J. Phys. Chem. 1992, 96, 6624-6629
6624
Geometrical Distortion Caused by Substituents. 5. Method of Evaluating Expllcative Hypotheses for Constrained Systems Luis Vbzquez* and Miguel A. Rios Departamento de Quimica Fhica, Facultad de Quimica, Universidad de Santiago de Compostela, Spain 15706 (Received: December 11, 1991)
The introduction of substituents into a substrate molecule alters both local quantities such as hybridization angles (explicative variables) and, secondary to the foregoing, molecular geometry. Though the interpretation of changes in the latter in terms of the former is sometimes clear, there is no general model for their analysis. Interpretation can be especially difficult in cases (e.g., cyclic molecules) in which geometrical constraints introduce transmission and damping phenomena. In this article we approach this problem using a model that postulates simple relationships between the explicativevariables and the primary distortion of geometrical parameters from their characteristic (Le., unstrained or natural) values and then calculates an estimate of molecular distortion from the primary distortions by means of a linear transformation derived from the geometrical constraints and an idealized force field for the unconstrained molecule. Since in practice the coefficients of the linear transformation appear to depend mainly on molecular structure rather than on the idealized force field used, it may be possible to facilitate research on the mechanisms of distortion by determining, for individual molecular fragments, coefficients that can be used generally for molecules containing those fragments. The application of this method to a number of benzene and cyclopropane derivatives and to heterocycles treated as benzene derivatives is described. The results confirm the importance of constraint effects and illustrate the potential of the method for the interpretation of distortion.
1. Introduction The geometrical distortion of molecular structure by substituents can be measured experimentally or obtained theoretically (as the difference between the calculated structures of the substituted and unsubstituted molecules), but neither procedure involves the use of any formal scheme of the mechanisms by which the distortion is generated (e.g., by the alteration of hybridization angles or of electrostatic or steric interactions). Though the formulation and testing of such schemes is a legitimate goal of chemical research, there is currently no general theory allowing geometrical distortion to be derived from parameters such as those mentioned above. The desirability of such a theory is particularly evident in the case of the many molecules for which the interpretation of distortion is complicated by internal geometrical constraints. Substituted benzene rings, whose distortion has been the subject of many statistical or qualitative studies,’-8 are a case in point whose study led us to the formulation presented here. This article extends and unites our earlier work on distortion and its interpretati~n”~by taking into account both (a) the relationship between explicative distorting variables considered as the immediate expression of substitution (e.g., changes in hybridization angles) and the primary distortion of geometrical parameters from their characteristic (unstrained or natural) values and (b) the relationship between the primary distortion and the final geometry of the substituted molecule, which depends on geometrical constraints. In the approach presented here, the latter relationship is represented by effective constraint cmfficients whose calculation we describe. We are currently applying these methods to the most common cyclic molecules so as to produce coefficient tables suitable for use by the working research chemist. 11. Conceptual Model
By the characteristic value of a geometric parameter (e.g., a C-C bond length or a C-C-C bond angle) we mean its ideal (natural, typical, unstrained) value; the value it would possess if it were possible to isolate it from external influences and constraints. We consider the immediate expression of substitution to be the alteration of explicative variables such as hybridization angles, bond path angles, etc. As the result of such alterations, the geometric parameters are considered to undergo primary geometric distortion. The resulting primarily distorted values are thus held to be functions of the explicative variables and to reflect the immediate effects of substitution but not those of other external influences or constraints. To a first approximation, the primary distortion of a bond angle, for example, may be considered
as equal to the change in the corresponding hybridization angle or bond path angle that is effected by substitution, or the primary distortion of a bond length may be related to the change in bond order by an empirical factor. If E is the space of all values of a set of geometric parameters that is sufficient to describe a particular molecule, we shall denote by c the point whose coordinates are the primarily distorted values in a generic substitution state, and by q,the zero-distortion case, Le., the point defined by the characteristic values. The actual geometry e of the substituted molecule depends, in the present model, on the force field of the unsubstituted molecule, the primarily distorted values of its geometric parameters, and geometric constraints such as ring closure. We shall see that, to a first approximation, an estimate Ae of the actual distortion can be calculated from the primary distortion Ac = c - co using a matrix of effective constraint coefficients (ecc’s) whose mathematical definition and derivation we describe below. By comparing Ae with the experimentally observed distortion Ap, we can test the hypotheses that have been made concerning the functional relationship between the experimentally unobservable distortion Ac and the explicative variables. 111. Effective Constraint Matrix Notation. We use row vectors and the following conventions. Scalar variables appear in italic or Greek fonts, vectors in lowercase bold, and matrices in uppercase bold. Where convenient, the dimensions of matrices are indicated by subscripts. Vector or matrix transposition is indicated by a superscript t . Other symbols employed are 0, = the null vector of dimension a; Ooxb = the null a X b matrix; I,,, = the unit a X a matrix; m = the number of dimensions of E, the space of descriptive geometric parameters; n = the number of geometric constraints; h = m - n = the number of degrees of freedom of the problem. Effective Coustraint Matrix and tbe Ecc’s. Let be the point in E whose coordinates are the m geometric parameters of the unsubstituted molecule at equilibrium, and let the molecule be subject to n geometric constraints that are mutually independent to fmt order, i.e., such that the matrix of their fmt-order expansion coefficients, QmXn, is of rank n. The point representing any geometry that satisfies the geometric constraints and involves only first-order distortion from equilibrium lies in an h-dimensional linear manifold Q that contains eo and satisfies (rl - r2)Qmxn = on (1) where r l and r2 are any two such points. Points in the neighborhood of a primarily distorted geometry c, whether or not they satisfy the geometrical constraints on the
