Remarks on the Theory of Aromatic Free Radicals

In the quantum-mechanical treatment of the benzene molecule (and of other aromatic molecules) the approximation is usually made of neglecting the pairs of electrons which are considered to form bonds in the plane of the ring. The problem then becomes one of discussing the wave function which can be constructed for the remaining electrons by use of the p, orbitals, one for each carbon atom. This discussion can be made either by the molecular orbital method or by the valence bond method, the results being in general agreement. The treatment of a free radical such as phenylmethyl is closely similar, seven p, orbitals and seven electrons being considered for this molecule. It is found convenient in applying the valence bond method' to introduce a "phantom orbital" and an additional electron, a procedure which simplifies the treatment without changing the results

The Quantum-Mechanical Treatment of Molecules by the Method of Spin Valence

The method of spin valence, developed by Born, Heitler, Rumer and Weyl for the treatment of molecular systems, becomes completely equivalent to the Slater method if the bonding forces are considered to be exerted between individual orbitals, and not between entire atoms in definite spectroscopic states. This change in the point of view eliminates several difficulties which were formerly present in the method (e.g., the obscuring of the directional properties of the valence bonds). The secular equation, for which a simplified derivation is given, is of quite a different form from the Slater equation, and the problem of its solution is considered in some detail. In addition to the rigorous methods, several approximate methods are given, of which two (applicable to unsaturated and aromatic hydrocarbons) do not require that the secular equation be set up at all. The method of spin valence is found to be more convenient than the Slater method for the treatment of systems of high multiplicity. Finally, the extension to cases of orbital, as well as spin degeneracy is briefly outlined

Solving quadratic equations over polynomial rings of characteristic two

We are concerned with solving polynomial equations over rings. More precisely, given a commutative domain A with 1 and a polynomial equation an tn + ··· + a0 = 0 with coefficients ai in A, our problem is to find its roots in A. We show that when A = B[x] is a polynomial ring, our problem can be reduced to solving a finite sequence of polynomial equations over B. As an application of this reduction, we obtain a finite algorithm for solving a polynomial equation over A when A is F[x1,... ,xN ] or F(x1,... ,xN ) for any finite field F and any number N of variables. The case of quadratic equations in characteristic two is studied in detail

A Calculation of the Equilibria in Keto-Enol Tautomerism

A calculation of the equilibria obtaining in some simple cases of keto‐enol tautomerism has been carried through with the aid of bond and resonance energies derived from empirical data. The results obtained are in qualitative and semi‐quantitative agreement with experiment

The Quantum Mechanics of Unsaturated and Aromatic Molecules: A Comparison of Two Methods of Treatment

A comparison is made of two quantum‐mechanical methods of treatment of unsaturated and aromatic organic molecules. The HLSP treatment, which is based upon the use of Slater valence‐bond eigenfunctions, seems to give results in somewhat better general agreement with experiment than the HMH, which is based upon the use of molecular orbitals of the Hund‐Mulliken type. The latter, however, can be extended to a wider variety of problems. The problems considered include the energy relations among hydrocarbons, the dissociation of aryl substituted ethanes, the electron‐affinity of free radicals and the acid strength of hydrocarbons