New Quantum Mechanics 5: Model as Schrödinger + Neumann

This sequence of posts presents an alternative Schrödinger equation for an atom with $N$ electrons starting from a wave function Ansatz of the form

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

as a sum of $N$ electronic complex-valued wave functions $\psi_j(x,t)$, depending on a common 3d space coordinate $x$ and a time coordinate $t$, with non-overlapping spatial supports $\Omega_j(t)$ filling 3d space, satisfying for $j=1,...,N$ and all time:

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

where $\Gamma_j(t)$ is the boundary of $\Omega_j(t)$, $\dot\psi =\frac{\partial\psi}{\partial t}$ and $H=H(x,t)$ is the (normalised) Hamiltonian given by

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

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

• $V_k(x)=\int\frac{\psi_k^2(y)}{2\vert x-y\vert}dy$, 

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 homogeneous Neumann boundary condition (2b) is complemented by the following global free boundary condition:

• $\psi (x,t)$ 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 homogeneous 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 spherical symmetry, together with ground state and dissociation energy for H2 and CO2 molecules in rotational symmetry.

In summary, the model is formed as a system of one-electron Schrödinger equations, or electron container model, on a partition of 3d space depending of a common spatial variable and time, supplemented by a homogeneous 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 homogeneous Neumann condition corresponds to vanishing kinetic energy of each electron normal to the boundary of its support as a condition of separation or interface condition between different electrons meeting with continuous charge density.

Here is one example: Argon with 2-8-8 shell structure with NIST Atomic data base ground state energy in first line (526.22), the computed in second line and the total energies in the different shells in three groups with kinetic energy in second row, kernel potential energy in third and repulsive electron energy in the last row. Note that the total energy in the fully filled first (2 electrons) and second shell (8 electrons) are nearly the same, while the partially filled third shell (also 8 electrons out of 18 when fully filled) has lower energy. The color plot shows charge density per unit volume and the black curve charge density per unit radial increment as functions of radius. The green curve is the kernel potential and the cyrano the total electron potential. Note in particular the vanishing derivative of charge density/kinetic energy at shell interfaces.