Hybrid atomic orbitals (Sections 1.12 and 1.15) An orbital that results from the mathematical combination of pure atomic orbitals, such as the combination of pure s and p orbitals in varying proportions to form hybrids such as s, sjp, and sp orbitals. [Pg.1159]

Hybridization (Section I.8B) The mathematical combination of two or more atomic orbitals (having different shapes) to form the same number of hybrid orbitals (all having the same shape). [Pg.1203]

Hybrid orbital (Section 1.8B) A new orbital that results from the mathematical combination of two or more atomic orbitals. The hybrid orbital is intermediate in energy compared to the atomic orbitals that were combined to form it. [Pg.1203]

In order to determine the mathematical form of the four sp hybrid orbitals, we can apply the projection operator method. Application of the projection operator for the A IRR on the cp basis function yields k[(p +02+ < 3+< 4] which Is identical to the result obtained in Equation (10.4) after the resulting wavefunction has been normalized and the appropriate s and p AOs have been substituted for the basis functions cp -cp. Likewise, application of the projection operator for the Tj IRR yields (after normalization) the three wavefunctions given by Equations (I0.5)-(I0.7). Thus, application of group theoretical methods to the methane molecule can determine not only which type of hybridization will occur but also the mathematical forms of the resulting hybrid orbitals. [Pg.265]

Use the projection operator method to determine the mathematical forms of the five dsp hybrid orbitals. [Pg.334]

Transformation properties of atomic orbitals 11-3. Hybrid orbitals for c-bonding systems 11 -i- Hybrid orbitals for >r-bonding systems 11-5. The mathematical form of hybrid orbitals [Pg.166]

To reduce the complex task of finding orbitals that fit VSEPR, we base their descriptions on mathematical combinations of standard atomic orbitals, a process called hybridization the orbitals thus formed are hybrid orbitals. The number of hybrid orbitals is equal to the number of standard valence atomic orbitals used in the mathematics. For example, combining two p-orbitals with one r-orbital creates three unique and equivalent sp s-p-two) hybrid orbitals pointing toward the vertices of a triangle surrounding the atom. [Pg.800]

The mathematical form of p AOs is such that any combination of p AOs gives an orbital of the same shape (i.e., another p AO), pointing in some intermediate direction. Each of the orbitals is therefore identical in shape, being drawn from the s AO and p AOs in similar proportions (i.e., 1 3). The shape of each of these hybrid orbitals is indicated in Fig. 1.19(a, b). The combination of the s AO with a p AO leads to a lopsided p-like orbital in which [Pg.28]

Molecular orbital (MO) theory describes covalent bond formation as arising from a mathematical combination of atomic orbitals (wave functions) on different atoms to form molecular orbitals, so called because they belong to the entire molecule rather than to an individual atom. Just as an atomic orbital, whether unhybridized or hybridized, describes a region of space around an atom where an electron is likely to be found, so a molecular orbital describes a region of space in a molecule where an electron is most likely to be found. [Pg.20]

So in all cases exactly four orbitals are present and add up to the equivalent of one s and three p, even though each type of hybridization leads to a form of bonding and molecular shape very different from any of the others. As these examples show, the mathematical nature of hybridization is very flexible, to maximize favorable bonding attractions and minimize unfavorable electron-electron repulsions. [Pg.533]

To begin with, we recall that atomic orbitals are mathematical functions that come from the quantum mechanical model for atomic structure. oao(Section 6.5) To explain molecular geometries, we often 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. [Pg.359]

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