Session L23: Classical and Quantum Monte Carlo II

A Quantum Monte Carlo Study of Molecular Titanium Systems

We present a quantum Monte Carlo study of molecular TiH$_2$ and Ti-ethylene-hydrogen complexes which have been of recent interest for their relation to systems that can reversibly adsorb hydrogen.\footnote{E. Durgun \textit{et al.}, Phys. Rev. Lett. \textbf{97}, 226102 (2006).}$^,$\footnote{J. A. Platts, J. Mol. Struct. \textbf{545}, 111 (2001).}$^,$\footnote{B. Ma, C. L. Collins, H. F. Schaefer, J. Am. Chem. Soc. \textbf{118}, 870 (1996).} We study these systems using diffusion Monte Carlo with the fixed-node approximation and pseudopotentials. The symmetry and nodal structure used are determined by trial wave functions constructed of molecular orbitals from DFT. In the TiH$_2$ system, the four lowest states have different symmetries and are very close in energy due to the fact that the d-states are almost decoupled from the bonding. We show that partially occupying the relevant d-states at the DFT level allows for the construction of symmetry classified trial functions that are more directly comparable at the DMC level. This procedure has potential to be useful in analogous systems where d-state occupation results in nearly degenerate states of different symmetry.   

High Pressure Phase Transitions in FeO from Quantum Monte Carlo

LDA+U and LDA+DMFT are successful methods for determining the electronic structure of FeO under pressure, but they suffer from two deficiencies. The extreme sensitivity of the spin collapse in MnO on the parameters $U$ and $J$ casts doubt upon the predictive power of the methods$^1$. Additionally, the symmetry of the occupation matrix has a profound effect on the electronic structure and magnetoelastic coupling in FeO$^2$, and is not easy to determine a priori. We perform diffusion Monte Carlo (DMC) calculations for FeO at a range of pressures using trial wavefunctions generated with LDA+U for several values of $U$. The limits of the Slater-Jastrow wavefunction we use causes Variational Monte Carlo to give different and erroneous predictions of the optimal value of $U$, so we use the variational property of the nodal surface in DMC to find the optimal value. The magnetoelastic coupling is shown to be insensitive to the symmetry of the occupation matrix. Thus we are able to make an accurate prediction of the pressure for the high spin low-spin transition in FeO as the possibility of a ferromagnetic state at high pressures. $^1$ Kasinathan, D. et al. New J. Phys. {\bf 9}, 235 (2007)\,\, $^2$ Gramsch. S. A. et al. American Mineralogist {\bf 88}, 257 (2003)   

Path Integral Monte Carlo calculation of hydrogen Hugoniot

We use restricted path integral Monte Carlo to calculate the compression of liquid Hydrogen/Deuterium between 50 and 600 GPa for a temperature ranging from 10,000 to 1 Million Kelvin. Different theories and experiments place the compression between 4 and 6. We attempt to tackle this inconsistency with a new fermion fixed node algorithm and careful finite size effect study. Our preliminary results place the compression at 4.3.   

Quantum Monte Carlo calculations of bulk Li

Using all-electron fixed-node quantum Monte Carlo methods we calculate equations of states for ambient and high pressure phases of Li using the structural data from experimental observations up to high pressures.\footnote[1]{M.~Hanfland, K.~Syassen, N.~E.~Christensen, and D.~L.~Novikov, Nature 408, 174--178 (2000).} We compare the suitability of orbital sets from several Density Functional Theory functionals for use in many-body trial Slater-Jastrow wave-functions. We reduce finite-size errors by utilizing twist-averaging\footnote[2]{C.~Lin, F.~H.~Zhong, and D.~M.~Ceperley, Phys. Rev. E 64, 016702 (2001).} and by structure factor correction for several sizes of simulation cell.\footnote[3]{S.~Chiesa, D.~M.~Ceperley, R.~M.~Martin, and M.~Holzmann, Phys. Rev. Lett. 97, 076404 (2006).}   

Diffusion quantum Monte Carlo study of Silicon Carbide

Silicon Carbide (SiC) is an important semiconductor used in high temperature electronic applications because of its large excitation energy. We use diffusion quantum Monte Carlo (DMC) to calculate some of it's electronic, physical, and optical properties. An analysis of the symmetry of the trial wave-function's single particle orbitals is required because SiC is an indirect gap semiconductor. In order to obtain an upper bound estimate of the energy of the excited state we symmetrize the exciton trial wave-function so that it belongs to an irreducible represention of dimension one. We report results on the equation of state of SiC, and the finite size scaling of the band gap obtained with DMC.   

