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The natural atomic orbitals for atom A in the molecular environment may be defined as those that diagonalize the DAA block, NAOs for atom B as those that diagonalize the DBB block, etc. These NAOs will in general not be orthogonal, and the orbital occupation numbers will therefore not sum to the total number of electrons. To achieve a well-defined division of the electrons, the orbitals should be orthogonalized.

The NAOs will normally resemble the pure atomic orbitals (as calculated for an isolated atom), and may be divided into a "natural minimal basis" (corresponding to the occupied atomic orbitals for the isolated atom), and a remaining set of natural "Rydberg" orbitals based on the magnitude of the occupation numbers. The minimal set of NAOs will normally be strongly occupied (i.e. having occupation numbers significantly different from zero), while the Rydberg NAO usually will be weakly occupied (i.e. having occupation numbers close to zero). There are as many NAOs as the size of the atomic basis set, and the number of Rydberg NAOs thus increases as the basis set is enlarged. It is therefore desirable that the orthogonalization procedure preserves the form of the strongly occupied orbitals as much as possible, which is achieved by using an occupancy-weighted orthogonalizing matrix. If all orbital occupancies are exactly 2 or 0, the orthogonalization is identical to the Lowdin method (eq. (9.8)). The procedure is as follows:

(1) Each of the atomic blocks in the density matrix is diagonalized to produce a set of non-orthogonal NAOs, often denoted "pre-NAOs".

(2) The strongly occupied pre-NAOs for each centre are made orthogonal to all the strongly occupied pre-NAOs on the other centres by an occupancy-weighted procedure.

(3) The weakly occupied pre-NAOs on each centre are made orthogonal to the strongly occupied NAOs on the same centre by a standard Gram - Schmidt orthogonalization.

(4) The weakly occupied NAOs are made orthogonal to all the weakly occupied NAOs on the other centres by an occupancy-weighted procedure.

Weakly occupied

(Rydberg)

Strongly occupied

Atom A Atom B

Figure 9.5 Illustration of the orthogonalization order in the NAO analysis

The final set of orthogonal orbitals are simply denoted NAOs, and the diagonal elements of the density matrix in this basis are the orbital populations. Summing all contributions from orbitals belonging to a specific centre produces the atomic charge. If is usually found that the natural minimal NAOs contribute 99+% of the electron density, and they form a very compact representation of the wave function in terms of atomic orbitals. The further advantage of the NAOs is that they are defined from the density matrix, guaranteeing that the electron occupation is between 0 and 2, and that they converge to well-defined values as the size of the basis set is increased. Furthermore, the analysis may also be performed for correlated wave functions. The disad

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