Exploring the potential-energy surface

The first derivative of the energy, E, with respect to the internal parameter, x, is then given by 

This equates to the gradient of the PES, but the product kx also equates to the forces on the atoms. Therefore, when the structure is far from optimized these forces will be appreciable. When the bond length corresponding to the minimum of the function is located the forces will fall to zero. 

The general first-principles geometry optimization scheme. 

Taking the second derivative of the harmonic potential yields  

The force constant is a measure of the stiffness of the bond or angle or, put another way, an indication of the energy penalty resulting from ehanging the stractural parameter away from its equilibrium value. This equates to the curvature of the PES, and in most geometry optimization sehemes the foree constants are also obtained to speed up the process of finding the equUibrium structure. 

---

In some situations we have performed finite temperature molecular dynamics simulations [50, 51] using the aforementioned model systems. On a simplistic level, molecular dynamics can be viewed as the simulation of the finite temperature motion of a system at the atomic level. This contrasts with the conventional static quantum mechanical simulations which map out the potential energy surface at the zero temperature limit. Although static calculations are extremely important in quantifying the potential energy surface of a reaction, its application can be tedious. We have used ah initio molecular dynamics simulations at elevated temperatures (between 300 K and 800 K) to more efficiently explore the potential energy surface. 

In this section we focused our attention to a rationalization of the factors determining the stereoselectivity through ab initio mixed quantum/classical (QM/MM) Car-Parrinello molecular dynamic simulations. We have used the same basic computational approach used in Section 3 to explore the potential energy surface of the reaction. Since the catalyst system, 1, is relatively large, we have used the combined QM/MM model system B as shown in Figure 3 and described in subsections 2.1 and 3.1. 

Density functional theory and a high-level cib initio procedure (G2+) have been used to explore the potential energy surface for the base-induced elimination reaction of fluoride ion with ethyl fluoride.11 The DFT barriers are smaller and looser than those predicted by the ab initio method but the nature of the transition state cannot be defined with confidence since the predictions are unusually sensitive to the choice of functional and basis set. The results suggest that improvement in density functional methods will require fundamental change in the functionals themselves. 

I. Matanovic et al., Exploring the potential energy surface for proton transfer in acetylacetone. Chem. Phys. 306, 201-207 (2004)... 

In this section we will present results of ab initio molecular dynamics simulations performed for more complex chemical reactions. Catalytic copolymerization of a-olefins with polar group containing monomers, chosen here as an example, is a complex process involving many elementary reactions. While for many aspects of such a process the standard approach by static quantum chemical calculations performed for the crucial reaction intermediates provides often sufficient information, for some aspects it is necessary to go beyond static computations. In the case of the process presented here, MD was priceless in exploring the potential energy surfaces for a few elementary reactions that were especially difficult for a static approach, due to a large number of alternative transition states and thus, alternative reaction pathways.77... 

Marti S, V Moliner (2005) Improving the QM/MM description of chemical processes A dual level strategy to explore the potential energy surface in very large systems. J. Chem. Theory Comput. 1 (5) 1008-1016... 

Ab initio molecular orbital theory is utilized to study the hydrogen abstraction reaction of n-bromopropane with hydroxyl radical and chlorine atom. The stability of the trans and gauche isomers of n-bromopropane is explored. The potential energy surface of both reactions is characterized by pre- and post-reactive complexes, as well as transition state structures in both trans and gauche isomeric forms. The importance of these two reactions relies on the ultimate product distribution from both reactions. Differences in the reactivity of 1-bromopropane toward OH and Cl are observed. The reaction of n-bromopropane with OH radical favors the abstraction of hydrogen atoms while the reaction with Cl atoms favors the abstraction of hydrogen atoms at the a and p carbon sites. 

To study CO2 on clean Pd(lll), two different clusters Pdio(7,3) and Pd 15(10,5) were selected to represent mono-coordinated and bi-coordinated adsorption modes respectively. The local/outer separation described above was employed, pseudopotentials and basis sets chosen according to this partition. The hybrid B3LYP density functional method was used to explore the potential energy surface. The different optimizations converged to three unique species corresponding to two coordination models only. For theri -C coordination two different species were found, one being a physisorbed and... 

Van der Waals Complexes a Tool to Explore the Potential Energy Surface in the Electron-transfer Region... 

The study of reaction paths, in DFT, is not a new 1114,115. Thus we have chosen to explore the potential energy surfaces (PES) introducing the possibility to rationalize the results through the computations of the global hardnesses along the whole reaction path, with the aim to verify if, for the studied processes, the maximum hardness principle (MHP) 53 is satisfied. 

Early LSDA static pseudopotential approaches to sodium microclusters date back approximately 20 years [122], see Appendix C. It would be misleading to consider LDA calculations as the natural extension of jellium models. However, the global validity of the latter cannot but anticipate the success of the former. Clearly, these should also clarify the role of the atomic structure in determining the electronic behavior of the clusters and the extent to which the inhomogeneity of the electron distribution is reflected in the measurable properties. Many structural determinations are by now available for the smaller aggregates, made at different levels of approximation and of accuracy (e.g. [110, 111], see Appendix C). The most extensive investigation of sodium clusters so far is the LDA-CP study of Ref. [123] (see Appendix C), which makes use of all the features of the CP method. Namely, it uses dynamical SA to explore the potential-energy surface, MD to simulate clusters at different temperatures, and detailed analysis of the one-electron properties, which can be compared to the predictions of jellium-based models. 

Modeling the inhibition activity of peptides and peptidomimetics containing epoxide ring against the cysteine protease Helter and co-workers explored the potential energy surface for interaction of oxirane (1), a, 3-epoxy carbonyl compounds (22,23) with methylthiolate-anion at BLYP/6-311 + G(d) and BLYP/TZV+P levels of theory [46, 47]. 

Exploring the potential-energy surface for solid-state structures... 

B. S. Jursic, Chem. Phys. Lett., 264, 113 (1997). Exploring the Potential Energy Surface of the Hydrogen Abstraction Reaction from Hydrochloric Acid with Hydrogen and Methyl Radicals Using Ab Initio and Density Functional Theory Methods. An Example of a Polar Radical Hydrogen Abstraction Reaction. 

Blomgren, R, 8c Larsson, S. (2005). Exploring the potential energy surface of retinal, a comparison of the performance of different methods. Journal of Computational Chemistry, 26(7), 738-742. 

The goal of quantum mechanical calculations is to explore the potential energy surface (PES) for a given atomic composition at r = OK. By augmenting the calculations with statistical mechanics approaches, the Gibbs free energy can be estimated at some required temperature, mostly at T =298K for the study of laboratory reactions, or at T = 310 K for the study of biochemical interactions in humans. 

---