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

$\psi=\sum_{n=0}^{\infty}c_n\phi_n$

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 2^nd 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:

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.

Linear combination of wavefunctions

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