J . Phys. Chem. 1987, 91, 3719-3720 energy. However, if so, it would correspond to an energy of 15-35 cm-I. According to these energy values, T, is not due to the spectral shift of the transmitted light which is about 200 cm-l. Another interpretation of B and T, has to be found. In our opinion, a more realistic interpretation of these phenomena could arise from the existence of an absorption band edge at hv,. The kT, value of 15-35 cm-I would be related to an energy difference h(vL- vo) in which vL is a mean effective laser frequency for this phenomenon. In the same manner, the value of the parameter B could depend on the same parameter XL - Xo and also on the sample concentration or total absorption. However, no real mathematical calculation has yet supported this analysis, and more detailed models based on the existence
3719
of a continuum of states are being performed in our lab in order to determine the real important parameters entering these phenomena. As a conclusion, if these temperature effects obtained from a self-diffraction experiment performed on the low-energy absorption edge of a dye molecule are not well understood, it seems to us that these experimental data are already in complete disagreement with a model based on a two-level system inhomogeneously broadened. Thus, more information on these effects is still needed to enlighten these stimulated and stimulating fields. Acknowledgment. M. T. Portella acknowledges the CNP, and FAPESP (Brazil) and the CNRS (France) for financial support.
Pyramidalization: Geometrical Interpretation of the .Ir-Orbital Axis Vector in Three Dimensions R. C. Haddon AT& T Bell Laboratories, Murray Hill, New Jersey 07974 (Received: May 4 , 1987)
The a-orbital axis vector (POAV) analysis provides a measure of the orientation (and hybridization) of a-orbitals in formally sp2-hybridizedorganic molecules which deviate from planarity. In the present Letter we provide a geometrical interpretation of the a-orbital axis vectors themselves, in terms of the underlying a-skeleton. It is shown that the POAV may be expressed in terms of the vector areas of the three triangles formed by the three o-bonds at the conjugated atom in question. The construction makes it clear that the term pyramidalization provides an apt description of the processes accompanying deviations from planarity in conjugated organic molecules. This analysis allows a connection to be established between the POAV approach and previous theories of molecular structure.
In previous publications on nonplanar conjugated organic molecules,'-5 we have introduced a nonparametric method based on orbital orthogonality for the calculation of the orientation (and hybridization) of a-orbitals in formally sp2-hybridized systems which deviate from planarity (Figure la). The a-orbital axis vector (POAV) analysis provides the most logical and natural bridge between the u-a separability assumed in planar conjugated systems and the realistics of a-bonding in nonplanar geometries. In the present Letter we provide a geometrical interpretation6 of the a-orbital axis vectors themselves, in terms of the geometry of the und_erlying_u-skeleton. Let VI, V, and V3be unit vectors lying along the three u-orbitals (taken to be directed along the internuclear axes to, the_adjacent %toms) radiating from a conjugated atom, and let V,,, V,( l ) , and Vr(2) be unit a-orbital axis vectors (general, POAVl and POAV2, respectively). Relationships involving the corresponding vectors of urbitrury length will be written as proportionalities; these vectors are a scalar multiple of the unit vectors, and their magnitude will be denoted N . Then we can recast the POAV2 orbital orthogonality relations (eq 15 of ref 3) in vector notation ( VI*V2)( P3*Pr) = ( P3*Pl)( P2*Pr)
PI. Pr)
Vr) PI*Pr)
( P2. V3)( Pff)= ( VI * P2)( P3. ( V3.VI ) ( VZ. = ( V y V3)(
(1)
c.q
where we have made use of the relations = cos e,, in rewriting the previous equations. From eq 1 we obtain ( 1 ) Haddon, R. C.; Scott, L. T. Pure Appl. Chem. 1986, 58, 137. (2) Haddon, R. C. Chem. Phys. Lett. 1986, 125, 231. (3) Haddon, R. C. J . Am. Chem. SOC.1986, 108, 2837. (4) Haddon, R. C.; Brud, L. E.; Raghavachari, K. Chem. Phys. Lett. 1986, 125, 459; 1986, 131, 165. (5) Haddon, R. C. J . Am. Chem. SOC.1987, 109, 1676. (6) The importance of the geometric and topological viewpoint in organic
chemistry has recently been stressed: Turro, N. J. Angew. Chem., Int. Ed. Engl. 1986, 25, 882.
