# Quantum mechanics/Molecular orbital theory

When it is not possible to calculate the wavefunction exactly (i.e. most of the time) several approximation methods are available. The most common is to express an approximate wavefunction as linear combination of basis functions, i.e.

$\Psi =c_{1}\varphi _{1}+c_{2}\varphi _{2}+...+c_{n}\varphi _{n}$ (Eq. 1)

Where the coefficients {c1, c2, ..., cn} are to be determined to make Ψ the best possible approximation of the ground or excited wavefunction. The electrons in molecules are described by molecular orbitals that are linear combinations of atomic orbitals (LCAO). A molecular orbital is expressed as Eq. 1 and {φ1, φ2, ..., φn} are just a set of atomic orbitals. Calculating a molecular orbital is equivalent to determining the coefficients in Eq. 1. This method is general and can be used to solve any quantum mechanical problem. The general theory is derived from a theorem called the variation theorem which can be applied to any quantum mechanical problem. Below we will see only a simplified version that illustrates how the coefficients {c1, c2, ..., cn} can be determined in the general case.

## What is the best linear combination that solves HΨ=EΨ ? Exercise

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