Angular-Momentum Conditions for a Correlated Wavefunction

We obtain the conditions under which a correlation wavefunction containing pair correlation functions for the 2pⁿ electrons of C(³P, ¹D, ¹S), N(⁴S), O(³P), and Ne will be an eigenfunction of the orbital and spin angular‐momentum operators. These pair functions contain quite general powers of electronic and interelectronic coordinates along with their variational parameters and should avoid the convergence difficulties of configuration interaction. In all these cases, except carbon which has just one pair of 2p electrons, one cannot obtain such a pair function, û_ij⁽¹⁾, just by minimizing its pair energy, ${\mathrm{\epsilon ̃}}_{\mathrm{ij}}^{\text{(1)}}$ , alone; i.e., minimizing the variational parameters of a û_ij⁽¹⁾. Minimization of one pair at a time corresponds to working with just a part of the correlation wavefunction which by itself cannot converge to the right symmetry state. One must minimize sums of pair energies by taking a correlation wavefunction containing enough pair functions so as to have the right symmetry. The sets of variational parameters in these different û_ij's are not independent but are coupled. These symmetry requirements now make the determination of pair correlation functions quite a bit more difficult. In carbon (³P), these conditions do not apply, but the û_ij⁽¹⁾ of the 2p² electrons of the ¹D multiplet must be made orthogonal to the unoccupied orbitals of the open shell if it is to contain general powers of the interelectronic coordinate. This is in addition to the required orthogonality to occupied orbitals. We also obtain a form of the correlation wavefunction for 2p²(¹S), a nonsingle‐determinant Hartree—Fock (HF) state. This has not been previously known. We do this by applying the angular momentum stepdown operator to the correlation wavefunctions of multiplet states with single Slater determinant HF states and studying the resulting forms of the correlation wavefunction which, in turn, belong to some nonsingle‐determinant states of these multiplets.