--- 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}