Structural parameter relaxation and pucker mode—internal mode coupling in the four-ring molecules cyclobutanol, cyclobutylfluoride and cyclobutylchloride

Specmchimica Printed

Acta, Vol. 42A, No

213. pp. 259-273,

1986.

0584-8539186 $3.00 + 0.00

0 1986Pergamon Press Ltd.

in Great Britain.

Structural parameter relaxation and pucker mode-internal mode coupling in the four-ring molecules cyclobutanol, cyclobutylfluoride and cyclobutylcbloride R. GUNDE,* T.-K. HA and H. H. GtiNTHARDt Physical

Chemistry

Laboratory

ETHZ,

ETH Zentrum,

(Received 8 July

8092 Ziirich, Switzerland

1985)

Abstract-The present work is directed towards the investigation of the pucker-internal rotation modes of cyclobutanol and cyclobutanol-OD by the aid of a semirigid two-dimensional model. Based on spherical four-ring coordinates and general geometrical relations expressing wagging, twisting and rocking coordinates for CHZ and CHX groups, experimental rotational constants of cyclobutanol, cyclobutylfluoride and cyclobutylchloride in the ground and excited pucker states are analyzed in terms of a two-dimensional fourring pucker model with structural relaxation of CHZ and CHX groups concerted with the ring puckering angles. The rotational data are found to be satisfactorily described by a one-dimensional ring pucker model including structural relaxation of the rocking angles of the CHX and the CHI groups. Then results of extended SCF computations (4-31G level) of the puckering-internal rotation potential of cyclobutanol are presented. These emphasize again the importance of structural relaxation of bond length and rocking type coordinates of the corner groups concerted with the puckering angle and serve as a basis for numerical solution of the two-dImensiona pucker-internal rotation problem within the framework of a semirigid model. Predicted transitions and energy eigenstates will be correlated with far i.r. spectral data of cyclobutanol and cyclobutanol-OD observed in the 30-300 cm _ I range and will be interpreted in terms of an alternative assignment, whose relation to an earlier analysis will finally be discussed.

from kinetic coupling of the pucker mode with infinitesimal group modes on the one hand and effects stemming from structural relaxation-which should be classified as predominant coupling through the potential function-involves an “exact” solution of the pucker motion-vibration-rotation problem of fourring systems [4,5]. Hints for pucker induced structure relaxation have been obtained from ab initio calculations of the puckering potential of cyclobutane. Earlier quantum chemical calculations for this molecule failed to produce a double minimum potential and it has been found that inclusion of relaxation of the CH2 rocking coordinates leads to a double minimum potential

1. INTRODUCTION

It has been suggested on several occasions that pucker type distortion of four-ring type molecules implies concerted variation of other structural parameters, in particular of corner groups like CH2 and CHX, i.e. internal coordinates like bond angles, wagging, twisting and rocking coordinates and possibly bond lengths determining the geometry of such groups vary as functions of the pucker angle. This relaxation of structural parameters is thought to occur in such a manner that the electronic potential function always is minimal with respect to variation ofall other structural parameters for any given puckering angle. Structural relaxation appears to be a general phenomenon in molecules featuring large amplitude motions. This naturally gives rise to the problem of the distinction between effects eventually caused in spectra by relaxation type variation of internal coordinates and effects originating directly from dynamical, in particular vibrational, effects. Both experimental and theoretical arguments might be useful in the solution of this largely open question. In the specific case of four-ring molecules dynamic interaction of the pucker motion with wagging, twisting and rocking type modes have at times been considered for interpretation of isotope shift [l-3]. A satisfactory discrimination between effects arising

[6,71. Based on rotational constants of four-ring molecules in excited states of the puckering mode, BALTAGI et al. made attempts to derive information on relaxation of the rocking coordinates [S-J. These authors used spherical coordinates to describe the kinematics of semirigid four-ring models and demonstrated that for many four-ring molecules a one dimensional treatment of the pucker motion is justified [9]. Furthermore they found for l,l-diffuorocyclobutane that reproduction of the rotational constants of the ground and six excited pucker states from a semirigid molecular model requires structural relaxation of the CF, group. However the available data did not allow them to derive information on relaxation of CH, groups, since the rotational constants proved to be too insensitive with respect to relaxation of their rocking coordinates. In this paper work will be reported aiming towards elucidation of the interaction of the four-ring pucker

Dedicated to Professor R. C. LORD in honor of his 75th birthday. *Present address: Technical Chemistry Laboratory, ETH Zentrum, 8092 Ziirich, Switzerland. t Author to whom correspondence should be addressed. 259

R. G~JNDE et al.

260

mode with other modes, either infinitesimal like typical CHs group modes or other large amplitude modes like OH torsion. In Section 2 the internal coordinates required for the description of the kinematics of fourrings and their corner groups will be presented. These then will be used in Section 3 for fitting the rotational constants of cyclobutylfluoride, cyclobutylchloride and cyclobutanol in excited states of the pucker mode as a function of the spherical ring coordinates 8,4 and the angle coordinates of the corner groups. It is found that the coordinate 4 % n/4 is practically independent of the pucker state and that only the rocking type coordinate of the /KHz groups and the CHX (CHOH) group experience signikant relaxation. Section 4 is devoted to the results of extended a6 inifio calculations at various levels for cyclobutanol, in particular the pucker-internal rotation potential function. Furthermore the relaxation of a number of internal coordinates as predicted quantum chemically is discussed. The nuclear motion eigenstates of a semirigid model of cyclobutanol with relaxation described by the puckering angle and the internal rotation angle of the OH group are considered in Section 5. New spectral data will be reported for the 300-30 cm- ’ range taken with a resolution of = 0.3 cm-‘. A comparison of the predicted electrical dipole transitions with experimentally determined far i.r. spectra of cyclobutanol and cyclobutanol-OD leads to a suggestion for the assignment of the far i.r. spectra.

