J. Phys. Chem. 1994, 98, 10440-10446
10440
Relationships between Torsional Angles and Ring-Puckering Coordinates. 5. Application to Six-Membered Rings Angel L. Esteban"and Enrique Ruiz Departamento de Quimica Fisica, Facultad de Ciencias, Universidad de Alicante, 03080 Alicante, Spain
Ernest0 Diez and Jesus San-Fabian Departamento de Quimica Fisica Aplicada, Facultad de Ciencias C-2, Universidad Autbnoma de Madrid, 28049 Madrid, Spain Received: January 4, 1994; In Final Form: June 13, 1994@
The DE relationships derived for infinitesimal puckering of a six-membered ring which describe the variation of the AS RP coordinates (@2, P2, @3) with the CP ones (qz, P'2, q3) and those giving the dependence of the endocyclic torsion angles @ upon both sets of RP coordinates have been applied to analyze a set of molecular mechanics geometries of rings (CH2)5X (X = CH2, 0, S ) , as well as a set of X-ray structures of derivatives. The parameters in the corresponding expressions have been estimated theoretically from the bond angles and bond lengths of planar reference conformations. The torsion angles @i are accurately described using the AS RP coordinates, the (T deviations amounting 0.3"for MM2 and 0.5" for X-ray geometries. The DE relationships describe worse the variation with the CP RP coordinates of the #j angles and of the AS RP coordinates. However, the accuracy of these relationships is greatly improved when the finite puckering effects are taken into account by substituting the 42 and q3 amplitudes for functions in both q2 and q3 with only four 1 parameters to be determined. Very accurate equations between the AS and CP RP coordinates are so obtained which permit one to analyze quantitatively the relation between the conformational spaces spanned by both kinds of RP coordinates.
Introduction The puckering of an N-membered ring may be specified by a set of N - 3 ring-puckering (RP) coordinates computed from either the 4j endocyclic torsion angles, the Altona, Sundaralingam, et al. or Fourier transform (FT)6-8RP coordinates (P, and P,, or the atomic Cartesian, the Cremer and Pople (CP)9310RP coordinates qm and P,. The definition of RP coordinates for a general monocyclic ring proposed by Cremer and Poplelo is a generalization of that introduced by Kilpatrick et al. for cy~lopentane.~ The three CP RP coordinates (q2, Pz,q3) for a six-membered ring (R-6) are computed from the atomic displacements zj perpendicular to a mean plane by using eqs 2 and 6 below. On the other hand, the definitions of RP coordinates for an N-membered ring based upon the torsion angles 4, are generalizations of that introduced by Altona, Sundaralingamet al. for the case of five-membered ring^.^,^ The three AS RP coordinates ((Pz, P2, (P3) for a R-6 are computed by using eqs 4 and 5 below. The general relationships between the AS and CP RP coordinates, and those between the torsion angles @j and both kinds of RP coordinates, were derived by Diez, Esteban, et al. (DE) assuming infinitesimal normal displacements from a nonequilateral planar reference conf~rmation.~,~ The particular DE relationships for a R-6 are given in the Methods section. The AS and CP E2P coordinates are related by the basic eq 11 from which the remaining eqs 12-19 can be easily derived. The variation of the torsion angles 4j with the CP and AS RP coordinates is described by eqs 20 and 21, respectively. The parameters (a,j, cmj) in eq 21, (a',j, d m j ) in eq 20, and bv in eq 11 can be estimated by parametrization from the corresponding ~
~
~~
~
~
~
~~~
~
~
~
* To whom correspondence should be addressed. @
Abstract pubhshed in Advance ACS Abstracts, September 1, 1994.