Electronic structure of azobenzene: ground and first excited singlet states

QMC techniques are used to obtain energies at selected points on the potential energy surfaces of a photoswitchable molecule, azobenzene (AB), along the torsion pathway (CNNC dihedral angle), in the ground and first excited singlet states. We study the excitation energies of well separable cis- and trans-conformers, and th Inst. Phys., Slovak Acad. Scie energy of the transition state located at 90\r{ }. By a careful QMC optimization of the Slater-Jastrow wavefunctions with up to 500 determinants, chemical accuracy is obtained. Our results not only outperform all the available quantum chemistry results such as CAS-SCF, CAS-PT2, as well as DFT results with proper spin symmetry taken into account (ROKS), but open also a credible window to possible correction/reinterpretation of the available experimental data.   

Release Node Calculation of Unpolarized Atomic Fermi Gas

Using release node techniques in diffusion Monte Carlo method, we study the unpolarized atomic fermi gas in the unitarity regime. We use several types of the trial and guiding wave functions to elucidate the size of fixed-node error. We also analyze the changes in the nodal surfaces for trial functions with varying degree of the pairing strength. We study the bosonic component in guiding functions in order to optimize the extraction of the fixed-node bias and corresponding error bars. The most accurate results for the fixed-node bias were obtained using the the BCS-based guiding functions.   

Study of pseudopotential errors in transition metal molecules:implications for quantum Monte Carlo

We investigate pseudopotential errors for the electron structure in transition-metal molecules with strong electron correlation. We compare high spin-low spin state energy differences within Density Functional Theory (DFT), Hartree-Fock and hybrid functionals. In particular, we compare results from small-core Dirac-Fock pseudopotentials and both relativistic and nonrelativistic all-electron calculations. The presence of exact exchange affects these errors and points towards the importance of exact exchange in this context. Since the accuracy of quantum Monte Carlo (QMC) results depends on pseudopotential in a crucial manner, we study the implications for QMC calculations.   

Improved Calculation of Vibrational Mode Lifetimes in Anharmonic Solids

We propose a formal foundation for practical calculations of vibrational mode lifetimes in solids. The approach is based on a recursion method analysis of the Liouvillian. From this we derive the lifetime of a vibrational mode in terms of moments of the power spectrum of the Liouvillian as projected onto the relevant subspace of phase space. In practical terms, the moments are evaluated as ensemble averages of well-defined operators, meaning that the entire calculation is to be done with Monte Carlo. These insights should lead to significantly shorter calculations compared to current methods.   

Dipole moment of ultra-cold polar molecules:A quantum Monte Carlo study

There has been recently a great interest in the ultra-cold heteronuclear molecules that have a large electric dipole moment interaction both theoretically and experimentally. In this work, we calculate the dipole moment of a two-atom alkaline molecule, LiSr. We use two approaches: the configuration interaction and the quantum Monte Carlo method. We take the wavefunction calculated by configuration interaction and add the correlated terms, optimize the wavefunction to get a good candidate for Quantum Monte Carlo calculation. In order to reach a better accuracy for dipole moment, not only a diffusion Monte Carlo but a repetition Monte Carlo method is also implemented to better treat the quantity which does not commute with the Hamiltonian. We are also trying to evaluate the modifications coming from the spin orbit coupling term using quantum Monte Carlo method   

Using Monte Carlo Simulations to Develop an Understanding of the Hyperpolarizability Near the Fundamental Limit

The hyperpolarizability governs all light-matter interactions. In recent years, quantum mechanical calculations have shown that there is a fundamental limit of the hyperpolarizability of all materials. The fundamental limits are calculated only under the assumption that the Thomas Kuhn sum rules and the three-level ansatz hold. (The three-level ansatz states that for optimized hyperpolarizability, only two excited states contribute to the hyperpolarizability.) All molecules ever characterized have hyperpolarizabilities that fall well below the limits. However, Monte Carlo simulations of the nonlinear polarizability have shown that attaining values close to the fundamental limit is theoretically possible; but, the calculations do not provide guidance with regards to what potentials are optimized. The focus of our work is to use Monte Carlo techniques to determine sets of energies and transition moments that are consistent with the sum rules, and study the constraints on their signs. This analysis will be used to implement a numerical proof of three-level ansatz.   

Continuum quantum Monte Carlo simulations of solids on GPUs

Continuum quantum Monte Carlo (QMC) has proved to be an invaluable tool for predicting the properties of matter from fundamental principles. The multiple forms of parallelism afforded by QMC algorithms make it an ideal candidate for acceleration in the many-core paradigm on graphical processing units (GPUs). We present the results of porting the QMCPACK code to the NVIDIA CUDA platform. Using mixed precision on G200 GPUs and MPI for intercommunication, we observe typical full-application speedups of approximately 10x to 15x relative to quad-core Xeon CPUs alone, while reproducing the double-precision CPU results within statistical error. We discuss the algorithm modifications necessary to achieve good performance on this heterogeneous architecture and summarize the results of applying our code to structural and electronic phase transitions in bulk materials. Based on our experience, we make projections for the applicability of GPUs to other electronic structure methods.