treat one or a few nuclei QM and remaining nuclei classically
: coordinates of the classical particles
: coordinates of the quantum particles
The motion of the quantum mechanical particles is described by the Hamiltonian
Example:
proton transfer reaction in solution
-treat hydrogen atom being transferred QM
-treat remaining nuclei (donor, acceptor, solvent) classically
: typically a double well (as a function of )
for fixed
For each configuration of the classical particles, we can solve the time-independent
Schrödinger equation to calculate the proton quantum states
-the lowest few states can typically be characterized as being localized in either
the reactant (covalent) or product (ionic) well
-a reaction is defined as starting localized in the reactant well and ending localized in the product well
-the average time for this to occur is the rate of the proton transfer reaction
A. Adiabatic methods
-system remains in a single adiabatic state (typically the ground state)
-assumes that the proton can respond instantaneously to changes in classical coordinates
-adiabatic limit: low narrow barrier, large tunnel splitting
Algorithm
Repeat the following 3 steps
1. For a given classical configuration, calculate the proton quantum states by
solving the Schrödinger equation:
2. Calculate the Hellmann-Feynman forces on the classical particles using the
ground proton quantum state:
3. Use these forces to move the classical particles using Newton's equations of motion
Disadvantage: adiabatic approximation not always valid (i.e. transitions to excited
proton quantum states can be significant)
B. Mixed state, TDSCF
-expand time-dependent wavefunction in terms of adiabatic states
-integrate time-dependent Schrodinger equation to obtain quantum amplitudes
-classical particles move according to average
path due to mixture of adiabatic states
1. For a given classical configuration, calculate the proton quantum states by solving the Schrödinger equation
2. Define a time-dependent wave function as a linear combination of these adiabatic states:
and integrate the time-dependent Schrödinger equation:
to get the time-dependent wave function (i.e. to calculate
at each time t)
3. Calculate the Hellmann-Feynman forces on the classical particles
using the time-dependent wave function
4. Use these forces to move the classical particles using Newton's equations of motion
Disadvantage: does not accurately describe branching processes (i.e. processes
involving multiple channels or pathways)
C. Surface hopping
S. Hammes-Schiffer and J.C. Tully, J. Chem. Phys. 101, 4657 (1994).
-system remains in single adiabatic state except for instantaneous switches from one state
to another
-average over an ensemble of trajectories
-integrate time-dependent Schrodinger equation to determine quantum amplitudes at all times
-phase coherence maintained (or can be damped if desired)
-probabilistic algorithm used to determine when switches occur:
ensures that fraction of trajectories in any state j at any time t is quantum probability
for state j (as determined by integration of time-dependent Schrodinger equation)
Algorithm:
1. For a given classical configuration, calculate the proton quantum states by solving the Schrödinger equation:
2. Define a time-dependent wave function as a linear combination of
these adiabatic states:
and integrate the time-dependent Schrodinger equation:
to get the quantum amplitudes at all times t
The probability of being in state n at time t is 
.
3. Calculate the Hellmann-Feynman forces on the classical particles
using the occupied adiabatic state k
4. Use these forces to move the classical particles using Newton's equations of motion
5. Use a probabilistic algorithm (that depends on the coefficients Cn and the
nonadiabatic coupling) to determine if the system should switch
adiabatic states. If a switch occurs, then adjust the velocities to conserve energy.
The probabilistic algorithm ensures that for a large ensemble of trajectories,
the fraction in any state n at any time t is .
Equilibrium and dynamical properties can be obtained by averaging over an ensemble of trajectories.
Advantages
-valid in adiabatic and nonadiabatic limits and intermediate regime
-can accurately describe branching processes
D. Path integral methods
Represent the hydrogen atom as a string of beads, and perform slightly altered
classical molecular dynamics simulations on these beads together with the heavier
particles in the system.
Problem: In general, these methods can be used to calculate only equilibrium properties.
Methods are currently being developed to use path integral methods to
calculate dynamical properties
See, for example,
J. Cao and G. A. Voth, J. Chem. Phys. 101, 6157 (1994).
Centroid molecular dynamics
uses the Feynman path integral formalism to produce an effective ``QM" potential on which the hydrogen atom moves classically.
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