Spin-orbit Density Research Articles

Method of Alternant Orbitals for Allyl

The π-electron correlation energy and ground-state wave function for allyl are calculated by the method of alternant orbitals. This method accounts for 98.8% of the correlation energy given by the configuration interaction treatment. The atomic orbital spin density matrix obtained with this approximation is also included.

Electron Spin Resonance of an Irradiated Single Crystal of Alanine: Second-Order Effects in Free Radical Resonances

The electron spin resonance of single crystals of l- and d-alanine has been observed at T=300°K and analyzed for different orientations of the crystal in the magnetic field and at several microwave frequencies ranging from 9 kMc/sec to 34 kMc/sec. The stable free radical produced by the irradiation is proved to be of the form CH3CHR, where R is a group which has no nuclei with detectable coupling. The hydrogens of the CH3 group of the radical are shown to have equivalent, isotropic coupling of 26 gauss each, essentially independent of the frequency of observation. This CH3 group coupling is interpreted as arising from s orbital spin density of the hydrogens, via hyperconjugation. The hydrogen of the CH group has both an isotropic, Fermi term, Af=20 gauss, arising from s orbital density on the hydrogen, and an anisotropic term Aμ=7 gauss arising from dipole-dipole interaction of the proton moment with the electron spin density, ρC, on the carbon. Although the signs of Af and Aμ could not be learned, they are shown to be of opposite sign. Hence the spin density is negative on either H or C. Principal values of the CH coupling are A1=7 gauss, A2=A3=27 gauss. The value of ρC is shown to be 0.80 approximately. For certain orientations of the crystal the CH coupling becomes equal to the CH3 coupling, and a quintet pattern is observed. Interesting second-order transitions are observed which for the [001] orientation become in the region of 24 kMc/sec as strong as the normal first-order transitions. A general theory is developed which accounts satisfactorily for these second-order effects which are probably of consequence in the electron spin resonance patterns of numerous other free radicals trapped within solids.

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 Szρ(x, y, z), where Sz 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 ψi, then the normalized spin density function ρ(x, y, z) can be expressed in terms of the ψi and the spin density matrix ρijρ(x,y,z)= ∑ ijρijψj*ψi.If the ψi are molecular orbitals, then ρij is the molecular orbital spin density matrix; if the ψi are atomic orbitals, then ρij is an atomic orbital spin density matrix. The equation of conservation of spin is Tr(Sρ)=1 where S is the non-orthogonality matrix, Sij=〈ψi|ψj〉. 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 aN and the π-electron atomic orbital spin density matrix ρNN′, involves a (hyperfine interaction)-(exchange interaction) matrix QN′N″″N such that aN=Tr(QNρ).However, for practical calculations, observed hyperfine splittings may be used to estimate the diagonal elements of the spin density matrix, aN≈QρNN,where Q has the semiempirical value of —63 Mc or —23 gauss