0022-3654 /92/2096-6624%03.00/0 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6625
Evaluation of Hypotheses on Constrained Distortion molecule, can be regarded as possessing, relative to c, an energy Vc(r) due to distortion over and above the primary distortion c - q,. Expansion of this hypothetical energy function up to second order in (r - c) yields
Vc(r) = g(r - c)'
+ Y2(r - c)K(r - c)'
n
m
(3)
m
n
J=l
i= 1
a = 1, ..., m (4)
or, in matrix form g
+ (e - c)K + AQ' = 0,
(5)
where A = (A,, ..., An). The effect of small changes in c upon e can be calculated as follows. From eq 5
(e - ro)K + AQ' = (c - ro)K - g
(6)
Combining eqs 6 and 1 yields
and multiplication by the inverse of the square symmetric matrix in eq 7 affords
(8)
where
M,,,
I,,,
- Q(Q'K--IQ)-'Q'K'
(9)
Hence
e - ro = (c - ro - gK-I)M
(10)
and since g and K (and hence M) are assumed independent of small changes in c:
Ae = (Ac)M
3
93
92 b2
4
Benzene
Cyclopropane m=6 n=3 h=3
r=(e,,,e3,b , , , 6 , ) N p e 1. Parameter notation and ordering.
where the Xi are Lagrange multipliers and ro is an arbitrary reference point satisfying the constraints on the molecule. At e
aG/ar, = g, + CKja(ej- cj) + EAiQai = 0
fj 1
(2)
where gm is the vector of first-order expansion coefficients and K,,, the matrix of second-order coefficients. The relationship between K and force field matrices for displacements from e, the corresponding equilibrium geometry of the constrained molecule, is discussed in part I of the Appendix. The equilibrium geometry e in a substitution state corresponding to c is that minimizing
G = V, + CXiCQji(rj - roil j=l
1
(11)
We term M the effective constraint matrix, and its elements effective constraint cwfficients (ecc's). Thus an ecc Muquantifies the distorting effect exerted upon f j by an alteration of the characteristic value cp Note that M is lnvariant under nonsingular column transformations of the constraint matrix Q (substitute QP for Q in eq 9, where P is an arbitrary nonsingular n X n matrix) and that M cannot have an inverse (because if it did, eq 11 would have a unique solution for all values of Ae, including those that fail to comply with the constraints of eq 1, which by hypothesis must be satisfied by Ae). The singularity of M means that we cannot, given M, work back from geometric distortions Ae to primary distortions Ac, as we would like to do; instead, we can test hypotheses concerning Ac only by using eq 11 to calculate values of Ae that we can then compare with experimental results.
Calculation of M, which may involve the transformation of a published force field (seebelow), is readily performed by computer. The program used for the results presented in this paper was written in c by one of us (L.V.) and covers both calculation of M and all other aspects of the problem.
IV. Force Field Matrix K Since K, by definition, determines the second-order contribution to the energy change associated with unconstrained displacements from c in E, then once we have specified that it must be symmetric (as we may without loss of generality) it suffers none of the indeterminacy associated with force field matrices F,,, for displacements from e (see Appendix, part 11). In general, K cannot of course be determined experimentally, because c is not a physically realizable codiguration of the molecule if it lies outside Q. In view of the nature of our model, however, it seems reasonable to assume that for the constants Ku we may use symmetric force field constants Fij that have been determined for the corresponding coordinate pairs of structurally analogous unconstrained molecular fragments. There are two general sources of Such Fij. Procedure I. The Fij taken as Kij are set to convenient values (respecting any symmetry) or are transported from other molecules (or other parts of the same molecule) in which the fragment in question is unconstrained. Procedure II. In keeping with the results of part I of the Appendix, K is assumed to have the same effect in Q as a particular experimentally determinable force field for the molecule considered. The indeterminacy in the matrix expression F,,,,, of this force field (see Appendix, part 11) is removed by imposing conditions following from or in keeping with the conceptual model. Firstly, the matrix elements Ki, for any homologous coordinate pairs (see Appendix, part 111) must clearly be equal. If, as is generally the case, this symmetry condition is insufficient to determine K fully, then (see Appendix, part IV) the symmetry criterion may be relaxed (i.e., coordinate pairs that are structurally similar but not strictly homologous are in fact treated as homologous), and/or certain Ki{s may be neglected or set directly in accordance with ad hoc criteria. The number and kind of such additional conditions may, of course, be limited by inherent mathematical restrictions. V. Examples We illustrate the use of the methods described above by applying them to a number of benzene or cyclopropane derivatives and heterocycles treated as benzene derivatives. The parameters used in each case are shown in Figure 1 (since the aim of this article is to present the method rather than to discuss data exhaustively, we omit coordinates for extraannular atoms; test calculations have shown that their inclusion has a negligible influence on ring fragment results, as expected, and extraannular a ' s are of no interest because extraannular atoms are not geometrically constrained). From the coordinates of each unsubstituted molecule at equilibrium, the corresponding constraint
6626 The Journal of Physical Chemistry, Vol. 96, No. 16, 199'2 matrices Q can be calculated (by hand or by computer) in terms of R, the C-C distance in the molecule, as a
1
0
,..