0022-365419412098-10440$04.50/0
data for appropriate sets of puckered rings. However, when handling the DE relationships, it is advisable to take advantage of the geometrical significance of all these parameters which may be calculated from the bond angles and bond lengths of a given planar reference conformation. The equations to be used to do these calculations for a R-6 are collected in the Appendix. In general, the bond lengths are known with the required accuracy so that the problem reduces to determine a set of independent angles to close to planar ring: three angles for an asymmetric R-6 and only two for an (CHZ)~X molecule. The DE relationships have been applied thoroughly to the puckering study of five-membered rings'." several years ago and recently of four-membered rings.12 On the other hand, the published applications to six-membered ring are scarce and the aim of this paper is to evaluate in detail the performance of such relationships for this important kind of rings. In the first application to R-6 of the DE relationships 21 for 4J((P2,P2, @3), a set of 96 unbridged pyranose rings from X-ray structures was used.' A standard deviation u of 0.37" for 4J was obtained, which was half of that resulting when using the eq 3, valid for equilateral rings. When the constraint was imposed that the parameters (a,,, cmJ)in eq 21 correspond to a symmetrical planar conformation with known bond lengths, the number of parameters to be optimized reduces to only two bond angles and u increases slightly to 0.40". The DE relationship for 4, (Q2, P2, (P3) was applied later by Ciarkowski13 and coworkers to the particular case of the R-6 of 2,5-dioxopiperazine using a total of 65 X-ray conformers. On the basis of their discussion of results, they recommend the use of the torsion angle formalism in any conformational study of a R-6 capable of existing in both chair (C) and flexible twisthoat (TB) forms at the same time. They conclude, "we have proved beyond any doubt that such a treatment ensures the maximum possible
0 1994 American Chemical Society
Torsional Angles and Ring-Puckering Coordinates
J. Phys. Chem., Vol. 98, No. 41, 1994 10441
The q2, q3, and q 2 values can be calculated by Fourier inversion as
3
q2 cos 1/1, = (1/3)lnxzj cos((2n/3)j) j
q2 sin ly2 = (1/3)”2xzj sin((2n/3)j) j
q3 = ( 1 / 6 ) ” ’ ~ z j cos nj I
j
/J5 0
On the other hand, the AS RP coordinates @2, @ 3 , and P2 for a R-6 are calculated from the torsion angles 4j and are based’s4 upon the expression for the torsion angles of a regular hexagonal derived for infinitesimal puckering:
Figure 1. Adopted notation for geometrical parameters in sixmembered rings.
accuracy both in describing the existing conformational states and in predicting any geometrically feasible states of a R-6 in terms of its torsion angles”. The description of R-6 conformers by means of the AS RPcoordinates has been analyzed in detail recently by Haasnoot. l4 A set of 8451 conformers of R-6 was extracted from the Cambridge Structural Database (CSD)15 for derivatives of the (CH&X molecules of cyclohexane (X = CH,), piperidine (X = NH), tetrahydropyran (X = 0),and thiane (X = S), as well as for derivatives of cyclohexene and 4-oxathiane. Likewise, sets of R-6 conformations are generated by molecular mechanics calculations. The validity of eq 3 for 4j (@2, P2, @3), which does not take into account effects of nonequilaterality of rings, was tested and the AS and CP RP coordinates were compared. In the present work sets of R-6 conformers were generated by molecular mechanics for the (CH2)sX (X = CH2, 0, S ) molecules and extracted from the CSD for derivatives. These sets have been used to test the validity of all the DE relationships for R-6. Since analytical expressions are used and the effects of nonequilaterality of rings are accounted for, a more accurate description and a deeper understanding of the puckered R-6 geometries have been achieved. The notation introduced in ref 1, which simplifies the writing of equations and facilitates comparisons, will be used hereafter. Equations quoted from references will be expressed directly in this notation which differs from that used in ref 4 (see page 54 in ref 1). The atoms are numbered from 0 to 5 and the endocyclic torsion angles around the bond joining atoms j and j 1 is denoted by 4, (see Figure 1). Symbols with zero superscript designate planar conformation values. Primed or unprimed symbols are used for parameters according to whether the CP or AS RP coordinates appears in the respective equations. If an atom numbering from 1 to 6 is preferred,j must be changed by j - 1 in equations, unless j is used as a subindex.
+
q5j = a, cos(P,
The
@2,
+ (2n/3)j) +
Pz,@ 3 may be
Q3 cos
nj
rewriting eq 3 as
4j = (a, cos P,) cos -j2 n - (a, sin P2) sin -j2 z 3
3
+
(a3) cos zj (4) The unknowns, in parentheses, can be determined from the 4, values by solving the linear equations system by a least-squares method. When all the six values of 4j are known the solution coincides with the Fourier inversion:
Gi (3 a3 (3
a2sin 1/1, = - - x 4 j sin -j j
= - x 4 , cos nj j
According to eq 8 in ref 1, for infinitesimal puckering of an equilateral R-6, the CP phase $Jz is 150“ shifted with respect to the AS phase Pz. Therefore, to facilitate comparisons of both kinds of RP coordinates, it is convenient to use, instead of v2, another phase P‘z defined as
P,= Q2 + 150”
(6)
The description of the R-6 conformations by three RP coordinates computed from the torsion angles 4, was fist carried out7 by Can0 et al. Then, they proposed6 to expand the 4j in a Fourier series: @j
= a.
