}{k^2}-\frac{\mathbf k\otimes\mathbf k}{2k^4}\partial(k^2)$$

$$\gamma:=\frac{2.5}{\sqrt[3]{V_\text{uc}}},\quad R_\text c:=\frac6\gamma,\quad k_\text c:=10\gamma$$

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

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)

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

$$\begin{aligned} \frac{\partial T_{ab}^\text{erfc}(\mathbf r,\gamma)}{\partial r_c} &=-\left( \frac{r_a\delta_{bc}+r_b\delta_{ca}}{r^5}- 5\frac{r_ar_br_c}{r^7} \right)C(r,\gamma)-3\frac{r_c\delta_{ab}}{r^5}B(r,\gamma) \\ &-\frac{r_ar_br_c}{r^6}\frac{\partial C(r,\gamma)}{\partial r}+\frac{r_c\delta_{ab}}{r^4}\frac{\partial B(r,\gamma)}{\partial r} \end{aligned}$$

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)