Variational quantum Monte Carlo

Ra]agopal G, Needs R J, James A, Kenney S D and Foulkes W M C 1995 Variational and diffusion quantum Monte Carlo calculations at nonzero wave vectors theory and application to diamond-structure germanium Phys. Rev. B 51 10 591-600... 

Eckstein FI, Schattke W, Reigrotzki M and Redmer R 1996 Variational quantum Monte Carlo ground state of GaAs Phys. Rev. B 54 5512-15... 

A method that avoids making the HF mistakes in the first place is called quantum Monte Carlo (QMC). There are several types of QMC variational, dilfusion, and Greens function Monte Carlo calculations. These methods work with an explicitly correlated wave function. This is a wave function that has a function of the electron-electron distance (a generalization of the original work by Hylleraas). 

Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. 

D. Bressanini, P.J. Reynolds, Between classical and quantum Monte Carlo methods Variational QMC. Adv. Chem. Phys. 105, 37 (1998)... 

The quality of a variational quantum Monte Carlo calculation is determined by the choice of the many-body wavefunction. The many-body wavefunction we use is of the parameterized Slater-Jastrow type which has been shown to yield accurate results both for the homogeneous electron gas and for solid silicon (14) (In the case of silicon, for example, 85% of the fixed-node diffusion Monte Carlo correlation energy is recovered). At a given coupling A, 4>A is written as... 

QMCcp Quantum Monte Carlo method with a core potential. SVM Stochastic variational method. 

Two coupling modes are considered for the Pdj CO cluster the first mode (denoted as h) represents vibration of the rigid CO molecule with respect to the transition metal surface. The second mode is either the Pd-Pd vibration wi in the plane of Pd surface atoms (r) or out-of-plane stretch of the surface/sub-surface Pd-Pd bond (z). The total energy surfaces (h,r) and (h,z) are calculated for discrete points and then fitted to a fourth order polynomial. Variational and Quantum Monte Carlo (QMC) methods were subsequently applied to calculate the ground and first excited vibrational states of each two-dimensional potential surfaces. The results of the vibrational frequences (o using both the variational and QMC approach are displayed in Table II. 

Table II. Vibrational frequencies (cm ) of the two coupled modes (h,r) and (h,z) obtained from a harmonic (Har), variational (Var) and Quantum Monte Carlo (QMC) approach...

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... 

In the investigation of Ortiz et al. [104], a stochastic method is presented which can handle complex Hermitian Hamiltonians where time-reversal invariance is broken explicitly. These workers fix the phase of the wave function and show that the equation for the modulus can be solved using quantum Monte Carlo techniques. Then, any choice for its phase affords a variational upper bound for the ground-state energy of the system. These authors apply this fixed phase method to the 2D electron fluid in an applied magnetic field with generalized periodic boundary conditions. 

X. W. Wang, J. Zhu, S. G. Louie, and S. Fahy (1990) Magnetic structure and equation of state of bcc solid hydrogen A variational quantum Monte Carlo study. Phys. Rev. Lett. 65, p. 2414... 

Quantum Monte Carlo (QMC) [41] is one of the most accurate methods for solving the time-independent Schrodinger equation. As opposed to variational ab initio approaches, QMC is based on a stochastic evaluation of the underlying integrals. The method is easily parallelizable and scales as 0(N3), however, with a very large prefactor. 

BETWEEN CLASSICAL AND QUANTUM MONTE CARLO METHODS VARIATIONAL QMC... 

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). 

Since the square of the wave function represent a probability function, the associated energy can be calculated by Quantum Monte Carlo (QMC) methods. For a (approximate) variational wave function, the energy can be re-written as in eq. (4.86). 

Keywords Electronic structure theory ab initio quantum chemistry Many-body methods Quantum Monte Carlo Fixed-node diffusion Monte Carlo Variational Monte Carlo Electron correlation Massively parallel Linear... 

The variational quantum Monte Carlo method (VMC) is both simpler and more efficient than the DMC method, but also usually less accurate. In this method the Rayleigh-Ritz quotient for a trial function 0 is evaluated with Monte Carlo integration. The Metropolis-Hastings algorithm " is used to sample the distribution... 

A well-known solution for this problem is correlated sampling. In this case all evaluations of finite differences are done within one QMC calculation. This approach reduces the statistical error and circumvents the computational costs of doing several quantum Monte Carlo calculations. So far correlated sampling was only investigated in a closed unbiased form for variational Monte Carlo. The VMC correlated sampling calculation involves wave functions of the reference structure ij/ and the perturbed structure... 

Table 5.2. Experimental vibrational redshifts for DF and HF with sequential addition of argon solvent atoms. Also shown are redshifts calculated using diffusion quantum Monte Carlo techniques from Ref. 66 and bound state variational calculations by Ernesti and Hutson from Refs. 9,11. The two columns reflect the values calculated within the approximation of pairwise additivity, and including the corrective three-body terms as described more fully in the text. 

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