AN ANALYSIS OF THE VIBRATIONS OF CERTAIN LARGE

MOLECULES INCLUDING SPIROPENTANE,

METHYLENECYCLOPROPANE} AND CYCLOPROPANE

DISSERTATION

Presented in Partial Fulfillment of the Requirements

for the Degree Doctor of Philosophy in the

Graduate School of The Ohio State

University

By

WALTER BERNARD LOEWENSTEIN, B.S.

The Ohio State University

195^

Approved by:

Adviser ACKNOWLEDGMENT

The author wishes to express his sincere grat itude to Professor W. H. Shaffer for his guidance, counsel and encouragement throughout the recent years. Pie would also like to express his thanks to the Texas Company for the award of a fellowship during the tenure of which this work was completed

He is also grateful for the constant help and en­ couragement given by his parents. i

TABLE OF CONTENTS >ag I INTRODUCTION 1

II THE SPIROPENTANE MOLECULE 5

Introduction 5 The Multiple Origin Method 5 The Structure of Spiropentane 7 Coordinates 9

III SYMMETRY PROPERTIES OF SPIROPENTANE 16

Group Theoretical Predictions 16 Covering Operations 19 Symmetry Coordinates 21

IV THE SECULAR DETERMINANT OF SPIROPENTANE 31

Introduction 31 The Kinetic Energy 31 The Potential Energy 3b The Secular Determinant k-5

V FREQUENCY ASSIGNMENTS AND CALCULATIONS 53

Possible Assignments 53 Calculations 59 More Definite Assignments 66

VI THE METHYLENECYC LOPROPANE MOLECULE 69

Introduction 69 The Geometry 69 Coordinate Systems Ik Symmetry Coordinates 78 Eckart Conditions 8 0 iii

TABLE OF CONTENTS (Continued) Page

CHAPTER VII THE SECULAR DETERMINANT OF METHYLENE- CYCLOPROPANE 85

A. Introduction 85 B. The Kinetic Energy 86 C. The Potential Energy 91 D. The Secular Determinant 95

CHAPTER VIII FREQUENCY ASSIGNMENTS AND CALCULATIONS 98

A. Assignments of Fundamental Vibrations 98 B. Further Assumptions 101 C. Revised Equations 102 D. Calculations 106 E. The Force Constants 111

CHAPTER IX THE VIBRATIONAL ANALYSIS OF PLANAR DISTRIBUTION OF IDENTICAL CHARACTERISTIC GROUPS 115

A. Introduction 115 B. The Boundary Conditions 117 C. Geometry 117 D. The Kinetic Energy 121 E. The Potential Energy 122 F. The Coordinate Axes 129 G. The Equations of Motion 132 H. The Secular Determinant lk-1 I. The Discrete Frequencies 1A5

CHAPTER X THE CYCLOPROPANE MOLECULE 150

A. Introduction 150 B. The Geometry of Cyclopropane 151 C . The Kinetic Energy 159 D. The Potential Energy 162 E. The Secular Determinant 163 F. The Fundamental Frequency Assignments 165 G. Calculation of Force Constants 167 SUMMARY AND CONCLUSIONS 171 BIBLIOGRAPHY 175 1

CHAPTER. I

INTRODUCTION

The proper interpretation of the infrared and Raman spectra

obtained from a macroscopic collection of molecular systems is one

of the most powerful methods of obtaining information concerning

the inherent physical properties of such systems. Such information

can include the structure of a molecule, the forces determining

the structure, thermodynamic properties and even some of the proper­

ties of nuclei present in the molecule. It is usually true that the

number of distinct observables is much smaller than the number of unknown parameters. To uniquely determine the unknown parameters with the observables it is necessary to introduce information from

other studies and experiments to reduce the number of unknowns.

This procedure leaves a problem with a reduced number of unknown parameters which still exceeds the number of distinct observables.

At this point the particular type of analysis may suggest simplifi­

cations which lead to a successful completion.

In a molecular vibration problem, the unknown parameters which may be immediately introduced are those relating to the geometrical structure of the molecule. These may be obtained from electron and x-ray diffraction data and to some extent from the rotational spec­ tra If they can be resolved for the molecules of Interest. The primary unknowns in a vibrational analysis are the parameters associated with the force field within a given molecule. The de­ termination of a potential function, from which a force field may 2

be obtained, for two related molecules Is the objective of this

dissertation.

There is a growing interest in determining the properties of

what may be, at present, called large molecules. It is true that

as the number of atoms is increased, the complexity of the analysis

increases. It is therefore desirable to describe the system in a

manner such that the number of unknown parameters is greatly de­

creased, thus simplifying the analysis. In i. vibration problem

this is equivalent to describing the system as vibrating in or

almost In a normal mode. It is found that the application of the

rotating axes theorem of classical mechanics will tend to give such

a description. A vibrational analysis utilizing the multiple origin method as developed by Deeds is the result. The details of the method are discussed In Chapter II.

The choice of the two molecules was governed by the fact that

there is a growing interest in investigating the properties of molecules containing supposedly strained rings. It is also true that, in general, many of the force constants associated with the multiple origin method cannot be freely transferred from molecule to molecule. However, the structure of these two molecules is such that it is geometrically permissible to attempt to transfer certain ones. It is of extreme interest to treat the two molecules inde­ pendently and compare such constants. Further, in the application of the multiple origin method to the study of a three member strained ring, portions of the analysis reduce to those arising in the study of the bent symmetrical XY2 molecule. Because of this reduction,,

it should be possible to incorporate those well known treatments to simplify the analysis.

The correct assignment of fundamental vibration frequencies is one of the major obstacles to be surmounted in the analysis.

As molecules of interest become larger it can be expected that associated moments of inertia become larger. As a result, one finds that the differences in frequency between adjacent lines in the rotation-vibration spectra decrease. For large molecules this usually results in an essentially unresolved rotation-vibration band, so much so that quite frequently it is very difficult to differentiate between a perpendicular and a parallel band. There­ fore the frequency assignments must rely more heavily on the basis of calculated results. It is apparent that a rigorous analysis of an unresolved vibration-rotation band is extremely unlikely. In line with the low resolution approach of the analysis, the centers of the bands will be assumed to correspond to the vibration frequen­ cies. This is a practice frequently used by those studying large molecules. It can be expected that an-harmonicity will be quite pronounced in vibrations involving a strained ring.

The description of a molecular system must be quantum mechanical in nature. However, such a description must be one covering all degrees of freedom, some having and others not having classical analogues. It Is well known that the total SchrtSdinger equation can, to a high degree of approximation, be separated into a number of separate equations. In a molecular vibrational analysis one

deals with one of the separated Schro&inger equations obtained by the assumption of separation of variables. Then the diagonal-

ization of the kinetic and potential energy expressions yields

the transformation to normal modes. With a knowledge of the nor­ mal modes it then becomes possible to obtain information concerning

energy levels, selection rules and intensities. 5 CHAPTER II

THE SPIROPENTANE MOLECULE

A . Introduction

The infrared and Raman spectra of spiropentane (see Fig. l)

were first obtained by Cleveland, Murray and Gallaway (l). Scott,

Finite, Hubbard, McCullough, Gross, Williamson, Waddington and

Huffman (2) carried out a vibrational analysis, assuming cyclopro­

pane force constants, in order to place the fundamental frequency

assignments on a stronger basis. Some of these assignments agreed

with the tentative assignments given by the former authors, others

differed considerably. Blau (3) used the calculated force con­

stants of the latter authors in his study of methylenecyclopropane

(C1).H^) but seriously questioned some of the assignments in spiro­

pentane . It will be seen that the method of analysis used here will considerably simplify the vibrational analysis and will make

some of the assignments more definite.

B. The Multiple Origin Method

The vibrational analysis will incorporate the Multiple Origin

Method, hereafter referred to as M.O.M.

The M.O.M. has been developed in the course of a program to

study the vibrations of large molecules carried on at this university under the direction of Professor W. H. Shaffer. The method was In­ troduced by H. S. Long (^l-), further developed by W. A. Pliskin (5) and R. E. Kidder (6 ). W. E. Deeds (7 ) improved the method and applied it to the paraffin chains. R. G. Breene(8 ) applied it to 6 dimethyl amine. B. Gurnutte (9 ) applied it to closed ring hydro­ carbons .

The essence of the M.O.M. lies in the repeated application of the !Moving Axes " Theorem (10) of classical mechanics. This approach is ordinarily used to separate the free rotation of the molecule as a whole from the internal motion of the constituent particles, but here it will be used to separate further the various kinds of motion occurring under a given vibration. It is generally known that a C-H bond is much stronger than a C-C bond. To apply the M.O.M. we seek ensembles of atoms within the molecule which are more tightly bound to each other than to any neighboring atoms or groups of atoms. Such an ensemble of atoms will then be consid­ ered a characteristic group and the motion of its constituent par­ ticles under a given vibration of the molecule as a whole will be considered as consisting of the hindered translation of the charac­ teristic group as a whole In the molecular framework, the hindered rotation of the characteristic group about a principal axis through its center of mass and the motion of the particles constituting the characteristic group relative to its center of mass. The logical characteristic groups for spiropentane are the four CH2 groups and the single carbon atom.

The result of thus describing the motion of the nuclei consti­ tuting a molecule Is that, coupled with judicious approximations, the vibrational analysis of a relatively large molecule can be handled, quite easily. For example, Scott, et al (2) solved at

least one sixth order determinant in their analysis of spiropen­

tane whereas it will he seen that the application of the M.O.M.

will require the solution of no greater than second order deter­

minants in this molecule. Recent work by Higgs (ll) indicates the

use of an approach similar to the M.O.M.

C . The Structure of Spiropentane

The most reasonable structure of spiropentane is that having

D2 ^ symmetry, this being confirmed by the electron diffraction data

of Donohue, Humphrey and Schomaker (12). Their values for equilib­

rium distances and angles will be incorporated in this work. These

are listed below and have meaning when referring to Figure 1.

C-H = 1.08 A (assumed)

c^-c^ = ± .03) A

ci_c2 = (1.51 ± .ok) A

< 02°3C1 = (6l*5 ± 2 )°

< HCH = (120 ± 8 )

(O-G)ave = (1-^9 ± .01) A

Generally it is assumed that the plane of the CHg group lies perpendicular to the plane of the adjoining C-C-C angle, and this practice is adopted here. It is generally also assumed that this plane bisects the C-C-C angle. In place of the latter assumption, it will here be assumed that the plane of the CHp group bisects the

M.O.M. framework angle as will be described later. This somewhat 3

Figure 1. The Spiropentane Molecule Symmetry 9

simplifies the treatment and has very little effect on the low

resolution calculations with which we are concerned.

In the application of the M.O.M. each CH^ group will be con­

sidered as a translating hindered rotator with internal degrees of

freedom. The essential motions are:

1. The translation of the center of mass of the characteristic group.

2. The rotation of the characteristic group as a rigid body about an axis passing through its center of mass.

3. Relative motion of particles within the charac­ teristic group relative to the center of mass of the characteristic group.

The framework equilibrium distances can be calculated, using the

previously listed bond distances and bond angles, and are listed

below with their meaning indicated in Figure 2 and 3* The masses

pertinent to this calculation may be found in Table 7-

© = 6^ ° 8 *

t = 57°56i

rQ= 1.55 A

D. Coordinates

A number of sets of coordinates will be used throughout this

analysis. The factors governing the choice of a particular set of

coordinates are determined by the simplicity they afford in carrying

out further operations. The kinetic energy will be set up in what will be called a ncarte sian" set of coordinates. This means that

the initial kinetic energy expression trill be diagonal. Portions 10

Figure 2. The Spiropentane Molecule With M,0*M# Framev/ork 11

of the potential energy will he set up in what Deed (7) has called

the "modified” valence coordinates because it is believed that the

potential energy is more nearly diagonal under such a description.

This is true because the "modified8 valence coordinates are not very

different from some choice of true valence coordinates. The descrip­

tions will then be combined through transformations to the M.O.M.

symmetry coordinates.

The choices will further be governed by the manner in which the

coordinates transform under the allowable covering operations. The

coordinate axes will in general be oriented in such a way that if R

represents some covering operation and f ^ represents a particular

type of coordinate in the i ^ characteristic group, then R \ ^ = ± f . J where j refers either to the same or another characteristic group.

In all cases, the orientation of the coordinate axes will be deter­ mined by the equilibrium configuration of the molecule.

For the internal motions of the CH2 groups, the familiar u,v,w

symmetry coordinates will be used. It was demonstrated by Shaffer

and Newton (13) that they satisfy the Eckart conditions (1*0 for the

Internal motions of bent symmetrical XY2 molecules.

With these factors in mind, the following coordinate axes and displacement coordinates will be chosen for this analysis. Reference to Figure 3 will be useful. Figure 3- M.O.M. Coordinate Systems in Spiropentane 13 For i =s l,2,k,5

= coordinate axis through C.M. of ittL CE q group in plane of

M.O.M. framework bisecting the angle 5^ , the positive sense

being outward from the framework.

z^ = coordinate axis through C.M. of i ^ CH^ group in plane of

M.O.M. framework, perpendicular to the positive sense

being toward the rotation-inversion axis.

x^ = coordinate axis defined by y.j_, and application of the right

hand rule.

^ ^ = translation of i^ 1 CHg group along x^ axis.

'Yj ^ = translation of i^ 1 CH2 group along the y^ axis.

^ = translation of i^ 1 CH2 group along the z-^ axis.

= rotation of i^ 1 CH2 group about the x^ axis, the positive

sense being that of a right hand screw advancing along the

positive x^ axis. (Wagging motion)

= rotation of i^*1 CH2 group about the y^_ axis, the positive

sense being that of a right hand screw advancing along the

positive y-^ axis. (Twisting motion)

7 p =* rotation of i^ 1 CH2 group about the Zj_ axis, the positive

sense being that of a right hand screw advancing along the

positive z± axis. (Rocking motion) lA

u-^ = symmetric mode (bending)

Vj_ = symmetric mode (stretching)

wi = nonsymmetric mode (stretching)

For i = 3 one must merely describe the three translational

degrees of freedom for the central carbon atom as indicated in

Figure 3 using coordinates

Using the modified valence coordinate systems the following

coordinates will be used.

Ifr-L = change from equilibrium of distance from central carbon atom

to C.M. of i^ 1 CH2 group.

S0-L = change from equilibrium of angle indicated as 0^ in Figure 3 .

In addition we have to define three additional coordinates for

further allowable framework motions.

SI = the angle between the base (of the triangle) bonds when, say,

the upper ring rotates clockwise and the lower ring rotates

counter clock1,hLse .

Aa and A^ = the angles observed when the upper (lower) ring rocks

perpendicular to the plane of the lower (upper) ring.

The angle used here will be defined as the change from

equilibrium of the lines joining the central carbon atom

with the center of mass of the two CHg groups in each

ring. In general the equilibrium value of the angle will be used

in the calculations rather than that of 6 since it is possible to

simplify a number of expressions more easily.

On the basis of these definitions, it can be shown that the following relations hold for infinitesimal changes from equilibrium.

A = sin j^/2 2.1

B = cos j^/2 2.2

^sinjzJ + '??2 cos^ 2.3

S r ^ = ^ |jB+ ^ A - ^ sirijzS - f ^ cos^ 2 . 5

S r 5 = <5^2sinS^+ $ 3 Cos^ 2.6

S©! =(l/^ ^A^+^2 )-(^+^2 )3 -2 ^ 0 0 3 ^- 2.7

S ©2 = ^ ^A(?J 14+^5 )b + 2 ^ c o s ^ 2.8

SL =[l/2 rcos^[j1+f2+^+f^j 2.9) Aa =[l/2rsln^j [f 2 .1 0 )

Ab =[l/2rsin^| [j 1^ * 5+ % + B ^ 1 “^ 2 > 2 .1 1 )

In addition the coordinates X, X, Z. will be used to donate translation of the center of mass of the molecule as a whole, the positive senses being to coincide with those of %■$> res­ pectively. 16

CHAPTER III

SYMMETRY PROPERTIES OF SPIROPENTANE

A. Group Theoretical Predictions

It has been established (12), as discussed previously, that spiropentane has D2 ^ symmetry. The character table obtained from Margenau and Murphy (15) is given below with other information to be used later.

I 2S^(Z) sg = cl1 2C2 2crd

Qxx>Qyy R(p) A1 + 1 + 1 +1 + 1 +1 azz

a 2 + 1 +1 + 1 -1 -1 R z - R(d) Bl + 1 -1 +1 +1 -1 QXx>Q'yy

R(d) b 2 + 1 - 1 +1 -1 +1 T z I.R.(ll) T T E' +2 0 -2 0 0 x y “xMy R(d) R R X V ^ X z ^ z I.R.(l)

Table 1

Characters and Symmetry TyPeB

The molecule consists of a total of 13 atoms. The framework consists of 5 characteristic groups. Hence we can expect

3.13 - 6 = 33 vibrational degrees of freedom for the molecule as a whole and 3»5 - 6 = 9 vibrational degrees of freedom for the frame­ work alone. It is next of interest to determine how these vibrational 17

degrees of freedom are distributed among the species of the group.

The number of the vibrational degrees of freedom associated with each species can be obtained from the character of a general vibra­

tion and use of the expansion theorem (l6)(l7). The results of such a calculation are given in Table 2.

Total Framework Trans Rot.

A1 5 2 0 0 a2 3 0 0 1

0 0 B1 1

B2 5 2 1 0

E 8.2 = 16 2.2 = 14- 2 2

Table 2

Number of Degrees of Freedom Associated with Each Species

It is easily seen that the characteristic groups containing more than one particle have C2v symmetry when considered alone.

Table 5 gives the character table of this group along with the species type of the familiar u,v,w coordinates. The latter will be explicitly defined later. 18

I C2 (y) er (yx) o' (yz ) xo A1 4*1 -hi +1 +i u >v r1+r2

A2 +i +i -1 -l

B1 +i -l +1 -l

b2 +i -l -1 4*1 w R-l- ^

Table 3

C2v Character Table and Symmetry Types

The coordinate axes are oriented consistently with those in­ dicated in Figure 3 and are indicated in Figure if.

H H

Figure if

Details of CHg Description

Also indicated in Figure if are the equilibrium bonds and angle which will aid in obtaining the behavior of u ;v,w under the cover­ ing operations of the D 2(i group. 19

B . The Covering Operations of the Group

There are eight covering operations distributed among the six

classes of the group. These are:

E: The identity element.

tr/2 . : Rotation by TT/2 around Z axis and reflection in plane perpen­

dicular to Z axis passing through the center of mass of the

molecule.

W 2 : Rotation by 3tt/2 around Z axis and reflection in plane per­

pendicular to Z axis passing through the center of mass of

the molecule.

Rotation by TT around Z axis.

CgC+Xj+Y): Rotation by K around an axis lying in the XY plane bi­

secting the angle formed by the +X and +Y axes.

C2(+Xj-Y): Rotation byaround an axis lying in the XY plane bi­

secting the angle formed by the +X and -Y axes. er'(XZ): Reflection in the XZ plane. cr'(YZ): Reflection in the YZ plane.

The symmetry elements are indicated in Figure 5* 20

C P. = S4

\ c 2 (x ,y )

X / \ / \ c 2 (x ,-y )

Figure 5* Symmetry Elements of D 2(^ Symmetry Group 21

C. Symmetry Coordinates

It is of extreme importance to see how the coordinates defined

In Chapter II-D transform under the covering operations allowed by

the Dga group. To do this, the definition of the u,v,w coordinates must be given explicitly. Reference to the work of Shaffer and

Newton (13) and Figure k indicate that these are:

u = (m^/2M ) ^/2 (dRQ (Sx ) + £ ( S r 1+Sr2 )) (3 -1 )

v = l/2 (^/M) 1//2 ( - i R 0Sx+d(SRi+&R2)) (3 .2 )

w = 1/2(u/m)1^ f(5R1-5R2 )/C (3-3)

x n / / where d = cos ' 2 ^ = 2m.^tic/M

f = sin(Xo/2 ) M = m c+2nijj

C = 1 + (mc/M)cot2 (XD/2)

It is apparent that u and v transform as either Sx or SRq+Sr2 since they transform in the same way under C2v* Wtransforms in the same way as Sr-|_-Sr2 .

Table h shows how the coordinates transform under the covering operations. It should be noted that the covering operations are performed on the molecule, leaving the coordinate axes unchanged.

A set of symmetry coordinates can be obtained by using the method of Nielsen and Berryman (l8 ), however to use this method, a knowledge of the irreducible representations of all the species is necessary.

For the D 2c^ SrouP these can be obtained from the work of Seitz (1 9 ).

