Calculation of kinetic energy terms for the vibrational Hamiltonian: Application to large-amplitude vibrations using one-, two-, and three-dimensional models

JOURNAL OF MOLECULAR SPECTROSCOPY

91, 300-324 (1982)

Calculation of Kinetic Energy Terms for the Vibrational Hamiltonian: Application to Large-Amplitude Vibrations Using One-, Two-, and Three-Dimensional Models M. A. HARTHCOCK AND J. LAANE Department of Chemistry, Texas A&M University, College Station, Texas 77843 One-, two-, and three-dimensional models involving large-amplitude vibrations have been used to calculate kinetic energy terms. Principle G matrix elements as well as cross terms in the kinetic energy were determined. Calculations were done on models involving the ringpuckering and PH inversion vibrations for 3-phospholene and the ring-puckering, ring deformation, and SiH2 in-phase rocking vibrations for 1,3-disilacyclobutane. Kinetic energy expansions for gti and g45 type terms were determined. Calculations show a coordinate dependence of the principle G matrix elements as well as of the g,, terms. The vectorial models used in these calculations make it possible to treat vibrations in a one-, two-, or three-dimensional model separate from the other ~brations without carrying out a coordinate transformation, which would be necessary for the Wilson GF high- or low-frequency separation. I. INTRODUCTION

In recent years the study of large-amplitude vibrations has drawn a considerable amount of attention. In particular, the ring-puckering motion in four- and fivemembered rings has been extensively investigated (Z-S). The far-infrared and Raman spectra recorded in the early studies were generally analyzed using a reduced potential function and little effort was made to obtain accurate kinetic energy functions. Chan et al. (6) were the first to carefully examine models for calculating the reduced mass for the ring-puckering motion in four-membered rings. However, the early spectroscopic data did not lend themselves to accurate analysis. Ueda and Shimanouchi (7) later examined reduced-mass models for the puckering of fivemembered rings. Their models, however, did not accurately represent the CH2 motions and, as a result, these workers found their kinetic energy expansion to yield poorer results than the fixed reduced-mass models. The calculations of Malloy (8) a few years later showed that properly calculated kinetic energy expansions would indeed lead to better results than fixed reduced-mass calculations. More recently, work in our laboratory (9, 10) has shown that the ring-puckering spectra for several isotopic forms of 1,3-disilacyclobutane or cyclopentene could be properly analyzed only if the puckering model was modified to incorporate SiHz or CH2 rocking motions. Most ring-puckering and other large-amplitude vibrations have been analyzed quantum mechanically with a one-dimensional vibrational Hamiltonian (1-9). The low-frequency separation was applied so that the low-frequency large-amplitude motion was assumed to be independent of the 3N-7 higher frequency modes, and 0022-2852/82/020300-25%02.00/0 Copyright 8 1982 by Academic Press, Inc. All rights of rcproduotionin any fomt rcSeWCd.

300

CALCULATION

301

OF KINETIC ENERGY TERMS

vice versa. While, in general, this assumption has proved to be quite reasonable, recent spectroscopic work has demonstrated that significant interaction may occur with the low-frequency mode. Thus, for example, the ring puckering in 1,3-disilacyclobutane interacts with both the ring deformation and in-phase SiHz rocking modes (9, 12, 13). Similarly, the PH inversion and ring-twisting motions of 3phospholene are coupled to the ring puckering (14, 15) and in cyclopentene the puckering and twisting interact (16). It is apparent then that for the proper analysis of these molecules two- or three-dimensional Hamiltonians are required with corresponding kinetic energy expansions. Several recent articles (17-23) have considered the interaction of other vibrations with large-amplitude motions. In order to arrive at an accurate potential energy function representing the interaction of two or more vibrations, it is first necessary to have a good model from which the kinetic energy terms may be determined. The previous paper (24) described how vector methods may be used to represent the vibrations. It is the purpose of this paper to describe the calculation of kinetic energy expansions for two- and three-dimensional problems and to relate this to one-dimensional situations. In addition, numerical results will be presented for both principle and interaction-type kinetic energy terms for several vibrations of I ,3-disilacyclobutane and 3-phospholene. Also, principle g matrix elements for cyclopentene will be presented. II. THEORY

Meyer and Giinthard have derived a general Hamiltonian for the internal motion of a molecule (25). The Hamiltonian in center-of-mass coordinates is given by

-t

ml, qzr* . . , q%v--61, (1)

where the qi’s and qj’s are the vibrational coordinates, the gjj’s are appropriate terms from the g matrix or expansions of G matrix elements in the vibrational coordinate(s), and g is the determinate of the G matrix. The kinetic energy part of the internal motion Hamiltonian for two vibrations may be written as 1 T,=--Zh2

dglla+ag12_d Ca41 acri aql

a aq,

V’ is described as the pseudopotential potential energy, and it is defined as

+-

aq2

+ $22d g,zL af?i aq2

a632>

-t-V’.

(2)

because some of the terms resemble the

HARTHCOCK

302

AND LAANE

Most calculations on large-amplitude vibrations utilizing a complete form of Eq. (1) have been concerned with the internal-rotation vibration. The reader is referred to Refs. (20, 21, 26) for details. Equation (1) was derived from a Hamiltonian which included all of the 3N degrees of freedom. Translation to center-of-mass coordinates then separated the three translational motions from the vibrations and rotations. We have further transformed our coordinates to the principal axis system although the vibrational terms are independent of the choice of axis system (27). In order to determine the g, terms in Eq. (2) we need to determine the vibrational-rotational G matrix. This may be done in a fashion similar to that used by Malloy in analyzing one-dimensional ring-puckering motions (8). The G matrix is given by G=

X’1

’ [

x-’ Y

’ (4)

where I is the 3 X 3 moment of inertia tensor, X is a 3 X (3N-6) matrix representing the interaction of the vibrations and rotations, and Y is a (3N-6) X (3N-6) matrix which is the vibrational contribution to the G matrix. For two vibrations the G matrix may be written in an expanded form as

-z*y

I -z:; G

=

zyy

-In

-4z

-Ix,

x1* x12 -’

-4z

&I

x22

I,,

X,,

X32

Xl,

x21

x3,

y,,

Yl2

Xl2

x22

x32

YL2

y22

,

I

(5) where the elements of the matrix are defined by Iit = $ llz,(r, *ra - &),

i = x, y, or z (x, y, and z are molecular fixed axes)

(6)

a-1

hk

=

5

a-1

%rairmk

,

i#k,

(7)

(8) and (9) N is the number of atoms in the molecule, m, is the mass of the ath atom, r, is the coordinate vector of the &h atom originating from the center of mass (see Fig. l), and rrr, and r,k are the ith or kth components of the cuth vector. The derivatives in Eq. (9) are numerically determined by calculating the structure of the molecule

CALCULATION

FIG. 1. Equilibrium coordinate vectors.

