Authorea

For example, for a one-dimensional Li atom with an external potential \[v_{\rm ext}(x)=-\frac{3}{\sqrt{x^2+1}}\] and the soft Coulomb interaction, we obtain the states and energy eigenvalues given in table \ref{tab:young}.

Eigenstates for a one-dimensional lithium atom. The first and the fourth eigenstates show norms that are smaller than \(10^{-13}\) and \(10^{-11}\), respectively, for all diagrams. Hence, these states are bosonic and removed from any further calculations. The second and third states are energetically degenerate and correspond to diagrams b) and c) in Fig. \ref{fig:young}. The same is true for the fifth and sixth states.\label{tab:young}
State	Energy	Young diagram	Norm
1	-4.721	bosonic	\(<10^{-13}\)
2	-4.211	b)	0.2
3	-4.211	c)	0.6
4	-4.086	bosonic	\(<10^{-11}\)
5	-4.052	b)	0.4
6	-4.052	c)	0.7

If certain state energies are degenerate, the Young diagram “projection” contains an additional loop, ensuring that the same diagram is not used to symmetrize successive states: this would yield the same spatial part for each wave function in the degenerate sub-space. A given diagram is only used once in the sub-space, on the first state whose projection has significant weight.

The implementation also allows for the treatment of bosons, in which case the total wave function has to be symmetric under exchange of two particles. Here one will use a spin part symmetrized with the same Young diagram (instead of the mirror one for fermions), such that the total wave function becomes symmetric.

In order for the (anti-)symmetrization to work properly one needs to declare each particle in the calculation to be a fermion, a boson, or an anyon. In the latter case, the corresponding spatial variables are not considered at all in the (anti-)symmetrization procedure. One can also have more than one type of fermion or boson, in which case the symmetric requirements are only enforced for particles belonging to the same type.