3191
J. Phys. Chem. 1980, 84, 3191-3196
Ring-Puckering Geometrical Models for Five-Membered Rlngs E.
Dkr," A. L. Esteban, F. J. Bermejo, and M. Rico
Departamentos de Electroqdmlca e Investlgaciones Qdmlcas, UnivershladAutenoma, Madrid, Facultad de Ciencks, Allcanfe and Instituto de Qdmica hsica, Serrano 119, Madrkl, Spain (Recelved: May 16, 1980)
Four geometrical models for simulation of puckered conformationsof five-membered rings with unequal bond lengths from the (q,4) pseudorotational coordinates are described and applied to the (CHZ),X(X = CHz, 0, S, Se) molecules. Model A, a generalization of that introduced for cyclopentane by Kilpatrick et al., leads to an unreasonable variation in bond lengths. Better results are obtained with models B1and Bz for which the invariance condition in bond lengths is imposed. Model B1presents the advantage that its (q,$) coordinates are consistent with those defiied by Cremer and Pople. Assuming infinitesimalvalues for q, analytical expressions of t,hetorsion angles, O j ( q , + ) , and bond angles, cxj(q,4), have been deduced, which define model L. The 8, and aj values, computed for the (CHZ),Xmolecules using the models A, B1, and Bz, are satisfactorily fitted to these explressions. Application of our Oj(q,+) formula to the determination of the pseudorotational coordinates of the furanose rings in nucleosides and nucleotides provides better results than those obtained by using the formula of Altona et al., to which the former is reduced when the ring atoms are equivalent.
Introduction The conformational analysis of saturated five-membered-ring molecules is complicated because of the existence of pseudorotation, that is, of dynamic puckering deformations moving around the ring.l The geometry of a puckered conformation is often described by two independent parameters: one, 4, representing the phase angle of pseudorotation, ithe other, q, the amounts of puckering. However, nine internal coordinates are needed to fix the geometry of the five atoms of the ring. Therefore, models to set up the internal coordinates from q and 4 and methods to calculate these from the former are required. The present paper attempts to give a critical examination as well as an extension of some of these models and methods. The puckering models have been applied to the (CH2),X molecules of cyclopentane (CP), tetrahydrofuran (THF), tetrahydrothiopherie (THT), and tetrahydroselenophene (THS). The conformational behavior of these molecules has been studied ink gaseous phase by different empirical methods (calorimetric,' electron diffraction?+ far-IR, IR, Raman, and MW7-13)as well as by theoretical computat i o n ~ . ' , ~ ~We ~ Jhave ~ ' ~carried out the complete analysis of the proton magnetic resonance spectra of THF, THT, and THS,1*20 trying to get reliable liquid-phase conformational idormation from the observed coupling constants. On these studies and in others, such as electron diffraction analysiei, suitable models for simulation of a large number of puckered conformationsmay be necessary. The simulation of the puckered conformations from the (q,4) pseudorotational coordinates is made in the present work using purely geometric models. The ring atoms are shifted out of a planar reference conformation under different constraints. Model A is a generalization, to cover any five-membered-ring molecule, of the model for an equilateral pentagon introduced by Kilpatrick et al. in their discussion of pseudorotation in the CP molecule.' Being the most simple of all, model A implies a physically unreasonable variatioin in bond lengths. In models B1 and B2 the invariance condition in bond lengths is imposed. Model Bz is a generalization of the one used by Adams et al. in their electron diffraction study of CP,2 whereas the model B1 is a modification of this in order to obtain the same atomic displacements zj, perpendicular to the plane of the reference conformation, as in model A. Model L is defined by the analytical expressions of the torsion angles, 0022-3654/80/2084-3191$01.00/0
flj(q,4),and bond angles, cxj(q,4),derived for infinitesimal atomic displacements perpendicular to the plane of the reference conformation. The determination of the pseudorotational coordinates from a given puckered conformation can be made by one of the two procedures here denoted as methods I and 11. Method I, proposed by Cremer and Pople,21is based on the definition of a unique ring mean plane. When it is applied to a conformation generated by using models A or B1for a given pair of (q,4) values, these same values are obtained. Method I1 is a generalization to any five-membered-ring molecule of the method for equilateral pentagons introduced by Altona et a1.,22and it is based on the analytical expression of the torsion angles of model L. This method has been applied in this work to the furanose rings in nucleosides and nucleotides and to the 8. values of the (CHZ),X rings computed with models A, E&, and B2, obtaining in this way improved results in comparison with those obtained by using the method of Altona et al. The adopted notation is illustrated in Figure 1. Symbols with zero eiuperscript will designate reference conformation values. For the (CH2),X molecules the ring atoms are numbered clockwise starting from the heteroatom. Pseudorotatioin Models Models A and L. Model A is a generalization of the model introduced by Kilpatrick et al.l for the CP molecule. From consideration of the two normal modes of out-ofplane motions of a planar regular pentagon, they described a displacement of the jth ring atom, perpendicular to the plane, as zj
=
(2/6)+
cos [2(4 + 27rj/5)]
j = 0, 1, 2, 3, 4
(I)
where q measures the amplitude and 4 the phase angle of the puckering. Such a model may be extended to any five-membered-ring structure by selecting properly a planar reference conformation. The Cartesian coordinates zj are given by eq 1, and the x i and y, are those of the reference conformation. An empirical formulation of pseudorotation involving the five intracyclic torsion angles 8, rather than out-ofplane displacements was made by Altona et a1.22 They proposed a relationship of the form 8, = Bm cos (A/2 + 47rj/5) (2) 0 1980 American Chemical Society
3192
The Journal of Physical Chemistry, Vol. 84, No. 24, 1980
a
Diez et al.
