Scholars' Mine Masters Theses Student Theses and Dissertations 1974 Models for molecular vibration Allan Bruce Capps Follow this and additional works at: https://scholarsmine.mst.edu/masters_theses Part of the Applied Mathematics Commons Department: Recommended Citation Capps, Allan Bruce, "Models for molecular vibration" (1974). Masters Theses. 3420. https://scholarsmine.mst.edu/masters_theses/3420 This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. MODELS FOR MOLECULAR VIBRATION BY ALLAN BRUCE CAPPS, 1950- A THESIS Presented to the Faculty of the Graduate School of the UNIVERSITY OF MISSOURI-ROLLA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE IN APPLIED MATHEMATICS 1974 T3068 Approved by 62 pages c .l ii ABSTRACT The purpose of the investigation is to determine if a classical model can be used to characterize molecular vibrational and librational (restricted rotation) frequencies. In general, the model treats a molecule as asymmet­ rical and rigid and simulates the intermolecular forces by springs along the bonds. Molecules constrained to a plane and molecules free to move in three dimensions are analyzed. The frequencies of water molecules are investigated, in particular. The model and analytic components are found to function well. Within the limits set for the model, the water molecule simulation is not successful as the motion becomes anharmonic for energy less than the lowest quantum mechanical energy level. An interesting result is that infinitesimal rotations in three dimensions can be described by two independent variables. Ill ACKNOWLEDGEMENT I wish to express my gratitude to the University of Missouri - Rolla, the Mathematics Department, and the Cloud Physics Research Center for supporting this thesis and for provision of the Author’s teaching and research assistantships. I especially am grateful to Dr. O. R. Plummer for his guidance, technical advice, and encouragement. I thank Dr. P. L. M. Plummer and Dr. A. J. Penico for serving on my committee and for their cooperation and suggestions. I thank Marlene Pau for typing this thesis. Last, but not least, I thank my wife, Nancy, for her understanding and patience during this research. This work has been supported in part by the Atmospheric Science Section of the National Science Foundation under grant GA-32388. iv TABLE OF CONTENTS Page ABSTRACT............................................................................................................................ii ACKNOWLEDGEMENT................................................................................................ iii LIST OF ILLUSTRATIONS.............................................................................................vi LIST OF T A B L E S .......................................................................................................... vii NOMENCLATURE.........................................................................................................viii I. INTRODUCTION........................................................................................... 1 II. PHYSICAL MODEL................................................................................................4 III MATHEMATICAL MODEL...................................................................................6 A. Introduction.......................................................................................... 6 B. Lagrange Equations in the General Ca s e .................................... 6 C. "Linearized” Lagrange Equations. ..............................................15 D. Normal Coordinate Development.................................................... 18 IV. SOLUTION TECHNIQUES............................................................................21 A. Numerical Integration...........................................................................21 B. Fourier Transform, Fourier Coefficients, and Pseudo-Power S p e c tra .......................................................................22 C. Spectrum Plotting.................................................................................. 24 D. Time Series Filtering...........................................................................24 E . Eigenvalues and Eigenvectors............................................................25 F. Programs: As a Unit and Individually.........................................25 V Table of Contents (continued) Page V. RESULTS AND DISCUSSION......................................................................... 29 A. Program Verification................................................................. .....29 B. Applications to Water M olecule ................. .............................. 35 C. Conclusions................................................................................................48 BIBLIOGRAPHY. ............................................................................................................... 50 VITA.......................................................................................................................................... 52 vi LIST OF ILLUSTEATIONS Figure Page 1. Two dimension development parameters............................................. 7 2. Three dimension development param eters......................................... 8 3. Geometry I at equilibrium ............................................................ 30 4. Geometry II at equilibrium....................................................................... 30 5. Spectra of linear and general two dimensional systems with initial displacement to same position..................................................37 6. Spectrum change for general two dimensional system as a function of initial displacement; displacement given as a percentage of the normalized normal co o rd in a te ...........................40 7. Shift of ’’small frequency” as a function of initial displacements along a normalized normal coordinate...................41 8. Shift of "104 cm-1'' frequency as a function of initial displacements along a normalized normal coordinate...................42 9. Spectrum of the general two dimensional case with 100 percent of the normalized normal coordinate as the initial displacement...............................................................................43 vii LIST OF TABLES Table Page I. Defining constants for problems I, II,III, and IV .............................................31 It. Defining constants for three dimensional analysis problem................................................................................................................... 45 viii NOMENCLATIVE Two dimensions = vector locating equilibrium position of ith spring attach­ ment to rigid body; |a | = a. = vector locating rotated position of ith spring attachment to rigid body; |b.i |i = b. I = moment of inertia about center of mass k. = ith spring constant 1 l. = equilibrium length of ith spring i m = m ass of rigid body O = equilibrium position of the center of mass -4 r. = vector locating rotated and translated position of ith i spring attachment to rigid body; |r. | = r s = number of springs in system -4 t. = vector locating translated position of ith spring attach- 1 ment to rigid body; |t | = t = vector locating position of wall relative to equilibrium position of the center of mass; |w | = w = change from equilibrium length of ith spring = angle from x-axis to W at equilibrium; positive counterclockwise = angle from W. to a. at equilibrium; positive counter­ clockwise ix Three dimensions 1 1 I. INTRODUCTION Recent studies involving the thermodynamics of aggregates of water molecules (Plummer, 1972, 1973, 1974; Hale, 1974), with the aim of pro­ viding a molecular basis for the understanding of various atmospheric phenomena, have required knowledge of the m olecular vibrational energy levels which are present in the systems of interest. It has been difficult to characterize the librational (restricted rotation) frequencies unambiguously from the experimental data. Detailed quantum mechanical calculations for the systems have not yet proved feasible. It is the objective of this thesis to explore the extent to which a simple classical model can provide the needed information about such systems. If the classical motion can be well represented as a superposition of simple harmonic motions, then the quantum mechanical energy levels can be simply related to the frequencies of this motion. The classical model employed treats the molecule as rigid and simulates the intermolecular forces by springs along the bonds. No bond bending forces are included. Systems constrained to a plane, relevant to a surface motion, as well as systems free to move in three dimensions were analyzed. The Lagrange equations for such a problem lead to a non-linear system of differential equations (especially non-linear in rotational variables) and closed form solutions appeared unlikely. Therefore, the problem was approached in two ways: (1) The equations were integrated numerically and 2 fundamental frequencies sought from the Fourier transform of the solutions; and (2) the small motions approximation was made, and the frequencies obtained from the appropriate eigenvalue problem. A set of computer routines was prepared