Response and Propagator Methods

The first four terms only involve derivatives of operators and AO integrals. However, for the last three terms we need the derivative of the density matrix and MO energies. These can be obtained by solving the first-order CPHF equations (Section 10.5). 

The perturbation and derivative approaches in sections 10.2 and 10.3 are not suitable for time-dependent properties since there is no well-defined energy function is such cases. The equivalent of eq. (10.20) for a time-dependent perturbation is eq. (10.114). 

The perturbation is usually an oscillating electric field, which we can write as in eq. (10.115). 

Here cOk is the frequency of the field, F is the corresponding field strength and Q is the perturbation operator. The QF term should again be interpreted as a sum over all products of components. In most cases the field can be represented by its linear approximation, i.e. Q is the dipole operator r and F is a vector containing the x,y and z components of the field. Concentrating on a uniform field of strength F with a single frequency, eq. (10.115) reduces to eq. (10.116). 

The expectation value of a given operator P can be expanded according to perturbations Q, R. 

---

J. Oddershede, Response and propagator methods, in S. Wilson, G. H. F. Dieicksen (Eds.), Methods in Computational Molecular Physics, Plenum Press, New York, 1992. 

In this paper, an overview of the origin of second-order nonlinear optical processes in molecular and thin film materials is presented. The tutorial begins with a discussion of the basic physical description of second-order nonlinear optical processes. Simple models are used to describe molecular responses and propagation characteristics of polarization and field components. A brief discussion of quantum mechanical approaches is followed by a discussion of the 2-level model and some structure property relationships are illustrated. The relationships between microscopic and macroscopic nonlinearities in crystals, polymers, and molecular assemblies are discussed. Finally, several of the more common experimental methods for determining nonlinear optical coefficients are reviewed. 

Reinforced concrete is a complex material to model due to the brittle nature of concrete and non-homogenous properties. Although sophisticated methods are available to model crack propagation and other responses, simplified methods are normally used in blast design of facilities. These methods are based on a flexural response and rely on elimination of brittle modes of failure. To achieve a ductile response for concrete, proper proportioning and detailing of the reinforcing is necessary. 

When Jens Oddershede was elected a Fellow of the American Physical Society in 1993, the citation read For contribution to the theory, computation, and understanding of molecular response properties, especially through the elucidation implementation of the Polarization Propagator formalism. Although written more than a decade ago, it is still true today. The common thread that has run through Jens work for the past score of years is development of theoretical methods for studying the response properties of molecules. His primary interest has been in the development and applications of polarization propagator methods for direct calculation of electronic spectra, radiative lifetime and linear and non-linear response properties such as dynamical dipole polarizabilities and... 

Heat is the most common product of biological reaction. Heat measurement can avoid the color and turbidity interferences that are the concerns in photometry. Measurements by a calorimeter are cumbersome, but thermistors are simple to use. However, selectivity and drift need to be overcome in biosensor development. Changes in the density and surface properties of the molecules during biological reactions can be detected by the surface acoustic wave propagation or piezoelectric crystal distortion. Both techniques operate over a wide temperature range. Piezoelectric technique provides fast response and stable output. However, mass loading in liquid is a limitation of this method. 

One somewhat displeasing detail in the approximate polarization propagator methods discussed in the previous section is the fact that concern needs to be made as to which formulation of wave mechanics that is used. This point has been elegantly resolved by Christiansen et al. in their quasi-energy formulation of response theory [23], in which a general and unified theory is presented for the evaluation of response functions for variational as well as nonvariational electronic structure methods. 

Electronic structure calculations have over the years almost solely been performed using state approaches. The basic approach has been to develop better and better methods to evaluate the wavefunction and/or the energy and other properties of the individual state. To a large extent the efforts have concentrated on ground-state properties. Clearly these methods have also dominated the literature and even very recent reviews of electronic structure calculations (Schaefer, 1984) maintain the view that only state function methods have yet demonstrated their usefulness in electronic structure calculations. It is the purpose of the present review to describe a different approach, the propagator methods. In these methods we compute state energy differences, transition probabilities and response properties directly without knowing the wavefunctions of the individual states. 

Eq. (58) represents the starting point for all approximate propagator methods. Even though in the derivation we only discussed the linear response functions or polarization propagators, a similar equation holds for the electron propagator. The equation for this propagator has the same form but there are differences in the choice of h and in the definition of the binary product (Eq. (52)), which for non-number-conserving, fermion-like operators should be... 

Structural features of disperse systems, in particular the existence of the electrical double layer (EDL), are responsible for a number of peculiar phenomena related to heat and mass transfer and electric current propagation in such systems. The description of electromagnetic radiation propagation is also included in this chapter. These features are utilized in numerous practical applications and underlie methods used to study disperse systems. These methods include particle size distribution analysis, studies of the surface structure and of near-surface layers, the structure of the EDL, etc. In the most general way the most transfer phenomena can be described by the laws of irreversible thermodynamics, which allow one to carry out a systematic investigation of different fluxes that originate as a result of the action of various generalized forces. 

We turn first to computation of thermal transport coefficients, which provides a description of heat flow in the linear response regime. We compute the coefficient of thermal conductivity, from which we obtain the thermal diffusivity that appears in Fourier s heat law. Starting with the kinetic theory of gases, the main focus of the computation of the thermal conductivity is the frequency-dependent energy diffusion coefficient, or mode diffusivity. In previous woik, we computed this quantity by propagating wave packets filtered to contain only vibrational modes around a particular mode frequency [26]. This approach has the advantage that one can place the wave packets in a particular region of interest, for instance the core of the protein to avoid surface effects. Another approach, which we apply in this chapter, is via the heat current operator [27], and this method is detailed in Section 11.2. 

This limitation on the cracking temperature does not apply to the highly nonequilibrium processes where concentrations of the reactive particles responsible for the reaction initiation and propagation and, therefore, initiation and propagation rates are not subjected to the thermodynamic limitations. It opens an opportunity of a considerable reduction of the cracking temperature in the methods for oil processing based on application of ionizing irradiation. 

For thin shell structures, the most promising methods are those based in the analysis of the propagation of elastic waves. The wave propagation methods have often used piezoelectric wafer active sensors (PWAS) as transmitters to generate waves and simultaneously as receivers to measure the echo signals due to the defects. A time-frequency analysis allows an estimation of crack size on the basis of the relationship between new and baseline response. The sensitivity of Lamb waves to defects depends largely on the frequency, and for complex structures the dispersive Lamb waves interact with reinforcements with partial reflections and refractions. These systems have not reached the level of maturity required for industrial applications. A full discussion with alternatives is presented in the book by Giurgiutiu (2008). 

The determination of stresses in the peel test is complex and warrants the use of finite element methods. Adams and Crocombe have applied finite element analysis to the peel test. They found that the principal tensile stress in the adhesive is responsible for propagating a crack through the bond, thereby causing failure. 

---