Applications of multireference perturbation theory to potential energy surfaces by optimal partitioning ofH: Intruder states avoidance and convergence enhancement

Knowles Partitioning at the Multireference Level

A multireference generalization of the partitioning introduced recently by Knowles (J. Chem. Phys. 2022 156, 011101) is presented with the aim to study electronic systems at a medium level of correlation. The multireference formalism applied is the general framework of multiconfiguration perturbation theory (MCPT).

Analysis and Assessment of Knowles' Partitioning in Many-Body Perturbation Theory

A detailed analysis of a new partitioning in many-body perturbation theory recently proposed by Knowles (J. Chem. Phys. 156, 011101, 2022), termed "perturbation adapted partitioning" (PAPT), is presented. Level shift and orbital rotation effects are identified as gears of the zero-order Hamiltonian. These two components are examined separately, revealing that, in themselves, neither of the two is competitive with the combined effect. The success of PAPT can be attributed to determining a set of molecular orbitals and corresponding orbital energies that can systematically outperform the canonical orbitals and Koopmans' energy-based Møller-Plesset partitioning. The self-consistent version of the method is also tested in terms of energy and convergence. Previous numerical studies are further complemented with an application to an inherent multireference example and an investigation of van der Waals interaction energies. In addition, a rigorous mathematical analysis of the consequence of the linear dependence of projection functions on the solution of the Knowles' equations is provided.

Many-Body Excited States with a Contracted Quantum Eigensolver

Calculating ground and excited states is an exciting prospect for near-term quantum computing applications, and accurate and efficient algorithms are needed to assess viable directions. We develop an excited-state approach based on the contracted quantum eigensolver (ES-CQE), which iteratively attempts to find a solution to a contraction of the Schrödinger equation projected onto a subspace and does not require a priori information on the system. We focus on the anti-Hermitian portion of the equation, leading to a two-body unitary ansatz. We investigate the role of symmetries, initial states, constraints, and overall performance within the context of the model strongly correlated rectangular H4 system. We show that the ES-CQE achieves near-exact accuracy across the majority of states, covering regions of strong and weak electron correlation, while also elucidating challenging instances for two-body unitary ansatz.

Dynamical Self-energy Mapping (DSEM) for Creation of Sparse Hamiltonians Suitable for Quantum Computing

We present a two-step procedure called the dynamical self-energy mapping (DSEM) that allows us to find a sparse Hamiltonian representation for molecular problems. In the first part of this procedure, the approximate self-energy of a molecular system is evaluated using a low-level method and subsequently a sparse Hamiltonian is found that best recovers this low-level dynamic self-energy. In the second step, such a sparse Hamiltonian is used by a high-level method that delivers a highly accurate dynamical part of the self-energy that is employed in later calculations. The tests conducted on small molecular problems show that the sparse Hamiltonian parameterizations lead to very good total energies. DSEM has the potential to be used as a classical-quantum hybrid algorithm for quantum computing where the sparse Hamiltonian containing only O(n²) terms on a Gaussian orbital basis, where n is the number of orbitals in the system, could reduce the depth of the quantum circuit by at least an order of magnitude when compared with simulations involving a full Hamiltonian.

Resource-Efficient Chemistry on Quantum Computers with the Variational Quantum Eigensolver and the Double Unitary Coupled-Cluster Approach

Applications of quantum simulation algorithms to obtain electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices require careful consideration of resources describing the complex electron correlation effects. In modeling second-quantized problems, the biggest challenge confronted is that the number of qubits scales linearly with the size of the molecular basis. This poses a significant limitation on the size of the basis sets and the number of correlated electrons included in quantum simulations of chemical processes. To address this issue and enable more realistic simulations on NISQ computers, we employ the double unitary coupled-cluster (DUCC) method to effectively downfold correlation effects into the reduced-size orbital space, commonly referred to as the active space. Using downfolding techniques, we demonstrate that properly constructed effective Hamiltonians can capture the effect of the whole orbital space in small-size active spaces. Combining the downfolding preprocessing technique with the variational quantum eigensolver, we solve for the ground-state energy of H₂, Li₂, and BeH₂ in the cc-pVTZ basis using the DUCC-reduced active spaces. We compare these results to full configuration-interaction and high-level coupled-cluster reference calculations.

Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms

In this paper, we discuss the extension of the recently introduced subsystem embedding subalgebra coupled cluster (SES-CC) formalism to unitary CC formalisms. In analogy to the standard single-reference SES-CC formalism, its unitary CC extension allows one to include the dynamical (outside the active space) correlation effects in an SES induced complete active space (CAS) effective Hamiltonian. In contrast to the standard single-reference SES-CC theory, the unitary CC approach results in a Hermitian form of the effective Hamiltonian. Additionally, for the double unitary CC (DUCC) formalism, the corresponding CAS eigenvalue problem provides a rigorous separation of external cluster amplitudes that describe dynamical correlation effects-used to define the effective Hamiltonian-from those corresponding to the internal (inside the active space) excitations that define the components of eigenvectors associated with the energy of the entire system. The proposed formalism can be viewed as an efficient way of downfolding many-electron Hamiltonian to the low-energy model represented by a particular choice of CAS. In principle, this technique can be extended to any type of CAS representing an arbitrary energy window of a quantum system. The Hermitian character of low-dimensional effective Hamiltonians makes them an ideal target for several types of full configuration interaction type eigensolvers. As an example, we also discuss the algebraic form of the perturbative expansions of the effective DUCC Hamiltonians corresponding to composite unitary CC theories and discuss possible algorithms for hybrid classical and quantum computing. Given growing interest in quantum computing, we provide energies for H₂ and Be systems obtained with the quantum phase estimator algorithm available in the Quantum Development Kit for the approximate DUCC Hamiltonians.

Correction for Triples in Reduced Multireference Coupled-Cluster Approaches

The performance of the recently proposed version of the reduced multireference (RMR) coupled-cluster (CC) method with singles and doubles (SD), which employs a modest-size configuration interaction wave function as an external source for a small subset of approximate connected three- and four-body cluster amplitudes that are primarily responsible for the nondynamic correlation effects, and which has been perturbatively corrected for the remaining triples along the same line as in the standard CCSD(T) method (Li X., Paldus J.: J. Chem. Phys. 2006, 124, 174101), referred to by the acronym RMR CCSD(T), is being tested by evaluating equilibrium spectroscopic constants for a demanding system of the beryllium dimer, as well as by computing atomization energies for several di- and triatomics. The focus is on the equilibrium properties, since it has been demonstrated earlier that the RMR CCSD method corrects well for the nondynamic correlation in bond-breaking situations. We find that in all the cases we have examined, the RMR CCSD(T) method does in fact improve the performance of CCSD(T) even in the vicinity of the equilibrium geometry. For states possessing a moderate multireference character, the improvement in computed thermochemical properties relative to CCSD(T) amounts to a few kJ/mol, a meaningful amount when striving for chemical accuracy.

Relativistic effective valence shell Hamiltonian method: Excitation and ionization energies of heavy metal atoms

The relativistic effective valence shell Hamiltonian H(v) method (through second order) is applied to the computation of the low lying excited and ion states of closed shell heavy metal atoms/ions. The resulting excitation and ionization energies are in favorable agreement with experimental data and with other theoretical calculations. The nuclear magnetic hyperfine constants A and lifetimes tau of excited states are evaluated and they are also in accord with experiment. Some of the calculated quantities have not previously been computed.

Generation of potential energy curves for the XΣg+1, BΔg+1, and B′Σg+1 states of C2 using the effective valence shell Hamiltonian method

Calculations of the ground and excited state potential energy curves of C2 using the third-order effective valence Hamiltonian (Hv3rd) method are benchmarked against full configuration interaction and other correlated single-reference perturbative and nonperturbative theories. The large nonparallelity errors (NPEs) exhibited even by state-of-art coupled cluster calculations through perturbative triples indicate a serious deficiency of these single-reference theories. The Hv method, on the other hand, produces a much reduced NPE, rendering it a viable approximate many-body method for accurately determining global ground and excited state potential energy curvessurfaces.

