Orbital Approximation to Spatial Eigenfunctions of the Many-Electron Hamiltonian

A new procedure is developed for calculating the best wavefunction for an N‐electron atom in terms of $N$ one‐electron orbitals which are eigenfunctions of $l^2{\mathrm{,\; l}}_z$ and $s_z$ . This is a special case of a method for calculating the best wavefunction in terms of $N$ orbitals unrestricted in angular momentum but restricted in spin component and is thus a partial generalization of Löwdin's EHF method. We calculate approximate spatial eigenfunctions of $H$ using the function $${\Phi }^{\mathrm{[\alpha ,\lambda ]}}{\mathrm{\hspace{0.17em}=\hspace{0.17em}D}}_{\mathrm{\lambda \lambda }}^{\mathrm{[\alpha ]}}{\displaystyle \sum _{\text{i\hspace{0.17em}=\hspace{0.17em}1}}^k}{\delta }_i{P}_i\mathrm{\phi ,}$$ where $\phi$ is a product of $N$ spatial orbitals and where the $P_i$ are the $k$ permutations out of which $k$ linearly independent projections onto the Young tableau $\mathrm{[\alpha ,\lambda ]}$ can be obtained. The method gives an improvement to each of Goddard's GI solutions in such a way that they all give the same energy when fully minimized, while only involving $\text{k−1}$ additional parameters. For Li atom there is only one additional parameter and calculations are performed using Goddard's GI and GF orbitals, obtaining small improvements in the energy. An energy of −7.4477076 a.u. is obtained.