Autoionizing doubly-excited states of 3Σg− symmetry of H2

We report R-matrix calculations of doubly-excited 3 − g states of molecular hydrogen corresponding to 3d̃ n̃ configurations. These states form Rydberg series converging to the 3d̃ series limit. They lie in the continuum of the doubly-excited states of 3 − g symmetry built on the 2p̃ ion core, and therefore they are autoionized. Calculations of resonance positions and widths are presented.


Introduction
It is well-established that doubly-excited states of H 2 play an important role in the dissociative recombination (DR) [1][2][3] of H + 2 .Such states are ideally described within the framework of multichannel quantum defect theory (MQDT), but in practice many MQDT studies have been hampered by the difficulty to determine the quantum defects directly from first principles, especially when the number of open channels is large and the adjustment of diagonal and off-diagonal quantum defect matrix elements based on experimental data becomes difficult or even impossible.During the last decade, we have been developing the so-called "halfium model" for two-electron molecular systems, which combines the variational R-matrix method with generalized multichannel quantum defect theory implemented in prolate spheroïdal coordinates [4].The method is designed for the determination of realistic molecular quantum defect matrices from first principles over a wide range of energies and molecular geometries, which should be directly applicable to the calculation of the rovibronic dynamics of higly excited molecular states (dissociative recombination, photoionization, photodissociation).Previous applications ranged from the double-minimum EF 1 + g state at low energy [5] to core-excited states of H 2 of − symmetry near 27 eV [6].
Notwithstanding any possible role of 3 − g doubly-excited states in the DR process, we are presently involved in a systematic investigation of electronic states of H 2 .Here we extend our previous study of 2 p n ˜ Rydberg series to higher energies above the 2 p threshold and up to 3 d threshold.In this EPJ Web of Conferences energy range, the core excited 3 d n ˜ 3 − g states may autoionize as a result of their coupling with 2 p ˜ continua that also have 3 −  g symmetry.After a brief sketch of the "halfium" formalism (Sect.2) in its updated v −symmetrized version (where v refers to the reflection of the electronic wave function at a plane containing the two nuclei), we present the results of a four-channel calculation that produces quantum defect matrices as functions of energy E and bond length R. Electronic energies and autoionization widths of (3 d )n d and (3 d )n g (n ≤ 8) doubly-excited states are reported.

