Static and Dynamical Correlation in Diradical Molecules by Quantum Monte Carlo Using the Jastrow Antisymmetrized Geminal Power Ansatz

Beyond Single-Reference Fixed-Node Approximation in Ab Initio Diffusion Monte Carlo Using Antisymmetrized Geminal Power Applied to Systems with Hundreds of Electrons

Diffusion Monte Carlo (DMC) is an exact technique to project out the ground state (GS) of a Hamiltonian. Since the GS is always bosonic, in Fermionic systems, the projection needs to be carried out while imposing antisymmetric constraints, which is a nondeterministic polynomial hard problem. In practice, therefore, the application of DMC on electronic structure problems is made by employing the fixed-node (FN) approximation, consisting of performing DMC with the constraint of having a fixed, predefined nodal surface. How do we get the nodal surface? The typical approach, applied in systems having up to hundreds or even thousands of electrons, is to obtain the nodal surface from a preliminary mean-field approach (typically, a density functional theory calculation) used to obtain a single Slater determinant. This is known as single reference. In this paper, we propose a new approach, applicable to systems as large as the C60 fullerene, which improves the nodes by going beyond the single reference. In practice, we employ an implicitly multireference ansatz (antisymmetrized geminal power wave function constraint with molecular orbitals), initialized on the preliminary mean-field approach, which is relaxed by optimizing a few parameters of the wave function determining the nodal surface by minimizing the FN-DMC energy. We highlight the improvements of the proposed approach over the standard single-reference method on several examples and, where feasible, the computational gain over the standard multireference ansatz, which makes the methods applicable to large systems. We also show that physical properties relying on relative energies, such as binding energies, are affordable and reliable within the proposed scheme.

Correlated pair ansatz with a binary tree structure

We develop an efficient algorithm to implement the recently introduced binary tree state (BTS) ansatz on a classical computer. BTS allows a simple approximation to permanents arising from the computationally intractable antisymmetric product of interacting geminals and respects size-consistency. We show how to compute BTS overlap and reduced density matrices efficiently. We also explore two routes for developing correlated BTS approaches: Jastrow coupled cluster on BTS and linear combinations of BT states. The resulting methods show great promise in benchmark applications to the reduced Bardeen-Cooper-Schrieffer Hamiltonian and the one-dimensional XXZ Heisenberg Hamiltonian.

Bridging physical intuition and hardware efficiency for correlated electronic states: the local unitary cluster Jastrow ansatz for electronic structure

A prominent goal in quantum chemistry is to solve the molecular electronic structure problem for ground state energy with high accuracy. While classical quantum chemistry is a relatively mature field, the accurate and scalable prediction of strongly correlated states found, e.g., in bond breaking and polynuclear transition metal compounds remains an open problem. Within the context of a variational quantum eigensolver, we propose a new family of ansatzes which provides a more physically appropriate description of strongly correlated electrons than a unitary coupled cluster with single and double excitations (qUCCSD), with vastly reduced quantum resource requirements. Specifically, we present a set of local approximations to the unitary cluster Jastrow wavefunction motivated by Hubbard physics. As in the case of qUCCSD, exactly computing the energy scales factorially with system size on classical computers but polynomially on quantum devices. The local unitary cluster Jastrow ansatz removes the need for SWAP gates, can be tailored to arbitrary qubit topologies (e.g., square, hex, and heavy-hex), and is well-suited to take advantage of continuous sets of quantum gates recently realized on superconducting devices with tunable couplers. The proposed family of ansatzes demonstrates that hardware efficiency and physical transparency are not mutually exclusive; indeed, chemical and physical intuition regarding electron correlation can illuminate a useful path towards hardware-friendly quantum circuits.