1 0 -2
Vizquez and Rlos TABLE I: Sets of Idealized Force Constants and the Corresponding Sets of Effective Constraint Coefficients (Ecc's)' Benzene El4 C14 DI4 E force constants (u,) rb 1.23 0.33 1.23 0.2 0, 1.66 0.0 0,+1 0.26 -0.03 -0.70 -0.03 0.006 -0.23 O.OOb 0.0 0,+2 0.00 or+, 0.00 0.OOb 0.006 0.006 0.0 0.43 0.71 0.0 0.43 b, 0.49 0.0 b,+, 0.00 0.006 0.006 0.14 0.006 0.006 0.006 0.0 b,+2 0.00
- b 1 -2b
0
~~
F
A
I
L'
6, b,+, bl+2 b,+3
(1 2)
where a = 3'I2R and b = 3-'I2R. The matrices S(K)and S(M) defined in part I11 of the Appendix are similarly calculated from the symmetry point group of the molecule. Effective Colrptrpint Matrices and Coefficients. We calculated M using various matrices K obtained by procedures I and I1 of section IV. Among the procedure I K s we included several very exaggerated fictitious fields with a view to testing the sensitivity of ecc's to K. For some procedure 11 Ks we made the simple, rather crude, assumption that certain off-diagonal KGs were zero. Table I lists ecc values and u, values (=fKU; see Appendix, part 111) for some of the K's used,the columns headed r, and rb indicate the nature (angle or bond length) and relationship (ipso, ortho, meta, or para) of the corresponding coordinate pairs. Semitivity of M to K. The smallness of the differences among corresponding ecc's in the various columns of Table I, and of their differences with respect to ecc's calculated for other fields (results not shown), shows that M calculated as above, is relatively insensitive to changes in K. This conclusion is further supported by the ecc's obtained from the fictitious fields. We infer that in these molecules M is determined chiefly by molecular geometry (via Q and the structure of K), rather than by the actual details of K. This is a pleasing result, since (a) it is in keeping with the assumption made in eq 11 that M is independent of small changes in c, and (b) it means that the conclusions we arrive at below concerning the effects of constraint are not subordinated to the question of which force field is "best". "he Ecc's. For each ring considered, there are four ecc classes Correspondingto the following four submatrices of M: M", which relates primary angular distortion to final angular distortion; Mbb, which relates primary bond distortion to final bond distortion; M*, which relates primary angular distortion to final bond distortion; and Mba,which relates primary bond distortion to final angular distortion. The effect of M" on a primary distortion involving just a single angle (i.e., Aci # 0 for some angle parameter ci, Ac, = 0 otherwise) is to dampen the distortion at angle i itself (Mii 'Y 0.5 for benzene, Mii N 0.1 for cyclopropane), to transmit a smaller distortion in the opposite direction to the ortho angles, and to transmit minor distortions to remaining angles. Note that for benzene the predicted ratio Aeo,,ho/AeiwN -0.7 agrees very well with experimental results for monosubstituted benzene. Mbbis approximately a unit matrix, primary distortion of a bond length is almost wholly apparent in the final geometry and has virtually no effect on other bond lengths. This reflects the fact that bonds are more "rigid" than angles, because of which the distortion of a bond length tends to be compensated for by alterations in angles rather than in other bond lengths. The coefficients of Mba show that the angular alterations by which primary bond length distortion is compensated for need not be very large. Assuming primary distortion of a single bond, the predicted relationship between the final distortion of the bond and that of its adjacent angles agrees qualitatively with experiment, but at the same time shows that primary bond distortion is not sufficient to account for observed angular distortions. All the coefficients of Mabare small, Le., primary distortion of angles has almost no effect on final bond lengths. This means
10.28 6.77 0.93 0.80 0.00 -0.49 0.23 0.00
rbc
0,
o,+, o,+* e,+, b,
b,+, b,+2 b,
b,+,
0.51 -0.33
0.50 -0.33 -0.01 -0.00 0 16 0.16 0.01 -0.00 0.02 0.01 0.03 0.02 1.00 0.99 -0.01 -0.00 0.01 0.00 0.01 0.00
