Dynamics in state space

The anharmonic coupling term, 122QlQ2 may be manipulated trivially so that the two dynamically important terms appear in the form discussed in Section 9.1.8, 

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The second topic is a simplified scheme for going from a spectrum, I(oj), to a potential surface, V(Q), to dynamics, ( ), t). The essential feature of this scheme is embodied in two complementary reduced-dimension pictures dynamics in configuration space and dynamics in state space. 

The finer structure within each feature state corresponds to the dynamics of the Franck-Condon bright state within a four-dimensional state space. This dynamics in state space is controlled by the set of all known anharmonic resonances. The state space is four dimensional because, of the seven vibrational degrees of freedom of a linear four-atom molecule, three are described by approximately conserved constants of motion (the polyad quantum numbers) thus 7-3 = 4. 

Some insights into the information content of the quantities that describe dynamics in state space are obtained by choosing the simplest possible pair of... 

When the state space spanned by /(0) includes more than 2 eigenstates, the dynamics in state space can begin to look very complicated. However, the concepts of bright state, dark state, state-selective detection, and... 

The fast-forward protocol can be regarded as a prescription for finding a shortcut in state space, [50] from the initial state to the target state. There are, of course, many possible shortcuts in state space but very few proposals to find those shortcuts. In this section, we generalize the fast-forward protocol in a two-level system, developing different shortcuts in which, in contrast to fast-forward field (FFE)-driven dynamics, the amplitude and the phase of the wave function of the intermediate state are modulated [50]. 

Since the reactive trajectories will in general slow down near the dynamical bottlenecks of the reaction, this allows one to identify the transition state regions roughly as the regions where mAsix) is peaked. Observe however that these regions can be multiple (i.e. there may be more than one dynamical bottleneck for a reaction) and quite wide (i.e. the dynamical bottleneck may be a rather extended region in state-space). 

It is the early time dynamics, while the system remains highly localized in state space, the specific coupled states are known a priori, and the coupling matrix elements are experimentally measurable or theoretically predictable, for which our study of diatomic molecule spectra and dynamics prepares us. At early time, the system evolves in a reduced dimensionality. The excitation initially is localized spatially on a part of the molecule (a chromophore) and restricted by approximate constants of motion to a tiny fraction of energetically-accessible state space. 

For static and (structural) dynamic analysis, for determination of eigenfre-quencies and eigenmodes, several different commercial tools exist such as NASTRAN, ABAQUS or ANSYS. Some of them are also able to handle actuators and piezoelectric materials, and also to carry out some types of model reduction techniques. Nevertheless, specific techniques might have to be established by the user via accessing the modal data base. These data are then also used to set up a modal or otherwise condensed state-space representation possibly including specific actuator and sensor models. A description of the transformation of finite-element models from ANSYS to dynamic models in state space form in MATLAB can be found in [20]. 

Car, R., and M. Parrinello. 1987. The unified approach to density functional and molecular dynamics in real space. Solid State Comm. 62 403-405. 

A system identification method is considered parametric if a mathematical dynamic model (often formulated in state-space) is realized in a first step and the dynamic properties of the system estimated from the realized model in the second step. Nonparametric system identification methods directly estimate the dynamic parameters of a system from transformation of data, e.g., Fourier transform or power-spectral density estimation. Time-domain identification methods estimate the dynamic parameters of a system by directly using the measured response time histories, while frequency-domain methods use the Fourier transformation or power-spectral density estimation of the measured time histories. There is also a class of time-frequency methods such as the short-time Fourier transform and the wavelet transform. These methods are commonly used for identification of time-varying systems in which the dynamic properties are time-variant Linear system identification methods are based mi the assumption that the system behaves linearly and... 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

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