State Preparation of Antisymmetrized Geminal Power on a Quantum Computer without Number Projection

The antisymmetrized geminal power (AGP) is equivalent to the number projected Bardeen-Cooper-Schrieffer (PBCS) wave function. It is also an elementary symmetric polynomial (ESP) state. We generalize previous research on deterministically implementing the Dicke state to a state preparation algorithm for an ESP state, or equivalently AGP, on a quantum computer. Our method is deterministic and has polynomial cost, and it does not rely on number symmetry breaking and restoration. We also show that our circuit is equivalent to a disentangled unitary paired coupled cluster operator and a layer of unitary Jastrow operator acting on a single Slater determinant. The method presented herein highlights the ability of disentangled unitary coupled cluster to capture nontrivial entanglement properties that are hardly accessible with traditional Hartree-Fock based electronic structure methods.

Static Electron Correlation in Anharmonic Molecular Vibrations: A Hybrid TAO-DFT Study

Hybrid thermally-assisted-occupation density functional theory is used to examine the effects of static electron correlation on the prediction of a benchmark set of experimentally observed molecular vibrational frequencies. The B3LYP and B97-1 thermally-assisted-occupation measure of static electron correlation is important for describing the vibrations of many of the molecules that make up several popular test sets of experimental data. Shifts are seen for known multireference systems and for many molecules containing atoms from the second row of the periodic table of elements. Several molecules only show significant shifts in select vibrational modes, and significant improvements are seen for the prediction of hydrogen stretching frequencies throughout the test set.

Variational Quantum-Neural Hybrid Eigensolver

The variational quantum eigensolver (VQE) is one of the most representative quantum algorithms in the noisy intermediate-scale quantum (NISQ) era, and is generally speculated to deliver one of the first quantum advantages for the ground-state simulations of some nontrivial Hamiltonians. However, short quantum coherence time and limited availability of quantum hardware resources in the NISQ hardware strongly restrain the capacity and expressiveness of VQEs. In this Letter, we introduce the variational quantum-neural hybrid eigensolver (VQNHE) in which the shallow-circuit quantum Ansatz can be further enhanced by classical post-processing with neural networks. We show that the VQNHE consistently and significantly outperforms the VQE in simulating ground-state energies of quantum spins and molecules given the same amount of quantum resources. More importantly, we demonstrate that, for arbitrary postprocessing neural functions, the VQNHE only incurs a polynomial overhead of processing time and represents the first scalable method to exponentially accelerate the VQE with nonunitary postprocessing that can be efficiently implemented in the NISQ era.

Improved Accuracy on Noisy Devices by Nonunitary Variational Quantum Eigensolver for Chemistry Applications

We propose a modification of the Variational Quantum Eigensolver algorithm for electronic structure optimization using quantum computers, named nonunitary Variational Quantum Eigensolver (nu-VQE), in which a nonunitary operator is combined with the original system Hamiltonian leading to a new variational problem with a simplified wave function ansatz. In the present work, as nonunitary operator, we use the Jastrow factor, inspired from classical Quantum Monte Carlo techniques for simulation of strongly correlated electrons. The method is applied to prototypical molecular Hamiltonians for which we obtain accurate ground-state energies with shallower circuits, at the cost of an increased number of measurements. Finally, we also show that this method achieves an important error mitigation effect that drastically improves the quality of the results for VQE optimizations on today's noisy quantum computers. The absolute error in the calculated energy within our scheme is 1 order of magnitude smaller than the corresponding result using traditional VQE methods, with the same circuit depth.

General Correlated Geminal Ansatz for Electronic Structure Calculations: Exploiting Pfaffians in Place of Determinants

We propose here a single Pfaffian correlated variational ansatz that dramatically improves the accuracy with respect to the single determinant one, while remaining at a similar computational cost. A much larger correlation energy is indeed determined by the most general two electron pairing function, including both singlet and triplet channels, combined with a many-body Jastrow factor, including all possible spin-spin, spin-density, and density-density terms. The main technical ingredient to exploit this accuracy is the use of the Pfaffian for antisymmetrizing a highly correlated pairing function, thus recovering the Fermi statistics for electrons with an affordable computational cost. Moreover, the application of the diffusion Monte Carlo, within the fixed node approximation, allows us to obtain very accurate binding energies for the first preliminary calculations reported in this study: C2, N2, and O2 and the benzene molecule. This is promising and remarkable, considering that they represent extremely difficult molecules even for computationally demanding multideterminant approaches, and opens therefore the way for realistic and accurate electronic simulations with an algorithm scaling at most as the fourth power of the number of electrons.

