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 Perturbation Theory with Localized Orbitals: Accounting for Localization Diagrams as Integral Dressing.

We argue that the so-called localization diagrams, originating from off-diagonal Fockian elements, do not have to be dealt with explicitly in the Davidson-Kapuy many-body perturbation theory with localized orbitals but can be accounted for by dressed two-electron integrals.

A note on perturbation-adapted perturbation theory.

The partitioning introduced recently by Knowles [J. Chem. Phys. 156, 011101 (2022)] is analyzed and its connections with the Adams partitioning and the Davidson-Kapuy partitioning are discussed. Davidson's partitioning is reformulated using the second quantized formalism. A relation is pointed out between the Knowles condition for the many-body perturbation theory zero order Hamiltonian and the CEPA0 equations.

Dressing of Vertices by Cumulants in Multi-Reference Coupled Cluster.

A new scheme is introduced in Multi-Reference (MR) Coupled Cluster (CC) based on the MR Generalized Normal Ordering (MRGNO) and the corresponding MR Generalized Wick Theorem (MRGWT) of Kutzelnigg and Mukherjee. The key element is the identification of a structure in MRGWT generated terms, facilitated by Goldstone diagram techniques. This allows for bundling the many terms of the MRGWT expansion and introduces a hierarchy in the equations that can be harnessed in devising approximations. One- and two-particle interaction vertices are found to be uniformly substituted for their counterpart dressed by density cumulants. This allows for a straightforward rewriting of the ordinary energy expression of CC with interaction dressed (id) one- and two-particle terms and reveals the presence of three- and higher-rank dressed interaction vertices too. Cumulants appearing out of dressed interaction vertices contribute to the amplitude equations and can be interpreted to have an amplitude dressing role. Dressing of one- and two-particle interaction vertices is the most straightforward to implement and does not hinder computational feasibility, provided that the reference function involves strictly limited active space sizes. The Generalized Valence Bond wave function, underlying pilot numerical tests, fulfills this criterion. Results on multiple bond breaking scenarios point to the need of stepping beyond one- and two-particle id. An extremely simple version of incorporating amplitude dressing in addition to one- and two-particle id is seen to cure the potential energy curves remarkably, stimulating further investigations along this line.

Improving half-projected spin-contaminated wave functions by multi-configuration perturbation theory.

Allowing triplet components of individual geminals, spin-contaminated strongly orthogonal geminal wave functions may emerge, which can be ameliorated by spin-projection techniques. Of the latter, half-projection was previously shown to be useful, offering a compromise between the amount of remaining spin-contamination and the violation of size consistency generated by projection. This paper investigates how a half-projected spin-contaminated geminal wave function can be improved by multi-configuration perturbation theory to incorporate dynamical correlation effects.

Bilinear Constraints upon the Correlation Contribution to the Electron-Electron Repulsion Energy as a Functional of the One-Electron Reduced Density Matrix.

Perturbative analysis of the functional U[n, ψ] that yields the correlation component U of the electron-electron repulsion energy in terms of the vectors ψ(1) and n of the natural spinorbitals and their occupation numbers (the 1-matrix functional) facilitates examination of the flaws inherent to the present implementations of the density matrix functional theory. Recognizing that the practical usefulness of any approximate 1-matrix functional hinges upon its capability of exactly reproducing the leading contribution to U at the limit of vanishing electron-electron interactions gives rise to asymptotic bilinear constraints for the (exact or model) 2-cumulant G2 that enters the expression for U. The asymptotic behavior of certain blocks of G2 is found to be equally important. These identities, which are obtained for both the single-determinantal and a model multideterminantal cases, take precedence over the linear constraints commonly enforced in the course of approximate construction of such functionals. This observation reveals the futility of designing sophisticated approximations tailored for the second-order contribution to G2 while neglecting proper formulation of the respective first-order contribution that in the case of the so-called JKL-only functionals requires abandoning the JK-dependence altogether. It has its repercussions not only for the functionals of the PNOF family but also for the expressions involving only the L-type two-electron repulsion integrals (in the guise of their exchange counterparts) that account only for the correlation effects due to electrons with antiparallel spins and are well-defined only for spin-unpolarized and high-spin systems (yielding vanishing U for the latter).

Application of the Cauchy integral formula as a tool of analytic continuation for the resummation of divergent perturbation series.

Previous attempts to the resummation of divergent power series by means of analytic continuation are improved applying the Cauchy integral formula for complex functions. The idea is tested on divergent Møller-Plesset perturbation expansions of the electron correlation energy. In particular, the potential curve of the LiH molecule is computed from single reference MPn results which are divergent for bond distances larger than 3.6 Å. Preliminary results for the Hartree-Fock molecule are also tabulated.

Effect of partitioning on the convergence properties of the Rayleigh-Schrödinger perturbation series.