However, for a relatively simple case such as this, It seems wiser, 22

E SjW 2 c i 1 c 2(x ,y ) Cg(X,-Y) CT-1

*1 ~ h -*4 ^2 h $4 - 4 - h ^2 *2 -1 4 - h f l h *5 -*2 - h h h °?3 "S3 v 3 -v3 'S3 3 3 U h - * i ~$2 h ?2 fi ~ h -54 $5 55 -52 -J l 3 4 Si 52 -fr ~$5 7 i 71 7 5 7 4 7 2 7 5 *94 Vi 7 2 7 2 772 7 4 7 5 7 i > 7 5 72 Vi V 3 ?3 f 3 "53 -73 1 3 3 V 3 - 7 3 Vi, 7 4 7 i 7 2 7 5 7 2 7 i **7 5 7 4 V 5 7 5 Vz 7 i 7 4 7 i ^ 2 ^ '4 7 5

? 1 <$5 ? 4 ^ 2 ^ 5 f 4 ^ 2 ^ 2 # 2 ? 4 ^ 5 ^ 1 ^ 4 ? 5 %2 %3

Table 4a Coordinate Transformations 23

n/2 . 3 V 2 c11 E Sl/ S^ 2 c2 (x,r) c 2 (x ,-y ) cr o-1

U1 U 1 u5 ult u2 u 5 u l+ ui u2

U2 U2 % u 5 U1 % U 5 U2 U1

u5 ul+ uk ul u2 u5 u2 U1

u 5 U5 U2 U1 % U1 U2 u5

v2 vl V1 v5 v5 vl+ V1 v2

V2 V2 vif V5 V1 V5 V2 V1

Vif V^ V1 V2 V5 V2 V1 v5 V k

V5 v5 V2 V1 V1 V2 vif V5

Wij. -w5 -wh -w2 W1 wi *5 w2 _wl 1 wr -w2 *1

W2 W2 Wl4- 5 W1 'W 5 H

-wi -w5 W1 W2 w5 "W2 -w>*

W5 W5 W2 W1 -W1 "V2 “w4 -”5

Table J+b

Coordinate Transformations 2h

though perhaps not as sophisticated, to give the Eckart condi­

tions (1*+) and let them be the guide in the choice of proper

s y m m e t r y coordinates. It will be seen that this method can be used

to obtain the irreducible representations which are often quite

difficult to obtain.

The Eckart conditions, for the molecule as a whole, are re­

quirements that, in a chosen body fixed framework, the motion of

the molecule has no total linear momentum and no total angular momen­ tum. There is no overall translation or rotation of the molecule in the body fixed system. This overall motion is considered with re­ spect to the already described XYZ system. The conditions can be written in the form:

( 1 l- S 2 ) + (pA $ 3 + (% "75 )b - ( V ^ 5 )a “ 0

+ (m/ ^ - (* V ? 2 )b + = 0 (3*5)

j^yi~v2+V h + y^] a + f+^ f B - ° ( 3 . 6 )

* i + *2 ~ V f 5 53 0

sin/ + A \ y 2 " ^ l ] + B 1^1“ = 0 (3-8)

sin^ 1^2” + A [ V \ j + B = 0 (3*9)

where m = mc and M =

It should be noted that the last 3 equations are approximate because they do not account for the contribution of the hindered rotations to the angular momentum. This is small and may be consid­ ered negligible at this point. 25 The apparent simplicity of these expressions is due to the fact that common non-zero terms have been factored out. This is because the central carbon atom, while not an inversion center, is in a sense a geometrical center at the center of mass.

With the latter equations as a guide, it is-not difficult to establish that the combinations listed in Table 5^ have the in­ dicated transformation properties.

To obtain the degenerate symmetry coordinates, more care must be taken, but if they satisfy the orthogonality requirement, and all pairs of symmetry coordinates transform in the same way satis­ fying the character requirements, then the transformation matrices should be just the irreducible representations of the degenerate species and this is indeed, found to be so. The degenerate symmetry coordinates are given by pairs in Table 5h. 2 6

E 2Si^(Z ) c j i 2C2 20“ d S p e c ie s

+1 +1 +1 + 1 4-1 aj+ag+a^+a:^ A1

v1+ v2+ v^ + v 5

ul +u2+u^-+u5

Sl+f32-^ -P 5

+1 4-1 4-1 -1 -1 7- i + / '2 ~ ^ " 75 A2

i i^+^+ii ^2 +£?5

+1 - 1 +1 + 1 - 1 7l + 7 2+ V 75 B1

V1+W2"V^ “W5

V ? a - V ? 5

no +1 - 1 4-1 -1 4-1 B2

al+a2“at -a5

ul +u2_uif“u5

v1+ v2 - v^ - v 5

Table 5a

Won-Degenerate Symmetry Coordinates ,w/2 S3k/2 E c21 1 c 2 (x ,y ) c 2 (x ,-y )

pr fi2

M s

7^7-z

V ? 2 7 if 7 j

1 o o 0 -l -10 0-1 0 +1 - 1 0 1 0 M s V u5 o -i o 1 0 0 -1 -1 0 +1 0 0 1 0-1 M a V U2

*3 V v5 vr v2 %

V\

V w5

Trace -2 0 0 0 0

Table 5b

Degenerate Symmetry Coordinates 28

The M.O.M. symmetry coordinates then follow by the introduc­ tion of normalizing factors and will be denoted by q^ and as indicated below.

3 -1 0 )

q 2 = Jl/^^+^+^+jfc; ) 3.11)

q3 = [1/^a.j+a^+a^-a^ ) Avl 3.12)

q*i- = 1^/^V1+V2+Vi4-+V5) 3.13)

q5 =[1/4(ul+u2+uif+u5^ 3.1*0

Pi =[i/|(Si +S2~ V i 5 ) 3-15)

q6 3 .1 6 ) A 2 q7 =CL/i(7l+72''/4"/5^ 3.17)

q# = [i/i{>i+w2+i^+u5 ) 3-18)

19 = t-/s}( S 3_+ f2+ J^+ fij ) 3.19)

^10 =[i/i(p 1+p2+p^ 5 ^ 3 .2 0 ) B1 qn = f r /ir i+^' 2+7 1O7 5 ) 3 .2 1 ) q12 = [l/i(vi+w2-vIrw5 )

qi3 = ) 3.23)

3.2*0 qi4 =& /ic9li+f2-V?5) P2 = ^3 B2 3.25)

q15 =[1/^o;1+a2-Q:i(.-oj5) 3 .2 6 )

ql 6 = [ l/% u 1+u2 -Ulf- U5) 3.27)

q17 = [l/iC^+Vg-v^-v^) 3 .2 8 ) 29

^ =&/^(Si-V (3 .2 9 )

pi,, (3-30)

P6 - §3 (3.31)

P5 =-73 (3-32)

qla “ f c / r c W ' V (3.33)

qlb (3-31*)

q2a (3 -35)

q2b = &•/$2] C?l"?2 ^ (3 -36)

q3a = [ V j 2] K - < V (3.37)

q3b (3 *38 )

qim “ t/^lCei-Pa) (3.39)

q7b “ L1 / W] <:pit-p5 ) (3 A 0 )

^5 a ==ri/’S^]^7l" 72 ^ E (3-^1)

15b 5 3 ^ (3A2)

96a =t1/d(ui>-u5) (3 A3)

96b (3.*A)

q7a =[1/^2'](v4 -t5 ) (3 A 5 ) q7b *0-/4sl(Tl-v2) e (3-W >

% a =I1/'l2l(w2 -wl) (3 A7 )

93b “tA^H-V (3A8) 30

For the framework a number of modified valence coordinates have been defined. By substituting the above equations it is found that the following can be considered modified valence symmetry co­ ordinated, denoted by .

Q1 j^Sr-j^+Srg+iSr^+Sr^j a Bq1+Aq2 (3*^9)

^2 s3^-/j2)(Sei+5e2) = -J— (Aqi-Bq2) Al (3 .5 0 )

Q13 = (l/^J^Sr-j+Sr^Sr^ - Bq^3+AqjLi4.+2p2sin/ (3*52 )

Ql4 = ( V ^ (S0i -5©2) = [Aq^-Bq^-apgCos^] (3-53)

Qla =(1/r2 )(K -Sr5) = B<1la+A(l2a- ^ P6cos^ (3*54)

Qlb = ( l / r £ } (Sr^Srg) = Bqlb+Aq2b- >T2 p^cos^ (3-55)

Q2a “ Aa = 2FT 5J5J [f2 Plf-1>P6+ J2 Bql a - Sz A q , J E (3-56)

Q2b = \ = 2r sin7 Bqlb- Jz fil2b] (3-57) 31

CHAPTER IV

THE SECULAR DETERMINANT FOR SPIROPENTANE

A . Introduction

Any description of the properties of a system of particles

such as this molecule must he of a quantum mechanical nature em­ bodying the Bohr frequency relation. It is well known that, to

facilitate the quantum mechanical problem, it is best to express

the Hamiltonian in as simple a form as possible. In a vibration problem this indicates that the Hamiltonian is to be expressed in normal coordinates. In zero order approximation it is assumed that only the quadratic terms of the potential energy are appreciable, the linear terms having to be zero because of the stability of the equilibrium configuration. It is in the process of finding the transformation to normal coordinates that the secular determinant

|v -*t | = 0 arises. The solution of this determinant, provided one knows the correct eigen values, can yield considerable information concerning the nature of the forces which maintain the equilibrium configuration.

B. The Kinetic Energy.

As previously indicated, the kinetic energy will first be written in "cartesian" or diagonal form. Let this be denoted by 2T = 5g5 .

Then the transformation to the M.O.M. symmetry coordinates and p^

Is defined by Equations (3-10-3.U8 ) and may be denoted by §= Fq. 32

Hence the equation:

2T = Jg | = 'tFgFq (if.l)

Then it will be vise to impose the Eckart conditions (lk) and

thus remove 6 of the degrees of freedom. This is a means of remov­

ing the six zero roots in the secular determinant and thereby sim­

plifying the problem slightly.

The kinetic energy is then no longer diagonal, but will be

further partially diagonalized by another transformation.

The initial kinetic energy expression may be written as:

(if.2 )

Transformed to the M.O.M. symmetry coordinates, the kinetic energy is:

M = mass of C R ^ group 2 2 P m => mass of carbon atom a ,b and c are

the principal radia of gyration

associated with a, P and y respec­

tively 33

2T = 2TAi + 2TAa + 2TBi + 2Tg2 + 2TE (It. 3) where 2Ta1 = M ^ q^ + q§ + a242 + 4^ + ( ^ ■ *0

2Ta 2 = M £ p | + b 2qg +' c2q2 + q§ J (^-5)

2TBl ^ M L^9 + ^ 0 + C^ ll + ^12 J (4’6)

2Tj32 “ M[_4^3 + \ \ h + g p | + a24| 5 + 4 ^ + -2 J (lt.7)

2TIS = M [ ' 4 b . + 54 + 4 b + 42b + + a2 (4!a+4lb^

,2 / * 2 *2 \ , „2/-2 ,*2 x *2 -2 + * ^ + 4 ^ ) + = h 4 | a-»!|b ) + 4 | a + 4gb (*.8)

• 2 .2 .2 .2 “ ] + *7a + *Jb + q8 a + q8b J

The Eclcart conditions, Eqns. 3 .4-3 .9 transformed to M.O.M. symmetry coordinatesare:

2P l = 0 B1 (^•9)

(k .1 0 ) M p 2 + 2 C A q l 3 + B q -lif] = 0 B2

m E (if.1 1 ) Pi<- + T t m P6 + BCLla ' Aq2a = ° m ( k .1 2 ) p3 + T 2 H P5 + Bcilb “ Ap2b = 0 E

Pi+sin^ - Aqla + Bq2a = 0 E (if.1 3 )

p^sinj^ - Aqlb + Bq2b = 0 E (if.lif) 34

Making the substitutions by eliminating the p-j_, the follow­

ing expressions result.

“ 2 JU = Mm LS* + 4-2q2 + a2^3: + + (^*15) . -2 = M Lb25|+ qy • (4.16) —1 1— 0 + p . = M *2 LB1 ~ “ L . ^ ‘ b“ ^10 qIo ‘ qH q f 2J (U . 1 7 )

2Tb 2 = M [_ C1* 1# 12) « 1 3 + (1+3^B2 ) °ik + a2415 + 4®t16 (4.18) • 2 + q 17 + (4“ BW ) q13411(.J

2TEe =- M ^ G l^la^ q ^ + qHlb ^ )1 T+ Gp(4pfl2 'H2a T+ ^4 2 ^ b 1) ~- tw2G^ 3 VHl(4n a H2 a TSLlb ^4Ph H2b )

+a2(43a+S2 -*-£2 43 b 'i) 4-+ v ^.2^ “tq^+^iib) 2 4-A2 'i a. + r c.2 /1?,2(45 a+45b)j-B2 ^.19

+(q6a+i|b) + (i2a+i7b) + (^ a+q6b^3 where G-, = 1 + . + 2^ (B + i —)2 1 T&S m 2B

G22 = 1 + T-irr4A2 + 2^ m (A + i-)22A

1 = 5 ab + ^ (B + §b > (A + 2ff)

It is of interest to diagonalize the kinetic energy again. It is found that diagonalizing 2TE is too cumbersome to be practical, but 2Tg^ may be diagonalized using essentially the same transforma­ tion used in diagonalizing the kinetic energy of the non-linear 35

bent XY2 molecule (13)* For reasons of symmetry;, and in anticipa­

tion of later calculations, similar transformations will be

applied to 2T^ . The transformation will be defined by:

Ul = ^ ArfSe^+Seg) + Bfir-L+Srg+Sr^+Sr^ )J-

4 h i - A q 0 ( 4 * 2 0 )

= 1 /2 C^ ) 1//2 Br(S9-j+£e2 ) + AfSr-L+Srg+Sr^+Sr^/j-

-R* LA^ 2] C^-2 1 )

2 = Ar(S©1-S©2 ) + B(Sr1+Sr2 -Sr^-Sr5 )J

( 4 '2 2 )

2 “ 1/ 2 Cg)1^2 ^ - KrfSej^-ieg) + A(8 r1+Sr2 -Srlt-Sr5 )J’

“ eI«S (Aql3+B

It can then be shown that the inverse of the transformation is:

qi = <$>1/2 L » i + AvJ ^ - 2t)

q2 = f^ )1 / 2 [-AU-l + B v J (A.25) 36

(^•2 6 ) M where e = 1 + 4— m

(4.27)

Making this transformation, the and Bg kinetic

energies becomes:

2TAi = % [u^V^] + M [a24§+qg + aj (4.28)

(4.29)

The reason for not making the coefficients of U*2 and Vp *? 2 equal to unity will be apparent when equations linking the modified

valence force constants with the U,V force constants are written.

C . The Potential Energy

To set up the potential energy in workable form, a number of

simplifying approximations must be made. The first is, as already

mentioned, that the true potential energy of the system, when

expanded in a Taylor series in many variables, is well represented

by only the quadratic terms in the expansion. This approximation

alone leaves the problem in an unsolvable situation because the

number of unknowns still far exceeds the number of observables.

Assumptions will have to be made concerning the magnitude of various potential interactions. 37

To properly set up the quadratic portion of the potential

function, the fact that the potential energy is a scalar, invari­

ant under all the symmetry operations, becomes important. This

means that products of coordinates belonging to two different

species cannot occur. Further no products between the a and b

coordinates in the degenerate species can occur. Hence the poten­

tial energy function will split into 6 distinct blocks under a

matrix representation, belonging to the species Ap, Ag, Bp, B2

and two identical ones belonging to the E species. The latter

transform into each other under the symmetry operations.

It is to be noted that three types of coordinates are being

used in the analysis, these being associated with internal motions

within the CHp group relative to its center of mass, hindered

rotation of the CH2 group about a principal axis through its center

of mass, and translations of the CH2 group and the carbon atom within

the framework of the molecule relative to its center of mass.

Qualitatively these correspond to, for example, such things as

C-H stretch or bend, methyl wag, rock or twist and ring stretch or

deformation respectively. In short, the initial description is one not far removed from the normal mode. This is especially true for a molecule having as much symmetry as this one.

Thus, the first assumption to be made is that the potential interactions between these three kinds of motion, while allowed by symmetry, are negligible. At this point it should be noted 1hat the 38 kinetic interaction between hindered rotation and translation of

groups within the framework has already been neglected by using 3

approximate Eckart equations.

With this assumption, each of the 6 blocks will be split into

three blocks except the A2 block which will be split into two blocks.

In the M.O.M. symmetry coordinates, these will appear as:

2V - 2 + 2V„ + 2V + 2V + 2V Bl E E (4.30)

If (4.31) -X

SAo [q 6 ,q T,q a ] (4.32)

C^-33) SB 1 tq9 ,ql0 ,qll^ 12]

(4.34) •SBo - [ U2 ' V2 J q1 5 ' qlb » ql7\|

(4.35) —E i ^ l a ’ q2a ’ q3a* q4a * q5a* ^a ’ q 7 a -' q8 a 3

(4.36) S e j^-lb' q2b' q3b' q2b> q5b ’ q6b > q7b' ^t Q

2VA 1 X 1 0 l Kuv, (4.37)

K3,3

l\ , k K4 ,5 0 K4,5 K 5,5 39

2Vhr ><6 ,6 *6 , 7 0 *6,7 *7,7 (^.38) 0 K.8,8

2V Bl *9,9 SB, o (^•39) K.10,10 Kn v10,ll K K 10,11 11,11

K12,12

2Vb _ = -Br- 0 {h.ho) Kav2 ^ 2 K 15,15 Kl6 ,l6 ki6 ,17 0 K16,17 K17,17

1 2V = q1 E 4 , 1 . 4 , 2 o tsE 9G 1,2 * 2,2 ( ^ l ) *3,3 * 3 A *3 * 3 A 4 t^-E *3,5 *?,5 5 *6,6 *6,7 *6,8

*6,7 *7,77e ,7 *?,*7,8 JD t,E *§,8 *7,8 *8,8

2V. ( ^ A 2 ) 40

Because, as Deeds (7 ) pointed out, further simplifying

assumptions may be justified when considering modified valence

coordinates, the potential energy will be considered in such a

system. Since the number of modified valence symmetry coordinates

chosen just equal the number of degrees of vibrational freedom

associated with the framework, the problem of a singular transforma­

tion which arose in the work of Curnutte (9 ) will not arise.

It is then instructive to restate Equations 3.49-3*57 with

the Eckart conditions imposed.

(4.45)

Q13 = (1/2) (Sr-j+Srg-Sr^-Sr ) = B(l+8 ^A.2 )q13 + A(l+8 ^jB2 )qlif (4.46)

$14 = (S©1-Se2 ) = (l-4~j-cos5z4 )q13 - ^ (l+^os^q^

(4.48) 41

Qlb = (1/ r2)(&r±-Sr2 ) = F ^Llb + F2q-2b (4.49)

w h e re F-, = B + cos^ -1- m b M(2A +1) ■^2 “ A m A cos

_ ^ (4-50) *^2a ~ r s±n$z4 < * % >

2b Ab - r s

Under such a description, the framework portions of the poten­ tial energy are: where l%1

S®2“ tQ13,Ql4. " *G?r II Kt ][%a'Q2a

ll r ~ Q = Q E \3lb 2b

F 2V - Q V V (4.52) 11 12 —A A1 A1 1 V12 V 22

(^.53) 1 M i

^B2 V13,13 V13,14 (^.5J0 v13,14 vi4,14 k-2

(^•55)

(^.56)

There is a transformation, not orthogonal since the Eckart

conditions have been imposed, which relates the modified valence

symmetry coordinates with the M.O.M. symmetry coordinates. This

transformation is defined by Equations: *4-.4-3-^ *51*

The large difference between the number of unknowns and the

number of observables leads to the previous author's adoption of the

principle that ICi, j = 0 ,i^j and refers to the interaction between

types of hindered rotation and to types of internal motion. The

justification of neglecting interaction between types of hindered

rotation is based on the qualitative description of CHg wag, rock

or twist. Neglecting the interactions between the types of internal motion is more difficult, and can be justified only insofar as result­

ing calculations compare favorably with those of previous authors and must be considered as calculations with the limitations of assump­

tion which all the authors have made.

(^-57) but the equations h .^3 -^. 51 define Q, = tq where t = L^ijl Hence it follows:

2Vs ^ %a W Ss = Ss~ [VB1 t q g (4.58) o r * [ K sl - ~ L v sl t (4.59)

The latter relation will be useful in obtaining the framework force constants.

For the hindered rotations andinbernal motions,the trans­ formation F will be useful. However, in transforming from the q^ to the f ^ the symmetry blocks vanish.

2V £ = j_ H r s [ k J q. = T [ k 1 q (4.60) s-

The transformation F was defined such that J = F(

It follows q = F"1! .

is defined by Equations 3-10-3.48

2W - I [Kq“] q = 1 r - 1 [k^] F ’1 * = i [Ka ] i

or;

Much use will be made of Equations 4.59"^-°2 to obtain the physically interesting force constants from the M.O.M. symmetry force constants.

