Partition electronic

Partitioning Electron Density The Theory of Atoms in Molecules... 

R F W Bader s theory of atoms in molecules [Bader 1985] provides an alternative way to partition the electrons between the atoms in a molecule. Bader s theory has been applied to many different problems, but for the purposes of our present discussion we will concentrate on its use in partitioning electron density. The Bader approach is based upon the concept of a gradient vector path, which is a cuiwe around the molecule such that it is always perpendicular to the electron density contours. A set of gradient paths is drawn in Figure 2.14 for formamide. As can be seen, some of the gradient paths terminate at the atomic nuclei. Other gradient paths are attracted to points (called critical points) that are... 

It is reasonable (but not necessarily correct ) to assign any electrons associated with a particular diagonal element, to that atom on which the basis function ( ), is located. It is also reasonable to assign electrons associated with off-diagonal elements P v, where both ( ), and ( )v reside on the same atom, to that atom. However, it is not apparent how to partition electrons from density matrix elements P, v where ( ) i and ( )vreside on different atoms. Mulliken provided a recipe. Give each atom half of the total. Very simple but completely arbitrary ... 

Assigning atom charges and bond orders involves calculating the number of electrons belonging to an atom or shared between two atoms, i.e. the population of electrons on or between atoms hence such calculations are said to involve population analysis. Earlier schemes for population analysis bypassed the problem of defining the space occupied by atoms in molecules, and the space occupied by bonding electrons, by partitioning electron density in a somewhat arbitrary way. The earliest such schemes were utilized in the simple Hiickel or similar methods [256], and related these quantities to the basis functions (which in these methods are essentially valence, or even just p, atomic orbitals see Section 4.3.4). The simplest scheme used in ab initio calculations is Mulliken population analysis [257]. 

Partitioning electron density in four electron systems... 

K. B. Wiberg and J. J. Wendoloski, Proc. Natl. Acad. Sci. U.S.A., 78, 6561 (1981). Effect of Basis Set on Electron Populations Calculated by Using Bader s Criterion for Partitioning Electron Density Between Atoms. For an interpretation of NMR data, see H. Boaz, Tetrahedron Lett., 55 (1973). Separable Contributions of Induction and Polarization to the Chemical Shift. I. Symmetrical, Saturated Hydrocarbons Having No Internal Rotation. 

Figure 5.39. The Mulliken scheme for partitioning electron density.

The potential energy of the electrons, V, which is a negative quantity that can be partitioned into bulk and surface contributions, as shown. 

This method, introduced originally in an analysis of nuclear resonance reactions, has been extensively developed [H, 16 and F7] over the past 20 years as a powerful ab initio calculational tool. It partitions configuration space into two regions by a sphere of radius r = a, where r is the scattered electron coordinate. 

The full system Hamiltonian is partitioned so as to define an electronic Hamiltonian,... 

Mciny of the theories used in molecular modelling involve multiple integrals. Examples include tire two-electron integrals formd in Hartree-Fock theory, and the integral over the piriitii >ns and momenta used to define the partition function, Q. In fact, most of the multiple integrals that have to be evaluated are double integrals. 

The electronic partition function of the transition state is expressed in terms of the activation energy (the energy of the transition state relative to the electronic energy of the reactants) E as ... 

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. 

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

---