A * linear combination* of wavefunctions is the weighted sum of a complete set of basis wavefunctions .

It is commonly used as a technique in quantum chemistry to approximate a multi-electron wavefunction , where

Because the Hamiltonian operator is a linear operator, a linear combination of is a solution to if are solutions to . However, it is only meaningful to solve the eigenvalue equation , not just finding solutions to . The problem is (e.g. ) is not an eigenfunction of :

The terms on the RHS of the 2^{nd} equality of the above equation cannot be expressed as , which implies that a state described by does not have a well-defined eigenvalue. Nevertheless, we can find the average value (or expectation value) of . This means that for wavefunctions that are not eigenfunctions of , we can use approximations like to find . The exception to eq42 is if all the basis functions describe a degenerate eigenstate, where :

In general, a linear combination of wavefunctions (or a linear combination of atomic orbitals, * LCAO*) is a method of finding an easy or solvable solution to an otherwise complicated or unsolvable one. LCAO is often used in:

- Finding the energy of a system using the Hartree-Fock-Roothan method.
- Eliminating complex wave functions to find the formula of hydrogenic
*p*/*d*/*f*-orbitals that describe their corresponding*p*/*d*/*f*degenerate states. - Constructing wavefunctions that satisfy Pauli’s exclusion principle.