The free-electron model for conjugated systems is consistently developed as the limiting case of a three-dimensional quantum-mechanical treatment of the π electrons in such systems. Joint conditions (for branching points) and boundary conditions (for free end points) are derived and the hermiticity of the Hamiltonian is shown. A matrix formulation of the theory is established which makes the application to large systems feasible, and at the same time leads to a close analogy with the LCAO model (LCAO MO treatment considering only nearest neighbor interactions). Quantities analogous to the quantities q (the charge in an atomic orbital) and p (the bond order) are defined, and special attention is given to alternant conjugated systems for which a population theorem, analogous to the one in LCAO theory, is valid.

The well-known perturbation expansion, £ ?tr w(Z)=Z 2 2 c-wz-S of the eigenvalues of the nonrelativistic Schrodinger equation for N electrons about a nucleus of charge Z, has been widely used in the past for the extrapolation and interpolation of atomic energies. The presence of many small effects not explicitly taken into account by the perturbation expansion analysis reduce such calculations to a process of empirical curve fitting of limited range and reliability. These small effects include relativistic effects, the mass polarization, and the Lamb terms; to a good approximation, these effect can also be expanded in a descending power series, but with a leading term containing Z 8 . On the basis of three plausible assumptions, theoretical approximations make it possible, in a semiempirical fashion, to remove a major portion of these small effects from the experimental data. In this way accurate values for e 2 (N) and good estimates for e s (N) have been obtained for 3^.N^ 10. These coefficients have been used to disclose inaccuracies and to fill gaps in the existing atomic energy data and to estimate electron affinities. GROUND STATE OF ISOELECTRON1CATOMIC SERIES 831

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A perturbation study of the NP states of two-electron atoms has been made. In particular, oscillator strength values for the (1*5, 2 i P) and (2 3 S, 2 3 P) transitions are obtained. The 2 i P and 2 3 P states are studied through ninth and tenth order, respectively. In addition, the N P and N Z Pstates are studied in first order through the 10P member of the series. Perturbation energy coefficients and other expectation values for several important operators have been computed. Perturbation energy coefficients for the 1*S state (through 25th order) and 2 3 S state (through 17th order) are also reported. Where comparison is possible, these results are in satisfactory agreement with the results obtained from variational calculations by C. L. Pekeris and co-workers. The variational-perturbation method for excited states requires auxiliary conditions on the perturbation wave functions. The condition on the «th-order wave function is derived here. This is a generalization of the first-order condition given by Sinanoglu.

The self-consistent-field molecular-orbital method in the LCAO (linear combination of atomic orbitals) approximation is applied to the ground-state configuration of the nitrogen molecule at the observed equilibrium internuclear separation. No other approximations are used; all electrons are included. The forms of the resulting molecular orbitals are discussed. The s-p hybridization is computed to be 26.1 percent in agreement with Mulliken's magic formula estimation of 21 percent. The quadrupole moment is computed to be —1.87 a.u. (experiment, 1.11 a.u. of undetermined sign). The ratio of the computed to the experimental total molecular energy is found to be 0.9905. Ionization potentials and single-electron excitation energies are computed and compared with experimental spectroscopic values. The spectral results are more encouraging than have been hitherto reported from calculations by the LCAO molecular-orbital method. The approximation of neglecting the inner-outer-shell mixing, as it has usually been done in computations of this type, somewhat changes the present results and is not recommended.