Diffusion Monte Carlo

The central difficulty is now to obtain the Green's function. In general, the exact Green's function is not known, for if it were the Monte Carlo method would not be necessary! In this program we use a short-time approximation to obtain analytic (but approximate) Green's functions. This method is commonly called diffusion Monte Carlo (DMC) because of its underlying connection to a diffusion problem. The mapping of the imaginary time-dependent Schrödinger equation onto a classical diffusion equation is a connection noted at least as far back as Fermi.

Consider the imaginary-time electronic Schrödinger equation introduced in the previous section with the Hamiltonian now explicitly written out, i.e.

$$-{ \partial {\phi}( {\bf x},\tau ) \over \partial \tau} = D... ...{\curly V}}( {\bf x}) - E_T \right ) {\phi}( {\bf x}, \tau) ~,$$ (14)

where $D$, the ``diffusion'' constant, is $\hbar^2 / 2 m_e = 1/2$ in atomic units and $\hbox{{\curly V}}$ is the potential energy operator. As $\tau \to \infty$, asymptotically stationary behavior is obtained, and $\partial {\phi}/ \partial \tau= 0$. In this limit Eq. 14 becomes the time-independent electronic Schrödinger equation. Without the second term on the right hand side, Eq. 14 is the usual diffusion equation. Alternatively, ignoring the first term on the right hand side of Eq. 14 and retaining the second term results in a first-order rate equation or branching process whose ``rate constant'' is $( E_T - \hbox{{\curly V}})$. Both diffusion and rate processes can be simulated separately by the Monte Carlo method. It is therefore reasonable to expect that the entire equation could be simulated by a combined stochastic process consisting of diffusion plus branching.