Ab initio geometries

Rigid-geometry ab initio MO calculations of 86 torsional isomers of the dimethylphosphate anion (CH30)2P02 led to the determination of parameters for the Lennard-Jones type of nonbonded interaction, two- and three-fold torsional, and electrostatic interaction potential functions (215). Extension of this approach to full relaxation ab initio and MM schemes will be extremely useful, not only for phosphorus but also for other heteroatoms. 

As we have noted, the geometry changes involved are symmetry allowed in molecules of d ions (Fig. 3) 46), and since the motions that interconvert D h and C4 are natural vibrations of the system, we should expect easy interchange between the two geometries. Ab initio MO calculations on [RhH(C2H4)(CO)2(PH3)] found a very small intrinsic activation barrier to such motion 60). 

For systems whose geometries are experimentally unknown (as Is the case of the majority of the large oligomers and polymers studied), we must make use of other techniques in order to obtain reasonable input geometries. Ab-Initio techniques, even with small basis sets, rapidly become too expensive when large compounds are considered. As a result we often use geometries optimized with a method such as MNDO. 

The most widely applied, and apparently the most practical in terms of theory/-experiment correlation and predictive capability, has been the Valence Effective Hamiltonian (VEH) method propounded by Br6das and others [18, 217], which also commonly uses an initial ab initio optimization of monomer or oligomer geometry ("ab initio parametrization"), or less frequently, experimental geometries, prior to calculation. The VEH method has been used to arrive at ionization potentials, bandwidths, and, in the coplanar approximation, bandgaps, which reasonably follow experimental trends. 

At the time the experiments were perfomied (1984), this discrepancy between theory and experiment was attributed to quantum mechanical resonances drat led to enhanced reaction probability in the FlF(u = 3) chaimel for high impact parameter collisions. Flowever, since 1984, several new potential energy surfaces using a combination of ab initio calculations and empirical corrections were developed in which the bend potential near the barrier was found to be very flat or even non-collinear [49, M], in contrast to the Muckennan V surface. In 1988, Sato [ ] showed that classical trajectory calculations on a surface with a bent transition-state geometry produced angular distributions in which the FIF(u = 3) product was peaked at 0 = 0°, while the FIF(u = 2) product was predominantly scattered into the backward hemisphere (0 > 90°), thereby qualitatively reproducing the most important features in figure A3.7.5. 

Pulay P 1969 Ab initio calculation of force constants and equilibrium geometries in polyatomic molecules. I. Theory/Wo/. Phys. 17 197... 

Pulay P, FogarasI G, Pang F and Boggs J E 1979 Systematic ab initio gradient calculation of molecular geometries, force constants and dipole moment derivatives J. Am. Chem. Soc. 101 2550... 

FogarasI G, Zhou X, Taylor P W and Pulay P 1992 The calculation of ab initio molecular geometries efficient optimization by natural Internal coordinates and empirical correction by offset forces J. Am. 

Eckert F, Pulay P and Werner H-J 1997 Ab initio geometry optimization for large molecules J. Comput. Chem. 18 1473... 

An excellent, up-to-date treatise on geometry optimization and reaction path algorithms for ab initio quantum chemical calculations, including practical aspects. 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

Now, we discuss briefly the situation when one or both of the adiabatic electronic states has/have nonlinear equilibrium geometry. In Figures 6 and 7 we show two characteristic examples, the state of BH2 and NH2, respectively. The BH2 potential curves are the result of ab initio calculations of the present authors [33,34], and those for NH2 are taken from [25]. 

Figure 8. Three-dimensional mean-potential surface for the X IT state of HCCS, (Pi, Pa, y), presented in form of its ID sections. Curves represent the function given by Eq. (75). (with Ati — 0.0414, k2 — 0.952, tt 2 — 0.0184) for fixed values of coordinates p, and P2 (attached at each curve) and variable y — 4 2 4t Here y — 0 corresponds to cis-planar geometry and Y = ft to trans-planar geometry. Symbols results of explicit ab initio computations.

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

Fig. 5.35 Geometry predicted by CASSCF ab initio calculations of the two possible transition structure geometries for the Diels-Alder reaction between ethene and butadiene. (Figure adapted from Houk KN, J Gonzalez and Y Li 1995. Pericyclic Reaction Transition States Passions and Punctilios 1935-1995. Accounts of Chemical Research 28 81-90.)...

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