Configuration space, definition

With these definitions we can now look for the necessary condition(s). Thus, we assume that at each point sq iu configuration space the diabatic potential matrix W(/l) [= W(s,so)] fulfills the condition ... 

Fock Space Representation of Operators.—Let F be some operator that neither creates nor destroys particles, and is a known function in configuration space for N particles. In symbols such an operator must by definition have the following matrix elements in Fock space ... 

In Eq. (13), the vector q denotes a set of mass-weighted coordinates in a configuration space of arbitrary dimension N, U(q) is the potential of mean force governing the reaction, T is a symmetric positive-definite friction matrix, and , (/) is a stochastic force that is assumed to represent white noise that is Gaussian distributed with zero mean. The subscript a in Eq. (13) is used to label a particular noise sequence For any given a, there are infinitely many... 

From the definitions given above it follows that the configurational space (continuous) of a peptide is infinite, even if its conformational space... 

The definition of the pair part or, equivalently the no-pair part of Hmat is not unique. The precise meaning of no-pair implicitly depends on the choice of external potential, so that the operator Hm t depends implicitly on the external potential, whereas the sum Hmat Hm t + Hm t is independent of the choice of external potential. Since the no-pair part conserves the number of particles (electrons, positrons and photons) we can look for eigenstates of Hm j in the sector of Fock space with N fermions and no photons or positrons. Following Sucher [18,26,28], the resulting no-pair Hamiltonian in configuration space can be written as... 

The double bracket notation in equation (3) implies the integration over the electronic configuration space By definition, i) f is the ath nonadiabatic... 

Various generalizations of syntopy are possible by replacing the configuration space distance by some alternative similarity measures. In fact, any of the similarity measures and symmetry deficiency measures may serve as a suitable parameter in the definition of fuzzy syntopy membership functions. 

This is the general idea behind the definition of a stable structure—an open region of nuclear configuration space, all points of which have the same molecular graph. A molecular graph is determined by the gradient vector... 

The discussion so far has demonstrated that the definition of structure is inextricably bound up with the definition of structural stability. We now show that the two definitions taken together lead to a partitioning of nuclear configuration space into a finite number of structural regions. 

As discussed above and illustrated in Fig. 3.4, the partitioning of nuclear configuration space obtained as a result of the definition of molecular structure leads to the concept of a structure diagram. The space R is partitioned into a finite number of structural regions with their boundaries, as defined by the catastrophe set, denoting the configurations of unstable structures. This information constitutes a system s structure diagram, a... 

A function/of the essential variables, is said to be of finite co-dimension if a small perturbation of/gives rise to only a finite number of topological types. It is in this case that a map / of co-rank k can be embedded in a family of deformations (/, x R), parametrized by q variables which form the control space a R. By definition, q, the dimension of the control space, is the co-dimension of the singularity represented by /. The family of functions (/, / X R R) thus defined, describes the universal unfolding of /, in which the function/itself is the member associated with the origin of control space. In our applications the control space is a subset of the nuclear configuration space. 

The B-matrix is, by definition, a unitary matrix (it is a product of two unitary matrices) and at this stage except for being dependent on T and, eventually, on So, it is rather arbitrary. Since the electronic eigenvalues (the adiabatic PESs) are uniquely defined at each point in configuration space we have u(0) = u(P) and therefore Eq. (14) implies ... 

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