br+2 b,+j 0, -0.20 -0.20 e,+* -0.00 -0.00 e)+, 0.20 0.20
rb
1 2 3 4 5 6
Y 1 2 3 4 5 6 7
8
0, 0, 0, 0, b! b,
0, 0,+1
b, b,+, b, b,+, rbC
0, 0, 0, 8, b, b, b, b,
@,
@,+,
b, b,+, b, b,+I 0,
e,+*
7.00 7.00 0.92 0.92 -0.61 -0.61 O.OOb 0.006 ecc's ( o Y ) 0.47 0.49 -0.35 -0.34 0.01 0.01 0.19 0.18 0.05 0.05 0.02 0.04 -0.01 0.02 1.03 1.01 0.01 0.01 -0.01 -0.01 -0.03 -0.01 -0.22 -0.21 -0.00 -0.00 0.22 0.21
1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0
7.0 0.0 0.0 0.0
0.0
0.51 -0.33
0.61 -0.28
-0.02
-0.06
0.16 0.05 -0.01 -0.14 0.00 0.00 0.01 0.14 1.00 0.89 -0.00 -0.06 0.00 0.06 0.01 0.11 -0.20 -0.14 -0.00
-0.00
0.20
0.14
Cyclopropane A E" C force constants (u,) 0.46 0.5 1.52 O.OOb 0.0 0.07 0.OOb 0.0 0.43 0.OOb 0.0 0.03 4.28 5.0 6.93 0.59 -0. I3 0.0 ecc's (u,) 0.08 0.08 0.13 -0.04 -0.04 -0.07 -0.04 -0.03 -0.02 0.07 0.07 0 15 0.92 0.92 0.87 0.04 0.04 0.07 -0.34 -0.34 -0.31 0.67 0.68 0 61
'Force-field references are given in column headings.
D 0.5
0.0 0.0 0.0 1 .o 0.0 0.27 -0.13 -0.12 0.23 0.74 0.13 -0.23 0.46
The units of
u, and uv are appropriate for displacements in angstroms and radians;
the potential energy will be in mdyn A. Idealized force fields were obtained for benzene ( A ) by procedure I, using free AM1 force constants transferred from hexatriene, (E-D) by procedure 11, using different sets of eqs 16, and ( E , F) by procedure I, using fictitious force constants. For cyclopropane, ( A ) by procedure I, using free AMI force constants for butane, ( E ) by procedure 11; (C, D)by procedure I, using fictitious force constants. bValues imposed by eq 16 in procedure I1 (see part IV of Appendix). indicates the nature of the primarily distorted parameter, rb that of the other member of the parameter pair kind.
that primary angular distortion is compensated for essentially by alteration of other angles and that observed bond length distortion can only to a very limited extent be accounted for by primary distortion of angles. Variations in Hybridization Angles as Explicative Variables. The hypothesis that substitution-inducedangle distortion is due to changes in hybridization angles (the "rehybridization hypothesis") is widely resorted to in spite of its limitations and was used by the present authors in studies of distortion in substituted benzeneg,'*and in six-membered rings with nitrogen heteroatom^.'^ In these studim we consciously ignored the question of constraints, which was partially tackled in other papers'OJI (see
The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6627
Evaluation of Hypotheses on Constrained Distortion
TABLE II: Primary Distortions (Ac,), "beoretical DidOrtiOM (Ae,), and observed Distortions (Apt) for the Angular Parameters of Benzene and Cyclopropane"
ring position (i) 1
2
3
4
1
2
3
4
Benzene Derivatives
0.2
-0.3 1.6 -0.2
7.5 4.4 5.0
-1 .o -2.8 -2.9
-0.8 -0.1 0.3
-0.4 1.6 0.2
-0.5 -1.8 -2.1
-0.3 0.0 0.5
-0.3 0.8 -0.2
2.2 1.5 1.8
-0.6 -0.9 -1 .o
-0.5 -0.1 0.1
-0.3 0.6 0.0
0.8 0.7 0.8
-0.4 -0.4 -0.2
-0.2 -0.0 -0.1
-0.1 0.2 -0.1
-3.9 -2.0 -2.3
0.1 1.4 1.4
-0.0 -0.1 -0.4
0.1 -0.6 0.4
1.4 1.6 0.3
-1.3 -3.1 -0.5
-0.8 1.5 0.3
1.6 0.5 0.2
-0.9 -0.9 -0.3
-0.8 0.4 0.2
2.2 0.6 1.1
-0.8 -1.1 -1.4
-0.6 0.5 0.3
1.9 2.0 2.1
-1.0 -1 .o -1 .o
8.4 4.1 1.7
-0.8 -3.1 -1.0
-0.6
4.7 2.1 3.4
-0.0
Heterocycles with Nitrogen as "Benzene Derivatives" -18.3 -7.4 -3.9
-6.0 8.9 8.3
-2.9 4.1 4.6
-0.6 0.3 -2.2
-0.2 -2.1 -0.9
-18.3 -6.7 -4.2
-3.1 2.0 1.2
-1.2 0.5 -2.2
6:N,l:N Ac (E) he AP 1:N,4:N Ac (E) Ae AP
-0.1
-3.4 5.1 3.1
-18.0 -9.1 -4.4
-3.5 4.8 2.2
1.6 0.2 0.3
-0.8 -0.1 -0.2
-21.2 -2.1
-0.8 -2.4 -2.9
Cyclopropane Derivatives 0.6 0.1 0.0
-0.3 -0.0 -0.0
1-F,l-F Ac (E) Ae APE
