--- trunk/multipole/Dielectric_Supplemental.tex	2016/04/10 12:57:41	4417
+++ trunk/multipole/Dielectric_Supplemental.tex	2016/04/10 15:13:19	4418
@@ -36,8 +36,13 @@ jcp]{revtex4-1}
 \usepackage{url}
 \usepackage{rotating}
 \usepackage{braket}
+\usepackage{array}
+\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+\newcolumntype{R}[1]{>{\raggedleft\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
 
 
+
 %\usepackage[mathlines]{lineno}% Enable numbering of text and display math
 %\linenumbers\relax % Commence numbering lines
 
@@ -63,22 +68,34 @@ of Notre Dame, Notre Dame, IN 46556}
 \date{\today}% It is always \today, today,
              %  but any date may be explicitly specified
 
+\begin{abstract}
+  This document includes useful relationships for computing the
+  interactions between fields and field gradients and point multipolar
+  representations of molecular electrostatics.  We also provide
+  explanatory derivations of a number of relationships used in the
+  main text. This includes the Boltzmann averages of quadrupole
+  orientations, and the interaction of a quadrupole with the
+  self-generated field gradient. This last relationship is assumed to
+  be zero in the main text but is explicitly shown to be zero here.
+\end{abstract}
+
 \maketitle
 
-This document contains derivations of useful relationships for
-electric field gradients and their interactions with point multipoles.
-We rely heavily on both the notation and results from Torres del
-Castillo and Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} In
-this work, tensors are expressed in Cartesian components, using at
-times a dyadic notation. This proves quite useful for our work as we
-employ toroidal boundary conditions in our simulations, and these are
-easily implemented in Cartesian coordinate systems.
+\section{Generating Uniform Field Gradients}
+One important task in performing out the simulations mentioned in the
+main text was to generate uniform electric field gradients. We rely
+heavily on both the notation and results from Torres del Castillo and
+Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} In this work,
+tensors were expressed in Cartesian components, using at times a
+dyadic notation. This proves quite useful for computer simulations
+that make use of toroidal boundary conditions.
 
 An alternative formalism uses the theory of angular momentum and
 spherical harmonics and is common in standard physics texts such as
-Jackson,\cite{Jackson98} Morse and Feshbach, and Baym. Because this
-approach has its own advantages, relationships are provided below
-comparing that terminology to the Cartesian tensor notation.
+Jackson,\cite{Jackson98} Morse and Feshbach,\cite{Morse:1946zr} and
+Stone.\cite{Stone:1997ly} Because this approach has its own
+advantages, relationships are provided below comparing that
+terminology to the Cartesian tensor notation.
 
 The gradient of the electric field,
 \begin{equation*}
@@ -100,26 +117,69 @@ written:
 electrostatic potential that generates a uniform gradient may be
 written:
 \begin{align}
-\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. \\
- & \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) .
+\Phi(x, y, z) =\; -\frac{g_o}{2} & \left(\left(a_1b_1 -
+                         \frac{cos\psi}{3}\right)\;x^2+\left(a_2b_2
+                         - \frac{cos\psi}{3}\right)\;y^2 +
+                         \left(a_3b_3 -
+                         \frac{cos\psi}{3}\right)\;z^2 \right. \nonumber \\
+ & + (a_1b_2 + a_2b_1)\; xy + (a_1b_3 + a_3b_1)\; xz + (a_2b_3 + a_3b_2)\; yz \bigg) .
 \label{eq:appliedPotential}
 \end{align} 
 Note $\mathbf{a}\cdot\mathbf{a} = \mathbf{b} \cdot \mathbf{b} = 1$,
 $\mathbf{a} \cdot \mathbf{b}=\cos \psi$, and $g_0$ is the overall
-strength of the potential.  
+strength of the potential.
 
