Explicit Spin Coordinates

Abstract

The recently established existence of spherical harmonic functions, $Y_\ell^{m}(\theta,\phi)$ for half-odd-integer values of $\ell$ and $m$, allows for the introduction into quantum chemistry of explicit electron spin-coordinates; i.e. spherical polar angles $\theta_s, \phi_s$, that specify the orientation of the spin angular momentum vector in space. In this coordinate representation the spin angular momentum operators, $S^2, S_z$, are represented by the usual differential operators in spherical polar coordinates (commonly used for $L^2, L_z$), and their electron-spin eigenfunctions are $\sqrt{\sin\theta_s} \exp(\pm\phi_s/2)$. This eigenfunction representation has the pedagogical advantage over the abstract spin eigenfunctions, $\alpha, \beta,$ that ``integration over spin coordinates'' is a true integration (over the angles $\theta_s, \phi_s$). In addition they facilitate construction of many electron wavefunctions in which the electron spins are neither parallel nor antiparallel, but inclined at an intermediate angle. In particular this may have application to the description of EPR correlation experiments.