Atomic and Molecular Orbitals

A convenient orbital method for describing eleetron motion in moleeules is the method of molecular orbitals. Molecular orbitals are defined and calculated in the same way as atomic orbitals and they display similar wave-like properties. The main difference between molecular and atomic orbitals is that molecular orbitals are not confined to a single atom. The crests and troughs in an atomic orbital are confined to a region close to the atomic nucleus (typieally within 1-2 A). The electrons in a molecule, on the other hand, do not stick to a single atom, and are free to move all around the molecule. Consequendy, the crests and troughs in a molecular orbital are usually spread over several atoms. 

The first three underbraced integrals are just unity because we have implicitly used normalized molecular and atomic orbitals, being able to hold just two electrons the last integral is the overlap integral S12 and the last term, 2C1C2S12, is called the overlap population according to Mulliken [83]. Mulliken also heuristically partitioned the above equation into three different parts, namely... 

More typically, however, a simplification is made. We can have two electrons share the same spatial wavefunction and achieve orthogonality through the spin part. We would have Vra(l) (l)v a(2)P(2) for a two-electron system. Writing out the entire wavefunction for an n electron system we obtain Eq. 14.22. This is termed a restricted total wavefunction—electrons are required to line up in pairs in spatial orbitals, with one a and the other p. Now the total number of orbitals is half the number of electrons, and the subscript on the last reflects this. This is the orbital theory with which we are all familiar. Molecular and atomic orbitals contain two electrons, with the electrons having opposite spins. Often, this is viewed as the definition of the Pauli principle. Actually, the Pauli principle is expressed in Eq. 14.16. We satisfy it by using a determinantal wavefunction, and if we have a determinantal wavefunction, two electrons with the same spin cannot occupy the same orbital. 

Mueller, C. R., J. Chem. Phys. 19, 1498, Scmilocalized orbitals. II. A comparison of atomic, molecular and semilocalized orbital methods for diatomic hydrogenfluoride. ... 

In general, all 17 s-primitives contribute to each s-derived molecular orbital. Obviously, the tighter Gaussians will contribute more strongly to the inner-shell molecular orbitals and the more diffuse Gaussians to the valence i-orbitals. Nevertheless, it is impossible (and also not desired) to make a connection between basis functions and atomic orbitals. 

Figure A.14 Energy diagram for the adsorption of an atom on a d-metal. Chemisorption is described with molecular orbitals constructed from the d-band of the metal and atomic orbitals of the adatom. The chemisorption bond in b) is weaker, because the antibonding chemisorption orbital is partially filled (compare Fig. A.5).

To introduce some of the basic ideas of molecular orbital theory, let s look again at orbitals. The concept of an orbital derives from the quantum mechanical wave equation, in which the square of the wave function gives the probability of finding an electron within a given region of space. The kinds of orbitals that we ve been concerned with up to this point are called atomic orbitals because they are characteristic of individual atoms. Atomic orbitals on the same atom can combine to form hybrids, and atomic orbitals on different atoms can overlap to form covalent bonds, but the orbitals and the electrons in them remain localized on specific atoms. 

The first computational consideration is that of obtaining the solutions of the unperturbed problem, Eq. (15), and the approach taken in the present study is to utilize the Crystal program [1] as it has been successfully used for studies in molecular crystals [10-12,15], A given crystalline orbital, (k,r), such as that required for the matrix elements necessary given by the integral in Eq. (16), is expressed as a linear combination of Bloch functions, a ,(k) and atomic orbitals, (k,r) [1]... 

We have outlined above the procedure for the construction of orthonormal molecular-orbital and atomic-orbital Gel fand states and for the conversion of the latter to the non-orthogonal valence bond states. We require, in addition, a freeon Hamiltonian to compute the spectra of the several polyene systems. For this we employ the freeon, reduced Hiickel-Hubbard Hamiltonian which has the following form ... 

Hybridization means the mixing of atom orbitals of different types of the given atom in one molecular (or atom) orbital. Hybridization principles are well-developed in accordance with the experimental data in the frames of general theories of valence bond (VB) and molecular orbitals (MO). 

Here d is the z projection of the dipole matrix elements for the spinless S PZ transition of an outer electron. The factor (— )sw in equation (38) characterizes the symmetry in the arrangement of atomic dipoles in mixed S-P atomic states. Spin S and parity w of a molecular state are relevant either to the exchange of electrons (with atomic orbital fixed at nuclei) or to the exchange both of electrons and atomic orbitals. The exchange of orbitals (with atomic electrons attached to corresponding nuclei) is accomplished by the product of these two transformations. This explains the appearance of the spin quantum number in (38). 

Fig. 5.26. X-ray photoelectron, uv photoelectron, and angle-resolved carbon K and oxygen K x-ray emission spectra for carbonates, shown along with molecular-orbital assignments and atomic-orbital populations derived from the calculations of Connor et al. (1972) and orbital assignments derived from the calculations of Tossell (1976b) (after Tegeler et al., 1980 reproduced with the publisher s permission).

The Molecular Orbital scheme for NiO is very similar to that of FIMn(CO)5. The six bonding o type orbitals are mainly loccdized on oxygen or sulfur and dpubly occupied. In NiO, as in HM(CO)4, and atomic orbitals are... 

The electron density itself is typically calculated from a molecular wavefunction, for example, one determined by the Hartree-Fock method, and expressed in terms of a density matrix and atomic orbital basis set, as follows. If an LCAO (Linear Combination of Atomic Orbitals) ab initio wavefunction is computed for a molecule A of some fixed conformation K, then the electronic density p(r) can be computed in a simple way. If one denotes by n the number of atomic orbitals cpt( r) (z = 1,2,..., ), ris the three-dimensional position vector variable, and P is the n x n density matrix, then the electronic density p(r) of the molecule can be computed as... 

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