*In degrees. See text for definitions. Molecular positions in Figure 1 . Notation of derivatives explained in text. Acis are substitution-induced variations of hybridization angles (7,) from STO-3G MO localized by Boys' (B) or Edmiston and Ruedenberg's (E) criteria. The 7 values in unsubstituted molecules are 121.47' (B) and 121.26' (E) for benzene, 117.19' (B) and 116.91' (E) for cyclopropane. Most Ac and Ap values were taken from refs 12 and 13. bSTO-6-31G* optimization. 'STO-3G optimization. section VI). Even when constraints were ignored, rehybridization proved to allow semiquantitative interpretation of angular distortion, except for certain substituents (notably NH2). We therefore employ the rehybridization hypothesis in this article to illustrate how the methods described in the previous sections can be used to examine explicative hypotheses. Specifically, we hypothesize that, to a first approximation, the contribution of changes in hybridization angles AT^) to the primary distortion Ac caused by substitution is zero for bond lengths and, for bond angles, equal to the changes in the hybridization angles themselves (i.e., with the parameter ordering of Figure 1, Ac,,, = (AT', ..., AT,, 0, ..., 0), a = 6 for benzene and a = 3 for cyclopropane), and we accordingly compare the angular coordinates of the experimental distortion Ap with the corresponding ones of Ae, the result of rehybridization calculations after transformation by M (by eq 11). To evaluate the utility of the transformation, we also compare Ap with the untransformed variation in hybridization, AT = ( A q , ..., Aca). Note that great precision in ecc values is not necessary, since (a) calculated hybridization angles depend on geometry, localization criteria, and quantum mechanical method and basis set, so that differences between the results of one method and another can sometimes exceed 1O, and (b) discrepancies in experimental values, though generally rather smaller, may also exceed this figure. For each ring atom, the corresponding internal hybridization angle ri can be calculated as the angle between the directions of pseudohybrid atomic orbitals obtained from localized molecular orbital^.^-^^*'^ In this work we used the localization criteria of BoysI6 and of Edmiston and Ruedenbergl' with appropriate (M
separation.'* To prevent angular distortion from affecting the rehybridization calculations and so ensure that the alteration in hybridization would be a possible cause of geometrical distortion rather than vice versa, we imposed the geometry of the unsubstituted molecule on homocycle fragments bearing substituents; in the case of the heterocycles, in which the deviation from 120° at the heteroatoms was considered too large to be ignored, optimized geometries were used. Treating heterocycles as benzene derivatives is of course a drastic simplification, since it ignores the fact that in these cases the nature of the "substituted positions" certainly changes, Le., the force field might change sufficiently to have a significant effect on the ecc's. The AT,%reported in ref 13 nevertheless exhibit good constancy and additivity from one ring position to another and are much greater than the deviations from 120' postulated for their calculation, which suggests that, as an approximation, it is worth seeing what the method described here makes of them. Table I1 lists, for the angular parameters of a number of benzene and cyclopropane "derivatives" denoted by the substitution position and substituent (e.g., 1-NO2for nitrobenzene or 1:N for pyridine), the following data: (a) the primary distortion Ac calculated from ab initio hybridization angles as described above; (b) the corresponding theoretical geometrical distortion Ae calculated from eq l l using ecc's from Table I (column B for both benzene and cyclopropane); (c) the experimentally observed geometrical distortion, Ap. Note that since the ecc's derived from different nonfictitious K s hardly differ (as we have already seen), variation in Ae from this source would be negligible in comparison with the uncertainty in Ac and Ap. The reader should also bear
VBzquez and Rim
6628 The Journal of Physical Chemistry, Vol. 96, No, 16, 19
and no attention was paid to the origin of the primary distortions
TABLE III: Examples of Structure M8Mces for the Cyclopropane Ring with the Coordiartes of Figure 1 (Hydrogens Omitted)
r
1 2 2 3 4 3 2 1 2 3 3 4 2 2 1 4 3 3
4 3 L 3 4
3 6 5 3 6 6
1
6
5J
r2
5.
