Nonlinear molecules electronic wave function

For diatomics and other linear molecules, only (4.1. IS) is satisfied, whereas, for nonlinear molecules, the wave function is an eigenfunction of the discrete symmetry operations R of the (finite) molecular point group. It should be emphasized that the enforcement of spin and point-group symmetries on approximate wave functions constitutes a restriction on the wave fuiKtion, which in variational ground-state calculations may raise the electronic energy above what would be obtained in an unrestricted treatmenL Symmetry restrictions are discussed in Section 4.4. 

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... 

As mentioned in the introduction, the above discussion of the small-, large-, and intermediate-molecule limits of electronic relaxation processes can also be utilized with very minor modifications to discuss the phenomena of intramolecular vibrational relaxation in isolated polyatomic mole-cules. ° Figure 4 is still applicable to this situation. The basis functions are now taken to be either pure harmonic vibrational states, some local-mode vibrational eigenfunctions, or some alternative nonlinear mode-type wave-functions. In the following the nomenclature of vibrational modes is utilized, but its interpretation as normal or local can be chosen to suit the circumstances at hand. 

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. 

For nonlinear polyatomic molecules, no orbital angular-momentum operator commutes with the electronic Hamiltonian, and the angular-momentum classification of electronic terms cannot be used. Operators that do commute with the electronic Hamiltonian are the symmetry operators Or of the molecule (Section 12.1), and the electronic states of polyatomic molecules are classified according to the behavior of the electronic wave function on application of these operators. Consider H2O as an example. 

There are two contributions to the polarizability of a molecule the distortion of the electronic wave function and the distortion of the nuclear framework. The major contribution is from the electrons, and can be considered to be the sum of contributions from the individual electrons. The contributions of the inner-shell electrons are nearly independent of orientation and these contributions can be ignored. The polarizability of electrons in a bond parallel to the bond direction is different from the polarizability perpendicular to that bond. As a diatomic molecule or linear polyatomic molecule rotates, the components of the polarizability in fixed directions are modulated (fluctuate periodically) as the ellipsoid of polarizability rotates. The rotation of a diatomic or linear polyatomic molecule will be Raman active (produce a Raman spectram). In a nonlinear polyatomic molecule, the polarizabilities of the individual bonds add vectorially to make up the total polarizability. If the molecule is a symmetric top, the total polarizability is the same in all directions and the ellipsoid of polarizability is a sphere. A spherical top molecule has no rotational Raman spectmm. Symmetric tops and asymmetric tops have anisotropic polarizabilities and produce rotational Raman spectra. 

In this chapter we treat those nonlinear optical processes in which the electronic wave functions of the liquid crystal molecules are significantly perturbed by the optical field. Unlike the nonelectronic processes discussed in the previous chapters, these electronic processes are very fast the active elections of the molecules respond almost instantaneously to the optical field in the form of an induced electronic polarization. Transitions from the initial level to some final excited state could also occur. 

Pj, is a projection operator ensuring the proper spatial symmetry of the function. The above method is general and can be applied to any molecule. In practical application this method requires an optimisation of a huge number of nonlinear parameters. For two-electron molecule, for example, there are 5 parameters per basis function which means as many as 5000 nonlinear parameters to be optimised for 1000 term wave function. In the case of three and four-electron molecules each basis function contains 9 and 14 nonlinear parameters respectively (all possible correlation pairs considered). The process of optimisation of nonlinear parameters is very time consuming and it is a bottle neck of the method. 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

The geminal ansatz still requires more effort than the standard one-electron approach of the independent particle model. It is therefore usually restricted to small molecules for feasibility reasons. As an example how the nonlinear optimization problem can be handled we refer to the stochastic variational approach [340]. However, the geminal ansatz as presented above has the useful feature that all elementary particles can be treated on the same footing. This means that we can actually use such an ansatz for total wave functions without employing the Born-Oppenheimer approximation, which exploits the fact that nuclei are much heavier than electrons. Hence, electrons and nuclei can be treated on the same footing [340-342] and even mixed approaches are possible, where protons and electrons are treated in the external field of heavier nuclei [343-346]. The integrals required for the matrix elements are hardly more complicated than those over one-electron Gaussians [338,339,347]. 

As we can see from this simple but general consideration of a multilevel molecule, nonlinear electronic polarizations occur naturally in all materials illuminated by an optical field. The differences among the nonlinear responses of different materials are due to differences in their electronic properties (wave functions, dipole moments, energy levels, etc.) which are determined by their basic Hamiltonian Hq. For Uquid crystals in their ordered phases, an extra factor we need to take into account are molecular correlations. 

Figure 11.6 gives a short introduction to nonlinear optics of OPVs [27]. A light wave polarizes the molecules in a mode which corresponds to the periodicity of the E vector of the lightwave however, the function P(t) is not so symmetrical as E t). If for example E is a donor and E an acceptor group, the electrons are more easily shifted in the direction of E than in the opposite direction. A Fourier transformation of the periodic function P(t) leads not only to the original frequency CO, but also to the double and triple frequency. Hence, nonlinear optics provide a... 

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