Nuclear parameter space

We denote the d-dimensional internal nuclear parameter space by A4 (d = 3 for triatomics) and the subspace of A4 for which a conical intersection occurs by 2 c A4. The subspace 2 is of dimension d — 2, and for triatomic molecules it is a one-dimensional curve in the three-dimensional nuclear parameter space Two conditions, tt(x) = 0 and v(x) = 0, define... 

The presence of this term can also introduce numerical inefficiency problems in the solution of Eq. (31). Since the ADT matrix U(qx) is arbitrary, it can be chosen to make Eq. (31) have desirable properties that Eq. (15) does not possess. The parameter U(q> ) can, for example, be chosen so as to automatically minimize W (Rx) relative to W l ad(R/ ) everywhere in internal nuclear configuration space and incorporate the effect of the geometric phase. Next, we will consider the structure of this ADT matrix for an ra-electronic-state problem and a general evaluation scheme that minimizes the magnitude of W Rx). 

In this expression p is a mass parameter associated to the electronic fields, i.e. it is a parameter that fixes the time scale of the response of the classical electronic fields to a perturbation. The factor 2 in front of the classical kinetic energy term is for spin degeneracy. The functional f [ i , ] plays the role of potential energy in the extended parameter space of nuclear and electronic degrees of freedom. It is given by. 

M. Lombardi What is not needed is the validity of the adiabatic approximation, that is, that there is no transition between adiabatic states. But the geometric phase is defined by following states along a path in parameter space (here nuclear coordinates) with some continuity condition. In the diabatic representation, there is no change of basis at all and thus the geometric phase is identically zero. Do not confuse adiabatic basis (which is required) and adiabatic approximation (which may not be valid). 

As it has been pointed out in Section 5.2, it is natural to formulate dynamic shape analysis aproaches in terms of the dynamic shape space D described earlier [158]. The reader may recall that the dynamic shape space D is a composition of the nuclear configuration space M, and the space of the parameters involved in the shape representation, for example, the two-dimensional parameter space defined by the possible values of the density threshold a, and the reference curvature parameter b of a given MIDCO surface. 

A temperature dependent X—X+N study (100, 135, 170, and 205 K) on naphthalene [66] addresses the problem of thermal de-convolution, that is, the efficiency of the pseudoatom model to decouple density deformations due to chemical bonding from those due to nuclear motion. The authors analyze the self-consistency of multipole populations, extracted from different temperature XRD data, in terms of statistical distances d ) in the parameter space of the same refinement model ... 

Another factor that favours oscillations is the steepness of the function characterizing the repression of per transcription by nuclear PER. Such steepness depends on the degree of cooperativity, n, of the repression process, which is described in eqn (11.1a) by a Hill function. In agreement with such a destabilizing role of cooperativity, oscillations in fig. 11.7 were obtained for a value of n = 4 however, periodic behaviour can also occur in the model for n = 2 and for n = 1, but the domain in parameter space where sustained oscillations occur is smaller than for n = 4. 

One may analyse the detailed variations of contributions of various nuclear configurations to each shape type Xj as a function of some continuous parameters, for example, as function of the contour density value a and reference curvature parameter b of isodensity contours G(a). This is equivalent to the analysis of the parameter dependence (for example, (a,b)-dependence) of the Ty subsets within the configuration space M, and in particular, in relaxed cross sections or within each catchment region C(X,i) [44]. These changes can be monitored within the dynamic shape space D, obtained as the product space of the nuclear configuration space M and the space of the actual continuous parameters. This approach has been described in some detail in ref. [44], and various applications can be found in refs. [45-47]. 

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