Content-Type: multipart/mixed; boundary="-------------0603070341271" This is a multi-part message in MIME format. ---------------0603070341271 Content-Type: text/plain; name="06-55.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-55.comments" 19 pages ---------------0603070341271 Content-Type: text/plain; name="06-55.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="06-55.keywords" MSC-class: 81V70,81V55 (primary);35B65,35J10,81Q05,35Q40 (secondary) ---------------0603070341271 Content-Type: application/x-tex; name="nonisotropic.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="nonisotropic.tex" \documentclass[12pt,reqno]{amsart} \NeedsTeXFormat{LaTeX2e}[1994/12/01] \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{amstext} \newcommand{\N}{{\mathbb N}} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \theoremstyle{plain} \newtheorem{thm}{Theorem}[section] \newtheorem{lemma}[thm]{Lemma} \theoremstyle{definition} \newtheorem*{acknowledgement}{Acknowledgement} \newtheorem{remark}[thm]{Remark}{\it}{\rm} \newenvironment{pf}{\par\medskip\noindent\textit{Proof}:\,}{\hspace*{\fill}\qed\medskip\par\noindent} \newenvironment{pf*}[1]{\par\medskip\noindent\textit{#1}\,:}{\hspace*{\fill}\qed\medskip\par\noindent} \numberwithin{equation}{section} \title[Regularity of the density at nuclei]{Non-Isotropic Cusp Conditions and Regularity of the Electron Density of Molecules at the Nuclei} \thanks{\copyright\ 2006 by the authors. This article may be reproduced in its entirety for non-commercial purposes.} \author[S. Fournais, M. and T. Hoffmann-Ostenhof, and T. \O. S\o rensen] {S. Fournais \and M. Hoffmann-Ostenhof \and T. Hoffmann-Ostenhof \and T. \O stergaard S\o rensen} \address[S. Fournais]{CNRS and Laboratoire de Math\'{e}matiques, Universit\'{e} Paris-Sud - B\^{a}t 425, F-91405 Orsay Cedex, France. } \email{soeren.fournais@math.u-psud.fr} \address[T. \O stergaard S\o rensen] {Institut des Hautes \'Etudes Scientifiques, Le Bois-Marie, 35, route de Chartres, F-91440 Bures-sur-Yvette, France} \address[T. \O stergaard S\o rensen, permanent address] {Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg East, Denmark.} \email{sorensen@math.aau.dk} \address[M. Hoffmann-Ostenhof] {Fakult\"at f\"ur Mathematik, Universit\"at Wien, Nordbergstra\ss e 15, A-1090 Vienna, Austria.} \email{maria.hoffmann-ostenhof@univie.ac.at} \address[T. Hoffmann-Ostenhof]{Institut f\"ur Theoretische Chemie, W\"ahringer\-strasse 17, Universit\"at Wien, A-1090 Vienna, Austria.} \address[T. Hoffmann-Ostenhof, 2nd address]{ The Erwin Schr\"{o}dinger International Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria.} \email{thoffman@esi.ac.at} \date{\today} \begin{document} \thispagestyle{empty} \begin{abstract} We investigate regularity properties of molecular one-electron densities $\rho$ near the nuclei. In particular we derive a re\-pre\-sentation $$\rho(x)=e^{\mathcal F(x)}\mu(x)$$ with an explicit function $\mathcal F$, only depending on the nuclear charges and the positions of the nuclei, such that $\mu\in C^{1,1}(\mathbb R^3)$, i.e., $\mu$ has locally essentially bounded second derivatives. An example constructed