conformation

OF KINETIC

of 3-phospholene

ENERGY

TERMS

303

showing the center of mass (CM) and two of the

before and after an infinitesimal amount of vibration based on the vector representations as described in the previous paper (24). Once the moment-of-inertia matrix is established, the principal axes are found by diagonalizing the moment-of-inertia tensor by the unitary transformation (27) Idi*g= U’IU,

(10)

where U is the transformation matrix and I is the matrix defined in Eq. (4). The coordinate vectors are then transformed to the principal axis system by rz = Ur, ,

(11)

where rz is the new set of coordinate vectors originating from the center of mass in the principal axis system. Although the vibrational part of the G matrix is axisindependent, the transformation to principal axes is done so that useful values for the elements of X (which lead to Coriolis coupling terms) are determined as well as calculating the principal moments of inertia for the molecule. After the matrix elements in Eq. (5) have been determined, the matrix is inverted to give the G matrix in the desired form:

In this expression the g, that are in the top left 3 X 3 portion of the matrix (i = l-3 and j = l-3) are pure rotational terms. The elements which are needed for the vibrational kinetic energy of Eq. (2) are g,,, g45, and g5,. It is important to note that since the first three index values have been used to represent rotations (following previous convention), the indexing of the g elements has changed. That is, gM, g4,, and g55 in Eq. (12) have the identical meaning of g,,, g12, and g,, in Eq. (2). The G matrix is symmetric about the main diagonal (gV = gji).

304

HARTHCOCKANDLAANE

As has been observed in previous work (8, 9, 14, 28) the vibrational G matrix is often coordinate dependent for a large-amplitude motion. The coordinate dependence may be taken into consideration by forming a polynomial expansion of the gi;S in Eq. (1) as a function of the vibrational coordinate(s). The form of the expansion in one coordinate is gij

=

5

&'qi

I=0

9

(13)

where 4;’ are the coefficients of the various terms in the expansion. Usually the expansion is taken to fourth power for asymmetric vibrations and sixth power for symmetric vibrations. The coefficients in Eq. (13) are found by forming the G matrix of Eq. (12) at various values of the vibrational coordinates. After each computation a gM, g,,, and g55 result for particular values of the coordinates q, and q2. Each of the gij’s may be expanded in terms of q1 or q2. The expansion of Eq. (13) is found using the Newton-Raphson technique (27). If the G matrix element is approximately independent of the coordinate, the expansion in Eq. (13) reduces to (14) where h is the reduced mass of the vibration. III. CALCULATIONS

All calculations were done on a PDP 1 l/34, a PDP 1 l/44, or an Amdahl 470 V/6 and V/7 computer system and were performed using double precision arithmetic. The calculations were done using the basic bisector model (8, 24). All bond distances were assumed to remain constant. For the one-dimensional calculations, the G matrix was determined for 21 positions of the vibrational coordinate. For the two-dimensional calculations the G matrix was determined for 21 positions of each vibrational coordinate. Table I shows the structural parameters used in determining the G matrix for 3-phospholene and 1,3-disilacyclobutane. A. 3-Phospholene

Two large-amplitude vibrations, the ring puckering and the PH inversion, for 3-phospholene will be considered. These are of interest because the molecule may change conformations through both vibrations. Also, the interaction of these two vibrations has been observed spectroscopically (I 4). Figure 2 shows the coordinates chosen for the two vibrations (where q1 = x and q2 = y in Eq. (2)) (24). The PH inversion coordinate is chosen as the distance traveled by the PH hydrogen undergoing its curvilinear motion. Because of the coordinate dependence of the vibrational G matrix elements upon large-amplitude motions, it was desirable to see the effect of one- and two-dimensional models on the value of the G matrix elements. Figure 3 shows the G matrix element g 44 for the ring-puckering vibration as a

CALCULATION

OF KINETIC ENERGY TERMS

305

TABLE I Structural Parameters of 3-Phospholene and 1,3-Disilacyclobutane Used for the Calculation of the G Matrix 3-Phospholenea

c=c c-c

1.33A

C-P

1.87A

CPC Angle

92.2'

1.085A

HCH Angle

109.5O

=C-H

-C-H

1.51A

P-H

1.095A 1.419A

1,3-Disilacyclobutaneb c - si

1.87A

csic Angle

87.0°

C-H

1.092A

HCH Angle

lll.o"

Si - H

1.48A

HSiH Angle

114.3O

aJ. R. Durig, B. J. Streusand, Y. S. Li, L. Richardson, and J. Lame, J. Chem. Phys., 2, 5564 (1980). b Assumed from J. Lame,

Spectrochim Acta, a,

517 (1970).

function of the ring-puckering coordinate at PH inversion coordinate values of 1.8 and 0.0 A. The ring-puckering coordinate was varied from +0.3 to -0.3 A in the calculations. From the figure it is apparent that g4 is a function of the vibrational coordinate and is also dependent on whether a one- or two-dimensional model is chosen. Table II shows the ring-puckering kinetic energy expansions g,, at various positions of the PH inversion coordinate. One should note that the dependence of gU on the value of the puckering coordinate is reflected by the relatively large higher-order terms of the expansions. Furthermore, the changes in sign of the g$!,)and g!! coefficients reflect whether the PH inversion coordinate is positive or negative. When the PH hydrogen is planar ( y = O.O), the puckering vibration is

FIG. 2. The ring-puckering and PH inversion coordinates for 3-phospholene.

306

HARTHCOCK

-0.24

AND

-0.12

LAANE

0.00 X,

RING

0.12

0.24

PUCKERING

FIG. 3. The dependence of gU for the ring puckering of 3-phospholene as a function of the puckering coordinate, calculated at PH inversion coordinate values of y = 1.8 (solid line) and 0.0 A (dashed line). Curve A: one-dimensional calculation at y = 0.0; B: one-dimensional at y = 1.8 A; C: two-dimensional at y = 0.0; D: two-dimensional at y = 1.8 A.

symmetric and g!!) and &) are zero. However, these coefficients are very sensitive to asymmetry and their magnitudes increase significantly when the hydrogen is moved out of the plane. Figure 4 shows the dependence of g5,, the PH inversion kinetic energy term, on the inversion coordinate y, which is allowed to vary from 3.0 to -3.0 A. The gss curves are shown for both one-dimensional and two-dimensional calculations, the latter resulting in higher values. As compared to the gM term for the ring puckering, g,, varies much less with the coordinate. This is due to the fact that the motion of the light hydrogen atom has only a minor effect on the coordinate vectors of the other atoms. If rlo is the coordinate vector of the hydrogen atom, Eq. (9) may be written in a modified form: (15) It can be seen that Yzz, which is inversely related to g5,, derives primarily from the first term while the summed terms only slightly affect its value. Since the derivatives ar,/ay only vary somewhat with y, not much coordinate dependence is expected or observed. On the other hand, for the ring puckering all of the terms in Eq. (15) contribute to Yi,, which is related to g 44. Since many of the derivatives change as a function of the puckering coordinate, a greater coordinate dependence of Yi, and, therefore, g,, is present.