In the notation of this paper the semirigid model will be of the type Cs,,-C,; the comer group CXH is assumed to reduce the covering symmetry group to G (0,#) = C,. Figure 1 gives a schematic view of the semirigid four-ring model, the frame coordinate system ef and the spherical coordinates 0,& In each of the corners Ci, Cs, Cs, C, a local coordinate system is introduced as depicted by Fig. 2 for the comer Ci . The local coordinate system consists of three orthogonal unit vectors eil, eil, e;s defined as follows for C1: e;s-normal defined by the two bond vectors to the adjacent ring nuclei (Xa+i,Xn-iX e&negative median unit vector spanned by the two bond vectors to the adjacent ring nuclei, obviously e;s I eis, e;i = e;s x e;s (tangent in comer i). In Table 1 coordinate vectors of the four-ring nuclei and unit bond vectors as expressed analytically by the spherical four-ring coordinates 0, r$~are given. From this information the bases of the local coordinate 6

2. INTERNAL COORDINATES FOR SEMIRIGIDFOURRING MODELS

In this work the spherical coordinates 0, Q suggested by BALTAGI et al. for the assessment of the kinematics of a semirigid four-ring model with two finite degrees of freedom and constant bond lengths L1, LZ will be used [9].

Fig. 1. Illustration of sphericalcoordinates 8.4 for semirigid four-membered ring models.

Fig. 2. Projected view of the local coordinate bases for comer groups CX2 (comer 1). e;,, e;,, eij denote, respectively, tangent, inner medii and normal unit vector lccaked at ring atom i = 1. Analog definitions hold for the coordinate basis defined by group C,X,, X,1, cf. Eqn. (3).

Pucker-internal Table 1. Relevant coordinate vectors of C,.C, systems

Corner 1 2 3 4

4

four-ring

e$ 0

0

L,sine sin 4

L,cosB

0 - u(+)L2 sin et

0 L, cos e

C,C*

to coincide

Angular motion of the (quasi) rigid CX2 groups may now be conceived as a rotation about the system {e’) (a,, R,, Q,, respectively, denote finite wagging, twisting and rocking angles)

{nimiti} ={eflelae13}D{eb,R,} x D{t%,Q,}

Bond unit vector $ GG GG GG

any Ci may be chosen

(llimiti} = (ei,ej2e&}.

Coordinate vector 4

L,sinl?cos+ 0 -L,sinfIcos+ 0

261

mode coupling in Cring molecules

sin e sin4 -u(+)sine sin e sin $J -u(4)sine

-sinecos4 -kin ecosd sin ecosd + lsin ecos4

come come COST come

D{&,Q,},

(1)

where D{ e,4} denotes the orthogonal matrix representing a rotation by angle 4 about the rotation axis e [IO, 111; it may conveniently be taken in the form (- denoting transposed, 1 3 x 3 unit matrix) D(e,#}

=cos41+(1

Molecule fixed coordinate system, cf. Fig. 1. z ==L,/L,. IBond unit vector e12 related to Cl-CI bond, etc

-cos4)ee

l

Tu(c#J) = (1 -Pcos2#‘~;

+ sin4r;fz:

systems may be expressed by the frame basis after elementary though lengthy calculation

group, m,--outer median defined by the two bond vectors of the group, tj = lli X Illi. For local Czv symmetry the two systems localized at

(2)

:;I.

Restriction of the angular motion to small wagging, twisting and rocking angles R,, R,, R, leads, by Taylor expansion of Eqn. (1) with center R, = R, = R, = 0 to

{nimrtij={e~j[l+ The results of this calculation, i.e. the matrices Ti, i = 1, 2, 3, 4, are collected in Table 2. For description of (infinitesimal) wagging, twisting and rocking motion the corner groups CH2 will be considered as (quasi) rigid systems, which themselves define local coordinate systems ni, mi, ti, defined as n,-normal spanned by the two bond vectors of the

ri

[i:

-iI

-zllj.

(3)

Hence the (infinitesimal) wagging, twisting and rocking coordinates determine the skew symmetric part of the rotation matrix in Eqn. (3). The latter will be applied in this work to solve two problems: (i) computation of wagging, twisting and rocking coordinates of nuclear configurations obtained from ab initio calculations by expressing the two coordinate systems {ej} and { ni m, ti} by the frame basis {el) = {e’} Ti,

Table 2. Basis systems localized in corners 14 of C,,&

{ni} = {e’} Ui.

four-ring systems (cf. Figs 1, 2)

Component Corner

Basis vector

1

e; e;

2

e\ 4

e;

e3

3

e; e2’

e; 4

e; e2 e;

4

4

4

(A- 1)cos 4/D* (44 (sin 4 + u(4))lD1(4,4 - (1 + I)sin e c0s414 (e,fj.2) (sin 4 -u(4))sin O/o,(O, 4, A)

(U (4) + sin 4k0s e/D, (e, 4.4 -1 0

0 (A- 1)cos d/D, (4 4 (l+I)sinecos 4/D,(e,&A) - (I+$)+ sin d)cos O/D, (e, 4, I) 1 0 0

(1 -I)cosecos~iD,(e,~,I)

(u(4)+Lsinq5)sinecosd/D3(e,+,1)

0 -sinesin+/D,(e,qj)

-cOse,&e,d)

-~0seiDde,4)

sine sin c#I/D,(B,c$)

-(sin 4 + u(+))lD, (4, A)

0

(sink-u(~))sine/D,(e,~,1) 2cose/D2(0,4J,4 (1 - 4~0s e cos 4/D, (e, 4,1)

(~(4)+Asin1#1)sinecos4/D,(ecp,A)

0 u($)sin elD,(e,h4 cos e/D, (e, 44

0 -cos e/D, (e, 44 u(4)sin e/D, (e, 4 4

D, @,A) = (2(1 -Icoszq5 + u(d)sin $)}“2; &(e, 4, A) = {2sin2e(l +2cos2+ -u(&sin~)+&os* ejl’*; Ds (e, 494 = {cos* O[ (u(d)+ sin 4)’ + (1 -I)%os2qb] + (u(4) + kin &2sin2ecos2 4}“‘; D4(e, 4)= {1 -sin’ e COS?qb)"*; D5(0, 4, A) = (1 -lZc~~Z~sin20}1~2; u($) = I(1 -22c~~2~)112~.

262

R. ch_JNDE

et al.

