Subsystems electron densities

In fact, in a precise sense, no molecular fragment is rigorously transferable, although approximate transferability is an exceptionally useful and, if used judiciously, a valid approach within the limitations of the approximation. In particular, it is possible to define non-physical entities, such as fuzzy fragment electron densities, which do not exist as separate objects, yet they show much better transferability properties than actual, physically identifiable subsystems of well-defined, separate identity. This aspect of specially designed, custom- made , artificial subsystems of nearly exact additivity has been used to generate ab initio quality electron densities for proteins and other macromolecules. 

The original Hohenberg-Kohn theorem was directly applicable to complete systems [14], The first adaptation of the Hohenberg-Kohn theorem to a part of a system involved special conditions the subsystem considered was a part of a finite and bounded entity regarded as a hypothetical system [21], The boundedness condition, in fact, the presence of a boundary beyond which the hypothetical system did not extend, was a feature not fully compatible with quantum mechanics, where no such boundaries can exist for any system of electron density, such as a molecular electron density. As a consequence of the Heisenberg uncertainty relation, molecular electron densities cannot have boundaries, and in a rigorous sense, no finite volume, however large, can contain a complete molecule. 

It is possible, however, to avoid any violation of these fundamental properties, and derive a result on the local electron densities of non-zero volume subsystems of boundaryless electron densities of complete molecules [159-161]. A four-dimensional representation of molecular electron densities is constructed by taking the first three dimensions as those corresponding to the ordinary three-space E3 and the fourth dimension as that representing the electron density values p(r). Using a compactifi-cation method, all points of the ordinary three- dimensional space E3 can be mapped to a manifold S3 embedded in a four- dimensional Euclidean space E4, where the addition of a single point leads to a compact manifold representation of the entire, boundaryless molecular electron density. 

Nevertheless, approximate transferability is a valid concept and in the next section a particular approach will be discussed, based on fuzzy subsystems of molecular electron densities. 

If the electron density partitioning results in subsystems without boundaries and with convergence properties which closely resemble the convergence properties of the complete system, then it is possible to avoid one of the conditions of the Holographic Electron Density Fragment Theorem , by generating fuzzy electron density fragments which do not have boundaries themselves, but then the actual subsystems considered cannot be confined to any finite domain D of the ordinary three-dimensional space E3. 

We now introduce a set of quantities that are closely related to the electron density p(r), but provide still more detailed information about one-particle subsystems of the /V-particle system described by [Pg.21]

Any subsystem wavefunction, refers only to Nr electrons with an effective field Hamiltonian, whose form depends on the forms of the 1-electron density matrices of all subsystems all such functions can be optimized, in an iterative manner, by standard methods and without the constraints implied by any a priori partitioning of the global basis... 

In Chapter 3 we introduce the formal construction and testing of an intermediate procedure bridging QM and MM procedures. This will be a mechanistic treatment, derived from the quantum description of the molecular system. Then this technique will be used to define the one-electron states of the frontier atoms - the key elements of the intersubsystem border/junction the shapes of the one-electron states at the frontier atoms, their electronic densities and the response of either subsystem to the variables characterizing each subsystem. 

In the QM part of the system, the variation of the bond orders can also take place. In variance with the pure SLG picture [11,12] used here as the QM method underlying the MM part of the system, the atoms in the QM part of the combined system may have off-diagonal elements of the one-electron density matrix between orbitals ascribed to the QM subsystem. The latter are obviously the (Coulson) bond orders for the QM part of the system. The corresponding contribution to the energy reads ... 

Second, these grids make the regular structure like domain in fact. Here domains are the regions with different electron density. The existence of domain electron structure will contribute to the receptivity of the nanotubes and due to domain structure one may hope to find out the effects of a memory in the electron subsystem of nanotubes. One should to note that the similar domains can play the determining role (due to Coulomb interaction of electrons) in physical properties of as nanotube ropes and multi wall carbon nanotubes. 

Another possible description is given by the 3D electron density pel(re, qnuc) which is a scalar function of re and contains qnuc as parameters. These two representations of the electron subsystem form the basis for the development of either conventional quantum chemistry methods or electron Density Functional Theory (DFT). The electron subsystem generates an effective potential, U(qnuc), acting on the classical nuclei, which can be expressed as an average of the full potential V over the electron wave function IP, and written as ... 

Accordingly, the modifications to the KS operator are twofold (i) a static contribution through the static multipole moments (here charges) of the solvent molecules and (ii) a dynamical contribution which depends linearly on the electronic polarizability of the environment and also depends on the electronic density of the QM region. Due to the latter fact we need within each SCF iteration to update the DFT/MM part of the KS operator with the set of induced dipole moments determined from Eq. (13-29). We emphasize that it is the dynamical contribution that gives rise to polarization of the MM subsystem by the QM subsystem. 

Subsystems of electron density distributions of molecules can be obtained by a variety of methods here we shall be concerned with two, fundamentally different approaches ... 

Note that the holographic electron density theorem has been proven for boundaryless electron densities, where the boundaryless, fuzzy nature of the complete electron density is an essential feature if realistic molecular representations are to be considered [24,25], Note that an earlier result on subsystems used models where both the subsystem and the complete system were assumed... 

If one follows approach (ii) to the study of subsystems of molecules, as outlined in Sect. 1.2, then it is natural to consider subsystems having properties analogous to complete molecules. In fact, it is possible to define molecular subsystems with electron densities whose convergence properties are analogous to those of complete molecules. 

The AFDF local molecular pieces can be used to build high quality approximate electron densities for large molecules, and also to study the local molecular subsystems themselves. 

System-independence Aiming at universal applicability, the interactions between the subsystems resulting from partitioning the total electron density are described using only system-independent quantities. 

Cortona introduced the total energy functional22 in which the terms of unknown and known analytic dependence on p are partitioned in yet another way - as it is done in neither EHK[p nor EKS[p. Compared to the Kohn-Sham case, this functional, depends not on one but on several functions - electron densities of subsystems... 

The effective potential VKohn-Sham effective potential for the isolated subsystem A at the electron density p = pA ... 

The first set of one-electron equations (Eq. 31) leads to the minimum of the functional Hs subject to the following constraint dps = 0 which corresponds to freezing the electron density of subsystem B. The second set (Eq. 32) represents a similar constrained minimization for the other subsystem. 

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