Order Correlation Energy Research Articles

Perturbation calculation of the uniform electron gas with a transcorrelated Hamiltonian.

With a transcorrelated Hamiltonian, we perform a many body perturbation calculation on the uniform electron gas in the high density regime. By using a correlation factor optimized for a single determinant Jastrow ansatz, the second order correlation energy is calculated as 1-ln⁡2π2ln(rs)-0.05075. This already reproduces the exact logarithmic term of the random phase approximation (RPA) result, while the constant term is roughly 7% larger than the RPA one. The close agreement with the RPA method demonstrates that the transcorrelated method offers a viable and potentially efficient method for treating metallic systems.

Phase Properties of Different HfO2 Polymorphs: A DFT-Based Study

Background: Hafnium Dioxide (HfO2) represents a hopeful material for gate dielectric thin films in the field of semiconductor integrated circuits. For HfO2, several crystal structures are possible, with different properties which can be difficult to describe in detail from an experimental point of view. In this study, a detailed computational approach has been shown to present a complete analysis of four HfO2 polymorphs, outlining the intrinsic properties of each phase on the basis of atomistic displacements. Methods: Density functional theory (DFT) based methods have been used to accurately describe the chemical physical properties of the polymorphs. Corrective Hubbard (U) semi-empirical terms have been added to exchange correlation energy in order to better reproduce the excited-state properties of HfO2 polymorphs. Results: the monoclinic phase resulted in the lowest cohesive energy, while the orthorhombic showed peculiar properties due to its intrinsic ferroelectric behavior. DFT + U methods showed the different responses of the four polymorphs to an applied field, and the orthorhombic phase was the least likely to undergo point defects as oxygen vacancies. Conclusions: The obtained results give a deeper insight into the differences in excited states phenomena in relation to each specific HfO2 polymorph.

Intermolecular interaction energies from fourth order many-body perturbation theory. Impact of individual electron correlation contributions.

The performance of Møller-Plesset perturbation theory methods for describing intermolecular interaction energies has been investigated with the focus on illuminating the impact of individual electron correlation energy contributions in fourth order. It is shown that a physically meaningful decomposition of the fourth order correlation energy can be obtained by grouping individual correlation energy terms that share the same diagrammatic loop structure. This decomposition of the fourth order singles (S), doubles (D), triples (T), and quadruples (Q) terms revealed that individual terms from each excitation class can have a huge impact on the energy that is much larger than the total fourth order correlation contribution. A partial summation of S, D, and Q terms has been derived that can reproduce the full fourth order interaction energies with a good accuracy and which does not include the computationally expensive triples energy term.

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.

The ground-state potential curve of the beryllium dimer

The ground-state potential curve for the beryllium dimer is calculated using different methods. In the most accurate calculation a minimum of 0.9 kcal/mole is found at a distance of 4.9 a.u. Corrections for deficiencies in the basis set lead to a predicted well depth of 2.0 kcal/mole. Three classes of methods have been used—MCSCF, CI, and MBPT methods—and calculations were performed with two different basis sets. The largest basis set has 7s, 3p, and 2d contracted Gaussian functions where the 2d functions were optimized in the molecule. Nearly complete CI calculations with this basis set recovers 97% of the valence correlation energy. The main conclusion from the calculations with the different methods and basis sets is that it is necessary to compute practically all the valence shell correlation energy in order to obtain a reliable, qualitatively correct, potential curve for Be2. The best illustration of this point is that large MCSCF calculations with 1832 configurations, obtained from a complete distribution of 4 electrons in 20 orbitals, give qualitatively wrong potential curves. This is particularly remarkable since 98.8% of the valence correlation energy obtainable with the basis set is recovered.

Violations of N-representability from spin-unrestricted orbitals in Møller–Plesset perturbation theory and related double-hybrid density functional theory

By examining the natural orbital occupation numbers, it was discovered that unrestricted Møller–Plesset perturbation theory (MP2) can violate N-representability by having natural occupation numbers greater than 2 or less than 0, even though the energy appears well behaved. Analytically, this problem stems from the fact that the MP2 effective one-particle density matrix is dependent on the inverse of the orbital hessian matrix (A′). When a molecular system goes through the point where spin-restricted orbitals become unstable to unrestriction, one of the eigenvalues of the A′ matrix will become 0, making it a singular matrix, thereby causing problems in the relaxed one-particle density matrix and discontinuities in the first order properties such as nuclear forces and dipole moments. Since the new ‘double-hybrid’ density functionals also involve a second-order perturbation expression for correlation using Kohn–Sham orbitals, these functionals also exhibit similar issues near the unrestriction point. The unphysical orbital occupations can be eliminated and the first derivative discontinuities removed by optimizing orbitals with inclusion of the second order correlation energy. Connections to the second derivative discontinuities in Hartree–Fock and Kohn–Sham density functional theory are also discussed.

