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%\usepackage[mathlines]{lineno}% Enable numbering of text and display math |
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\date{\today}% It is always \today, today, |
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% but any date may be explicitly specified |
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\begin{abstract} |
| 72 |
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This document includes useful relationships for computing the |
| 73 |
+ |
interactions between fields and field gradients and point multipolar |
| 74 |
+ |
representations of molecular electrostatics. We also provide |
| 75 |
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explanatory derivations of a number of relationships used in the |
| 76 |
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main text. This includes the Boltzmann averages of quadrupole |
| 77 |
+ |
orientations, and the interaction of a quadrupole density with the |
| 78 |
+ |
self-generated field gradient. This last relationship is assumed to |
| 79 |
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be zero in the main text but is explicitly shown to be zero here. |
| 80 |
+ |
\end{abstract} |
| 81 |
+ |
|
| 82 |
|
\maketitle |
| 83 |
|
|
| 84 |
< |
This document contains derivations of useful relationships for |
| 85 |
< |
electric field gradients and their interactions with point multipoles. |
| 86 |
< |
We rely heavily on both the notation and results from Torres del |
| 87 |
< |
Castillo and Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} In |
| 88 |
< |
this work, tensors are expressed in Cartesian components, using at |
| 89 |
< |
times a dyadic notation. This proves quite useful for our work as we |
| 90 |
< |
employ toroidal boundary conditions in our simulations, and these are |
| 91 |
< |
easily implemented in Cartesian coordinate systems. |
| 84 |
> |
\section{Generating Uniform Field Gradients} |
| 85 |
> |
One important task in carrying out the simulations mentioned in the |
| 86 |
> |
main text was to generate uniform electric field gradients. To do |
| 87 |
> |
this, we relied heavily on both the notation and results from Torres |
| 88 |
> |
del Castillo and Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} |
| 89 |
> |
In this work, tensors were expressed in Cartesian components, using at |
| 90 |
> |
times a dyadic notation. This proves quite useful for computer |
| 91 |
> |
simulations that make use of toroidal boundary conditions. |
| 92 |
|
|
| 93 |
|
An alternative formalism uses the theory of angular momentum and |
| 94 |
|
spherical harmonics and is common in standard physics texts such as |
| 95 |
< |
Jackson,\cite{Jackson98} Morse and Feshbach, and Baym. Because this |
| 96 |
< |
approach has its own advantages, relationships are provided below |
| 97 |
< |
comparing that terminology to the Cartesian tensor notation. |
| 95 |
> |
Jackson,\cite{Jackson98} Morse and Feshbach,\cite{Morse:1946zr} and |
| 96 |
> |
Stone.\cite{Stone:1997ly} Because this approach has its own |
| 97 |
> |
advantages, relationships are provided below comparing that |
| 98 |
> |
terminology to the Cartesian tensor notation. |
| 99 |
|
|
| 100 |
|
The gradient of the electric field, |
| 101 |
|
\begin{equation*} |
| 117 |
|
electrostatic potential that generates a uniform gradient may be |
| 118 |
|
written: |
| 119 |
|
\begin{align} |
| 120 |
< |
\Phi(x, y, z) =\; -g_o & \left(\frac{1}{2}(a_1\;b_1 - \frac{cos\psi}{3})\;x^2+\frac{1}{2}(a_2\;b_2 - \frac{cos\psi}{3})\;y^2 + \frac{1}{2}(a_3\;b_3 - \frac{cos\psi}{3})\;z^2 \right. \\ |
| 121 |
< |
& \left. + \frac{(a_1\;b_2 + a_2\;b_1)}{2} x\;y + \frac{(a_1\;b_3 + a_3\;b_1)}{2} x\;z + \frac{(a_2\;b_3 + a_3\;b_2)}{2} y\;z \right) . |
| 120 |
> |
\Phi(x, y, z) =\; -\frac{g_o}{2} & \left(\left(a_1b_1 - |
| 121 |
> |
\frac{cos\psi}{3}\right)\;x^2+\left(a_2b_2 |
| 122 |
> |
- \frac{cos\psi}{3}\right)\;y^2 + |
| 123 |
> |
\left(a_3b_3 - |
| 124 |
> |
\frac{cos\psi}{3}\right)\;z^2 \right. \nonumber \\ |
| 125 |
> |
& + (a_1b_2 + a_2b_1)\; xy + (a_1b_3 + a_3b_1)\; xz + (a_2b_3 + a_3b_2)\; yz \bigg) . |
| 126 |
|
\label{eq:appliedPotential} |
| 127 |
|
\end{align} |
| 128 |
|
Note $\mathbf{a}\cdot\mathbf{a} = \mathbf{b} \cdot \mathbf{b} = 1$, |
| 129 |
|
$\mathbf{a} \cdot \mathbf{b}=\cos \psi$, and $g_0$ is the overall |
| 130 |
< |
strength of the potential. |
| 130 |
> |
strength of the potential. |
| 131 |
|
|
| 132 |
< |
An alternative to this notation is to write an electrostatic potential |
| 133 |
< |
that generates a uniform gradient using the notation of Morse and |
| 134 |
< |
Feshbach, |
| 132 |
> |
Taking the gradient of Eq. (\ref{eq:appliedPotential}), we find the |
| 133 |
> |
field due to this potential, |
| 134 |
> |
\begin{equation} |
| 135 |
> |
\mathbf{E} = -\nabla \Phi |
| 136 |
> |
=\frac{g_o}{2} \left(\begin{array}{ccc} |
| 137 |
> |
2(a_1 b_1 - \frac{cos\psi}{3})\; x & +\; (a_1 b_2 + a_2 b_1)\; y & +\; (a_1 b_3 + a_3 b_1)\; z \\ |
| 138 |
> |
(a_2 b_1 + a_1 b_2)\; x & +\; 2(a_2 b_2 - \frac{cos\psi}{3})\; y & +\; (a_2 b_3 + a_3 b_3)\; z \\ |
| 139 |
> |
(a_3 b_1 + a_3 b_2)\; x & +\; (a_3 b_2 + a_2 b_3)\; y & +\; 2(a_3 b_3 - \frac{cos\psi}{3})\; z |
