Monte Carlo Configuration Interaction Research Articles

Solving the Schrödinger Equation in the Configuration Space with Generative Machine Learning.

The configuration interaction approach provides a conceptually simple and powerful approach to solve the Schrödinger equation for realistic molecules and materials but is characterized by an unfavorable scaling, which strongly limits its practical applicability. Effectively selecting only the configurations that actually contribute to the wave function is a fundamental step toward practical applications. We propose a machine learning approach that iteratively trains a generative model to preferentially generate the important configurations. By considering molecular applications it is shown that convergence to chemical accuracy can be achieved much more rapidly with respect to random sampling or the Monte Carlo configuration interaction method. This work paves the way to a broader use of generative models to solve the electronic structure problem.

Machine learning matrix product state ansatz for strongly correlated systems.

Machine learning (ML) has been used to optimize the matrix product state (MPS) ansatz for the wavefunction of strongly correlated systems. The ML optimization of MPS has been tested for the Heisenberg Hamiltonian on one-dimensional and ladder lattices, which correspond to conjugated molecular systems. The input descriptors and output for the supervised ML are lattice configurations and configuration interaction coefficients, respectively. Efficient learning can be achieved from data over the full Hilbert space via exact diagonalization or full configuration interaction, as well as over a much smaller sub-space via Monte Carlo Configuration Interaction. We show that this circumvents the need to calculate energy and operator expectation values and is therefore a computationally efficient alternative to variational optimization.

Active Learning Assisted MCCI to Target Spin States.

Strongly correlated systems and their accurate solutions have been challenging to quantum chemistry. Several methods have been developed over the years for the accurate understanding of such systems, and selected configuration interaction and Monte Carlo configuration interaction (MCCI) form important classes of systems in this category. However, MCCI is plagued by slow convergence. This is further exacerbated by the fact that most of the current MCCI implementations do not target specific spin states. In our work, we use active learning assisted MCCI to speed up the convergence manyfold and also develop a method for spin targeting. This method has been tested with several model Hamiltonian systems akin to molecular systems and has shown improved convergence and accuracy.

Efficient Computation of Two-Electron Reduced Density Matrices via Selected Configuration Interaction.

We create an approach to efficiently calculate two-electron reduced density matrices (2-RDMs) using selected configuration interaction wavefunctions. This is demonstrated using the specific example of Monte Carlo configuration interaction (MCCI). The computation of the 2-RDMs is accelerated by using ideas from fast implementations of full configuration interaction (FCI) and recent advances in implementing the Slater-Condon rules using hardware bitwise operations. This method enables a comparison of MCCI and truncated CI 2-RDMs with FCI values for a range of molecules, which includes stretched bonds and excited states. The accuracy in energies, wavefunctions, and 2-RDMs is seen to exhibit a similar behavior. We find that MCCI can reach sufficient accuracy of the 2-RDM using significantly fewer configurations than truncated CI, particularly for systems with strong multireference character.

Chembot: A Machine Learning Approach to Selective Configuration Interaction.

We introduce a machine learning-based approach to selective configuration interaction, dubbed Chembot, that utilizes many novel choices for its model design and training. These choices include the use of a support vector machine to select important configurations, the use of the charge density matrix and configuration energy as features, and heuristics to improve the quality of training data. We test Chembot's ability to obtain near full configuration interaction quality energies and find that it definitively outperforms its purely Stochastic cousin Monte Carlo configuration interaction by requiring fewer iterations to converge, fewer determinants in the variational space, and fewer important configurations to achieve the same energy. In addition, Chembot at times requires fewer determinants in its variational space than the heat-bath configuration interaction method to achieve the same energy. We demystify Chembot's innards and then showcase our claims on the set of small but challenging systems: the hydrogen ring (H4), stretched methylene (H2C), and stretched water (H2O).

Configuration interaction trained by neural networks: Application to model polyaromatic hydrocarbons.

The main bottleneck of a stochastic or deterministic configuration interaction method is determining the relative weights or importance of each determinant or configuration, which requires large scale matrix diagonalization. Therefore, these methods can be improved significantly from a computational standpoint if the relative importance of each configuration in the ground and excited states of molecular/model systems can be learned using machine learning techniques such as artificial neural networks (ANNs). We have used neural networks to train the configuration interaction coefficients obtained from full configuration interaction and Monte Carlo configuration interaction methods and have tested different input descriptors and outputs to find the more efficient training techniques. These ANNs have been used to calculate the ground states of one- and two-dimensional Heisenberg spin chains along with Heisenberg ladder systems, which are good approximations of polyaromatic hydrocarbons. We find excellent efficiency of training and the model this trained was used to calculate the variational ground state energies of the systems.

