DQMC method

The FN-DQMC calculation in the table uses the diffusion quantum Monte Carlo method. quantum Monte Carlo (QMC) method uses a random process to solve the Schrftdinger equation. Many QMC methods exist, but the diffusion QMC (DQMC) method is most commonly used for molecular calculations. Defining the imaginary time variable t s itjh. 

Because of the Pauli principle antisymmetry requirement, the ground-state wave function has nodal surfaces in 3n-dimensional space, and to ensure that the walkers converge to the ground-state wave function, one must know the locations of these nodes and must eliminate any walker that crosses a nodal surface in the simulation. In the fixed-node (FN) DQMC method, the nodes are fixed at the locations of the nodes in a known approximate wave function for the system, such as found firom a large basis-set Hartree-Fock calculation. This approximation introduces some error, but FN-DQMC calculations are variational. (In practice, the accuracy of FN-DQMC calculations is improved by a procedure called importance sampling. Here, instead of simulating the evolution of with t, one simulates the evolution off, where / = where is a known accurate trial variation function for the ground state.)... 

To properly apply the DQMC method, one must allow for the nodes produced by the Pauli antisymmetry requirement, (a) Consider a system of three electrons in a onedimensional box, where we shall pretend that the interelectronic repulsions are small enough to... 

The DQMC method is based on the similarity of the Schrodinger equation and the diffusion equation. It has its roots in the Monte Carlo simulation of neutron diffusion and capture by Fermi and others at Los Alamos in the 1940s. Metropolis and Ulam first outlined the method in 1949 ... 

The DQMC method is basically a simple game of chance involving the random walks of particles through space and their occasional multiplication or disappearance. It may be viewed as based on the similarity between the Schrodinger equation and the diffusion equation (i.e.. Pick s second law of diffusion) and the use of the random walk process to simulate the diffusion process. Following the early discussions in the 1940s by Metropolis and Ulam and by King, " a number of related techniques were proposed and discussed, but applications to multicenter chemical systems were not practical until fast computers became available. ... 

The fourth method used for quantum chemical calculations is the quantum Monte Carlo (QMC) method, in which the Schrodinger equation is solved numerically. There are three general variants of QMC variational MC (VMC), diffusion QMC (DQMC), and Green s function QMC (GFQMC), all of which... 

The Diffusion Quantum Monte Carlo (DQMC) algorithm and related methods such as the Vibrational Quantum Monte Carlo approach have the important property of scahng well with system size (number of degrees of freedom). At the same time the method can be pursued in principle to yield a numerically exact energy. DQMC was introduced... 

QMC methods have given very accurate results in some calculations on small systems, but the method can require very long calculation times, and no efficient method has yet been found to allow geometry optimization in a QMC calculation. For more on DQMC, see Problems 15.52 and 15.53, and J. B. Anderson, Int Rev. Phys. Chem., 14,85 (1995) K. Raghavachari and J. B. Anderson,/ Phys. Chem., 160,12960 (1996). 

Figure 1 A family tree of quantum chemistry DFT, density functional theory QMC, quantum Monte Carlo RRV, Rayleigh-Ritz variational theory X-a, X-alpha method KS, Kohn-Sham approach LDA, BP, B3LYP, density functional approximations VQMC, variational QMC DQMC, diffusion QMC FNQMC, fixed-node QMC PIQMC, path integral QMC EQMC, exact QMC HF, Hartree-Fock EC, explicitly correlated functions P, perturbational MP2, MP4, Maller-Plesset perturbational Cl, configuration interaction MRCI, multireference Cl FCI, full Cl CC, CCSD(T), coupled-cluster approaches. Other acronyms are defined in the text.

Carlo method (VQMC), the diffusion quantum Monte Carlo method (DQMC), the Green s function quantum Monte Carlo method (GFQMC), and the path integral quantum Monte Carlo method (PIQMC). These methods are by their nature strongly related and each has its own peculiar advantages and disadvantages relative to the others. 

The PIQMC method is the result of coupling of Feynman s path integral formulation of quantum mechanics with Monte Carlo sampling techniques to produce a method for finite temperature quantum systems. The calculations are not much more complicated than DQMC and produce a sum over all possible states occupied as for a Boltzmann distribution. In the limit of zero temperature... 

The diffusion quantum Monte Carlo method (DQMC) approaches the solution of the Schrodinger equation in a way completely different from that of variational methods. The basic ideas were given above in the succinct description quoted from the original report by Metropolis and Ulam. Here we give a more complete description. 

The fixed-node method was first applied in DQMC calculations for the systems H P, H2 H4 and Be The results indicated that good energies could be obtained with node locations of relatively poor quality. Because the nodal surfaces of ground state systems are typically located in regions of low electron density (i.e., according to hAq), one might expect the calculated energies to be insensitive to small departures in node locations from those of the true wavefunctions. 

Both DQMC and GFQMC provide the lowest energy solution to the Schrodinger equation subject to any constraints that may be imposed on the solution. For excited states, one must impose the necessary constraints. In some cases, this is relatively easy to do, but in others it is difficult or as yet impossible. For many cases, alternate methods are available in particular, a matrix procedure may be applied to the simultaneous evolution of several states in imaginary time. ... 

QMC methods were first applied to the case of the electron gas by Ceper-ley in the late 1970s,and the results have been widely used in density functional theory. Only recently have these early calculations been extended by others to provide greater detail. Pickett and Broughton carried out VQMC calculations for the spin-polarized gas. Ortiz and Ballone used both VQMC and fixed-node DQMC for the spin-polarized gas in the density range most important to density functional theory. Kenny et al. performed VQMC and DQMC calculations for the nonpolarized homogeneous electron gas, incorporating relativistic effects via first-order perturbation theory. 

Methane was one of the first molecules used to illustrate the effectiveness of fixed-node DQMC calculations relative to standard ab initio methods. The first fixed-node DQMC calculations for methane recovered 97% of the correlation energy and gave a total electronic energy 30 kcal/mol below the lowest energy variational result (at the time) and only 3 kcal/mol above the experimental value. Since then many more calculations for a large variety of carbon and hydrocarbon systems have shown similarly impressive results. These systems ranged from methylene to graphite and diamond structures and were treated with and without pseudopotentials. 

Figure 6 Binding energies (in eV) for seven hydrocarbons as determined from DFT-LDA, DQMC, and HF methods. Energies are shown as the differences from experimental values. (From Ref. 123.)...

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