Sturmian method

The most straightforward way to solve the time dependent Schrodinger equation (6.2.1) with the Hamiltonian (6.2.2) is to expand the wave function I t)) in a complete set of square normalizable states. This method 

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 are composed of the matrix elements of the potential energy 1/rc and the kinetic energy — /d/jp . For the potential matrix elements we obtain 

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 

Although it is, in principle, possible to evaluate (6.2.11) analytically, it is numerically more advantageous to evaluate the integral in (6.2.11) with the help of a suitable Gauss-Laguerre integration formula (see, e.g., Stroud and Secrest (1966)). 

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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. 

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... 

Fig. 6.8. Ionization rates computed with the Sturmian method for three different Sturmian labels, (a) a = 3, (b) a = 4, (c) a = 5.

Can the Generalized Sturmian Method be applied to /V-electron molecules This is another question that we are starting to explore. Let us consider a single electron... 

In the preceding discussion, the creation and annihilation operators always referred to a set of orthonormal spin-orbitals. We can ask how much of the formalism holds in cases where complete orthonormality can no longer be assumed, for example, in valence-bond calculations or in the generalized Sturmian method. 

The generalized Sturmian method for solving the iV-electron Schrodinger equation [11 -20] is a direct configuration interaction method, using basis functions which are antisymmetrized isoenergetic solutions of the approximate iV-electron equation... 

The examples discussed here show that even in the case where the generalized Sturmian method is applied to atoms, a case in which radial orthonormality between different configurations is sometimes lost, simplified formulas for the matrix elements can often be derived. However, even when this is not possible, the... 

The generalized Sturmian method [1-22] for solving the Schrodinger equation of an /V-particle system is a direct configuration interaction method, in which the configurations are chosen to be isoenergetic solutions to the approximate many-particle Schrodinger equation... 

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. 

Having used the generalized Sturmian method to calculate the wave function for an A-electron atom, we are in a position to derive both the corresponding density distribution and the first-order density matrix [36-52], However, because we cannot assume orthonormality between the one-electron spin-orbitals of different configurations, it is necessary to use expressions analogous to the generalized Slater-Condon rules. If we let... 

The generalized Sturmian method is particularly well suited for calculating the spectra of few-electron atoms in strong external fields. In the case of a uniform electric field, the additional term in the potential (in atomic units) is... 

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