Schrodinger equation time-dependent wavepacket

The key to performing a wavepacket calculation is the propagation of the wavepacket forward in time so as to solve the time-dependent Schrodinger equation. In 1983, Kosloff proposed the Chebyshev expansion technique [5, 6, 7, 8] for evaluating the action of the time evolution operator on a wavepacket. This led to a huge advance in time-dependent wavepacket dynamics [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Several studies have compared different propagation methods [30, 31, 32] and these show that the Chebyshev expansion method is the most accurate. 

In a number of cases, it can be advantageous to compute the absorption spectrum without any reference to the eigenstates of the system but rather using the time-dependent wavepacket computed through the solution of the time-dependent Schrodinger equation. Using the integral form of the Dirac delta function, Eq. (4.55) can be recast as... 

A1.6.2.1 WAVEPACKETS SOLUTIONS OF THE TIME-DEPENDENT SCHRODINGER EQUATION... 

If the PES are known, the time-dependent Schrodinger equation, Eq. (1), can in principle be solved directly using what are termed wavepacket dynamics [15-18]. Here, a time-independent basis set expansion is used to represent the wavepacket and the Hamiltonian. The evolution is then carried by the expansion coefficients. While providing a complete description of the system dynamics, these methods are restricted to the study of typically 3-6 degrees of freedom. Even the highly efficient multiconfiguration time-dependent Hartree (MCTDH) method [19,20], which uses a time-dependent basis set expansion, can handle no more than 30 degrees of freedom. 

The time-dependent Schrodinger equation governs the evolution of a quantum mechanical system from an initial wavepacket. In the case of a semiclassical simulation, this wavepacket must be translated into a set of initial positions and momenta for the pseudoparticles. What the initial wavepacket is depends on the process being studied. This may either be a physically defined situation, such as a molecular beam experiment in which the particles are defined in particular quantum states moving relative to one another, or a theoretically defined situation suitable for a mechanistic study of the type what would happen if. .. 

Our main concern in this section is with the actual propagation forward in time of the wavepacket. The standard ways of solving the time-dependent Schrodinger equation are the Chebyshev expansion method proposed and popularised by Kossloff [16,18,20,37 0] and the split-operator method of Feit and Fleck [19,163,164]. I will not discuss these methods here as they have been amply reviewed in the references just quoted. Comparative studies [17-19] show conclusively that the Chebyshev expansion method is the most accurate and stable but the split-operator method allows for explicit time dependence in the Hamiltonian operator and is often faster when ultimate accuracy is not required. All methods for solving the time propagation of the wavepacket require the repeated operation of the Hamiltonian operator on the wavepacket. It is this aspect of the propagation that I will discuss in this section. 

The initial wavepacket, described in Section III.B is intrinsically complex (in the mathematical sense). Furthermore, the solution of the time-dependent Schrodinger equation [Eq. (4.23)] also involves an intrinsically complex time evolution operator, exp(—/Ht/ ). It therefore seems reasonable to assume that aU the numerical operations involved with generating and analyzing the time-dependent wavefunction will involve complex arithmetic. It therefore comes as a surprise to realize that this is in fact not the case and that nearly all aspects of the calculation can be performed using entirely real wavefunctions and real arithmetic. The theory of the real wavepacket method described in this section has been developed by S. K. Gray and the author [133]. 

This equation is exact and constitutes an iterative equation equivalent to the time-dependent Schrodinger equation [185,186]. The iterative process itself does not involve the imaginary number i therefore, if h(f) and )( — x) were the real parts of the wavepacket, then (f + x) would also be real and would be the real part of the exact wavepacket at time (f + x). Thus, if )( ) is complex, we can use Eq. (4.68) to propagate the real part of 4>(f) forward in time without reference to the imaginary part. 

Taken together, the use of the real part of the wavepacket and the mapping of the time-dependent Schrodinger equation lead to a very significant reduction of the computational work needed to accompfish the calculation of reactive cross sections using wavepacket techniques. 

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 time-dependence of the wavepacket evolving on any potential surface can be numerically determined by using the split operator technique of Feit and Fleck [10-15]. A good introductory overview of the method is given in Ref. [12]. We will discuss a potential in two coordinates because this example is relevant to the experimental spectra. The time-dependent Schrodinger equation in two coordinates Qx and Qy is... 

In practice one does not proceed as we did in the above derivation. Instead of calculating first all stationary wavefunctions and then constructing the wavepacket according to (4.3), one solves the time-dependent Schrodinger equation (4.1) with the initial condition (4.4) directly. Numerical propagation schemes will be discussed in the next section. Since 4 /(0) is real the autocorrelation function fulfills the symmetry relation... 

The wavepacket 4>/(t) on the other hand is a function of time and therefore contains all energies, weighted by the matrix elements t(Ef,n) defined in Equation (2.68). It is a solution of the time-dependent but not the time-independent Schrodinger equation. 

Therefore, we can use this somewhat artificial wavepacket, when it has been determined by direct integration of the time-dependent Schrodinger equation, to extract the transition amplitudes and hence the photodissociation cross sections for all energies. 

The time-dependent Schrodinger equation for a nuclear wavepacket on a single potential energy surface in a laser field is written as... 

As long as the photodissociation reaction is fairly direct, the time-dependent formulation is fruitful and provides insight into both the process itself and the relationship of the final-state distributions to the absorption spectrum features. Moreover, solution of the time-dependent Schrodinger equation is feasible for these short-time evolutions, and total and partial cross sections may be calculated numerically.5 Finally, in those cases where the wavepacket remains well localized during the entire photodissociation process, a semi-classical gaussian wavepacket propagation will yield accurate results for the various physical quantities of interest.6... 

A bright state is prepared at t = 0. Lee and Heller (1979) derive a rigorous description of the wavepacket that is actually prepared when the excitation pulse is of finite duration. It is not a single eigenstate. It evolves in time subject to the spectroscopic time-independent Heff, as specified by the time-dependent Schrodinger equation Eq. (9.1.2). 

---