Extension Of Random Phase Approximation Research Articles

Multireference Correlated Oscillator Strengths from Adiabatic Connection Approaches Based on Extended Random Phase Approximation.

We show that accurate oscillator strengths can be obtained from adiabatic connection (AC) approaches based on the extended random phase approximation (ERPA) combined with multireference (complete active space, CAS) wave functions. The oscillator strengths calculated using the perturbation-corrected ERPA transition density matrices, proposed in this work, and the excitation energies calculated with recently introduced AC correlation energy methods, AC0 and AC0D, compete with accuracy in the perturbational CASPT2 approach and require less computational effort. AC0 and AC0D methods scale more favorably with the number of active orbitals than multiconfigurational perturbation approaches like CASPT2 and NEVPT2 thanks to their dependence on reduced density matrices up to the order of 2. Importantly, the newly developed approach for computing correlated transition dipole moments does not entail any additional costs, as all intermediate quantities become available when AC0 energies are being computed. We also test the performance of the recently proposed AC method corrected for the negative-transition contributions to the correlation energy, AC0D, for triplet excitation energies. Similarly, as for the singlet excitations, the correction improves the performance of the AC0 method, particularly for the low-lying excited states.

Excited States by Coupling Piris Natural Orbital Functionals with the Extended Random-Phase Approximation.

In this work, we explore the use of Piris natural orbital functionals (PNOFs) to calculate excited-state energies by coupling their reconstructed second-order reduced density matrix with the extended random-phase approximation (ERPA). We have named the general method PNOF-ERPA, and specific approaches are referred to as PNOF-ERPA0, PNOF-ERPA1, and PNOF-ERPA2, according to the way the excitation operator is built. The implementation has been tested in the first excited states of H2, HeH+, LiH, Li2, and N2 showing good results compared to the configuration interaction (CI) method. As expected, an increase in accuracy is observed on going from ERPA0 to ERPA1 and ERPA2. We also studied the effect of electron correlation included by PNOF5, PNOF7, and the recently proposed global NOF (GNOF) on the predicted excited states. PNOF5 appears to be good and may even provide better results in very small systems, but including more electron correlation becomes important as the system size increases, where GNOF achieves better results. Overall, the extension of PNOF to excited states has been successful, making it a promising method for further applications.

Toward more accurate adiabatic connection approach for multireference wavefunctions.

A multiconfigurational adiabatic connection (AC) formalism is an attractive approach to compute the dynamic correlation within the complete active space self-consistent field and density matrix renormalization group (DMRG) models. Practical realizations of AC have been based on two approximations: (i) fixing one- and two-electron reduced density matrices (1- and 2-RDMs) at the zero-coupling constant limit and (ii) extended random phase approximation (ERPA). This work investigates the effect of removing the "fixed-RDM" approximation in AC. The analysis is carried out for two electronic Hamiltonian partitionings: the group product function- and the Dyall Hamiltonians. Exact reference AC integrands are generated from the DMRG full configuration interaction solver. Two AC models are investigated, employing either exact 1- and 2-RDMs or their second-order expansions in the coupling constant in the ERPA equations. Calculations for model molecules indicate that lifting the fixed-RDM approximation is a viable way toward improving the accuracy of existing AC approximations.

Accurate non-covalent interaction energies on noisy intermediate-scale quantum computers via second-order symmetry-adapted perturbation theory.

The calculation of non-covalent interaction energies on noisy intermediate-scale quantum (NISQ) computers appears to be challenging with straightforward application of existing quantum algorithms. For example, the use of the standard supermolecular method with the variational quantum eigensolver (VQE) would require extremely precise resolution of the total energies of the fragments to provide for accurate subtraction to the interaction energy. Here we present a symmetry-adapted perturbation theory (SAPT) method that may provide interaction energies with high quantum resource efficiency. Of particular note, we present a quantum extended random-phase approximation (ERPA) treatment of the SAPT second-order induction and dispersion terms, including exchange counterparts. Together with previous work on first-order terms (Chem. Sci., 2022, 13, 3094), this provides a recipe for complete SAPT(VQE) interaction energies up to second order, which is a well established truncation. The SAPT interaction energy terms are computed as first-level observables with no subtraction of monomer energies invoked, and the only quantum observations needed are the VQE one- and two-particle density matrices. We find empirically that SAPT(VQE) can provide accurate interaction energies even with coarsely optimized, low circuit depth wavefunctions from a quantum computer, simulated through ideal statevectors. The errors of the total interaction energy are orders of magnitude lower than the corresponding VQE total energy errors of the monomer wavefunctions. In addition, we present heme-nitrosyl model complexes as a system class for near term quantum computing simulations. They are strongly correlated, biologically relevant and difficult to simulate with classical quantum chemical methods. This is illustrated with density functional theory (DFT) as the predicted interaction energies exhibit a strong sensitivity with respect to the choice of functional. Thus, this work paves the way to obtain accurate interaction energies on a NISQ-era quantum computer with few quantum resources. It is the first step in alleviating one of the major challenges in quantum chemistry, where in-depth knowledge of both the method and system is required a priori to reliably generate accurate interaction energies.

