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 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? Serhan N. Altunata, Stephen L. Coy, and Robert W. Field MIT Department of Chemistry

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

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, δΨ/Ψ, the logaritmic derivative of the wavefunction at r 0. Use a variational method for solution.

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

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.

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

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

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

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 Δ states 2 Σ states 2 Π states

June 19, 2006Altunata, Coy, Field, MI1110 Extend into the continuum: The energy dependence of the  + Eigenquantum defect is too large  + 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, 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 (Mullikan’s rule or “recapitulation” ) is lost. What is a Shape Resonance? (a.k.a. Accordion Resonance) Shape resonance is a distorted accordion

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

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

June 19, 2006Altunata, Coy, Field, MI1114 Lifetime matrix eigenvector for its max eigenvalue shows that Σ + 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   Excited State:

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

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!

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

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

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

June 19, 2006Altunata, Coy, Field, MI1120 Photo-ionization anisotropy from  + 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.

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

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), (2003) S.N Altunata, S.L. Coy and R.W. Field J. Chem. Phys. 123, (2005) S.N Altunata, S.L. Coy and R.W. Field J. Chem. Phys. 123, (2005) S.N Altunata, S.L. Coy and R.W. Field J Chem. Phys., 124, (2006)

June 19, 2006Altunata, Coy, Field, MI1123 BaF

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)   Excited State High Rydberg member of the 0.55 series, built on the X state terminus. Also polarized behind the Ca nucleus.

June 19, 2006Altunata, Coy, Field, MI1125 Rydberg State Wavefunctions in the = 10 Region ( II ) (C)   Excited State High Rydberg member of the 0.88 series. The electronic wavefunction is polarized behind the F - (D)   Excited State High Rydberg member of the 0.16 series. The wavefunction is d-f mixed.

June 19, 2006Altunata, Coy, Field, MI1126 Small value of the quantum defect corresponds to the hydrogenic limit (D)   Excited State Non-penetrating Rydberg state. The wavefunction is a dominant f state. Spherical Symmetry is recovered