Zero-order reference

Of course, in the actual computation, one first has to go to all orders to establish the exact Xco, of Eq. (4) from which Eq. (7) follows. Nevertheless, in practice, for atomic and molecular ground states where the shell model holds well and the zero-order reference is a closed-shell, single deferminan-tal HE wavefunction, it remains true that the dominant contribution to the Ecorr comes from the double excitations (electron pair correlations), although for a given state, the exact magnitude of each term of Eq. (4) depends on the computational method and on the function spaces that are used. 

Silverstone and Sinanoglu [77] wrote the cluster expansion of fhe nonrela-f ivisfic N-elecf ron eigenfunction in terms of a zero-order reference wavefunc-fion fhaf is multicmfigurational, in accordance wifh the earlier suggestion of Wafson [31] and the study of H-ND in Be [75, 76]. In their formalism, the one-, two-, three-, etc. correlation functions (i.e., the virtual electron-excitations in the language of Cl) are linked fo spin orbitals from an extended zero-order set of occupied and unoccupied spin orbitals. This set was named the Hartree-Fock sea (H-F sea). Optimally, the H-F sea spin orbitals are supposed to be computed self-consistently. 

For the multitude of cases in quantum chemistry in which the omnipresence of interelectronic interactions render the use of one-electron models inadequate, simple consideration of the formalism and the meaning of perturbation theory that is based on well-defined zero-order reference wave-functions indicates that not all terms play the same role as regards their contribution to the eigenfunction of each state. Therefore, depending on the problem under consideration, it may be possible, to a good and practical approximation, to partition the total wavefunction in such a way to... 

The average value of the dipole moment will be calculated by means of Dirac s perturbation theory for nonstationary. states, up to third order the zero order refers to the free molecules in the absence of the field. Let the wave function of the system of the two interacting molecules in- the external field be specified by y, an eigenfunction of the total Hamiltonian H. This wave function y> may be expanded in a complete set of the energy eigenfunctions unperturbed system the index n labels the various unperturbed eigenstates characterized by the energy En. We may then write... 

The second type of absorption model that is commonly applied is zero-order absorption. Zero-order refers to the case where the absorption rate is... 

This is the zero-order (reference) multiconfigurational description, with configurations consisting of the self-consistently optimized "Fermi-Sea" (FS) orbitals [40-42,131]. In other words. 

The basic concept behind MBPT is to model the effects of electron correlation by treating them as a perturbation of a zero-order reference wavefunction, Pq- To this end, the correlated wavefunction Tq is expanded in a series, Tq -i- p(i) + p(2) + p(3) +. . . where the superscripts in parentheses indicate the order of the perturbative expansion. Solving the resulting Schrodinger equation yields energies, Eq + -I- -i- > + . ... 

Nondynamical Correlation Quality of the Zero-order Reference... 

Sink. A flux (S) of material out of a reservoir -very often this flux is assumed to be proportional to the content of the reservoir (S = kM). In such cases the sink flux is referred to as a first-order process. If the sink flux is constant, independent on the reservoir content, the process is of zero order. Higher-order fluxes, i.e. S = kM with a > 1, also occur. 

Refers to initial, inlet, external. or zero order... 

If a reaction that must be investigated follows a reaction sequence as in Scheme 10.1, and if the reaction order for the substrate equals unity, it means that (with reference to Eq. (4 b)), the observed rate constant (k0bs) is a complex term. Without further information, a conclusion about the single constants k2 and fCM is not possible. Conversely, from the limiting case of a zero-order reaction, the Michaelis constant cannot be determined for the substrate. For particular questions such as the reliable comparison of activity of various catalytic systems, however, both parameters are necessary. If they are not known, the comparison of catalyst activities for given experimental conditions can produce totally false results. This problem is described in more detail for an example of asymmetric hydrogenation (see below). 

The subscript "s" refers to experimental values. The plot of the self-heat rate as a function of the reciprocal temperature (at the start of the reaction) may result in a straight line with a slope of Ea/R which is the zero-order line. 

FIGURE 3.1 7. Section of the zero-order ( ) and first-order (-) potential energy surfaces along the reaction coordinate in cases where stretching of the cleaving bond is the dominant factor of nuclei reorganization. Adapted from Figure 1 of reference 24a, with permission from the American Chemical Society. 

Referring to reactions in which the reaction velocity is independent of the reactant under consideration. For example, for the reaction A + B C, if the empirical rate expression is v = A [B], the reaction is first order with respect to B but zero order with respect to A. See Chemical Kinetics Rate Saturation Michaelis-Menten Equation... 

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