Gaussian wavepackets

The general form of a Gaussian wavepacket is given by (Heller 1975 Child 1991 ch.ll) 

The propagation of the wavepacket is thereby reduced to the solution of coupled first-order differential equations for the parameters representing the Gaussian wavepacket, with the true potential being expanded about the instantaneous center of the wavepacket [i2( ),f( )]. This propagation scheme is very appealing and efficient provided the basic assumptions are fulfilled. The essential prerequisite is that the locally quadratic approximation of the PES is valid over the spread of the wavepacket. This rules out bifurcation of the wavepacket, resonance effects, or strong an-harmonicities. 

The Gaussian wavepacket approach works best for direct and fast dissociation (Lee and Heller 1982) several extensions and improvements of the original idea have been proposed (Coalson and Karplus 1982 Skodje 1984 Sawada, Heather, Jackson, and Metiu 1985 Heather and Metiu 1986 Huber and Heller 1988 Henriksen and Heller 1988, 1989 Kucar and Meyer 1989 Huber, Ling, Imre, and Heller 1989). 

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Figure Al.6.1. Gaussian wavepacket in a hannonic oscillator. Note tliat the average position and momentum change according to the classical equations of motion (adapted from [6]).

It is not difficult to show that, for a constant potential, equation (A3.11.218) and equation (A3.11.219) can be solved to give the free particle wavepacket in equation (A3.11.7). More generally, one can solve equation (A3.11.218) and equation (A3.11.219) numerically for any potential, even potentials that are not quadratic, but the solution obtained will be exact only for potentials that are constant, linear or quadratic. The deviation between the exact and Gaussian wavepacket solutions for other potentials depends on how close they are to bemg locally quadratic, which means... 

Gaussian wavepackets are very special fiinctions which, in a sense, bridge the gap between classical and quantum... 

To remedy this diflSculty, several approaches have been developed. In some metliods, the phase of the wavefunction is specified after hopping [178]. In other approaches, one expands the nuclear wavefunction in temis of a limited number of basis-set fiinctions and works out the quantum dynamical probability for jumping. For example, the quantum dynamical basis fiinctions could be a set of Gaussian wavepackets which move forward in time [147]. This approach is very powerfLil for short and intemiediate time processes, where the number of required Gaussians is not too large. 

In this seiniclassical calculation, we use only one wavepacket (the classical path limit), that is, a Gaussian wavepacket, rather than the general expansion of the total wave function. Equation (39) then takes the following form ... 

A different approach is to represent the wavepacket by one or more Gaussian functions. When using a local harmonic approximation to the trae PES, that is, expanding the PES to second-order around the center of the function, the parameters for the Gaussians are found to evolve using classical equations of motion [22-26], Detailed reviews of Gaussian wavepacket methods are found in [27-29]. 

To add non-adiabatic effects to semiclassical methods, it is necessary to allow the trajectories to sample the different surfaces in a way that simulates the population transfer between electronic states. This sampling is most commonly done by using surface hopping techniques or Ehrenfest dynamics. Recent reviews of these methods are found in [30-32]. Gaussian wavepacket methods have also been extended to include non-adiabatic effects [33,34]. Of particular interest here is the spawning method of Martinez, Ben-Nun, and Levine [35,36], which has been used already in a number of direct dynamics studies. 

The Gaussian wavepacket based spawning method, mentioned above, has also been used in direct dynamics where it is called ab initio multiple spawning... 

Finally, Gaussian wavepacket methods are described in which the nuclear wavepacket is described by one or more Gaussian functions. Again the equations of motion to be solved have the fomi of classical trajectories in phase space. Now, however, each trajectory has a quantum character due to its spread in coordinate space. 

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... 

One drawback is that, as a result of the time-dependent potential due to the LHA, the energy is not conserved. Approaches to correct for this approximation, which is valid when the Gaussian wavepacket is narrow with respect to the width of the potential, include that of Coalson and Karplus [149], who use a variational principle to derive the equations of motion. This results in replacing the function values and derivatives at the central point, V, V, and V" in Eq. (41), by values averaged over the wavepacket. 

Sawada et al. [26] made a detailed study of the methodology and numerical properties of the method. They paid paiticular attention to the problem of using a superposition of Gaussian wavepackets... 

The standard semiclassical methods are surface hopping and Ehrenfest dynamics (also known as the classical path (CP) method [197]), and they will be outlined below. More details and comparisons can be found in [30-32]. The multiple spawning method, based on Gaussian wavepacket propagation, is also outlined below. See [1] for further infomiation on both quantum and semiclassical non-adiabatic dynamics methods. 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

A more comprehensive Gaussian wavepacket based method has been introduced by Martinez et al. [35,36,218]. Called the multiple spawning method, it has already been used in direct dynamics studies (see Section V.B), and shows much promise. It has also been applied to adiabatic problems in which tunneling plays a role [219], as well as the interaction of a... 

A technical difference from other Gaussian wavepacket based methods is that the local hamionic approximation has not been used to evaluate any integrals, but instead Maiti nez et al. use what they term a saddle-point approximation. This uses the localization of the functions to evaluate the integrals by... 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

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