Calculating potential energy surfaces

As pointed out earlier, to calculate the PES for a given system requires a solution of the Schrodinger equation, which is a difficult task even for the simplest H + H2 — H2 + H prototype three-electron system. In general, the methods currently used to calculate PESs can be classified in the following three major categories  

Ab-initio methods. These solve the Schrodinger equation with the highest accuracy possible. 

Semi-empirical methods. Normally, they use ab initio theory, but the theory is adjusted to fit known experimental results from spectroscopy or kinetic data. 

Empirical methods. These build the PES using models and formulae whose parameters are optimized by fitting the surface to empirical data. 

In the following part of this section, a short description of each category, with examples, is presented. For a more detailed insight into these methods, specialized books and monographs are available (see the Further Reading list at the end of the book). 

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The first basic approximation of quantum chemistry is the Born-Oppenheimer Approximation (also referred to as the clamped-nuclei approximation). The Born-Oppenheimer Approximation is used to define and calculate potential energy surfaces. It uses the heavier mass of nuclei compared with electrons to separate the... 

Figure 15. Calculated potential energy surface and geometries of intermediates of the V" + CO2 reaction. The energy of the lowest energy state for the quintet (solid hnes) and triplet (dotted lines) stationary points are shown. Energies are calculated at the CCSD(T)/6-311+G(3df) level, at the B3LYP/6-311+G(d) geometry and include zero-point energy at the B3LYP/6-311+G(d) level.

In another work, a series of substituted 2-aminoindans was analyzed using MM2-85108. The calculated potential energy surface for rotation around the CH—CH—CH2— N and CH—CH2—N—lp in model compound 70 is presented in Figure 9 and shows 9 minima. These were scrutinized as potential candidates for dopamine receptor agonists, according to several criteria The highest-energy conformation f (Erei = 7.3 kcalmol-1)... 

As expected from the calculated potential energy surfaces, the barriers to intramolecular rearrangement are generally higher for [M(bidentate)2(unidentate)] complexes than for [M(uniden-tate)3] complexes.74-77... 

Therefore, two parts of any molecular mechanics package that have a direct influence on a particular optimized structure, i. e., on the nuclear coordinates of a specific energy minimum on the calculated potential energy surface, are the mathematical functions and the corresponding parameters. The potential energy functions and the force field parameters are interrelated and, therefore, the parameters may not, in general, be transferred from one force field into another. 

Reactions between light molecules have been extensively studied in the last two decades, generally by molecular beam techniques (see Chapter 4, Section 4.2), and these have allowed detailed testing of the predictions made from calculated potential energy surfaces. There are three typical mechanisms for gas phase reactions. 

These are only rough guides to the true entropy change on activation. Detailed calculations using spectroscopic data for the reactants and a calculated potential energy surface for the activated complex will yield accurate partition functions for the translational, rotational and vibrational terms involved. Since the quantities contributing to the partition functions for each molecule will be different, then accurate calculations will be able to differentiate between such reactions as... 

P. L. Fast, M. L. Sanchez, and D. G. Truhlar, Chem. Phys. Lett., 306, 407 (1999). Multi-Coefficient Gaussian-3 Method for Calculating Potential Energy Surfaces. 

Fig. 15.6. Calculated potential energy surfaces for the two lowest electronic states of 1A" symmetry of H2S, (a) 1 lA" and (b) 2lA" the 1 lA" PES is the lower one. The bending angle is fixed at 92°. The arrows schematically indicate the initial motions of the nuclear wavepackets started in the dissociative and in the binding states after the photon has promoted the system to the excited states. Adapted from Heumann, Diiren, and Schinke (1991).

If a species has one calculated frequency very close to 0 cm 1 what does that tell you about the (calculated) potential energy surface in that region ... 

Tetrahedral N4 is expected to dissociate into two N2 molecules, but this reaction is forbidden by orbital symmetry. Dunn and Morokuma [33] characterized a transition state for the exothermic dissociation of tetrahedral N4 into two N2 and estimated the activation barrier to be 63 kcal/mol at the CASSCF(12e,12o) level, which indicates that N4 is a metastable species with significant kinetic stability. The calculated potential energy surface of N4 suggests that the low-lying triplet state might cross with the singlet surface (Fig. 3), which could reduce the activation energy barrier to about 30 kcal/mol [29,31,32],... 

Apart from spectroscopic constants, other observable spectroscopic features may be studied by consideration of calculated potential-energy surfaces. Notable in this respect is the understanding of observed Jahn-Teller splittings, a good example being the methane ion, CH4+.ia... 

Tresadern G, H Wang, PF Faulder, NA Burton, IH Hillier (2003) Extreme tunnelling in methylamine dehydrogenase revealed by hybrid QM/MM calculations potential energy surface profile for methylamine and ethanolamine substrates and kinetic isotope effect values. Mol. Phys. 101 (17) 2775-2784... 

Calculations of - reaction rates by the transition-state method and based on calculated - potential-energy surfaces refer to the potential-energy maximum at the saddle point, as this is the only point for which the requisite separability of transition-state coordinates may be assumed. The ratio of the number of assemblies of atoms that pass through to the products to the number of those that reach the saddle point from the reactants can be less than unity, and this fraction is the transmission coefficient , k. (There are also reactions, such as the gas-phase colligation of simple radicals, that do not require activation and which therefore do not involve a transition state.) See also - Gibbs energy of activation, - potential energy profile, - Poldnyi. 

A somewhat modified MO LCAO scheme, without restriction on the identity of spin orbitals (p and

unrestricted Hartree-Fock (UHF) method and is usually used to treat open-shell systems (free radicals, triplet states, etc.). Electron correlation is partially taken into account in this method, and therfore it can be expected to be more efficient than the RHF method when applied to calculate potential energy surfaces of chemical rearrangements whose intermediate or final stages may involve the formation of free- or bi-radical structures. The potentialities of the UHF method are now under active study in organic reaction calculations. Also, it is successfully coming into use in chemisorption computations (6). 

Fig. 16. The calculated potential energy surface for the interconversion of HSi[(p-(NH2)]2SiH, 13, and...

A series of DFT calculations see Molecular Orbital Theory) on Rh() -C3H5)3 indicate that the ground-state structure has no symmetry. Calculated ionization energies agree well with values obtained from photoelectron spectra. The calculated potential-energy surface indicates the presence of three transition states, one of which involves an n] -allyl ligand between the several minima that are found, and variable-temperature NMR measurements appear consistent with there being three distinct fluxional processes see Stability Constants their Determination) ... 

Fig. 14. Calculated potential energy surface of the MLCT state of Mn(H)(CO)3(DAB) as a function of the MnH and MnCO bond lengths (DAB, 1,4-diaza-l,3-butadiene). Adapted from Ref. 128).

Figure 10. The calculated potential energy surface of the Cu-Smci bond in the Cu(Im)2(SCH3)-(SCCHs) ) complexes [35]. Two curves are given for each oxidation state, one in vacuum and one in water (calculated with the CPCM method). The actual potential in any protein can be expected to found in between these two extreme cases. Reduction potentials can be found by forming the difference between the curves of the oxidised and reduced complex together with a hypothesis whether the Cu-Smo bond is constrained in the oxidised, reduced, or both states [68]. Note that 1 kJ/mole = 10.4 mV.

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