(4)

latter are collected in Table 3. The coefficients for the rotational part follow immediately from the equation (m, n = 1,2, 3)

Comparison with Eqn. (3) for small distortions yields directly the angular coordinates R,, R,, R, (radian). (ii) expression of the kinetic energy contribution of the corner groups CX2 by expressing coordinates and velocities of the X atoms taken with respect to the center of mass basis {efCM}. Choosing the latter parallel to {ef} and denoting the center of mass coordinate vector with respect to the origin of ef as XL4 one has the general relation for ring nuclei (is[L4])

where the nuclear coordinate vectors X, refer to the frame system with origin in the center of mass. This implies that the inertia tensor appears in semidiagonal form and has to be diagonalized for derivation of the rotational constants Al, Al, A,, which then are functions of 8 and Q,and the relaxed parameters. In Table 4 a survey is given of the structural parameters which were varied in the fitting process; all other parameters were kept

Elimination of {e’} yields {b} = {ef)T;‘U1.

fixed.

X/CM = x:--x;.,

(5)

for CX2 or CXY group [B stands for the 2nd term in the matrix of Eqn. (3)] YlcM = XjcM +Ti(O, 4)(1 +R)Y~m~‘),

3.1.1. Computational details. The fitting used in this work aimed at minimization quantity

(5’)

Q=

Y {Q* 4) denoting the coordinate vector of nucleus

Xr of the group CiXz, referred to (‘4 mi ti}. By the last two equations and their time derivative the rotational constants and the internal kinetic energy may be expressed as a function of the internal coordinates 0,4, R,, R,, R, and their time derivations. If R,, R,, R, are approximated as functions of 8 alone (structural relaxation), one has R, = dR,/dB f$i = w, t, r). The foregoing equations simplify considerably for special cases of practical interest, e.g. for 1 = 1 (equilateral fourring) and 4 = n/4 (one dimensional pucker problem) [9].

(d$l (4 - A,,,l>“z,

minimization was actually effected by a modified golden cut technique with variable interval boundaries. The golden cut technique consists in choosing initial intervals for the individual parameters, partitioning these intervals by two inner points obtained by the

Table 3. Structuralparametersof Czv-C, four-ringsystems Parameter*

C&OH t

L, (Fig. 1)

HYDROXY-,

OF ROTATIONAL

CHLORO-

CONSTANTSOF

AND FLUOROCYCLOBUTANE

In this section the foregoing formulae will be used to analyze the rotational constants of progressions of excited states of the pucker mode by fitting rotational constants obtained from microwave spectroscopy as functions of the ring pucker angles tI and 4 and possibly the CX1 (CXY) group coordinates R,, Cl, and 0,. 3.1. Classical kinetic energy matrix In principle the kinetic energy matrix (g,,,,,)of the rotation-internal motion problem may be obtained analytically from the coordinate expressions collected in Table 2. Since the equations for the elements g,,,,,(g, I$, a,, a,, a,) of the internal kinetic energy expression are quite complex, none of these will be presented here. Instead the following procedure will be adopted: (i) the coordinates Q,, a,, f2, will not be considered as independent dynamical coordinates but as structural parameters relaxing with 0 in a concerted manner, R = Q(e). (ii) the coefficients g,.(O, 4, Q) will be calculated numerically for typical structural parameters. The

Molecule W-W 3

C&F 0

1.525

1.525

1.55 1.085 112

1.525 1.55

L2 (Fig. 1) 3. ANALYSIS

process of the

4 (WAH)) a,( %(HC,W) 4(4C,W)

1.085 112 1.100

1.55 1.085 112 1.100

26, ( t (HGH)) L(d(CX))

110 1.42

110 1.775

1.10

IZH (d(C2W)

1.10

97 137 0.965 109

262 ( * W2X))

* (m2, ec,-dll don cCOH

1.100 110 1.37

114 135 -

1.10 114 132 -

l Bond length & bond angles ‘. tRef. [13]. t,:; [;;I. e. . 1)Angle between median at atom C1 and bond unit vector from C2 to substituent X.

Table 4. Independent fitting parameters for four-ring molecules Parameter (CH,),CHOH

: *

1: (m2,

nr

ec,x)

(CH,),CHCI

(CH&CHF

1:

z:

+* ;:

+*

;:

0,

+

+

+

%

+

+

+

l

Relevant fit parameters.

Pucker-internal mode coupling in Cring molecules golden cut of the interval and calculation of Q on the 4’ (4’ if rocking angles are included) parameter points. If Q has a single minimum on the initial interval, the computation localizes its position in subintervals of each initial interval. The process is repeated using these subintervals until each interval length lies below a preselected value. Starting from the structural parameters given in Table 3 and values of 0 E (90°, lOO”), 4~ (40”, 45”) and Q(m,, ec,_x)c (Cl, - 5, R2 + S’), where R, denotes the value of 4: (m2, ec2_,) in Table 3, the fitting process was carried out for several sets of initial values. By this it should be investigated whether the fit leads to unique values of the parameters. Final values and rms estimates of the parameters were obtained as mean and rms value of the middle points of all final intervals obtained from different choices of initial intervals; the number of such choices is given in Tables 5, 6, 7 (last column). 3.2. Results First it should be mentioned that no substitution structure data are available for cyclobutanol and cyclobutylfluoride. For cyclobutylchloride substitution coordinates of H,, C, and Cl are available. Further structural parameters were derived from a fit of over 20 rotational constants [12]. A number of the ring parameters were subsequently transferred to the cyclobutylfluoride structure. Therefore structural parameters collected in Table 3 cannot be rendered more reliable by fitting the rotational constants. Nevertheless the accuracy of the reproduction of the rotational constants from assumed structural parameters and from fitted parameters 0, 4 etc. supports the essential correctness and the transferability of the former parameters. It therefore may rather safely be assumed that all three molecules belong to the 1 z 1 type. It can be said that the structural parameters 0,# and +Z(m,, ec,_x) are well determined rather generally by observed rotational constants. However, the parameters L?,, fir and R, are not well determined. Therefore neither value nor their dependence on the pucker state can be derived unambiguously from rotational constants. Quantum chemical calculations of the pucker potential function indicate that relaxation of the rocking angle R, (concerted with 0) is relevant for a reproduction of a qualitatively correct pucker potential function. 3.2.1. Cyclobutanol. For cyclobutanol only two states of the pucker mode have been observed [13] and the rotational spectrum is complicated further by the internal rotation of the hydroxyhc group. Up to now the rotational spectrum of only one OH group conformer has been assigned and identified as belonging to the tram conformer characterized by the internal rotation angle r = 0, in which hydroxyhc and a-H atoms are in the tram position. Table 5 shows that 8, 4 and +z((m,, ec2,) are relevant fitting parameters for two pucker states, with @Jbeing nearly constant near 44.25”. The angle a2