In general, for a 2x2 matrix, if:

2vd = 1 K ] a = H i *q = s D'd ] a (it--63)

where Q, = tq (it .6k) with = Vva,a V a,b

Va,b Vb,b

(h,6 5 ) and Qa = ^a,a "^a_,b da

,a %

where ta b = tb ^a

it follows:

2 - 2 Va,ata ;a+ ^D,btbja Va Jata Jata/D+Vb Jb l:b Jat'b ,b rvD i = u .v^ ] t - +^ a 7b^a,a^b,a +^a,b ,a^b>b+tb ,a^a,b

^a,ata;a taJb+Vb ;btbJa^b,b va a , b+Vb , b "^b , b

+^a,b ^a;a^b;b+^ b , a \ ;b + 2Va,bt’aJb'tb,b

from which follows; in general:

('*.67) K l = if K l

and if V , = 0 a,b

(f .68) K l - If T t v a ,a a J+5

D. The Secular Determinants

The next task in the analysis is that of diagonalizing the

Hamiltonian, so that the Schrodinger equation is in workable form.

This is the problem of simultaneously diagonalizing two positive

definitive quadratic forms. In a sense one is looking for

'•principal axes’* or normal modes. The problem is perhaps more

physically treated by the quite well known approach indicated

below.

Classically, the system of particles is conservative and the

Lagrangian L may be defined as L = T(q) - V(q). The resulting

familiar Lagrangian equations of motion are:

dL _(Jl d.T B v dt S?5d dqi dt 8 qi

where T = i m. -q q V = ^ ^ K, .^q. 2 i,j J 2 x, j 1 J

The equations of motion become:

+ K..q. f = 0 i = 1 ...f

ico t For small oscillations it is assumed q^ = q ,.e J oj

• t 2 It follows q, = -CO iq.. «] 0 *3 2 2 where A = oo = (2 n v )

= Ca3*" = (2 A c )2 V 2- 46

Making these substitutions:

f 4 6 { - A m u + K 1;j^ q. = 0 i - 1 * * • f d

In order that these f equations have a non-trivial solution

for the q., it is necessary that the determinant of the coeffi- d cients vanish. This determinant is the secular determinant:

K-;-ij . - /'-Xj.A inj 0

In this analysis, it is assumed that the rtm ^ are known as veil as to the various X which are obtained from the infrared and Raman spec­

tra. With these data, some information can then be obtained concern­

ing the It; . which are the force constants.

'With the separation already effected in sections B and C, the

determinants to be solved are the following, in the M.O.M. sym­ metry coordinates:

Ku V1 0 A, (4.69) 1 KVl -X

Ku2 ^UVp = 0 B (4.70) IW V - A ?

= o Bj_ (4.71) ^9 > 9 ^ ^ i+7

K ^ l - > G XM ldj^ 2 + X G ^ M

= 0 E(2) (IK 72) 2 1 ^ G3M g — GgM

A1 (7-73)

k 6,6 ~ ^ Mb ~ ^5,7 - 0 A.o (7.77) p *6,7 K'7 , 7 -^ Me

P K-i10,10 “ K1 0 ,ll

= 0 B1 (7.75) *<10,11 Kll, 11- >Mc':

K - XMa^ = 0 ■15A5 Be (7.76)

E p E E K3 3 - X M a K 3,7 K3,5 E E ;rE K3 ,7 K ^ - X M b 7,5 0 E(2) (7.77) 2 K13 K, *3,5 % 5 5,5 "*Mc

K 7,7 “ * M K7,5 = 0 a l (7 .7 8 ) K, K, - X M 7,5 5,5

K, A 2 (7.79) h Q

K B (if.8 0 ) 12 , 12 “ 0

Kl6 ,17 Ki6 ,i6 “ = 0 Bp (if.8 1 ) K- Kl o , 17 17,17

E E E -AM k ; *6,6 Ao,7 v 6,8 k g K? 7 - = 0 E(2) (if.8 2 ) ^Oj Vf ( 3 I 7,8 K§ j8 - > M o ,8 *7 ,8

As previously mentioned, following previous authors:

k6 ,7 “ K1 0 ,ll K?3,7 h = 3,5 C = K 7,5), c = K)7 ,5 cr = K10,17t " -1.7 = o, -

= 0 = ^6 8 " .8

This procedure greatly simplifies the solution of the determin­

ants, none heing of order greater than 2. It may he indicated that

it is not yet wise to transfer diagonal force constants from other molecules in order to gain insight into the interactions considered

negligible here. Those diagonal constants were obtained using the

same assumptions and thus would not give a correct picture of the

interactions should one wish to include them. Should one be satis­

fied that those diagonal constants give a time description in the molecules studied previously, one is certainly forced to conclude that the electronic structure of the moecule under consideration is k 9

quite different, and thus the reliability of transfer to this

molecule is very much open to question. It will also be assumed

that V® 2 = 0 since an(i Q2b mus_i: close to the normal

coordinate. With these approximations in mind, it is then possible

to give additional equations relating the M.O.M. symmetry force

constants with those of interest.

= ''f,1*1,1 + ^2,2*2,1 + 2V1,2*1,1*2,1

4,2 ~ ^1,1*1,2 + ^,2*2,2 + 2Vf,2*l,2*2,2

E E E E , I<1 ,2 - Vi^it^tq^ + V2 ^2t2 ,2t2 ,l + vl,2 (‘bl,2t2 ,l+tl,lt2 ,2 ) (^-85)

Kg 9 = % = — 57 = V|>a 22 (^.8 6 ) s s r'-cos‘-p r^cos p where;

■%1 = Fi = B + “ (2B2+1 ) ^ ^ (^.87)

tl,2 = F2 = A " i (2A2+ 1 ) £ £ ^ (it.88) A

•4^JIT X I l-L Mm! U(1+2B2J ) ~ v t2 A = L* l Jf + mj= J ---- B~~ (4 '8 9)

t o o = - - M - r +|0 ll±p£l (^-90) J2>2 rsin^? |T+ [ t +i mj ] A

v?^2 ,2P = kAk a (^.91) 50

(T.9 2 ) = ^ A ) [ K5,5 + Ki 6,i 6 + 2K6,s]

(7.93) V = (lA ) [ K5 , 5 + Kl6,l6 - 2 K6 ,6^

(7.97) ^ 2 = (1 '/1t)]jS,5 " k±6 ,ig\

(7.95) k Vq =(1A)l]Ki4.,it. + A y ,i t + 2Kt ,7~1

(7 .9 6 ) ICV! =(1 /0)[jV7 + k17,1T ~ 2K7,t1

A..97) \ 2 = (1A)[^A,t " kit,itJ k\!0 + IC12,12 + 2k8,8^1 (7.98)

KV1 =(1A)jj%^8 + K12,12 “ < , 0 (7.99)

KW2 =(i A)[_KQj8 - K12j12] (7.100)

(7.101) % o =(1A)[i%,3 + k15 ,15 + 2K3,3~1

(7.102) A l = M [ k 3 ,3 + k 15,15 “ 2K3,3jj

=(i A)[ k 3j3 - K15A5*] (7.103)

(7.107) Ao 4^)13-0,10 + %,6 + 2Ki,J

(7.105) Kpi =(i A)[ k 1 0 A 0 + A , 6 - 2KS , J

*%> ^^-A^io^io ~ A,6^[ (7.106) 51

v7 0 + ^7,7 + 2 If,5] (4-107)

V - 2Kf;5] (4.108)

K 72 W [ KU , 1 1 " K",7I (4.109) vhere

K = diagonal term

= interaction in ring

Kq = interaction from ring to ring

= j~(l^Jll+KU 2 )sin2^ + (It ^+K ^)cos2^

,1 (ij-.llO)

= + (KVl-Kv 2 ) c o s ^

a (ll-.lll)

v2 r Kr0 = + ?i_ (Ku1+^u2)cos2^ + (Kv-j+K^sin2^ (U- .1 1 2 ) ^ + (Ku v +*„ Va)«in^J 'tira ■ «r0 - vf.i J (4.1X3)

K r 1 ri^ = K K U 1 - K u 2 ) c ° s 2 i(! + (^-Kygjsin2^

+ (I

K*■ J-'-'X riei - M Q ^ , 1+I

— ^ ^ i - KU2 -Kvi+Kv2 )Sii ^ ' (Ku^-Kbvgjoos^ J (^.1 1 6 )

Here Kq = diagonal term

a _ = interaction between rings when i=£j

interaction within ring when i=j 53

CHAPTER V

FREQUENCY ASSIGNMENTS AND CALCULATIONS

A. Possible Assignments

A set of tentative assignments with respect to symmetry types is given in the work of Cleveland, Murray and Galloway (l). These are by no means definite; especially insofar as it is very difficult to differentiate between the B2 and E species, that is to say with certainty whether one has a parallel pr perpendicular band. Scott et al (2 ) made a normal coordinate calculation, assuming cyclopro­ pane force constants, and based assignments on this calculation.

Some of the assignments are open to question, in particular the A^_ ring stretching which should only be Raman active, but was assigned to a band also strong in the Infrared. It is also possible that the assignments In cyclopropane are not as definite as might be under­ stood. Recent work by H. Neill (20) suggests the reassignment of some of the fundamental frequencies as listed in Herzberg (2l). A reasonable set of C-H stretching assignments is given by Blau (3)*

He also gives some strong arguments concerning the magnitude of the aforementioned A^ ring stretching fundamental.

First it may be best to list the assignments of Cleveland,

Murray and Galloway. These are found in Table 6 . I.R. Raman Liquid Gas Tentative 1 . Av 1 fi a ,d. m.d. % lA Assignment 305 2 .86 .03 .07 Fundamental, E, 30? CO U"\ 1— 1 16 .69 .02 .02 Fundamental, A]_, 581

613 4 .71 .02 .11 Fundamental E, 613

779 2 .66 .07 .12 783 s 778 m Fundamental, E, 779

798 13 97-613 = 782, BXXE

86 l vs P Branch or Il63-305=853(B2 or E)XE 872 30 .82 .01 .02 871 vs 870 vs Fundamental, B2 or E, 870 S79 vs R Branch

096 v 305+581=686, Aj_XE

(928) VW 929 w 3(305)=915,E3

920 vw 305+6l3=91&, EX3

983 VS P Branch 990 vs 993 vs Fundamental, E, 993 1003 vs R Branch

1033 50 .02 .01 .03 Fundamental, Ax, 1033

1029 s 1053 S Fundamental, B2 or E, 1053

1150 1 - 1155 S 1151 s Fundamental, Bp or E, 1151 a \ji I.R. Raman Liquid Gas Tentative Av I (0 a.d. m.d. vg 7 Assignment

ilo3 s Fundamental, 32 or E, U 63

1260 vw 2(613)=122o,E2

1280 vw 305+993=l£98; EXE

1325 m Fundamental, B2 or E, 1325

1340 w 13^0 m Fundamental, B2 or E, 1370

1397 5 .81 .03 .07 Fundamental, Bq, 1397 l72o 10 .82 .02 .05 1720 s 1730 s Fundamental, Bg or E, 1730

1775 w 531+870=1751, A1X(B2 or E)

1530 w 2 (778 }=1556, 2

1630 w 305+1325=1630 EX(B2 or E)

1675 w 613 +1053=1666 EX(B2 or E) 1760 w 613+1151=1767 EX(B2 or E) 1775 w 1765 m 613 +1163=1776 EX(Bg or S)

1815 w 1033+778=1811 AqXE

1825 vw 1053+778=1331 EX(B2 or E) I.R Raman Liquid Gas Tentative Av I p a.d. m.d. 1^ Vg Assignment

i860 vw 870+993=1863 EX(B2 or E)

1885 vw 1880 870+1033=1903 AxX(B2 or E)

1915 vw 581+1340=1921, A1X(B2 or E)

1985 vw 2(993)=1986, E2

2025 m 2030 993+1033=2026, AqXE

2085 m 2085 1033+1053=2086, A1X(B2 or E)

2175 w 2190 1033+1151=213^ AXX(B2 or E)

2275 w 870+1397=2267 B q X ^ or E)

2320 m 2345 993+13^0=2333, EX(B2 or E)

2415 m 993+13^0=2423, EX(B2 or E)

2450 w 2435 1053+1397=2450, B1X(B2 or E)

2545 V 1151+1397=2548, BiX(B2 or E)

2565 1163+1397=2560, B]X(B2 or E)

2836 .4 Fundamental, Aq or Bq, 2836 1 .K. Raman Liquid Gas Tentative Av I p a.d. m.d. VJL V g Iff Assignment CO 2860 m C\J uo, 0 s Fundamental^ Bq or E, 2850

2831 vw - - Fundamental; Bq or Aq:, 2381

2935 vs 2985 vs Fundamental; Bq or E, 2985

2991 100 .15 -03 .05 Fundamental; Aq; 299I

3050 vs 3050 vs Fundamental; Bq or E ; 3050

3065 30 .80 .02 .03 Fundamental; Bq; 3065

3370 V 305+3065=3370, BqXE

3^00 V

Table 6

Observed Frequencies and Tentative Assignments

p = mean of six values of depolarization factor a.d.= average deviation from mean value of p m.d.= maximum deviation from mean value of p

vn -0 58

The calculated and observed frequencies of Scott et al are below:

Class Motion Calculated Observed

CH2 bend 1480

Ring Str. A1 1150 1150 R

R(p ), CH2 wag 1025 1033 R(p) Ring def. 581 581 R ( p )

CHg twist II98

CHp rock 853 852 (from comb.)

Bl CH2 twist 1203 R(d),. CHg rock 8'/2 872 (R(d)

Frame twist 272 272 (from comb.)

CHp bend 1481 B2 Ring Str. 1397 1397 R(a) R(d),I.R.(ll) CH2 wag 1024 993 I.R.(1 1 ) Ring def. 87O 870 I.R.(1 1 )

CH2 bend 1425 1430 R(d),I.R.(l) E CHp twist 1157 1157 I.R. R(d),I.R.(l) CH2 wag 1087 1053 I.R.(l) Clip rock. 872 897 (I.R.) Ring def. 810 778 R(?),I.R.(1) Frame bend 305 305 R(d) It is to be noted that no use is made of the 613 Raman line

which is as intense as the 305 line. Blau*s assignments of the

C-H stretching frequencies are:

Class Observed

299I

B 2881 1 B 2 2985 E 2850 3050

The listed C-H stretching frequencies are certainly reasonable,

if in the E species one considers the lower frequency to belong to

the symmetric stretch.

The large difference between the A-j_ and Bg ring deformation

frequencies as assigned by Scott et al, 581 and 870 respectively

raises the question of whether the 870 might be replaced by the

613 line which is by no means definitely assigned to the E species.

The fundamental occurs only in the Raman spectrum.

B. Calculations

In Chapter II-C the following were obtained for M.O.M. equil­

ibrium parameters based on the electron diffraction data of

Donohue, Humphrey and Schomaker (12).

e0 = 6A° 8* $ = 57° 56' r0 = 1 .5 5 A O RQ ±s I .08 A (assumed) XQ = 118° 12' 60

XD is adopted on the basis of Blau's (3 ) discussion.

Next, one must obtain values for the masses of the constit­ uent particles. These are given in Table 7 as well as the cal­ culated principal radii of gyration of the dig groups.

mH = 1 .6 7 3 2 10_2l)' gm.

m = mQ = 1 9 .9 2 3 0 1 0 “2if gm.

M = 2m^ + mc = 2 3 .269^ lO-2^ gm.

2M + m/2 = 56.5003 lO-2^ gm.

o 0 2 a2 = .038 A

p o p b = ,12k A

c2 = .162 A2

Table 7

Masses and Principal Radii of Gyration

It is then possible to proceed vrith the calculations if the assignments are to be considered correct. In order to make the assignments more definite, a number of alternate calculations will be carried out to see what further insight may be gained.

In general the calculations are quite simple and straightforward,

After the symmetry force constants are determined, the Equations

*4-.92-4.116 define the force constants of interest. First calcu­ lations are made, using the assignments of Scott and Blau. In the solution of the second order equations, use is made of the polynomial expansion theorem. It might be noted that this theorem 6l

is the source of the famous Teller-Redlich Product Rule (22) which

is not applicable in this analysis. The constants for the hin­

dered rotations and internal motions for CH2 groups are listed in

Table 8 with some calculations duplicated for different possible

assignments.

The interesting portion of the calculation will be to gain

insight concerning the framework forces.

The framework twisting force constant is easily evaluated.

Then the 3 second order determinants will be solved to give addi­

tional information.

For the Ap and determinants, the following is the method

of solutions. From the secular equation and the polynomial expan­

sion theorem, it follows:

(5.1) fcie —= 1 1for OX A e = 1 + if ^ for B m

‘If (X1+X ) e ^ | e +

Thus there are two equations in 3 unknowns, and K ^ y

One assumption though not wise, is to set KuV = 0 . Perhaps a

better assumption will be to set Krp©g = 0 in Equations if-,115

and if-.116. This will give the desired equations relating the K^y with the Ku and Kv . This assumption is also made reluctantly, but

is probably better than the former.

The E determinant can be solved unambiguously by setting

Vp 2 = 0* This leaves the following determinant to be solved. 62

Assignments A A B B E Evaluated Constants 1 2 1 2

? - - 7 lU-30 KUq = 1 6 .6 8 105 + dyne/cm.

^ 2 ’ 1/k<-K5,5-Kl6,i6'> ? - - V 1397 ^ = 16.111^ 105 + KUl

2991 - - 2985 2850 Ky = 7 0 .3 8 105 dyne/cm.

Kiy = 3*321 10-'’ dyne/cm.

= .0693 10^ dyne/cm.

? 2881 - .3050 KLr = 72.66 lO^ dyne/cm. + KA. w0 1 = -^.137 105 + dyne/cm.

1033 - 993 1053 K = .33^9 10 -11 erg/rad2 a 0 K = -.0129 10 11 erg/rad2 al K = +.0064 lO-1^- erg/rad2

? ? 1157 = 1.3703 10“11 + erg/rad^

? ? 1 1 5 0 Kj3o = 1 .3 5 i4-10-1 1 + K^erg/rad2

852 872 - 896 K a 1.0338 erg/rad2 70 k7i = - .0 3 9 9 -8 *- erg/rad2 -1 1 K y = +.01153 10 erg/rad2

852 872 - 870 K/o = 1.003 10 i-n erg/rad2

r 11 erg/rad2 -11 K = +.0115 10 erg/rad2 ?2

Table 8

Alternate Calculations vf ,1 tf, 1+V2 , 2^ 2, vf ^ 2. bl, ltl , 2+V2 , 2 b2 , lt 2 , 2+* MG3

E E % E 2 E , 2 v =° V1 , lbl , 1^1,2+'^2,2 't2 , 2t2 ,1+XMG3 Vi , ^ 2+ ^ 2 , 2 ^ , 2 " ^ 1G2 (5-3) which can he solved using the polynomial expression method, where

V 2,2 = “a •

Assuming Kuvp “ ^uvp = °* This means Aj_ and B2 are lxl de­

terminants to be solved.

Aj_: 581 anG alternate below.

B2 : 870,1397, Ei 778,305, B1= 272

1033______1150

°l 2 .14.11.3 0^ 2 .5 9 1 6 2

%L e 2 -.2 0 0 9 2 2 -.05197 r72

Kr 3 .8 8 3 2 5 k.2 6 l8 x lo5 dyne/cm. 0 Krir 2 1.5732 1.9517 .8276 KrlrJ+ 1 .20o2

Kri©i -.29lfl -.5312 r

Krl92 -I.O607 -1.2979 r KA .3^09 .£ = ; 2 ? Kj l = .1722 r^

Table 9 Framework Constants Using Scott's Assignments with Kuv2yKuv2=G

These values for the modified valence force constants are not unrea­ sonable, but the assumption Kuvi=Kuv2=0 cannot really be justified. 61+

Assuming Kr-t@-; = 0 i = j i + j

Assignments Al Ap Bi B2 E Force Constants 1033 Kuv! = -6.579 105 Ku± = 1 0 .8 8 8 105 KV1 - 1 7 .2 6 8 105

1150 KUV1 = -8.8825 10^ Ku± = 12.3313 105 KV1 = 2O.9I+I+6

- - - 870 - Complex 1397

— - — 870 — Complex 1^30

- — 779 — Complex 1397

— — 779 — Complex 1^30

613 *W 2 = -3 . ^ 7 8 or -.7921 105 1397 Ku2 = 9*83^51 or 7 .6^53 io5 KV2 = if. 15019 or 6.339^1 10 5

613 11+30 — if.07137 or --3381+9 105 KU2 = 3.92929 or 6 .991+66 105 Ky = 2 6 .2 3 6 or 8 .81+92 105

778 305 = .3AO9 105 = V2 ,2 ^ 2 V1 1 = 2 *3101 105 rsin/

- - 272 - - K_n_ s — 2 — = .1722 1 C? r^

Table 10

Symmetry Force Constants Assuming Kji^q ^ O (Given in dyne/cm) 6 5

A1 1033 1150 1033 1150 A1 581 581 581 581 b2 1*1-30 1*1-30 1397 1397 B2 613 613 613 613

®o 2 .7 2 1 3.1*48 3.772 2.733 3.165 r2 3-3*45

K>L « 2 ... 1 .1 1 0 .*i-866 1-5372 .913*4 1.093 1 .5 2 0 r£ K r 2 .96*4 3 .8 1 5 3 .06*46 3.915 2.756 2 .8 8 6 x o .65*4-2 .*4*45*4- .5*458 ^r lr2 1.505 • 75*45 1 .6 0 5

Krqr4 • 1573 -.6935 .2576 -.5931 .3660 . *400*4

Table 11

Force Constants Assuming Kr_^0 ^ = 0 (Given in 105 dyne/cm.)