| 140 |
> |
\end{array} \right), |
| 141 |
> |
\label{eq:CE} |
| 142 |
> |
\end{equation} |
| 143 |
> |
while the gradient of the electric field in this form, |
| 144 |
> |
\begin{equation} |
| 145 |
> |
\mathsf{G} = \nabla\mathbf{E} |
| 146 |
> |
= \frac{g_o}{2}\left(\begin{array}{ccc} |
| 147 |
> |
2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\ |
| 148 |
> |
(a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_3) \\ |
| 149 |
> |
(a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3}) |
| 150 |
> |
\end{array} \right), |
| 151 |
> |
\label{eq:GC} |
| 152 |
> |
\end{equation} |
| 153 |
> |
is uniform over the entire space. Therefore, to describe a uniform |
| 154 |
> |
gradient in this notation, two unit vectors ($\mathbf{a}$ and |
| 155 |
> |
$\mathbf{b}$) as well as a potential strength, $g_0$, must be |
| 156 |
> |
specified. As expected, this requires five independent parameters. |
| 157 |
> |
|
| 158 |
> |
The common alternative to the Cartesian notation expresses the |
| 159 |
> |
electrostatic potential using the notation of Morse and |
| 160 |
> |
Feshbach,\cite{Morse:1946zr} |
| 161 |
|
\begin{equation} \label{eq:quad_phi} |
| 162 |
< |
\Phi(x,y,z) = - \left[ a_{20} \frac{2 z^2 -x^2 - y^2}{2} |
| 162 |
> |
\Phi(x,y,z) = -\left[ a_{20} \frac{2 z^2 -x^2 - y^2}{2} |
| 163 |
|
+ 3 a_{21}^e \,xz + 3 a_{21}^o \,yz |
| 164 |
< |
+ 6a_{22}^e \,xy + 3 a_{22}^o (x^2 - y^2) \right] . |
| 164 |
> |
+ 6a_{22}^e \,xy + 3 a_{22}^o (x^2 - y^2) \right]. |
| 165 |
|
\end{equation} |
| 166 |
|
Here we use the standard $(l,m)$ form for the $a_{lm}$ coefficients, |
| 167 |
|
with superscript $e$ and $o$ denoting even and odd, respectively. |
| 168 |
|
This form makes the functional analogy to ``d'' atomic states |
| 169 |
< |
apparent. The gradient of the electric field in this form is: |
| 169 |
> |
apparent. |
| 170 |
> |
|
| 171 |
> |
Applying the gradient operator to Eq. (\ref{eq:quad_phi}) the electric |
| 172 |
> |
field due to this potential, |
| 173 |
> |
\begin{equation} |
| 174 |
> |
\mathbf{E} = -\nabla \Phi |
| 175 |
> |
= \left(\begin{array}{ccc} |
| 176 |
> |
\left( 6a_{22}^o -a_{20} \right)\; x &+\; 6a_{22}^e\; y &+\; 3a_{21}^e\; z \\ |
| 177 |
> |
6a_{22}^e\; x & -\; (a_{20} + 6a_{22}^o)\; y & +\; 3a_{21}^o\; z \\ |
| 178 |
> |
3a_{21}^e\; x & +\; 3a_{21}^o\; y & +\; 2a_{20}\; z |
| 179 |
> |
\end{array} \right), |
| 180 |
> |
\label{eq:MFE} |
| 181 |
> |
\end{equation} |
| 182 |
> |
while the gradient of the electric field in this form is: |
| 183 |
|
\begin{equation} \label{eq:grad_e2} |
| 184 |
|
\mathsf{G} = |
| 185 |
|
\begin{pmatrix} |
| 188 |
|
3a_{21}^e & 3a_{21}^o & 2a_{20} \\ |
| 189 |
|
\end{pmatrix} \\ |
| 190 |
|
\end{equation} |
| 191 |
< |
which can be factored as |
| 191 |
> |
which is also uniform over the entire space. This form for the |
| 192 |
> |
gradient can be factored as |
| 193 |
|
\begin{gather} |
| 194 |
|
\begin{aligned} |
| 195 |
|
\mathsf{G} = a_{20} |
| 226 |
|
\label{eq:intro_tensors} |
| 227 |
|
\end{gather} |
| 228 |
|
The five matrices in the expression above represent five different |
| 229 |
< |
symmetric traceless tensors of rank 2. The trace corresponds to |
| 169 |
< |
$\nabla \cdot \mathbf{E} = 0$, consistent with being in a charge-free |
| 170 |
< |
region. Using the Cartesian notation of |
| 171 |
< |
Eq. (\ref{eq:appliedPotential}), this tensor is written: |
| 172 |
< |
\begin{equation} |
| 173 |
< |
\mathsf{G} =\nabla\bf{E} |
| 174 |
< |
= \frac{g_o}{2}\left(\begin{array}{ccc} |
| 175 |
< |
2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\ |
| 176 |
< |
(a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_3) \\ |
| 177 |
< |
(a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3}) |
| 178 |
< |
\end{array} \right). |
| 179 |
< |
\label{eq:GC} |
| 180 |
< |
\end{equation} |
| 229 |
> |
symmetric traceless tensors of rank 2. |
| 230 |
|
|
| 231 |
< |
It is useful to find vectors $\mathbf a$ and $\mathbf b$ that generate |
| 232 |
< |
the five types of tensors shown in Eq. (\ref{eq:intro_tensors}). If |
| 233 |
< |
the two vectors are co-linear, e.g., $\psi=0$, $\mathbf{a}=(0,0,1)$ and |
| 234 |
< |
$\mathbf{b}=(0,0,1)$, then |
| 231 |
> |
It is useful to find the Cartesian vectors $\mathbf a$ and $\mathbf b$ |
| 232 |
> |
that generate the five types of tensors shown in |
| 233 |
> |
Eq. (\ref{eq:intro_tensors}). If the two vectors are co-linear, e.g., |
| 234 |
> |
$\psi=0$, $\mathbf{a}=(0,0,1)$ and $\mathbf{b}=(0,0,1)$, then |
| 235 |
|
\begin{equation*} |
| 236 |
|
\mathsf{G} = \frac{g_0}{3} |
| 237 |
|
\begin{pmatrix} |
| 265 |
|
\end{equation*} |
| 266 |
|
The pattern is straightforward to continue for the other symmetries. |
| 267 |
|
|
| 268 |
< |
Using Eq. (\ref{eq:quad_phi}) the electric field is written: |
| 269 |
< |
\begin{equation} |
| 270 |
< |
\mathbf{E} |
| 222 |
< |
= \left(\begin{array}{ccc} |
| 223 |
< |
\left(-a_{20} + 6a_{22}^o \right) x + 6a_{22}^e y + 3a_{21}^e z \\ |
| 224 |
< |
6a_{22}^e x+(-a_{20} - 6a_{22}^o) y + 3a_{21}^e z \\ |
| 225 |
< |
3a_{21}^e x +3a_{21}^o y + 2a_{20} z |
| 226 |
< |
\end{array} \right). |
| 227 |
< |
\label{eq:MFE} |
| 228 |
< |
\end{equation} |
| 229 |
< |