Nonunitary Operations for Ground-State Calculations in Near-Term Quantum Computers

We introduce a quantum Monte Carlo inspired reweighting scheme to accurately compute energies from optimally short quantum circuits. This effectively hybrid quantum-classical approach features both entanglement provided by a short quantum circuit, and the presence of an effective nonunitary operator at the same time. The functional form of this projector is borrowed from classical computation and is able to filter out high-energy components generated by a suboptimal variational quantum heuristic Ansatz. The accuracy of this approach is demonstrated numerically in finding energies of entangled ground states of many-body lattice models. We demonstrate a practical implementation on IBM quantum hardware up to an 8-qubit circuit.

Singlet-triplet gaps in diradicals obtained with diffusion quantum Monte Carlo using a Slater-Jastrow trial wavefunction with a minimum number of determinants

Diradicals are essential species in a wide range of chemical processes, whereas the computational study of their electronic structure often remains a challenge due to near-degeneracy of the frontier molecular orbitals. The fixed-node diffusion quantum Monte Carlo (FN-DMC) method is employed to calculate adiabatic energy gaps of some typical diradicals with the Slater-Jastrow trial wavefunction. The antisymmetrized part of the trial wavefunction is taken to be a linear combination of a minimum number of determinants using RB3LYP orbitals from the closed-shell singlet state or ROB3LYP orbitals from the triplet state. Our results show that using the two-determinant-Jastrow trial wavefunction is necessary to achieve reliable energy differences between closed-shell singlet states. The energy of the triplet state with MS = 1 is calculated to be lower than that with MS = 0 with FN-DMC even using trial wavefunctions with spin-pure states as their antisymmetrized parts and this difference is reduced with better orbitals. This indicates that the fixed-node error is smaller for the triplet state with MS = 1. Adiabatic energy gaps obtained from the present FN-DMC calculations are in reasonable agreement with available experimental values. Compared with results of the high level EOM-SF-CC method, energy gaps of FN-DMC with RB3LYP orbitals are slightly better than those using ROB3LYP orbitals and results of EOM-SF-CCSD. The present FN-DMC calculations using the simplest ansatz for the trial wavefunction can achieve reasonable results for these diradicals and they can readily be applied to large diradicals.

The Awareness of the Fascial System

Fascia is a cacophony of functions and information, a completely adaptable entropy complex. The fascial system has a solid and a liquid component, acting in a perfect symbiotic synchrony. Each cell communicates with the other cells by sending and receiving signals; this concept is a part of quantum physics and it is known as quantum entanglement: a physical system cannot be described individually, but only as a juxtaposition of multiple systems, where the measurement of a quantity determines the value for other systems. Fascial continuum serves as a target for different manual approaches, such as physiotherapy, osteopathy and chiropractic. Cellular behaviour and the inclusion of quantum physics background are hardly being considered to find out what happens between the operator and the patient during a manual physical contact. The article examines these topics. According to the authors' knowledge, this is the first scientific text to offer manual operators’ new perspectives to understand what happens during palpatory contact. A fascial cell has not only memory but also the awareness of the mechanometabolic information it feels, and it has the anticipatory predisposition in preparing itself for alteration of its natural environment.

