\section{Introduction}

Consider a molecule of composition $AB$ with parts $A$ and $B$ having different ionization potentials when isolated. A long-standing problem is how to associate charges with each part as the parts are separated. At intermediate separations, one expects that, as bonding electrons would spend unequal time in the vicinity of each part, one would have to assign non-integer average numbers of electrons to each, numbers which become integers at infinite separation. Density-functional theory (DFT) is defined only for integer electron numbers as originally developed \cite{HK64, KS65}. If the dependences of the energy functionals of integer DFT on electron density were continued to densities containing noninteger electron numbers and applied to the separation of $AB$ into $A+B$, at infinite separation $A$ and $B$ would have unphysical noninteger electron numbers, as pointed out by Perdew et. al. (PPLB) \cite{PPLB82,P85}. Instead, PPLB argued that an ensemble generalization of ground-state DFT should be used for systems with noninteger electron number.

With how to treat noninteger electron number resolved by PPLB, the issue of how rigorously and systematically to decompose a system into its parts remains \cite{PAN05,GCGV06}. Two of the present authors have proposed an exact scheme, partition theory (PT) \cite{CW03,CW06,CW07}, based on the PPLB ensemble DFT. In \cite{CW07}, their PT was brought to full formal development and used for a reconstruction of chemical reactivity theory which eliminated the inconsistencies of earlier formulations and enriched them. Applying PT to the case introduced above, $AB\rightarrow A + B$, the parts would obviously be $A$ and $B$.

To illustrate the conceptual structure and physical content of PT, a very simple system was studied in \cite{CWB07}, a caricature of the hydrogen molecule consisting of two electrons moving independently in one dimension under the influence of two attractive delta-function potentials of equal strength, 1D$H_2$. In the present paper, a corresponding model of a heteronuclear diatomic molecule $AB$ is studied via PT (1D$AB$). The model once again consists of two electrons moving independently in one dimension under the influence of two attractive delta-function potentials of unequal strengths $-Z_A$ and $-Z_B$ with $Z_B<Z_A$. Since a one-electron $B$-atom would tend to donate its electron to the more electronegative $A$-atom when brought together, $A$ can be thought of as a Lewis acid and $B$ as a Lewis base.

In the limit $Z_A=Z_B=Z$, the model becomes that treated in \cite{CWB07}. Moreover, reducing the number of electrons from 2 to 1 requires little modification of that analytic theory, and its numerical results can be used to examine the dependence of charge transfer on $Z_A-Z_B$ and inter``nuclear'' separation. The resulting theory can also be used to explore how symmetry breaking localizes the single electron of $A_2^+$ when it is separated into $A$ and $A^+$, a subtler problem than the localization of both electrons on $A$ when $AB\rightarrow A^-+B^+$.

In Section 2, the formalism developed for the 1D$H_2$ problem in ref.\cite{CWB07} is extended to the present 1D$AB$ problem. Numerical results are given in Section 3 for the dependence of the electron densities of the parts, for the charge transfer from $A$ to $B$, and for the dipole moment as functions of $Z_A$, $Z_B$, and the inter``nuclear'' separation $R$. Results are also given there for the partition potential of PT, and how it induces electronegativity equalization between the parts is discussed. The transferability of the properties of the atoms is discussed as well. The united-atom limit, of more academic interest than chemical relevancy, is discussed separately at the end of Section 3. Section 4 is devoted to the one-electron molecule 1D$AB^+$. The case $Z_A\downarrow Z_B$ is used to show how trivial symmetry breaking, $(Z_A-Z_B)<<Z_A$, is sufficient to localize the electron on $A$, illustrating how real $H_2^+$ separates into $H$ + $H^+$ because of small environmental perturbations. We conclude in Section 5 with a brief discussion of the significance of these very simple illustrations of the power and utility of PT for population analysis and for the transferability of fragments with their properties between different molecular contexts. Detailed derivations of all analytic results are presented in an Appendix.

\section{1D$AB$; independent electrons moving in unequal $\delta$-function potentials in one dimension}

\subsection{The Molecule}

In \cite{CWB07}, we considered an analogue of the $H_2$ molecule in which two electrons move independently in one dimension under the influence of two $\delta$-function potentials of equal strength $-Z$. In the present section, we consider the heteronuclear analogue 1D$AB$ in which the nuclear $\delta$-function of the acid $A$ is of strength $-Z_A$ and that of the base of strength $-Z_B$, with $Z_A>Z_B$. These ``nuclear charges'' are allowed to vary continuously. The ground-state wave functions $\psi_{\alpha}^0(x)$ and energies $E_{\alpha}^0$ of the isolated ``atoms'' ${\alpha}=A,B$ are (atomic units are used throughout):
\begin{equation}
\psi_{\alpha}^0(x)=\sqrt{Z_{\alpha}}\exp[-Z_{\alpha}|x|]~~,\label{e:3.1}\end{equation}
\begin{equation} E_{\alpha}^0=-Z_{\alpha}^2/2~~.\label{e:3.2}\end{equation}

The ground-state energy $E_M(N_M=1)$ of one electron moving independently in the two $\delta$-functions, that of strength $-Z_A$ at $x=-R/2$ and that of strength $-Z_B$ at $x=R/2$, is $-\kappa^2/2$, where
\begin{equation}
\left(\kappa-Z_A\right)\left(\kappa-Z_B\right)=e^{-2\kappa R}Z_AZ_B~~.
\label{e:3.3}
\end{equation}
The solutions of Eq.(\ref{e:3.3}) are plotted as a function of internuclear separation in Subsection A of the Appendix. From here on we are concerned only with the lowest-energy solution, denoted simply as $\kappa$, which corresponds to a bonding state that is doubly occupied when $N_M=2$.

The corresponding ground-state wave function, $\psi_M(x)$ is
\begin{eqnarray}
\nonumber\psi_M(x)&=&Ce^{\kappa(R/2+x)}~~~~~~~~~~,~~x<-R/2~~,\\
&=&De^{\kappa x}+Fe^{-\kappa x}~~,~~-R/2<x<R/2~~,\label{e:3.4}\\
\nonumber &=& G e^{\kappa(R/2-x)}~~~~~~~~~~,~~R/2<x~~,
\end{eqnarray}
where $C$, $D$, $F$, and $G$ are constants whose explicit expressions in terms of $Z_A$, $Z_B$, $R$, and $\kappa$ are given in subsection A of the Appendix (Eqs.\ref{e:3.5}, \ref{e:A2}).
In the combined-atom limit $R\downarrow 0$, $G=C=\kappa^{1/2}$, and $D$ and $F$ are irrelevant. In the limit $R\uparrow \infty$, $C=\kappa^{1/2}$, $D=e^{-\kappa R/2}\kappa^{1/2}$, and $G=F=0$ so that $\psi_M(x)$ is localized on $A$ only.

The two-electron molecular density is
\begin{equation}
n_M(x)=2|\psi_M(x)|^2~~;\label{e:3.7}
\end{equation}
the total energy of the molecule $M$ is
\begin{equation} E_M(N_M=2)=2E_M(N_M=1)=-\kappa^2~~;\label{e:3.8} \end{equation}
and the chemical potential of the molecule is
\begin{equation} \mu_M=E_M(2)-E_M(1)=-\kappa^2/2~~,\label{e:3.9} \end{equation}
all just as for the 1D$H_2$ case of ref.\cite{CWB07}.
Fig.\ref{f:separation} shows the dependence of $n_M(x)$ on internuclear distance $R$ for $Z_A=1.02$, $Z_B=0.98$. For these only slightly different values of $Z_A$ and $Z_B$ the transition from primarily ionic character (right panel) to mixed ionic-covalent character (center panel) occurs at relatively large bond length $R\sim 3.2$. The left panel of Fig.\ref{f:separation} shows a density belonging to the interesting but less chemically meaningful united-atom regime, a regime that we discuss separately in Subsection 2D because it allows us to draw conclusions regarding the limits of utility of Partition Theory.

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{separation_new2.eps}
\caption{Molecular density for three different internuclear separations: $R=0.4$ (left), $R=2.0$ (center), and $R=4.0$ (right). In this plot, $Z_A=1.02$, and $Z_B=0.98$.}\label{f:separation}
\end{center}
\end{figure}

\subsection{The parts}

Our task is to partition $n_M(x)$ into contributions from the two parts of $M$, fragments $A$ and $B$,
\begin{equation} n_M(x)=n_A(x)+n_B(x)~~,\label{e:3.10}\end{equation}
with $n_{A,B}$ localized primarily around $-R/2$, $R/2$, respectively. Because $\psi_M$ and therefore $n_M$ is larger near $A$ than near $B$ (recall that $Z_A>Z_B$), the electron numbers of the fragments,
\begin{equation} N_{A,B}=\int dx n_{A,B}(x)~~,\label{e:3.11} \end{equation}
are unequal with $N_A > N_B$ and
\begin{equation} N_A + N_B = N_M =2~~.\label{e:3.12} \end{equation}
$N_{A,B}$ are nonintegers in general so that Eq.(\ref{e:3.1}) implies that
\begin{equation} N_A = 2- \nu~~,~~N_B = \nu~~, ~~0\leq \nu\leq 1~~,\label{e:3.13}\end{equation}
and the PPLB ensemble \cite{PPLB82} must be used for the fragments. In this use of the PPLB ensemble lies the main difference with Parr's atoms-in-molecules approach
based on a minimum promotion energy criterion \cite{P84}. For $A$, a singly-occupied state of either spin occurs with probablility $\nu/2$, and a doubly-occupied state occurs with probability $1-\nu$ in the PPLB ensemble. For $B$, an unoccupied state occurs with probability $1-\nu$ and a singly-occupied state of either spin occurs with probability $\nu/2$.
The densities of the fragments are, accordingly,
\begin{equation}
n_A(x)=(2-\nu)\psi_A^2(x)~~,~~n_B(x)=\nu\psi_B^2(x)~~,\label{e:3.14} \end{equation}
where $\psi_{\alpha}(x)$ (${\alpha}=A,B$) is a real one-electron wavefunction localized around $-R/2$, ${\alpha} = A$, or $+R/2$, ${\alpha}=B$. Analytic expressions for the $\psi_{\alpha}(x)$ are derived in subsections B through D of the Appendix. They can be written most simply as:
\begin{equation}
\psi_A(x)=(2-\nu)^{-1/2}\psi_M(x)\left(\cos\beta(x)+\sin\beta(x)\right)~~,
\label{e:psiA}
\end{equation}
\begin{equation}
\psi_B(x)=\nu^{-1/2}\psi_M(x)\left(\cos\beta(x)-\sin\beta(x)\right)~~,
\label{e:psiB}
\end{equation}
where $\beta(x)$ is an auxiliary polar-angle function that determines $\nu$ according to
\begin{equation} \nu = 1-\int dx \psi_M^2(x)\sin{2{\beta}(x)}~~,\label{e:3.22}\end{equation} as shown in detail in Subsection B of the Appendix. The corresponding measure of electron transfer $q$ is
\begin{equation}
q=1-\nu=\int dx \psi_M^2(x)\sin{2{\beta}(x)}
\label{e:2.15p}
\end{equation}

In Partition Theory, the wave
functions and densities of the parts are found by minimizing the sum of the
energies of the individual parts subject to the constraints that the electron
densities and numbers of the assembly of parts be identical to those of the
molecule $M$.  The expressions for $\psi_A(x)$ and $\psi_B(x)$ of Eqs.(\ref{e:psiA})-(\ref{e:psiB})
together with Eqs.(\ref{e:3.13})-(\ref{e:3.14}), guarantee that the constraints are met for 1D$AB$.
Only the determination of $\beta(x)$ remains, and the procedure for determining
it by minimizing the sum of the energies of the fragments is described in Subsections C and D of
the Appendix.

Numerical evaluation of the quantities of interest, $\nu$, ${\beta}(x)$, and $\psi_{\alpha}(x)$, is straightforward for representative $Z_\alpha$ and various $R$. For example, Fig. {\ref{f:density}} shows the fragments' atomic densities for $Z_A=1.05$, $Z_B=0.95$, and $R=2$. Interestingly, in spite of having high electron density in the bonding region around the molecule's center of mass, the fragment densities resemble true atomic densities. This is one of the most significant results of our work. In order to investigate the extent to which this is true, and quantify it, results for different parameter regimes are reported and discussed in the next Section. First, however, we discuss in the following subsection how the partition potential $v_P(x)$ of PT can be extracted from these results.

