Splined potentials

Another way of representing atom-pair interactions, which lends itself particularly well to computational methods is the spline potential. A least-squares calculation fits various accurately known sets of numerical interaction energies at certain interatomic regions (perhaps from various theoretical and/or empirical investigations) into a functional form under conditions of matching first and second derivatives, so that a potential and its derivatives may be invoked at an arbitrary atom separation. 

It is also important to recognize that a particular potential form may impose a significant computational or algorithmic burden in practical calculations. Trigonometric and exponential functions are evaluated slowly, and lookup tables or splined potentials see, e.g.. Ref. 65) needed to speed some computation requires development effort on the part of the prt ammer. These... 

The functional forms of the potentials are the same as in a number of previous cases in which the same scheme was employed (Ackland, et al. 1987 Vitek, et al. 1991). They are cubic splines so that the functions which make up the potentials are ... 

The two computational methods, CMS-Xa and LCAO B-spline DPT, for now provide consistent, comparable results [57] with little to choose between them in comparison with experiment in those cases presented here (Sections I.D. 1. a and I. D.a.2). The B-spline method holds the upper hand aesthetically by its avoidance of a model potential semiempirically partitioned into spherical atomic regions. More importantly it olfers greater scope for future development, particularly as the inevitable increases in available computing power open new doors. 

A common problem for both methods lies in the use of potentials that do not possess the correct net attractiveness. This can have the consequence that continuum feamres appear shifted in energy. In particular, there is evidence that the LB94 exchange-correlation potential currently used for the B-spline calculations, although possessing the correct asymptotic behavior for ion plus electron, is too attractive, and near threshold features can then disappear below the ionization threshold. An empirical correction can be made, offsetting the energy scale, but this can mean that dynamics within a few electronvolts of threshold get an inadequate description or are lost. There is limited scope to tune the Xa potential, principally by adjustment of the assumed a parameter, but for the B-spline method a preferable alternative for the future may well be use of the SAOP functional that also has correct asymptotic behavior, but appears to be better calibrated for such problems [79]. 

Unfortunately, experimental difficulties precluded measurements closer to threshold, and the B-spline calculation also does not properly span this near threshold region down to the onset [57]. However, the general trend rising above 5 eV is for the dichroism to become attenuated, easily rationalized as the ejected photoelectron displaying less sensitivity to the chiral molecular potential as it acquires more energy. 

A number of other analytic potential functions such as Rotated-Morse-Curve-Spline (RMCS), Bond-Energy-Bond-Order (BEBO) have been used for fitting of ab initio surfaces. 

In conclusion, the repulsive interactions arise from both a screened coulomb repulsion between nuclei, and from the overlap of closed inner shells. The former interaction can be effectively described by a bare coulomb repulsion multiplied by a screening function. The Moliere function, Eq. (5), with an adjustable screening length provides an adequate representation for most situations. The latter interaction is well described by an exponential decay of the form of a Bom-Mayer function. Furthermore, due to the spherical nature of the closed atomic orbitals and the coulomb interaction, the repulsive forces can often be well described by pair-additive potentials. Both interactions may be combined either by using functions which reduce to each interaction in the correct limits, or by splining the two forms at an appropriate interatomic distance . 

The intermolecular potential term is represented by a simple Lennard-Jones function that is attenuated at short interatomic distances by a cubic spline so that at small (covalent) intemuclear distances, the description of the interaction is that of the intramolecular term only. The original form of... 

In spite of the flexibility introduced into equation (62) by using 2D splines to describe DM(a, v), a comparison of classical trajectories run on a model potential with those run on a rotated Morse-cubic spline fitted to that model does not show good point-by-point agreement (see ref. 54 and references therein). 

Figure 4. (a) The electric potential (circles) as a function of the distance from the surface for the system described in the text. The continuous line represents a Spline interpolation, (b) The average polarization of a water molecule (squares) as a function of the distance from surface for the system described in text the macroscopic electric field (triangles) obtained through the numerical derivative of the potential is not proportional to the average polarization. 

I. The Cubic Splines for Expression of the Potential Energy Functions 1. Introduction... 

---