As noted before, the origins of the molecular mechanics method can be traced to work done by eg Barton (see reprints on van der Waals, electrostatic or the joined up analysis), Bright-Wilson (Molecular Vibrations, 1938. DOI: 10.1063/1.1750232) and others in the late 1940s and early 1950s and e.g. Allinger (starting in 1959, DOI: 10.1021/ja01530a049) and the MM2/3/4 force fields and Kollman/Amber in the 1970s-80s.
Organic chemists are taught that carbonium ion stability is predominantly due to electronic factors such as whether they are primary, secondary, tertiary, allylic, benzylic etc. What is mentioned much less is that their relative stability is also very sensitive to their geometries, and in particular the angles of the three substituents at the carbon. Geometries of carbonium ions are particularly inaccessible; it is not easy at all to get crystal structures! So how can Molecular Mechanics model the following carbonium ions? They are arranged in the expected order of increasing stability. Notice that the MM2 method gradually decreases the predicted angle energies. Notice also however how it predicts that the tertiary carbonium ion is actually LESS stable than the preceeding primary ion, which is surprising to say the least. Clearly, electronic factors must be playing an important role as well. Suffice to say that even without ANY consideration of electronic factors, the MM2 method does not do too badly in its prediction of relative stabilities. This re-enforces the conclusion that bond angles in carbonium ions are just as important as substitution! See also DOI: 10.1021/ed800058c for more detail.
stretch = 0.868 angle = 19.173 stretch bend = -0.672 dihedral = 6.497 improp torsion= 0.155 van der Waals = 2.663 electrostatics= -1.341 hydrogen bond = 0.000 Energy of the final structure is 27.342 kcal/mol. (QM B3LYP/cc-pVTZ → rearranges to final product directly by breaking/forming bonds)
stretch = 0.447 angle = 18.758 stretch bend = -0.078 dihedral = 6.916 improp torsion= 1.124 van der Waals = 1.622 electrostatics= 0.000 hydrogen bond = 0.000 Energy of the final structure is 28.788 kcal/mol. (Relative QM B3LYP/cc-pVTZ free energy = 25.7 Kcal/mol)
stretch = 0.571 angle = 5.694 stretch bend = -0.027 dihedral = 7.179 improp torsion= 0.009 van der Waals = 5.678 electrostatics= -0.545 hydrogen bond = 0.000 Energy of the final structure is 18.559 kcal/mol. (Relative QM B3LYP/cc-pVTZ free energy = 23.3 Kcal/mol)
stretch = 0.512 angle = 3.439 stretch bend = -0.153 dihedral = 6.662 improp torsion= 0.002 van der Waals = 3.872 electrostatics= 0.683 hydrogen bond = 0.000 Energy of the final structure is 15.017 kcal/mol. (Relative QM B3LYP/cc-pVTZ free energy = 0 Kcal/mol)
The H-bond energy functions in the Mechanics method are normally specifically parameterised against standard models, which involve three atoms, ie X...H...Y. The model of hydrogen bonding in this molecule is unusual to say the least, since it involves a whole ring and not a single atom as the H acceptor, and hence the Mechanics method (in theory) has no functional form which can reproduce it. In particular, aromatic rings are attractive to electrophiles by virtue of their quadrupole moment, which is a higher order property of the charge distribution (thus the so-called multipole expansion goes as monopole charge, dipole, quadrupole, octapole and hexadecapole. The equations above in effect only include the first and possibly second term of this expansion).
The optimised geometry does predict what looks like H-bonding, but its directionality is not quite correct. The steric energy is -49 compared with -18 if the two components are infinitely separated. This makes the binding energy = -31 kcal/mol. In this case, it is predicted to arise almost entirely from the non-bonded van der Waals term rather than electrostatic or hydrogen bonding (dipole-dipole) terms.
Stretch: 1.3257 Bend: 2.8090 Stretch-Bend: 0.2180 Torsion: -24.8762 Non-1,4 VDW: -3.1495 1,4 VDW: 10.7361 Dipole/Dipole: 4.1782 Total Energy: -8.7587 kcal/mol
Stretch: 2.5669 Bend: 6.6539 Stretch-Bend: 0.3319 Torsion: -54.0415 Non-1,4 VDW: -29.6591 1,4 VDW: 15.6989 Dipole/Dipole: 9.4549 Total Energy: -48.9942 kcal/mol
There is one final aspect which must be considered, namely the entropy of the interaction. The equation
ΔG = ΔH - T.ΔS
implies that if entropy decreases (i.e. ΔS is -ve) as a result of a reaction (or in this case a binding, in which two free molecules are reduced to one bound complex), then ΔG must increase (two -ve signs = a positive result) as a result by the amount T.ΔS. The Mechanics method could in principle provide an estimate of this term via calculation of the normal modes of vibration (and solution of the appropriate partition function equations), but few Mechanics programs actually perform this calculation (which requires the second derivatives to be evaluated, something only available using MM4, accurate to between 25-50 cm-1). An approximate estimate of T.ΔS is about -(12-15) kcal/mol at 298K (i.e. approx 2 kcal/mol per degree of lost freedom), which has to be added (-/- = +) to the binding energy obtained above, to obtain a final value of -(15) kcal/mol. This value is certainly in the correct ballpark.