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{density.eps}
\caption{Molecular density $n_M(x)$ (solid), and fragment
densities $n_A(x)$ (dotted) and $n_B(x)$ (dashed) for $Z_A=1.05$, $Z_B=0.95$ and $R=2$. For these values, $\nu=0.6$,
indicating substantial charge transfer even with only a 10\% difference between $Z_A$ and $Z_B$ and a small $R$.
}\label{f:density}
\end{center}
\end{figure}

\subsection{The partition potential}

Within the framework of PT \cite{CW07}, the wavefunctions $\psi_{\alpha}(x)$ for the parts ${\alpha}=A,B$ satisfy the Schr\"{o}dinger equation
\begin{equation}
\left(H_{\alpha}+v_P(x)\right)\psi_{\alpha}(x)=\mu_M\psi_{\alpha}(x)~~,~~{\alpha}=A,B~~,
\label{e:3.41}
\end{equation}
where the Hamiltonians $H_{\alpha}$ for the parts are defined by Eq.({\ref{e:3.24}}). The partition potential $v_P(x)$ is {\em the same} for both parts, as is the eigenvalue $\mu_M$ corresponding to the molecular chemical potential, Eq.(\ref{e:3.9}). We construct $v_P(x)$ in subsection E of the Appendix, obtaining:
\begin{eqnarray}
 v_P(x)&=&-\frac{1}{2}\frac{Z_A^2\psi_M^4(-R/2)\cos^2{2{\beta}_A}}{\psi_M^4(x)}\theta(R/2-|x|)~~\label{e:3.46}\\
\nonumber &&~~~+\frac{1}{2}\sum_{\alpha} v_{\alpha}\left(1-s_{\alpha}\tan{\beta}_{\alpha}(1+\cos{2{\beta}_{\alpha}})\right)~~,
\end{eqnarray}
where $\beta_{\alpha}$ (${\alpha}=A,B)$ are constants that determine the upper and lower bounds of $\beta(x)$ (derived in subsection D of the Appendix), $s_{\alpha}=\pm 1$ (plus sign for part $A$, minus sign for part $B$), and $\theta(y)$($=0$ for $y<0$, $1$ for $y>0$) is the Heaviside step function. Numerical results for $v_P(x)$ are presented and discussed in the next Section.

\section{Illustrative Results}

For purposes of discussion, we define three critical parameters ${R_{\rm IP}}$, ${R_{\rm den}}$, and ${R_{\rm occ}}$ as the values of $R$ at which significant changes in the ionization potential, the density, and the occupation of a fragment, respectively, take place. Our definitions for ${R_{\rm IP}}$ and ${R_{\rm den}}$ are immediately applicable to any diatomic molecule, and our definition for ${R_{\rm occ}}$ could be easily generalized to be applicable to any diatomic molecule. We start by defining ${R_{\rm occ}}$ in Subsection A in order to discuss population analysis and charge transfer, and continue in Subsection B with ${R_{\rm den}}$ and ${R_{\rm IP}}$ to address the issue of transferability.

\subsection{Population Analysis; Charge Transfer}
 First, note that $q=1-\nu$ of an electron is transferred from $B$ to $A$. Figure {\ref{f:nu_vs_a}} shows how $\nu$ changes when $R$ is varied for 4 different values of $\Delta Z$ (and $\bar{Z}=1$) for $R>1$, and Fig.\ref{f:nu_vs_a_complete} does for smaller $R$ as well. Define ${R_{\rm occ}}$ as the larger of the two values of $R$ for which $N_A=2N_B$ (and therefore $\nu=N/3$), a reasonable criterion for the ionic to mixed ionic-covalent crossover.  The solid curve in Fig.\ref{f:critical_a} displays the behavior of ${R_{\rm occ}}$ vs. $\Delta Z$. We observe that ${R_{\rm occ}}$ is a sensitive function of $\Delta Z$, especially as $\Delta Z\to 0$. (We come back to this point in the next paragraph and in Section 4 when discussing electron localization in H$_2^+$ as the molecule dissociates). This is a simple illustration of the utility of PT to quantify the degree of charge transfer taking place as a heteronuclear diatomic molecule is stretched out. That charge transfer occurs is readily seen from the sequence of molecular densities plotted in Fig.\ref{f:separation}, and PT provides an unambiguous way to characterize it.

This $R$-dependence of $\nu$ is interpretable as a sequence of crossovers in the electronic structure of the molecule $M$. At large $R$, $M$ is ionic with $\nu\downarrow 0$ and $q\uparrow 1$. As $R$ decreases, there is a crossover around ${R_{\rm occ}}$ from that ionic to a mixed ionic-covalent state. As $R$ decreases further the covalent character of that mixed state increases. At still smaller $R$, there is another crossover to a combined-atom state around the maximum in $\nu$. Finally, at very small $R$, there is a rapid crossover back to the ionic state, an interesting fact that we discuss further in Subsection E.

It is also of interest to examine how well the product of transferred charge with the internuclear separation agrees
with the actual electronic component of the dipole moment of the molecule.  Figure \ref{f:dipoles} compares this point-charge dipole (PC)
\begin{equation}
d_{\rm PC}=-qR~~,
\label{e:dipole_PT}
\end{equation}
with the actual electronic part of the dipole moment $d=\int dx~xn_M(x)$ for various values of $\Delta Z$. If the
fragment densities were inversion-symmetric about their ``nuclei",
there would be exact agreement between $d_{\rm PC}$ and $d$. But there is in addition a dipole moment
on each fragment from the polarization of its electron cloud, $d_{\alpha}=\int dx~ xn_\alpha(x)$, $d=d_{\rm PC}+d_A+d_B$. Figure \ref{f:atomic_dipole} shows $d(R)$ along with the fragment dipoles $d_{\alpha}(R)$, $\alpha=A,B$, for fixed $\Delta Z$. Clearly, when $\Delta Z=0.04$ the fragment dipoles are small compared to the molecular dipole only for separations larger than $R\sim 3$. The estimate (\ref{e:dipole_PT}) shows excellent agreement whenever the dipole is significant, and only fails (error $> 20\%$) when the dipole is small (Fig.\ref{f:atomic_dipole}).
Note also from the inset of Figure \ref{f:dipoles} that the $R$-dependence of the percent difference changes little when $\Delta Z$ changes from 0.01 to 0.12. Figure \ref{f:atomic_dipole} shows that $d_A$ is of opposite sign to $d_B$ and $d_{\rm PC}$. This sign reversal arises from the distortion of $n_A$ by covalency, which increases it in the bonding region. However, in 1D$AB$ $d_A$ is not large enough in magnitude to overcome $d_B+d_{\rm PC}$ even at small $R$. If it were, $d$ would be of sign opposite to that expected from the electronegativity difference of $A$ and $B$. In real diatomic molecules containing small bases, such sign reversals are observed \cite{HMS93}, and partition theory promises a simple explanation.

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{nu_vs_R_trunc.eps}
\caption{Population of fragment $B$ (=$\nu$) for various values of $\Delta Z$, for $\bar{Z}=1$.}\label{f:nu_vs_a}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{dipoles_inset_smallR.eps}
\caption{Electronic dipoles as a function of inter``nuclear'' separation, both from the point-charge value of Eq.(\ref{e:dipole_PT}) (solid lines), and exactly (dashed), for 3 different values of $\Delta Z$. The inset shows the absolute percent difference. ($\bar{Z}=1$).}\label{f:dipoles}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{PC_dipoles.eps}
\caption{Electronic dipole moment for the 1D$AB$ molecule (solid), and fragment dipole moments, for fixed $\Delta Z = 0.04$ and $\bar{Z}=1$.} \label{f:atomic_dipole}
\end{center}
\end{figure}

\subsection{Transferability}
The mixed covalent to ionic crossover occurs at quite small $R$ for very small $\Delta Z$ when $\bar{Z}=1$. For the crossover to occur at an internuclear separation of about 1${\rm \AA}$, $\Delta Z$ need only be about 0.09.
The {\em shape} of each atom, however, is not as sensitive. Figure {\ref{f:atoms_vs_a}} shows $\psi_A(x)$ (solid) and $\psi_B(x)$ (dotted) for two different separations and three different values of $\Delta Z$ covering the same range as that of Fig.{\ref{f:nu_vs_a}}. It is apparent that orbitals corresponding to different values of $\Delta Z$ start differing significantly only close to the region where $\psi_A(x)$ and $\psi_B(x)$ overlap (the ``bonding'' region), and only for large internuclear separations $R\gtrsim 3$. Since at large internuclear separations the orbitals $\psi_\alpha(x)$ become equal to the isolated solutions $\psi_\alpha^0(x)$ of Eq.(\ref{e:3.1}), it is interesting to examine how and when the $\psi_\alpha(x)$ start departing from the $\psi_\alpha^0(x)$. Figure {\ref{f:a_psi}} shows that for $\Delta Z=0.08$, $\psi_A(x)$ is almost identical to $\psi_A^0(x)$ when $R=4.8$, differing only slightly in its right-hand tail, and that the shape is still preserved for chemically-relevant values like $R\sim 1.6$. In order to appreciate the differences more clearly, Fig.{\ref{f:differences}} displays the differences of the squares $D(x)\equiv \psi_A(x)^2-(\psi_A^0(x))^2$. We observe that the two orbitals depart appreciably when the spatial integral of the absolute value of this quantity reaches a value of $\int dx |D(x)| \sim 0.2$, so we define ${R_{\rm den}}$ as the corresponding separation. As shown in the inset of Fig.{\ref{f:a_psi}}, ${R_{\rm den}}$ remains almost constant at ${R_{\rm den}}\sim 0.66$, small compared to ${R_{\rm occ}}$ in the corresponding range of $\Delta Z$. A similar behavior is observed for ${R_{\rm IP}}$, which we define as the value of $R$ at which the ionization potential of the molecule begins to differ significantly [20\%] from the ionization potential of the most electronegative (isolated) atom. Figure {\ref{f:critical_a}} compares ${R_{\rm occ}}$, ${R_{\rm IP}}$ and ${R_{\rm den}}$. The preservation of the shape of the orbitals in a range of $\Delta Z$ where there is signficant charge transfer (${R_{\rm den}}<<{R_{\rm occ}}$) is a strong indication of the transferability of the fragments emerging from Partition Theory \cite{A00}. The feature of transferability can also be seen for fixed $\Delta Z$ by comparing fragments corresponding to different values of $R$. Figure {\ref{f:comparison}} shows such a comparison for $\Delta Z=0.06$ by overlapping the $\psi_A(x)$'s corresponding to different $R$'s (this requires shifting the origin of $A$-atoms to the left and $B$-atoms to the right). We note that even though the {\em size} of the fragments (controlled by $\nu$) changes substantially as $R$ varies from $1$ to $4$, their shape is quite insensitive to relatively large changes in $R$ with concomitantly large changes in $n_M(x)$ and $\mu_M$. For the value of $\Delta Z$ chosen in Fig.\ref{f:comparison}, $\nu$ is only about 0.2 when $R=4$, and yet the shape of the corresponding fragments only differ slightly from those obtained when $R=1$ and $\nu$ is close to 0.9.

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{atoms_vs_a_dall_new.eps}
\caption{Fragment orbitals $\psi_A(x)$ (solid) and $\psi_B(x)$ (dotted) for different values of $R$ (1 and 3). The fragment-$A$ orbitals are plotted for three different values of $\Delta Z$ (0.04, 0.08, and 0.12).  Orbitals corresponding to different values of $\Delta Z$ start to differ in the overlapping region only for large internuclear separations $R\gtrsim 3$. Differences due to changes in $\Delta Z$ are indistinguishable to the eye for $R\lesssim 3$.}\label{f:atoms_vs_a}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{psi_vs_psi0_2.eps}
\caption{Comparison of fragment orbital $\psi_A(x)$ (solid line) and isolated atomic orbital $\psi_A^0(x)$ (dotted) for $R=4.8$ and $R=1.6$. The inset shows the slow variation of ${R_{\rm den}}$ with $\Delta Z$ (see text).}\label{f:a_psi}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{differences_new.eps}
\caption{Difference between the fragment density on $A$ and the isolated atomic density, $(\psi_A^0(x))^2-\psi_A^2(x)$, for different values of $R$ and fixed values of $\Delta Z=0.08$ and $\bar{Z}=1$. }\label{f:differences}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{critical_Rs.eps}
\caption{Comparison of ${R_{\rm occ}}$ (value of $R$ at which the occupation on $A$ is twice the occupation on $B$), ${R_{\rm IP}}$ (at which the ionization potential of the molecule is 20\% larger than the ionization potential of an isolated $A$ atom), and ${R_{\rm den}}$ (at which the fragment densities change significantly as compared to the corresponding isolated-atom densities - see text). All curves with $\bar{Z}=1$.}\label{f:critical_a}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{comparison_2.eps}
\caption{Fragment orbitals $\psi_A(x)$ and $\psi_B(x)$ for $R=4$ (solid), $R=2$ (dotted), and $R=1$ (dashed), when $Z_A=1.03$ and $Z_B=0.97$. The A-fragment corresponding to $R=2$ was shifted to the left by 1a.u. and the B-fragment was shifted to the right by 1a.u. The A-fragment corresponding to $R=1$ was shifted to the left by 1.5a.u., and the B-fragment was shifted to the right by 1.5a.u.}\label{f:comparison}
\end{center}
\end{figure}

\subsection{Partition Potentials}
The corresponding partition potentials are shown in Fig. {\ref{f:partition_potential}}. Both parts $A$ and $B$ must have equal electronegativities, sharing the same HOMO eigenvalue, Eq.(\ref{e:3.41}), which must be equal to the overall chemical potential $\mu_M$. The partition potential ensures that this happens by acquiring a specific form, with an asymmetric negative well in between the fragments and two negative delta-functions at $\pm R/2$. In the limit of infinite separation, when $\kappa\to Z_A$, the external potential of ``nucleus'' $A$ does not require any correction in order to reach $\mu_M$, but the external potential of ``nucleus'' $B$ does. Accordingly, the delta-function component of $v_P$ vanishes on $A$ at infinite separation, but {\em not} on $B$ (fig.\ref{f:weights}). It is negative there, and has a magnitude smaller than $\Delta Z$ because the smooth negative well persists at large separations, getting further from $A$, and closer to $B$ (Figs.\ref{f:partition_potential}-\ref{f:vR_vs_a}), thereby contributing to the lowering of the eigen-energy of part $B$ towards $\mu_M$.  Figure {\ref{f:vR_vs_a}} shows a closer view of the dependence of the smooth part of the partition potential with $\Delta Z$ and with $R$, and Figure {\ref{f:weights}} illustrates the same dependences for the amplitudes of its delta-function components.

