Many-electron wave functions and fermion nodes.

The cube in Fig. below is a supercell of a nitrogen solid structure. It contains N electrons which are described by the wave function Psi(r₁, r₂,...,r_N). The wave function fulfills the Schrodinger equation:

H Psi = E Psi ,          where H is the Hamiltonian and E is the total energy.

For some configurations of electrons the wave function is zero because of fermion antisymmetry. The set of such configurations forms a hypersurface which is called a fermion node and the green isosurface below is a 3D cut through it. The isosurface is found by scanning the cube with coordinates of one electron while keeping the rest of electrons fixed at some positions. The shades of blue color represent values of negative part of the wave function while the positive part (see a movie) is located in the complementary (empty) space. The complete nodal hypersurface has (3N-1)-dimensions, where N is the number of electrons with the same spin (in this case N=20). Note that the wave function fulfills the periodical boundary conditions as the infinite solid is represented by the periodically repeating supercell.

The fermion node represents a boundary condition in the so-called fixed-node quantum Monte Carlo methods which solve the Schrodinger equation by exploring the power and efficiency of a combination of analytical insights and stochastic techniques. Although the fermion node is extremely complex, the stochastic methods can deal with these and other many-body effects very efficiently. We develop and use quantum Monte Carlo methods for high accuracy electronic structure calculations of various systems such as clusters, molecules and solids. The quantum Monte Carlo methods are unique because of genuine many-body framework, applicability to large systems and high accuracy (read more in our reviews).

Fig.: 3D cut through 59-dimensional fermion node of the electronic wave function in nitrogen solid.