Sturmian state

Because of the simple analytical structure of the Sturmian states (6.2.3) the matrix elements of Ho and x can be computed analytically. For the... 

The matrix elements known, the set (6.2.7) can now be integrated with any of the standard methods for the numerical solution of coupled differential equations (see, e.g., Milne (1970)). In order to be able to interpret the results obtained by the Sturmian method in the physical space of SSE states, we need the overlaps between the Sturmian states (6.2.3) and the SSE states (6.1.24). The result is... 

First, we try the Sturmian method outUned in Section 6.2.1. The computations are performed in a basis of 50 Sturmian states as defined in (6.2.3) for three different choices of the Sturmian label, a = 3,4,5. The resulting ionization probabiUties after 100 cycles of the field, P (IOO), are shown in Figs. 6.8(a) - (c) for the three different Stm-mian labels, respectively. Our first impression is that the results depend strongly on the choice of the Sturmian label. This dependence can in principle be reduced, but only at the cost of increasing the basis size substantially. 

Basis sets of the type discussed in this paper can only be applied to bound-state problems. It is interesting to ask whether it might be possible to constmct many-electron Sturmian basis sets appropriate for problems in reactive scattering in an analogous way, using hydrogenlike continuum functions as building-blocks. We hope to explore this question in future publications. 

For systems interacting through Coulomb forces, as defined by equation (19), is independent of po. Notice that the eigenvalues of the Sturmian secular equation (20) are not values of the energy but values of the parameter po, which is related to the binding energy of bound states by equation (2). 

Table 1 shows analogous equations for po for the ground states of higher isoelec-tronic series, derived in the crude approximation where only one many-electron Sturmian basis function is used. Figure 1 shows the dementi s values [10] for the Hartree-Fock ground state energies of the 6-electron isoelectronic series... 

Figure 1 This figure shows the ground-state energies of the 6-electron iso-electronic series of atoms and ions, C, iV, 0 +, etc., as a function of the atomic number, Z. The energies in Hartrees, calculated in the crudest approximation, with only one 6-electron Sturmian basis function (as in Table 1), are represented by the smooth curve, while dementi s Hartree-Fock values [10] are indicated by dots. 

Tables 2, 3 and 4 show the first few excitation energies for the ions and again calculated in the crudes approximation Only one many-electron Sturmian basis function is used for the ground state, and only one for the excited state. As can be seen from the tables, where the experimental values [13] are also listed, even this very crude approximation gives reasonable results.

A new development is the notion of Molecular Sturmians, recently published /6/. It is stated there that ... 

We now use the method defined in (6.2.58) to compute microwave ionization probabiUties for the same field strength and frequencies as were used above in connection with the Sturmian method. Using the exact decay rates A = 7re g /2 according to (6.2.48) and (6.2.49), and retaining only the first five SSE states in (6.2.58), we obtain the ionization probabilities shown in Fig. 6.10. They compare favourably with the probabiUties... 

We choose 2 = 2 so that io( ) is the radial orbital uio(r) of the ground state (4.27). The radial orbitals UnAA are known as the Sturmians (Roten-berg, 1962). They are sometimes called pseudostates . 

The configuration-interaction representation of the lower-energy states of an atom is the IV-electron analogue of the Sturmians in the hydrogen-atom problem. We choose an orbital basis of dimension P, form from them a subset of all possible A/ -electron determinants pk),k = 0,Mp, and use these determinants as a basis for diagonalising the IV-electron Hamiltonian. It may be convenient first to form symmetry configurations kfe) from the pfe). 

The total cross section for exciting discrete target states is subtracted from it. This is obtained by projecting the Sturmians n) onto exact target eigenstates n). 

Apart from these one-particle effects , additional complications arise in the case of two electrons, which may dissolve from a bound state to the positronic and electronic continua. A mathematically rigorous approach to avoid this so-called continuum dissolution is to use as basis functions the relativistic Coulomb Sturmians... 

Because of the scaling factor pK, which is different for each state, the Sturmian basis functions adjust in scale automatically For tightly bound states they are contracted, while for highly excited states they are diffuse. 

Fig. 2 This figure shows the Sturmian molecular orbital corresponding to the ground state of the Hj ion, withS = 6, k = 1.16885, and/J = 5.13325 Bohrs...

In the present paper, we shall discuss a method for generating many-electron states of a given symmetry using Kramers pair creation operators and other symmetry-preserving pair creation and annihilation operators. We will first develop the formalism for the case where orthonormality between the orbitals of different configurations can be assumed. Afterwards we will extend the method to cases where this orthonormality is lost, so that the method also can be used in generalized Sturmian calculations [11-13] and in valence bond calculations. 

In making this table, the basis set used consisted of 63 generalized Sturmians. Singlet and triplet states were calculated simultaneously, 0.5 s of 499 MHz Intel Pentium III time being required for the calculation of 154 states. Experimental values are taken from the NIST tables (http //physics.nist.gov/asd). Discrepancies between calculated and experimental energies for the ions may be due to experimental inaccuracies, since, for an isoelectronic series, the accuracy of the generalized Sturmian method increases with increasing atomic number. 

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