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{atomA_2.eps}
\caption{Left panels: A and B fragments: $\psi_A(x)=\sqrt{n_A(x)/(2-\nu)}$ (solid) and $\psi_B(x)=\sqrt{n_B(x)/\nu}$ (dashed) for $R=4$ (upper), $R=2$ (center), and $R=1$ (bottom). Right panels: corresponding partition potentials ($\delta$-functions at $\pm R/2$ not shown). For all these plots, $Z_A=1.03$ and $Z_B=0.97$.}\label{f:partition_potential}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{vR_vs_a_1_3.eps}
\caption{Smooth part of the partition potential for 2 different values of $R$ and $\Delta Z = 0.04$ (solid), $\Delta Z = 0.08$ (dashed) and $\Delta Z=0.12$ (dotted). For all these curves, $\bar{Z}=1$.}\label{f:vR_vs_a}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{weight_vs_a_2.eps}
\caption{Delta-function weights of the partition potential, for $\Delta Z=0.04$ (solid), $\Delta Z=0.08$ (dashed), and $\Delta Z=0.12$ (dotted). The 3 upper curves correspond to atom $A$, and the 3 bottom ones to atom $B$. For all these curves, $\bar{Z}=1$.}\label{f:weights}
\end{center}
\end{figure}

\subsection{United-atom limit}

In order to focus attention on the chemically relevant range of internuclear separations, we started Fig.\ref{f:nu_vs_a} at $R=1$. The small-$R$ behavior of $\nu$ can be seen in Fig.\ref{f:nu_vs_a_complete}. Perhaps counterintuitively, as $R$ decreases below a given (small) value, a rapid crossover takes place from a mixed ionic-covalent state to an essentially ionic state. In the 1D$H_2$ model of ref.\cite{CWB07}, this reversion to the ionic state is absent, the sequence of crossovers being atomic to covalent to combined-atom state. There is a singularity in the 1D$AB$ model at $\Delta Z=0$, $R=0$. If $\lim_{R,\Delta Z\downarrow 0}\left\{\frac{\Delta Z}{R}\right\}\downarrow 0$, the combined-atom state persists to $R=0$. If, however, $\lim_{R,\Delta Z\downarrow 0}\left\{\frac{R}{\Delta Z}\right\}\downarrow 0$, a reversion to the ionic state occurs. This counterintuitive feature illustrates an important limitation to the utility of partition theory. When $\kappa R$ becomes significantly less than unity as $R$ decreases, the $\psi_\alpha(x)$ overlap substantially, and the primary motivation of partition theory, decomposition of the electron density into distinct {\em localized} components, is frustrated.

The shape of the fragment densities also departs significantly in this limit from pure atomic densities. For example, for the same value of $\Delta Z = 0.08$ used in Fig.\ref{f:differences}, the maximum value of the difference between $\psi_A(x)^2$ and $\psi_A^0(x)^2$ when $R=0.4$ is about 6 times larger than the corresponding maximum when $R=1.6$.

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{nu_vs_R.eps}
\caption{Same as Fig.\ref{f:nu_vs_a} with added small-$R$ range to illustrate striking united-atom behavior of $\nu$ (see text).}\label{f:nu_vs_a_complete}
\end{center}
\end{figure}

\section{Electron localization in Dissociating H$_2^+$}

\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{a_nu1_2.eps}
\caption{Behavior of ${R'_{\rm occ}}$ for 1D$AB^+$ when $\bar{Z}=1$. The inset shows ${R'_{\rm occ}}$ vs. $\ln\Delta Z$. The line corresponds to the best linear fit in the range $-9\leq\ln\Delta Z\leq -5$ (slope: -0.185; intercept: -0.502, agreeing with the tight-binding formula derived in the Appendix, Eq.(\ref{e:tight_binding}))}\label{f:a_nu1}
\end{center}
\end{figure}

Even though the ground-state wavefunction of H$_2^+$ is symmetric, with 50\% of its amplitude on the right atom, and 50\% on the left, the slightest asymmetry due to environmental perturbations forces the electron to localize onto one of the two nuclei as the molecule dissociates. Since this symmetry breaking can now be studied experimentally via intense few-cycle laser pulses with controlled field evolution \cite{KSVK06}, there is resurgent interest in theoretical models to describe electron localization during molecular dissociation (see for example ref.\cite{KRG08} for a recent study of dissociation and ionization of small molecules steered by external noise). Our simple theory of the preceeding sections can be used as such a model, provided we interpret small differences between the magnitudes of $Z_A$ and $Z_B$ as due to the effect of an inhomogeneous environment. In fact, since we have dealt with 2 {\em non-interacting} electrons, only minor modifications of our results are needed in order to analyze the one-electron case, 1D$AB^+$ \cite{CD38}: the chemical potential is still identical to that given by Eq.(\ref{e:3.9}). The number constraint of Eq.(\ref{e:3.12}) is modified to $N_M'=1$ (we use primed symbols to represent 1-electron quantities to distinguish them from their 2-electron analogs), and Eq.(\ref{e:3.13}) goes to:
\begin{equation}
N_A'=1-\nu'~~,~~N_B'=\nu'~~,~~0\leq\nu'\leq\frac{1}{2}~~.
\label{e:4.1}
\end{equation}
The densities of the atoms are:
\begin{equation}
n_A'(x)=(1-\nu')\psi_A^2(x)~~,~~n_B'(x)=\nu'\psi_B^2(x)~~,
\end{equation}
and following the same steps leading to Eq.(\ref{e:3.22}), we find
\begin{equation}
\nu'=\frac{1}{2}\left[1-\int dx \psi_M^2(x)\sin{2\beta'(x)}\right]~~.\label{e:nu_1dAB+}
\end{equation}
Since the Euler equation for $\beta(x)$, Eq.(\ref{e:3.28}), as well as the boundary conditions, remain unchanged, $\beta'(x)=\beta(x)$ and the values that $\nu'$ takes as a function of $R$ and $\Delta Z$ are simply half of those calculated for 1D$AB$, $\nu'=\nu/2$. As before, we define ${R'_{\rm occ}}$ as the value of $R$ for which $N_A' = 2N_B'$, corresponding to $\nu'=1/3$ (from Eq.(\ref{e:4.1}) and $N_A'+N_B'=1$). The behavior of ${R'_{\rm occ}}$ as a function of $\Delta Z$ is then identical to that of ${R_{\rm occ}}(\Delta Z)$. The inset of Fig.(\ref{f:a_nu1}) shows that for small $\Delta Z$, ${R_{\rm occ}}$ is proportional to $\ln\Delta Z$. We conclude that the more inhomogeneous the environment (the larger $\Delta Z$), the earlier electron localization occurs along the dissociation pathway, with a logarithmic dependence in this case. As
${R_{\rm occ}}$ becomes very large when $\Delta Z$ becomes small, this striking relation between ${R_{\rm occ}}$ and
$\Delta Z$  follows from a simple tight-binding argument outlined in the Appendix (Subsection F).

\section{Concluding Remarks}

We have applied Partition Theory \cite{CW07} to a simple model of a heteronuclear diatomic molecule. We found analytic expressions for the densities of the parts, the charge associated with each of the molecular fragments, and the partition potential that guarantees electronegativity equalization. Numerical calculations for various parameter regimes allow us to reach important conclusions: (1) ${R_{\rm occ}}$ has a strikingly different bahavior than ${R_{\rm IP}}$ and ${R_{\rm den}}$ as $\Delta Z\to 0$ (Fig.\ref{f:critical_a}). Since ${R_{\rm occ}}$ measures the value of inter``nuclear'' separation at which significant charge transfer occurs, and ${R_{\rm den}}$ and ${R_{\rm IP}}$ measure the value of $R$ at which signifcant change in the {\em shape} of the fragment wavefunctions takes place, we conclude that the fragments of Partition Theory, at least within this simple model, are to large extent transferrable. (2) Environmental fluctuations (modeled by small finite $\Delta Z$) localize the single electron of H$_2^+$ onto one of the two nuclei. As H$_2^+$ dissociates, the more inhomogeneous the environment, the earlier localization occurs along the dissociation pathway. The explicit results reported both here and in ref.\cite{CWB07} both illustrate important features of Partition Theory \cite{CW07} and support the proposed use of PT for a broad range of applications to real systems including the sharp definitions of parts of a larger system, population and charge-transfer analysis, and the examination of transferability.

 \renewcommand{\theequation}{A.\arabic{equation}}

  \setcounter{equation}{0}
  \section*{APPENDIX}

\subsection{The Molecule}

Eq.(\ref{e:3.3}) has two solutions for all $R<(Z_A+Z_B)/2Z_AZ_B\equiv R^*$: one, $\kappa_+$, belonging to a bonding state doubly occupied when $N=2$, and another one, $\kappa_0=0$, belonging to a state at the bottom of the continuum. For all $R>R^*$ there is another solution, $\kappa_-$, corresponding to an unoccupied antibonding state. As $R\uparrow\infty$, $\kappa_+$ and $\kappa_-$ give rise to the two energies $E_{A,B}^0$ of Eq.(\ref{e:3.2}) corresponding to the two states $\psi_{A,B}^0(x)$ of Eq.(\ref{e:3.1}) of which the lower state $\psi_A^0(x)$ localized to the Lewis acid $A$ is doubly occupied, the Lewis base $B$ having donated its electron. As $R\downarrow 0$, $\kappa_+$ approaches $(Z_A+Z_B)$ and gives rise to a doubly occupied combined-atom state with the $Z_{\alpha}$ of Eqs.(\ref{e:3.1}) and (\ref{e:3.2}) replaced by $Z_A+Z_B$. The value of $\kappa_+$ decreases monotonically from $Z_A+Z_B$ at $R\downarrow 0$ with finite derivative at zero and vanishing derivative at infinity. $\kappa_-$ increases monotonically from zero at $R\downarrow R^*$ to $Z_B$ at $R\uparrow \infty$. $\kappa_0$ remains zero throughout (see Fig.{\ref{f:kappa_vs_a}}). In the text following Eq.(\ref{e:3.3}), we referred only to $\kappa_+$, dropping the subscript $+$ for notational simplicity.

