A study of the variation of the vibrational potential energy contribution with interbond angles in XY 3 pyramidal molecules confirms the observation previously made for XY 2 bend symmetric systems that the actual equilibrium configuration lies in the premises of minimum F bend and zero y stretch-bend * An analysis of the variation of the vibrational potential energy with geometry in simple molecules can be of fundamental interest and a mathematical formalism for this purpose has been developed recently [1]. The plots of various contributions to the potential energy V with semi interbond angle 9 in XY 2 bend symmetric systems seem to suggest that the actual equilibrium configuration lies in the premises of minimum for ^bend and zero for F bend _ stretch . Extension of the analysis to XY 3 pyramidal systems is discussed below.XY 3 pyramidal molecules belong to the C 3v point group, have the vibrational representation T = + 2E, and contributions to the potential energy come from stretching, bending, and different mutual interactions between the two. The recipe for plotting the potential energy contributions as a function of the semi interbond angle 9 is almost the same as given in the earlier work [1] except that here one has to deal with two vibrational species viz., A : and E, both of order two (as against one vibrational species of order two and one vibrational species of order one viz., A : and B ! in XY 2 bend symmetric systems).Fortunately, there are a few XY 3 pyramidal molecules in the literature for which the interbond angles are uniquely fixed and the normal and isotopic frequencies have been exactly determined [2], Moreover, these happen to be hydrides where the isotopic fre-1050
A method is developed to investigate the variation of potential energy contributions (F stretch , F bend , ^stretch-bend an d Ktretch-stretch) > n simple molecules when their inter-bond angles 6 are varied arbitrarily. Applied to XY 2 bend symmetric systems, the V-6 plots suggest that the actual equilibrium configuration in these molecules lies in the premises of minimum F bend and zero F stretch . bend .In terms of internal symmetry coordinates, the average potential energy of a vibrating molecule may be written as [1] <2 V)=ZF ij Z ij .(1)Here the matrices F and I define the intramolecular force field and the mean square amplitudes of vibration, respectively. They are given byThe elements A x and A i of the diagonal matrices A and A can easily be calculated from the values of the vibrational frequencies. The elements of the L matrix are significant in the sense that they relate the normal coordinates Q, to the internal symmetry coordinates S, in the form S = LQ. The evaluation of the L matrix is a basic problem in molecular dynamics, since one has to determine n 2 elements using the n vibrational frequencies as the input data. One most useful condition in this context has been provided by Wilson [2] as LL=G, (4) where G is the inverse kinetic energy matrix obtainable from the molecular geometry and atomic masses. Thus the potential energy of a vibrating molecule depends on its vibrational frequencies and geometry.It will be of fundamental interest to investigate the variation of the potential energy with the geometry of Reprint requests to Dr. M. K. R. Warier, Department of Physics, Maharajas College, Kochi 682011, Indien. a molecule since every molecule has its own preferential geometry for stability. A mathematical formalism for this kind of investigation is developed here. As an illustration, the method is applied to XY 2 bend symmetric systems. A parametric approach developed by to analyse vibrational problems in molecules basically involves splitting up of the L matrix into the formHere L 0 is written in the lower triangular form and is easily calculated using (4). C is an orthogonal matrix of the n(n -1)/2 parameters associated with each vibrational species of order n. Many simple molecules of types XY 2 bend symmetric, XY 3 planar, XY 3 pyramidal and XY 4 tetrahedral possess vibrational species of order n = 2, and in such cases the orthogonal matrix C of (5) can be written asAlong with (5), the invariance of the F matrix under isotopic substitution enables us to write (2) asHere the asterisk refers to the case after isotopic substitution. Since C^ is orthogonal,This reduces to a simple quadratic equation of the form pc 2 + qc + r = 0 .0932-0784 / 92 / 0600-781 $ 01.30/0. -Please order a reprint rather than making your own copy. Unauthenticated Download Date | 5/13/18 12:22 AM
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.