Representation unique

Figure 10.1 Cell representation of a small region of disordered (a) and ordered (b) configurations at full occupation each cell contains a particle. Each cell representation uniquely defines a potential well or basin in the potential energy landscape. Observe that while...

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

The occurrence of the argument pj2 shows that these eigenvectors are defined up to a sign only. For a unique representation we have to cut the plane along a half-axis. By this, vector fields uniquely defined on the cut plane. They cannot, however, be continued over the cut, but change their roles there instead. Thus, we have the situation of a crossing at which the eigenvector field is discontinuous and Assumption (A) of Thm. 3 is hurt. 

A nomenclature or notation is called unambiguous if it produces only one structure. However, the structure could be expressed in this nomenclature or notation by more than one representation, all producing the same structure. Moreover, uniqueness" demands that the transformation results in only one - unique -structure or nomenclature, respectively, in both directions. 

The ROSDAL syntax is characterized by a simple coding of a chemical structure using alphanumeric symbols which can easily be learned by a chemist [14]. In the linear structure representation, each atom of the structure is arbitrarily assigned a unique number, except for the hydrogen atoms. Carbon atoms are shown in the notation only by digits. The other types of atoms carry, in addition, their atomic symbol. In order to describe the bonds between atoms, bond symbols are inserted between the atom numbers. Branches are marked and separated from the other parts of the code by commas [15, 16] (Figure 2-9). The ROSDAL linear notation is rmambiguous but not unique. 

A special extension of SMILES is USMILES (sometimes described as Broad SMILES) [23-25]. This Unique SMILES of Daylight is a canonical representation of a structure. This means that the coding is independent of the internal atomic numbering and results always in the same canonical, unambiguous, and unique description of the compound, granted by an algorithm (see Section 2.5.2). 

If the indexing of the atoms is changed, the CT will have a different appearance. Thus, the representation of a chemical structure in a CT is unambiguous but not unique, which can only be achieved by canonicalization (see below). 

A SMILES code [22], MDL Molfile [50], or JME s own compact format (one-line representation of a molecule or reaction including the 2D coordinates) of created molecules may be generated. The created SMILES is independent of the way the molecule was drawn (unique SMILES see Section 2.3.3). Extensions to JME developed in cooperation with H. Rzepa and P. Murray-Rust also allow output of molecules in the CML format [60]. 

The problem of perception complete structures is related to the problem of their representation, for which the basic requirements are to represent as much as possible the functionality of the structure, to be unique, and to allow the restoration of the structure. Various approaches have been devised to this end. They comprise the use of molecular formulas, molecular weights, trade and/or trivial names, various line notations, registry numbers, constitutional diagrams 2D representations), atom coordinates (2D or 3D representations), topological indices, hash codes, and others (see Chapter 2). 

The teehniques used earlier for linear moleeules extend easily to non-linear moleeules. One begins with those states that ean be straightforwardly identified as unique entries within the box diagram. For polyatomie moleeules with no degenerate representations, the spatial symmetry of eaeh box entry is identieal and is given as the direet produet of the open-shell orbitals. For the formaldehyde example eonsidered earlier, the spatial symmetries of the nji and nn states were A2 and Ai, respeetively. 

Now write structural formulas for the remaining three isomers Be sure that each one IS a unique compound and not simply a different representation of one writ ten previously... 

The reduction of the x representation, like all such reductions, gives a unique set of irreducible representations which is... 

Here N is the number of bonds or molecules of a given type in the crystal, and is a geometric tensor associated with a particular microscopic polarizabiHty P this tensor is related to the crystallographic orientation of the bond. In extended systems such as covalent soHds it becomes difficult to define a species to which one can assign a unique value of P, and thus the value of P for a given group can only be an approximate representation. In... 

Figure 13.12 Schematic representation of the structure of the complex anion LSbjCIiiO] " showing the two different coordination geometries about Sb and the unique quadruply bridging Cl atom.

