Session W24: Focus Session: Configuration interaction Quantum Monte Carlo techniques

Stochastic Coupled Cluster Theory

In an extension of the Full Configuration Interaction Monte Carlo method of Alavi et al.[1], I describe a stochastic algorithm to perform Coupled Cluster Theory[2] which represents excitation amplitudes as populations discrete excitation particles (excips) in the space of excitation operators (excitors). Re-expressing the Coupled Cluster equations as the dynamics of excips in this space, we show that a simple set of rules consisting of spawning, death, and annihilation steps suffice to evolve a distribution of in the space of excitors to sample the Coupled Cluster solution and correctly evaluate its energy. These rules are extremely simple to implement and not truncation-specific and thus this method can calculate solutions to an arbitrary level of truncation. I present results of CCSDTQ calculations on the neon atom with basis sets up to cc-pV6Z as well as calculations on the uniform electron gas beyond the capability of other present methods. \\[1pt] [1] GH Booth, AJW Thom, A Alavi, J. Chem. Phys. (2009) 131, 054106 \\[1pt] [2] AJW Thom, Phys. Rev. Lett. (2010) 105, 263004   

Full Configuration Interaction Quantum Monte Carlo and Diffusion Monte Carlo: A Comparative Study of the 3D Homogeneous Electron Gas

Full configuration interaction quantum Monte Carlo$^1$ (FCIQMC) and its initiator adaptation$^2$ allow for exact solutions to the Schr\"odinger equation to be obtained within a finite-basis wavefunction \textit{ansatz}. In this talk, we explore an application of FCIQMC to the homogeneous electron gas (HEG). In particular we use these exact finite-basis energies to compare with approximate quantum chemical calculations from the VASP code$^3$. After removing the basis set incompleteness error by extrapolation$^{4,5}$, we compare our energies with state-of-the-art diffusion Monte Carlo calculations from the CASINO package$^6$. Using a combined approach of the two quantum Monte Carlo methods, we present the highest-accuracy thermodynamic (infinite-particle) limit energies for the HEG achieved to date. $^1$ G. H. Booth, A. Thom, and A. Alavi, J. Chem. Phys. 131, 054106 (2009). $^2$ D. Cleland, G. H. Booth, and A. Alavi, J. Chem. Phys. 132, 041103 (2010). $^3$ www.vasp.at (2012). $^4$ J. J. Shepherd, A. Gr\"uneis, G. H. Booth, G. Kresse, and A. Alavi, Phys. Rev. B. 86, 035111 (2012). $^5$ J. J. Shepherd, G. H. Booth, and A. Alavi, J. Chem. Phys. 136, 244101 (2012). $^6$ R. Needs, M. Towler, N. Drummond, and P. L. R\'ios, J. Phys.: Condensed Matter 22, 023201 (2010).   

Development of multicomponent semistochastic quantum Monte Carlo method for variational solution of molecular Hamiltonian without invoking the Born-Oppenheimer approximation

We present the multicomponent extension of the semistochastic quantum Monte Carlo (mc-SQMC) method for treating electron-nuclear correlation in the molecular Hamiltonian. All particles in the molecule are treated quantum mechanically and the variational solution is obtained with the SQMC method. The key feature of this approach is that the BO and separation-rotor approximation are not assumed. The application of the SQMC method for multicomponent systems involves many formidable challenges and this talk will focus on strategies to address these challenges including, appropriate coordinate system for the molecular Hamiltonian, separation of the center of mass kinetic energy, construction of the 1-particle basis functions for electrons and nuclei, construction of the multicomponent CI space and evaluation of connected configurations needed during propagation step in the SQMC method. Results from mc-SQMC will be presented for H$_2$, He$_2$, and H$_2$O systems. The H$_2$ system has been extensively studied using various methods, such as QMC and PIMC, making it an ideal system to test and compare the mc-SQMC implementation. The impact of the BO approximation and vibration-rotation coupling will be discussed by comparing mc-SQMC results with reported values for the weakly bound He$_2$.   