while using Eq. (\ref{eq:appliedPotential}), we find: |
| 230 |
< |
\begin{equation} |
| 231 |
< |
\mathbf{E} |
| 232 |
< |
=\frac{g_o}{2} \left(\begin{array}{ccc} |
| 233 |
< |
2(a_1\; b_1 - \frac{cos\psi}{3})\;x \;+ (a_1\; b_2 \;+ a_2\; b_1)\;y + (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
| 234 |
< |
(a_2\; b_1 \;+ a_1\; b_2)\;x + 2(a_2\; b_2 \;- \frac{cos\psi}{3})\;y + (a_2\; b_3 \;+ a_3\; b_3)\;z \\ |
| 235 |
< |
(a_3\; b_1 \;+ a_3\; b_2)\;x + (a_3\; b_2 \;+ a_2\; b_3)\;y + 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
| 236 |
< |
\end{array} \right). |
| 237 |
< |
\label{eq:CE} |
| 238 |
< |
\end{equation} |
| 239 |
< |
We find the notation of Ref. \onlinecite{Torres-del-Castillo:2006uo} |
| 240 |
< |
to be helpful when creating specific types of constant gradient |
| 241 |
< |
electric fields in simulations. For this reason, |
| 268 |
> |
We find the notation of Ref. \onlinecite{Torres-del-Castillo:2006uo} |
| 269 |
> |
helpful when creating specific types of constant gradient electric |
| 270 |
> |
fields in simulations. For this reason, |
| 271 |
|
Eqs. (\ref{eq:appliedPotential}), (\ref{eq:GC}), and (\ref{eq:CE}) are |
| 272 |
< |
used in our code. |
| 272 |
> |
implemented in our code. In the simulations using constant applied |
| 273 |
> |
gradients that are mentioned in the main text, we utilized a field |
| 274 |
> |
with the $a_{22}^e$ symmetry using vectors, $\mathbf{a}= (1, 0, 0)$ |
| 275 |
> |
and $\mathbf{b} = (0,1,0)$. |
| 276 |
|
|
| 277 |
|
\section{Point-multipolar interactions with a spatially-varying electric field} |
| 278 |
|
|
| 291 |
|
\end{align} |
| 292 |
|
where $\mathbf{r}_k$ is the local coordinate system for the object |
| 293 |
|
(usually the center of mass of object $a$). Components of vectors and |
| 294 |
< |
tensors are given using Green indices, using the Einstein repeated |
| 295 |
< |
summation notation. Note that the definition of the primitive |
| 296 |
< |
quadrupole here differs from the standard traceless form, and contains |
| 297 |
< |
an additional Taylor-series based factor of $1/2$. In Ref. |
| 298 |
< |
\onlinecite{PaperI}, we derived the forces and torques each object |
| 299 |
< |
exerts on the other objects in the system. |
| 294 |
> |
tensors are given using the Einstein repeated summation notation. Note |
| 295 |
> |
that the definition of the primitive quadrupole here differs from the |
| 296 |
> |
standard traceless form, and contains an additional Taylor-series |
| 297 |
> |
based factor of $1/2$. In Ref. \onlinecite{PaperI}, we derived the |
| 298 |
> |
forces and torques each object exerts on the other objects in the |
| 299 |
> |
system. |
| 300 |
|
|
| 301 |
|
Here we must also consider an external electric field that varies in |
| 302 |
|
space: $\mathbf E(\mathbf r)$. Each of the local charges $q_k$ in |
| 309 |
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma|_{\mathbf{r}_k = 0} r_{k \delta} |
| 310 |
|
r_{k \varepsilon} + ... |
| 311 |
|
\end{equation} |
| 312 |
< |
Note that once one shrinks object $a$ to point size, the ${E}_\gamma$ |
| 313 |
< |
terms are all evaluated at the center of the object (now a |
| 314 |
< |
point). Thus later the ${E}_\gamma$ terms can be written using the |
| 315 |
< |
same global origin for all objects $a, b, c, ...$ in the system. The |
| 316 |
< |
force exerted on object $a$ by the electric field is given by, |
| 285 |
< |
|
| 312 |
> |
Note that if one shrinks object $a$ to a single point, the |
| 313 |
> |
${E}_\gamma$ terms are all evaluated at the center of the object (now |
| 314 |
> |
a point). Thus later the ${E}_\gamma$ terms can be written using the |
| 315 |
> |
same (molecular) origin for all point charges in the object. The force |
| 316 |
> |
exerted on object $a$ by the electric field is given by,\cite{Raab:2004ve} |
| 317 |
|
\begin{align} |
| 318 |
|
F^a_\gamma = \sum_{k \textrm{~in~} a} E_\gamma(\mathbf{r}_k) &= \sum_{k \textrm{~in~} a} q_k \lbrace E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
| 319 |
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
| 322 |
|
+ Q_{a \delta \varepsilon} \nabla_\delta \nabla_\varepsilon E_\gamma + |
| 323 |
|
... |
| 324 |
|
\end{align} |
| 325 |
< |
Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc. |
| 325 |
> |
Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc. |
| 326 |
|
|
| 327 |
|
Similarly, the torque exerted by the field on $a$ can be expressed as |
| 328 |
|
\begin{align} |
| 354 |
|
\begin{equation} |
| 355 |
|
U(\mathbf{r}) = \mathrm{C} \phi(\mathbf{r}) - \mathrm{D}_\alpha \mathrm{E}_\alpha - \mathrm{Q}_{\alpha\beta}\nabla_\alpha \mathrm{E}_\beta + ... |
| 356 |
|
\end{equation} |
| 357 |
< |
The results has been summarized in Table I. |
| 357 |
> |
These results have been summarized in Table \ref{tab:UFT}. |
| 358 |
|
|
| 359 |
|
\begin{table} |
| 360 |
|
\caption{Potential energy $(U)$, force $(\mathbf{F})$, and torque |
| 361 |
< |
$(\mathbf{\tau})$ expressions for a multipolar site $\mathrm{r}$ in an |
| 362 |
< |
electric field, $\mathbf{E}(\mathbf{r})$. |
| 363 |
< |
\label{tab:UFT}} |
| 364 |
< |
\begin{tabular}{r|ccc} |
| 361 |
> |
$(\mathbf{\tau})$ expressions for a multipolar site at $\mathbf{r}$ in an |
