Molecular wave functions, symmetry properties

Qualitative information about molecular wave functions and properties can often be obtained from the symmetry of the molecule. By the symmetry of a molecule, we mean the symmetry of the framework formed by the nuclei held fixed in their equilibrium positions. (Our starting point for molecular quantum mechanics will be the Born-Oppenheimer approximation, which regards the nuclei as fixed when solving for the electronic wave function see Section 13.1.) The symmetry of a molecule can differ in different electronic states. For example, HCN is linear in its ground electronic state, but nonlinear in certain excited states. Unless otherwise specified, we shall be considering the symmetry of the ground electronic state. 

Symmetry considerations have long been known to be of fundamental importance for an understanding of molecular spectra, and generally molecular dynamics [28-30]. Since electrons and nuclei have distinct statistical properties, the total molecular wave function must satisfy appropriate symmehy... 

In this chapter, we discussed the permutational symmetry properties of the total molecular wave function and its various components under the exchange of identical particles. We started by noting that most nuclear dynamics treatments carried out so far neglect the interactions between the nuclear spin and the other nuclear and electronic degrees of freedom in the system Hamiltonian. Due to... 

We begin by considering the diatomic molecule as an example. The overall symmetry of the molecular wave function must include the properties of the nuclear wave function when the nuclei are permuted within the molecule. For an heteronu-clear diatomic molecule with nuclei a and b the only permutation of nuclei allowed... 

The Hartree-Fock or self-consistent-field approximation is a simplification useful in the treatment of systems containing more than one electron. It is motivated partly by the fact that the results of Hartree-Fock calculations are the most precise that still allow the notion of an orbital, or a state of a single electron. The results of a Hartree-Fock calculation are interpretable in terms of individual probability distributions for each electron, distinguished by characteristic sizes, shapes and symmetry properties. This pictorial analysis of atomic and molecular wave functions makes possible the understanding and prediction of structures, spectra and reactivities. 

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. 

Even though (6.105) is only an approximation, its symmetry properties (e.g., its parity) are the same as those of the true molecular wave function, since the symmetry properties of the wave function follow rigorously from the symmetry of the true Hamiltonian H the perturbation H cannot change the overall symmetry of p. 

In the following three sections we shall discuss four applications of quantum mechanics to miscellaneous problems, selected from the very large number of applications which have been made. These are the van der Waals attraction between molecules (Sec. 47), the symmetry properties of molecular wave functions (Sec. 48), statistical quantum mechanics, including the theory of the dielectric constant of a diatomic dipole gas (Sec. 49), and the energy of activation of chemical reactions (Sec. 50). With reluctance we omit mention of many other important applications, such as to the theories of the radioactive decomposition of nuclei, the structure of metals, the diffraction of electrons by gas molecules and crystals, electrode reactions in electrolysis, and heterogeneous catalysis. 

In this section we shall discuss the symmetry properties of molecular wave functions to the extent necessary for an under-... 

There is no evidence that any classical attribute of a molecule has quantum-mechanical meaning. The quantum molecule is a partially holistic unit, fully characterized by means of a molecular wave function, that allows a projection of derived properties such as electron density, quanmm potential and quantum torque. There is no operator to define those properties that feature in molecular mechanics. Manual introduction of these classical variables into a quantum system is an unwarranted abstraction that distorts the non-classical picture irretrievably. Operations such as orbital hybridization, LCAO and Bom-Oppenheimer separation of electrons and nuclei break the quantum symmetry to yield a purely classical picture. No amount of computation can repair the damage. 

The wave functions and levels in polyatomic molecules are described in terms of their symmetry. (See Sections 23.14 and 23.16.2). For example, if we consider the water molecule, its symmetry requires that any molecular wave function either be invariant or change only in algebraic sign under any symmetry operation. This requirement severely restricts the form of the wave functions. These wave functions can be of only four types—denoted by the letters 2 5 and Z 2 ach of which belongs to a particular symmetry species. The symmetry properties of each type are summarized in the character table of the group C2y, Table 23.5. 

The matrix elements (a,7) depend on the symmetry characteristics of the molecular states. While the theoretical evaluation of the magnitude of (a,7) demands a knowledge of the corresponding wave functions, the question whether atj) is zero or not depends on the symmetry properties of the molecular wave functions... 

There are additional designations possible in certain linear molecules, depending on the symmetry properties of the molecular wave function. For example, the complete symbol for the ground state of H2 is A discussion of the complete notation is... 

Wigner rotation/adiabatic-to-diabatic transformation matrices, 92 Electronic structure theory, electron nuclear dynamics (END) structure and properties, 326-327 theoretical background, 324-325 time-dependent variational principle (TDVP), general nuclear dynamics, 334-337 Electronic wave function, permutational symmetry, 680-682 Electron nuclear dynamics (END) degenerate states chemistry, xii-xiii direct molecular dynamics, structure and properties, 327 molecular systems, 337-351 final-state analysis, 342-349 intramolecular electron transfer,... 

In order to apply group-theoretical descriptions of symmetry, it is necessary to determine what restrictions the symmetry of an atom or molecule imposes on its physical properties. For example, how are the symmetries of normal modes of vibration of a molecule related to, and derivable from, the full molecular symmetry How are the shapes of electronic wave functions of atoms and molecules related to, and derivable from, the symmetry of the nuclear framework ... 

If we imagine the nuclei to be forced together to = 0, the wave function Is A + Iss will approach, as a limit, a charge distribution around the united atom that has neither radial nor angular nodal planes. This limiting charge distribution has the same symmetry as the Is orbital on the united atom, Helium. On the other hand, the combination Isa Iss has a nodal plane perpendicular to the molecular axis at all intemuclear separations. Hence its limit in the united atom has the symmetry properties of a 2p orbital. A simple correlation diagram for this case is ... 

In the molecular orbital theory and electronic spectroscopy we are interested in the electronic wave functions of the molecules. Since each of the symmetry operations of the point group carries the molecule into a physically equivalent configuration, any physically observable property of the molecule must remain unchanged by the symmetry operation. Energy of the molecule is one such property and the Hamiltonian must be unchanged by any symmetry operation of the point group. This is only possible if the symmetry operator has values 1. Hence, the only possible wave functions of the molecules are those which are either symmetric or antisymmetric towards the symmetry operations of the... 

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