The choice of solutions arises because if a quadratic had one real solution, the other must be real also, and may lie very close to the original solution. There is very little physical evidence to guide the choice of roots if they are of the same order of magni­ tude as in Table 11. 66

C. More Definite Assignments

The relatively weak Raman activity of the 1150 line indi­ cates that it probably does not belong to the Ap species and the arguments of Blau (3 ) indicate that the very strong 1033 Raman line most probably is that associated with the Ap ring stretching.

It is felt that the relatively strong 613 line probably corres­ ponds to the B2 ring deformation mode. Whether the 1397 or 1^30 line corresponds to the B2 ring stretch is decided on the basis of intensity. Neither the Ap nor B2 CH2 bending fundamental is observed; hence it is unlikely that the very strong 1730 band corresponds to the E CH2 bend; hence the weaker 1397 Raman line is assigned to this fundamental, which leaves the 17-30 band free to be assigned to the B2 ring stretching mode. The relatively strong 1130 band is reluctantly assigned to the E CH2 twist. The reluctance arises because of the relatively strong infrared activ­ ity since neither the A2 nor Bp CH2 twist are observed. The ques­ tion arises as to what is to be done with the very strong 8 7 0 band.

This may well be the E CIDj rocking mode assigned by Scott et al at

8 9 6 . These assignments are based on the premise that it is not likely that the difference between a perpendicular and a parallel band can be detected from the spectra of Cleveland, Murray and

Galloway (l). The tentative assignments to be used are listed in

Table 12. 6?

Glass Motion Assignment

A1 CI^ bend Ring stretch 1033 CH2 wag Ring def. 581

A t CH2 twist CH2 rock 8 5 2 (comb.)

B i Cl-h, twist CH2 rock 872 Frame twist 2 7 2 (comb.)

B, CH bend Ring stretch 114-30 CH2 wag 993 Ring def. 613

£ CH2 bend 1397 CHg twist 1150 CIIq wag 1053 CH2 rock Q70 Ring def. 778 Frame bend 305

Table 12

Assignments of Fundamental Vibration

Using these assignments, the force constants obtained are the following (Table 13).

Ken 2 .7 2 1 r^ 3.31+5

Ke1 e2 1 .1 1 0 . A866 r

11 2 .96A JT' 3-315 xlO^ dyne/cm. 0 Krir 2 = .654 2 1 .5 0 5 •1573 -.8935 I

Ka .3^09 105 dyne/cm = rsin/ Kjl. .1722 1 0 5 dyne/ cm

K,Uq = -I* l6 .ll 10^ dyne/cm

?

Ku2 = 0 (assumed)

1C. = 7 0.^ 6 10^ dyne /cm o Kv = 3.32 lo5 dyne/cm KVg = .0693 lo5 dyne/cm

KWq - 7 2 .7 1 0 5 + Kv dyne/cm

= ?

Kw2 = •1 ^- 105 dyne /cm

% Q = Kq^ + -3286 10"11 erg/rad2 ^ 2. = ^^2 “ *^385 10~21 erg/rad2 ^ = ?

Kpo = Kp-^ + 1 .35^ 10~21 erg/rad2

Kp! = ?

Kp2 = 6 (assumed)

Ky = 1 .0 0 3 1 0 “11 erg/rad2 ICy-^ - -.0 0 9 2 1 0 -11 erg/rad2

K72 = + .0 1 2 1 0 -11 erg/rad2

Table 13

I'orce Constants Based on Assignments in Table 12. 69

CHAPTER VI

THE METHYLENECYCLOPROPANE MOLECULE

A. Introduction

Methylene cyclopropane (Cl^Hg), see Figure 6 , was first syn­

thesized by Gragson (23). The first infrared and Raman spectra

of what is believed to be methylene cyclopropane, were obtained

by Blau (3)* working under William J. Taylor. A normal coordin­

ate calculation was made to guide the assignments of fundamental

frequencies. In this calculation the spiropentane force con­

stants of Scott et al (2) were used. The interest in this molecule

with respect to the M.O.M. is to investigate the possibility of

force constant transfer from spiropentane to C^Hg since the M.O.M. ring geometry in these molecules is quite similar. If the ring

dimensions are different,, it may well be possible to apply Bodger's

Rule (2) to M.O.M. force constants.

B. The Geometry of Methylenecyclopropane

As yet, there are no electron diffraction data concerning the structure of the molecule. Blau assumed the ring dimensions of spiropentane and the double bond distance from ethylene. The theory of directed valence would lead one to assume C2 V symmetry as Blau did. Figure 6 shows the structure of the molecule, while

Figures 7 and 8 show the M.O.M. framework. As in spiropentane, it will be assumed that the plane of the methylenic CHp groups is Figure 6. The Methylenecyclopropane Molecule, Symmetry 71

Figure 7» The Methylenecyclopropane Molecule With M.O.M. Framevrork 72

perpendicular to the plane of the ring and bisects the M.O*M. ring

angle Q. The ethylenic CH2 group will be assumed to lie in the

plane of the M.O.M. framework. This is the only assumption lead­

ing to G2v symmetry which has reasonable chemical basis.

The assumed equilibrium distances used by Blau are the fol­

lowing :

X(l) = X(2) = 118° 12*

X(4) = 120°

R = C-H = 1,08 A

C3 -C4 = 1.34 A

C1 -C3 = C2 -C^ = 1.48 A

C1 -C2 = 1.51 A

< C1 "C3"C2 = 6l° 21'

< c3"ci"c2 = < c3 ~c2 _cl = 20'

Using the masses listed in Table 7 , the M.O.M. equilibrium parameters can be calculated and are:

Q = 57° 5 6 '

eQ = 64° 8*

rQ = 1.55 A

JO = 1.418 A 1 o

a2 = .038 A2

b2 = .124 A 2 for methylenic CHg, groups

c2 - .036 A2 73

a 4 = *°3 ^ a 2

bj^ = .126 A2 for ethylenic CHg group

= .162 A2

The character table (15) for the C2v symmetry group is given

in Table 14. Blau’s c.alculations indicate that the molecule is not too far removed from a symmetric top.

I =* 4 .1 9 7 10"39 gm cm^ 2 IB = 12.593 10"39 gm cm 2 Ic = 1 5 .6 2 10"39 gm cm

Species E C2 (z) O'(XZ) or(YZ)

1 1 1 1 A1 a K x a y: y ^ z z 'Az 1 1 -1 -1 A 2 “ XY 1 -1 1 -1 B1 MX “xz

b 2 1 -1 -1 + 1 “ y “ yz

Table 14

Character Table for C2v Symmetry Group

The origin of the XYZ coordinate axes is to be located at the center of mass of the equilibrium configuration of the molecule.

The positive Z> axis is to lie along the double bond, the positive

Y axis being perpendicular to it and in the plane of the M.O.M. framework pointing in the ’^direction” of the methylenic group '(k labeled No. 2. The positive X axis is then defined, by the right hand rule.

By use of the character of a general vibration and application of the expansion theorem it is possible to obtain the number of vibrations, rotations and translations associated with each sym­ metry species. The molecule has 10 atoms.

No. of Species No. of framework Translations Notations vibrations vibrations

A 1 3 3 Tz Ag If X R

B1 5 1 TX % Bg / 2 Ty Ry

The framework vibration is not a pure framework made insofar as the hindered rotation of the ethylenic CHg group must be considered.

C . Coordinate Systems

The manner in which the coordinate sytems will be set up is quite similar to that employed in the spirotane molecule. The characteristic groups are the CHg groups and the single ’'central" carbon atom. The M.O.M. framework appears in Fig. 8 and the coordinates are defined below. Referral to Fig. 7 may be useful. 75

Figure S. M.O.M. Coordinate Systems in

Methylenecyclopropane 76

The yp and y2 axes bisect the M.O.M. framework equilibrium angle ^ in the plane of the M.O.M. framework the positive sense being away from the center of mass of the molecule. The zp and 22 axes are perpendicular to the yp and y2 axes respectively,and also lie in the plane of the M.O.M. equilibrium framework., the positive sense pointing toward the C2 axis of the molecule. The Xp and X2 axes are then fixed by the right hand rule. The X3 axis is parallel to the xp and X axes, having the same positive sense.

The Z3 axis is parallel to the Z axis having the same positive sense. The y3 axis is then fixed by the right hand rule. The yij. axis is parallel to the Z3 or z axes with the same positive sense.

The zjj. axis is parallel to the X3 and Z axes with the same positive sense. The x^ axis is then specified by the right hand rule.

The coordinates associated with the coordinate axes described are:

% ± - rigid translation of C . of M. of i^h characteristic

group along Xp axis.

^ - rigid translation of C . of M. of i^*1 characteristic

group along yp axis.

^ P - rigid translation of C . of M. of i ^ characteristic

group along zp axis.

Op - rigid rotation of i^*1 characteristic group about the

xp axis from equilibrium.

(3p - rigid rotation of i^h characteristic group about the

yp axis from equilibrium. 77

7 ^ - rigid rotation of* i ^ characteristic group about the

zp axis from equilibrium.

Hi, vp, wp, - the internal vibrational coordinates of the

ith CIi2 group.

The modified valence coordinates to be used in this analysis are:

& - change from equilibrium of the angle ©, the angle sub­

tended at the central carbon atom by the C. of M. of the

methylenic group.

- change from equilibrium of rp, the distance between

the C. of M. of the i'th methylenic group and the central

carbon atom.

5 jo - change from equilibrium of °, the distance from the C.

of M. of the ethylenic CH2 group and the central carbon

atom.

- T L - the framework twisting angle, essentially ethylenic

twist, defined as the sum the ethylenic twist and the

framework twist, consistent with the requirements of a

pure vibration.

- essentially the out of plane double bond bend, actually

the out of plane double bond bend against the framework.

A - essentially the in plane double bond bend, but defined

to be the angle made between the line joining the ethyl­

enic group and the central carbon atom and the line 78

joining the central carhon atom "with the C. of M. of both

methylenic groups.

The transformation between the modified valence coordinates and the M.O.M. coordinates are given below.

Se = i_ -T flh+$,)A - « 1+ f2 )B - a?3< w } (6 .1 )

-t-^?2 cos^ + 9i B + ? X A (6 .2 )

&r2 = ^sinj^ -'^cosji + ^ 2 B + ^2 A (6.3)

S jO = V k - f 3 (6-0

_ n _ = Pl+ ' 2r ’cos$ (^l+-£7 0-5)

'f' = + ^ 1 -^ 2 (6 .6 ) /°Q 2r0sinj!

A = ( - V ^ ) + 2?j+B(-)1 --yg)-A('ii1 -g2 ) /Oe 2 r sin^ (8 .7 ) where A = sing B = cosk

D. Symmetry Coordinates

Since the molecule belongs to the Cgv symmetry group, with no degenerate representations, it should not require too much effort to obtain a set of M.O.M. symmetry coordinates. To this end, the transformation table of the M.O.M. coordinates is given in Table 15. 79

E C2CZ.) O' (XZ) «r (YZ)

Si Si +52 - h f t f 2 $ 2 Si ~s± f t ^ 1 V i ^ 2 V 2 y ±

^ 2 7 i V i V 2

% 1 ^ 2 f• y» t f t % 2 ^ 2 f t 7 1 f t S 3 -J3 5 3 “53 ? 3 °?3 “?3 373 ^ 3 ^ 3 f t f t f t f t f t ■ V k 7 1 + 7 1 + «i al OCq 02 «i ° 2 ° 2 al al Pi Pi P2 -P2 “Pi P2 P2 Pi "Pi -P2 7l 71 72 “72 -71 72 72 71 “71 “72 aij. aij. -ai+ «!+ -cti^ Pi+ Pit- Pl+ -pi+ -Pl+

71+ 71+ “71+ ~7k 71+ U1 U1 u2 u2 U1 u2 u2 U1 U1 u2 VI vi v2 v2 V1 v2 v2 V1 V1 v2 Wi W1 w2 “v2 -W1 -w2 w2 w2 V1 -w -l U4 U1+ ui+ ul+ vk vl+ vi+ V++ Wl,. -wij. +wj+ - ^ j k f t 1+ ** %

Table 15

Transformations of M.O.M. Coordinates under Covering Operations 80

It should be noted that here; as in the previous molecule,

the covering operations are performed on the molecule, leaving

the coordinate systems unchanged. It can be ascertained by in­

spection of Table 16, that the coordinates having subscripts 3

and U are already symmetry coordinates, while plus and minus com­ binations of like coordinates having subscripts 1 and 2 will also be symmetry coordinates. This is to be expected since some coordi­ nate in a molecule belonging to the Cpv symmetry group, can only be transformed into plus or minus itself or plus or minus only one other like coordinate provided the coordinate axes are chosen' to be adapted to the symmetry of the molecule. A set of M.O.M. symmetry coordinates is given in Table 16.

E. Eckart Conditions

The Eckart conditions have been described in the analysis of spiropentane and for methylenecyclopropane have the following form.

For reasons which will soon become apparent, the contribution of the angular momentum of the hindered rotations is included.

(6.8)

(6.9)

(6 .1 0 )

0 ( 6 . 1 1 ) 8 l

E C2 (Z) O'(XZ) cr(yz) Species

<11 (1/ rs) toi+Vz) <12 (1 / ^ 2) (fll+^2 ) — > <13 i 3 h % + + 4- 'I* (l/ vTp) (ap+Og) <122 C1/ ^ 2) (u l+ u 2 ) <123 (1/ 4 2 ) (v1+ v2 ) T2)(-W 1+ W 2 )

116 (l/ I17 (l/ T 2 ) C “^ l +^ 2 ) 4- + b2 — > 118 ^ 3 —> q19 fk q20 (1 / 4 ^ ) (-cei+ag) q.21 7k 1 2 8 C1 / ^ " ui+u2 ) q29 t1 / ^ ) ( - vl+v2 ) q30 vk

Table l6 A Set of M.O.M. Symmetry Coordinates indicates coordinates to be removed by application of Eckart conditions, Qg will later be reintroduced). 8 2

-b2B(Pi-f32 ) + c2A(7 i-72) -l- a^cuij. -t(Si-jg)

+ ( p + s )^j.+^ s ^3 = 0 (6 .1 2 )

rcos^-j + ^g) + - b2A(P1+P2 ) - c2 B(71 +7 2 ) = ° (6.13)

where t = -fit( ^ S t ^ L 3+2 M

s = rsin^-t and all other designations are those indicated previously. The assumption that the contribution of the hindered rotations to the total angular momentum is negligible can easily be made in two of the three equations (6.11,, 12). However, making this assumption in

Equation 6.13 would require ^q+f2 = 0, not a likely situation, hence this equation will be retained as is.

Transforming these equations to the M.O.M. symmetry coordin­ ates, and solving for the coordinates to be eliminated they become:

q3 = 1 (A

q19 - >J2 £ (Aqqg - Bq±7) b 2 ( 6 .1 9 )

The definitions of the modified valence framework coordin­

ates are such that all except rq and r2 transform into themselves

under the allowable covering operations, the latter transforming

into each other. Taking plus and minus combinations of the

latter, the modified valence symmetry coordinates are given below

and are obtained from Equations 6 .1-6.7, by transforming these to

the qq and then imposing the Eckart conditions.

Aq (6.20)

Aq (6.21)

Ax (6.22)

Aq (6.23)

Bq (6 .2*0

Q l6 = 7 y ( Srl“£r2 ) = ‘ ^(A^-B)cos^+bJ ql 6

b2 (6 .2 5 ) + " A1 ^ 1 7 -4 co

VO H a1

no H ICVl

<3 II t— & 85

CHAPTER VII

THE SECULAR DETERMINANT FOR METHYLENECYCLOPROFANE

A . Introduction

The method of analysis giving rise to the secular determinant is discussed at length in Chapter IV in connection with the spiro- pentane analysis. Briefly, again, it arises through the efforts of seeking the transformation to normal coordinates or the simul­ taneous diagonalization of two positive definite quadratic forms.

Both the kinetic and potential energy functions are positive definite, the former by definition, the latter by the requirement that the equilibrium configuration of the molecule is stable.

Suffice it to state then that the secular determinant to be solved here,as in spiropentane,is the one with the six zero roots already removed by imposing the Eckart conditions. The matrix

[ v - m ] has been rigorously reduced to four symmetry blocks, each of which has been reduced to three blocks by assuming no potential interaction between the three distinct types of motion.

Hence the determinant to be solved appears as;

m . 0 ( 7 .±)

and it follows then | V^~ = 0 86

B. The Kinetic Energy

The kinetic energy will he expressed in the same manner as

was done for spiropentane. First it will he written in the

diagonal form using M.O.M. coordinates. Then it will he trans­

formed to M.O.M. symmetry coordinates. The Eckart conditions will he imposed and off diagonal terms will thereby he introduced.

Some of these off diagonal terms will he removed hy further trans­

formations .

In the M.O.M. coordinate system, the kinetic energy is:

(7.2)

al~a2=a at~al(- etc.

Applying the transformation defined by Table lo (let it he denoted by F such that %= Fq) the kinetic energy yields:

A / » • 2T = = q F W F| = 2TAAj_ +2TA A 2 +2TB B-l +2T- Bg (7.3) where: L ^ + 4l'iW- 3 + ^ + a 2 4 f ] + M [412+923+421++^

[q|+b24|+bgq|+c24§] + M [ q | 6] 87

2Tb1 = M [ 4 10+n^ll+cli2+b2 4i3 +c2 4i^+al;ii5j + M [q2T] (7-6)

„ f »2 .2 m»2 .2 2*2 2»2 »2. »2 «2 1 r„ 2Tb2 = M Lcl1 6+cll7+Stil8 +

After elimination of the coordinates q_^} q^, q^p, 3-12>

qj_8 ^19 ~by imposing the Eckart conditions, the resulting

kinetic energy terms are:

+ (1+2i )4i + (1+a>^ + 3i«a 2Ta „ = M 1 ■2 ^ “2 ^ iB^ 2 % + a2^5+^22+^23+4li++4 l y

(7 .8 )

.2 2.2 2.2 / [l+2 ^ ( l +£sin^)2+ (|)2 sin2^ 2 TBn ^10 'ill'*'0 °-l7 C

+a|qf5+q2 7 j (7.9)

^ ^ . ] •2 ^ 1 [1 + B 2 % J q8 2 T ft = M' 2 ^ 2 "^—jy ArcosjzJ q^q^-1^ BrcosjzS q^qQ+2c2^ sinjzi q b if ^ b4 — /

R [R^)] !>< I^i)] 4 ? t 1

2Tr,B 2 = M ^ [t3!02^ 2^ ] 4l6»17+a2920+ci;^l+4l9+5|9+ifo (7 .1 1 ) 88

L p J

In principle all of the off diagonal terms can be removed by

suitable transformations. However, often this is not practical.

It would be desirable to simplify both the A-^ and A2 kinetic ener­

gies since both of these contain 3 off diagonal terms each. Prac­

tical considerations force the reduction of the A2 kinetic energy

to the approximate form:

(7 .12 )

This is done by invoking the approximate Eckart condition

The reason for doing this is prompted by the fact that very few of the A2 fundamentals will be identified experimen­ tally, and by thus simplifying the kinetic energy some information may be obtained. In any event, this approximation is not less valid than using the other two approximate Eckart conditions.

To remove the off diagonal terms from 2 T ^ , one can again resort to the transformations of Shaffer and Newton (13) in the treatment of the bent symmetric XY2 molecule. Those transforma­ tions, while not really applicable to the problem, do give a start toward the reduction. 8 9

The transformation defined by Equations 7*13-7*15 give a partial reduction.

j*r£©sin£$ + (Sr^-t-Srg JcosjzTj = QLqB-qgA (7*13)

Sr - JgLf-r&e cosjzi + (Sr1 +Sr2 )sinj^ + 2fy\ = TjF ]Aqi+Bq2+ \Tf qiQ

(7*1*0

S3 = t -r£©cos^ + (Sr-L+Sr^sin/^ -2(l+M)5/T[ ==

S I jAq1+Bq2 - ^ 2 ^ ~ \ (7*15)

All coordinates not involved above are subjected to the identity transformation.