R-matrix approach
The variational R-matrix theory of H 2 distinguishes between short-range and long-range interactions of the two electrons, which it treats separately by combining two complementary theoretical tools.Our approach, which at the outset was inspired by the work of Yoo and Greene [7], has been presented in detail in Refs.[4,6], and only a brief outline will be given here.
The electronic Schrödinger equation is solved first inside a finite ellipsoidal volume for fixed nuclei separated by the internuclear distance R. The variational two-electron basis functions y ij ( − → r 1 , − → r 2 ) are v -symmetrized combinations of functions y i p j m ( − → r 1 , − → r 2 ), which in turn are spin-symmetrized products of one-electron wave functions i ( − → r ).Thus one writes: (1) Here − → r 1 and − → r 2 are the position vectors of the two electrons, the indices i and j refer to the one-electron wave functions of H + 2 ion, and p, m = +/−.S and q refer, respectively, to the total electron spin and the symmetry of the electronic wave function when the symmetry operation v is applied.N ij and N ij are normalization factors.A general variational solution ( − → r 1 , − → r 2 ) of the two-electron problem inside the reaction volume is written in terms of the two-electron basis functions y ij ( − → r 1 , − → r 2 ) as [6,8]: where the index runs over the variational eigenchannels and the double index ij over the basis set.Each eigensolution is characterized by a stationary logarithmic derivative with respect to the radial coordiate on the reaction surface max( 1 , 2 ) = 0 .
Outside the reaction volume the wave function is developed as an expansion over products of target wave functions multiplied by radial functions of the outermost electron.The target wave functions k (E, ) are defined for each channel k which is represented explicitly in the asymptotic zone.They include in fact all coordinates except the radial coordinate of the outermost electron, that is, they also include the angular part of the latter, and for this reason are often referred to as "surface harmonics".The radial part of the external wave function in turn is written in scattering form, as a linear combination of regular and irregular two-center Coulomb radial functions f ( ) and g( ) for any given total energy E [9]: where c stands for the electron energy, within a given channel, with respect to the threshold.The two forms Eq. ( 3) and ( 4) of the solutions are matched on the reaction surface and thus yield the coefficients I k and J k and hence the short-range reaction matrix K = JI −1 .The quantum defect matrix l(E, R) is related to K according to K(E, R) = tan (E, R).Quite often quantum defects turn 04003-p.2DR2013 out to exhibit smooth variations both with the energy E and with the bond length R, as one might expect for quantities reflecting short-range interactions.
Once the reaction matrix K is known on an appropriate mesh of E and R values spanning the range of interest, the physical MQDT boundary conditions are applied in order to obtain observable quantities such as bound state positions, cross sections etc.The boundary conditions are cast in the form of a determinant whose zeros are to be found: where k is the asymptotic phase parameter appropriate for each channel k depending on the channel energy k : for k < 0 it is equal to the phase accumulated by the oscillating asymptotic radial function before it diverges as → ∞, whereas for k > 0 it is equal to the negative of the open-channel eigenphase, − .The N o eigenphases (with N o the number of open channels) are obtained by solving Eq. ( 5) with k (E) as stated.
A useful theoretical tool for the analysis of a continuum containing resonant structures is the state density N(E), which is defined by where the sum is taken over the N o open channels.In the case of a single closed core-excited Rydberg channel that is only weakly coupled to the continua, N (E) will consist of a Rydberg series of Breit-Wigner profiles centered on the positions of the core-excited Rydberg states, which are readily interpreted in terms of resonance positions E n and autoionization decay widths n .Conversely, when the channel interactions are strong and many channels are present, "overlapping" resonances occur, and the concepts of resonance position and width become blurred to some extent.Such a situation was encountered for instance in the u resonances of H 2 studied in an earlier publication [10].

Results
The continua (2 p ) ˜ of 3 − g symmetry present above the 2 p threshold correspond to odd ˜ , while core-excited Rydberg series of 3 −  g symmetry associated with the 3 d ion core state correspond to (3 d )n ˜ with even ˜ .In the present calculations we have explicitly included four channels in the asymptotic expansion of Eq. ( 4), namely two continua, (2 p ) p , and (2 p ) f and two core-excited Rydberg channels, (3 d )n d and (3 d )n g .Effectively we thus have included all contributing partial waves up to ˜ = 4.
The calculations have been carried out using a reaction volume parametrized as a function of the internuclear distance R (in a.u.) according to 0 = 1 + C/R [7], with the constant C = 20 a.u.chosen such that the energy of the confined core orbital 3 d deviates from the exact value of the free H + 2 ion by less than about 10 −3 a.u.over the whole range of 1 ≤ R ≤ 8 a.u.considered [4].The variational basis included about 200 "closed" functions (that vanish on the reaction boundary), with two "open-type" basis functions (whose derivative vanishes on the reaction boundary) added for each asymptotic channel.We ascertained in all calculations that the basis was well balanced including roughly equivalent contributions from each asymptotic channel.The quantum defect matrix was evaluated for typically 50 energies on a grid spanning the range between the 2 p and 3 d thresholds.n d and n g resonances appear as pairs with unequal widths, illustrating how for a given n value the higher partial wave autoionizes much more slowly than the lower one.The three R− values documented by Fig. 1, namely R = 4.0, 4.5 and 5.0 a.u., have been selected because they illustrate another interesting phenomenon: it may be seen that the (3 d ) 2 resonance approaches the 2 p ion threshold from above as R increases, and crosses into the discrete range near R = 5 a.u.This is characteristic of an antibonding configuration, and is analogous to the well-known case of the (2 p ) 2 1 + g resonance at much lower energy, which crosses the 1s ion threshold near R = 2.6 a.u. and eventually forms the outer well of the discrete EF  interaction, is a prototype example of a molecular Rydberg-valence interaction and induces a so-called "direct" DR process.