+ a1cos Q3 cos
Methods Ring-Puckering Coordinates. The CP RP coordinates q2, and $J2 for a R-6 are calculated from the Cartesian coordinates of the ring atoms using the method devised by Cremer and Poplelo which defines an appropriate mean plane through the ring so that the atomic displacements z, perpendicular to this plane fulfill the condition 43,
( j = 0, 1, ..., 5 ) (1)
(3)
zj (7)
and to use the a,,and P,, values, computed by Fourier transform (FT), for the description of conformations. When the usually small amplitudes @o and @1 are not taken into account and all the six values of 4, are known the FT and AS methods become equivalent. The term @2 cos(P2 (n13)j) in eq 3 correspond to BIT pseudorotational path and the term @3 cos nj to C conformations. Therefore, any conformation of R-6 may be considered as a linear combination of a BIT and a C forms.16 The cylindrical coordinates (@z, P2, @3) may be replaced by a spherical polar set (a, 0, P2) where @ represents the total
+
10442 J. Phys. Chem., Vol. 98, No. 41, I994
Esteban et al.
puckering amplitude:
and 8 is an angle (0 I8
Iz ) such
For the case of the equilateral ring of cyclohexane b22 is equal to bll and the by are equal to zero when i f j . In consequence, the above equations reduce to
that
In the particular case of chair conformations ( q 2 = 0) the amplitude 0 3 is proportional to q3: Using this coordinate system, all types of puckering for a given amplitude @ can be mapped on the surface of a sphere. The poles correspond to C forms and the equator to the BIT forms. Likewise, the CP RP coordinates (q2, P'2, q3) may be replaced'O by the set (Q, 8',P'2):
Q3
= b33q3
On the other hand, for the pseudorotational pathway of boat/ twist-boat conformations (a3 = 0) the amplitude @2 depends upon both q 2 and P'2:
@, = (co
+ c2 cos 2P', + s2 sin 2P',)'l2q2
(18)
while P2 depends upon P5 only:
q2 = Q sin 8'
b,, sin P', P, = arctan b,, cos P', Relationships between the AS and CP RP Coordinates. For infinitesimal puckering the RP coordinates ( @ 2 , P2, @ 3 ) and (q2, P'z, q3) are related by the equations'
+ b,, cos P', + b,, sin P',
(19)
Relationships between Torsional Angles and RP Coordinates. The general relationships between torsion angles I$J and RP coordinates derived for infinitesimal puckering's4 reduce in the case of six-membered rings to
4l = ~ ' , ~cos(P, q,
+ + ( 2 d 3 ) j )+ a'3Jq3cos n1
(20)
for the CP RP coordinates and to
I$J = u2JQ2cos(P, where the bq terms may be computed from the bond angles and bond lengths of a planar reference conformation by means of eqs A9-AlO and Al-A5. According to eqs 11 each of the AS RP coordinate ( @ 2 , P2, C J ~depends ) upon all the three CP RP coordinates (q2, P'z, q3):
b,, sin P',
P, = arctan bll
cos P'2
+ b,, cos P', + b,,(q3/q,) + b,2
sin P'2
+ b13(q,/q2)
(12)
+ e5 + (2d3)j) + a3]@.,cos zj
(21)
for the AS RP coordinates. The parameters a'zJ, ~ ' 3 and ~ , e'zJ in eq 20 can be computed from the bond angles eoJand bond lengths lo, of a planar reference conformation (eqs Al-A5). The parameters a,,, a3,, and in eq 21 can be computed from the a'5, a'3], and e'zJ ones, (eqs A6-Al2). The eighteen parameters ag, a3,, and €5 in eq 21 are related by nine equations of constraint (eqs B2 in ref 1). This allows us to rewrite the general eq 21 for I$J as the eq 3 for an equilateral hexagon with an additional extension term which correspond to the contribution @O @ I cos(P1 zjj13)in eq 7. This terms depends upon nine parameters only being
+
+ Q3r3 cos P I = @,(s, cos P, + v2 sin P,) + Q3s3 sin P I = Q2(t2 cos P, + w2 sin P,) - Q3t3
Q0 = @,(r2 cos Ql
+
P,
+
24,
sin P,)
(22)
The parameters r2, u2, 1 3 , s2, v2, s3, t 2 , w2, and t 3 in eq 22 can be computed from the ag, a3,, and ones (eq A14). Sets of R-6Conformations. The relationships for the torsion angles 20 and 21, and between the AS and CP RP coordinates, 11, have been applied to the (CH&X (X = CH2, 0, S) molecules, using sets of molecular mechanics geometries, as well as to derivatives using X-ray structures. Sets of R-6 conformations, corresponding to sets of given values of the CP RP-coordinates ( q 2 , P'2, q3) were generated for the (CH2)sX molecules using the well-known MM2 program.17 For each molecule the input geometry for the MM2 program corresponding to given values of (q2. P'2, q3) was generated using a geometrical model. The MM2 optimization of each imput geometry was carried out by keeping unchanged
J. Phys. Chem., Vol. 98, No. 41, 1994 10443
Torsional Angles and Ring-Puckering Coordinates
TABLE 1: Geometrical Parameterf for the Planar Reference Conformation of the (CH2)sX (X = CH2, 0, S) Molecules X = CH2 x=o x=s MM2 X-ray MM2 X-ray MM2 X-ray 1.414 1.429 1.797 1.813 1.522 1'0 1.532 1.522 1.523 1.526 1.523 1.532 1.522 1'1 1.527 1.524 1.522 1.523 1.521 1'2 1.532 eo,, 120.00 120.00 122.32 122.32b 109.53 109.53b eo1 120.00 120.00 120.89 121.39 119.99 119.67 eo2 120.00 120.00 119.88 117.26 123.31 124.39
b
QI
Bond length lol (angstroms) and bond angles eo, (degrees). Fixed values.