CALCULATION

OF KINETIC

307

ENERGY TERMS

TABLE II Kinetic Energy Expansions for the Rung-Puckering Vibration (g&x)) in 3-Ph~pholene as a Function of the Ring-Puckering Coordinate at Various Values of the PH Inversion Coordinate” PH Inversion

Coordinate

One

Dimensional 2.7

0.833502

-0.525359

-0.256005

0.254253

1.8

0.819921

-0.513276

-0.255472

0.250042

-0.224563 -0.201176

0.0

0.805893

0.0

-0.255267

0.0

-0.175065

-1.8

0.819921

0.513276

-0.255472

-0.250042

-0.201176

-2.7

0.833502

0.525359

-0.256005

-0.254253

-0.224563

Two Dimensional

a

2.7

0.062447

-0.063038

-0.267712

0.020026

-0.235646

1.8

0.862373

0.058134

-0.267694

-0.018466

-0.235679

0.0

0.866283

0.0

-0.268654

0.0

-0.233963

-1.8

0.862373

-0.050134

-0.267694

0.018466

-0.235679

-2.7

0.862247

0.063038

-0.267712

-0.020026

-0.235646

The kinetic energy

(0) 844 = 844 wits

expansion are

+ gttjx + $'x'

of the form:

+ 9:2)x3 + gjijx4

where the coefficients

p$'

have units af amu-lA-i and

of x are in A.

Table III presents the kinetic energy functions for g,, as a function of inversion coordinate. These expansions are given for three different positive values of the ring-puckering coordinate (i.e., three different conformations) and for both oneand two-dimensional models. Expansions for negative values of the puckering coordinate are not given since they are identical to those for the positive values except that the signs change on the coefficients g$i) and g@. As can be seen the degree of puckering has only a minor effect on the gs5 values although it is responsible for the asymmetry in the expansions. Interestingly, the asymmetry is reversed in going from the one-dimensional to the two-dimensional model. Figure 5 shows how the ring-puckering kinetic energy term gM is influenced by changes in the inversion coordinate, and Fig. 6 shows the effect of changing the puckering coordinate on gS5. As is evident these coordinate dependences are small even for the one-dimensional model. For the two-dimensional model any dependence vanishes for practical purposes. As can be seen in Eq. (2), the kinetic energy term g4, must be considered in addition to g4, and g,, for the two-dimensional Hamiltonian. This interaction can be found by inverting the matrix shown in Eq. (5). The Y1,, Yz2,Xjk, and Zikvalues used here are identical to those for one-dimensional computations, but in addition Ylz must be calculated from Eq. (9). Unless Y,, is zero g4, is nonzero and the twodimensional gM and gs5 values are different from the one-dimensional values. The Wilson GF method (30) provides formulas for computing g values for various types of internal coordinates. These may be applied when the complete set of 3N6 vibrations are considered. If only a partial set of vibrational coordinates is used

308

HARTHCOCK

AND LAANE

Y, P H INVERSION FIG. 4. The dependence of gsS for the PH inversion of 3-phospholene as a function of the inversion coordinate y at ring-puckering coordinate values of x = 0.12 (solid line) and 0.0 A (dashed line). Curve A: one-dimensional calculation for x = 0.0; B: one-dimensional at x = 0.12 A; C: two-dimensional at x = 0.0; D: two-dimensional at x = 0.12 A.

(on the basis of the high-low frequency separation), the g values must be modified by “freezing” the vibrations which are not considered. In order to solve the secular equation involving G (rather than G-l) it is necessary to do a coordinate transforTABLE III Kinetic Energy Expansion for the PH Inversion Vibration (gSs(x)) in 3-Phospholene as a Function of the PH Inversion Coordinate at Various Values of the Ring-Puckering Coordinate” Ring Puckering Coordinate

One Dimensional

Two

a

0.27

0.106839

-0.900346

-0.120680

0.890545

0.18

0.106822

-0.606518

-0.126481

0.608890

0.701446

0.0

0.106816

0.0

-0.131142

0.0

0.742098

0.27

0.115078

0.462928

-0.227638

-0.295146

1.15401

0.18

0.114984

0.381795

-0.226752

-0.233118

1.17110

0.0

0.114760

0.0

-0.224909

0.651267

Dimensional

The kinetic energy expansions

(0) =55 = %5

units

+

&

+ 855(2) Y 2 +

of y are in A.

0.0

1.18440

are of the form: (3) 3 (4) 4 855 y + 855 Y

where

the coefficients

(i) g55 have units of amu-l*-i and

CALCULATION

_----

_-_----

? g

~3 L

OF KINETIC

ENERGY

3- PHOSPHOLENE ---_

309

TERMS

---

---__ 7’0

t

0.80

a

07

0.76

I

I

I

I

I

-1.2

-2.4

I

I

0.00

Y, PH

1.2

I

I

2.4

INVERSION

FIG. 5. The dependence of g,, for 3-phospholene on the PH inversion coordinate at puckering values of 0.12 (solid line) and 0.0 A (dashed line). Curve A: one-dimensional at x = 0.12 A; B: one-dimensional at x = 0.0; C: two-dimensional at x = 0.12 A; D: two-dimensional at x = 0.0.

mation to a smaller set of G$: G% = G,,, - 2 G,A,G,~,~ , SS’

(16)

where s refers to the coordinates to be “frozen” and the elements X,,. must satisfy the relations 2 X,,~G,~,~ = 6,,. . (17) S’ The G,, are the tabulated Wilson G matrix values (30). This transformation procedure may be avoided by using the vector methods presented by this paper and the previous one (24).

3- PHOSPHOLENE

-0

24

-012 X, RING

0.00 PUCKERING

0 12

0 24

FIG. 6. The dependence of g,, for 3-phospholene as a function of the ring-puckering coordinate at inversion values of y = 1.8 (solid line) and 0.0 A (dashed line). Curve A: one-dimensional at y = 1.8 A; B: one-dimensional at y = 0.0; C: two-dimensional at y = 1.8 A; D: two-dimensional at y = 0.0.

310

HARTHCOCK

AND LAANE

7

0 x

t”

m

-0.23

-0.27 -0.12

-0.24

0.00

X, RING I

I -2.4

I

I -1.2

0.12

I

I 0.00

Y, PH

0.24

PUCKERING

1.2

I

2.4

INVERSION

FIG. 7. The dependence of the kinetic energy cross term ge5 for 3-phospholene as a function of puckering and PH inversion coordinates. Curve A: gd5 vs x with y = 0.0, B: g,, vs x with y = 1.8 A; C: g,, vs y with x = 0.12 A; D: ga5 vs y with x = 0.0.

