Density Born Oppenheimer

We expect our proposed approach to the understanding of chemically important issues to be rejected by many readers who operate in a comfort zone defined by probability densities, Born-Oppenheimer systems, hybrid orbitals, potential-energy surfaces, ab initio theory and DFT simulations—all of them Copenhagen spinoffs. We realize, of course, that these models have been developed to standards, where they produce a very accurate optimization of structures and properties of molecular compounds and materials in many areas [11], and for application-oriented... 

The usual way chemistry handles electrons is through a quantum-mechanical treatment in the frozen-nuclei approximation, often incorrectly referred to as the Born-Oppenheimer approximation. A description of the electrons involves either a wavefunction ( traditional quantum chemistry) or an electron density representation (density functional theory, DFT). Relativistic quantum chemistry has remained a specialist field and in most calculations of practical... 

One may consider the above equation as a generalization of Born-Oppenheimer dynamics in which electrons always stay on the Born-Oppenheimer surface. For a given conformation of nuclei, the numerical value of the fictitious mass associated with electronic degrees of freedom determines how far the electron density is allowed to deviate from the Born-Oppenheimer one. Each consecutive step along the trajectory, which involves electronic and nuclear degrees of freedom, can be obtained without determining the exact Born-Oppenheimer electron density. 

The electronic wave function of an n-electron molecule is defined in 3n-dimensional configuration space, consistent with any conceivable molecular geometry. If the only aim is to characterize a molecule of fixed Born-Oppenheimer geometry the amount of information contained in the molecular wave function is therefore quite excessive. It turns out that the three-dimensional electron density function contains adequate information to uniquely determine the ground-state electronic properties of the molecule, as first demonstrated by Hohenberg and Kohn [104]. The approach is equivalent to the Thomas-Fermi model of an atom applied to molecules. 

If one is interested in spectroscopy involving only the ground Born Oppenheimer surface of the liquid (which would correspond to IR and far-IR spectra), the simplest approximation involves replacing the quantum TCF by its classical counterpart. Thus pp becomes a classical variable, the trace becomes a phase-space integral, and the density operator becomes the phase-space distribution function. For light frequency co with ho > kT, this classical approximation will lead to substantial errors, and so it is important to multiply the result by a quantum correction factor the usual choice for this application is the harmonic quantum correction factor [79 84]. Thus we have... 

It is clear that arbitrary one-particle densities of a molecular system need not have the same topology. In fact, only those belonging to the same structural region will share this property. To make these concepts clearer, consider two nuclear configurations X and Y belonging to the nuclear configuration space in the context of the Born-Oppenheimer approximation. The corresponding one-electrons densities are p r X) and p(r T), respectively. Consider the... 

We restrict ourselves to the clamped-nucleus or Born-Oppenheimer approximation [30,31] because essentially all the work done to date on electron momentum densities has relied on it. Therefore we focus on purely electronic wavefunctions and the electron densities that they lead to. 

Eq. (5.34). However, it is possible to construct approximate wavefunctions that lead to electron momentum densities that do not have inversion symmetry. Within the Born-Oppenheimer approximation, the total electronic system must be at rest the at-rest condition... 

Within the Born-Oppenheimer approximation, the nuclei are at rest and have zero momentum. So the electron momentum density is an intrinsically one-center function that can be expressed usefully in spherical polar coordinates and expanded as follows [162,163]... 

Here p is the density of vibrational levels of states Sj and Sf at the energy of the electronic transition E. The overlap of the electronic wavefunctions 0i5 0f and of the vibrational wavefunctions (0i 0f) are factorized according to the Born-Oppenheimer approximation just as in the case of radiative transitions. The density of vibrational levels is greater for the lower (final) state Sf... 

The relevant question regarding secondary IEs on acidity is the extent to which IEs affect the electronic distribution. How can an inductive effect be reconciled with the Born-Oppenheimer approximation Although the potential-energy function and the electronic wave function are independent of nuclear mass, an anharmonic potential leads to different vibrational wave functions for different masses. Averaging over the ground-state wave function leads to different positions for the nuclei and thus averaged electron densities that vary with isotope. This certainly leads to NMR isotope shifts (IEs on chemical shifts), because nuclear shielding is sensitive to electron density.16... 

In contrast to the above situation, based on an average charge density (pa), one may identify another dynamical regime where the solvent electronic timescale is fast [50-52] relative to that of the solute electrons (especially, those participating in the ET process). In this case, H F remains as in Equation (3.106), treated at the Born-Oppenheimer (BO) level (i.e., separation of electronic and nuclear timescales), but HFF is replaced by an optical RF operator involving instantaneous electron coordinates [52] ... 

In the related work of Kim and Hynes [50], Equations (3.107) and (3.112) have been designated, respectively, by the labels SC (self-consistent or mean field) and BO (where Born-Oppenheimer here refers to timescale separation of solvent and solute electrons). More general timescale analysis has also been reported [50,51], Equation (3.112) is similar in spirit to the so-called direct RF method (DRF) [54-56], The difference between the BO and SC results has been related to electronic fluctuations associated with dispersion interactions [55], Approximate means of separating the full solute electronic densities into an ET-active subspace and the remainder, treated, respectively, at the BO and SC levels, have also been explored [52],... 

For a direct absorption, x and g label the excited and ground states, respectively. These states are, in turn, coupled by a dipole moment operator p, and are assumed to be of Born-Oppenheimer type, i.e. the electronic contributions are separated from those of the nucleus. With these assumptions, the interactions between these states result from vibrational overlaps between the ground and excited state with the transition probability Wg x and the density of the final states pf. In general terms, we now can evaluate the transition probability, which mainly depends upon two parameters ... 

It is hardly necessary to discourse on the importance of determining the structure of many-electron systems. If one focuses on molecules and adopts a Born-Oppenheimer viewpoint1, it is often sufficient to analyze the ground state of such a system, and this will be our domain of interest. To this end, the density functional approach2 has become an increasingly effective tool. What we want to do is to obtain information on the structural characteristics of valid density functionals in order to more reliably construct the parametrized empirical... 

The classical idea of molecular structure gained its entry into quantum theory on the basis of the Born Oppenheimer approximation, albeit not as a non-classical concept. The B-0 assumption makes a clear distinction between the mechanical behaviour of atomic nuclei and electrons, which obeys quantum laws only for the latter. Any attempt to retrieve chemical structure quantum-mechanically must therefore be based on the analysis of electron charge density. This procedure is supported by crystallographic theory and the assumption that X-rays are scattered on electrons. Extended to the scattering of neutrons it can finally be shown that the atomic distribution in crystalline solids is identical with molecular structures defined by X-ray diffraction. 

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