using Hydrogenic eigenfunctions shows that this regularity result is sharp. For atomic eigenfunctions which are either even or odd with respect to inversion in the origin, we prove that \(\mu\) is even \(C^{2,\alpha}(\R^3)\) for all \(\alpha\in(0,1)\). Placing one nucleus at the origin we study $\rho$ in polar coordinates $x=r\omega$ and investigate $\frac{\partial}{\partial r}\rho(r,\omega)$ and $\frac{\partial^2}{\partial r^2}\rho(r,\omega)$ for fixed $\omega$ as $r$ tends to zero. We prove non-isotropic cusp conditions of first and second order, which generalize Kato's classical result. \end{abstract} \maketitle \section{Introduction and statement of the results} We consider a non-relativistic $N$-electron molecule with the nuclei fixed in \(\R^3\). The Hamiltonian describing the system is given by \begin{equation}\label{Hbis} H=\sum_{j=1}^N\Big(-\Delta_j-\sum_{k=1}^K\frac{Z_k}{|x_j-R_k|}\Big) +\sum_{1\le i0$. By \cite[Theorem 1.2]{AHP} (see also \cite[Remark 1.7]{AHP}) this implies the existence of constants \(C_{1}, \gamma_{1}>0\) such that \begin{align} \label{eq:dec_grad_psi} \big|\nabla\psi(\mathbf x)\big|\leq C_{1}\,e^{-\gamma_{1}|\mathbf x|} \quad\text{ for almost all } \mathbf x\in\mathbb R^{3N}. \end{align} Since \(\psi\) is continuous, \eqref{eq:exp-dec} is only an assumption on the behaviour at infinity. For references on the exponential decay of eigenfunctions, see e.g.\ Agmon~\cite{Agmon}, Froese and Herbst~\cite{froese-herbst}, and Simon~\cite{simon1}. The proofs of our results rely (if not indicated otherwise) on some kind of decay-rate for \(\psi\); exponential decay is not essential, but assumed for convenience. Note that \eqref{eq:exp-dec} and \eqref{eq:dec_grad_psi} imply that \(\rho\) is Lipschitz continuous in \(\R^3\) by Lebesgue's theorem on dominated convergence. In \cite{ArkMat} we showed that $\rho$ is real analytic away from the nuclei ($\rho\in C^{\omega}(\mathbb R^3\setminus\{R_1,\dots, R_K\})$); for earlier results see also \cite{AHP}, \cite{CMP1} and \cite{COM}. Note that the proof of the analyticity does {\em not} require any decay of $\psi$ (apart from $\psi\in W^{2,2}(\mathbb R^{3N})$). That $\rho$ itself is not analytic in all of $\R^3$ is already clear for the ground state of the Hydrogen atom ($N=K=1; R_1=0, Z_1=1$): $\psi(x)=e^{-|x|/2}$ so that the associated $\rho$ (up to a normalization constant) equals $e^{-|x|}$; hence $\rho$ is just Lipschitz continuous near the origin. For the atomic case ($K=1; R_1=0, Z_1=Z$) a quantity studied earlier is the spherical average of $\rho$ which, in polar coordinates $x=r\omega$ with $r=|x|$ and $\omega=x/|x|$, is defined by \begin{equation}\label{def:tilderho} \widetilde\rho(r)=\int_{\mathbb S^2}\rho(r\omega)\,d\omega\ , \quad r\in[0,\infty). \end{equation} The above mentioned analyticity result implies that $\widetilde \rho\in C^{\omega}((0,\infty)).