Calculated g4, values for 3-phospholene are plotted in Fig. 7 as a function of both the ring-puckering and PH inversion coordinates. These values show a large amount of asymmetry and coordinate dependence as reflected in the asymmetric expansion terms in Table IV. The minimum value occurs when both coordinates are at their zero values, i.e., when the PH hydrogen lies in the plane of the planar ring. When either coordinate is nonzero, the minimum g45 value as a function of the other coordinate also occurs at a nonzero value. The g45 term for this case is negative and its magnitude is greater than that of the gU term. Studies of largeamplitude vibrations by Quade (I 7) on the effect of freezing out the other vibrations show that this sort of result is not unexpected. B. l,3-Disilacyclobutane 1. One- and two-dimensional results. The S!CH2 molecule is nonplanar and has C,, symmetry (9); in its planar conformation the symmetry becomes D2,,. The three low-frequency vibrations of interest, which are described in the previous paper (24), have for Dzh symmetry the following symmetry species: Bi, for ring puckering, A, for the ring deformation, and B,, for the SiH2 in-phase rock (32). For the CZUnonplanar structure symmetry species for each vibration becomes A,. These symmetry properties have a significant influence on the kinetic energy terms, as will be seen later. Since spectroscopic evidence has shown that the ring puckering vibration interacts significantly with the other two motions, the

CALCULATION

OF KINETIC

311

ENERGY TERMS

TABLE IV Kinetic Energy Expansion for the Cross Term in the Kinetic Energy, g,,, in 3-Ph~pholene Function of Both the Ring-Puckering and PH Inversion Coordinates PH Inversion

(1) g45

(0) 845

Coordinatea

(4) g45

(3) 845

(2) %5

as a

2.7

-o.174157x1o-1

-0.143715x10-1

o.397555x1O-1

0.468018~10 -1

0.757564~10 -1

1.8

-o.2147o9xlo-1

-0.148682x10-1

-1 0.339938X10

0.483659x10-1

1.24192x10-1

0.0

-0.263412x10-1

0.272664x10-1

0.0

1.81573x10-1

0.174521~10-2 0.173762~10 -2

0.149776x10-3 -0.651121x10-4 -0.648763x10 -4 0.117300~10 -3

0.168080x10-2

0.0

0.0

Ring Puckering Coordinateb

0.27 0.18

-0.233047x10-1 -0.220341x10-2 -0.7.52312x10-1 -0.172705x10-2

0.0

-0,262577x10-1

0.0

-0.628231x10-4

aExpansionsare of the form: (2) 2 (0) (3) 3 (4) 4 where the coefficients (1) gig' have units of ar~u-~~-~ and units g45 = g 45 + %45 x + 845 * + g45 x + g45 x of x are in A. b 12) 2 (4) 4 where the coefficients (0) (1) pi;' have units as describedin footnotea. 845 = 845 + 845 Y + 845 Y + g::)Y3+ &+45Y

evaluation of the appropriate kinetic energy terms in two- and three-dimensional models was necessitated. The vibrational motions considered for the calculations are described in the previous paper (24). The ring puckering is described by Table I, the in-phase SiHz rocking by Eqs. (14)-( 16), and the ring deformation by Eqs. (17)-(20). For the calculations the ring-puckering coordinate x was varied between plus and minus 0.3 8, &to,, between kO.378 rad (+21.6”), and SD,, between kO.175 radians (deformation AZ 5” larger than equilibrium value). Values of 0.0 and 0.2 for the bending parameter w were both considered. The kinetic energy expression (Eq. ( 1)) for the three vibrations to be considered for disilacyclobutane may be written as

r,-;~\~g44a+i ax a

a

- a

-a

~a

asRoCK g55 asRocK + asD,g66 as,,,

~ a

+ aXg45 asRocK

+

a a a - a -a -a asRoCK g4s -ax+ ax - g46 asDEF+ dsDEF g46 dx a ~ a -a - a ____ + asRoCK gs6 asDEF + asDEF gs6 asRocK+ “-

1

For a two-dimensional

calculation

involving the ring-puckering

(’ ‘)

and ring defor-

312

HARTHCOCK

,,02 _1,3-

0.70-

DISILACY

AND LAANE

CLOGUTANE

\

4

I

I -0.24

I

I

I

0.00

-0.12 X,

RING

I 0.12

I 0.24

PUCKERING

FIG. 8. The dependence of the kinetic energy term g,, for the ring puckering of 1,3-disilacyclobutane as a function of the puckering coordinate (x) at various values of the bending parameter o and of the ring deformation (Su,,) and SiH2 in-phase rocking (S socK) coordinates. Several curves are coincident. Curve A: one-dimensional calculation at S oaF = -0.105 rad, Su,k = 0.0, and w = 0.0, B: one-dimensional at SRocK = 0.233 and 0.0, SDEF= 0.0, and w = 0.0, C: one-dimensional at Suer = -0.105 rad and 0.0, S aocs = 0.233 and 0.0, and w = 0.2. Also two-dimensional with the second coordinate So,, = -0.105 and 0.0, SRocK = 0.0, and w = 0.2; D: two-dimensional with the second coordinate Ssock = 0.233 and 0.0, Sosr = 0.0, and o = 0.0 and 0.2.

mation vibrations, for example, the g elements in the second, fourth, fifth, eighth, and ninth terms of Eq. ( 18) would not be considered. Then, Eq. (5) is used for the two-dimensional calculation. Figure 8 graphically presents g4, as a function of the ring-puckering coordinate for several different situations. Four different curves are shown but these actually represent 13 different calculations since a number of curves are coincident. Curves A, B, and C represent one-dimensional calculations for different values of SDEF, SRWK, and o; it is evident that g 44 is only slightly affected by these variables. However, it does change by about 13% at x = +0.30 A, thus showing greater dependence on the extent of ring puckering. The comparable change for the g,, value of 3-phospholene was considerably more, about 30%. Curve C is also coincident with that from a two-dimensional calculation where SDEF represents the second coordinate. This is because the YrZterms calculated from Eq. (9) are either exactly zero (for a planar ring) or very close to it. Thus when the matrix in Eq. (5) is inverted the gM value differs significantly from the one-dimensional value. The two-dimensional calculation involving S RocKas the second coordinate, however, is significantly different. Because both the puckering and SiHz in-phase rocking have the same symmetry, the Y,, terms are significant and the gU values calculated