-An alternative to this notation is to write an electrostatic potential
-that generates a uniform gradient using the notation of Morse and
-Feshbach,
+Taking the gradient of Eq. (\ref{eq:appliedPotential}), we find the
+field due to this potential,
+\begin{equation}
+\mathbf{E} = -\nabla \Phi
+=\frac{g_o}{2} \left(\begin{array}{ccc}
+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 \\
+ (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 \\
+(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 
+\end{array} \right),
+\label{eq:CE}
+\end{equation} 
+while the gradient of the electric field in this form,
+\begin{equation}
+\mathsf{G} = \nabla\mathbf{E} 
+= \frac{g_o}{2}\left(\begin{array}{ccc}
+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) \\
+ (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) \\
+(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})
+\end{array} \right),
+\label{eq:GC}
+\end{equation}  
+is uniform over the entire space.  Therefore, to describe a uniform
+gradient in this notation, two unit vectors ($\mathbf{a}$ and
+$\mathbf{b}$) as well as a potential strength, $g_0$, must be
+specified. As expected, this requires five independent parameters.
+
+The common alternative to the Cartesian notation expresses the
+electrostatic potential using the notation of Morse and
+Feshbach,\cite{Morse:1946zr}
 \begin{equation} \label{eq:quad_phi} 
-\Phi(x,y,z) = - \left[ a_{20} \frac{2 z^2 -x^2 - y^2}{2}
+\Phi(x,y,z) = -\left[ a_{20} \frac{2 z^2 -x^2 - y^2}{2}
 + 3 a_{21}^e \,xz + 3 a_{21}^o \,yz  
- + 6a_{22}^e \,xy +  3 a_{22}^o (x^2 - y^2) \right]  .
+ + 6a_{22}^e \,xy +  3 a_{22}^o (x^2 - y^2) \right].
 \end{equation}
 Here we use the standard $(l,m)$ form for the $a_{lm}$ coefficients,
 with superscript $e$ and $o$ denoting even and odd, respectively.
 This form makes the functional analogy to ``d'' atomic states
-apparent. The gradient of the electric field in this form is:
+apparent. 
+
+Applying the gradient operator to Eq. (\ref{eq:quad_phi}) the electric
+field due to this potential,
+\begin{equation}
+\mathbf{E} = -\nabla \Phi
+= \left(\begin{array}{ccc}
+\left( 6a_{22}^o -a_{20} \right)\; x &+\; 6a_{22}^e\; y &+\; 3a_{21}^e\;  z  \\
+6a_{22}^e\; x & -\; (a_{20} + 6a_{22}^o)\; y & +\; 3a_{21}^o\; z \\
+3a_{21}^e\; x & +\; 3a_{21}^o\; y & +\; 2a_{20}\; z
+\end{array} \right),
+\label{eq:MFE}
+\end{equation}
+while the gradient of the electric field in this form is:
 \begin{equation} \label{eq:grad_e2}
 \mathsf{G} = 
 \begin{pmatrix}
@@ -128,7 +188,8 @@ which can be factored as
 3a_{21}^e  &  3a_{21}^o & 2a_{20} \\
 \end{pmatrix} \\
 \end{equation}
-which can be factored as
+which is also uniform over the entire space.  This form for the
+gradient can be factored as
 \begin{gather}
 \begin{aligned}
 \mathsf{G}  = a_{20} 
@@ -165,24 +226,12 @@ symmetric traceless tensors of rank 2.  The trace corr
  \label{eq:intro_tensors}
 \end{gather}
 The five matrices in the expression above represent five different
-symmetric traceless tensors of rank 2.  The trace corresponds to
-$\nabla \cdot \mathbf{E} = 0$, consistent with being in a charge-free
-region.  Using the Cartesian notation of
-Eq. (\ref{eq:appliedPotential}), this tensor is written:
-\begin{equation}
-\mathsf{G} =\nabla\bf{E} 
-= \frac{g_o}{2}\left(\begin{array}{ccc}
-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) \\
- (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) \\
-(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})
-\end{array} \right).
-\label{eq:GC}
-\end{equation}
+symmetric traceless tensors of rank 2. 
 
-It is useful to find vectors $\mathbf a$ and $\mathbf b$ that generate
-the five types of tensors shown in Eq. (\ref{eq:intro_tensors}).  If
-the two vectors are co-linear, e.g., $\psi=0$, $\mathbf{a}=(0,0,1)$ and
-$\mathbf{b}=(0,0,1)$, then
+It is useful to find the Cartesian vectors $\mathbf a$ and $\mathbf b$
+that generate the five types of tensors shown in
+Eq. (\ref{eq:intro_tensors}).  If the two vectors are co-linear, e.g.,
+$\psi=0$, $\mathbf{a}=(0,0,1)$ and $\mathbf{b}=(0,0,1)$, then
 \begin{equation*}
 \mathsf{G} = \frac{g_0}{3}
 \begin{pmatrix}
@@ -216,31 +265,14 @@ Using Eq. (\ref{eq:quad_phi}) the electric field is wr
 \end{equation*}
 The pattern is straightforward to continue for the other symmetries.
 
