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Adam
Kirrander
*^{a},
Christian
Jungen
^{b},
Robert J.
Donovan
^{a} and
Kenneth P.
Lawley
^{a}
^{a}EaStCHEM, School of Chemistry, University of Edinburgh, David Brewster Road, EH9 3FJ Edinburgh, UK. E-mail: Adam.Kirrander@ed.ac.uk; Tel: +44 (0)131 6504716
^{b}Department of Physics and Astronomy, University College London, London WC1E 6BT, UK

Received
21st May 2018
, Accepted 21st June 2018

First published on 21st June 2018

Extremely large vibrational amplitude (≈8700 a.u.) heavy Rydberg levels in the H^{1}Σ^{+}_{g} state, located only 25 cm^{−1} below the ion-pair dissociation limit, are reported. The calculations are done using a hybrid log derivative/multichannel quantum defect approach that accounts for predissociation and is capable of dealing with any number of long-range closed channels, and of providing positions and widths for the heavy Rydberg resonances. In this case, resonance positions can be reproduced qualitatively using a simple diabatic model (however, the resonance widths cannot). Absolute quantum defects are derived for the vibrational series ranging from ν = 0 to ν = 2010. The influence of the Coulomb potential and continuity of heavy Rydberg behavior throughout the ^{1}Σ^{+}_{g} manifold of states is demonstrated.

The vibrational levels of ionically bound states in diatomic molecules are known to form Rydberg series^{2,3} and such vibronic states are now sometimes referred to as ‘heavy Rydberg’ states.^{4–8} Heavy Rydberg (HR) states are analogous of electronic Rydberg states but with the Rydberg electron replaced by a negative ion. Thus both the center of mass and the reduced mass of HR states differ significantly from those of electronic Rydberg states. However, in either case a system bound by a Coulomb potential at long range will support an infinite number of bound levels and the vibrational levels of a HR state will form a Rydberg series with well-defined effective quantum numbers n* and quantum defects. The vibrational quantum number ν is a measure of the number of nodes in a given vibrational state and simply replaces the radial quantum number n normally associated with electronic Rydberg states;^{3} note that ν is a node count so ν + 1 = n.

HR states have attracted much recent interest and substantial progress has been made in their identification. The ion-pair (IP) states of molecular hydrogen, the hydrogen halides, diatomic halogens and several other molecules exhibit extended vibrational progressions that have been shown to be HR in character over much of their range.^{8–25} Heavy Rydberg behaviour is diagnostic of the dominance of the Coulomb potential in a progression of vibrational levels. The large electronic transition moments associated with valence ↔ IP state transitions (which must always be parallel), when combined with good Franck–Condon factors, makes ion-pair states effective doorway states for photochemical reactions involving charge transfer. In homonuclear HR states a large transition dipole connects g/u pairs of HR states that correlate with the same ionic products and this can lead to amplified spontaneous emission.

There is an intimate connection between ion-pair states and electronic Rydberg states that lie in the same energy region. Their electronic configurations are related by a one or two-electron transfer between molecular orbitals. Asaro and Dalgarno pointed out in 1985 (ref. 2) that the vibrational states in LiF would acquire Rydberg character with increasing excitation. The year before, Mies pioneered the idea that multichannel quantum defect theory (MQDT, see e.g.ref. 26) could be applied to vibrational states in diatomics,^{27,28} and in 1988 published a paper that explicitly discussed the Rydberg-like properties of rotational–vibrational states in alkalihalides.^{3} Around the same time, Zewail and co-workers published their wave packet experiments on NaI,^{29,30} which were almost immediately followed up by time-dependent calculations.^{31,32} In the experiment, a surprisingly strong dependence on the excitation energy was observed for the lifetime of the wave packet. Chapman and Child eventually demonstrated that in the energy domain, this could be understood in terms of a periodic interferometric modulation of the widths of the states constituting the wave packets.^{33} Later, this modulation, which can span several orders of magnitude, was shown to cause Fano q-parameter reversals in lineshapes,^{34,35} and more recently, was proposed as the basis for the excitation of long-range ion-pair states in ultracold Rb gas.^{36}

