Wavepackets computational methods

When two or more normal coordinates are coupled, the wavepacket dynamics depends on all of the coupled coordinates simultaneously. Thus, the (t) s for each coordinate computed individually and Eq. (3) cannot be used. Instead, the multidimensional wavepacket must be calculated by using the time dependent Schrodinger equation. In this section we show how to calculate the wavepacket for two coupled coordinates. The computational method discussed here removes all of the restrictive assumptions used in deriving Eq. (4). Any potential, including numerical potentials, can be used. 

The ab initio atom-centered density matrix propagation (ADMP) and the quantum wavepacket ab initio molecular dynamics (QWAIMD) computational methods are briefly described. Studies on vibrational and electronic properties obtained utilizing these methods are highlighted. 

The collinear A+BC dissociative collision can be treated in a straightforward manner,using the Time Dependent WavePacket (TDWP) method. The reason is that the dissociative continuum of the BC molecule is handled automatically within the space discretization scheme of the grid. As the basic method has already been described in detail elsewhere ,it will be only outlined here,emphasizing the technical points and some new features which lead to a significant reduction in computation time. 

Since the first experiments on the I-CN [21] bond cleavage and the wavepacket oscillations between the ionic and covalent potentials in the photodissociation of Nal [22, 23], pump-probe techniques have been applied to a wide range of important photochemical processes. However, the data obtained Ifom such experiments are often difficult to interpret and theoretical modeling is needed to get further insight into the excited state dynamics of the systems of interest at the atomistic level. In this context, the development of efficient and accurate computational methods for the description of ground and excited electronic states of mid-size molecular systems in a balanced way [24, 25], has greatly facilitated the theoretical study of photochemical processes. 

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

In Chapter VI, Ohm and Deumens present their electron nuclear dynamics (END) time-dependent, nonadiabatic, theoretical, and computational approach to the study of molecular processes. This approach stresses the analysis of such processes in terms of dynamical, time-evolving states rather than stationary molecular states. Thus, rovibrational and scattering states are reduced to less prominent roles as is the case in most modem wavepacket treatments of molecular reaction dynamics. Unlike most theoretical methods, END also relegates electronic stationary states, potential energy surfaces, adiabatic and diabatic descriptions, and nonadiabatic coupling terms to the background in favor of a dynamic, time-evolving description of all electrons. 

Since the classical treatment has its restrictions and the applicability of the quantum OCT is limited to low-dimensional systems due to its formidable computational cost, it would be very desirable to incorporate the semiclassical method of wavepacket propagation like the Herman-Kluk method [20,21] into the OCT. Recently, semiclassical bichromatic coherent control has been demonstrated for a large molecule [22] by directly calculating the percent reactant as a function of laser parameters. This approach, however, is not an optimal control. 

Equation (20) was also used to compute the acoustic response of fluid cylinders immersed in water and insonified normal to their axis with a sinusoidal wavepacket. The examples shown here can be considered by other techniques ( 5 ) but serve as appropriate tests for the accuracy of the model which can then be used to compute the acoustic responses of systems which cannot be readily treated by other methods. The material properties of the cylinder are shown in Table 1 and were chosen to enable the calculated echo structure of the cylinders to be compared with previously published analytical work ( 5 ). ... 

Of the dynamical techniques available the most rigorous and informative are the quantum mechanical dynamics methods. These methods are, however, the most sophisticated and computationally intensive to employ. Two of the most widely used quantum dynamics techniques are quantum scattering (QS) [35] and wavepacket (WP) [125] analysis. 

Other examples of wavepacket calculations applied to photofragmentation may be found in Refs. [34], [57] and [58]. Most recently, time-dependent methods have been used to compute vector correlations and alignment parameters [59, 60]. 

From a practical point of view the classical method using directly the probability density function is not convenient, and it is computationally preferable to use an approach that involves trajectory calculations. A derivation of such formulation can be made by starting from the quantum-mechanical TDSCF, and using semiclassical (gaussian) wavepackets. Here we merely quote the final result. In analogy to (62), the single-mode classical SCF potentials are given by... 

In the TSH method, the molecule always feels the potential energy surface of only one of the two adiabatic states. (For a comparison of TSH with quantum wavepackets see, e.g.. Worth et al. .) The populations a(t) are monitored to decide which adiabatic state shornd be used to compute the gradient and to propagate the nuclei. When the occupation number of a state reaches a given threshold (e.g.,... 

In order to compute the evolving state i(/)), Tal-Ezer and Kosloff (77) were the first to propose an expansion of the evolution operator in terms of Chebyshev polynomials. They initially developed this method for wavepacket calculations on spatial grids. More recently, this procedure has been adapted and applied to bound systems (20). It involves breaking the total integration time (for instance 2 ps) into smaller time steps At (each about 25 fs), and using a polynomial expansion of the evolution operator (/(Af) over each small time step. This efficient method provides all the transition probabilities Plf(t) from initial state i) in one calculation because it directly provides the evolving state /(r)). 

The method consists in computing the time evolution of a wavepacket ijj, which represents initially an atom A impinging on a molecule BC in a given vibrational state v ... 

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