Convergence features of the Rayleigh-Schrödinger perturbation theory (PT) strongly depend on the partitioning applied. We investigate the large order behavior of the Møller-Plesset and Epstein Nesbet partitionings in comparison with a less known partitioning obtained by level shift parameters minimizing the norm of operator Q^W^, with W^ being the perturbation operator while Q standing for the reduced resolvent of the zero order Hamiltonian H^0. Numerical results, presented for molecular systems for the first time, indicate that it is possible to find level shift parameters in this way which convert divergent perturbation expansions to convergent ones in some cases. Besides numerical calculations of high-order PT terms, convergence radii of the corresponding perturbation expansions are also estimated using quadratic Padé approximants.

Resonance Raman Optical Activity of Single Walled Chiral Carbon Nanotubes.

Resonance (vibrational) Raman Optical Activity (ROA) spectra of six chiral single-walled carbon nanotubes (SWCNTs) are studied by theoretical means. Calculations are performed imposing line group symmetry. Polarizability tensors, computed at the π-electron level, are differentiated with respect to DFT normal modes to generate spectral intensities. This computational protocol yields a ROA spectrum in good agreement with the only experiment on SWCNT, available at present. In addition to the conventional periodic electric dipole operator we introduce magnetic dipole and electric quadrupole operators, suitable for conventional k-space calculations. Consequences of the complex nature of the wave function on the scattering cross section are discussed in detail. The resonance phenomenon is accounted for by the short time approximation. Involvement of fundamental vibrations in the region of the intermediate frequency modes is found to be more notable in ROA than in Raman spectra. Calculations indicate exceptionally strong resonance enhancement of SWCNT ROA signals. Resonance ROA profile of the (6,5) tube shows an interesting sign change that may be exploited experimentally for SWCNT identification.

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).

Energy error bars in direct configuration interaction iteration sequence.

A computational scheme for approximate lower bound to eigenvalues of linear operators is elaborated, based on Löwdin's bracketing function. Implementation in direct full configuration interaction algorithm is presented, generating essentially just input-output increase. While strict lower bound property is lost due to approximations, test calculations result lower bounds of the same order of magnitude, as the usual upper bound, provided by the expectation value. Difference of upper and lower bounds gives an error bar, characterizing the wavefunction at the given iteration step.

Vibrational optical activity of chiral carbon nanoclusters treated by a generalized π-electron method.

Cross sections of inelastic light scattering accompanied by vibronic excitation in large conjugated carbon structures is assessed at the π-electron level. Intensities of Raman and vibrational Raman optical activity (VROA) spectra of fullerenes are computed, relying on a single electron per atom. When considering only first neighbor terms in the Hamiltonian (a tight-binding (TB) type or Hückel-model), Raman intensities are captured remarkably well, based on comparison with frequency-dependent linear response of the self-consistent field (SCF) method. Resorting to π-electron levels when computing spectral intensities brings a beneficial reduction in computational cost as compared to linear response SCF. At difference with total intensities, the first neighbor TB model is found inadequate for giving the left and right circularly polarized components of the scattered light, especially when the molecular surface is highly curved. To step beyond first neighbor approximation, an effective π-electron Hamiltonian, including interaction of all sites is derived from the all-electron Fockian, in the spirit of the Bloch-equation. Chiroptical cross-sections computed by this novel π-electron method improve upon first-neighbor TB considerably, with no increase in computational cost. Computed VROA spectra of chiral fullerenes, such as C76 and C28, are reported for the first time, both by conventional linear response SCF and effective π-electron models.

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.

Linearized Coupled Cluster Corrections to Antisymmetrized Product of Strongly Orthogonal Geminals: Role of Dispersive Interactions.

A linearized Multireference Coupled Cluster (MR-LCC) theory is formulated based on the Antisymmetrized Product of Strongly Orthogonal Geminals (APSG) reference state. The role of dispersive interbond interactions is discussed. The presented theory has led to qualitatively correct potential curves for the case when both OH bonds dissociate in H2O, a result that cannot be achieved by adding only perturbative corrections to APSG. The potential curve obtained for the He···He problem practically coincides with the full CI (FCI) result, showing the unexpected accuracy of the MR-LCC approach in this case.

Sensitivity analysis of state-specific multireference perturbation theory.

State-specific multireference perturbation theory (SS-MRPT) developed by Mukherjee et al. [Int. J. Mol. Sci. 3, 733 (2002)] is examined focusing on the dependence of the perturbed energy on the initial model space coefficients. It has been observed earlier, that non-physical kinks may appear on the potential energy surface obtained by SS-MRPT while related coupled-cluster methods may face convergence difficulties. Though exclusion or damping of the division by small coefficients may alleviate the problem, it is demonstrated here that the effect does not originate in an ill-defined division. It is shown that non-negligible model space coefficients may also be linked with the problem. Sensitivity analysis is suggested as a tool for detecting the coefficient responsible. By monitoring the singular values of sensitivity matrices, orders of magnitude increase is found in the largest value, in the vicinity of the problematic geometry point on the potential energy surface. The drastic increase of coefficient sensitivities is found to be linked with a degeneracy of the target root of the effective Hamiltonian. The nature of the one-electron orbitals has a profound influence on the picture: a rotation among active orbitals may screen or worsen the effect.