-Using Eq. (\ref{eq:quad_phi}) the electric field is written:
-\begin{equation}
-\mathbf{E} 
-= \left(\begin{array}{ccc}
-\left(-a_{20} + 6a_{22}^o \right) x + 6a_{22}^e y + 3a_{21}^e  z  \\
-6a_{22}^e x+(-a_{20} - 6a_{22}^o) y + 3a_{21}^e z \\
-3a_{21}^e x +3a_{21}^o y + 2a_{20} z
-\end{array} \right).
-\label{eq:MFE}
-\end{equation}
-while using Eq. (\ref{eq:appliedPotential}), we find:
-\begin{equation}
-\mathbf{E}
-=\frac{g_o}{2} \left(\begin{array}{ccc}
-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 \\
- (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 \\
-(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 
-\end{array} \right).
-\label{eq:CE}
-\end{equation}
 We find the notation of Ref. \onlinecite{Torres-del-Castillo:2006uo}
-to be helpful when creating specific types of constant gradient
-electric fields in simulations. For this reason,
+helpful when creating specific types of constant gradient electric
+fields in simulations. For this reason,
 Eqs. (\ref{eq:appliedPotential}), (\ref{eq:GC}), and (\ref{eq:CE}) are
-used in our code.
+implemented in our code.  In the simulations using constant applied
+gradients that are mentioned in the main text, we utilized a field
+with the $a_{22}^e$ symmetry using vectors, $\mathbf{a}= (1, 0, 0)$
+and $\mathbf{b} = (0,1,0)$.
 
 \section{Point-multipolar interactions with a spatially-varying electric field}
 
@@ -259,12 +291,12 @@ tensors are given using Green indices, using the Einst
 \end{align}
 where $\mathbf{r}_k$ is the local coordinate system for the object
 (usually the center of mass of object $a$).  Components of vectors and
-tensors are given using Green indices, using the Einstein repeated
-summation notation. Note that the definition of the primitive
-quadrupole here differs from the standard traceless form, and contains
-an additional Taylor-series based factor of $1/2$.  In Ref.
-\onlinecite{PaperI}, we derived the forces and torques each object
-exerts on the other objects in the system.
+tensors are given using the Einstein repeated summation notation. Note
+that the definition of the primitive quadrupole here differs from the
+standard traceless form, and contains an additional Taylor-series
+based factor of $1/2$. In Ref.  \onlinecite{PaperI}, we derived the
+forces and torques each object exerts on the other objects in the
+system.
 
 Here we must also consider an external electric field that varies in
 space: $\mathbf E(\mathbf r)$.  Each of the local charges $q_k$ in
@@ -277,12 +309,11 @@ Note that once one shrinks object $a$ to point size, t
 + \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma|_{\mathbf{r}_k = 0}  r_{k \delta}
 r_{k \varepsilon} + ... 
 \end{equation}
-Note that once one shrinks object $a$ to point size, the ${E}_\gamma$
-terms are all evaluated at the center of the object (now a
-point). Thus later the ${E}_\gamma$ terms can be written using the
-same global origin for all objects $a, b, c, ...$ in the system. The
-force exerted on object $a$ by the electric field is given by,
-
+Note that if one shrinks object $a$ to a single point, the
+${E}_\gamma$ terms are all evaluated at the center of the object (now
+a point). Thus later the ${E}_\gamma$ terms can be written using the
+same (molecular) origin for all point charges in the object. The force
+exerted on object $a$ by the electric field is given by,
 \begin{align}
 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} 
 + \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta}
@@ -291,7 +322,7 @@ Thus in terms of the global origin $\mathbf{r}$, ${F}_
 + Q_{a \delta \varepsilon} \nabla_\delta \nabla_\varepsilon E_\gamma +
 ... 
 \end{align}
-Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc.
+Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc. 
   
 Similarly, the torque exerted by the field on $a$ can be expressed as
 \begin{align}
@@ -323,14 +354,14 @@ The results has been summarized in Table I.
 \begin{equation}
 U(\mathbf{r}) = \mathrm{C} \phi(\mathbf{r}) - \mathrm{D}_\alpha \mathrm{E}_\alpha - \mathrm{Q}_{\alpha\beta}\nabla_\alpha \mathrm{E}_\beta + ...
 \end{equation}
-The results has been summarized in Table I.
+These results have been summarized in Table \ref{tab:UFT}.
 
