Complete-Graph Tensor Network States: A New Fermionic Wave Function Ansatz for Molecules

We present a new class of tensor network states that are specifically designed to capture the electron correlation of a molecule of arbitrary structure. In this ansatz, the electronic wave function is represented by a Complete-Graph Tensor Network (CGTN) ansatz which implements an efficient reduction of the number of variational parameters by breaking down the complexity of the high-dimensional coefficient tensor of a full-configuration-interaction (FCI) wave function. We demonstrate that CGTN states approximate ground states of molecules accurately by comparison of the CGTN and FCI expansion coefficients. The CGTN parametrization is not biased towards any reference configuration in contrast to many standard quantum chemical methods. This feature allows one to obtain accurate relative energies between CGTN states which is central to molecular physics and chemistry. We discuss the implications for quantum chemistry and focus on the spin-state problem. Our CGTN approach is applied to the energy splitting of states of different spin for methylene and the strongly correlated ozone molecule at a transition state structure. The parameters of the tensor network ansatz are variationally optimized by means of a parallel-tempering Monte Carlo algorithm

Excited states with selected CI-QMC: chemically accurate excitation energies and geometries

We employ quantum Monte Carlo to obtain chemically accurate vertical and adiabatic excitation energies, and equilibrium excited-state structures for the small, yet challenging, formaldehyde and thioformaldehyde molecules. A key ingredient is a robust protocol to obtain balanced ground- and excited-state Jastrow-Slater wave functions at a given geometry, and to maintain such a balanced description as we relax the structure in the excited state. We use determinantal components generated via a selected configuration interaction scheme which targets the same second-order perturbation energy correction for all states of interest at different geometries, and we fully optimize all variational parameters in the resultant Jastrow-Slater wave functions. Importantly, the excitation energies as well as the structural parameters in the ground and excited states are converged with very compact wave functions comprising few thousand determinants in a minimally augmented double-$\zeta$ basis set. These results are obtained already at the variational Monte Carlo level, the more accurate diffusion Monte Carlo method yielding only a small improvement in the adiabatic excitation energies. We find that matching Jastrow-Slater wave functions with similar variances can yield excitations compatible with our best estimates; however, the variance-matching procedure requires somewhat larger determinantal expansions to achieve the same accuracy, and it is less straightforward to adapt during structural optimization in the excited state.Comment: 11 pages, 4 figure

Bond-slip analysis via a cohesive-zone model simulating damage, friction and interlocking

A recently proposed cohesive-zone model which effectively combines damage, friction and mechanical interlocking has been revisited and further validated by numerically simulating the pull-out test, from a concrete block, of a ribbed steel bar in the post-yield deformation range. The simulated response is in good agreement with experimental measurements of the bond slip characteristics in the post-yield range of deformed bars reported in the literature. This study highlights the main features of the model: with physically justified and relatively simple arguments, and within the sound framework of thermodynamics with internal variables, the model effectively separates the three main sources of energy dissipation, i.e. loss of adhesion, friction along flat interfaces and mechanical interlocking. This study provides further evidence that the proposed approach allows easier and physically clearer procedures for the determination of the model parameters of such three elementary mechanical behaviours, and makes possible their interpretation and measurement as separate material property, as a viable alternative to lumping these parameters into single values of the fracture energy. In particular, the proposed approach allows to consider a single value of the adhesion energy for modes I and II

Lecture Notes of Tensor Network Contractions

Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.Comment: 134 pages, 68 figures. In this version, the manuscript has been changed into the format of book; new sections about tensor network and quantum circuits have been adde

A Perturbative Density Matrix Renormalization Group Algorithm for Large Active Spaces

We describe a low cost alternative to the standard variational DMRG (density matrix renormalization group) algorithm that is analogous to the combination of selected configuration interaction plus perturbation theory (SCI+PT). We denote the resulting method p-DMRG (perturbative DMRG) to distinguish it from the standard variational DMRG. p-DMRG is expected to be useful for systems with very large active spaces, for which variational DMRG becomes too expensive. Similar to SCI+PT, in p-DMRG a zeroth-order wavefunction is first obtained by a standard DMRG calculation, but with a small bond dimension. Then, the residual correlation is recovered by a second-order perturbative treatment. We discuss the choice of partitioning for the perturbation theory, which is crucial for its accuracy and robustness. To circumvent the problem of a large bond dimension in the first-order wavefunction, we use a sum of matrix product states (MPS) to expand the first-order wavefunction, yielding substantial savings in computational cost and memory. We also propose extrapolation schemes to reduce the errors in the zeroth- and first-order wavefunctions. Numerical results for Cr 2 with a (28e,76o) active space and 1,3-butadiene with a (22e,82o) active space reveal that p-DMRG provides ground state energies of a similar quality to variational DMRG with very large bond dimensions, but at a significantly lower computational cost. This suggests that p-DMRG will be an efficient tool for benchmark studies in the future

Remarkable Hydrogen Storage on Beryllium Oxide Clusters: First Principles Calculations

Since the current transportation sector is the largest consumer of oil, and subsequently responsible for major air pollutants, it is inevitable to use alternative renewable sources of energies for vehicular applications. The hydrogen energy seems to be a promising candidate. To explore the possibility of achieving a solid-state high-capacity storage of hydrogen for onboard applications, we have performed first principles density functional theoretical calculations of hydrogen storage properties of beryllium oxide clusters (BeO)$_{n}$ (n=2 -- 8). We observed that polar BeO bond is responsible for H$_{2}$ adsorption. The problem of cohesion of beryllium atoms does not arise, as they are an integral part of BeO clusters. The (BeO)$_{n}$ (n=2 -- 8) adsorbs 8--12 H$_{2}$ molecules with an adsorption energy in the desirable range of reversible hydrogen storage. The gravimetric density of H$_{2}$ adsorbed on BeO clusters meets the ultimate 7.5 wt% limit, recommended for onboard practical applications. In conclusion, beryllium oxide clusters exhibit a remarkable solid-state hydrogen storage.Comment: This document is the Accepted Manuscript version of a Published Work that appeared in final form in JPCC, copyright American Chemical Society after peer review and technical editing by the publisher. To access the final edited and published work see , see http://pubs.acs.org/doi/abs/10.1021/jp410994