1. INTERFERENCE AND QUANTIZATION IN SEMICLASSICAL VIBRATIONAL RESPONSE FUNCTIONS Scott Gruenbaum Department of Chemistry and Chemical Biology Cornell University

2. Response Theory Use perturbation theory! ? = + + + + … equilibrium dynamics R (3) (t 1,t 2,t 3 ) R (1) (t) R (2) (t 1,t 2 ) t2t2 t t1t1 t2t2 t3t3 t1t1

3. Quantum vs. Classical Quantum –Solve vibrational Schrödinger equation –Correct, but challenging for large systems Classical –Solve classical equations of motion (molecular dynamics) –Possible for very large systems (e.g. proteins) –Works well for heavy atoms

4. Example: Morse Oscillator   Quantum Classical

5. Semiclassical Methods Problem: quantum response theory is difficult, but the classical theory can be incorrect! We want to approximate the quantum response function using only classical information One solution: use a semiclassical approximation to K(t) ^

6. Semiclassical Methods Herman - Kluk propagator: Good: Bad: 1)Only uses classical inputs 2)Exact for harmonic systems 3)Exact in classical limit 1)Wildly oscillatory 2)Unbounded increase with time Herman, Kluk, Chem Phys, 1984.

7. Semiclassical Response Function Quantitative agreement! Noid, Ezra, Loring, J Chem Phys, 2003.

8. Semiclassical Response Function Noid, Ezra, Loring, J Chem Phys, 2003. R (3) (t,0,t) Quantitative agreement for nonlinear response functions too!

9. How does it work?  The height between steps in the oscillatory phase generates quantization of energy! Total phase of R (1) Gruenbaum, Loring, J Chem Phys, 2008. Classical mechanics

10. Time Dependent Semiclassical Quantization Integration over the phase generates peaks in the energy distribution of the classical trajectories. Classical energy distribution time ≈ 0

11. Time Dependent Semiclassical Quantization Integration over the phase generates peaks in the energy distribution of the classical trajectories. Semiclassical energy distribution time > 0

12. Mean Trajectory Approximation By analyzing the semiclassical response function, we can simplify the calculation: Numerical trajectories Approximate quantization Classical mechanics

13. By analyzing the semiclassical response function, we can simplify the calculation: Quantum Classical (k=0) Mean Trajectory Approximation

14. By analyzing the semiclassical response function, we can simplify the calculation: Quantum First recurrence (k=0,1) Mean Trajectory Approximation

15. By analyzing the semiclassical response function, we can simplify the calculation: Quantum Two recurrences (k=0,1,2) Mean Trajectory Approximation

16. By analyzing the semiclassical response function, we can simplify the calculation: Quantum Three recurrences (k=0,1,2,3) Mean Trajectory Approximation

17. What’s next? Analyze and calculate higher order response functions e.g. the vibrational echo R (3) (t,0,t) quantum classical semiclassical Morse oscillator

18. Conclusions Semiclassical propagators give quantitatively accurate response functions However, the calculation is numerically challenging We have simplified the semiclassical response function without sacrificing accuracy

19. Acknowledgments Roger Loring NSF Grants CHE0413992 and CHE0743299

20. Response Functions-- Math Quantum: dipolestate of system time evolution Classical: distributionclassical trajectory