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Appendix A: Wave packet MD equations of motion

The Gaussian wave packets used in our wave packet molecular dynamics contain a radial momentum term that we have only found once in the literature [38]. We give our motivation for this term, and a derivation of the corresponding equation of motion, below.

Consider the wave packet $ \Psi = \exp({i \mathbf{p}_{x} \cdot \mathbf{x}}) \cdot \exp(-a(r-x)^{2})$. Heller [5] showed that substituting this wavefunction into the time dependent Schrodinger equation gives the Hamilton equations of motion $ \mathbf{p_{x}} = m \, \dot{\mathbf{x}}$ and $ \dot{\mathbf{p_{x}}} = -\nabla V$, consistent with Ehrenfest's theorem, which states that the average position of a wavefunction follows a classical trajectory.

In the above wave packet, $ x$ and $ p_{x}$ are real variables that are conjugate to each other. Heller derived an equation of motion for $ a$ as well, but only for complex $ a$; substituting $ \Psi = \exp(-a x^{2})$ into the time-dependent Schrodinger equation with a harmonic potential gives (taking $ \hbar = 1$)

$\displaystyle i \frac{d\Psi}{dt}$ $\displaystyle =$ $\displaystyle -\frac{1}{2m} \frac{\partial^{2} \Psi}{d x^{2}} + \frac{1}{2} k x^{2} \Psi$ (3.17)
$\displaystyle -i \dot{a} x^{2}$ $\displaystyle =$ $\displaystyle -\frac{1}{2m}(4 a^{2} x^{2} - 2 a) + \frac{1}{2} k x^{2}$ (3.18)
$\displaystyle i \dot{a}$ $\displaystyle =$ $\displaystyle \left(\frac{2}{m}\right) a^{2} - \frac{1}{2} k.$ (3.19)

To begin, we examine the time evolution of the wavefunction $ \Psi(t=0) = exp(-a x^{2})$ when there is no external potential:

$\displaystyle i \dot{a} = \frac{2}{m} a^{2} \Rightarrow a = \frac{a_{0}}{1 + (2 a0/m) i t}.$ (3.20)

Then define
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle Re(a) = \frac{a_{0}}{1 + (4 a_{0}^{2} / m^{2}) t^{2}}$ (3.21)
$\displaystyle p_{\alpha}$ $\displaystyle =$ $\displaystyle \frac{m}{4} \dot{\alpha} = -\frac{a_{0} m / 4}{(1 + (4 a_{0}^{2}/m^{2}) t^{2})^{2}} \cdot \frac{8 a_{0}^{2}}{m^{2}} \cdot t = \alpha Im(a)$ (3.22)

which gives us
$\displaystyle \Psi$ $\displaystyle =$ $\displaystyle \exp(-(Re(a) + i\,Im(a)) x^{2}) = \exp(\alpha + i \, p_{\alpha} / \alpha)$  
  $\displaystyle =$ $\displaystyle \exp\left(-\left(\frac{1}{s^{2}} - \frac{2 p_{s}}{s} i\right) x^{2}\right)$ (3.23)

where in the last step we have made the change of variables $ \alpha = 1/s^{2}$.

We substitute $ a = 1/s^{2} - 2 p_{s} / s$ into equation 3.19 to derive equations of motion for $ s$ and $ p_{s}$:

$\displaystyle -\frac{2}{s^{3}} \dot{s}\,i + \frac{2 \dot{p_{s}}}{s} - \frac{2 p...
...- \frac{4 p_{s}}{s^{3}}\,i - \frac{4 p_{s}^{2}}{s^{2}} \right) - \frac{1}{2} k.$ (3.24)

Equating imaginary parts:

$\displaystyle -\frac{2}{s^{3}}\,\dot{s}\,i = -\frac{2}{m}\,\frac{4 p_{s}}{s^{3}}\,i \ \Rightarrow \ \boxed{p_{s} = \frac{m}{4} \dot{s}}$ (3.25)

Equating real parts:

$\displaystyle \frac{2 \dot{p_{s}}}{s} = \frac{2}{m} - \frac{1}{2} k \ \Rightarr...
...\ \textrm{where}\ E = \frac{1}{2} \frac{1}{r^{2}} + \frac{1}{8} k r^{2},\ m = 1$ (3.26)

which gives us back the Hamilton relations of equation 3.7. In a three-dimensional spherical Gaussian wave packet, there is one radial coordinate but three dimensions affecting its variation. The end effect is that each dimension contributes a mass factor of $ 1/4$ to equation 3.7. From the equations of motion, it also follows that if we define a kinetic energy $ T$ as

$\displaystyle T = \sum_{i} \frac{1}{2} m_{i} v^{2} + \frac{1}{2} \left(\frac{3}{4} m_{i}\right) v_{s}^{2}$ (3.27)

that the total energy $ T + V(x, s)$ is a constant of motion. The kinetic energy of motion $ T$ is not to be confused with the electronic kinetic energy $ 3/2 r_{e}^{-2}$ which appears in $ V(x, s)$.

The wave packet equations of motion are exact for harmonic potentials, but we do not know how well they describe the wave packets in anharmonic potentials. To test our approximation, we propagated wave packets numerically on a 1D line using a discretized version of the time-dependent Schrodinger equation [39], and compared the average position and width of the wave packet to a pure Gaussian wave packet propagated with WPMD. As expected, the expansion of a free electron and the oscillations of a harmonic oscillator matched in both models to within the error of the simulation (Figure 3.6).

Figure 3.6: Gaussian and exact wave packet dynamics match for free particle and harmonic oscillator potentials.

We then tested the double well potential $ V = 1/20 x^{4} - 1/2 x^{2}$, giving the wave packet enough energy to traverse the center barrier. In the exact simulation, the wave packet bounced back and forth twice, but quickly spread out and delocalized over both wells, so that both the position and size reached a constant value. In contrast, the Gaussian wave packet showed no signs of damping, and had more rapid radial oscillations than in the exact case (Figure 3.7).

Figure 3.7: Exact and Gaussian wave packet dynamics for a double well potential.

Over a short time interval, the two models matched well. We conclude that in systems where electrons are well-localized, wave packet molecular dynamics should describe well how electrons move; but it may overemphasize radial oscillations that in a real system would be damped out by quantum interference. Our practical choice of an artificially heavy electron mass may compensate somewhat for this difference.

next up previous contents
Next: Appendix B: Adiabatic excited Up: The electron force field, Previous: History and current progress   Contents
Julius 2008-04-29