$ The existence of $\widetilde\rho\,'(0)$ and the so-called \textbf{cusp condition} \begin{equation}\label{eq:KatoCusp} \widetilde \rho\,'(0)=-Z\widetilde\rho(0) \end{equation} follow from a similar result of Kato \cite{kato1} for $\psi$ itself; see also \cite{seiler} and \cite[Remark~1.13]{AHP}. The existence of $\widetilde\rho\,''(0)$ and an implicit formula for it was proved in \cite[Theorem 1.11]{AHP}; see \eqref{eq:secondAHP} below for the exact statement. In \cite{CMP2} the present authors generalized the results of Kato for $\psi$ considerably for the Hamiltonian in \eqref{Hbis}. In the present paper we obtain results, partly in the spirit of these findings, for the density $\rho$. In particular, we prove results on the regularity of the density $\rho$ at the nuclei and derive identities which the first and second radial derivatives of \(\rho\) satisfy. These identities can be interpreted as cusp conditions (analogously to \eqref{eq:KatoCusp}). The methods developed in \cite{AHP} play an essential role in the proofs of these results. We indicate the importance of the electron density in quantum mechanics. From the eigenfunction \(\psi\) it is, in principle, possible to calculate the energy, various expectation values, etc.; but \(\psi\) depends on \(3N\) variables. Physicists and chemists usually aim at understanding atomic and molecular properties by means of the electron density which is just a function on \(\R^3\) and can be visualized. The density also has an immediate probabilistic interpretation. In computational chemistry density functional methods are of increasing importance for calculations of ground state energies of large molecules. Thereby the energy is approximated by minimizing a `density functional' which depends nonlinearly and nonlocally upon the density. The minimizing function is believed to be a good approximation to the density itself. The relationship between most of these functionals and the full \(N\)-electron Schr\"odinger equation remains unclear though. One exception is of course the archetype density functional theory, the Thomas-Fermi theory, which is mathematically and physically interesting, and very well understood, see \cite{LiebSimon} and \cite{Lieb1}. For an interesting recent review on various mathematical problems related to the many models in computational chemistry, see \cite{BrisLions}. For some work on the density \(\rho\) from a numerical point of view, related to regularity questions, see \cite{flad1}. Questions concerning the one-electron density \(\rho\), as defined by \eqref{rhohat}, pose some challenging mathematical problems. Results as given in the present paper contribute to a better understanding of the physics of atoms and molecules and in addition should have relevance for computational quantum chemistry. In the following we use the standard definition and notation for H\"older continuity and Lipschitz continuity, see e.g. \cite{GandT}. Let $f:\mathbb R^n \supset\Omega\to \mathbb R$, then $f\in C^{k,\alpha}(\Omega)$ means, for $\alpha =0$, that $f$ is $k$ times continuously differentiable, for $\alpha\in (0,1]$ that the $k$-th partial derivatives of $f$ are H\"older continuous with exponent $\alpha$. In the case \(k=0\), we often write \(C^{\alpha}(\Omega):=C^{0,\alpha}(\Omega)\) when \(\alpha\in(0,1)\). The main result of the present paper is the following. \begin{thm}\label{thm:ThmC-1-1} Let $\psi\in L^2(\mathbb