Population Analysis Based on the Wave Function

Perhaps the most rigorous way of dividing a molecular volume into atomic subspaces is the Atoms In Molecules (AIM) method of Bader.The electron density is the square of the wave function integrated over N — coordinates (it does not matter which coordinates since all electrons are identical). 

The border between two three-dimensional atomic basins is a two-dimensional surface. Points on such dividing surfaces have the property that the gradient of the electron density is perpendicular to the normal vector of the surface, i.e. the radial part of the derivative of the electron density (the electronic flux ) is zero. 

Once the molecular volume has been divided, the electron density may be integrated within each of the atomic basins to give a net atomic charge. As the dividing surface is 

The other problem in the AIM approach is the presence of non-nuclear attractors in certain metallic systems, such as lithium and sodium clusters. While these are of interest by themselves, they spoil the picture of electrons associated with nuclei forming atoms within molecules. 

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 

There are two other problems with AIM atomic charges. One is the magnitude of the calculated atomic charges for polar bonds, the calculated internal redistribution is often significantly-larger than commonly accepted values. This may qualitatively be- 

The other problem in the AIM approach is the presence of non-nuclear attractors in 

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WAVE FUNCTION ANALYSIS 9.3 POPULATION ANALYSIS BASED ON THE WAVE FUNCTION 223... 

Two general techniques are used to extract charges from wave functions - the Mulliken population analysis, based on partitioning the electron distribution, and the ESP method, based on fitting properties which depend on the electron distribution to a model which replaces this distribution by a set of atomic... 

Population analysis in semiempirical methods fall into two categories. Methods including overlap in the Fock equations use the Mulliken population analysis. The majority of semiempirical methods uses the ZDO approximation, and the net charges are interpreted on the basis of symmetrically orthog-onalized AOs. It is pointed out that this interpretation is not exactly valid, because of truncation and empirical adjustment. But the corresponding nonsymmetrical orthogonalization is not uniquely defined. Charge models based on semiempirical wave functions play an important role in the calculation of molecular electrostatic potentials for reactivity. 

One more trend in laser control is based on the use of the property of coherence of the laser light. To effect coherent laser control, it is necessary that not only the light, but also the atom (or molecule) should be in a coherent state during the interaction. For atoms in a beam or in a low-pressure gas, the phase relaxation time of their wave functions depends on spontaneous decay or on collisions and can be comparatively long (from 10 to 10 s). It was for precisely this reason that the main experiments on coherent interaction were conducted with atoms. These experiments led in the final analysis to the discovery of new effects, such as coherent population trapping (Arimondo 1996), electromagnetically induced transparency (Harris 1997), and the slow-light effect (Hau et al. 1999 Kash et al. 1999). 

A plane wave basis set with an orbital cutoff distance of 3.9 A was used. All-electron calculations and a double numerical basis set with polarization functions (DNP) were employed. Wave function integration in reciprocal space was performed via fine grid sampling of k points with a separation of 0.02 A . For the calculation of the density of states (DOS), a 5 X 8 X 3 Monkhorst-Pack grid was used. Charge transfer was calculated based on the Mulliken Population Analysis (MPA) (Mulliken 1955). 

A new set of charge models based on AMI and PM3 wave functions was introduced by Cramer, Truhlar and co-workers. The idea is to transform a set of charges, obtained by a usual population analysis, to a new set of charges, so that the physical observables, e.g.. dipole moments, are more accurate than those calculated from the original charges. These models are called AM I-CM IA and PM3-CM1P. All these schemes can serve to generate efficient and accurate MESPs from semiempirical wave functions. These MESPs are needed to study solvation effects in continuum solvation models. 

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