 \begin{table}
 \caption{Potential energy $(U)$, force $(\mathbf{F})$, and torque
-  $(\mathbf{\tau})$ expressions for a multipolar site $\mathrm{r}$ in an
-  electric field, $\mathbf{E}(\mathbf{r})$.
-\label{tab:UFT}}
-\begin{tabular}{r|ccc}
+  $(\mathbf{\tau})$ expressions for a multipolar site at $\mathbf{r}$ in an
+  electric field, $\mathbf{E}(\mathbf{r})$ using the definitions of the multipoles in Eqs. (\ref{eq:charge}), (\ref{eq:dipole}) and (\ref{eq:quadrupole}).  
+  \label{tab:UFT}}
+\begin{tabular}{r|C{3cm}C{3cm}C{3cm}}
   & Charge & Dipole & Quadrupole \\ \hline
 $U(\mathbf{r})$ &  $C \phi(\mathbf{r})$ & $-\mathbf{D} \cdot \mathbf{E}(\mathbf{r})$ & $- \mathsf{Q}:\nabla \mathbf{E}(\mathbf{r})$ \\
 $\mathbf{F}(\mathbf{r})$ & $C \mathbf{E}(\mathbf{r})$ & $\mathbf{D} \cdot \nabla \mathbf{E}(\mathbf{r})$ &  $\mathsf{Q} : \nabla\nabla\mathbf{E}(\mathbf{r})$ \\
@@ -338,9 +369,8 @@ $\mathbf{\tau}(\mathbf{r})$ & & $\mathbf{D} \times \ma
 \end{tabular}
 \end{table}
 
-
 \section{Boltzmann averages for orientational polarization}
-The dielectric properties of the system is mainly arise from two
+The dielectric properties of the system  mainly arise from two
 different ways: i) the applied field distort the charge distributions
 so it produces an induced multipolar moment in each molecule; and ii)
 the applied field tends to line up originally randomly oriented
@@ -412,13 +442,14 @@ the body fixed co-ordinates to the space co-ordinates,
 \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)},
 \end{equation}
 where $\eta_{\alpha\alpha'}$ is the inverse of the rotation matrix that transforms
-the body fixed co-ordinates to the space co-ordinates,
-\[\eta_{\alpha\alpha'} 
-= \left(\begin{array}{ccc}
-cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\
-cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\
-sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta
-\end{array} \right).\]
+the body fixed co-ordinates to the space co-ordinates.
+% \[\eta_{\alpha\alpha'} 
+% = \left(\begin{array}{ccc}
+% cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\
+% cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\
+% sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta
+% \end{array} \right).\]
+
 Integration of 1st and 2nd terms in the denominator gives $8 \pi^2$
 and $8 \pi^2 /3\;{\nabla}.\mathbf{E}\; Tr(q^*) $ respectively. The
 second term vanishes for charge free space, ${\nabla}.\mathbf{E} \; = \; 0$. Similarly integration of the
@@ -437,34 +468,6 @@ where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a
 where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular quadrupole polarizablity  and  ${\bar{q_o}}^2=
 3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ is a square of the net quadrupole moment of a molecule. 
 
-% \section{External application of a uniform field gradient}
-% \label{Ap:fieldOrGradient}
-
-% To satisfy the condition $ \nabla \cdot \mathbf{E} = 0 $, within the box of molecules we have taken electrostatic potential in the following form
-% \begin{equation}
-% \begin{split}
-% \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. \\
-% & \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),
-% \end{split}
-% \label{eq:appliedPotential}
-% \end{equation} 
-% 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,
-% \[\mathbf{E} 
-% = \frac{g_o}{2} \left(\begin{array}{ccc}
-% 2(a_1\; b_1 - \frac{cos\psi}{3})\;x \;+  (a_1\; b_2 \;+ a_2\; b_1)\;y + (a_1\; b_3 \;+ a_3\; b_1)\;z \\
-%  (a_2\; b_1 \;+ a_1\; b_2)\;x + 2(a_2\; b_2 \;- \frac{cos\psi}{3})\;y +  (a_2\; b_3 \;+ a_3\; b_2)\;z \\
-% (a_3\; b_1 \;+ a_3\; b_2)\;x +  (a_3\; b_2 \;+ a_2\; b_3)y + 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z 
-% \end{array} \right).\] 
-% The gradient of the applied field derived from the potential can be written in the following form,
-% \[\nabla\mathbf{E} 
-% = \frac{g_o}{2}\left(\begin{array}{ccc}
-% 2(a_1\; b_1 - \frac{cos\psi}{3}) &  (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\
-%  (a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_2) \\
-% (a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3})
-% \end{array} \right).\] 
-
-
-
 \section{Gradient of the field due to quadrupolar polarization}
 \label{singularQuad}
 In this section, we will discuss the gradient of the field produced by
@@ -566,6 +569,7 @@ sphere.
 polarization produces zero net gradient of the field inside the
 sphere.
 
+\section{Geometric Factors for Two Embedded Point Charges}
 
 \bibliography{dielectric_new}
 \end{document}