TABLE 2: Parameters bi. and 3. for Relationships 11 (with q*, from Eq 24 Instead odq,) between the AS and CP RP Coordinates Based in MM2 Structures for (CH2)sX Molecules and on X-ray Structures for Derivatives" x=o x=s X = CH2 MM2 X-ray MM2 X-ray MM2 X-ray 89.63 90.11 80.71 80.90 bil 86.37 86.94 4.88 4.12 -0.52 -1.88 biz 0.00 0.00 12.69 13.64 -6.64 -5.67 bl3 0.00 0.00 4.88 4.52 -1.88 0.00 -0.52 bzi 0.00 87.94 86.34 86.07 86.94 89.02 b22 86.37 -1.87 3.27 -7.32 b23 0.00 0.00 3.83 -4.18 9.40 10.05 0.00 -4.91 b31 0.00 2.42 -5.42 -5.62 0.00 2.83 b32 0.00 bj3 -105.78 -106.48 -109.37 -109.01 -102.10 -102.01 LZ4 -0.16 -0.16 -0.18 -0.18 -0.14 -0.14 -0.48 -0.48 -0.51 -0.54 -0.54 A.23' -0.51 -0.25 -0.26 -0.26 -0.24 -0.24 d3' -0.25 -0.19 -0.18 -0.18 -0.18 -0.19 132' -0.18 a ( P 2 ) 0.00 0.03 0.03 u(@z) 0.29 0.30 0.20 ~ ( @ j ) 0.13 0.49 0.32 0.95 0.43 3.25 a The bi, in eq 11 were computed from the planar geometries in Table 1 by means of eqs A9, A10, and Al-A5. The 1 in eq 24 were obtained from least-squares treatments for MM2 geometries and fixed to the respective MM2 values for derivatives.
the zj values1* (restricted motion option in the program), maintaining so the initial values of (q2, P'2, q3). The (q2, P'2, q3) values were selected so that the total puckering Q, eqs 10, was varied in steps of 0.1 8, from 0.0 to 0.8 8, and P'2 and 8' were varied in steps of 30". Data sets of X-ray structures were extracted from the CSD for free derivatives of the (CH2)5X (X = CH2,0, S ) , molecules, i.e., for rings which were not constrained by the presence of extra fused or bridged rings. Those entries containing errors andor with a R factor greater than 0.1 andor with reported u (C-C) greater than 0.031 8, were removed. A total of 1366 X-ray structures in near chair conformation (0" I 8' I 10" or 170" I 8' I 180°), was retrieved comprising 855 derivatives of cyclohexane, 499 of tetrahydropyran, and 12 of thiane. Bond angles and bond lengths for planar conformations are given in Table 1. The MM2 values correspond to optimized planar rings. The empirical values were determined using eq 21. as described below.
