q*****o
发帖数: 438 | 1 有一个题目，物理进入化学，所谓切入点（entry point）可能有用。就是把分子作为
弱耦合振子，那么每个振动模式上都有一个（量子）零点能。命题是如何让经典力学计
算也保持零点能。因为生计我现在不再做这些了。写了一些没写完。希望会有一点点用。
Introduction
For years molecules are modeled by balls and springs as an oscillator system
. Consequently, Newton's equations, or in more sophisticated forms of
Hamilton's equations of classical mechanics, are integrated numerically to
obtain chemically interesting properties such as spectra, reaction cross
sections, and reaction rate constants. This is often referred to as the
classical trajectory study of molecular systems.
There can be several weakly coupled oscillators in a molecule. Each
oscillator corresponds to a vibrational mode. While quantum mechanics
requires that each mode maintain a minimal amount of energy during the
entire time evolution (the zero point energy - ZPE), classical mechanics
does not require that. Therefore, in the classical trajectory study, energy
may completely flow out of one mode to be injected to another mode. In a
reasonably large molecule, ZPE energies flown out of several modes into one
mode can be large enough to break that chemical bond – causing so large an
amplitude of oscillation that dissociates the two connecting atoms.
Existing methods
The question arises on how to make a quantum ZPE correction in the classical
trajectory study. The existing methods are divided into three categories as
follows.
1. Active method
Enforce a hard wall around coordinate x=0 and momentum p=0 for each mode, i.
e. for all i’s trajectories are not allowed to enter into the region
bounded by
x2/2 + p2/2 = ei_min
where ei_min corresponds to the quantum ZPE of the ith mode. If a trajectory
gets close to entering the region, its momentum is artificially forced to
reverse direction or its momentum is artificially changed to let the
trajectory slide on the above minimum energy ellipse until it departs
outside of the ellipse. This method has been shown to lead to chaotic
classical dynamics, where dynamics is otherwise integrable, as this method
actually changes the Hamiltonian of the system.
2. Passive method
No hard wall of minimum energy is set, so that the Hamiltonian is unchanged.
However, the trajectories that enter the minimum energy ellipse are treated
as unphysical and are thrown away, unaccounted for in physical quantity
calculations. However, since the total energy of all modes is conserved,
this method tends to overestimate the mode energy of the specific mode where
many trajectories are tossed and underestimate the energies of other modes.
3. Semiclassical method
This method either treats some or all of oscillators semi-classically and
the rest of the oscillators still classically, by way of a constant width (
frozen) Gaussian or a variable width (thawed) Gaussian. Often times, the
weakest force constant(s) (shallowest potential(s)) oscillators are treated
semi-classically to prevent unphysical large amplitude oscillations (
dissociations - breaking of a chemical bond). This method can be
computationally costly for molecules of biological sizes.
Choice of initial conditions
Traditionally, the initial conditions of the classical trajectory study are
selected from the constant energy shell. That is, in a study corresponding
to the quantum ZPE, points are randomly sampled from
x2/2+p2/2=ei_min for all i’s.
Geometrically, the sampling is uniform on a multidimensional torus. This
method is reminiscent of the semi-classical action quantization of
quantizing the classical actions of circular, elliptic, and perhaps other
periodic trajectories to obtain bound eigen-energies of Bohr, Sommerfeld,
Masloff, and so on. The implicit assumption is that quantum eigenfunctions
correspond to classical periodic orbits with the corresponding quantized
actions. Due to overall energy conservation, all sampled points have an
equal total energy with each another. Therefore, in the subsequent time
evolution, some modes develop mode energy lower than the ZPE (ZPE violating)
and other modes develop mode energy higher than the ZPE. Geometrically, for
some modes the trajectory enters inside the torus and for other modes the
trajectory goes outside the torus.
A periodic trajectory in a truly isolated system would remind periodic
forever, as is a quantum eigenfunction. In reality, no system is truly
isolated and any periodic orbit is not forever, just as an eigenfunction is
not forever.
According to decoherence theory, a pure state of linear combination of
eigenfunctions decoheres to a definite probability distribution of non-
interfered classical phase space distribution with definite probabilities
corresponding to each eigenfunction. Therefore, the ground state
eigenfunction decoheres to a Gaussian distribution in phase space with
probability 1. For a system of multiple oscillators, the distribution is a
product of Gaussians of each mode. Sampling of initial conditions becomes a
Gaussian-weighted sample of each mode, with the most probable points around
x=p=0, the classical fixed point.
Applying a classical canonical transformation, the phase space Gaussian
distribution can generate the mode energy distribution of each mode,
therefore showing the percentage of the sampling points with an energy below
that of the ZPE. It is noted that the total energy constraint of all points
having an equal total energy, as in the torus sampling scenario, is relaxed
. Essentially, all points have different energies, only with an average
energy corresponding to the mode ZPE. A point with a lower-than ZPE in some
modes does not lead to its mode energies in other modes being higher than
the ZPE. This is where the relaxation from the stringent torus sampling lies
.
Work to do
A plot of a 2-torus
A plot of a 2-Gaussian
Promised canonical transformation from (x,p) to e_i
Perhaps Gaussian sampling of Danny Gerber's shallow potential |
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