Fixed-node Diffusion Monte Carlo method

The FN-DMC method [RCAJ82] modifies the DMC method to allow an approximate treatment of excited states of many-body systems. The idea of the FN-DMC method is to treat excited states by ``enforcing'' the positive definiteness of the probability distribution $f({\vec r},\tau)=\psi_T({\bf R})\psi({\vec r},\tau)$. The function $f({\vec r},\tau)$ is positive definite everywhere in configuration space, and can hence be interpreted as a probability distribution, if $\psi_T({\bf R})$ and $\psi({\vec r},\tau)$ change sign together, and thus share the same (high-dimensional) nodal surface. To ensure positive definiteness of $f({\vec r},\tau)$, the trial wave function $\psi_T({\bf R})$ imposes a nodal constraint, which is fixed during the calculation. Within this constraint, the function $f({\vec r},\tau)$ is propagated (following a scheme very similar to that outlined in Sec. 2.3), and reaches an asymptotic distribution for large $\tau\to\infty$, $f({\vec r},\tau)=\psi_T({\bf R})\,\psi({\vec r},\tau)$. In the FN-DMC method, $\psi_T({\bf R})$ is an approximation to the exact excited eigenfunction of the many-body Schrödinger equation (and not the groundstate wave function as in the DMC method). It can be proven that, due to the nodal constraint, the fixed-node energy is a variational upper bound to the exact eigenenergy for a given symmetry [RCAJ82]. In particular, if the nodal surface of $\psi_T({\bf R})$ were exact, then $\psi({\vec r},\tau)$ would be exact. Thus, the FN-DMC energy depends crucially on a good parameterization of the many-body nodal surface.

Thus, in a FN-DMC simulation the function $f({\bf R},\tau)=\psi_T({\bf R})\Psi({\bf R},\tau)$, where $\Psi({\bf R},\tau)$ denotes the wave function of the system and $\psi_T({\bf R})$ is a trial function used for importance sampling, is evolved in imaginary time according to the Schrödinger equation

$$-\frac{\partial f({\bf R},\tau)}{\partial\tau}= - D\nabla_{\bf R}^2 f({\bf R},\tau) + D \nabla_{\bf R}[{\bf F}({\bf R}) f({\bf R},\tau)] + [E_L({\bf R})-E_{ref}]f({\bf R},\tau)$$ (2.36)

In the above equation ${\bf R}=({\bf r}_1,...,{\bf r}_N)$, $E_L({\bf R})= \psi_T({\bf R})^{-1}H\psi_T({\bf R})$ denotes the local energy, ${\bf F}({\bf R})=2\psi_T({\bf R})^{-1}\nabla_{\bf R} \psi_T({\bf R})$ is the quantum drift force, $D=\hbar^2/(2m)$ plays the role of an effective diffusion constant, and $E_{ref}$ is a reference energy introduced to stabilize the numerics. The energy and other observables of the state of the system are calculated from averages over the asymtpotic distribution function $f({\bf R},\tau\to\infty)$. To ensure positive definiteness of the probability distribution $f$ for fermions, the nodal structure of $\psi _T$ is imposed as a constraint during the calculation. It can be proved that, due to this nodal constraint, the calculated energy is an upper bound to the eigenenergy for a given symmetry [RCAJ82]. In particular, if the nodal surface of $\psi _T$ were exact, the fixed-node energy would also be exact.

Construction of the trial wave function, as well as evaluation of the energy, is described in Sec. 2.6.