Quadratic function Surface fitting

In 1970, Bender and Schaefer " reported afc initio computations of triplet methylene. Employing the CISD/DZ method, they computed the energy of triplet methylene at 48 different geometries, varying the C-H distance and H-C-H angle. Fitting this surface to a quadratic function, they predicted that the H-C-H angle is 135.1°, and emphatically concluded that the molecule is not linear. 

Specimen 5 was particularly problematic because it contained significant distortion (both cup and bow). This specimen had obvious regions of bondline voids that should have been readily detected by the laser-scanning method- but were instead obscured by the specimen distortion. For this specimen, fitting a parabolic function to a single scan and subtracting it from all the scans was not sufficient to reproduce a flat bondline. Rather, to improve the analysis, it may have been useful to initially fit a bi-quadratic function to the data to completely remove all surface distortion, and then apply the relevant tolerance to better resolve these surface features. 

Figure 18. The centers of mass of the induced electronic charge (xe), and induced polarization charge (x,) as a function of the amount of induced surface charge, in units of 10-3 e/(a.u.)3. The black filled circles show the calculated values of xe, and the solid black line is a quadratic fit to these values. The open circles indicate the calculated center of mass of the xs (equal to the edge position of an equivalent, classical uniform dielectric), and the line labeled Exp. dielectric edge indicates where they would need to be in order to reproduce the experimental compact capacity. The line labeled Shifted Oxygen dist. is the position of the oxygen surface layer as a function of charge, shifted downward by 2.4 a.u. From Ref. 52, by permission.

Most ab initio analyses of vibrational spectra invoke a double-harmonic assumption wherein the potential energy surface in the vicinity of the minimum is fit to a function that involves only quadratic dependence of the energy with respect to the nuclear motions. The intensities of the normal vibrational modes are extracted from the derivatives of the dipole moment, taken as linear with respect to nuclear coordinates. Within this approximation, the intensities of the fundamentals are proportional to the square of the dipole moment derivatives with respect to normal coordinates ". ... 

Interface Tracking in Immersed Boundary Method The immersed boundary or interface denoted by C t) (a curve in two dimensions or a surface in three dimensions), see Fig. 1 la, is represented by K markers of coordinates Xk s) with k= 1,2, The markers are uniformly distributed along C(f) at some fraction of the grid spacing, 0.5h < ds < 1.5h. The interface is parameterized as a function of the arclength s by fitting quadratic polynomials Xjt(s) = -I- bj(S + Ck through three consecutive marker points of coordinates... 

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