| 362 |
> |
electric field, $\mathbf{E}(\mathbf{r})$ using the definitions of the multipoles in Eqs. (\ref{eq:charge}), (\ref{eq:dipole}) and (\ref{eq:quadrupole}). |
| 363 |
> |
\label{tab:UFT}} |
| 364 |
> |
\begin{tabular}{r|C{3cm}C{3cm}C{3cm}} |
| 365 |
|
& Charge & Dipole & Quadrupole \\ \hline |
| 366 |
|
$U(\mathbf{r})$ & $C \phi(\mathbf{r})$ & $-\mathbf{D} \cdot \mathbf{E}(\mathbf{r})$ & $- \mathsf{Q}:\nabla \mathbf{E}(\mathbf{r})$ \\ |
| 367 |
|
$\mathbf{F}(\mathbf{r})$ & $C \mathbf{E}(\mathbf{r})$ & $\mathbf{D} \cdot \nabla \mathbf{E}(\mathbf{r})$ & $\mathsf{Q} : \nabla\nabla\mathbf{E}(\mathbf{r})$ \\ |
| 369 |
|
\end{tabular} |
| 370 |
|
\end{table} |
| 371 |
|
|
| 341 |
– |
|
| 372 |
|
\section{Boltzmann averages for orientational polarization} |
| 373 |
< |
The dielectric properties of the system is mainly arise from two |
| 374 |
< |
different ways: i) the applied field distort the charge distributions |
| 375 |
< |
so it produces an induced multipolar moment in each molecule; and ii) |
| 376 |
< |
the applied field tends to line up originally randomly oriented |
| 347 |
< |
molecular moment towards the direction of the applied field. In this |
| 348 |
< |
study, we basically focus on the orientational contribution in the |
| 349 |
< |
dielectric properties. If we consider a system of molecules in the |
| 350 |
< |
presence of external field perturbation, the perturbation experienced |
| 351 |
< |
by any molecule will not be only due to external field or field |
| 352 |
< |
gradient but also due to the field or field gradient produced by the |
| 353 |
< |
all other molecules in the system. In the following subsections |
| 373 |
> |
If we consider a collection of molecules in the presence of external |
| 374 |
> |
field, the perturbation experienced by any one molecule will include |
| 375 |
> |
contributions to the field or field gradient produced by the all other |
| 376 |
> |
molecules in the system. In subsections |
| 377 |
|
\ref{subsec:boltzAverage-Dipole} and \ref{subsec:boltzAverage-Quad}, |
| 378 |
< |
we will discuss about the molecular polarization only due to external |
| 379 |
< |
field perturbation. The contribution of the field or field gradient |
| 380 |
< |
due to all other molecules will be taken into account while |
| 358 |
< |
calculating correction factor in the paper. |
| 378 |
> |
we discuss the molecular polarization due solely to external field |
| 379 |
> |
perturbations. This illustrates the origins of the polarizability |
| 380 |
> |
equations (Eqs. 6, 20, and 21) in the main text. |
| 381 |
|
|
| 382 |
|
\subsection{Dipoles} |
| 383 |
|
\label{subsec:boltzAverage-Dipole} |
| 384 |
|
Consider a system of molecules, each with permanent dipole moment |
| 385 |
< |
$p_o$. In the absense of external field, thermal agitation orients the |
| 386 |
< |
dipoles randomly, reducing the system moment to zero. External fields |
| 387 |
< |
will tend to line up the dipoles in the direction of applied field. |
| 388 |
< |
Here we have considered net field from all other molecules is |
| 389 |
< |
considered to be zero. Therefore the total Hamiltonian of each |
| 390 |
< |
molecule is,\cite{Jackson98} |
| 385 |
> |
$p_o$. In the absense of an external field, thermal agitation orients |
| 386 |
> |
the dipoles randomly, and the system moment, $\mathbf{P}$, is zero. |
| 387 |
> |
External fields will line up the dipoles in the direction of applied |
| 388 |
> |
field. Here we consider the net field from all other molecules to be |
| 389 |
> |
zero. Therefore the total Hamiltonian acting on each molecule |
| 390 |
> |
is,\cite{Jackson98} |
| 391 |
|
\begin{equation} |
| 392 |
< |
H = H_o - \mathbf{p_o}\cdot \mathbf{E}, |
| 392 |
> |
H = H_o - \mathbf{p}_o \cdot \mathbf{E}, |
| 393 |
|
\end{equation} |
| 394 |
|
where $H_o$ is a function of the internal coordinates of the molecule. |
| 395 |
< |
The Boltzmann average of the dipole moment is given by, |
| 395 |
> |
The Boltzmann average of the dipole moment in the direction of the |
| 396 |
> |
field is given by, |
| 397 |
|
\begin{equation} |
| 398 |
< |
\braket{p_{mol}} = \frac{\displaystyle\int d\Omega\; p_o\; cos\theta\; e^{\frac{p_oE\; cos\theta}{k_B T}}}{\displaystyle\int d\Omega\; e^{\frac{p_oE\;cos\theta}{k_B T}}}, |
| 398 |
> |
\langle p_{mol} \rangle = \frac{\displaystyle\int p_o \cos\theta |
| 399 |
> |
e^{~p_o E \cos\theta /k_B T}\; d\Omega}{\displaystyle\int e^{~p_o E \cos\theta/k_B |
| 400 |
> |
T}\; d\Omega}, |
| 401 |
|
\end{equation} |
| 402 |
< |
where $\bf{E}$ is selected along z-axis. If we consider that the |
| 403 |
< |
applied field is small, \textit{i.e.} $\frac{p_oE\; cos\theta}{k_B T} << 1$, |
| 404 |
< |
\begin{equation} |
| 405 |
< |
\braket{p_{mol}} \approx \frac{1}{3}\frac{{p_o}^2}{k_B T}E, |
| 406 |
< |
\end{equation} |
| 407 |
< |
where $ \alpha_p = \frac{1}{3}\frac{{p_o}^2}{k_B T}$ is a molecular |
| 402 |
> |
where the $z$-axis is taken in the direction of the applied field, |
| 403 |
> |
$\bf{E}$ and |
| 404 |
> |
$\int d\Omega = \int_0^\pi \sin\theta\; d\theta \int_0^{2\pi} d\phi |
| 405 |
> |
\int_0^{2\pi} d\psi$ |
| 406 |
> |
is an integration over Euler angles describing the orientation of the |
| 407 |
