Gaussian wave packets

Heather R and Metiu H 1985 Some remarks concerning the propagation of a Gaussian wave packet trapped in a Morse potential Chem. Phys. Lett. 118 558-63... 

Sawada S and Metiu H 1986 A multiple trajectory theory for curve crossing problems obtained by using a Gaussian wave packet representation of the nuclear motion J. Chem. Phys. 84 227-38... 

Braun M, Metlu H and Engel V 1998 Molecular femtosecond excitation described within the Gaussian wave packet approximation J. Chem. Phys. 108 8983-8... 

The big advantage of the Gaussian wavepacket method over the swarm of trajectory approach is that a wave function is being used, which can be easily manipulated to obtain quantum mechanical information such as the spechum, or reaction cross-sections. The initial Gaussian wave packet is chosen so that it... 

To deal with the problem of using a superposition of functions, Heller also tried using Gaussian wave packets with a fixed width as a time-dependent basis set for the representation of the evolving nuclear wave function [23]. Each frozen Gaussian function evolves under classical equations of motion, and the phase is provided by the classical action along the path... 

A typical initial condition in ordinary wave packet dynamics is an incoming Gaussian wave packet consistent with particular diatomic vibrational and rotational quantum numbers. In the present case, of course, one has two diatomics and with the rotational basis representation of Eq. (30) one would have, for the full complex wave packet. 

The wave packets <()( ) and x(0 to be propagated forward and backward, respectively, are expanded in terms of the frozen Gaussian wave packets as (see also Section II.B)... 

The similar expansion applies to x(f). The frozen Gaussian wave packets gy,q p, are explicitly given by... 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

Figure 3. Infinite resolution spectrum (sticks) generated from a Gaussian wave packet launched at the inner turning point of PO 1 in Fig. 2, and low resolution version of it (full line). 

In Fig. 3 we present the stick spectrum [corresponding to infinite resolution see eq. (12)] generated from a Gaussian wave packet of the form... 

Figure 4. Diabatic (solid lines) and adiabatic (dashed lines) potential-energy curves of Model IVa. The Gaussian wave packet indicates the initial preparation of the system at time t = 0.

We see from Fig. 2.5 that the Gaussian wave packet has its intensity, F 2, centred on x0 with a half width, W, whereas (k) 2 is centred on k0 with a half width, 1/W. Thus the wave packet, which is centred on x0 with a spread Ax — W, is a linear superposition of plane waves whose wave vectors are centred on k0 with a spread, A = jW. But from eqn (2.8), p = Hk. Therefore, this wave packet can be thought of as representing a particle that is located approximately within Ax = W of x0 with a momentum within Ap = h/W of po = hk0. If we try to localize the wave packet by decreasing W, we increase the spread in momentum about p0. Similarly, if we try to characterize the particle with a definite momentum by decreasing 1/W, we increase the uncertainty in position. 

Fig. 2.5 The relation between a Gaussian wave packet Y, and its Fourier transformation, . The quantity, ( 2, has a half width of W, where 2 has a half width of 1.

Henriksen, N.E. and Heller, E.J. (1988). Gaussian wave packet dynamics and scattering in the interaction picture, Chem. Phys. Lett. 148, 567-571. 

Consider a (complex-valued) Gaussian wave packet of the form... 

We consider now the dynamics of the Gaussian wave packet within the framework of a time-dependent local harmonic approximation (LHA) to the exact potential V(x) around xt. ... 

The LHA is exact for potentials that contain, at most, quadratic terms but obviously an approximation for anharmonic potentials. Thus, a single Gaussian wave packet within a local harmonic approximation can, e.g., not tunnel or bifurcate, i.e., there will be no simultaneously reflected and transmitted part in scattering off barriers. 

When the LHA is invalid, the Gaussian wave packet will not keep its simple analytical form. One must then solve Eq. (4.101) numerically. To that end various methods have been developed [12,13]. 

It is instructive to consider the momentum-space representation of the Gaussian wave packet. In this representation, the states are projected onto the eigenstates of the momentum operator, i.e., P p) = p p), which in the coordinate representation takes the form... 

Fig. 4.2.1 The probability density associated with the Gaussian wave packet. The most probable position is at x = xt, which also coincides with the expectation (average) value of the time-dependent position. The width is related to the time-dependent uncertainty (Ax)t, i.e., the standard deviation of the position.

Equation (4.173) displays clearly how the cross-section is determined from the scattering dynamics via the time evolution of the initial channel state U(t — to) 4>n) and a subsequent projection onto the final channel state. In practice, the plane wave of the initial state in Eq. (4.169) can be replaced by a Gaussian wave packet, as illustrated in Fig. 1.1.1. When this wave packet is sufficiently broad, it will be localized sharply in momentum space. 

In the control scheme [13,17] that we have focused on, the time evolution of the interference terms plays an important role. We have already discussed more explicit forms of Eq. (7.75). One example is the Franck-Condon wave packet considered in Section 7.2.2 another example, which we considered above, is the oscillating Gaussian wave packet created in a harmonic oscillator by an (intense) IR-pulse. Note that the interference term in Eq. (7.76) becomes independent of time when the two states are degenerate, that is, AE = 0. The magnitude of the interference term still depends, however, on the phase S. This observation is used in another important scheme for coherent control [14]. 

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