Exciting progress has been achieved in understanding the fundamental approximation of the DMC method, namely, the fixed-node bias. In particular, a conjecture by Ceperley [ref] that the fermion node hypersurface divides the space of electron configurations into the minimal number of two domains for non-degenerate ground states has been actually explicitly demonstrated for a range of non-interacting systems and for important trial functions such as those based on pairing orbitals in BCS and pfaffian forms. This work shows that the topologies of the nodal surfaces are generically (i.e., almost always) rather simple except possibly for complicated cases of quantum phase transitions and large-scale/infinite degeneracies or nonlocal interactions. This research prompted further investigations into the accurate shapes of the nodal surfaces that are important for minimization of fixed-node errors using, for example, multi-pfaffian expansions [Bajdich1,2]. This line of research has also opened a new direction which introduces concepts of topology and quantum geometry into electronic structure theory. Although much needs to be done, further research based on ideas well-established in field theory and quantum statistical mechanics is anticipated to influence both fundamental understanding of quantum systems and practical applications.
Coupling of QMC and molecular dynamics (MD) approaches.
The first attempt to combine the strength of QMC with ab intio MD methods was a study by Grossman and Mitas [Grossman2]. This effort was based on efficient calculations of DMC energies along the nuclear trajectories generated by the ab initio DFT/MD method. The key gain was obtained by updating the walker distribution and DMC trial wave function along the nuclear paths. Nuclei in ab initio MD require very small timestep displacements that are of the order of 0.001-0.0001 a.u., which is sufficient to provide an opportunity for the electronic walker distribution to relax in a few DMC steps, because DMC step is typically one to two orders of magnitude larger (0.1 – 0.01 a.u.). The authors found that about three DMC steps are enough to update the wave function for each step in nuclear position and the QMC part of the calculation increased the overall computation by only a factor of two. Further, they found that although the detailed wave function information at any given step is determined by the walker distribution so that the error bars are significant at any single point in the MD simulation, over the period of the entire MD run they were able to obtain thermodynamic averages with very small error bars. One of the key results was an estimation of the heat of water evaporation, a dynamical and finite temperature quantity, that was found to be 6.5XXX kcal/mol, in excellent agreement with the experimental value of 6.1 kcal/mol (the DFT estimate is too large by about 30 %). Very recent MD/QMC calculations were carried out by Sorella et al. [Sorella1] for hydrogen systems included also forces from QMC in the study of liquid hydrogen and its properties at varying thermodynamical conditions.
TOTALLY NEW DIRECTION – Christov, combines QMC with Bohmian dynamics, all in real time! Only 1-D application to date. Possible some mention, what do you think?
Applications
Following is a list of calculations with brief comments that provide insight on the types of DMC calculations that have been carried out to date. The list is biased towards calculations of the energy for which DMC has been found to yield highly accurate results, although findings regarding other properties are given.
Molecular systems involving only first row atoms
ETHANE
A further validation of the DMC method for small molecules was recently reported in the form of the singlet-triplet splitting in ethane. This property was computed using a large basis set with the CCSD(T) method and compared the result of a DMC calculation with a Slater-Jastrow trial function. Excellent agreement between the methods was obtained. [Dixon].
OZONE-I’ll have to double check these results by others (???)
O4 – there is a recent PRL by Caffarel and collaborators that looks interesting.
The O4 Molecule
The O4 species has relevance for atmospheric processes. In this regard the barrier to dissociation and the heat of formation have particular importance. Owing to its small size the expectation is that state-of-the-art computational approaches would provide a full understanding of the electronic properties of the system, but complications due to the open-shell nature of O2, a proper description to bond breaking, and the complicated spin recoupliongs that occur in this system present essential complications.. A recent DMC study has addressed these difficulties in describing the passage from a singlet O4 reactant to two ground state O2 triplet products [Caffarel5].
The authors have carefully addressed the issue of accurate differences of total energies for barriers, enthalphies, affinities, etc. required for these calculations, an issue which pervades all electronic structure methods. In this study, the exceptional step was taken of varying the multireference character of the nodes that made possible very high accuracy for the energy differences.
C20 - there is the comparison of results that Peter Taylor and
Porphyrin – large system; singlet-triplet and singlet-singlet energy differences (Bill ?)
Free Base Porphyrin
Accurate DMC calculations of allowed and nonallowed transitions in porphyrin have been reported [Aspuru-Guzik4]. The vertical transition between the ground state singlet and the second excited state singlet as well as the adiabatic transition between the ground state and the lowest triplet state were computed for this 162-electron system. The DMC results were found to be in excellent agreement with experiment.
Photosynthesis study – size systems that are being treated that no other ab initio approach can address.
Lubos – what studies in condensed matter, as well as your work on atoms and molecules?