Results and Discussion
RP Coordinates. The values of coefficients bv in the relationship 11 between the AS and CP RP coordinates are given in Table 2. The b,, were computed from the planar geometries in Table 1. In the simplest case of the equilateral cyclohexane ring eq 11 reduces to eq 16. The AS phase Pz coincides with the CP phase P'z and the AS amplitudes @ 2 and @3 are proportional to the CP amplitudes q 2 and q 3 . respectively, but
Figure 2. Variation of @2 with q 2 for C forms and of @ 3 with q 3 for BIT forms as determined from cyclohexane conformations generated by MM2. with different coefficients being lblll < lb33(. Scrutiny of the AS and CP RP coordinates from MM2 geometries shows that the deviations from eq 16 are smaller than 0.3" for the phase P2 but increase with the puckering for the amplitudes @2 and @3. The plots in Figure 2 of @2 versus q 2 for C forms and of @ 3 versus q3 for BIT forms show that, for finite puckering, the functional dependence between Qm and qm may be described by a polynomial expression with two or three terms rather than by a simple proportional relationship. For ring conformations which are combination of C and BIT forms @ 2 and 0 3 also depend somewhat upon q3 and q2, respectively. The variation of Qm with both qmand qn, (m,n)= (2,3) or (m,n)= (3,2), may be described by the expression resulting after substitution of qm in eq 11 for q*,:
s=l
t-1
A series of least-squares treatments was carried out for the RP coordinates from MM2 geometries in which various numbers of terms were admitted into eq 23 in order to evaluate the relative importance of these. When the series 23 is truncated in the form
and the qm in eq 11 are submitted for the corresponding q*m the results given in Table 2 are obtained. The (5 deviation are smaller than 0.1" for P2 and do not surpass 0.5" for @ 2 and @3. The values of each ilparameter are very close for the different molecules. In fact when using for thiane the il values found for cyclohexane the u deviations increase less than 0.1". Therefore, the deviations from the DE relationship 11 between the AS and CP RP coordinates may be effectively reduced 1" for the molecules under study using q*m in eq 11 instead of qm with the values A24, A232, A32, and A3z2, from cyclohexane geometries. For finite puckering of cyclohexane the variation of the AS with the CP RP coordinates may be described by eq 16 after substituting qmfor q*m. This does not affect to phases, i.e., for any conformation the AS and CP phases P2 and P'2 are equal or near equal in value. Likewise, the C and BIT forms are
Esteban et al.
10444 J. Phys. Chem., Vol. 98, No. 41, 1994
TABLE 3: Parameters for Relationships 21 and 20 Giving the Dependence of the Torsion Angles $j upon the AS and CP FW Coordinates Based on M M 2 Structures for ( C H Z ) ~ ~ Molecules and on X-ray Structures for Derivative#
300
a"
X = CH2 MM2 X-ray
200
100
0
-1oo~,,,,,.,~,,,,,,,,,,,,,,,,,,,,,,,,,,,l 0 100 200 300 PI' (degrees)
Figure 3. Variations of P2 with P'2 for two MM2-generated pseudorotation circuits of thiane of equal total amplitude Q (0.6 A) and for different phases 0'. described as such by both the AS and CP RP coordinates. However, for mixed forms the AS amplitudes for the contributions of C and BIT forms, and Q 3 , depend somewhat on the CP amplitudes for the contributions of BIT and C forms, q2 and q3, respectively, because of the presence of the terms &22@4z2 and i1232q2q32in eq 24. In the case of a nonequilateral ring (CH2)sX the interrelation between the AS and CP descriptions becomes more complicated since each (@2, Pz, @3) AS RP coordinates is a function of all the three (q2, P'2, q3) CP RP coordinates. For infinitesimal puckering these functions are given by eqs 11-13, for finite puckering each qm amplitude in this equations should be substituted for the corresponding q*m. Equations 11-13 reduce to eq 17, for C forms, and to eqs 18 and 19, for BIT forms. The C and BIT forms are described as such by both the AS and CP RP coordinates but in the BIT forms P2 shows a periodic variation of small amplitude with P'2 according to eq 19. Both phases coincide for symmetric forms, Le., for P'z(=Pz) = 60", 150", 240", and 330", because the following relation is fulfilled:
The analytical expressions 11-13, and q*m instead of qm, allow us to analyze quantitatively the differences between the conformational spaces spanned by the AS and CP RP coordinates, respectively. In particular, the changes with the ratio q3I q2 for the functional dependence between the phases P2 and P'2 are well described by the DE eq 12, which gives root-meansquare deviations of 0.84" for tetrahydropyran and 2.10" for thiane. These changes are illustrated in Figure 4 of ref 14 for MM2 geometries of 4-oxathiane as well as in Figure 3 of this paper where plots of Pz versus P'z are displayed for two pseudorotational circuits of thiane. When the relation q3Iq2 is small, the terms b23(q3/q2) in the numerator and b13(q3/q2) in the denominator of eq 12 become negligible and eq 12 reduces to eq 19 for BIT forms. The differences between P2 and P'2 are then moderate. Even for q3 = qz (e' = 45"), where the two forms C and BIT combine in similar proportions, the differences P2 - PI2 are smaller than 8", see in Figure 3 the plot for 8' = 45" and Q = 0.6 A. On the other hand, for near chair forms the relation q3Iq2 is large and the effect of terms b23(q3/qZ)and b13(q3/q2)in eq 12 is notorious replacing the near linear variation of P2 with P'2 found for BIT forms by a near periodic
