Variational path integral

Variational Path Integral Molecular Dynamics Study of Small Para-Hydrogen Clusters... 

In this section, we briefly describe the variational path integral molecular dynamics method. We start to consider a system consisting of N identical particles whose coordinates are collectively represented to be i . The Hamiltonian of the system is written by H = f + V where f and V are the kinetic and potential energy operators, respectively. The system is assumed to be well described by a trial wavefunction exact ground state of the system, 4 o), can be obtained using the trial wavefunction 7 ) by the following relation [1,4]... 

As in the standard path integral method for finite temperature systems [19], the above pseudo partition function can be regarded as a configurational integral of classical polymers. However, in the variational path integral, the classical isomorphic systems consist of open-chain polymers. Furthermore, distributions of end-point coordinates at = 0 and M are affected by the trial wavefunction and respectively. 

This review article is divided into two major sections, the first of which details the theoretical basis of RQMC (Sect. 18.2). Initially we describe quantum Monte Carlo sampling from the pure distribution and mixed distribution F, showing that the RQMC approach to sample from the pure distribution rests on Metropolis-Hastings (MH [25,26]) sampling, as does the variational path integral (VPI [27]) method. As already mentioned, RQMC proposes reptafion-type moves while... 

Wong K-Y, Gao J (2007) An automated integration-free path-integral method based on Kleinert s variational perturbation theory. J Chem Phys 127(21) 211103... 

Kleinert H (2004) Path integrals in quantum mechanics, statistics, polymer physics, and financial markets. 3rd edition. World Scientific Singapore River Edge, NJ, p xxvi, 1468 p. For the quantum mechanical integral equation, see Section 1.9 For the variational perturbation theory, see Chapters... 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

J.A. Poulsen and G. Nyman and P.J. Rossky. Practical evaluation of condensed phase quantum correlation functions A Feynman-Kleinert variational linearized path integral method. J. Chem. Phys., 119 12179, 2003. 

Warshel and Chu [42] and Hwang et al. [60] were the first to calculate the contribution of tunneling and other nuclear quantum effects to PT in solution and enzyme catalysis, respectively. Since then, and in particular in the past few years, there has been a significant increase in simulations of quantum mechanical-nuclear effects in enzyme and in solution reactions [16]. The approaches used range from the quantized classical path (QCP) (for example. Refs. [4, 58, 95]), the centroid path integral approach [54, 55], and variational transition state theory [96], to the molecular dynamics with quantum transition (MDQT) surface hopping method [31] and density matrix evolution [97-99]. Most studies of enzymatic reactions did not yet examine the reference water reaction, and thus could only evaluate the quantum mechanical contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (for example. Refs. [4, 58, 95]) concluded that the quantum mechanical contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

Variations on this surface hopping method that utilize Pechukas [106] formulation of mixed quantum-classical dynamics have been proposed [107,108]. Surface hopping algorithms [109-111] for non-adiabatic dynamics based on the quantum-classical Liouville equation [109,111-113] have been formulated. In these schemes the dynamics is fully prescribed by the quantum-classical Liouville operator and no additional assumptions about the nature of the classical evolution or the quantum transition probabilities are made. Quantum dynamics of condensed phase systems has also been carried out using techniques that are not based on surface hopping algorithms, in particular, centroid path integral dynamics [114] and influence functional methods [115]. 

This review is a brief update of the recent progress in the attempt to calculate properties of atoms and molecules by stochastic methods which go under the general name of quantum Monte Carlo (QMC). Below we distinguish between basic variants of QMC variational Monte Carlo (VMC), diffusion Monte Carlo (DMC), Green s function Monte Carlo (GFMC), and path-integral Monte Carlo (PIMC). 

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