and substitution of eq 1 and 3 yields 0, = -(2/5)1/2[q/(sin a,O sin ado)](R, cos 7, Sa sin 7,) (10) with
+
R, = [cos (87~/5)- cos (4r/5)][(sin ado/lCbo) (sin a,O/le$)] (Ila)
+
Flgure 1. Adopted notation for geometrlcal parameters in five-membered rings.
for these torsion angles, where again Om is an amplitude and A is a phase angle. Dunitz showedz3that eq 2 is correct for infinitesimal displacements from a reference plane of a regular pentagon, yielding in that particular case A = 44 + r and Om = 162q. However, for finite q in the range 0.1 < q < 0.3 A, better results are obtained by taking 0, = 150q. We have derived formulas of the torsion angles, Oj(q,4), and bond angles, aj(q,c$),for infinitesimal displacements from planarity of atoms in a nonregular pentagon. These formulas define the model L that, for infinitesimal q values, coincide with the model A. The analytical expressions of intracyclic torsion and bond angles derived for models A and L are conveniently written if the following definitions are introduced: 7j = 2($ 2~j/5) (34
+
(34
pI. = lk.0/l..O I 11
Intracyclic bond angles are expressed in model A by cos = (cos Y:( - z a e z b a ) ( l + zae2)-1'2(1 + Zba2)-1/2(4) as can be easily derived from the iea.iab dot product. If a, is written in terms of the decrease 6, resulting from out-of-plane motions, i.e. a, = a,O - 6,, we can express 6,, after a power series expansion of cos a, and neglecting higher powers of 6, and 2, as
+ &a2) + 2z,,zb,](2
6, = -[cos a,0(Za,2
sin a,O)-l
(5)
from which straightforward substitution of eq 1and 3 leads to 6, = N , + ( M , - N,) cos2 7,+ Pasin 27, (6) with Ma = [cos (4?r/5) - 1]2[(p,-1
+ pa) cos a,O - 2]Q, N , = sin2 (47r/5)[(p11 + p,) cos a,O + 2]&,
Pa = sin (4~/5)[cos(47/5) - l][(p,-l Q, = -q2/51,>lba0 sin :a
- pa)
(7a) (7b)
cos a,O]Qa (74
Sa = 2 sin (8a/5) sin (a> + adO)/ld> [sin (4r/5) - sin (8r/5)l[(sin ad0/&bo)
+ (sin .,0/led)]
Ulb) Formulas 5 and 9 are reduced to eq 4 and 6 of the Dunitz paperz3 if the reference conformation considered is an equilateral pentagon with unit bond length. In order to compare with expression 2, first given by Altona et al., formula 10 can be rewritten as o, = e,, COS [(A + €,)/2 + 4 r a / 5 ~ a = 0, i , 2 , 3 , 4 (12) showing that for each function 0, there is a maximum value, Om, and a correction, e,, to be applied to the phase angle A. Om, and e, are given by 8,, = (2/5)'/2[q/sin a : sin ad0)](R,2 S2)1/2 (13a)
+
e, = 2 arcsin
[R,(Ra2+ Sa2)-1/2]
(13b) Model L is defined by formulas 6 and 12 parameterized according to eq 7 and 13. When the reference conformation is an equilateral pentagon, eq 12 is reduced to eq 2 and the last term in eq 6 is zero. Models B1and Bz. Application of model A leads to unreasonable variation in bond lengths because the ringatom Cartesian coordinates xj and y j are constant whereas the z j values vary according to eq 1. This fact is overcome on models Bl and B2 by introduction of the invariance condition in bond lengths. This is imposed by shifting the atoms along the lines joining the atomic positions (x,,yj,zj) given by model A and the points (xc,yc,mj). Models $1 and B2 then result from consideration of mi = z j and mj = 0, respectively. The point (xc,yoO)is the center of the planar reference conformation which may be specified as the point for which the sum of square deviations between its distances to the five ring atoms and the mean distance is a minimum. In a puckered conformation generated by using model B1 for a given pair of (q,c$) values, the zivalues are given by eq 1. Therefore, these same (q,4)values are obtained when that conformation is analyzed by means of method I. This occurs since method I is based on the definition of a unique ring mean plane such that eq 1is fulfilled and the (q,+) values are obtained by solving them. Model Bz, however, provides smaller zivalues than formula 1, and, when method I is applied to a particular conformation, smaller q values are obtained, which furthermore are dependent on the angle. Numerical computations were carried out on a UNIVAC 1108 computer by means of a Fortran V program developed by us. For B-type computations, the equations defining the bond length invariance conditions were expanded in Taylor series. If one neglects the nonlinear terms, five linear relations are obtained