Where e = 1+3 — m

g = 3 + — m

Under this transformation the A-^ kinetic energy becomes:

= M + s| + s| + 2 J S2 S3 + a2 4^+q22+q|3+42 t+4|5^

(7 *1 6 )

To remove the one remaining off diagonal term, the solution of the eigen value equation yields the roots 1 ± VJ ® 3g 90

The further transformation which totally diagonalizes the

A^ kinetic energy is:

U1 = S 1 “ ‘ll3 ~ ^2a

(7. 17) U i = (r$©sinjz£ + [5r^+Sr2J cosj^)

= "|= (S2+S3 ) = W. [Aqi+BqJ - (l .1 8 ) U2 = h! L -rS©cosj^ + (^rl+^r2 ) + h2 ^>

.19) U3 = T f {"S2+S3 ) = JT LAcil+B

where Pj = vfi + {3S

P2 “ “ nTg + 'I3e

U 3 = h^ Ij-rSscosjzi + (6 r-^+6 r2 ) sin^fj -h^-Sp

where

^ ] _ n z * nJ| (i-sO

h2 = -J4 P- LL_ 1® - >P’• e m -4

h3 = # L " 1 5 - \ J f ( i - s j

h k = “IT^ I> + H (^ )j

Then the final Aj_ kinetic energy has the form:

2TAl = M S U? + (1+ (7.20 + ( 1 + l f F )d2 + (l'\||^)0 3 + a 2 4 5 + 4 l 2 +

reduce the coefficients of Ihj? to 1 would add complicated factors

to the transformations.

C . The Potential Energy

The problem of correlating a definite and limited number of

observables with a much greater number of unknowns has been dis­

cussed during the analysis of spiropentane. Symmetry properties require the separation of the potential energy into four portions.

This requirement is due to the fact that the potential energy must remain invariant under all symmetry operations. It follows then that:

(7 .2 1 )

As in the study of spiropentane, each of these contributions will be assumed to consist of three parts.

F H I 2Vs 2VS + 2VS + 2VS (7 .22)

This is the statement which assumes that the potential interac­ tions between framework motion (F), hindered rotation (H), and in­ ternal motion (i) is negligible. Kinetic interaction between these kinds of motion has been neglected by use of the approximate Eckart equations. It follows:

WB1 ' B2 F ' H>1

s i 92

On this basis, it is then possible to write the quadratic

portions of the potential energy as below, utilizing the indica­

ted M.O.M. symmetry coordinates.

^ + + IC3,3u3 + £k 2,3u 2°3 + 2Kl,3ylu3 + 2 IC2 ^3U2U3 (7.2*0

2 ^ 1 = S . ' & j (7'25)

I 2 P 2 2 2V = I{22,22^22 + ^3,23^23 + + *<25,25^25

+ 2K2 2 ,23^22^23 + 2K2 2 ,2 ^2 2 qL2lf + 2K2 2 , 25^ 22^25

+ 2K23 ,24223^2^ + 2K23,25^23^25 + 2 IC2if,25(3-24^25 (7-26)

2V^2 - 0 (7.27)

2 vA 2 = ^ 7 ,7^7 + + ^ 9 , 9 ^ + 2 K7,8q7q8 + 2K7 ,9^7^ + 2 K8,9q8q9

( 7 . 2 8 )

2 VA 2 = I<[2o,26q-26 (7-29)

2 VB1 - K10,10q10 (7-30)

H 2V-Bt = K-i o n n*!?-] + K_ 1 . 1 q2 +K qf- + 2K q q . "1 13,13*13 1.4,14*14 1 5 ,1 5 * 1 5 1 3 , 1 4 13 14

+2K13,15^13^15 + 2Kl4,15ql4a-15 (7.31) 93

(7-32)

2vb 2 - Ici6 ,i6 qi6 + % 7 ,i7 qi7 + 2 Ki6 ,I7 qi6qi7 (7-33)

H 2 2 2 VB2 = ^20,20q20 + K21,21q21 + ^0,, 21^2 0^ 1 (7.3*0

2 VB2 = K28,28q2 8 + K29Jf29q29 + + 2K2 8 ,29928929

+ 2K2 8 ,3 0q2 8 q3 0 + 2K29,30q29q30 (7.35)

Here, the Kj_ * are the potential constants associated with

the M.O.M. symmetry coordinates. Even with the assumption that

the potential interactions between the different kinds of motion

are negligible, it is apparent that some further ideas must be

introduced to obtain information from the problem.

The framework portions of the potential energy can also be written in terms of the valence symmetry coordinates. These are:

(7.36)

(7-37)

2VB1 “ V10,10Q10 (7-38)

2vb2 = vi6 ,i6qi6 + vi7 ,17^17 + 2Vi6 ,17^16^17 (7-39) 9k

■where the V± j are the valence symmetry potential constants. Here

again, as in spiropentane, the number of coordinates used to de­

scribe the system is equal to the number of vibrational degrees

of freedom of the system and thus the transformation to the valence

coordinates from the valence symmetry coordinates is a nonsingular

one .

In the B2 framework potential energy Deed's assumption that

the potential energy is nearly diagonal in valence coordinates will be adopted. However, in the Aj_ framework potential energy, some1

interaction terms must remain, as preliminary calculations show

specifically that the ^00 and pr interactions are not negligible.

This is because it will be quite difficult to qualitatively separate the symmetric ring stretching mode from the double bond stretching mode. These constants qualitatively correspond to the

Kr^rij. and K02.02 spiropentane.

To obtain the force constants of interest, a number of trans­ formations must be considered.

In general: Q, (T-^o)

with _Q = uq defined by Equations 6.20-6.26 and possibly

7-17-7.19

Hence: 2vf = ^ § = t 't Evj., j] t q = t q (7.W.)

It follows 95

For the hindered rotations and the internal motions, the

transformation F defined by Table l6 'will be useful in perform­

ing this transformation, the symmetry blocks will vanish.

,H 2V“ = f q = I f I/i . ^ F ! (T.^2) where q_ = F i

and ^ represents the 2V1 = q [%,j7 a = S F ’[Ki,7Fi (7^3) appropriate column vec­

tor of the hindered

rotations and the in­

ternal motions in the

M.O.M. coordinate sys­

tem respectively.

It follows D O = F [ K i , j ] F

where 1^ = K^l^ * • •Kuo>K-ul • • ■e'tc

D. The Secular Determinant

The determinants to be solved then are the following:

K- K- K ■1,1 " XM 1,2 1,3

K •1,2 Kg, 2 - "2,3 = 0

K1,3 k3,3 - (7.MO

^ - / Ma = 0 C 7 • ^5 ts) td td tri > > > ro*j P H P W H ^ i o h ro k H K

1^ PP* r^ -0 CO O s» »—i ro v* P P o* v- -T — 4 rvT r^ o» ro CO CO P P V* Oj o* o< Co ro ro ro ro -VI o 9^e9 ^ VO CD t 0* v* ro P P ro ro ro i on 4=" 1 1 y on •r CO i V V V y g g ro g ro 1 I p ti + Ph ro ro d T CO

o 0 = - wv' r*i F*5 V* 0 r ^ -s CO CD - 4 r£ ro r^ P 4=” SIS U3 Co CO ro -©| 4 M ✓-N VO 1 00 s» Si II P PP |.. . l ro ro i ro -r 4=" CO . + V vn 4=n CO 1 s* U |4 V H i P In £ v n 4=” p ro P o ro ro r\!P vcf1 + V* £ r^ ixT VO - f ■r 00 ro P v* ** on T>|4 I VO VO ro I ro ro s* on 4=- •r P P* ro V on 4=~ CO cn v* P g i P P P •rro ro OH on ro r^ % i on P ,Ur^, 4^ro ro n on r^ ivT r^ 4=- oo ro o v* II ro io ro ■0|4 O Y on on on O g

I! -'J -0 — 4 p O « « on on 4=~ ■r 4=" -4 ii P VO • O pi OA on o (8ViO VO ro Ov 97

B2 ^2 0 ,2 0 " X Ma2 IC2 0 ,2 1 (7*53) K2 0 ,2 1 ^■2 1 ,2 1 “ > Mc§

K. s i K2 8 ,2 8 “ > m K2 8 j 29 28,30 0 ^ 8 ,2 9 k29,29 " > M *29,30 (7-5^)

^ 8 , 3 0 ^ 9 , 3 0 *30,30 "

It must still be apparent that in the study of this 10 atomic molecule with a total of 2b fundamental vibrations, not all of which cam be assigned, there remain more than 2b parameters to be

determined in this already highly simplified potential function. 98

CHAPTER VIII

FREQUENCY ASSIGNMENTS AND CALCULATIONS

A. Assignments of Fundamental Vibrations

The assignments of Blau (3) are essentially those used in this analysis. They are given in cm~-*-

Ring deformation possibly 598 or 723

Ring sti*etch 1032 (I.R.)

Ai Double bond stretch I7 6 O (i.R.)

Methyl wag 1002 (I.R.) Methyl deformation ikjG (R)

Methyl stretch 2994 (I.R.) 2985 (R)

Ethyl deformation 1407 (E) 1403,1410 (I.R.)

Ethyl stretch 2994 (I.R.) 2935 (R)

Ethyl twist

A2 Methyl twist

Methyl rock 663 (R) possibly 596

Methyl stretch

Ap Double bond bend 287 (R)

Methyl twist

Methyl rock 695 (I.R.) Ethyl wag 890 (I.R.)

Methyl stretch 3073 (I.R.) 3065 (R) 99

B2 Double bond bend 35 7 (R)

Ring deformation possibly 969 or 723

Methyl wag 1125 (I*R.)

Ethyl rock 1333 (I.R.)

Methyl deformation 1*735 (R)

Methyl stretch 2886 or 2 9 0 8 (I.R.)

Ethyl stretch 3065 (R)

Table 17

Assignments of Blau

Preliminary calculations indicate that, with the assumptions made, the 723 assignment is favored for the Ap ring deformation rather than the B 2 ring deformation. This is based on the fact that no real values for force constants can be obtained in the solution of the B2 framework determinant when the 723 line is assigned as the B2 ring deformation fundamental. Transferral of various possible values of Kr0 - Kr^ from spiropentane predicts the B2 frequency as > 8 0 0 c m " 1 . In assigning the very strong 723 line to the Ap ring deformation, one can in a sense assume that the very very weak 59^ is the first harmonic of the 287 line for

2 (2 8 7 ) = 574

Blau also remarked that the 1 7 6 0 line is very high for a double bond stretching frequency and since there are no lines in 100

the normal double bond sti-etching frequency range (s5sl6 5 0 ) might

indicate the absence of a double bond in the molecule. However

it may be mentioned that recent studies by Edgell and Ultee (2 5 )

show a C=C stretching fundamental frequency assignment in CFp=CD2

and CF2 =CHD around 1725 cm"^.

Further it must be considered that the qualitative double bond stretching and Aq ring stretching modes interact quite

strongly, insofar as a normal mode predominantly one of the motions will contain a non-negligible amount of the motion of the other.

In making his normal coordinate calculations Blau estimated that the B2 methyl wag and ethyl rock interact quite strongly.

This is due to a Kinetic interaction since the potential inter­ action can be expected to be quite small. The manner in which the

M.O.M. is generally applied by use of the approximate or framework

Eckart conditions does not account for this, and the calculated values of force constants will reflect this. A possible method of alleviating this situation will be discussed later.

In summary, the only addition to the assignments of Blau is that of the 723 line to the Aq ring deformation. It is felt that the 9 6 9 line is too weak to be considered as the Bp ring stretching fundamental. 101

B. Further Assumptions

As in the work on spiropentane , cyclopentane, pyrrole and

the paraffin chains (1 ,7 >9 ) > ^ "will be assumed that the poten­

tial interaction constants between unlike coordinates, describing

the hindered rotations as well as the internal motions of the CH2

groups are negligible. This was discussed somewhat in connection

with spiropentane and the same discussion applies here. The

diagonal force constants have previously been obtained on the

basis of these assumptions and hence cannot be confidently sub­

stituted in this molecule to obtain insight concerning the inter­

action terms.

In the B2 framework determinant, it will be assumed that the potential energy is diagonal in the valence symmetry coordinates

(Q). This assumption is largely based on Deed’s (7) discussions.

Preliminary calculations on the Ap framework determinant in­ dicate that there are non-negllgible interactions, specifically

Kp0 and Kprp = Kjpr2 . This is because the solution for force con­ stants yields imaginary values if these interactions are neglected.

If these interactions are to be included, judicious approximations must be made elsewhere. Examination of the definitions of UpjU2 and U3 and comparison with the definitions of the internal coordi­ nates u,v of the CH2 groups gives a correspondence, which may be indicated as: Up —* u Up ^ v U^ v 102

Hence it is not unreasonable to suppose that Kq 2 = ^1,3 ~ ®

since this is essentially equivalent to the assumption of setting

ICuv = 0 f which Deeds and Curnutte (7*9) used quite freely for the

internal motions of the CH2 groups. It is true that the physical

system of the ring is quite different from that of the CH2 group,

but apparently the assumption is as good a one as can be made at

this point.

C . The Revised Equations

Following the assignments of fundamental frequencies and the

discussion of the previous section^ the equations to be solved are

the following.

Al K2 2 , 2 2 ‘ X M = 0

K23 ,23 - > M = 0 (8 .1 )

^24 r2k ~ > M = 0

^5,25 ** > M “ 0 HOJ < (8 .2 ) K2 6 , 2 6 " = 0

^ 7 , 2 7 " = 0 (8.3)

b I ^ 8 , 2 8 “ 3 0

^29 y 29 “ = 0 (8.4) K30,30 “ = 0

A? K5 5 - >Ma2 = 0 (8.5) 103

- *Mb£ a 2 107,7 0 K3^8 " > M C 0 (8.6)

0 *9,9

B k13,13 ■ * Mb2 0 0 Kllf,llf ■ *mc2 (8-7)

K15,15 " 0

b I ^2 0 , 20 “ *Ma “ 0 (8.8) K2l ,2 1 " ^ Mci+ = 0

Al Kl,l - >M = 0

K 2j2 - XM(1+ J ~ ) *2,3 = 0 (8 .9 ) *2,3 3,3 M3g

B 0 (8 .1 0 ) K1 0 ,1 0 ~ >M L 1+2im^1+psin^ 2 +

F B % 6 ,16- > 4 . 1 + a 1<^)2aS+hg7}] W 7 - 2H l ^ lG2-i(fi)2sin3 =0 Kl 6 )1 7 - 2>M i :5lG2-|^)2sln^] k17 ,17- > M [1+ 2 ^ (F )2b £ ^ 2a. (8 .1 1 )

On the basis of the definition of the transformation matrices it can be shown that the following relations are valid and consis­ tent with the statements in Section D. 101+

K a.'o _ IC J U s l ^ c o ^ £ l | K 2 ,s+*§K3 <3+2K ^ 3Vl3^ (8 .1 2 ) ~rZ ~

K 'ro ~ ^1 >1 cos2^+ ^ 1lK2 ^2+h3K3 3+2;K2 3lllll3^ (8.13)

^ > 0 = (8 .11+)

- p “ |j(2,2hih2_K3,3h3hi++K2,3^h2l:13"hlhl+i] cos^ (8.15)

“ +j^2,2hlh2+K3^3h3lll4.+K:2J3^h2h3-hlhl+0 Gin^ (8 ,lo)

i K .1? - f r (8.17) r -%,2hl K 3 ,3h!-2K2,3hlh3l ^

(8 .1 8 ) K^01 = | LK5^5 + K20,2o{

V* % n :a^ [ 1K5>5 K2 0 ,2 o\ (8 .1 9) ‘ 301 refers to diagonal methyl group con­

stant

“ K15,15 (8 .2 0 ) ^sqIi refers to diagonal ethyl group con­

stant

^ 0 1 - 2 1^7 (8 .2 1 ) refers to methyl- methyl interaction

constant 105 »— 4 • * ro I ------1 OCO CO I n | n I *>. | si MCM CM CM W » & —1 11 i o- to • rH H ro O H A

1 (CM H o HHkS III II * \ o -H 1 i S ? r |CM H >■—\ CO CM w Itlv-< + 0 ^ -H* * HH ?H r ~1 o rH »\ £ ICM H ? r s - r 0 0 LA CM !2 III II r—J CO i * rH N -4- rH •N |CM H ^ CD -O 0 0 — CM CM i CM i N -M- s 1 — 1 — H CM j rH l i i l ® r —s / CM r— ►—t 2? J- II 00 CM MC CM CM CM CM CM CM o rH *\ i S J CVIH —N ^— O CO QO* CO CM !2 III II 1 CO CM OJ CM 3 rH r\ CM c O — hH K* -H- >. CM 3 o — U cd |c H CO o r o * J + n » > (O CM o r CG - C GC OC CM O H > 2 - L H | | OJ H CO o r H * J II l CO CM o r CM H m CO ro !*?” OJ II ■ r\ ir LA CM -H" O m 1 rH | CM | rH — CO* to ro •-M ■i? + 1! CO O CM H CM 1 ,26 1 CM | H ------CO o r II CO ■'MO i CM K. - t CM H f ro LT\ 2P iJ* II o o CO o CM CM CM P P -P > > ■P > + II VQ r'-'N VO VO CO VO OCO CO CO CO VO CO 'O CO CO m rH H rH rH r>- H H H I i H IH i—! rH H H t- i rH rH —! *C %•\ •% *\

CM CM CT) CM > P P -P > -P > + + II VO VO '-O CO CO s H rH H H H rH - O H rH rH - S '- E S'— rH H H rH H C— H H *v *\ »\ 1 0 6

Kl6 ,1 7 = vl6 ,1 6 ^1 6 , i6tl6 , ir7+vi7 ,17^1 7 ,17 ^1 7 , i6+

+vl6,17 1^16,17t 1 7 ,lb+ti6 ,l6tl7 ,17] (8.38)

2 K1 0 ,1 0 " ^OjlO^OjlO (8-39) where: t-^Q

*16,16 =-[b^(B 4|a)cos/| - -2.6173 ■

*16,17 = -[a - ^ C a + ^ J c o s ^] = 1.3028

^ 7 ,1 6 ■ ~ ^ [ p + f e f F =

tl7 A 7 - 1 5 [ p i f (1+i)B+ri + = r i ^ j - +^ 2

As was previously indicated V^5 17 ^ ®

D . Calculations

Using the assignments and the equations in the previous sec­ tions, the following constants associated with the symmetry coor­ dinates of the internal CHp motions and the hindered rotations can be easily calculated.

n-1 1 ___ /— ^2 K5 ,5 = *315 l O ^ e r g / W

K q 3 = .588 lO’^erg/rad^ 107

^ = .646 10“11 erg/rad2

K = .236 10-11 erg/rad2 i5,15

^2 0 ,2 0 = *397 10-11 erg/rad2

■^2 1 ,2 1 = 1.99 IQ' 11 erg/rad2

^-2 2 ,2 2 = 17*99 105 dyne/cm.

^2 3 ,2 3 = 74.02 1 0 ^ dyne/cm.

K-2 k,2 k = 16.35 105 dyne/cm.

^2 5 ,2 5 = 74.02 105 dyne/cm.

K2 7 f2'7 = 77*97 105 dyne/cm.

4?8, 28 = 17*0 1 0 ^ dyne/cm.

I<[2 9 ,2 9 “ ^9*3 1 0 ^ dyne/cm.

K3 o ,3 0 = 77*57 105 dyne/cm.

To complete the analysis, a number of transformation factors must be numerically evaluated. These are: g = ^ .16797 e = 4.50391

1 + 51 = 2.16797 m

1 - M = -.16797 m

= .360200 3g 108

hl = .336665 ho = .096233

ho = - .385134 *3 1.34737

Upon evaluation of the known factors, the framework deter­

minants become:

F Ai 2 - *M(1.6OOI6 7 ) 2,3 = 0 (8.40) IC2,3 *3 ,3 -399833)

F B K1 0 ,1 0 ->M(11.382) ^ 0 (8.41)

F Be Kl6 ,l6 ->M(6.i64) Kl6 ,17 + >*M (5.737)

= 0 (8.42) Kl6 ,17 + >M(5-737) K:11 A 1 - ^m (7 .6 7 8 )

The solution of the second order determinants is based on polynomial expansion theorem already cited in connection with the

spiropentane analysis.

Since the B2 ring stretching frequency is not assigned, the constant appropriate to ^ 2.6 , 1 6 6 e transferred from spiropen­ tane. This is = 2.3101 10^ dyne/cm. Substituting in the determinant, a solution will be obtained for Vj_7 17 = Ka and the order of magnitude of the B 2 ring stretching frequency will be in­ dicated. 109

The Aj_ framework determinant will be solved once assuming

K2 ,3 = 0 to gain insight concerning the justification of assum­

ing Kv© = 0 when Kp^ = 0 .

The handling of the equations is quite simple and will not

be treated in detail. Implicit in the treatment is the concept

of mass normalization, but because of the simplicity the trans­

formations inherent in doing this will also not be treated in

detail.

Solution of the equation yields

Vl0 ,1 0 = = .210 lO"-^ erg/rad2

F Solution of the Bp equation, using KrQ - Kr-^rp from spiro­

pentane yields,

V17,17 KA s -- — a = .3519 10^ dyne/cm. rsin^p rsin^-jo

or KA = .6073 10- -1-1 erg/rad2

The predicted ring deformation frequency is 822 cm-'*'.

Solution of the A^_F equations, assuming K p ^ = 0 yields:

K1 1 ~ 105 dyne/cm.

&2,2 ~ 1^*069 105

K3 j3 15 1 0 *2 2 5 l o 5

Here it is assumed that on the basis of the numerical values of the hithat U2 corresponds most closely to the predominantly 110 ring stretching mode and corresponds most closely to the pre­ dominantly double bond stretching mode. The calculated potential constants then are:

__§° = 2.426 10 "’ dyne/cm. r2

Kr = 3-997 105 o

Kxir2 = 1q5

= 3.059 io5 r

= - (cot/ )Kq r

!

