Localized atomic orbitals

There exists no uniformity as regards the relation between localized orbitals and canonical orbitals. For example, if one considers an atom with two electrons in a (Is) atomic orbital and two electrons in a (2s) atomic orbital, then one finds that the localized atomic orbitals are rather close to the canonical atomic orbitals, which indicates that the canonical orbitals themselves are already highly, though not maximally, localized.18) (In this case, localization essentially diminishes the (Is) character of the (2s) orbital.) The opposite situation is found, on the other hand, if one considers the two inner shells in a homonuclear diatomic molecule. Here, the canonical orbitals are the molecular orbitals (lo ) and (1 ou), i.e. the bonding and the antibonding combinations of the (Is) orbitals from the two atoms, which are completely delocalized. In contrast, the localization procedure yields two localized orbitals which are essentially the inner shell orbital on the first atom and that on the second atom.19 It is thus apparent that the canonical orbitals may be identical with the localized orbitals, that they may be close to the localized orbitals, that they may be identical with the completely delocalized orbitals, or that they may be intermediate in character. 

Fio. 1. Orbital energy level diagram for carbon oxysulphide using theoretical values (Clementi, 1962). The deeper levels are essentially localized atomic orbitals. The energy scale is in electron volts, expanded on the right to show the valence shell structure. 

VB Wave Functions with Localized Atomic Orbitals... 

The only electrons that might be useful in the kind of attraction we have discussed so far are the lone pair electrons on bromine. But we know from many experiments that electrons flow out of the alkene towards the bromine atom in this reaction—the reverse of what we should expect from electron distribution. The attraction between these molecules is not electrostatic. In fact, we know that reaction occurs because the bromine molecule has an empty orbital available to accept electrons. This is not a localized atomic orbital like that in the BF3 molecule. It is the antibonding orbital belonging to the Br-Br G bond the c orbital. There is therefore in this case an attractive interaction between a full orbital (the Jt bond) and an empty orbital (the o orbital of the Br-Br bond). The molecules are attracted to each other because this one interaction is between an empty and a full orbital and leads to bonding, unlike all the other repulsive interactions between filled orbitals. We shall develop this less obvious attraction as the chapter proceeds. 

The principles of calculation of the size of the spin-orbit interaction are straightforward in the case of a free atom or for truly localized atomic orbitals of atoms within molecules, e.g. the 4f orbitals in rare earth compounds. Thus Hund accurately predicted values of the relevant moments L, J and S for rare earth ions in solid salts and aqueous solutions and showed that their magnetic moments were given by Van Vleck s equation (equation 15) ... 

For a group orbitals, therefore, it suffices to find the local magnitudes of the harmonic at the vertices. For n we need the local gradients [as first derivatives] and for 5 the local concavities [as second derivatives]. These determine the combination coefficients of the local resultant functions in the group orbitals, which local functions are found by making the appropriate linear combinations oi a ne and See and 5 combinations of the local atomic orbitals decorating the orbit vertices. 

The electronic structure of solids and surfaces is usually described in terms of band structure. To this end, a unit cell containing a given number of atoms is periodically repeated in three dimensions to account for the infinite nature of the crystalline solid, and the Schrodinger equation is solved for the atoms in the unit cell subject to periodic boundary conditions [40]. This approach can also be extended to the study of adsorbates on surfaces or of bulk defects by means of the supercell approach in which an artificial periodic structure is created where the adsorbate is translationally reproduced in correspondence to a given superlattice of the host. This procedure allows the use of efficient computer programs designed for the treatment of periodic systems and has indeed been followed by several authors to study defects using either density functional theory (DFT) and plane waves approaches [41 3] or Hartree-Fock-based (HF) methods with localized atomic orbitals [44,45]. 

Ochsenfeld C, Head-Gordon M. A reformulation of the coupled perturbed self-consistent field equations entirely within a local atomic orbital density matrix-based scheme. Chem Phys Lett 1997 270 399 405. 

The only drawback to the NPA methodology is that it rests on allocating electrons to atoms through an orbital occupancy basis, not in terms of the actual locations of the electrons. The latter property is a physical observable, whereas orbital occupancy is not an observable. The NPA method does attempt to define as well as possible localized atomic orbitals, but completely localized atomic orbitals are impossible. Some electrons in an orbital centered on atom A will, in fact, be closer to atom B. Natural orbitals will have tails that penetrate zero-flux surfaces. Therefore, the NPA populations will differ from the rigorous spatially defined topological values. 

Although the title has an almost magical sound to it, the nature of the chemical bond was truly the domain Pauling began to explore. He formulated the concept of hybridization to explain how localized atomic orbitals best overlap to form two-electron bonds. The Kossel-Lewis-Langmuir picture explained ionic and covalent bonding in terms of the octet rule. An interesting question was... 

Since the mathematics is the same, we might want to rewrite Bloch s theorem such that it resembles the definition of the structure factor as closely as possible, and we can do this by expanding the extended, delocalized, k-dependent crystal orbital tp k,r) over a series of localized atomic orbitals atomic positions Vj inside the crystallographic unit cell. The orbitals might belong to different atoms but this does not necessarily have to be the case. Thus, a Bloch expansion reads... 

This simple model gives good quantitative results for energetic bands derived from strongly localized atomic orbitals, where eigenfunctions practically vanish at half the distance to the neighboring atom. 

The system that we are interested in is divided into three partitions a molecular part (M) and left/right electrodes (L/R). The total Hamiltonian (//) and overlap (5) matrix within the localized atomic orbital basis representation are written as follows ... 

In sect. 1, we considered how particular mechanisms affect the behaviour of d and f orbitals in free atoms. We now turn to the question of how these atomic properties are modified when dealing with the atom in the solid. A crucial aspect of rare earth physics is the persistence of quasi-atomic spectral multiplet structure in the solid. To explain this, we first note that a collapsed orbital, localized in the inner reaches of the atom, will tend to survive as a localized atomic orbital in the solid, whereas an orbital which lies in the outer reaches of the atom (or in the outer well of the double-well potential described above) will be completely modified, and may hybridize with the conduction band. 

From the two localized atomic orbitals, and 2, one can form, by linear combination, two delocalized molecular orbitals. The symmetric combination leads to a bonding molecular orbital Of gerade symmetry (i.e., symmetric with respect to inversion about the point centered between the nuclei)... 

In molecular orbital (MO) theory, the electrons in a molecule are placed, not in localized atomic orbitals, but into molecular orbitals, so called because they are associated with the entire molecule. In contrast, an atomic orbital is associated with only one atom. 

For the present studies, we have used state-of-the-art DFT-based methods with plane wave basis set, viz., VASP [30] with PAW potentials [31] for extended systems, and with localized atomic orbital or Gaussian basis set, viz., DMol3 [32] or GAUSSIAN03 [33] for molecular or cluster systems. In all our calculations, the ions are steadily relaxed toward equilibrium until the Hellmann-Feynman forces are converged to less than 10 eV/A. Available experimental structural data have been used as input for some of the hydrides whenever they are available. 

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