Potential function parametrisation

Methods relying on parametrised potential functions for the description of energy hypersurfaces are commonly referred to as molecular mechanics (MM) or classical mechanics and these methods have a long tradition in computational chemistry. Entire data sets of balanced potential functions and their respective parameters are referred to as force fields [23,24,25], The key advantage of MM methods is the low computational demand compared to quantum mechanical computations. 

The representation of chemical systems on the basis of parametrised potential functions is by no means trivial and endless efforts have been devoted to balance and refine the employed parameters and functions yielding a considerable number of force fields along with various different parameter sets. These parameter sets are aimed at distinct classes of chemical systems, basically organic compounds, nucleic acids or proteins, whereas the treatment of compounds involving metal atoms is more difficult. 

The evaluation of interactions between particles inside and outside the quantum mechanical region is usually achieved on the basis of molecular mechanics, i.e. by the application of parametrised potential functions. Thus, parameters for partial charges and non-Coulombic interactions are required for all QM particles although these species are treated by quantum mechanics. The constmction of these functions is a time-consuming and tedious task requiring the evaluation of thousands of solute-solvent interaction points, which afterwards have to be fitted to an analytical representation in agreement with all other MM functions like the solvent-solvent interactions. As mentioned earlier the accuracy of these functions is in many cases insufficient for the treatment of polarisable compounds such as solvated ions [4,5,6,7,8], Sometimes these insufficiencies can be partially compensated by the inclusion of correction potentials as discussed above, but the accuracy is still not always satisfactory. 

We shall first introduce the parametrisation in the simplest possible manner, using some of the parameters already defined. It turns out that three parameters are needed to describe the energy dependence of the potential function P (E) with reasonable accuracy over an energy range of the order of A -B. Hence, on the basis of hindsight (Sect.3.4) we construct the following expression... 

Fig.2.8. Illustration of how the canonical bands i along one symmetry direction are transformed into unhybridised energy bands by the non-linear scaling prescribed by (2.12). The process is most easily understood if instead of the potential function P (E) one considers its inverse parametrised by (2.25)...

It follows from the energy scaling outlined in Sect.2.6 that the slope of the potential function P (E) provides a measure of the inverse of the width of the i band. To parametrise P (E) and establish a local approximation to the bandwidth we therefore wish to calculate... 

So far we have obtained parametrisations of the logarithmic derivative and potential functions which are appropriate when the Schrodinger equation is regarded as a differential equation, and which allow us to find and whenever E is given. In the ASA, however, Schrodinger s equation is treated as an eigenvalue problem subject to boundary conditions in the form of specified logarithmic derivatives at the sphere. Therefore, we need to find a parametrisation of the function E (D) inverse to D (E), valid around E. ... 

Currently, a wide variety of methods exists for calculating the molecular structure of large liquid crystal molecules which make use of pre-determined functional forms for the interactions in a molecule and semi-empirical information to parametrise the potentials. In general the interaction terms represent the energy cost of distorting bonds and bond angles from equilibrium. These can be expressed as... 

The local density approximation is highly successful and has been used in density functional calculations for many years now. There were several difficulties in implementing better approximations, but in 1991 Perdew et al. successfully parametrised a potential known as the generalised gradient approximation (GGA) which expresses the exchange and correlation potential as a function of both the local density and its gradient ... 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

It should be mentioned that there are only a few restrictions on the choice of the matrices C/j. Firstly they have to be unitary and analytical (holomorphic) functions on a suitable domain of, and secondly they have to permit a decomposition of Hm in even terms of definite order in the external potential according to Eq. (73). It is thus possible to parametrise them without loss of generality by a power series expansion in an odd and antihermitean operator Wi of ith order in the external potential. In the following, the physical consequences of this freedom in the choice of the unitary transformations will be investigated. Therefore we shall start with a discussion of all possible parametrisations in terms of such power series expansions. Afterwards the most general parametrisation of Ui is applied to the Dirac Hamiltonian in order to derive the fourth-order... 

Both the Finnis-Sinclair and the embedded-atom potentials (together with others that we have not considered here) can be represented using a very similar functional form. However, it is important to realise that they differ in the way that they connect to the first-principles, quantum mechanical model of bonding. They also differ in the procedures used to parametrise the models, so that different parametrisations may be reported for the same material. 

In what follows, we propose a phenomenological model of the chemisorbed radical-anion standing in the electrochemical double-layer. We shall hence detail the reasons why this chemisorbed radical anion is intrinsically unstable but most probably has a finite lifetime on the polarized metallic surface. We outline the procedure through which we expect that an order of magnitude of the lifetime of the chemisorbed radical-anion may be evaluated numerically via this model. The model potential felt by the radical-anion as it is formed on the polarized electrode is described as the sum of three terms, for which a parametrisation is proposed. One of these terms is meant to include both the surrounding solvent and the repulsion by the polarized electrode, thanks to a mean, locally uniform, electric field. In the present paper, the intensity of this uniform electric field is calibrated on the basis of a conparison between experimental Stark-Tuning shifts for CO chemisorbed on palladium surfaces in solution and Density Functional Theory calculations of field-induced vibrational shifts for CO chemisorbed on palladium clusters. The shape of the resulting model potential is then discussed. 

The results are remarkably alike, the only significant difference being the near disappearance of E(< )) in PEP401. This is one example of the impossibility of ascribing conformational differences to any one term of a potential energy function it all depends on the underlying model and its parametrisation. The total energy difference of 20 - 25 kJmol " corresponds to what is found by other... 

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