Perspective: Explicitly correlated electronic structure theory for complex systems

No abstract

We analyze the effect of increasing charge density on the Fixed Node Errors in Diffusion Monte Carlo by comparing FN-DMC calculations of the total ground state energy on a 4 electron system done with a Hartree-Fock based trial wave function to calculations by the same method on the same system using a Configuration Interaction based trial wave function. We do this for several different values of nuclear charge, Z. The Fixed Node Error of a Hartree-Fock trial wave function for a 4 electron system increases linearly with increasing nuclear charge. *

We study lithium systems over a range of number of atoms, specifically, atomic anion, dimer, metallic cluster, and body-centered cubic crystal by the fixed-node diffusion Monte Carlo method. The focus is on analysis of the fixed-node errors of each system, and for that purpose we test several orbital sets in order to provide the most accurate nodal hypersurfaces. The calculations include both core and valence electrons in order to avoid any possible impact by pseudopotentials. To quantify the the fixed-node errors, we compare our results to other highly accurate calculations, and wherever available, to experimental observations. The results for these Li systems show that fixed-node diffusion Monte Carlo achieves accurate total energies, recovers 96-99% of the correlation energy and estimates binding energies with errors bounded by 0.1 eV/atom.

We outline the basic notions of nodal hypersurface and domain averages for antisymmetric wave functions. We illustrate their properties and analyze the results for a few electron explicitly solvable cases and discuss possible further developments.PACS numbers: 61.46.+w, 36.40.Cg Quantum Monte Carlo is one of the most effective many-body methodologies for the study of quantum systems. It is based on a combination of analytical insights, robustness of stochastic approaches, and performance of parallel architectures [1][2][3][4][5][6][7][8][9][10]. The approach has been applied to a variety of challenging problems in electronic structure of atoms, small molecules, clusters, solids, ultracold condensates, and beyond [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. The two most commonly used QMC methods are variational Monte Carlo and diffusion Monte Carlo (DMC). Let us briefly recapitulate the basics of the DMC method.It is straightforward to show that for τ → ∞, the operator exp(−τ H) projects out the ground state of a given symmetry from any trial function with nonzero overlap. We assume that the Hamiltonian H is time-reversal symmetric so that the eigenstates can be chosen to be real. This projection is most conveniently carried out by solving the Schrödinger equation in an imaginary time integral form so that the product f (R, t) = Φ(R, t)Ψ T (R) obeysThe Green's function is given bywhere R = (r 1 , ..., r N ) denotes positions of N particles and E T is an energy offset. In the DMC method, the function f (R, t) is represented by a set of 10 2 -10 4 random walkers (sampling points) in the 3N -dimensional space of electron configurations. The walkers are propagated for a time slice τ by interpreting the Green's function as a transition probability from R ′ → R. The kernel is known for small τ , and the large time t limit is obtained by iterating the propagation. The method is formally exact provided that the boundary conditions, i.e., the fermion nodes of the antisymmetric solution defined as Φ(R, ∞) = 0, are known [1,5,14]. Unfortunately the antisymmetry does not specify the nodes completely, and currently we have to use approximations. The commonly used fixed-node approximation [14] enforces the nodes of f (R, t) to be identical to the nodes of Ψ T (R) which then implies that f (R, t) ≥ 0 everywhere. It is therefore clear that the accuracy of the fixed-node DMC is determined by the quality of the trial wave function nodes. The commonly used form for Ψ T is the Slater-Jastrow wave function given aswhere U corr is the correlation factor explicitly depending on interparticle distances thus describing pair or higher order correlations explicitly. The typical number of Slater determinants is between 1 and 10 3 , and the corresponding weights {d n } are usually estimated in multi-reference Hartree-Fock (HF) or Configuration Interaction (CI) calculations and then re-optimized in the variational framework.It is quite remarkable that the nodes of such SlaterJastrow wave functions (often with a single-determinant p...