x=o
x=s
MM2 X-ray MM2 X-ray 1,0000 1.oooO 1.0141 1.0145 0.9620 0.9594 1.OOOO 1.0000 1.0082 1.0021 0.9979 0.9976 1.0000 1.0000 0.9781 0.9838 1.0419 1.0448 -0.91 0.54 1.08 0.00 -0.11 0.00 3.01 -0.50 3.26 0.00 0.00 -1.50 -1.14 0.85 1.35 0.00 -0.68 0.00 1.OOOO 1.0000 0.9962 1.0065 0.9800 0.9783 1.0000 1.OOOO 1.0076 0.9871 1.0398 1.0434 1.0000 1.OOOO 0.9962 1.0065 0.9802 0.9783 0.21 0.35 0.46 0.23 0.63 0.47 86.37 86.94 95.85 95.63 68.80 68.15 89.35 88.71 89.48 87.12 86.37 86.94 86.37 86.94 82.78 84.47 93.26 94.43 1.75 -6.88 -7.13 0.00 0.00 2.11 0.00 0.00 2.13 2.61 -4.94 -5.19 1.07 0.70 0.88 -0.78 0.00 0.00 -86.58 -105.78 -106.48 -115.69 -115.41 -87.28 -105.78 -106.48 -109.99 -107.54 -106.37 -105.61 -105.78 -106.48 -102.42 -104.07 -112.65 -113.84 0.79 1.58 0.29 0.90 0.58 0.71 The a ' , and E'*, parameters in eq 20 and a , and em, in eq 21 were computed from the planar geometries in Table 1 by using the eqs A1A5 and All-A12, respectively. bThe ucp deviations correspond to eq 20 with q*mfrom eq 24 instead of qm and the 1 values in Table 2.
dependence of amplitude -30" in the case of the plot in Figure 3 for 9' = 5" and Q = 0.6 A. The planar reference conformations determined from X-ray structures are close to those computed using the MM2 force field; see Table 1. In consequence, the differences between the b,Jcoefficients in eq 11 from X-ray and from MM2 geometries are small. The I parameters in eq 24 could not be determined by empirical parameterization because all the used X-ray structures are near chairs with puckering amplitudes q3 in a close range, 0.5 < q3 < 0.7 A. The a deviations for a3 from X-ray geometries are larger than those for MM2 geometries and increase with the nonequilaterity of rings amounting to 3.25" for thiane derivatives. The main contributions to u arise from the changes in bond angles and bond lengths caused by the exocyclic substituents andor packing effects. Torsion Angles. The values for the a, and cmJparameters in the DE relationship 21 between the torsion angles r$J and the AS RP coordinates (a,,P ' 2 , @3) are given in Table 3 together with the values for the dmJand clmJparameters in the DE relationship 20 between r$J and the CP RP-coordinates (q2, P'2, q3). These values were computed from the planar geometries in Table 1. In the simplest case of the equilateral cyclohexane ring the eq 21 in the AS RP coordinates reduces to eq 3 and the root-mean-square deviation u for from MM2 geometries is 0.21". Similar low-a values result for the studied nonequilateral molecules when eq 21 is applied. However, the deviations from eq 3 increase with the nonequilaterity of the ring being u equal to 0.72" for tetrahydropyran and 1.14" for thiane. In the latter case u is 5 times larger for eq 3 than for eq 21. When the eq 20 in the CP RP coordinates is applied directly to the MM2 r$J angles, the a deviations are high, amounting to ca. 4". However, when the qm in eq 20 are substituted for the q*m in eq 24, with the I values in Table 2, the o deviations reduce to the values given in the bottom of Table 3. Even so, these u are larger than those from eq 21 and increase with the nonequilaterity of the ring.
Torsional Angles and Ring-Puckering Coordinates The planar reference conformations given in Table 1 for X-ray structures were determined as those that have the symmetry of the parent molecules, giving the minimum u deviations for 4, by applying to the empirical data sets eq 21 in the AS RP coordinates. The bond lengths for the planar rings were computed as the average of the corresponding experimental data. The bond angles are all 120" in the case of cyclohexane derivatives, but for nonequilateral rings two independent angles should be determined. However, as the X-ray structures are all near chair forms, it is not possible to determine both angles independently, and therefore the angle 00was fixed to the MM2 value. The so-calculated planar rings are close to the MM2 ones, and consequently the differences between the parameters a,, em,, almj, and d m j , from X-ray and from MM2 geometries are small; see Table 3. The u deviations for 4, from X-ray are somewhat larger than those from MM2 geometries due mainly to substituents and/or packing effects. When eq 21 is used in the AS RP coordinates the u are smaller than 0.7". On the other hand, for eq 20 in the CP RP coordinates the u are larger, even using q*minstead of qm,amounting 1.58"for thiane derivatives.