(7d) Intracyclic torsions are given in model A by the formula cos 0, = [-cos (a> + ado)Zdc2+ cos (Y>ZdcZed + cos .do ZdcZcb + ZcbZed - sin a> sin a d 0 ] /[ (zd: + 2 cos ff>zdczcb+ Z,b2 + sin2 a>)'/'(Zd: + 2 COS adoZdcZed+ z e d 2 + sin2 Cud0)li2] (8) 15 which can be derived from the (tbcX~cd).~~dX~de).products. Afk = (afk/a&)A& (14) After power series expansion of cos 0, and high power i=l truncation of 0, and 2, one obtains where symbols & and f j are Cartesian coordinates and bond 8, = -[sin (ace + ado)z&+ sin a> z e d + sin adozcb] x lengths, respectively. Ten further independent equations (sin a> sin (9) are those defining the lines joining the points (xj,yj,zj) and
+
The Journal of Physical Chemistry, Vol. 84, No. 24, 1980 3183
Rlng-Puckering Geometrical Models TABLE I : Bond Lengths (A), Bond Angles (deg), and Atom-Center Distances ( A ) , of the Reference Conformations, and q o Values ( A ) , for the Indicated Molecules parameter CP THF THT THS C-CY
c-xa
C-Ha C-5-Cb
1.546 1.546 1.114
1.538 1.428
108.00
111.66 1.271 114.16 1.259 1.289 1.265 0.366 0.380
dl
1.315
C-X-Cc
108.00
doC
1.315 1.315 1.315 0.416 0.435
dIC d ;z 40 4Qe
'
1.536 1.839 1.120 98.79 1.413 94.55 1.460 1.368 1.432 0.474 0.500
1.110
1.537 1.975 1.116 95.08 1.463 90.21 1.520 1.405 1.490 0.497 0.525
a From ref 2 4, 5, and 6 for CP, THF, THT, and THS, respectively. For first reference conformations. For second reference conformations. For models L, A, and B,. e For model B 1 .
'
(xc,yc,mj).The system of these 15 linear equations can be solved iteratively with a least-squares criterion. The system can be represented by the matrix equation AA = B (15) where A is the Jacobian matrix (dfk/d&), A the vector of corrections to be applied to the Cart,esian coordinates, and B the vector of residuals. Solution of eq 15 leads to A = (ATA)-'ATB (16) Results and Discussion Application of Geometrical Models to (CHJ4X Molecules. The first step to set up a puckered conformation from a pair of given (q,+) values is to choose a suitable planar reference conformation. In the most simple case of molecules belonging to the D B hsymmetry group, such as CP, only one boind length is needed to specify the reference conformation, whereas for molecules with Czusymmetry, such as (CI-[z)4Xrings, one more parameter (e.g., the CXC angle) is needed in addition to the bond lengths. Geometric data defining the reference conformations for the four molecules studied are listed in Table I. Bond lengths are values from electron diffraction analysis. In order to evaluate the effects of a given starting reference conformation, we chose two planar conformations for each (CH,),X molecule, which will be referred to as first and second reference conformations, having the CXC angle values shown in the fourth and sixth rows of Table I, respectively. The first reference conformations are pentagons, for which an equidistant point to the five ring atoms exist such as Geise et alS3assumed for the THF planar conformation. The second reference conformations for T H T and THS are those that provided the best fit of whereas the experimental geometries58by using model B1, for THF the CXC angle value was arbitrarily selected as 2.5" greater than thle one of the first reference conformation. The q values used in this work for all models, except for model B,, are given in the penultimate row of Table I. These q values are the result of applying method I to the experimental data.2* As said above, model B2 is not consistent with method I with respect to the q values. Thus, these have been selected (last row of Table I) so that, by applying rnethodl I to the conformations generated by model B2,the average q values in the penultimate row of Table I are obtained. The torsion and bond angle values computed by applying the different, models to the symmetrical conformations C, (envelope, q5 = 0') and C2 (twist, q5 = 45') of
TABLE 11: Torsion and Bond Angles ( A ) from the Different Models for Two Symmetrical C, and C, Conformations in CP 6 , deg 0
a l , a4 a,, a 3 0 0
el,- e4 e2,-e,
45
La
Aa
B, a
BZb
100.52 103.25 107.66 0.00 -25.64 41.48 108.42 105.69 101.68 -43.62 35.29 -13.48
101.28 103.60 107.67
102.32 103.79 106.19
102.13 103.95 106.13
angle 010
CYQ
al,a4 a * , as 0 0
el, e4 e2,ea
0.00
0.01
0.01
-24.22 38.06 108.42 105.80 101.91 -39.81 32.81 -12.90
-24.98 40.29 106.61 105.12 102.73 -42.34 34.32 -13.15
-25.01 40.27 106.43 105.26 102.68 -42.29 34.34 -13.16
a For q values in penultimate row of Table I. values in the last row of Table I.