The present article is inspired by the observation of photoexcited HR states with high principal quantum number (n > 1000) by Ubachs et al.,^{4,5,7,8} predominantly in H_{2}. Molecular hydrogen has attracted intense interest from both theoreticians and spectroscopists over many decades.^{8,9,11,12,14,15,19–21,24,25} The potential energy curves for many of the electronic states of H_{2} below the ionisation energy can now be calculated to high precision and there is close agreement between experiment and theory for the rovibronic term-values of most of the isotopomers which include mass-dependent non-adiabatic corrections.^{15} Mixing between HR and electronic Rydberg states provides doorway states that give access to the HR manifold. We present close-coupled channel calculations in H_{2}, using a hybrid log derivative/MQDT method^{19,20} in an energy range that corresponds to principal quantum numbers n ≈ 2000. These HR states are embedded in the ionization and dissociation continua. We account for the non-adiabatic couplings to the neutral states of H_{2}, leading to dissociation into neutral atoms, and focus on the high energy long-range states between the n = 3 and the ion-pair dissociation limit. This is the region employed for threshold ion-pair production spectroscopy (TIPS) and zero ion kinetic energy (ZIKE) techniques, and is of interest to astrophysicists, especially as it closely relates to the process of mutual neutralization, H^{+} + H^{−} → H(nl) + H(n′l′), recently discussed by Larson et al.^{37}

(1) |

(2) |

All scaling laws for Rydberg states originate from the properties of the Coulomb potential and the Rydberg formula. Since the energy levels scale as n^{−2}, the level spacing scales as n^{−3} (with the same scaling implied for line widths and conversely n^{3} for lifetimes). The size of the Rydberg states scales as the classical outer turning point R_{tp},

(3) |

It is important to note the effect of the reduced mass M_{AB} in eqn (2) and (3), which is illustrated in Fig. 1. As a consequence of this mass scaling, heavy Rydberg states which by definition have (M_{AB}/m_{e}) ≫ 1 are more compact and have a larger proportion of the wave function contained in the short-range interaction region for a given n. The mass scaling also has an obvious influence on the distribution of energy levels, as illustrated for the H-atom and H^{+}H^{−} in Fig. 1. In electronic Rydberg states, represented here by the H-atom, almost all energy levels occupy a narrow band just below the ionization limit, while the HR states in H^{+}H^{−} occupy a much wider energy band.

The quantum defect μ(E) in eqn (2) can be related to the asymptotic phase shift in the scattering of the two opposite-charge fragments,^{26} with non-zero quantum defects μ(E) ≠ 0 indicative of shorter range interactions. Consider for instance the first electronically excited state of molecular hydrogen, the H_{2}(B^{1}Σ^{+}_{u}) state, which has an ionic configuration H^{+}H^{−} derived from the molecular orbital configuration σ^{1}_{g}σ^{1}_{u} for radial separations R < 10 a.u. where it crosses the atom-pair asymptote H(1s) + H(2s) leading to dissociation. The potential below this crossing is thus coulombic albeit modified by a small polarization term, until, on the inner branch of the potential, nuclear repulsion and electron-pair repulsion begin to dominate at short range. The resulting inner wall, present in all molecular ion-pair potentials, restricts the available phase space for vibration and results in large negative quantum defects.^{3} This is in contrast to the positive quantum defects for electronic Rydberg series in which the Rydberg electron can penetrate through inner valence shells to experience an increased effective nuclear charge giving a positive quantum defect. The energy dependence of the quantum defect generally decreases with increasing principal quantum number, as the Coulomb waves become increasingly similar in the interaction region, although the onset of this effect is slower in HR states due to mass scaling.^{19} As a general remark in the context of H_{2}, note that HR states can only appear in the singlet manifolds of states, since the singlet H^{−}(1s^{2}) negative ion is the only stable form of the anion.

