Zeroth-order approximation states

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch.(7) In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Bom-Oppenheimer states. Henry and Kasha(8) and Jortner and co-workers(9-12) have pointed out that the Bom-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

Figure 10. Low-energy vibronic levels in the X2II state of HCCS computed in various approximations [152]. Hq zeroth-order approximation (both vibronic and spin-orbit couplings neglected). Hi. vibronic coupling taken into account, spin-orbit interaction neglected. Hi + Hs0 both vibronic and spin-orbit couplings taken into account. Solid horizontal lines K = 0 vibronic levels dashed line K — 1 dash-dotted lines K = 2 dotted lines K — 3. Values of the quantum numbers V4, N of the basis functions dominating the vibronic wave function of the level in question are indicated. Approximate correlation of vibronic states computed in various approximations is indicated by thin lines. In all cases the stretching quantum numbers are assumed to be zero.

The first order perturbation energy is calculated by the following zeroth-order approximation to the ground-state wave function P° of the two molecules ... 

A remarkable number of organic compounds luminesce when subjected to consecutive oxidation-reduction (or reduction-oxidation) in aprotic solvents1-17 under conditions where anion radicals are oxidized or cation radicals are reduced. In many instances, the emission is identical with that of the normal solution fluorescence of the compound employed. In these instances the redox process has served to produce neutral molecules in an excited electronic state. These consecutive processes which result in emission are not special examples of oxidative chemiluminescence, but are more properly classified as electron transfer luminescence in solution since the sequence oxidation-reduction can be as effective as reduction-oxidation.8,10,12 A simple molecular orbital diagram, although it is a zeroth-order approximation of what might be involved under some conditions, provides a useful starting... 

Just as the expansion in the zeroth-order states can describe the exact molecular eigenstates, likewise an expansion in the exact states can be used to prepare, for a short time, a zeroth-order state. If the perturbation V is small, and the model Hamiltonian Ho is a good approximation to //, then the initially prepared superposition of eigenstates will resemble a zeroth-order state. The dephasing of the exact molecular eigenstates in the wave packet superposition subsequently leads to an evolution of the initial zeroth-order electronic character, transforming into a different zeroth-order electronic state as a function of time. 

In the zeroth order approximation, the ground state is given by... 

The second problem centers about the use of an approximate ground-state wave function that eminates from a multiconfigurational zeroth-order approximation. The N2 calculations in Section III.C suggest that the restriction to a single configuration zeroth-order ground state imposes a fundamental limitation on the quality of the calculated EOM ionization potentials for that system. 

Studies of Alcohol Oxidation. When the reaction was investigated from the direction of alcohol oxidation under pre-steady-state conditions in the presence of IBA, the time-resolved spectra obtained from RSSF measurements again show evidence for the formation of a transient intermediate in the NAD -mediated oxidation of benzyl alcohol.Data collected at pH values of 9.0, 5.6, and 4.8 are shown in Figs. 7 and 8. In the wavelength region 300 to 450 nm and at pH 9.0, the time-resolved spectra are characterized by a fast, pre-steady-state (exponential) phase dominated by the appearance of bound NADH. This process is followed by an approximately zeroth-order (steady-state) phase in which free NADH is generated by multiple turnovers. The difference spectra in Fig. 7C,D compare the changes which occur in the pre-steady-state phase with those in the steady... 

MP perturbation theory is generally used only for the ground state energy. The zeroth-order approximation for the wavefunction is taken to be the lowest-energy HF determinant. Nevertheless, the function E z) obtained from this perturbation series, when considered as a function over the complex z-plane, contains within it the full energy spectrum of eigenstates with the same symmetry as the ground state. This is the consequence of a theorem presented by Katz [14] ... 

A liquid state theory has been developed on the basis of an ideal liquid, which is a hard-sphere liquid. Usually, thus, a random disordered structure of liquid has been assumed. This is the basis for the description of liquid by the two-body density correlator, or the radial distribution function g r). Recent studies indicate this picture is not sufficient even for a hard-sphere liquid [46,47], The assumption of a disorder structure of a liquid is always correct as the zeroth order approximation. However, we believe that a physical description beyond this is prerequisite for understanding unsolved fundamental problems in a liquid state, which include thermodynamic and kinetic anomalies of water type liquids, liquid-liquid transition, liquid glass transition, and crystal nucleation. 

Here C ° and E denote a zeroth-order approximation for the quasi-particle states. In our Si calculation this zeroth-order approximation was extracted from an empirically fitted pseudopotential band-structure (see ref.4 and 35). This bandstructure is fitted in terms of a fourth-nearest neighbor (in the fcc-lattice sites) overlap model of bonding and antibond ng orbitals as described n our earlier work on optical properties and impurity screening. Also the calculation of the two-particle Green s function is based on this bandstructure and follows closely the impurity studies (for details see in particular, ref.35). 

In our previous diamond work we determined the bare HF part of E by making a Slater-Koster fit to an existing HF band calculation, while the correlation part was determined, as in the present Si work, by evaluating the matrix elements in an explicit basis set, which represents a zeroth-order approximation to the actual quasi-particle states. 

Zeroth-order approximations to higher excited states in H2 may be obtained from linear combinations of higher-energy hydrogen atom AOs, subject to the symmetry and normalization constraints of Eqs. 4.20 and 4.22. Like the states +> formed from the Is AOs, the higher lying LCAOs yield inaccurate trial energies—but their nodal patterns do furnish useful illustrations of the... 

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