Zeroth-order energy

In all methods, the first-order interaetion energy is just the differenee between the expeetation value of the system Hamiltonian for the antisyimnetrized produet fiinetion and the zeroth-order energy... 

The zeroth-order vibronic wave function is ut- Uy 0 0-I-). The zeroth-order energy is... 

The zeroth-order energy level is twofold degenerate. The corresponding vibronic basis functions are ur K+2 0 0 —) = 11) and luj- A"—2 0 0 +) = 2). The first-order energy correction is... 

In other cases, the zeroth-order vibronic levels are generally more than twofold degenerate and the perturbative handling is much more complicated. An exception is the case 07= , U( =li K = 0 with the twofold degenerate zeroth-order level. The basis functions are 1 1 1 1 —) = 1) and 1 —1 1 —l- -)s 2). The zeroth-order energy is... 

The system to be treated consists of two nuclei A and B and two electrons 1 and 2. In the unperturbed state two hydrogen atoms are assumed, so that the zeroth-order energy is 2WH-If the first electron is attached to nucleus A and the second to nucleus B, the zeroth-order eigenfunction is ip (1)

[Pg.48]

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. 

This matrix is diagonal in the zero-field interaction, so the zeroth-order energy levels can be directly seen to be... 

The zeroth-order energy level is >j K 100 ) = 14) twofold degenerate—... 

This quasidegeneracy is the analogue of the classical condition of the intersection of the resonance surface (3.27) with the constant (zeroth-order) energy surface. 

In order to define Hq such that 4tsoAGP is an eigenfunction, Hq was split into a sum of operators Hk- These operators Hk are defined so that HkSk = Ef gx (Eq. (14)). Then the zeroth-order energy is merely W = H Hq. 

Using (4.39) for the zeroth-order energies, and including only the (aq + cq3) part of H we have... 

The first term in this expression is the zeroth order energy E(0,0). The two next terms gives the first derivatives with respect to the parameters Ty (equation (3 37)) and SK0 (equation (3 42)). We obtain the following result for the Erst derivative with respect to the orbital rotation parameters ... 

Zeroth-order energies and wavefunctions may be obtained by neglecting the coupling between different j- and different 17-states and solving the... 

Fig. 26. Spectral function A4p(E) (Eq. (59)) for 4SPd to 54Xe. Arrows and thresholds refer to JSCF plus Fermi sea correlation energies and represent zeroth-order energies entering the calculation of the spectral function. — frozen 4d2 ion -------relaxed 4d2 ion (see text)...

Complex adiabatic energies (a) compared to resonance energies (b) for a model of a closed and an open channel described in reference (30). The zeroth order energy has been subtracted out. Unit cm , Interchannel coupling 200 cm . Crossing diabatic energy equal to the energy of the level with v = 18. 

For the latter two representations, an approximate solution of Eq. (3) may be obtained by assuming that the difference in zeroth-order energy between the metal orbitals and the ligand valence orbitals. Hum - Hll, is large while the interaction between them is relatively small, or, Gml 0 and Hml (H m Hll)-... 

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