(4) |

(5) |

Ψ′′(R) = W(R)Ψ(R), | (6) |

(7) |

Eqn (6) is solved using the log derivative method,^{42–44} which propagates the log derivative matrix Y, defined as

Y(R) = Ψ′(R)Ψ^{−1}(R), | (8) |

Y′(R) = W(R) − Y^{2}(R), | (9) |

The matrix is propagated out to the matching radius R_{f}, at which point we assume that the channel interactions have vanished. At that point, the matrix is used to calculate the wave function in the form

Ψ = F − GK, | (10) |

K = G^{−1}F + G^{−1}B^{−1}G^{−1}Ω, | (11) |

In essence, the elimination procedure uses the N linearly independent solutions given by the column vectors of Ψ in eqn (10) to form N_{o} superpositions with correct asymptotic boundary conditions. This can be expressed as

Φ = ΨZ, | (12) |

Rewriting the matrix K as

(13) |

(14) |

(15) |

The component of the wave function given by eqn (15) for channel i and solution j is

(16) |

(17) |

(18) |

For asymptotically open channels, eqn (16) has ϕ_{i}(ε_{i},R → ∞) = k_{i}R + δ_{i}, assuming the asymptotic potential is constant. We can rewrite the sum as

(19) |

To obtain a condition similar in form to the closed channel condition in eqn (17), while ensuring that each open channel has the same asymptotic phase shift πτ, we can define

(20) |

(21) |

The unknowns in eqn (18) and (21) are the N × N_{o} coefficient matrix B and the N_{o} eigenphases τ. A numerically convenient method to obtain all these in a single calculation is to combine eqn (18) and (21) in the form of a generalized eigenvalue equation,

ΓB = tanπτΛB, | (22) |

(23) |

The main remaining steps are to normalize the continuum eigenvectors in B and to calculate the scattering matrix S. According to the definitions of T in eqn (19), we have that

(24) |

(25) |

The matrix T can be shown to be orthogonal^{46} and becomes unitary, TT^{†} = 1, if it is normalised according to . This condition fixes the overall normalization of the N coefficients B_{ij} for each j.

We can now obtain the N_{o} × N_{o} scattering matrix S^{−} for incoming wave boundary conditions, appropriate for photodissociation. Combining the final wave function in eqn (15) with the definition of the coefficients T in eqn (19) gives the wave function for solution j in channel i (i, j ∈ N_{o}) as follows,

(26) |

Henceforth we will use the asymptotic value of the open channel phase, ϕ_{i}(ε_{i},R → ∞) = k_{i}R + δ_{i}.

The following superposition corresponds to an asymptotic wave function incoming boundary condition,

(27) |

(28) |

Fig. 3 Potential energy curves for H_{2} of ^{1}Σ^{+}_{g} symmetry. The potential energy curves are taken from Wolniewicz et al. for states 2–6 (ref. 12 and 38) and from Detmer et al. for states 7–9.^{47} The H^{+}_{2} ground-state potential is also included, as is the ion-pair potential. The position of the n = 2000 HR state is indicated by a vertical line, with the classical turning point R_{tp} at 8700 a.u. and the position of R_{f} (see Fig. 2). The unusual barrier shape in the region of R ≈ 10 on the excited potential energy curves can be understood using a simple Fermi model as contact potential interactions between a colliding H(1s) atom and an almost-Rydberg H(3l) atom.^{48,49} Note that the scale of the axis for the internuclear distance R is logarithmic. For R > 60 a.u. the potential energy curves are continued analytically. |

