Broad Shape Resonance Effects
in the Rydberg Molecule, CaF (and BaF)
Serhan N. Altunata, Stephen L. Coy, and Robert W. Field
MIT Department of Chemistry
•
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 Mulliken’s 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?
June 19, 2006
Altunata, Coy, Field, MI11
1
Overview of Rydberg States – The Quantum Defect Matrix
Ec
Rydberg Series
Effective principal
quantum number:
1

2( Ec  E )
“Rydberg-Ritz formula” with smoothly varying quantum defects
En  Ec 
1
2(n   ( En ))
2
where  ( En )     En
Knowledge of the eigen-quantum defects, (E), yields the complete
electronic spectrum.
The full quantum defect matrix has information about channel
coupling / l-mixing.
June 19, 2006
Altunata, Coy, Field, MI11
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Solving the nearly-one-electron problem: R-Matrix Theory
Internal region(Core): r < ro
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.
R-matrix connects the complex inner solution to the
simple longer-range solutions. It is a Ratio, δΨ/Ψ,
the logaritmic derivative of the wavefunction at r0.
Use a variational method for solution.
June 19, 2006
Altunata, Coy, Field, MI11
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From R-matrix boundary to the long-range solutions:
Propagation using a one-sided Green function
r 
  r  
At long range only the
monopole remains:
r  ro
  ro 
1
Vii ' (r  )   ii '
r
Monopole solutions are
analytically known.
Outer Core:
Multipole-Moment Interaction Region
(Monopole + Dipole / Quadrupole)
 fi (r ) 


 gi ( r ) 
Regular
Coulomb Wave
Irregular
Coulomb Wave
r 
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Altunata, Coy, Field, MI11
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R-matrix theory unifies continuum and bound state calculations
The log-derivative R-matrix hides the complexity of the core.
Propagating the wavefunction outward from the core yields the K
matrix.
 (r  , E )  f (r , E )  g (r , E ) K ( E )
ij
i
ij
i
ij
The K (reaction) matrix contains all dynamics – bound state energies
to scattering phase shifts.
“REACTION MATRIX”
Resonances
“Scattering Matrix”
S
1  iK 1  iK 
1
K
1
( ) Arc tan( K )

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.
June 19, 2006
Altunata, Coy, Field, MI11
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Long-range and Short-range Reaction Matrices
Spherically Symmetric Coulomb
Potential:
Separation in
spherical harmonics at
long range ( l )
r 
r  rc
Long-Range Reaction
Matrix, K,
in the l - representation
r  ro
Coulomb
+ Dipole potential:
Short-Range Reaction
Matrix, K,
in the l - representation
Separation in
Dipolar harmonics at
short range ( l )
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The entire calculation:
From short range to long range
core
dipole+coulomb+core
r  ro
R-matrix
Variational
condition for the Rmatrix reduces to a
generalized
eigenvalue problem
based on the full
core Hamiltonian.
June 19, 2006
r  rc
dipole+coulomb
K-matrix
Self-consistent shortrange representation
of the wavefunction
beyond the core
(Dipole-reduced,
fractional-l,
reaction matrix).
Smooth energy
dependence, accurate
quantum defects.
Altunata, Coy, Field, MI11
r 
K-matrix
Valid only in the long
range integer-l
representation.
Determines long-range
properties like photoionization
cross sections.
7
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
June 19, 2006
Altunata, Coy, Field, MI11
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Results: K Eigenquantum defects vs Energy agree with experiment
2Σ
states
2Π
states
2Δ
states
• Experimental quantum defects are calculated by deperturbing data
for rotating molecule effects.
•Using dipole-reduced K matrix is essential.
June 19, 2006
Altunata, Coy, Field, MI11
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Extend into the continuum: The energy dependence of the
0.88 2S+ Eigenquantum defect is too large
0.88 2 S+ 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.
June 19, 2006
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What is a Shape Resonance?
(a.k.a. Accordion Resonance)
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+
Shape resonance is a
distorted accordion
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
(Mullikan’s rule or “recapitulation” )
is lost.
June 19, 2006
Altunata, Coy, Field, MI11
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Locating the Shape Resonance: The Lifetime Matrix
dS
Q  iS
dE
†
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.
qmax
d 
 TrQ  2
dE
En  Ec 
,
1
2(n   ( En )) 2
d dE
dn/dE = Rate of
change of energy shift
with respect to
excitation energy
June 19, 2006
Continuum
Excitation
Altunata,
Coy, Energy
Field, MI11
12
CaF Shape Resonance from the Lifetime Matrix
  0.15
ER  0.013
Broad
Resonance
The peak in qmax locates the shape resonance in the molecular
potential.
June 19, 2006
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Lifetime matrix eigenvector for its max eigenvalue
shows that 0.88 2Σ+ is most affected
Branching ratios:
Re-expand the shape
resonance in the CaF
eigenchannels
10.88 2S
Excited State:
Ca 2 F 
June 19, 2006
Excluded volume
about F- leads
to large time delay:
Resonance in the
0.88 series
z
Altunata, Coy, Field, MI11
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Understanding the Shape Resonance:
A super-simple adiabatic approximation
Assume the electronic motion in a collision
channel is governed by a central potential
CaF Adiabatic Potential Curves
with centrifugal barrier.
For Electronic Motion, R=3.1
Vl (r )  Vl
core
l (l  1)
(r ) 
2r 2
Make an adiabatic
approximation for the
electronic radial
coordinate to define a 1D potential:
 i
lˆ 2 
i
V
(
r
)


