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Masters Theses Student Theses and Dissertations
1974
Models for molecular vibration
Allan Bruce Capps
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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 to deal with the numerical
solution and its Fourier transform for any asymmetrical rigid body subject to forces arising from springs attached to any point on the body. The pseudo power spectrum was found from the above analysis and graphed interactively with a computer routine, which drives a Tektronics T-4002 graphics terminal.
From these plots the fundamental frequencies were ascertained.
The small motions approximation yields a linear system of differential equations with solutions which are pure harmonic motions along normal coordinates. In the three dimensional case, it was found that small rotations can be described with two independent variables. This implies that such rota tions have but two degrees of freedom.
Two significant relations were found between the small motions approx imation and the general case. (1) The small motions approximation represented the general case for sufficiently small amplitudes except for the occurence of zero frequencies. This suggested that additional springs, perhaps to model the bond-bending forces, would be necessary to produce non-zero frequencies.
(2) The motions along normal coordinates served as useful guides for the general case, in that the simplification of the spectrum was most significant with initial displacements along the normal coordinates. This however, did not sufficiently simplify the spectrum for analysis. 3
In the general cases, it was found that although the motion remains
periodic, the frequencies are shifted as a function of amplitude and the motion
becomes anharmonic. The shifted frequencies can serve as an optimization
as they permit better estimates of quantum mechanical energy levels than
those from the small motions approximation. A singularity of the three
dimensional general case occurred. It is believed that an interpolative tech
nique can treat the singularity.
As for the water molecule simulation, within the limits set by our
model, the model was not successful. This was due to the result that the motion becomes anharmonic when the energy is as large as the lowest quantum
mechanical energy level. 4
II. PHYSICAL MODEL
Through this section, reference will be made specifically to a water molecule. Analogous results will follow for other molecular forms.
The problem, as situated in the real world, involves a water molecule executing motion about its equilibrium position within a small (less than 200) cluster of water molecules. Such clusters, in either the liquid or solid state, appear to exist in a regular lattice long enough to have a well defined vibrational spectrum (Plummer, P. L. M ., 1972, 1973, 1974; Hale, 1974).
We model the system as follows: (1) the environment of an individual molecule is modeled as a fixed universe, (2) chemical interaction between the molecule and its environment is modeled with springs, and (3) the molecule is modeled as a rigid body. In short, the model consists of a rigid body attached to a fixed universe with springs.
A justification of the model's components follows. (1) The fixed universe aspect is feasible considering the following. A disturbance of one molecule within the cluster perturbs the adjoining molecules, causing a collective motion of the cluster. These collective motions are only weakly coupled to that of the individual molecule, thus allowing the fixed universe aspect (Plummer, 1972). (2) As the molecule executes motion about equi librium it is constrained by intermolecular chemical bonds. This constrained movement can be modeled using springs which, obviously, constrain the motion while allowing for vibrational motion. It is also common to use the harmonic oscillator potential with its associated force constant in molecular 5
physics (Tipler, 1969, p. 370). (3) The rigid body aspect seems plausible as the intramolecular bond forces are greater than the intermolecular bond forces, Thus, internal motions will be of higher frequency and will average to zero over a period of the intermolecular motions. Consequently, the average motion of the molecule will move as a rigid body with its atoms fixed at their equilibrium values as opposed to moving and changing shape
(Plummer, O. R ., 1974). (In passing, it is to be noted that the intramole cular forces could be modeled by other springs of considerably greater force constant. See section V, B2.) 6
IE. MATHEMATICAL MODEL
A. Introduction
Our goal is to determine if the motion is approximately harmonic, and, if so, the frequencies of oscillation. The stretching and compressing of the springs give rise to a scalar potential energy V = V(q^,... ,q^), where the q.’s are generalized coordinates and d is the number of degrees of freedom of the system, from which the generalized forces Q^, k = 1 ,..., d, may be derived. This allows us to define the Lagrangian function L = T - V, where
T = T(q^,... , q^, q^, „.. , q^) is the kinetic energy and the q.’s are generalized velocities. Thus, the Lagrange equations
BL ( B L = 0 , i = 1, ... d dt \dq. / dq. may be found (Symon, 1971, p. 355-366). From these second order equations a system of 2d first order differential equations may be found whose solutions yield the dependence on time of the q., i = 1 ,..., d.
B. Lagrange Equations in the General Case
Before proceeding, it may be helpful to refer to Figures 1 and 2 and the nomenclature listing.
1. Two Dimensional Case
Consider first the case of a rigid body confined to move in two dimen sions. There being three degrees of freedom, we seek the three associated
Lagrange equations. We choose to describe two dimensional motion by the generalized coordinates q^ = x, q^ = y* an(^ Tg =
where x and y form a 7
Figure 1. Two dimension development param eters Figure 2. Three dimension development parameters 9
cartesian coordinate system whose origin O is the center of mass at the
equilibrium position and oo is the angle of rotation about the cartesian coordi nate system origin.