Correlation-Driven Dimerization and Topological Gap Opening in Isotropically Strained Graphene

The phase diagram of isotropically expanded graphene cannot be correctly predicted by ignoring either electron correlations, or mobile carbons, or the effect of applied stress, as was done so far. We calculate the ground state enthalpy (not just energy) of strained graphene by an accurate off-lattice quantum Monte Carlo correlated ansatz of great variational flexibility. Following undistorted semimetallic graphene at low strain, multideterminant Heitler-London correlations stabilize between ≃8.5% and ≃15% strain an insulating Kekulé-like dimerized (DIM) state. Closer to a crystallized resonating-valence bond than to a Peierls state, the DIM state prevails over the competing antiferromagnetic insulating state favored by density-functional calculations which we conduct in parallel. The DIM stressed graphene insulator, whose gap is predicted to grow in excess of 1 eV before failure near 15% strain, is topological in nature, implying under certain conditions 1D metallic interface states lying in the bulk energy gap.

Towards a formal definition of static and dynamic electronic correlations

Some of the most spectacular failures of density-functional and Hartree-Fock theories are related to an incorrect description of the so-called static electron correlation. Motivated by recent progress in the N-representability problem of the one-body density matrix for pure states, we propose a method to quantify the static contribution to the electronic correlation. By studying several molecular systems we show that our proposal correlates well with our intuition of static and dynamic electron correlation. Our results bring out the paramount importance of the occupancy of the highest occupied natural spin-orbital in such quantification.

Theoretical description of protein field effects on electronic excitations of biological chromophores

Photoinitiated phenomena play a crucial role in many living organisms. Plants, algae, and bacteria absorb sunlight to perform photosynthesis, and convert water and carbon dioxide into molecular oxygen and carbohydrates, thus forming the basis for life on Earth. The vision of vertebrates is accomplished in the eye by a protein called rhodopsin, which upon photon absorption performs an ultrafast isomerisation of the retinal chromophore, triggering the signal cascade. Many other biological functions start with the photoexcitation of a protein-embedded pigment, followed by complex processes comprising, for example, electron or excitation energy transfer in photosynthetic complexes. The optical properties of chromophores in living systems are strongly dependent on the interaction with the surrounding environment (nearby protein residues, membrane, water), and the complexity of such interplay is, in most cases, at the origin of the functional diversity of the photoactive proteins. The specific interactions with the environment often lead to a significant shift of the chromophore excitation energies, compared with their absorption in solution or gas phase. The investigation of the optical response of chromophores is generally not straightforward, from both experimental and theoretical standpoints; this is due to the difficulty in understanding diverse behaviours and effects, occurring at different scales, with a single technique. In particular, the role played by ab initio calculations in assisting and guiding experiments, as well as in understanding the physics of photoactive proteins, is fundamental. At the same time, owing to the large size of the systems, more approximate strategies which take into account the environmental effects on the absorption spectra are also of paramount importance. Here we review the recent advances in the first-principle description of electronic and optical properties of biological chromophores embedded in a protein environment. We show their applications on paradigmatic systems, such as the light-harvesting complexes, rhodopsin and green fluorescent protein, emphasising the theoretical frameworks which are of common use in solid state physics, and emerging as promising tools for biomolecular systems.

Correlation Effects in Scanning Tunneling Microscopy Images of Molecules Revealed by Quantum Monte Carlo

Scanning tunneling microscopy (STM) and spectroscopy probe the local density of states of single molecules electrically insulated from the substrate. The experimental images, although usually interpreted in terms of single-particle molecular orbitals, are associated with quasiparticle wave functions dressed by the whole electron-electron interaction. Here we propose an ab initio approach based on quantum Monte Carlo to calculate the quasiparticle wave functions of molecules. Through the comparison between Monte Carlo wave functions and their uncorrelated Hartree-Fock counterparts we visualize the electronic correlation embedded in the simulated STM images, highlighting the many-body features that might be observed.

