Second-order Correlation Energy Research Articles

Repartitioned Brillouin-Wigner perturbation theory with a size-consistent second-order correlation energy.

Second-order Møller-Plesset perturbation theory (MP2) often breaks down catastrophically in small-gap systems, leaving much to be desired in its performance for myriad chemical applications such as noncovalent interactions, thermochemistry, and dative bonding in transition metal complexes. This divergence problem has reignited interest in Brillouin-Wigner perturbation theory (BWPT), which is regular at all orders but lacks size consistency and extensivity, severely limiting its application to chemistry. In this work, we propose an alternative partitioning of the Hamiltonian that leads to a regular BWPT perturbation series that, through the second order, is size-extensive, size-consistent (provided its Hartree-Fock reference is also), and orbital invariant. Our second-order size-consistent Brillouin-Wigner (BW-s2) approach can describe the exact dissociation limit of H2 in a minimal basis set, regardless of the spin polarization of the reference orbitals. More broadly, we find that BW-s2 offers improvements relative to MP2 for covalent bond breaking, noncovalent interaction energies, and metal/organic reaction energies, although rivaling coupled-cluster with single and double substitutions for thermochemical properties.

Энергия корреляции двухатомных молекул в пределе полного базиса из дистрибутивных гауссовых s-функций

In this paper possibilities of distributive s-Gaussian functions to estimate the second-order correlation energy (МР2) are investigated to the complete basis set limit. In contrast to standard atom-centered, the proposed bases consist of functions distributed along the molecular axis and off-axis functions. We propose and investigate the possibilities of one of the models for choosing the alignment of such functions. The model is characterized by a set of subsets of off-axis s-functions uniformly distributed along circles whose planes are perpendicular to the molecular axis. The function parameters are determined by minimizing the Hartree-Fock energy (for functions located along the molecular axis) and minimizing the Hylleraas functional (for off-axis s-functions). The resulting basis sequences, combined with known extrapolation models, are used to predict the MP2 energy in the complete basis set limit. The efficiency of the models is demonstrated by calculating the energy MP2 of simple H2 and LiH molecules.

The complete basis set limits of the correlation energy of diatomic molecules using distributive s-Gaussian functions

In this paper possibilities of distributive s-Gaussian functions to estimate the second-order correlation energy (MP2) are investigated to the complete basis set limit. In contrast to standard atom-centered, the proposed bases consist of functions distributed along the molecular axis and off-axis functions. We propose and investigate the possibilities of one of the models for choosing the alignment of such functions. The model is characterized by a set of subsets of off-axis s-functions uniformly distributed along circles whose planes are perpendicular to the molecular axis. The function parameters are determined by minimizing the Hartree-Fock energy (for functions located along the molecular axis) and minimizing the Hylleraas functional (for off-axis s-functions). The resulting basis sequences, combined with known extrapolation models, are used to predict the MP2 energy in the complete basis set limit. The efficiency of the models is demonstrated by calculating the energy MP2 of simple H2 and LiH molecules. Keywords: distributive off-axis functions, correlation energy, extrapolation models.

Second-order perturbative correlation energy functional in the ensemble density-functional theory

We derive the second-order approximation (PT2) to the ensemble correlation energy functional by applying the G\"{o}rling-Levy perturbation theory on the ensemble density-functional theory (EDFT). Its performance is checked by calculating excitation energies with the direct ensemble correction method in 1D model systems and 3D atoms using numerically exact Kohn-Sham orbitals and potentials. Comparing with the exchange-only approximation, the inclusion of the ensemble PT2 correlation improves the excitation energies in 1D model systems in most cases, including double excitations and charge-transfer excitations. However, the excitation energies for atoms are generally worse with PT2. We find that the failure of PT2 in atoms is due to the two contributions of an orbital-dependent functional to excitation energies being inconsistent in the calculations. We also analyze the convergence of PT2 excitation energies with respect to the number of unoccupied orbitals.

Generalizing Double-Hybrid Density Functionals: Impact of Higher-Order Perturbation Terms.

Connections between the Görling–Levy (GL) perturbation theory and the parameters of double-hybrid (DH) density functional are established via adiabatic connection formalism. Moreover, we present a more general DH density functional theory, where the higher-order perturbation terms beyond the second-order GL2 one, such as GL3 and GL4, also contribute. It is shown that a class of DH functionals including previously proposed ones can be formed using the present construction. Based on the proposed formalism, we assess the performance of higher-order DH and long-range corrected DH formed on the Perdew–Burke–Ernzerhof (PBE) semilocal functional and second-order GL2 correlation energy. The underlying construction of DH functionals based on the generalized many-body perturbation approaches is physically appealing in terms of the development of the non-local forms using more accurate and sophisticated semilocal functionals.