 The $R$-dependences of all three solutions of Eq.(\ref{e:3.3}) for $\kappa$ are shown in Fig.\ref{f:kappa_vs_a} for $Z_A=1, Z_B=0.9$. Note that $\kappa_+$ begins to differ significantly from $\kappa_+(\infty)$ only at separation
$R<{R_{\rm IP}}\sim 1.8$, less than
relevant for real molecules, except for H$_2$. For example, for $Z_A=1$ and $Z_B=0.9$, one gets from Eqs.(\ref{e:3.3}) and (\ref{e:3.9}) an ionization energy equal to that of lithium hydride (LiH), for which the equilibrium bond distance is $R_0=3.05$ ($>>{R_{\rm IP}}$).
\begin{figure}
\begin{center}
\epsfxsize=80mm \epsfbox{kappa_vs_a_2.eps}
\caption{$\kappa$ versus $R$ for $AB$ with $Z_A=1$ and $Z_B=0.9$. The upper curve shows the dependence of $\kappa$ for the doubly occupied orbital of the ground state which goes over from ionic on $A$ at infinite $R$ through mixed ionic-covalent to combined-atom at small $R$. The lower curve shows that of $\kappa$ for the empty orbital of the excited state which changes from atomic on $B$ at infinite $R$ to antibonding at intermediate $R$ and disappears for $R<R^*$; $\kappa_0=0$ is also a solution of Eq.(\ref{e:3.3}). The vertical dotted line indicates the separation ${R_{\rm IP}}$ at which $\kappa$ begins to differ significantly (by 10\%) from $\kappa_+(\infty)$.
}\label{f:kappa_vs_a}
\end{center}
\end{figure}

Finally, the constants of Eq.(\ref{e:3.4}) are given in terms of $Z_A$, $Z_B$, $R$, and $\kappa$, by:
\begin{equation}
\left.
\begin{array}{lll}
D&=&e^{\kappa R/2}\left(1-Z_A/\kappa\right) C\\
F&=& e^{\kappa R/2}\left(Z_A/\kappa\right)C\\
G&=& e^{\kappa R}\left[(\kappa-Z_A)/Z_B\right]C
\end{array}
\right\}~~,
\label{e:3.5}
\end{equation}
\begin{eqnarray}
C=&&(2\kappa)^{1/2}\left\{\frac{Z_A(\kappa-Z_A)}{Z_B(\kappa-Z_B)}\left(1+\frac{Z_B^2}{\kappa^2}\right)\right.\label{e:A2}\\
\nonumber&&\left.-\frac{e^{-2\kappa R}Z_A^2}{\kappa^2}+\frac{2Z_A}{\kappa}\left[1+2R(\kappa-Z_A)\right]\right\}^{-1/2}~~.
\end{eqnarray}

\subsection{The polar angle ${\beta}(x)$}

Substituting Eqs.(\ref{e:3.14}) and (\ref{e:3.7}) into Eq.(\ref{e:3.10}) leads to
\begin{equation} (2-\nu)\psi_A^2(x)+\nu\psi_B^2(x)=2\psi_M^2(x) \label{e:3.15}~~.\end{equation}
Eq.(\ref{e:3.15}) can be rewritten as
\begin{equation}
\chi_A^2(x)+\chi_B^2(x)   = \psi_M^2(x)~~,\label{e:3.16}\end{equation}
where
\begin{equation}
\chi_A(x)=(1-\nu/2)^{1/2}\psi_A(x)~~,~~\chi_B(x)=(\nu/2)^{1/2}\psi_B(x)~~,~~\label{e:3.17}\end{equation}
which permits us to take over the analytic procedures of ref.\cite{CWB07}. We first rotate $\chi_A(x)$ and $\chi_B(x)$ by $\pi/4$ in the
function space in which they are defined, introducing
\begin{equation} \chi_{\pm}(x)=\frac{1}{\sqrt{2}}\left(\chi_A(x)\pm\chi_B(x)\right)~~\label{e:3.18}\end{equation}
and leaving ``lengths'' within that space invariant so that
\begin{equation}
\chi_+^2(x)+\chi_-^2(x)=\psi_M^2(x)~~.\label{e:3.19}\end{equation}
Finally, we introduce the polar angle ${\beta}={\beta}(x) $ in the function space,
\begin{equation} \chi_+(x)=\psi_M(x)\cos{\beta}(x)~~,~~\chi_-(x)=\psi_M(x)\sin{\beta}(x)~~,\label{e:3.20}\end{equation}
\begin{equation} \chi_{A,B}(x)=\frac{1}{\sqrt{2}}\psi_M(x)\left(\cos{\beta}(x)\pm\sin{\beta}(x)\right)~~.\label{e:3.21}\end{equation}
Because $\chi_{A,B}(x)$ are non-negative, $|{\beta}|$ cannot exceed $\pi/4$.

Inserting Eq.(\ref{e:3.21}) into Eq.(\ref{e:3.17}) and subtracting the squares of the two resulting equations leads to an expression for $\nu$, Eq.(\ref{e:3.22}):
\begin{equation} \nu = 1-\int dx \psi_M^2(x)\sin{2{\beta}(x)}~~,\label{e:App_nu}\end{equation}
after integrating over $x$. Determination of the polar angle ${\beta}(x)$ is thus sufficient for the determination of $\nu$ (charge transfer) and the electron population of the fragments (population analysis). For the symmetric 1D$H_2$ case, ${\beta}(x)$ is odd and $\psi_M(x)$ even so that the integral in Eq.(\ref{e:3.22}) vanishes, yielding $\nu=1$ and equally populated fragments. In the present case, since $\psi_M(x)$ is normalized to unity and $|\sin 2{\beta}(x)|\leq 1$ with the domain of positive ${\beta}(x)$ weighted more heavily than that of negative ${\beta}(x)$, the integral in Eq.(\ref{e:3.22}) lies in $(0,1)$, as must $\nu$ in accordance with Eq.(\ref{e:3.13}).

\subsection{The Euler equation for ${\beta}(x)$}

In partition theory (PT) \cite{CW07}, a Hamiltonian is assigned to each part for each integer number of electrons entering into its PPLB ensemble. Since our ``electrons'' do not interact, it is sufficient to assign a one-electron Hamiltonian to each part,
\begin{equation} H_{\alpha}=\frac{p^2}{2}+v_{\alpha}(x)~~,~~v_{A,B}(x)=-Z_{A,B}\delta(x\pm R/2)~~.\label{e:3.24}\end{equation}
The PPLB energy functional of the collection of parts is then
\begin{eqnarray}
\nonumber{\cal E} &=& (2-\nu)\left(\psi_A,H_A\psi_A\right)+\nu\left(\psi_B,H_B\psi_B\right)\\
&=&2\left[\left(\chi_A,H_A\chi_A\right)+\left(\chi_B,H_B\chi_B\right)\right]~~.\label{e:3.25}
\end{eqnarray}
Inserting the transformation (\ref{e:3.21}) and the definition (\ref{e:3.24}) of $H_{\alpha}$ into Eq.(\ref{e:3.25}) results in
\begin{eqnarray}
\nonumber
{\cal E}=\int dx \left\{\psi_M'^2(x)+\psi_M^2(x)\left[{\beta}'(x)^2+v_A(x)+v_B(x)+\right.\right.\\
\left.\left.+(v_A(x)-v_B(x))\sin{2{\beta}(x)}\right]\right\}~~~~~~~\label{e:3.26}
\end{eqnarray}
for the energy functional ${\cal E}$, now a functional only of ${\beta}(x)$. In Eq.(\ref{e:3.26}) and in the following, primes indicate derivatives with respect to $x$.

Varying ${\cal E}$ with respect to ${\beta}(x)$ yields
\begin{eqnarray}
\nonumber
\delta{\cal E}=2\int dx \psi_M^2(x)\left\{{\beta}'(x)\delta{\beta}'(x)+(v_A(x)-v_B(x))\right.\\
\times\left.\cos{2{\beta}(x)}\delta{\beta}(x)\right\}~~.\label{e:3.27}
\end{eqnarray}
The usual integration by parts leads to
\begin{eqnarray}
\nonumber
\delta{\cal E}=&&2\left\{\left.\psi_M^2(x){\beta}'(x)\delta{\beta}(x)\right|_{x=-\infty}^{x=\infty}+\right.\\
&&+\left.\int dx\left[-\frac{d}{dx}\left(\psi_M^2(x)\frac{d{\beta}(x)}{dx}\right)\right.\right.\\
\nonumber
&&+\left.\left.\psi_M^2(x)(v_A(x)-v_B(x))\cos{2{\beta}(x)}\right]\delta{\beta}(x)\right\}~~.\label{e:3.28}
\end{eqnarray}
The Euler equation
\begin{eqnarray}
\nonumber
-\frac{d}{dx}\left(\psi_M^2(x)\frac{d{\beta}(x)}{dx}\right)+\psi_M^2(x)(v_A(x)-v_B(x))\\
\times\cos{2{\beta}(x)}=0~~~~\label{e:3.29}
\end{eqnarray}
and the boundary conditions
\begin{equation} \left.\psi_M^2(x){\beta}'(x)\delta{\beta}(x)\right|_{-\infty}^{\infty}=0\label{e:3.30}\end{equation}
result from imposing stationarity on ${\cal E}$. At first glance it might seem that Eq.(\ref{e:3.30}) is satisfied automatically since $\psi_M^2(x)\downarrow 0$ as $|x|\uparrow\infty$. However, unless the boundary condition
\begin{equation} {\beta}'(x)=0~~,~~|x|=\infty \label{e:3.31}\end{equation} is imposed, ${\beta}'(x)$ diverges unacceptably as $\psi_M^{-2}(x)$ as $|x|\uparrow\infty$.

\subsection{Solving for ${\beta}(x)$}

Because the $v_{\alpha}(x)$ are $\delta$-function potentials, Eq.(\ref{e:3.29}) reduces to
\begin{equation} \frac{d}{dx}\left(\psi_M^2(x)\frac{d{\beta}(x)}{dx}\right)=0~~,\label{e:3.32}\end{equation}
with the additional boundary conditions
\begin{equation} \left.\begin{array}{l
l}{\beta}(-\frac{1}{2} R^+)={\beta}(-\frac{1}{2} R^-)\equiv{\beta}_A\\{\beta}'(-\frac{1}{2} R^-)-{\beta}'(-\frac{1}{2} R^+)=Z_A\cos{2{\beta}_A}\end{array}\right\}~x=-\frac{1}{2} R~,\label{e:3.33}
\end{equation}
\begin{equation} \left.\begin{array}{l
l}{\beta}(\frac{1}{2} R^-)={\beta}(\frac{1}{2} R^+)\equiv{\beta}_B\\{\beta}'(\frac{1}{2} R^-)-{\beta}'(\frac{1}{2} R^+)=-Z_B\cos{2{\beta}_B}\end{array}\right\}~x=\frac{1}{2} R~.\label{e:3.34}
\end{equation}
The general solution of Eq.(\ref{e:3.32}) is
\begin{eqnarray}
\frac{d{\beta}(x)}{dx}&=&\frac{{\alpha}_1}{\psi_M^2(x)}~~,\label{e:3.35}\\
{\beta}(x)&=&\int^x dx'\frac{{\alpha}_1}{\psi_M^2(x')}+{\alpha}_2~~,\label{e:3.36}
\end{eqnarray}
with the constants ${\alpha}_1$ and ${\alpha}_2$ taking on different values in the three domains $|x|>R/2$, $-\frac{1}{2} R<x<\frac{1}{2} R$. The condition (\ref{e:3.31}) implies that ${\alpha}_1$ and ${\beta}'(x)$ vanish for $|x|>R/2$ so that
\begin{equation}
{\beta}(x)={\beta}_A~~,~~x<-R/2~~{\rm and}~~{\beta}(x)={\beta}_B~~,~~x>R/2~~.\label{e:3.37}
\end{equation}
The conditions (\ref{e:3.33}) and (\ref{e:3.34}) imply that
\begin{eqnarray}
\nonumber
{\alpha}_1&=&-Z_A\psi_M^2(-R/2)\cos{2{\beta}_A}\\
&=&-Z_B\psi_M^2(R/2)\cos{2{\beta}_B}~~,~~|x|<R/2\label{e:3.38}
\end{eqnarray}
From Eq.(\ref{e:3.33}) ${\alpha}_2$ is ${\beta}_A$ when the lower limit in Eq.(\ref{e:3.36}) is set at $-R/2$, yielding a second
relation between ${\beta}_A$ and ${\beta}_B$,
\begin{equation}
{\beta}_B-{\beta}_A={\alpha}_1\int_{-R/2}^{R/2}\frac{dx}{\psi_M^2(x)}~~,\label{e:3.39}
\end{equation}
Inserting the explicit form (\ref{e:3.4}) for $\psi_M(x)$ into the integral in Eq.(\ref{e:3.39}) results in
\begin{equation}
\int_{-R/2}^{R/2}\frac{dx}{\psi_M^2(x)}=\frac{1}{2\kappa D}\left[\frac{e^{\kappa R/2}}{\psi_M(-R/2)}-\frac{e^{-\kappa R/2}}{\psi_M(R/2)}\right]\label{e:3.40}
\end{equation}
Taken together, Eqs. (\ref{e:3.38}-\ref{e:3.40}) fix the values of ${\beta}_{A,B}$, determining ${\beta}(x)$ for $|x|>R/2$, and Eq.(\ref{e:3.36}) then determines ${\beta}(x)$ for $|x|<R/2$.