J. Phys. Chem., Vol. 98, No. 41, 1994 10445 tion. The particular expressions for a six-membered ring are given below. The parameters ~ ' z j a',,, , 6'2, in the relationship 4j (q2, P'2, q3), eq 20, can be computed by means of the following equations: a',, = (4/3)"2(sin
eoj sin Boj+,)-'
sin(n/3)[(R2,)
+ (S2,,)]'/* (All
d2,= arctan(S,j/R,,) R, = sin(eoj
(A2)
+ t9°j+l)/loj+l+ cos(2n/3)[sin OoJloj+2,,+l + sin 13~,+,/l~,,-~] (A3)
S2, = sin(2n/3)[sin Ooi/10j+2j+, - sin Ooj+,/lojj-,] a',,
[sin(eoj
= -(2/3)"'(sin
(A4)
eojsin O O ~ + , ) - I
+ 8°j+l)/lojj+,- sin B0J10j+2j+l - sin 8°j+l/lojj-,] ('45)
Conclusion An important objective of the ring-puckering theory is to derive relationships as accurate as possible between the geometrical parameters in a ring (intracyclic torsion angles $j, bond angles O,, and bond lengths lj) and the RP coordinates spanning the conformational space, as well as between the two kinds of these coordinates used in practice (the (@,,,,Pm) of AS and the (qm.P") of CP). The accurate description of the torsion angles using RP coordinates is of particular interest because the 4j angles may be derived from NMR coupling constants. The results reported in this paper confirm, for saturated sixmembered rings, the accuracy of the DE relationship between the 4, angles and the AS RP coordinates (a,, P2, a,),eq 21, despite the fact that the DE equations were derived for infinitesimal puckering. The worse performance of the DE relationship for the 4Jwhen using the CP RP coordinates ( q 2 , P'2, q3),eq 20, may be efficiently palliated by taking into account finite puckering effects. This is accomplished using, instead of the qmamplitudes, the q*mexpressions with four A parameters (eq 24). The DE relationship between the phases P2 of AS and P'2 of CP, eq 12, describes well the conspicuous changes with the ratio q 3 / q 2 in the functional dependence between both phases; see Figure 3. On the contrary, the DE relationships of proportionality between the amplitudes amof AS and the corresponding qm of CP for cyclohexane (eq 16) are not well fulfilled for high amplitudes, as Figure 2 illustrates. However, accurate relations between the AS and CP RP coordinates are obtained when the qm in eq 11 are substituted for the q*m in order to account for finite puckering effects.
Acknowledgment. This work was supported by the Direccidn General de Investigacidn Cientifica y TCcnica of Spain (Project PS90-0 157). Appendix The general relationships for an N-membered ring between the endocyclic torsion angles @j and the AS and CP RP coordinates (eqs 21 and 20), as well as that between both kinds of RP coordinates (eq I l ) , were derived in refs 1 and 4 for infinitesimal puckering from a planar conformation. The parameters in all these relations can be computed from the bond angles Ooj and bond lengths lo, of a planar reference conforma-
These equations correspond to eqs 4a-e in ref 4 and 15 in ref 1 (see page 54 in ref 1 for changes in notation with respect to ref 4). The AS and the CP RP coordinates are related by eq 11, which may be written in matrix notation as
R=jD
('46)
where E and 6 are column vectors with terms given by
r', = Qj, cos P2,
r', = a,sin P,,
rf3= a3(A7)
d', = q2 cos P'2,
d', = q2 sin Pf2,
d', = q, (A8)
and is a square matrix of order 3 with terms by which depend only on the 4,a',,, and ~ ' 2 parameters: j 'In
= '/jc.) C j ( n cos(2d3)j j
b3n
c.) j n cos nj
=
('49)
j
where the (c,)" are the terms of a vector ( c ~= ) ~a'2j cos(C2, (cj)' =
e, given by
+ (2d3)j) + (2n/3)j)
(c,), = a'3j cos nj
The values for the a,,, a3,, and c2j parameters in the relationships 4,(@2,@3,P2), eq 21, are obtained from the a'zj, a',,, and 6'2, ones by solving the equations
+ (2n/3)j) = (e,)] sin(e2, + (2d3)j) = -(ej),
a', cos(e,, a,,
a3,cos nj = (e,),
(All)
where the (e,), are the terms of a vector E defined as -?---I E,T-- CjB
(A 12)
10446 J. Phys. Chem., Vol. 98, No. 41, 1994
Esteban et al.
