Quantum Monte Carlo for Atoms, Molecules and Solids
Pseudopotentials and Effective Core Potentials
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Pseudopotentials and Effective Core PotentialsPseudopotentials, or Effective Core Potentials (ECPs) as they are often referred to in quantum chemical literature, are important for increasing efficiency of calculations for heavier atoms. As was shown previously [Ceperley, Hammond2] the cost of calculations grows proportionally to  , where Z is the atomic number. Clearly, such Z dependence makes calculations of heavy atom systems computationally demanding and inefficient with most of the computer time used for sampling large energy fluctuations in the atomic core region, which has relatively little affect on the chemistry. Despite this shortcoming all-electron calculations have been carried out for systems containing atoms as large as Xe [Towler]. When the important chemistry and physics occurs in the valence space, it is possible to replace the atomic core with accurate ECPs. The construction of quality ECPs is therefore of prime importance, and often electrons occupying the shell below the valence electrons are included in the valence space to take into account "core polarization" effects, e.g. including the 3s and 3p electrons in the valence shell for the 3d-trasition metals. The ability of QMC to include ECPs has induced further developments in ECPs, such as new ECP forms with smooth and bounded behavior at the nucleus [Greef, Lester, Needs, Filippi].
Evaluation of ECPs in the VMC method is straightforward, however DMC and other exact QMC methods are more difficult. The ECP is in the form of a non-local integral operator, hence the value of the ECP at a single point depends on the entire wave function. In DMC the wave function is unknown, and so we must evaluate the ECP using Calculations of excited statesThere is no difficulty in carrying out QMC calculations of excited states that are the lowest state of a given symmetry. One simply requires a trial wave function of the appropriate symmetry. Excited states of the same symmetry as a lower state are more difficult, in principle, owing to the absence of a theorem that insures that the calculation evolves to the appropriate excited state. Early calculations of the E state of H2 confirmed the capability of the DMC method to accurately calculate the energy of the state for which a small MCSCF trial wave function was used. [Grimes]. Rather challenging calculations of excited states for small hydrocarbon system have been carried out in Filippi’s group [Filippi3]. The difficulty comes from the fact that these systems exhibit a number of excited states which are competitive in energy and sensitive to the treatment of electron correlation. Using careful optimization of the variational wave functions for both ground states and excitations the authors were able to identify the correct excitations within the given symmetry sectors and relative differences. Computation of properties other than the energy: moments, forces, equilibrium geometries, transition states,.[Caffarel1,2,3, Rappe2, Schuetz, Liu, Hammond1, Liu] .Calculations of properties that do not commute with Hamiltonian are more difficult to carry out with DMC methods because the distribution is the mixed distribution  , not the pure fixed-node solution. The calculation of pure expectation values requires sampling from the pure distribution which has been addressed by Barnett, et al. [Barnett1, Barnett2] in studies of QMC expectation values of moments, and by by Rothstein, et al. [Rothsteina,b, Caffarela] for differential operators and relativistic corrections. All such “pure” methods require the computation of weighting factors (either explicitly or by random walk) which correct for the mixed distribution. This adds significant computation above and beyond energy, and adds variance to the final result.
A second difficulty in computing properties other than the energy is that their estimators can have substantially larger fluctuations. The use of importance sampling can in principle reduce the variance of the local energy to zero for the case where the trial function is exact trial. This zero-variance principle does not hold for other properties, e.g. the dipole moment operator, and the variance is particularly significant for calculations of atomic forces because the estimators involve derivatives of the total energy. Various aspects of these complications have been tackled in recent years. As suggested recently, the variance issue can adressed by modifying the estimator so that the fluctuations decrease significantly without biasing the resulting estimate [Caffarel1]. ??(need more on this or leave out) The issue of divergences of potentials and associated large fluctuations has been addressed by Chiesa et al [Chiesa] by filtering out the corresponding terms and thereby decreasing the corresponding variance. The mixed estimator issue has been also studied and a promising method was introduced by Moroni and Baroni [Moroni] called reptation Monte Carlo. The approach eliminates the branching step from DMC and enables one to compute pure estimators effectively using forward walking (or, more precisely, backward averaging) without carrying the weights. To date the method has not been efficient for large systems, however, recent developments [Moroni, private] indicate that the method can be improved to remove this shortcoming. Another approach that has yielded very good results is the phaseless auxiliary field quantum Mont Carlo method [Zhang1]. Except for special cases, two-body interactions require auxiliary fields that are complex. As a result, the single-particle orbitals become complex, and the MC averaging over auxiliary field configurations becomes an integration over complex variables in many dimensions which leads to a phase problem that defeats the algebraic scaling of Monte Carlo and makes the method scale exponentially. This is analogous to but more severe than the sign problem with real auxiliary fields or in real-space methods. The sign problem in the complex plane is eliminated by projection onto the real axis introducing thus an approximation analogous to fixed-node approximation in the usual DMC method. The new approach enables the use of any one-particle basis and projects out the ground state by random walks in the space of Slater determinants. Some of the less desireable features of this method is the fact that the energy is not necessarily variational and the amount of correlation energy is limited by the size of the used basis. Nevertheless, excellent results have been obtained with the method using a plane-wave basis and nonlocal pseudopotentials [Zhang2, Zhang3]. [LM-what additional difficulties did you have in mind?] Download 242 Kb. Share with your friends:
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, introducing a so called “localization approximation” [Christiansen, Reynolds, Mitas]. This approximation enabled a number