JCG’s G1 benchmark study
A systematic study of QMC performance on the G1 set of molecules has been carried out by Grossman [Grossman1]. He used the simplest QMC single determinant Slater-Jastrow wave functions and found mean error/max error …. in comparison with CCSD(T) … . The largest error was found for P_2 molecule and further improvements of the trial function using multi-reference MCSCF antisymmetric part decreased the error to ….. This provided an important systematic insight into the QMC performance at that time. It showed great potential of QMC method but also pointed out that in difficult cases the simple Slater-Jastrow wave function might not be sufficient for desired accuracy. It pointed out a clear need for improvements of trial wave functions and directed the effort towards overcoming the limits of simple wave functions.
Transition Metal Systems
Significant progress has been achieved with DMC in the calculation of transition metal systems since the pioneering studies of Fe [Mitas1], of CuSin[Ovcharenko], and CO adsorption on a Cr(110) surface [El Akramine]. Recent developments include all-electron FN-DMC calculations of the low-lying electronic states of Cu and its cation. The states considered were the most relevant for organometallic chemistry of Cu-containing systems, namely, the 2S, 2D, and 2P electronic states and the 1S ground state of Cu+. The FN-DMC results presented in this work provide, to the best of our knowledge, the most accurate nonrelativistic all-electron correlation energies for the lowest-lying states of copper and its cation. To compare results to experimental data the authors included relativistic contributions for all states through numerical Dirac-Fock calculations, which for Cu provide almost the entire relativistic effects. It was found that the fixed-node errors using HF nodes for the lowest transition energies and the first ionization potential of the atom cancel out within statistical fluctuations. The overall accuracy achieved with DMC for the nonrelativistic correlation energy (statistical fluctuations of about 1600 cm–1 and near cancelation of fixed-node errors) is good enough to reproduce the experimental spectrum when relativistic effects are included. These results illustrated that, despite the presence of the large statistical fluctuations associated with core electrons, accurate all-electron FN-DMC calculations for transition metals are presently feasible using extensive, but accessible computer resources.
For transition-metal-containing molecules, recent developments include calculations of transition metal-oxygen molecules such as TiO, MnO for binding energies, equilibrium bond lengths and dipole moments [Wagner 1, Wagner2]. Calculations of energy related quantities showed the strength of QMC quite clearly since with the simplest Slater-Jastrow trial functions one could get results on par with quite large CCSD(T) calculations and within a few percent from experiments or better. In addition, calculations of bond-breaking is not much more complicated than in the equilibrium unlike in CCSD(T) method where special care has to be taken to obtain reliable results for such cases. The dipole moments showed somewhat larger discrepancies for CrO and MnO, even when taking into account less realiable experimental data. Dipole moments are very sensitive to the contribution of higher excitations, while energy is less so, and show the missing pieces in the trial wave functions and points out a direction for improvements. Very recently, large-scale QMC calculations has been carried out of FeO solid at high pressures providing equilibrium properties such as equilibrium lattice constant, bulk modulus, cohesive energy of 9.66(6) eV per FeO (experiment 9.7(1) eV per FeO), all in excellent agreement with experiment [Kolorenc]. In addition, the calculations have produces also very good estimation of the band gap and, finally, enabled to construct equations of state and estimation of transition pressure into another phase at the lower end of available experimental data. Clearly, this is an important accomplishment showing that calculations of complicated systems with several hundreds of valence electrons are possible and will become more routine with the onset of the next generation of parallel architectures and multi-core processors.
Solids
Weak interactions
Noteworthy DMC studies of weakly bound molecules have established the capability of FNDMC to yield highly accurate interaction energies [Anderson2, Luechow1]. It was shown that FNDMC does not suffer from basis set superposition error owing to the convergence of monomer(s) with basis set, unlike ab initio post-Hartree-Fock methods. However, it was found for large systems that special attention must be paid to the ability of the atomic orbital basis set to describe the asymptotic behavior of the wave function to avoid sampling errors. Accurate results were reported for water, ammonia, and two benzene dimers – one T shaped and the other displaced parallel to the first.
Some of the QMC methods matured and are implemented in several existing codes and packages which are available for use by communities at large. We will mention the packages which are most familiar to us: CASINO [NeedsCASINO], QMCPACK [KimQMCPACK], QWalk [WagnerQWALK] and CHAMP [CHAMP]. QMCPACK and QWalk are distributed under Open Source licences.
Acknowledgments
WAL was supported by the Director, Office of Science, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098, and by the CREST Program of the U. S. National Science Foundation. LM is supported by NSF/CMG grant EAR-0530110 and by the DOE grant DE-FG05-08OR23337 and some of the projects mentioned in the paper have been carried out using INCITE and NCMS allocations at DOE ORNL computing facilities.
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