(
r
,

)

V
( r ) l ( r , )

l
l
2 

2
r


June 19, 2006
Altunata, Coy, Field, MI11
Ca
2
F
15
WKB methods can be used to decide if there is a quasi-bound
level inside the adiabatic potential barrier
R=3.54 Bohr
potential
WKB phase
Lifetime
R=3.1 Bohr
A quasi-bound state exists with a lifetime like that from the accurate calculation!
June 19, 2006
Altunata, Coy, Field, MI11
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CaF Shape Resonance is due to F- – e- repulsion:
Adiabatic breakdown at the F position
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.
June 19, 2006
Altunata, Coy, Field, MI11
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K ↔ K: Long range field effects in photo-ionization
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 Defects for Ca F
Dipole-reduced quantum defects are
nearly linear with energy, as predicted
by the Rydberg-Ritz formula,
extrapolate smoothly, and match
experimental quantum defects.
3.54
1
Eigenquantum Defects
0.8
0.6
0.4
0.2
0.02
0.04
June 19, 2006
0.06
0.08
Energy au
0.1
0.12
Altunata, Coy, Field, MI11
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An anti-crossing in K quantum defects
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
Partial l Character
A
1
0.8
0.55s 2
0.6
0.55p 2
0.4
0.88s 2
0.2
0.88p 2
0
0
0.02 0.04 0.06 0.08 0.1 0.12
Energy au
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.
June 19, 2006
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Photo-ionization anisotropy from 10.55 2S+
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.
E  0.01
E  0.15
E  0.03
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Conclusions
•
•
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 Fion.
•
•
•
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.
June 19, 2006
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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)
June 19, 2006
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BaF
June 19, 2006
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Rydberg State Wavefunctions in the
n = 10 Region ( I )
(A) X 2S Ground State
The X state has no amplitude
close to F. The wavefunction
is confined within 5 Bohr of the
Ca nucleus.
June 19, 2006
(B) 10.55 2S Excited State
High Rydberg member of the
0.55 series, built on the X state
terminus. Also polarized behind
the Ca nucleus.
Altunata, Coy, Field, MI11
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Rydberg State Wavefunctions in the
n = 10 Region ( II )
(C) 10.88 2S Excited State
High Rydberg member of the
0.88 series. The electronic
wavefunction is polarized
behind the FJune 19, 2006
(D) 10.16 2S Excited State
High Rydberg member of the
0.16 series. The wavefunction is
d-f mixed.
Altunata, Coy, Field, MI11
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Small value of the quantum defect corresponds to the hydrogenic
limit
(D) 10.08 2S Excited State
Non-penetrating Rydberg
state. The wavefunction is
a dominant f state.
1
E  Ec  2
2n
Spherical Symmetry is recovered
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