For q
CP are given in Table 11. The values obtained for THF, THT, and THS show trends similar to those of CP. Differences between the results from models B1and B2are small, and, for the first reference conformations of CP, THF, THT, and THS, the maximum values are respectively 0.19, 0.17, 0.45, and 0.53' fo'r the bond angles and 0.05, 0.15,0.47, and 0.67O for the torsion angles. Torsion angle values arising from models B are closer to the ones computed with model L than are those provided for model A, whereas an opposite behavior is followed by the bond angle values. The torsion and bond angle values computed for model B1 can be satisfactorily fitted to the analytical expressions 12 and 6. Optimal parameter values for sets (8ma,ea) and (MJVJ',) in these formulas have been obtained from the correspondent least-squares fittings. The torsional parameters (8mo,ta)obtained from model B1 data for the two reference conformations are given in Table 111. Comparison of the parameter values for the two reference conformations shows that the only differences greater than 1' are 1.8' for BmO in THT and 2.5 and 1.2O for Bnt0 and t 1 in THS, respectively. On the other hand, the maximum differences between the parameter values for models B1and L are 3.0" foro,8 and 2.0' for in THS. Formula 12 provides far better results than formula 2, for the CP ring, when they are proposed by Altona et alaz2 applied to the torsion angle fitting of the CZumolecules. This is mainly due to the fact that the condition 8, = 8, is not fulfilled and the 8ma values of the THF, THT, and THS molecules are spread over ranges of 5.5, 15.1, and 21.4', respectively. Another condition not fulfilled is c1 = c2 = O', and the fitting of model B1data to formula 12 with e, = 0' provides 8, values close to the ones reported in Table 111, but the maximum deviations dl and dz increased in significant amounts reaching 3 . 5 O for dl in THS. Equation 13 states for model L that maximum torsions 8ma are proportional to q and phase corrections t, are independent of q. For model B1 the following quadratic expressions give a reasonable fit to the values obtained from the fittings of the torsion angles according to eq 12 ern, E,
-
= Caq + Daq2
E,* = E,q
+ Fa$
(17)
(18)
where E,* is the E @ value in model L. When the linear and quadratic coefficients are computed from the data in Table I11 for model B1 imd the first reference conformations, the resulting equations supply B,, and E , values that, for q 5 1.5q0,differ less than 0.5' from the ones obtained by fitting the torsional angles.
3194
The Journal of Physical Chemlstry, Vol. 84, No. 24, 1980
Die2 et
TABLE 111: Valuesa of Parameters e ma and ea and of Their Derivatives de ,,/dq ema
molecule angle
CP
e,
~b
B,
al.