\subsection{The partition potential}

Since the $\chi_{\alpha}(x)$ are proportional to the $\psi_{\alpha}(x)$, Eq.(\ref{e:3.17}), they satisfy the same Schr\"{o}dinger Eqs.(\ref{e:3.41}). Summing over ${\alpha}$ and dividing by $\chi_A(x) +\chi_B(x)$ yields
\begin{eqnarray}
\nonumber
v_P(x)&=&\mu_M-\frac{1}{\chi_A(x)+\chi_B(x)}\frac{p^2}{2}(\chi_A(x)+\chi_B(x))+\\
&~&~~~-\frac{v_A(x)\chi_A(x)+v_B(x)\chi_B(x)}{\chi_A(x)+\chi_B(x)}~~.
\label{e:3.42}
\end{eqnarray}
Expressing $\chi_A(x)$ and $\chi_B(x)$ in terms of $\chi_+(x)$ and $\chi_-(x)$ via  Eq.(\ref{e:3.18}), and using Eq.(\ref{e:3.20}), results in
\begin{eqnarray}
\nonumber
v_P(x)&=&\mu_M+\frac{1}{2\psi_M(x)\cos{\beta}(x)}\frac{d^2}{dx^2}(\psi_M(x)\cos{\beta}(x))+\\
&~&~~~-\frac{1}{2}\sum_{\alpha} v_{\alpha}(x)(1+s_{\alpha}\tan{\beta}_{\alpha})~~,
\label{e:3.43}
\end{eqnarray}
where $s_{\alpha}=1$ for ${\alpha}=A$ and $s_{\alpha}=-1$ for ${\alpha}=B$. The $\delta$-function character of $v_{\alpha}(x)$ and the definitions of ${\beta}_A$ and ${\beta}_B$ of Eqs.(\ref{e:3.32})-(\ref{e:3.33}) were also taken into account in arriving at Eq.(\ref{e:3.43}).
Since the molecular wavefunction $\psi_M(x)$ satisfies
\begin{equation}
-\frac{1}{2}\frac{d^2\psi_M(x)}{dx^2}+(v_A(x)+v_B(x))\psi_M(x)=\mu_M\psi_M(x)~~,
\label{e:3.44}
\end{equation}
Eq.(\ref{e:3.43}) can be transformed to
\begin{eqnarray}
\nonumber v_P(x)&=&-\frac{1}{2}\left\{\tan{\beta}(x)\left[\frac{2}{\psi_M(x)}\frac{d\psi_M(x)}{dx}\frac{d{\beta}(x)}{dx}+\right.\right.\\
&~&~~~\left.\left.+\frac{d^2{\beta}(x)}{dx^2}\right]+\left(\frac{d{\beta}(x)}{dx}\right)^2\right\}\\
\nonumber
&&~~~+\frac{1}{2}\sum_{\alpha} v_{\alpha}(x)(1-s_{\alpha}\tan{\beta}_{\alpha})~~.
\end{eqnarray}
Using the Euler Eq.(\ref{e:3.29}) for ${\beta}(x)$, $v_P(x)$ can be further expressed as
\begin{eqnarray}
\nonumber
v_P(x)=-\frac{1}{2}\left(\frac{d{\beta}(x)}{dx}\right)^2+\frac{1}{2}\sum_{\alpha} v_{\alpha}(x)\\
~~~~~\times(1-s_{\alpha}\tan{\beta}_{\alpha}(1+\cos{2{\beta}_{\alpha}}))~.
\end{eqnarray}
Finally, by using Eqs.(\ref{e:3.36}) and (\ref{e:3.38}), we note that in the internuclear region $|x|<R/2$, $d{\beta}(x)/dx$ is simply proportional to $\psi_M^{-2}(x)$, yielding
\begin{eqnarray}
 v_P(x)&=&-\frac{1}{2}\frac{Z_A^2\psi_M^4(-R/2)\cos^2{2{\beta}_A}}{\psi_M^4(x)}\theta(R/2-|x|)~~\label{e:3.46}\\
\nonumber &&~~~+\frac{1}{2}\sum_{\alpha} v_{\alpha}\left(1-s_{\alpha}\tan{\beta}_{\alpha}(1+\cos{2{\beta}_{\alpha}})\right)~~,
\end{eqnarray}
where $\theta(y)\left(=0\right.$ for $y<0$, $1$ for $\left.y>0\right)$ is the Heaviside step function. Eq.(\ref{e:3.46}) correctly reduces to the partition potential of 1D$H_2$ \cite{CWB07} when $Z_A=Z_B$.

\subsection{Proof that ${R_{\rm occ}}$ is proportional to $\ln\Delta Z$ for small $\Delta Z$}

We expect that ${R_{\rm occ}}$, the nuclear separation at the crossover from covalent to ionic behavior, goes to infinity as $\Delta Z \downarrow 0$. There, the fragment wavefunctions must approach the free-atom wavefunctions, and the tight-binding LCAO must be a good approximation to the molecular orbital. We can thus write
\begin{equation}
\psi_M(x)=A\psi_A^0(x)+B\psi_B^0(x)~~
\end{equation}
where the $\psi_{\alpha}^0(x)$ are the orbitals of the isolated atoms, Eq.(\ref{e:3.1}). Taking matrix elements of the molecular Hamiltonian yields equations for $A$, $B$, and the molecular energy $E_M$:
\begin{subequations}
\begin{eqnarray}
\label{e:A.1a}\\
\nonumber
\left[E_A^0-E_M+v_B^{AA}\right]A+\left[(E_B^0-E_M)S_{AB}+v_A^{AB}\right]B=0\\
\nonumber
\left[(E_A^0-E_M)S_{AB}+v_B^{AA}\right]A+\left[E_B^0-E_M+v_A^{BB}\right]B=0 \\
\label{e:A.1b}
\end{eqnarray}
\end{subequations}
where the $E_\alpha^0$ are the energies of Eq.(\ref{e:3.2}), $v_{A/B}^{\alpha\beta}$ are matrix elements of the $A/B$ potentials of Eq.(\ref{e:3.24}), and $S_{AB}$ is the overlap $\left(\psi_A^0,\psi_B^0\right)$.
In evaluating all quantities except $E_A-E_M$ and $E_B-E_M$, we can take the limit $\Delta Z\downarrow 0$. The result is
\begin{eqnarray}
\nonumber
S_{AB}=S_{BA}&\longrightarrow& S\sim ZRe^{-ZR}\\
\nonumber
v_B^{AA}\to v_A^{BB}&\longrightarrow& v_d\sim -Z^2e^{-2ZR}\\
\nonumber
v_A^{AB}\to v_B^{BA}&\longrightarrow& v_0\sim -Z^2e^{-ZR}
\end{eqnarray}
The complete solution of Eqs.(\ref{e:A.1a})-(\ref{e:A.1b}) shows that $S$ enters into $E_M$, $A$, and $B$ as $1-S^2$ and $Sv_d$ which become exponentially small corrections as $\Delta Z\downarrow 0$, $R\uparrow \infty$. Similarly $v_d$ enters only in the combination $Sv_d$ and can be neglected as well. Thus the equations for $A$ and $B$ simplify to the classic bonding-antibonding equations
\begin{subequations}
 \begin{eqnarray}
  (E_A-E_M)A+v_0B&=&0\label{e:A.2a}\\
v_0A+(E_B-E_M)B&=&0\label{A:2b}
 \end{eqnarray}
\end{subequations}
The bonding eigenvalue is
\begin{equation}
E_M=\frac{1}{2}(E_A+E_B)-\left\{\left[\frac{1}{2}(E_A-E_B)\right]^2+v_0^2\right\}^{1/2}
\label{e:A.3}
\end{equation}
and
\begin{equation}
\frac{B}{A}=\frac{E_M-E_A}{V_0}
\label{e:A.4}
\end{equation}
Inserting Eq.(\ref{e:A.3}) into Eq.(\ref{e:A.4}) and rearranging gives
\begin{equation}
\frac{B}{A}=\sqrt{1+\left(\frac{E_B-E_A}{2v_0}\right)^2}-\left|\frac{E_B-E_A}{2v_0}\right|~~.
\end{equation}
Now, since $E_B-E_A=Z\Delta Z$ and $v_0=-Z^2e^{-ZR}$ then
\begin{equation}
\frac{B}{A}=\sqrt{1+\left(\frac{\Delta Z e^{ZR}}{2Z}\right)^2}-\frac{\Delta Z e^{ZR}}{2Z}\leq 1~~.
\end{equation}
Thus
\begin{eqnarray}
\nonumber
\frac{B}{A}&=&f(y)\leq 1~~~,~~~y\equiv \frac{\Delta Z}{2Z} e^{ZR}\\
\nonumber
A&=&(1+f(y)^2)^{-1/2}
\end{eqnarray}
The molecular density is
\begin{eqnarray}
\nonumber
n_M(x)&=&N_M\psi_M(x)^2\\
\nonumber
&=&N_M \left[A^2\psi_A^2(x)^2+B^2\psi_B^2(x)+2AB\psi_A(x)\psi_B(x)\right]\label{e:A.5}\\
&=&\frac{N_M Z^2}{1+f(y)^2}(g_1(x)+g_2(x))~~,
\end{eqnarray}
where
\begin{eqnarray}
\nonumber
 g_1(x)&=&e^{-2Z|x|}+f^2(y)e^{-2Z|x-R|}\\
\nonumber
g_2(x)&=&2f(y)e^{-Z(|x|+|x-R|)}
\end{eqnarray}
It can be checked that $g_1(x)$ always exceeds $g_2(x)$ and becomes exponentially larger than $g_2(x)$ as $x$ departs from $\frac{1}{2} R-\frac{1}{2}(\ln f(y))/Z$. The cross term in Eq.(\ref{e:A.5}) can then be neglected and $n_M(x)$ is thus of the form obtained from Partition Theory: $n_M(x)=N_M A^2\psi_A(x)^2+N_M B^2\psi_B^2(x)$, so that $2A^2$ can be identified with $2-\nu$ and $2B^2$ can be identified with $\nu$, implying that
\begin{equation}
f^2(y)=\frac{\nu}{N_M-\nu}~~.
\end{equation}
We have chosen $\nu=N_M/3$ to define ${R'_{\rm occ}}$. That implies that $f(y)=1/\sqrt{2}$ at ${R_{\rm occ}}'$ for $N_M=1$ (or $f(y)=1/\sqrt{5}$ for $N_M = 2$). This leads to the observed behavior of ${R'_{\rm occ}}$ in Fig.(\ref{f:a_nu1}):
\begin{equation}
{R'_{\rm occ}}=\frac{1}{Z}\ln Z -\frac{1}{Z}\ln(\sqrt{2}\Delta Z)
\label{e:tight_binding}
\end{equation}


\section{Consistency check}

The calculation approach of \cite{Fosco Lombardo Mazzitelli 08} is
based on expressing the Casimir energy as a trace of a logarithm
of an integral operator, and this step is widely used in Casimir
calculations. The trace of the operator can be calculated directly
as a sum of its eigenvalues. For their determination authors of
\cite{Fosco Lombardo Mazzitelli 08} propose particular
Sturm-Liouville problem (on a finite interval) which is obtained
using a specific coordinate transformation, see Section II.A
\cite{Fosco Lombardo Mazzitelli 08}.

As a consistency check of this method we use  it to calculate the
trace of the integral operator $D(z,z')$, defined in (32) of
\cite{Fosco Lombardo Mazzitelli 08}, and compare it with result of a
straightforward alternative calculation. Expressing the trace of
$D(z,z')$ as a sum of its eigenvalues $\mu_l$ and
using (33) \cite{Fosco Lombardo Mazzitelli 08} we obtain
\begin{equation}
 \textrm{Tr} D(z,z')
    = \sum_l \mu_l=\sum_l \frac{\alpha_l-1}
        {\tilde\lambda(\omega,\textbf{k}_\parallel)}.
    \label{Tr_mu}
\end{equation}
For the limit of piecewise constant profile being approached,
$\alpha_l$ is given in (66) \cite{Fosco Lombardo Mazzitelli 08}
\begin{equation}
    \alpha_l=\alpha_l(\omega,k_\parallel, \epsilon)
         = 1+\frac{2\epsilon\tilde{\lambda}(\omega,\textbf{k}_\parallel)}
            {l^2\pi^2+(2\epsilon\kappa)^2}.
\label{alph}
\end{equation}
Then one easily finds
\begin{equation}
 \textrm{Tr} D(z,z')
    = \sum_{l=1}^\infty \frac{2\epsilon}{l^2\pi^2+(2\epsilon\kappa)^2}
    =\frac{2\epsilon\kappa\coth(2\epsilon\kappa)-1}{4\epsilon \kappa^2}
    \label{Tr a}.
\end{equation}

On the other hand, it is straightforward to calculate the trace
of $D$ without appealing to any eigenvalue problem
\begin{eqnarray}
 \textrm{Tr} D(z,z')
    &=& \int_{-1/2}^{1/2} dz D(z,z)=\nonumber\\
    &=&\int_{-\infty}^{\infty} dx D(x,x)\sigma_\epsilon(x)
        =\frac{1}{2\kappa}.
    \label{Tr b}
\end{eqnarray}
Evidently, (\ref{Tr b}) contradicts  to (\ref{Tr a}).
At the same time it is straightforward to prove the equivalence of
the trace definitions used in (\ref{Tr a}) and (\ref{Tr b}) taking
into account that  $\psi_\alpha$  (33) \cite{Fosco Lombardo
Mazzitelli 08} constitute a complete set of eigenfunctions of
Sturm-Liouville problem (39) \cite{Fosco Lombardo Mazzitelli 08}.
Moreover, it must be emphasized here that (\ref{Tr b}) is valid
independently of any particular profile of $\sigma_\epsilon(x)$
provided it satisfies the normalization condition (7)[1]. It is
due to the fact that $D(z,z)$ (or equivalently $D(x,x)$) is
position-independent.

We see that the spectral problem used for calculation of the
eigenvalues of $D$ in the case of (approximated) piecewise
constant profile is either ill posed, or not equivalent to the
original problem. It gives a wrong answer (\ref{Tr a}) for the
trace of $D$. In virtue of (31) \cite{Fosco Lombardo Mazzitelli
08} the same must also hold for the trace of the operator $\ln
\tilde{{\cal K}}$ which defines the Casimir energy (30)
\cite{Fosco Lombardo Mazzitelli 08}. Hence, it is not legitimate
to use the eigenvalues $a_l$ (\ref{alph}) for the (correct) calculation of
the Casimir energy (43) \cite{Fosco Lombardo Mazzitelli 08}.

Thus, we must conclude that the calculation method proposed in
\cite{Fosco Lombardo Mazzitelli 08} contains an internal
inconsistency when applied to (approximations of) piecewise
constant profile of the mirror. One can also check that an
expansion of (67) \cite{Fosco Lombardo Mazzitelli 08} in a power
series in constant $\lambda$ (acknowledged in Section III.C
\cite{Fosco Lombardo Mazzitelli 08}) does not coincide with the
usual perturbation theory.

These arguments unambiguously show that
the result for the Casimir
energy of a single slab (68) \cite{Fosco Lombardo Mazzitelli 08}
is presumably incorrect.

\section{Alternative approach}
In \cite{Pismak Fialkovsky 08} we presented a detailed treatment
of the similar system of piecewise constant profile without
reference to more general cases. We restricted ourselves to
consideration of homogenous and isotropic infinite plane slab of
thickness $\epsilon$, placed in the $x_1x_2$ plane.

Casimir interaction of (multi) layered systems has been actively
studied for dielectric materials (e.g., within surface modes
formalism \cite{modes}, or macroscopical field operators method
\cite{fields}). However, the mathematical formulation of these
problems differ from one considered here, and no direct comparison
of the results is possible.

For modeling of the interaction of massive scalar field with
volume defects we followed the Symanzik approach
\cite{Symanzik'81}. The complete action of the model is following
\begin{eqnarray*}