Convergence of third order correlation energy in atoms and molecules

We have investigated the convergence of third order correlation energy within the hierarchies of correlation consistent basis sets for helium, neon, and water, and for three stationary points of hydrogen peroxide. This analysis confirms that singlet pair energies converge much slower than triplet pair energies. In addition, singlet pair energies with (aug)-cc-pVDZ and (aug)-cc-pVTZ basis sets do not follow a converging trend and energies with three basis sets larger than aug-cc-pVTZ are generally required for reliable extrapolations of third order correlation energies, making so the explicitly correlated R12 calculations preferable.

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.

Kohn-Sham perturbation theory: Simple solution to variational instability of second order correlation energy functional

The orbital-dependent correlation energy functional resulting from second order Kohn-Sham perturbation theory leads to atomic correlation potentials with correct shell structure and asymptotic behavior. The absolute magnitude of the exact correlation potential, however, is greatly overestimated. In addition, this functional is variationally instable, which shows up for systems with nearly degenerate highest occupied and lowest unoccupied levels like Be. In this contribution we examine the simplest resummation of the Kohn-Sham perturbation series which has the potential to resolve both the inaccuracy and the instability problem of the second order expression. This resummation includes only the hole-hole terms of the Epstein-Nesbet series of diagrams, which has the advantage that the resulting functional is computationally as efficient as the pure second order expression. The hole-hole Epstein-Nesbet functional is tested for a number of atoms and ions. It is found to reproduce correlation and ground state energies with an accuracy comparable to that of state-of-the-art generalized gradient approximations. The correlation potential, on the other hand, is dramatically improved compared to that obtained from generalized gradient approximations. The same applies to all quantities directly related to the potential, as, for instance, Kohn-Sham eigenvalues and excitation energies. Most importantly, however, the hole-hole Epstein-Nesbet functional turned out to be variationally stable for all neutral as well as all singly and doubly ionized atoms considered so far, including the case of Be.

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.

Gaussian basis sets of 5 zeta valence quality for correlated wave functions

Contracted Gaussian basis set of 5 zeta valence quality plus polarization functions for the atoms from H to Ar are presented. We assess the performance of the basis set comparing total HF and second order correlation energies of a sample of diatomic molecules with results obtained numerically or using basis sets reported in the literature. By fitting the directly calculated values through an extrapolation scheme, estimates of the complete basis set limits for second order correlation energy have been obtained. In addition, using some correlated methods, dissociation energies are evaluated and compared with experimental data.

Study of gas-phase O–H bond dissociation enthalpies and ionization potentials of substituted phenols – Applicability of ab initio and DFT/B3LYP methods

In this paper, the study of phenol and 37 compounds representing various ortho-, para-, and meta-substituted phenols is presented. Molecules and their radical structures were studied using ab initio methods with inclusion of correlation energy and DFT in order to calculate the O–H bond dissociation enthalpies (BDEs) and vertical ionization potentials (IPs). Calculated BDEs and IPs were compared with available experimental values to ascertain the suitability of used methods, especially for the description of the substituent induced changes in BDE and IP. MP2, MP3, and MP4 methods do not give reliable results, since they significantly underestimate substituent induced changes in BDE and do not reflect distinct effect of substituents related to para and meta position correctly. DFT/B3LYP method reflects the effect of substituents on BDE satisfactorily, though ΔBDEs are in narrower range than experimental values. BDE of phenol was calculated also using CCSD(T) method in various basis sets. Both, DFT and HF methods describe the effect of substituents on IP identically. However, DFT considerably underestimates individual values. HF method gives IPs in very good agreement with experimental data. Obtained results show that dependences of BDEs and IPs on Hammett constants of the substituents are linear. Linearity of DFT BDE vs. IP dependence is even better than the dependences on Hammett constants and obtained equations allow estimating of O–H BDEs of meta- and para-substituted phenols from calculated IPs.

Gaussian basis sets of triple and quadruple zeta valence quality for correlated wave functions

Contracted basis sets of triple and quadruple zeta (TZ and QZ, respectively) valence quality for the atoms from H to Ar are presented. They have been determined from fully-optimized basis sets of primitive Gaussian-type functions generated in atomic Hartree–Fock (HF) calculations. Sets of Gaussian polarization functions optimized at the Møller–Plesset second-order (MP2) level were added to the TZ and QZ basis sets. This extends earlier work on segmented contracted double zeta valence basis sets. The performance of the basis sets are assessed in molecular HF and MP2 calculations for a sample of diatomic molecules by a comparison of energies, dissociation energies, and dipole moments with results obtained numerically or using basis sets reported in the literature. By fitting the directly calculated values through two extrapolation schemes, estimates of the complete basis set limit for second order correlation energy have been obtained. In addition, results for MP2-R12/A calculations to establish the basis set convergence for the standard calculations are also presented.

