## Tight binding model & LCAO intro Before we really enter the regime of [[Hartree-Fork method|Hartree-Fork method]], [[Secular equation|LCAO]] or [[Begin of DFT, Hohenberg-Kohn theorems|DFT]], we will first have a brief review on quantum mechanics. - Hamiltonian satisfies $\hat{H}\Psi = E\Psi$ - $H$ is a five term operator if we do not consider external field $\hat{H}=-\sum_{i} \frac{\hbar^{2}}{2m_{k}}\nabla_{i}^{2} - \sum_{k} \frac{\hbar^2}{2m_{k}}\nabla_{k}^{2} - \sum_{i} \sum_{k} \frac{e^{2}Z_{k}}{r_{ik}} +\sum_{i<j} \frac{e^{2}}{r_{ij}}+\sum_{k<l} \frac{Z_{k}Z_{l}e^{2}}{r_{kl}}$- We assume the wave functions of the problem are complete and orthonormal, $\int \Psi_i \hat{H} \Psi_j = \delta_{ij} E_i$- With [variational principle](https://en.wikipedia.org/wiki/Variational_method_(quantum_mechanics)) $\frac{\int\Psi \hat{H}\Psi dr}{\int\Psi^{2}dr}\geq E_{0}$ this means the **best** trial wave function to obtain the ground state wave function is the one with the lowest possible $\frac{\int\Psi \hat{H}\Psi dr}{\int\Psi^{2}dr}$, or computed energy. >[!Note] >In above variational principle, we are changing the wavefunction $\Psi$. This gives the foundation of LCAO, we use a linear combination of atomic orbitals with different coefficients, and by optimizing these coefficients we get the best wave function for a molecule. i.e.,$\psi =\sum_{i=1}^{N}a_{i}\psi_{i}$