Reduced-Density-Matrix Theory: The 2-Matrix of Four-Electron Systems

The theory of reduced density matrices for polyelectronic systems is formulated in a manner such that the reduced density matrix of any order $p$ is characterized by a coefficient matrix. This matrix of coefficients, resulting from expressing the polyelectronic wave function in the appropriate bilinear form, is sufficient to allow one to find the eigenvalues and the transformation to natural form. This formalism is a generalization of the work of Löwdin and Shull on the natural orbitals of two-electron systems. The second-order reduced density matrix, the 2-matrix, is obtained exactly from the approximate solutions $\Psi$ of the Schrödinger equation for the Be-atom functions of Weiss, Watson, and Boys, and the LiH function of Ebbing. The important eigenfunctions and complete eigenvalue spectra of the integral operator ${\Gamma }^{\mathrm{}\left(2\right)\mathrm{}}$, which has the 2-matrix as kernel, are reported here. The degeneracies of the eigenvalue spectra of ${\Gamma }^{\mathrm{}\left(2\right)\mathrm{}}$ and the properties of the natural geminals, the eigenfunctions of ${\Gamma }^{\mathrm{}\left(2\right)\mathrm{}}$, are discussed in detail. The multiplicities 1, 2, 3, 4, and 6 are the only nonaccidental degeneracies that can occur in the 4-electron problem when the one-electron basis of $\Psi$ is considered in symmetry-adapted spin-orbital form. The natural geminals can always be obtained in symmetry-adapted form and can be completely described by a set of numbers ($\lambda ,s,m_s,m_l,\dots$), eigenvalues for the operators ${\Gamma }^{\mathrm{}\left(2\right)\mathrm{}},S^2,S_z,L_z,\dots$, respectively. The identity of the eigenvalue spectra and the equivalence of the two operators ${\Gamma }^{\mathrm{}\left(p\right)\mathrm{}}$ and ${\Gamma }^{\mathrm{}(N-p)\mathrm{}}$ are demonstrated in the case where the one-electron basis of $\Psi$ is finite. The natural expansion of $\Psi$ is defined as the expansion in eigenfunctions of ${\Gamma }^{\mathrm{}\left(p\right)\mathrm{}}$ and ${\Gamma }^{\mathrm{}(N-p)\mathrm{}}$. In the case $2p=N$, the phase of the two sets of eigenfunctions can be chosen as equal and the signs of the natural expansion coefficients are uniquely determined by the function $\Psi$.