The kinetic energy T is given by
T = i m(x2 + y2) + i I b 2 (1) and the potential energy V is given by
s k v = S ~ (A i)2. (2) 1=1 2 1
The change in equilibrium length of the ith spring, A.l, is found using
Chasles' Theorem (Goldstein, 1971, p. 124): the most general displacement of a rigid body is a rotation plus a translation. Consequently, it can be seen in the following that A.l = | r. - W. | - 1., A general rotation cp changes the length of the ith spring to |b. - W. |. Following this with a general displace ment the length becomes I Id. + £ - W. I = I r. - W. I. Whence 1. + A.l= i* l i i i i i i I r - W L from which the conclusion follows. W. and 1. are assumed known. 1 i i 1 i i The vector a. in two dimensions is given by a. = (a. cos(0. + £.), a. sin(0.+£.)), i 1 1 1 1 1 1 a while the general rotated and translated vector r. is found to be r. = (x + a. cos (id + 0. + £.)» y + a- sin(^ + 0. + £ .))• Thus, i ' i i i i l l
2 2 2 2 A 1 = [ x +y + w. + a. - 2 w.a. cos(
+ 2x{ a. cos(cp + 0. + cos 0j] (3)
i + 2y{ a. cos (cp + 0. + £ .) - w. sin 0.} ] 2- 1. » J i i v i i i l 10
Using Equations (1), (2), and (3), we can write the Lagrangian function
L = T - V as
-r 1 2 .2 1 . 2 L = i m(x + y ) + | I 2 2 2 2 - Z J k. {x + y + w. + a. - 2 w. a. cos(o + £ ) - l i i i l l i = l + 2x[ a. cos(cn + 0. + £.) - w. cos 0.] l l ’’ l l l _ _ 2 + 2y[ a. sin{ 2 2 2 2 - 21.[ x + y + w. + a. - 2w. a. cos( + 2x(a. cos( l + 2y(a. sin(co + 0. + £.) - w. sin 0.)]?}. l l ^ i i i The Lagrange equations for x, y, and o become, respectively: s A 1 mx + Z) k.[ x + a. cos(o + P. + £ .) - w. cos 0. ] [; ------t-= ] = 0 . „ L i v i ' i; i l 1. + A.1 i=l i l s A.1 my + E k.[ y + a. sin( 2. Three Dimensional Case For brevity, in this subsection we will denote the trigonometric functions sin 0 and cos 0 by s0 and c0, respectively. 11 Consider now the ease of a rigid body moving in three-dimensional space. There are six degrees of freedom which we choose to describe by the generalized coordinates q = x, q = y, q0 = z, q = The translational motion of the center of mass is described by the cartesian coordinates x, y, z, and the rotational motion about the center of mass is described by to, 0, 0 , the Euler angles (Goldstein, 1950, pp. 107-8). We proceed to find the six Lagrange equations. The kinetic energy can be written as the translational energy of the center of mass plus the rotational energy about the center of mass. That is, i ,.2 .2 .2 2 2 2 T = Jm(x + y + z ) 4(i 03 , + I 03 , + i , 03 / (5) X X y z z where it is to be noted that I ,, I , , and I / are the moments of inertia x y z about the fixed body axes x', y', z , respectively. Goldstein (1950, p. 134) finds = (psQsij) + $c0, = Using (5) and (6), T can be found. The potential energy is again given by Equation (2). Using the same reasoning as for two dimensions, A.1 = ] r. - [ - 1., where r. and W. are three dimensional vectors in this case. The general rotated and translated vector r will now be found. Goldstein (1950, p. 109) shows, using Euler i angles Cip c A = cipscp + cQccpsip -Sip SKp + CQC(pClp -S0c S0S0 sQcip c0 The vector a. is defined in this three dimensional development by a. = (a. s£.c(0. + £.), a.s^.s(0. + £.), a.c£.), Thus, 1?., the result of a general -+ rotation of vector a. is given by operating on a. with A, which yields: b. a.c *4 b. = b a.s b. _ a.s0s£.s(+) + a.c0c£. i,3 1 1 1 1 where (+) = 0 + 0. + £ . A general rotation and translation of vector a., denoted by r ., is given by ri =(X + bi,l ’ y +bi,2* Z+bi,3>- (9) Using Equations (7), (8), and (9), A.1 is 2 2 2 2 2 A.1 = [ x + y + z + a. + w. - 2w.a. { s f ,c(+)cr?. c(cp - 0.) i L J i i i i vi ' 7 i i c 0sr.s(+)cr? s( + c 0cr .S77.} + 2x{ a.C(ps£. c(+) - a.s + a.s<£S0c£ . - W.C77,c0.} + 2y{ a.s given by Equations (2) and (10), the Lagrangian function L = T - V can be found. The Lagrange equations for the generalized coordinates x, y, z, 0, 0 were found to be, respectively: (Note the terms in square brackets [ ] denote, for each Lagrange equation, —■ ( ^, while the remainder of each dt V a q / B L equation corresponds to - ) aqk [m x] + Z) k.[x + a.c [my] + E k. { y + a.sc/js^.c(+) + a.c [lsr{ips2 0s20 + (obs2\l)S(2Q) + (ptys20s(20) + i 0S0S(20) + $0s0c(20) + i 02c0s(20) 1 + 1 , {(0S2e c 2ib + • 2 - 00s0c(20) - | 0 c0s(20) } + I / [ + £ k {w a (sg c(+)cr? s(cp - 0 ) + c0s£.s(+)cr7.c( - x(a.s A.1 + y(a.c 2 2 • • «• [0 { I /C 0+ 1 ,s 0 1 + { I , - I ,1 { 00 S(2j/) ) - J OS0S(2 0) x y y x > 2 - £ (I /{ ^2s(20)s^(0) + 0 . 2 + 1 ,{- + £ k.{ - w.a.{s0s^.s(+)cr?.s( + c 0s£.s(+) B77. - s 0c^.s?7.1 + x{ a.s A.1 + y{-a.c [ I ,1 0 + c2 eip - 2^0c0s0} ]+ £{ I , - I ,} { 02s(20)-^52s20s(20)-20^c(20)s0l z x y s + £ k.{ w .a.{ s^.s(+)c?7.c( - x{ a.c A.l + z{ a.s0s£.c(+) 1 1 i 1 j - rj } = 0 , i i where (+) = 0 + 0. + £ . and A.l is given by Equation (10). 15 C. "Linearized" Lagrange Equations We now have at our disposal the Lagrange equations for both the two and three dimensional cases. We develop from them the "linearized" Lagrange equations which give the equations of motion near an equilibrium configuration (Symon, 1971, p. 467). The reasons for considering this topic are twofold: (1) the results are of interest in themselves, as they indicate the possibilities of optimizing the linear case to describe results of the general case, and (2) the results can be used in a verification procedure for the analytic com ponents of the model. It was found that by "linearizing" the Lagrange equations in the following manner, the same results are obtained as would be by Symon’s approach (Symon, 1971, p. 469): (1) approximate sine and cosine with the first order terms in the Taylor series expansion; (2) approximate (1 ± x)n, where x is small, by 1 ± nx; and (3) neglect all products of variables or their first derivatives. For two dimensions, the "linearized" Lagrange equations are, in matrix form, M2^2 + K2X2 = w^ere “ *■ i 2 i i i i m 0 0 ( c j C C c c X 0 3 4 2 3 s K. i i i 2 i i II o M2= 0 m 0 c c y * 0 3 4 C4 C2°4 ’ X2 \ to 1=1 1 i i i i , 1 2 0 0 1 , For three dimensions, the "linearized" Lagrange equations are, in matrix form, M X + K X = 0 , where O u O O O m 0 0 0 0 X 0 0 m 0 0 0 y 0 m3 = 0 0 m 0 0 z 0 , and > X3 = •°3 = 0001,0 0 z % 0 0 0 0 1 , e 0 x _ i 2 i i b \ = w.a. sin £. cos 77. sin £. 