Optimization of atomic density-fitting basis functions for molecular two-electron integral approximations.

A general procedure for the optimization of atomic density-fitting basis functions is designed with the balance between accuracy and numerical stability in mind. Given one-electron wavefunctions and energies, weights are assigned to the product densities, modeling their contribution to the exchange and second-order correlation energy, and a simple weighted error measure is minimized. Generally contracted Gaussian auxiliary basis sets are optimized to match the wavefunction basis sets [D. N. Laikov, Theor. Chem. Acc. 138, 40 (2019)] for all 102 elements in a scalar-relativistic approximation [D. N. Laikov, J. Chem. Phys. 150, 061103 (2019)].

Optimal power series expansions of the Kohn\u2013Sham potential

A fundamental weakness of density functional theory (DFT) is the difficulty in making systematic improvements to approximations for the exchange and correlation functionals. In this paper, we follow a wave-function-based approach [N.I. Gidopoulos, Phys. Rev. A 83, 040502 (2011)] to develop perturbative expansions of the Kohn–Sham (KS) potential. Our method is not impeded by the problem of variational collapse of the second-order correlation energy functional. Arguing physically that a small magnitude of the correlation energy implies weak perturbation and hence fast convergence of the perturbative expansion for the interacting state and for the KS potential, we discuss several choices for the zeroth-order Hamiltonian in such expansions. Our first two choices yield KS potentials containing only Hartree and exchange terms: the exchange-only optimized effective potential (xOEP), also known as the exact-exchange potential (EXX), and the Local Fock exchange (LFX) potential. Finally, we choose the zeroth order Hamiltonian that corresponds to minimum magnitude of the second order correlation energy, aiming to obtain at first order the most accurate approximation for the KS potential with Hartree, exchange and correlation character.

Statistical Electronic Structure Calibration Study of the CCSD(T*)-F12b Method for Atomization Energies

In the explicitly correlated CCSD(T)-F12b coupled cluster method only the singles and doubles component of the energy benefits from inclusion of terms involving the interelectronic distance. Consequently, only that component exhibits accelerated convergence with respect to the 1-particle basis set. The smaller perturbative triples component converges at the same rate as the corresponding piece in standard CCSD(T). With the alternative CCSD(T*)-F12b method the triples correlation energy is scaled up by the ratio of explicitly correlated to standard second-order perturbation theory correlation energies in an attempt to better approximate the basis set limit. An extensive and diverse 212 molecule collection of reference total atomization energies, developed with large basis sets (up to aug-cc-pV9Z in some cases) and standard CCSD(T), was used to calibrate the performance of CCSD(T*). Scaling of the (T) energy led to improved results relative to raw F12b values but only provided a statistical advantage over previously proposed complete basis set extrapolation techniques for the smallest basis sets. With larger sets, scaling (T) produced noticeably poorer results, sometimes by a factor of 2. In agreement with earlier studies, basis set extrapolated CCSD(T)-F12b was found to exhibit a systematic bias toward overestimating reference atomization energies with an error that increases with the magnitude of the valence correlation energy.

Orbital-dependent second-order scaled-opposite-spin correlation functionals in the optimized effective potential method.

The performance of correlated optimized effective potential (OEP) functionals based on the spin-resolved second-order correlation energy is analysed. The relative importance of singly- and doubly- excited contributions as well as the effect of scaling the same- and opposite- spin components is investigated in detail comparing OEP results with Kohn-Sham (KS) quantities determined via an inversion procedure using accurate ab initio electronic densities. Special attention is dedicated in particular to the recently proposed scaled-opposite-spin OEP functional [I. Grabowski, E. Fabiano, and F. Della Sala, Phys. Rev. B 87, 075103 (2013)] which is the most advantageous from a computational point of view. We find that for high accuracy, a careful, system dependent, selection of the scaling coefficient is required. We analyse several size-extensive approaches for this selection. Finally, we find that a composite approach, named OEP2-SOSh, based on a post-SCF rescaling of the correlation energy can yield high accuracy for many properties, being comparable with the most accurate OEP procedures previously reported in the literature but at substantially reduced computational effort.

A density difference based analysis of orbital-dependent exchange-correlation functionals

We present a density difference based analysis for a range of orbital-dependent Kohn–Sham functionals. Results for atoms, some members of the neon isoelectronic series and small molecules are reported and compared with ab initio wave function calculations. Particular attention is paid to the quality of approximations to the exchange-only optimised effective potential (OEP) approach: we consider both the localised Hartree–Fock as well as the Krieger–Li–Iafrate methods. Analysis of density differences at the exchange-only level reveals the impact of the approximations on the resulting electronic densities. These differences are further quantified in terms of the ground state energies, frontier orbital energy differences and highest occupied orbital energies obtained. At the correlated level, an OEP approach based on a perturbative second-order correlation energy expression is shown to deliver results comparable with those from traditional wave function approaches, making it suitable for use as a benchmark against which to compare standard density functional approximations.

