Energy profiles

PEP theory has also been applied to modelling the free energy profiles of reactions in solution. An important example is the solvent effect on the SN2 reaction... 

Figure A2.3.21 Free energy profile of the SN2 reaction Cl +CH2CI— [Cl-CHg-Cl]— CICH +Cl in the gas phase, dimethyl fonnamide and in water (from [93]).

Figure A2.4.8. Potential energy profile at the metal-vacuum interface. Bulk and surface contributions to Vare shown separately. From [16].

Figure A3.4.8. Potential energy profiles for reactions without barrier. Complex fomiing bimolecular reaction (left) and direct barrierless bimolecular reaction (right).

Figure A3.12.1. Schematic potential energy profiles for tluee types of iinimolecular reactions, (a) Isomerization, (b) Dissociation where there is an energy barrier for reaction in both the forward and reverse directions, (c) Dissociation where the potential energy rises monotonically as for rotational gronnd-state species, so that there is no barrier to the reverse association reaction. (Adapted from [5].)...

The rupture force measured in AFM experiments is given, therefore, by the average slope of the energy profile minus a correction related to the effects of thermal fluctuations. Equation (11) demonstrates that the rupture force measured in AFM experiments grows linearly with the activation energy of the system (Chilcotti et ah, 1995). A comparison of (10) and (11) shows that the unbinding induced by stiff springs in SMD simulations, and that induced by AFM differ drastically, and that the forces measured by both techniques cannot be readily related. 

The free energy profile or potential of mean force along a conformational coordinate may be defined as... 

Example Solvation can have a profound effect on the potential energy profile for a reaction. Jorgensen s research group provided important insights into the role of solvation. Consider the nucleophilic addition of the hydroxide anion to formaldehyde ... 

Fig. 5.21 The energy profile for the gas-phase Cl + MeCl reaction. (Adapted in part from Chandrasekhar J, S F Smith and PV L Jorgensen 1985. Theoretical Examination of the S 2 Reaction Involving Chloride Ion and Methyl Chloride in the Gas Phase and Aqueous Solution. Journal of the American Chemical Society 107 154-163.)...

Potential of mean force for the Cr+MeCl reaction in various solvents. (Figure redrawn from Chandrasekhar ] and W L Jorgensen 1985. Energy Profile for a Nonconcerted S>]2 Reaction in Solution. Journal of tho American Chemical Society 107 2974-2975.)... 

Fig. 11.37 Free energy profile for the nucleophilic attack of water on CO2 (a) in aqueous solution and (b) in the enzyme carbonic anhydrase. (Graphs redrawn from Aqvist J, M Fothergill and A Warshel 1993. Computer Simulai of the COj/HCOf Interconversion Step in Human Carbonic Anhydrase I. Journal of the American Chemical Society 115 631-635.)...

A potential energy diagram for nng inversion m cyclohexane is shown m Figure 3 18 In the first step the chair conformation is converted to a skew boat which then proceeds to the inverted chair m the second step The skew boat conformation is an inter mediate in the process of ring inversion Unlike a transition state an intermediate is not a potential energy maximum but is a local minimum on the potential energy profile... 

Figure 12 11 compares the energy profile for nitration of benzene with those for attack at the ortho meta and para positions of (trifluoromethyl)benzene The presence of the electron withdrawing trifluoromethyl group raises the activation energy for attack at all the ring positions but the increase is least for attack at the meta position... 

Fig. 4. Schematic representation of energy profiles for the pathways for the hydrogenation of a prochiral precursor to make L-dopa (19). The chiral...

Fig. 1. Free-energy profile for a kinetic resolution depicted by equation 1 that follows Michaelis-Menten kinetics.

The twist form is also called the flexible form, because without angle strain a left-handed twist (c) can be converted into a right-handed one (d), by way of a third form ca. 7 kJ mol higher in energy (and actually not strictly an intermediate, since it occupies a maximum on the energy profile) which is the boat form (e). 

Application of Eq. (30) corrects the free energies of the endpoints but not those of the intermediate conformations. Therefore, the above approach yields a free energy profile between qp and q-g, that is altered by the restraint(s). In particular, the barrier height is not that of the namral, unrestrained system. It is possible to correct the probability distributions P,. observed all along the pathway (with restraints) to obtain those of the unrestrained system [8,40]. Erom the relation P(q)Z, = P,(q)Z, cxp(UJkT) and Eqs. (6)-(8), one obtains... 

Eor many problems, the ideal umbrella potential would be one that completely flattens the free energy profile along q, i.e., UXq) = W(q). Such a potential cannot be determined in advance. However, iterative approaches exist that are known as adaptive... 

Figure 8 An accurate estimate of the barrier height can be found by adding a sufficient number of intermediate points in the discretized transition pathways. The solid line in the graph represents the energy profile for a reaction path described by 11 intermediate configurations of the system. The dashed line shows a coarse pathway described by only two intermediate configurations. The latter path underestimates the true energy ban ier.

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