> |
molecule. |
| 408 |
> |
|
| 409 |
> |
If the external fields are small, \textit{i.e.} |
| 410 |
> |
$p_oE \cos\theta / k_B T << 1$, |
| 411 |
> |
\begin{equation} |
| 412 |
> |
\langle p_{mol} \rangle \approx \frac{{p_o}^2}{3 k_B T}E, |
| 413 |
> |
\end{equation} |
| 414 |
> |
where $ \alpha_p = \frac{{p_o}^2}{3 k_B T}$ is the molecular |
| 415 |
|
polarizability. The orientational polarization depends inversely on |
| 416 |
< |
the temperature and applied field must overcome the thermal agitation. |
| 416 |
> |
the temperature as the applied field must overcome thermal agitation |
| 417 |
> |
to orient the dipoles. |
| 418 |
|
|
| 419 |
|
\subsection{Quadrupoles} |
| 420 |
|
\label{subsec:boltzAverage-Quad} |
| 421 |
< |
Consider a system of molecules with permanent quadrupole moment |
| 422 |
< |
$q_{\alpha\beta}$. The average quadrupole moment at temperature T in |
| 423 |
< |
the presence of uniform applied field gradient is given |
| 424 |
< |
by,\cite{AduGyamfi78, AduGyamfi81} |
| 421 |
> |
If instead, our system consists of molecules with permanent |
| 422 |
> |
\textit{quadrupole} tensor $q_{\alpha\beta}$. The average quadrupole |
| 423 |
> |
at temperature $T$ in the presence of uniform applied field gradient |
| 424 |
> |
is given by,\cite{AduGyamfi78, AduGyamfi81} |
| 425 |
|
\begin{equation} |
| 426 |
< |
\braket{q_{\alpha\beta}} \;=\; \frac{\displaystyle\int d\Omega\; e^{-\frac{H}{k_B T}}q_{\alpha\beta}}{\displaystyle\int d\Omega\; e^{-\frac{H}{k_B T}}} \;=\; \frac{\displaystyle\int d\Omega\; e^{\frac{q_{\mu\nu}\;\partial_\nu E_\mu}{k_B T}}q_{\alpha\beta}}{\displaystyle\int d\Omega\; e^{\frac{q_{\mu\nu}\;\partial_\nu E_\mu}{k_B T}}}, |
| 426 |
> |
\langle q_{\alpha\beta} \rangle \;=\; \frac{\displaystyle\int |
| 427 |
> |
q_{\alpha\beta}\; e^{-H/k_B T}\; d\Omega}{\displaystyle\int |
| 428 |
> |
e^{-H/k_B T}\; d\Omega} \;=\; \frac{\displaystyle\int |
| 429 |
> |
q_{\alpha\beta}\; e^{~q_{\mu\nu}\;\partial_\nu E_\mu /k_B T}\; |
| 430 |
> |
d\Omega}{\displaystyle\int e^{~q_{\mu\nu}\;\partial_\nu E_\mu /k_B |
| 431 |
> |
T}\; d\Omega }, |
| 432 |
|
\label{boltzQuad} |
| 433 |
|
\end{equation} |
| 434 |
< |
where $\int d\Omega = \int_0^{2\pi} \int_0^\pi \int_0^{2\pi} |
| 435 |
< |
sin\theta\; d\theta\ d\phi\ d\psi$ is the integration over Euler |
| 436 |
< |
angles, $ H = H_o -q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of |
| 437 |
< |
a quadrupole in the gradient of the |
| 438 |
< |
applied field and $ H_o$ is a function of internal coordinates of the molecule. The energy and quadrupole moment can be transformed into body frame using following relation, |
| 439 |
< |
\begin{equation} |
| 440 |
< |
\begin{split} |
| 441 |
< |
&q_{\alpha\beta} = \eta_{\alpha\alpha'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
| 442 |
< |
&H = H_o - q:{\nabla}\mathbf{E} = H_o - q_{\mu\nu}\;\partial_\nu E_\mu = H_o -\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu. |
| 443 |
< |
\end{split} |
| 444 |
< |
\label{energyQuad} |
| 445 |
< |
\end{equation} |
| 434 |
> |
where $H = H_o - q_{\mu\nu}\;\partial_\nu E_\mu $ is the energy of a |
| 435 |
> |
quadrupole in the gradient of the applied field and $H_o$ is a |
| 436 |
> |
function of internal coordinates of the molecule. The energy and |
| 437 |
> |
quadrupole moment can be transformed into the body frame using a |
| 438 |
> |
rotation matrix $\mathsf{\eta}^{-1}$, |
| 439 |
> |
\begin{align} |
| 440 |
> |
q_{\alpha\beta} &= \eta_{\alpha\alpha'}\;\eta_{\beta\beta'}\;{q}^* _{\alpha'\beta'} \\ |
| 441 |
> |
H &= H_o - q:{\nabla}\mathbf{E} \\ |
| 442 |
> |
&= H_o - q_{\mu\nu}\;\partial_\nu E_\mu \\ |
| 443 |
> |
&= H_o |
| 444 |
> |
-\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu |
| 445 |
> |
E_\mu. \label{energyQuad} |
| 446 |
> |
\end{align} |
| 447 |
|
Here the starred tensors are the components in the body fixed |
| 448 |
< |
frame. Substituting equation (\ref{energyQuad}) in the equation (\ref{boltzQuad}) |
| 449 |
< |
and taking linear terms in the expansion we get, |
| 448 |
> |
frame. Substituting equation (\ref{energyQuad}) in the equation |
| 449 |
> |
(\ref{boltzQuad}) and taking linear terms in the expansion we obtain, |
| 450 |
|
\begin{equation} |
| 451 |
< |
\braket{q_{\alpha\beta}} = \frac{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)q_{\alpha\beta}}{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)}, |
| 451 |
> |
\braket{q_{\alpha\beta}} = \frac{\displaystyle \int q_{\alpha\beta} \left(1 + |
| 452 |
> |
\frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu |
| 453 |
> |
E_\mu }{k_B T}\right)\; d\Omega}{\displaystyle \int \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)\; d\Omega}. |
| 454 |
|
\end{equation} |
| 455 |
< |
where $\eta_{\alpha\alpha'}$ is the inverse of the rotation matrix that transforms |
| 456 |
< |
the body fixed co-ordinates to the space co-ordinates, |
| 457 |
< |
\[\eta_{\alpha\alpha'} |
| 458 |
< |
= \left(\begin{array}{ccc} |
| 459 |
< |
cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
| 460 |
< |
cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
| 461 |
< |
sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
| 462 |
< |
\end{array} \right).\] |
| 463 |
< |