81
1 2 2 7 8 7
2 3 4 L 3
2 1 8 7 7 3 4 5 6 6 3 3 6 5 6 4 3 6 6 5J
I
in mind that the following paragraphs are intended to show how our model can be used, rather than as a conclusive discussion of the molecules referred to. Benzene 'Derivatives". In 1-C1, the value of Ac, is very much greater than the observed deviation, Ap,. Though the large value of Acl (=AT,) may be an artifact of the STO-3G calculations, it seems more likely to be a valid consequence of the presence of the second-row atom. The fact that .hel is still much greater than Apl (though less than Ac,), and, in particular, the fact that the predicted distortion of ortho and para angles deviates much more from the observed distortion than does that predicted by the corresponding Ac, themselves, suggests that rehybridization is not a sufficient explanation. Other mechanisms should therefore be investigated in subsequent work; such exploration is not the aim of the present paper. In l-NOz, 1-F, 1-CN, and 1-BH2the Ae, are closer than the Ac, to the Ap, for all angles except para. This does not, of course, definitively prove the validity of the rehybridization hypothesis for distortion by monosubstitution by these substituents, but is certainly strong evidence in its favour. In 1-CCH, the Ac,'s and Ap,'s are relatively small and agree well. Because they are small,the Ac,'s differ little from the Ae,'s, which accordingly do not approximate the Ap,'s better than do the Ac,'s themselves. For the diderivatives, Ae is a considerableimprovement on Ac. The Ae,'s agree in sign with the Ap,'s and are almost all closer to them than the Ac,'s are. The difference between Ac, = 7.4' and Ae, = 1.6O at the substituted positions of o-dichlorobenzene is particularly striking. For the heterocycles, too, the Ae's are much better estimates of distortion than the Ac's, which deviate greatly from the Ap's. Many Ae,'s have reasonably good values even when the corresponding Ac, appears not to be even qualitatively related to Ap,. Cycloproppne Derivatives. The bond path angle calculated for cyclopropane using the 6-31G* basis set, 78.8O,l8 implies that near the C atoms each bond path deviates by 9.4" from the corresponding C-C line. The calculated hybridization angles appear to overestimate this strain: T = 116.9", which is 56.9O wider than the C-C-C angle of 60" (implying a deviation of 28.4" between hybrid orbitals and the corresponding C - C lines) and is in fact wider than the tetrahedral sp3hybridhtion angle of 109.5". These values of e - c mean that for real (constrained) cyclopropane Mgs r - c is much too large for the approximation (terms up to the second order only) assumed in eq 2 to be valid. Even so, our model appears to work at a qualitative or semiquantitative level, since in both cyclopropane derivatives, Ae agrees remarkably well with Ap despite the significant deviation of Ac. Particularly striking is the improvement from 1.6" to 0.2" (for Ap, = 0.3") in 1-F,l-F.
VI. Final Remarks and Conclusions In ref 10 we described a first step toward the present approach. In effect, we considered single primary distortions only, and (in terms of the present notation) imposed the condition he, = Ac, when the primarily distorted parameter was e,. The transmission coefficients defined in that work can be shown to approximate M,,/M,, when the molecule is only slightly strained, as in the benzene derivatives considered here. A second step toward the present approach was described in ref 11, in which the consequences of special (deforming) coordinates s, (in the present notation, the AcJ were analyzed jointly rather than singly, and without the condition Ae, = Ac, used in ref 10. Final distortion Ae was predicted using a matrix equation, Ae = SA,that was similar in form to eq 11 of this work. However, A differed from M, being constructed from the transmission coefficients of ref 10,
We suggested above that the relative insensitivity of ecc's to changes of force field was indicative of their depending chiefly on structural features. As already noted, this, together with the fact that the uncertainty in Ac and Ap values will generally be greater than any change in Ae arising from ecc alterations due to force-field modification, justifies neglect of the effect of substitution on the force field. Furthermore, it turned out that most of the ecc's in this study were negligible, a result that, in view of the discussion given in section V, seemslikely to be generalizable to other ring systems. These two considerations open the prospect that once calculations have been performed for a sufficient number of ring systems-including planar and nonplanar, simple and condensed, aromatic and saturated, homocyclic and heterocyclic types-then it will be possible to use this data base to obtain "typical" ecc's that are approximately valid for similar molecular fragments of reasonably wide classes of compounds. This would make it unnecessary to m p u t e M for individual compounds when semiquantitative results are sufficient for the objective pursued. We are currently working to this end. Coociusiow. If the effects of intramolecular geometrical constraints are not taken into account, they can impede evaluation of hypotheses as to what explicative variables mediate substitution-induced distortion. For small distortions, Le., distortions that do not amount to conformational changes, the methods described in this article constitute a valid means of taking geometrical constraints into account. Given a hypothesis concerning the dependence of primary distortion Ac on explicative variables, good agreement between the observed distortion Ap and the theoretical distortion Ae can be regarded as evidence that the hypothesis is an acceptable one. Significant disagreement between Ae and Ap can be taken to disprove the hypothesis. In the case of the benzene and cyclopropane derivatives and heterocycles used as examples in this article, the hypothesis that substitution-induced angle distortion can largely be explained in terms of rehybridization was found to be tenable for some substituents but not for others (notably NH2 and Cl). The methods described in this article are not primarily designed to improve actual prediction of distortion, for which accurate energy minimization methods are available. Their purpose is to allow testing of explicative hypotheses formulated in terms that are ignored by energy minimization methods.