CALCULATION

313

OF KINETIC ENERGY TERMS TABLE V

Kinetic Energy Expansion for the Ring-Puckering Vibration in 1,3-Disilacyclobutane as a Function of the Ring-Puckering Coordinate at Various Values of the SiH2 In-Phase Rock Coordinate”

one Dimensional

w-o.0

w=o.2

-0.230559

0.3403

0.807774

-0.612443

-0.114065

0.205478

0.3375

0.807413

-0.409731

-0.114038

0.137526

-0.229180

0.0

II.807125

-0.114016

0.0

-0.228069

0.0

0.1594

0.807775

-0.282446

-0.101267

0.100609

-0.586578

0.3488

0.807415

-0.189250

-0.101215

0.064604

-0.585686

0.0

O.8O7127

0.0

-0.101173

0.0

-0.584965

Two Dimensional

w-0.0

w-O.2

0.3403

0.139373

-0.19cl514

-0.029428

0.884647

0.2275

0.094066

-0.190517

-0.198743

0.884423

0.0

0.0

-0.190519

0.0

0.884241

0.3594

I.02130

0.049242

-0.189445

-0.011689

0.889374

0.3488

1.02141

0.033201

-0.189457

-0.007882

0.889217

0.0

I.02149

0.0

-0.189467

0.0

0.889089

for the two-dimensional model (curve D) differ substantially from the one-dimensional case. Expansions for the ring-puckering kinetic energy term g4, are presented in Table V. These expansions are, of course, symmetric when S,oc, = 0 but became increasingly asymmetric when the amount of SiHz rocking is increased. The asymmetry is reflected in the magnitudes of the g $) and $da terms. It should be noted that these terms change sign when S RoCKis made negative but that the magnitudes remain the same. The expansions vary somewhat with w since the SRoc, coordinate is modified by changing the CSiC ring angle (see Eqs. ( 14) and ( 15) of the previous paper (24)). Table VI shows the gU expansion at various values of the ring deformation coordinate. Since changes in S bEF only affect the ring angles without changing the symmetry, these expansions are symmetric (i.e., g!!!,) = g@ = 0). Furthermore, the expansions are very similar in all cases and the values of g’,” for the one-dimensional and two-dimensional models are identical. The kinetic energy terms gs5 for the SiH2 in-phase rock and g6, for the ring deformation are both for vibrations of amplitude smaller than that of the puckering so their values are expected to vary less. In fact, changes of only about 4 and 7% are calculated, respectively, for the extreme values. The G element for the deformation is given the name g66 although the two-dimensional calculation is done only with a 5 X 5 matrix. This is done only to distinguish among the three vibrations. Table VII lists the expansion coefficients for gss as a function of the rocking coordinate. These are given for both one- and two-dimensional calculations, for three

314

HARTHCOCK

AND LAANE

TABLE VI Kinetic Energy Expansion for the Ring-Puckering Vibration in 1,3-Disilacyclobutane as a Function of the Ring-Puckering Coordinate at Various Values of the Ring Deformation Coordinate”

-0.157 -0.105 0.0

WO.0

wo.2

0.105 0.157

0.807536 0.807271

-0.115771 -0.114907 -0.113195 -0.111504 -0.110665

-0.157 -0.105 0.0 0.105 0.157

0.803971 0.805337 0.807087 0.807536 0.807271

-0.095891 -0.097364 -0.100311 -0.103301 -0.104824

-0.094039 -0.090270 -0.085279 -0.083603 -0.084013

0.231755 0.220722 0.202687 0.189491 0.184563

-0.157 -0.105 0.105 0.157

0.803971 0.805337 0.807087 0.807536 0.807271

-0.088345 -".092332 -0.098250 -0.102713 -0.104380

-0.108219 -0.09RR52 -0.084213 -0.074194 -0.072919

0.244151 0.299853 0.208697 0.197999 0.196936

-0.157 -0.105 0.0 0.105 0.157

0.802971 0.805337 0.807087 0.807536 0.807271

-0.087526 -0.091306 -0.097787 -0.102813 -0.104764

-0.139961 -0.127647 -0.107168 -0.092193 -0.086822

0.314643 0.292140 0.254788 0.227499 0.217722

w-o.0

0.0

w-o.2

-0.047401 -0.046932 -0.048321 -0.052833 -0.056304

0.195041 0.193090 0.196776 0.200728

0.198210

(6) 6 + g44 x .

844

TABLE VII Kinetic Energy Expansion for the SiH2 In-Phase Rocking Vibration in 1,3-Disilacyclobutane as a Function of SiH2 in-Phase Rocking Coordinate at Various Values of the Ring-Puckering Coordinate” Ring Puckering Coordinate

w-o.0

w=o.z

0.27

0.402538

-0.207432

-0.100668

0.828702

-0.089663

0.18

0.402961

-0.147892

-0.101030

0.592540

-0.089064

0.0

0.403336

-0.101344

0.0

-0.088526

0.27

0.448641

-0.216963

-0.095367

0.652920

-0.100768

0.18

0.423274

-0.150604

-0.098772

0.491571

-0.095780

0.0

0.403336

-0.101344

0.0

-0.088526

Two-Dimensional

w=o.o

opo.2

0.0

0.0

(with ring-puckering)

0.27

0.494173

0.049030

-0.128008

0.012621

-0.100738

0.18

0.502792

0.040592

-0.130455

0.077944

-0.101902

0.0

0.510484

0.0

-0.132619

0.0

-0.102980

0.27

0.545642

-0.127229

-0.120124

0.440882

-0.115168

0.18

0.525724

-0.086187

-0.126938

0.325729

-0.110704

0.0

0.510484

-0.132619

0.0

-0.102966

%~ansions

0.0

and units take on the previously

described

forms (see Table III).

CALCULATION

315

OF KINETIC ENERGY TERMS

1,3- DISILACYCLDBUTAWE T;

1.08-

0 5

z

1.04 -

F 2

iI

io.90:r0.86 -

B

I

I -0.140

I

I -0.0699

0.00

I 0.0699

I

I 0.140

%EF FIG. 9. The dependence of the ring deformation kinetic energy term g,, for 1,3-disilacyclobutane as a function of &,sr at various values of the puckering coordinate x. Curve A: one-dimensional calculation at x = 0.18 A and o = 0.0; B: same as A except w = 0.2, C: two-dimensional calculation with the second coordinate x = 0.18 A and w = 0.2; D: same as A except w = 0.0; E: one- and two-dimensional calculations for x = 0 and o = 0.0 and 0.2.

different puckering coordinate values, and for two different w values. When the ring-puckering coordinate is zero (planar ring), these expansions are symmetric but rocking in the positive and negative directions is not equivalent for puckered ring structures. For puckered rings the value of w can be seen to influence the expansion terms demonstrating that the deformation, puckering, and rocking are all kinematically interrelated. The two-dimensional values differ significantly from the one-dimensional ones reflecting the presence of nontrivial YiZ values for Eq. (5). Figure 9 and Table VIII summarize the results for the ring deformation vibration showing how g6, varies with Sn,,. The change is slow and nearly linear. Curves A, B, C, and D are very similar and are calculated for a puckering value of 0.18 A. The value of o or whether a one- or two-dimensional calculation is performed makes little difference. However, as seen from curve E, which corresponds to a planar structure, the extent of puckering has a greater effect on the g6, term. The expansions in Table VIII are the same at positive and negative puckering coordinate values. The effect of puckering the ring on g,, and g& values can be seen from Figs. 10 and 11. The g,, values for the two-dimensional calculations are considerably higher than those for the one-dimensional calculations, reflecting the fact that g.,, (and Y,J differs significantly from zero. The g5, values are also sensitive to the parameter w; this parameter affects how far the SiHz groups are from the center of mass for a particular value of the puckering coordinate and thus influence the magnitudes

