Atomic orbitals mathematical origin

Fig. 4.5 Hybridization is forming new atomic orbitals, on an atom, by mathematically mixing (combining) original atomic orbitals on that atom. Mixing two orbitals gives two hybrid orbitals, and in general n AOs give n hybrid AOs. Orbitals are mathematical functions and so can be added and subtracted as shown in the figure...

The use of frozen orbitals, such as the bond orbitals connecting the quantum to the classical part of the system, can be extended to nonempirical quantum methods such as ab initio Hartree -Fock, post Hartree Fock, or DFT. In these cases, the overlap between atomic orbitals is taken into account and the orthogonality conditions are more difficult to fulfill. The mathematical formulation of the method has been developed in the original papers [26 28] and the process can be summarized as follows. 

To begin with, we recall that atomic orbitals are mathematical functions that come from the quantum mechanical model for atomic structure. (Section 6.5) To explain molecular geometries, we can assume that the atomic orbitals on an atom (usually the central atom) mix to form new orbitals called hybrid orbitals. The shape of any hybrid orbital is different from the shapes of the original atomic orbitals. The process of mixing atomic orbitals is a mathematical operation called hybridization. The total number of atomic orbitals on an atom remains constant, so the number of hybrid orbitals on an atom equals the number of atomic orbitals that are mixed. 

Although we must not lose sight of the fact that wave-functions are mathematical in origin, most chemists find such functions hard to visualize and prefer pictorial representations of orbitals. The boundary surfaces of the s and three p atomic orbitals are shown in Figure 1.9. The... 

Similar concerns apply to molecular orbitals. One constructs molecular orbitals and populates them with electrons is a manner analogous to an individual atom by adopting the linear combination of atomic orbitals (LCAO) approximation. While this might lend the impression that molecular orbitals are merely an extension of atomic orbitals, they are conceptually distinct. An atomic orbital is a description of the state of motion of an electron subject to the influence of a single nucleus plus other electrons. But molecular orbitals describe electron motions in the field of two or more nuclei plus the other electrons and the use of the LCAO method is merely a matter of mathematical convenience (Gavroglu and Simoes 2012, p. 83). The delocalized character of molecular orbitals is conceptually quite distinct from the idea of atomic orbitals, and Mulliken - one of the originators of the molecular orbital approach - was at pains to distinguish his conceptual scheme from the methods employed to compute them (ibid, pp. 84—85). 

The s and p orbitals used in the quantum mechanical description of the carbon atom, given in Section 1.10, were based on calculations for hydrogen atoms. These simple s and p orbitals do not, when taken alone, provide a satisfactory model for the tetravalent— tetrahedral carbon of methane (CH4, see Practice Problem 1.22). However, a satisfactory model of methane s structure that is based on quantum mechanics can be obtained through an approach called orbital hybridization. Orbital hybridization, in its simplest terms, is nothing more than a mathematical approach that involves the combining of individual wave functions for r and p orbitals to obtain wave functions for new orbitals. The new orbitals have, in varying proportions, the properties of the original orbitals taken separately. These new orbitals are called hybrid atomic orbitals. 

Orbital hybridization (Section 1.12) A mathematical (and theoretical) mixing of two or more atomic orbitals to give the same number of new orbitals, called hybrid orbitals, each of which has some of the character of the original atomic orbitals. 

Shell corrections can also be evaluated without recourse to an expansion in powers of v, but existing calculations such as Refs. [13,14] are based on specific models for the target atom and, unlike equation (19), do not end up in expressions that would allow to identify the physical origin of various contributions. It is clear, however, that orbital motion cannot be the sole cause of shell corrections The fact that the Bethe logarithm turns negative at 2mv /I< 1 cannot be due to the neglect of orbital motion but must be of a purely mathematical nature. Unfortunately, the uncertainty principle makes it impossible to eliminate orbital motion in an atom from the beginning. 

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