June 19, 2006Altunata, Coy, Field, MI111 Broad Shape Resonance
Effects in the Rydberg Molecule, CaF (and BaF) For Rydberg
molecules like CaF and BaF, all bound electronic states are members
of some Rydberg series. The Rydberg series should be quite regular
and predictable with nearly constant quantum defects. The
wavefunctions in a Rydberg series should follow Mullikens rule, or
recapitulation, where the inner lobes of the wavefunction scale,
but do not change shape. They should stretch in and out like an
accordion with energy. BUT One CaF Rydberg series, with quantum
defect 0.88, shows evidence of a broad and strong interaction, with
effects from the lowest Rydberg levels into the continuum on
eigenstate energies, on photo-electron distributions, and on
scattering. The effect is both general, over a wide energy range,
and specific, affecting one series. What is the cause? Serhan N.
Altunata, Stephen L. Coy, and Robert W. Field MIT Department of
Chemistry Slide 2 June 19, 2006Altunata, Coy, Field, MI112 Overview
of Rydberg States The Quantum Defect Matrix Rydberg Series
Effective principal quantum number: Knowledge of the eigen-quantum
defects, (E), yields the complete electronic spectrum. The full
quantum defect matrix has information about channel coupling /
l-mixing. Rydberg-Ritz formula with smoothly varying quantum
defects Slide 3 June 19, 2006Altunata, Coy, Field, MI113 Solving
the nearly-one-electron problem: R-Matrix Theory An effective
one-electron potential based on ab-initio calculations is used.
This is complex in the core region, but gets progressively simpler
at longer range. Internal region(Core): r < r o R-matrix
connects the complex inner solution to the simple longer-range
solutions. It is a Ratio, /, the logaritmic derivative of the
wavefunction at r 0. Use a variational method for solution. Slide 4
June 19, 2006Altunata, Coy, Field, MI114 From R-matrix boundary to
the long-range solutions: Propagation using a one-sided Green
function Multipole-Moment Interaction Region (Monopole + Dipole /
Quadrupole) Outer Core: At long range only the monopole remains :
Monopole solutions are analytically known. Regular Coulomb Wave
Irregular Coulomb Wave Slide 5 June 19, 2006Altunata, Coy, Field,
MI115 The log-derivative R-matrix hides the complexity of the core.
Propagating the wavefunction outward from the core yields the K
matrix. The K (reaction) matrix contains all dynamics bound state
energies to scattering phase shifts. R-matrix theory unifies
continuum and bound state calculations REACTION MATRIX K S
Scattering Matrix Resonances Cross Sections Quantum Defect Matrix
Bound States But this is still too complicated! Long range dipole
fields force the core to be LARGE and makes K energy dependent. We
remove the dipole contribution analytically. K is dipole-reduced
reaction matrix. Slide 6 June 19, 2006Altunata, Coy, Field, MI116
Separation in spherical harmonics at long range ( l ) Spherically
Symmetric Coulomb Potential: Long-Range Reaction Matrix, K, in the
l - representation Coulomb + Dipole potential: Separation in
Dipolar harmonics at short range ( l ) Short-Range Reaction Matrix,
K, in the l - representation Long-range and Short-range Reaction
Matrices Slide 7 June 19, 2006Altunata, Coy, Field, MI117 The
entire calculation: From short range to long range R-matrixK-matrix
Variational condition for the R- matrix reduces to a generalized
eigenvalue problem based on the full core Hamiltonian.
Self-consistent short- range representation of the wavefunction
beyond the core (Dipole-reduced, fractional-l, reaction matrix).
Smooth energy dependence, accurate quantum defects. Valid only in
the long range integer-l representation. Determines long-range
properties like photo- ionization cross sections.
coredipole+coulomb+coredipole+coulomb Slide 8 June 19,
2006Altunata, Coy, Field, MI118 The ab-initio-based one-electron
effective potential for CaF Hamiltonian terms 1.Coulomb 2.e -
-induced dipole 3.nuclei-induced dipole 4.induced dipole induced
dipole 5.Ca core correction Slide 9 June 19, 2006Altunata, Coy,
Field, MI119 Results: K Eigenquantum defects vs Energy agree with
experiment Experimental quantum defects are calculated by
deperturbing data for rotating molecule effects. Using
dipole-reduced K matrix is essential. 2 states 2 states 2 states
Slide 10 June 19, 2006Altunata, Coy, Field, MI1110 Extend into the
continuum: The energy dependence of the 0.88 2 + Eigenquantum
defect is too large 0.88 2 + quantum defect rises significantly
across a wider E region. What is the origin of the energy
dependence? A hint: 0.88 precursor orbital is polarized behind F -.