m

+ 3

263

Pucker-internal

= <(m,, ec,o) decreases from 138” by approx. 0.5” going from the ground state to the first excited pucker states. If in addition the rocking coordinate of the /ICH, groups is admitted as a fitting parameter, its values R,, (0, = 96.2”) and R,, (0, = 94.7”) are nearly the same for both pucker states but significantly different from zero ( - 3.7”), correspondingly the values of 0, 4 and Rz change slightly. However, the reproduction of the rotational constants is not significantly improved and remains within some 10-20 kHz. Admission of R, or R, as relaxing parameters instead of R, yields practically vanishing values for both R, and R, and does not lead to improvement of observed rotational constants. 3.2.2. C~dobutykhloride. For this molecule the ground state and two excited pucker states have been studied [ 121. In Table 6 the results of the analysis of the rotational constants in terms of ($4, + (mz, ecZc,) and R, (corners 1,3) are collected. They may be commented upon as follows: (i) the relevant fitting parameters are 0, 4, ~(m,, ec--_cI1,whereas R,, R, , R, prove unimportant. (ii) 8 and 0, = ~(m,, eczmc,) vary systematically with excitation of the pucker state, whereas 4 remains unaffected. (iii) the CH2 group structural parameters are not well determined, though for R, (corners 1 and 3) the fitting process bears out a change from approx. - 4” to + 0.5” with increasing pucker excitation. Parameters R, and R, for all corners and R, of corner 4 remain nearly zero. 3.2.3. Cyclobutylfuoride. For this molecule rotational spectra of three excited pucker states have been measured [12]. Table 7 contains the results of the fit of rotational constants in the 0, 4, + (m,, eC2_F), R, parameterization. Similar comments as made for cyclobutylchloride apply: (i) 0 and R, = %:(m,, ec,_F) show a dependence on pucker excitation; however, for R2 a indication of slight zigzagging seems to exist. Again 4 is essentially constant with a very weak trend for decrease with excitation. (ii) the CH, group parameters are not well determined, though for R, at corners 1 and 3 a wide variation from + 3” (ground state) to - 13” (u = 3) is found. R, and R, are found to be small. In view of the indeterminateness of the CH, group parameters, the variation of R, on pucker excitation cannot be considered as a strong support for relaxation. 4. AB INI TIO SCF COMPUTATIONS In this work the electronic potential function V (0, z) of a semirigid model of cyclobutanol defined by the pucker angle 0 and the OH internal rotation angle r (4 z 44.25”) has first been computed within the ab initio SCF approximation employing the split-valence 4-31G basis set [14]. In a first step computations were out for a grid {(f&,,,T.)}: {(f?,, m)} carried SA(A)42:2/3-N

265

mode coupling in Cring molecules

= [76(3)106”] x [0(30)180”], using structural parameters determined by microwave spectroscopy, cf. Table 3 [13]. In a step 2 the rocking coordinates of the fl- and y-CH2 groups and the angle %(ei2, eclo) were relaxed. Finally computation on the grid 0 E (80, loo”) with relaxation of all other internal parameters were carried out (step 3) using the program package MONSTERGAUSS [lS]. Complete geometry optimization with respect to all internal coordinates has been carried out by the force method with ana lytical gradient [16], as implemented in the MONSTERGAUSS program system. The geometry optimization was terminated when the gradient length, 112 g =

i

@Elaqi)2/n I

Ii

was reduced below 5.10-4 mdyn/A. The results of these computations may be illustrated as follows. In Fig. 3 a perspective view of the potential function as obtained after step 2 is shown, whereas sections of V(0, t) for T = 0 (tram OH with respect to a-CH) and r = 120” (gauche) and 0 = 0, are reproduced in Fig. 4. The results of step 2 were further used to determine by least squares and by Lagrange interpolation the coefficients V,” ofa numerical expression for V (6,~) of the form 3 v (8,~)

=

c

v,(e)

cos

~7,

0

(6) the polynomial coefficients are collected in Table 8. The values of the relaxed angle coordinates at corner s also may be interpolated in polynomial form (s = 1,3, 4, R,, rocking angles)

(7) R, = 3:(e;,,

e,,o)

= 1 rZpep. r

Typical data so obtained are listed in Table 9 for the tram OH conformer: these data are only very slightly dependent on T, furthermore R, has similar values for /?- and y-CH, groups. As the data show relaxation of the CH, groups is approximately symmetric with respect to 0 = 90” (planar ring). Furthermore relaxation of R, = c(ei2, ec,o) amounts to 126.5” and 135.5” for 0 = 76” and 106” respectively, i.e. is surprisingly large and underlines the relevance of relaxation for the puckering potential and the fitting of rotational constants. This is further illustrated by the influence of relaxation on the pucker potential of the gauche OH conformer: the function V(0, 120”) now exhibits two local minima at z 99” and x 82”: the latter being rather shallow: V(82, 120) - V(99, 120) x 1.1 kcal/mol, with a barrier of x V(88, 120) - V(82, 120) % 150 cal/mol (cf. Fig. 4). A number of comments appear in order:

266

R. GUNIX et al.

Fig. 3. Perspective view of the ring pucker-internal rotation potential function V(0, T) of cyclobutanol obtained by ub initio SCF.

(i) step 1 computations lead to a single minimum pucker potential with eti,, *: 97”, slightly depending on the torsional angle, (ii) for 8 z f&, the OH torsional potential exhibits two local minima at t = 0 (trans) and T = 120 (gauche) with energy difference V(6,,,k,, 120) - V (6, , 0) % 3.5 kcal/mol, (iii) by the relaxation modes admitted in step 2 the value of V(6, r) for fixed 7 is lowered by about 1 kcal/mol generally, (iv) the relaxation of the CH, rocking angles tends to be slightly asymmetric with respect to 0 = W, (v) the quantum chemical data indicate the torsional potential to be only very slightly influenced by relaxation of the type used above. By step 3 type computations in which besides 0 and r all internal coordinates are relaxed, the results of the simpler approaches step 2 and of earlier work [6,7] are confirmed and extended. In Table 10 a few results concerning relaxation of internal coordinates are reproduced. Some of these should be commented upon as follows: (4 the rocking data found in step 2 computations are essentially confirmed, (ii) wagging and twisting angles relax to a much lower extent than the rocking angles, (iii) axial CH bonds of the CHI groups rather systematically seem to be longer than equatorial bonds. This observation may have a correlation with the frequency of the v (CH) modes. According to findings published by MCKEAN et al. [17,18]

