Quantum defect functions

For small nonpolar species such as H2 and N2 the dominant interaction between the Rydberg electron and the nuclear vibrational and rotational motion occurs within a small radius around the ionic core, which is traversed in a fraction of a femtosecond. This short encounter justifies the sudden treatment of vibration and rotation in MQDT theory, while also permitting Bom-Oppenheimer estimates of the necessary quantum defect functions. It is also central to the n-3 scaling law because the core transit time is almost energy independent, while the Rydberg orbit time increases as n3. 

Ross and Jungen (1999) have used differences between ab initio ion-core and Rydberg state potential curves to determine quantum defect functions. Thus the independent quantum defect functions associated with the atom-like Hel(°) and the molecular H 1) are, respectively, ai(R) and ai (R). 

Formulae for calculating transition probabilities in both the LS and Jcl coupling schemes, within the context of the Relativistic Quantum Defect Orbital (RQDO) formalism, which yields one-electron functions, are given and applied to the complex atomic system Ar 11. The application of a given coupling scheme to the different energy levels dealt with is justified. 

We shall now discuss this matrix element in more detail. Several approximations—namely the quantum defect model (e.g., Bebb, 1969), the use of a S function (Lucovsky, 1965), or of a Coulomb potential for the impurity— have long been well known and have been reviewed earlier (e.g., Milnes, 1973 Stoneham, 1975). Here we emphasize various recent developments. First, we present some work that analyzes the cross section in terms of the contributing matrix elements and of their symmetry. Results from this are important, since they show certain aspects to be relatively independent of the details of the deep state. Subsequently we give other recent papers, some of which consider various modifications to the impurity wave functions, others to the band wave functions. 

The 3-photon ionization photoelectron spectrum, plotted as the photoelectron signal intensity as a function of the binding energy of the Rydberg states, is shown in Fig. 2. Three series of Rydberg peaks, with quantum defects of 0.93, 0.76, and 0.15 are obtained. From the quantum... 

Ch. Jungen The vibronic coupling is included through the R dependence of the diagonal and off-diagonal quantum defect matrix elements. The effective principal quantum number, or more precisely the quantum defect, gives a handle on the electronic wave function. The variation with R then contains the information concerning the derivative with respect to R of the electronic wave function. 

Johnson and Rice used an LCAO continuum orbital constructed of atomic phase-shifted coulomb functions. Such an orbital displays all of the aforementioned properties, and has only one obvious deficiency— because of large interatomic overlap, the wavefunction does not vanish at each of the nuclei of the molecule. Use of the LCAO representation of the wavefunction is equivalent to picturing the molecule as composed of individual atoms which act as independent scattering centers. However, all the overall molecular symmetry properties are accounted for, and interference effects are explicitly treated. Correlation effects appear through an assigned effective nuclear charge and corresponding quantum defects of the atomic functions. 

There are two parameters in the atomic coulomb functions, the effective nuclear charge and the quantum defect. The values of these were taken by Johnson and Rice from available spectral data. The effective atomic charge was adjusted to give the correct ionization potential of the molecule, 9.25 eV, requiring thereby z = 0.8243. The quantum defects of carbon were taken from the appropriate atomic series and were 1.04 for the 5-state and 0.73 for the p-states. It is interesting to compare the calculated molecular quantum defects (i.e., those corresponding to the Johnson-Rice LCAO function) with those which can be obtained from the various benzene Rydberg series.218 The asymptotic form of the elu orbital constructed from s atomic functions is... 

Fig. 4.2 (a) Radial matrix element of the K 4s—>n p transition with n a continous variable. The quantum defect of the 4s state is 2.23. The quantum defects of the np states are 1.71, so the np states fall at the locations shown by the arrows near where the matrix element crosses zero, (b) H 2s-n p radial matrix element as a function of n. Note that the maximum amplitudes of the matrix element occur at integer values of n. ... 

Fig. 23.10 Quantum defects of the Ba msns and msnd states as a function of the core principal quantum number m (from ref. 36).

Solutions of this equation are related to the Kummer functions (21). The parameter 6, is the quantum defect and c is an integer chosen to ensure the normalizability of the orbitals and their correct nodal pattern -the number of radial nodes is equal to n-l-c-1. 

Recapitulation will be discussed in more detail in section 3.4. As n increases, the number of nodes also increases and, as n — oo, the inner nodes coincide with those of a continuum functions. In fact, the positions of the nodes determine the phase of the continuum function (which is oscillatory) at threshold. There is a simple relation between the phase shift above threshold and the quantum defect of the bound states, which will be explained in chapter 3. If the eigenfunctions recapitulate, i.e. the positions of the nodes are nearly constant, then it follows that the... 

If such a wavepacket were formed in H, then the wavepacket would remain intact, with a fixed orientation in space, until some incoherent process (either spontaneous emission (see chapter 4) or collisions, discussed above) destroys the coherence. This arises because conservation of angular momentum for the excited electron applies strictly in this case. However, the experiment is performed in an alkali atom, which possesses a core, and there is a back reaction of the excited electron on this core (core polarisation), which depends on the degree of penetration of the excited electron into the core, i.e. on the quantum defect, which itself is a function of the angular momentum. Thus, the wavepacket precesses under the influence of a small potential due to the quantum defect of the alkali. It is found to follow a classical trajectory determined by the core polarisation potential. 

Recapitulation is an important property because it provides us with an immediate interpretation of the physical meaning of the quantum defect p for large enough n, the bound state wavefunctions possess an oscillatory inner part which defines a phase, and is nearly independent of energy if p is nearly constant in energy. A change in the value of p corresponds to a shift in the radial position of all the nodes. As one tends to the series limit, the oscillatory part grows. Continuum functions, of course, become... 

In molecules, starting from a case (d) basis set, the part of the electronic energy that must be added corresponds to the energy difference between the states of different A due to the different Coulomb and exchange interactions with the core [see Section 3.3.2], a difference which can be expressed, like the example shown by Herzberg and Jungen (1972) for a p-complex, as a function of the difference between the quantum defects of a and 7r. 

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