As E, it expands. Slide 11 June 19, 2006Altunata, Coy, Field,
MI1111 A Shape Resonance results from a Double-Well Potential The
inner well can have either atomic origin (e - subshell
oscillation), or molecular origin (in CaF, excluded volume around F
- ). The inner well is largest for a particular l or for a range of
l values because the centrifugal barrier modifies the potential. An
Atomic Example Ba+ At right, radial wavefunctions in Ba+ show a BIG
phaseshift between 4f and 5f. This is why shape resonances are also
called accordion resonances. Above the resonance, the connection to
the inner core (Mullikans rule or recapitulation ) is lost. What is
a Shape Resonance? (a.k.a. Accordion Resonance) Shape resonance is
a distorted accordion Slide 12 June 19, 2006Altunata, Coy, Field,
MI1112 Life Time Matrix Q: Largest eigenvalue of the lifetime
matrix shows a Lorentzian lineshape centered at the resonance. We
use the dipole-reduced S matrix to isolate core effects. Continuum
d n /dE = Rate of change of energy shift with respect to excitation
energy, Locating the Shape Resonance: The Lifetime Matrix Slide 13
June 19, 2006Altunata, Coy, Field, MI1113 CaF Shape Resonance from
the Lifetime Matrix The peak in q max locates the shape resonance
in the molecular potential. Broad Resonance Slide 14 June 19,
2006Altunata, Coy, Field, MI1114 Lifetime matrix eigenvector for
its max eigenvalue shows that 0.88 2 + is most affected Branching
ratios: Re-expand the shape resonance in the CaF eigenchannels
Excluded volume about F - leads to large time delay: Resonance in
the 0.88 series 10.88 2 Excited State: Slide 15 June 19,
2006Altunata, Coy, Field, MI1115 Understanding the Shape Resonance:
A super-simple adiabatic approximation Assume the electronic motion
in a collision channel is governed by a central potential with
centrifugal barrier. Make an adiabatic approximation for the
electronic radial coordinate to define a 1- D potential: CaF
Adiabatic Potential Curves For Electronic Motion, R=3.1 Slide 16
June 19, 2006Altunata, Coy, Field, MI1116 WKB methods can be used
to decide if there is a quasi-bound level inside the adiabatic
potential barrier R=3.54 Bohr R=3.1 Bohr potentialWKB phase
Lifetime A quasi-bound state exists with a lifetime like that from
the accurate calculation! Slide 17 June 19, 2006Altunata, Coy,
Field, MI1117 The partial l character of the adiabatic modes show a
strong r dependence in the vicinity of the F - nucleus. Adiabatic
approximation in the electronic radial coordinate breaks down at F
-. Electron escapes potential well by tunneling across the barrier
or by decaying to other degrees of freedom via non-adiabatic
coupling. CaF Shape Resonance is due to F - e - repulsion:
Adiabatic breakdown at the F position Slide 18 June 19,
2006Altunata, Coy, Field, MI1118 Dipole-reduced quantum defects are
nearly linear with energy, as predicted by the Rydberg-Ritz
formula, extrapolate smoothly, and match experimental quantum
defects. Without the dipole reduction, features due solely to the
long-range dipole field appear. The anti-crossing below is dipolar,
and has little information about the structure of the core, but
affects photo-ionization X-sections. K K: Long range field effects
in photo-ionization Slide 19 June 19, 2006Altunata, Coy, Field,
MI1119 State are s-p mixed below and above the dipolar scattering
resonance. A ~180 degree flip in mixing angle occurs across
resonance. We saw that the shape resonance actually occurs in the d
channel. There is little d-channel activity in this resonance. The
anti-crossing in K quantum defects on the previous slide is a
largely a dipolar effect that operates on the shape
resonance-modified levels. Dipolar resonance mixes s and p
partial-l characters in the wavefunction An anti-crossing in K
quantum defects Slide 20 June 19, 2006Altunata, Coy, Field, MI1120
Photo-ionization anisotropy from 10.55 2 + disappears at the
dipolar resonance in the continuum. PI differential cross section
is calculated by transforming to lab frame, and averaging over
spatial orientations of the nuclei. Slide 21 June 19, 2006Altunata,
Coy, Field, MI1121 The R-matrix theory presented here has produced
the global electronic spectrum of CaF in good agreement with the
experiment. A broad shape resonance was identified from the
collective behavior of bound and continuum state wavefunctions. The
shape resonance is due to the trapping of the electron between a
centrifugal barrier on the Ca atom and the excluded volume on the F
- ion. The shape resonance explains values and trends in electronic
energies, photo-ionization properties and molecular constants. The
generic properties of the resonance indicate the possibility of
similar behavior in other alkaline earth mono-halides (e.g. BaF).
Influence of resonances on electronic structure is firmly
established in the current framework. Conclusions Slide 22 June 19,
2006Altunata, Coy, Field, MI1122 Acknowledgements Chris H. Greene
(U. of Colorado) References S.N Altunata and R.W.Field Phys. Rev. A
67 (2), 022507 (2003) S.N Altunata, S.L. Coy and R.W. Field J.
Chem. Phys. 123, 079903 (2005) S.N Altunata, S.L. Coy and R.W.
Field J. Chem. Phys. 123, 079918 (2005) S.N Altunata, S.L. Coy and
R.W. Field J Chem. Phys., 124, 194302 (2006) Slide 23 June 19,
2006Altunata, Coy, Field, MI1123 BaF Slide 24 June 19,
2006Altunata, Coy, Field, MI1124 Rydberg State Wavefunctions in the
= 10 Region ( I ) (A) X 2 Ground State The X state has no amplitude
close to F. The wavefunction is confined within 5 Bohr of the Ca
nucleus. (B) 10.55 2 Excited State High Rydberg member of the 0.55
series, built on the X state terminus. Also polarized behind the Ca
nucleus. Slide 25 June 19, 2006Altunata, Coy, Field, MI1125 Rydberg
State Wavefunctions in the = 10 Region ( II ) (C) 10.88 2 Excited
State High Rydberg member of the 0.88 series. The electronic
wavefunction is polarized behind the F - (D) 10.16 2 Excited State
High Rydberg member of the 0.16 series. The wavefunction is d-f
mixed. Slide 26 June 19, 2006Altunata, Coy, Field, MI1126 Small
value of the quantum defect corresponds to the hydrogenic limit (D)
10.08 2 Excited State Non-penetrating Rydberg state. The
wavefunction is a dominant f state. Spherical Symmetry is
recovered