Feature | n* |
R
_{tp} (a.u.) |
---|---|---|

H(1s) + H(2l) dissociation limit | 68.72 | 11.08 |

H(1s) + H(3l) dissociation limit | 128.52 | 36.05 |

H(1s) + H(4l) dissociation limit | 362.30 | 286 |

Outer well state 7 (at R = 33.8 a.u.) | 128.68 | 36.13 |

Barrier on state 7 (at R = 8.7 a.u.) | 193.53 | 81.56 |

At n = 100 energy | 100 | 22 |

At n = 200 energy | 200 | 87 |

At n = 1000 energy | 1000 | 2177 |

At n = 2000 energy | 2000 | 8709 |

At large internuclear distances (R > 10 a.u.) the ion-pair (HR) states are accurately described by the Coulomb interaction energy between the positive and negative ion given by eqn (1) with the ion-pair dissociation energy D_{H+H−} = −0.527751014 a.u. (ref. 38 and 50) and the polarizability of H^{−}(1s^{2}) α_{H−} = 211.897 (a.u.)^{3} (see ref. 51 and also footnote§). The reduced mass for H_{2} is M_{H2} ≈ 918.0764 a.u.^{53}

For the model diabatic calculation, we construct an effective diabatic potential E^{ion,diab}(R) from the H^{1}Σ_{g}ab initio adiabatic state calculated by Wolniewicz^{38} and the asymptotic ion-pair potential in eqn (1).^{50,51} The potentials are merged as follows,

E^{ion,diab}(R) = γ(R)E^{ion}(R) + (1 − γ(R))E^{H}(R), | (29) |

The main physics underpinning the progression of line positions across the range n = 160–230 is assessed by comparison to the simple diabatic model constructed from H^{1}Σ^{+}_{g} and the ion-pair potential (see Section 4). A comparison between the line positions resulting from the full log derivative–MQDT hybrid calculation and the model in this energy region show that the model follows the pattern of line positions closely, with only a comparatively small and almost constant off-set (Fig. 4).

It therefore appears that a large portion of the energy dependence in the quantum defect, μ(E), can be attributed to the vibrational wavefunction accumulating phase on the diabatic H potential. The model diabatic potential and a bound state on that potential are shown in Fig. 5. Note that states on the model potential are technically bound states, and that the model in its current form cannot make predictions of line widths. Given the importance of interferometric lifetime modulation in HR states, and the complexity of the H_{2} potentials in the interaction region, it is doubtful that a meaningful simplified model could be constructed for the lifetimes (widths).

Fig. 5 Effective potential (merged H and ion-pair potential) together with the bound n = 160 radial wavefunction. |

We now proceed to the part of the HR spectrum just below the ion-pair dissociation limit. Using the hybrid log derivative–MQDT approach we calculate a narrow range with principal quantum number 1990 < n < 2010 as shown in Fig. 6 for total angular momenta J = 0, 1, and 2. The line positions are quite regular, with nearly constant quantum defects μ(E) across the range. The widths, on the other hand, exhibit modulation across the range, with different variation in line widths for the different angular momenta J = 0, 1, 2. The calculated width for the state n = 2000 is 5 × 10^{−4} cm^{−1}, corresponding to a lifetime of approximately 10 ns. This is almost an order of magnitude shorter than the observed lifetimes reported by Ubachs et al.^{4} Since one would normally expect the calculations to over-estimate lifetimes on the basis that not all decay channels are included in the calculations, this lends support to the notion that such high-n HR states have a strong tendency to undergo J-mixing by external fields, analogous to the l-mixing that occurs in electronic zero electron kinetic energy (ZEKE) spectroscopy.

Fig. 6 Progression of heavy Rydberg resonances around the principal quantum number n = 2000 for J = 0, 1 and 2. These states have classical turning points at around R_{tp} = 8700 a.u. |

Term values of the vibrational levels supported by the adiabatic H potential of Wolniewicz have been calculated up to ν = 39 (E_{b} = 8000 cm^{−1}) and we include a complete table of resonances in the energy region E_{b} = 1800–6000 cm^{−1}(ESI†) calculated by solving the close-coupled equations for seven ^{1}Σ^{+}_{g} states. If the vibrational numbering is known, the absolute quantum defect follows from −μ(E_{b}) = n* − ν − 1 − J, where . Such μ(E_{b}) plots, both from model potentials and experimentally, for a wide range of ion-pair states have been shown to be nearly linear over a wide range of E_{b} values as the dissociation limit is approached. Thus, if experimental T_{v} values are used from fragmentary spectra in which the absolute vibrational numbering is not known, ν is adjusted so that the ν(E_{b}) values for a partial progression lie on the μ(E_{b}) plot extrapolated from regions of known numbering. If the numbering is changed by ±1, clear discontinuities in the sections of the μ(E_{b}) plot result.

