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Reference methods for ground and excited states

In parameterizing and validating the electron force field, it is necessary to assemble reference data from experiments, and theoretical methods that are applicable to ground states and excited states. Most of the theoretical methods are well known, e.g., Hartree-Fock for computing uncorrelated energies, CCSD(T) for high-accuracy correlated energies. In two cases, though, we have studied systems beyond the scope of those methods: the uniform electron gas and dense hydrogen plasma under high pressure.

For those cases, we have used reference data obtained from stochastic methods, which are very expensive computationally but very accurate and in principle general to any system. Diffusion Monte Carlo [36] is a method for computing high-accuracy energies for ground states. We write the time-independent Schrodinger equation as a diffusion equation in imaginary time:

$\displaystyle \frac{\partial \Psi}{\partial t} = \frac{1}{2} \nabla^{2} \Psi + (E - V(\mathbf{x})) \Psi.$ (3.12)

Then the Green's function for this equation can be approximated for small time steps
    $\displaystyle \Psi(\mathbf{y}) = \int G(\mathbf{y}, \mathbf{x}; \tau) \Psi(\mathbf{x}) d\mathbf{x}$ (3.13)
    $\displaystyle G(\mathbf{y}, \mathbf{x}; \tau) \approx e^{-(\mathbf{y} - \mathbf{x})^{2}/2\tau}\ e^{-(V_{avg}(\mathbf{x})-E)}$ (3.14)

(which becomes exact as $ \tau \rightarrow 0$) and applied iteratively to form a probability distribution that converges to the true distribution over time. The energy is exact to within the position of the nodes, which must be specified in advance through a trial wavefunction; this trial wavefunction is also used to sample the distribution preferentially at places where the electron density is highest.

Path integral Monte Carlo [37] is a useful method to compute thermodynamic averages of quantum operators at finite temperature. We write the position density matrix operator as an integral over successive paths:

$\displaystyle \braket{\mathbf{y}\vert e^{-\mathcal{H}/k T}\vert\mathbf{x}} = \i...
...e^{-\tau \mathcal{H}}\vert\mathbf{y}}\,d\mathbf{R_{1}} \cdots d\mathbf{R_{N-1}}$ (3.15)

where $ \tau = 1 / (k T N)$ is the time step. Then each density matrix element can be evaluated in the short time limit:
$\displaystyle \braket{\mathbf{R_{i}}\vert e^{-\tau (\mathcal{T}+\mathcal{V})}\vert\mathbf{R_{i+1}}}$ $\displaystyle \approx$ $\displaystyle \braket{\mathbf{R_{i}}\vert e^{-\tau \mathcal{T}}\,e^{-\tau \mathcal{V}}\vert\mathbf{R_{i+1}}}$  
  $\displaystyle \propto$ $\displaystyle e^{-(\mathbf{R_{i}}-\mathbf{R_{j}})^{2}/2\tau} \cdot e^{-\tau V(\mathbf{R_{i}}) \delta(\mathbf{R_{i}}-\mathbf{R_{j}})}.$ (3.16)

The path is varied in a Monte Carlo procedure to evaluate the expectation value of the desired operator. The procedure works best for high temperatures; at lower temperatures, the number of path links N must be increased to keep $ \tau$ small.


next up previous contents
Next: History and current progress Up: The electron force field, Previous: Wave packet molecular dynamics   Contents
Julius 2008-04-29