and dea/dq (Angles in deg and q in A)
Ea
B,
Lb
B,'
Bl
demab/dq dcab/dq
dab,d
42.40 0.00 0.00 0.00 96.9 0.00 0.06 36.40 0.00 0.00 0.00 94.8 0.00 0.02 O1 37.54 -2.79 -2.43 -2.07 99.2 1.93 0.08 0, 41.76 -2.71 -2.57 -3.34 108.5 1.04 0.08 52.66 THT e, 0.00 0.00 0.00 106.1 0.00 0.32 8, 49.06 6.48 5.13 4.18 96.1 -5.22 0.29 e, 39.74 7.25 6.39 7.10 78.5 -2.59 0.30 THS e, 57.17 0.00 0.00 0.00 110.2 0.00 0.54 51.55 8.87 6.82 5.65 96.3 -7.56 0.55 e2 38.92 9.84 8.92 9.49 72.8 -3.98 0.44 a The parameter data for model B, are optimal values from torsion an le fitting. For model L have been computed with eq 13. The q values used are those in the penultimate row of Table I. For first reference conformations. For second reference conformations. Maximum deviations from torsion angle fitting. THF
e,
43.62 36.70 38.98 42.38 56.82 49.77 39.89 62.66 52.40 39.43
42.40 35.94 38.01 41.38 54.47 48.27 39.41 59.71 50.73 38.31
The fitted (Ma,Na,Pa) parameter values, referring to bond angles, found from model B1 data for the two reference conformations are given in Table IV. Only a small dependence of the parameter values on the reference conformation is found again, the larger differences between the results for the two reference conformations being smaller than 1'. On the other hand, differences between the parameter values for models B1 and L are significant, the Ma for the later model being generally greater than the corresponding ones for the former model. An opposite situation concerning the N , values is found. This is in agreement with the smaller bond angle value range that results for model B1. Furthermore the P1 and Pzvalues of the C2"molecules are negligible for model L but not for model B1. When models B1 and L are compared, the greatest differences correspond to the a2function of THS. The range of values for this angle is 13.2' in model L and 5.9' in model B1.The extremum location shift from the CP ones is -15' in the latter model. A dependence of (Ma,Na,P,) on q2 is predicted by eq 7 for model L, and a similar dependence is found with model B1.If the proportionality constants are determined by using data from Table IV for the first reference conformations, the resulting equations supply (Ma,N,,Pa)values that, for q I 0.7 A, differ less than 0.5' from the ones obtained by fitting the bond angles. Application of Method II. The @,,A) parameter values in eq 2 can be obtained from the five intracyclic torsion angles of a given puckered five-membered ring by following the method reported by Altona et al.22924The phase angle may be computed by applying eq 19, which is easily de@(A/2) = l(& + 84) (81 + 83)(/[280(sin36' + sin 72')] (19) rived from eq 2. Once the A value is known, the torsional amplitude 8, can be obtained from eq 20. The calculated 80 = ern COS (A/2) (20) parameters (8,,A) depend somewhat on which atom is chosen as number one. To lessen this difficulty the five torsional angles are chosen in turn as Bo, and the average results are taken as the final (8,,A) values. If this analysis is carried out on the puckered conformations resulting from application of models B, the correct (8,,A) values are found for the CP molecule, whereas for the CZumolecules a noticeable dependence of Brn on A occurs and the A values only are correct for the symmetrical conformations. So, for the THF conformations simulated with model B1 using the first reference conformation the 8, values decrease from 40.9 for 4 = 0' (A = 180') to 36.9' for 4 = 45' (A = 360'), and for input A values of 216,252, 288, and 324' one computes A values
'
of 212.6, 246.3, 282.1, and 320.2', respectively. These disagreements are partly a consequence of using eq 2 instead of eq 12 for the analysis of nonequilateral pentagons. On the other hand, the (d,,A) values can be obtained from eq 12, provided that the phase corrections ej and the amplitude ratios drno/8rnj are known. The phase angle can be computed by a least-squares fitting of the 8 j / 8 0 data set to eq 21. Once the A value is known, the amplitudeo,8 sin ( e j / 2 + 4nj/5)tg(A/2) = cos (ej/2 + 4nj/5) (orno/drnj)(dj/do) .I' = 1,2,3,4 (21) can be obtained by following a similar procedure by application of eq 22. cos [(A
+ ej)/2 + 4 ~ j / 5 ] ( 8 , j / 8 ~ ~ ) 8=, ~0, (22)
j = 1, 2, 3, 4
As a test of the two procedures discussed herein, 59 ribose and deoxyribose rings in nucleosides and nucleotides, which had been analyzed in terms of eq 2 by Altona and S~ndaralingam,~~ have been analyzed in this work by means of eq 12. We obtain an average deviation between experimental and computed 8, values of 0.231, which compares favorably with the one of 0.48 resulting from the Altona et al. procedure.26 In this particular application of eq 21 and 22, we used for the parameters e . and 8rnj/8,0 the set of values corresponding to model $, with the first reference conformations (see Table 111). The obtained results depend on the values of these parameters and therefore on the chosen reference conformation. The average deviation increases to 0.339 for the second reference conformation and shows a minimum of 0.225 for a planar conformation with a COC angle of 112.1'. A smaller minimum of 0.19 is found when the parameters ej and 8mj/8rno,instead of corresponding to one reference conformation, are obtained applying the equations COS
( 4 2
+ 144°)(8,1/8,0)
COS
(9/2
+ 288")(8,2/8,o)
+ 84)/280 = (82 + 83)/Z80 = (0,
(23a)
(23b)