Reduced Density Matrices in Full Configuration Interaction Quantum Monte Carlo

Reduced density matrices are a powerful construct in quantum chemistry, providing a compact representation of highly multi-determinantal wavefunctions, from which the expectation values of important physical properties can be extracted, including multipole moments, polarizabilities and nuclear forces$^{1,2}$. Full configuration interaction quantum Monte Carlo (FCIQMC)$^3$ and its initiator extension (\emph{i}-FCIQMC)$^4$ perform a stochastic propagation of signed walkers within a space of Slater determinants to achieve FCI-quality energies \emph{without} the need to store the complete wavefunction. We present here a method for a stochastic calculation of the 1- and 2-body reduced density matrices within the framework of (\emph{i})-FCIQMC, and apply this formulation to a range of archetypal molecular systems. Consideration is also given to the source and nature of systematic and stochastic error, and regimes to effectively alleviate these errors are discussed$^5$. $^1$ P.-O. L\"{o}wdin, Phys. Rev. 97, 1474 (1955). $^2$ C. A. Coulson, Rev. Mod. Phys. 32, 170 (1960). $^3$ G. H. Booth, A. Thom, and A. Alavi, J. Chem. Phys. 131, 054106 (2009). $^4$ D. Cleland, G. H. Booth, and A. Alavi, J. Chem. Phys. 132, 041103 (2010). $^5$ D. Cleland, PhD thesis, University of Cambridge, 2012.   

Evaluation of expectation values in full configuration interaction quantum Monte Carlo

The full configuration interaction quantum Monte Carlo (FCIQMC) method[1-3] provides access to the exact ground state energy. However, like diffusion Monte Carlo, it is hard to precisely calculate expectation values of operators which do not commute with the Hamiltonian due to the stochastic representation of the wavefunction. Following related work on diffusion Monte Carlo[4], we have formulated an approach to stochastically sample additional operators in FCIQMC by using the Hellmann-Feynman theorem and sampling pumped equations of motion coupled to the standard equation of motion used to evolve the wavefunction. Our approach requires only minor modifications to existing FCIQMC programs and can be used to evaluate expectation values of arbitrary operators. We will present example calculations on the Hubbard model and molecular systems. \\[1pt] [1] G.H. Booth, A.J.W. Thom, A. Alavi, J. Chem. Phys. 131, 054106 (2009). [2] D. Cleland, G.H. Booth, A. Alavi, J. Chem. Phys. 132, 041103 (2010). [3] J.S. Spencer, N.S. Blunt, W.M.C. Foulkes, J. Chem. Phys. 136, 054110 (2012). [4] R. Gaudoin, J.M. Pitarke, Phys. Rev. Lett. 99, 126406 (2007).   

Improvements and Applications of Semistochastic Quantum Monte Carlo

Fully stochastic quantum Monte Carlo (QMC) methods, such as the full configuration interaction quantum Monte Carlo (FCIQMC) [1,2] allow one to compute the ground state of a Hamiltonian in a far larger Hilbert space than is possible using deterministic iterative diagonalization techniques. However, QMC methods suffer from the sign problem and may have large statistical errors. Recently we have shown [3] that these problems can be greatly alleviated by using a semistochastic quantum Monte Carlo (SQMC) approach, wherein the iterative projector is applied deterministically for a small subset of the Hilbert space states and stochastically elsewhere. In addition, the initiator bias, which is introduced to tame the sign problem in FCIQMC, is often greatly reduced. We explore further improvements to SQMC and apply it to a subset of the G2 set of molecules [4]. [1] George Booth, Alex Thom, Ali Alavi. J Chem Phys 131, 050106, (2009). [2] Deidre Cleland, George Booth, and Ali Alavi. J Chem Phys 132, 041103 (2010). [3] F. R. Petruzielo, A. A. Holmes, Hitesh J. Changlani, M. P. Nightingale, and C. J. Umrigar. Phys Rev Lett (Accepted 5 Oct 2012). [4] L. A. Curtiss, K. Raghavachari, G. W. Trucks, and J. A. Pople, J Chem Phys 94, 7221 (1991).   

Investigating the Singlet-Triplet Gap in Tetramethyleneethane using Quantum Monte Carlo Techniques

Tetramethyleneethane (TME) is an organic molecule composed of two allyl subunits that is the simplest disjoint diradical. The ground state according to experimental and theoretical evidence is a singlet state with $^{1}$A symmetry.~\footnote{Clifford, E. P.; Wenthold, P. G.; Lineberger, W. C.; Ellison, G. B.; Wang, C. X.; Grabowski, J. J.; Vila, F.; Jordan, K. D. J. Chem. Soc., Perkin Trans. 2 1998, 1015.} Due to the near degeneracy of the frontier orbitals, however, this state is inherently two-configurational. As the molecule is twisted through torsional angles about the central C-C bond, we compute the singlet-triplet gap using quantum Monte Carlo (QMC). In its diffusion Monte Carlo (DMC) variant, QMC is an exact method for solving the Schr\"{o}dinger equation within the bounds of the fixed-node approximation.~\footnote{ J.B. Anderson, J. Chem. Phys. 63, 1499 (1975).} DMC calculations using a multi-configurational trial wave function produce the correct ordering of the singlet and triplet states. We also investigate an alternate approach, full configuration interaction quantum Monte Carlo (FCIQMC). We compare the FCIQMC singlet-triplet energy gap as a function of torsional angle with the different theoretical methods.   