   &S=&S_0+S_I+J \phi\\
  &&S_0=\frac12\int d^4x\, (\partial_\mu\phi(x)\partial_\mu\phi(x)+m^2 \phi^2(x)),\\
  &&S_I= \frac{\lambda}2\int d^4x\, \theta(\epsilon, x_3)\phi^2(x).
\end{eqnarray*}
The distribution function $\theta(\epsilon, x_3)$ represents the
piecewise constant profile being equal to $1/\epsilon$ when
$|x_3|<\epsilon/2$, and zero otherwise. Then there is no need for
introduction of implicit variables' change defined by (21)
\cite{Fosco Lombardo Mazzitelli 08}, and we can proceed explicitly
in Cartesian coordinates.

We consider the generating functional of Green's functions and
similarly to \cite{Bordag'98} introduce auxiliary fields defined
on the support of the defect (this line is also followed in
\cite{Fosco Lombardo Mazzitelli 08} with minor generalizations).
These fields satisfy free boundary conditions on the edges of the
layer (in other words there are no constrains imposed
on the fields). Following this approach one introduces integral operators
acting on the auxiliary fields with support in the finite
interval, such as $\lambda^{-1}$ in (12) of \cite{Fosco Lombardo
Mazzitelli 08}. In our case it reduces to a unity operator and
there is no need for any special treatment of its boundary
conditions.

Functional integral takes then explicitly gaussian form and we are
able to derive both the Casimir energy and the propagator,
arriving at
\begin{equation}
\begin{array}{c}
  \mathcal{Z}[J]=({\rm Det} Q)^{-1/2}\,
    \exp{\left\{\frac12J\hat S J\right\}}, \\
  \hat S=D-\eta (\Delta{\cal O}) Q^{-1}({\cal O} \Delta),\ \
    Q=\textbf{1}+\eta({\cal O} \Delta {\cal O}).   \\
\end{array}
\label{Z(j)}
\end{equation}
here $\eta={\lambda}{\epsilon}^{-1}$. The definitions of $\hat S$ and $Q$
must be understood in terms of integral operators.
With  $\Delta=(-\partial^2+m^2)^{-1}$ we denote the standard free
propagator of scalar field, and the projecting operator ${\cal O}$ acts as
$$
    \psi{\cal O}\phi\equiv
        \int d\vect x\int_{-\epsilon/2}^{\epsilon/2} dx_3\psi\phi,
$$
The unity operator $\textbf{1}$ is also defined on the defect only.
We shall note here that
our operator $Q$ is the direct analog of the ${\cal K}$
(24) \cite{Fosco Lombardo Mazzitelli 08}. However, the trace calculation which
defines the Casimir energy and is presented below, differs significantly
in our approach.

The Casimir energy density per unit area of the layer is given
by
\begin{equation}
{\cal E}
    =\int\frac{d^{3}\vect p}{2(2\pi)^{3}}
        \mbox{Tr}\ln[Q(\vect p; x_3,y_3)],
    \label{E_TrLn2}
\end{equation}
where the Fourier transformation of the  coordinates
parallel to the defect (i.e.  $\vect x=(x_0, x_1, x_2)$) was performed,
$\vect p=(p_0,p_1,p_2)$.

Introducing operator $U\equiv Q^{-1}-1$ and using the definition
of $Q$ we can express the $\eta$-derivative
of the integrand of (\ref{E_TrLn2}) in the following form
$$
    \partial_\eta \ln Q=-\frac{U}{\eta}.
$$

For explicit calculation of $U$ we note
that it is proportional to a Green function
of an ordinary differential operator:
\begin{eqnarray}
  &&K_\rho U= -\eta   \label{s1} \\
  && K_{V}(x,y)\equiv
        \left(-\frac{\partial^2}{\partial x^2}+V^2\right)\delta(x-y).\nonumber
\end{eqnarray}
where $\rho=\sqrt{\eta+E^2}$, $E=\sqrt{\vect p^2+m^2}$.

Employing the symmetry conditions $U(x,y)=U(y,x)=U(-x,-y)$ which follow
from the definition of $U$ we can write
\begin{equation}
U=-\eta \frac{e^{- \rho |x-y|}}{2\rho}
    +2a\cosh\((x+y) \rho\)
    +2b\cosh\((x-y) \rho\).
\end{equation}
From (\ref{Z(j)}) and definition of $U$ it follows
that
$$
U+\eta {\cal O}\Delta {\cal O}(1+U)=0.
$$
Then one can derive the coefficients $a$, $b$ as
\begin{eqnarray}
  a&=&-\frac{\xi \eta^2 e^{\epsilon {\rho}}} {2 {\rho}},\qquad
    b=-\frac{\xi \eta (E-{\rho})^2}{2  \rho},
    \label{abc}\\
  &&\xi =\(e^{2\epsilon \rho}(E+ \rho)^2-(E- \rho)^2\)^{-1}.\nonumber
\end{eqnarray}

For the energy density we have
\begin{equation}
\label{res1}
{\cal E}
    =-\mu^{3-d}
        \int\frac{d^{d}\vect p}{2(2\pi)^{d}}\int_0^\eta \frac{d\eta}{\eta}\,
        \mbox{Tr} U.
\end{equation}
where we introduced dimensional regularization to free oneself
from ultraviolet divergencies ($d=3$ corresponds to removing of
regularization), and  an auxiliary mass parameter $\mu$. We have
chosen the lower limit of integration over $\eta$ to satisfy the
energy normalization condition ${\cal E}|_{\eta=0}=0$. It can be
shown that the integral is convergent at $\eta=0$.

The calculation of trace of the operator $U$ is straightforward
\begin{equation}
\mbox{Tr} U\equiv \int_{-\epsilon/2}^{\epsilon/2} dx U(x,x)
    =2 b\epsilon+\frac{4 a \sinh(\epsilon \rho)-\epsilon\eta}{2
    \rho}.
    \label{Tr U}
\end{equation}

Next, putting (\ref{abc}) into (\ref{Tr U}) we can prove directly that
\begin{equation}
\mbox{Tr} U
    =-\eta\frac{\partial}{\partial\eta}
        \ln\left[ \frac{e^{-\epsilon(E+\rho)}}{4E\rho\xi}\right].
    \label{Tr U1}
\end{equation}
Thus, from (\ref{res1}) and  (\ref{Tr U1}) we obtain
the following expression for the Casimir energy
\begin{widetext}
\begin{equation}
{\cal E}=\mu^{3-d}\int \frac{d^{d}\vect p}{2 (2\pi)^{d-1}}
    \ln\[
        \frac{e^{- \epsilon E}}{4 E \rho}
            \(e^{ \epsilon \rho }(E+ \rho)^2-e^{- \epsilon \rho }(E- \rho)^2\)
    \],\quad \rho=\sqrt{E^2+\lambda\epsilon^{-1}}
    \label{E}
\end{equation}
\end{widetext}
It can be shown that (\ref{E})  is in full agreement
with the usual perturbation theory.

It is easily to generalize (\ref{E}) for non-local translation
invariant $\lambda\equiv\lambda(\vect x-\vect y)$ as considered in
\cite{Fosco Lombardo Mazzitelli 08} (see equation (4) there). In this
case the final expression for ${\cal E}$ (\ref{E}) remains the
same provided that $\lambda$ is replaced there with $\tilde\lambda
(\vect p)$ -- Fourier image of $\lambda(\vect x)$ defined in (26)
\cite{Fosco Lombardo Mazzitelli 08}. In the massless model (with
$m=0$, $E=|\vect p|$) the energy ${\cal E}$ obtained in such a way can
be compared with $\mu^{3-d}{\cal E}_0$ for ${\cal E}_0$ presented
in (67) \cite{Fosco Lombardo Mazzitelli 08} for
plane slab. One can easily check that these results
do not coincide.

To compare (\ref{E}) with results of \cite{Bordag'95} and
\cite{Vass} one needs to replace the zeta-function regularization
used there with dimensional one applied in our approach. Using
(15) of \cite{Bordag'95} one can obtain a generalization of (8)
\cite{Bordag'95} (written for $d=3$) to $d$-dimensional case
in the form
\begin{equation}
V_{\textit{eff}}
    =
        \frac{\Omega_d \mu^{3-d}}{2(2\pi)^{d} }
        \int_m^\infty dk (k^2-m^2)^{d/2}
            \frac{\partial}{\partial k}\ln s_{11}(ik)
        \label{Veff}
\end{equation}
where $\Omega_d=2\pi^{d/2}/\Gamma(d/2)$ is the volume of the
$(d-1)$-dimensional sphere in the $d$-dimensional space, and
$s_{11}(ik)$ for considered case of plane slab is defined by (22)
\cite{Bordag'95}. Integrating by part in (\ref{Veff}) and changing
integration variable $k=\sqrt{|\vect p|^2+m^2}$, one can check that the
right hand sides of (\ref{E}) and (\ref{Veff}) coincide (up to
change of notation $L=\epsilon$, $V_0=\eta$). In a similar way
one can verify that result obtained in \cite{Vass} for the plane
slab for $d=2$ with help of zeta-function regularization agrees
with (\ref{E}).

\vskip 3mm
We hope that the arguments given above are sufficient to conclude
that the calculation methods proposed in \cite{Fosco Lombardo
Mazzitelli 08} need to be thoroughly verified in order to expose
and eliminate theirs defects. Then it may be possible to employ effectively
the basic ideas of \cite{Fosco Lombardo Mazzitelli 08}
in calculation of Casimir energy in the models of
quantum field theory with nontrivial background.

\begin{acknowledgments}
V.M. and Yu.P. are grateful to the Russian Foundation of Basic Research  for
financial support in framework of the grant RFRB $07$--$01$--$00692$. The work
of I.F. was partly supported by FAPESP.

\end{acknowledgments}


\section{Introduction}
In all standard models for ultra-high energy cosmic ray (UHECR) propagation,
their range is ultimately limited by the opacity of the
cosmic microwave background radiation.
The UHECR energy above which this becomes significant is
about $6 \times 10^{19}$~eV in the current epoch. This cuts off
their travel beyond distances of order 50~Mpc as first noted by
Greisen~\cite{Greisen}, and Zatseptin and Kuzmin~\cite{ZK} (GZK). As a result
of this absorption, the UHECR energy above this GZK cutoff
is ultimately converted to photons, neutrinos, and lower energy hadrons.
The resulting neutrinos were first described by
Berezinsky and Zatsepin~(BZ)~\cite{BZ}. In standard UHECR source models
the BZ neutrino fluxes peak at energies about 2 orders of magnitude below the
GZK energy. Thus a ``guaranteed'' flux of neutrinos at energies of
$E_{\nu} = 10^{17-20}$~eV exists. Its detection is one of the clearest ways to
reveal the nature and
cosmic distribution of the UHECR sources~\cite{Seckel05}, which is
one of the longest-standing problems in high energy astrophysics.

The ANITA-1 Long Duration Balloon experiment was designed specifically to search for this
cosmogenic BZ neutrino flux. ANITA-1 exploits the Askaryan effect, in which
strong coherent radio emission arises from electromagnetic showers in
any dielectric medium~\cite{Ask62}.
The effect was first observed in 2000~\cite{Sal01}, and
has now been clearly
confirmed and characterized for ice as the medium, as part of the pre-flight calibration
of the ANITA-1 payload~\cite{slac07}. A prior flight of a prototype payload
called ANITA-lite in 2003-2004 led to validation of the technique and initial
neutrino flux limits that ruled out several UHE neutrino models~\cite{anitalite}.

In a previous paper~\cite{ANITA-1}, we describe
in detail the ANITA-1 instrument, payload, and flight system. Reference ~\cite{ANITA-1}
also includes
details of the instrument performance during the flight, estimates of the overall
sensitivity of the instrument to neutrino fluxes, and discussions of possible
backgrounds. Because of the complexity of the flight system and methodology,
we refer the reader to ref.~\cite{ANITA-1} for more detail when necessary.

The ANITA-1 payload (Fig.~\ref{payload})
launched from Williams Field, Antarctica
 near McMurdo station, on December 15, 2006, and executed
more than three circuits of the continent. The payload landed on the Antarctic
plateau about 300~miles from Amundsen-Scott (South Pole) Station, after 35 days aloft. Anomalous
stratospheric conditions led to a misalignment of the polar
vortex for the 2006-2007 season, and as a result the ANITA-1 trajectory spent
an unusually large fraction of the time over West Antarctica where the ice sheet
is smaller and shallower. In addition, the payload field-of-view to the horizon
(at a distance of about 650~km at typical altitudes of 35-37~km above mean sea level,
or 33-35~km above the ice surface),
often included the two largest occupied stations in Antarctica, McMurdo and
Amundsen-Scott, and thus was subject to higher-than-expected levels of anthropogenic
electromagnetic interference (EMI).
Despite these effects, the payload
accumulated a net exposure livetime of 17.3 days with a mean ice depth in the
field of view of 1.2~km, comparable to the attenuation length of the ice at sub-GHz radio
frequencies~\cite{icepaper}. ANITA-1 was
thus able to synoptically view a volume of ice of $\sim 1.6$~M~km$^3$.
Our volumetric acceptance to a diffuse