Kr — ® = -.4290 105 r

= 1 8 .6 9 10 5

On the basis of these latter calculations^ it seems reason­ able to assume Krg = 0 , provided in the subsequent calculations many of the other constants; particularly Kp ; have smaller mag­ nitudes . F Assuming Kj. =0; = solution of the A^ equations are, remembering that a quadratic equation has two real roots if it has one: Ill

+ — Kl,l = 4 *315 Kg 2 = 2 3 .1 5 6 18.496 1 0 ^ dyne/cm. K3 3 = 7.954 9.118

Kg#3 = 6.3499 4 . 9 7 9

It follows:

= 2.158 2 .1 5 8

- 1 3 .0 0 8 15.434 V K_ P© -.9355 -1-331 r x lQl dyne/cm. = 1.493 2.447

K*. o = 3-313 3-313 K1 r = 1.003 1.003

It appears that the column labeled + gives values of real physical significance. This is based on the idea that one would expect diagonal force constants to be greater than interaction force constants.

E. The Force Constants

The calculation of the physically significant force constants for the hindered rotations and internal motions from the M.O.M. symmetry force constants is defined by Equations 8.18-8.35• This results in the following values. 112

.356 IQ-11 erg/rad^ • w ■ .236 10"11 K c w =

11 - .014-1 10 ”11

%C >1 .617 1 0 -11

1.99 1 0 -1 1

.053 1 0 - 11 f 1 !

O i-i 17.5 1 0 ^ dyne/cm. if II -d'

o 16.4 io5

II • 50 1 0 5 H

71.7 10 5 1

O "(k. 0 105 J" II

H 2.k 105

Kwul - V = 7 7 .9 7 1 0 5 if II 77.6 105 o

Table 13

Force Constants for Hindered .Rotation and Internal Motions

The quite high value for is due to the fact that the sup posedly high kinetic interaction between the ethyl rocking and methyl wagging (K^^) motions not as yet accounted for.

Presumably the best framework constants are the following, obtained from the previous section. 113

io5 Kl,1 ~ J+.315 105 “2, 2 = 23-15 105 K3, 3 ~ 7.95^ 6 .3 5 0 105 *2, 3 =

11 2 .1 6 1 0 5

0^ • J1'1 105 - 1 .0 0 lo5

II 1 3 .0 1 105

V e . - .9 3 6 lo5 r 1 .4 9 10 5 -

Table 19

Framework Symmetry and. Modified Valence Force Constants.

To be surej Kp has a higher valence than might be expected, but this is partly due to the 1760 assignment. It may be noted that by including L, ^ in the solution, K was lowered from f>o 18x10^ to 13x105 dyne/cm. Hence it may be expected that if the constants K-^ 2 and ^ are not negligible, the value of would be depressed more, though a much lower value of IC, might not be o acceptable. Ilk

SPIROPENTANE Diagonal Calculation Using METHYLENE- Scott's Assignments Revised CYCLOPROPANE Except for 1150 Band Assignments

% 2.kk io5 2 .7 2 or 3.35 2 .1 6 1 0 5 r2

3 .8 8 105 2 .9 6 or 3 .8 1 3-31 io5 Kro

1 .5 7 lo5 .85 t ox1 1.51 1 .0 0 i q 5 Krlr 2

.328 io5 .157 or -.69^ -

- 1 .1*9 i05 *rp

Ko 0 - .2 1 0 105 1..1 1 or .kd'J - 1 2 r2 npe - - -.936 105 r

Table 20

Spiropentane and Methylene Cyclopropane Frame­ work Force Constants (those which can be compared) 115

CHAPTER IX

THE VIBRATIONAL ANALYSIS OF A PLANAR DISTRIBUTION

OF IDENTICAL CHARACTERISTIC GROUPS

A. Introduction

The vibrational analysis of crystals has been treated by many investigators. One of the earliest methods is that due to

Debye (26) based on a continuum theory. However, certain specific heat anomalies required a more detailed treatment and such a one was given by Born and von Karman (2 7 ). This treatment is applica­ ble to linear, planar, and three-dimensional crystals as well as large molecules with some modifications. The major modification is perhaps the substitution of molecular force constants for the bulk properties of the crystal in the equations of motion.

Blackman (28) made a number of detailed calculations in applying the method, principally to cubic crystals. Recently, the method was applied by Hsieh (2 9 ) in specific heat calculations of

Germanium and Silicon, and Krumhansl and Brooks (30) on the speci­ fic heat of granite.

The first serious application of the Born-von Karman method to the vibrational analysis of molecules was made by Kirkwood (31)

In treating the paraffin chains. In this analysis, the major features of the Born-von Karman Method were retained, including the method of obtaining the discrete frequencies from the contin­ uous spectrum by use of the full wavelength, assumption. Shortly 116

thereafter, Whitcomb, Nielsen and Thomas (32) and Pitzer (33)

carried out similar analyses of the paraffin chains. In these

treatments, the full wavelength assumption was rejected and the

half wavelength assumption introduced as one which at least qual­

itatively satisfied molecular requirements. This assumption was

placed on a firmer foundation by the work of Deeds (7) who com­ bined the M.O.M. and the Born-von Karman treatment in the vibra­

tional analysis of the paraffin chains. In his treatment, Pitzer points out that the treatment is essentially an approximate one, and as such it is probably true that the lowest chain bending frequency really corresponds to a free rotation of the molecule.

This statement is at least indicated if not considered by Deeds' frequency distributions.

An approach similar to the M.O.M. was introduced by Higgs (11) in his work on the vibrational analysis of long coiled chains.

This work was undertaken in the hope of leading toward the vibra­ tional analysis of the protein chains.

At this point it seems worthwhile to extend the combination of the M.O.M. and the Born-von Karman treatment in two ways. One would be to consider a planar distribution of identical character­ istic groups. The other would be to consider a linear chain com­ posed of two distinct characteristic groups alternately situated along the chain. The latter will present some interesting problems 117

when one attempts to pick the discrete frequencies from the con­

tinuous distribution. The method used by Born and von Karman (26)

on the diatomic chain will not be applicable.

B . The Boundary Conditions

The above discussion indicates that the half wavelength

assumption is to be preferred. Because of the approximate nature

of the solution it is necessary to introduce another parameter.

This parameter may be introduced in a number of ways, one of

which is discussed by Deeds (7)* Another method of introducing

the parameter will be discussed in Section G of this chapter.

By introducing such a parametex- it may well be possible to reduce

to zero the freqxiencies of the modes which correspond to the free

rotations.

C . Geometry

Suppose one has a distribution of characteristic or identical

groups of atoms or particles such that when in equilibrium the

center of mass of each group lies in a given plane. This does not

require all the atoms to lie within the given plane when in equil­

ibrium. To reduce the problem to practical considerations one

assumes that the groups as well as the constituent atoms form some kind of periodic lattice. The M.O.M. fi-amework lattice will be determined by lines joining the centers of mass of the constituent groups. This may bear little, if any resemblance to the lattice 118

formed, by the constituent atoms. In some cases it may well be that

a complicated lattice has a simple cubic M.O.M. lattice. A typical

jjlanar lattice of this type appears in Figure 9 . This lattice

bears similarity to the graphite lattice, but it must be remembered

that here the lattice points refer to the centers of mass of the

characteristic groups. The parameters rQ , and 9Q have the in­

dicated meaning, and will be defined later. If the characteristic

group consists of N particles then associated with these W par­

ticles, there are 3N degrees of freedom. In applying the M.O.M.

3N-6 of these refer to the internal motions of the group relative

to its center of mass. Three degrees of freedom are associated

with the translation of the C.M. of the characteristic group and

3 degrees of freedom are associated with the 3 Independent rota­

tions to which the characteristic group can be subjected.

The lattice points will be labeled as indicated on Figure 9 , and as yet are quite arbitrary. A typical lattice point n,p will have associated with it a definite set of xn n ; yn T,: zn n coordi- nate axes. The orientation of these axes will be determined by the principal axes of the characteristic groups in order to remove products of inertia from consideration. The positive sense of the axes will be determined by the symmetry of the particles making up the group. For instance suppose the characteristic group is a bent, symmetrical XYp group. One of the principal axes of this group 1 1 9

©-

Figure 9. The M.O.M. Lattice of Interest

Figure 10. The Cubic M.O.M. Lattice 120

lies along the line joining the C. of M. of the group and the X

atom. Assume then that one assigns the n,p axis to this line, the positive sense being from the G. of M. toward the X atom. In the n+ H, p+r group, the positive sense of the xn+^£p+r axis is defined in the same way.

Assuming that one has a well-defined system of M.O.M. co­

ordinate axes, the 6 coordinates associated with the overall

motion of the characteristic group are:

>n,p translation of C . of M . along the positive xn,p axis.

translation of C . of M. along the positive yn,p axis .

II translation of C . of M. along the positive axis . b

k zn,p

II rotation of charactex-istic group :about xn ^p, axis the $

positive sense being that of a right-hand screw ad­

vancing along the positive xn ^p axis.

Pn ^p = rotation of characteristic group about yn,p axis the

positive sense being that of a right-hand screw ad­

vancing along the positive yrijp axis.

7n,p = rotation of characteristic group about ^xi,p axis the positive sense being that of a right-hand screw ad­

vancing along the positive zn ^p axis.

The internal coordinates cannot be specified until the exact nature of the characteristic group is known. 121

For the purpose of* setting up a potential function, parameters of the lattice will be useful.

rn ^p = distance from C. of M. of group n,p to n+l,p

p = distance from C. of M. of group n,p to n,p+l

0n ^p = largest angle between rHjp and rn-1

D. The Kinetic Energy-

Following the approach of the M.O.M. and its inherent rota­ ting axes theorem of classical mechanics it is relatively easy to write the kinetic energy of the system described when the constit­ uent particles are slightly removed from equilibrium. It is assumed that there Is no internal angular momentum due to the in­ ternal motion of the characteristic groups. This latter statement is equivalent to neglecting coriolis interactions.

The kinetic energy of the group labeled n,p becomes:

Tn -p. = i m f”?2 +ii2 +a2a^ +b2$2 JT1 +T^ ~ (9*1) n >P 2 [_7n,p /n,p 7n,p n,p n,p n ^PJ n >P where T^ is the contribution of the internal motions n,p m is the mass of the characteristic group

a,b and c are the principal radii of gyration.

The total kinetic energy of the lattice is then

T = ^ Tn n p n

+2T^n,p (9*2) where the summation extends over all the possible values of n and p. 122

To write -the equations of motion, it will become necessary to obtain the expressions for *9

5L_^L_L_ = *

E. The Potential Energy

To write a correct expression for the potential energy of the system described is an almost insurmountable task on the basis of present methods of analysis. Since the system has an equilibrium configuration, the linear terms in the Taylor Series expansion vanish. It will then be assumed that the quadratic portion of the expansion gives a much greater contribution to the potential energy than all succeeding terms. As in the study of molecules where the M.O.M. is applied, it will be assumed that the potential interaction between the different kinds (framework, hindered rota­ tion, internal motion) of motion is negligible. This sort of assumption is somewhat discussed in the work of Dennison and

Sutherland (3*0* £n principle, interactions between the like co­ ordinates of all the lattice points should be considered, however, practical considerations force the inclusion of only nearest neigh­ bor interactions. This sort of assumption is justified by the values Deeds obtained for his interaction constants. 1 2 3

These considerations lead to the following potential energy

expression

V = V (9*^) n > P ’ ^

Vn ,p = v n.(P'-3' (!.■>?) " + v n,pv7/ (?) + v n,p (a,p) + v n^px,// (7 ) + v n^.pv ( 1 ) (x9 .5 )

It is assumed in the previous equation that the interaction

between the out of plane framework motion and the in plane frame­ work motion is negligible. Similar considerations apply to the hindered rotations.

Vn p(i) -i*s potential energy associated with the internal motions of the characteristic groups. In general V .-.(fj'V) can be constructed in a number of ways. These depend on the nature of the assumed force field. In general these assumptions are central force, valence and cartesian fields. Ultimately a cen­ tral force field will be assumed such that Vn ~ Vn p(£r,5p)

If q and s are combinations of the coordinates f, , a ;

Pj y, it will be assumed:

Vn,p(l,s) - Vn ,p(q) + Vn;p(s) + Kq ,s

2

+2N^ K , p )qn+l,p+l> + 2V^n,p-M>K+p)

(9-t ” 12if

where: IfIn (9.8) <11

2n n : (9-9) <11 - ^ ‘v )

and. N111 + h2n = K3 + (9 .1 0 ) ql

(-l)n J Y n - Nlnl = K3 - (9 .1 1 ) v L qiq] q.-1-J q*1

l(n±l) 2n If = N (9 .1 2 ) 2 (nil) = ^ln IT (9.13)

Kj i = 1,2,3,!+ are assumed to he constants which is equivalent

to the assumption of negligible end effects. This is justified

if the planar distribution of matter is what might be considered

of infinite extent.

If vnjP(S,7 ) = vnjp(Sj.,5r) (9.1!+) then

S r ) _ 1 f ^ o ^ f n . p )2 + Vn ,p($P + 2Hln [(Srnjp)(Spn p ) + (SrnjP+1 )(Spn+1 )P2l

+ 2H2n[_(Srn;p)($pn+1;p) + (Spn>p)(Srnjp+i)]

(9-15) where + [l+(-l)3 >& p [ (9.16)

H211 = Kip + Krp| (9-17) H2 (n±l) = ^ (9 *2 1 )

In summary for the M.O.M. lattice where 6 < -a, the

potential function to be used is:

V = -Z!n IE. p V n,p (9-22)

where Vn ^p = V ^ p (Sp,Sr) + V ^ p (^) + Vn ^ ( a ) + V ^ p (p)

(9.23)

+ Vn , p ^ ) * ^q ^ ^ jP )(^n,p )+Vn,p(l)

with the terms contributing being defined by Equations 9*7 an

9.15* For simplicity it will be assumed that is small,

for this will greatly simplify the equations of motion.

Ultimately, the equations of motion must be written and in

that development it is necessary to obtain ^ ^

For q = 7 , a, 0

3 'q^p = + 1<^i^qn+l,p+^n-l,p^ + I^q1^n,p+1+I^n,p-1^

+ ^ ( ^ n 4-l,p+l + *xi-l,p+l) + Nq ^ W , p - l +^n-l,p-I^

+ Kqssn,p % a ^s0 + Kqssn,p ^q0^sa (9 *2i0 where gf.j_ j is the Kronecker delta. 126

'dv _ ifc. 3fn,p °J n 'p' + ¥ h t f H

+ 1 SPn,p ‘Spn+1,p-i )S^ r t) ; p } + < SPr., p+SPn -1, p -1

, H2n f7 &o +&o , (§« +gp ^■(.%;n.-i),.pTl 4- H ^pn+l,p+^njP-lJ + ^Pn-l^p+TVp-l' 9 j n p J

(9.25)

S v _ K,, V/5L, + fSr , ^ C ^ rn-l,p)] ^ ----- ro P ® rn,pJ ^ 07 ^ rn-l,p'^*, d 7 n,p i - ^ >n,p y n ,p J

+ ^ , P ^ f +

A j & r n.p^.p-xj^f ^ W ^ P -I]^ ^

Q ^ rn,p+&rn+ljp- ] b + [^r‘‘,P-l ^ ri,-1 ,0-

+ i f j j}pn+l,P+fyr +{fPn-l,p+Sf>n,p-i]^n’p 5 ^

(9 -2 6 )

In the writing of the last two equations it is assumed that

Sp = fyiy) Sr = Sr (f,y)

As yet the orientation of the coordinate system Is fixed only in

so far that the yn ^p axis lies parallel to po . 127

It may be of interest to obtain Vn p(J,^) in terms of what may be called a modified valence description. However, because

of additional complexity, this potential function will only be

investigated for the case © = since under such circumstances o it, the central force equations reduce to a particularly simple form.

Consideration of a typical lattice section in Fig. 10 leads to the adoption of the indicated angles. In this analysis it is assumed that

&r = 5r(^)

However, the positive senses of % and ^ are not yet specified.

It is readily apparent that this analysis introduces the redundancy

Sa + + Siz5 + S ¥ = 0 . w n,p ^n,p n,p /n,p 128

If 0 ^ it, the in plane framework potential energy might

appear as:

Vn,p(f>y) = VniP(Sr,£p,Sy>SV',&i,S0)

= | ^ | ^ Kro(Srn,p)2+KPc,(SPn,p)2+4 0l?V'n,p+^ n Jp]

| > | d, S x 2 "1 nQ L n ,pJ

+2 KrA £_ Srn_1 ^pSAn ^ + S r n ^pS x n ^^|

+2K^[Srn-l,pi ^„,p+Srn,p^n,^]

+2I^yfiPn,P(S'!in,p+ S rn,p)+2^A% n ,p.1( S X n;p+SAn)P)j

(9-27)

nil &

The latter definitions are in line with the definitions of the N^n and H^n in Eqns . 9.3, 9-9j 9*16 and 9*17* q

If © = it, K^o = k-A0-’ > ^PY ~

The where q = %, "*) can he written, but because of the " ^ P 7 complexity will be deferred until the valence coordinates are ex­ plicitly defined. 129

F. The Coordinate Axes

As has already been indicated, the y_ri,p axes are to lie parallel to the pQ . It is conceivable that for many character­

istic groups this will not be a principal axis, but from the

symmetry of the M.O.M. lattice it can be expected that it will

be one for many distributions.

The x n,p axis will lie in the plane, but a number of possi- bilities occurs. These are indicated by Fig. 11 which are * typical lattice sections. Once the and yn ^p axes are

chosen, the zn ^p axis is fixed by the right hand rule.

The symmetry of the actual system under consideration will

govern which of the possibilities one might use in an analysis.

It should be indicated that 2 Is the choice made by Deeds (7 )

in the paraffin problem. The axes have the same orientation as

one runs through the values of p with n fixed.

It is useful to give the transformation relating Sr, £p, and the changes in the bond angles in terms of the % and .

Sf>n,p = %, p + l - (9 .2 8 )

Srn ,p = D [S n+l,p-Sn ,^] + <-1 >n ['yn+l,p--S1)p]E (9-29) 130

n, p

n+1, p

1

n, p

■n-l,P

n - 1 , p n + 1 , p 2

n,p

n-l,p n + 1 , p

3

Figure 11. Orientation of Coordinate Axes in Lattice 131

For *Pn,p = C-l>n fc7n,p+l (9-3°)

fcn.p - [ln+l,p - Sn,p] » + [7n+1,p + 7n J E (9-3D

For 3 V . P = <-l)n+1 l7 n jP+l -7n,p] (9-32)

fcn,p “ (‘1)n D » l , p + Jn,p] D* h +l,P +^ n , 3 E (9'33)

6 where D = sin-

E = cos^ 2

If Gq = it, choice 1 will be used and the parameters are:

V n , p = % , p + l ~Jfn,p (9*3‘0

^rn,p = /n+l,p’ ~Jn,p (9*35)

S ^n,p - J « n >P+rJn,p^ + r <9-36)

S ^ , P ~ ? <*n,plnjP+d * ? Vn.g-jLl.J ^

1 Sa = i (f ) + - C7 „ -'V ) (9-33) n,p y> ■’n^p-l fn/p r 'n-l^p /n,p e

^ n . p “ p « n , p - A , p . l > + ?

With this definition of parameters it becomes possible to write the equation of motion for a typical characteristic group. 132

G . The Equations of Motion

In a vibration problem such as this it is useful to define

the Lagrangian L = T-V.

The Lagrangian Equations of Motion are:

_-- = o Cf — 1 * • *f but since T = T(q) and V = V(q) it follows:

(I3L) + = o at

Combining the results of Sections B_, C, D, it follows:

For q = ^

*1.. 1 - 2 M *n,P + Kqo(qn,p) + ^ i t ^ ^ p + V ^ p ) + Kql K , p +l+cVp-l) In 2n + ^qi(t3-n+l,p+l+qn-l,p+l) + ^ql(qn+l,p-l+

+ ^

To be sure this can be simplified if IC^S is small and the method of variation of parameters can be used as discussed by

Deeds (7 ) or (35)* 133

The in plane framework equations have the form:

For coordinate axis choice 1 m

+H D ["^+l,p+?n-l,p+?n+l,p-l"7n-l,p-l]

2 n +H D j^n-l,p+l“7n+l,p+l+^n+l,p“%i-l,p] = ° (9-^1)

%,p + V E(-l)n+1 (l ’+l>P‘^-l,P) + ’^o*a(2 JS>.p-^l,p-3fn:l.P) + kPo [ 23,^ ^n>p+1 /^n>p_J

n+1 + Hlnj 2E(-l) ^ n ^ p+1 “ 2 y?n ^ p+ ^ p - l+?n+l ^ p+?n -1 ^ p'/:n+l, p - 1

-7n-l,p-l ^ J^^*n+l,p-l 5n-l,p-l ^n+l^p+^n-l,p^J J H2 n ^ 2E^ n+1 2 ?n;P 7 n ?P- 1 7n,p+l"%i+l,p 7n-l,p+7n+l,p+l

+7n-l,p+l

^ji>n+l,p ^n-l,p ^n+l,p+l+Jn-l.,p+lj V.