2 1 1 1 1 1 B* = a. sin £. cos(0. + £.) - w. cos 77. cos 0. 17 B1. = a. sin F sin(0. + £.) - w. cos 77. sin 0. 4 1 i ' i l l i B* = w.a. sin F. cos 77. sin £. 5 1 1 *i 'i *1 B* = w.a. cos F. cos 77. sin 0. - w.a. sin F. sin 77. sin(0. + £.) 6 1 1 *1 '1 1 1 1 1 '1 v 1 V B* = a. cos F. - w. sin 77. 7 1 1 1 fi As a check, we note that if £ = 77/2 and 77. = 0, the B. = C., 1 = 1, 2, 3, 4, and C. = 0, i = 5, 6 , 7. Thus, the three dimensional case reduces to the two dimensional case. Notice the three dimensional case reduces to a 5 by 5 matrix. This results from the Theorem: Infinitesimal rotations in three dimensions can be described by two independent variables. Proof: The matrix of the infinitesimal rotation in three dimensions, obtainable directly from matrix A (Section B) by retaining only first order terms, is 1 i (P+0 1 -6 = 0 10 + (cpf 0) 1 0 0 +e 00-1 0 e 1 0 0 1 0 0 0 010 m * - - = I + P< i+ 8e2 18 Thus p = D. Normal Coordinate Development Should the normal coordinates, defined later, be found from the "linearized” Lagrange equations, they afford a handy and useful tool for our verification procedure. We now develop the normal coordinates (Symon, 1971, p. 469-472). Observe that the kinetic energy matrices M and M are diagonal and Z o the potential energy matrices K and K are symmetric. If we define a new Z u vector q' with components q' such that i 1,2,3,4,5 for three dimensions we can write our systems of "linearized" Lagrange equations in the simpler form 4, *4 q + W q = 0 (11) where for two dimensions l i i — ______1 q/ - (m2x, m2y, I2/ and for three dimensions i I 1 I 1 K3kl q' = (msx, m 2y, m2z, I *,(0 + 0), ^ A) and Wgk, = (m ^ „ >* • The potential matrices W and W remain real and symmetric, hence, each ^ O can be diagonalized by an orthogonal transformation yielding the eigenvalues X. and normalized eigenvectors e. = (e^, e% y * ,e dj^* the coinPonent s of the vector q ' along e^, e^t. . . , e^ be f^, f^,... ,f^; that is, q' = 2 f. e.. In terms of components, q, = 2 e. . f, and f. = 2 e. . q' . k j=l kj 3 J k=l kj k In the f„ ,..., f , coordinate system, 1 d d « d V = 2 i X. C and T = S | r . (12) J 3=1 J J 3=1 In view of Equation (12), the "linearized” Lagrange equations separate into the equations for each f : f. + X. f. = 0, j = 1 ,... ,d. (13) J J J The f. are referred to as normal coordinates. The solution to (13) is the J harmonic oscillator solution f. = A. cos /X .t + B. sin /X.t , j = 1 ,... ,d, J J J J J _ l where A. = f.(0) and B. = X? f (0) are arbitrary constants. Thus, the physical J J J J j significance of the eigenvalues is they are the square of the frequency of oscillation in radians per second, and the eigenvectors correspond to normal modes. Whence, for our two and three dimensional cases, we have 20 d q', = 2 e .(A. cos /X . t + B. sin /X . t) K i=1 J J 3 J d d - J where A. These specific relations J ekj qk <0) 8ndv £ 1xj % ^ {0)- will be used later in the verification process. 21 IV. SOLUTION TECHNIQUES A. Numerical Integration Returning to our primary goal, we now describe mathematical tech niques for solving the Lagrange equations and extracting the frequencies. It is apparent by observing the extent to which the Lagrange equations are coupled, that no closed form solution is probable. Symon (1971, p. 454) notes that: " . . .for the general problem of the motion of an unsymmetrical body under the action of external torques., .there are no generally applicable meth ods of solution, except by numerical integration of the equations of motion." The Runge-Kutta numerical integration method modified by Gill was chosen (Ralston, 1962, p. 110). This method approximates the solution for a system of first order differential equations. It is a fourth order integration procedure which requires the functional values at a single point (our initial conditions) to obtain the functional values at the next point. Each functional 4 value is calculated by a Taylor series through terms of order h , where h is the increment of the independent variable. Our systems, being second order, were of the form q. = f(q., q.), i = 1, 2 ,... ,d. To transform such a system to a first order system, Gear (1971, p. 47) suggests defining y2 i-l = % and y2i = V 1 = lf * * *,d * Doing this, our system of d second order equations become the system of 2d first order equations 22 »• • •» where the q.Ts are obtained from the Lagrange equations. B. Fourier Transform, Fourier Coefficients, and Pseudo-Power Spectrum Using the above process, the solutions are theoretically accessible. Supposing the solutions were simple enough, the frequencies could be dis cerned by inspection. However, the envisioning of complicated superpositions of the frequencies suggested finding a method capable of readily finding the frequencies. Each solution of the Lagrange equations can be viewed as a time series that is, q. = q.(t), i = 1,... ,d. In fact, each q.(t) can be interpreted as a real, discrete, finite time series. It is therefore possible to Fourier transform each time series q.(t) and find the associated power spectrum (Enochson, 1968, p. 81,82). This means, in effect, our time series solutions can be transformed in such a way that the results can be used to discern the fundamental frequencies of oscillation. We give some details, first, about the Fourier transform. The trans form of interest to us is the Fourier transform F of real, finite, discrete data { x(t)} = [x(tQ), x ^ ), . . .• , ,x(tN 1)} : N-l -i ’At F(oi) = A t S x(t.) e la5J , - co < w < oo. J=0 } In an effort to reduce computation time, selected frequencies are chosen at which to evaluate the Fourier transform F . These being where = kAco= k ( , o . • , N/2. 