Integration of 1st and 2nd terms in the denominator gives $8 \pi^2$ |
| 423 |
< |
and $8 \pi^2 /3\;{\nabla}.\mathbf{E}\; Tr(q^*) $ respectively. The |
| 424 |
< |
second term vanishes for charge free space, ${\nabla}.\mathbf{E} \; = \; 0$. Similarly integration of the |
| 425 |
< |
1st term in the numerator produces |
| 426 |
< |
$8 \pi^2 /3\; Tr(q^*)\delta_{\alpha\beta}$ and the 2nd term produces |
| 427 |
< |
$8 \pi^2 /15k_B T (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
| 428 |
< |
{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta$, |
| 429 |
< |
if ${\nabla}.\mathbf{E} \; = \; 0$, |
| 430 |
< |
$ \partial_\alpha E_\beta = \partial_\beta E_\alpha$ and |
| 431 |
< |
${q}^*_{\alpha'\beta'}= {q}^*_{\beta'\alpha'}$. Therefore the |
| 432 |
< |
Boltzmann average of a quadrupole moment can be written as, |
| 455 |
> |
Recall that $\eta_{\alpha\alpha'}$ is the inverse of the rotation |
| 456 |
> |
matrix that transforms the body fixed co-ordinates to the space |
| 457 |
> |
co-ordinates. |
| 458 |
> |
% \[\eta_{\alpha\alpha'} |
| 459 |
> |
% = \left(\begin{array}{ccc} |
| 460 |
> |
% cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
| 461 |
> |
% cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
| 462 |
> |
% sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
| 463 |
> |
% \end{array} \right).\] |
| 464 |
|
|
| 465 |
+ |
Integration of the first and second terms in the denominator gives |
| 466 |
+ |
$8 \pi^2$ and |
| 467 |
+ |
$8 \pi^2 ({\nabla} \cdot \mathbf{E}) \mathrm{Tr}(q^*) / 3 $ |
| 468 |
+ |
respectively. The second term vanishes for charge free space (where |
| 469 |
+ |
${\nabla} \cdot \mathbf{E}=0$). Similarly, integration of the first |
| 470 |
+ |
term in the numerator produces |
| 471 |
+ |
$8 \pi^2 \delta_{\alpha\beta} \mathrm{Tr}(q^*) / 3$ while the second |
| 472 |
+ |
produces |
| 473 |
+ |
$8 \pi^2 (3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'} - |
| 474 |
+ |
{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'})\partial_\alpha E_\beta / |
| 475 |
+ |
15 k_B T $. |
| 476 |
+ |
Therefore the Boltzmann average of a quadrupole moment can be written |
| 477 |
+ |
as, |
| 478 |
|
\begin{equation} |
| 479 |
< |
\braket{q_{\alpha\beta}}\; = \; \frac{1}{3} Tr(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
| 479 |
> |
\langle q_{\alpha\beta} \rangle = \frac{1}{3} \mathrm{Tr}(q^*)\;\delta_{\alpha\beta} + \frac{{\bar{q_o}}^2}{15k_BT}\;\partial_\alpha E_\beta, |
| 480 |
|
\end{equation} |
| 481 |
< |
where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular quadrupole polarizablity and ${\bar{q_o}}^2= |
| 482 |
< |
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ is a square of the net quadrupole moment of a molecule. |
| 481 |
> |
where $\alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular |
| 482 |
> |
quadrupole polarizablity and |
| 483 |
> |
${\bar{q_o}}^2= |
| 484 |
> |
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ |
| 485 |
> |
is the square of the net quadrupole moment of a molecule. |
| 486 |
|
|
| 440 |
– |
% \section{External application of a uniform field gradient} |
| 441 |
– |
% \label{Ap:fieldOrGradient} |
| 442 |
– |
|
| 443 |
– |
% To satisfy the condition $ \nabla \cdot \mathbf{E} = 0 $, within the box of molecules we have taken electrostatic potential in the following form |
| 444 |
– |
% \begin{equation} |
| 445 |
– |
% \begin{split} |
| 446 |
– |
% \phi(x, y, z) =\; &-g_o \left(\frac{1}{2}(a_1\;b_1 - \frac{cos\psi}{3})\;x^2+\frac{1}{2}(a_2\;b_2 - \frac{cos\psi}{3})\;y^2 + \frac{1}{2}(a_3\;b_3 - \frac{cos\psi}{3})\;z^2 \right. \\ |
| 447 |
– |
% & \left. + \frac{(a_1\;b_2 + a_2\;b_1)}{2} x\;y + \frac{(a_1\;b_3 + a_3\;b_1)}{2} x\;z + \frac{(a_2\;b_3 + a_3\;b_2)}{2} y\;z \right), |
| 448 |
– |
% \end{split} |
| 449 |
– |
% \label{eq:appliedPotential} |
| 450 |
– |
% \end{equation} |
| 451 |
– |
% where $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ are basis vectors determine coefficients in x, y, and z direction. And $g_o$ and $\psi$ are overall strength of the potential and angle between basis vectors respectively. The electric field derived from the above potential is, |
| 452 |
– |
% \[\mathbf{E} |
| 453 |
– |
% = \frac{g_o}{2} \left(\begin{array}{ccc} |
| 454 |
– |
% 2(a_1\; b_1 - \frac{cos\psi}{3})\;x \;+ (a_1\; b_2 \;+ a_2\; b_1)\;y + (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
| 455 |
– |
% (a_2\; b_1 \;+ a_1\; b_2)\;x + 2(a_2\; b_2 \;- \frac{cos\psi}{3})\;y + (a_2\; b_3 \;+ a_3\; b_2)\;z \\ |
| 456 |
– |
% (a_3\; b_1 \;+ a_3\; b_2)\;x + (a_3\; b_2 \;+ a_2\; b_3)y + 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
| 457 |
– |
% \end{array} \right).\] |
| 458 |
– |
% The gradient of the applied field derived from the potential can be written in the following form, |
| 459 |
– |
% \[\nabla\mathbf{E} |
| 460 |
– |
% = \frac{g_o}{2}\left(\begin{array}{ccc} |
| 461 |
– |
% 2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\ |
| 462 |
– |
% (a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_2) \\ |
| 463 |
– |
% (a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3}) |
| 464 |
– |
% \end{array} \right).\] |
| 465 |
– |
|
| 466 |
– |
|
| 467 |
– |
|
| 487 |
|
