Approximation techniques, second-harmonic

The broad spectrum of recently developed experimental techniques together with traditional measurements of the surface tension has been applied during the last decade to surface layers of micellar solutions [66-76]. For example, Lehmann et al. studied the correlation between the results of a non-linear optieal technique (second-harmonic generation) and surface tension measurements [66], The concentration dependence of the second-order susceptibility component exhibits a kink point in the vicinity of the CMC with a subsequent levelling off (Fig. 5.5). Such behaviour can be explained by the approximate constancy of the adsorption above the CMC. 

Vibrational sum-frequency spectroscopy (VSFS) is a second-order non-linear optical technique that can directly measure the vibrational spectrum of molecules at an interface. Under the dipole approximation, this second-order non-linear optical technique is uniquely suited to the study of surfaces because it is forbidden in media possessing inversion symmetry. At the interface between two centrosymmetric media there is no inversion centre and sum-frequency generation is allowed. Thus the asynunetric nature of the interface allows a selectivity for interfacial properties at a molecular level that is not inherent in other, linear, surface vibrational spectroscopies such as infrared or Raman spectroscopy. VSFS is related to the more common but optically simpler second harmonic generation process in which both beams are of the same fixed frequency and is also surface-specific. 

The first common method for molecular first hyperpolarizability determination is the electric field-induced second harmonic generation (EFISH) technique in solution [6-10]. This technique can be applied only to dipolar molecules. Under an applied external electric field, molecules in solution orient approximately in the direction of the field giving rise to second harmonic generation. The measured third-order nonlinear optical susceptibility is given by the following expression ... 

Second-harmonic generation is a nonlinear optical process through which two photons at a fundamental frequency are converted to one photon at twice the fundamental frequency. In the electric dipole approximation, this process is forbidden in media with inversion symmetry and therefore only occurs at interfaces where the inversion symmetry is broken. As a main drawback to such a surface specificity, the process is very weak, its efficiency being in the range of 10 %. However, with the availability of high intensity lasers and highly sensitive detectors, signal levels of few photons per pulse are routinely detected in laboratories The suitability of the technique to the study... 

In contrast to the free surface, solid substrates can induce the polar orientational order in surface nematic layers. This has been shown by the optical second harmonic generation (SHG) technique which is very sensitive to any kind of polar ordering (in the dipole approximation SHG is forbidden in centro-symmetric media). The polar order exists at the interface between cyano-substituted compounds like 8CB and glasses covered by surfactants and polymer films [22, 23]. 

Basically the perturbative techniques can be grouped into two classes time-local (TL) and time-nonlocal (TNL) techniques, based on the Nakajima-Zwanzig or the Hashitsume-Shibata-Takahashi identity, respectively. Within the TL methods the QME of the relevant system depends only on the actual state of the system, whereas within the TNL methods the QME also depends on the past evolution of the system. This chapter concentrates on the TL formalism but also shows comparisons between TL and TNL QMEs. An important way how to go beyond second-order in perturbation theory is the so-called hierarchical approach by Tanimura, Kubo, Shao, Yan and others [18-26], The hierarchical method originally developed by Tanimura and Kubo [18] (see also the review in Ref. [26]) is based on the path integral technique for treating a reduced system coupled to a thermal bath of harmonic oscillators. Most interestingly, Ishizaki and Tanimura [27] recently showed that for a quadratic potential the second-order TL approximation coincides with the exact result. Numerically a hint in this direction was already visible in simulations for individual and coupled damped harmonic oscillators [28]. 

In a second step, in order to determine the influence of the anharmonicity in the exact potential we will expand the term up to higher powers of the components of r and treat them as small perturbations to the harmonic approximation of the Hamiltonian by means of first order perturbation theory. These perturbative calculations offer insight into the effects of the anharmonic parts of the potential onto the energies and the form of the wave functions. For a discussion of the basis set method and the computational techniques used for the numerical calculation of the exact eigenenergies and eigenfunctions in the outer potential well we refer the reader to [7]. In the following we discuss the results of these numerical calculations of the exact eigenenergies and wave functions and... 

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

Another technique to obtain the effects of the anharmonic terms on the excitation frequencies and the properties of molecular crystals is the Self-Consistent Phonon (SCP) method [71]. This method is based on the thermodynamic variation principle, Eq. (14), for the exact Hamiltonian given in Eq. (10), with the internal coordinates not explicitly considered. As the approximate Hamiltonian one takes the harmonic Hamiltonian of Eq. (18). The force constants in Eq. (18) are not calculated at the equilibrium positions and orientations of the molecules as in Eq. (19), however. Instead, they are considered as variational parameters, to be optimized by minimization of the Helmholtz free energy according to Eq. (14). The optimized force constants are found to be the thermodynamic (and thus temperature dependent) averages of the second derivatives of the potential over the (harmonic) lattice vibrations ... 

The primitive approximation contains all the physics involved in the treatment of quantum many-body systems at nonzero temperature. It is simple, intuitive, and highly flexible. Moreover, it reveals clearly the so-called classical isomorphism, Eqs. (25)-(27), a correspondence that has important consequences on a range of different issues (e.g., formal study of structures and computational techniques). Nevertheless, the computational efficiency of the primitive scheme is generally poor as pointed out in earlier applications. A couple of examples will help to understand these drawbacks. First, the kinetic energy, given by the first two terms on the right-hand side of Eq. (31), which shows increasing variances with P, a fact associated with the stiffness of the harmonic links in [63]. Second, the P con-... 

---