What Do We Mean by "Electronic Structure"? Jeffrey J. Kay Massachusetts Institute of Technology Christian Jungen Centre National de la Recherche Scientifique (CNRS) Robert W. Field Massachusetts Institute of Technology June 20, 2007 NSF

Scattering (MQDT) vs. Born- Oppenheimer Multichannel Quantum Defect Theory:  (Q) –What, where, why, and how much? –Global: all states, all dynamics Born-Oppenheimer: H(q,Q) –Stuff happens –State by state

π * quantum defect is the radial phase shift: Phase shift (and dependence on R, etc) tells you everything about electronic wavefunction Example: R-dependence of QD gives R-dependence of electronic wavefunction and governs strength of vibrationally nonadiabatic interactions μ = 0 μ = 0.25 nucleus Quantum Defect & Radial Wavefunction

What Does the Rydberg Electron See? Z=1 ion-core at long-range –Multipole moments and polarizabilities Z(r)>>1: Z eff ℓ >1 and not-round at short- range – phase shift  quantum defect matrix –l-mixing Electronic excitation of ion-core at short- range: permanent vs. temporary –Main subject of this talk

F–F– M 2+ x r C.O.M. r = 2.70 Å (r = 2.70 Å for CaF + ) Turning Points } Penetrating vs. Nonpenetrating Rydberg Orbitals

DIRECT INDIRECT A B + e-e- + e-e- B A A* B* e-e- AB + Electronically Excited Temporary Core Excitation

DIRECT INDIRECT A B + e-e- + B A A* B* e-e- AB + Electronically Excited Permanent Core Excitation

Pattern-Breaking = Resonance 3 kinds of broken patterns: from local to global –Local perturbation –Curve crossing –Channel crossing Each accommodated by a matrix deperturbation model

Reduced Term Values Level Shifts Two-Level Avoided Crossing

Two-State Avoided Crossing

K 11 K 22 K 12 μ α (1) μ α (2) μ 11 μ 22 (R-R e + ) Energy One parameter controls an infinite number of interactions Reaction Matrix Elements Quantum Defects Energy Levels (R-R e + ) Two-Channel Avoided Crossing

Lowest Electronic States of CaF X A B C D E E’ Ion RKR

Quantum Defect Functions AB + AB * Vertical energy difference = E ion - E nL Invert R-dependent Rydberg formula

Surprising Similarity to Ion: with one exception! RKR One deviation from linearity

Eigenquantum Defects vs. R RKR Note the absence of pairwise interactions. C 2 π A 2 π

Eigenquantum Defects vs. R RKR C 2 π “Extra” repulsion at small R “Relaxation” to smaller QD derivative at large R

Overlap Repulsion Fix this by mixing some F  orbital into Ca  orbitals to enforce orthogonality. Effectively transfers charge from F to Ca. At low n*, n* -3/2 scaled amplitude in inner lobes results in overlap with F . Violates exclusion principle! Ca + F-F- Ca δ+ Fδ-Fδ- This looks like configuration interaction. Ca + F-F- Ca δ+ Fδ-Fδ- Ca 0 F0F0 +

Core Excitation So the solution is to add covalent channels to our electronic reaction, K, and phase, P, matrices. There are two: Ca + F 0 ( 1 Σ + ) + e - Ca + F 0 ( 1 Π ) + e - These are: 1.The Sigma and Pi repulsive valence states 2.Low-n* termini of Rydberg series converging to the first and second electronically-excited states of the CaF + ion- core.