R^{3N})$ be a molecular or atomic N-electron eigenfunction, i.e., \(\psi\) satisfies \eqref{Hpsi}, with associated density $\rho$. Define $\mathcal F:\mathbb R^3\to \mathbb R$ by \begin{equation}\label{def:F} \mathcal F(x)=-\sum_{k=1}^KZ_k|x-R_k|. \end{equation} Then \begin{equation}\label{mu} \rho(x)=e^{\mathcal{F}(x)}\mu(x) \end{equation} with \begin{equation}\label{muC11} \mu\in C^{1,1}(\mathbb R^3). \end{equation} This representation is optimal in the following sense: There is no function $\widetilde{\mathcal{F}}:\R^3\to\R$ depending only on \(Z_1,\ldots,Z_K\), \(R_1,\ldots,R_K\), but neither on $N$, $\rho$, nor \(E\), with the property that \(e^{-\widetilde{\mathcal{F}}}\rho\) is in \(C^2(\R^3)\). Furthermore, $\mu$ admits the following representation: \par\noindent There exist $C_1,\dots, C_K\in \mathbb R^3$ and $\nu:\mathbb R^3\to \mathbb R$ such that \begin{equation}\label{munu} \mu(x)=\nu(x)+\sum_{k=1}^K|x-R_k|^2\big( C_k\cdot\frac{x-R_k}{|x-R_k|}\big), \end{equation} with \begin{equation}\label{nualpha} \nu\in C^{2,\alpha}(\mathbb R^3) \text{ for all }\alpha\in (0,1). \end{equation} \end{thm} \begin{remark}\label{at} In the case of atoms (\(K=1; R_1=0, Z_1=Z\)), the statement of the theorem reads: There exists \(C\in\R^3\) such that \begin{equation}\label{eq:1} \rho(x)=e^{-Z|x|}\mu(x), \quad \mu(x)=\nu(x)+|x|^2\big( C\cdot\frac{x}{|x|}\big) \end{equation} with \begin{equation}\label{nuatom} \nu\in C^{2,\alpha}(\mathbb R^3) \text{ for all } \alpha\in (0,1). \end{equation} To simplify the exposition, we shall give the proof of Theorem~\ref{thm:ThmC-1-1} only in the case of atoms. The proof easily generalizes to the case of several nuclei. \end{remark} \begin{remark}\label{thm:j-s} It will be evident from the proof that the result (appropriately reformulated) also holds for each \(\rho_j\) seperately (see \eqref{rhohat}). The same is true for the results below. \end{remark} \begin{pf*}{Proof of the optimality} We study `Hydrogenic atoms' ($N=K=1; R_1=0, Z_1=Z$) and use the notation (contrary to the rest of the paper) $x = (x_1, x_2, x_3) \in {\mathbb R}^3$, $r=|x|$. In this case, the operator in \eqref{Hbis} reduces to \(H_Z=-\Delta_x-Z/|x|\). We will present an example where, no matter what the choice of \(\widetilde{\mathcal{F}}\) (as in the theorem), $\mu=e^{-\widetilde{\mathcal{F}}}\rho$ cannot be \(C^2\). The argument resembles the proof of the corresponding result in \cite{CMP2}. The $1s$ eigenfunction is \(\psi_{1s}(x)=e^{-Zr/2}\) with \(H_Z\psi_{1s}=-(Z^2/4)\psi_{1s}\) and the associated density is \(\rho_{1s}(x)=e^{-Zr}\). The $2s$ and $2p$ eigenfunctions are \begin{align*} \psi_{2s}(x) &= (1-\tfrac{Z}{4}r)e^{-Zr/4} , & \psi_{2p}(x) &= x_1 e^{-Zr/4}. \end{align*} % Both satisfy \(H_Z\psi=E\psi\) with $E=-Z^2/16$. The associated densities are \begin{align*} \rho_{2s}(x) &= \psi_{2s}^2(x) = (1-\tfrac{Z}{4}r)^2e^{-Zr/2}, & \rho_{2p}(x) &= \psi_{2p}^2(x) = x_1^2 e^{-Zr/2}. \end{align*} Consider now $\psi_{{\rm mixed}}= \psi_{2s} + \psi_{2p}$ and \begin{align*} \rho_{{\rm mixed}} = \psi_{{\rm mixed}}^2 = \rho_{2s}+ \rho_{2p} + 2 \psi_{2s} \psi_{2p}. \end{align*} A simple calculation