Of the zeroth order

At this point it is worth comparing the different techniques of contrast enliancements discussed so far. They represent spatial filtering teclmiques which mostly affect the zeroth order dark field microscopy, which eliminates the zeroth order, the Schlieren method (not discussed here), which suppresses the zerotii order and one side band and, finally, phase contrast microscopy, where the phase of the zeroth order is shifted by nil and its intensity is attenuated. 

Furthennore, we obtain the equation of motion of the zeroth-order nuclear wave function ... 

The eigenfunctions of the zeroth-order Hamiltonian are written with energies. ground-state wavefunction is thus with energy Eg° To devise a scheme by Lch it is possible to gradually improve the eigenfunctions and eigenvalues of we write the true Hamiltonian as follows ... 

The sum of the zeroth-order and first-order energies thus corresponds to the Hartree-Fock energy (compare with Equation (2.110), which gives the equivalent result for a closed-shell system) ... 

The first-order energy eorreetion is given in terms of the zeroth-order (i.e., unperturbed) wavefunetion as ... 

The explieit form of the zeroth-order vibrational wavefunetions and energy levels, F j V and E j y, depends on the deseription used for the eleetronie potential energy surfaee... 

In experiments on the nitration of benzene in acetic acid, to which urea was added to remove nitrous acid (which anticatalyses nitration 4.3.1), the rate was found to be further depressed. The effect was ascribed to nitrate ions arising from the formation of urea nitrate. In the same way, urea depressed the rate of the zeroth-order nitration of mesitylene in sulpholan. ... 

The dependence of the zeroth-order rate constants on the concentration of acetyl nitrate is shown in fig. 5.1 in the absence of added acetic acid the rate increases according to the third power of the concentration of acetyl nitrate, but when acetic acid is added the dependence becomes... 

Nitrations of the zeroth order are maintained with much greater difficulty in solutions of acetyl nitrate in acetic anhydride than in solutions of nitric acid in inert organic solvents, as has already been mentioned. Thus, in the former solutions, the rates of nitration of mesi-tylene deviated towards a dependence on the first power of its concentration when this was < c. o-05-o-i mol 1 , whereas in nitration with nitric acid in sulpholan, zeroth-order kinetics could be observed in solutions containing as little as 10 mol 1 of mesitylene ( 3.2.1). 

The kinetic effect of increased pressure is also in agreement with the proposed mechanism. A pressure of 2000 atm increased the first-order rates of nitration of toluene in acetic acid at 20 °C and in nitromethane at 0 °C by a factor of about 2, and increased the rates of the zeroth-order nitrations of p-dichlorobenzene in nitromethane at 0 °C and of chlorobenzene and benzene in acetic acid at 0 °C by a factor of about 559. The products of the equilibrium (21a) have a smaller volume than the reactants and hence an increase in pressure speeds up the rate by increasing the formation of H2NO. Likewise, the heterolysis of the nitric acidium ion in equilibrium (22) and the reaction of the nitronium ion with the aromatic are processes both of which have a volume decrease, consequently the first-order reactions are also speeded up and to a greater extent than the zeroth-order reactions. 

Elements of the zeroth-order, inverse-propagator matrix are... 

One way in which we can solve the problem of propagating the wave function forward in time in the presence of the laser field is to utilize the above knowledge. In order to solve the time-dependent Schrodinger equation, we normally divide the time period into small time intervals. Within each of these intervals we assume that the electric field and the time-dependent interaction potential is constant. The matrix elements of the interaction potential in the basis of the zeroth-order eigenfunctions y i Vij = (t t T(e(t)) / ) are then evaluated and we can use an eigenvector routine to compute the eigenvectors, = S) ... 

If the small time interval in question extends from f to f + f then the matrix elements of the propagator for this short time interval may be written as a matrix in the basis of the zeroth-order eigenfunctions ... 

The perturbation operators must be modified in accordance with the new definition of the zeroth-order electron states ... 

Kinetic Parameters for the Symmetric Transition Taking Account of the Modulation of the Zeroth-Order Electron Densities (Beyond the Condon Approximation)... 

The physical mechanism of entirely nonadiabatic and partially adiabatic transitions is as follows. Due to the fluctuation of the medium polarization, the matching of the zeroth-order energies of the quantum subsystem (electrons and proton) of the initial and final states occurs. In this transitional configuration, q, the subbarrier transition of the proton from the initial potential well to the final one takes place followed by the relaxation of the polarization to the final equilibrium configuration. 

The quantum approaches of direct and indirect dampings, respectively by Rosch and Ratner [58] and by Boulil et al. [90], may be considered as the well-behaved extension of the zeroth-order approach of Marechal and Witkowski [7]. 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

Definition of the Zeroth Order Hamiltonian in Multiconfigurational Perturbation Theory (CASPT2). 

Here dt = 1/2 except for bending modes in linear molecules, for which dt = 1. The frequencies of the zeroth-order modes are given by... 

The value of P is important only with respect to technical considerations. The presence of finite orbital overlap between the initial and final states shghtly alters the p dependence of the relationship between Ft and A, The overlap corrections for the instanton analysis arise from the slightly different normalization of the symmetric and antisymmetric eigenstates, which can be obtained from the simple model in Fig. 2. The coefficients of the symmetric and antisymmetric eigenstates in terms of the zeroth order states, nominally + become instead... 

The explicit form of the zeroth-order vibrational wavefunctions and energy levels, FOj v and EQj v, depends on the description used for the electronic potential energy surface... 

In general, a given electron configuration has several wave functions, each corresponding to the same eigenvalue of the zeroth-order Ham-... 

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