Transcript How H 3 + and its isotopomers recombine efficiently at low energies

How H3+ and its isotopomers recombine efficiently at low
energies
Chris Greene
Department of Physics and JILA
University of Colorado at Boulder
Main Collaborator: Viatcheslav
Kokoouline(U. Central Florida)
also with assistance from: Brett Esry
support: DOE, NERSC, and NSF
The dissociative recombination process:

e  H3
H H H
 H2  H
Outline of this talk
1. Overview of dissociative recombination theory, direct versus
indirect processes
2. Theoretical techniques for the description of molecular
Rydberg states
3. Incorporation of the Jahn-Teller effect and polyatomic
dissociation in its full 3D dimensionality
4. Comparisons between theory and experiment
5. Remaining problems with the theory to address and
overcome.
A question of general chemical importance: How does
electronic energy convert into bond-breaking energy?
….
….
Bates’ 1950 article points out that DR can
explain why ionized molecular gases can
neutralize so rapidly, even in 10-13 sec or
faster.
And…. 43 years later… still at the helm:
D. R. Bates (1993 “Enigma of H3+ Dissociative Recombination”):
“It is concluded that the evidence that the recombination coefficient at
300K is around 1.5 x 10-7 cm3/s is overwhelming. However, such a
high rate coefficient has appeared irreconcilable with theory, there
being no crossing of potential energy curves favourable to DR at low
temperature.”
The “direct” DR theory:
Systematic inclusion of Rydberg state physics
through multichannel quantum defect theory
DIRECT
INDIRECT via
Rydberg states
An early calculation of DR for H2+,
compared with experiment (Giusti,
Derkits, Bardsley 1983):
Another foray, slightly later, by Nakashima, Takagi, and
Nakamura:
The beginnings of understanding electron-ion resonances
Prehistory of general resonance physics theory:
O. K. Rice, 1933 JCP; Fano, 1935 Nuovo Cimento, 1961 Phys. Rev.;
Breit & Wigner, 1936 Phys. Rev.; Blatt & Weisskopf textbook, 1952;
Feshbach, 1958,1962, Ann. Phys.;
Systematic method for treating Coulomb field aspects of the electron-ion interaction:
(Otherwise known as MQDT – multichannel quantum defect theory)
Ham, 1955 Solid State Physics; Seaton, 1958 M. Not. R. Astron. Soc.;
Seaton, 1983 Rep. Prog. Phys.
Fano, Lee, Lu, Johnson, Cheng, 1970s and 1980s PRAs, PRLs.
Extensions to molecular physics:
Fano, 1970 Phys. Rev. A; Jungen and Atabek, 1977 JCP. Jungen & Dill, 1980 JCP;
Including dissociation in molecular MQDT: Lee 1977 PRA; Giusti-Suzor, 1980 J Phys. B;
Jungen, 1984 PRL
Triatomic Rydberg states: Fano and Lu, 1984 Can. J. Phys; Bordas & Helm, Jungen & Child,
JCP; Stephens & Greene, early 1990s;
The characteristic Fano resonance lineshape
U. Fano 1935, and 1961. H. Feshbach, 1958, 1962
E
Eres
2
Lorentzian or “Breit-Wigner” limit
q
2
q
Two typical resonance plots of an observable versus
the energy rescaled by the resonance width.
The picture of resonance physics using multichannel quantum defect theory (MQDT)
Seaton, Fano, Jungen …; View bound or quasi-bound states as “scattering at E < 0”
ionic rovibrational
channel thresholds Ev’
Energy(a.u.)
closed channels c (E<Ec )
total energy E
open channels o (E>Eo )
r / aB 
multichannel quantum defect theory “channel
elimination” formula – imposes exponential decay
Effects of complicated level perturbations and
interlopers, are now fully understood, as in this
example of strontium ground state photoionization:
Brown, Longmire,
Ginter, JOSA B 1983,
expt.