shows that \begin{align*} e^{Zr} \rho_{2s}, e^{Zr} \rho_{2p} \in C^{2,1}({\mathbb R}^3), \end{align*} but \(e^{Zr}\rho_{\rm mixed}\) is just \(C^{1,1}\), since the mixed derivative \(\partial_{x_2}\partial_{x_1}\) of \begin{align*} e^{Zr} \psi_{2s} \psi_{2p} = x_1 e^{Zr/2}(1-\tfrac{Z}{4}) \end{align*} does not exist at \(x=0\). But if \(\rho=e^{\widetilde{\mathcal{F}}}\mu\) with \(\mu\in C^2\), then \begin{align*} \frac{\mu_{\rm mixed}}{\mu_{1s}}=\frac{e^{-\widetilde{\mathcal{F}}}\rho_{\rm mixed}}{e^{-\widetilde{\mathcal{F}}}e^{-Zr}} =e^{Zr}\rho_{\rm mixed} \end{align*} should also be \(C^2\), a contradiction. \end{pf*} Note that $\psi_{2s}(x)=\psi_{2s}(-x)$ and $\psi_{2p}(x)=-\psi_{2p}(-x)$, but their linear combination $\psi$ is neither even nor odd. \begin{remark} The representation of $\rho$ as a product $\rho=e^{\mathcal F}\mu$ with a fixed `universal' $\mathcal F$ such that $\mu$ is by one degree smoother than $\rho$ corresponds to Theorem 1.1 in \cite{CMP2} where a similar result was obtained for the eigenfunction \(\psi\) itself. In that case though, the correponding $\mathcal F$ is more complicated since many-particle interactions have to be taken into account. For some interesting recent investigation in connection with Jastrow factors from a numerical point of view, see \cite{flad2}. \end{remark} The proof of Theorem \ref{thm:ThmC-1-1} will be given in the next section. Here we just mention that $\rho$ satisfies an inhomogeneous Schr\"odinger equation whose investigation is crucial for regularity results like the above, as well as it was for the results in \cite{AHP}. Let $H$ be given by \eqref{Hbis} and consider an eigenfunction $\psi$ satisfying \eqref{Hpsi}. To simplify notation we assume without loss that \(\psi\) is real. The equation \begin{equation}\label{hatint} \int_{\mathbb R^{3N-3}}\psi(x, \mathbf{\hat x}_j)(H-E)\psi(x,\mathbf{\hat x}_j)\,d \mathbf{\hat x}_j=0 \end{equation} leads to an equation (in the sense of distributions) for $\rho_j$, namely, \begin{equation} \label{eq:first-rho} \Big(-\frac{1}{2}\Delta -\sum_{k=1}^K\frac{Z_k}{|x-R_k|}\Big)\rho_j+h_j=0. \end{equation} Summing \eqref{eq:first-rho} over $j$ we obtain the equation for $\rho$, \begin{equation}\label{eq:firstRho} \Big(-\frac{1}{2}\Delta -\sum_{k=1}^K\frac{Z_k}{|x-R_k|}\Big)\rho+h=0, \end{equation} with $h=\sum_{j=1}^Nh_j$. The functions $h_j$ will be given explicitely in Section~\ref{sec:proofs}; see \eqref{eq:defh}. In \cite{AHP} we considered the spherically averaged density \(\widetilde\rho\) (as defined by \eqref{def:tilderho}) for the atomic case. The regularity of \(\widetilde h\) (the spherical average of \(h\) above) was crucial for the results obtained there. Here we study the {\it non-averaged} density \(\rho\) for the general case of molecules. Again, the regularity of \(h\) is essential for our results. We continue to consider $\rho$ in the neighbourhood of one nucleus with charge $Z$. Without loss we can place this nucleus at the origin. The equations \eqref{munu} and \eqref{nualpha} show that it is natural to consider the behaviour of $\rho(r\omega)$ for fixed $\omega$ as $r$ tends to