Electronic states time-dependent wave functions

If the time-dependent wave function of electrons Vl(t) is known, the current density approach outlined in Section 2 can be used directly to calculate the energy loss and also to analyze the spatial distribution of related effects. This is the case when the energy loss to a free electron is considered. Assuming that the electron is initially at rest, we can describe its state in the projectile frame as a state of scattering in the projectile Coulomb field. The wave function in the laboratory frame is then given by the expression... 

It describes the radiated power, while the atomic electron undergoes the dynamics. Another important physical quantity of interest is the ATI spectrum, which corresponds to the energy density spectrum of the electron in the continuum. It can be calculated by projecting the simulated, time-dependent wave function Itf (O) that evolves from the initially unperturbed ground state of the atom I 0) = < (r — — oo)) under the influence of the external laser pulse onto free-electron continuum states transition amplitudes involved are obtained via... 

A fundamental theoretical issue for computational photochemistry is the treatment of the hop (nonadiabatic) event. One needs to add the time propagation of the solutions of the time-dependent Schrodinger equation for electronic motion to the classical propagation of the nuclei, thus obtaining the populations of each adiabatic state. The time-dependent wave function for electronic motion is just a time-dependent configuration interaction vector ... 

The time-dependent theory of spectroscopy bridges this gap. This approach has received less attention than the traditional time-independent view of spectroscopy, but since 1980, it has been very successfully applied to the field of coordination chemistry.The intrinsic time dependence of external perturbations, for example oscillating laser fields used in electronic spectroscopy, is also expKdtly treated by modern computational methods such as time-dependent density functional theory, a promising approach to the efficient calculation of electronic spectra and exdted-state structures not based on adjustable parameters, as described in Chapter 2.40. In contrast, the time-dependent theory of spectroscopy outlined in the following often relies on parameters obtained by adjusting a calculated spectrum to the experimental data. It provides a unified approach for several spectroscopic techniques and leads to intuitive physical pictures often qualitatively related to classical dynamics. The concepts at its core, time-dependent wave functions (wave packets) and autocorrelation functions, can be measured with femtosecond (fs) techniques, which often illustrate concepts very similar to those presented in the following for the analysis of steady-state spectra. The time-dependent approach therefore unifies spectroscopic... 

Let us assume that the molecule is promoted from the ground electronic state, with initial wave function xo and energy Eq, to two coupled excited states. The corresponding wave packets are denoted by xi(t) and X2(t)- For simplicity of the notation, the dependence on the coordinates is omitted. The wave packets describing the motion in the excited states are solutions of the two-state time-dependent Schrodinger equation... 

Here l (t)) is the time-dependent wave-function of the system and denotes a continuum state with a free electron of energy Ek-... 

In the quantum-mechanical theories the intersection of the potential energy surfaces is deemphasized and the electron transfer is treated as a radiationless transition between the reactant and product state. Time dependent perturbation theory is used and the restrictions on the nuclear configurations for electron transfer are measured by the square of the overlap of the vibrational wave functions of the reactants and products, i.e. by the Franck-Condon factors for the transition. Classical and quantum mechanical description converge at higher temperature96. At lower temperature the latter theory predicts higher rates than the former as nuclear tunneling is taken into account. 

Quantum mechanics is based on the wave nature of all atomic particles. In a H atom, an electron orbits around the nucleus (a proton) electron energies, or energy states, can be conveniently described in terms of a wave function, P(x , y, z, t), which depends on particle space coordinates, x, y, z, and time t. Stable states having well-defined (discrete) energies can be represented as the product of a sinusoidal time-dependent term of angular frequency co, and a time-independent wave function l/(x, y, z) ... 

Abstract We review and further develop the excited state structural analysis (ESSA) which was proposed many years ago [Luzanov AV (1980) Russ Chem Rev 49 1033] for semiempirical models of r r -transitions and which was extended quite recently to the time-dependent density functional theory. Herein we discuss ESSA with some new features (generalized bond orders, similarity measures etc.) and provide additional applications of the ESSA to various topics of spectrochemistry and photochemistry. The illustrations focus primarily on the visualization of electronic transitions by portraying the excitation localization on atoms and molecular fragments and by detaiUng excited state structure using specialized charge transfer numbers. An extension of ESSA to general-type wave functions is briefly considered. 

For systems with just one or two degrees of freedom, the explicit consideration of time-dependent wave packets as defined in Eqs. (22) and (24) can be illuminating. The book by Schinke on photodissociation dynamics, for example, contains nice examples.If the problem involves three or more nuclear degrees of freedom, on the other hand, a reduced description, which condenses the information carried by the wave packet, is desirable. Such reduced descriptions are obtained by integrating the probability density over part of the degrees of freedom. The relevance of reduced descriptions for complex systems is based on the fact that an experimental measurement will not yield the complete quantum mechanical wave function, but rather partially integrated information, e.g. the population of an electronic state or... 

Time-dependent density-functional theory (TDDFT) extends the basic ideas of ground-state density-functional theory (DFT) to the treatment of excitations and of more general time-dependent phenomena. TDDFT can be viewed as an alternative formulation of time-dependent quantum mechanics but, in contrast to the normal approach that relies on wave-functions and on the many-body Schrodinger equation, its basic variable is the one-body electron density, n(r,t). The advantages are clear The many-body wave-function, a function in a 3A-dimensional space (where N is the number of electrons in the system), is a very complex mathematical object, while the density is a simple function that depends solely on the 3-dimensional vector r. The standard way to obtain n r,t) is with the help of a fictitious system of noninteracting electrons, the Kohn-Sham system. The final equations are simple to tackle numerically, and are routinely solved for systems with a large number of atoms. These electrons feel an effective potential, the time-dependent Kohn-Sham potential. The exact form of this potential is unknown, and has therefore to be approximated. 

Besides the apparent similarities. Table 8.1 illustrates also the obvious formal differences between bras and kets and their second quantized counterparts. Namely, the corresponding symbols are mathematically very different. The bra and ket vectors are elements of a linear vector space over which quantum-mechanical operators are defined, while the creation and annihilation operators are defined over the abstract space of particle number represented wave functions serving as their carrier space. This carrier space leads to the concept of the vacuum state, which has no analog in the bra-ket formalism. Moreover, an essential difference is that the effect of second quantized operators depends on the occupancies of the one-electron levels in the wave function, since no annihilation is possible from an empty level and no electron can be created on an occupied spinorbital. At the same time, the occupancies of orbitals play no role in evaluating bra and ket expressions. Of course, both formalisms yield identical results after calculating the values of matrix elements. 

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

The expression for the force on the nuclei, Eq. (89), has the same form as the BO force Eq. (16), but the wave function here is the time-dependent one. As can be shown by perturbation theory, in the limit that the nuclei move very slowly compared to the electrons, and if only one electronic state is involved, the two expressions for the wave function become equivalent. This can be shown by comparing the time-independent equation for the eigenfunction of H i at time t... 

To use direct dynamics for the study of non-adiabatic systems it is necessary to be able to efficiently and accurately calculate electronic wave functions for excited states. In recent years, density functional theory (DFT) has been gaining ground over traditional Hartree-Fock based SCF calculations for the treatment of the ground state of large molecules. Recent advances mean that so-called time-dependent DFT methods are now also being applied to excited states. Even so, at present, the best general methods for the treatment of the photochemistry of polyatomic organic molecules are MCSCF methods, of which the CASSCF method is particularly powerful. 

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