Groundstate Energy in Spherical Sym

Below you find the ground state energies produced by realQM with electrons filling a  spherical shell structure homogenised to spherical symmetry.  realQM then reduces to a wave equation in one radial variable r in terms of a wave function \psi (r) defined on a sequence of intervals/spherical shells with a certain total charge density /number of electrons in each interval/shell.  The ground state is computed iteratively by energy minimisation over each shell and and update of the free boundary separating shells to reach continuity. See below for details.

The basic shell structure is given by the sequence 2, 8, 18, 32,…, as the number of electrons in fully filled shells of increasing radius, that is with 2\times n^2 electrons i shell n. Here the factor 2 reflects the structure of Helium with two electrons filling two half-spherical shells meeting a separating plane as free boundary with a Bernoulli free boundary, with subsequent regular subdivision into n^2 subregions. As an example, the electron shell structure for Neon then shows to be 2+4+4 with the second full shell subdivided into two subshells with 4 electrons in each.

In general computed energies agree with observed  energies up to about three digits. Here you can inspect sample results with the shell structure indicated.

For an atom/ion with kernel charge Z and N electrons realQM takes the following form in spherical coordinates with the electrons filling an expanding sequence of spherical shells and homogenize the electron charge distribution in each shell into spherical symmetry of the same total charge. Find the wave function \psi (r) as a function of distance r>0 to the atom kernel of the form

  • \psi (r)=\sum_{j=1}^J\psi_j(r)

supported by a partition \Gamma :0=r_0<r_1<r_2<...<r_J=\infty of the interval (0,\infty) into intervals S_j=(r_{j-1},r_j) with \psi_j^2>0 in S_j and \psi_j = 0 outside, satisfying

  • \psi \,\,\mbox{is continuous,}\,\, \frac{\partial\psi}{\partial r}(r_j) = 0\,\mbox{for}\, j=1,...,J-1,\mbox{ and}\,\frac{\partial\psi}{\partial r}(0) = -Z\psi (0)    (1)

and the normalization condition with \sum_{j=1}^Jn_j=N and n_j>0 the number of electrons in shell $S_j$:

  • 4\pi\int_{S_j}\psi_j^2\, r^2dr = n_j\,\mbox{ for } j=1,...,J,   (2)

which minimises the total energy

  • TE(\psi )\equiv K(\psi )+PK(\psi )+PE(\psi ),


  • K(\psi ) =4\pi\,\int^\infty_0\frac{1}{2}(\frac{\partial\psi}{\partial r})^2r^2\, dr
  • PK(\psi )=- 4\pi\int_0^\infty\frac{Z}{r}\psi^2(r)r^2\, dr
  • PE(\psi )=4\pi\,\sum_j\int_0^\infty\sum_{k\neq j} V_k(r)\psi_j^2(r)r^2\, dr,


  • V_k(r)=2\pi\,\int_0^\infty min(\frac{1}{r},\frac{1}{s})c_k(r,s)\psi_k^2(s)s^2\,ds \quad\mbox{for}\quad r>0,       (3)

where c_k(r,s)=\frac{n_k-1}{n_k} for r,s\in S_k is a reduction factor due to lack of self-repulsion, and c_k(r,s)=1 else.

Recall that the potential P(r) generated by a spherically symmetric charge distribution of total charge C(s) at kernel distance s, is given by the formula P(r)=min(\frac{1}{r},\frac{1}{s})C(s) which motivates (1).

The boundary condition at r=0 reflects balance of \frac{1}{r}-terms in Schrödinger’s equation in spherical coordinates.

The ground state is computed iteratively by minimization of TE(\psi ) over \psi satsifying (1) and (2) by a gradient method with vanishing gradient condition

  • -\frac{1}{2}\frac{\partial^2\psi_j}{\partial r^2}-\frac{1}{r}\frac{\partial\psi_j}{\partial r}+W\psi_j+2\sum_{k\neq j}V_k\psi_j-E_j\psi_j=0\quad\mbox{in }S_j,

for j=1,...,J, where the E_j are Lagrange multipliers for the charge conservation (2) combined with  update of the free boundary \Gamma to reach continuity of \psi.