Spin Density Matrices for Paramagnetic Molecules

The density of electron‐spin angular momentum (erg‐sec/cm³) at a point x, y, z in a paramagnetic molecule is ℏ S_zρ(x, y, z), where S_z is the z component of the total electron spin angular momentum in units of ℏ. If the exact wave function for the molecule Ψ is built up from one‐electron basis functions ψ_i, then the normalized spin density function ρ(x, y, z) can be expressed in terms of the ψ_i and the spin density matrix ρ_ij $$\mathrm{\rho (x,y,z)=}{\displaystyle \sum _{\mathrm{ij}}}{\rho }_{\mathrm{ij}}{\psi }_j^{*}{\psi }_i.$$ If the ψ_i are molecular orbitals, then ρ_ij is the molecular orbital spin density matrix; if the ψ_i are atomic orbitals, then ρ_ij is an atomic orbital spin density matrix. The equation of conservation of spin is $\mathrm{Tr(}\mathfrak{S}\text{ρ)=1}$ where $\mathfrak{S}$ is the non‐orthogonality matrix, ${\mathfrak{S}}_{\mathrm{ij}}{\mathrm{=〈\psi }}_i{\mathrm{|\psi }}_j〉$ . The elements of the spin density matrices are sensitive to overlap and electron correlation effects. This is illustrated by calculations of the spin density matrices of the ethylene positive ion radical, and of the allyl radical. The spin density matrix for the latter molecule is calculated using (a) the simple molecular orbital approximation, (b) the simple valence bond approximation, and (c) the complete π‐electron interaction configurational treatment given by Chalvet and Daudel. It is shown that the general connection between σ‐proton hyperfine splittings a_N and the π‐electron atomic orbital spin density matrix ρ_NN′, involves a (hyperfine interaction)‐(exchange interaction) matrix Q_N′N″″^N such that $$a_N{\mathrm{=Tr(Q}}^N\mathrm{\rho ).}$$ However, for practical calculations, observed hyperfine splittings may be used to estimate the diagonal elements of the spin density matrix, $$a_N{\mathrm{\approx Q\rho }}_{\mathrm{NN}},$$ where Q has the semiempirical value of —63 Mc or —23 gauss