23 Thus, the Fourier transform actually used is of the form N-l 2ffjk F = F(kAco) = E x(t.) e N , k = 0,1,..., N/2 . k j=o 3 From the finite Fourier transform, the associated power spectrum G, is simply obtained by K Gk = G H> =" F lFk |2* k = N/2 (Enochson, 1968, p. 167). Thus, we have a discrete set of numbers, { G(co^)} » k = 0 , 1 , . . . , N/2, given as a function of the frequency. We could discern, by inspection or plotting, at what frequencies the largestelement(s) of { G, } occur. These would be the frequencies of oscillation we seek. This !v theory would remain valid should we plot the pseudo-power spectrum. By this, we simply mean, when finding the power spectra G^, omit any multiplicative 2At , ,2 scaling factors, such as , and find | F | , which we refer to as pseudo- power spectrum, P^. Enochson (1968, p. 81) notes that the classical Fourier series is computationally identical to the discrete Fourier transform. This means, i ,2 in effect, that the spectrum can be found using the relation P^ = | F^ | , or 2 2 the relation P^ = + b^, where a^ and b^ are the Fourier coefficients. The latter approach was taken, and the coefficients were found using a program based on the Cooley-Tukey algorithm for finding the discrete Fourier transform (Enochson, 1968, p. 89). 24 C. Spectrum Plotting To examine the spectrum , we interactively plotted a continuous curve through the points P with a spline fit interpolation program (Pennington, K 1970, p. 448). Basically, the spline fit is accomplished by connecting the adjacent elements P with a third-degree polynomial such that the curve and xv its first and second derivatives are continuous at each point oo^. In this plotted form, fundamental frequencies can be read from the ordinate values corresponding to sharp spikes. Figure 9 is an excellent illustration. D. Time Series Filtering Enochson (1968, p. 160), with regard to the initial time series, suggests: "it is desirable to taper a random time series at each end to enhance the characteristics of the spectral estimates [ prior to applying the Fourier transform] . Tapering is a ’data window’. .. [ and] is equivalent to applying a convolution operation to th e.. .Fourier transform. The purpose of tapering is to suppress large side lobes in the effective filter obtained with the .. .transform. ’’ ( 2) The taper suggested, and used by us, was a cosine taper, u over N ’ one-tenth of each end of the data. In equation form (and selecting discrete values), 2 5jrt 2N J 2N cos if -N/2 £ t ^ - and — <; t N/2 N 5 < 1 0 elsewhere 25 This filter and ’’the effective filter”, the box car filter 1 11 I £ N/2 u( 1) N/2 0 elsewhere already inherent in the finite Fourier transform were the only filters examined. E. Eigenvalues and Eigenvectors When the eigenvalues and eigenvectors were found for matrices W z and W , the Jacobi diagonalization method was used (Ralston, 1962, p. 84). Basically, this procedure annihilates, one at a time, the off diagonal elements of our symmetric matrices by orthogonal transformations (rotations). The products of the orthogonal transformations yield the eigenvectors. F . Programs: As a Unit and Individually The programs as a unit function in the following manner. The solutions to the Lagrange equations are obtained using the Runge-Kutta method, yielding i time series { q.(t)}, where i is the number of generalized coordinates. For each time series, effective filtered and cosine taper filtered, the Fourier coefficients, a and b , are calculated using the fast Fourier transform n n method of Cooley and Tukey0 The pseudo-power spectrum P^ is found for 2 2 each transformed time series from the relation P. = a, + b, . p. is then k k k k plotted interactively at a graphics terminal, as a continuous curve, from which the fundamental frequencies are read. Should the eigenvalues and eigenvectors be needed, they are found using the Jacobi diagonalization method. 26 The individual programs will be examined now. 1. Runge Kutta The purpose of the Runge-Kutta program is to obtain an approximate dyi solution of a system of first order differential equations -jjj- = f.(x, y , y^t . . . , y ), i = 1 , . . . ,n with given initial conditions y.(x ) = y , i = 1 , . . „ ,n. The n i o io actual program used was a program from the Scientific Subroutine Package, (1968, p. 333), hereafter referred to as SSP. The physical quantities necessary for input are the initial displace ments and velocities of the generalized coordinates (or normal coordinates for the linearized case), a description of the geometry corresponding to constants shown in Tables I or n , length of the time interval, and the initial increment of the independent variable time. Other necessary input is upper error bound on the local truncation erro r, number of first order equations, and "weights” for the local truncation error in each component of the dependent variables q. or f.. l l Output is user controlled. It includes the time value; the position, velocity, and acceleration at that time; initial increment; number of bisections of the initial increment; lower and upper bound of total time interval, and local truncation error bound for each evaluated point, which is never greater than the initial increment. The fourth order Runge-Kutta method was chosen as it is a commonly used method (McCracken, 1964, p. 325), it is stable and self-starting, and it allows changing the independent variables increment at any evaluation point in the calculations (Ralston, 1962, p. 111). 27 2. DFHABM and the Spectrum The program called DFHARM, from the SSP (1968, p. 281), directly computes the Fourier coefficients of our real time series. Within DFHABM is a subroutine DHAFM (SSP, 1968, p. 276), which uses the algorithm of Cooley and Tukey to perform discrete complex Fourier transforms. As input, the unfiltered or cosine taper filtered time series is used. The DBHARM program, given these N real numbers x., computes the Fourier coefficients a , . . . , a^T /n - , b„, . . . ,b T .i n the equation o N/2 - 1 1 N/2 - 1 N/2-1 cos 2 T3k + bfe sin + i aN /2(-l)\ j= 0 , 1 ...... N - l . = K + N k=l In order to find the Fourier coefficients, the subroutine DHAFM is called to compute the complex coefficients N /2-1 i4ffjk/N N E (x0.