Exciting Determinants in Quantum Monte Carlo: Loading the Dice with Fast, Low-Memory Weights.

High-quality excitation generators are crucial to the effectiveness of coupled cluster Monte Carlo (CCMC) and full configuration interaction Quantum Monte Carlo (FCIQMC) calculations. The heat bath sampling of Holmes et al. [Holmes, A. A.; Changlani, H. J.; Umrigar, C. J. J. Chem. Theory Comput. 2016, 12, 1561-1571.] dramatically increases the efficiency of the spawn step of such algorithms but requires memory storage scaling quartically with system size which can be prohibitive for large systems. Alternatively, Alavi et al. [Smart, S. D.; Booth, G. H.; Alavi, A. Unpublished results.] approximated these weights with weights based on Cauchy-Schwarz-like inequalities calculated on-the-fly. While reducing the memory cost, this algorithm scales linearly in system size computationally. We combine both of these ideas with the single-reference nature of many systems studied and introduce a spawn-sampling algorithm that has low memory requirements (quadratic in basis set size) compared to the heat bath algorithm and only scales either independently of system size (CCMC) or linearly in the number of electrons (FCIQMC) that works especially well on localized orbitals. Tests on small water chains with localized orbitals with CCMC and with an initiator point sample in FCIQMC indicate that it can be equally efficient as the other excitation generators. As the system gets larger, calculations with our new algorithm converge faster than the on-the-fly weight algorithm while having a much more favorable memory scaling than the heat bath algorithm.

Machine Learning Configuration Interaction.

We propose the concept of machine learning configuration interaction (MLCI) whereby an artificial neural network is trained on-the-fly to predict important new configurations in an iterative selected configuration interaction procedure. We demonstrate that the neural network can discriminate between important and unimportant configurations, that it has not been trained on, much better than by chance. MLCI is then used to find compact wave functions for carbon monoxide at both stretched and equilibrium geometries. We also consider the multireference problem of the water molecule with elongated bonds. Results are contrasted with those from other ways of selecting configurations: first-order perturbation, random selection, and Monte Carlo configuration interaction. Compared with these other serial calculations, this prototype MLCI is competitive in its accuracy, converges in significantly fewer iterations than the stochastic approaches, and requires less time for the higher-accuracy computations.

Development of spin-orbit coupling for stochastic configuration interaction techniques.

To perform spin-orbit coupling calculations on atoms and molecules, good zeroth-order wavefunctions are necessary. Here, we present the software development of the Monte Carlo Configuration Interaction (MCCI) method, to enable calculation of such properties, where MCCI iteratively constructs a multireference wavefunction using a stochastic procedure. In this initial work, we aim to establish the efficacy of this technique in predicting the splitting of otherwise degenerate energy levels on a range of atoms and small diatomic molecules. It is hoped that this work will subsequently act as a gateway toward using this method to investigate singlet-triplet interactions in larger multireference molecules. We show that MCCI can generate very good results using highly compact wavefunctions compared to other techniques, with no prior knowledge of important orbitals. Higher-order relativistic effects are neglected and spin-orbit coupling effects are incorporated using first-order degenerate perturbation theory with the Breit-Pauli Hamiltonian and effective nuclear charges in the one-electron operator. Results are obtained and presented for B, C, O, F, Si, S, and Cl atoms and OH, CN, NO, and C2 diatomic radicals including spin-orbit coupling constants and the relative splitting of the lowest energy degenerate state for each species. Convergence of MCCI to the full configuration interaction result is demonstrated on the multireference problem of stretched OH. We also present results from the singlet-triplet interaction between the X3Σg- and both the a1Δg and b1Σg+ states of the O2 molecule. © 2017 Wiley Periodicals, Inc.

Excitation variance matching with limited configuration interaction expansions in variational Monte Carlo.

In the regime where traditional approaches to electronic structure cannot afford to achieve accurate energy differences via exhaustive wave function flexibility, rigorous approaches to balancing different states' accuracies become desirable. As a direct measure of a wave function's accuracy, the energy variance offers one route to achieving such a balance. Here, we develop and test a variance matching approach for predicting excitation energies within the context of variational Monte Carlo and selective configuration interaction. In a series of tests on small but difficult molecules, we demonstrate that the approach is effective at delivering accurate excitation energies when the wave function is far from the exhaustive flexibility limit. Results in C3, where we combine this approach with variational Monte Carlo orbital optimization, are especially encouraging.

