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 weaknesses. We review a range of recent applications of casino.
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The use of the terms “static correlation” and “dynamic correlation” in the quantum chemistry literature is less clearcut than suggested by our QMC-focused definition in terms of qualitative and quantitative errors in the nodal surface of the wave function. Nevertheless, the definition we have used captures the idea that static correlations are due to the errors in the Hartree–Fock state, while dynamical correlations are due to the electrons avoiding each other.
For a strictly one-dimensional system in which the particles interact via the Coulomb 1/r potential, the wave function must go to zero at all coalescence points in order for the energy expectation value to be well-defined. In practice, the easiest way of achieving this requirement in a study of a one-dimensional electron system is to treat all the electrons as indistinguishable particles.253 Equation (9) continues to be valid for a one-dimensional system, provided the plus sign is selected, irrespective of the spins of the coalescing particles.
The approximation only makes sense under an integral sign. At any instant, the error in this approximation is proportional to , where W is the number of walkers. The error term is, by construction, zero on average.
If the trial wave function does not describe the lowest-energy state that transforms as a 1D irreducible representation of the symmetry group (e.g., if one were trying to calculate the energy of the 2s state of a hydrogen atom with an approximate trial wave function whose nodal surface is at the wrong radius), then the required equilibration imaginary-time period can be much larger than the subsequent decorrelation period in the statistics-accumulation phase. In this case, the equilibration imaginary-time scale is the time taken for the walker populations in high-energy nodal pockets to die out, which is given by the reciprocal of the difference of the pocket ground-state energy eigenvalues.
For a narrow-gap semiconductor, on the other hand, the sharpest features in deviate from free-electron behavior is , where a is the lattice constant of the primitive cell. Hence, the longest length scale for narrow-gap semiconductors is .
For metals, the longest length scale grows with system size; however, shorter length scales make the largest energy contributions.