Conformational geometry functions

Schafer, L., M. Cao, M. Ramek, B. J. Teppen, S. Q. Newton, and K. Siam. Conformational Geometry Functions Additivity and Cooperative Effects. J. Mol. Struct., in press. 

The dependence of bond lengths and angles on associated torsional angles can be described by conformational geometry functions (CGF) which have the property of being approximately additive (L. Schafer et al. 1986G, in press, G). CGF additivity arises from the fact that the interactions encountered during torsional motion in a complex molecule can be approximately represented as the sum of the interactions encountered by individual structural components. For the case of ALA, for example, it is shown in Fig. 7.19 that... 

Fig. 7.18 Plots of relative N-C(a)-C angle values (surfaces of differences, in degrees, relative to the values at < > = / = 180°) for the ( ), /-space of ALA. The top surface represents values directly calculated for ALA as a whole by HF/4-21G geometry optimizations the center surface represents simulated parameter values which were obtained using the conformational geometry function additivity principle as described in the text. The bottom surface is the difference, top minus center. All surfaces were plotted with the same scale factor, but offset by arbitrary and constant amounts for the sake of graphical clarity. The numerical values used to construct this Figure were taken from L. Schafer, M. Cao, M. Ramek, B. J. Teppen, S. Q. Newton, and K. Siam, J. Mol. Struct., in press.

Fig. 7.19 Illustration of the additivity of conformational geometry functions for the -ro-tation in (CH3CO)(H)N-C(CH3)(H)(CONHCH3) (ALA). During the torsional motion about the N-C(a) bond of ALA, the interactions within the system are the same as those encountered during the N-C torsion in N-ethyl acetamide (NEA), plus those encountered during the N-C(a) torsion in N-acetyl N -methyl glycine amide offset by 120° as shown (GLY), minus those encountered during the N-C torsion in N-methyl acetamide (NMA).

In order to study this question in a more systematic way, we have recently optimized 144 different structures of ALA at the HF/4-21G level, covering the entire 4>/v )-space by a 30° grid (Schafer et al. 1995aG, 1995bG). From the resulting coordinates of ALA analytical functions were derived for the most important main chain structural parameters, such as N-C(a), C(a)-C, and N-C(a)-C, expanding them in terms of natural cubic spline parameters. In fact, Fig. 7.18 is an example of the type of conformational geometry map that can be derived from this procedure. 

The key to get a diabatic electronic state is a strict constraint i.e. keep local symmetry elements invariant. For ethylene, let us start from the cis con-former case. The nuclear geometry of the attractor must be on the (y,z)-plane according to Fig.l. The reaction coordinate must be the dis-rotatory displacement. Due to the nature of the LCAO-MO model in quantum computing chemistry, the closed shell filling of the HOMO must change into a closed shell of the LUMO beyond 0=n/4. The symmetry of the diabatic wave function is hence respected. Mutatis mutandis, the trans conformer wave function before n/4 corresponds to a double filling of the LUMO beyond the n/4 point on fills the HOMO twice. At n/4 there is the diradical singlet and triplet base wavefunctions. 

Figure 6. An energy surface as in Figure 2, but based on alternative assumptions about skeletal geometry and conformational energy functions as described in the text...

Finding the minimum of the hybrid energy function is very complex. Similar to the protein folding problem, the number of degrees of freedom is far too large to allow a complete systematic search in all variables. Systematic search methods need to reduce the problem to a few degrees of freedom (see, e.g.. Ref. 30). Conformations of the molecule that satisfy the experimental bounds are therefore usually calculated with metric matrix distance geometry methods followed by optimization or by optimization methods alone. 

---