-ix0. .) k = 0 , 1 ,... - 1 . 3=0 v 2j 2j+ l7 4 These are then used to compute the Fourier coefficients which the program gives as output. This program was selected as it provided directly the pseudo-power spectrum. We were able to use the Fourier transform approach as a result of the time-saving algorithm of Cooley and Tukey. 3. Plotting of Spectrum Each spectrum was found and the first 150 elements were plotted using the spline fit. The first 150 values were chosen since, by inspection of P^, there appeared no large elements beyond this number. The plot was done 28 interactively at a graphics terminal, with the following possible interactions: (1 ) selection of spectrum by variable, (2 ) filtered or unfiltered spectrum, (3) range of P and domain of frequencies, (4) number of evaluation points IV to use in constructing the continuous curve, and (5) branching to (1) or (3). Those plots that were of interest were then hard-copied. Some such hard copies were used to produce Figures 5 through 9. This latter capability, along with the ease of examining several spectrum at any domain or range, are the reasons for using this program. 4. TWODEVVS and THREDEW These programs calculated the eigenvectors and eigenvalues for the two and three dimensional cases, respectively. As subroutines, they used the SSP (1968, p. 164) program EIGEN. The input necessary is the values of the matrix entries. These were general enough to allow calculating them within the program. The input necessary for this was the geometry constants of Tables I and n for two and three dimensions, respectively. The output is the non-diagonalized m atrix, eigenvalues, and corresponding eigenvectors. The program was used because (1) the process is iterative to a desired accuracy, (2) known to converge, (3) adapted for symmetric matrices, and (4) yields both eigenvalues and eigenvectors. 29 V. RESULTS AND DISCUSSION Ao Program Verification It was deemed necessary by sound scientific principles to construct a systematic technique to determine if the analytic components of the model function properly. To this end the following verification procedure was con structed. The verification process was done by using the general procedure to solve the two dimensional linearized systems by normal mode methods. Two geometrical configurations were examined, one simple enough to allow deter mination of normal coordinates by symmetry (see Figure 3) and one, more complicated, simulating a water molecule constrained to the surface (see Figure 4). The particular results that directed us to the following con clusions were found from examining problems I, n, HI, and IV as defined on geometries I and n by the constants given in Table I. 1. Assertion 1: The eigenvalues and eigenvectors for the linearized Lagrange equations are correctly found by TWODEVVS using m atrix W0 . The evidence is the following: Li (a) agreement with hand calculations of eigenvalues and eigen vectors using matrix Kg (b) agreement with eigenvalues from an independent program using matrix (c) agreement of the eigenvalue-eigenvector equation 30 Figure 3. Geometry I at equilibrium Figure 4. Geometry n at equilibrium 31 TABLE I: DEFINING CONSTANTS FOR PROBLEMS I, II, m , AND IV Constants (Units) Problem I Problem n Problem III Problem IV -23 m(xlO gm) 1 . 0 2.98975 2.98975 2.98975 -39 2 I(xlO gm-cm ) 1 . 0 0.33818 3.38177 0.33818 5 k1(xlO dynes/cm) 1 . 0 0.18 0.18 0.18 k2 (xl0 dynes/cm) 1 . 0 0.18 0.18 0.18 ™* 3 a^(xlO cm) 1 . 0 0.95718 0.85531 0.85531 / -i a~8 a2 (xl0 cm) 1 . 0 0.95718 0.85531 0.85531 1^ x 1 0 8cm) 2.2385 2.03531 1 . 8 1 . 8 l2 (xl0 8cm) 2.2385 2.03531 1 . 8 1 . 8 W1(xlO-i cm) 2 . 0 1 . 8 2.64652 2.64652 , - 8 v w2(xl0 cm) 2 . 0 1 . 8 2.64652 2.64652 0 1 (radians) 0 . 0 0 .0 0 .0 0 .0 0 2(radians) 3.14159 3.14159 1.93599 1.93599 ^(radians) -1.57078 -1.57078 -0.11819 -0.11819 4 2(radians) -1.57078 -1.57078 0.11819 0.11819 Geometry I I n n 32 The eigenvalues for problem IV were found by methods (a) and (b) above. They were in agreement with TWODEVVS to five significant figures. It is noteworthy that methods (a) and (b) and TWODEVVS are performed on different matrices K and W , respectively. The agreement in the eigen- values is expected, since the matrices model the same physical system. Results for the eigenvalues of problems I and IE found by method (a) were in similar agreement with TWODEVVSo As for the eigenvectors, those found for problem I by method (a) were in agreement to six significant figures with TWODEVVSo The eigenvectors for the remaining problems were checked using method (c) and were accurate to six significant figures. 2. Assertion 2: The linearized Lagrange equations were correctly derived. The evidence is the following: (a) same eigenvalues from two different systems The results pertaining to eigenvalues in Assertion 1 indicate the small motions approximation was correctly found from the "linearized" Lagrange equations, since the potential matrices of each system have the same eigen values. (b) normal coordinates from TWODEW S agree with those from symmetry arguments. The normal coordinates as found by TWODEVVS for problem I are y = x /2 , y = -2 x , and (c) independent approach of Symon (1971, p. 467) agrees with our linearization approach In the standard small motions approximation the matrix W is found A from W_,, = — 7 r—r . These values were developed and found to agree 2kl dq kdqt with those found by our "linearization" process. (d) three dimensional case reduces to two dimensional case The substitution of £ = fr/2 and 77. = 0 in the three dimensional "linearized" Lagrange equations reduces them to the two dimensional system. 3. Assertion 3: The Runge-Kutta program works correctly. The evidence is the following: (a) detailed examination of q.