R = 3.6 a0 R = 3.2 a0 R = 4.0 a0 Driving Force for CI

n* = 3 Diagonal Elements R-Independent With channel coupling Without channel coupling Quantum Defect (R-R e + ) [a0] Two-Channel Problem

n* = 5 Diagonal Elements R-Independent With channel coupling Without channel coupling Quantum Defect (R-R e + ) [a0] Two-Channel Problem

Electronic Structure: Causal vs. Descriptive Electronic structure of cation embedded in MQDT description of neutral Unified view of influence of cation excited states –Permanent effects: autoionization and predissociation –Indirect effects: breakdown of propensity rules –New insight: low-n* distortion of potential curves

Multichannel Quantum Defect Theory (1) is phase shift (relative to H) n* δ ℓ is quantum defect independent of n dependent on ℓ independent of ℓ Eigenquantum defect R-dependent mixed ℓ 1+1+ Hydrogen e-e- Sodium e-e- CaF 20 + + e-e- reaction matrix phase matrix quantum defect matrix

- EigenvalueCondition: Reaction Matrix Phase Matrix N (rotation) and p (parity): Rigorously good Quantum Numbers Elements of Reaction Matrix: Non-Diagonal Reaction Matrix: Effective Principal Quantum Number Elements of Diagonal Phase Matrix: Quantum Defects Quantum Defect Theory [2]

Shape resonance

Diagonal qds and eqds Sigma n=3 and infinity Range 1.5 Idea is to show several E dependent channel crossings, make connection to shape resonance?

Diagonal qds and eqds Pi n=3 and infinity, range 1.5 Idea is to show some E dependent channel crossings

Diagonal qds and eqds Delta n=3 range of 1.5 n=infinity range of 1.5 Idea is to show absence of any E dependent channel crossings

Polarized Orbitals p - d p + d pπpπpπpπ dπdπdπdπ + Ca 2+ F-F- F-F- F-F- F-F- “normal polarization” “reverse polarization”

Beyond Ligand Field Theory What does the ligand field picture neglect? Two effects due to overlap of Ca + and F - orbitals: “Pauli Repulsion” (Destabilizing) Wavefunctions of two fermions with parallel spin must be orthogonal Metal and ligand orbitals mix to “orthogonalize” individual orbitals Exchange (Stabilizing) Exclusion principle: Result: Can be treated as a strong “repulsion” in an effective one-electron treatment. Coulomb (Mixing)

Core Excitation DIRECT INDIRECT B A + e-e- + e-e- A B B* A* e-e- (AB + + e - ) Electronically Excited

Orbitals that shield the Ca  from the F  have positive d   /dR

Quantum Defects Hydrogen AtomSodium Atom Principal Quantum Number (Integer) Effective Principal Quantum Number (Noninteger) QUANTUM DEFECT

Quantum Defects Sodium AtomMolecule Rydberg Series converging to each ion core energy level

Two-State Avoided Crossing

- Eigenvalue Condition: Reaction Matrix Phase Matrix N (rotation) and p (parity) are Good Quantum Numbers Elements of Non-Diagonal Reaction Matrix: Effective Principal Quantum Number Elements of Diagonal Phase Matrix: MQDT (2)

Two-Level Avoided Crossing

Quantum Defects Hydrogen Atom Sodium Atom Diatomic Molecule No Quantum Defect Nonzero Quantum Defect R-Dependent Nondiagonal Quantum Defect Matrix p+p+ Na + AB +

Core Excited Channels n* = 4 n* = 5 n* = 6 n* = 7 X 1 Σ + “n*” = 1.5 A 1 Σ + Red: Repulsive state as “core-excited” Rydberg state Black: Normal Rydberg states Ca 0 ( 1 S) + F 0 ( 2 P) Ca 2+ F - ( 1 Σ + ) + e - Ca 2+ F - ( 1 Σ + ) Ca + ( 2 S) + F 0 ( 2 P)

Cubic Fit The C 2 Π state Has a cubic R-dependence! RKR

Quantum Defect & Radial Wavefunction π * quantum defect is the radial phase shift: Phase shift (and dependence on R, etc) tells you everything about electronic wavefunction Example: R-dependence of QD gives R-dependence of electronic wavefunction and governs strength of vibrationally nonadiabatic interactions μ = 0 μ = nucleus

Two-Channel Avoided Crossing