Investigation of challenging spin systems using Monte Carlo configuration interaction and the density matrix renormalization group.

We investigate if a range of challenging spin systems can be described sufficiently well using Monte Carlo configuration interaction (MCCI) and the density matrix renormalization group (DMRG) in a way that heads toward a more "black box" approach. Experimental results and other computational methods are used for comparison. The gap between the lowest doublet and quartet state of methylidyne (CH) is first considered. We then look at a range of first-row transition metal monocarbonyls: MCO when M is titanium, vanadium, chromium, or manganese. For these MCO systems we also employ partially spin restricted open-shell coupled-cluster (RCCSD). We finally investigate the high-spin low-lying states of the iron dimer, its cation and its anion. The multireference character of these molecules is also considered. We find that these systems can be computationally challenging with close low-lying states and often multireference character. For this more straightforward application and for the basis sets considered, we generally find qualitative agreement between DMRG and MCCI. © 2017 Wiley Periodicals, Inc.

Open‐shell systems investigated with Monte Carlo configuration interaction

AbstractWe investigate the open‐shell systems of the methyl radical, the allyl radical and molecular oxygen using Monte Carlo configuration interaction. We look at whether modifying Monte Carlo configuration interaction to use the increased flexibility of unrestricted Hartree‐Fock orbitals offers any benefits in the description of these systems by the resulting compact wavefunctions. The expectation of the total spin squared when using Slater determinants is calculated to investigate if pure spin states can be evolved by Monte Carlo configuration interaction and how this might be accelerated. We consider the multireference character of these open‐shell systems. To this end we demonstrate how a previously introduced way [J. P. Coe and M. J. Paterson, J. Chem. Theory Comput. 11, 4189 (2015)] of viewing a multireference indicator of P.‐O. Löwdin [P.‐O. Löwdin, Phys. Rev. 97, 1474 (1955)] in terms of the spatial entanglement can be generalized to open‐shell systems.

Positronic molecule calculations using Monte Carlo configuration interaction

We modify the Monte Carlo configuration interaction procedure to model atoms and molecules combined with a positron. We test this method with standard quantum chemistry basis sets on a number of positronic systems and compare results with the literature and full configuration interaction when appropriate. We consider positronium hydride, positronium hydroxide, lithium positride and a positron interacting with lithium, magnesium or lithium hydride. We demonstrate that we can capture much of the full configuration interaction results, but often require less than 10% of the configurations of these multireference wavefunctions. The effect of the number of frozen orbitals is also discussed.

Excited States of the Nickel Carbonyls Ni(CO) and Ni(CO)4: Challenging Molecules for Electronic Structure Theory.

We apply a wide range of correlated electronic structure approaches to the excited states of Ni(CO)4 and Ni(CO) as model complexes of saturated and unsaturated transition metal carbonyls respectively to understand the performance of each method, in addition to setting benchmark data for these metal carbonyls. In particular, we apply the coupled-cluster linear response hierarchy, complete-active-space self-consistent field theory, N-electron valence state multireference perturbation theory, Monte Carlo configuration interaction, and time-dependent density functional theory with a range of functionals and basis sets. We find that although the systems can qualitatively be described by a single configuration, electron correlation effects are sufficiently strong to give large single amplitudes in cluster expansions, which cause spurious solutions to the response equations for the intermediate CCn methods. DFT also performs well if care is taken to choose an appropriate functional, although for Ni(CO) several popular functionals give the incorrect ground spin-state, depending on the amount of Hartree-Fock exchange.

Investigating Multireference Character and Correlation in Quantum Chemistry.

We review a range of multireference diagnostics for quantum chemistry and discuss them in terms of choices of the molecular orbitals. We show how an approach1 of P.-O. Löwdin can also be viewed as quantifying the electron correlation via the spatial entanglement relative to a single determinant. We consider three example systems from quantum chemistry that exhibit three different combinations of multireference character and correlation: not strongly multireference and not strongly correlated, strongly multireference but not strongly correlated, and strongly multireference together with strong correlation. We find that a multireference measure (MR) does not change substantially with the cutoff used for a Monte Carlo configuration interaction calculation and investigate the effect of using natural orbitals. We see that a coupled-cluster singles and doubles diagnostic and a density-functional theory diagnostic give a correct general prediction of the multireference character for these systems. We also look at the issue of multireference character for a collection of noninteracting hydrogen molecules and the effect of basis size on the multireference character of a stretched hydrogen molecule.

