In output of CASTEP population calculation, there are Mulliken analysis and Hirshfeld analysis, both show values of spin of atoms. These values are not the same. So which one should we use to conclude about values of spin of atoms?
Dear Nguyen TRUNG Hieu ,
From a chemical point of view, Hirshfeld analysis is better as it takes implicitly the electronegativity of atoms into account via pro-atomic (promolecular) densities, that Mulliken does not. As a clear example a diatomic molecule, for Mulliken, if you have an atom A and a atom B, described by phi_A and phi_B wavefunctions, the resulting density is (phi_A + phi_B)^2. the decomposition yields to phi_A^2 + 2 phi_A phi_B + phi_B^2. Mulliken analysis then gives to atom A the "phi_A^2" contribution and to the atom B the "phi_B^2" contribution. Concerning the mixed term "2 phi_A phi_B", it is equally divided to the two concerned atoms ; thus a "phi_A phi_B" on atom A, and, a "phi_A phi_B" on atom B, without taking account of the electronegativity of the atoms!
As an information, I know ORCA is performing Löwdin analysis for atomic spin, but this method resulting from orthogonalization just "hide" the crossed-terms and has, a priori, no physical value.
The Hirshfeld analysis the molecular density into atomic density contributions is tested. Mulliken analysis is the most important method to calculate atomic charges and is still widely used.
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