zero. \begin{thm}\label{thm:dirDerI} Let $\psi\in L^2(\mathbb R^{3N})$ be a molecular or atomic eigenfunction, i.e., \(\psi\) satisfies \eqref{Hpsi}, with associated density $\rho$. Assume without loss that \(R_1=0\) and write $Z$ instead of \(Z_1\). Let $r_0=\min_{k>1}|R_k|$ (\(r_0=\infty\) for atoms) and let \(\omega\in\mathbb{S}^2\) be fixed. \begin{itemize} \item[(i)] The function $r\mapsto \rho(r,\omega):=\rho(r\omega)$, $r\in [0,r_0)$, satisfies \begin{align}\label{eq:rhoOmega} \rho(\cdot, \omega)\in C^{2,\alpha}([0,r_0))\text{ for all }\alpha\in (0,1). \end{align} \item[(ii)] Denote by \({}'\) the derivative \(\frac{d}{dr}\), and define \begin{align} \eta(x)=e^{Z|x|}\rho(x)\ , \ \chi =\eta - r^2( C\cdot\omega), \end{align} where \(C\in\R^3\) is the constant \(C_1\) in \eqref{munu} (resp.\ \(C\) in \eqref{eq:1}). Then \begin{align}\label{reg:eta&chi} \eta\in C^{1,1}(B(0,r_0))\ , \ \chi\in C^{2,\alpha}(B(0,r_0))\text{ for all }\alpha\in(0,1), \end{align} and \begin{align}\label{eq:fixedW3} \rho'(0,\omega)& =-Z\rho(0)+ \omega\cdot (\nabla \eta)(0), \\ \label{eq:fixedW4} \rho''(0,\omega) &=Z^2\rho(0)+2\omega\cdot[C-Z(\nabla \eta)(0)]+ \omega\cdot\big((D^2\chi)(0)\omega\big). \end{align} Here $(D^2\chi)(0)$ is the Hessian matrix of $\chi$ evaluated at the origin. \end{itemize} \end{thm} \begin{remark}\label{rem:new cusps} \(\, \) \begin{enumerate} \item[\rm (i)] For atoms, \(\eta\) equals \(\mu\) from Theorem~\ref{thm:ThmC-1-1} and \(\chi\) equals \(\nu\) from Remark~\ref{at}. \item[\rm (ii)] Note that \eqref{eq:rhoOmega} trivially implies that \(\rho(r,\omega)=e^{-Zr}\eta(r,\omega)\) with \(\eta(\cdot,\omega) \in C^{2,\alpha}([0,r_0))\) for all \(\alpha\in(0,1)\). Compare with \eqref{mu}, \eqref{muC11}. \item[\rm (iii)] In \cite[Theorem~1.11]{AHP} it was proved that \(\widetilde \rho\) defined by \eqref{def:tilderho} belongs to \(C^2([0,r_0))\cap C^{2,\alpha}((0,r_0))\) for all \(\alpha\in(0,1)\). (The proof in \cite{AHP} for the atomic case easily generalizes to the molecular case.) Reading the proof of \cite[Theorem~1.11]{AHP} carefully, one sees that it in fact yields \(\widetilde \rho\in C^{2,\alpha}([0,r_0))\). The statement in \eqref{eq:rhoOmega} shows that for fixed \(\omega\in\mathbb{S}^2\) this holds already for \(\rho(\cdot,\omega)\), i.e., without averaging. \item[\rm (iv)] The identities \eqref{eq:fixedW3} and \eqref{eq:fixedW4} can be considered as \textbf{non-isotropic cusp conditions of first and second order}. They generalize the cusp condition \eqref{eq:KatoCusp}, as well as the previously mentioned result in \cite{AHP} for \(\widetilde\rho\,''(0)\); more on this in Remark~\ref{mu21} (ii) below. See also the second order cusp conditions obtained in \cite{CMP2} for the eigenfunction \(\psi\) itself. \item[\rm (v)] It is worth noting that \eqref{eq:fixedW3} and \eqref{eq:fixedW4} can be interpreted as a structural result for the density $\rho$: From Theorem~\ref{thm:dirDerI} it follows that in a neighbourhood of a nucleus (which is at the origin), $\rho$ satisfies (for all \(\alpha\in(0,1)\)) \begin{equation}\label{formalrho} \rho(r,\omega)=\rho(0)+r\phi_1(\omega)+r^2\phi_2(\omega)+O(r^{2+\alpha})\ ,\ r\downarrow0, \end{equation} and \eqref{eq:fixedW3}, \eqref{eq:fixedW4} show that $\phi_1$ is a linear and $\phi_2$ a quadratic polynomial restricted to $\mathbb S^2$.