The effects of AO hybridization on gross AO and overlap populations for LCAO‐MO electron configurations are discussed in terms of some simple examples, using equations and graphs. The validity of ``gross AO populations'' as true measures of the population in various AOs is critically discussed. It is shown that the degree of hybridization in the AOs of an LCAO MO does not in general give the true amount of s or p character in the MO; this is given instead by the gross s or p population in the MO. Nevertheless, ``forced hybridization'' among the AOs, although not contributing to gross AO populations, leads to important negative contributions to overlap populations, hence to bond energies. For example, forced 2s—2p hybridization has the result that the total overlap population for two pairs of electrons occupying bonding σ LCAO MOs built from 2s and 2pσ AOs in a homopolar diatomic molecule is actually less than the sum of the overlap populations which one would have if one pair could be in an MO built from pure 2s AOs and the other in an MO built from pure 2pσ AOs. This sort of forced hybridization with resultant loss of bond strength may explain why double bonds seldom if ever consist of two σ bonds.

1.
R. S.
Mulliken
,
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23
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1883
 
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(a) The elements of the R matrix of
P. O.
Löwdin
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J. Chem. Phys.
19
,
1572
(
1951
) are the same as the “atom” and “bond charges” of McWeeny (1951, reference 3 of Paper I) and the “net‐atomic” and “overlap” populations here, whose prototypes are used in reference 2 of I.
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(b) The quantities here called “gross atomic populations,” used by the writer in 1949 (reference 6 of I), were called “formal charges” by Chirgwin and Coulson (1950, reference 6 of I) and by McWeeny (1951, reference 3 of I).
(c) For some applications, see
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2.
It is not absolutely necessary even to use orthogonal MOs, but if they are not used, the same consequences for molecular properties arise as if they were used, but arise in forms which are less convenient to deal with, and which in practice have often suggested approximations that have resulted in misleading conclusions. SCF‐LCAO MOs are always mutually orthogonal.
3.
R. S.
Mulliken
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900
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4.
This statement is subject, however, to the reservation that, since all these states are of the same (1∑g+) symmetry, one must take into account strong configuration interactions among them in seeking better approximations to the actual molecular states. However, these interactions are not of interest here and will be neglected in the following discussion.
5.
All σu MOs are orthogonal by symmetry to all σg MOs, hence both states (mσu)2 are automatically orthogonal to both states (mσg)2.
6.
R. S.
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19
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912
(
1951
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7.
See reference 6, Figs. 1 and 2. It seems possible that this exclusion of unreasonable N(rk) values may be rigorous for states of actual molecules, but an investigation and possible proof is beyond the scope of the present paper.
8.
The closure is of course an arbitrary simplification. More accurately, all possible σ AOs should be admitted, and must be admitted if one goes to excited states requiring higher MOs 4σg, 5σg,⋯or 4σu, 5σu⋯.
9.
In the paragraph following Eq. (8), two alternative assumptions for simplifying the mσg and mσu MOs were discussed, of which the second (⁠μ = 0 for m = 2 and 3, which as a corollary requires α and β not zero for m = 1⁠) was adopted. In a previous paper,6 the first alternative (⁠α = β = 0 for m = 1, which requires μ≠0 for m = 2 and 3) was chosen. For any given complete electron configuration, the computed effect of 1s, 2s and 1s, 2pσ forced hybridization on the overlap population is necessarily the same by either choice.
10.
They are, however, subject to the requirement that the energy of the state in which they occur shall be minimized; with the further requirement, in the case of excited states, that their wave functions shall be orthogonal to those of any lower‐energy states of the same species.
11.
Here, as in general for one of any fully occupied complete closed set of MOs of a given species, there is no energy‐minimization limitation on the form of 2σg. The total energy is independent of the hybridization in 2σg, provided only that 1σg, 2σg, and 3σg are all orthogonal.
12.
C. W.
Scherr
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569
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1955
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13.
As a simple example of extra‐valence‐shell free hybridization, that of 2pπ‐3dπ hydridization in Cl2 may be cited [see
R. S.
Mulliken
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© 1955 American Institute of Physics.
1955
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