0022-3654/87/2091-3719$01.50/0
0 1987 American Chemical Society
3720
N(1) = [(212*212)+ (j23’223)
POAV
4
A d
(C 1 (d) Figure 1. (a) %-orbital axis vector (POAV) shown for a nonplanar conjugated carbon atom ( 0 )bonded to atoms 1,2, _and 3 (pyramidalization has been exaggerated for clarity). (b) Vi, V,, and V, are unit vectors lying along the internuclear axes to the adjacent atoms 1, 2, and 3, whereas V, is thegeneral unit POAV. (e) A construction for POAVl . I (d) Vector areas ( A ) of the triangles formed by the faces of the tetrahedron with vertex at the conjugated atom.
These equations allow a straightforward geometrical interpretation of the a-orbital axis vectors. In the original derivation of the POAVl , the a-orbital axis vector was constructed by taking the cross pr$ucf of two yectors lying in the plane defined by the terminii of VI, V,, and V3(Figure IC) P*(l) lx (P2- VI) x (P3- V , )
=
(PIx
P2) a
Pr(1) =
+ (2’31’231)
+ 2(21+ 2(A23’”1)1’’~
This result is a reflection of the fact that the sum of the vector areas of the faces of closed polyhedra are identically zero. Thus, the orientation of POAVl may be interpreted as the sum of the vector areas of the three triangles formed by the three a-bonds at the conjugated atom in question (Figufe I$) or asthe normal to the plane formed by the terminii of VI, V2,and V3. In the POAVl analysis OI2 = 623 = &, and it may be seen that eq 5 and 8 are equivalent in this circumstance. In fact, it may be seen from the preceding analysis (eq 4 and 8) that the two treatments differ only in tke fact that_the POAV2 analysis does not utilize the unit vectors VI, V2,and V3 directly in the derivatizn of the v@or areas, but r$her their scaled counterparts: cos 023V1, cos 031V2,and cos OI2V3. Thus, the tetrahedron (Figure Id) employed in the POAVl analysis has three sides of common length (from the conjugated atom), whereas the tetrahedron employed in the POAVZ construction may lose all symmetry. It is clear that the term pyramidalization provides an apt description of the processes accompanying deviations from planarity in conjugated organic molecules. It should be noted that no sense has been associated with the POAV in this discussion. At this point the sense of the vector is arbitrary, although in connection with the 3D-HMO theory5 it was found convenient to take the sense of the POAV shown in Figure l a as positive. The POAV analysis may be interpreted in terms of previous theories of molecular s t r u c t ~ r e . ~The - ~ ~relationship between the POAV2 approach and hybridization theory’ is clear-in fact, the POAV2 analysis is based on a rigorous treatment of rehybridization within the constraints of the orbital orthogonality relationships. Although we have previously interpreted the POAV 1 analysis in terms of rehybridization, there is a clear analogy with the valence shell electron pair repulsion (VSEPR) theory,8 for in the POAVl treatment the a-orbital makes equal angles with the three a-orbitals. The above analysis makes clear that a rather simple geometrical construction underlies all of these treatments.
+ (P2 x P3)+ (P3x Pi)
212 + A 2 3 + 231
and where
Letters
The Journal of Physical Chemistry, Vol. 91, No. 14, 1987
(212 + 2 2 3 + 23l)/N(I)
(8)
(7) Pauling, L. Nature of the Chemical Bond, Cornell University: Ithaca, NY, 1960. (8) Gillespie, R. J. Molecular Structure; Van Nostrand Reinhold: New York, 1972. (9) Cotton, F. A.; Wilkinson, G. Advanced Inorganic Chemistry; Wiley: New York, 1980; Chapter 5 . (10) Burdett, J. K. Molecular Shapes; Wiley: New York, 1980.
