Reduced representations, definition

The underlaying assumption in the evaluation of both information, the Sij and the representation of a group, is that compounds with similar properties have similar structural features thus producing similar spectra. In view of the fact that no strict rule for quantitative definition of similarity between structures exists it is hard to justify the above assumption. However, many valuable results can be obtained using the correlation between the similarity of properties (structures) and similarity of spectra. If the reduced representation of a spectrum i is written in a vector form Ri as... 

Before we start our analysis of the quality of the energy approximation by graph invariants, it is reasonable to look at their total numbers for different systems. Table 1 shows these numbers for some ice nanotubes INT . In particular, the dependence of the numbers of distinct graph invariants on the order of the invariant and the indices m and n is revealed. The values in the table can be found numerically by applying the graph invariant definition to a specific water cluster. Alternatively, one can determine these numbers analytically on the basis of symmetry considerations. Transformation of bond variables defines a (reducible) representation T of the symmetry group G of the cluster. The number of the h order invariants is simply the number of the totally symmetric representations in the h symmetric power of T. Thus, only characters of T are necessary to determine numbers in Table 1. 

For any application of the reduction formula we will always find that the number of objects in the irreducible set of representations is equal to the number used in the definition of the reducible representation, i.e. the number of basis functions. 

In this framework the definition of the metrics jj — (/) plays a key role. An arbitrary reduced representation could be used in the spirit of the string method in CV space. In general, the Cartesian coordinates of a subset of atoms are widely used and compared via mean square deviation after optimal alignment ... 

Each of these columns of this symmetrical matrix may be seen as representing a molecule in the subspace formed by the density functions of the N molecules that constitute the set. Such a vector may also be seen as a molecular descriptor, where the infinite dimensionality of the electron density has been reduced to just N scalars that are real and positive definite. Furthermore, once chosen a certain operator in the MQSM, the descriptor is unbiased. A different way of looking at Z is to consider it as an iV-dimensional representation of the operator within a set of density functions. Every molecule then corresponds to a point in this /V-dimensional space. For the collection of all points, one can construct the so-called point clouds, which allow one to graphically represent the similarity between molecules and to investigate possible relations between molecules and their properties [23-28]. 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

As shown in the second line, like the expression for the energy as a function of the 2-RDM, the energy E may also be expressed as a linear functional of the two-hole reduced density matrix (2-HRDM) and the two-hole reduced Hamiltonian K. Direct minimization of the energy to determine the 2-HRDM would require (r — A)-representability conditions. The definition for the p-hole reduced density matrices in second quantization is given by... 

The interelectronic interactions W are defined using constrained search [21, 22] over all A-representable 2-RDMs that reduce to R g). Since the set of 2-RDMs in the definition of W contains the AGP 2-RDM of g, that set is not empty and W is well defined. Through this construction, E still follows the variational principle and coincides with the energy of a wavefunction ip, which reproduces R g) = D[ T ] and = W[g]. The latter is due to... 

To complete the definition of the renormalization step for the left block, we also need to construct the new matrix representations of the second-quantized operators. In the product basis Z <8> p, matrix representations can be formed by the product of operator matrices associated with left, p j and the partition orbital p separately. Then, given such a product representation of O say, the renormalized representation O in the reduced M-dimensional basis / of LEFIi. p is obtained by projecting with the density matrix eigenvectors L defined above,... 

Definition 6.1 A representation (G, V, p) is irreducible z/to only invariant subspaces are V itself and the trivial subspace 0. Representations that are not irreducible are called reducible. 

Representation of the density n(r) [or, effectively, the electrostatic potential — 0(r)] near any one of the sinks as an expansion in the monopole and dipole contribution only [as in eqn. (230c)] is generally, unsatisfactory. This is precisely the region where the higher multipole moments make their greatest contribution. However, the situation can be improved considerably. Felderhof and Deutch [25] suggested that the physical size of the sinks and dipoles be reduced from R to effectively zero, but that the magnitude of all the monopoles and dipoles, p/, are maintained, by the definition... 

The definition of a reduced dimensionality reaction path starts with the full Cartesian coordinate representation of the classical A-particle molecular Hamiltonian,... 

The expressions eqs. (1.197), (1.199), (1.200), (1.201) are completely general. From them it is clear that the reduced density matrices are much more economical tools for representing the electronic structure than the wave functions. The two-electron density (more demanding quantity of the two) depends only on two pairs of electronic variables (either continuous or discrete) instead of N electronic variables required by the wave function representation. The one-electron density is even simpler since it depends only on one pair of such coordinates. That means that in the density matrix representation only about (2M)4 numbers are necessary to describe the system (in fact - less due to antisymmetry), whereas the description in terms of the wave function requires, as we know n 2m-n) numbers (FCI expansion amplitudes). However, the density matrices are rarely used directly in quantum chemistry procedures. The reason is the serious problem which appears when one is trying to construct the adequate representation for the left hand sides of the above definitions without addressing any wave functions in the right hand sides. This is known as the (V-representability problem, unsolved until now [51] for the two-electron density matrices. The second is that the symmetry conditions for the electronic states are much easier formulated and controlled in terms of the wave functions (Density matrices are the entities of the second power with respect to the wave functions so their symmetries are described by the second tensor powers of those of the wave functions). 

Note the appearance of the AB equilibrium geometry (Rm, which can be obtained by a self-consistent procedure136) and Le Roy s parameter142 [R0, which represents the smallest value of the internuclear distance for which the asymptotic series of the dispersion energy is still a good representation of the damped series (49)] in the definition of the reduced coordinate x represents the expectation value of the square of the radial coordinate for the outermost valence electrons, which is tabulated in the literature143 for atoms with 1 120. Other important parameters in the dispersion damping... 

A group of researchers in Budapest continued the line of Yoneda [16-21] but avoided the combinatorial explosion of the number of products by the preliminary definition of acceptable reaction products. Thus, the species to be included in the mechanism were fixed a priori, and the program provided the list of reactions. They used the matrix technique of Yoneda for the representation of reactions and species structures, but the number of generated reactions was limited by applying certain restrictions. The most important restriction was that bimolecular reactions were considered only with a maximum of three products. The number of generated reactions was kept low based on reaction complexity and thermochemical considerations. The mechanism obtained was reduced by qualitative and quantitative comparisons with experimental results, including contributions of elementary reactions to measured rates. The method proposed 538 reactions for the liquid phase oxidation of ethylbenzene. The reaction-complexity investigation approved only 272 reactions and the reaction heats were feasible in the cases of 168 reactions. This mechanism was reduced to a 31-step final mechanism. 

This review shows how the photochemistry of ketones can be rationalized through a single model, the Tunnel Effect Theory (TET), which treats reactions of ketones as radiationless transitions from reactant to product potential energy curves (PEC). Two critical approximations are involved in the development of this theory (i) the representation of reactants and products as diatomic harmonic oscillators of appropriate reduced masses and force constants (ii) the definition of a unidimensional reaction coordinate (RC) as the sum of the reactant and product bond distensions to the transition state. Within these approximations, TET is used to calculate the reactivity parameters of the most important photoreactions of ketones, using only a partially adjustable parameter, whose physical meaning is well understood and which admits only predictable variations. 

In the calculation of frictional pressure drop it is advantageous to define a few parameters that are suitable for the representation of two-phase frictional pressure drop and the volumetric quality. The frictional pressure drop is often reduced to the pressure drop for single phase flow, using the definitions from Lockhart and Martinelli [4.84]... 

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