We use sets of de Broglie-Bohm trajectories to describe the quantum correlation effects which take place between the electrons in helium atom due to exchange and Coulomb interactions. A short-range screening of the Coulomb potential is used to modify the repulsion between the same spin electrons in physical space in order to comply with Pauli's exclusion principle. By calculating the electron-pair density for orthohelium, we found that the shape of the exchange hole can be controlled uniquely by a simple screening parameter. For parahelium the interelectronic distance, hence the Coulomb hole, results from the combined action of the Coulomb repulsion and the nonlocal quantum correlations. In this way, a robust and self-interaction-free approach is presented to find both the ground state and the time evolution of nonrelativistic quantum systems.

The electronic many-body problem is of key importance for the theoretical treatments of physics and chemistry. A typical manifestation of the quantum many-body effects is the electron correlation which results from the Coulomb and exchange interactions between the electrons combined with the underlying quantum nonlocality. Since in general the electron correlation reshapes the probability density in configuration space, it is difficult to elucidate this effect for higher dimensions. Therefore, to better understand the effects of electron correlation in atoms and molecules, one needs, besides one-particle quantities such as the electron density function, to consider also extensions which explicitly incorporate many-body effects. Such an appropriate quantity is the electronic pair-density function which represents the probability density of finding two electrons at distance

The importance of the electron-pair density, also known as electron position intracule, comes from the fact that it can be associated with experimental data obtained from X-ray scattering, and it can also be used to visualize the notion of exchange and correlation holes which surround the quantum particles. However, the calculation of the many-body wave function in (

Here we calculate the electron-pair densities for helium atom in 2 ^{1}S and 2 ^{3}S states using the recently proposed time-dependent quantum Monte Carlo (TDQMC) method which employs sets of particles and quantum waves to describe the ground state and the time evolution of many-electron systems [

The TDQMC is an

For Hamiltonians with no explicit spin variables the exchange effects can be accounted for efficiently using screened Coulomb potentials as described in [

In our calculation a Coulomb potential screened by an error function is used [

In the approach outlined previously, a self-interaction-free dynamics in physical space is achieved, where the separate walkers do not share guiding waves which represent different distributions. In order to calculate the many-body probability distribution in configuration space, a separate auxiliary set of walkers with primed coordinates

The two major sources of electron-electron correlation are due to the symmetry of the quantum state and due to the Coulomb repulsion. Here we consider first the effect of the exchange correlation on the pair-density function of helium atom. Although the electron-pair densities for helium have been analyzed by different techniques, they have never, to the author’s knowledge, been studied using time-dependent methods.

In order to examine the electron correlation which is due to the exchange interaction, we consider the spin-triplet ground state of helium (orthohelium). The preparation of the ground state is described elsewhere [^{3}S state

Radial electron density for the ground state of orthohelium, for MC walkers guided in physical space (blue and green lines), and for MC walkers guided in configuration space (red line). The inset shows the projection of the coordinates of the MC walkers in the

The electron-pair density for the ground state was calculated very efficiently by simply performing kernel density estimation over the ensemble of distances between the primed walkers. The result is shown in Figure ^{3}S state of helium (see also e.g., [

Electron-pair density as function of the interelectronic distance, for the ground state of orthohelium. (a) Red line—no screening (no exchange), blue line—short-range screened Coulomb potentials. Exchange hole (b) for screened Coulomb potentials (black) and for Hartree-Fock exchange (green).

For the ground state of the 2 ^{1}S (para)helium, the quantity of interest is the Coulomb hole which occurs due to the repulsion of the closely spaced walkers. Figure

Radial electron density for the ground state of parahelium, for MC walkers guided in physical space (red line), and from the Hartree-Fock approximation (blue line). The inset shows the projection of the coordinates of the MC walkers in the

Electron-pair density as function of the interelectronic distance for the ground state of parahelium. (a) Red line—correlated result, blue line—Hartree-Fock approximation. The Coulomb hole (b).

In this paper, it has been shown that for charged particles, the quantum correlation effects which occur due to the exchange and Coulomb correlations can adequately be described by sets of de Broglie-Bohm walkers within the time-dependent quantum Monte Carlo framework. A short-range screening of the Coulomb potential ensures that each replica of a given electron interacts with only those replicas of the rest of the same spin electrons which are sufficiently apart to respect Pauli’s exclusion principle in space. On the other hand, the electron-electron interaction is modified by the quantum nonlocality which demands that each replica of a given electron interacts with the replicas of the other electrons which are within the range of the nonlocal quantum correlation length. This concept allows one to build a robust, self-consistent, and self-interaction-free approach to find both the ground state and the time evolution of quantum systems. It is demonstrated here that the otherwise awkward procedure for calculating the pair distribution functions of para- and orthohelium atom can be simplified to the level of finding the ground state probability distributions of the corresponding Monte Carlo walkers.

Besides the relative ease of its implementation, another advantage of using TDQMC is the affordable time scaling it offers which is almost linear with the system dimensionality. This is especially valid when using multicore parallel computers where little communication overhead between the different processes can be achieved, thus utilizing the inherent parallelism of the Monte Carlo methods. This nears the TDQMC to other efficient procedures for treating many-body quantum dynamics such as the time-dependent density functional approximation which, however, suffers systematic self-interaction problems due to the semiempirical character of the exchange-correlation potentials.

The author gratefully acknowledges support from the National Science Fund of Bulgaria under Grant DCVP 02/1 (SuperCA++). Computational resources from the National Supercomputer Center (Sofia) are gratefully appreciated.