Wave equation symmetry property

The projection operator formalism also gives interesting aspects on the correlation problem. Previously one mainly used the secular equation (Eq. III.21) for investigating the symmetry properties of the solutions, and one was often satisfied with those approximate wave functions which were the simplest linear combinations of the basic functions having the correct symmetry. In our opinion, this problem is now better solved by means of the projection operators, and the use of the secular equations can be reserved for handling actual correlation effects. This implies also that, in place of the ordinary Slater determinants (Eq. III.17), we will essentially consider the projections of these functions as our basis. 

Those who applied quantum mechanics to atoms and molecules had a wealth of chemists data at hand well-defined bond properties including dipole moments, index of refractions, and ultraviolet absorption qualities and polarizability as well as well-defined valence properties of atoms in molecules. If one attempted to set up a wave equation for the water molecule, for example, there were 39 independent variables, reducible to 20 by symmetry considerations. But the experimental facts of chemistry implied or required certain properties that made it possible to solve equations by semiempirical methods. "Chemistry could be said to be solving the mathematicians problems and not the other way around," according to Coulson. 148... 

Equations 3.3 and 3.4 seem to imply that both atomic wave functions j/A and contribute fully (100%) to both molecular wave functions 0, and 02 The impossibility of this is taken care of in the section on normalization For the purposes of this section, normalization may be ignored as it does not alter the symmetry properties of the molecular wave functions. 

An interesting and useful method of theoretical treatment of certain properties of complexes and crystals, called the ligand field theory, has been applied with considerable success to octahedral complexes, especially in the discussion of their absorption spectra involving electronic transitions.66 The theory consists in the approximate solution of the Schrddinger wave equation for one electron in the electric field of an atom plus a perturbing electric field, due to the ligands, with the symmetry of the complex or of the position in the crystal of the atom under consideration. 

It will be obvious from the content of Chapter 5 why such combinations are desired. First, only such functions can, in themselves, constitute acceptable solutions to the wave equation or be directly combined to form acceptable solutions, as shown in Section 5.1. Second, only when the symmetry properties of wave functions are defined explicitly, in the sense of their being bases for irreducible representations, can we employ the theorems of Section 5.2 in order to determine without numerical computations which integrals or matrix elements in the problem are identically zero. 

The numerical evaluation of the energies of orbitals and states is fundamentally a matter of making quantum mechanical computations. As indicated in Chapter 1, quantum mechanics per se is not the subject of this book, and indeed we have tried in general to avoid any detailed treatment of methods for solving the wave equation, emphasis being placed on the properties that the wave functions must have purely for reasons of symmetry and irrespective of their explicit analytical form. However, this discussion of the symmetry aspects of ligand field theory would be artificial and unsatisfying without some... 

The symmetry properties of the electronic wave function and the energy of the system are two determining factors in chemical behavior. The relationship between the wave function characterizing the behavior of the electrons and the energy of the system—atoms and molecules—is expressed by the Schrodinger equation. In its general and time-independent form, it is usually written as follows,... 

The kind of statistics obeyed by the system depends on the symmetry properties of the quantum-mechanical wave functions describing the molecules composing the system [3-7], For example, in some cases the a values may be taken as either integers (0, 1,. . . ) or half-integers (, f,. . . ) the choice is based on the nature of the particular Schrodinger equation describing the molecule. 

From symmetry properties of the wave function and from the Coulomb matrix elements it is possible to distinguish between two classes of basis states, namely gerade (denoted by +) and ungerade (denoted by -) states. The corresponding wave functions and 0 may be obtained by replacing the spherical har monics T in equation (14) by... 

Note that the above discussion does not depend on knowing anything about the detailed functional form of wave functions. We do not have to solve the Schrodinger equation. The statements are based solely on the symmetry properties and the consequent mathematical properties of the symmetry group. 

Next, in Equation (2.3), there is the Coulomb attraction between the electrons (charged —e) and the nuclei (charged Za) being ViA apart from each other, and this is what brings the system into "motion". In addition, we find the electrostatic repulsion between the electrons (fourth term) and also the electrostatic repulsion between the nuclei (fifth term) separated by Rab i r>ofe that we have excluded the double-counting of these interactions (i.e., / > i, B > A). We must also keep in mind that the electronic coordinates of this nonrelativistic H only contain spatial and no spin coordinates, and a spin-dependent description is eventually achieved by requiring a certain symmetry property for the many-electron wave function Y (see Sections 2.9 and 2.11.3). 

Molecular orbitals are a model, a simplified picture that we use as a first, big step toward understanding a complicated system. For any many-electron molecule, there is no one true set of equations for its MOs. Instead, we are free to decide how we want to build them, because they are only an approximation of the many-electron wavefunction, and even the wave-function itself is just a mathematical construct. With a few exceptions, for the rest of the text we choose to write our MOs as symmetry orbitals, one-electron wavefunctions that share the symmetry properties of an irreducible representation in the molecule s point group. For example, F2O in Fig. 6.10 has C2V symmetry, and so we label all of its MOs by the representations Uj, Uj, b, and b2, which are the four representations of C2V appearing in Table 6.3. This bears closer examination, however, so let us return to the simplest molecules for a start. 

Equation 144 is just the equation for the electronic energy which was discussed in Chapter XI. The electronic wave functions F x, y, z, r) therefore have the symmetry properties of the various irreducible representations of the groups Dooa or Coop according as the nuclei are identical or different. The vibrational wave function R r) depends only on the distance between the two nuclei and therefore belongs to the totally symmetrical representation. The complete wave fimction will thus have the symmetry properties of the product F x, y, z, r)U(6, x)-In order to discuss the nature of the solutions of 14 6 it will be con-... 

We now consider higher-order modes of fibers with circulariy symmetric cross-sections and profiles If we express in cylindrical polar coordinates r, (j> as in Table 30-1, page 592, there are two separable solutions of Eq. (13-8) for each value of These are 4 = f, (r) cos l(j) and V — F, (r) sin Icj), where / is a positive integer and f, (r) satisfies the equation in Table 13-1. Because of symmetry, any pair of orthogonal x- and y-axes may be chosen as optical axes in the fiber cross-section. It also follows from symmetry that there are four possible directions for e, depending on the particular combination of the two solutions of the scalar wave equation used in Eq. (13-7) [1,2]. This is discussed further in Section 32-7. Hence, for each value of / > 0, there are four modes with the fields shown in Table 13-1. These combinations can also be derived without recourse to symmetry properties using the formal methods of Section 32-6. In general, this representation for is the simplest possible, for reasons explained in Section 32-8. 

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