Knowledge Bank

The great difficulties inherent in a very accurate determination of the potential function of a polyatomic molecule from its infrared spectrum, have led us to develop an original method allowing the treatment of pure numbers as well as non-commutative quantum operators and to adhere very closely to the internal structure of modern computers in order to reduce computation time to a minimum. This method is based upon the principles of general algebra and has been applied to carbon dioxide. The passage from the expansion of the potential function with respect to exact internal coordinates to the vibrational levels is performed by means of a very large and extensive ALGOL program. The principal steps performed by this program are: 1)---The first expansion of the potential function with respect to internal coordinates is reexpanded with respect to normal coordinates (to the fourth order of approximation). 2)---The kinetic energy (vibrational contribution only) is then expanded with respect to normal coordinates using the commutation rules existing between these coordinates and their conjugate momenta. By adding this expansion to the preceding one we get the untransformed Hamiltonian (H) expansion with respect to normal coordinates. 3)---The perturbation treatment of this operator is then carried out by means of two successive contact transformations which are automatically performed taking into account all of the accidental resonances one wishes to consider. At the end of this step we obtain the normal-coordinates expansion of the twice-transformed Hamiltonian $H†$. 4)---Then the program computes the matrix elements of $H†$, and determines the spectroscopic constants of the molecule. 5)---With these constants the program constructs the factorized energy matrix and its diagonalization gives us the calculated vibrational levels and the initial norm. 6)---Having determined the partial derivatives of each calculated level with respect to the coefficients appearing in the original expansion of the potential function, the program computes the corrections to the initial values of the potential coefficients. The convergence is very fast. For most of the levels the differences between experimental and calculated energy values are within the limits of the experimental errors. Let us notice that: a)---the potential function we have presently at our disposal (and which was determined by using only $\mathrm{<mrow\; data-mjx-texclass="ORD">12</mrow>}C16O2$ data) allows a very good determination of the vibrational levels for $\mathrm{<mrow\; data-mjx-texclass="ORD">13</mrow>}C16O2$, and more generally, for all of the other symmetrical or non-symmetrical isotopic species. b)---despite the size of the program (5000 cards), one iteration is performed in two minutes with a 3600 C.D.C. computer (three times faster than a 7090 I.B.M. computer). This is accounted for by the fact that the program structure is very well suited to the computer structure. A very large part of the computation needs to be done only once and, moreover, gives results which are valid for all the triatomic linear molecules. The experimental results we have used in this computation were obtained in the past or more recently by C. P. Courtoy, T. K. McCubbin, Jr., K. Narahari Rao, and by W. S. Benedict who studied very extensively the Venus spectra recorded by J. and P. Connes two years ago.