Holographic electron density theorem

Before discussing molecular quantum similarity as a basis for so-called quantum QSP/AR, a typical example of the usage of molecular quantum similarity will be discussed. 

Chirality plays a key role in many fields of chemistry. Examples include the area of catalysis and especially medicinal chemistry, where two molecules that differ solely in their absolute configuration can exhibit substantially different biological activities. Consider the case of enantiomers. One enantiomer can exhibit the desired medicinal activity, whereas the other may have other characteristics. The latter can simply lack this biological activity, or it can have an undesirable effect, and in extreme cases, it may be poisonous and even deadly. Knowledge of the absolute configuration of a molecule is therefore of prime importance. Chirality is often considered to be a black/white topic, which means that a molecule is either chiral or achiral. As will be shown below, from a molecular quantum similarity point of view, this opinion is not the case, because we can express a degree of chirality. To highlight the utility of molecular quantum similarity, two case studies of molecular chirality will be discussed in some detail. 

Boon et al. also studied several chiral molecules, which included again two amino acids (Ala and Leu) and CHFClBr, a prototype of chiral molecules. Ab initio total molecular electron densities yielded both Euclidean distances and Carbo indices between the enantiomers of these molecules. Molecular superposition was performed with, on the one hand, a manual alignment based on chemical intuition and the QSSA method, on the other hand. When analyzing the tables of the work by Boon et al. and comparing the results to the work by Mezey et al., similar values for the Euclidean distances between the two enantiomers appear for Ala and Leu, which once again illustrates the power of the ASA promolecular densities to yield quantum similarity measures in good agreement with those obtained from ab initio calculations. 

Boon et al. ° rely on the Hirshfeld picture of an atom in a molecule. As we mentioned, probably the most obvious way to proceed is via a product of two Hirshfeld coefficients. Boon et al., however, opted to use only one coefficient to retain the spirit of the stockholder idea. First a superposed electron 

Both in the case of Mezey et al. and Boon et al. the calculation of involves coalescence of the chiral atom in both molecules, similar to the atoms-in-mole-cules work by Cioslowski et mentioned earlier. This coalescence pro- 

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The former approach is subject to the constraints imposed by the recently proven Holographic Electron Density Theorem [24,25] ... 

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... 

The Role of the Holographic Electron Density Theorem and the Predictability of Differences in the Reactivities of Functional Groups... 

The holographic electron density theorem does not exclude the possibility that two MIDCOs, for example, the two MIDCOs GpA (K, ciab) and GFb(K, uAb) on the boundaries of the two density domains DDpA (K, Uab) and DDff,(F, aAB) coincide,... 

In fact, according to the holographic electron density theorem, no density threshold interval [aA, aA] of positive width exists where Gfa(K, a) = G iK, a) for every value a from this interval. In other words, any MIDCO pair fulfilling the condition... 

Specific aspects of the quantum chemical concept of local electron densities and functional groups of chemistry have been discussed, with emphasis on the Additive Fuzzy Density Fragmentation (AFDF) Principle, on the Adjustable Density Matrix Assembler (ADMA) Method of using a local density matrix formalism of fuzzy electron density fragments in macromolecular quantum chemistry, and on the fundamental roles of the holographic electron density theorem, local symmetry, and symmetry deficiency. 

Holographic Electron Density Theorem and Existence of First-Principles for Open Systems... 

Mezey P (1999) The holographic electron density theorem and quantum similarity measures. Mol Phys 96(2) 169-178... 

Zheng X, Yam CY, Wang F, Chen GH (2011) Existence of time-dependent density-functional theory for open electronic systems time-dependent holographic electron density theorem. Phys Chem Chem Phys 13 14358... 

Based on the tools employed in the Hohenberg-Kohn theorem, also in part on the result of Riess and Miinch, and on a four-dimensional version of the Alexandrov one-point compactification method of topology applied to the complete three-dimensional electron density, it was possible to prove recently that for nondegenerate ground-state electron densities, the Holographic Electron Density Theorem applies any nonzero volume part of the nondegenerate ground-state electron density cloud contains all information about the molecule [4,5]. 