(t) agrees with normal mode method when several modes excited (b) q.(t), i = 1 ,... ,d, are periodic (c) displacement along one normal mode excites one frequency; general displacement excites all frequencies Using the normal coordinate development of section III, D, q^, i = 1 ,... ,d are found using initial conditions with displacements only and no d initial velocities. The q'’s are then given by q' = Z) e A. cos /X .t . The 1 k kj j J Runge-Kutta when programmed to solve the same initial value problem, approximates qj, i = 1 ,..., d, also. A comparison showed the values were in agreement to four significant figures. More importantly, the period expected from TWODEVVS was accurately produced for several periods to one tenth of a time unit, the smallest time division used in the integration. The particular 34 location of the initial displacement affected the motion. If initial displacement was along a normal coordinate, the computed motion oscillated along that normal coordinate with the proper frequency. If the initial position was a general point, a superposition of non-zero frequencies occurred in the time series. 4. Assertion 4: The Fourier transform - spectrum unit functioned correctly. The evidence is the following: (a) peaks of plotted spectrum agreed with the frequencies obtained from other methods When the spectrum was plotted for various problems, those frequencies found by inspection or the eigenvalue program were reproduced to the same degree of resolution 5. Assertion 5: The general Lagrange equations are correct. The evidence is the following: (a) general case agrees with linear system results for small motions. A sequence of programs was run using the general Lagrange equations in which the initial displacements were increased gradually along a normal coordinate. It was found that the general equations agreed with the "linearized” system results for small, approximately 4 percent of the normalized normal coordinate, displacements. This was an indication the general equations were formulated correctly. (b) three dimensional case reduces to two dimensional case. 35 The substituting of £ = 77/2 , 77. = 0, and z = 0 = ip = 0, in the three dimensional general equations reduces them to the two dimensional case. B. Applications to Water Molecule 1. Time Interval Before turning to specific results of the water molecule, we explain a few details about the time interval for calculating time series. It was believed, due to the normal coordinate verification above and actual Runge- Kutta values, that the Runge-Kutta would perform accurately over long time intervals. In programs used for the final results given below, a time interval of 819.2 time units was used with ten evaluations per unit time for a total of 8192 data points. There were several reasons for these choices. Programs were run using 20 evaluations per unit time, but no better results in regard to frequency were obtained. The Fourier transform program required the number of data points be of the form 2^t p an integer. We had concluded, from observing shorter time programs, 256, 1024, 2048, and 4096 points, that a longer interval yet was necessary. In one instance, a frequency had been completely unsuspected for 512 points, appeared as a small spike for 1024 points, and was apparent as a fundamental frequency for 2048 points. Thus, a time interval of 819.2 was chosen because of the 2^ criterion, feasibility of cost, and there existed sufficient assurance that the time interval was long enough to obtain the fundamental frequencies. 36 2. Two Dimensions We turn now to the application of our techniques. The geometry, mass, and moments of inertia, appropriate to water constrained by two bonds to a surface, are those of problem IV. For the frequencies, in reciprocal centimeters, of the ’linearized” Lagrange equations we obtained 104.7, 70.8, and 0 with respective normalized normal coordinates (0 . 71945 0, -0.495163 , 0.487035), (0.566949 , 0.82 3 752, 0.0), and (-0.401197, 0.276124, 0.873381). The zero frequency indicates that, to lowest order, a motion along the corresponding normal coordinate "stretches no spring"; that is, the motion is force free. We infer that, in the real system, the potential is extremely flat along this axis in configuration space and that the general motion should exhibit one small and two large frequencies. It was found that a spring put along the -y axis of Figure 4 would produce a third non-zero frequency. It was previously argued that the bond-bending force could be ignored; however, the need for a third non-zero frequency may necessitate a spring being used to model such a force. The plotted results for the "linearized" problem IV explicitly illustrate the frequencies with initial displacements along normal coordinates or at any general point. Figure 5 illustrates the spectrum for an initial displacement to the general point x = 0.05, y = 0.072, and spond to 70 and 104 cm * , where cm * are found using k. = oo./c, c the speed of light. Note that displacement to a general point excites both non-zero frequencies. For problem m , identical with problem IV except for the inertia, cm Figure 5. Spectra of linear and general two dimensional systems with initial displacement to same position 38 a superposition of the two non-zero frequencies was visible in the x and y time series. This superposition made the individual time series appear disorderly, but was easily resolved in the Fourier analysis. For the two dimensional general case, a general initial displacement produces a much less clear indication as to the fundamental frequencies. Figure 5 illustrates the spectrum of the general case for initial displacements to x = 0.05, y = 0.072, and "linearized” example above (also shown in Figure 5). Clearly, a much more complicated and less concise result in comparison. This and results like it prompted a search for means to reduce the complexity of the general spectrum. Displacements along the first normal coordinate (0. 