which are a consequence of eq 12 if the assumptions Oml = em4,Brn2 = em3,el = -e4, and t2 = -e3 are introduced. Once the right-hand sides of eq 23, a and b, have been computed from the values of the 59 ring structures reported in Table I of ref 24, the ratios Brn1/8,0 and 8rn2/8,0 can be obtained by solving these equations for given pairs of el and e2 values. Operating in this way, we found the minimum average deviations when €2 (e1 - 1)/2. Physical Reliability of Ring-PuckeringModels. Models A, B1, and B2 provide a means of computing the atomic Cartesian coordinates of the five ring atoms from a given
The Journal of Physical Chemistry, Vol. 84, No. 24, 1980
Ring-Puckering Geometrical Models
3195
TABLE IV: Valued' of Parameters Mu,Nu,Pa (deg)
MU molecule angle
CP THF THT THS
NU
I
~b
B,&
BIC
Lb
7.48 7.27 6.38 5.60 5.79 8.02 11.04 5.19 8.13 12.70
5.68 4.76 4.88 4.59 6.35 6.32 7.09 6.52 6.63 7.69
5.68 4.85 4.86 4.59 6.42 6.52 6.86 6.73 6.86 7.29
-0.42 -0.35 -0.35 -0.34 -0.45 -0.47 -0.47 -0.56 -0.48 -0.51
PU
Bib
BIC
1.40 1.06 1.71 0.68 1.80
1.40 1.27 1.55 0.75 1.12 0.48 3.31 0.99 0.06 4.30
- 0.01
3.44 1.91 -0.60 4.44
Lb
Bl
B,C
dabpd
0.00 0.00
0.00 0.00
0.00 0.00
-0.51 - 0.45
-0.58 - 0.32
0.00
0.00
1.58 1.69
1.89 1.29
0.00
0.00
2.27 2.52
2.74 1.97
0.01 0.02 0.02 0.01 0.09 0.10 0.07 0.15 0.17 0.10
-0.04
0.00 0.00 0.10 0.00 0.00 0.14
0.00
The parameter data for model B, are optimal values from bond angle fitting, For model L have been computed with For second For first reference conformations. eq 7. The q .values used are those in the penultimate row of Table I. reference conformation. Maximum deviations from bond angle fitting.
(q,+) pair of values, introducing simple geometrical assumptions that not always can be physically justified. Application of model A leads to physically unreasonable variations in bond lengths, while in models B the bond length invariance condition is fulfilled. The geometry of the ring can be specified by nine internal coordinates, and, if the bond lengths are constrained to fixed values, the number of variables is reduced to four. Attempting to build up the ring geometry from the two (q,4) pseudorotational coordinates implies a simplifying reduction in two degrees of frleedom. This is the first qualification to be taken into account regarding the physical reliability of models B. In general four coordinates, for example, (q,$,ao,al),must be used, and the reduction to the (g,$) pair would be justified if the potential energy function E(q,4,ao,al)shows a well-defined local minimum for each pair of (q,$) values. A model B would be then physically reasonable if, for given (q,C$)values, the computed (ao,al) values are close to the ones corresponding to minimum energy. The check of these conditions is limited by the difficulty of computing the molecular energy accurately. A number of papers have appeared where four variables are used and the energy, computed by means of molecular mechanics force field1~5~6J6-17 or quantum chemistry14 methods, has been minimized with respect to some of them for given values of the others. These energetical models of ring puckering arie physically more justifiable than the geometrical models of the present work, but the numerical calculations are more laborious, and, in addition, approximate methods are )used in the energy computations. Using ab initio molecular orbital theory, Cremer and Pople14 have studied the ring puckering of the CP and T H F molecules as follows. First the angle 4 was fixed. Then the STO-3G energy was minimized with respect to the (q,ao,al)variables. Finally a partial geometry reoptimization was carried out at the 4-31G A list of the computed structural parameters for four puckered conformations of CP and eight of THF, as well as for the respective planar conformations, were reported in Tables I1 and I11 of ref 14. An examination of these puckered structures by meanti of model B1,using the same (g,C$) values and reference conformations, yielded bond and torsion angles rather close to the theoretically predicted ones, showing average differences for CP and THF, respectively, of 0.02 and 0.09' for the torsion angles and 0.11 and 0.52' for the bond angles. The respective maximum differences are 0.07, 0.34, 0.24, and 1.83O. From another point of view, models B can be checked against the experimental molecular geometries. The THT and THS molecules have been analyzed by electron diffraction by assuming C2symmetry,5v6and the results can be simulated by applying model B1to the second reference
conformations,with differences between experimental and computed bond and torsion angles within 0.1O. This very good agreement can be attributed to the fact of having selected a suitable reference conformation. The CP molecule has been analyzed by electron diffraction by assuming free pseudorotation.' Model B2 was used to simulate the puckered conformations. If model B1had been used instead of model B2, the same fitting between experimental and theoretical intensity and radial distribution functions surely would have been attained because, as can be seen in the last columns of Table 11, only small differences exist between the results of models B1 and B2 for the CP molecule. Similar considerations can be made for THF. In short, the physical reliability of models B cannot be rigorously checked, but conformations close to the theoretical and experimental ones can be simulated with these models.
