On the solutions of the Schrodinger equation with some molecular potentials: wave function ansatz

Making an ansatz to the wave function, the exact solutions of the $D$% -dimensional radial Schrodinger equation with some molecular potentials like pseudoharmonic and modified Kratzer potentials are obtained. The restriction on the parameters of the given potential, $\delta$ and $\eta$ are also given, where $\eta$ depends on a linear combination of the angular momentum quantum number $\ell$ and the spatial dimensions $D$ and $\delta$ is a parameter in the ansatz to the wave function. On inserting D=3, we find that the bound state eigensolutions recover their standard analytical forms in literature.Comment: 14 page

ON THE MULTIPLICITY OF SOLUTIONS IN FORCE CONSTANTS AND NMR CALCULATIONS.

Author Institution: Czechoslovak Academy of Scienc

ACETYLENE SPECTRA IN THE 5 MICRON REGION

Author Institution: Department of Physics, The Pennsylvania State UniversityThe $\Pi_{u}-\Sigma^{+}_{g}$ bands $(2\nu_{4}+\nu_{3})^{1}$ at 1940.002 and $1959.697 cm^{-1}$ and $3\nu^{1}_{5}$ at $2169.166 cm^{-1}$ together with fourteen accompanying “hot” bands of $^{12}C_{2}H_{2}$ were measured using a high-resolution grating spectrometer equipped with an InSb photo-voltaie detector. The “hot” bands are the $\Sigma^{+}-\Pi, \Sigma^{-}-\Pi$ and $\Delta-\Pi$ transitions of $3\nu_{4}+\nu^{1}_{4}, 2\nu_{4}+2\nu_{5}-\nu^{1}_{5}\nu_{4}+3\nu_{5}-\nu^{1}_{4}$ and $4\nu_{5}-\nu^{1}_{5}$; in addition, the R-branches of the $\nu_{2}+\nu^{1}_{5}-\nu^{1}_{4}$ and $\nu_{2}+\nu^{1}_{5}-\nu^{1}_{4}\Pi-\Pi$ bands were also observed. The information obtained on the strongly interacting upper states involved in the transitions analyzed has made it possible to arrive at a reliable set of spectroscopic constants (anharmonic and rotation-vibration coupling constants) for the bending modes of the acetylene molecule

ANHARMONCITY OF VALENCE FORCES IN SOME LINEAR TRIATOMIC MOLECULES

$^{*}$On leave from Institute of Chemistry, Czechos lovak Academy of Science, Prague, Czechoslovakia. $^{1}$J. Pliva, Collection Czechoslov, Chem. Communs, 23, 777 (1958). $^{2}$ E. R. Lippincott and R. Schroeder. J. Chem. Physics 23, 1131 (1955).Author Institution: Division of Pure Chemistry, National Research Council OttawaIt was previously $shown^{1}$ that an empirical valence-force type potential function set up in terms of general valence-force displacement coordinates may be used for an adequate representation of the potential energies of simple polyatomic molecules an