Quantum Monte Carlo Treatment of the Charge Transfer and Diradical Electronic Character in a Retinal Chromophore Minimal Model

The penta-2,4-dieniminium cation (PSB3) displays similar ground state and first excited state potential energy features as those of the retinal protonated Schiff base (RPSB) chromophore in rhodopsin. Recently, PSB3 has been used to benchmark several electronic structure methods, including highly correlated multireference wave function approaches, highlighting the necessity to accurately describe the electronic correlation in order to obtain reliable properties even along the ground state (thermal) isomerization paths. In this work, we apply two quantum Monte Carlo approaches, the variational Monte Carlo and the lattice regularized diffusion Monte Carlo, to study the energetics and electronic properties of PSB3 along representative minimum energy paths and scans related to its thermal cis–trans isomerization. Quantum Monte Carlo is used in combination with the Jastrow antisymmetrized geminal power ansatz, which guarantees an accurate and balanced description of the static electronic correlation thanks to the multiconfigurational nature of the antisymmetrized geminal power term, and of the dynamical correlation, due to the presence of the Jastrow factor explicitly depending on electron–electron distances. Along the two ground state isomerization minimum energy paths of PSB3, CASSCF calculations yield wave functions having either charge transfer or diradical character in proximity of the two transition state configurations. Here, we observe that at the quantum Monte Carlo level of theory, only the transition state with charge transfer character can be located. The conical intersection, which becomes highly sloped, is observed only if the path connecting the two original CASSCF transition states is extended beyond the diradical one, namely by increasing the bond-length-alternation (BLA). These findings are in good agreement with the results obtained by MRCISD+Q calculations, and they demonstrate the importance of having an accurate description of the static and dynamical correlation when studying isomerization and transition states of conjugated systems.

Theory and practice of uncommon molecular electronic configurations

The electronic configuration of the molecule is the foundation of its structure and reactivity. The spin state is one of the key characteristics arising from the ordering of electrons within the molecule's set of orbitals. Organic molecules that have open-shell ground states and interesting physicochemical properties, particularly those influencing their spin alignment, are of immense interest within the up-and-coming field of molecular electronics. In this advanced review, we scrutinize various qualitative rules of orbital occupation and spin alignment, viz., the aufbau principle, Hund's multiplicity rule, and dynamic spin polarization concept, through the prism of quantum mechanics. While such rules hold in selected simple cases, in general the spin state of a system depends on a combination of electronic factors that include Coulomb and Pauli repulsion, nuclear attraction, kinetic energy, orbital relaxation, and static correlation. A number of fascinating chemical systems with spin states that fluctuate between triplet and open-shell singlet, and are responsive to irradiation, pH, and other external stimuli, are highlighted. In addition, we outline a range of organic molecules with intriguing non-aufbau orbital configurations. In such quasi-closed-shell systems, the singly occupied molecular orbital (SOMO) is energetically lower than one or more doubly occupied orbitals. As a result, the SOMO is not affected by electron attachment to or removal from the molecule, and the products of such redox processes are polyradicals. These peculiar species possess attractive conductive and magnetic properties, and a number of them that have already been developed into molecular electronics applications are highlighted in this review. WIREs Comput Mol Sci 2015, 5:440-459. doi: 10.1002/wcms.1233 For further resources related to this article, please visit the WIREs website.

Ground State Geometries of Polyacetylene Chains from Many-Particle Quantum Mechanics