719450, -0.495163, 0.487035) were examined with three objectives in mind: (1) determine if initial displacements along the normal coordinate aid in simplifying the spectra, (2) discern to what extent the "linearized” approximation represents the general case, and (3) examine the relationship between initial energy (potential) and frequency of oscillation. The displacements were a percentage of the normalized normal coordinate. Hence, we shall, for example, refer to the spectrum of a system whose initial displacement was four percent of the normalized normal coordinate as "the four percent curve", or "a four percent displacement." In regard to the first objective, the conclusion is both positive and negative. The displacement along the normal coordinate does reduce the complexity; however, the spectrum remains difficult to interpret, Figure 9 is 39 such a spectrum. We offer an explanation. The "artificial” period imposed by our 819.2 time interval forces the Fourier transform to determine what frequencies are present in that period. Consequently, it is probable the spectrum does describe those frequencies existing in the 819.2 time interval, but the motion in the interval is not periodic. One time series plot was examined over a 819.2 time interval and no periodic motion was visible. The results pertinent to objective two are shown in Figure 6 . We discuss this objective using two norms: (1 ) existence of expected frequency, and (2) appearance of small frequency. If the former is accepted as the norm, the "linearized" approximation represents the general case for displacements equalling approximately 4.25 percent of the normalized normal coordinate. This is evidenced by the drastic change in spectrum between a displacement of 4.25 percent of the normal coordinate and one of 4.5 percent. The linear frequency of 104 cm 1 is clearly present in the form er, while missing in the latter. A displacement of the same proportion along the other normal coordinate corresponding to a non-zero frequency give similar results. If the latter is accepted as the norm, the "linearized" approximation represents the general case for displacements less than 3 percent of the normalized normal mode, as the small frequency appears in the 3 percent displacement as a small "blip" and progressively becomes more dominant. Nevertheless, the small frequency does appear as was expected by our above discussion. The results of the last objective are dramatically illustrated by Figures 7 and 8, which show the frequency increase as the initial energy is 40 Figure 6 . Spectrum change for general two dimensional system as a function of initial displacement; displacement given as a percentage of the normalized normal coordinate 41 3.00% initial displacement 4.25% initial displacement Figure 7. Shift of "small frequency" as a function of initial displacements along a normalized normal coordinate 42 Figure 8 . Shift of ”104 cm frequency as a function of initial displacements along a normalized normal coordinate 43 Figure 9. Spectrum of the general two dimensional case with 100 percent of the normalized normal coordinate as the initial displacement 44 increased for the small frequency and the 104 cm spike, respectively. It is this result which could be used as an optimization. Also note in Figures 7 and 8 the curves corresponding to a 4.25 percent displacement. In particular notice its shape to the left. It is known that when a harmonic function is Fourier transformed, it will appear, similar to the 3.00 percent displacement curve, as a spike with small side lobes. The fact that the 4.25 percent curve does not appear this way indicates the motion remains periodic but is not harmonic. A calculation was done to find the lowest quantum mechanical energy level E = J tied and to compare it with the initial energy in the system for a o 4 025 percent displacement. The former was found to be much larger than the latter. Whence, it was concluded that the motion becomes anharmonic when the energy is as large as the lowest quantum mechanical energy level. Thus, within the limits set by our model, the model, as pertaining to the water molecule simulation, is not successful because of the above result. For this reason we offer no frequencies of oscillation for the general two dimensional case, 3. Three Dimensions We consider now the three-dimensional case. The geometry, mass, and moments of inertia used were those appropriate to a water molecule centered at the center of a tetrahedron by four bonds. Table n contains the values of the constants used. 45 TABLE II: DEFINING CONSTANTS FOR THREE DIMENSIONAL ANALYSIS PROBLEM Constants (Units) Spring 1 Spring 2 Spring 3 Spring 4 k (x l0 dynes/cm) 0.18000 0.18000 0.18000 0.18000 —8 a(xl0 cm) 0.86713 0.86713 1.06916 1.06916 l(x l0 §cm) 1.80000 1.80000 1.80000 1.80000 —8 w;(xl 0 cm) 2.65893 2.65893 2.86300 2.86300 9 (radians) 0 .0 0 0 0 0 2.01860 4.09654 4.09654 £ (radians) -0.11302 0.11302 0 .0 0 0 0 0 0 .0 0 0 0 0 £ (radians) 1.57079 1.57079 2.39055 0.75104 77 (radians) 0 .0 0 0 0 0 0 .0 0 0 0 0 -0.90483 0.90483 _22 m = 2.98975 x 10 g -39 2 I , = 0.14831 x 10 gm-cm x -39 2 I , = 0.18643 x 10 gm-cm y -99 2 I , = 0.33331 x 10 gm-cm z 46 The frequencies, in reciprocal centimeters, for the linearized case were found to be 112.6, 107.0, 94.3, 1.2, and 0.0, with respective normal ized normal coordinates (0.0, 0.0 , 0.8389 8 7, 0.0, -0.544150), (0.790 502, -0.397986, 0.0, 0.465523, 0.0), (0.472690, 0.879778, 0.0, -0.050530, 0.0), (-0.389447, 0.259993, 0.0, 0.883591, 0.0), and (0.0, 0.0, 0.544150, 0.0, 0.838987). The latter normal coordinates are of the form (x, y, z, An analogous analysis of the frequencies as was done for two dimensions follows for three dimensions, also. As with two dimensions, we seek as many non-zero frequencies as degrees of freedom to use for optimization. It was found that if an additional fifth spring were put in a plane perpendicular to the two planes defined at equilibrium by the original four springs, five non-zero frequencies were