\section{Gradient of the field due to quadrupolar polarization} |
| 488 |
|
\label{singularQuad} |
| 489 |
< |
In this section, we will discuss the gradient of the field produced by |
| 490 |
< |
quadrupolar polarization. For this purpose, we consider a distribution |
| 491 |
< |
of charge ${\rho}(\mathbf r)$ which gives rise to an electric field |
| 492 |
< |
$\mathbf{E}(\mathbf r)$ and gradient of the field ${\nabla} \mathbf{E}(\mathbf r)$ |
| 493 |
< |
throughout space. The total gradient of the electric field over volume |
| 494 |
< |
due to the all charges within the sphere of radius $R$ is given by |
| 495 |
< |
(cf. Jackson equation 4.14): |
| 489 |
> |
In section IV.C of the main text, we stated that for quadrupolar |
| 490 |
> |
fluids, the self-contribution to the field gradient vanishes at the |
| 491 |
> |
singularity. In this section, we prove this statement. For this |
| 492 |
> |
purpose, we consider a distribution of charge $\rho(\mathbf{r})$ which |
| 493 |
> |
gives rise to an electric field $\mathbf{E}(\mathbf{r})$ and gradient |
| 494 |
> |
of the field $\nabla\mathbf{E}(\mathbf{r})$ throughout space. The |
| 495 |
> |
gradient of the electric field over volume due to the charges within |
| 496 |
> |
the sphere of radius $R$ is given by (cf. Ref. \onlinecite{Jackson98}, |
| 497 |
> |
equation 4.14): |
| 498 |
|
\begin{equation} |
| 499 |
< |
\int_{r<R} {\nabla}\mathbf{E}\;d^3r = -\int_{r=R} R^2 \mathbf{E}\;\hat{n}\; d\Omega |
| 499 |
> |
\int_{r<R} \nabla\mathbf{E} d\mathbf{r} = -\int_{r=R} R^2 \mathbf{E}\;\hat{n}\; d\Omega |
| 500 |
|
\label{eq:8} |
| 501 |
|
\end{equation} |
| 502 |
|
where $d\Omega$ is the solid angle and $\hat{n}$ is the normal vector |
| 503 |
|
of the surface of the sphere, |
| 483 |
– |
$\hat{n} = sin[\theta]cos[\phi]\hat{x} + sin[\theta]sin[\phi]\hat{y} + |
| 484 |
– |
cos[\theta]\hat{z}$ |
| 485 |
– |
in spherical coordinates. For the charge density ${\rho}(\mathbf r')$, the |
| 486 |
– |
total gradient of the electric field can be written as, ~\cite{Jackson98} |
| 504 |
|
\begin{equation} |
| 505 |
< |
\int_{r<R} {\nabla}\mathbf {E}\; d^3r=-\int_{r=R} R^2\; {\nabla}\Phi\; \hat{n}\; d\Omega =-\frac{1}{4\pi\;\epsilon_o}\int_{r=R} R^2\; {\nabla}\;\left(\int \frac{\rho(\mathbf r')}{|\mathbf{r}-\mathbf{r}'|}\;d^3r'\right) \hat{n}\; d\Omega |
| 505 |
> |
\hat{n} = \sin\theta\cos\phi\; \hat{x} + \sin\theta\sin\phi\; \hat{y} + |
| 506 |
> |
\cos\theta\; \hat{z} |
| 507 |
> |
\end{equation} |
| 508 |
> |
in spherical coordinates. For the charge density $\rho(\mathbf{r}')$, the |
| 509 |
> |
total gradient of the electric field can be written as,\cite{Jackson98} |
| 510 |
> |
\begin{equation} |
| 511 |
> |
\int_{r<R} {\nabla}\mathbf {E}\; d\mathbf{r}=-\int_{r=R} R^2\; {\nabla}\Phi\; \hat{n}\; d\Omega =-\frac{1}{4\pi\;\epsilon_o}\int_{r=R} R^2\; {\nabla}\;\left(\int \frac{\rho(\mathbf r')}{|\mathbf{r}-\mathbf{r}'|}\;d\mathbf{r}'\right) \hat{n}\; d\Omega |
| 512 |
|
\label{eq:9} |
| 513 |
|
\end{equation} |
| 514 |
|
The radial function in the equation (\ref{eq:9}) can be expressed in |
| 520 |
|
If the sphere completely encloses the charge density then $ r_< = r'$ and $r_> = R$. Substituting equation (\ref{eq:10}) into (\ref{eq:9}) we get, |
| 521 |
|
\begin{equation} |
| 522 |
|
\begin{split} |
| 523 |
< |
\int_{r<R} {\nabla}\mathbf{E}\;d^3r &=-\frac{R^2}{\epsilon_o}\int_{r=R} \; {\nabla}\;\left(\int \rho(\mathbf r')\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\frac{{r'^l}}{{R^{l+1}}}\;{Y^*}_{lm}(\theta', \phi')\;Y_{lm}(\theta, \phi)\;d^3r'\right) \hat{n}\; d\Omega \\ |
| 524 |
< |
&= -\frac{R^2}{\epsilon_o}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\int \rho(\mathbf r')\;{r'^l}\;{Y^*}_{lm}(\theta', \phi')\left(\int_{r=R}\vec{\nabla}\left({R^{-(l+1)}}\;Y_{lm}(\theta, \phi)\right)\hat{n}\; d\Omega \right)d^3r |
| 523 |
> |
\int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} &=-\frac{R^2}{\epsilon_o}\int_{r=R} \; {\nabla}\;\left(\int \rho(\mathbf r')\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\frac{{r'^l}}{{R^{l+1}}}\;{Y^*}_{lm}(\theta', \phi')\;Y_{lm}(\theta, \phi)\;d\mathbf{r}'\right) \hat{n}\; d\Omega \\ |
| 524 |
> |
&= -\frac{R^2}{\epsilon_o}\sum_{l=0}^{\infty}\sum_{m=-l}^{m=l}\frac{1}{2l+1}\;\int \rho(\mathbf r')\;{r'^l}\;{Y^*}_{lm}(\theta', \phi')\left(\int_{r=R}\vec{\nabla}\left({R^{-(l+1)}}\;Y_{lm}(\theta, \phi)\right)\hat{n}\; d\Omega \right)d\mathbf{r} |
| 525 |
|
' |
| 526 |
|
\end{split} |
| 527 |
|
\label{eq:11} |
| 535 |
|
\end{split} |
| 536 |
|
\label{eq:12} |
| 537 |
|
\end{equation} |
| 538 |
< |
Using equation (\ref{eq:12}) we get, |
| 538 |
> |
where $Y_{l,l+1,m}(\theta, \phi)$ is a vector spherical |
| 539 |
> |
harmonic.\cite{Arfkan} Using equation (\ref{eq:12}) we get, |
| 540 |
|
\begin{equation} |
| 541 |
|
{\nabla}\left({R^{-(l+1)}}\;Y_{lm}(\theta, \phi)\right) = [(l+1)(2l+1)]^{1/2}\; Y_{l,l+1,m}(\theta, \phi) \; \frac{1}{R^{l+2}}, |
| 542 |
|
\label{eq:13} |
| 543 |
|
\end{equation} |
| 544 |
< |
where $ Y_{l,l+1,m}(\theta, \phi)$ is a vector spherical harmonics \cite{Arfkan}. Using Clebsch-Gorden coefficients $C(l+1, 1, l|m_1,m_2,m) $, equation \ref{eq:14} can be written in spherical harmonics, |
| 544 |
> |
Using Clebsch-Gordan coefficients $C(l+1,1,l|m_1,m_2,m)$, the vector |