Acknowledgment. We thank I. C. Coleman for his contribution to the drafting of this article.
Appendix L Form of K on Q. To see the relationship between K (assumed symmetric) and symmetric matrices of force field constants for displacements from the molecular equilibrium geometry e, we substitute (r - e) (e - c) for (r - c) in eq 2, obtaining Vc(r) = Y2(r - e)K(r - e)r + g(e - c)' + V2(e - c)K(e - c)l + g(r - e)l + Y2(r - e)K(e - c)l+ !/?(e - c)K(r - e)' (13)
+
The last two terms on the right-hand side here are equal, being 1 X 1 matrices which, K being symmetric, are the transposes of each other. Hence the sum of the last three terms is (g + (e c)K)(r - e)'. If we introduce here the expression for g + (e - c)K obtained by substitution of eq 9 in eq 10, we arrive at
-(c - ro - gK-l)Q(QfK-'Q)-'Qf(r- e)r = 0 for r in Q, since Q'(r - e)' = ((r - e)Q)' = (on),by eq 1. Thus for real molecules, Le., molecules complying with the constraints Q, VJr) reduces to V,(r) = &(r - e)K(r - e)' + g(e - c)' + X(e - c)K(e - c)l Here the first term on the right-hand side has the form of the potential energy due to displacements from e in a quadratic force field defined by K, and the sum of the second and the third terms is a constant, the energy difference between e and c, which can
The Journal of Physical Chemistry, Vol. 96, No. 16, 1992 6629
Evaluation of Hypotheses on Constrained Distortion be ignored when it is only energy differences in Q that are in question. U. Imletermioacy of Farce FieMs in Redudant coordinrte SeQ. Once we have specified that they be symmetric, force field matrices Fhxhdefined, with respect to coordinate sets for Q, for displacements from e in Q are well defined, i.e., for a given force field and coordinate set there is a single h X h matrix. However, if the number of coordinates used is greater than the number of real degrees of freedom of the molecule (i.e., m > h), then matrices F,,, defined with respect to these larger coordinate sets are indeterminate, Le., given a force field and a coordinate set there are an infinite number of m X m matrices that have the same effect on vectors in Q: for any force-field matrix (F,,, + AF,,,) is equivalent to F,,, if, for r in Q, AF satisfies
(r - q)(AF)(r - e,# = 0 and by eq 1 this relation is certainly satisfied by AF of the form Q(X,,,,), where the elements of X are completely arbitrary. When using a force field matrix F for K (section IV,procedure 11), it is necessary to remove the indeterminacy of F because M is not the same for all equivalent F's. To see this, note that, by eq 9, dM = (I- - M)(dF)FIM, so that for dF of the form Q dx,dM = Q ( d x ) F I M - MQ(dx)F'M; but the second term on the right here is zero because MQ = Om,, (from eq 9), so that dM = Q(dX)FIM, which is not in general zero. III. Structure of K and M. Because of molecular symmetry, the set of all coordinate pairs ( i j ) (1 Ii, j Im) is the union of a number, say T, of mutually exclusive subsets ("pair kinds"), in each of which the members of each coordinate pair are geometrically related in the same way (Le., the pairs are mutually homologous); we arbitrarily index the pair kinds by a variable t (1 If I7'). For a given pair kind t, the elements Kij corresponding to its member pairs must all have the same value u, (or -u,, depending on the definition of coordinates); pairs ( i j ) for which Kij = 0 by symmetry are grouped in a generic null pair kind ( t = 0). Largely for programming purposes, the structure of K (and indeed of the molecule) can be represented by a symmetric m X m matrix S(K) defined by T
origin at e, whose first h basis vectors lie in (and span) Q. In a coordinate system of this kind (whose individual coordinates are of course no longer the values of our original geometric parameters), all points in Q are represented by coordinate vectors of the form ( a l ,...,ah, 0, ..., 0 ) = (a,o,). If A is the matrix performing this coordinate transformation, Le., if (a,o,,) = (r - q ) A for r in Q,then F is transformed to FA = A-'F(A-'),, and the potential V of displacements from q that satisfy the geometrical constraints is given by