dv(CH)/dd,(CH) x - 10 cm- i/r&; it therefore seems possible to provide experimental evidence for C-H bond length relaxation either by detailed study of the i.r. bands of v(CH) modes of cyclobutane type molecules at low temperature or by observing coalescence of v(CH) modes caused by the dynamics of the ring pucker mode with elevating temperature. Experiments in this direction have been reported by LASCOMBEet al. [ 19,203. (iv) step 2 computations resulted in a minimal total energy E, = V(99,O) z - 230.60728 AEU whereas step 3 computation with completely E, = V (98,0) % yielded structure relaxed - 230.61300 AEU, illustrating that optimization of all internal structural parameters leads to a significantly lower total SCF energy minimum. 5. LARGE

AMPLITUDE NUCLEAR MOTION PROBLEM

AND FAR INFRARED SPECTRA OF CYCLOBUTANOL

In this section an investigation of the coupling of the ring pucker mode with the OH torsional mode within the framework of a two-dimensional semirigid mode1 will be presented. The results of this study will be confronted with new measurements of the far i.r. spectra of cyclobutanol and cyclobutanol-OD. 5.1. Symmetry considerations and selection rules If the large amplitude puckering-internal rotation problem of cyclobutanol is approximated by the aid of

Pucker-internal

mode coupling

in 4-ring molecules

26-l

B

Ei u-l-\

A

a--+76

0--c

Theta,DEG

76

106

i% N

Theta,DEG

---w

0-L

0

-180

Tau,DEG

Fig. 4. Sections of the ob initio SCF potential function V (0, T) for fixed r or 8. Upper: pucker potential function. A: V(0, 120) OH gauche conformation; B: V(0,O) OH trans conformation. Energy levels and density of eigenstates as obtained by numerical solution of the one-dimensional pucker problem including relaxation. Lower: OH internal rotation potential for 0 = 8,. A: Energy levels and eigenstates of internal rotation problem of cyclobutanol-OD; B: same for cyclobutanol.

R. GUNDE et

268

al.

Table 8. Polynomial coefficients V,, (kcal/mol) of the ab initio electronic potential function of cyclobutanol [Eqn. (@I*

c

0

1

;

3

4

0 1 2 3

7.533 - 0.589 -0.087 -0.330

4.849 -0.712 -0.218 -0.366

4.473 -0.714 -0.358 - 0.335

3.195 -0.892 - 0.438 - 0.282

7.398 - 1.014 -0.382 - 0.267

[;I=

*Values of V (0, T)as obtained upon relaxation of the CHIrocking angles and of 2:(e;,, ec,a) (step 2 computations). Table 9. Relaxation polynomials of CHI rocking angles R, and of c (eh, ec,a) (deg)’

I]

[;I.

F (0, T) = c S,(e) (a, cos FT + b, sin PT), P

Q2

-4.1197 10.1165

r43

7.3175E-6 - 1.89454E-3 -2.68441E-3 -9.6453E-6

0.156964 -0.259616

*Cyclobutanol trolls OH conformer. tCf. Eqn. (7); the relaxation coefficients are equal within errors for corners 1. 3, 4. *dim r,, = deg-“+‘.

Denoting the two irreducible representations ofJIP(B, r) ‘2 *v, by l-O+ and r”- we find the selection rules (for the internal problem) (x f polarization),

ML: To+++ r”+, TO-+, r”Table 10. Structural relaxation of some internal coordinates of cyclobutanol, trans OH conformation (step 3 computations) 79.88.

Spherical angle 0 90.0

109.10 108.18 1.27 1.26 - 0.42 - 0.67 4.51 -0.70 (-0.4) (5.0) 1.0820 1.0803(e) 1.0805 1.0815 (a) 1.0820 1.0801 (e) 1.0805 1.0814(a) 1.0821 (a) 1.0806 1.0794 (e) 1.0802 1.0747 1.0755

c (C,HIH~)* J&t R, R,

d(C,H,)§ d(GHz) d(C,H,) d(GH,) d(GH,) d tC4H2) d(CzH,)

99.37 108.86 0.66 0.06 - 5.25

( - 4.W 1.0838 (a) 1.0794 (e) 1.0838(a) 1.0794 (e) 1.0789 (e) 1.0811(a) 1.0766

*Angles in degrees, bond length in A. fR,, R,, R,, respectively, denote wagging, twisting and rocking angles [cf. Eqn. (3)]. $Values in parentheses indicate step 2 results. §a and e denote axial and equatorial CH bonds.

a semirigid model (including structural relaxation or not) with spherical pucker angle 0 and internal rotation angle 7 as finite coordinates, its symmetry is adequately described by the isometric group &’ (0, z) %V2, represented by the two (isometric) substitutions* [21-231 *Suppressing the primitive period isometric substitution, which is not relevant for the internal motion problem: r:\

= [’

L’l

L

-;

“1 [:I. 11

LlJ

(9’)

k = 2, 3.

MI = 1 mjfk)cos pt,

Mf:l-‘+,+I-‘-

Internal coordinate

(9)

this implies that

M : = C m$l) sin pr, R,

(8)

v = C v,(e)~~~fit,

r3

r2

_;

If both the potential function V(0, ~5)and the components of electric dipole vector M:(0, t) (taken in the frame system) are chosen in the form (f,(e) denotes polynomials in e - e,)

Polynomial coefficientst rl

[’

(~5, X$ polarization),

other dipole transitions being forbidden. Taking the values of the rotational constants into account one predicts for predominantly puckering and torsional transitions A/C and B type contours, respectively (see below).