Potential energy curves
The resonance positions E n of the lowest ten doubly-excited states, (3 d )n d with n = 3 − 8 and (3 d )n g with n = 5 − 8, have been extracted from the state density plots and converted into potential energies in eV.They are shown in Fig. 2 as solid red lines.The 2 p and 3 d threshold curves are also represented as dashed black curves.They delimit the energy range of interest here, which varies from 29 to 45 eV.In addition, the doubly-excited state (2 p ) 2 , previously studied by Martin [11], Komasa [12] and ourselves [6] is shown as a dotted-dashed blue curve.This state has a minimum at R = 5.6 a.u. and may be regarded as a "second ground state" of H 2 .
The values of the energies and the effective quantum numbers of the doubly-excited states are given in Tables 1 and 2. As Fig. 2 shows, the potential curves run almost parallel to the 3 d ion potential, indicating that the effective principal quantum numbers vary little with R.This fact emerges clearly from the n * values shown in Table 2.The corresponding quantum defects are given by = n − n * and are in fact remarkably constant for n ≥ 4, as functions of both n and R. Inspection of Table 2 shows that the ˜ = 4 resonances have very small quantum defects as expected for a non-penetrating outer electron.On the other hand, the quantum defects for ˜ = 2 are rather large for a d electron in a system with just a few electrons, ≈ 0.25 ± 0.04 for n ≥ 4, and even ≈ 0.55 for the (3 d ) 2 configuration.This behavior reflects the fact that the n d orbitals are penetrating in the present situation because they have a precursor in the core, 3 d (which is a core orbital of the same symmetry as the series of Rydberg orbitals under consideration).The jump of the quantum defect by a factor of two between n = 3 and 4 is related to the fact that for n = 3 one has a (3 d ) 2 configuration which is only formally of Rydberg type.Finally, the weak R− dependences of all these series are in sharp contrast with the strong dependences generally found in Rydberg series associated with ground state H + 2 .This in turn appears to be related to the fact that the shape of the core orbital 3 d changes rather little over the range of R studied here, as compared to the 1s orbital, cf. for instance the contour plots given in the paper by Bates, Ledsham and Stewart [13] (their Figs. 1 and 8).

Autoionization widths
Table 3 lists the autoionization widths for the (3 d )n d and (3 d )n g doubly-excited Rydberg series as extracted from the state densities N (E).The widths of the d resonances are seen to be several orders of magnitude larger that those of the g resonances.This is indeed what one qualitatively expects based on the fact that the Rydberg electron with higher ˜ penetrates less into the 3 d ion core.Table 3 further shows that the widths decrease with n -one expects them to decrease roughly proportional to (n * ) −3 -and they increase quite significantly with R, in line for what is known in the case of the lower core excited resonances (see e.g.[5], in particular Figs.5b and 7b).

Figure 1
Figure 1 displays the state density N (E) obtained for the range between the 2 p and 3 d thresholds and for three different R−values as functions of the energy.It may be seen that the (3 d ) 2 and (3 d )4 d resonances appear well isolated at the low energy end of the spectrum, whereas starting with n = 5 the 04003-p.3

Figure 1 .
Figure 1.State density plot of (3 d )n d and (3 d )n g doubly-excited 3 − g states of H 2 for internuclear distances (a) R = 4.0 a.u., (b) R = 4.5 a.u. and (c) R = 5.0 a.u.).Energies, in Rydbergs, refer to the 3 d threshold.The 2 p and 3 d H + 2 ion thresholds are indicated by dashed blue lines.

Table 1 .
Potential energy curves of doubly-excited 3 − g states in H 2 : the (3 d )n d and (3 d )n g autoionizing resonance series are indicated by full red lines.The H + 2 thresholds 2 p , 2 p and 3 d are represented by dashed black lines.The lowest state of 3 − g symmetry, (2 p ) 2 (from [6]), is indicated by the dashed-dotted blue line.The energy scale, in eV, refers to the X 1 + g ground state minimum.Electronic energies (in Rydbergs) of doubly- 1+ g double minimum state of H 2 .This latter crossing, and the associated configuration Figure 2.

Table 2 .
Effective principal quantum numbers n * of the (3 d )n d and (3 d )n g resonance series.