The representation of the solution must have a unique interpretation, computers having less tolerance for ambiguity than humans, so when executing the steps with the same input data, the same outputs are obtained. 

Notice that in this example, the speed of the packet is inversely proportional to the packet s spatial size. While there is certainly nothing unique about this particular representation, it is interesting to speculate, along with Minsky, whether it may be true that, just as the simultaneous information about position and momentum is fundamentally constrained by Heisenberg s uncertainty relation in the physical universe, so too, in a discrete CA universe, there might be a fundamental constraint between the volume of a given packet and the amount of information that can be encoded within it. 

For concreteness, let us suppose that the universe has a temporal depth of two to accommodate a Fi edkin-type reversibility i.e. the present and immediate past are used to determine the future, and from which the past can be recovered uniquely. The RUGA itself is deterministic, is applied synchronously at each site in the lattice, and is characterized by three basic dimensional units (1) digit transition, D, which represents the minimal informational change at a given site (2) the length, L, which is the shortest distance between neighboring sites and (3) an integer time, T, which, while locally similar to the time in physics, is not Lorentz invariant and is not to be confused with a macroscopic (or observed) time t. While there are no a priori constraints on any of these units - for example, they may be real or complex - because of the basic assumption of finite nature, they must all have finite representations. All other units of physics in DM are derived from D, L and T. 

The second axiom, which is reminiscent of Mach s principle, also contains the seeds of Leibniz s Monads [reschQl]. All is process. That is to say, there is no thing in the universe. Things, objects, entities, are abstractions of what is relatively constant from a process of movement and transformation. They are like the shapes that children like to see in the clouds. The Einstein-Podolsky-Rosen correlations (see section 12.7.1) remind us that what we empirically accept as fundamental particles - electrons, atoms, molecules, etc. - actually never exist in total isolation. Moreover, recalling von Neumann s uniqueness theorem for canonical commutation relations (which asserts that for locally compact phase spaces all Hilbert-space representations of the canonical commutation relations are physically equivalent), we note that for systems with non-locally-compact phase spaces, the uniqueness theorem fails, and therefore there must be infinitely many physically inequivalent and... 

Les differents puits sont representes sur le schema joint (p. 62), qui correspond uniquement a la Classification par objectifs". 

Theorem C establishes the fact that there exists essentially only one irreducible representation of the y-matrices (all others are related to it by a similarity transformation). As a corollary to this theorem, theorem B allows us to assert that 8 is uniquely defined up to a factor. For suppose there were two such 8% say Sx and S2 such that y = Stf8x 1 and y u = 82yu82 1 then by equating y M in these equations... 

Since the operators P commute with one another we can choose a representation in which every basis vector is an eigenfunction of all the P s with eigenvalue It should be noted that the specification of the energy and momentum of a state vector does not uniquely characterize the state. The energy-momentum operators are merely four operators of a complete set of commuting observables. We shall denote by afi the other eigenvalues necessary to specify the state. Thus... 

The approach presented above is referred to as the empirical valence bond (EVB) method (Ref. 6). This approach exploits the simple physical picture of the VB model which allows for a convenient representation of the diagonal matrix elements by classical force fields and convenient incorporation of realistic solvent models in the solute Hamiltonian. A key point about the EVB method is its unique calibration using well-defined experimental information. That is, after evaluating the free-energy surface with the initial parameter a , we can use conveniently the fact that the free energy of the proton transfer reaction is given by... 

In the anion radical of XXIII, the starting distorted structures belonging to the Og, bjg and b2u irreducible representations of point group D2k all converge into the unique set of bond lengths corresponding to... 

Structure searching is the chemical equivalent of graph isomorphism, that is, the matching of one graph against another to determine whether they are identical. This can be carried out very rapidly if a unique structure representation is available, because a character-by-character match will then suffice to compare two structures for identity. However, connection tables are not necessarily unique, because very many different tables can be created for the same molecule depending upon the way in which the atoms in the molecule are numbered. Specifically, for a molecule containing N atoms, there are N ... 

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