Due to the crucial role played by electron correlation, the accurate determination of ground state geometries of π-conjugated molecules is still a challenge for many quantum chemistry methods. Because of the high parallelism of the algorithms and their explicit treatment of electron correlation effects, Quantum Monte Carlo calculations can offer an accurate and reliable description of the electronic states and of the geometries of such systems, competing with traditional quantum chemistry approaches. Here, we report the structural properties of polyacetylene chains H-(C₂H₂)(N)-H up to N = 12 acetylene units, by means of Variational Monte Carlo (VMC) calculations based on the multi-determinant Jastrow Antisymmetrized Geminal Power (JAGP) wave function. This compact ansatz can provide for such systems an accurate description of the dynamical electronic correlation as recently detailed for the 1,3-butadiene molecule [J. Chem. Theory Comput. 2015 11 (2), 508-517]. The calculated Bond Length Alternation (BLA), namely the difference between the single and double carbon bonds, extrapolates, for N → ∞, to a value of 0.0910(7) Å, compatible with the experimental data. An accurate analysis was able to distinguish between the influence of the multi-determinantal AGP expansion and of the Jastrow factor on the geometrical properties of the fragments. Our size-extensive and self-interaction-free results provide new and accurate ab initio references for the structures of the ground state of polyenes.

Ab initio molecular dynamics simulation of liquid water by quantum Monte Carlo

Although liquid water is ubiquitous in chemical reactions at roots of life and climate on the earth, the prediction of its properties by high-level ab initio molecular dynamics simulations still represents a formidable task for quantum chemistry. In this article, we present a room temperature simulation of liquid water based on the potential energy surface obtained by a many-body wave function through quantum Monte Carlo (QMC) methods. The simulated properties are in good agreement with recent neutron scattering and X-ray experiments, particularly concerning the position of the oxygen-oxygen peak in the radial distribution function, at variance of previous density functional theory attempts. Given the excellent performances of QMC on large scale supercomputers, this work opens new perspectives for predictive and reliable ab initio simulations of complex chemical systems.

π-Conjugation in trans-1,3-butadiene: static and dynamical electronic correlations described through quantum Monte Carlo

We investigate the effects of the static and dynamical electronic correlations on the level of conjugation of the trans-1,3-butadiene molecule through Quantum Monte Carlo methods applied to an Antisymmetrized Geminal Power (AGP) wave function, with a Jastrow factor similar to the Gutzwiller ansatz. The degree of conjugation is measured through the convergence of the structural properties of 1,3-butadiene and in particular of the Bond Length Alternation (BLA), that is the difference between the lengths of the single and double carbon bonds. After verifying the different roles of the Fermionic AGP part of our wave function and of the Jastrow factor in recovering electronic correlation, we study the effects of a constrained Active Space AGP (AGPAS), similar to that used in the Complete Active Space (CAS) representation. Through this AGPAS, we are able to identify the effect of the limited active space on the degree of conjugation, showing that in the limit of infinite active space the structural properties converge exactly to those of the atomic AGP, giving a BLA for 1,3-butadiene around 0.1244(5) Å.

Compact two-electron wave function for bond dissociation and Van der Waals interactions: a natural amplitude assessment

Electron correlations in molecules can be divided in short range dynamical correlations, long range Van der Waals type interactions, and near degeneracy static correlations. In this work, we analyze for a one-dimensional model of a two-electron system how these three types of correlations can be incorporated in a simple wave function of restricted functional form consisting of an orbital product multiplied by a single correlation function f (r12) depending on the interelectronic distance r12. Since the three types of correlations mentioned lead to different signatures in terms of the natural orbital (NO) amplitudes in two-electron systems, we make an analysis of the wave function in terms of the NO amplitudes for a model system of a diatomic molecule. In our numerical implementation, we fully optimize the orbitals and the correlation function on a spatial grid without restrictions on their functional form. Due to this particular form of the wave function, we can prove that none of the amplitudes vanishes and moreover that it displays a distinct sign pattern and a series of avoided crossings as a function of the bond distance in agreement with the exact solution. This shows that the wave function ansatz correctly incorporates the long range Van der Waals interactions. We further show that the approximate wave function gives an excellent binding curve and is able to describe static correlations. We show that in order to do this the correlation function f (r12) needs to diverge for large r12 at large internuclear distances while for shorter bond distances it increases as a function of r12 to a maximum value after which it decays exponentially. We further give a physical interpretation of this behavior.