Energetic consequence, electron densities

Summing up, the structure of the effective Hamiltonian of Equation (1.107) makes explicit the nonlinear nature of the QM problem, due to the solute-solvent interaction operator depending on the wavefunction, via the expectation value of the electronic density operator. The consequences of the nonlinearity of the QM problem may be essentially reduced to two aspects (i) the necessity of an iterative solution of the Schrodinger Equation (1.107) and (ii) the necessity to introduce a new fundamental energetic quantity, not described by the effective molecular Hamiltonian. The contrast with the corresponding QM problem for an isolated molecule is evident. 

The examples presented in this work by no means cover the subject of the C-H bond activation on a spectrum of catalytic media. Interaction of methane with the small clusters discussed here obviously cannot pretend to fully mimic catalytic centers in reality. Nevertheless, they seem to justify drawing generalized conclusions regarding the mechanism of catalytic activation in terms of electron withdrawal or donation to the interacting hydrocarbon molecule. A variety of properties contribute consequently to the emerging scheme (electronic density redistribution, geometry evolution in critical points, energetical factors, vibrational analyses) which substantially increases credibility of the conclusions. 

As a result of static disorder, often alternate positions of atoms can be resolved in electron density maps. In macromolecular crystals, some of the flexible regions possess variable conformers and as a consequence none of the individual ones can be detected in electron density maps. Thus, dynamics between several energetically similar states and/or larger amplitude makes to vanish more mobile sequential subunits. In practice, more than two or three conformer makes detection and/ or assignment impossible. In other words lower than 25-33% of relative population of conformers is unseen by diffraction methods. 

There is another physical phenomenon which appears at the correlated level which is completely absent in Hartree-Fock calculations. The transient fluctuations in electron density of one molecule which cause a momentary polarization of the other are typically referred to as London forces. Such forces can be associated with the excitation of one or more electrons in molecule A from occupied to vacant molecular orbitals (polarization of A), coupled with a like excitation of electrons in B within the B MOs. Such multiple excitations appear in correlated calculations their energetic consequence is typically labeled as dispersion energy. Dispersion first appears in double excitations where one electron is excited within A and one within B, but higher order excitations are also possible. As a result, all the dispersion is not encompassed by correlated calculations which terminate with double excitations, but there are higher-order pieces of dispersion present at all levels of excitation. Although dispersion is not necessarily a dominating contributor to H-bonds, this force must be considered to achieve quantitative accuracy. Moreover, dispersion can be particularly important to geometries that are of competitive stability to H-bonds, for example in the case of stacked versus H-bonded DNA base pairs. ... 

The relaxation or deformation of the density cloud has an energetic consequence. The redistribution of electron density that accompanies the formation of the complex stabilizes the system, thus contributing to the interaction energy. In order to... 

Inspection of the porphyrin HOMOs and LUMOs show that only the b2 orbital has no electron density over the central nitrogens. Consequently, this will be the only orbital to be energetically unaffected by metal chelation or dication formation. Electrons on the metal will electrostatically repel the electrons in the remaining orbitals and so raise their energies. Conversely, the two positive charges on the central nitrogens (which arise on dication formation, from the addition of two extra protons) will electrostatically attract the electrons in the bj and ba orbitals and so lower their energy. 

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