Computational observables

Once numerical estimates of the weight of a trajectory and its variance (2cr ) are known we are able to use sampled trajectories to compute observables of interest. One such quantity on which this section is focused is the rate of transitions between two states in the system. We examine the transition between a domain A and a domain B, where the A domain is characterized by an inverse temperature - (3. The weight of an individual trajectory which is initiated at the A domain and of a total time length - NAt is therefore... 

Training courses are available in analytical methods of fault-tracing. Computers are also in use which monitor a number of parameters and draw attention to any observed abnormality. The control/ monitoring device may then make a judgement as to the cause, or this may rely on the interpretation of the operator. Considerable use is now made of logic control/monitoring devices which can oversee the operation of a large number of installations from a central computer/observation terminal. 

Density functional theory (DFT) is an entrancing subject. It is entrancing to chemists and physicists alike, and it is entrancing for those who like to woik on mathem cal physical aspects of problems, for those who relish computing observable properties from theory and for those who most enjoy developing correct qualitative descriptions of phenomena in the service of the broader scientifrc community. 

Once MD simulations for the CC have been carried out the Eqs. (52) and (53) for the expansion coefficients can be easily solved (also in the presence of an external radiation field). Moreover, the coefficients are used to compute observables of interest. The excited state population follows as... 

Thus far we have focussed on the dynamics of quantum-classical systems. In practice, we are primarily interested instead in computing observables that can be compared eventually to experimentally obtainable quantities. To this end, consider the general quantum mechanical expression for the expectation value of an observable,... 

The focus of this chapter is exploration of the ability of mixed quantum classical approaches to capture the effects of interference and coherence in the approximate dynamics used in these different mixed quantum classical methods. As outlined below, the expectation values of computed observables are fundamentally non-equilibrium properties that are not expressible as equilibrium time correlation functions. Thus, the chapter explores the relationship between the approximations to the quantum dynamics made in these different approaches that attempt to capture quantum coherence. 

With our realization that dynamical variables are represented as operators, we can also raise the question of how to compute observed quantities within the framework of quantum mechanics. The essential rule is that if we are interested in obtaining the expectation value of the operator 0(r, p) when the system is in the state xj/, then this is evaluated as... 

What we have produced so far is an approximate Hamiltonian designed to study chemical reactions in complex condensed phases. We also have a mathematical method to evaluate quantum propagation using this Hamiltonian. We as yet have no practical method to compute observables such as rates. The flux correlation... 

Specific Force Constant Analysis and Computational Observables... 

Figure 11 Differential dihedral potential in ethane. The ab initio second derivatives of the energy with respect to the dihedral angle t h are plotted vs. Thh> showing the periodic asymmetric behavior of this coordinate. Although a computational observable, this curve is not experimentally accessible. Recalculated from ref. 118.

We stress that V" is a computational observable that cannot be directly obtained from experiment. The technique outlined above enables us therefore to extract from ab initio calculations pairwise dihedral interactions that are otherwise inaccessible. 

The arrow intentionally goes in only one direction, because the left-hand side is a well-defined quantum mechanical entity whereas the right-hand side is not. Thus, the gradient of the dipole moment determines the charge, but the latter cannot be derived independently to predict the computationally observed" dipole gradient. 

Then Q iv in Eq. [58] represents a computational observable having the dimension of the charge squared. If we plot this variable versus 1/Kab we should obtain a line whose intercept is the atomic charge product and whose slope will be determined by both the charge and the dipole. 

Figure 2.30 shows the accumulated instantaneous excited state lifetime (t) and cis trans conversion ratio (p) for 1 and Im computed at the CASSCF/3-21G level of theory. For each individual series of calculations, the computed observables reach a limiting value after 50 trajectories. The swarm seems to be already of reasonable size to draw qualitative conclusions. For example, one has confidence in the conclusion that the methylation of the model chromophore reduces (by 20 to 60%) the efficiency of the conversion and extends (by about 40 to 50 fs) the lifetime of the excited state. 

The most intriguing computational observation was the decrease in insertion barrier caused by the addition of the naphthyl substituent. Addition of a steric encumbrance decreased the barrier Visual examination of the insertion saddle point provided an explanation for this observation see Figure 7. At the insertion saddle point the methyl group of the propylene was found to be within van der Waals contact of the aromatic ring and placed in the attractive well of the interaction rather than the inner repulsive wall. In the reactants this stabilizing van der Waals interaction was absent thus, the saddle point was differentially stabilized. 

Two significant aspects of the symmetry observed in the analysis of periodicity are the inverted electronic energy levels and the approach of Z/N —> 1 for all nuclides. The inversion is explained by the computational observation that electronic sub-levels respond differently to compression of an atom. 

In this instance, S is smaller than previously computed. Observe also that whereas in the previous example the feed point is situated on a point on the boundary of the stoichiometric subspace, the feed point lies within the new region. 

If the Hartree-Fock equations associated with the valence pseudo-Hamiltonian (167) are solved with extended basis sets, then all the above F are almost basis-set-independent. At the present time, and for practical reasons, most of the ab initio valence-only molecular calculations use coreless pseudo-orbitals. The reliability of this approach is still a matter of discussion. Obviously the nodal structure is important for computing observable quantities such as the diamagnetic susceptibility which implies an operator proportional to 1/r. From the computational point of view, it is always easy to recover the nodal structure of coreless valence pseudo-orbitals by orthogonalizing the valence molecular orbitals to the core orbitals. This procedure has led to very accurate results for several internal observables in comparison with all-electron results. The problem of the shape of the pseudoorbitals in the core region is also important in relativity. For heavy atoms, the valence electrons possess high instantaneous velocities near the nuclei. Schwarz has recently investigated the compatibility between the internal structure of valence orbitals and the representations of operators such as the spin-orbit which vary as 1/r near the nucleus. ... 

Q.3.7 Error discrepancy between a computed, observed, or measured value or condition and the true, specified, or theoretically correct value or condition. 

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