# Mathematical/Physical Model

realQM is a physical model in terms of classical continuum mechanics of an atom or ion consisting of a pointlike kernel of positive charge $Z$ surrounded by $N$ electrons of negative unit charge, as a free boundary problem for a system of $N$ distributed non-overlapping electron unit charge densities in 3d Euclidean space $\Re^3$. For a neutral atom $N=Z$, while $N for an ion with positive charge $Z-N$ and $N>Z$ for an ion with negative charge $N-Z$, as an atom which has lost or gained electrons. The chemical properties of an atom is largely determined by the energy required to remove an electron (ionisation energy) and the energy released by capturing an electron (electron affinity), as prime tasks for an atom model.

The electron charge distribution of the ground state of an atom/ion is characterised by minimising a total energy as the sum over electrons of kernel potential energy, inter-electron potential energy and so-called kinetic energy as a measure of charge compression. Each electron minimises its contribution to the total energy under a free boundary condition of continuity of charge density.

realQM is a many-electron model as a collection of one-electron models over a partition in space coupled by electron potentials and satisfying a free boundary condition of charge continuity. As such realQM combines simplicity, generality and physicality.

In mathematical terms realQM starts from a wave function Ansatz

• $\psi (x) = \sum_{j=1}^N\psi_j(x)$        (1)

as a sum of $N$ real-valued electron wave functions $\psi_j(x)$ depending on a common Euclidean 3d space coordinate $x\in \Re^3$ and having non-overlapping spatial supports $\Omega_j$ with boundaries $\Gamma_j$ for $j=1,...,N$, together filling $\Re^3$.

We assume that $\psi_j\in H^1(\Omega_j )$ for $j=1,...,N$ ,where $H^1(\Omega )$ is the set of real-valued functions defined on the domain $\Omega$ in $\Re^3$ which are square integrable along with first derivatives.

We ask the electron wave functions $\psi_j$ to statisfy the normalization condition

• $\int_{\Omega_j}\psi_j^2\, dx = 1\quad\mbox{for}\quad j=1,..,N,$       (2)

attributing unit charge to each electron with $\psi_j^2(x)$ representing the charge density of electron $j$.

We consider $\Gamma$ as the union of $\Gamma_i\cap\Gamma_j$ for $i,j=1,...,N$ to be a free boundary to be determined along with the wave function $\psi$ and as a free boundary condition, we shall ask $\psi$ to be continuous across $\Gamma$, that is that $\psi_j$ and $\psi_k$ agree on $\Gamma_i\cap\Gamma_j$. We express satisfaction of the free boundary condition of continuity by asking that $\psi\in H^1(\Re^3)$.

We start considering real-valued wave functions $\psi (x)$ depending on the space coordinate $x$, and later extend to complex-valued wave functions with time dependency. We seek the neutral ground state of the atom with $N=Z$ as a real-valued function $\psi\in H^1(\Re^3)$ of the form (1) satísfying (2), which with the $\psi_j$ varying freely over $H^1(\Omega_j)$ minimizes the total energy:

• $TE(\psi )\equiv K(\psi )+PK(\psi )+PE(\psi )$          (3)

as the sum of kinetic energy (with Planck’s constant $h$ normalized to 1):

• $K(\psi )=\frac{1}{2}\int_{\Re^3}\vert\nabla\psi (x)\vert^2\, dx,$

attractive kernel potential energy:

• $PK(\psi )=\int_{\Re^3}W(x)\psi^2(x)\, dx,$

and repulsive electronic potential energy:

• $PE(\psi )=\sum_j\int_{\Omega_j}\sum_{k\neq j} V_k(x)\psi_j^2(x)\, dx,$

where $W(x)=-\frac{Z}{\vert x\vert}$ is the potential generated by a pointlike positive kernel of charge $Z$ and $V_k(x)$ is the potential generated by electron $k$ defined by

• $V_k(x)=\int_{\Omega_k}\frac{\psi_k^2(y)}{2\vert x-y\vert}dy \quad\mbox{for}\quad x\in\Re^3$.

We see that the total energy $TE(\psi )$ of an electronic configuration defined by the wave function $\psi (x)$ has a negative contribution from Coulombic attractive kernel potential energy $PK(\psi )$, a positive contribution from Coulombic repulsive electronic potential energy $PE(\psi )$ without self-repulsion, and a positive contribution from $K(\psi )$ as a measure of concentration of electron charge.

We see that

• $PE(\psi )=\sum_{k\neq j}\int_{\Omega_j}\int_{\Omega_k} \frac{\psi_k^2(y)\psi_j^2(x)}{2\vert x-y\vert}\, dx\, dy,$

and understand that $k\neq j$ expresses that an electron does not interact with itself and that the factor 2 accounts for the double presence in the sum over all $k\neq j$.

A minimising wave function $\Psi =\sum_j\Psi_j\in H^1(\Re^3)$ satisfies the following system of one-electron Schrödinger equations expressing vanishing of the gradient of $TE(\psi )$ with respect to free variation of $\psi_j$ over $H^1(\Omega_j)$ under the side condition $\int\psi_j^2\,dx=1$:

• $H_j\Psi_j\equiv -(\frac{1}{2}\Delta+W+2\sum_{k\neq j}V_k)\Psi_j = E_j\Psi_j\,\mbox{in }\Omega_j,\, \frac{\partial\Psi_j}{\partial n_j}=0\,\mbox{on }\Gamma_j$    (4)

for $j=1,...,N$, where $E_j$ acts as a Lagrange multiplier for $\int\Psi_j^2\,dx=1$.

We observe the presence of the homogeneous Neumann condition

• $\frac{\partial\Psi_j}{\partial n_j} = 0 \,\mbox{on}\,\Gamma_j$,

as a variationally imposed condition reflecting free variation of $\psi_j$ in $\Omega_j$. Further, the factor 2 in $2\sum_{k\neq j}V_k$ reflects the presence of $\Psi_j$ in the equations for $\Psi_k$ with $k\neq j$ through the potential $V_j$.

We can thus formulate the effective boundary condition to be satisfied on the free boundary $\Gamma$ as follows:

• $\Psi \,\mbox{is continuous and}\, \frac{\partial\Psi}{\partial n} = 0$,

with $n$ normal to $\Gamma$. The free boundary thus carries both a homogeneous Neumann condition and a Dirichlet condition asking continuity which makes a connection to what is referred to as a Bernoulli free boundary problem for the Laplacian in a domain with a combined Neumann and Dirichlet condition on a part of the boundary.

We observe that the total energy $TE(\Psi )$ of a minimising $\Psi =\sum_j\Psi_j$ with eigenvalues $E_j$ is not given as $\sum_jE_j$ because of the factor 2 of the electronic potential in (4).