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.

Correlation filters with controlled scale response

Correlation filtering methods are becoming increasingly popular for image recognition and location. The recent introduction of optimal tradeoff circular harmonic function filters allowed the user to specify the response of a correlation filter to in-plane rotation distortion. In this paper we introduce a new correlation filter design that can provide a user-specified response to in-plane scale distortion. The design is based on the Mellin radial harmonic (MRH) transform and incorporates multiple harmonics into the correlation filter for improved discrimination capability. Additionally, the filter design minimizes the average correlation energy in order to achieve sharp correlation peaks, and thus we refer to these filters as minimum average correlation energy Mellin radial harmonic (MACE-MRH) filters. We present underlying theory, a MACE-MRH filter design method, and numerical simulation results.

Accurate Gaussian basis sets for the ground state of the CS molecule

Sequences of increasing size atom centered basis sets of Gaussian-type functions for the ground state of the CS molecule are generated with the molecular improved generator coordinate Hartree-Fock (HF) method. At the HF level, total and orbital energies and electric dipole moment and, at the second-order Mφller-Plesset (MP2) level, correlation and dissociation energies and electric dipole moment were calculated and compared with the results obtained with other Gaussian basis sets reported in the literature. Considering our largest basis set, the HF energy is in error by 56.2 µhartree and the second order correlation energy corresponds to » 80% of an estimate of the limiting value. At the MP2 level, the dipole moment and the dissociation energy computed with our largest basis set are in good agreement with the corresponding experimental values. The CS molecule is considered a prototype for systems containing atoms from different rows of the periodic table.

Ab Initio Benchmark Study of (2-Pyridone)2, a Strongly Bound Doubly Hydrogen-Bonded Dimer

The 2-pyridone dimer, (2PY)2, has two antiparallel N−H···O H-bonds analogous to nucleobase dimers. The gas-phase rotational constants and all six intermolecular vibrational frequencies of (2PY)2 have been previously measured, providing benchmarks for theory. The structure, rotational constants, vibrational frequencies, and binding and dissociation energies of (2PY)2 were calculated at the correlated level using second-order Møller−Plesset perturbation theory (MP2) with medium to very large basis sets. The MP2 binding energy limit was extrapolated to the complete basis set (CBS) as De,CBS = −22.62 ± 0.07 kcal/mol. Higher order correlation energy contributions to De at the CCSD(T) level are destabilizing (+0.77 kcal/mol). This implies that (2PY)2 is the most strongly bound doubly hydrogen-bonded dimer known so far. The Hartree−Fock contribution to De,CBS is only ≈65%. Several medium-size basis sets yield MP2 De's within ±5% of the CBS value, as well as structure, rotational constants, and intermolecular vibrations in good agreement with experiment. The PW91 density functional method also shows very good performance with regard to all properties calculated, comparable to MP2. The results imply that correlated methods combined with carefully chosen medium-size basis sets may give near-quantitative results for the structures, binding energies, and intermolecular vibrational frequencies of nucleic acid base dimers.

Many-body perturbation theory for spatially extended systems

An ab-initio self-consistent field method with standard basis sets was applied to normal saturated hydrocarbons. The occupied and virtual canonical orbitals were localized separately by Boys' procedure and the third canonical order correlation energy corrections in the localized representation of the many-body perturbation theory have been calculated. The integrals are partitioned using a “new concept” of neighbourhood order which is based on the distances of localized orbitals. The speed of convergence of the localization correction in the canonical second and third order terms was investigated. It is shown that the convergence is rapid but slows down with increasing canonical order. It is found that the computational dependence is significantly reduced with the size of the system.

Ab initio transition state theory calculations of the reaction rate for OH+CH4→H2O+CH3

We have carried out ab initio calculations using Mo/ller–Plesset perturbation theory, scaling all correlation energy in second order (MP-SAC2) with several large basis sets, for the reaction OH+CH4→H2O+CH3. We found that correlation has a large effect on the geometry, barrier height, and vibrational frequencies of the transition state. The final calculated values, obtained with a correlation-balanced basis set, for the forward and reverse classical barrier heights are 7.9 and 21.2 kcal/mol, respectively. We have used these with transition state theory and an Eckart model for semiclassical tunneling calculations of the rate constants for the above reaction in the temperature range from 200 to 2000 K. We found that the present model, which requires information only at the reactants, transition state, and products, predicts rate constants of the same order of magnitude as the experimental data for this wide temperature range.

Applications of the MBPT in the localized representation the behaviour of the localization terms

The behaviour of the localization corrections of the MBPT is investigated. It is shown that calculating the third and fourth order localization corrections we obtain sufficiently accurate results to the second order correlation energy both for cyclic polyenes and for saturated hydrocarbons. The evaluation of the localization diagrams does not require significant extra computer time. The extra computer time can be recovered if small non-local effects will be neglected.