5.2. One-dimensional subproblems Based on quantum chemical potential functions discussed in Section 4 the two one-dimensional subproblems with 7 kept constant (puckering mode) or 8 kept constant (torsional mode) were treated by means of a numerical procedure developed by MEYER [24,25-J. The one-dimensional models should mainly provide for a qualitative picture of pure pucker and internal rotation states and transitions, respectively. Both levels and density of the lowest eigenstates are shown in Fig.4. The potential function (Figs 3, 4 and Table 8) shows the pucker and internal rotation modes to be coupled, hence no numerical values f&r the single mode systems will be given. Qualitatively from Fig. 4 it may be concluded: (i) the tram conformation of the OH group (T = 0) is predicted to be more stable than the gauche conformation (T = 120”). Experimentally so far only the former has been detected [13]. (ii) O(I’O+)-+ l(rO’) pucker and O(rO’) + 1 (row) OH torsional transitions are predicted at 230 cm - 1 and 3OOcm-I, respectively, indicating the SCF potential to overestimate the force constants appreciably. These findings are in agreement with the general experience that SCF computations at the 431G level tend to overestimate diagonal valence

mode coupling in Cring molecules

Pucker-internal

force constants and that comparison with experimental values requires scaling of the potential [26]. Therefore tentative scaling factors for the two-dimensional potential V(0, r) may be extracted from the predictions of the onedimensional problems. 5.3. Two-dimensional

pucker-internal

rotation problem

In order to take coupling of pucker and internal rotation into account, a numerical treatment of the puckering-internal rotation problem was carried out. Employing a procedure and a computer program developed by MEYER [27,28], the kinetic energy expression for the semirigid model of cyclobutanol (cyclobutanol-OD) mentioned in Section 2 (including relaxation of rocking angles) and the SCF potential V(& T) given by Eqn. (6) and Table 8, the lowest 10 energy eigenstates could be calculated with some accuracy. The potential function was then resealed such that the lowest predominantly puckering and predominantly torsional transition occurred at 178 and 244cm-‘. This leads to the predictions collected in Tables 11 and 12. Furthermore three-dimensional views of the probability densities of the two lowest eigenstates and of an “axial” and a mixed pucker-internal rotation state are shown in Fig. 5. In the tables transitions in the range 20-300 cm- ’ Table 11. Cyclobutanol: Frequency (cm - ‘) SCF* Obst

23 53 66 84 93 107 137 150 157 159 159 178 178

2ocl 230 234 241.6 243.0 243.6 244.2 266 296

98 B 87 (Q) 89 (Q) 93 (Q) 121(Q) 128 (Q)? 151 (Q) 132B? 153 (Q) 162 (Q) 178 (Q) 182 (Q)

188 B? 195 205 vb

239 (Q) 244B

far infrared Initial state1

269

are collected, with lower states not exceeding 420 cm- ’ (360 cm- ’ for OD). They also give information about rotational envelopes and relative approximate intensities. Band envelopes were simulated using the rotational constants reported by MACDONALD et al. [13], cf. Table 5 [29]. Coarse features of these simulations are: -neither A-, B- or C-type envelopes exhibit resolved rotational structure, if simulated with spectral slit width x 0.3 cm-‘, -A-type bands should possess a strong Q branch and z 20 cm-’ P-R separation, -B-type bands should have a Q gap approx. 20 cm ’ wide, X-type bands should have a very strong Q branch and relatively weak P and R branches -A/C-type hybrids of any mixing ratio will possess strong to very strong Q branches. Estimates for intensities were derived from the twodimensional numerical wave functions and the electrical dipole moment and its derivatives aM f/dB and dM ‘/ar by numerical integration; the latter quantities were derived from a bond moment model for cyclobutanol. The entries in the tables are proportional to products of the dipole matrix elements squared and the Boltzmann population factors exp (- (E, - E,)/kT). A few comments should be added: pucker-internal Final state1

rotation

Band envelope 4

transitions Relative intensity11

Stleqrt 5Oeqrt 2Beq 40eq 3rt 40eq

6rt 7th 3rt 5eeqst 4Oeq 6rt

B B B B B ‘WC

6E-5 2E-5 8E-5 4E-5 3E-5 2E-6

4tIeq

7th

AIC

1E-3

5eeqrt

87g

AIC

6E-6

B NC AIC

3E-5 3E-4 1E-3 3E-4 lE-3

50eqrt 5eeq7t 2eeq 3rt 1 3rt 37t 4&q

4&q 4eeq 2eeq 1 2&q 2eeq

9th lOeeq7t 4tkq 50eqrt 2&q

6Tt 7th 8v 9eax 1Weqrt

5eeq7t 37t 6rt 7th

NC NC B

B B AIC B B B

AIC NC

2E-1

3E-5 lE-3 3E-5 5E-2 l.lE-1 2.7E-1 lE-5 2E-5

*Prediction by semirigid pucker-internal rotation model. t B:B-type band envelope, (Q): Q branch. $@eq, lJax, Tt, sg denote states with noticeable density localized near pucker angle 0 = 98”. 0 = 82”. torsion angle T = 0 (tram) and r = 120” (gauche), respectively. §A/C- or B-type predicted envelope. 11Squared dipole derivative x Boltzmann factor of initial state (relative).

R. GUNDEet al. Table 12. Cyclobutanol-OD: far infrared pucker-internal rotation transitions Frequency (cm-‘) SCF* 0bst

aeq

27 100 121 129 137 140 150 155 158 159 164 165 170 174 176 176.8 177.1 185.1 185.3 185.6 191

Initial state1

l&Q) -

150B -

l&Q) 165 B? -

1;(Q)

176 (Q)

178(Q) 185(Q) 188 B

Final state1

6tleqrt 5rt

60eqrt 70ax 70ax

6tIeqrt

87r

aeq 57t

78ax

Rand envelope 4 B B

NC

Relative intensity11 lE-4 lE-4 4E-6

NC NC

87~

B

lE-5 lE-3 l.lE-1

3rt

48eq

4@eq

85t

B B

lE-3 2E-3

Zfkqrt

NC

4E-4 1E-3 7E-2

60eqrt 28eq 6Beqrt 3rt 57t 2ee.q

5rt 37t

1 4&q 2&q 1 aeq

4@eq lOOeq(2)rt 5rt 9(2)Oeqrt 57t

1Weq (2)st 60eqrt 2eeq 9(2)8eqrt 60eqst 37t

lWeq(2)rt

AjC

B B B

NC A/C NC

AIC B B B

AIC

1.6E-1 lE-3 3E-5 lE-4 4E-4 lE-3 4E-2 8E-2 1.9E-1 7E-6

*t$$IISee footnotes for Table 11.

