Many-Body Nodal Hypersurface and Domain Averages for Correlated Wave Functions

Abstract

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...

Weighted nodal domain averages of eigenstates for quantum Monte Carlo and beyond

“…After introducing the nodal domain averages [17], we later found that the nda expressions correspond to a special case of identity which was derived idependently by Sogge and Zelditch [28]. We will present here the identity in a form that includes potentials which is a straightforward generalization (the original paper [28] assumes Laplacian eigenstates, i.e., with no potentials).…”

“…i) The nda components differ for degenerate eigenstates. For example, building upon previous work [17], we were able to analytically derive that for the noninteracting Be atom ground and excited state the components show clearly different values (in previous work the values for the state 1 S (1s 2 2p 2 ) were estimated numerically). These values are shown in Table 1.…”

“…In the previous paper [17], we have tried to look at the nodal biases from a new angle. In order to shed some light on such cases, we suggested so-called nodal domain averages that actually enabled us to reveal new aspects of node properties in a quantitative manner.…”

“…Let us recall our recent results [17] where we have shown that for any exact fermionic (or excited bosonic) eigenstate one can express the total energy as a sum of kinetic and potential components, that we dubbed nodal domain averages (nda, in short). They are defined as follows…”

“…In particular, we show that the nodal behavior of eigenstates encodes information about eigenvalues and it can also provide a possible, if difficult, route for finding the eigenstates themselves. This understanding is being built through nodal domain averages which we introduced before [17] and which we generalize here by introducing a weight function. We explore a few possibilities that are offered by this generalization including new identities which reveal that the energy differences between states can be recast as integrals of wave function gradients over nodal surfaces normalized by overlaps with the bosonic ground state.…”