Spurious dipole mode in random-phase approximation and in models based on this approximation

The problems related to the existence of the spurious dipole mode (SDM) in self-consistent nuclear-structure models are considered. A method is formulated that allows one to eliminate the coupling of the SDM with the physical modes in the extended random-phase approximation (ERPA) theories, in particular, in the time blocking approximation (TBA), which is a model of the ERPA type. It is shown that the application of this method in realistic TBA calculations of the $E1$ excitations gives results which are very close to the results of TBA calculations without using this method if the bare external-field $E1$ operators are replaced by effective ones.

Effects of ground-state correlations on the damping of giant dipole resonances in LS closed-shell nuclei

Abstract The effects of ground-state correlations on the damping of isovector giant dipole resonances in the LS closed-shell nuclei 16O and 40Ca are studied using extended random phase approximation (RPA) approaches derived from the time-dependent density matrix theory. It is pointed out that unconventional two-body amplitudes of one-particle–three-hole and three-particle–one-hole types that are neglected in most extended RPA theories play an important role in the fragmentation of isovector dipole strength.

Symmetry-Adapted Perturbation Theory Based on Multiconfigurational Wave Function Description of Monomers.

We present a formulation of the multiconfigurational (MC) wave function symmetry-adapted perturbation theory (SAPT). The method is applicable to noncovalent interactions between monomers which require a multiconfigurational description, in particular when the interacting system is strongly correlated or in an electronically excited state. SAPT(MC) is based on one- and two-particle reduced density matrices of the monomers and assumes the single-exchange approximation for the exchange energy contributions. Second-order terms are expressed through response properties from extended random phase approximation (ERPA). The dispersion components of SAPT(MC) have been introduced in our previous works [HapkaM.J. Chem. Theory Comput.2019, 15, 1016−102730525591; HapkaM.J. Chem. Theory Comput.2019, 15, 6712–672331670950]. SAPT(MC) is applied either with generalized valence bond perfect pairing (GVB) or with complete active space self-consistent field (CASSCF) treatment of the monomers. We discuss two model multireference systems: the H2 ··· H2 dimer in out-of-equilibrium geometries and interaction between the argon atom and excited state of ethylene. Using the C2H4* ··· Ar complex as an example, we examine second-order terms arising from negative transitions in the linear response function of an excited monomer. We demonstrate that the negative-transition terms must be accounted for to ensure qualitative prediction of induction and dispersion energies and develop a procedure allowing for their computation. Factors limiting the accuracy of SAPT(MC) are discussed in comparison with other second-order SAPT schemes on a data set of small single-reference dimers.

Excited states in the adiabatic connection fluctuation-dissipation theory: Recovering missing correlation energy from the negative part of the density response spectrum.

The adiabatic connection (AC) theory offers an alternative to the perturbation theory methods for computing correlation energy in the multireference wavefunction framework. We show that the AC correlation energy formula can be expressed in terms of the density linear response function as a sum of components related to positive and negative parts of the transition energy spectrum. Consequently, generalization of the adiabatic connection fluctuation-dissipation theory to electronically excited states is obtained. The component of the linear response function related to the negative-transition energy enters the correlation energy expression with an opposite sign to that of the positive-transition part and is non-negligible in the description of excited states. To illustrate this, we analyze the approximate AC model in which the linear response function is obtained in the extended random phase approximation (ERPA). We demonstrate that AC can be successfully combined with the ERPA for excited states, provided that the negative-excitation component of the response function is rigorously accounted for. The resulting AC0D model, an extension of the AC0 scheme introduced in our earlier works, is applied to a benchmark set of singlet excitation energies of organic molecules. AC0D constitutes a significant improvement over AC0 by bringing the excitation energies of the lowest excited states to a satisfactory agreement with theoretical best estimates, which parallels or even exceeds the accuracy of the n-electron valence state perturbation theory method. For higher excitations, AC0D is less reliable due to the gradual deterioration of the underlying ERPA linear response.

An adiabatic connection for doubly-occupied configuration interaction wave functions.

