Fixed-node calculations

The problem of node locations—the sign problem in quanmm Monte Carlo —remains one of the major obstacles to obtaining exact solutions for systems of more than a few electrons. In analytic variational calculations and in VQMC, the locations of the nodal smfaces of a trial wavefunction may be and usually are optimized along with the rest of the wavefunction in the attempt to reach a minimum in the expectation value of the energy. In DQMC and GF-QMC, the node locations are not so easily varied. 

the node problem can be overcome by exact cancellation methods (described below), and exact solutions can be obtained. For systems of as many as 10 electrons,released-node or transient estimate methods (also described below) can provide excellent approximate solutions. But, in general, the method of choice for systems of more than about 10 electrons is the fixed-node method. Although the fixed-node method is variational and does not yield exact results, it is the only choice available for quantum Monte Carlo calculations on many larger systems. The fixed-node method is remarkably accurate and generally yields energies well below those of the best available analytic variational calculations. 

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. 

Unless the assumed nodal surface is exactly correct, the overall wavefunction will not be exactly correct, and the energy obtained will be an upper bound to the true energy. The fixed-node method is thus variational with respect to 

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The calculation of excited-state energies has been attempted only occasionally with QMC methods. The simplest situation is to determine the excitation energy from the state of one symmetry to a state of different symmetry (e.g., the ls-to-2p excitation in hydrogen). Since both states are ground states within their symmetries, one can do fixed-node calculations for each state individually and get individual upper bounds to their energies. 

We call attention to several prior reviews in the QMC area which give different insights and additional details. These include one book of general coverage, several review articles of a general nature, a review of exact methods, a discussion of fixed-node calculations, a review of applications to solids, and a review of treatments of vibrational states in molecules and clusters. 

For systems containing two or more electrons of the same spin or other indistinguishable particles, an additional problem appears the node problem. For these systems, it is necessary to restrict the form of the total wavefunction (space and spin parts) such that it is antisymmetric to the exchange of electrons. For any electronic state other than the ground state, it is necessary to restrict further the properties of the wavefunction. The effect of these restrictions is the imposition of nodal surfaces, on which V /(X) = 0, in the space part of the wavefunction. The topic of nodal surfaces is discussed later in the section on Fixed-Node Calculations. 

The imposition of additional boundaries corresponding to nodes for fixed-node calculations has been described by Ceperley, by Skinner et al., and by Moskowitz and Schmidt.The procedures involve conditional sampling, in which the steps chosen for walkers are accepted depending on a property of the new position, together with smaller steps for walkers in the vicinity of the nodes. 

The fixed-node method may be used for excited states when the nodes are known in advance as in the case of the P helium atom for which the nodal surface occurs at r- = r2- For electronic systems of more than two electrons, such a specification cannot be made in advance, but for vibrations of diatomic and polyatomic molecules, the nodes for many modes of vibration can be specified from geometric considerations. Thus, fixed-node calculations have a place in calculations for excited states—especially for the first few states of small systems. 

Preliminary results have successfully shown the initial repulsion and de-stabilisation of CO adsorbed on the (3u (100) surface. Skin depth effects are limited to a two-layer copper slab, with periodic boundary conditions in two dimensions. The CO molecule transfers electronic charge to the surface and subsequently presents a partial positive charge on the carbon atom, near the copper surface. This has then be shown to provide a site for nucleophilic attack by the hydride ion (H-). This adsorbed reaction has been compared to its gas-phase counterpart and shown to have a lower activation barrier. All these systems require a DMC fixed node calculation to obtain sufficiently low QMC variance energies to argue reliably the case for a surface catalyst effect. Other nucleophiles are now being considered. 

Dilfusion and Green s function QMC calculations are often done using a fixed-node approximation. Within this scheme, the nodal surfaces used define the state that is obtained as well as ensuring an antisymmetric wave function. 

Although the previous discussion has focused on ground states, the DMC method can also be applied to the calculation of electronically excited states. This is most simply achieved using the fixed-node approximation. Note that the ground state of a fermion system is itself an excited state. It is the lowest antisymmetric state of the system. 

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... 

We will not exhaustively review previous applications and methods, as there is a recent book on the subject [10] as well as reviews [11,12] with details of methods and overviews of many applications. There are also very recent reviews by Anderson on rigorous QMC calculations for small systems [13] and on fixed-node applications [14]. The focus here is on examining to what extent QMC could perform calculations of chemical... 

Figure 1. Error in the nonrelativistic total energy (in millihartrees) for the first-row atoms of atomic charge Z. The upper line is from coupled cluster calculations [27]. The dashed lines with symbols and error bars denote VMC calculations [30], with a single Slater determinant and the correlation factor with 17 variational parameters. The dotted lines with symbols are results of fixed-node DMC calculations with a single Slater determinant. The circles are Cl calculations [83,84],...

A number of VMC and DMC calculations of atomic, molecular, and solid systems have been carried out by this approach. This includes sp and transition element atoms [32, 46, 52] silicon and carbon clusters [55, 58], nitrogen solids [59], and diamond [60]. Our experience indicates that with a sufficient number of valence electrons one can achieve a high final aecuracy. This, however, requires using 3s and 3p in the valence space for the 3d elements and, possibly, 2s and 2p states for elements such as Na. Once the core is sufficiently small, the systematic error of the fixed-node... 

The calculation model of two-dimensional model, considering the angle of coal seam, set up as shown in Figure 1 and trapezoidal model, a total of 2800 mesh, 5858 nodes. The stress boundary condition, the model surface applied uniform vertical compressive stress, the model under the surface of the vertical displacement fixed. The calculation of the model using Mohr—coulomb criterion is used as a rock mass failure criterion (Qian et al. 1991, Li et al. 2000). 

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.)... 

The first description of QMC is attributed to E. Fermi in a classic paper by Metropolis and Ulam [2]. Some years later, Kalos [3] proposed Green s Function QMC which was applied to the calculation of the ground state of three- and four-body nuclei. In retrospect this paper was indicative of the versatility of QMC for studies of nuclear, condensed matter as well as atomic and molecular systems. It remained for Anderson [4] to make the initial significant foray into atomic and molecular systems in the mid-seventies. These classic papers introduced the fixed-node approximation and served as a bellwether for the DMC described below. 

It is important to mention that the displacement of the interfacial nodes, U/, are not degrees of freedom in the continuum calculation Because the interfacial nodes coincide with interface atoms, r/, they are moved as atoms and appear as fixed nodes with prescribed displacements to the other elements of the continuum region. Lastly, it must be remarked that this formulation of the continuum/atomistic coupling does not allow for the use of a unique energy functional because E includes the elastic energy of the pad atoms, whose energy is already implicitly contained in the continuum energy, E. ... 

DQMC calculations for atoms and molecules such as H2, H4, Be, H2O, and HF made by means of fixed-node structures obtained from optimized single-determinant SCF calculations typically recover more than 90% of the correlation energies of these species and yield total electronic energies lower than the lowest energy analytic variational calculations. These results suggest that optimized single-determinant wavefunctions have node structures that are reasonably correct. 

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