Configuration space dimensionality

Fig. 9. Two-dimensional sketch of the 3N-dimensional configuration space of a protein. Shown are two Cartesian coordinates, xi and X2, as well as two conformational coordinates (ci and C2), which have been derived by principle component analysis of an ensemble ( cloud of dots) generated by a conventional MD simulation, which approximates the configurational space density p in this region of configurational space. The width of the two Gaussians describe the size of the fluctuations along the configurational coordinates and are given by the eigenvalues Ai.

Usually the space over which the objective function is minimized is not defined as the p-dimensional space of p continuously variable parameters. Instead it is a discrete configuration space of very high dimensionality. In general the number of elements in the configuration space is exceptionally large so that they cannot be fully explored with a reasonable computation time. 

As alluded to above, the method relies upon the identification of a 3A-dimensional vector in configuration space, es, which points along a favored direction for the motion of the system. We then choose the initial momenta for the trajectory ensemble from a Gaussian distribution artificially extended in the direction of es, as illustrated in the right panel of Fig. 8.2. In the case of free energy reconstructions from (8.49), we wish to induce motion along a predefined pulling direction, and so es can be found by inspection. 

The electronic wave function of an n-electron molecule is defined in 3n-dimensional configuration space, consistent with any conceivable molecular geometry. If the only aim is to characterize a molecule of fixed Born-Oppenheimer geometry the amount of information contained in the molecular wave function is therefore quite excessive. It turns out that the three-dimensional electron density function contains adequate information to uniquely determine the ground-state electronic properties of the molecule, as first demonstrated by Hohenberg and Kohn [104]. The approach is equivalent to the Thomas-Fermi model of an atom applied to molecules. 

In this paper we examined quantum aspects of special classical configurations of two-electron atoms. In the doubly excited regime, we found quantum states of helium that are localized along ID periodic orbits of the classical system. A comparison of the decay rates of such states obtained in one, two and three dimensional ab initio calculations allows us to conclude that the dimension of the accessible configuration space does matter for the quantitative description of the autoionization process of doubly excited Rydberg states of helium. Whilst ID models can lead to dramatically false predictions for the decay rates, the planar model allows for a quantitatively reliable reproduction of the exact life times. 

During the last five decades, an alternative way of looking at the quantum theory of atoms, molecules, and solids in terms of the electron density in three-dimensional (3D) space, rather than the many-electron wave function in the multidimensional configuration space, has gained wide acceptance. The reasons for such popularity of the density-based quantum mechanics are the following ... 

If the transition state is assumed to have zero thickness, it is of a dimensionality of one lower than the configuration space of the system. If entropy effects were to be neglected, the natural choice would be to let the transition state follow the... 

In Kirkwood s original formulation of the Fokker-Planck theory, he took into account the possibility that various constraints might apply, e.g., constant bond length between adjacent beads. This led to the introduction of a chain space of lower dimensionality than the full 3A-dimen-sional configuration space of the entire chain and it led to a complicated machinery of Riemannian geometry, with covariant and contravariant tensors, etc. 

Long progressions of feature states in the two Franck-Condon active vibrational modes (CC stretch and /rani-bend) contain information about wavepacket dynamics in a two dimensional configuration space. Each feature state actually corresponds to a polyad, which is specified by three approximately conserved vibrational quantum numbers (the polyad quantum numbers nslretch, "resonance, and /total, [ r, res,fl)> and every symmetry accessible polyad is initially illuminated by exactly one a priori known Franck-Condon bright state. 

The polyad model for acetylene is an example of a hybrid scheme, combining ball-and-spring motion in a two-dimensional configuration space [the two Franck-Condon active modes, the C-C stretch (Q2) and the tram-bend (Q4)] with abstract motion in a state space defined by the three approximate constants of motion (the polyad quantum numbers). This state space is four dimensional the three polyad quantum numbers reduce the accessible dimensionality of state space from the seven internal vibrational degrees of freedom of a linear four-atom molecule to 7 - 3 = 4. 

Carlo run constrained to make moves on the 3N-1 dimensional dividing surface in configuration space, and supplying momenta from the appropriate multidimensional Maxwell distribution. Alternatively, the dividing surface may be sampled by an unconstrained Monte Carlo run that is encouraged to remain near S by adding to the potential a holding term that is constant on S but increases rapidly as moves away from S. 

In three-dimensional (3-D) configuration space (Figure 3.1) a position vector r is the sum... 

When these matrices are applied in an N-particle configuration space (N k) the moments of the hamiltonian behave on ensemble averaging in the limit of large dimensionality as in the case of noninteracting particles without averaging. This reflects the dominance of binary correlations in the operator products HP when the ensemble averaging is performed in the dilute system (k N). 

Before continuing with an evaluation of the surface integral in Eq. (5.25), let us briefly consider the evaluation of such an integral in ordinary three-dimensional configuration space. 

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