Electronic wave function symmetry properties

For atoms, electronic states may be classified and selection rules specified entirely by use of the quantum numbers L, S and J. In diatomic molecules the quantum numbers A, S and Q are not quite sufficient. We must also use one (for heteronuclear) or two (for homonuclear) symmetry properties of the electronic wave function ij/. ... 

In the case of atoms (Section 7.1) a sufficient number of quantum numbers is available for us to be able to express electronic selection rules entirely in terms of these quantum numbers. For diatomic molecules (Section 7.2.3) we require, in addition to the quantum numbers available, one or, for homonuclear diatomics, two symmetry properties (-F, — and g, u) of the electronic wave function to obtain selection rules. 

For the orbital parts of the electronic wave functions of two electronic states the selection rules depend entirely on symmetry properties. [In fact, the electronic selection rules can also be obtained, from symmetry arguments only, for diatomic molecules and atoms, using the (or and Kf point groups, respectively but it is more... 

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 ... 

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... 

For nonlinear molecules, each electronic wave function is classified according to the irreducible representation (symmetry species) to which it belongs the symmetry properties of i cl follow accordingly. For example, for the equilibrium nuclear configuration of benzene (symmetry ), the... 

In essence, jSel is the interaction between the electronic wave functions of A and B. Obviously there must be some spatial overlap between the two in order to give rise to a finite value. Furthermore, if 0A. and />B fall in the same symmetry class or, in complex molecules, have similar local symmetry, the value of j8 may be relatively large. This will depend upon whether or not the perturbation operators, H, have preferred symmetry properties. The Woodward-Hoffman rules suggest that these operators can... 

The amplitude fu is antisymmetric with respect to interchange of the nuclei, which is a direct reflection of the symmetry property of the corresponding electronic wave function. This implies that the cross section need not be symmetric about 0CM = 9O°. We can define a scattering amplitude fd(6) for direct scattering... 

The most direct way to represent the electronic structure is to refer to the electronic wave function dependent on the coordinates and spin projections of N electrons. To apply the linear variational method in this context one has to introduce the complete set of basis functions k for this problem. The complication is to guarantee the necessary symmetry properties (antisymmetry under transpositions of the sets of coordinates referring to any two electrons). This is done as follows. 

In this set the functions can be classified into two types in the right column the spatial multiplier is symmetric with respect to transpositions of the spatial coordinates and the spin multiplier is antisymmetric with respect to transpositions of the spin coordinates in the left column the spatial multiplier is antisymmetric with respect to transpositions of the spatial coordinates and the spin multipliers are symmetric with respect to transpositions of the spin coordinates. Because in the second case the spatial (antisymmetric) multiplier is the same for all three spin-functions, the energy of these three states will be the same i.e. triply degenerate - a triplet. The state with the antisymmetric spin multiplier is compatible with several different spatial wave functions, which probably produces a different value of energy when averaging the Hamiltonian, thus producing several spin-singlet states. From this example one may derive two conclusions (i) the spin of the many electronic wave function is important not by itself (the Hamiltonian is spin-independent), but as an indicator of the symmetry properties of the wave function (ii) the symmetry properties of the spatial and spin multipliers are complementary - if the spatial part is symmetric with respect to permutations the spin multiplier is antisymmetric and vice versa. 

Similar to quantum mechanics, which can be formulated in terms of different quantities in addition to the traditional wave function formulation, in quantum chemistry a number of alternative tools are developed for this purpose, which may be useful in the context of the present book. We have already described different approximate models of representing the electronic structure using (many-electronic) wave functions. The coordinate and second quantization representations were employed to get this. However, the entire amount of information contained in the many-electron wave function taken in whatever representation is enormously large. In fact it is mostly excessive for the purpose of describing the properties of any molecular system due to the specific structure of the operators to be averaged to obtain physically relevant information and for the symmetry properties of the wave functions the expectation values have to be calculated over. Thus some reduced descriptions are possible, which will be presented here for reference. 

Application of the projection operator will also be demonstrated pictorially in forthcoming chapters. These representations will emphasize the results of summation of symmetry-sensitive properties while the absolute magnitudes will not be treated rigorously. Thus, for example, the directions of vectors will be summed in describing vibrations, and the signs of the angular components of the electronic wave functions will be summed in describing the electronic structure. 

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,... 

As mentioned before, the symmetry properties of the one-electron wave function are shown by the simple plot of the angular wave function. But, what are the symmetry properties of an orbital and how can they be described We can examine the behavior of an orbital under the different symmetry operations of a point group. This will be illustrated below via the inversion operation. 

Full exploitation of the symmetry properties of electronic wave functions requires the use of group theory, which we briefly introduce in a later section of this chapter. However, we state some facts ... 

Group theory is a branch of mathematics that involves elements with defined properties and a single method to combine two elements called multiplication. The symmetry operators belonging to any symmetrical object form a group. The theorems of group theory can provide useful information about electronic wave functions for symmetrical molecules, spectroscopic transitions, and so forth. 

However, this is not a satisfactory description, especially, for electronic configurations in which there are unpaired electron spins, since this, so-called Hartree product form, does not comply with the anti-symmetry requirement of the Pauli Principle. In general, many-electron wave functions are written as Slater Determinants, which do exhibit the necessary anti-symmetry properties for electron exchange. ... 

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