Another property of this molecule can also be studied using molecular mechanics, namely the rotation and preferred orientation of the aryl-C bond, for which mechanics is very well suited. The following "dihedral driving" calculation shows the MM2 energy as a function of the dihedral angle between the trifluoromethyl group and the anthracene ring. The barrier (about 10-11 kcal/mol) compares reasonably well with the measured value of ΔG (from NMR lineshape analysis) of about 14 kcal/mol. The discrepancy may well be due to ignoring solvation of the OH group, which increases its size, and hence the barrier.
Reference: M. L. Webb and H. S. Rzepa, Chirality, 1994, 6, 245-250. DOI: 10.1002/chir.530060406
Fast forward to next installment of this story
Now that we have a quantitative method for estimating energies of molecules, we can tackle a problem such as shown below. An application of "shape design" is the recent area of designing catalysts to promote reactions by ensnaring them in a cavity, or binding pocket. Whilst designing enzymes to do this is somewhat beyond the scope of the current course, a simple example of small molecule design can be shown here. A resorcinol-anthracene based aromatic molecule was designed in the form of spacers (the aromatic rings) and links (in the form of hydrogen-bonds) to create in the solid state a system with well defined cavities, which are large enough to accommodate small organic molecules, such as ethyl methacrylate and cyclohexadiene. The former is held within the cage by further hydrogen bonds to the cavity lattice. In particular, various changes to the structure can be made, to increase or decrease steric bulk, etc and the effects of these changes can be modelled, ie predicted.
The previous systems were modelled by drawing/creating the model using the built-in editor (ChemBio3D, etc). Here we will be importing the X-ray derived coordinates. There are some important considerations to doing so.
When a standard MM2 mechanics force field is applied to the resorcinol-anthracene anvil, the following results are obtained:
|Supermolecule without substrates (Anvil)||-140.1|
|Anvil computed as four components||-92.8|
|Three Substrates together, without anvil||+14.2|
|Three separated Substrates||+23.4|
|Anvil + Substrates separately||-140.1+14.2=-125.9|
|Binding Energy of substrates to anvil||-157.2-(-125.9)=-31.3|
These energies are NOT corrected for entropy (i.e. they are closer (but not quite) enthalpic, and not free energies) a significant limitation. Very likely this sort of catalyst functions by converting enthalpic energies (the binding of the catalyst components) into entropic energies (i.e. pre-organising the reactants in the cavity, which of course consumes entropy. The free energy cost of assembling the three substrates is likely to be around +24-30 kcal/mol, but this is still likely to result in a win!).
What IS useful with the mechanics approach is the ability now to make minor variations to the environment, such as changing the nature of the substrates included in the centre of the anvil to see how the binding energy responds. Its a tuning procedure!
Literature Citation. Catalysis by Organic Solids. Stereoselective Diels-Alder Reactions Promoted by Microporous Molecular Crystals Having an Extensive Hydrogen-Bonded Network, K. Endo, T. Koike, T. Sawaki, O. Hayashida, H. Masuda, and Y. Aoyama, J. Am. Chem. Soc., 1997, 4117 - 4122; DOI: 10.1021/ja964198s
An interesting alternative to the above anvil is the molecular "tennis ball" (e.g. DOI) elaborated by Julius Rebek. Quite large molecules can be enclosed in these balls, and once in, they organize themselves very specifically, a phenomenon Rebek has termed molecular "social isomerism" (DOI: 10.1002/anie.200462839)
Just as with the anvil, the tennis ball is held together with hydrogen bonds, and this can be modelled using the Mechanics method, as can the organisation of the ligands inside the cavity.
Stretch: 5.1147 Bend: 49.5243 Stretch-Bend: 0.4564 Torsion: -3.2591 Non-1,4 VDW: -20.0160 1,4 VDW: 21.6595 Dipole/Dipole: 25.2888 Total Energy: 78.7685 kcal/mol
Stretch: 9.8872 Bend: 103.0995 Stretch-Bend: 1.2134 Torsion: -3.8372 Non-1,4 VDW: -87.3993 1,4 VDW: 43.0785 Dipole/Dipole: 29.6762 Total Energy: 95.7182 kcal/mol
Here, the energy liberated upon dimerisation is 62 kcal/mol, of which 41 is due to the non-bonded dispersion terms and 21 to hydrogen bonds. This is more than enough to overcome the entropic terms which acrue when substrates are trapped inside.