Conclusions Owing to the existence of pseudorotation, the conformational analysis of saturated five-membered rings requires consideration of a large number of puckered conformations. These can be suitably simulated, for several purposes, from the (q,$) pseudorotational coordinates by using the purely geometrical models B1or B2 examined in this work. Alternatively, energetical models can be applied, but, in addition to implying far more laborious computations, the use of approximate methods in the calculation of energies can override the advantage of having a better physical basis. Models B preserve bond lengths. Conformations computed with model Bz are very similar to the ones simulated with model Bl provided that smaller q values are used for the former. Model B1presents the advantage that its (q,$) coordinates are consistent with those defined by Cremer and Pople.21 In both models a planar reference (conformationmust be chosen, but small changes on it have a limited influence on the resulting values of the torsion angles and of the variations of bond angles with respect to those of the reference conformation. The relation between the intracyclic torsion angles and the @,,A) pseudorotational coordinates (eq 2), proposed is quite accurately satisfied in saturated by Altona et al.,22 pentacarbocyclic sings. For rings with very unequal bond lengths expression 2 is clearly not applicable. To deal with these cases, the more general expression 12 has been deduced in this work. Our expression is correct for infinitesimal q values and shows that for each torsion angle 0, there is a maximum value, Om, instead of a unique Om, and a correction, e,, to be applied to the phase angle A. Expression 12 is consistent with torsion angles computed with models B, and its application to the determination of (q,r$)
3196
J. Pbys. Cbem. 1980, 84, 3196-3198
coordinates in a large series of nucleosides and nucleotides provides improved results over those obtained by using the formula of Altona et al.
Acknowledgment. This study was supported by the Comisidn Asesora de Investigacidn Cientifica y TBcnica, Madrid. References and Notes (1) J. E. Kilpatrick, K. S. Pitzer, and R. Spitzer, J. Am. Cbem. Soc., 69, 2483 (1947). (2) . . W. J. Adams, H. J. Geise, and L. S. Bartell. J. Am. Cbem. Soc.. 92, 5013 (1970). (3) H. J. Geise, W. J. Adams, and L. S. Bartell, Tetrahedron, 25, 3045 (1969). (4) A. Almenningen, H. M. Seip, and T. Willadsen, Acta Cbem. Scand., 23, 2748 (1989). ( 5 ) Z. Nahlovska, B. Nahlovsky, and H. M. Seip, Acta Cbem. Scand., 23. 3534 (1969). (6) Z. Nahlovska, B: Nahlovsky, and H. M. Seip, Acta Cbem. Scand., 24, 1903 (1970). (7) J . R. Durig and 0. W. Wertz, J . Cbem. Pbys., 49, 2118 (1968). (8) L. A. Carreira, G. J. Jiang, W. B. Person, and J. N. Willis, J. Cbem. Phys., 56, 1440 (1972). (9) W. J . Lafferty, D. W. Robinson, R. V. St. Louis, J. W. Russell, and H. L. Strauss, J. Chem. Pbvs.. 42. 2915 (1965). (10) J. A. Greenhouse and H. L. Sirauss, J. Chen;. Pbys., 50, 124 (1969). (11) G. G. Engerholm, A. C. Luntz, W. D. Gwinn, and D.0. Harris, J. Chem. Pbys., 50, 2446 (1969). (12) D. W. Wertz, J. Cbem. Phys., 51, 2133 (1969).