Systematically improvable auxiliary-field quantum Monte Carlo for strongly correlated systems

The quest for an accurate and scalable many-body method for strongly correlated systems is still ongoing despite many years of efforts. The auxiliary-field quantum Monte Carlo (AFQMC) method is an exact many-body method, but it suffers from a sign problem that limits its usefulness. The phaseless AFQMC (ph-AFQMC) has been introduced\footnote{Zhang and Krakauer, Phys. Rev. Lett. 90, 136401 (2003)} to control the sign problem. In this work we employ the unconstrained (exact) AFQMC method on massively parallel supercomputers to systematically improve ph-AFQMC results. Applications to strongly correlated systems, including transition-metal compounds, will be presented.   

Acceleration of Self Healing Diffusion Monte Carlo for nearly degenerate eigenstates

The Self-Healing Diffusion-Monte-Carlo method (SHDMC) recurisvely applies an evolution operation for a finite imaginary time. SHDMC and finds the full configuration interaction coefficients of the many-body ground state by projecting out excited states. The convergence of the SHDMC, being a projection method, is dictated by the energy separation between the ground and excited states. In this talk we explore methods to accelerate the convergence of SHDMC for nearly degenerate states using the dynamical information of the excited states accumulated over the recursive iterations and to compute ground and excited states simultaneously.   

Excited states and spectral functions within full configuration interaction quantum Monte Carlo

Here we consider a modified propagator in order to obtain stable convergence to excited states within the full configuration interaction quantum Monte Carlo framework.\footnote{G. H. Booth and G. K.-L. Chan, ArXiv:1210.6643 (2012)} By working with a Gaussian propagator, the dominant eigenstate is one which is closest to an initial guess energy for the state. Issues with the speed of convergence compared to the ground state propagator are discussed, with results presented for pilot applications, and potential improvements for the algorithm considered.   

Quantum Monte Carlo Study of $\pi $-bonded Transition-metal Organometallic Sandwiches

Accurate quantum Monte Carlo (QMC) calculations enabled us to determine the structure, spin multiplicity, ionization energy, dissociation energy, and spin-dependent electronic gaps of neutral and positively charged vanadium-benzene and cobalt-benzene half-sandwich and sandwich systems. The most intriguing application of these systems is as spin filters. For this purpose we have used a multi-stage combination of techniques with consecutive elimination of systematic biases except for the fixed-node approximation in QMC. The-fixed node approximation was treated at different levels from quantum chemistry (CAS-SCF) to various DFT schemes such as GGA, meta-GGA, hybrid, double-hybrid and local-hybrid functionals. While QMC results indicate a very limited predictive power of mean field DFT methods for this class of systems, QMC results are quite stable with fixed-node approximation based on several classes of DFT orbitals. Our results significantly differ from the established picture based on previous less accurate calculations and point out the importance of high-level many-body methods for predictive calculations of similar transition metal-based organometallic systems.   

Symmetry in Auxiliary-Field Quatnum Monte Carlo Calculations

We discuss how symmetry properties can be preserved rigorously to improve the accuracy and efficiency in auxiliary-field quantum Monte Carlo (AFQMC) calculations. Using the Hubbard model as an example, we study the effect of symmetry in two aspects of ground-state AFQMC calculations, the Hubbard-Straonovich transformation and the form of the trial wave function. In unconstrained calculations, the implementation of symmetry often leads to shorter convergence time and much smaller statistical errors, thereby resulting in a substantial reduction of the sign problem and allowing exact calculations for larger and more strongly correlated systems. Moreover, certain excited states become possible to calculate which are otherwise beyond reach. In calculations with constraints,\footnote{S.~\ Zhang, J.~\ Carlson, and J.~\ Gubernatis, Phys. Rev. B {\bf 55}, 7464 (1997)}$^,$\footnote{C.~\ Change, S.~\ Zhang, Phys. Rev. B {\bf 78}, 165101 (2008)} it is shown that the use of symmetry can often reduce the systematic error significantly. Results are presented for the two-dimensional repulsive Hubbard model.