Pair occupations

To first investigate the effects of H-bonding, we compare lone pair occupancies of monomers and clusters in gas phase a significant decrease is found passing from the free pyridazine to either the one-water or the two-water cluster. This decrease, which indicates that electron population has been removed from the lone pair orbital, is accompanied by an increase of the diamagnetic contribution. 

An sp hybridized nitrogen centre is consistent with the trigonal planar shape of [N03]. Allow the hybrid orbitals to overlap with suitable orbitals from oxygen a choice of sp hybridization on the O atom provides suitable orbitals to accommodate the oxygen lone pairs. Occupation of each bonding orbital by a pair of electrons gives three equivalent N—O f7-bonds ... 

To further examine hyperconjugation, NBO analysis [30, 57] was carried out on stmctures I-IV of both fluoropi-peridine and trifluoromethylpiperidine. The key results at the (oB97X-D/aug-cc-pVTZ level are summarized in Table 6. The nitrogen lone pair occupancy, n, is lowest for stmcture IV at 1.872 electrons for FP and 1.908 electrons for TP. Since the nitrogen lone pair population is less than two, we conclude that the endo-anomeric effect is operative for stmcture IV, and we will show below that this is due to hyperconjugation in the NBO analysis framework. 

Figure 8.6. Features of the BCS model Left Cooper pair occupation variables m p and vp as a function of the energy e. Right superconducting density of states ) as a function of the energy gf is the density of states in the normal state at the Fermi level.

Finally, we consider the situation at finite temperature. In this case, we will take the temperature-dependent occupation number of the single-particle state k to be k for each spin state. This leads to the assignment of (1 — 2 k) as the Cooper pair occupation number of the same wave-vector. Consequently, each occurrence of a Cooper pair will be accompanied by a factor (1 — 2 k). while the occurrence of an electronic state that is not part of a Cooper pair will be accompanied by a factor 2 k taking into account spin degeneracy. With these considerations, the total energy of the ground state at finite temperature will be given by... 

As shown in the output, this particular NBO search terminated successfully after only a single cycle, which satisfied the default search criteria. The search yielded a Lewis structure with one core (CR), one bond (BD), and three lone pair (LP) Lewis-type (L) NBOs, which described about 99.95% of the total electron density (i.e., 9.995 of the 10 electrons). These five L-type NBOs easily satisfied the default threshold (1.90e) for pair occupancy [the 0 under Low occ (L) ] and the remaining 17 non-Lewis (NL) NBOs were all well below the O.le occupancy threshold [ High occ (NL) ] to be considered a satisfactory Lewis structure. [The Dev entry refers to deviations from the initial guess that steers multiple cycles of the search algorithm (if required), beyond the scope of this book consult the NBO... 

The threshold in (8.16) and (8.17) generally leads to assigned BD (two-center) versus LP (one-center) labels that are consistent with common chemical usage. The 5%-limit on delocalization tail density is also consistent with the default 1.90e threshold for satisfactory NBO pair occupancy (i.e., missing no more than 5% of 2e). Although somewhat arbitrary. 

Since the ON operators are projectors, we may now interpret the diagonal elements mpq as simultaneous occupations of pairs of spin orbitals (pair occupations), noting that topg represents the squared amplitude of the part of the wave function where spin orbitals 0/> and [Pg.22]

The reader may wish to verify that a weaker upper bound to the pair occupations... 

The orbital occupations (2.7.18) may exceed 1 since each orbital may be occupied by two electrons of opposite spins. For two distinct orbitals, the pair occupations (2.7.19) may be as laige as 4, representing a situation where both orbitals are doubly occupied in all detominants. The pair occupations are defined for a single orbital p = q since two electrons may simultaneously occupy the same orbital. Thus, (Opp - dpppp measures the simultaneous occupati< i of electrons in orbitals pa and 0pp and is equal to 2 if both orbitals are occupied in each determinant. 

The relationship between tlie lattice gas and the Ising model is also transparent in the alternative fomuilation of the problem, in temis of the number of down spins [i] and pairs of nearest-neighbour down spins [ii]. For a given degree of site occupation [i]. 

NAOs for an atomic block in the density matrix which have large occupancy numbers (say >1.90) are identified as lone pair orbitals. Their contributions to the density matrix are also removed. 

Each pair of atoms (AB, AC, BC,...) is now considered, and the two-by-two subblocks of the density matrix (with the core and lone pair contributions removed) are diagonalized. Natural bond orbitals are identified as eigenvectors which have large eigenvalues (occupation numbers larger than say 1.90). 

We define a fee lattice and affect at each site n, a spin or an occupation variable <7 which takes the value +1 or —1 depending on whether site n is occupied by a A or B atom. Within the generalized perturbation method , it has been shown that substitutional binary alloys AcBi-c may be described within a Ising model with effective pair interactions with concentration dependence. Thus, the energy of a configuration c = (<7i,<72,- ) among the 2 accessible configurations for one system can be written... 

In this chapter we meet three increasingly sophisticated models of molecular shape. The first considers molecular shape to be a consequence merely of the electrostatic (coulombic) interaction between pairs of electrons. The other two models are theories that describe the distribution of electrons and molecular shape in terms of the occupation of orbitals. 

Fig. 1. Electronic states [or iron-group atoms, showing number of states as qualitative [unction of electronic energy. Electrons in band A are paired with similar electrons of neighboring atoms to form bonds. Electrons in band B are d electrons with small interatomic interaction they remain unpaired until the band is half-filled. The shaded area represents occupancy of the states by electrons in nickel, with 0.6 electron lacking from a completely filled B band. (States corresponding to occupancy of bond orbitals by unshared electron pairs are not shown in the diagram.)...

With this assumption, and with 3.22 other orbitals available for occupancy by odd electrons or electron pairs, the ferromagnetic moment would be predicted to rise to a maximum of 3.22/u,B at Co and then to decrease steadily with slope -1 to 0 for the alloy Zn7gGa22- This prediction agrees only in part with observation the values of Pb for alloys Fe-Co, Co-Ni, and Ni-Cu (Fig. 1) reach a rounded-off maximum of about 2.50 and then decrease to 0 at... 

However, this uninhibited resonance requires that the atom receiving a bond (M+ or M) have an orbital available for its reception (occupied in M-). It is the possession of this orbital (the metallic orbital), in addition to the orbitals required for occupancy by unshared electron pairs and bonding electrons, by all or many of the atoms in a... 

---