Orbital hybridisation

A hybrid orbital is an orbital that arises mathematically from a linear combination of the wave functions of the basic atomic orbitals. This modeling process is called hybridization of orbitals. The concept was developed by Linus Pauling in 1931.

Mechanism

The orbitals generated by calculating the probability of each of the electrons do not always coincide with the result of chemical properties then unknown forms. Thus it is found that the carbon atom two s and two p- electrons in the outer electron shell has. Accordingly these orbitals would at the CH bonds in methane ( CH4) lead to different bonds. In fact, however, it is found that the four bonds are identical and indistinguishable. This can be explained by sp3 hybridization: The doubly occupied, spherical 2s orbital is with the dumbbell-shaped 2p orbitals ( singly occupied 2, one vacant ) to four equal, clavate sp3 hybrid orbitals combined, which are occupied with one electron each. These are based on tetrahedral in space and form with the 1s electron of the hydrogen atom similar bonds. The resulting model is equivalent to the observed properties of methane.

Carbocations ( carbenium ions ) are basically sp2 hybridized, so the empty orbital has 100 % p character and thus "wasted" no s- fraction ( Bencheh rule).

Physical interpretation

Orbitals are obtained as solution of the Einelektronenproblems ( hydrogen atom) in quantum mechanics. The orbitals of the same cup are degenerate with respect to the principal angular momentum. This means that every superposition of wave functions of the same dish with different angular momentum is again a solution, and thus describes a possible orbital. While the degeneracy is lifted in a free atom by the spin-orbit interaction and thus orbitals with different total angular momentum have different energy eigenvalues ​​, raises in molecules, the influence of the electric fields of neighboring atoms, the orbital degeneracy. In molecules thus appear as hybrid orbitals energetically more favorable overlapping of orbitals with different orbital angular momentum.

The hybrid orbitals form as all orbitals involved in atomic bonds by mixing with the orbitals of neighboring molecules molecular orbitals.

Mathematical viewing

To solve the one-electron Schrödinger equation the product approach is usually done, a procedure which is often useful for solving differential equations of 2nd order. The wave function Ψ, which is first arbitrarily from spherical coordinates (r, θ, φ ) is composed written as a product.

This will erase all solutions, which can not be written as such a product. Solutions that are lost when product approach can be recovered by a linear combination of the resulting solution. Any linear combinations at the same energy (and thus the same principal quantum number ) are also exact solutions of the one-electron Schrödinger equation because of the linearity of the Hamiltonian. They are called hybrid orbitals.

Be, and the wave functions of s-, p-and d- orbitals of the same quantum number. The wave function of a hybrid orbital is then formed as follows.

The superscript indicates the squared proportion of the atomic orbital on the hybrid orbital again.

The wave functions in more complex systems behave similarly. There, however, the wave functions of different principal quantum number are combined. Similar energy is crucial.

If a complete set of hybrid orbitals are formed, it should be noted that the transformation matrix is ​​a unitary matrix (real Special case: orthogonal matrix ) has to be. This means that the hybrid orbitals back must form an orthonormal basis. The associated scalar product is:

Examples

• A linear (180 °)
• Sp hybridization
• Eg CO2

• Bend (90 °)
• Sd hybridization
• Eg VO2

• Trigonal planar (120 °)
• Sp2 hybridization
• Eg, BF 3, graphite

• Trigonal- pyramidal (90 °)
• Sd2 hybridization
• Eg CrO3

• Tetrahedral ( 109.5 ° )
• Sp3 hybridization
• For example, diamond

• Tetrahedral ( 109.5 ° )
• Sd3 hybridization
• Eg MnO4 -

• Tetragonal- pyramidal (66 °, 114 ° )
• Sd4 hybridization
• Eg Ta ( CH3) 5

• Trigonal- prismatic (63 °, 117 ° )
• Sd5 hybridization
• For example, W ( CH3) 6

• Non-integer hybridization ( s and d orbitals )
• For example, Fe ( CO) 5

• Non-integer hybridization ( s and three d orbitals )
• For example, V ( CN) 74 -

• Non-integer hybridization ( s and three p orbitals )
• Eg IF8 -

• Non-integer hybridization ( s and four d orbitals )
• For example, Re ( CN) 83 -