Roothaan equation is the matrix version of Hartree-Fock equations. The derivation is direct as follow. Hartree-Fock equation for the \(i^\mathrm{th}\) spatial orbital \(\varphi_{i}\) \(\hat{f}(1)\varphi_{i}(1)=\epsilon_{i}\varphi_{i}(1)\) Given a set of basis functions \(\left\{\chi_{\nu}\right\},\nu=1,2,\dots,m\), the HF orbitals can be expanded as \(\varphi_{i}=\sum_{\nu=1}^{m}{C_{{\nu}i}\chi_{\nu},i=1,2,\dots,m}\) Denote overlap matrix and Fock matrix as follows \(S_{\mu\nu}=\langle\chi_{\mu}|\chi_{\nu}\rangle\) \(F_{\mu\nu}=\left\langle\chi_{\mu}\left|\hat{f}\right|\chi_{\nu}\right\rangle\) HF equations are then transformed into […]
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