Time dependent Quantum Theory

As in the case of photodissociation, the quantum theory of reactive molecular scattering was initially entirely based on time-independent scattering theory [4-7,100-123]. There were several early attempts to apply time-dependent quantum theory to reactive scattering processes [124—131]. But the modern era of the field really began with the seminal work of Kosloff et al. [37] and the subsequent application of his grid-based methods to the reactive scattering problem by Neuhauser and Baer and coworkers [45,132]. There have been many developments in the field [93,133-138], and several reviews and a book have been written on the topic [10,139-141]. My aim in the next section will be to outline the basic methods of time-dependent quantum theory used in reactive scattering calculations. While the review will cover many aspects of the theory, it will not cover all the approaches currently in use (as of 2003). 

The physical properties and chemical reactivity of molecules may be and often are drastically changed by a surrounding medium. In many cases specific complexes are formed between the solvent and solute molecules whereas in other cases only the non-bonded intermolecular interactions are responsible for the solvational effects. By one definition, the environmental effects can be divided into two principally different types, i.e. to the static and dynamic effects. The former are caused by the coulombic, exchange, electronic polarization and correlation interactions between two or more molecular species at fixed (close) distances and relative orientation in space. The dynamic interactions are due to the orientational relaxation and atomic polarization effects, which can be accounted for rigorously only by using time-dependent quantum theory. 

Computational procedures have been developed by Allison (1972) for single-channel optical potentials, and by Wolken (1972) for multi-channel (but a single rearrangement channel) optical potentials. White et al. (1973) have discussed these potentials within time-dependent quantum theory. 

Chapter 3 treats nuclear motions on the adiabatic potential energy surfaces (PES). One of the most powerful and simplest means to study chemical dynamics is the so-called ab initio molecular dynamics (or the first principle dynamics), in which nuclear motion is described in terms of the Newtonian d3mamics on an ab initio PES. Next, we review some of the representative time-dependent quantum theory for nuclear wavepackets such as the multiconfigurational time-dependent Hartree approach. Then, we show how such nuclear wavepacket d3mamics of femtosecond time scale can be directly observed with pump>-probe photoelectron spectroscopy. 

Time dependent quantum theory of reactive molecular collisions... 

The foundations of the modem tireory of elementary gas-phase reactions lie in the time-dependent molecular quantum dynamics and molecular scattering theory, which provides the link between time-dependent quantum dynamics and chemical kinetics (see also chapter A3.11). A brief outline of the steps hr the development is as follows [27],... 

Quack M 1992 Time dependent intramolecular quantum dynamics from high resolution spectroscopy and laser chemistry Time Dependent Quantum Molecular Dynamics Experiment and Theory. Proc. NATO ARW 019/92 (NATO ASI Ser. Vol 299) ed J Broeckhove and L Lathouwers (New York Plenum) pp 293-310... 

Maciejko J, Wang J, Guo H (2006) Time-dependent quantum transport far from equilibriam an exact nonlinear response theory. Phys Rev B 74 085324... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

In this chapter we will focus on one particular, recently developed DFT-based approach, namely on first-principles (Car-Parri-nello) molecular dynamics (CP-MD) [9] and its latest advancements into a mixed quantum mechanical/molecular mechanical (QM/MM) scheme [10-12] in combination with the calculation of various response properties [13-18] within DFT perturbation theory (DFTPT) and time-dependent DFT theory (TDDFT) [19]. 

Time-dependent perturbation theory shows that the linewidth of an n-quantum transition, generated by a single pumping frequency, should be 1/n of the corresponding... 

Time-Dependent Quantum Molecular Dynamics Experiment and Theory,... 

S. U. M. Khan, P. Wright, and J. O M. Bockris, Elektrokhimya 13 914 (1977). The first application of time-dependent perturbation theory to quantum electrode kinetics redox reactions. 

Transition probabilities. The interaction of quantum systems with light may be studied with the help of Schrodinger s time-dependent perturbation theory. A molecular complex may be in an initial state i), an eigenstate of the unperturbed Hamiltonian, Jfo I ) = E 10- If the system is irradiated by electromagnetic radiation of frequency v = co/2nc, transitions to other quantum states /) of the complex occur if the frequency is sufficiently close to Bohr s frequency condition,... 

Equations 2.85 and 2.86 may be considered the Schrodinger representation of the absorption of radiation by quantum systems in terms of spectroscopic transitions between states i) and /). In the Schrodinger picture, the time evolution of a system is described as a change of the state of the system, as implemented here in the form of the time-dependent perturbation theory. The results hardly resemble the classical relationships outlined above, compare Eqs. 2.68 and 2.86, even if we rewrite Eq. 2.86 in terms of an emission profile. Alternatively, one may choose to describe the time evolution in terms of time-dependent observables, the Heisenberg picture . In that case, expressions result that have great similarity with the classical expressions quoted above as we will see next. 

Application of time-dependent quantum mechanical theory then gives the rate of the change. 

Quantum Mechanics Time Dependent Perturbation Theory 342... 

The discussion in the previous section was helpful in identifying the factors at the molecular level which are involved when electron transfer occurs. Two different theoretical approaches have been developed which incorporate these features and attempt to account for electron transfer rate constants quantitatively. The first, by Marcus34 and Hush,35 is classical in nature, and the second is based on quantum mechanics and time dependent perturbation theory. The theoretical aspects of electron transfer in chemical36-38 and biological systems39 have been discussed in a series of reviews. 

Returning to equation (38), in the limit that ve vn, Ke = 1 and vet = vn. Electron transfer reactions that fall into this domain where the probability of electron transfer is unity in the intersection region have been called adiabatic by Marcus. Reactions for which Kei < 1, have been called non-adiabatic . In the limit that ve 2vn and e = vjvn, the pre-exponential term for electron transfer is given by vet = ve. This was the limit assumed in the quantum mechanical treatment using time dependent perturbation theory. 

B. Simon, Resonances in n-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory, Ann. Math. 97 (2) (1973) 247. 

Zhang, J.Z.H. (1990). New method in time-dependent quantum scattering theory Integrating the wave function in the interaction picture, J. Chem. Phys. 92, 324-331. 

The general theory for the absorption of light and its extension to photodissociation is outlined in Chapter 2. Chapters 3-5 summarize the basic theoretical tools, namely the time-independent and the time-dependent quantum mechanical theories as well as the classical trajectory picture of photodissociation. The two fundamental types of photofragmentation — direct and indirect photodissociation — will be elucidated in Chapters 6 and 7, and in Chapter 8 I will focus attention on some intermediate cases, which are neither truly direct nor indirect. Chapters 9-11 consider in detail the internal quantum state distributions of the fragment molecules which contain a wealth of information on the dissociation dynamics. Some related and more advanced topics such as the dissociation of van der Waals molecules, dissociation of vibrationally excited molecules, emission during dissociation, and nonadiabatic effects are discussed in Chapters 12-15. Finally, we consider briefly in Chapter 16 the most recent class of experiments, i.e., the photodissociation with laser pulses in the femtosecond range, which allows the study of the evolution of the molecular system in real time. 

---