= 0 (9.^2) 13b

For coordinate choice 2

m *n,P + L 2 ^n,pln+1 , p l n - 1 ,p] + K r 0D E [ ^ n . 1 ) p -7 n + 1 ) g

+ H2 aD(-l)n^n+l,p+l"^n-lJp+l’i*?n-lJp"^n+l,p] 0 (9 3

in X , p * + KroEl 2^ , P +^ l - P + ^ - 1 - 3

+ ^PoL^njP 3fc,p+l

Inf + H -V E(-l) 2 ?,n,p+l+ /Jn,p-l“2 ^n,p^n+lJp ?n-l,p+ /n-l,p

+%i~l>p-lj

+ ( - 1 ^ [j n-l,p~3n+l,p+Jn+l,p~l~/n-l,p~l]

•bH2 n V 2E( -l)n 2^n,p "^n^p+1 7n^p~l+”?n+l,p+,^n-l,p ?n+i,p+i

"^n-l^p+aj

1 +(-l)nD n+ 1 ,j>~$>n - 1 ,p+ £n-l,p+±~$ n+l^p+ll (9-^) “* —J

= 0 135

For coordinate choice 3•

m 'fn,p + K^oD2 ^2 ^nJp+ ^n+l,p+ ^n-l,p^+ ^“1 ^nKroDE^n+l,P_^n-l,p^

+ H D \^Vn+x}jp ~ V n - l , v - l + %L-l,p-l]

=» 0 + H D£^n+l,p+l“^n-l,p+l“%H-l,p+ 2n-l,pj (9.^5)

^ n,p + Kr0DE(-l) (£n+l,p”^n-l,p^ + ^ o 12 ^2 ^n,p+^n+l,p+%-lj,p^

+ Kjpo ^^Vxxyp~Vn,p+l~Vn,p-l ^

+h1 ■/'+e(-1 ) [ 2 Vn,p“^n,p+l"7n,p-l+^n+l,p+2n-l,p >^n+l,p-ll-

- % ■ l,p-l

+ D L ^n+l,p-l+^n-l,p-l+Jn+l,p /n-i,p| ~J J

+H ■f 2E(-l)n 2^ , p + /'n,p+i+^n,p-l %i+l,p Xl-l,p+^n+l,p+l ^

'h7n-l,p+l

+D £~Jn+l,p+Jn-l,p+£n+l,p+l 5n-l,p+jQ (9^6)

= 0 136

For © = it and coordinate system 1

m ^n,p + 2 ICrot,^njP 2 ^n+l,p+5n-l,p^j

r 1 + KAm. ^-j2ln,p fn,p+l + F (Vn -lJp+l+ "^n+l^p-l ^n+l,p+l

£i,p-l ”?n-l,p-l)

+ j" _i^n,p+2/n+l,p+2 Jn-l,p+ 2£n,p+l+ 2/n,p-l = 0 “5n+l,p+l_£n-l,p+l ^n-l^p- 1 Xn+l,p-l __ (9-^7) m % , V + 2Yfo]yn,A (Vn,^%,:,P- 1^ ‘

+ KAo h r ^2 3^n_,p 7n+l,p /n-l_,p^ + p ^n-l,p+l+^n+l,p-l

n+1 , p+l-Jn -l, p - 1 ^

2KjoA “J+%i, p+ 2 %, Pfl+2^n j p - l+2^n+l 7 p+2^n -1 , p| = 0

~/^n+l,p+l %+l,p-l /^n-l;,p+l /^n-ljp- 1 _ O . b B ) To solve these equations of motion it becomes necessary to

introduce a new concept. It will be assumed that the lattice is

of infinite, or large, extent in both the n and p directions. An

exact solution of this problem would then require the solution of

an infinite, or large, determinant.

The procedure toward the solution employed is based on that

employed by Deeds as well as the discussion given by Seitz (3^).

Since the periodic or full wavelength boundary condition has been

rejected in favor of the half wavelength assumptions, the actual

writing of the solution will follow a different form with the same

ultimate result.

First it will be assumed that any coordinate q satisfies the

equation

.. 2 from which it follows Q.n^p consistent with the results of

vibration studies.

Substitution of this relation into the equations of motion

leaves them in terms of the unknown time independent functions A^ ^

It is then of interest to obtain relations between the A^- when considering different lattice points. In line with the assump tion of travelling waves as discussed by Deeds, it becomes pos­ sible to write , , c. \ An,p = Q0e P q with Qo real and Eg, = constant. 136

This relation indicates that all of the like coordinates have

the same amplitude. This is not strictly true, for examination

of the diagrams of Whitcomb, Nielsen and Thomas (32) will quali­ tatively indicate this. Especially some low frequency half wave­ length standing wave fundamentals will not be pure vibrations, but will include some overall rotation under this description.

Making the indicated substitutions in the equations of motion one obtains:

j -M-M*1602+K 60 +K +KT^ cos S+2K2 cosA+2 (k6 )cosScosA ° (_ '!o lo 11 11 v n !i

The in plane framework equations for coordinate choice 1 are:

]{_(I<^p+Kr ) s in A c o s 8 +(-l)nsin£ £(K^-K^)(l-cosA )-KrQE]

(l-cosS)+2I^,o (l-cos A)h-EE(K^. -K^)(l-cos£-cosA+cosScosA V

0 (9-51) 139

For coordinate choice 2 the in plane framework equations are;

(E^-T-mO)2 + 2 K j . D2(1-cos£)^

+% L- 2Dsin£|^(-l)n (K^; +K2 )cos£ sinA-sin£ j(K^; -K2 )(l-cosA)

+KroE]j = ° (9*52)

£ [_2DsinSj^ (-l)n (K^ +K2 )cos£ sinA-sin£ -Kp^)(l-cosA)

+KroE|A "

-rnO) 2+2Kr qE2 (1+co sS )+2Kpo (1-c o s A ) - ( 1 ^ - ^ (1+c o sS - c os A

-cos8cosA)j|

= 0 ( 9 * 5 3 )

For coordinate choice 3 the in plane framework equations are:

(5*q 2+2Kr D2(1+c o s

-2Dsin (i%* % )cos£ sinA

-(-1)nsin£ R -K2 )(l-cosA)-Kr e J f = 0 P P -9 (9.5*0 1^0

£ [,-2DsinS] n (Kj +K^ )cos£ sinA L P P

+ ( -l)nsin£. [ ( 4 p-I^p ) (l-c°sA )-KI-0E]^

-^/o ^ -0100^+21^ q E^(1+ c o s S)+2I^0o (c o s A -i )

-4E(ld; **»IC^ )(1+cos£~cosA“Cos6cos^ j = 0 (9*55) o O —'

For 6 = it and coordinate choice 1, the equations become:

The valence angle A has been replaced by the angle 3C to

eliminate confusion with the phase angle A.

£ o ‘j^-m.ui^ + 2 Kro(l-cos&) + 8 (l-cosA)

-8 (l-cosS-cosA+cos&osAjJ

+2?o ^ cos£ sinSsinA ^ = 0 (9*56)

£ o 1 — ■ eos£ sin^sinA ( L rp J

-m£4j^+2 Kp0 (l-cosA) + 8 (1 -cosS)

- s V , (1-cos£-cosA+cos&cosA) c =0 (9-57) r J

It is important to note the manner in which the last step was q carried out, that of substituting the A J n,p 1*1-1

Substituting the the in plane framework equations take the form

£■/*«!(*,A) + % e±^ fl ^ A> = 0 (9-58)

£ 0e*** g2(S,b) +2?0e±£? f2(^A) = 0 (9-59) where and f2 are the diagonal terms. a n d ^ are real.

If 'fc*ie resulting equations are;

£ 0 3 i ( ^ a ) + | 20ei£ ^ ( 5 , a) = 0 (9 *8 0 )

foe'1* G2(S,A) + ^ of2 ($,A) = 0 (9*81)

Taking reals, the Equations 9 -^9-9*57 follow.

II. The Secular Determinant

The equations of motion are linear homogeneous equations of degree one In the unknown amplitudes. In order that there be a non-trivial solution for the unknown amplitudes, it is necessary for the determinant of the coefficients to vanish. Thus arises the secular determinant, and on expansion, the equations for the continuous frequency distributions.

These equations are;

60 | +2K^cos§+2KyicosA+2(K|i+K^i )co6&oBA^‘ (9 *82) 142

GO^ = i-r 4. ma£■7 ^ + 2 4 , cosS+2If~ cosA+£^6 +K^ JcosScosA c +Gcos£ (9*63)

2 _ j: ^Kpo+2Kp^cos&+2K^_^cosA+2(K^^+Kp )cos5cosA^ -Geos “ 5 ' ^ (9.64)

2 1 CJp = - osS+2K^ cosA+2(K~’ 4-K^; )cos^cosA^ (9.65) o me

where

K, G = cosA 5 £ V 'W +2(^r4i>oosS+2<4 mab •1 - 1 +2c41+4 1-4 1-4 1)oos5cosi3

(9 .6 6 ) £ = f e~ £a

It is assumed that Kq^ < <

The secular equation for the % , y motion takes the form

A1X -<"2 A12 = 0 A 1 A - to£ 21 22

IT CjO 1 ,2 2 A1 1 + A22 ± B All"^2 ^ + ^ 1 2 0 (9.67)

i = 1 ,2 ,3 , 1 depending on choice of coordinate JT axes:

(9 .6 8 ) Ai i - £ j] 114-3

E2 (l-cosS)+2I^ (l-cosA )+lj-E(l^ - K ^ ) ( 1 - c o s S- c o s A + c o s 6 g o s A p ^ p (9 .6 9 )

A7n=- +K^ )sinAcos£ -(-l)nsin£ |(k J^-K^)(1- c o s A )-Kro^ | P > (9.70) 1 _2Dsin< -Tl ' +K^ )sinAcos£. + (-l)n sin£ )(l-cosA )-KrQE| 21 m I "p (9.71)

A11 = m t Icr'0D2C1-cosS (9-72)

e 2 ( i + c o s 8 )+2KpJl-cosAME(K^-K| )( i + c o s S- c o s A- c o s S c o s A ^

(9*73)

^--n~ S - ( -1 )n ( +I<^ )cos£sinA+sin£ j(l

p 2 . 2 Psin' ^+(-l)n (K^+K^)cos£sinA+sin£ J( K ^ - K ^ ) (l-cosA )+Kr^ ^ | ’ 2 1 m ^ (9*75)

A11 = E [^(l+cosS)] (9 .7 6 )

A 22=^ •^2Kr^E^(l+cos£)+2K^(l-cosA )-lt-E(K^-K^ ) (1+ c o s 8 - c o s A- c o s 5 c o s A

, 0 . £ r r -19*77) A1 2 =~~^~~:^^*|^(K^+K^^)cos£slnA- ( -l)n sin£ ]^Kj^-Kp^) (l-cosA )-Kr^Ejj

A ^ = - ::22|isi[(l^;)+K2f))0os£sin4+(-l)nsin£[(I!lp -K2p )(l-cosA)-KroS C9‘78) (9*79) IMA

=~

A2 2 = m^Vo^""C°S^ ^“^1 _cos^^_—— — (i-cosS-ccisA+cosScosA )!■ (9 .8 1 )

lit r A., = — ^ cos£sinosinA (9.82) 12 pr '

„ lit C / n \ A„.. -— cos£sinosinA (9*83) pr

It is to be noted that in general ^ ± A;L . _L CL C—JL

However if £ = 0 A^-0 = A^:-, -Li-

For all values of £ A^2 = A ^

In general Cof = 0 )^(8 , A) where S and A are independent variables., i = 1**6. That is cO^ describes a surface, a result which is expec­ ted on the basis of the work done by Blackmann (28). Further, for

£ = 0 , it is true that W?(it + s^1 , n -fr jS2) = «-P2 ) *

Distinct values of U )2 lie in the region 0 < 5, A ^ it. For arbi- 2 trary £ , **> is independent of y\ , as one might expect, though the equations of motion do depend on n.

The equations must contain the six zero roots corresponding to the ovex-all translation and overall rotation of the lattice.

For 0 the surfaces are such that co^(jt+^i, it+j^g)

-^2 )* This Is to be expected since the introduction of 114-5

here can afford the corrections introduced "by Deeds in his discus­

sion of end effects. The latter corrections also serve to destroy

the symmetry about the line n: in the discrete chain frequency

values chosen for the vibrations of a one dimensional lattice.

I. The Discrete Frequencies

The next task is that of obtaining the discrete frequencies

front the continuous distributions developed in the previous sec­

tions. These distributions are functions of cosA, cosS and powers

thereof. Therefore relative maxima and minima occur at S=0, A=0;

A=0; 8 = 0 , A=jt; and 8 =jt, A=it.

If one is dealing with a crystal which may be considered of

infinite extent it is not likely that any of the bands can be

resolved into discrete frequencies. However, analysis of maximum

and minimum observed frequencies as well as strong and weak inten­

sity portions of the non-overlapping bands could lead toward the

assignment of small frequency intervals to a few, preferably one,

combinations of o and A for each such interval. This sort of

analysis could yield a number of unknown parameters, the force

constants, in terms of the assigned band increments.

If one Is dealing with an essentially molecular problem where

the framework Is to be considered of finite extent, some assumption must be made about the nature of the framework boundary. For this

lattice It will be assumed that the boundary is a rectangle. The 1*1-6 lines of the boundary can be indicated through the lattice points, n = 1, n = N, p = 1, p = P such that 2(N+P) < < NP where NP is the total number of lattice points. O Each of the 60!r surfaces will have to contribute NP degrees of freedom resulting in 3 NP-6 vibrational degrees of freedom.

In any event NP distinct frequencies will have to be taken from each of the surfaces with 6 of these 3 NP choices having the zero value.

Deeds (7 ) gives rather extensive arguments in relation.to the one dimensional lattice, and these results appear to be well suited to the problem here under consideration.

Neglecting end effects these results would appear as

S = pj-i m = 1*•• N-l O-^o^it (9*8*4-)

A = |£_ s = I- • • P-l 0 ^ A ^ it (9*65)

These relations substituted into Equations 9 . 6 2 - 9 . 6 7 would yield the 3NP values of , six of which should be zero. The 2 2 six zero roots can be expected to occur as two each in Ctf and to ^.

To gain some insight as to the manner in which the Equations

9.8*4- and 9 .8 5 are obtained, suppose there is an Y axis located parallel to the boundary p = 1 and P and a y axis located paral­ lel to the boundary n = 1 and N. 147

The assumed solutions had the form

where Aq = Q e i

The striking comparison with procedure of solving the prob­ lem of the wave equation for a vibrating rectangular membrane will guide the procedure of the quantization conditions for the problem under consideration.

For a displacement amplitude u(x,y,t) the wave equation is

(9 .8 6 )

icut Then one generally assumes u = T(t)w(x,y) = w(x,y)e

The equation for w becomes;

For a rectangular membrane one then assumes:

w(xJJy) = X(x)y(y) yielding the solution

where K2 + K2 = K2 * y The analogy can be considered.

a Hn,p u(x,y,

A q w(x,y) n,p inS eiK*X ^ e

eipA

X n

1

y p

Ky ~ A

The analogy is not valid insofar as the various kinds of motion considered in the lattice vibrations are more varied than that of the standard vibrating membrane, though the motion may closely approximate the latter.

In any event the solution assumed for Is essentially n,p that of separation of variables in the differential equation.

As such, the boundary conditions for the constituent functions of

A^ , namely exp(inS) and exp(ipA) can be investigated separately.

These lead directly to Deeds' considerations of the essentially one-dimensional lattice whose results are given In Equations 9*8^- and 9 *8 5 •

Whenever the framework is subjected to a normal vibration, standing waves are set up, which for the motion can appear much like those of a central rectangular section of a rectangular

stretched membrane. The framework must be assumed free, that is

there are no external forces of any kind acting on the system.

Therefore, in order to have standing waves the boundaries must be

at antinodes of the standing wave pattern. If the requirement is

that the boundaries are at antinodes, it is necessary that there

be an integral number of half wavelengths in the standing wave.

This requirement appears as:

(N-l)L = m(|)

w h e re L - |©o for ^ associated with A 0 L = r„sin —2. for X associated with s o 2 o N = N for associated with 6

N = P for X associated with A

2tcL with

for "transverse" waves

© = S for L = rQ sin ■—

j - A for L =

This then gives Deeds' discrete equations as previously listed. 150

CHAPTER X. THE CYCLOPROPANE MOLECULE

A . Introduction

The infrared and Raman spectra of cyclopropane, Fig. 12, have been studied by many investigators including Linnett (3 8 )>

Kistiakowsky and Rice (39)> Smith (40), Anathakrishnan (*J-l), Harris,

Ashdown and Armstrong (12), Bonner (^3); and King, Armstrong and

Harris (Mj-). An extensive vibrational analysis has been given by

Saksena (^5)* The potential function used by the latter author has recently been appropriately modified and applied to the ethyl­ ene oxide molecule by Stone (^-6). Recent high resolution studies by Neill (20) on the infrared spectrum suggest the reassignment of some of the fundamental vibrations as given in summarized form by

Herzberg.

The vibrational analysis will Incorporate the previously de­ scribed M.O.M. The close structural relation of the cyclopropane ring to the ring structures in spiropentane and methylenecyclopro­ pane is evident. Although the geometric structures are quite similar, the M.O.M. framework will be different since the character­ istic groups will be CH2 groups. While not rigorously transferable, the associated force constants should be of the same order of mag­ nitude in all three of the molecules.

Since cyclopropane is not too large a molecule, but large enough to apply the M.O.M., one could hope to obtain information 151

about some of the assumptions made in the previous studies. Par­

ticular emphasis will be placed on determining the effect of

applying the already described approximate Eckart conditions as

opposed to the exact Eckart conditions. This analysis will most

emphatically indicate the manner in which application of the

M.O.M., coupled with judicious approximations, simplifies the

vibrational analysis of a relatively large molecule.

B. The Geometry of Cyclopropane

The cyclopropane molecule is believed to have 0 3 ^ symmetry.

The carbon atoms are believed to form an equilateral triangle.

The plane of the CHg group is assumed to bisect the C-C-C neigh­ boring C-C-C angle. If the characteristic groups are again chosen to be the CH2 groups, it is then evident that the M.O.M. framework also forms an equilateral triangle. The M.O.M. coordi­ nate axes are indicated in Fig. 13 and associated coordinates are defined in the same manner as was done in the spiropentane and methylene cyclopropane analyses.

The character table for the symmetry group obtained from

Margenau and Murphy (15) is given in Table 21. 152

Figure 12. The Cyclopropane Molecule Uith M.O.M. Framev/ork

Figure 13. M.O.M. Coordinate Systems in Cyclopropane 153

Species E 2C3 3C2 ^n 2s6 3crv _ 1 A1 1 1 1 l 1 1 1 1 - 1 A1A11 -1 1 -1 A1 1 1 1 1 -1 -1 2 A- 1 -1 1 -l -1 -1

E1 2 2 -1 -l 0 0

J7]H 2 -2 -1 l 0 0

Table 21 Character Table of Symmetry Group

The molecule consists of 9 atoms. The framework consists of 3 characteristic groups. Therefore there are 3*9-6 = 21 vibrational degrees of freedom for the molecule and 3*3-6 = 3 vibrational degrees for the framework alone. To determine how these vibrational degrees of freedom are distributed among the species of the group, one applies the theorems concerned with the character of a general vibration and the expansion theorem

(16)(17). The results of this calculation are given in Table

22 . 15^

Species Total Framework Trans. Rot.

A 1 1 0 0 A1 3

A11 1 0 0 0 1

A1 1 0 0 1 2

A11 2 0 1 0 2

mr-,1 2.k » 8 2.1 a 2 2 0

2^11 2.3 = 6 0 0 2

Table 22

Number of Degrees of Freedom Associated with Each Species

It is then of interest to determine how the coordinates transform under the covering operations of the D^h Symmetry group.

