- $\psi (x,t) = \sum_{j=1}^N\psi_j(x,t)$ (1)

- $i\dot\psi_j + H\psi_j = 0$ in $\Omega_j$, (2a)
- $\frac{\partial\psi_j}{\partial n} = 0$ on $\Gamma_j(t)$, (2b)

- $H = -\frac{1}{2}\Delta - \frac{N}{\vert x\vert}+\sum_{k\neq j}V_k(x)$ for $x\in\Omega_j(t)$,

with $V_k(x)$ the repulsion potential corresponding to electron $k$ defined by

- $V_k(x)=\int\frac{\psi_k^2(y)}{2\vert x-y\vert}dy$, for $x\in R^3$,

and the electron wave functions are normalised to unit charge of each electron:

- $\int_{\Omega_j(t)}\psi_j^2(x,t) dx=1$ for $j=1,..,N$ and all time. (2c)

The differential equation (2a) with Neumann boundary condition (2b) is complemented by the following global free boundary condition:

The ground state is determined as a the real-valued time-independent minimiser $\psi (x)=\sum_j\psi_j(x)$ of the total energy

- $\psi (x,t)$ is continuous across inter-electron boundaries $\Gamma_j(t)$. (2d)

The ground state is determined as a the real-valued time-independent minimiser $\psi (x)=\sum_j\psi_j(x)$ of the total energy

- $E(\psi ) = \frac{1}{2}\int\vert\nabla\psi\vert^2\, dx - \int\frac{N\psi^2(x)}{\vert x\vert}dx+\sum_{k\neq j}\int V_k(x)\psi^2(x)\, dx$,

under the normalisation (2c), the Neumann boundary condition (2b) and the free boundary condition (2d).

In the next post I will present computational results in the form of energy of ground states for atoms with up to 54 electrons and corresponding time-periodic solutions.

In summary, the model is formed as a system of one-electron Schrödinger equations on a partition of 3d space depending of a common spatial variable and time, supplemented by a homogenous Neumann condition for each electron on the boundary of its domain of support combined with a free boundary condition asking continuity of charge density across inter-element boundaries.

We shall see that for atoms with spherically symmetric electron partitions in the form of a sequence of shells centered at the kernel, the Neumann condition corresponds to vanishing kinetic energy of each electron on the boundary of its support as a condition of separation between different electrons meeting with continuous charge density.