An adiabatic connection (AC) is developed as an electron correlation correction for doubly occupied configuration interaction (DOCI) wave functions. Following the work of Pernal [Phys. Rev. Lett. 120, 013001 (2018)], the working equations of the approach, termed AC-DOCI, are rooted in the extended random phase approximation (ERPA) and require knowledge of only the ground-state two-electron reduced density matrix (2RDM) from the DOCI. As such, the AC is naturally suited to modeling electron correlation in variational 2RDM (v2RDM)-based approximations to the DOCI. The v2RDM-driven AC-DOCI is applied to the dissociation of molecular nitrogen and the double dissociation of water; the approach yields energies that are similar in quality to those from second-order multireference perturbation theory near equilibrium, but the quality of the AC-DOCI energy degrades at stretched geometries. The exact adiabatic connection path suggests the assumption that the one-electron reduced-density matrix is constant along the AC path is invalid at stretched geometries, but this deficiency alone cannot explain the observed behavior. Rather, it appears that the ERPA's single-particle-transition ansatz cannot, in general, provide good approximations to the 2RDM along the AC path. The AC-DOCI is also applied to a set of 45 reaction energies; for these systems, the approach has an average accuracy that is comparable to that of single-reference second-order many-body perturbation theory.

Second-Order Exchange-Dispersion Energy Based on a Multireference Description of Monomers.

We present a method for calculation of the second-order exchange-dispersion energy in the framework of the symmetry-adapted perturbation theory (SAPT) for weakly interacting monomers described with multiconfigurational wave functions. The proposed formalism is based on response properties obtained from extended random phase approximation (ERPA) equations and assumes the single-exchange (S2) approximation. The approach is applicable to closed shell systems where static correlation cannot be neglected or to systems in nondegenerate excited states. We examine the new method in combination with either generalized valence bond perfect pairing (GVB) or complete active space self-consistent field (CASSCF) description of the interacting monomers. For model multireference dimers in ground states (H2···H2, Be···Be, He···H2), exchange-dispersion energies are reproduced accurately. For the interaction between the excited hydrogen molecule and the helium atom we found unacceptably large errors which is attributed to the neglect of diagonal double excitations in the employed approximation to the linear response function.

Modeling core-level excitations with variationally optimized reduced-density matrices and the extended random phase approximation.

The information contained within ground-state one- and two-electron reduced-density matrices (RDMs) can be used to compute wave functions and energies for electronically excited states through the extended random phase approximation (ERPA). The ERPA is an appealing framework for describing excitations out of states obtained via the variational optimization of the two-electron RDM (2-RDM), as the variational 2-RDM (v2RDM) approach itself can only be used to describe the lowest-energy state of a given spin symmetry. The utility of the ERPA for predicting near-edge features relevant to x-ray absorption spectroscopy is assessed for the case that the 2-RDM is obtained from a ground-state v2RDM-driven complete active space self-consistent field (CASSCF) computation. A class of killer conditions for the CASSCF-specific ERPA excitation operator is derived, and it is demonstrated that a reliable description of core-level excitations requires an excitation operator that fulfills these conditions; the core-valence separation (CVS) scheme yields such an operator. Absolute excitation energies evaluated within the CASSCF/CVS-ERPA framework are slightly more accurate than those obtained from the usual random phase approximation (RPA), but the CVS-ERPA is not more accurate than RPA for predicting the relative positions of near-edge features. Nonetheless, CVS-ERPA is established as a reasonable starting point for the treatment of core-level excitations using variationally optimized 2-RDMs.

Multiple bond breaking with APSG-based correlation methods: comparison of two approaches

Antisymmetrized product of strongly orthogonal geminals (APSG) Ansatz is a computationally economic wavefunction class with favorable formal properties. These include extensivity, variational determination of the wavefunction parameters or qualitatively correct description of single bond dissociation. Breaking multiple bonds or non-isolated single bonds results in fragments of incorrect spin state when computed by APSG. This has been identified as a potential problem in APSG-based linearized coupled cluster approach (LCC). An alternative correction scheme based on the extended random phase approximation (ERPA) is investigated from this point of view, in parallel with LCC. The two methods are compared formally. Potential energy curves and atomic spin by APSG-based LCC and ERPA are presented on illustrative examples for multiple bond breaking. Origin of the marked difference between the behavior of LCC and ERPA is explored.

Correlation Energy from the Adiabatic Connection Formalism for Complete Active Space Wave Functions.

Recently, the adiabatic connection (AC) formula for the electron correlation energy has been proposed for a broad class of multireference wave functions (Pernal, K. Electron Correlation from the Adiabatic Connection for Multireference Wave Functions. Phys. Rev. Lett. 2018, 120, 013001). We show that the AC formula used together with the extended random phase approximation (ERPA) can be efficiently applied to complete active space (CAS) wave functions to recover the remaining electron correlation. Unlike most of the perturbation theory approaches, the proposed AC-CAS method does not require construction of higher than two-electron reduced density matrices, which offers an immediate computational saving. In addition, we show that typically the AC-CAS systematically reduces the errors of both the absolute value of energy and of the energy differences (energy barrier) upon enlarging active spaces for electrons and orbitals. AC-CAS consistently gains in accuracy from including more active electrons. We also propose and study that the performance of the AC0 approach resulting from the first-order expansion of the AC integrand at the coupling constant equal to 0. AC0 avoids solving the full ERPA eigenequation, replacing it with small-dimension eigenproblems, while retaining good accuracy of the AC-CAS method. Low computational cost, compared to AC-CAS or perturbational approaches, makes AC0 the most efficient ab initio approach to capturing electron correlation for the CAS wave functions.

