Geometry optimisation

The Hellmann-Feynman forces are also sensitive to the effect of moving ions on the basis set (pj) of the electronic wave function (, = This 

Recent applications of this method have shown that isolated benzene retains perfect six-fold symmetry to within numerical accuracy of the calculation [69] whereas in 5CB subtle symmetry breaking in the constituent phenyl rings occurs by contraction of the C—C bonds along the long axis of the molecule [69]. 

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A drawback of the SCRF method is its use of a spherical cavity molecules are rarely exac spherical in shape. However, a spherical representation can be a reasonable first apprc mation to the shape of many molecules. It is also possible to use an ellipsoidal cavity t may be a more appropriate shape for some molecules. For both the spherical and ellipsoi cavities analytical expressions for the first and second derivatives of the energy can derived, so enabling geometry optimisations to be performed efficiently. For these cavil it is necessary to define their size. In the case of a spherical cavity a value for the rad can be calculated from the molecular volume ... 

W C, A Tempcz)rrk, R C Hawley and T Hendrickson 1990. Semianalytical Treatment of Solvation for Molecular Mechanics and Dynamics. Journal of the American Chemical Society 112 6127-6129. ensson M, S Humbel, R D J Froese, T Matsubara, S Sieber and K Morokuma 1996. ONIOM A Multilayered Integrated MO + MM Method for Geometry Optimisations and Single Point Energy Predictions. A Test for Diels-Alder Reactions and Pt(P(t-Bu)3)2 + H2 Oxidative Addition. Journal of Physical Chemistry 100 19357-19363. 

It is feasible to carry out Hartree-Fock calculations on our available computer resources (an SGI Crimson Elan Workstation) using an STO-3G basis set with full geometry optimization of CeoMu but only partial geometry optimisations of the ChoMu isomers. Fig. 1 shows planar graphs of Ceo and C70 with the carbon atoms suitably labelled for future reference. 

Table 2 Restricted Hartree-Fock energies (Hartrees) for Ceo and C70 and their muon adducts. AE is the difference in energy between the carbon allotrope and its adduct. In all cases, except where indicated by f, only the six carbon atoms in the immediate vicinity of the muon have had there positions optimised, f means that a full geometry optimisation has been carried out. The type specifies the defect and for C70 is identified in Table 1. is the spin density at the muon in atomic units (and the hyperfine coupling constant in MHz). JMuon constrained to lie in equatorial plane. indicates geometry not fully optimized.

This prompted us [111 to try to represent CeoMu by clusters of carbon atoms, CisHuMu and C30H12MU, the external atoms being constrained to lie on a part of a spherical surface with the same radius as Ceo- The results were very similar to the CeoMu calculations with partial geometry optimisation to suggest that this adduct did not depend on the full structure but corresponded to a locdised defect , both structurally and electronically. 

This seems to imply that the associated distortion of the cage is only large for one atom (not two as for the type a structures) and therefore possibly better accomodated by the defects defined in Table 4. It must be concluded that either structures b and c are chemically unrealistic or there is a subtle, but significant, stabilisation provided by a complete geometry optimisation. 

Johnston, R.L., et al., Application of genetic algorithms in nanoscience Cluster geometry optimisation. Applications of Evolutionary Computing, Proceedings, Lecture Notes in Computer Science, Springer, Berlin, 2279, 92, 2002. 

The matrix elements of the momentum operator in (12) are now written in terms of derivative overlap integrals which can be obtained from the gradient package. These integrals are standard available in Ab Initio programs capable of performing a geometry optimisation. 

The SCF-MI algorithm, recently extended to compute analytic gradients and second derivatives [18,41], furnishes the Hartree Fock wavefunction for the interacting molecules and also provides automatic geometry optimisation and vibrational analysis in the harmonic approximation for the supersystems. The Ml strategy has been implemented into GAMESS-US package [42]. 

As regards SCF and SCF-MI calculations, the GAMESS-US program was employed, in which the SCF-MI algorithm including evaluation of analytic gradient, geometry optimisation and force constant matrices computation is available [18,41,42]. 

Cq of 2.40MHz, the unoptimised structure calculation gave Cq= 1.86MHz, whereas full geometry optimisation led to a much better calculated value of Cq=2.36MHz. These facts illustrate that the choice of the best structure to be used in calculations of Mg NMR parameters is still an open issue that needs to be addressed case by case. 

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