Propagation of double Rydberg wave packets

Double Rydberg wave packets for He electronic states are propagated in time using fully quantum mechanical calculations. The wave packets are constructed so that the two electrons are simultaneously excited up to nRyd ∼ 15 states and coupled to total orbital angular momentum equal to zero and to total spin equal to zero. We attempt to construct a wave packet to isolate symmetric stretch motion. Classical and quantum ideas are used to interpret several features of the time-dependent wave function. We briefly discuss some of the interesting problems that can be addressed. A recent area of research is the time development of electron wave packets in two or more spatial dimensions [1–14]. This interest has been sparked by the ability of experimentalists to initiate and measure the time-dependent behaviour of quantum systems and by the increase in computational power and numerical sophistication that allows calculations for complex systems. Another reason for this interest is that it is instructive to observe quantum systems in ways that are quite similar to analogous classical systems. In atomic physics, the motion of one electron wave packets has proved to be an interesting playground for enhancing our ideas about the flow of energy and probability through different degrees of freedom [9–14]. There have been a few steps in the next obvious direction in which two electrons participate in the wave packet motion in a nontrivial manner [1–8]. In the early studies, both electrons are excited above the ground state although at least one of the electrons has been restricted to a quite small distance, typically less than 5 Bohr radii. Thus, quantum effects completely dominate the behaviour of one of the electrons and it is not possible to establish a correspondence with classical dynamics. Very recently, Pisharody and Jones [8] have observed wave packet behaviour for two electrons when they can both be simultaneously at large distances. The results of this measurement were interpreted using classical trajectories and simplified quantum models. Although the simplified models reproduce the main features of the experiment, it is clearly time for a strong theoretical effort to perform fully quantum calculations of double Rydberg wave packets. In this paper, we show how it is possible to extend the range of motion that can be theoretically investigated by performing accurate and fully quantum calculations where both 0953-4075/05/020363+09$30.00 © 2005 IOP Publishing Ltd Printed in the UK S363 S364 F Robicheaux and R C Forrey electrons are simultaneously ∼300 Bohr radii from the nucleus. The wave packets are for electrons simultaneously in nRyd ∼ 15 states; both electrons are in the semiclassical limit and some correspondence with classical motion may be observed. For all the calculations presented in this paper, the angular momentum and spin of the two electrons are coupled together so that both the total orbital angular momentum and the total spin are zero. This reduces the 6-degrees of freedom to 3; but this is the only restriction and we attempt to account for the remaining three dimensions as accurately as possible using an efficient basis set expansion. The wave packets discussed here are true He wave packets and are not wave packets for a simplified model. The main result presented here is the knowledge that double Rydberg wave packets can be generated and investigated with quite modest resources; all the calculations for this paper were performed on a personal computer. We discuss the various possible choices for the computational techniques and the physics that determines which techniques might be best. The other main result pertains to one of the simplest investigations into the dynamics of two electron states. This investigation confirms some of our expectations but also demonstrates that many aspects of the time-dependent wave function can be understood at a qualitative level. The ability to obtain accurate time-dependent wavefunctions depends on how efficiently it can be represented and on the level of complexity of the wavefunction. Typically, the level of difficulty increases with the number of nodes in the wavefunction and with the number of spatial dimensions. There are two generic possibilities for representing the wavefunction: basis function techniques and a grid of spatial points. Basis functions can often represent the wavefunction with relatively few functions but sometimes the resulting representation of the Hamiltonian is dense; thus the number of Hamiltonian matrix elements scale with the square of the number of basis functions. A spatial grid of points usually gives a sparse representation of the Hamiltonian, but sometimes the number of points needed for an adequate description of the wavefunction is large. Wintgen and co-workers [15–17] showed that a Sturmian basis set in perimetric coordinates gave an extremely efficient representation of highly excited resonance states of He. The perimetric coordinates are q12 = r1 +r2−r12, q1 = −r1 +r2 +r12 and q2 = r1−r2 +r12. The basis functions are chosen to be yn12,n1,n2 = φn12(2βq12) [ φn1(βq1)φn2(βq2) + φn1(βq2)φn2(βq1) ]/√ 1 + δn1,n2 (1) with φn(x) = Ln(x) exp(−x/2) and where Ln(x) are the usual Laguerre polynomials. When the electrons’ spins are coupled to total spin 1, the + is replaced by −. The parameter β is a scale parameter which sets the size of the wavefunction. When trying to model the motion of two electrons both with principal quantum number nRyd, a decent choice for β is 2/nRyd. The symmetry of the functions means that only n1 n2 is needed in the basis set. Our basis set includes all functions with ω ≡ n12 + n1 + n2 ωmax. The number of basis functions is the nearest integer to ω3 max / 12 + 5ω2 max / 8 + 17ωmax/12 + 7/8. A very nice property of this basis set is that the Hamiltonian and overlap matrices are extremely sparse in this representation. Another advantageous property is that a complex scaling of the coordinates may be employed; thus, in principle it is not necessary to include the electron continuum states since any outgoing flux is àbsorbed’ in the complex plane. The resonance states have complex energies and their norm decreases exponentially with time. As discussed below, we could not take advantage of complex scaling in our time-dependent calculations. For the time-dependent wavefunctions, we found that the sparseness of the Hamiltonian and overlap matrix contributed greatly to the ability to increase the size of the basis set and to increase the speed of the calculation. We were thus able to obtain converged results for both electrons in nRyd = 16 states on a small PC with 250 Mb of RAM. The nondiagonal Propagation of double Rydberg wave packets S365 overlap matrix does not have a negative effect on the calculations presented here since we used an implicit method to propagate the wavefunction as discussed below; however, the overlap matrix effectively eliminates the possibility of using a direct time propagation of the wavefunction since it is necessary to invert the overlap matrix at each time step. The only unforeseen difficulty arose when we tried to use complex scaling to take care of the outgoing waves. There is more than enough energy for one of the electrons to escape the atom; these outgoing waves reflect from the edge of the basis set. Although complex scaling works perfectly well for time-independent calculations of resonance parameters, it works very poorly for this time-dependent problem. The reason is that unless |β| is larger than ∼1.7 there are always some eigenvalues of the time-independent problem that have positive imaginary parts which causes an exponential divergence of the wavefunction with time. Unfortunately, such a large value of |β| completely destroys the usefulness of the basis set for describing Rydberg states. Thus, we could not use complex scaling and were forced to use a mask at the edge of the basis set. The mask was a diagonal function of ω = n1 + n2 + n12 instead of a diagonal function of the distances. All the results presented here were tested for their dependence on the range and strength of the masking function. The wavefunction is expanded in a time-independent basis set. Letting N stand for the triple index n12, n1, n2 we write