Variational and Diffusion Quantum Monte Carlo Calculations with the CASINO Code

TL;DR

The paper surveys variational and diffusion quantum Monte Carlo (VMC and DMC) methods as implemented in the CASINO code, emphasizing methodological advances over the last decade, including advanced wave-function forms (generalized Jastrow factors, multideterminants, and geminals), and their impact on condensed-matter and model-system calculations. It discusses critical aspects such as fixed-phase/fixed-node constraints, imaginary-time propagation, time-step control, and finite-size corrections, along with computational efficiency and scaling on modern HPC architectures. The authors review extensive applications to excitonic complexes, HEGs, van der Waals systems, solid hydrogen, and electron-hole bilayers, illustrating QMC’s accuracy for dispersion interactions and excited-state properties, while acknowledging limitations in systems with strong static correlation or very large electron counts. The article also outlines future directions, including synergy with FCIQMC, improved pseudopotentials and spin-orbit/vibrational treatments, and the potential to extract forces, all framed within ongoing developments in algorithms and computer architectures. Overall, the work underscores QMC’s niche as a highly accurate, system-size-limited but scalable tool that complements DFT and GW in challenging electronic-structure problems and benchmark studies.

Abstract

We present an overview of the variational and diffusion quantum Monte Carlo methods as implemented in the CASINO program. We particularly focus on developments made in the last decade, describing state-of-the-art quantum Monte Carlo algorithms and software and discussing their strengths and their weaknesses. We review a range of recent applications of CASINO.

Figures (27)

• Figure 1: Reblocked standard error in the VMC energy of an all-electron lithium atom as a function of blocking length. The blocking length is the number of successive original data points averaged over to obtain each new data point).
• Figure 2: Local energy as a function of the $x$ coordinate of an electron in a carbon atom using a trial wave function which does not satisfy the electron-nucleus cusp condition ("no cusp") and wave functions which impose the electron-nucleus cusp condition by modifying the Gaussian orbitals near the nucleus ("orbital cusp") and via the Jastrow factor ("Jastrow cusp"). The nucleus is at $x=0$, and the other five electrons are at random positions.
• Figure 3: Decay of the average local energy of a set of walkers whose dynamics is governed by the fixed-phase imaginary-time Schrödinger equation for the ground state and an excited state of a 610-electron 2D HEG at density parameter $r_{\rm s}=10$. Results are shown for (i) a real, closed-shell, ground-state, Slater--Jastrow trial wave function with plane-wave orbitals and (ii) a complex, excited-state wave function in which a single electron has been added to a plane-wave orbital outside the Fermi surface without reoptimization of the ground-state Jastrow factor. The DMC time step is $0.4$ Ha$^{-1}$ and the target population is 12,000 walkers, so that the statistical error on the energy at each time step is small.
• Figure 4: Population-control bias in the $\Gamma$-point DMC energy per electron of a 3D paramagnetic electron gas of density parameter $r_{\rm s}=4$ at three different system sizes using a fixed time step of $0.1$ Ha$^{-1}$. Two different DTN Jastrow factors were used: $J_1$, consisting of a $u$ term with cutoff length $L=L_{\rm WS}/2$, and $J_2$, consisting of a better-quality $u$ term with $L=L_{\rm WS}$. The lines are linear fits to the data, and the bias is measured with respect to the value of the corresponding fit at infinite target population.
• Figure 5: Time-step bias in the $\Gamma$-point DMC energy per electron of a 3D paramagnetic electron gas of density parameter $r_{\rm s}=4$ at three different system sizes using a fixed target population of $512$ walkers. Two different DTN Jastrow factors were used: $J_1$, consisting of a $u$ term with a fixed cutoff length $L=7.82$ bohr, which equals $L_{\rm WS}/2$ at $N=114$, and $J_2$, consisting of a better-quality $u$ term with $L=L_{\rm WS}$. The lines are quadratic fits to the data, and the bias is measured with respect to the value of the fit at zero time step. The shaded areas indicate the range of time steps for which the bias is expected to be linear from length-scale considerations ($\tau \lesssim 0.01r_{\rm s}^2$). Lee_2010