v = f/z(~,O,,)F~(~,0,)' = y2aFIaf where FI is the symmetric h X h matrix formed by the first h rows and columns of FA. Note that the requirement that the last n new coordinates of points in Q all be zero is satisfied if Q is used as the right-hand m X n submatrix of A. In terms of the new coordinates, K becomes K A = A-lK(A-'),, and by eq 14 T KA
,=I
T
CU~B'~ = FI 1-1
(15)
where each B', is the submatrix of B, corresponding to F,. Equation 15 contains '/,(h(h + 1)) linear equations in Tunknowns (the u,). It can be solved using elementary column op erations to convert to column-canonical form and suitable thresholds as safeguards against rounding errors and the imprecision of published (FI),isi its equations must of course be consistent if there is no error in FI, the U,, or A. Let its rank be D. In general, D < T, Le., the number of independent equations is insufficient to provide a unique solution for the u,, showing that the symmetry considerations represented by ScK)are insufficient to remove all the indeterminacy from F. The remaining degrees of freedom, X = T - D,must be eliminated by introducing further conditions (of the kinds mentioned in section IV) in the form of additional linear equations: T
Cz,u, = 2
where the matrices U,are defined by
f=l
(16)
that are appended to eq 15. To set a particular ui to the value a, for example, the constants of eq 16 are zi = 1, Z, = 0 ( t # i), 2 = u
qj)= 1 if ( i j ) is of the tth pair kind and Kij = u, = -1
q)
The final expanded system must be of rank T.
if ( i j ) is of the tth pair kind and Kij = -uf
References and Notes
q)= 0
(1) Domenicano, A.; Vaciago, A.; Coulson, C. A. Acfa Crysfallogr. 1975, 831 (l), 221.
(2)Domenicano, A.; Vaciago, A.; Coulson, C. A. Acfa Crysfallogr.1975,
otherwise. K itself can be expressed as the sum
831 (6), 1630.
T f=1
I=l
where B, = A-lUf(A-l)f. The system of equations that identifies the idealized field represented by K with the force field represented by F, FA,and FI (in different coordinate systems) is therefore
,=I
K = Cu,U,
T
= A-'(EufU,)(A-I)' = EufBf
(14)
It can be shown, though the proof is lengthy, that M has the form Y
M = CoyVy y= 1
where y , Y,and V, are analogous to t, T,and U,,respectively, can also be defined. with Y 2 T; a matrix)'(S analogous to S(K) M is generally nonsymmetric, since in general some coordinates are "softer" than others (e.g., in benzene, primary distortion of a bond length has a greater effect on adjacent angles than primary distortion of angles has on adjacent bond lengths). Table I11 shows examples of matrices SK) and SM). IV. Expression of Force-Field Matrices with Respect to Coordinates Adapted to Q. To carry out procedure I1 of section IV, it is convenient to work with coordinates relative to a basis with
( 3 ) Domenicano, A.; Vaciago, A. Acta Crysfallogr.1979,835 (6), 1382. (4) Domenicano. A.; Mazzeo, P.; Vaciago, A. TefrahedronLen. 1976.13, 1029. (5) Domenicano, A.; Murray-Rust, P. Tetrahedron Leu. 1979,21, 2283. (6)Norrestam, R.; Schepper, L. A c f a Chem. Scand. 1981, ,435, 91. (7)Murray-Rust, P.Acra Crystallogr. 1982, 838, 2818. (8) Domenicano, A.;Murray-Rust, P.; Vaciago, A. Acfa Crysfallogr. 1983, 839, 457. (9)C a m p , J. A.; Casado, J.; Rios, M. A. J. Am. Chem. Soc. 1980, 102, 1501. (10)Rios, M.A.; Vfizquez, L. An. Quim. 1986, 82 (2), 170. (11) Vilzquez, L.; Ria, M. A. Mol. Phys. 1988.65 (l), 129. (12) Vizquez, L.; EstEvvez, C. M.; Rios, M. A. J. Chim. Phys. 1990.87, 565. (13) Vizquez, L.; Estbez,C. M.; Rios, M. A. J . Chim. Phys. 1992,89, 595. (14)Ozkabak, A. G.; Goodman, L. J . Chem. Phys. 1987.87 (5). 2564. (15)Zakharieva-Pencheva, 0.; FBrster, H.; Secbode, J. J . Chem. Soc., Faraday Trans 1 1986, 82, 3401. (16) Boys, S.F. Reo. Mod. Phys. 1960, 32 (2),296. (17) Edmiston, C.; Ruedenberg, K. Reo. Mod. Phys. 1963, 35. 457. (18)Wiberg, K. B.; Bader, R. F. W.; Lau, C. D. H. J . Am. Chem. Soc. 1987, 109 (4),985.
