Suspicious energy and eigenvalue for complex-valued wavefunction

[zhouf]

I am running VASP (compiled without "realmode") with LDA+U type 1 (Liechtenstein, rotationally invariant version) for an isolated Zr3+ ion with only one valence electron. I got several self-consistent solutions in difference atomic orbitals.

In the first two cases, the occupation matrix in the INCAR (LDAUPRINT = 1) is diagonal with (1,1) or (3,3) being 1 and all other elements 0. In other words, just one electron occupying one atomic state xy or z^2.
o = 0.93107 v = 1.0000 0.0000 0.0000 0.0000 0.0000 (first case, xy)
o = 0.92996 v = 0.0000 0.0000 -0.9878 0.0000 0.0000 (second case, z^2)

The third case however, has a complex-valued wavefunction xy+ I(x^2-y^2), or complex spherical harmonic function Y22.
o = 0.92891 v = -0.7071 0.0000 0.0000 0.0000 0.0000 (0.0x5) 0.0000 0.0000 0.0000 0.0000 0.7071 (0.0x5) (third case, Y22)

The LDA+U double counting energy for eigenvalues are respectively
E(eignv.) = 1.30401113885989
E(eignv.) = 1.29970211640269
E(eignv.) = 1.28569901280122
Therefore the eigenvalue shift due to LDA+U are essentially the same for real or complex valued wavefunctions, which is reasonable. And I expect the calculated eigenvalue and total energy to be not too different.


However, the energy eigenvalue in these three cases are
1 -20.9075 1.00000
1 -20.8413 1.00000
1 -20.5606 1.00000
and total energy
26.6999
26.7272
26.9098

Compared to real wavefunctions (xy or z^2), the complex wavefunction (Y22) is too high in energy. If anything, Y22 should be lower since its self-interaction is smaller. So what is wrong? All LDA+U double counting energies, including e-e, LSDA and eigenv. are very close, but the calculated eigenvalue and PAW double counting energies do not make sense ...

I have tested this problem on several ions with one d or f electrons and the complex valued wavefunctions always give solutions about 200-300 meV above the real-valued orbitals, while degeneracy or even the reserve is expected.

thanks for your help



<span class='smallblacktext'>[ Edited ]</span>

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