Separability theorem

Alternative integral equations for the cavity functions of hard spheres can be derived [61,62] using geometrical and physical arguments. Theories and results for hard sphere systems based on geometric approaches include the scaled particle theory [63,64], and related theories [65,66], and approaches based on zero-separation theorems [67,68]. These geometric theories have been reviewed by Stell [69]. 

Finally, using the zero-separation theorem, when particles n — 1 and n coincide we have... 

Many problems simplify significantly by choosing a suitable coordinate system. At the heart of these transfonnations is the separability theorem. If a Hamilton operator depending on N coordinates can be written as a sum of operators which only depend on one coordinate, the corresponding N coordinate wave function can be written as a product of one-coordinate functions, and the total energy as a sum of energies. 

Fig. 5. Graphical illustration of separation theorem all roots are assumed to be nondegenerate.

This theorem is easily derived from the separation theorem quoted in Section III.D(la) see Figure 5. 

The separation of a reactant system (solute) from its environment with the consequent concept of solvent or surrounding medium effect on the electronic properties of a given subsystem of interest as general as the quantum separability theorem can be. With its intrinsic limitations, the approach applies to the description of specific reacting subsystems in their particular active sites as they can be found in condensed phase and in media including the rather specific environments provided by enzymes, catalytic antibodies, zeolites, clusters or the less structured ones found in non-aqueous and mixed solvents [1,3,6,8,11,12,14-30],... 

Then the separability theorem [48,49] states that the strong orthogonality is equivalent to dividing the complete space of one-electron functions into mutually orthogonal subspaces ... 

This closure contains three parameters that will have to be determined thanks to consistency conditions and theorems (the Zero-Separation Theorems) connecting the properties of the bulk fluid with the correlation functions at coincidence (r = 0). This last relation reduces to (38) if ( and numerical implementation of consistency conditions with more than one parameter in the context of an iterational theory is far from being trivial. 

Thus, another approach consists in selecting some boundary conditions and properties. It is obvious that all exact correlation functions must satisfy and incorporate them in the closure expressions at the outset, so that the resulting correlations and properties are consistent with these criteria. These criteria have to include the class of Zero-Separation Theorems (ZSTs) [71,72] on the cavity function v(r), the indirect correlation function y(r) and the bridge function B(r) at zero separation (r = 0). As will be seen, this concept is necessary to treat various problems for open systems, such as phase equilibria. For example, the calculation of the excess chemical potential fi(iex is much more difficult to achieve than the calculation of usual thermodynamic properties since one of the constraints it has to satisfy is the Gibbs-Duhem relation... 

Consider the hard-sphere fluid and establish the zero separation theorem... 

Eg, jBj,]. a further useful property is expressed by the separation theorem, which guarantees that between two successive zeros of P E) there is only one zero of P . l(E). 

Separability theorem, 309 SHAKE algorithm, 385 SHAPES force field, 40 Simulated Annealing (SA), global optimization, 342 Simulation methods, 373 Supidfiidiil, iulcs, 3j6 Susceptibility, 237 Symbolic variables, for optimizations, 416 Symmetrical orthogonalization of basis sets, 314 Symmetry adapted functions, 75 Symmetry breaking, of wave functions, 76 ... 

Extension of the basis set (by considering more than n terms) leads to a systematic shift of the eigenvalues. According to the separation theorem any basis set extension may lead only to lowering the energy eigenvalues... 

To associate a self-adjoint operator to the temperature in U the KMS states separation theorem for the factors of type 111 for various temperatures (Takesaki Theorem) makes this construction as legitimate. The KMS states were described in the beginning of this Chapter. 

Let us consider the secular Eq. (26) in which the concept of the adiabatic potential originates. If the diagonal matrix elements are arranged in ascending order, Uii < U22..., the non-zero off-diagonal elements cause at least one eigenvalue to fulfil the inequality Ej < Uu. This result follows from the well known separation theorem A proof for the 2 x 2 problem is trivial since... 

Use the separation theorem to show how the discussion of the ground and first excited states of the helium atom (p. 15) could be modified in order to obtain upper bounds on the energies, still using orbital products as expansion functions. 

---