Order Zero

If only zero-order states from the same polyad are conpled together, this constitutes a fantastic simplification in the Flamiltonian. Enonnons compntational economies result in fitting spectra, becanse the spectroscopic Flamiltonian is block diagonal in the polyad nnmber. That is, only zero-order states within blocks with the same polyad number are coupled the resulting small matrix diagonalization problem is vastly simpler than diagonalizing a matrix with all the zero-order states conpled to each other. 

The ability to assign a group of resonance states, as required for mode-specific decomposition, implies that the complete Hamiltonian for these states is well approxmiated by a zero-order Hamiltonian with eigenfunctions [M]. The ( ). are product fiinctions of a zero-order orthogonal basis for the reactant molecule and the quantity m. represents the quantum numbers defining ( ).. The wavefimctions / for the compound state resonances are given by... 

Resonance states in the spectra, which are assignable in temis of zero-order basis will have a predominant expansion coefficient c.. Hose and Taylor [ ] have argued that for an assignable level r /,j>0.5... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

In many cases the variation is not very strong for reasonable displacements from equilibrium, and it is sufficient to use only the zero-order temi in die expansion. If diis is inserted hito equation (B 1.1.6) we get... 

If we can get by with using only the zero-order tenn of (B 1.1.7 ). we can take out of the integral and use the fact that ( ) q is nonnalized. The last equation then simplifies fiirther to... 

If we can use only the zero-order tenn in equation (B 1.1.7) we can remove the transition moment from the integral and recover an equation hrvolving a Franck-Condon factor ... 

We now discuss the lifetime of an excited electronic state of a molecule. To simplify the discussion we will consider a molecule in a high-pressure gas or in solution where vibrational relaxation occurs rapidly, we will assume that the molecule is in the lowest vibrational level of the upper electronic state, level uO, and we will fiirther assume that we need only consider the zero-order tenn of equation (BE 1.7). A number of radiative transitions are possible, ending on the various vibrational levels a of the lower state, usually the ground state. The total rate constant for radiative decay, which we will call, is the sum of the rate constants,... 

The synnnetry selection rules discussed above tell us whether a particular vibronic transition is allowed or forbidden, but they give no mfonnation about the intensity of allowed bands. That is detennined by equation (Bl.1.9) for absorption or (Bl.1.13) for emission. That usually means by the Franck-Condon principle if only the zero-order tenn in equation (B 1.1.7) is needed. So we take note of some general principles for Franck-Condon factors (FCFs). 

The potential matrix elements are then obtained by making Taylor expansions around 00, using suitable zero-order diabatic potential energy functions,... 

If the solution of the zero-order Schiodinger equation [i.e., all teiins in (17) except V(r,Ro) are neglected] yields an/-fold degenerate electronic term, the degeneracy may be removed by the vibronic coupling tenns. If F) and T ) are the two degenerate wave functions, then the vibronic coupling constant... 

The sum of the properties of the diatomic species X and Y2 is the same as twice the property of XY. This is the zero-order approximation to additivity rules. 

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

As a naive or zero-order approximation, we can simply ignore the V12 term and allow the simplified Hamiltonian to operate on the Is orbital of the H atom. The result is... 

Interestingly, at very low concentrations of micellised Qi(DS)2, the rate of the reaction of 5.1a with 5.2 was observed to be zero-order in 5.1 a and only depending on the concentration of Cu(DS)2 and 5.2. This is akin to the turn-over and saturation kinetics exhibited by enzymes. The acceleration relative to the reaction in organic media in the absence of catalyst, also approaches enzyme-like magnitudes compared to the process in acetonitrile (Chapter 2), Cu(DS)2 micelles accelerate the Diels-Alder reaction between 5.1a and 5.2 by a factor of 1.8710 . This extremely high catalytic efficiency shows how a combination of a beneficial aqueous solvent effect, Lewis-acid catalysis and micellar catalysis can lead to tremendous accelerations. 

Relativistic density functional theory can be used for all electron calculations. Relativistic DFT can be formulated using the Pauli formula or the zero-order regular approximation (ZORA). ZORA calculations include only the zero-order term in a power series expansion of the Dirac equation. ZORA is generally regarded as the superior method. The Pauli method is known to be unreliable for very heavy elements, such as actinides. 

It may even be possible to adjust conditions such that measurements are made under pseudo-zero-order conditions where... 

The data shown in the following table were collected for a reaction known to follow pseudo-zero-order kinetics during the time in which the reaction was monitored. 

Eor a pseudo-zero-order reaction a plot of [A]( versus time should be linear with a slope of -k, and a y-intercept of [A]o (equation 13.8). A plot of the kinetic data is shown in figure 13.7. Linear regression gives an equation of... 

Since the reaction is carried out under conditions in which it is pseudo-zero-order in creatinine and OH , the rate constant, k, is... 

Use zero-order Markov statistics to evaluate the probability of isotactic, syndio-tactic, and heterotactic triads for the series of p values spaced at intervals of... 

Figure 7.9 Fractions of iso, syndio, and hetero triads as a function of p, calculated assuming zero-order Markov (Bernoulli) statistics in Example 7.7.

The sample labeled atactic in Fig. 7.10 was prepared by a free-radical mechanism and, hence, is expected to follow zero-order Markov statistics. As a test of this, we examine Fig. 7.9 to see whether the values of p, P, and Pj, which are given by the fractions in Table 7.9, agree with a single set of p values. When this is done, it is apparent that these proportions are consistent with this type... 

In the research described in the preceding problem, Randall was able to assign the five peaks associated with tetrads in the C-NMR spectrum on the basis of their relative intensities, assuming zero-order Markov (or Bernoulli) statistics with Pm = 0.575. The five tetrad intensities and their chemical shifts from TMS are as follows ... 

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