Full electron density

The distribution of electric charges in a molecule plays a central role in all discussions of intermolecular interactions. No wonder, a very large amount of conceptual and computational effort has been and is being dedicated to its expression and interpretation. 

Consider molecule A identified by an electron density pa (r i), a charge distribution function q iri) and a number Nk of nuclei of charge Zt(A). The electron density is 

Recalling the definition of the density matrix in Box 3.1, for a molecule with Na nuclei of charge Zk at locations Rk and whose wavefunction is expressed in the usual LCAO form 3.46, the quantum mechanical (QM) electric potential at point r is given by [5] 

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Figure 3.51 also contains a dissection of the total energy ( totai) into Lewis (ii(L)) and non-Lewis (ElSL>) components. The localized Lewis component E" corresponds to more than 99.3% of the full electron density, and so incorporates steric and classical electrostatic effects in nearly exact fashion. Yet, as shown in Fig. 3.51, this component predicts local minima (at 70° and 180°) and maxima (at = 0° and 130 ) that are opposite to those of the full potential. In contrast, the non-Lewis component E (NL) exhibits a stronger torsional dependence that is able to cancel out the unphysical behavior predicted by (L), leading to minima correctly located near 0° and 120°. Thus, the hyperconjugative interactions incorporated in E(SL> clearly provide the surprising stabilization of 0° and 120° conformers that counter the expected steric and electrostatic effects contained in ElL>. 

The Coulomb field defined here is not to be confused with the physical Coulomb field calculated using the full electron density. The Coulomb field used here is based on point charges and point multipoles, not on extended electron densities. 

This potential is illustrated as a contour plot in the molecular plane in Fig. 4a, whereas Fig. 4b corresponds to a plane perpendicular to the molecule, rotated 90° from the first. The red contours indicate regions of negative potential (attractive to a positive charge) and blue represents positive. The red and blue contours occur in the vicinity of the O and H atoms, respectively, as expected, based on their respective electronegativities. It is reiterated that this potential is a function of the full electron density and has no dependence on any arbitrary assignment of charges to atoms or other sites. 

Figure 4 Molecular electrostatic potential of water molecule, represented as a contour plot with intervals of 0.025 au. Red contours indicate regions of negative potential and blue represents positive, (a-b) Potential generated from full electron density, in and perpendicular to the molecular plane, respectively (c d) potential generated from point charges situated at three atomic positions (e-f) potential generated from point charges and dipoles situated at three atomic positions. (See color plate at end of chapter.)...

It is easy to see that, when a more general P is expressed in an orthogonal basis and diagonalised, there is no set of doubly occupied MOs that generate the full electron density. The occupation numbers of the resulting orbitals range from close to 2 to much smaller values depending on the number of determinants and the care with which they have been chosen and optimised. 

In order to show the effect that inclusion of the correlation energy in a self-consistent way has on the electron density, as in ref. , we show in figures 4-7 the difference between the SCF and post-SCF densities (the post-SCF density is just the GVB-pp density). Figures 4,5 show the full electronic density difference for LiH and Li2, and figures 6,7 the valence bond-pair electron density difference for LiH and FH. 

Attempts Avere made recently to develop a set of deformation parameters, transferable between chemically similar molecules. Their application is much less time-consuming than a full electron density study and may give more realistic least-squares refinement (reduction of displacement parameters) and better predictions of molecular properties than the atomic approximation. 

DFT in its usual form uses the full electron density [5,6,16] function in 3-D space to describe the systan which itself is a tremendous simpMcation. However, accurate... 

All molecules were described using each of the four following smoothed molecular fields, that is, the promo-lecular atomic shell approximation (PASA) of the full electron density (ED) [35], a charge density (CD) calculated using the Poisson equation [33], the Coulomb electrostatic potential [34], and the Atomic Property Fields (APF) described by Totrov [15]. 

Leherte L (2004) Hierarchical analysis of promolecular full electron-density distributions description of protein structure fragments. Acta Crystallogr Sect D 60 1254—1265... 

These expressions are only correct for wave functions that obey the Hellmann-Feynman theorem. Flowever, these expressions have been used for other methods, where they serve as a reasonable approximation. Methods that rigorously obey the Flellmann-Feynman theorem are SCF, MCSCF, and Full CF The change in energy from nonlinear effects is due to a change in the electron density, which creates an induced dipole moment and, to a lesser extent, induced higher-order multipoles. 

At a physical level. Equation 35 represents a mixing of all of the possible electronic states of the molecule, all of which have some probability of being attained according to the laws of quantum mechanics. Full Cl is the most complete non-relativistic treatment of the molecular system possible, within the limitations imposed by the chosen basis set. It represents the possible quantum states of the system while modelling the electron density in accordance with the definition (and constraints) of the basis set in use. For this reason, it appears in the rightmost column of the following methods chart ... 

Thus v t) is a unique functional of the electron density since u(r) fixes the Hamiltonian we see that the full many-particle ground state is a unique functional of the electron density. 

Over the years, many workers have addressed the problem of choice of cavity and the reaction field. Tomasi s polarized continuum model (PCM) defines the cavity as a series of interlocking spheres. The isodensity PCM (IPCM) defines the cavity as an isodensity surface of the molecule. This isodensity surface is determined iteratively. The self-consistent isodensity polarized continuum model (SQ-PCM) gives a further refinement in that it allows for a full coupling between the cavity shape and the electron density. 

These treatments of periodic parts of the dipole moment operator are supported by several studies which show that, for large oligomeric chains, the perturbed electronic density exhibits a periodic potential in the middle of the chain whereas the chain end effects are related to the charge transfer through the chain [20-21]. Obviously, approaches based on truncated dipole moment operators still need to demonstrate that the global polarization effects are accounted for. In other words, one has to ensure that the polymeric value corresponds to the asymptotic limit of the oligomeric results obtained with the full operator. 

If the original macromolecular density matrix is already available, then such approximate macromolecular electron densities for slightly distorted nuclear geometries are simpler to calculate than the full recalculation of an ADMA macromolecular density matrix that involves a new fragmentation procedure. 

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