It is important to realize that in the proofs of the Hohenberg-Kohn theorem and the Holographic Electron Density Theorem, some very natural properties of molecular electron densities have been assumed. Two of these assumptions are i) the very existence of a ground-state electron density function and ii) the assumption of continuity of this function in the space variable r. 

By combining this theorem and the original Holographic Electron Density Theorem, one may obtain a single statement ... 

Since molecular recognition typically involves two or more molecules, it is useful to phrase the problem in terms of a generalization of the Holographic Electron Density Theorem to supermolecular and supramolecular structures involving several interacting, but formally individual molecules. Such a generalization is the Supramolecular Holographic Electron Density Theorem. 

In order to apply the Holographic Electron Density Theorem to both the independent molecules and the supramolecular object, consider a nonzero volume part P of the electron density of independent molecule E. For example, select a spherical volume about a specific nucleus X of molecule E. As what follows from the Holographic Electron Density Theorem, this volume P contains all the information about the independent molecule E, assumed to be infinitely removed from any other molecule. 

In the next step, bring the two molecules into some mutual position where some interaction occurs between them, and consider the same nonzero volume part P in the supramolecular object ED, for example, the spherical volume of the same radius about the same nucleus X. Whereas this volume P was originally specified for the independent molecule E, nevertheless, by applying the Holographic Electron Density Theorem to the entire supramolecular object ED, now, this volume P now contains all the information about the supramolecular object. 

This result is a rather trivial consequence of the original Holographic Electron Density Theorem nevertheless, it can be viewed as a basic aspect of supramolecular chemistry, where the components of the supramolecular structure may retain a sufficient degree of their original, individual autonomy to justify a reference to their original electron density. 

Mezey PG. The holographic electron density theorem and quantum similarity measures. 

In the above argument we have relied on the assumption that the information concerning a molecule is localized in the following sense the molecule, as an entity distinguishable from the rest of the universe, has its own identity, and the molecule fully characterizes all of its own properties. In fact, a similar argument is the basis of the holographic electron density theorem, discussed in a later section. 

Here G is the n-fold direct sum of an arbitrary, simply connected region g in the ordinary 3-space 1, z> g, the physical region of the system where at this stage there is no boundedness requirement for g. Note, however, that in the strict quantum mechanical sense g should be taken as the entire 3-space R, a fact that will require special attention in the proof of the holographic electron density theorem. 

First we shall describe the argument followed in ref. 49 describing a treatment where the complete physical system is confined to a finite domain of the space. Next, we shall extend the treatment to infinite systems, leading to the holographic electron density theorem. ... 

Nevertheless, by choosing an alternative approach to the proof of a more general problem, based on the natural convergence properties of electron densities, this difficulty can be circumvented [50], leading to the holographic electron density theorem on quantum mechanically correct, boundaryless molecular electron densities [50d. This also implies some fundamental relations between local and global symmetries and local and global chirality properties of electron densities [50b of molecules. 

Analyticity of p (r ) almost everywhere on the sphere is simply inherited from p(r). On the closed and bounded sphere 5 there exists a unique correspondence between the electron density of a subsystem and the electron density of the complete system. This proves the following holographic electron density theorem [50] ... 

Some of the fundamental relations of fuzzy set theory and the actual formulation of fuzzy set methods [59-63] appear ideally suitable for the description of the fuzzy, low-density electron distributions [64. A molecular electron density exhibits a natural fuzziness, in part due to the quantum mechanical uncertainty relation. Evidently, a molecular electron distribution, when considered as a formal molecular body, is inherently fuzzy without any well-dehned boundaries. A molecule does not end abruptly, since the electron density is gradually decreasing with distance from the nearest nucleus, and becomes zero, in the strict sense, at infinity. In fact, the same consideration has been the motivation for the compactihcation technique described in the proof of the holographic electron density theorem. For a correct description of molecules, the models used for electron densities must exhibit the natural fuzziness of quantum chemical electron distributions [64. ... 

The density functional approach is used for a study of the fundamental properties of electron densities, and the role of subsystems in determining the electron density of a complete system. The holographic electron density theorem is discussed that provides a more general justification of the determining role of subsystems than earlier approaches. 

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