316

HARTHCOCK

AND LAANE

TABLE VIII Kinetic Energy Expansion for the Ring Deformation in 1,3-Disilacyclobutane as a Function of the Ring Deformation Coordinate at Various Values of the Ring-Puckering Coordinate” Ring-Puckering Coordinate

One-Dimensional 0.27

0.072958

0.111002

-u.159057

-0.517454

-0.383647

0.18

0.089694

0.153088

-0.069581

-0.300727

-0.269954

0.0

0.105925

0.195593

0.033545

-0.024489

-0.019831

0.27

0.073313

0.100154

-0.164501

-0.569630

-0.395399

0.18

0.089920

0.151000

-0.070953

-0.316933

-0.274739

0.0

0.105952

0.195593

0.033545

-0.024489

-0.019831

Two-Dimensional (with ring-puckering) 0.27

0.073872

0.070749

-0.121417

-0.382895

-0.339998

0.18

0.090215

0.130689

-0.049054

-0.246480

-0.217316

0.0

0.105952

0.195593

0.033545

-0.024489

-0.019831

0.27

0.073401

0.087508

-0.121448

-0.415914

-0.353557

0.18

0.089991

0.142683

-0.047888

-0.237154

-0.218242

0.0

0.105952

0.195593

0.033545

-0.024489

-0.019831

=Expansions and units take on the previously described forms (see Table II).

of the ra and &,/L&ocx vectors. For similar reasons the value of ge6 shows marked dependence on the puckering coordinate. The dependence on the value of Su,, can also be noted but the other effects are minor. We now direct our attention to the cross kinetic energy terms, the presence of which, in fact, is responsible for the differences between one- and two-dimensional kinetic energy calculations. Figure 12 shows the coordinate dependence of g,, with respect to both the ring-puckering (curves E and F) and to SiHz rocking coordinates (curves A, B, C, and D). The value changes markedly (18% at +0.3 A) with the puckering but it stays nearly constant (22%) as SRocK varies. Table IX lists the expansion coefficients for g,,. The top half of the table represents the expansion in terms of the puckering coordinate for various values of Snack; the bottom is for expansion in terms of SRoCK at various values of the puckering. If desired, these data could be combined to produce the g4, polynomial expansions as a function of both coordinates. The same kind of symmetry properties as before are involved here. That is, when both coordinates are zero, the expansions are symmetric. Motion from the origin destroys this symmetry. In the expansions it is to be noted that the bending parameter o has a nontrivial influence on the coefficients. Thus the relative positions of the bending, rocking, and puckering all affect the value of g,,. Figure 13 presents the variation of g46 vs the ring-puckering coordinate (curves A, B, C) and SDEF (curves D, E, F). When the ring-puckering coordinate is zero

CALCULATION

i

I

I -0.24

I

I

317

OF KINETIC ENERGY TERMS

I

-0.12

I

I

0.00

X, RIRG

I 0.12

I

I

I

0.24

TUCKERING

FIG. 10. The dependence of the in-phase SiH2 rocking kinetic energy term gs5 for 1,3-disilacyclobutane as a function of the puckering coordinate x. Curve A: one-dimensional calculation at SRoCK= 0.233 and o = 0.0; B: same as A except o = 0.2; C: same as B except Snocx: = 0.0, D: same as A except twodimensional; E: same as D except o = 0.2; F: same as E except .SRocK= 0.0.

-0.24

-0.12

0.24

X, R,NG”‘~~~KER,R~12 FIG. 11. The dependence of g,, for 1,3~isilacyclobutane as a function of x. Curve A: essentially coincident for one- and two-dimensional at S nar = -0.105 and w = 0.0 and 0.2; B: essentially coincident for one- and two-dimensional at SoEF = 0.0 and w = 0.0 and 0.2.

318

HARTHCOCK

-0.27

-

7 -0.29 0x

-

AND LAANE

1,3- DlSllACYClOGUTANE

z CI) -0.31 -

-0.33

I

I

I

I

I

I -0.310

I

I

I

0.12

I

0.24

PUCKERING

I

-0.156

I

I

0.00 x, RING

I

I

I

-0.12

-0.24

I

I

0.00

1

I

0.156

I

0.310

‘ROCK FIG. 12. The interaction (puckering-rocking) kinetic energy term g4s for 1,3-disilacyclobutane as a function of x and SRocK. Curve A: g,, vs S s~~x at x = 0.0 and w = 0.2; B: same as A except x = 0.18 A and w = 0.2; C: same as B except o = 0.0; D: same as A except o = 0.0; E: g45vs x at SRocK = 0.233 and 0.0 and w = 0.2; F: same as E except w = 0.0. TABLE IX Kinetic Energy Expansions for the Cross Term in the Kinetic Energy, gd5, in 1,3-Disilacyclobutane a Function of Both the Ring Puckering and of the SiH2 In-Phase Rock Coordinates”

w-o.0

w-o.2

as

0.340

-0.325124

-0.606102

0.662682

0.125593

-0.622964

0.228

-0.328297

-0.408679

0.668181

0.084726

-0.627246

0.0

-0.330826

0.672508

0.0

-0.630624

0.0

0.359

-0.325125

-0.204764

0.526302

0.027020

-0.338632

0.240

-0.328298

-0.137774

0.532266

0.018257

-0.341277

0.0

-0.330826

0.537076

0.0

-0.343457

0.0

Ring Puckering Coordinate

w-o.0

UFO.2

0.27

-0.285161

-0.414295

0.428150

0.547017

0.18

-0.309693

-0.303620

0.459129

0.403345

0.547957

0.0

-0.330833

0.485457

0.0

0.588981

0.27

-0.293507

-0.140894

0.376654

0.161176

0.467527

0.18

-0.313781

-0.102538

0.433774

0.129626

0.537572

0.0

-0.330833

0.485457

0.0

0.588927

*expansions

0.0

0.0

and units take on the previously

described

form (see Table IV).

0.500722

CALCULATION

319

OF KINETIC ENERGY TERMS

1,3-DISILACYClOBUTANE

I

I

I

I

-0.12

-0.24

0.00

X, RING

I

I

-0.0699

I

I

0.24

PUCKERING I

I

I

- 0.140

I

0.12

0.00

I

0.0699

I

0.140

%EF FIG. 13. The interaction (puckering-deformation) kinetic energy term g,, for 1,3-disilacyclobutane as a function of x and Sosr. Curve A: g,, vs x at So,, = -0.105 and w = 0.0; B: same as A except Soar = 0.0; C: same as A except w = 0.2; D: g,, vs S out at x = -0.18 A and w = 0.0, E: same as D except x = 0.18 A and w = 0.2; F: same as D except x = 0.18 A.

(planar ring), the motions for all of the atoms for the puckering motion are exactly perpendicular to the motions of the atoms for the in-plane ring deformation. Consequently, g4, is then zero. When the amount of puckering is increased g& takes on small values. The sign of this term is the same as the sign of the puckering coordinate so that g4 changes sign in going through the origin. This can be demonstrated by examining the component of the derivatives used in Eq. (9) for calculating the Yik.