neutrino flux, accounting for the small solid angle of acceptance for any given volume element, is
several hundred km$^3$ water-equivalent steradians at $E_{\nu} \simeq 10^{19}$~eV~\cite{ANITA-1}.

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=3.35in]{Payload06_228.eps}
\caption{View of the ANITA-1 payload in launch configuration, with
photovoltaics at the top and bottom, and antenna clusters between. The
side of each square antenna mouth is about 0.9~m, and the payload stands
about 8~m tall.}
\label{payload}
\end{center}
\end{figure}

ANITA-1's antennas are 32 dual-linear-polarization,
quad-ridged horns, each
with a field of view which averages about $50^{\circ}$ angular diameter over their 200-1200~MHz
working bandwidth. The antennas are arranged in upper and lower rings,
each with 16 antennas at azimuthal intervals of $22.5^{\circ}$. All antennas
point at $10^{\circ}$ below the horizontal, to maximize sensitivity to
the largest portion of the volume near the horizon at $6^{\circ}$ below the
horizontal. The combined view of all antennas covers the entire lower hemisphere
down to nadir angles of about $55^{\circ}$, comprising 99.4\% of the area
within the horizon.  Radio impulses that exceed the ambient thermal noise by about
$5\sigma$ in at least
four antennas in coincident upper- and lower-ring pairs produce a trigger~\cite{ANITA-1},
and the entire antenna set of waveforms are then digitized and stored for
later analysis. Thermal noise fluctuations produce
random triggers at a rate of about 4-5~Hz, yielding a continuous monitor of
instrument health. These events are incoherent in phase and produce
a completely negligible background to actual coherent radio impulses.

The event analysis is conceptually simple, but requires detailed calibration
of the instrument to achieve good precision. In the results reported here,
we accepted only events having at least six adjacent
antennas with detectable signals. The six antenna
signals are analyzed using a method of pulse-phase interferometry to
determine the best arrival direction of the radio impulse plane wave,
and this direction and its associated uncertainty is then mapped onto
the Antarctic ice surface by reference to
onboard payload navigation instruments, with an angular precision
of $0.2^{\circ} \times 0.8^{\circ}$ in elevation and azimuth~\cite{ANITA-1}.

\begin{table}[htb!]
\caption{\label{event-table}Event totals vs. analysis cuts and estimated signal
efficiencies for unblinded ANITA-1 data set.
}
\begin{center}
\begin{tabular}{lrrrc}
\hline
Cut requirement~~~~~~~~~~passed: & total & Hpol & Vpol & Efficiency\\ \hline
(0) Hardware-Triggered& $\sim 8.2$M & ... & ... & ... \\
(1) Upcoming plane wave & 32308 & 15997 & 16311 & 0.93 \\
(2) Impulsive broadband~~ & 19695 & 10095 & 9600 & 0.98\\
(3) Isolated from other events & 9 & 8 & 1 & 0.94 \\
(4) Isolated from camps & 6 & 6 & 0 & 0.96 \\
(5) Vpol dominant & 0 &  0 & 0  & 0.99\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{figure*}[ht!]
\begin{center}
\centerline{\includegraphics[width=3.3in]{Recons.eps}\includegraphics[width=3.3in]{Hpol_Final.eps}}
\caption{Left: Plot of all reconstructed events, in both horizontal and vertical polarization;
major Antarctic stations are indicated on the map.
Right: events remaining after cuts to remove anthropogenic interference.
6 events remain in the horizontally
polarized group, but these are non-candidates for neutrino events, as discussed in the text.}
\label{recons}
\end{center}
\end{figure*}

To minimize bias, the analysis cuts were
optimized on a 10\% randomized sample of the entire data set, and the remaining
90\% was blinded from the analysts until the cuts were fixed.
The cuts proceed as follows:
{\bf (1)} Events that do not reconstruct to a coherent plane wave in arrival direction
are rejected as random thermal noise;
events that reconstruct from directions above the horizon are also rejected.
{\bf (2)} Events that reconstruct but have non-impulsive waveforms from relatively narrow-band
sources ($\leq 100$~MHz) are rejected.
{\bf (3)} Events that cluster with one another in source location
to within reconstruction errors projected onto
the ice, or 50 km radius, whichever is greater, are rejected.
True source candidates must be single, isolated events. Note
that this cut, and the ``camp cut'' that follows it, are largely but not completely redundant.
{\bf (4)} Events that coincide in source location with any known active or inactive
station, camp, aircraft flight path, or expedition traverse path, to within reconstruction
angular errors projected onto
the ice, or 50 km radius, whichever is greater, are rejected as being associated with
anthropogenic activity. Even inactive camps or those long-abandoned are considered
a risk, since left-over equipment might serve as a site for charge accumulation and
associated electromagnetic discharges which could be mistaken for signals.
{\bf (5)} Events whose radio waveforms are not predominantly vertically polarized (Vpol)
are rejected because,
from considerations of the Askaryan impulse generation process, and the Fresnel transmission
through the ice surface, they cannot originate from a
particle shower within the ice sheet. Conversely, strongly horizontally
polarized (Hpol) events are likely to originate from above the ice from similar considerations.

Table~\ref{event-table} shows the results of the total event sample after unblinding,
including the signal efficiency for each cut separately. The
10\% initial sample is included in the totals. Note that the isolation cut (3)
is the single most stringent criterion in rejecting impulsive events, and this shows that the
vast majority of triggers are not single, isolated events.  Signal efficiency in each
case was tested with a simulated event sample injected randomly into the data
stream, and the final energy-averaged efficiency of all cuts is estimated to be 81\%.

In Figure~\ref{recons} we show the before-and-after maps
of reconstructed ANITA-1 events superposed on the
Antarctic continent. The strong correlation to a small number of stations is evident.
The 6 surviving Hpol
events are by contrast widely distributed across the continent,
with no known camps or bases, either current or former, anywhere in their locale.
We have investigated the possibility of impulsive signals from earth-orbiting satellites
seen in reflection off the ice surface as a source for these events.
This hypothesis is ruled out because the waveforms for these events do not show any evidence of
differential group delay from ionospheric dispersion
which is several ns per MHz in the 200-400~MHz frequency range where
these events have most of their
spectral power. In fact the signals are
all of durations less than 10~ns.  We know of no other anthropogenic sources
for these events.

With regard to possible physics sources, our simulations of the high-frequency tail
of impulsive geo-synchrotron radio emission~\cite{Huege_Falcke05,Falcke,Suprun}
from  ultra-high energy cosmic ray extensive air showers
(EAS) suggest that these signals may be EAS events
seen in reflection off the ice surface~\cite{ANITA-1}.
Such events are expected to be predominantly
Hpol because of the strong Fresnel reflectivity in the
region near Brewster's angle, and the overall
initial preference for Hpol because of the more vertical polar magnetic fields.
Our simulations
predict a handful of such events for the flight, all of which arise from UHECR EAS with energies
above $10^{19}$~eV; however, the uncertainties are large~\cite{ANITA-1}.
While these events do not constitute a background for our neutrino search because
of their incorrect polarization, they are a potentially interesting signal in their own right.
Further analysis, including a search for similar events from above the horizon,
is in progress.

\begin{figure}[ht!]
\begin{center}
\includegraphics[width=3.2in]{Lim08_rev4.eps} \\
\caption{ANITA-1 limits based on no surviving candidates for 18 days of livetime. Other
limits are from AMANDA~\cite{AMANDA08}, RICE~\cite{RICE06}, ANITA-lite~\cite{anitalite},
Auger~\cite{Auger07}, HiRes~\cite{Hires08}, FORTE~\cite{FORTE04}. The BZ (GZK) neutrino
model range is determined by a variety of
models~\cite{PJ96,Engel01,Kal02,Kal02a,Aramo05,Ave05,Barger06}.}
\label{lim08}
\end{center}
\end{figure}

\begin{table}[hbt!]
\caption{Expected numbers of events $N_{\nu}$ from several UHE neutrino models, and
confidence level $CL=100(1-exp(-N_{\nu}))$ for exclusion by ANITA-1 observations.
\label{table2}}
\vspace{3mm}
 \begin{footnotesize}
  \begin{tabular}{lcr}
\hline \hline
{ {\bf Model \& references}}   &  predicted {$N_{\nu}$}~~      &  ~~{\bf CL,\%} \\ \hline
{\it Baseline BZ models} &  &  \\
~~~~~~Protheroe \& Johnson 1996~\cite{PJ96} & 0.22  & 19.7 \\
~~~~~~Engel, Seckel, Stanev 2001~\cite{Engel01} &  0.12 & 11.3 \\
~~~~~~Barger, Huber, \& Marfatia 2006~\cite{Barger06} & 0.38 & 31.6\\
{\it Strong source evolution BZ models}&  &  \\
~~~~~~Engel, Seckel, Stanev 2001~\cite{Engel01} & 0.39 & 32.3\\
~~~~~~Kalashev {\it et al.} 2002~\cite{Kal02} &  1.03 & 64.3 \\
~~~~~~Aramo {\it et al.} 2005~\cite{Aramo05} & 1.04 & 64.6\\
~~~~~~Barger, Huber, \& Marfatia 2006~\cite{Barger06} & 0.89 & 58.9\\
~~~~~~Yuksel \& Kistler 2007~\cite{Yuksel07} & 0.56 & 42.9\\
{\it BZ Models that saturate all bounds}: & & \\
~~~~~~Kalashev {\it et al.} 2002~\cite{Kal02} & 10.1 & $>99.99$\\
~~~~~~Aramo {\it et al.} 2005~\cite{Aramo05} & 8.50 & $>99.98$\\
{\it Waxman-Bahcall fluxes}: & & \\
~~~~~~Waxman, Bahcall 1999, evolved sources~\cite{WB}~~ & 0.76 & 53.2 \\
~~~~~~Waxman, Bahcall 1999, standard~\cite{WB} & 0.27 & 23.7\\ \hline
  \end{tabular}
 \end{footnotesize}
\end{table}

Based on the approach described in Refs.~\cite{Anch02,FORTE04}, the
resulting model-independent 90\%~CL limit on neutrino fluxes with Standard
Model cross-sections~\cite{Gan00} is shown in Fig.~\ref{lim08}.
Here we have included the net livetime and 81\% analysis efficiency.
Exclusion of the volume of ice near all camps and events
reduces the net effective volume by a
few percent. We estimate that experimental
systematics such as variations in ice properties and
calibration uncertainties in
the absolute radio signal strength lead to uncertainties
of order a factor of two on the limit.
These model-independent limits are calibrated such that a model spectrum
that matched the limit over one decade of energy would yield approximately
2.3 events; this choice is appropriate to smoothly varying models.
The limits are an average over all three neutrino flavors, as ANITA-1 had
roughly equal sensitivity to $\nu_e,~\nu_{\mu},~\nu_{\tau}$, and
the flavors should be equally mixed to first order via oscillations for all
BZ neutrino models.
We plot only an approximate set of bands for the BZ neutrino models,
which are too numerous to
individually plot here.

In Table~\ref{table2} we give the total number of events expected from
selected individual ultra-high energy neutrino models which
are representative of the range of BZ neutrino expectations; we also
include the case of a model which saturates
the canonical Waxman-Bahcall flux bounds for both evolved and standard
UHECR sources~\cite{WB}. ANITA-1 strongly limits the highest
BZ neutrino models, which require extremely high-energy
cutoffs in the parent cosmic-ray sources spectral energy distribution.
Our limits thus suggest that UHECR source spectra extending to
$10^{23}$~eV are disfavored. ANITA-1 sensitivity approaches
a class of models here denoted as ``strong-source evolution'' models,