Aymar, 1987 JPB,
theory, using Rmatrix theory,
multichannel
quantum defect
theory, and a frame
transformation
Examples of molecular photofragmentation, for diatomics
…An important 1980 JCP
article
Ch. Jungen and coworkers developed some clever ways to
include predissociation on the same footing as preionization of
molecular Rydberg state resonances:
Recent
variations
work even
better:
The “high energy” resonant recombination process is direct, with a strong
resonance at 9 eV, accounted for nicely by the Orel-Kulander theoretical
treatment. See also A. Larson’s talk Wednesday morning at UCL on the
ion-pair formation channel mediated by this resonance.
But we’re interested in far cooler temperatures, in the sub-1 eV
range. What do experiments say about H3+ dissociative
recombination at low-T?
The situation was the following, around the year 2000:
STORAGE RING EXPERIMENTS (@ 300K):
• Larsson et al.: 1.15 x 10-7 cm3/s
• Mitchell et al.: 1.2 x 10-7 cm3/s
AFTERGLOW EXPERIMENTS (300K):
•Gougousi, Johnsen,& Golde, 1995: 1 x 10-8 cm3/s
•Laube’, Le Padellec, Rebrion-Rowe, Mitchell,
D. R. Bates (1993 “Enigma of H3+
Dissociative Recombination”):
“It is concluded that the evidence that
the recombination coefficient at 300K is
around 1.5 x 10-7 cm3/s is
overwhelming. However, such a high
rate coefficient has appeared
irreconcilable with theory, there being
no crossing of potential energy curves
favourable to DR at low temperature.”
and Rowe, 1998: 7.8 x 10-8 cm3/s (+/- 2.3)
•Smith & Spanel, 1993 1-2 x 10-8 cm3/s
•Plasil, Glosik et al. 2002, < 3 x 10-9 cm3/s
An important
contribution in 2000,
by Schneider, Orel,
and Suzor-Weiner!
Inclusion of Rydberg “indirect
pathways, reduces the disagreement to
only 3 orders of magnitude!
5 orders of magnitude discrepancy
between the “direct pathway” and
experiment!
In the face of these persistent discrepancies between
theory and experiment, and in the face of disagreements
between storage ring experiments and afterglow
experiments, everyone went back to the drawing board …
Experimental improvements to the storage ring
experiments: colder ion sources, better energy resolution,
especially at CRYRING and TSR
Theoretical improvements: Consider Jahn-Teller
coupling in the electron-molecule collision, indirect
Rydberg pathways, and the full dimensionality of nuclear
vibrational motion
Our proposed (2001 Nature) mechanism
for H3+ dissociative recombination: Jahn-Teller-mediated
recombination via Rydberg pathways
Two degenerate in-plane p-orbitals
are coupled by Jahn-Teller symmetrydistortion physics:
Conical intersections,
where the non-BornOppenheimer
couplings blow up
real good.
Potential Surfaces for dissociative H3 2p,3p states,
from M. Jungen
Qualitative picture:
conversion of the conical
intersection problem in 3D
to coupled hyperradial
potential curves in a single
coordinate, the
hyperradius, R.
These hyperradial
adiabats are derived
by solving the fixed-R
Schroedinger
equation on the
dissociative 2p pi
surfaces of H3, from
Truhlar et al.
H+H+H
H2+H
H3+
Why don’t we just do coupled nuclear dynamics on the lower two (H+H2
and H+H+H) dissociative 2p pi surfaces? Because we also have to
account for all these Rydberg resonances at low incident electron energies.
Here are the np Rydberg series arising from vibrations only.
THE BIG PICTURE
Hyperspherical
representation of the
relevant pathways for
dissociative recombination
of H3+.
Notice that in this
representation, the DR
pathways DO OCCUR AS
CURVE CROSSINGS!
Clamped-hyperradius
electron-H3+ scattering
resonances. Shown are
the ionic potential
curves, and at several
hyperradii, the electron
scattering time delay as
a function of energy.
Comparison of the “simplified theoretical treatment” with low resolution experiment
Kokoouline, Greene &
Esry, Nature 2001
Taken from Kokoouline & Greene,
presented in Mosbach:
J. Phys.: Conf. Series 4, 74-82 (2005).
Next for our “advanced treatment”, a more ambitious approach.
