Atomic orbitals relationship between quantum

Table 1.4 summarizes the relationship between quantum numbers and hydrogenlike atomic orbitals. When Z = 0, (2/ + 1) = 1 and there is only one value of m/, so we have an s orbital. When / = 1, (2/ + 1) = 3, so there are three values of m , giving rise to three p orbitals, labeled p, Py, and p. When / = 2, (2Z + 1) = 5, so there are five values of mi, and the corresponding five d orbitals are labeled with more elaborate subscripts. In the following sections we discuss the s, p, and d orbitals separately. 

Relationship between Quantum Numbers and Atomic Orbitals... 

The relationship between orbital size and quantum number for the hydrogen atom. 

The explanation of the periodic system by quantum mechanics, for example, is only partial. The possible lengths of the various periods in the table follow deductively from the solution of the SchrOdinger equation for the hydrogen atom and the relationship between the four quantum numbers, which is also obtained deductively. However, the repetition of all but the first period length remains a source of debate (/). The repetition of all the other period lengths has not been deduced from first principles however (2). Stated more precisely, the empirical order in which the atomic orbitals are filled has not been deduced. If this were possible the explanation for the lengths of successive periods, including the repetitions, would follow trivially. 

Despite the quantitative victory of molecular orbital (MO) theory, much of our qualitative understanding of electronic structure is still couched in terms of local bonds and lone pairs, that are key conceptual elements of the valence bond (VB) picture. VB theory is essentially the quantum chemical formulation of the Lewis concept of the chemical bond [1,2]. Thus, a chemical bond involves spin-pairing of electrons which occupy valence atomic orbitals or hybrids of adjacent atoms that are bonded in the Lewis structure. In this manner, each term of a VB wave function corresponds to a specific chemical structure, and the isomorphism of the theoretical elements with the chemical elements creates an intimate relationship between the abstract theory and the nature of the... 

We can see the relationship between these terms and quantum numbers by looking carefully at Table 4.3. First, each unique value of n represents an energy level. Each 7 value represents a specific sublevel within an energy level. Recall from the previous section that these sublevels are typically referred to using their common names s, p, d, and f. Each unique combination of n and 1 values corresponds to a different sublevel. For example, for n = 3 and 7=2, this corresponds to the 3dsublevel of the atom. The rn values tell us how many orbitals are found in a given sublevel. For instance, in the 3d sublevel there are 5 orbitals possible (for 3d, rn —2, — 1, 0, 1, 2). The spin quantum number tells us that there can be no more than 2 electrons in any orbital, which you will learn more about later in this chapter. Let s summarize what we know in a new chart. 

Many solid-state physicists discuss the structure and properties of metals and alloys with use of the band theory, in its several modifications. This theory is also a quantum mechanical theory, which starts with a solution of the wave equation for a single electron, and introduces electron-electron correlation in one or another of several ways. The resonating-valence-bond theory introduces electron-electron correlation in several stages, one of which is by the formation of covalent bonds between adjacent atoms, and another the application of the electroneutrality principle to restrict the acceptable structures to those that involve only M+, M°, and M-. It should be possible to find a relationship between the band-theory calculations and the resonating-covalent-bond theory, but I have been largely unsuccessful in finding such a correlation. I have, for example, not been able to find any trace of the metallic orbital in the band-theory calculations, which thus stand in contrast to the resonating-valence-bond theory, in which the metallic orbital plays a predominant role."... 

In Bohr s model of the hydrogen atom, the circular orbits were determined by the quantum number more accurately, by the square of the quantum number n. No other orbits were allowed. By changing the orbits from circles to ellipses, Sommerfeld introduced a second radius, which gave him another variable to play with. So it was that Sommerfeld generalized Bohr s quantum condition for electron orbits in terms of the two quantum numbers n and k. His analysis led him to establish a relationship between the two quantum numbers namely, the quantum number n set the upper limit on the quantum number k, but k could have smaller values as follows ... 

For the higher quantum numbers the relationship between the energy values of the orbitals is more complicated (see Figure 7). Thus for example, the 4J orbital ( == 4, / = m) is more stable than the orbital (n = 3, / = 2). This complexity docs not permit a construction of the electronic distribution of the elements on the basis only of the analogy to a hydrogen like atom and it is necessary for each element to use the spectroscopic data to determine the electronic states. The sequence of the distribution thus obtained can, with only a few exceptions, be expressed by the data given in Figure 7. 

Whereas, for non-Coulombic potentials, one can define nr and , n is then no longer related simply to the binding energy. Indeed, for a complex, many-electron atom, it is not at all obvious how one should set about quantising the system, since there is no guarantee that the orbits of individual electrons will close.6 In fact, conservation of the angular momentum for individual electrons is, at best, only an approximation. It would hold exactly for central fields. Even then, the same simple, precise relationship between n and as for H is not to be expected for many-electron atoms. As we shall see, the very meaning of n (the principal or most important quantum number) becomes less clear-cut for many-electron systems. In a nutshell the n and quantum numbers of... 

The concept of atomic orbitals is based on Bohr theory. In order to understand the atomic structure in the quantum theory framework, it is important to characterize the relationship between angular momentum and quantum number. There is no doubt that to grasp the real meaning of atomic orbitals one has to comprehend the concept of the angular momentum of electrons. 

This obvious need for clarifying the relationship between the electronic and geometric stmcture of paramagnetic systems and their g values nowadays can be met with the help of first-principles quantum chemical methods. A theoretical description of electronic g values, based on a relativistic method which accurately treats spin-orbit interaction even in cases when it is too strong to be considered as a perturbation, will be uniformly applicable to systems with both light and heavy atoms. 

Scheme 16.2 Relationship between absorption and emission energies for molecules (discrete, left) and for semiconductor quantum dots (right). Upon the absorption of a photon, an electron is lifted from the ground state (1) to an electronically excited state (2). The bond order decreases, because excited states are antibinding and the atoms relax to larger intemuclear distances (3). From the lowest excited state (only one is shown here), emission of a photon (4) and relaxation to the ground state occurs. In semiconductor quantum dots (e.g. CdSe), more possibilities for the absorption of photons exist, because several/many orbitals can be found. (Reproduced with permission from E. M. Boatman et al., 2005. J. Chem. Ed. 82 1697-1699. Copyright 2005 American Chemical Society.)...

---