Optimized effective potential method based on spin-resolved components of the second-order correlation energy in density functional theory

A correlated optimized effective potential method based on scaled-opposite-spin second-order correlation (OEP2-SOS) is presented. This approach is based on the finding that the same-spin- and opposite-spin-correlation potentials are almost proportional to each other at each point in the real space. The performance of the OEP2-SOS method is validated for benchmark atomic and molecular systems, and we find that all the OEP2-SOS results largely outperform those from second-order G\"orling-Levy perturbation theory and, additionally, the presented method can converge also when quasidegeneracy is present (e.g., in the Beryllium atom). The OEP2-SOS approach is thus an accurate and efficient method to supplement exact exchange with an ab initio correlation and, importantly, with small additional computational cost.

MP2/CBS atomic and molecular benchmarks for H through Ar

We extrapolate to the MP2/CBS limit with a sequence of optimized n-tuple-zeta augmented polarized basis sets (n=4, 5, 6, and 7) for the entire set of 72 atoms, positive and negative atomic ions, homonuclear diatomic molecules, and hydrides representing the first two rows of the Periodic Table. The second-order correlation energies agree with accurate (+/-0.01 mE(h)) numerical values (He, Be, Ne, Mg, Ar, Zn(+2), and Kr) to within +/-0.1%. These MP2/CBS limits of the 72 species can now be used as benchmarks to calibrate more approximate calculations using smaller basis sets.

Assessment of basis sets for F12 explicitly-correlated molecular electronic-structure methods

One-electron basis sets for F12 explicitly-correlated molecular electronic-structure methods are assessed by analysing the accuracy of Hartree–Fock energies and valence-only second-order correlation energies of a test set of 106 small molecules containing the atoms H, C, N, O and F. For these molecules, near Hartree–Fock-limit energies and benchmark second-order correlation energies (accurate to within 99.95% of the basis-set limit) are provided. Absolute energies are analysed as well as the Hartree–Fock and second-order correlation contributions to the atomisation energies of the molecules. Standard basis sets such as the Karlsruhe def2-TZVPP and def2-QZVPP sets and the augmented correlation-consistent polarised valence X-tuple zeta (aug-cc-p VXZ, X = D, T, Q, 5) sets are compared with the specialised cc-pVXZ-F12 (X = D, T, Q) sets that were recently optimised by Peterson and co-workers [J. Chem. Phys. 128, 084102 (2008)] for use in F12 theory. The results obtained from F12 explicitly-correlated molecular electronic-structure calculations are compared with those that are obtained by standard electronic-structure calculations followed by basis-set extrapolation based on the X −3 convergence behaviour of the aug-cc-pVXZ basis sets. The most important conclusions are that the cc-pVXZ-F12 sets are the preferred basis sets for F12 theory and that the X −3 extrapolation from the aug-cc-pVQZ and aug-cc-pV5Z is slightly more accurate than F12 theory in the cc-pVTZ-F12 basis but less accurate than F12 theory in the cc-pVQZ-F12 basis.

Application of Dyson-corrected second-order perturbation theories

Previously [Y. Mochizuki, Chem. Phys. Lett. 453 (2008) 109], the self-energy shift obtained from the Dyson equation was introduced to modify the energy denominator of the second-order perturbational estimate for orbital relaxations in the method of configuration interaction singles with perturbative doubles, CIS(D). This Dyson correction technique is then applied also to the differential second-order correlation energy in CIS(D) calculations, in conjunction with the partial renormalization (PR) of amplitudes by using the topological factor. The potential of Dyson/PR-corrected second-order perturbation itself is separately examined with a couple of small systems having elongated bonds at the ground state.

The principle-quantum-number (and the radial-quantum-number) expansion of the correlation energy of two-electron atoms

The second-order correlation energy of two-electron ions is studied in terms of an expansion in minimal approximations to the first-order natural orbitals (NOs). The non-linear parameters of these NOs are determined by minimization of the second-order energy. An approximation to the total second-order correlation energy is obtained as a sum of increments e(lp), depending on the angular quantum number l and the radial quantum number p. (Either l or p can be eliminated in favor of the principal quantum number n = l + p.) Closed expressions for these energy increments are derived. For fixed p the increments go as (l + 1)(-5). This is consistent with the behavior of the exact partial wave increments (that depend on the parameter l only) as (l + 1/2)(-4). While the partial wave increments correspond to a summation of e(lp) over p, other partial summations of the two-parameter increments lead to either the principal-quantum-number expansion (PQNE) with energy increments approximately n(-4), or the radial-quantum-number expansion, with a less transparent convergence pattern. Unfortunately these partial summations can neither be done in closed form nor from the asymptotic expansion, but some insight is obtained from a numerical summation. The hope to find a rigorous derivation of the PQNE has not been fulfilled.

