THE DANIELEWSKY METHOD FOR OBTAINING THE CHARACTERISTIC POLYNOMIAL OF A MATRIX: APPLICATION TO NORMAL CO-ORDINATE TREATMENTS OF $C_{3v}$ SUBSTITUTED METHANKS*
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Date
1963
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Ohio State University
Abstract
“Eigenvalues of the secular determinants arising in normal coordinate treatments of molecules are commonly obtained by the Jacobi method, or by the considerably faster Frame method. An IBM 1620 electronic computer requires 15 sec to compute the characteristic polynomial of a $6 \times 6$ matrix by the Danielewsky $method,^{1}$ compared with 75 see for the Frame method. Since the number of multiplications (and hence the time) required by the Danielewsky method is proportional to $N^{a}$ (N =dimension of the matrix), complied with $N^{a}$ for the Frame method, the former method is increasingly advantageous as the size of the matrix increases. Also, the Danielewsky method requires less computer memory space than the Frame method. It has the disadvantage that the algorithm by which it proceeds is more complex than those of either of the other methods. Using this and other computer proceeds, the effect of assuming tetrahedral structure in normal coordinate treatment of substituted methanes has been investigated. Using potential energy constants previously obtained in this laboratory, the wave numbers of 13 substituted methanes of symmetry $C_{2v}$ were recalculated, using (a) the observed internuclear distances and interbond angle obtained from microwave spectra, and (b) the same internuclear distances together with tetrahedral interbond angles. The average and maximum differences in the corresponding wave numbers in (a) and (b) were $8.7 cm^{-1}$ (0.67 per cent) and $53 cm^{-1}$ (4.1 per rent), respectively. It thus appears that the use of the observed interbond angles in normal coordinate treatments of such molecules is desirable.”
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$^{*}$Aided by the National Science Foundation. $^{\dag}$National Science Foundation Predoctoral Fellow. $^{1}$H. Wayland, Quart. Appl. Math. 81, 282 (1945).
Author Institution: Department of physics, Illinois Institute of Technology
Author Institution: Department of physics, Illinois Institute of Technology