Perturbations I Have Known and Loved Robert W. Field June 21, th International Symposium on Molecular Spectroscopy Columbus, Ohio

Outline Assignment depends on textbook patterns A perturbation is a broken pattern Albin Lagerqvist and Richard Barrow: collectors Patterns of broken patterns o Grouping perturbers together o Electronic Symmetry o Vibrational Assignments Polyatomic Molecules: Vibrational Polyads o Membership, Scaling, and Emergence of New Stuff Large Amplitude Motion States vs. “Ergodic” States o “Isomerization States” as example of LAM states Broken Polyad Patterns: Isomerization Barrier Phase Space Structures: Bifurcations and Chaos

A Spectrum is more Regular than it Looks

Perturbation-Free and Perturbed Bands of SiO (0,0) (1,0)

R(44) and P(46) lines are strong: Why?

Proc. Phys. Soc. A 63, 1132 (1950)    Goldilocks

Patterns of Broken Patterns Level Crossings: Grouping Perturbers Together Multiple Level Crossings: Multiplet States

Proc. Phys. Soc. A 63, 1132 (1950) Heavy lines: perturbed state Light lines: perturber states Tie lines: same-Δv perturbations

Arkiv f. Fysik 18, 543 (1960) Thin curves: predicted Heavy curves: observed 1 Π~ 3 Σ -

Restoring Order Deperturbation Model Lifetimes from Eigenvectors Interference Effects

No Systematic Residuals, Accurate Mixing Fractions

A(0)~e(1) e-SymmetryRadiative Decay RatesCO Rates calculated from eigenvectors are more accurate and complete than measured 1/τ values

R, P Intensity Anomalies at J’= 48 but not J’=18   J’=18 J’=48

Behind the Broken Patterns Numerical Values of Perturbation Matrix Elements The “Matrix Element Method” for Absolute Vibrational Assignments A Lucky (?) Break! Stationary Phase

J. Chem. Phys. 60, 2400 (1974)

Patterns of Pattern-Breakers Inter-relationships between the electronic factors of perturbation matrix elements between pairs of states that belong to the σπ 4 π* and σ 2 π 3 π* configurations (all 10 valence electron molecules)

Based on a suggestion by Bill Klemperer

Self-Replicating Massively-Broken Patterns Vibrational Polyads

Nondegenerate Perturbation Theory Fails when the n,m matrix element is not small with respect to the n,m energy denominator Then we must diagonalize a 2x2 (or much larger) matrix of quasi-degenerate basis states This is the effective Hamiltonian  Polyad

Matrix Element and Membership Scaling Suppose ω 1 ≈2ω 2 The “polyad quantum number” is N=2v 1 +v 2 Basis states with the same polyad quantum number are automatically near-degenerate H v1,v2 ~ k 122 Q 1 Q 2 2 matrix elements scale as v 1 1/2 v 2 N=3 contains (v 1,v 2 )=(0,3),(1,1) N=4 contains (0,4),(1,2),(2,0) Membership and matrix elements both increase with N—MAGIC HAPPENS!

Scaling of Anharmonic Vibrational Matrix Elements

Polyad Examples Emergence of “Isomerization States” in HCP o Large Amplitude Motion wavefunctions are arranged along isomerization path Messy in one basis set, neat in another o HCCH S 0 normal mode to local mode Each class of isomerization barrier has a unique signature o HCCH S 1 trans-cis isomerization Classical Mechanics from Quantum Mechanics o HCCH S 0 structure of phase space

Isomerization State Falls Out of the Bottom of the Polyad 

r r r r [(0,16)] [(0,18)] [(0,20)] [(0,22)]

r r r r  r [(0,32)] Polyad Highest E Lowest E

Isomerization State: Large Amplitude Motion Along the Isomerization Path (0,40,0) I

Broken Patterns of Broken Patterns Emergence of a better basis set Violation of matrix element scaling rules Indirect interactions Interference effects

Order Out of Chaos: S 0 HCCH Spaghetti Diagram Normal modes are good Local modes are ok

Quanta in Mode 3 Make B 2 Polyad Explode

Example of polyad breakdown: K 4466 B2B2 B3B3 v 3 = v 3 = v 3 =

From Quantum to Classical The Structure of Phase Space Classical Chaos from QM H eff Emergence of new classes of regular vibrational motions

Heisenberg Correspondence Principle Go from Quantum Mechanical H eff to a Classical Mechanical Action( I )-Angle( ϕ ) H Run Trajectories: Compute Surface of Section Periodic Orbits, Bifurcations, Chaos: Structure of Phase Space

Onset of Classical Chaos Near Middle of Polyad

Classical Dynamics Near 15,000 cm –1 N bend =22 polyad bottom top

Interesting Things Happen at High Excitation Energy And molecules like nothing better than to tell us about them!

Guides on My Magical Mystery Tour Bill Klemperer Richard Barrow Albin Lagerqvist Herb Broida Hélène Lefebvre-Brion Anthony Merer Mike Kellman Howard Taylor Rick Gottscho Haruki Ishikawa David Jonas Kaoru Yamanouchi Matt Jacobson Adam Steeves Josh Baraban NSF, DOE, AFOSR, ACS- PRF

Some References The Spectra and Dynamics of Diatomic Molecules, Elsevier 2004 Effective Hamiltonians Handbook of High Resolution Spectroscopies, M. Quack and F. Merkt (eds) (2011) HCP-CPH: Caught in the Act! H. Ishikawa, Ann. Revs. Phys. Chem. 50, 443 (1999) Acetylene at the Threshold of Isomerization M. Jacobson, J. Phys. Chem. 104, 3086 (2000)

Proc. Phys. Soc. A 63, 1132 (1950)

Homogeneous vs. Heterogeneous Perturbation

Elsevier 2004

Predissociation is Prominent in Emission but Absent in Absorption: Why?

Christmas Tree Diagram

Arkiv f. Fysik 14, 387 (1958)

Intensity Interference Effects: Vanishing Point Depends on μ 10 /μ 20

Lambda Doubling in a Π State is Caused by Remote Σ States