OBSERVATION OF LARGE VIBRATIONAL l-TYPE RESONANCES IN THE S1 STATE OF BENZENE

Author Institution: Institut f\""{u}r Physikalische Chemie, TU M\""{u}nchen, Lichtenbergstrasse 4; Department of Physics, The Pennsylvania State UniversityImproved experimental methods have recently allowed us to analyze the structure of the fundamental bands in both the IR and UV spectrum of benzene with rotational resolution. Deviations from regular behavior could be well accounted for by anharmonic, Coriolis, and rotational l-type interactions. However, the spectra of multiply degenerate states are still incompletely understood. Large splittings of the various vibrational angular momentum components and the observed relative intensities remain to be explained. With sub-Doppler spectroscopy in a collimated beam, rotationally resolved spectra of the two vibronic bands which were previously assigned to the $6_{0}{^{1}} (\ell_{0}^{\pm 1})(l0_{0}1)10_{0}^{2}(\ell_{0}^{0})$ and $6_{0}{^{1}} (10_{1})10_{0}^{2}(\\ell_{0})$ transitions (Wilson numbering) were measured in the Munich laboratory. Each band could be separately described in terms of effective rotational and Coriolis $\zeta$-constants. However, the values of these effective band constants would imply unrealistically large deviations from planarity $(C_{\nu}-B^{\prime}_{v}/2$ of $-5.46 \times 10^{-4}$ and $6.72 \times 10^{-4}$ respectively for the two bands), and effective $\zeta$'s of -0.0083 and $+0.0656$ instead of the expected values $\mp \zeta_{0} = \mp 0.578$. We have therefore carried out a simultaneous analysis of the two bands with inclusion of a vibrational l-type interaction of the form $=C[(v+\ell+2)(v-\ell)(v\ell+2)(v-\ell)]^{1/2}$ in the Hamiltonian. The resulting constants, $C^{\prime}_{v}-B^{\prime}v$, of $1.6 \times 10^{-5} cm^{-1}$, and $\zeta_{eff}$ of -0.5248 and +0.5822, appear very reasonable; the coupling parameter C has the value $1.575 cm^{-1.}$ This result, and a similar one for the $6_{0}{^{1}} 16_{0}{^{2}}$ bands, shows the importance of vibrational $\ell$-type resonances in the multiply degenerate combination states of benzene. The vibrational angular momentum character of such states, which comprise the majority of all vibrational states of benzene at higher excess energies, is therefore not well defined. Important implications for the intramolecular vibrational redistribution and for the nonradiative decay of benzene states are expected to arise

CORRELATIONS AND ACCURACY OF ESTIMATION OF SPECTROSCOPIC CONSTANTS

Author Institution: Department of Physics, The Pennsylvania State UniversityA comparison and evaluation of the various methods for reducing spectroscopic data to spectroscopic constants or term values is made with the aid of an analysis of a number of ``synthetic” $\sum - \sum$ bands generated from fixed sets of constants with random noise superimposed on the line positions. It is shown that the strong correlations that exist between the upper state constants B^{\prime}, D^{\prime} and the lower state constants $B_{o}$, $D_{o}$ can be effectively broken up by using the difference constants $\Delta B$ = B^{\prime} - $B_{o}$ and $\Delta D$ = D^{\prime} - $D_{o}$ along with $v_{o}$, for representing the upper states. The lower state constants $B_{o}$ and $D_{o}$ and their standard errors calculated from the combination differences $\Delta_{2}$F$^{\prime\prime}$ (J) are shown to be as good as chose obtained from direct polynomial fits. If data for a number of bands originating in the same lower state are available, a considerable increase in accuracy of estimating the lower state constants can be attained by analyzing the bands simultaneously, e.g., using combination differences, provided the data are free from systematic errors. The dependence of the accuracy of determining the constants $B_{o}$, $D_{o}$, $\nu_{o}$, $\Delta$B, and $\Delta D$ on the extent of the band analyzed was investigated by varying the minimum and maximum J-values. The plots showing this dependence for both the actual errors and standard errors can be used e.g., to assess the band size necessary to attain a desired accuracy for a given constant

DECONVLVING INFRARED SPECTRA BEYOND THE DOPPLER LIMIT

Author Institution:In high-resolution molecular spectroscopy, the Doppler line width often poses a limitation on the resolution and thereby on the amount of information one can extract form the spectra. to enhance the effective resolution and attain effective line width smaller than the Doppler width, we have applied the deconvolution technique to the infrared spectra of medium size molecules (e.g., $C_{6}H_{6}$) measured with a difference frequency laser system, as well as to spectra of light molecules (e.g., $CH_{4}$) measured on a grating spectrometer. Using a Gaussian line shape function, we were able to enhance the resolution by a factor of 3 to 3.5 obtaining a linewidth of better than $1.5 \times 10^{-3} cm^{-1}$ in the deconvolved difference frequency laser spectrum. To test the reliability of the deconvolution routine, a synthetic many-line spectrum was generated and convolved with Gaussians of widths $4.3 \times 10^{-3} cm^{-1} (= 15$ points) and $1.4 \times 10^{-3} cm^{-1} (=5$ points) to match the linewidths of the measured and deconvolved spectra respectively. Deconvolution of the 15-point curve yielded a result almost indistinguishable from the 5-point curve. The effect of random noise and of slight background variations on the deconvolved curve was also studied. We conclude that, with proper care, quite reliable results can obtained by deconvolution of reasonably noise-free spectra