Linear combination of wavefunctions

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

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


Because the Hamiltonian operator \hat{H}=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx}+V(x) is a linear operator, a linear combination of \phi_n is a solution to \hat{H} if \{\phi_n\} are solutions to \hat{H}. However, it is only meaningful to solve the eigenvalue equation \hat{H}\psi=E\psi, not just finding solutions to \hat{H}. The problem is \psi (e.g. \psi=c_1\phi_1+c_2\phi_2) is not an eigenfunction of \hat{H}:

\hat{H}(c_1\phi_1+c_2\phi_2)=c_1\hat{H}\phi_1+c_2\hat{H}\phi_2=c_1E_1\phi_1+c_2E_2\phi_2\neq E\psi\; \; \; \; \; \; \; \; 42

The terms on the RHS of the 2nd equality of the above equation cannot be expressed as E\psi, which implies that a state described by \psi does not have a well-defined eigenvalue. Nevertheless, we can find the average value (or expectation value) of \psi. This means that for wavefunctions that are not eigenfunctions of \hat{H}, we can use approximations like \langle H\rangle to find E. The exception to eq42 is if all the basis functions describe a degenerate eigenstate, where E_1=E_2=E:

\hat{H}(c_1\phi_1+c_2\phi_2)=c_1E\phi_1+c_2E\phi_2= E(c_1\phi_1+c_2\phi_2)

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:

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



Next article: postulates of quantum mechanics
Previous article: eigenfunctions and eigenvalue equations
Content page of quantum mechanics
Content page of advanced chemistry
Main content page

Leave a Reply

Your email address will not be published. Required fields are marked *