دانلود رایگان مقاله حل مسیر عددی انتگرال به همبستگی قوی کولن در اتم هوک

Abstract

We present a new approach based on real time domain Feynman path integrals (RTPI) for electronic structure calculations and quantum dynamics, which includes correlations between particles exactly but within the numerical accuracy. We demonstrate that incoherent propagation by keeping the wave function real is a novel method for finding and simulation of the ground state, similar to Diffusion Monte Carlo (DMC) method, but introducing new useful tools lacking in DMC. We use 1D Hooke’s atom, a two-electron system with very strong correlation, as our test case, which we solve with incoherent RTPI (iRTPI) and compare against DMC. This system provides an excellent test case due to exact solutions for some confinements and because in 1D the Coulomb singularity is stronger than in two or three dimensional space. The use of Monte Carlo grid is shown to be efficient for which we determine useful numerical parameters. Furthermore, we discuss another novel approach achieved by combining the strengths of iRTPI and DMC. We also show usefulness of the perturbation theory for analytical approximates in case of strong confinements.

5. Conclusions

We have introduced a new approach based on the incoherent real time path integrals (iRTPI) for ground and excited state electronic structure calculations. It includes correlations between electrons exactly, within the numerical accuracy, which can be made better than the systematic error from the kernel simply by using Monte Carlo technique. Here, we use Hooke’s atom, a two-electron system with very strong correlation, as our test case, which we solve with both iRTPI and diffusion Monte Carlo (DMC) for comparison. The high accuracy and stability of iRTPI is demonstrated, and the improved Trotter kernel is shown to be useful with large enough number of Monte Carlo walkers. In addition, useful numerical parameters for the present case have been determined for stable and self-consistent simulations. In its present form the computational cost of iRTPI is significantly higher than that of DMC. However, one of the advantages of iRTPI is that it provides one with the wave function explicitly, and thus, the evaluation of local multiplicative expectation values becomes straightforward. Moreover, it is also capable of locating excited states, and thus, the related nodal surfaces, the technical details of which were not considered here. In addition, incoherent dynamics can be turned to coherent dynamics, in case quantum dynamics is relevant. We also showed that another novel approach obtained by combining the iRTPI and DMC methods allows a more straightforward means for evaluation of various observables within the robust framework of DMC. Due the capability of iRTPI for locating the nodal surfaces, it will be interesting to further test this combination method in a released node fashion of DMC. This would mean a trial wave function free DMC also for fermions. Perturbation theory was shown to be useful for analytical solutions in case of strong confinements, which may become more challenging for numerical methods and available approximate solutions. On the other hand, for weak confinements, e.g. in quantum dots, the presented numerical iRTPI method is expected to be robust.