Wave function analysis electron density

Chemists are able to do research much more efficiently if they have a model for understanding chemistry. Population analysis is a mathematical way of partitioning a wave function or electron density into charges on the nuclei, bond orders, and other related information. These are probably the most widely used results that are not experimentally observable. 

Alternatively, we can base our analysis on the electron density, which as we have seen, is readily obtained from the wave function. The advantage of analyzing the electron density is that, unlike the wave function, the electron density is a real observable property of a molecule that, as we will see in Chapter 6, can be obtained from X-ray crystallographic studies. At the present time however, it is usually simpler to obtain the electron density of a molecule from an ab initio calculations rather than determine it experimentally. Because this analysis is based on a real physically observable property of a molecule, this approach appears to be the more fundamental. It is the approach taken by the atoms in molecules (AIM) theory, which we discuss in Chapters 6 and 7, on which we base part of the discussion in Chapters 8 and 9. 

Whereas the concepts and method described in this contribution are equally applicable to various approximate and more advanced quantum-chemical representations, the basic concepts will be discussed and illustrated within the framework of the conventional Hartree-Fock-Roothaan-Hall SCF LCAO ab initio representation of molecular wave functions and electronic densities, as can be computed, for example, using the Gaussian family of computer programs of Pople and co-workers. The essence of the shape analysis methods will be discussed with respect to some fixed nuclear arrangement K note, however, that the generalizations will involve changes in the nuclear arrangement K. 

The examples in Section 9.1 illustrate that it would be desirable to base a population analysis on properties of the wave function or electron density itsell and not on the basis set chosen for representing the wave function. The electron density is the square of the wave function integrated over Aeiec - 1 coordinates (it does not matter which coordinates since the electrons are indistinguishable). 

Wave Function Analysis Based on the Electron Density... 

The division of the molecular volume into atomic basins follows from a deeper analysis based on the principle of stationary action. The shapes of the atomic basins, and the associated electron densities, in a functional group are very similar in different molecules. The local properties of the wave function are therefore transferable to a very good approximation, which rationalizes the basis for organic chemistry, that functional groups react similarly in different molecules. It may be shown that any observable... 

Density functional theory, direct molecular dynamics, complete active space self-consistent field (CASSCF) technique, non-adiabatic systems, 404-411 Density operator, direct molecular dynamics, adiabatic systems, 375-377 Derivative couplings conical intersections, 569-570 direct molecular dynamics, vibronic coupling, conical intersections, 386-389 Determinantal wave function, electron nuclear dynamics (END), molecular systems, final-state analysis, 342-349 Diabatic representation ... 

The IAM model further assumes the atoms in a crystal to be neutral. This assumption is contradicted by the fact that molecules have dipole and higher electrostatic moments, which can indeed be derived from the X-ray diffraction intensities, as further discussed in chapter 7. The molecular dipole moment results, in part, from the nonspherical distribution of the atomic densities, but a large component is due to charge transfer between atoms of different electronegativity. A population analysis of an extended basis-set SCF wave function of HF, for example, gives a net charge q of +0.4 electron units (e) on the H atom in HF for CH4 the value is +0.12 e (Szabo and Ostlund 1989). 

The analysis first given by these authors depends on the proof that the one-electron density of states vanishes at F, showing the so-called soft Coulomb gap, illustrated in Fig. 1.29. This analysis is limited to systems far from the transition, where the overlap between wave functions is small. Here there is rather... 

On the other hand, the results of an MO analysis might seem at odds with the obvious fact that in molecules such as CH4 or SFb all of the a bonds are equivalent. Actually, there is no inconsistency. If the electron density in the molecule is computed from the wave functions for all of the filled MOs (e.g., the A i and T2 MOs in CH4) the equivalence of all the bonds will be evident. At the same time, the fact that these equivalent bonds are a result of the presence of electrons in nonequivalent MOs is also experimentally verifiable by the technique of photoelectron spectroscopy. 

The electrostatic potential F(r,) at a given point i created in the neighboring space by the nuclear charges and the electronic distribution of a molecule can be calculated from the molecular wave function (strictly speaking from the corresponding first-order density function). As this quantity is directly obtainable from the wave function, it does not suffer from the drawbacks inherent in the classical population analysis. 

An alternate model has been proposed (296-298) that attempts to interpret the data in terms of a single mixed state. For this model S is no longer a good quantum number. The unpaired electron spin density at the metal and the wave function that represents the single state are functions of temperature and pressure. This latter model was considered as a possibility by Leipoldt and Coppens. In their structure of the Fe(Et2Dtc)3 complex at 297 and 79°K they attempted an analysis of the temperature parameters at 297°K. The analysis could not distinguish between a mixed-spin state and a mixture of two different spin states (403). 

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