Comparison of low-order multireference many-body perturbation theories

Tests have been made to benchmark and assess the relative accuracies of low-order multireference perturbation theories as compared to coupled cluster (CC) and full configuration interaction (FCI) methods. Test calculations include the ground and some excited states of the Be, H(2), BeH(2), CH(2), and SiH(2) systems. Comparisons with FCI and CC calculations show that in most cases the effective valence shell Hamiltonian (H(v)) method is more accurate than other low-order multireference perturbation theories, although none of the perturbative methods is as accurate as the CC approximations. We also briefly discuss some of the basic differences among the multireference perturbation theories considered in this work.

Improved perturbative treatment of electronic energies from a minimal-norm approach to many-body perturbation theory

We propose a zeroth-order Hamiltonian for many-body perturbation theory based on the unitary decomposition of the two-particle reduced Hamiltonian. For the zeroth-order Hamiltonian constrained to be diagonal in the Hartree-Fock basis set, the two-particle reduced perturbation matrix is chosen to have a minimal Frobenius norm. When compared with the Møller-Plesset partitioning, the method yields more accurate second-order energies.

Convergence Enhancement in Perturbation Theory

The Frobenius norm of operator QW is minimized with respect to level shift parameters applied to the zero-order spectrum, where W is the perturbation while Q is the reduced resolvent of the zero-order Hamiltonian. The stationary condition leads to a simple formula for the level shifts which eliminates degeneracy-induced singularities. Such level shifts may increase the radius of convergence of the perturbation series, and may improve low-order perturbative estimations - as it is found in the cases of a simple matrix eigenvalue problem and the one-dimensional quartic anharmonic oscillator.

Multireference perturbation theory with optimized partitioning. II. Applications to molecular systems

The second-order multireference perturbation theory using an optimized partitioning, denoted as MROPT(2), is applied to calculations of various molecular properties-excitation energies, spectroscopic parameters, and potential energy curves-for five molecules: ethylene, butadiene, benzene, N(2), and O(2). The calculated results are compared with those obtained with second- and third-order multireference perturbation theory using the traditional partitioning techniques. We also give results from computations using the multireference configuration interaction (MRCI) method. The presented results show very close resemblance between the new method and MRCI with renormalized Davidson correction. The accuracy of the new method is good and is comparable to that of second-order multireference perturbation theory using Møller-Plesset partitioning.

Intruder state avoidance multireference Møller-Plesset perturbation theory

A new perturbation approach is proposed that enhances the low-order, perturbative convergence by modifying the zeroth-order Hamiltonian in a manner that enlarges any small-energy denominators that may otherwise appear in the perturbative expansion. This intruder state avoidance (ISA) method can be used in conjunction with any perturbative approach, but is most applicable to cases where small energy denominators arise from orthogonal-space states-so-called intruder states-that should, under normal circumstances, make a negligible contribution to the target state of interests. This ISA method is used with multireference Møller-Plesset (MRMP) perturbation theory on potential energy curves that are otherwise plagued by singularities when treated with (conventional) MRMP; calculation are performed on the 1(3)Sigma(-)(u) state of O(2); and the 2(1)Delta, 3(1)Delta, 2(3)Delta, and 3(3)Delta states of AgH. This approach is also applied to other calculations where MRMP is influenced by intruder states; calculations are performed on the (3)Pi(u) state of N(2), the (3)Pi state of CO, and the 2(1)A' state of formamide. A number of calculations are also performed to illustrate that this approach has little or no effect on MRMP when intruder states are not present in perturbative calculations; vertical excitation energies are computed for the low-lying states of N(2), C(2), CO, formamide, and benzene; the adiabatic (1)A(1)-(3)B(1) energy separation in CH(2), and the spectroscopic parameters of O(2) are also calculated. Vertical excitation energies are also performed on the Q and B bands states of free-base, chlorin, and zinc-chlorin porphyrin, where somewhat larger couplings exists, and-as anticipated-a larger deviation is found between MRMP and ISA-MRMP.