One-Particle Operator Representation over Two-Particle Basis Sets for Relativistic QED Computations.

This work is concerned with two-spin-1/2-fermion relativistic quantum mechanics, and it is about the construction of one-particle projectors using an inherently two-particle, "explicitly correlated" basis representation necessary for good numerical convergence of the interaction energy. It is demonstrated that a faithful representation of the one-particle operators, which appear in intermediate but essential computational steps, can be constructed over a many-particle basis set by accounting for the full Hilbert space beyond the physically relevant antisymmetric subspace. Applications of this development can be foreseen for the computation of quantum-electrodynamics corrections for a correlated relativistic reference state and high-precision relativistic computations of medium-to-high-Z helium-like systems, for which other two-particle projection techniques are unreliable.

The Bethe-Salpeter QED Wave Equation for Bound-State Computations of Atoms and Molecules.

Interactions in atomic and molecular systems are dominated by electromagnetic forces and the theoretical framework must be in the quantum regime. The physical theory for the combination of quantum mechanics and electromagnetism, quantum electrodynamics has been "established" by the mid-twentieth century, primarily as a scattering theory. To describe atoms and molecules, it is important to consider bound states. In the nonrelativistic quantum mechanics framework, bound states can be efficiently computed using robust and general methodologies with systematic approximations developed for solving wave equations. With the sight of the development of a computational quantum electrodynamics framework for atomic and molecular matter, the field theoretic Bethe-Salpeter wave equation expressed in space-time coordinates, its exact equal-time variant, and emergence of a relativistic wave equation, is reviewed. A computational framework, with initial applications and future challenges in relation with precision spectroscopy, is also highlighted.

Relativistic two-electron atomic and molecular energies using LS coupling and double groups: Role of the triplet contributions to singlet states.

The triplet contribution is computed to the 1 and 2 S0e1 states of the He atom, to the 1S0e1 state of the Li+ and Be2+ ions, and to the X1Σg + ground state of the H2 molecule by extensive use of double-group symmetry (equivalent to LS coupling for the atomic systems) during the course of the variational solution of the no-pair Dirac-Coulomb-Breit (DCB) wave equation. The no-pair DCB energies are converged within sub-parts-per-billion relative precision, using an explicitly correlated Gaussian basis optimized to the non-relativistic energies. The α fine-structure constant dependence of the triplet sector contribution to the variational energy is α4Eh at leading order, in agreement with the formal perturbation theory result available from the literature.

Variational vs perturbative relativistic energies for small and light atomic and molecular systems.

Variational and perturbative relativistic energies are computed and compared for two-electron atoms and molecules with low nuclear charge numbers. In general, good agreement of the two approaches is observed. Remaining deviations can be attributed to higher-order relativistic, also called non-radiative quantum electrodynamics (QED), corrections of the perturbative approach that are automatically included in the variational solution of the no-pair Dirac-Coulomb-Breit (DCB) equation to all orders of the α fine-structure constant. The analysis of the polynomial α dependence of the DCB energy makes it possible to determine the leading-order relativistic correction to the non-relativistic energy to high precision without regularization. Contributions from the Breit-Pauli Hamiltonian, for which expectation values converge slowly due the singular terms, are implicitly included in the variational procedure. The α dependence of the no-pair DCB energy shows that the higher-order (α4Eh) non-radiative QED correction is 5% of the leading-order (α3Eh) non-radiative QED correction for Z = 2 (He), but it is 40% already for Z = 4 (Be2+), which indicates that resummation provided by the variational procedure is important already for intermediate nuclear charge numbers.

On the Breit interaction in an explicitly correlated variational Dirac-Coulomb framework.

The Breit interaction is implemented in the no-pair variational Dirac-Coulomb (DC) framework using an explicitly correlated Gaussian basis reported in the previous paper [P. Jeszenszki, D. Ferenc, and E. Mátyus, J. Chem. Phys. 156, 084111 (2022)]. Both a perturbative and a fully variational inclusion of the Breit term are considered. The no-pair DC plus perturbative Breit and the no-pair DC-Breit energies are compared with perturbation theory results including the Breit-Pauli Hamiltonian and leading-order non-radiative quantum electrodynamics corrections for low Z values. Possible reasons for the observed deviations are discussed.