The various calculated μ(E_{b}) values for the H state are summarized in Fig. 7. The high vibrational levels presented in this paper cover the range E_{b} = 25.5–25 cm^{−1} and, if positioned on the extrapolated μ(E_{b}) plot, indicate a quantum defect of −63.8 at E_{b} 25 cm^{−1}. The condition n ≫ μ is sometimes taken to define HR behaviour, and this is certainly fulfilled by the levels in Fig. 7. The broader μ(E_{b}) dependence in Fig. 7 indicates that there is no abrupt transition from a HR region to one deeper in the potential well in which the vibrational spacing becomes sensitive to the position of the inner wall of the potential, even after interruptions by avoided crossings that predissociate parts of the progression. In the H state, the potential has a potential maximum at E_{b} ∼ 12000 cm^{−1} and the vibrational numbering to larger E_{b} is that of the outer well only, whose outer wall continues initially to have a dominantly coulombic potential.

Fig. 7 Quantum defect plot for the vibrational levels of the H^{1}Σ^{+}_{g} state for ν′ = 0 up to ν′ = 2010. The absolute quantum defects, −μ(E_{b}), are plotted as a function of the binding energy E_{b} below the ion-pair dissociation limit. Previously calculated term values for ν′ = 0–40 are taken from ref. 54. The break in slope at E_{b} ≈ 13000 cm^{−1} corresponds to the top of the barrier between the electronic and heavy Rydberg potentials (ν′ = 16). Calculated resonances lie between E_{b} = 2000 and 6000 cm^{−1} (see the ESI†). The highest energy resonances, close to the ion-pair dissociation limit, are indicated by the extrapolated point close to the vertical axis (E_{b} ≈ 25 cm^{−1}). Note that the point on the y-axis corresponds to the states shown in detail in Fig. 6. Finally, the position of the crossings for n = 3 and n = 4 limits are indicated by arrows. |

The calculations presented here complete the sequence of HR vibrational levels associated with the EF^{1}Σ^{+}_{g} and H^{1}Σ^{+}_{g} states, the observed resonance structure and the higher n^{1}Σ^{+}_{g} states that are bound by the Coulomb potential. This is the first report of HR states above the n = 4 limit. HR behavior has now been observed from ν = 0 of the outer F^{1}Σ^{+}_{g} potential, through the vibrational levels of the H^{1}Σ^{+}_{g} state and up to within ≈25 cm^{−1} of the ion-pair dissociation limit.

The analogous vibrational systems in the ^{1}Σ^{+}_{u} manifold have been shown to have HR character from ν = 0 of the B^{1}Σ^{+}_{g} state, through the B^{1}Σ^{+}_{u} state, and diabatically through the n = 3 asymptote up to the limit of the published resonance structure. However, the region between the n = 4 asymptote and the ion-pair dissociation limit for the ^{1}Σ^{+}_{u} manifold remains to be investigated.

Work is underway to include ionisation by coupling the inner-most region (R < 10 a.u.) to the electronic continuum via a generalised R-matrix approach. The conceptual framework for incorporating ionisation and dissociation within an R-matrix formalism is already established.^{55–58} Judging by the present calculations, the effect of including ionisation on the line positions will be comparatively minor. In contrast, one should expect the effect on the line widths to be more significant, and a full account of the mixing with electronic Rydberg states will provide an understanding of how valence and electronic Rydberg states channel excitation intensity to heavy Rydberg states.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8fd00096d |

‡ Calculated numerically by inwards integration, see Jungen and Texier.^{45} |

§ Note that the electron affinity can be calculated as E_{ea} = D_{H+H−} − E_{H−} − E_{H} and that the value of the electron affinity obtained this way differs somewhat from the value −0.0277196153 a.u. reported by Radtzig et al.^{52} |

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