Electronic Systems Path Integral Simulations

Condensed-phase Electronic Systems Path Integral Simulations 

Compared with other areas such as ab initio electronic structure theory and molecular dynamics and Monte Carlo calculations, path integral simulation is a relative latecomer to the field of computational chemistry. While the analytical advantages of formulating quantum mechanics in terms of path integrals have influenced modem physics profoundly for the past 40 years, its computational advantages in areas of chemistry were not appreciated until rather late. 

A very brief discussion of Feynman path integrals is given here to set up the notations used below. For details, see the original work of Feynman (see also Path Integral Methods). 

All equilibrium properties of a quantum system can be obtained from the partition function 

P = IcbT (ke is the Boltzmann constant and T is the temperature) and H is the Hamiltonian. The sum in the first equality is carried out over all eigenstates v of the system, whereas the second equality is truly independent of the basis set used, since the trace is invariant under any rotation in Hilbert space. Most 

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CONDENSED-PHASE ELECTRONIC SYSTEMS PATH INTEGRAL SIMULATIONS 475... 

CONDENSED-PHASE ELECTRONIC SYSTEMS. PATH INTEGRAL SIMULATIONS 477... 

Condensed-phase Electronic Systems Path Integral Simulations Monte Carlo Quantum Methods for Electronic Structure Rates of Chemical Reactions Wave Packets. 

Condensed-phase Electronic Systems Path Integral Simulations 1 474... 

It is of considerable interest to use the electron bubble as a probe for elementary excitations in finite boson quantum systems—that is, ( He)jy clusters [99, 128, 208, 209, 243-245]. These clusters are definitely liquid down to 0 K [46 9] and, on the basis of quantum path integral simulations [65, 155], were theoretically predicted (see Chapter II) to undergo a rounded-off superfluid phase transition already at surprising small cluster sizes [i.e., Amin = 8-70 (Table VI)], where the threshold size for superfluidity and/or Bose-Einstein condensation can be property-dependent (Section II.D). The size of the ( He)jy clusters employed in the experiments of Toennies and co-workers [242-246] and of Northby and coworkers [208, 209] (i.e., N lO -lO ) are considerably larger than Amin- In this large cluster size domain the X point temperature depression is small [199], that is, (Tx — 2 X 10 — 2 X 10 for V = lO -lO. Thus for the current... 

While Section 4.4 considers electron transport in the framework of a single-charge tunneling problem, such a theory would not explain the conductivity of polyacetylene. It turns out that the conductivity of many ID conductors often involves many-electron effects, and to study ID electronic systems with path integral simulations, one has to first solve the fermion sign problem. 

Virtually all of the successful path integral simulations of 2-d models for electronic systems have been carried out by the auxiliary field MC method, sometimes called the determinantal method. The only thing that complicates the computation of the fermion partition in equation (8) is the interaction action 5i. As explained in Section 5.3, without 5i, the sum over exchanges can be performed analytically. Therefore, if the two-electron interaction term can be eliminated or at least decoupled, the fermion sign problem could be partially removed. This can be accomplished by a so-called Hubbard-Stratonovich transformation. The details can be found in the original paper. Briefly, two electrons (of opposite spin) on the same site i experience a repulsion of strength U and add a term —eUni ni to the action Si, where = 0, 1 is the occupation number of an f-spin electron on site i, and n, is the same for a -spin electron. To decouple the two-electron interaction, the following transformation (correct up to a multiplicative constant) can be used. 

This article offers a brief overview of applications of path integral simulations to various aspects of condensed-phase electronic systems. Despite the success of some of the methods described here, many challenges remain for path integral simulations. In particular, a general solution to the sign problem in fermion and real-time problems is of fundamental importance, and such a general method has yet to be found. This search will no doubt dominate research activities in the field of path integral simulations for the next few years. 

R. W. HaU (2005) Simulation of electronic and geometric degrees of freedom using a kink-based path integral formnlation Application to molecular systems. J. Chem. Phys. 122, p. 164112(1-8]... 

In this short review we have pointed out only very few of the basic issues involving the simulation of chemical systems with Quantum Monte Carlo. What has been achieved in the last few years is remarkable very precise calculations of small molecules, the most accurate calculations of the electron gas, silicon and carbon clusters, solids, and simulations of hydrogen at temperatures when bonds are forming. New methods have been developed as well high-accuracy trial wavefunctions for atoms, molecules, and solids, treatment of atomic cores, and the generalization of path-integral Monte Carlo to treat many-electron systems at positive temperatures. 

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