which assume that the UHECR source evolution follows the cosmic
evolution of more energetic sources, for example
gamma-ray burst host galaxies~\cite{Yuksel07};
these mid-range models are constrained at about the 60\% CL
but none are ruled out yet.
The ANITA-1 90\% CL integral flux limit on a pure $E^{-2}$ spectrum
for the energy range $10^{18.5}$~eV $\leq E_{\nu} \leq 10^{23.5}$~eV is
$E_{\nu}^2 F \leq 2 \times 10^{-7}$ GeV cm$^{-2}$ s$^{-1}$ sr$^{-1}$.

In summary, we have set the strongest bounds to date on the ultra-high
energy neutrino flux at energies above $3 \times 10^{18}$~eV, using
the radio Cherenkov method via synoptic observations of the Antarctic
ice sheets from stratospheric altitudes. Our methodology appears
to have no observed physics backgrounds, but may have detected events
due to cosmic-ray extensive air showers that are easily
separated from the neutrino events we seek.

This work has been supported by the National Aeronautics and Space
Administration, the National Science Foundation Office of Polar Programs,
the Department of Energy Office of Science High Energy
Physics Division, and the UK Science
and Technology Facilities Council. Special thanks to the staff of
the Columbia Scientific Balloon Facility.


\section{The Color Glass Condensate and the Glasma}

The Color Glass Condensate is the matter which composes the low x part of a hadron wavefunction at high energy.\cite{cgc}  It is  an ensemble of color electric and color magnetic fields which are
in the form of Lienard-Wiechart potentials, $\vec{E} \perp \vec{B} \perp \hat{z}$, where $\hat{z}$ is in the direction of the beam.

These fields change their character after the collisions, where in a very short time,
$t \sim Q_{sat}^{-1} e^{-\kappa/\alpha_S}$, longitudinal color  electric and magnetic fields are produced.  There are both color electric and magnetic fields because of the duality of QCD under $E \leftrightarrow B$, and the symmetry of the fields in the CGC under this transformation.  Since there is both a longitudinal electric and magnetic field, these fields carry topological charge.  They have a variation in the transverse
size scale of $\Delta r_\perp \sim Q_{sat}^{-1}$, and may be thought of as an ensemble of
flux tubes.  This ensemble of flux tubes is the Glasma.\cite{glasma}

Flux tubes have a long extent in longitudinal coordinate and therefore should have a long range in rapidity.  If a system undergoes 1 dimensional Hubble flow until a time $\tau_F$, and has an extent
in rapidity $y = ln\{(t+z)/(t-z)\}$, then it must have been produced at a time before $\tau_i = \tau_f e^{-y/2}$.  (For example, if $\tau_f$ is a nuclear size, and $y$ is the total rapidity, then $\tau_i$ is the Lorentz contracted nuclear size in the center of mass frame.)

The multiplicity of flux tubes is
\begin{equation}
  {{dN_{FT}} \over {dy}} \sim R^2/R^2_{FT} \sim Q_{sat}^2 R^2
\end{equation}
where $R$ is the radius of the nucleus, and $R_{FT}$ is the radius of the fluxtube.  The flux tube is composed of high intensity color fields, and when these flux tubes decay their multiplicty per flux tube per unit rapidity should be of order $1/\alpha_S$, a factor typical of classical fields.  This yields the familiar Kharzeev-Nardi formula for the multiplicity of initially produced gluons,\cite{kn}
\begin{equation}
{{dN_{gluons}} \over {dy}} \sim {1 \over {\alpha_S}} Q_{sat}^2 R^2
\end{equation}

\section{Flux Tubes and Long Range Rapidity Correlations}

To compute the long range rapidity correlation associated with a flux tube, one needs an expression for the two gluon emission amplitude.  The leading order classical contribution is given by Fig. 1a. \cite{dgmv} Note that if this were emission from purely classical source, then the connected two particle emission amplitude,
\begin{equation}
        {{d^2N} \over {d^2p^1_Tdy^1d^2p^2_Tdy^2}} = \left<{{dN} \over {d^2p^1_Tdy^1}} {{dN} \over {d^2p^2_T dy^2}}\right> -  \left<{{dN} \over {d^2p^1_Tdy^1}} \right> \left< {{dN} \over {d^2p^2_T dy^2}}
        \right>
\end{equation}
\begin{figure}[htbp]
\begin{center}
\begin{tabular} {l l l}
\includegraphics[width=0.15\textwidth] {2pclass.jpg}  & &
\includegraphics[width=0.25\textwidth] {2pcorr.jpg} \\
& & \\
a & & b \\
\end{tabular}
\end{center}
\caption{a:  The diagram contributing to the two particle correlation induced by the classical field.
b:  The short range correlation which arises as a quantum correction to the long range classical correlation.}
\label{emission}
\end{figure}
would vanish.
In emission from a Glasma, on the other hand, one has to average the sources, which are treated as a stochastic variable.
If the sources are Gaussian correlated, then one of the contractions of the sources from the nuclei indeed contributes to the disconnected diagram, but there are remaining contributions associated with the different contractions of the sources.\cite{dgmv}  These contributions are associated with quantum interference in the two gluon emission amplitude, and may be difficult to precisely model in simulations which have flux tubes as sources of gluons.  (In experiment, the disconnected contribution is subtracted by the effects of mixed events, since a disconnected contribution corresponds to no event by event correlation.)

In Fig. 1b, the correlated two particle emission amplitude is shown.  This arises as a quantum correction
to the classical emission.  It has a short range correlation in rapidity, and a correction to the long
range term.  (In fact there can in principle be an interference between the classical two particle
emission and the  quantum correction.)
Note that the classical contribution of Fig. 1a, is of order $1/\alpha_S^2$, since it arises from the amplitude squared corresponding to  two classical fields. The quantum correction is of order 1.

Star has measured the forward backward asymmetry in heavy ion collisions.\cite{srivastava}  The forward backward correlation strength is defined  to be
\begin{equation}
  b = {{<N_FN_B> -<N_F><N_B>} \over {<N_B^2> - <N_B>^2}}
\end{equation}
If the separation in rapidity for the forward backward correlation is large, the numerator should measure the strength of the long range correlation, but the denominator is a sum over short and long range pieces.  This gives,\cite{mk}-\cite{hidaka}
\begin{equation}
  b = {1 \over {1+ \kappa \alpha_S^2}}
\end{equation}
Since $\alpha_S$ should decrease as centrality increases, this correlation strength should increase as a function of centrality.
\begin{figure}[htbp]
\begin{center}

\includegraphics[width=.99\textwidth] {star_fb.jpg}
\end{center}
\caption{The forward backward correlation strength measured for Gold-Gold collisions in STAR}
\label{starfb}
\end{figure}
The measured strength of the correlation is indeed very strong for central collisions.  It is also claimed that the correlation is much larger than can be expected from impact parameter fluctuations.

The STAR experiment has also measured directly the two particle correlation.  If one plots the correlation as a function of the azimuthal angle between the two particles, and the rapidity between the two particles, a structure elongated in rapidity and collimated in angle appears.  This is the so called ridge.
It appears either when one or particles have a lower momentum cutoff or when the momenta is integrated over.

In the Glasma description, the two particle correlation should be very strong since essentially all the produced particles arise from flux tubes.  It should also be noted that the two particle correlation predicted by the Glasma extends into the hard regime.  As pointed out by Shuryak, at high momenta, the Glasma flux tubes are the fragmentation jets which appear when one perturbatively computes hard scattering.\cite{shuryak} When two high $p_T$ particles are produced, the charges which induce the scattering are scattered out of the beam, leaving image charges in the fragments of the receding hadrons which produced the jets.  The color fields associated with these image charges are responsible for the beam fragmentation jets, arising from the decay of the flux tube.

One of the interesting features of the ridge is the collimation in azimuthal angle.  As pointed out by S. Voloshin,\cite{voloshin} this may arise from flow effects.  The position in the transverse plane of the colliding nuclei for a flux tube is localized.  This structure acquires transverse momentum due to radial flow.  The flowing flux tube, when it decays its distribution of decay products will be  preferentially along the direction of motion of the flux tube.  The amplitude of the flux tube distribution will also acquire a non-trivial dependence upon centrality from flow.

With Sean Gavin, we estimated these flow effects for the Glasma using a blast wave model to incorporate the effects of flow.\cite{gavin}  We describe the ridge which is integrated over the energies of both the particles in the two particle correlation function.  Detail of the experimental method for extracting the ridge are given in the Quark Matter 2008 contribution of STAR presented by Daugherity.\cite{daugherity}
\begin{equation}
        {{d^2N} \over {d^2p^1_Tdy^1d^2p^2_Tdy^2}} = \left<{{dN} \over {d^2p^1_Tdy^1}} {{dN} \over {d^2p^2_T dy^2}}\right> -  \left<{{dN} \over {d^2p^1_Tdy^1}} \right> \left< {{dN} \over {d^2p^2_T dy^2}}
        \right>
\end{equation}
\begin{figure}[htbp]
\begin{center}
\begin{tabular} {l l l}
\includegraphics[width=0.50\textwidth] {ridge_amp.jpg}  & &
\includegraphics[width=0.46\textwidth] {ridge_ang.jpg} \\
& & \\
a & & b \\
\end{tabular}
\end{center}
\caption{a:  The amplitude for the ridge as a function of centrality at two different energies.  The formula without a factor of $1/\alpha_S$ which arises in the Glasma description is shown as dashed line.  b:  The angular width of the ridge as a function of centrality}
\label{emission}
\end{figure}
In Fig. 3 a, the dependence on the amplitude of the ridge as a function of centrality is shown.  The red points are data.  The solid line is our description.  There is a one parameter ambiguity due to an overall normalization for the emission amplitude, which is not yet computed.    The dashed line curve, is fit to the data not including a factor of $1/\alpha_S$ which arises from the classical field description of the ridge.
In Fig. 3b, the angular width of the ridge is shown as a function of centrality.

At this meeting, Takeshi Kodama informed me that the SPHERIO collaboration from Brazil have seen the ridge in their computations.  This arises because of the string-like configurations in the initial state.  An excellent talk has been presented at the meeting RANP 2008 by Jun Takahashi.\cite{takahashi}

Clearly the next stage of theoretical analysis of the ridge will be to expand upon the SPHERIO treatment of the ridge.  Some effort need be taken to properly account for the interference diagrams in the two particle correlation function which yield the ridge.  In addition, one needs to understand the effect of energy-momentum conservation, and how it appears in the backward direction from the ridge.

\section{Acknowledgements}  I thank the organizers of the International Symposium on Miltiparticle Dynamics for their kind hospitality.  I also thank Takeshi Kodama for telling me of the recent results from SPHERIO.  I thank my colleagues Adiran Dumitru, Sean Gavin, Francois Gelis, Geroge Moschelli and Raju Venugopalan for their efforts and keen insight during our collaboration.