Linear molecules symmetry properties

For many-electron systems such as atoms and molecules, it is obviously important that approximate wavefiinctions obey the same boundary conditions and symmetry properties as the exact solutions. Therefore, they should be antisynnnetric with respect to interchange of each pair of electrons. Such states can always be constmcted as linear combinations of products such as... 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

We therefore conclude that, for a combination of model, numerical and conceptual reasons the OHAO basis is well-adapted to a theory of valence. The hybrid orbital basis (for simple molecules) has a distinctive symmetry property it carries a permutation representation of the molecular symmetry group the equivalent orbitals are always sent into each other, never into linear combinations of each other. This simple fact enables the hybrid orbital basis to be studied in a way which is physically more transparent than the conventional AO basis. 

C2u character table because when x2 and y2 are of the same symmetry, any linear combination of the two will also have that symmetry. Note that although both the and d - orbitals transform as at in this point group, they are not degenerate because they do not transform together, ft would be a worthwhile exercise to confirm that the s, p, and d orbitals do have the symmetry properties indicated in a Cu molecule. Keep in mind, in attempting such an exercise, that the signs of orbital lobes are important... 

Various methods (described in Chapter 4) can be used to determine the symmetry of atomic orbitals in the point group of a molecule, i. e., to determine the irreducible representation of the molecular point group to which the atomic orbitals belong. There are two possibilities depending on the position of the atoms in the molecule. For a central atom (like O in H20 or N in NH3), the coordinate system can always be chosen in such a way that the central atom lies at the intersection of all symmetry elements of the group. Consequently, each atomic orbital of this central atom will transform as one or another irreducible representation of the symmetry group. These atomic orbitals will have the same symmetry properties as those basis functions in the third and fourth areas of the character table which are indicated in their subscripts. For all other atoms, so-called group orbitals or symmetry-adapted linear combinations (SALCs) must be formed from like orbitals. Several examples below will illustrate how this is done. 

Vibrational degrees of freedom One must also specify the displacements of the atoms from their equilibrium positions (vibrations). The number of vibrational degrees of freedom is 2>N — 5 for linear molecules and 3AA— 6 for nonlinear molecules. These values are determined by the fact that the total number of degrees of freedom must be 3A. For each vibrational degree of freedom, there is a normal mode of vibration of the molecule, with characteristic symmetry properties and a characteristic harmonic frequency. The vibrational normal modes for CO2 and H2O are illustrated schematically in Fig. 1. 

The different symmetry properties considered above (p. 131) for macroscopic susceptibilities apply equally for molecular polarizabilities. The linear polarizability a - w w) is a symmetric second-rank tensor like Therefore, only six of its nine components are independent. It can always be transformed to a main axes system where it has only three independent components, and If the molecule possesses one or more symmetry axes, these coincide with the main axes of the polarizability ellipsoid. Like /J is a third-rank tensor with 27 components. All coefficients of third-rank tensors vanish in centrosymmetric media effects of the molecular polarizability of second order may therefore not be observed in them. Solutions and gases are statistically isotropic and therefore not useful technically. However, local fluctuations in solutions may be used analytically to probe elements of /3 (see p. 163 for hyper-Rayleigh scattering). The number of independent and significant components of /3 is considerably reduced by spatial symmetry. The non-zero components for a few important point groups are shown in (42)-(44). 

In order to find the character of the representation of rotational motions and librations, the transformation properties of an axial vector (Fig. 2.1-lb) have to be taken into account. This vector is defined not only by its length and orientation, but also by a definite sense of rotation inherent to it. For linear molecules with only two degrees of rotational freedom, the Xr R) values for Cj and a depend on the orientation of the molecular axis to the symmetry elements. All values of XriR) tire also included in Table 2.7-1. 

Linear transformations of the integrals previously listed, in order to prepare for the determination of the effective Hamiltonian and take advantage of the symmetry properties of the molecule. The computing time needed for this increases at least as w even if the molecule has no symmetry at all, the presence of exchange terms in the Hartree-Fock theory makes a part of the transformation still necessary, namely the construction of the integral list. 

Linear transformations, including symmetry transformations in configuration space are analogous to those in Cartesian space (see Sections 1.2.2 and 1.2.3). For symmetrical reference structures, it is usually better to use not the internal coordinates themselves but to choose a new coordinate system in which the basis vectors are symmetry adapted linear combinations of the internal displacement coordinates with the special property that they transform according to the irreducible representations of the point group of the idealized, reference molecule (symmetry coordinates, see Chapter 2). 

As in the earlier three-dimensional example, where it was convenient to use hexagonal axes involving linear combinations of the rhombohedral basis vectors, it may also be useful here to use an alternative coordinate system to bring out certain symmetry properties. For molecules with a Tj frame, one choice is to take ... 

A.I.Maergoiz, E.E.Nikitin, and J.Troe, Correlation diagrams and symmetry properties of adiabatic states of a system composed of two linear dipole molecules, Khim. Fizika 11, 814 (1992)... 

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