Fig. 5. Probability density of eigenstates of pucker-internal rotation model of cyclobutanol. H2-H5, cyclobutanol; D5-D6, cyclobutanol-OD, H2, H3, first excited pucker and torsional states 2r”+ 177 cm-’ and 3r”- 244 cm-l, respectively; H4, doubly excited pucker state 4r”+ 338 cm-‘; H5, doubly excited pucker-torsional state 5T”- 422 cm- ‘; D5, doubly excited torsional state 5r”+ 351 cm-‘; D6, doubly excited pucker-torsional state 6r”- 362 cm-‘.

Pucker-internal

mode coupling in Cring molt%ules

(i) the two-dimensional model predicts further transitions with high intensity in the range 3&300 cm ’ from higher levels than 420 cm- I, e.g. for the OH modification from level 6 at 445 cm- ’ above the ground state level: 6 + 8, 127 cm-’ B, relative intensity factor 8E-2,6 + 10, 136 cm-’ B, relative intensity factor 2E-3, (ii) even with the 10 levels investigated in this work, there are many transitions predicted above 300 cm I, which might be relevant for assignment of the absorption bands in the mid i.r, predicted numerically obviously (iii) the transitions form a complex system of overlapping bands, in particular in the neighborhood of the lowest pucker and internal rotation transitions, (iv) by the nature of numerical treatment (truncation of the energy matrix of the pucker-internal rotation problem) the higher levels are subject to systematic errors and should be used with caution, Fig. 5 exemplifies the behavior of the twodimensional energy eigenstates: whereas the levels at 178 cm-’ (2) and 244 cm-’ (3) predominantly are pucker and torsional levels, the levels at 338 cm-’ (4r0’0eq) and 422 cm-’ (5r0-Beqst) with strong “doubly” excited levels are pucker-internal rotation mixing. In order to sketch in Tables 11 and 12 qualitatively typical properties of the density functions, the states are characterized as pucker states (0) with equatorial (eq) or axial (ax) CZ-0 bond and/or torsional states (T) with predominant tram (t) or gauche (g) conformation.

271

A \

5.4. Experimental Fourier transform far i.r. spectra of gaseous cyclobutanol-do and -0D were taken in the range 30&30 cm-’ on a RIIC Beckmann Model 720 interferometer equipped with a White type variable multipass cell of our own design and a PDPS/E minicomputer. Usually path lengths of x 20 m were used. Both sample and background interferograms were taken with 2 x 4096 points (symmetric total path length 2 x 1.67 cm), leading to a spectral resolution of z 0.3 cm-‘. Interferograms were processed by a program package developed by K~~HNE [30]. Since the measurement of interferograms took approximately 8 h per data collection, it proved essential to achieve constant gas pressure in the long path cell by low leakage rate and accurate pressure measurement (MKS Baratron, 10 Torr). All FT-far i.r. spectra shown in Fig. 6 consist of a complex system of transitions. Though a number of sharp peaks may be identified and indications of band contours are recognizable, disentangling the spectra meets considerable problems. This also applies to the spectra of the OD isotopomer. These are superpositions of the spectra of both OH and OD modifications. In spite of considerable efforts spectra of the pure OD compound could not be produced. Most of the prominent discernable features are collected in Tables 11 and 12, where they are contrasted with

90

150

210

270cm-1

Fig. 6. Fourier transform far i.r. gas spectra of cyclobutanol. Upper spectrum: cyclobutanol, p z 0.3 Torr, 1z 20 m, spectral resolution z 0.3 cm- ‘, nine sample and eight background interferograms averaged. Middle spectrum: upper trace cyclobutanol-OD, lower trace, cyclobutanol; p = 0.15 Torr, I z 20 m, spectral resolution = 0.6 cm- ‘, each spectrum derived from six sample and six background interferograms. Lower spectrum: cyclobutanol-OD, p P 0.8 Torr, I z 20 m, spectral resolution z 0.3 cm-‘, spectrum derived from four sample and four background interferograms.

predictions above.

by the nuclear motion treatment

described

5.5. Assignment of far infrared spectra 5.5.1. Cyclobutanol. From the predicted scheme of transitions collected in Table 11 and the envelope computations one expects to observe complex superpositions of A/C-type hybrid bands and of B-type bands, respectively, particularly in the range where the

212

R. GUNDE et al.

first predominantly pucker type and internal rotation type transitions (localized at T k: 0” and 8 = 98”) are foreseen. This is borne out experimentally. In the spectrum of the OH compound B-type bands may be located near 244, 188 cm-‘, eventually near 100 and 132 cm- ’ and, vaguely, in the strong absorption region around 200 cm- i. Furthermore a number of sharp Q branches associated’ with A/C-hybrid-type bands are recognizable, as listed in Table 11, in which these features are contrasted with transitions following from the model computations in the sense of tentative assignment. No attempt has been made to fit the twodimensional potential to reproduce observed spectral features, mostly since the model computations rather clearly show that the presently available data are barely suflicient for a detailed assignment. Though SCF potential functions based predictions should be considered mainly as a guide line for assignments, correlation with the experimental spectrum seems rather clearly be established. In particular some of the observed clusters of (weak) Q branches also appear in the prediction, furthermore the difficulties encountered in assignment of B-type bands seem to originate from clusters of overlapping B-type bands. Analysis of qualitative aspects of the levels are indicated in Table 11 and indicate that some of the transitions involve mixing of pucker and torsional modes and/or multiply excited modes.

As shown in Fig. 6 and listed in Table 11 there exist states with considerable density in a neighborhood of 0 m 82” (with axial orientation of the (Jr-0 bond), but they cannot be classified as a second, localized, pucker conformation. 5.52. CyclobutanoLOD. Table 12 contains transitions as predicted by the two-dimensional semirigid model, contrasted with the observed spectra; only transitions up to initial level 6, which is computed 362 cm- 1above the ground state, are included. If higher states are considered, many more (weak) transitions fall into the 30-300 cm-’ range. Again one can establish a correlation to the experimental spectral features in a rather straightforward way, which may be considered as a tentative and coarse assignment. Remarks similar to those made for the OH species also apply to both predicted and experimental spectra of the OD mod& cation. The qualitative classification of the eigenstates also is similar, but there occur some reversals in the sequence of eigenstates when classified qualitatively as pucker, internal rotation or mixed states with single or double excitation; this may be seen from the information given in Table 12 and Fig. 5.

