Concepts from Density Functional Theory

The success of FMO theory is not because the neglected terms in the second-order perturbation expansion (eq. (15.1)) are especially small an actual calculation will reveal that they completely swamp the HOMO-LUMO contribution. The deeper reason is that the shapes of the HOMO and LUMO resemble features in the total electron density, which determines the reactivity. 

A reaction will in general involve a change in the electron density, which may be quantified in terms of the Fukui function.  

The Fukui function indicates the change in the electron density at a given position when the number of electrons is changed. We may define two finite difference versions of the function, corresponding to addition or removal of an electron. 

The /+ function is expected to reflect the initial part of a nucleophilic reaction, and the / function an electrophilic reaction, i.e. the reaction will typically occur where the / function is large. For radical reactions the appropriate function is an average of /+ and /- 

In the frozen MO approximation the last terms are zero and the Fukui functions are given directly by the contributions from the HOMO and LUMO. The preferred site of attack is therefore at the atom(s) with the largest MO coefficients in the HOMO/LUMO, in exact agreement with FMO theory. The Fukui function(s) may be considered as the equivalent (or generalization) of FMO methods within Density Functional Theory (Chapter 6). 

The change in the electron density for each atomic site can be quantified by using the change in the atomic charges, although this of course suffers from the usual problems of defining atomic charges, as discussed in Chapter 9. The /+ functions may also be written in terms of orbital contributions. 

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Qualitative Theories 15.1 Frontier Molecular Orbital Theory 15.2 Concepts from Density Functional Theory 15.3 Qualitative Molecular Orbital Theory 34, 347 351 353 ... 

There are several cogent reasons to include a chapter on the solid state in a treatise devoted to chemical hardness, and other concepts, derived from density functional theory. One is that DFT has been the theoretical method of choice in dealing with solid-state problems for a number of years. 

Recently there have been important new developments, both with respect to acid-base interactions and with the concept of hardness (1). These developments come from density functional theory (DFT). a branch of quantum mechanics which focuses on the one-electron density function of a molecule, rather than its wave function (2). 

K5ster, A. M. Leboeuf, M. Salahub, D. R. Molecular Electrostatic Potentials from Density Functional Theory. In Molecular Electrostatic Potentials, Concepts and Applications, Murray, J. S. Sen, K., Eds. Elsevier Amsterdam, 1996 pp 105-142. 

In the last three decades, density functional theory (DFT) has been extensively used to generate what may be considered as a general approach to the description of chemical reactivity [1-5]. The concepts that emerge from this theory are response functions expressed basically in terms of derivatives of the total energy and of the electronic density with respect to the number of electrons and to the external potential. As such, they correspond to conceptually simple, but at the same time, chemically meaningful quantities. 

As we have seen, an atom under pressure changes its electron structure drastically and consequently, its chemical reactivity is also modified. In this direction we can use the significant chemical concepts such as the electronegativity and hardness, which have foundations in the density functional theory [9]. The intuition tells us that the polarizability of an atom must be reduced when it is confined, because the electron density has less possibility to be extended. Furthermore, it is known that the polarizability is related directly with the softness of a system [14], Thus, we expect atoms to be harder than usual when they are confined by rigid walls. Estimates of the electronegativity, x and die hardness, tj, can be obtained from [9]... 

In order to clarify these points and to verify the practicability of our concept, we investigated the course of the overall transformation by density functional theory (DFT) as depicted in Scheme 16. As expected, both the ring opening of 40 to 41 and the 5-exo cyclizations to 42 are exothermic. The calculated Ti - 0 bond lengths (1.86 A) are in excellent agreement with values obtained from crystallographic structures (1.85-1.89 A) [67-69]. 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

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