Mulliken rules

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 chemistry of unsaturated silicon compounds, i.e. silylenes and molecules having (p-p)ic-sili-con element multiple bonds >Si=E (E = C, Si, Ge, Sn, N, P, As, O, S), is an interesting field of research for the theoretician as well as for the preparative chemist because of the unexpected and fascinating results which can be obtained. Yet 30 years ago, such compounds were considered "non existent" because of the classical "double bond rule", established by Pitzer and Mulliken in the early fifties. Since then, the chemistry of unsaturated silicon compounds proceeded from the investigation of small" species in the gas phase to the synthesis and isolation of stable species with bulky substituents at the > Si =E moiety, and to the determination of their structural features. 

However, we have shown how the 18-electron rule is commonly satisfied in the absence of any significant p-orbital participation, on the basis of hypervalent 3c/4e cu-bonding interactions wholly within the framework of normal-valent sd" hybridization. Results of NBO and Mulliken analyses of high-level wavefunctions for transition-metal complexes commonly exhibit only paltry occupation of the outer p orbitals (comparable in this respect to the weak contributions of d-type polarization functions in main-group bonding). 

The Woodward-Hoffmann rules have intellectual roots that can be traced back to Wigner-Witmer correlation rules (E. Wigner and E. E. Witmer, Z. Phys. 51 [1928], 859) and general correlation-diagram concepts (R. S. Mulliken, Rev. Mod. Phys. 4 [1932], 1) as employed, e.g., by K. F. Herzfeld, Rev. Mod. Phys. 41 (1949), 527. Alternative MO... 

The photocyclodehydrogenation of thienyl ethylenes is well-defined when both thiophene rings are bound via a C(2) atom to the ethylenic bond as in (70). In other cases, however, more cyclization products are possible. To predict the photocyclization mode for heterohelicenes the F s rule fails in many cases, because correction factors for the hetero atoms in the Huckel MO calculation have to be introduced and the systems are not well comparable with carbocyclic diaryl ethylenes. A better reaction parameter in these cases is the Mulliken overlap population (nrs)51), introduced by Muszkat52) for these cases. The overlap populations of the atoms r and s in ground and excited state (nIiS and n s), are calculated using the extended Huckel method. Cyclizations should not occur when nr>s and An s (= nr>s — n s) have negative values. (This method can also be used for diaryl olefins, but in these cases calculation of F s is more simple.). 

The symbols used for the representations are those proposed by Mulliken. The A representations are those which are symmetric with respect to the C2 operation, and the Bs are antisymmetric to that operation. The subscript 1 indicates that a representation is symmetric with respect to the ov operation, the subscript 2 indicating antisymmetry to it. No other indications are required, since the characters in the o column are decided by another rule of group theory. This rule is the product of any two columns of a character table must also be a column in that table. It may be seen that the product of the C2 characters and those of gv give the contents of the The representations deduced above must be described as irreducible representations This is because they... 

Electronic states of an atom or a molecule are obtained by considering the properties of all the electrons in all the orbitals. The properties of the electrons in the unfilled shells are the main contributors. It is useful to classify electronic states in terms of their symmetry properties as defined by the group operations pertinent to that particular molecular species. Mulliken s terminology is based on the following rules (small letters are used for one electron orbitals and capital letters for molecular states) ... 

If necessary, numerical subscripts are added to the labels to distinguish the non-equivalent irreducible representations which are not distinguished by the foregoing rules. Except for the fact that the totally symmetric representation (one-dimensional unit matrices) is numbered and listed first, the numbering is arbitrary and the reader is referred to Appendix I or Mulliken s report for the internationally accepted conventions. 

Other groups may be handled in a similar manner to O in Example 8.1-1. For improper rotations, the two rules formulated previously hold also for double groups (Box 8.1). If the group contains the inversion operator, even or odd parity is indicated by a superscript of + or —in Bethe s notation and by a subscript g or u in Mulliken-Herzberg notation. 

Two vibronic states which happen to occur at nearly the same energy will perturb one another if certain conditions are fulfilled. The interacting states may be derived from different electronic states, but in polyatomic spectra are often merely different vibrational states of the same electronic states. The perturbations, which are either homogeneous (AK = 0) (e.g. Fermi resonance) or heterogeneous (AK = 1) (Coriolis resonance) (Mulliken, 1937) are then analogous to the perturbations observed in infrared and Raman spectra. Such perturbations are commonplace in electronic bands where the completely unperturbed band is the exception rather than the rule. 

Table 1 gives the orbital and spin-orbital selection rules appropriate for a D41, copper(II) ion in a compressed and elongated environment. Here the Mulliken... 

As mentioned before, carbon and silicon mostly differ in their ability to from multiple p,-p, bonds E=Y with suitable partners (E = C, Si Y = element of group 14 to 16 ). While the p orbital overlap in compounds >C=Y is sufficient to yield stable multiple bonded species, this overlap is strongly reduced in the case of silicon (classical double bond rule of Pitzer and Mulliken). Consequently, under comparable conditions the equivalents of many unsaturated monomeric compounds of carbon, such as H2C=CH2, R2C=0 or CO2 are silicon single bonded polymeric products, e g. polysilanes (-H2Si-SiH2-)n, silicones (-R2Si-0-) and silicon dioxide (Si02)n... 

The 7/ / terms are the Coulomb functions introduced in eq.(104), P are the relaxed (i.e. polarized by the solvent in a self-consistent manner) elements of the density matrix P = CnC, and F are the elements of the in vacuo Fock matrix, in the form used in the semiempirical approach, but referred to the relaxed P matrix. The charges qk are drawn from P, according to the Mulliken s rules, or according to a new definition given by... 

For the dipole transition, Laporte selection rule (Af = 1) for the emitting or absorbing atom is valid, when the probability of interatomic transition is very small. In such a case, the atomic orbital component (or partial density of states) obtained from Mulliken population analysis is useful for a rough estimation of... 

Symmetry and stability analysis. The semi-empirical unrestricted Hartree-Fock (UHF) method was used for symmetry and stability analysis of chemical reactions at early stage of our theoretical studies.1,2 The BS MOs for CT diradicals are also expanded in terms of composite donor and acceptor MOs to obtain the Mulliken CT theoretical explanations of their electronic structures. Instability in chemical bonds followed by the BS ab initio calculations is one of the useful approaches for elucidating electronic structures of active reaction intermediates and transition structures.2 The concept is also useful to characterize chemical reaction mechanisms in combination with the Woodward-Hoffman (WH) orbital symmetry criterion,3 as illustrated in Figure 1. According to the Woodward-Hoffmann rule,3 there are two types of organic reactions orbital-symmetry allowed and forbidden. On the other hand, the orbital instability condition is the other criterion for distinguishing between nonradical and diradical cases.2 The combination of the two criteria provides four different cases (i) allowed nonradical (AN), (ii) allowed radical (AR), (iii) forbidden nonradical (FN), and (iv) forbidden radical (FR). The charge and spin density populations obtained by the ab initio BS MO calculations are responsible for the above classifications as shown in Fig. 1. 

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