Hybrid wave functions

There is no unequivocal answer to the question as to which is the better method. Calculations by the VB method are likely to be more reliable than those by the MO method, but in practice are much more difficult to carry out. For many-electron molecules the MO procedure is simpler to visualize because we combine atomic orbitals into molecular orbitals and then populate the lower-energy orbitals with electrons. In the VB method, atomic orbitals are occupied, but the electrons of different atoms are paired to form bonds, a process that requires explicit consideration of many-electron wave functions. To put it another way, it is easier to visualize a system of molecular orbitals containing N electrons than it is to visualize a hybrid wave function of N electrons. 

It is easy to show that hybrid wave functions are orthonormal, that is,... 

Pj I //1 Pj> these values could differ somewhat in the solid from the corresponding values defined for the free atom but we will use free-atom values. The measurement of the energy of an electron described by such a hybrid wave function would give fig 25 percent of the time and Sp 75 percent of the time. 

To do so, a hybrid wave function (the augmented plane wave) is constructed, composed of atomic functions (called partial waves, with spherical harmonics Yii ) inside the atomic, muffin-tin sphere Tmt and a single plane wave outside the atomic sphere, that is... 

We now come to excited states. Using a hybrid wave function-density approach and an interpretation of exchange proposed by other workers, we have been able to calculate excited state energies and densities of several hundred excited states of various atoms. These were singly, doubly, and triply excited states, autoionizing states, satellite states, etc., and involved both small and large energy differences. 

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 un hybridized 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 electrons are most likely to be found. 

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