Variational Dirac-Coulomb explicitly correlated computations for atoms and molecules.

The Dirac-Coulomb equation with positive-energy projection is solved using explicitly correlated Gaussian functions. The algorithm and computational procedure aims for a parts-per-billion convergence of the energy to provide a starting point for further comparison and further developments in relation with high-resolution atomic and molecular spectroscopy. Besides a detailed discussion of the implementation of the fundamental spinor structure, permutation, and point-group symmetries, various options for the positive-energy projection procedure are presented. The no-pair Dirac-Coulomb energy converged to a parts-per-billion precision is compared with perturbative results for atomic and molecular systems with small nuclear charge numbers. Paper II [D. Ferenc, P. Jeszenszki, and E. Mátyus, J. Chem. Phys. 156, 084110 (2022).] describes the implementation of the Breit interaction in this framework.

Lower Bounds for Nonrelativistic Atomic Energies.

A recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput. 2021, 17, 1535) is further developed and applied to the highly accurate calculation of the ground-state energy of two- (He, Li+, and H-) and three- (Li) electron atoms. The method has been implemented with explicitly correlated many-particle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilli-Hartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained. Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory.

All-order explicitly correlated relativistic computations for atoms and molecules.

A variational solution procedure is reported for the many-particle no-pair Dirac-Coulomb and Dirac-Coulomb-Breit Hamiltonians aiming at a parts-per-billion (ppb) convergence of the atomic and molecular energies, described within the fixed nuclei approximation. The procedure is tested for nuclear charge numbers from Z = 1 (hydrogen) to 28 (iron). Already for the lowest Z values, a significant difference is observed from leading-order Foldy-Woythusen perturbation theory, but the observed deviations are smaller than the estimated self-energy and vacuum polarization corrections.

NECI: N-Electron Configuration Interaction with an emphasis on state-of-the-art stochastic methods.

We present NECI, a state-of-the-art implementation of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) algorithm, a method based on a stochastic application of the Hamiltonian matrix on a sparse sampling of the wave function. The program utilizes a very powerful parallelization and scales efficiently to more than 24 000 central processing unit cores. In this paper, we describe the core functionalities of NECI and its recent developments. This includes the capabilities to calculate ground and excited state energies, properties via the one- and two-body reduced density matrices, as well as spectral and Green's functions for ab initio and model systems. A number of enhancements of the bare FCIQMC algorithm are available within NECI, allowing us to use a partially deterministic formulation of the algorithm, working in a spin-adapted basis or supporting transcorrelated Hamiltonians. NECI supports the FCIDUMP file format for integrals, supplying a convenient interface to numerous quantum chemistry programs, and it is licensed under GPL-3.0.

Spin Symmetry and Size Consistency of Strongly Orthogonal Geminals.

An overview of geminal-based wavefunctions is given, allowing for singlet-triplet mixing within the two-electron units. Spin contamination of the total wavefunction (obtained as an antisymmetrized product) is restored by spin projection. Full variation after projection is examined for two models. One is the long known spin-projected, extended Hartree-Fock (EHF). The other is a yet unexplored function, termed spin-projected, extended antisymmetrized product of strongly orthogonal geminals (EAPSG). Studies on size consistency are presented for both models. Numerical evaluation of EHF and EAPSG is performed for small test systems (H4 and H8).

Spin-adaptation and redundancy in state-specific multireference perturbation theory.

Spin-adaptation of virtual functions in state-specific multireference perturbation theory is examined. Redundancy occurring among virtual functions generated by unitary group based excitation operators on a model-space function is handled by canonical orthogonalization. The treatment is found to remove non-physical kinks observed earlier on potential energy surfaces. Sensitivity analysis of the new approach confirms the elimination of the drastic increase in singular values of sensitivity matrices, reported earlier.