Fixednode

Fixed-node refers to an approximation used in quantum Monte Carlo methods to address the fermion sign problem. It constrains the simulation to regions of configuration space defined by the nodes of a trial wavefunction, i.e., the surfaces where the wavefunction changes sign. In diffusion Monte Carlo and related projector methods, the fixed-node constraint forbids walkers from crossing these nodal surfaces, effectively imposing Dirichlet boundary conditions on each nodal region. The resulting energy is the fixed-node energy, which is an upper bound to the true fermionic ground-state energy; if the trial nodes were exact, the fixed-node result would be exact.

Nodes are typically obtained from a Slater determinant of single-particle orbitals, such as those from Hartree-Fock or density functional theory calculations, and may be improved with multi-determinant expansions or backflow transformations to better approximate the exact nodal surface. The fixed-node energy is variational with respect to the nodal surface: changing the trial wavefunction alters the nodes and thus the fixed-node energy.

Strengths of the fixed-node approach include its ability to yield highly accurate ground-state energies for many-electron systems by leveraging established bosonic QMC techniques while mitigating the sign problem. Limitations arise from the dependence on nodal quality, as inaccurate nodes introduce bias that cannot be removed without a better trial function. The method also faces challenges for accessing excited states and can be computationally demanding when using sophisticated nodal structures.

Applications are widespread in quantum chemistry and condensed matter physics, where fixed-node diffusion Monte Carlo is used to benchmark energies of atoms, molecules, and solids.