Eigenstates development

Consider an ensemble composed of constituents (such as molecules) per unit volume. The (complex) density operator for this system is developed perturbatively in orders of the applied field, and at. sth order is given by The (complex). sth order contribution to the ensemble averaged polarization is given by the trace over the eigenstate basis of the constituents of the product of the dipole operator, N and = Tr A pp... 

The next significant development in the history of the geomebic phase is due to Mead and Truhlar [10]. The early workers [1-3] concenbated mainly on the specboscopic consequences of localized non-adiabatic coupling between the upper and lower adiabatic elecbonic eigenstates, while one now speaks... 

S, then these f2 S2 Lz Sz eigenfunctions must be coupled (i.e., recombined) to generate J2 Jz eigenstates. The steps needed to effect this coupling are developed and illustrated for the above p2 configuration case in Appendix G. 

Perturbation theory also provides the natural mathematical framework for developing chemical concepts and explanations. Because the model H(0) corresponds to a simpler physical system that is presumably well understood, we can determine how the properties of the more complex system H evolve term by term from the perturbative corrections in Eq. (1.5a), and thereby elucidate how these properties originate from the terms contained in //(pertJ. For example, Eq. (1.5c) shows that the first-order correction E11 is merely the average (quantum-mechanical expectation value) of the perturbation H(pert) in the unperturbed eigenstate 0), a highly intuitive result. Most physical explanations in quantum mechanics can be traced back to this kind of perturbative reasoning, wherein the connection is drawn from what is well understood to the specific phenomenon of interest. 

At the basis of the theoretical developments leading to the MCSEs lie a set of relations reported by Tel et al. [81], which link CMs of different orders among themselves. These C-relations establish a set of necessary conditions that the CMs must satisfy when they correspond to a Hamiltonian eigenstate. The... 

Here we describe the development of the coherent-control toolbox with gas-phase iodine molecules [37 1, 48]. The gas-phase molecules are isolated from each other, so that they have long coherence lifetime, serving as an ideal platform to observe and control quantum coherence. First, we describe our experiments to observe and control the temporal evolution of the WP interference. Second, the eigenstate picture of the WP interference is presented. Finally, we demonstrate the application of WPI to ultrafast molecular computing. 

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

Of course, the time development of this state is given by incorporating the phase development of the eigenstates, i.e.,... 

The wavefunction developed in the preceding subsection is, of course, only a first approximation to the true molecular continuum wavefunction. We have thus far considered only the nonvibrating molecule, and treated the resulting Born-Oppenheimer states as if they were the true eigenstates. In a real system vibrational motion of the nuclei, configuration interaction between vibronic states, etc., must be included in the description. 

Variational optimization of equation (11.9), where we are concerned with only one projection of tp corresponding to a particular electronic eigenstate, has been extensively studied. There are at least two well-developed techniques for such situations, namely, the multiconfiguration SCF (MCSCF) and iterative natural spin-orbital (INSO) approaches. 

Electron transfer (ET) is generally cast in terms of charge-localized diabatic states and charge densities (the initial and final state densities referred to above as pa) [28-30], These states differ from the corresponding eigenstates (the adiabatic states), which may be probed experimentally (spectroscopy) or theoretically (e.g., Cl or TD DFT), and various procedures have been developed for transforming adiabatic to diabatic states [60]. In one... 

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