Fixed-node methods

A further important property of the Fixed node method is the existence of a variational theorem the FN-RQMC energy is an upper bound of the true ground state energy Et oo) > Eq, and the equality holds if the trial nodes coincide with the nodes of the exact ground state [6]. Therefore for fermions, even projection methods such as RQMC are variational with respect to the nodal positions the nodes are not optimized by the projection mechanism. The quality of the nodal location is important to obtain accurate results. 

Particle statistics come in rather differently in PIMC. A permutation operation is used to project Bose and Fermi symmetry. (Remember that in DMC the fixed-node method with an antisymmetric trial function was used.) The permutations lead to a beautiful and computationally efficient way of understanding superfluidity for bosons, but for fermions, since one has to attach a minus sign to all odd permutations, as the temperature approaches the fermion energy a disastrous loss of computational efficiency occurs. There have been many applications of PIMC in chemistry, but almost all of them have been to problems where quantum statistics (the Pauli principle) were not important, and we do not discuss those here. The review article by Berne and Thirumalai [10] gives an overview of these applications. 

Recently, there has been some progress in generalizing the path-integral method to treat fermion systems, which is called restricted PIMC (RPIMC) [24]. One can also apply the fixed-node method to the density matrix. The fermion density matrix is given by... 

The fixed-node method does not qualify as a solution since the systematic errors are not under control, so they cannot be made arbitrarily small. There have been some recent attempts to parameterize the nodal surface and then determine the parameters dynamically. This will reduce the fixed-node errors, probably by an order of magnitude. However, it does not solve the problem since the error, even if it is smaller, is still uncontrolled. For a satisfactory solution one would have to parameterize, in a completely arbitrary fashion, the nodal surface. 

He ), 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... 

Figure 4 Illustration of fixed-node method for the first excited state of a particle in a two-dimensional box solid line, correct nodal line dotted and dashed lines, approximately correct nodal lines with inversion symmetry.

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. 

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. 

There are several distinct methods that that are referred to as QMC methods. Here we consider the fixed-node diffusion QMC method, or the DMC method other members of the QMC family are described in several recent reviews [7, 8, 9]. DMC has been widely applied to ground state systems in the Born-Oppenheimer approximation [10], but is not limited to such cases. 

The ground state wavefunction of a bosonic system is positive everywhere, which is very convenient in a Monte Carlo context and allows one to obtain results with an accuracy that is limited only by practical considerations. For fermionic systems, the ground-state wave function has nodes, and this places more fundamental limits on the accuracy one can obtain with reasonable effort. In the methods discussed in this chapter, this bound on the accuracy takes the form of the so-called fixed-node approximation. Here one assumes that the nodal surface is given, and computes the ground-state wavefunction subject to this constraint. 

For both bosonic systems and fermionic systems in the fixed-node approximation, G has only nonnegative elements. This is essential for the Monte Carlo methods discussed here. A problem specific to quantum mechanical systems is that G is known only asymptotically for short times, so that the finite-time Green function has to be constructed by the application of the generalized Trotter formula [6,7], G(r) = limm 00 G(z/m)m, where the position variables of G have been suppressed. 

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... 

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. 

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.)... 

In order to maintain the wave function antisymmetry, the diffusion QMC is normally used within the fixed node approximation, i.e. the nodes are fixed by the initial trial wave function. Unfortunately, the location of nodes for the exact wave function is far from trivial to determine, although simple approximations such as HF can give quite reasonable estimates. The fixed node diffusion QMC thus determines the best wave function with the nodal structure of the initial trial wave function. If the trial wave function has the correct nodal structure, the QMC will provide the exact solution to the Schrodinger equation, including the electron correlation energy. It should be noted that the region near the nuclei contributes most to the statistical error in QMC methods, and in many apphcations the core electrons are therefore replaced by a pseudopotential. 

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 nodes of the exact wave function are known for only a few simple systems, so the fixed-node constraint is an approximation. Unlike time step error, it is not possible to correct for fixed-node error by extrapolation. The error introduced by FNA is the only uncontrolled factor in the FN-DMC method. Fortunately, the FNA is found to perform well, even when modest trial functions are used. Errors due to the FNA are typically less than 3 kcal/mol, even when simple single-determinant trial functions are used [38]. 

The above mentioned DMC method is applicable to bosons only, since it requires that the function I (R t) can be considered as a (non-normalized) probability density. For the fermionic system, the fixed-node (FN) approximation with a trial function is often used. By using a time-independent trial function 4 t(R), the time-independent Schrbdinger equation is rewritten as... 

Relativistic QMC ZORA-VMC and fixed-node ZQRA-DMC. The lORA version of the QMC method is under development. 

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