This paper examines the vibronic energy levels of a symmetrical non-linear molecule in a spatially doubly degenerate electronic state which is split in first order by a doubly degenerate vibrational mode. The vibronic levels are classified by a quantum number, which in certain cases is formally related to the combined angular momentum of electronic and vibrational motion, and numerical values are obtained for the energies of these levels as functions of this quantum number and a dimensionless parameter measuring the magnitude of the electronic-vibrational coupling. It is shown that the selection rules for transitions from these vibronic levels to those of a non-degenerate electronic state allow changes in vibrational energy of any integral number of quanta, as though the Jahn-Teller effect were equivalent to a distortion which makes allowed vibrational transitions which would otherwise be forbidden. Numerical values are given for the oscillator strengths of vibronic absorption or emission bands involving transitions between a Jahn-Teller distorted state and an electronically non-degenerate state. It is found that in transitions from the latter to the former the vibrational structure of the electronic band exhibits two intensity maxima if the distortion is large.

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In analogy to Laplace's expansion, an arbitrary power rn of the distance r between two points (rI' {h,

Summary. An algebraic algorithm, the long quotient-modified difference (LQMD) algorithm, is described for the Gaussian quadrature of the one-dimensional product integral fl(x)w(x)dx when the weight function w(x) is known through modified moments v~ = f P/(x) w (x) d x; the P/(x) are any polynomials of degree l satisfying 3-term recurrence relations with known coefficients. The algorithm serves to establish the co-diagonal matrix, the eigenvalues of which are the Gaussian abscissas. Applied to ordinary moments it requires far fewer divisions than the quotient-difference algorithm; if the _P/(x) are themselves orthogonal with a kernel wo(;~), there is no instability due to rounding errors. For smooth kernels w (x) it is safe to use secondorder interpolation in determining the eigenvalues by Givens' method. The Christoffel weights can be expressed as ratios of two terms which are most easily calculated in a Sturm sequence beginning with the highest value of l. A formula for the Christoffel weights applicable for rational versions of the QR algorithm is also derived. Convergence and the propagation of rounding errors are illustrated by several examples, and an ALGOL procedure is given.

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