Mulliken approach

Several forms of wf have already been used within the field of MQS. These methods include the Hirshfeld partitioning [30], Bader s partitioning based on the virial theorem within atomic domains in a molecule [64], and the Mulliken approach [65]. For more information on all the three methods, refer to Chapter 15. 

Fig. 6.9 Atom charges and bond orders calculated using the AMI, PM3 and HF/3-21G( ) methods. In a and b the charges and bond orders are all from the Mulliken approach. In c and d the charges are all electrostatic potential charges, and the bond orders are Mulliken for AMI and PM3, and Lowdin for HF/3-21G< ) (Lowdin bond orders were not available for AMI and PM3 from the Spartan program used). Note that charges and bond orders involving hydrogens have been omitted...

Molecular orbitals are characterized by energies and amplitudes expressing the distribution of electron density over the nuclear framework (1-3). In the linear combination of atomic orbital (LCAO) approximation, the latter are expressed in terms of AO coefficients which in turn can be processed using the Mulliken approach into atomic and overlap populations. These in turn are related to relative charge distribution and atom-atom bonding interactions. Although in principle all occupied MOs are required to describe an observable molecular property, in fact certain aspects of structure and reactivity correlate rather well with the nature of selected filled and unfilled MOs. In particular, the properties of the highest occupied MO (HOMO) and lowest unoccupied MO (LUMO) permit the rationalization of trends in structural and reaction properties (28). A qualitative predictor of stability or, alternatively, a predictor of electron... 

The Mulliken approach to population analysis has certain problems for example, it sometimes assigns more than two elecuons, and sometimes a negative number of elecuons, to an orbital. It is also fairly basis-set dependent (Hehre, Radom, Schleyer and Pople compare Mulliken charges for a variety of molecules using the STO-3G,... 

One objection to the Mulliken approach, however, is that the results of the analysis depend (sometimes quite strongly) iq)on the particular localised basis set chosen as the reference. There is no a priori reason why atomic orbitals, for instance, should be a better basis set into rfiich to deconq>ose the system eigenfunctions than, say, a set of Muffin-tin orbitals, or a set of Gaussian orbitals. Once the atoms in a system are brought together to form a molecule or a solid, the localised orbitals of the isolated atoms cease to have any special meaning except as a convenient localised basis set, and we should not interpret a... 

Mulliken analysis based on such a set as if it provided a unique result. Furthermore, one should recall that the one-electron DFT eigenfunctions are only a good approximation to the true many-body eigenfunctions of the system, and to this extent the Mulliken approach is only ever an approximate treatment. 

Nevertheless, at this point an opportunity for confusion and misunderstanding can arise. This because there is a conceptual difference between the DFT values of IP and EA, that are for the ground state of a system, in definition (4.247), and their averaged values on the supposed valence or excited states, as displayed in Eq. (3.1). Such dichotomy can be transposed at the potential level in the Parr picture, since V(r) is a non-zero constant the almost vertical values are involved, whereas in the Mulliken approach the almost adiabatic case is fixed by setting V r) = 0, as no further electrons are attached to the system. 

Ldwdin population analysis avoids the problem of negative populations or populations greater than 2. Some quantum chemists prefer the Ldwdin approach to that of Mulliken as the charges are often closer to chemically intuitive values and are less sensitive to basis set. 

A simpler way of setting up a scale of electronegativities was devised by another American chemist, Robert Mulliken. In his approach, the electronegativity is the average of the ionization energy and electron affinity of the element (both expressed in electronvolts) ... 

In retrospect, we believe that Professor Mulliken was perhaps somewhat overstating his case in the remark quoted in the Introduction, concerning the status of computational chemistry in 1965. However, considered as a prophetic remark, the quotation certainly applies today, and it is a pleasure to dedicate this account to Professor Yngve Ohrn acknowledging his many important contributions to this development. These contributions include not only his own work on theoretical and computational approaches to central questions in chemistry, but also his continued engagement in the Sanibel Conferences and in the International Journal of Quemtum Chemistry. 

An example of quantum mechanical schemes is the oldest and most widely used Mulliken population analysis [1], which simply divides the part of the electron density localized between two atoms, the overlap population that identifies a bond, equally between the two atoms of a bond. Alternatively, empirical methods to allocate atomic charges to directly bonded atoms in a reasonable way use appropriate rules which combine the atomic electronegativities with experimental structural information on the bonds linking the atoms of interest. A widely used approach included in many programs is the Gasteiger-Hiickel scheme [1]. 

The authors [33] have elucidated the linear dependence of Ao0 (z-dep) on E for the polyanions by a quantum chemical consideration. A model Hamiltonian approach to the charge transfer (CT) interaction between a polyanion and solvents has been made on the basis of the Mulliken s CT complex theory [34]. 

In more recent years, additional progress and new computational methodologies in macromolecular quantum chemistry have placed further emphasis on studies in transferability. Motivated by studies on molecular similarity [69-115] and electron density representations of molecular shapes [116-130], the transferability, adjustability, and additivity of local density fragments have been analyzed within the framework of an Additive Fuzzy Density Fragmentation (AFDF) approach [114, 131, 132], This AFDF approach, motivated by the early charge assignment approach of Mulliken [1, 2], is the basis of the first technique for the computation of ab initio quality electron densities of macromolecules such as proteins [133-141],... 

All the different AIM methods that will be discussed below basically use this same approach but quite different in the nature of n (. Chronologically, we will discuss the Mulliken AIM, the Hirshfeld AIM, and the Bader AIM. This last approach will henceforth be called quantum chemical topology (QCT). There are more AIM methods, but most of them can be easily understood by the three selected emblematic approaches. 

Details on the numerical evaluation of the descriptors will be given in the individual cases but in most cases a computational DFT approach is used, with a hybrid functional of the B3LYP type [32]. Condensation of f(r) or sir) is done with conventional population analysis techniques (Mulliken [33], Natural Population Analysis (NPA) [34]) or with the Hirshfeld technique [35], often used by our group [36]. 

The nitrosonium cation bears a formal relationship to the well-studied halogens (i.e. X2 = I2, Br2, and Cl2), with both classes of structurally simple diatomic electron acceptors forming an extensive series of intermolecular electron donor-acceptor (EDA) complexes that show well-defined charge-transfer absorption bands in the UV-visible spectral region. Mulliken (1952a,b 1964 Mulliken and Person, 1969) originally identified the three possible nonbonded structures of the halogen complexes as in Chart 7, and the subsequent X-ray studies established the axial form II to be extant in the crystals of the benzene complexes with Cl2 and Br2 (Hassel and Stromme, 1958, 1959). In these 1 1 molecular complexes, the closest approach of the... 

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