Molecular electron densities

If the functional form of a molecular electron density is known, then various molecular properties affecting reactivity can be determined by quantum chemical computational techniques or alternative approximate methods. 

From the early advances in the quantum-chemical description of molecular electron densities [1-9] to modem approaches to the fundamental connections between experimental electron density analysis, such as crystallography [10-13] and density functional theories of electron densities [14-43], patterns of electron densities based on the theory of catastrophes and related methods [44-52], and to advances in combining theoretical and experimental conditions on electron densities [53-68], local approximations have played an important role. Considering either the formal charges in atomic regions or the representation of local electron densities in the structure refinement process, some degree of approximate transferability of at least some of the local structural features has been assumed. 

In representations of electron densities, the presence or lack of boundaries plays a crucial role. A quantum mechanically valid electron density distribution of a molecule cannot have boundaries, nevertheless, artificial electron density representations with actual boundaries provide useful tools of analysis. For these reasons, among the manifold representations of molecular electron densities, manifolds with boundaries play a special role. 

In a rigorous sense, non-transferability of molecular parts has profound implications on chemical conclusions based on electron densities. Since some of the original results on the utility and reliability of transferred electron densities have been derived within the framework of density functional theory, here we shall follow this approach, and describe a recent result on a general, holographic property of electron density fragments of complete, boundaryless molecular electron densities. 

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. 

The actual properties of this transformation combined with the convergence properties of molecular electron densities implies analyticity almost everywhere on the compact manifold. Consequently, this four-dimensional representation of the molecular electron density satisfies the conditions of a theorem of analytic continuation, that establishes the holographic properties of molecular electron densities represented on the compact manifold S3. 

This result, in turn, implies the following holographic properties of complete, boundaryless molecular electron densities within the ordinary three-dimensional space [159-161],... 

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. 

This method relies on the use of an auxiliary gaussian basis set to fit the molecular electron density obtained from an ab initio one-electron density matrix ... 

An early attempt to obtain insight from the molecular electron density was to subtract a reference density from it. The resulting difference density, Ap(r), introduced by Daudel and others is then simply ... 

Clearly the form of a deformation density depends crucially on the definition of the reference state used in its calculation. A deformation density is therefore meaningful only in terms of its reference state, which must be taken into account in its interpretation. As we will see shortly, the theory of AIM provides information on bonding directly from the total molecular electron density, thereby avoiding a reference density and its associated problems. But first we discuss experimentally obtained electron densities. 

If we are interested only in the determination of a molecular structure, as most chemists have been, it suffices to approximate the true molecular electron density by the sum of the spherically averaged densities of the atoms, as discussed in Section 6.4. A least-squares procedure fits the model reference density preKr)t0 the observed density pobs(r) by minimizing the residual density Ap(r), defined as follows ... 

As mentioned above and discussed in Chapter 2, atomic charges were often obtained in the past from dipole moments of diatomic molecules, assuming that the measured dipole moment equal to the bond length times the atomic charge. This method assumes that the molecular electron density is composed of spherically symmetric electron density distributions, each centered on its own nucleus. That is, the dipole moment is assumed to be due only to the charge transfer moment Mct. and the atomic dipoles Malom are ignored. 

Understanding and Interpreting Molecular Electron Density Distributions... 

Walker, P. D., and P. G. Mezey. 1993. Molecular Electron Density Lego Approach to Molecule Building, J. Am. Chem. Soc. 115,12423-12430. 

From the perspective of MQS, this means that the similarity needs to be computed between fragments of a molecule. This requires methods to obtain a fragment density from a molecular electron density. Generally speaking, use is made of some operator wf acting on the molecular density pMol(r) to yield the fragment density as pt(r) ... 

Kleywegt, G. J. and Jones, T. A. (1997). Template convolution to enhance or detect structural features in macro-molecular electron-density maps. Acta Crystallogr. D 53, 179-185. 

Mezey, P. G. (1998) The proof of the metric properties of a fuzzy chirality measure of molecular electron density clouds. J. Molec. Struct. (Theochem.) 455, 183-190. 

Crystals of pronase-released heads of the N2 human strains of A/Tokyo/3/67 [44] and A/RI/5+/57 were used for an x-ray structure determination. The x-ray 3-dimensional molecular structure of neuraminidase heads was determined [45] for these two N2 subtypes by a novel technique of molecular electron density averaging from two different crystal systems, using a combination of multiple isomorphous replacement and noncrystallographic symmetry averaging. The structure of A/Tokyo/3/67 N2 has been refined [46] to 2.2 A as has the structures of two avian N9 subtypes [47-49]. Three influenza type structures [50] have also been determined and found to have an identical fold with 60 residues (including 16 conserved cysteine residues) being invariant. Bacterial sialidases from salmonella [51] and cholera [52] have homologous structures to influenza neuraminidase, but few of the residues are structurally invariant. 

The essential result of the model calculation is that in phase I the molecular electron density is delocalized cylindrically symmetric in space and time average. This is achieved by a complete revolution of every CF2 unit around the chain axis within an axial range of 40 A. 

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