Symmetry constraints linear variation method

The virial ratio is, as we noted above, 1.3366 for the separate-atom AO basis MO calculation, i.e. not 1.0. Now within the confines of the linear variation method (the usual LCAO approach) there is no remaining degree of freedom to use in order to constrain the virial ratio to its formally correct value (or indeed to impose any other constraint). Thus imposing the correct virial ratio on the linear variation method is, in this case, not possible without simultaneously destroying the symmetry of the wave function. Only by optimising the non-linear parameters can we improve the virial ratio as the above results show. Even at this most elementary level, the imposition of various formally correct constraints on the wave function is seem to generate contradictions. 

Instead of using repeated solution of a suitable eigenvalue equation to optimize the orbitals, as in conventional forms of SCF theory, we have found it more convenient to optimize by a gradient method based on direct evaluation of the ener functional (4), ortho normalization being restored after every parameter variation. Although many iterations are required, the energy evaluation is extremely rapid, the process is very stable, and any constraints on the parameters (e.g. due to spatial symmetry or choice of some type of localization) are very easily imposed. It is also a simple matter to optimize with respect to non-linear parameters such as orbital exponents. 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

Three important and related technical points remain which are not standardized but central for tribological experiments they are related to the optical detection method used for most of the AFM. First, the laser beam must be centered on the cantilever end. The centring can easily be checked by scanning parallel and perpendicular to the cantilever symmetry axis. Secondly, for each apparatus, the four photodiodes quadrants orientation must be carefully checked (one method is described in fig lb). Lastly, one has to work within the linear response domain of the photodiodes, which implies a limited variation domain for the vertical movement of the sample when the vertical difference from the photodiodes signal is used. This constraint can be detoured by using horizontal difference signal, see ref (28,52). 

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