\par\noindent It is a natural question whether \eqref{formalrho} extends to higher orders. \end{enumerate} \end{remark} We continue with the atomic case. In view of Remark~\ref{at}, \eqref{eq:1} and the considerations after the proof of the optimal regularity of \(\mu\) in Theorem \ref{thm:ThmC-1-1}, the following theorem is natural. \begin{thm}\label{thm:ThmC-2-alpha} Let $\psi\in L^2(\R^{3N})$ be an atomic eigenfunction with associated density $\rho$. Suppose that \begin{equation}\label{symmetry} |\psi(\mathbf x)|=|\psi(-\mathbf x)|\text{ for all }\mathbf x\in \mathbb R^{3N}. \end{equation} Then $\rho$ satisfies \begin{equation}\label{main3} \rho(x)=e^{-Z|x|}\mu(x),\:\: \mu\in C^{2,\alpha}(\mathbb R^3)\text{ for all } \alpha \in (0,1). \end{equation} Furthermore, \begin{equation}\label{cusp0} \rho'(0,\omega)=-Z\rho(0)\ , \quad \rho''(0,\omega)=Z^2\rho(0)+ \omega\cdot\big((D^2\mu)(0)\omega\big). \end{equation} We also have \begin{align}\label{eq:Marias} \rho''(0,\omega)=\frac23\big( Z^2\rho(0)+h(0,\omega)\big) +\frac13\lim_{r\downarrow0}\frac{(\mathcal{L}^2\rho)(r,\omega)}{r^2}, \end{align} with \(h\) from \eqref{eq:firstRho}, and \(\mathcal{L}^2/r^2\) the angular part of \(-\Delta\), i.e., \(\Delta=\partial^2/\partial r^2+(2/r)\partial/\partial r-\mathcal{L}^2/r^2\). \end{thm} \begin{remark}\label{mu21} \(\, \) \begin{enumerate} \item[\rm (i)] In this case \(\mu=\nu=\chi=\eta\) , as can be seen from Remark~\ref{rem:new cusps} (i) and the proof of the theorem. \item[\rm (ii)] Note that \eqref{cusp0} shows that the cusp condition \eqref{eq:KatoCusp} in this case holds for fixed angle \(\omega\in\mathbb{S}^2\) without averaging. Further, taking the spherical average of \eqref{eq:Marias}, we get the formula for \(\widetilde\rho\,''(0)\) obtained in \cite[Theorem~1.11 (iv)]{AHP}: \begin{align}\label{eq:secondAHP} \widetilde\rho\,''(0)=\frac23\big(Z^2\widetilde\rho(0)+\widetilde h(0)\big). \end{align} To see this note that for all \(r>0\) \begin{align*} \int_{\mathbb{S}^2}1\cdot({\mathcal{L}}^2\rho)(r,\omega)\,d\omega =\int_{\mathbb{S}^2}(\mathcal{L}^21)\cdot\rho(r,\omega)\,d\omega=0. \end{align*} Note that \(\widetilde\rho\,''(0)\geq0\), since \begin{align*} \widetilde h(r)\geq \epsilon\widetilde\rho(r) \end{align*} for some \(\epsilon\geq0\) \cite[Theorem 1.11]{AHP}. This positivity is not an obvious consequence of the formula in \eqref{cusp0}. \item[\rm (iii)] As can be seen from the proof of Theorem~\ref{thm:ThmC-2-alpha}, \(h\in C^{\alpha}(\R^3)\) for all \(\alpha\in(0,1)\) in this case. \end{enumerate} \end{remark} \section{Proofs} \label{sec:proofs} \begin{pf*}{Proof of Theorem~\ref{thm:ThmC-1-1}} As noted in Remark~\ref{at}, we shall give the proof only in the case of atoms (\(K=1; R_1=0, Z_1=Z\)). For the regularity questions concerning $\rho$ defined in \eqref{rhohat} it suffices to consider the (non-symmetrized) density $\rho_1$ defined by \begin{align} \label{eq:rho}\nonumber \rho_1(x)&= \int_{{\mathbb