Time-independent methods

In this section we review three simple approximate time-independent methods which are specifically designed for calculating absorption spectra. Time-independent quantum scattering calculations, which can give exact results, will not be discussed. Such methods have been reviewed for instance by Nyman and Yu 2000 [81] and Althorpe and Clary 2004 [82]. In Section 6 we instead describe the time-dependent wavepacket approach, which can also give exact results. In the present section we review (i) a zero point energy model, (ii) the simple reflection principle model and (iii) the reflection principle model [3]. The accuracy of these models, which should only be applied to direct or near-direct reactions, will 

FIGURE 7.2 (A) Prediction of HCl spectrum using the simple reflection principle. 

It is usually assumed that the interaction between the electromagnetic field and the absorbing matter only involves two electronic states and that the interaction is weak and of long duration. This simplifies the expressions for the absorption spectrum cr co) from an initial state (z, v) which may be approxi- 

Here co is the angular frequency of the incoming light, c is the velocity of light and eo is the vacuum permittivity /, i, v and v are quantum numbers for the final (/, v ) and initial (2, v) electronic (f, i) and vibrational (v, v) states. The electronic transition dipole moment is defined in equation (3). iA(R) 

In the reflection principle, the kinetic energy is approximated by (1/2)Ezpe, where Ezpe is the harmonic ZPE of the ground electronic state. The absorption cross section in (13) can then be further simplified. 

In this chapter we outline the evaluation of multi-dimensional bound-free matrix elements of the type (2.68), 

H(Q) is the nuclear Hamiltonian in the corresponding electronic state at short distances it describes the motion of the complex and at large intermolecular separations it describes the free fragments. The matrix elements (3.1) are needed for the calculation of photodissociation cross sections. In this chapter we discuss numerically exact and approximate methods that are directly based on the solution of (3.2). The complementary time-dependent view follows in the next chapter. 

In order to keep the formulation as simple as possible we confine the discussion to systems with only two degrees of freedom. The extension to more complex problems is — formally at least — straightforward. We will treat triatomic molecules ABC dissociating into products A+BC. First, we again consider in Section 3.1 the linear model, outlined in Sections 2.4 and 2.5, in which the diatomic fragment vibrates while its rotational degree of freedom is frozen. Subsequently, we treat in Section 3.2 the 

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This section is divided into two sections the first concerned with time-dependent methods for describing the evolution of wavepackets and the second concerned with time-independent methods for solving the time independent Sclirodinger equation. The methods described are designed to be representative of what is in use. 

The approaches discussed so far are generally called time-independent methods, since they start from the time-independent Scln-ddinger equation, (//-E)l f(t. An alternative is to use the time-dependent Scln-ddinger equation [28, 29, 59, and M. Conceptually, tlie time-dependent approach is... 

Neuhauser D and Baer M 1990 A new accurate (time independent) method for treating three-dimensional reactive collisions the application of optical potentials and projection operators J. Chem. Phys. 92 3419... 

The five studies of hydrate formation given in Section 8.1 are of two types. The first three case studies show thermodynamic (time-independent) methods to prevent plug formation. However, the second type provides a closer, mechanistic look at the physical kinetics (time-dependent) hydrate formation and agglomeration. A goal of this section is to show how these two methods provide two different methods of plug prevention. 

Figure 4 shows the dependence on final rotational state of the C1-hH2(v=0, j =0)—> HCl(v=0jO DCSs at two different coUision energies. We observe that the DCSs all have a very similar shape. Likely, this implies that the overaU shape of the DCS is determined by the reactive probabdity (opacity) as a function of impact parameter and by an overall deflection function (h), while the product final state distribution is governed by the shape of the PES in the exit region of the arrangement channels. We remark also that the determination of fully final-state resolved DCS does not present any particular difficulty for our time-independent method, but would not be feasible with some of the time-dependent methods which have been applied recently to the X-hH2 reactions. [7, 16]... 

Because of the very large well in the i)otential energy surface, very few accurate quantum dynamical results were available five years ago, either by tiirK dcpendcnt or by time-independent methods. Indeed, a very largo nunit)cr of channels or of grid points are necessary to converge reaction probabilities and cross sections. 

Different theoretical methods have been used to calculate the complex energies, Eq. (8.1), for compound-state resonances. They can be divided into time-independent and time-dependent methods. A standard quantum mechanical time-independent method is a close-coupling calculation (Stechel et al., 1978) which considers resonant state formation as a result of a collision such as A + BC —> ABC AB + C. Determined... 

These a posteriori corrections are based on a very simple idea which is suggested by the work of Brandow [10]. Brandow used the Brillouin-Wigner perturbation theory as a starting point for a derivation of the Goldstone linked diagram expansion by elementary time-independent methods . At a NATO Advanced Study Institute held in 1991, Wilson wrote [112] ... 

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