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From the formal point of view, it is important to stress that Mulliken’s overlap population is the simplest quantity that is linear in the interatomic D-matrix elements and has the correct rotational–hybridizational invariance. 3 L¨owdin charges and the problem of rotational invariance One often turns to a L¨owdin-orthogonalized basis set permitting to avoid the complications caused by the overlap and to keep at the same time the connection with the original (overlapping) AO basis: the orthonormalized functions obtained by using L¨owdin’s “symmetric orthogonalization”  are those which are the closest in least-square sense to the original (overlapping) basis [5,35].
18): both S−1 and L are Hermitian but their product is usually not. In other words, Hermitian operators are represented by non-Hermitian matrices in non-orthogonal basis sets. However, for these non-Hermitian matrices the eigenvalue equations of type Tqi = εi qi have all the properties which are characteristic of Hermitian matrices in orthonormalized basis sets: the eigenvalues are real and the eigenvectors form an orthogonal set spanning the whole linear space defined by the original basis orbitals.
In the present case Lˆ = ̺ˆ , its integrals form matrix L = SPS, as discussed in case (iii) above. Then the transformation matrix T describing the effect of ̺ˆ in the non-orthogonal LCAO framework can be obtained by multiplying with S−1 from the left, yielding matrix T = PS, in full accord with that described under point (ii) above. A characteristic feature of non-orthogonal basis sets is that the matrix T describing a linear transformation induced by a Hermitian operator Lˆ is generally not Hermitian, as can be seen from Eq.