There are infinite sets of values of a'2j, a'3j, and ~ ' 2 jwhich give the same set of values of a2,, a3,, and €2,. These sets can be obtained by means of eq A12, rewritten as
ay = (k2j2
+ z~?)"~,
- sin(2njl3) cos( 2njl3)
cos(2njl3) sin( 2njl3)
+
r2 = 1 / 6 C a 2COS(E~, j 2nj13) j
+ 2nj13)
u2 = - l l 6 C a y j
r3 = '16Ca3,.cos nj i s2 = 113Ca2,cos(ey i
v2 = -'/3&j
+ 2nj13) cos nj13
sin(cy
+ 2nj13) cos nj13
j
s3 = '13Caycos nj cos nj13 j
t2 =
w2 =
+
1
I3CayC O S ( E ~2nj13) ~ sin nj13
-1~3Za2j sin(ry + 2nj13) sin nj13 j
t3 = 1I3Ca3, cos nj sin nj13
(A14)
j
From these nine parameters, the 18 a2j, a3j, and €2, ones can be calculated under the constraints of eq B2 in ref 1:
1. +
+ s2 cos(nj13) t2 sin(nj/3) + v2 cos(nj13) + w2 sin(njl3) a3, = 1 + r3 + t3 sin nj (A15)
1
r2
by giving arbitrary values to the terms of matrix B. The relationship 4,(@2,@3,P2) can be rewritten as eq 21 with 18 parameters or as eq 7 with terms @o, @ l cos P I ,and @ I sin P I given by eqs 22. In the latter case the number of parameters is reduced to the following nine:
= arctan(~~/k,~)
u2
I
References and Notes (1) Diez, E.; Esteban, A. L.; Bermejo, F. J.; Altona, C.; de Leeuw, F. A. A. M. J . Mol. Struct. 1984, 125, 64. (2) Altona, C.; Sundaralingam, M. J . Am. Chem. SOC.1972,94, 8205. (3) Rao, S. T.; Westhof, E.; Sundaralingam, M. Acta Crystallogr., Sect. A 1972, 37, 421. (4) Diez, E.; Esteban, A. L.; Guilleme, J.; Bermejo, F. J. J. Mol. Struct. 1981, 70, 61. (5) Esteban, A. L.; Galiano, C.; Diez, E.; Bermejo, F. J. J. Chem. SOC., Perkin Trans. 2 1982, 657. (6) Cano, F. H.; Foces-Foces, C.; Garcia-Blanco, S . Acta Crystallogr., Sect. A 1978, A34, S91. (7) Cano, F. H.; Foces-Foces, C.; Garcia-Blanco, S. Tetrahedron 1977, 33, 797. (8) Cano, F. H.; Foces-Foces, C. J . Mol. Struct. 1983, 94, 209. (9) Kilpatrick, J. E.; Pitzer, K. S.; Spitzer, R. J . Am. Chem. Soc. 1947, 69, 2483. (10) Cremer, D.; Pople, J. A. J . Am. Chem. Soc. 1975, 97, 1354. (11) de Leeuw, F. A. A. M.; van Kampen, P. N.; Diez, E.; Esteban, A. L.; Altona, C. J . Mol. Struct. 1984, 125, 67. (12) Esteban, A. L.; Galache, M. P.; Ruiz, E.; Diez, E.; Bermejo, F. J. J . Mol. Struct. 1991, 245, 333. (13) Jankowska, R.; Ciarkowski, J. Int. J . Peptide Protein Res. 1987, 30, 61. (14) Haasnoot, C. A. G. J . Am. Chem. SOC. 1992, 114, 882. (15) Allen, F. H.; Kennard, 0.;Taylor, R. Acc. Chem. Res. 1983, 16, 146. Allen, F. H.; Bellard, S.; Brice, M. D.; Cartwright, B. A.; Doubleday, A.; Higgs, H.; Hummelink, T.; Hummelink-Peters, B. G.; Kennard, 0.; Motherwell, W. D. S . ; Rodgers, J. R.; Watson, D. G. Acta Crystallogr. 1979, B35, 2331. (16) Cremer, D. Isr. J . Chem. 1980, 20, 12. (17) Allinger, N. L.; Yuh, Y. H. QCPE 1980, 12, 395. (18) Diez, E.; Palma, J.; San-Fabian, J.; Guilleme, J.; Esteban, A. L.; Galache, M. P. J . Comput. Chem. 1988, 9, 189.
