Complex energies and the polyelectronic Stark problem: II. The Lin= 4 levels for weak and strong fields

Abstract

We computed nonperturbatively, via the state-specific matrix complex eigenvalue Schrödinger equation (CESE) theory, the energy shifts and widths of all ten Li n = 4 levels which are produced by electric fields of strength, F , in the range 0.0-0.0012 au (6.17 × 10 6 V cm −1 ). By establishing the nature of the state vector to which every physically relevant complex eigenvalue corresponds, we delineated the field strength region where it is possible to characterize the perturbed levels in terms of small superpositions of unperturbed Li states (called here the weak field region) from the region where the mixing of the discrete and the continuous states renders such an identification impossible (the strong field region). For both regions, systematic and accurate CESE calculations have produced a complete adiabatic spectrum of perturbed energies. It turns out that the behaviour of the energies of the m = ±1, ±2 and ±3 levels is smooth. In contrast, the Stark spectrum of the m = 0 levels contains not only all the n = 4 levels but also parts of the n = 3 and 5 manifolds, and is characterized by regions of crossing as well as of avoided crossing of the real part of the complex energies as a function of F .In the former case, the widths of the crossing (diabatic) levels differ considerably-even by two orders of magnitude. In the latter case, there are abrupt changes in the widths, a result of significant changes in the perturbed wavefunctions. For weak fields, which, for example, for the m = 0, n = 4 levels correspond to values up to about 2 × 10 −4 au, the one-electron semiclassical Ammosov, Delone and Krainov (ADK) formula for the widths produces reasonable trends. However, when the wavefunction mixing increases with field strength, it fails completely.For the four m = 0 levels, comparison is made with the results of Sahoo and Ho, obtained nonperturbatively by applying the complex absorbing potential (CAP) method. For a range of relatively weak fields, the CAP results for the widths do not follow a physically meaningful curve and deviate from

Time asymmetry, nonexponential decay, and complex eigenvalues in the theory and computation of resonance states

“…(12)], the original CESE is transformed into a non-Hermitian complex eigenvalue problem with a two-part square-integrable eigenfunction whose first part, ⌿ 0 , is real and the second part is complex. The convenience and efficacy of this form for the high accuracy calculation of resonance states have been verified repeatedly, [5,7,9,10,[23][24][25] and references therein]. In fact, because of the explicit form of the coefficient in Eq.…”

“…It is then necessary to introduce the notion of wavepacket localization inside the continuum at t ϭ 0, which becomes a point of discontinuity in the time domain, and to emphasize the importance of an accurate calculation of a corresponding square-integrable wavefunction, ⌿ 0 . This requirement has been used as a cornerstone for advanced theory and computation of resonance (autoionizing) states in polyelectronic atoms, through the solution of statespecific CESEs [5,7,9,10,[23][24][25], as well as through the solution of the TDSE [4 -6, 26, 27]. Starting from this property of ⌿ 0 , three results that are connected to the issues quoted above emerge:…”

“…However, it is possible to implement manageable many-body nonperturbative methods by first converting the problem into a non-Hermitian one with L 2 real and complex function spaces via suitable procedures of regularization and analytic continuation [5, 7, 9, 10, 23-25, 28, 29 and references therein]. Of critical value to this approach is the fact that ⌿ 0 and E 0 , where most of the intelectronic interactions occur, are computed once, on the real axis [5,7,9,10,[23][24][25]. The form of the Hamiltonian operator does not change as it does, for example, in the complex coordinate rotation (CCR) method, where the coordinates in the Hamiltonian operators are made complex via the scaling r 3 ϭ re i .…”

Transition metal and nitrogen doped carbon nanostructures

“…The zero-field 186-th MO, depicted in Fig. 12, moves the fastest, at a rate of E ≈ 0.7 eV per V/nm, which is close to that of the 4s Li-orbital in such fields, equal to E ≈ 1.1 eV per V/nm [152]. These MOs eventually reach the HOMO and become populated.…”

“…In the fields of several V/nm, these extended MOs could have field-emission lifetimes of tens of fs, in analogy to other 4-6s atomic orbitals [152]. On the other hand, CNTs in the field of E = 3 V/nm have field-emission currents of j ≈ 1 nA [150], i.e.…”