INTERFERENCE AND QUANTIZATION IN SEMICLASSICAL VIBRATIONAL RESPONSE FUNCTIONS Scott Gruenbaum...
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INTERFERENCE AND INTERFERENCE AND QUANTIZATION IN QUANTIZATION IN SEMICLASSICAL VIBRATIONAL SEMICLASSICAL VIBRATIONAL RESPONSE FUNCTIONS RESPONSE FUNCTIONS Scott Gruenbaum Department of Chemistry and Chemical Biology Cornell University
Transcript of INTERFERENCE AND QUANTIZATION IN SEMICLASSICAL VIBRATIONAL RESPONSE FUNCTIONS Scott Gruenbaum...
INTERFERENCE AND INTERFERENCE AND QUANTIZATION IN QUANTIZATION IN
SEMICLASSICAL VIBRATIONAL SEMICLASSICAL VIBRATIONAL RESPONSE FUNCTIONSRESPONSE FUNCTIONS
INTERFERENCE AND INTERFERENCE AND QUANTIZATION IN QUANTIZATION IN
SEMICLASSICAL VIBRATIONAL SEMICLASSICAL VIBRATIONAL RESPONSE FUNCTIONSRESPONSE FUNCTIONS
Scott Gruenbaum
Department of Chemistry and Chemical Biology
Cornell University
Response TheoryResponse Theory
Use perturbation theory!
? =
+ +
+ + …
equilibrium dynamics
)()(ˆ)( ttHtt
i
)0()(ˆ)( tKtR(3)(t1,t2,t3)
R(1)(t)
R(2)(t1,t2)
t2
t
t1
t2
t3
t1
Quantum vs. ClassicalQuantum 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
Example: Morse OscillatorExample: Morse Oscillator
Quantum
Classical
Semiclassical MethodsSemiclassical 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)^
Semiclassical MethodsSemiclassical Methods
Herman - Kluk propagator:
dzzetzCztK tziSt 0
/),(),(2
1)(ˆ
Good:
Bad:
1) Only uses classical inputs2) Exact for harmonic systems3) Exact in classical limit
1) Wildly oscillatory2) Unbounded increase with time
Herman, Kluk, Chem Phys, 1984.
Semiclassical Response FunctionSemiclassical Response Function
)()(
),(),()2(
211221
/)(2
*12
21
tzzzzzf
etzCtzCdzdz
ttcl
tSi
)()1( tRHK
Quantitative agreement!
Noid, Ezra, Loring, J Chem Phys, 2003.
Semiclassical Response FunctionSemiclassical Response Function
Noid, Ezra, Loring, J Chem Phys, 2003.
R(3)(t,0,t)
Quantitative agreement for nonlinear response functions too!
How does it work?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
Time Dependent Semiclassical Quantization
Time Dependent Semiclassical Quantization
Integration over the phase generates peaks in the energy distribution of the classical trajectories.
Classical energy distribution
time ≈ 0
Time Dependent Semiclassical Quantization
Time Dependent Semiclassical Quantization
Integration over the phase generates peaks in the energy distribution of the classical trajectories.
Semiclassical energy distribution
time > 0
Mean Trajectory ApproximationMean Trajectory ApproximationBy analyzing the semiclassical response function, we can simplify the calculation:
R(1)(t) p(0)q(t) fcl (z) 1 2 cos k k
dz
Numerical trajectories
Approximate quantization
Classical mechanics
By analyzing the semiclassical response function, we can simplify the calculation:
R(1)(t) p(0)q(t) fcl (z) 1 2 cos k k
dz
Quantum Classical (k=0)
Mean Trajectory ApproximationMean Trajectory Approximation
By analyzing the semiclassical response function, we can simplify the calculation:
R(1)(t) p(0)q(t) fcl (z) 1 2 cos k k
dz
Quantum First recurrence (k=0,1)
Mean Trajectory ApproximationMean Trajectory Approximation
By analyzing the semiclassical response function, we can simplify the calculation:
R(1)(t) p(0)q(t) fcl (z) 1 2 cos k k
dz
Quantum Two recurrences (k=0,1,2)
Mean Trajectory ApproximationMean Trajectory Approximation
By analyzing the semiclassical response function, we can simplify the calculation:
R(1)(t) p(0)q(t) fcl (z) 1 2 cos k k
dz
Quantum Three recurrences (k=0,1,2,3)
Mean Trajectory ApproximationMean Trajectory Approximation
What’s next?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
ConclusionsConclusions
• Semiclassical propagators give quantitatively accurate response functions
• However, the calculation is numerically challenging
• We have simplified the semiclassical response function without sacrificing accuracy
AcknowledgmentsAcknowledgments
• Roger Loring
• NSF Grants CHE0413992 and CHE0743299
Response Functions-- MathResponse Functions-- Math
)(ˆˆ,ˆ)(ˆˆ)()1( tKxtKxTri
tR
Quantum:
dipole state of systemtime evolution
Rcl(1)(t) x(t)p(0) fcl (z) dzClassical:
distributionclassical trajectory
