Atomic Orbitals and Terms

The present section is offered as a review of the jargon of the theory of free-ion spectroscopy with little in the way of any free-standing explanation.  

A transition metal with the configuration t/ is an example of a hydrogen-like atom in that we consider the behaviour of a single (d) electron outside of any closed shells. This electron possesses kinetic energy and is attracted to the shielded nucleus. The appropriate energy operator (Hamiltonian) for this is shown in Eq. (3.4). 

Solutions to the Schrodinger equation (3.5) are called one-electron wavefunctions or orbitals and take the form in Eq. (3.6) 

The radial functions, R depend only upon the distance, r, of the electron from the nucleus while the angular functions, (6,(p) called spherical harmonics, depend only upon the polar coordinates, 6 and Examples of these purely angular functions are shown in Fig. 3-11. 

This topic is described fully, but at the same level as adopted in the present book, in Orbitals, Terms and States . 

---

The technique for this calcu latioii in volves two steps. Th e first step computes the Hamiltonian or energy matrix. The elem en ts of this matrix are integrals involving the atomic orbitals and terms obtained from the Schrddiiiger equation. The m ost importan t con -... 

In the familiar standard Huckel treatment2174 of planar 7r-systerns, the basis functions are atomic orbitals (AO) 2pz /x = (plL, z, being the coordinate perpendicular to the molecular plane. In analogy to the previous model, the basis energies of the atomic orbitals and the cross terms between neighbouring pairs of AOs, [Pg.203]

Now, the most direct interpretation of Eq. (11.5) follows from the observation, suggested by Eqs. (11.1) and (11.2), that/is essentially a relaxation term. In fact, Vk — represents the difference between the electrostatic potential at the h nucleus in the given molecule and the potential that the same nucleus would feel if the atomic orbitals and the equilibrium distances remained the same as in the reference molecule in spite of the change in electron populations. 

The delta function corresponds to Einstein s equation, which says that the kinetic energy of the emitted electron Ef equals the difference of the photon energy h(a and the energy level of the initial state of the sample, The final state is a plane wave with wave vector k, which represents the electrons emitted in the direction of k. Apparently, the dependence of the matrix element 1 j) on the direction of the exit electron, k, contains information about the angular distribution of the initial state on the sample. For semiconductors and d band metals, the surface states are linear combinations of atomic orbitals. By expressing the atomic orbital in terms of spherical harmonics (Appendix A),... 

It is of interest to enquire how the electrons are redistributed during an interaction and how a bond is affected. We use a simplified Mulliken population analysis [see Appendix A, equations (A.77)-(A.79)]. The simplification consists of dropping all terms involving the overlap of atomic orbitals and assuming that, in any given MO, there is only one atomic orbital on any given center). Thus, we may assume that the following relations hold ... 

These may be obtained in terms of the simple diatomic overlaps8 (5 y) by expressing the central atom orbitals in terms of a linear combination of equivalent orbitals oriented relative to the axes on the oxygen atoms. 

In more detail, we have three molecular orbitals to create from three atomic orbitals, and the linear combination is Equation 1.8, like Equation 1.1 but with three terms ... 

A basic assumption made earlier in the discussion of diatomic molecules and crystalline CsCl is that electronic states can be written as linear combinations of atomic orbitals. We do not need to depart from that assumption now as we begin to describe electronic states in solids as linear combinations of bond orbitals, since bond orbitals can be written as linear combinations of atomic orbitals, and vice versa. Bond orbitals and atomic orbitals arc equivalent representations, but thinking in terms of bond and antibonding orbitals, which can be made to correspond with occupied and empty states of the covalent solid (as was shown in Fig. 2-3), is essential in making approximations. 

Considering B4H8 or B R in terms of localized molecular orbitals (LMO), the 24 available atomic orbitals and the 20 available valence electrons must be organized in four (3c2e) and six (2c2e) molecular bonds. The only bicyclobutane-type structure in accord with these simple requirements is the one found by theory and by experiment (Fig. 5). 

In a computation of the kind of clustering reactions, between molecules you need the positions of the atoms for each ligand, that is the distances and the angles (and even that you can nowadays optimize if you wish). That is one input the other input is the basis set, namely the linear combination of atomic orbitals in terms of which you decide to develop your molecular orbitals. Now this is not an empirical parameter, it is a choice you decide that you ate going to express your molecular orbitals as a linear combination of a certain set of... 

In philosophy, the term naive realism is generally taken to mean a belief in macroscopic objects for what they appear to be and independently of any views on what lies below the surface. I will be using the term in this sense but will also use it to mean the adoption of superficial views about microscopic entities when discussing atomic orbitals and configurations. 

The second part of Pariser and Parr s theory is concerned with the matrix elements between the terms 1Xo, These depend ultimately upon the matrix elements of unity, of the core Hamiltonian, and of the electron repulsion in the system of atomic orbitals. Pariser and Parr, like Pople, introduced the zero differential overlap approximation which abolishes overlap between different atomic orbitals and includes all electron repulsion integrals except those of the type... 

---