Multireference X-ray emission and absorption spectroscopy calculations from Monte Carlo configuration interaction

We adapt the method of Monte Carlo configuration interaction to calculate core-hole states and use this for the computation of X-ray emission and absorption values. We consider CO, CH$_{4}$, NH$_{3}$, H$_{2}$O, HF, HCN, CH$_{3}$OH, CH$_{3}$F, HCl and NO using a 6-311G** basis. We also look at carbon monoxide with a stretched geometry and discuss the dependence of its results on the cutoff used. The Monte Carlo configuration interaction results are compared with EOM-CCSD values for X-ray emission and with experiment for X-ray absorption. Oscillator strengths are also computed and we quantify the multireference nature of the wavefunctions to suggest when approaches based on a single reference would be expected to be successful.

Approaching exact hyperpolarizabilities via sum-over-states Monte Carlo configuration interaction.

We propose using sum-over-states calculations with the compact wavefunctions of Monte Carlo configuration interaction to approach accurate values for higher-order dipole properties up to second hyperpolarizabilities in a controlled way. We apply the approach to small systems that can generally be compared with full configuration interaction (FCI) results. We consider hydrogen fluoride with a 6-31g basis and then look at results, including frequency dependent properties, in an aug-cc-pVDZ basis. We extend one calculation beyond FCI by using an aug-cc-pVTZ basis. The properties of an H4 molecule with multireference character are calculated in an aug-cc-pVDZ basis. We then investigate this method on a strongly multireference system with a larger FCI space by modelling the properties of carbon monoxide with a stretched geometry. The behavior of the approach with increasing basis size is considered by calculating results for the neon atom using aug-cc-pVDZ to aug-cc-pVQZ. We finally test if the unusual change in polarizability between the first two states of molecular oxygen can be reproduced by this method in a 6-31g basis.

Applying Monte Carlo configuration interaction to transition metal dimers: Exploring the balance between static and dynamic correlation

We calculate potential curves for transition metal dimers using Monte Carlo configuration interaction (MCCI). These results, and their associated spectroscopic values, are compared with experimental and computational studies. The multireference nature of the MCCI wavefunction is quantified and we estimate the important orbitals. We initially consider the ground state of the chromium dimer. Next we calculate potential curves for Sc2 where we contrast the lowest triplet and quintet states. We look at the molybdenum dimer where we compare non-relativistic results with the partial inclusion of relativistic effects via effective core potentials, and report results for scandium nickel.

Monte Carlo configuration interaction with perturbation corrections for dissociation energies of first row diatomic molecules: C2, N2, O2, CO, and NO

Dissociation energies for the diatomic molecules C2, N2, O2, CO, and NO are estimated using the Monte Carlo configuration interaction (MCCI) and augmented by a second order perturbation theory correction. The calculations are performed using the correlation consistent polarized valence "triple zeta" atomic orbital basis and resulting dissociation energies are compared to coupled cluster calculations including up to triple excitations (CCSDT) and Full Configuration Interaction Quantum Monte Carlo (FCIQMC) estimates. It is found that the MCCI method readily describes the correct behavior for dissociation for the diatomics even when capturing only a relatively small fraction (∼80%) of the correlation energy. At this level only a small number of configurations, typically O(10(3)) from a FCI space of dimension O(10(14)), are required to describe dissociation. Including the perturbation correction to the MCCI estimates, the difference in dissociation energies with respect to CCSDT ranges between 1.2 and 3.1 kcal/mol, and the difference when comparing to FCIQMC estimates narrows to between 0.5 and 1.9 kcal/mol. Discussions on MCCI's ability to recover static and dynamic correlations and on the form of correlations in the electronic configuration space are presented.

Scaling in the correlation energies of two-dimensional artificial atoms

We find an unexpected scaling in the correlation energy of artificial atoms, i.e., harmonically confined two-dimensional quantum dots. The scaling relation is found through extensive numerical examinations including Hartree–Fock, variational quantum Monte Carlo, density functional, and full configuration interaction calculations. We show that the correlation energy, i.e., the true ground-state total energy minus the Hartree–Fock total energy, follows a simple function of the Coulomb energy, confinement strength and number of electrons. We find an analytic expression for this function, as well as for the correlation energy per particle and for the ratio between the correlation and total energies. Our tests for independent diffusion Monte Carlo and coupled-cluster results for quantum dots—including open-shell data—confirm the generality of the scaling obtained. As the scaling also applies well to ≳100 electrons, our results give interesting prospects for the development of correlation functionals within density functional theory.