This information is found in Table 23. 155

E c3 Co c2 Co Co ft S3 s3 1 2 0 21+6 Y tX^+Y +Xj -Y 1 2 0 21+0 1 2 3

ft h ft % -*L ~h -fc ft ft 4 "ft -h -ft * 2 ft ft h ~h “ft - ? 1 * 2 Jl ft -h “ft "ft ft ft ft ft -ft “ f t ft ft Jl -ft “ ft -ft ft ft % ft h ft f t ft ?3 ft ft % ft f t ft ft ft ft ft ft ft ft ft ft ?2 ft ft ft ft ft ft ft ft ft ft ft ft ? 1 ft ft % ft “ft ~ % 3 -ft -ft “ft ft % ft % ft ft ft “ft “ ^ 2 "f t " f t “ft -ft ft % f t ft ft ft ft -ft 'ft “ ft "ft “ft -ft ft % -a^ - a 2 - o 2 Oj^ o D o 2 ft ai °3 a 2 ' a 3 “ft ~ a 3 3 0 2 -a3 - 0 2 -Oi -a3 0 2 0 1 0 2 al a 3 - 0 2 “ft 0 0 2 a 3 a? 0=2 “ 1 - 0 2 "a l - 0 3 -O 3 -0 ^ -ax 0 2 ft ft -p3 Pi Pi P3 P2 Pi P3 P 2 “ Pi -P2 “ Pi “PS “P2 f32 -P2 P2 Pi p3 P3 P2 Pi -Pi -ft -ft -ft -Pi _p2 P 3 P3 P 2 Pi P 2 Pi P3 “ P3 -Pi -P2 -Pi -P3 7i 7l ft 72 -71 -73 -72 7l ft 72 -7i “ft -72 72 7 2 /l ft - 7 3 -7 2 "7l 72 71 ft “ft -72 -7l y3 7 3 72 71 -72 “7i - 7 3 ft 72 71 -72 -71 -ft a2 u l U 1 u3 U1 u 3 u 2 Ui u 3 u 2 U1 u3 u 2 u 2 u2 U 1 u 3 u3 u 2 U1 u 2 U1 u 3 u3 u 2 U1 u 3 u 3 u 2 U1 u2 U 1 u3 u3 u 2 ft u 2 U1 u 3 v i vl v3 v 2 V1 v3 v2 V1 V 3 v 2 ft v3 ft v2 v2 vl v3 v3 v2 V1 v 2 V1 v3 v3 v2 ft v 3 v 3 v2 V1 ft ft v3 v3 v 2 ft ft V1 v3 -w2 —w ft ft w3 ft -ft “w 3 -Wjl “ft 2 ft ft ft -vr ■Wj_ ft ft ft W3 ■-v3 -w2 -ft “ft -ft 3 ft ++2 v-* w3 ■w2 ft -w2 -ft «-Wq -W 3 — 'Wg -ft ft ft W3

Table 23 Coordinate Transformation Table. 1 5 6

The Eclcart conditions (l^) i-ri.ll be useful in determining a set of symmetry coordinates. For this molecule, they are:

^ 1 + ?2 + % = 0 (10.1)

(2 £l-*2V3> " ^ = 0 ^10-2)

+ ^ (^2 -^3 ) = 0 (10.3)

c2 ^l+ ^2+ ^ " ^ 53 0 (10-^)

(f2-?3 ) + ~ (2 Pi-3 2 -P3 ) + § ^ 4 3 * (0 ,-0 3 ) = 0 (1 0 .5 )

( 2 ^ ^ - ^ ) + ^ ^3 (2^-0^-a3) - 3^f. (P2-P3 ) = 0 (10.6) a,b,c are the principal radii of gyration

Inspection of the above equations in conjunction with Table

23 indicates that the following equations define a set of sym­ metry coordinates.

1 q-L = (7i + 72 (10.7) 43 + 7 3 }

1 (10.8) q2 = & (Pi + P2 + P3 )

1 q3 - (1 0 .9 ) >13* ^ l + * 2 + *3> 1 = + $,) (1 0 .1 0 ) >13 (? 1 + % 2 1 q5 = («l + a2 + a3 ) (1 0 .1 1 ) 157 q6 = ^ <7i + 72 + 73) (10.12)

*La = T S 1 (2^1 " 5 2 ~ h } (10.13)

(10.lit-) qib - ^ (fg - $ 3 )

(1 0 .1 5 ) 92 a - ^2* ^ 3 " ?2 ^

92b “ ^ ( 2^1 2 “ V (1 0 .1 6 )

^3a = "jf "?3^ (10.17)

93b = vTg* (2fx - ? 2 - ^ 3 ) (1 0 .^.8 )

91m = ^ ( ^ 1 “ ^2 “ P3 } (IO.1 9 )

(1 0 .2 0 ) 94b = ~7 f ^ 3 ~

9-5a ~ ^ (a 2 C^) (1 0 .2 1 )

(1 0 .2 2 ) = ~\fo (2al a 2 ~

96a “ ,fg (2 7l 72 7^) (10.23)

96b = “~|jf (72 " 73 ) (1 0 .21*-) q7 = 1_ (Ui + u2 + u3 ) (1 0 .2 5 ) q8 = ^ y (V1 + v2 + v3 ) (1 0 .2 6 )

99 = ^j= (W1 + w2 + V (1 0 .2 7 ) 1 5 8

l__ q7 b = \f£T ^2ul " u 2 “ u3 ) (1 0 .2 8 )

(3-'7a = ^u3 " u 2 ^ (1 0 . 2 9 )

q.8 b = (2 vi“ v2 “ v3 ^ (1 0 .3 0 )

% a = ^ ( v3 " v2 ) (10.31)

^ (2W1 " W2 " V3 } .(10-32)

0.9a = ^ 2 (wo " w3 ^ (10.33)

The Eckart conditions can then be expressed in terms of the

just defined symmetry coordinates: 2 q3 " '0$ = 0 (10.3*0 qij. = 0 (10.35)

^la + ^2a = 0 (1 0 .3 6 )

^lb + ^2b = 0 (10-37)

b2 a2 [3 — q5a =* 0 (1 0 .3 8 )

b2 a2 Q-3b + ^ ~

The coordinates to be eliminated by these equations are

93; ^la' qlb

^ b 159

To set up the framework potential energy it will be necessary to

have a system of valence or central force symmetry coordinates.

It can easily be shown that the following equations are correct.

Sq = i ^ ( S r 1 +Sr2 +Sr3) = ^ ( ^ + 3^ + ^ )B = 2Bq1 (10.40) s2a = ^ “(^r2 “^r3 ) = 3J”2 “ Bq-2 a"A ^ ^■la (10.41)

S£b “ 7 j ( M r3'*i> = T b ^ ’V ^ t f ^ V f a ) = B42b-A (10.42)

^ 1 where A = sin^ = 2

/ JT B = cos-P = 2 since p1 — — jc

After substitution of the Eckart conditions, Equations

10.40-42 become:

S 1 = ^ qi (10.43)

S2a = & ^2 a (10.44)

Sgb = 9-2b (10.45)

C . The Kinetic Energy

The procedure followed in setting up the Kinetic energy will closely follow that used in the spiropentane and methylene-cyclo­ propane problems. l6o

In the M.O.M. coordinates the Kinetic energy is;

2T = M ^ ~ J ^ + ^ ? + ^ + a 2Q^+o2^?+c27?+u?+v?+vlr^| (10.46)

where M = mc + 2mjj

Transformation to M.O.M. symmetry coordinates leaves the

Kinetic energy in the following form:

r.2 . 2 . 2 . 2 . 2 .2 .2 . 2 . 2 2T = M Jq.ii- q x a+ Ho +^2 a+^2 u+13+!h+q3 a+ 13 b

2 / - 2 .2 .2 \ o /-2 .2 .2 \ 2 /*2 *2 .2 \ +a^(q5+q.5a+q5b) + h^ (q2+q4a+q4b) + c Cq6+q6a+% h )

7 ’a ’9 2 2 2 I + 2 1 (4t+4la+«lb) (10A7) i —

In order to remove the 6 coordinates corresponding to the zero roots it becomes necessary to Impose the Eckart conditions.

The Kinetic energy then is:

2T = ^ 5 " 2TS i = F, H, I (10.48)

1 11 1 11 1 -LL S — } A } Ag, Ag, E > E wnere:

2T*X = Mq^ (10.49)

2TI1 = M(a2+q2) (10.50) A^ 7 8 lol

2t“ii = (10.51)

2t£l = Mc2 ( i + ^ ) q ? (1 0 .5 2 )

H 2*0 2T711 = Ma q~ (10.53) 2 5

2i*ii = m4 2 (1 0 .5*0

F „ .*2 -2 n 2TE1 “ 'q2 a+q2 b) (10-55)

(1 0 .5 6 ) 2 te1

2Te1 ■ (10-57)

2Te11 " M I 49a+4 o ] <10'58>

2l“ll = M[ b2(1+^ H i j ; a+ C ) + a2 (l+3s|)(42a+52b)

(10.59) + 2 ( ^ _ (4irt,45b+5,)a45a)~]

It may be noted that the Eckart conditions imposed are the exact ones. They include the contributions of the hindered rota­ tions of the characteristic groups to the angular momentum of the molecule. However, the symmetry of the molecule is such that, under the M.O.M. description, the kinetic energy portion associ­ ated with a given coordinate species is immediately separated 1d2 into portions associated with the framework motions, the hin­ dered rotations and the internal motions of the CHg groups.

D. The Potential Energy

To set up the quadratic portion of the potential energy function it is initially assumed that interactions between frame­ work motions, hindered rotations and internal motions are negli­ gible .

The force constants associated with a given coordinate system are indicated as:

vl j ~ sisi IC, a? e tc. Qo i ^i^^i^j etc*

The total quadratic portion of the potential energy then is:

2 V = 2V1 (lO.bO) S i 3 where:

2vl i “ YPi = 3Vi,i4? = Ki,i4i (10-6l) P j? 2 2 E , 2 2 . E . 2 2 - 2V - “ 3V2j2(q2a+qa ) = U0.O2)

2vt = k7,rJr^{ + K o / 6q ti+ 2K'1!,8'l7q8 (10-63)

2VAll = 1C9.9q9 (10'61° 2 1 6 3

(1 0 .6 5 ) 2VEl = Ky , T (q7a+q7b )+K8 ; 8 (q8 a+q8b )+2Ky, 8 <<17a'13a+<17 b % u ) zv£ll = ^ ;9(q|a+q2b ) (io.6 6 )

2 vfll = i t ^ q 2 (1 0 .6 7 )

2 2V«1 (10.63) r~ ' K6 ,6 l6 4* C,

(1 0 .6 9 ) 2VEl

H p 2V.il = Kc q (1 0 .7 0 ) A 5 j 5 5 2

•VV 1-IT (1 0 .7 1 ) E1-1 = ■

E. The Secular Determinant

The equations to be solved then are the following:

* 2 1;1 A1 (1 0 .7 2 ) Mb = K 1

2t.lw2 = K^ j2 E1 (10.73)

k7,7 - M “ 2 *7,8 = 0 A (10.74) 1 k7,8 K8 ,8-Mt" 2

Al! (10.75) M “ 2 = *9,9

KE - M CO 2 K? o T j T = 0 E1 (1 0 .7 6 ) T71 _E 2 K „ ib o - M W 7,8 16k

Moo2 = E (10.77) 9,9

2 2 Mb oo = 2 (10.78) A'1

p 3c2 Me ( 1-1— p )co>2 = K Ar (10.79) r 6,6

Ma^,2^ CO 2 = IC, Ar (10.80) 5,5

M c 2 w »2 = 6 E (10.81)

itf - Mb2(l+3b2 )wl2 J S ----3M a2b2u)2 1 ,1 k,5 r2 -0 E (10.82) E 3M 2-, 2 2 K - ■—-■pa b ou KE - Ma2 (l-i-3±_)ct>2 ^,5 r 5,5 2 '

These equations arise in the analysis when the transforma­ tion to normal coordinates is sought as described in Chapter

IV and VII.

Letting the Ki, j ^ 0 as discussed in the previous analyses, the following equations relating force constants are valid.

Kr = i L 1 - KE 1 O = I K x + 2^ , 2] 1 9 L I, 1 2 ,2J

: = - [ K - K13 1 - | [K5,5 + < 5] 1 3 L 5,5 5,5J

KjP = - KP 0 3 [K2,2 + 2< , J 1 - f [*2 .8 - ltol "I [*6.6 ' Vl 165

Kuo° = 3 ; *- [ 7 V,1 ? + 2kE7 ,7 1J

Kvc = J [I(8 , 8 - 2 Ke U ]

Kw - - l~K + 2K2 1 ° 3 L 9.9 9.9J

F. Fundamental Frequency Assignments

The first complete frequency assignments for cyclopropane were given by Linnett (38)* Modifications in the assignments were given by Kistiakowsky and Rice (39) an& Smith (^-0).

Hersberg (2 l) gives a summarized assignment scheme based on the work of numerous authors. More recently, Neill (20) has made a high resolution study of the infrared absorptionspectrum. This study indicates a number of changes in fundamental frequency assignments. Table 2^ is taken from the dissertation of

Howard Neill (20) and gives a summary of existing frequency assignments. Associated Desig. Species M.O.M. Symmetry- 1 2 3 5 6 7 Coordinate

3000 3029 3028 Vl 4 % 1485 150V 1504 V2 q7 1187 1189 1189 V3 *1 Vl, 4 (1000) (1000) (1040) 1019 (1070) (1070) (1110) 1070 v5 4 % A 11 3050 3103 3103 v6 Ag q9 3103 v7 q5 104l ■ 872 872 853 V8 E ' 3000 3024 3024 3024 1^35 1435 1438 V9 q7a,q7b V10 1022 1028 1028 1028 860 868 868 740 870 vu ^2a ’ q-2b 3080 3030 3080 V12 e “ ^ a ^ b (1250) (1120) 740 866 740 v13 q5a;Q*5b 1125 740 740 (1140) 1122 vl4 q4a,q4b

Table 24 Assignments as Listed in Neill's Dissertation 1. Linnett 4. Herzberg 7* Neill 2. Kistiakowsky and Rice 5- Beckett 3. Smith 6. Scott 166 167

Particularly interesting is the 7^-0 cm""'' assignments to the

degenerate framework mode by Scott.

G. Calculations of Force Constants

The force constants are first calculated using the exact

Eckart conditions. Then they will be calculated using the

approximate Eckart conditions by neglecting the angular momentum

contributions of the characteristic group hindered rotations.

The parameters necessary to carry out the calculations are given

in Table 25•

M = 2 3 .2 6 9)4- 10 gm.

a2 - .162 A2 °o b2 = .124 A2

c2 = .0 3 8 A2

Ro = 1.53 A = C-C distance

ro = I .6 7 A = CHp-CH2 M.O.M. distance.

Table 25 Physical Constants Used in the Calculations

The C-C distance used is perhaps the most probable of those

listed by Neill (20). Actually the C-C distance is not too sen­

sitive to the low resolution calculations made here because the only terms involving rQ are a/r0 , b/r0 and c/rQ - Because of the 1 6 8 symmetry properties rQ will not occur in calculations involving the approximate Eckart equations.

The details of the calculation are quite simple and will not he considered in detail. The only complexity arises in the solution of the second degree E determinant. There the poly­ nomial expansion theorem, already described, is used to great advantage.

Exact Eckart Exact Eckart Approximate Equations using Equations using Eckart Equations R Q r0______

Ka0 1 . k6 1.63 1 .6 1

Ko^ - .2 2 0 - .3 0 8 - .2 9 7

KPo ■715 .783 •771 KPf .15*+ .1 2 0 .1 2 6

• 3*4-6 K/o .3^1 .3^6 k 7i .0 0 9 .0 1 5 .0 1 k

x 1 0 erg/rad^

Table 2o

Force Constants Associated with Hindered Rotations using Herzberg's Summarized Assignments 169

Herzberg Neill's Scott1s Assignments Modification

K u q 17-5

Kuj_ .5 8 2

Kv 75.6

Kvx + .077

Kwo 78.7 .

Kwp •391

Kf0 k . 06 3.31 CO 0 ir\ Kri 1 - .7 1 2

xlO^ dyne/cm

Table 27

Force Constants not Sensitive to Diffex'ence between Exact and Approximate Eckart Conditions

Neill Scott

Ka0 1 .6 0 1.58 Kh^ -.317 -.305 .7 8 k .951 KPX .IkO .0 5 6 • 3k6 • 3ko K?o K7X .0 1 k ,0 1 k X 1 0 ™ 11 erg/rad‘ Table 28 Force Constants Associated with Hindered Rotations Using Neill's and Scott's Assignments -using Exact Eckart Equations 170

It is to be noted that Scott’s modified assignments yield a value for Kr0 which is lower than those of splropentane and methylene cyclopropane. This is what may be qualitatively expec­ ted. However, the 7^0 cm--'- line which enters into the calculations is very weak and it is not likely that such a weak line can really be assigned to the degenerate framework vibration. With the 868 or 8 7 O cm"-*- assignment, the value for KrQ is still of the proper order of magnitude, though slightly larger than may have been expected on the basis of previous studies. 171

SUMMARY AM) CONCLUSIONS

In this dissertation a vibrational analysis has been devel­

oped lor three strained ring molecules, spiropentane, methylene-

cyclopropane, and cyclopropane. The analysis included the

application of the multiple origin method in each of these cases.

In discussing the results it must be remembered that the frequency

assignments are of a tentative nature based on the best evidence

at this time and are not likely to be correct in all cases.

It was found that the quadratic potential function is not

nearly a diagonal one expressed in modified valence symmetry co­

ordinates. This is to be expected on the basis of two considera­

tions. First it is apparent that the interactions should be of a

non-negligible nature because of the strong coupling between the

characteristic groups. The rings are closed in terms of the

valence bonds. Second, It is true that the potential function used Is not based on strict valence considerations. If the poten­

tial function were based on the latter, a number of redundancies would arise, which would further complicate the problem without yielding any move in formation.

It is very possible that the framework interaction constants for spiropentane in the B2 and E determinants and for methyleneey- clopropane in the B2 determinant are non-negligible thus requiring

some reassignments. 172

A major feature of interest in the analysis of these two molecules is to compare the calculated frame-work force constants which might be transferred on geometric considerations. The agreement is encouraging in that the constants of interest cer­ tainly do more than belong to the same order of magnitude. The agreement is perhaps not to as high a degree of transferability as might have been desired. This is due to the fact that a sym­ metry force constant from spiropentane was transferred to complete the solution of the other problem as well as possible errors in frequency assignments, and no reliable data on the structure parameters of the molecules. The framework constants of interest are compared in Table 3.

A major difficulty in the analysis of methylenecyclopropane is the strong kinetic interaction between the methyl wagging motion and the ethyl rocking motion. This has not been accounted for in the analysis through the use of the approximate Eckart equations. This coupling is most certainly through the framework.

It may be suggested that to partially account for this, one could apply the complete Eckart equations. Then to simplify the problem one might arbitrarily set the kinetic coupling factors between framework motion and hindered rotation equal to zero, leaving coup­ ling terms between the hindered rotations. The unfortunate factor in this development is that the neglected terms can be expected to be larger than those retained. 173

It may be mentioned, that the irreducible representations of

the Dpd symmetry group were obtained, by a straightforward, though

not particularly sophisticated., manner. This consisted, of finding

a model (the molecule) which could be subjected to the covering

operations permitted by the molecule. The M.O.M. coordinates were

then combined in a manner suggested by the Eckart equations. The

non-degenerate Irreducible representations were easily obtained by

constructing the nondegenerate symmetry coordinates. It was found

that the degenerate symmetry coordinates transformed by pairs,

but not all pairs transformed in the same way though satisfying

the character requirements. All pairs were then adjusted by re­

defining the constant coefficient of the symmetry coordinates to

transform in the same way, the transformation matrices being the

irreducible representations.

The application of transformations developed for the bent

symmetrical XYg molecule simplified the analysis considerably when

slightly modified to apply to the problem under consideration. It

is apparent that these transformations can be useful in the study

of any molecule containing a three member strained ring framework

when the multiple origin method is applied. A particular resulting

simplification was the reduction of the third order A^ kinetic energy in methylenecyclopropane without solving a cubic secular

determinant. 17^

To place the work on molecules containing strained rings on a firmer foundation it appeared of interest to perform a vibrational analysis of cyclopropane utilizing the multiple origin method.

This vibrational analysis of cyclopropane most effectively demon­

strates the simplifications obtained when the multiple origin method is applied.

The spectra of wholly and partially deuterated spiropentane and methylenecyclopropane would yield much information concerning the frequency assignments even though the product rule is not strictly applicable. The preliminary investigation on the application of the mul­ tiple origin method to the vibrational analysis of an essentially planar distribution of identical characteristic groups shows promise of being quite useful in the analysis of planar molecular crystals.

The method can be extended to non-planar distributions. The major advantage of the M.O.M.to the crystal problem lies in the fact that the lattice problem is greatly simplified in such an analysis. 175

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AUTOBIOGRAPHY

I, Walter Bernard Loewenstein, was born in Gensungen;

Hesse, Germany on December 23, 1 9 2 6. I received my secondary

school education in the public schools of Tacoma, Washington.

After starting my academic training, I served in the United

States Wavy where most of the time was spent in attending

electronic technician training school„ Following my discharge

I resumed my undergraduate training which was obtained at the

College of Puget Sound in Tacoma from which I received the

degree Bachelor of Science with Honors in 1 9^9 . That same year my graduate work began by enrolling in the Graduate School of

the University of Washington. In 1950 I entered the Graduate

School of The Ohio State University where I have specialized

in the Department of Physics. While completing the requirements for the degree Doctor of Philosophy I have held the positions of graduate assistant, research assistant, Research Foundation

Fellow and Texas Company Fellow. During the summers of 1952 and 1953 I was a research assistant at the Los Alamos Scientific

Laboratory.