produced. However, if the additional fifth spring was put in one of the original planes, only four non-zero frequencies were produced. Consequently, care must be taken when attempting to model for an additional fifth frequency. In the three dimensional case, an inherent and undesirable character istic of Euler angles appeared. In the process of putting our system of general Lagrange equation into the form q. = f(q., q.), the inverse of the following matrix (which does not exist when 0 = 0) was needed to obtain the correct form for the left hand side: • « Ix,s2 e s 2j/)+Iy/S2 0c2j/)+Iz,c20 |lx/S0s(20)-Jl^/s0s(2j/)) Iz,c 0 2 2 Jl ,S0S(20)-Jl /S0S(20) I ,c 0+1 ,s 0 0 x y x y i ,c2e 0 I , z z 47 Note, before proceeding, that when 0=0, the Lagrange equations reduce to a system of five distinct equations in x, y, z, "linearized" case, the The difficulty could not be overcome by the use of Hamilton’s equations (Symon, 1971, p. 392). (We concentrate here on the Euler angles, as we believe them to be causing the difficulty.) To develop Hamilton’s equations, the generalized coordinate velocities q^ were needed in term s of the generalized coordinates q^ and conjugate momenta p . A relationship was • • found between p , p , and ip, 0 , 0 . This relationship in matrix form is cp 0 0 given by ‘ p ' * ip * where P is the same m atrix as above. Consequently, both the Lagrange equations and Hamilton’s equations degenerate to rotation about a fixed axis z when 0 = 0. An interpolative procedure past the singularity offers another possibility to find the general solution. One such procedure is the following. It is known that when 0 = 0, ip + ip = a and 0 = f , where a is the right hand side of the cp La equation with 0 = 0 and divided by I ,, and f is the right side of the 0 equation z z 2 2 with 0 = 0 and divided by 1^, c 0+1 ,s 0 . We estimate the slope of ip from the estimated slope of 0 and the relation ip + ip = a . The 0 equation needs no 48 adjusting. If we let t , t , and t denote, respectively, the times at which c b a the next Runge-Kutta point is to be calculated, the last Runge-Kutta point evaluated (0=0), and the next to last Runge-Kutta point evaluated, then the slope m of the ip equation between t and t^ is * ( * b > - $ < y m = V*. Using ip = m(t^ - t&), No further results were attainable as a result of this singularity. C . Conclusions We list the conclusions found in this thesis. 1. The Model a. The model is theoretically correct. b. The analytic components of the model function very well. c. Infinitesimal rotations in three dimensions can be described by two independent variables. 2. Applications a. Two dimensional water molecule (1) The frequencies for the linearized case are 104. 7, 70. 8, and 0 cm (2) Three non-zero frequencies are obtainable if three springs are used. 49 (3) Displacements along the normal coordinates from the "linearized'’ case are the "best first choices" for initial displacements in the general case, but still do not sufficiently simplify the spectrum. (4) The frequencies increase as a function of increasing energy, allowing for possible optimization. (5) The motion is anharmonic within the limits set for the model, b. Three dimensions (1) The frequencies for the "linearized" case are 112.6, 107.0, 94.3, 1.2, and 0 cm \ (2) It is proposed that the singularity in the Euler angles at 0 = 0 can be treated by an interpolation procedure. 50 BIBLIOGRAPHY 1. Enochson, Loren D ., and Robert K. Otnes. Programming and Analysis for Digital Time Series Data, Washington D.C.: Navy Publication and Printing Service Office, 1968. 2. Gear, C. William. Numerical Initial Value Problems in Ordinary Differential Equations. Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1971. 3. Goldstein, Herbert. Classical Mechanics. Reading, Massachusetts: Addison-Wesley Publishing Company, In c., 1950. 4. Hale, B. N., and P. L. M. Plummer. Journal of Atmospheric Science, September, 1974. 5. International Business Machines Corporation. System/360 Scientific Subroutine Package (360A-CM-03X) Version m Programmers Manual. White Plains, New York: International Business Machines Corporation, 1968. 6 . McCracken, Daniel D ., and William S. Dorn. Numerical Methods and Fortran Programming. New York: John Wiley and Sons, Inc., 1964. 7. Pennington, Ralph H. Introductory Computer Methods and Numerical Analysis. Toronto, Canada: The Macmillan Company, Collier- Macmillan Canada, Ltd., 1970. 8. Plummer, O. R ., (1974) personal communication. 9. Plummer, P . L. M ., and B. N. Hale. Journal of Chemical Physics, 56 , 4329 (1972). 10. Plummer, P . L. M ., and B. N. Hale. International Conference on Nucleation. Leningrad, U.S. S.R., September, 1973. 11. Plummer, P. L. M ., and B. N. Hale. Journal of Weather Modification, 6 , 161 (1974). 12. Ralston, Anthony, and Herbert S. Wilf (eds.). Mathematical Methods for Digital Computers. New York: John Wiley and Sons, Inc., 1962. 13. Symon, Keith R. Mechanics. Third edition. Reading, Massachusetts: Addison-Wesley Publishing Company, Inc., 1971. 51 14. Tipler, Paul A. Foundations of Modern Physics. New York: Worth Publishers, Inc., 1969. 52 VITA Allan Bruce Capps was born May 31, 1950, in Granby, Missouri. He received his primary and secondary education in Granby graduating from East Newton High School in May, 1968. He received his college education from the University of Missouri - Rolla, in Rolla, Missouri, and the University of Missouri - Columbia, in Columbia, Missouri. He received a Bachelor of Science degree in Applied Mathematics from the University of Missouri - Rolla and a Bachelor of Science degree in Secondary Education from the University of Missouri - Columbia in December 1972. He has been enrolled in the Graduate School of the University of Missouri - Rolla since January 1973. He has held teaching assistant ships while being so enrolled. He held a National Science Foundation research assistantship during the summer of 1974. On July 24, 1971, he was married to Nancy Joan Mary Travis. 240810