Examples of Geometry Optimizations

Example Jensen and Gorden calculated the potential energy surface of glycine using ab initio and semi-empirical methods.This study is of special interest to developers of molecular mechanics force fields. They frequently check their molecular mechanics methods by comparing their results with ab initio and semi-empir-ical calculations for small amino acids. 

Jensen, J.H. Gordon, M.S. The conformational potential energy surface of glycine A 

The researchers established that the potential energy surface is dependent on the basis set (the description of individual atomic orbitals). Using an ab initio method (6-3IG ), they found eight Cg stationary points for the conformational potential energy surface, including four minima. They also found four minima of Cg symmetry. Both the AMI and PM3 semi-empirical methods found three minima. Only one of these minima corresponded to the 6-3IG conformational potential energy surface. 

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We have carried out a series of geometry optimizations on nanotubes with diameters less than 2 nm. We will present some results for a selected subset of the moderate band gap nanotubes, and then focus on results for an example chiral systems the chiral [9,2] nanotube with a diameter of 0.8 nm. This nanotube has been chosen because its diameter corresponds to those found in relatively large amounts by Iijima[7] after the synthesis of single-walled nanotubes. 

We use relativistically optimized geometries throughout. (See, e.g., ref. (52) for an example of the optimization procedure that was used for the uranium compounds.) The transition metal complexes M(CO)6 and [MO4]2-, M = Cr, Mo, W, are the only exception experimental geometries have been used in these cases (7). 

With respect to the question of symmetry of Pt(2-thpy)2, it is also of interest to compare the energy difference for different geometries of Pt(2-thpy)2. For example, a geometry optimization based on a DFT (density functional theory) method resulted in a distorted ground state structure, in which the ligands are... 

While all reliable calculations agree that the inversion barrier for (HF)2 is rather small, there is a good deal of sensitivity to the precise theoretical approach. For example, using geometries optimized for the minimum and the C2, transition state at the SCF level and with various standard basis sets, Del Bene et al. [94, 98] found that the SCF barrier varies between 1.1 and 5.0 kJ/mol. However, when correlation is applied directly to these same geometries, the results become quite erratic in some cases the barrier is negative, i.e. the C2h geometry is more stable than Cg. Inclusion of correlation effects directly into the optimization leads to inversion barriers of 4.5 [97] and 4.3 [96] kJ/mol. 

In this section we will try to explain why one would want to use quantum mechanics. We will make a brief foray into the theory, so as to gain an appreciation of the approximations involved. Geometric input and molecular orbital (MO) output will be illustrated by examples. We will then comment on basis sets and other terminology. Factors to consider in selecting an MO method will be presented. Some pitfalls of geometry optimization (energy minimization) will be pointed out. In the final part of this seaion, literature examples will show the quality of results expected. 

In this section, we will discuss general optimization methods. Our example is the geometry optimization problem, i.e., the minimization of (q). However, the results apply to electronic optimization as well. There are a number of usefiil monographs on the minimization of continuous, differentiable fimctions m many variables [6, 7]. 

FIGURE 8.3 Example of paths taken when an angle changes in a geometry optimization. (a) Path taken by an optimization using a Z-matrix or redundant internal coordinates. (A) Path taken by an optimization using Cartesian coordinates. 

Transition structures can be dehned by nuclear symmetry. For example, a symmetric Spj2 reaction will have a transition structure that has a higher symmetry than that of the reactants or products. Furthermore, the transition structure is the lowest-energy structure that obeys the constraints of higher symmetry. Thus, the transition structure can be calculated by forcing the molecule to have a particular symmetry and using a geometry optimization technique. 

Because geometry optimization is so much more time-consuming than a single geometry calculation, it is common to use different levels of theory for the optimization and computing hnal results. For example, an ah initio method with a moderate-size basis set and minimal correlation may be used for opti-... 

Usually the constants involved in these cross terms are not taken to depend on all the atom types involved in the sequence. For example the stretch/bend constant in principle depends on all three atoms. A, B and C. However, it is usually taken to depend only on the central atom, i.e. = k , or chosen as a universal constant independent of atom type. It should be noted that cross tenns of the above type are inherently unstable if the geometry is far from equilibrium. Stretching a bond to infinity, for example, will make str/bend go towards — oo if 0 is less than If the bond stretch energy itself is harmonic (or quartic) this is not a problem as it approaches +oo faster, however, if a Morse type potential is used, special precautions will have to be made to avoid long bonds in geometry optimizations and simulations. 

A comparison between Gl, G2, G2(MP2) and G2(MP2,SVP) is shown in Table 5.2 for the reference G2 data set the mean absolute deviations in kcal/mol vary from 1.1 to 1.6 kcal/mol. There are other variations of tlie G2 metliods in use, for example involving DFT metliods for geometry optimization and frequency calculation or CCSD(T) instead of QCISD(T), with slightly varying performance and computational cost. The errors with the G2 method are comparable to those obtained directly from calculations at the CCSD(T)/cc-pVTZ level, at a significantly lower computational cost. ... 

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