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Linear Combinations of Atomic Orbitals:

Molecular orbital theory concerns itself with finding the best possible representations of the electron density surrounding the nuclei of molecules. The reason approximate solutions are used, instead of the exact solutions as derived from the Schrödinger Equation, is a result of the difficulty involved in analytically solving the Schrödinger Equation for increasingly more complex systems. The more nuclei and electrons present in any given molecule, the more interactions which must be accounted for in order to derive good molecular orbitals (MO's) and energy states for the molecule. The general hamiltonian for any molecule:

For anything larger than the hydrogen molecule ion (H₂⁺), the eigenfunctions of this hamiltonian cannot be solved for symbolically. One needs an approximation which is close to the actual structure. One solution which has proven to be a reasonable start in predicting molecular electronic structure is the Linear Combination of Atomic Orbitals model (LCAO). This one assumes the electron orbitals for a given molecule are linear combinations of the atomic orbitals surrounding each of the nuclei of the molecule. As a first example, molecular hydrogen is considered. These are the bonding and antibonding molecular orbitals of molecular hydrogen, the most simple molecule which can be constructed.

Notice the wave functions of each of the hydrogen atoms (1s) can either interfere constructively or destructively.

s: E(742 pm)= -2.8 eV(12)

The total energy for the bonding state approaches a minimum when the atoms are 742 pm apart. In contrast, the energy of the antibonding state does not approach a minimum for finite separations of the atoms. This means that in the case of the antibonding state, the hydrogen atoms will not remain at a defined distance relative to each other, and will, upon close approach, repel each other away.

This is the probability distribution for the ground state of hydrogen fluoride.

H-F: 0.918x10^-10 m, 565 kJ/mol (9)

The fluorine 2px and 2s orbitals have the correct symmetry to overlap with the hydrogen 1s orbital. Fluorine is the most electronegative element of all, strongly withdrawing electron density through any bond it may form. Due to this, fluorine is not nucleophilic and makes a poor leaving group. Hydrogen fluoride is a gas at standard conditions. It has been referred to as the most non-ideal gas known, readily forming polymers of various molecular weights in the vapor phase.

This is the probability distribution for the ground state of BCl₃.

B-Cl: 1.75x10^-10 m, 456 kJ/mol (9)

The 3px, 3py, and 3s orbitals on each of the chlorine atoms overlap to produce sp2 hybridized orbitals. Each of these in turn overlap with atomic orbitals on boron forming s bonds which is the blue area in the contour diagram. Due to the electronegativity difference between chlorine and boron, most of the electron density resides close to the halogens.