& _ aa,. _--+?!kj+%kk. %i

a4i

a4i

(19) &7i

When the ring-puckering coordinate is positive, the derivatives for the ring atoms during the puckering motion &,/ax, &,/ax, &,/ax, and da4/dx are positive while &/ax, &,/ax, a&/ax, and dcJdx are negative (Fig. 2 in the previous article (24) is useful for reference). For the ring deformation db,/dSDEF, daz/dSDEF, and all &,/d&r are negative while db3/dSDEF and aa4/aSDEF are positive. For a negative ring-puckering coordinate the ring-puckering derivatives ab,/ax, da,/ax, ab3/ax, and aa,/axall change sign as well as all &,/dSr,EF for the deformation. Consequently, the YIs values calculated from Eq. (9) have the same magnitude but different sign when calculated for the different directions of the puckering coordinate. The sign and magnitude of g,$, are inversely related to Y,J. Table X lists

320

HARTHCOCK

AND LAANE

TABLE X Kinetic Energy Expansions for the Cross Term in the Kinetic Energy, ga6, in 1,3-Disilacyclobutane as a Function of Both the Ring-Deformation and Ring-Puckering Coordinates” s

g~~+xlo-l)

-0.157

-0.105 0.0 0.105 0.157

0.166454 0.153099 0.125749 0.097367 0.082719

-0.583671 -0.542310 -0.457778 -0.368260 -0.320586

0.840819 0.773522 0.632050 0.475810 0.389744

-0.157 -0.105 0.0 0.105 0.157

0.092810 0.079357 0.051747 0.023017 o.ooa161

-0.481948 -0.439456 -0.352731 -0.261116 -0.212431

0.918282 0.844672 0.698379 0.548335 0.469970

DEFa

w=o.o

w=o.z

g$XlO-l)

g;;+xlo-l)

Ring-Puckering Coordinateb

o=o.o

-0.27 -0.18 0.0 0.18 0.27

0.258518 0.200824 0.0 -0.200824 -0.258518

-0.573289 -0.432454 0.0 0.432454 0.573289

-0.912086 -0.724637 0.0 0.724637 0.912086

-0.084901 -0.318302 0.0 0.318302 0.084901

0.387136 -0.264254 0.0 0.264254 -0.387136

w=o.2

-0.27 -0.18 0.0 0.18 0.27

0.080365 0.073852 0.0 -0.073852 -0.080365

-0.576397 -0.436280 0.0 0.436280 0.576397

-0.929564 -0.781642 0.0 0.781642 0.929564

-0.064121 -0.319425 0.0 0.319425 0.064121

0.942173 -0.164850 0.0 0.164850 -0.942173

'The expansions

for g46 as a function of the ring-puckering

coordinate have the form:

(5) 5 (1) (3) 3 g46(x) = g46 x + g46 x + g46 x b

The expansions

for g46 as a function of the ring deformation

coordinate

is analogous

f~ that for g45.

(See Table IV).

the g,, kinetic deformation

energy terms expanded in terms of both ring-puckering and ring coordinates. For the former expansion the only nonzero terms are the

asymmetric ones. These results are in agreement with symmetry principles which dictate that vibrations of different symmetry species should have no cross kinetic energy terms for small-amplitude vibrations. For a planar ring of DZh symmetry the B,, puckering should not interact with the A, ring deformation. The final cross kinetic energy term to be considered is g,, for interaction between the in-phase SiHz rocking and ring deformation. Table XI shows the resulting expansions in terms of both deformation (top half of table) and rocking (bottom half) coordinates for various values of the ring dihedral angle, of SRocK, and SDEF. As should be evident g,, is generally small but changes dramatically as a function of the puckering and rocking coordinates. Even the sign of the cross term is a function of the structure. The value of SDEF, which only affects the internal ring angles of the molecule, has a considerably smaller affect on gSa. As with g&, when the ring is planar and when S,oc, = 0 so that the molecule has DZh symmetry, gs6 is zero reflecting the fact that the B,, SiH2 rock does not interact with the A, ring deformation. 2. Three-dimensional results. A three-dimensional calculation was carried out

CALCULATION

321

OF KINETIC ENERGY TERMS TABLE XI

Kinetic Energy Expansion for the Cross Term, gs6, in 1,3-Disilacyclobutane as a Function of Both the SiHz In-Phase Rock and Ring Deformation Coordinates” Dihedral Angle

g;;)(xlo-l)

g;;)(xlo-l)

g~;)(x10-2)

g;;)(xlo-3)

g;;)(x10-3)

-5.OO

0.3406 0.2277 0.0 -0.2277 -0.3406

-0.017460 -0.031827 -0.060719 -0.088550 -0.101587

0.025631 0.027021 0.029525 0.031567 0.032378

-0.116532 -0.109775 -0.097750 -0.078500 -0.070138

0.418707 0.442660 0.486478 0.523002 0.537800

-0.157412 -0.154718 -0.147596 -0.138574 -0.133484

0.0

0.3406 0.2277 0.0 -0.2277 -0.3406

0.042475 0.028639 0.0 -0.028639 -0.043475

-0.003355 -0.002260 0.0 0.002260 0.003355

-0.020489 -0.013812 0.0 0.013812 0.020489

-0.057940 -0.039084 0.0 0.039084 0.057940

-0.006689 -0.004517 0.0 0.004517 0.006689

0.3406 0.2277 0.0 -0.2277 -0.3406

0.279668

-0.139760

0.272598

30.00

-0.130652 -0.137739 -0.133716 -0.130955

-0.298020 -0.270423 -0.209743 -0.144337 -0.110839

-0.036870 -0.010738 0.043907 0.099338 0.126401

-0.938660 -0.922584 -0.878028 -0.819332 -0.785492

Dihedral Angle

%e

S ROCK

S

DEF

-5.00

0.157 0.105 0.0 -0.105 -0.157

0.0

0.157 0.105 0.0 -0.105 -0.157

30.00

0.157 0.105 0.0 -0.105 -0.157

expansions

0.254605 0.232361 0.219953

g:;)(xlo-l)

are analogous

g;;)(xlo-3)

g;;)(xlo-2) 0.123435 0.124059 0.125187 0.126163 0.126598

0.098696 0.097027 0.101512 0.106264 0.108756

-0.163269 -0.164016 -0.165350 -0.166490 -0.166992

0.092832

0.0 0.0 0.0 0.0 0.0

0.125029 0.125640 0.126756 0.127737 0.128182

0.0 0.0 0.0 0.0 0.0

-0.166454 -0.167182 -0.168497 -0.169637 -0.170146

0.0 0.0 0.0 0.0 0.0

0.232132 0.239751 0.254605 0.268957 0.275942

0.078803 0.079808 0.081372 0.082361 0.082645

-0.056297 -0.057726 -0.060719 -0.063921 -0.065611

-0.313241 -0.322769 -0.341138 -0.358614 -0.367018

-0.083303 -0.084402 -0.086143 -0.087296 -0.087658

0.095473 0.101027 0.106992 0.110148

-0.302516 -0.313141 -0.333987 -0.354288 -0.364226

to g45 and g46 (see Table I" and X).