| 545 |
> |
spherical harmonics can be written in terms of spherical harmonics, |
| 546 |
|
\begin{equation} |
| 547 |
< |
Y_{l,l+1,m}(\theta, \phi) = \sum_{m_1, m_2} C(l+1,1,l|m_1,m_2,m)\; {Y_{l+1}}^{m_1}(\theta,\phi)\; \hat{e}_{m_2}. |
| 547 |
> |
Y_{l,l+1,m}(\theta, \phi) = \sum_{m_1, m_2} C(l+1,1,l|m_1,m_2,m)\; Y_{l+1}^{m_1}(\theta,\phi)\; \hat{e}_{m_2}. |
| 548 |
|
\label{eq:14} |
| 549 |
|
\end{equation} |
| 550 |
|
Here $\hat{e}_{m_2}$ is a spherical tensor of rank 1 which can be expressed |
| 555 |
|
\end{equation} |
| 556 |
|
The normal vector $\hat{n} $ is then expressed in terms of spherical tensor of rank 1 as shown in below, |
| 557 |
|
\begin{equation} |
| 558 |
< |
\hat{n} = \sqrt{\frac{4\pi}{3}}\left(-{Y_1}^{-1}{\hat{e}}_1 -{Y_1}^{1}{\hat{e}}_{-1} + {Y_1}^{0}{\hat{e}}_0 \right). |
| 558 |
> |
\hat{n} = \sqrt{\frac{4\pi}{3}}\left(-Y_1^{-1}{\hat{e}}_1 - Y_1^{1}{\hat{e}}_{-1} + Y_1^{0}{\hat{e}}_0 \right). |
| 559 |
|
\label{eq:16} |
| 560 |
|
\end{equation} |
| 561 |
|
The surface integral of the product of $\hat{n}$ and |
| 562 |
< |
${Y_{l+1}}^{m_1}(\theta, \phi)$ gives, |
| 562 |
> |
$Y_{l+1}^{m_1}(\theta, \phi)$ gives, |
| 563 |
|
\begin{equation} |
| 564 |
|
\begin{split} |
| 565 |
< |
\int \hat{n}\;{Y_{l+1}}^{m_1}\;d\Omega &= \int \sqrt{\frac{4\pi}{3}}\left(-{Y_1}^{-1}{\hat{e}}_1 -{Y_1}^{1}{\hat{e}}_{-1} + {Y_1}^{0}{\hat{e}}_0 \right)\;{Y_{l+1}}^{m_1}\; d\Omega \\ |
| 566 |
< |
&= \int \sqrt{\frac{4\pi}{3}}\left({{Y_1}^{1}}^* {\hat{e}}_1 +{{Y_1}^{-1}}^* {\hat{e}}_{-1} + {{Y_1}^{0}}^* {\hat{e}}_0 \right)\;{Y_{l+1}}^{m_1}\; d\Omega \\ |
| 565 |
> |
\int \hat{n}\;Y_{l+1}^{m_1}\;d\Omega &= \int \sqrt{\frac{4\pi}{3}}\left(-Y_1^{-1}{\hat{e}}_1 -Y_1^{1}{\hat{e}}_{-1} + Y_1^{0}{\hat{e}}_0 \right)\;Y_{l+1}^{m_1}\; d\Omega \\ |
| 566 |
> |
&= \int \sqrt{\frac{4\pi}{3}}\left({Y_1^{1}}^* {\hat{e}}_1 +{Y_1^{-1}}^* {\hat{e}}_{-1} + {Y_1^{0}}^* {\hat{e}}_0 \right)\;Y_{l+1}^{m_1}\; d\Omega \\ |
| 567 |
|
&= \sqrt{\frac{4\pi}{3}}\left({\delta}_{l+1, 1}\;{\delta}_{1, m_1}\;{\hat{e}}_1 + {\delta}_{l+1, 1}\;{\delta}_{-1, m_1}\;{\hat{e}}_{-1}+ {\delta}_{l+1, 1}\;{\delta}_{0, m_1} \;{\hat{e}}_0\right), |
| 568 |
|
\end{split} |
| 569 |
|
\label{eq:17} |
| 570 |
|
\end{equation} |
| 571 |
< |
where ${Y_{l}}^{-m} = (-1)^m\;{{Y_{l}}^{m}}^* $ and |
| 572 |
< |
$ \int {{Y_{l}}^{m}}^*\;{Y_{l'}}^{m'}\;d\Omega = |
| 571 |
> |
where $Y_{l}^{-m} = (-1)^m\;{Y_{l}^{m}}^* $ and |
| 572 |
> |
$ \int {Y_{l}^{m}}^* Y_{l'}^{m'}\;d\Omega = |
| 573 |
|
\delta_{ll'}\delta_{mm'} $. |
| 574 |
|
Non-vanishing values of equation \ref{eq:17} require $l = 0$, |
| 575 |
|
therefore the value of $ m = 0 $. Since the values of $ m_1$ are -1, |
| 578 |
|
modified, |
| 579 |
|
\begin{equation} |
| 580 |
|
\begin{split} |
| 581 |
< |
\int_{r<R} {\nabla}\mathbf{E}\;d^3r = &- \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;{Y^*}_{00}(\theta', \phi')[ C(1, 1, 0|-1,1,0)\;{\hat{e}_{-1}}{\hat{e}_{1}}\\ &+ C(1, 1, 0|-1,1,0)\;{\hat{e}_{1}}{\hat{e}_{-1}}+C( |
| 582 |
< |
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d^3r' \\ |
| 583 |
< |
&= -\sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;d^3r'\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right)\\ |
| 584 |
< |
&= - \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\;C_{total}\;\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right). |
| 581 |
> |
\int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} = &- \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;{Y^*}_{00}(\theta', \phi')[ C(1, 1, 0|-1,1,0)\;{\hat{e}_{-1}}{\hat{e}_{1}}\\ &+ C(1, 1, 0|-1,1,0)\;{\hat{e}_{1}}{\hat{e}_{-1}}+C( |
| 582 |
> |
1, 1, 0|0,0,0)\;{\hat{e}_{0}}{\hat{e}_{0}} ]\; d\mathbf{r}' \\ |
| 583 |
> |
&= -\sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\int \rho(r')\;d\mathbf{r}'\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right)\\ |
| 584 |
> |
&= - \sqrt{\frac{4\pi}{{3}}}\;\frac{1}{\epsilon_o}\;C_\mathrm{total}\;\left({\hat{e}_{-1}}{\hat{e}_{1}}+{\hat{e}_{1}}{\hat{e}_{-1}}-{\hat{e}_{0}}{\hat{e}_{0}}\right). |
| 585 |
|
\end{split} |
| 586 |
|
\label{eq:19} |
| 587 |
|
\end{equation} |
| 588 |
< |
In the last step, the charge density was integrated over the sphere, yielding a total charge $\mathrm{C_total}$.Equation (\ref{eq:19}) gives the total gradient of the field over a sphere due to the distribution of the charges. |
| 589 |
< |
For quadrupolar fluids the total charge within a sphere is zero, therefore |
| 590 |
< |
$ \int_{r<R} {\nabla}\mathbf{E}\;d^3r = 0 $. Hence the quadrupolar |
| 588 |
> |
In the last step, the charge density was integrated over the sphere, |
| 589 |
> |
yielding a total charge $C_\mathrm{total}$.Equation (\ref{eq:19}) |
| 590 |
> |
gives the total gradient of the field over a sphere due to the |
| 591 |
> |
distribution of the charges. For quadrupolar fluids the total charge |
| 592 |
> |
within a sphere is zero, therefore |
| 593 |
> |
$ \int_{r<R} {\nabla}\mathbf{E}\;d\mathbf{r} = 0 $. Hence the quadrupolar |
| 594 |
|
polarization produces zero net gradient of the field inside the |
| 595 |
|
sphere. |
| 596 |
|
|
| 569 |
– |
|
| 597 |
|
\bibliography{dielectric_new} |
| 598 |
|
\end{document} |
| 599 |
|
% |
