Quantum wavepacket methods

S. C. Althorpe,/. Chem. Phys., 114,1601 (2001). Quantum Wavepacket Method for State-to-... 

Althorpe, S.C. (2001) Quantum wavepacket method for state-to-state reactive cross sections,, 7. Chem. Phys. 114. 1601-1616. 

Figure 5. Total reactive cross section, calculated using the quantum wavepacket method, for the reaction N( D) + 02(X E ) 0( P) + NO(X n). The initial vi-...

However, at the time of their publication the exact TD quantum wavepacket methods were not viewed as... 

In contrast to the above shown ab initio molecular dynamics approach, quantum wavepacket methods can provide a rigorous description for any chemically important properties, thereby properly taking account of the quantum effects of nuclear dynamics. However, the direct applications of the full quantum theories are often prohibitively difficult for many dimensional systems. Also, the time evolution of quantum wavefunctions generally requires the knowledge of the global PES beforehand, in contrast to the trajectory-based methods such as ab initio molecular dynamics, which demands only local information along the paths used. The latter is often referred to as on-the-fiy method. 

We compute the photoelectron spectra using a quantum wavepacket method described in [15] with enhancements to handle the nonadiabatic interactions. Briefly, the system is expanded in terms of the three relevant electronic states 1 (Vi state), 2 (V2 state), and (ion state), as... 

A further model Hamiltonian that is tailored for the treatment of non-adiabatic systems is the vibronic coupling (VC) model of Koppel et al. [65]. This provides an analytic expression for PES coupled by non-adiabatic effects, which can be fitted to ab initio calculations using only a few data points. As a result, it is a useful tool in the description of photochemical systems. It is also very useful in the development of dynamics methods, as it provides realistic global surfaces that can be used both for exact quantum wavepacket dynamics and more approximate methods. 

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... 

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... 

Full quantum wavepacket studies on large molecules are impossible. This is not only due to the scaling of the method (exponential with the number of degrees of freedom), but also due to the difficulties of obtaining accurate functions of the coupled PES, which are required as analytic functions. Direct dynamics studies of photochemical systems bypass this latter problem by calculating the PES on-the-fly as it is required, and only where it is required. This is an exciting new field, which requires a synthesis of two existing branches of theoretical chemistry—electronic structure theory (quantum chemistiy) and mixed nuclear dynamics methods (quantum-semiclassical). 

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. 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross-sections. Using the method of spectral quantization (SQ), it is possible to extract a Seigert state wavefunction from time-dependent quantum wavepackets using the Fourier relation Eq. (21). The state obtained in this way for J = 0 is shown in Fig. 7 this state is localized in the collinear F — H — D arrangement with 3-quanta of excitation in the asymmetric stretch mode, and 0-quanta of excitation in the bend and symmetric stretch modes. If the state pictured in Fig. 7 is used as an initial (prepared) state in a wavepacket calculation, one observes pure... 

How well do these quantum-semiclassical methods work in describing the dynamics of non-adiabatic systems There are two sources of errors, one due to the approximations in the methods themselves, and the other due to errors in their application, for example, lack of convergence. For example, an obvious source of error in surface hopping and Ehrenfest dynamics is that coherence effects due to the phases of the nuclear wavepackets on the different surfaces are not included. This information is important for the description of short-time (few femtoseconds) quantum mechanical effects. For longer timescales, however, this loss of information should be less of a problem as dephasing washes out this information. Note that surface hopping should be run in an adiabatic representation, whereas the other methods show no preference for diabatic or adiabatic. 

Key topics covered in the review are the analysis of the wavepacket in the exit channel to yield product quantum state distributions, photofragmentation T matrix elements, state-to-state S matrices, and the real wavepacket method, which we have applied only to reactive scattering calculations. 

In Fig. 2 we compare results using e = 0.4 for the two mixed quantum-classical methods outlined in this chapter with exact results obtained from MCTDH wavepacket dynamics calculations. To make a reliable comparison the approximate finite temperature calculations were performed at very low temperatures (/ = 25), though a product of ground state wave functions for the independent harmonic oscillator modes could have been used to make the initial conditions identical to those used in the MCTDH calculations. 

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. 

This article is organized as follows. In Section 2 ab initio molecular dynamics methods are described. Specifically, in Section 2.1 we discuss the extended Lagrangian atom-centered density matrix (ADMP) technique for simultaneous dynamics of electrons and nuclei in large clusters, and in Section 2.2 we discuss the quantum wavepacket ab initio molecular dynamics (QWAIMD) method. Simulations conducted and new insights obtained from using these approaches are discussed in Section 3 and the concluding remarks are given in Section 4. 

Often the integration steps have to be very small because it is not possible to evaluate the second derivative (Hessian) matrix of such systems. In such methods, the real nuclear (quantum) wavepacket must be emulated by a swarm of trajectories. Such trajectories are generated by sampling, which should be extensive enough (i.e., the swarm contains a sufficient variety of trajectories) to ensure that all relevant geometries involved in the chemical event have been explored. 

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 the full-quantum dynamics method, the distribution of nuclear positions is accounted for in nuclear wavepacket form, that is, by a function that defines the distribution of momenta of each atom and the distribution of the position in the space of each atom. In classical and semi-classical or quasi-classical dynamics methods, the wavepacket distribution is emulated by a swarm of trajectories. We now briefly discuss how sampling can generate this swarm. 

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