for the three low-frequency motions of 1,3-disilacyclobutane and the resulting individual gij elements are given in Table XII. The procedure used was analogous to that previously described for the two-dimensional computation except that the matrix to be inverted has one row and column in addition to those in Eq. (5). Thus the additional elements Xij, Xzj, Xj3, Yij, Yz3, and YJ3 must be calculated. The other Xij and Yj remain the same. In the principal axis coordinate system the G matrix takes on the form g,, 0

0 g,,

0 0

0 0

0 0

0 0

o

o

o

g45

&?SS

gS6

o

o

o

g46

gS6

g66

The gij are defined as before. From Table XII it can be seen that the selection of the dimension, the coordinates, and the values of the coordinates all are critical

322

HARTHCOCK

AND LAANE

TABLE XII Kinetic Energy Terms for 1,3-Disilacyclobutane for the Ring-Puckering, SiH2 In-Phase Rock, and Ring Deformation Using One-, Two-, and Three-Dimensional Models” Ring-Puckering Coordinate

0.18

844

0.0 -0.18

0.18 gs5

0.0 -0.18

0.18

866

0.0 -0.18 0.18

845

0.0 -0.18 0.18

g46

0.0 -0.18 0.18

'56

0.0 -0.18

=gDEF

8

ROCK

One

Two

Dimensional

Dimensional

0.233 0.0 0.228 0.0 0.233 0.0

0.007737 0.007738 0.008074 0.008071 0.007743 0.007738

with deformation 0.007746 0.007744 0.008074 0.008071 0.007747 0.007744

0.233 0.0 0.228 0.0 0.233 0.0

0.417530 0.423275 0.398067 0.403336 0.418220 0.423275

with puckering 0.518592 0.525725 0.503592 0.510484 0.518986 0.525725

0.233 0.0 0.228 0.0 0.233 0.0

0.089784

0.233 0.0 0.228 0.0 0.233 0.0 0.233 0.0 0.228 0.0 0.233 0.0 0.233 0.0 0.228 0.0 0.233 0.0

0.089920

0.105952 0.105952 0.090061 0.089920 ___ --_ _--

_-__-_-_-__ --_ -__ _-_____

with puckering 0.089881 0.089991 0.105956 0.105952 0.090108 0.089991

with rock 0.009610 0.009610 0.010214 0.010215 0.009609 0.009610 with deformation 0.421703 0.426459 0.398145 0.403336 0.420454 0.426459 with rock 0.090682 0.090597 0.105973 0.105952 0.090542 0.090597

Three Dimensional

0.009612 0.009612 0.010214 0.010215 0.009610 0.009612 0.523290 0.529334 0.503676 0.510484 0.521541 0.529334 0.090695 0.090608 0.105973 0.105952 0.090551 0.090608

-0.0311644 -0.0313781 -0.0328304 -0.0330833 -0.0311169 -0.0313781

-0.031248 -0.031445 -0.032831 -0.033083 -0.031168 -0.031445

0.0008670 0.0007385 0.0001584 0.0 -0.0006076 -0.0007385

-0.000366 -0.000336 -0.000034 0.0 0.000304 0.000336

0.0194530 0.0169845 0.0028639 0.0 -0.0142215 -0.0169845

0.020641 0.018083 0.002975 0.0 -0.015208 -0.018083

- 0.0 and w = 0.2.

for determining the value of g. For a planar structure Y,3, Yzs, and, therefore, g+j and gS6, are zero. Consequently, the two-dimensional g,, and g,, (puckering and rocking) values are identical to the three-dimensional, g6, values are equal to the three-dimensional member and the g4, values are also identical. For either puckered molecules or those for which S,,, # 0 symmetry no longer holds, the above equivalence no longer holds exactly but the members remain very similar. C. Cyclopentene A number of kinetic energy calculations have previously been carried out on cyclopentene and three of its deuterated species (10). These have been concerned with the g4, terms for the ring-puckering motion involving various amounts of CH2 rocking. Recently a two-dimensional potential energy calculation has been done (16) involving also the ring twisting vibration. Thus it was necessary to calculate the gS5value for the twisting motion. Table XIII lists the reduced masses (P = l/

CALCULATION

OF KINETIC ENERGY TERMS

323

TABLE XIII Reduces Masses for the Ring-Puckering and Ring-Twisting Motions of Cyclopentene and Several Deuterated Derivatives

117.88

119.78 dD=C”-CH2-CH2-C;12

24.02

28.56

CD=CD-CD2-CH2-CH2

138.95

35.08

I ICD=CD-CD2-CD2-CD2

184.51

36.13

g) for the puckering and twisting of a planar cyclopentene molecule calculated using the models of the previous paper. Since the g4, terms are exactly zero for a planar structure and very small for puckered structures, the one- and two-dimensional results are essentially identical. IV. CONCLUSION

Based on the models of the previous paper we have carried out extensive calculations to determine the kinetic energy terms of 3-phospholene and 1,3-disilacyclobutane and reduced masses for cyclopentene. We have first shown that the g term (reciprocal reduced mass) is not independent of coordinate and often changes substantially with it. Other structural changes affect the g values also. Most importantly we have demonstrated that the kinetic energy expansions for a two- and three-dimensional analysis differ significantly from those for one-dimensional systems. Although this was recognized by Wilson et al. (3~9, many researchers tend to ignore this fact. The differences among one-, two-, and three-dimensional calculations may be compensated for by carrying out the transformations described by Eqs. (16) and (17). However, this is a rather cumbersome procedure. The methods we have described in this paper will directly yield the correct g values for the problem of interest, no matter what the dimension. Since many spectroscopic analyses lend themselves better for partial vibrational analyses than for complete studies, the techniques of this paper should often prove useful even for smallamplitude motions. When the vibrations considered are large amplitude, there is no choice but to use this approach. In previous two-dimensional analyses of large-amplitude motions the cross kinetic energy terms (e.g., g45) have been neglected. The results of this paper show that such assumptions are in general not justified and, moreover, will lead to erroneous values of the gii values. While the calculations described in these two papers may seem rather tedious, they can be carried out very quickly for virtually any type of vibration once the heart of the computer program (which calculates the derivatives, Xij terms, Y,

324

HARTHCOCKANDLAANE

terms, and moments of inertia and then inverts the matrix) has been set up. All that is necessary is to describe the motion of interest in a vectorial fashion (see, for example, Table I of the previous paper for a description of the ring-puckering motion). Since most vibrations are much less complicated than those described in these papers, the description of the motion is almost trivial. ACKNOWLEDGMENT This work was supported by the National Science Foundation. RECEIVED:

July 17, 1981 REFERENCES

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