Spin component scaling in multiconfiguration perturbation theory.

We investigate a term-by-term scaling of the second-order energy correction obtained by perturbation theory (PT) starting from a multiconfiguration wave function. The total second-order correction is decomposed into several terms, based on the level and the spin pattern of the excitations. To define individual terms, we extend the same spin/different spin categorization of spin component scaling in various ways. When needed, identification of the excitation level is facilitated by the pivot determinant underlying the multiconfiguration PT framework. Scaling factors are determined from the stationary condition of the total energy calculated up to order 3. The decomposition schemes are tested numerically on the example of bond dissociation profiles and energy differences. We conclude that Grimme's parameters determined for single-reference Møller-Plesset theory may give a modest error reduction along the entire potential surface, if adopting a multireference based PT formulation. Scaling factors obtained from the stationary condition show relatively large variation with molecular geometry, at the same time they are more efficient in reducing the error when following a bond dissociation process.

Generalized Møller-Plesset Partitioning in Multiconfiguration Perturbation Theory.

Two perturbation (PT) theories are developed starting from a multiconfiguration (MC) zero-order function. To span the configuration space, the theories employ biorthogonal vector sets introduced in the MCPT framework. At odds with previous formulations, the present construction operates with the full Fockian corresponding to a principal determinant, giving rise to a nondiagonal matrix of the zero-order resolvent. The theories provide a simple, generalized Møller-Plesset (MP) second-order correction to improve any reference function, corresponding either to a complete or incomplete model space. Computational demand of the procedure is determined by the iterative inversion of the Fockian, similarly to the single reference MP theory calculated in a localized basis. Relation of the theory to existing multireference (MR) PT formalisms is discussed. The performance of the present theories is assessed by adopting the antisymmetric product of strongly orthogonal geminal (APSG) wave functions as the reference function.

Comparative study of multireference perturbative theories for ground and excited states.

Three recently developed multireference perturbation theories (PTs)-generalized Van Vleck PT (GVVPT), state-specific multireference PT (SS-MRPT), and multiconfiguration PT (MCPT)-are briefly reviewed and compared numerically on representative examples, at the second order of approximations. We compute the dissociation potential curve of the LiH molecule and the BeH(2) system at various geometries, both in the ground and in the first excited singlet state. Furthermore, the ethylene twisting process is studied. Both Møller-Plesset (MP) and Epstein-Nesbet partition are used for MCPT and SS-MRPT, while GVVPT uses MP partitioning. An important thrust in our comparative study is to ascertain the degree of interplay of dynamical and nondynamical correlation for both ground and excited states. The same basis set and the same set of orbitals are used in all calculations to keep artifactual differences away when comparing the results. Nonparallelity error is used as a measure of the performance of the respective theories. Significant differences among the three methods appear when an intruder state is present. Additionally, difficulties arise (a) in MCPT when the choice of a pivot determinant becomes problematic, and (b) in SS-MRPT when there are small coefficients of the model function and there is implicit division by these coefficients, which generates a potential instability of the solutions. Ways to alleviate these latter shortcomings are suggested.

A sparse matrix based full-configuration interaction algorithm.

We present an algorithm related to the full-configuration interaction (FCI) method that makes complete use of the sparse nature of the coefficient vector representing the many-electron wave function in a determinantal basis. Main achievements of the presented sparse FCI (SFCI) algorithm are (i) development of an iteration procedure that avoids the storage of FCI size vectors; (ii) development of an efficient algorithm to evaluate the effect of the Hamiltonian when both the initial and the product vectors are sparse. As a result of point (i) large disk operations can be skipped which otherwise may be a bottleneck of the procedure. At point (ii) we progress by adopting the implementation of the linear transformation by Olsen et al. [J. Chem Phys. 89, 2185 (1988)] for the sparse case, getting the algorithm applicable to larger systems and faster at the same time. The error of a SFCI calculation depends only on the dropout thresholds for the sparse vectors, and can be tuned by controlling the amount of system memory passed to the procedure. The algorithm permits to perform FCI calculations on single node workstations for systems previously accessible only by supercomputers.

Theoretical interpretation of Grimme's spin-component-scaled second order Møller-Plesset theory.

It is shown that spin-component-scaled second order Møller-Plesset theory proposed by Grimme [J. Chem. Phys. 118, 9095 (2003)] can be interpreted as a two-parameter scaling of the zero order Hamiltonian, a generalization of the approach reported by Feenberg [Phys. Rev. 103, 1116 (1956)].