Wave function analysis natural orbitals

WAVE FUNCTION ANALYSIS Qfl NATURAL ATOMIC ORBITAL AND NATUFIAL BOND ORBITAL ANALYSIS 231... 

A wave function can often be expressed in different orbital bases. This flexibility may be exploited to express the optimized Cl wave function in an orbital basis with particularly attractive features for the analysis of the wave function. Thus, it is often useful to express the wave function in the natural-orbital basis. The transformation to this basis can of course be achieved by repeating the whole calculation in the new basis, but this procedure is rather cumbersome. We here present an alternative, inexpensive scheme for expressing the Cl wave function in the new basis, avoiding the reoptimization of the wave function 117]. 

The results of a valence bond treatment of the rotational barrier in ethane lie between the extremes of the NBO and EDA analyses and seem to reconcile this dispute by suggesting that both Pauli repulsion and hyperconjugation are important. This is probably closest to the truth (remember that Pauli repulsion dominates in the higher alkanes) but the VB approach is still imperfect and also is mostly a very powerful expert method [43]. VB methods construct the total wave function from linear combinations of covalent resonance and an array of ionic structures as the covalent structure is typically much lower in energy, the ionic contributions are included by using highly delocalised (and polarisable) so-called Coulson-Fischer orbitals. Needless to say, this is not error free and the brief description of this rather old but valuable approach indicates the expert nature of this type of analysis. 

At the same time, the formally independent particle nature of DFT allows the application of standard interpretative tools developed for the HF approach. This is true not only for the standard MuUiken population analysis, but also for more sophisticated schemes, like the Natural Bond Orbital (NBO) analysis [9], the Atomic Polarizable Tensor population [10], or the Atom in Molecule (AIM) approach [11]. These tools allow the use of familiar and well known models to analyze the molecular wave function and to rationalize it in terms of classical chemical concepts. In short, DFT is providing very effective quantum... 

Decomposition of interaction energies is desired for qualitative chemical analyses of complicated multi-valent interactions in supramolecular aggregates but such a decomposition cannot be uniquely defined within fundamental physical theory. A popular semi-quantitative decomposition method with nice formal features to be mentioned in this context is Weinhold s natural bond orbital (NBO) approach to intermolecular interactions [232, 233]. Comparable is the recently proposed energy decomposition analysis by Mo, Gao and Peyerimhoff [234, 235] which is based on a block-localized wave function. Other energy decomposition schemes proposed are the energy decomposition analysis (EDA) by Kitaura and Morokuma [236] and a similar scheme by Ziegler and Rauk [237]. 

Several schemes for the analysis of the wave function have been proposed. The most commonly used are those proposed by Mulliken and Lbwdin, those based on natural bond orbital theory (NBO), the Bader AIM theory, and the fitting of the electrostatic potential. 

This type of analysis of the natural orbitals in terms of electron correlation can be performed for any wave function that is accurate enough to include the appropriate orbitals. In this chapter we shall describe molecular systems and chemical processes where the NOs are not trivially partitioned into strongly and weakly occupied and we shall show that in such cases the HF wave function is not a good starting point. We need more than one configuration for a qualitatively correct description of the electronic stmcmre. As a first example we shall consider the simplest of all chemical bonds the hydrogen molecule. 

At the end of the reaction we have two new bonding orbitals from the ring. They are single bonds, which typically have occupation numbers close to two. The importance of this analysis is that it is valid for the exact wave function. Whether it remains true for approximate methods depends on the method. Below we shall discuss an approach that takes these features of the electronic structure explicitly into account. But first, we shall look more closely at the situation where all occupied orbitals have occupation numbers close to two. This situation is common for most molecules in their ground electronic state, close to their equilibrium geometry. It is a natural first approximation to assume that the occupation numbers are exactly two or zero, which can be... 

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