Orbital-optimized opposite-spin scaled second-order correlation: An economical method to improve the description of open-shell molecules

Coupled-cluster methods based on Brueckner orbitals are well known to resolve the problems of symmetry breaking and spin contamination that are often associated with Hartree-Fock orbitals. However, their computational cost is large enough to prevent application to large molecules. Here the authors present a simple approximation where the orbitals are optimized with the mean-field energy plus a correlation energy taken as the opposite-spin component of the second-order many-body correlation energy, scaled by an empirically chosen parameter (recommended as 1.2 for general applications). This "optimized second-order opposite-spin" (abbreviated as O2) method requires fourth-order computation on each orbital iteration. O2 is shown to yield predictions of structure and frequencies for closed-shell molecules that are very similar to scaled second-order Moller-Plesset methods. However, it yields substantial improvements for open-shell molecules, where problems with spin contamination and symmetry breaking are shown to be greatly reduced.

The self‐energy of the uniform electron gas in the second order of exchange

AbstractThe on‐shell self‐energy of the homogeneous electron gas in second order of exchange, Σ2x = Re Σ2x (kF, k2 F/2), is given by a certain integral. This integral is treated here in a similar way as Onsager, Mittag, and Stephen [Ann. Physik (Leipzig) 18, 71 (1966)] have obtained their famous analytical expression e2x = (in atomic units) for the correlation energy in second order of exchange. Here it is shown that the result for the corresponding on‐shell self‐energy is Σ2x = e2x. The off‐shell self‐energy Σ2x (k, o) correctly yields 2e2x (the potential component of e2x) through the Galitskii‐Migdal formula. The quantities e2x and Σ2x appear in the high‐density limit of the Hugenholtz‐van Hove (Luttinger‐Ward) theorem.

From RHIC to LHC: a relativistic diffusion approach

The on-shell self-energy of the homogeneous electron gas in second order of exchange, Σ2x = Re Σ2x (kF, k2 F/2), is given by a certain integral. This integral is treated here in a similar way as Onsager, Mittag, and Stephen [Ann. Physik (Leipzig) 18, 71 (1966)] have obtained their famous analytical expression e2x = (in atomic units) for the correlation energy in second order of exchange. Here it is shown that the result for the corresponding on-shell self-energy is Σ2x = e2x. The off-shell self-energy Σ2x (k, o) correctly yields 2e2x (the potential component of e2x) through the Galitskii-Migdal formula. The quantities e2x and Σ2x appear in the high-density limit of the Hugenholtz-van Hove (Luttinger-Ward) theorem.

The self-energy of the uniform electron gas in the second order of exchange

The on-shell self-energy of the homogeneous electron gas in second order of exchange, $\Sigma_{2{\rm x}}= {\rm Re} \Sigma_{2{\rm x}}(k_{\rm F},k_{\rm F}^2/2)$, is given by a certain integral. This integral is treated here in a similar way as Onsager, Mittag, and Stephen [Ann. Physik (Leipzig) {\bf 18}, 71 (1966)] have obtained their famous analytical expression $e_{2{\rm x}}={1/6}\ln 2- 3\frac{\zeta(3)}{(2\pi)^2}$ (in atomic units) for the correlation energy in second order of exchange. Here it is shown that the result for the corresponding on-shell self-energy is $\Sigma_{2{\rm x}}=e_{2{\rm x}}$. The off-shell self-energy $\Sigma_{2{\rm x}}(k,\omega)$ correctly yields $2e_{2{\rm x}}$ (the potential component of $e_{2{\rm x}}$) through the Galitskii-Migdal formula. The quantities $e_{2{\rm x}}$ and $\Sigma_{2{\rm x}}$ appear in the high-density limit of the Hugenholtz-van Hove (Luttinger-Ward) theorem.

CALCULATIONS WITH CORRELATED MOLECULAR WAVE FUNCTIONS: HF, MP2 AND DFT CALCULATIONS ON SECOND-ROW DIATOMIC HYDRIDES

The performance of the previously proposed double zeta valence quality plus polarization functions (DZP) and augmented DZP (ADZP) basis sets is tested at the Hartree–Fock, second-order Møller–Plesset, and density functional levels of theory for the electronic ground state of the second-row diatomic hydrides. Total energy, second-order correlation energy, dissociation energy, bond length, vibrational frequency, and dipole moment are calculated and compared with results obtained with popular basis sets reported in the literature. It is shown that the DZP and ADZP basis sets in general provides better accuracy for a similar number of basis functions.