0.5 cm-’ resolution) these authors concluded the pucker potential to have two local minima with energy difference = 100 cm- 1 separated by a barrier of k: 750 cm-’ height and, as a consequence, assume cyclobutanol to exist in two conformations with respect to ring puckering, besides the tram and gauche OH conformation. The two large amplitude modes were treated as one-dimensional problems. The far i.r. data obtained in this work essentially confirm the earlier measurements. However, though frequency range and resolution have been extended, the new data appear not sufficient to provide a basis for decision between the earlier model and the alternative arrived at in this work. According to the latter the pucker potential-as derived from SCF computations--is found to have a second local minimum only for the gauche OH conformation, which is separated from the dominant pucker potential minimum by a barrier of z 50 cm-l. This low barrier and the fact that the SCF potential V(8, r) is found to contain appreciable pucker mode-internal rotation coupling do not give rise to two-dimensional) eigenstates with clearcut localization of pucker conformations with 8 k: 82-83”, i.e. with an axial C&O bond. Since the ab initio potential cannot be considered sufficiently reliable, the analysis presented above cannot rule out the earlier interpretation. In recent work, GUNDE et al. attempted to decide the presence of a second puckering conformer using the temperature dependence of v(OH) Raman band and thermal molecular beam i.r. matrix spectroscopy [32], but no unique experimental arguments in favor of one of the alternatives could be derived. At present it appears that far i.r. data with signi6cantly improved resolution and-particularly-signal-to-noise ratio might best be suited for resolution of this interesting problem. Acknmledgements-This work was supported by the Swiss National Foundation (Project No. 2.612-0.80, 2.079481). Theauthors wish to thank Dr. R. MEYER for making available his computer programs for treatment of two-dimensional large amplitude problems, J. KELLERfor help with programming and numerical problems,and the ETH Zurich computer center for generously providing free computer time. We gratefully acknowledge GONTHARD.

REFERENCES [l]

[Z]

6.RELATIONTOOTHERWORK

the typing of the manuscript by K.

ANDCONCLUDING

REMARKS

[3]

The assignment suggested above should be considered as an alternative to the results of an analysis presented by DURIG et al. [31]. From Raman (4OOO-100cm- ‘) and far i.r. spectra (300-100 cm- ‘,

[S]

[S]

L. A. CARREIRA,R. C. LORD and R. B. MALLOY,JR., in Topics in Current Chemistry, Vol. 82, p. 1. Springer, Berlin (1979). C. S. BLACKWELLand R. C. LORD, in Yi6rationul Spectra cad Structure, Vol. 1, p. 1 (edited by J. R. DURIG). Dekker, New York (1972). J. R. DURIG and H. W. GREEN, Spcctrochim. Acto 25, 849 (1969). A. C. LEGON,Chem.Rev. 80,231 (198Ok the reader may consult this paper for a detailed review of four-ring molecule spectral data, potential functions and treatment of the four-ring pucker motion. F.BALTAGI,F. HENRICI, A. BAUDER,T. DEDA and H. H.

Pucker-internal mode coupling in Cring molecules GUNTHARD,Molec. Phys. 24, 945 (1972). [6] J. L. NELSONand A. A. FROST, .I. Am. them. Sot. 94, 3727 (1972). [7] D. CREMER,J. Am. them. Sot. 99, 1307 (1977). [S] F. BALTAGI,A. BAUDERand H. H. G~NTHARD,J. molec. Struct. 62, 275 (1980). [9] F. BALTAGI,A. BAUDERand H. H. G~NTHARD,J. molec. Spectrosc. 42, 112 (1972). [lo] E. P. WIGNER, Group Theory, p. 150. Academic Press, New York (1959). [ 1l] A. DuscHEK and A. HOCHRAINER, Tensorrechnung, part 1, p. 78. Springer, Vienna (1954). r121 H.Kl~and W. D. Gw~~~,J.chem. Phys. 44,865 (1966). 5131 J. N. MACDONALD, D. NORBURYand J. SHERIDAN, Soectrochim. Acta 34A. 815 (1978). [ 141 d. DITCHFIELD,W. J. HEHRE‘andJ. A. POPLE,J. them. Phys. 54, 714 (1971). [15] M. R. PETERSENand R. A. POIRIER, MONSTERGAUSS program, University of Toronto, Canada (1981). [16] H. B. SCHLEGEL,Ph.D. thesis, Queen’s University, Kingston, Canada (1975). [ 171 D. C. MCKEAN, EUCMOS XVI, Abstract p. 38, Sofia (1983). [18] D. d. MCKEAN, J. L. DUNCAN and L. BATT, Spectrochim. Acta 29A, 1039 (1973). [19] J. C. LASS~GUES,M. BESNARD,D. CAVAGNATand J.

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on Raman LASCOMBE, 9th Int. Conference Spectroscopy (ICORS), Abstract p. 50, Tokyo (1984). J. LASCOMBE,D. CAVAGNAT, J. C. LAS&CUES, C. RAFILIPOMANANA and C. BIRAN,J. molec. Strut?. 113, 179 (1984). H. FREI, R. MEYER,A. BAUDERand H. H. G~NTHARD, Molec. Phys. 32, 443 (1976). H. FREI, R. MEYER,A. BAUDERand H. H. G~NTHARD, Molec. Phys. 34, 1198 (1977). H. FREI,P. GRONER,A. BAUDERand H. H. G~NTHARD, Molec. Phys. 36, 1469 (1978). R. MEYER, J. them. Phys. 52, 2053 (1970). R. MEYER, program FMXOl (unpublished); this program has been modified for application to four-ring systems and inclusion of rocking angle relaxation. C. E. BLOM, Dissertation, University of Leyden (1976). R. MEYER, Habilitationsschrift, ETH Ziirich (1978). R. MEYER, J. molec. Spectrosc. 76, 266 (1979): F. PFEIFFER.Dinlomarbeit. ETH Ziirich (1968). nrograms ROTENV’ 1,2,3, modified by R. G&DE anh H. J. KELLER,unpublished. H. K~HNE, Thesis No. 5713, ETH Ztirich (1976). J. R. DURIG, G. A. GUIRGIS, W. E. BUCY, D. A. C. COMF-TONand V. F. KALASINSKY, J. molec. Struct. 49, 323 (1978). R. GUNDE and H. H. G~NTHARD, Specfrochim. Acta 39A, 315 (1983).