Our goal here has been an attempt to:
(1) Perform the first polyatomic dissociative recombination
calculation ever that includes ALL 9 degrees of freedom
quantum mechanically. (Hopefully resolve the orders-of-
magnitude discrepancy between theory and experiment that
has existed for this H3+ system for decades.)
(2) Obtain spectroscopic accuracy that can also be compared
resonance-by-resonance with H. Helm’s photoionization
measurements of metastable H3. (i.e. use the same
wavefunctions, but apply them to an observable different from
DR.)
(3) Include enough physics to predict the position of most Rydberg
state resonances to within 3 meV = 40K, since the
astrophysicists want to know this DR rate at T=40K.
Scope of the 9D 4-body problem we are faced with:
Test of the Staib-Domcke parameterization of the Jahn-Teller fixed-nuclei K-matrix, fitted
to ‘undiagonalized’ ab initio calculations of M. Jungen. Figures are taken from Mistrik,
Reichle, Muller, Helm, M. Jungen, and J. Stephens, 2000 Phys. Rev. A.
Diabatic
quantum
defect model
fitted to the
ab initio
surfaces,
compared
with the raw
surface
values at
various
geometries
Fixed-R S-matrix
Approximated as Eindependent over 0-2 eV
incident energies
Smooth MQDT S-matrix for interactions between the electron and
the vibrational motion. Note that it is a projection akin to the
infinite-order sudden approximation, but we include closed
channels in order to get the Rydberg resonance physics. Then
add the rotational frame transformation (L-uncoupling physics),
and impose exponential decay in the closed ionization channels
via MQDT channel elimination:
Adiabatic hyperspherical method:
Solve for the hyperspherical adiabatic
functions using a potential surface for the 3
nuclei in H3+, plotted below for a fixed
hyperradius.
Adiabatic hyperspherical potential curves for H 3+ vibrational motion:
Nicely parallel near the equilibrium geometry
=> Adiabatic approximation should work well!
H3+ ab initio surface used from
How do we treat molecular Rydberg states
while including processes such as
photoionization, autoionization, or
photodissociation, which involve departures
from the Born-Oppenheimer
approximation?
Answer: Multichannel quantum defect
theory (Seaton), combined with a
rovibrational frame transformation (Jungen,
Fano, Dill, Raoult, Chang, Chase,
Arthurs&Dalgarno…) is the only way at
present, for many problems, to cope with the
immense number of competing channels,
and fragmentations of a qualitatively
different nature.
Concept of an electron-diatomic molecule collision, viewed
as a vibrational frame transformation
Key Tools in
Understanding Rydberg
Molecules:
Seaton’s multichannel
quantum defect theory
The Fano-Dill-Jungen
rovibrational frame
transformation theory
 short-range electron-molecule
scattering matrix is diagonal in the
“quantum number” R.
A more complete solution using multichannel quantum defect theory
Idea of Fano’s frame transformation: Consider the physical
meaning of a scattering matrix in the representation where it is
diagonal:
=> If we can identify in advance the representation | α 
in which the S-matrix is diagonal, then we just need to
find the eigenphaseshifts μ α and the unitary
transformation matrix connecting the eigenchannels | α 
to the fragmentation channels | i  .
Born-Oppenheimer adiabaticity in electron-diatomic scattering:
=> the “quantum number” that is conserved during the shortrange electron-molecule scattering is R, which gives an S-matrix:
An open channel version of this
concept was pioneered by
Chase 1956, and by Arthurs &
Dalgarno 1960.
In these states | i > are buried many symmetry considerations, and the full dependence of
the wavefunction on vibrational and rotational coordinates, the angular and spin
wavefunction of the electron(s), and the nuclear spin wavefunctions. The details are in:
A problem: The S-matrix
just discussed has only
ionization channel indices.
How can we represent the
dissociation channels? (One
way – R-matrix idea of Jungen
and Ross.)