R}^{3N-3}} \,|\psi(x,x_2,\ldots,x_N)|^2\,dx_2\cdots dx_N \\& =\int_{\mathbb R^{3N-3}}|\psi(x,\hat{\bf x}_1)|^2\,d\hat{\bf x}_1 \end{align} with \(x\in\mathbb R^3, \hat{\bf x}_1=(x_2,\ldots,x_N)\in\mathbb R^{3N-3}\). As explained in \eqref{hatint}--\eqref{eq:firstRho} \(\rho_1\) satisfies the Schr\"{o}dinger-type equation \begin{align}\label{eq2:rho} {}-\Delta \rho_1 - \frac{2Z}{|x|}\rho_1 + 2h_1=0, \end{align} % where the function $h_1$ is given by \begin{align} \label{eq:defh} h_1(x) &= J_1 - J_2 + J_3 - E\rho_1(x),\\ J_1(x) &= \sum_{j=1}^N \int _{{\mathbb R}^{3N-3}} |\nabla_j \psi |^2 \,d\hat{\bf x}_1\ ,\quad J_2(x) = \sum_{j=2}^N \int _{{\mathbb R}^{3N-3}} \frac{Z}{|x_j|} \psi^2 \,d\hat{\bf x}_1,\nonumber\\ J_3(x) &= \sum_{k=2}^N \int _{{\mathbb R}^{3N-3}} \frac{1}{|x-x_k|} \psi ^2 \,d\hat{\bf x}_1 +\sum_{2\leq j0. \end{align} Note that, by l'H\^{o}pital's rule and \eqref{cusp0}, \begin{align*} \lim_{r\downarrow0}\frac{2}{r}\big(\rho'(r,\omega)+Z\rho(r,\omega)\big) =2\big(\rho''(0,\omega)+Z\rho'(0,\omega)\big), \end{align*} and so \eqref{eq:inPolar} implies that \(\mathcal{R}(0,\omega):=\lim_{r\downarrow0}\mathcal{R}(r,\omega)\) exists, and \begin{align*} \mathcal{R}(0,\omega)=3\rho''(0,\omega)+2Z\rho'(0,\omega)-2h(0,\omega). \end{align*} The existence of \(h(0,\omega):=\lim_{r\downarrow0}h(r,\omega)\) follows from Lemma~\ref{lem:formH}. Therefore, using \eqref{cusp0}, we obtain \eqref{eq:Marias}. \end{pf*} \appendix \section{A useful lemma}\label{app:lem} The following lemma is Lemma~2.9 in \cite{CMP2}; we include it, without proof, for the convenience of the reader. (The proof is simple, and can be found in \cite{CMP2}). \begin{lemma}\label{usefullemma} \label{lem:XdotG} Let \(G:U\to\mathbb R^{n}\) for \(U\subset\mathbb R^{n+m}\) a neighbourhood of a point \((0,y_{0})\in\mathbb R^{n}\times\mathbb R^{m}\). Assume \(G(0,y)=0\) for all \(y\) such that \((0,y)\in U\). Let \begin{align*} f(x,y)=\left\{\begin{array}{cc} \frac{x}{|x|}\cdot G(x,y)& x\neq 0, \\ 0& x=0. \\ \end{array}\right. \end{align*} Then, for \(\alpha\in(0,1]\), \begin{align} \label{eq:lem_G=0} G\in C^{0,\alpha}(U;\mathbb R^{n})\Rightarrow f\in C^{0,\alpha}(U). \end{align} Furthermore, $\| f \|_{C^{\alpha}(U)} \leq 2\| G \|_{C^{\alpha}(U)}$. \end{lemma} \begin{acknowledgement} Parts of this work have been carried out at various institutions, whose hospitality is gratefully acknowledged: Mathematisches Forschungs\-institut Ober\-wolfach (SF, T\O S), The Erwin Schr\"{o}\-dinger Institute (SF, T\O S), Universit\'{e} Paris-Sud (T\O S), and the IH\'ES (T\O S). Financial support from the European Science Foundation Programme {\it Spectral Theory and Partial Differential Equations} (SPECT), and EU IHP network {\it Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems}, contract no.\ HPRN-CT-2002-00277, is gratefully acknowledged. T\O S was partially supported by the embedding grant from The Danish National Research Foundation: Network in Mathematical Physics and Stochastics, and by the European Commission through its 6th Framework Programme {\it Structuring the European Research Area} and the contract Nr. 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