Self-consistency in the phonon space of the particle-phonon coupling model

In the paper the non-linear generalization of the time blocking approximation (TBA) is presented. The TBA is one of the versions of the extended random-phase approximation (RPA) developed within the Green-function method and the particle-phonon coupling model. In the generalized version of the TBA the self-consistency principle is extended onto the phonon space of the model. The numerical examples show that this non-linear version of the TBA leads to the convergence of the results with respect to enlarging the phonon space of the model.

Effects of ground-state correlations on dipole and quadrupole excitations of 40Ca and 48Ca

The effects of ground-state correlations on the dipole and quadrupole excitations are studied for $^{40}$Ca and $^{48}$Ca using the extended random phase approximation (ERPA) derived from the time-dependent density-matrix theory. Large effects of the ground-state correlations are found in the fragmentation of the giant quadrupole resonance in $^{40}$Ca and in the low-lying dipole strength in $^{48}$Ca. It is discussed that the former is due to a mixing of different configurations in the ground state and the latter is from the partial occupation of the neutron single-particle states. The dipole and quadrupole strength distributions below 10 MeV calculated in ERPA are in qualitatively agreement with experiment.

Role of small-norm components in extended random-phase approximation

The role of the small-norm amplitudes in extended RPA theories such as the particle-particle and hole-hole components of one-body amplitudes and the two-body amplitudes other than two particle - two holes components are investigated for the one-dimensional Hubbard model using an extended RPA derived from the time-dependent density-matrix theory. It is found that these amplitudes cannot be neglected in strongly interacting regions where the effects of ground-state correlations are significant.

Ring coupled-cluster doubles correction to geminal wavefunctions

ABSTRACTA ring approximation in coupled-cluster (CC) theory is worked out at the multireference (MR) level. The underlying CC framework is based on generalised normal ordering and applies the corresponding generalised Wick's theorem. Contractions among cluster operators are avoided by adopting a normal ordered exponential Ansatz. The MR ring CC doubles (MR rCCD) theory is found to represent a companion to the previously introduced extended random phase approximation (ERPA). Equations for wavefunction parameters are derived in parallel and compared. Condition for the ground state consistent with ERPA is obtained. This paves the way towards comparison with the ERPA-based correction to the antisymmetrised product of strongly orthogonal geminal (APSG) reference, ERPA-APSG.

Quantum chaos on a critical Fermi surface

We compute parameters characterizing many-body quantum chaos for a critical Fermi surface without quasiparticle excitations. We examine a theory of [Formula: see text] species of fermions at nonzero density coupled to a [Formula: see text] gauge field in two spatial dimensions and determine the Lyapunov rate and the butterfly velocity in an extended random-phase approximation. The thermal diffusivity is found to be universally related to these chaos parameters; i.e., the relationship is independent of [Formula: see text], the gauge-coupling constant, the Fermi velocity, the Fermi surface curvature, and high-energy details.

Variational optimization of the two-electron reduced-density matrix under pure-state N-representability conditions.

The direct variational optimization of the ground-state two-electron reduced-density matrix (2-RDM) is typically performed under ensemble N-representability conditions. Accordingly, variationally obtained 2-RDMs for degenerate ground states may not represent a pure state. When considering only ground-state energetics, the ensemble nature of the 2-RDM is of little consequence. However, the use of ensemble densities within an extended random phase approximation (ERPA) yields astonishingly poor estimates of excitation energies, even for simple atomic systems [H. van Aggelen et al., Comput. Theor. Chem. 1003, 50-54 (2013)]. Here, we outline an approach for the direct variational optimization of ground-state 2-RDMs that satisfy pure-state N-representability known as generalized Pauli constraints. Within the ERPA, 2-RDMs that satisfy both ensemble conditions and the generalized Pauli constraints yield much more reliable estimates of excitation energies than those that satisfy only ensemble conditions.

Application of an extended random-phase approximation to giant resonances in light-, medium-, and heavy-mass nuclei

We present results of the time blocking approximation (TBA) on giant resonances in light, medium and heavy mass nuclei. The TBA is an extension of the widely used random-phase approximation (RPA) adding complex configurations by coupling to phonon excitations. A new method for handling the single-particle continuum is developed and applied in the present calculations. We investigate in detail the dependence of the numerical results on the size of the single particle space and the number of phonons as well as on nuclear matter properties. Our approach is self-consistent, based on an energy-density functional of Skyrme type where we used seven different parameter sets. The numerical results are compared with experimental data.