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While the square root of Dirac’s $$\delta$$ δ is not defined in any standard mathematical formalism, postulating its existence with some further assumptions defines a generalized function called $$\gamma (x)$$ γ ( x ) which permits a quasi-classical treatment of simple systems like the H atom or the 1D harmonic oscillator for which accurate quantum mechanical energies were previously reported. The so-defined $$\gamma (x)$$ γ ( x ) is neither a traditional function nor a distribution, and it remains to be seen that any consistent mathematical approaches can be set up to deal with it rigorously. A straightforward use of $$\gamma (x)$$ γ ( x ) generates several paradoxical situations which are collected here. The help of the scientific community is sought to resolve these paradoxa.

Perturbative correction to a wave function built from singlet-triplet mixed two-electron functions (geminals) is derived in the context of symmetry-adapted schemes, applying partial spin-projection. Imposing the constraint of strong orthogonality of geminals results in a reference function that captures static correlation in a computationally feasible way. In case of a lack of spin purification, the product of spin-unrestricted geminals is generally spin-contaminated, potentially undermining performance of a subsequent dynamic correlation treatment. In this work, spin symmetry of the reference is partially restored by half-projection in a variation-after-projection scheme. Applying perturbation theory (PT) to recover the missing part of electron correlation is hampered by the fact that an obvious choice for a zero-order Hamiltonian is not provided. The situation is amended by adopting the formalism of symmetry-adapted PT. The resulting framework is tested on singlet-triplet gaps of biradicaloids, and it is found to perform well in situations where its unprojected counterpart fails because of spin contamination.

Half-projection of a wave function built as a product of singlet-triplet mixed two-electron fragments (geminals) is explored. The condition of strong orthogonality is imposed between geminals, the initial Ansatz accommodating the unrestricted Hartree-Fock (UHF) wave function as a special case. The here explored geminal product is more general than UHF, it allowing for single covalent bond breaking in a spin pure manner. The strong correlation present in two geminals simultaneously is accompanied by spin contamination at the geminal product level. Variation after half-projection leads to an eigenvalue problem of an effective two-electron Hamiltonian, conserving the computational economy of the initial model. While the resulting expressions are shown to violate size consistency, quantification reveals a suppressed error with half-projection as compared to the full spin projected then varied scheme. Spin contamination is also found to be diminished compared to the initial, unprojected wave function.

While the square root of Dirac's