(13) W. H. Green, A. B. Harvey, and J. A. Greenhouse, J. Chem. Pbys., 54, 850 (1971). (14) D. Cremer and J. A. Pople, J. Am. Cbem. Soc., 97, 1358 (1975). (15) K. S. Pitzer and W. E. Donath, J. Am, Cbem. Soc., 81, 3213 (1959). (16) J. 8. Hendrlckson, J. Am. Cbem. Soc., 83, 4537 (1961). (17) H. M. Seip, Acta Cbem. Scand., 23, 2741 (1969). (18) E. Dbz, A. L. Esteban, and M. Ria, J. Magn. Reson., 16, 136 (1974). (19) A. L. Esteban and E. Dhz, J. Magn. Reson., 36. 113 (1979). (20) A. L. Esteban, Ph.D. Thesis, Universklad Complutense, Madrid, Spain, 1976. (21) D. Cremer and J. A. Pople, J. Am. Cbem. Soc., 97, 1354 (1975). (22) C. Altona, H. J. Geise, and C. Romers, Tetrahedron, 24, 13 (1968). (23) J. D. Dunitz, Tetrahedron, 26, 5459 (1972). (24) C. Altona and M. Sundaralingam, J. Am. Chem. Soc., 94, 8205 (1972). (25) On the inltial analysis by means of eq 12 of compound 11 in Table I of ref 24, a 35.9' value was computed for the torsion angle Bo, whlch is in disagreement wlth the experimentalvalue of 42.2' quoted there. This later value is erroneous, and a more rellable experimental value of 36.2' was obtained by us from the positional parameters given in Table I1 of J. Rubin, T. Brennan, and M. Sundarallngam, Biocbemisfry, 11, 3112 (1972). (26) It is worth noting that Cremer and Pople" made a comparslon of their 0.37- and 0.394-A 9 values for CP, which result by applying method I to the minimum energy conformations computed for the STO-3G and 4-31G basis sets, with the 0.43 A value obtained by analyzing the electron diffraction data by means of model B2? The agreement between the experlmental and theoretical 9 values is actually better because the comparison must be made against the average 9 value of 0.41 A which results when method I is used to analyze the conformations computed wlth model B, for q = 0.43
A.
H2 Formation in the Reaction of O('D) with H20 R. Zellner," G. Wagner, and B. Hlmme Institut fur Pbysikaliscbe Cbemie der Universifat,3400 Giittingen, Wesf Germany (Received: June 18, 1980; I n Final Form: August 5, 1980)
Hydrogen formation in the reaction of O(lD) atoms with water has been determined via the branching ratio k 2 / k l ,where (1)O(lD) + HzO 20H and (2) O(lD) + HzO Hz + 02, by direct measurement of the yields of Hz and OH in the flash photolysis of O3/HZ0/Hemixtures. kz/kl is found to be 0.01 (+0.005, -0.01) at 298 K. -+
-+
Introduction Reactions of electronically excited oxygen atoms O(1D) have been a matter of great interest both from the viewpoint of chemical kinetics and as important initiators of hee-radical chemistry in the upper &nosphere. The interactions of O(lD) with the trace constituents H20, H2, and CHI result primarily in the formation of OH, which tonether with H and HO, are dominantlv responsible for t h i destruction of odd oxygen (0,03) in"the Gpper atmosphere. Of particular interest in reactions of excited atoms is the magnitude of the branching ratio into various exothermic product channels and the ratio of total reactive to inelastic (physical quenching) interactions. In this note we report a measurement of the (reactive) branching ratio k z / k l for the interaction of O(lD) atoms with H20 O(lD) + HzO 20H AH = -28.4 kcal/mol (1)
O('D)
+ HzO
--
Hz + O2
AH = -47.2 kcal/mol (2)
The predominance of channel 1 has previously been suggested' and has henceforth been used as the exclusive reaction channel in models of atomospheric HO, chemistry. However, this suggestion has, to our knowledge, never been tested, and it appears that a direct determination of k2/kl 0022-3654/80/2084-3196$01.OO/O
-
should be made. This is important in view of a contradictorY theoretical Prediction (kz/ki 0.6 at 300 N2and because a branching ratio of k d k i > 0.05 would make O(lD) + H2O an important H2 source in the upper stratosphere. The rate coefficient for this reaction is well known k = 2.3 X 10-l' cm3/(molecule s), and is independent of temPeratureS3 Moreover, the extent of Physical quenching
o ( ~ D )+ H,O
-
+
o ( 3 ~ )H
~O
which has been reinvestigated in parallel with this work, was found to be completely negligible! Therefore, together with the branching ratio k z / k l ,the O(lD) + HzO reaction appears now to be well characterized. Experimental Section The investigation of the product channels in the O('D) H20 reaction was carried out in a static flash photolysis system using O3 photolysis in the Hartley continuum as a source of O(lD) atoms. The experimental arrangement of the flash photolysis system is similar to the one described r e ~ e n t l y . ~If one chooses a sufficiently large H 2 0 / 0 3ratio, predominant removal of O(lD) in the reaction with HzO is ensured. Direct photolysis of water vapor, which would also lead to OH formation via HzO +
+
0 1980 American Chemical Society