Our solution:
Siegert state methods
adapted from Tolstikhin,
Ostrovsky, and Nakamura,
Phys. Rev. A 58, 2077
(1998).
Modified form of the rovibrational frame transformation using a Siegert
state vibrational basis to account for the possibility of dissociation.
Experiment (red): McCall, Huneycutt, Saykally, Djuric, Dunn, Semaniak, Novotny, Al-Khalili,
Ehlerding, Hellberg, Kalhori, Neau, Thomas, Paal, Oesterdahl, Larsson
Theory (green), with corrected convolution over , [thanks to Andreas Wolf pointing out its
importance]: Kokoouline and Greene (2004) unpublished (no toroidal correction applied yet)
This discrepancy is
not yet understood
This discrepancy is
understood – the
higher experimental
DR rate here is from
the toroidal
correction
Resonance
modulations are
overestimated by
theory because the
convolution over delta
E(parallel) was not
performed (yet).
Sharp drop
of DR cross
section
followed by
plateau region
0.4 eV – 0.7 eV
shows good
agreement
between theory
and
experiment.
Blowup of the
comparison
between theory
and experiment to
better test the
crucial region from
about 0.05 eV up
to 1 eV.
Experiment:
Origin of the toroidal correction effect
Comparison of the ortho H3 photoionization spectrum with the new experimental DR
spectrum
DR
Partial
DR,
ortho
Another prediction of the rotational frame
transformation: Average rotational excitation
probabilities (squared scattering matrix elements) per pwave collision, for a low-energy incident electron. Note
that many of these are comparable to the unitarity limit
(unity).
Thermally-averaged rates
suggest a possible
difference at low
temperatures between the
destruction rates of ortho
and para-H3+.
D3+ dissociative
recombination
calculation,
compared with
two experiments
at different
resolutions.
Overall, H3+
recombines at a
rate about 3
times higher
than D3+.
H2 D +
Illustration of the
stronger
nonadiabatic
hyperradial coupling
in H2D+ compared to
H3+.
H3+ hyperspherical
adiabats
Vibrational energy levels for H2D+, with
and without nonadiabatic hyperradial
couplings included
H3+ levels
Storage ring experiment
Calculation in adiabatic approx.
Best DR calculation
D2H+ dissociative recombination rate versus parallel component of energy
This factor of
3-5
discrepancy
is not yet
understood
Comparisons shown on a
linear-linear scale for
sharper assessment of the
extent of disagreement
between theory and
experiment.
CONCLUSIONS
(1) Inclusion of the Jahn-Teller coupling mechanism increases the low energy
dissociative recombination cross section by 2-3 orders of magnitude
compared to previous theory (Schneider et al., 2000 PRL) which omitted
Jahn-Teller symmetry distortion effects.
(2) This is the first polyatomic species for which a full-dimensionality quantal
description has been carried out for the DR process, and it can serve as a
benchmark meeting ground for experiment and theory.
(3) At T=300K, our theoretical thermally averaged DR rate is
7.2  1.1 108 cm3 / s
(4) The newest rate in the Larsson et al. storage ring experiment, at Te=300K
electron temperature (but Trot=40K rotational temperature) is
8
6.8  10 cm / s
3
(5) Our calculated value is consistent with some of the flowing afterglow
experiments (e.g. Laube’, LePadellec, Sidko, Rebrion-Rowe, Mitchell, and
Rowe, 1998 JPB):
but seemingly inconsistent with the stationary afterglow expt of Glosik et al.
Summary:
•The theory of dissociative recombination of the triatomic
hydrogen ion is now in encouraging shape, but some
potentially important discrepancies remain to be understood.
•The C2v asymmetric isotopomers show poorer agreement
between theory and experiment, even though one of the key
approximations – the adiabatic hyperspherical approximation
– has been improved compared to the D3h symmetric ions.
•Other observables still need to be calculated, such as the
branching ratio between 2-body and 3-body dissociation, and
the distribution of rovibrational levels of H2 that is produced,
in order to further test and improve the theory
•Also, should explore effects of the energy-dependence of the
quantum defect parametrization, the