Approximate energy-evaluating schemes for a system of weakly overlapping group functions

The energy of weakly overlapping group functions can be written as a series according to the powers of the (σ – I) matrix, where σ is the molecular overlap matrix and I is the unit matrix [1,2]. This power series of the energy is studied by investigating the importance of different order terms to obtain accurate energies and to predict equilibrium bond lengths. It is found that the series is truncated advantageously at an even-order term. Approximate formulas for the first- and second-order terms are proposed in order to reduce computational work. Numerical examples are presented to illustrate the effect of these terms to the energy. The relation of the projection energy to the approximate first- and second-order terms is also discussed. It is found that, by choosing appropriate projection factors, the projection energy corrects the zeroth-order energy more efficiently than does the first-order term. The inclusion of the approximate second-order term represents a slight improvement with respect to the use of the projection energy at the expense of some extra computation. © 1995 John Wiley & Sons, Inc.     

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.     

Theoretical study on the second-order nonlinear optical properties and reorganization energy of silafluorenes and spirobisilafluorenes derivatives

The electronic absorption and emission spectra, second-order polarizability and reorganization energy of the twenty silafluorenes and spirobisilafluorenes derivatives have been studied at the density functional theory level. The results show that the second-order polarizability (β) increases with increase in the number of the branches due to cooperative enhancement of the charge transfer, whereas the reorganization energy (λ) follows the opposite trend for the studied compounds. The properties (β and λ) of the compounds at the 3, 6-positions substitution are much better than those of compounds at the 2, 7-positions substitution. The effects of donor/acceptor (D/A) substitution and different spiroatoms (silicon or carbon) on second-order polarizability and reorganization energy are also discussed. It is noted that the charge transport properties can be tuned by changing the donor/acceptor (D/A) substitution, and the acceptor substitution can greatly reduce the reorganization energy. The electronic absorption spectra show that all studied compounds can meet the requirement of nonlinear optical (NLO) transparency. Thus, increasing the number of branches and acceptor substitution can remarkably enhance performance of this kind of compounds. Based on larger β, smaller λ and excellent optical transparency, this kind of compounds have a possibility to be excellent second-order NLO or charge transport materials.     

Second-order energy components in basis-set-superposition-error-free intermolecular perturbation theory

 Recently a basis-set-superposition-error-free second-order perturbation theory was introduced based on the “chemical Hamiltonian approach” providing the full antisymmetry of all wave functions by using second quantization. Subsequently, the “Heitler–London” interaction energy corresponding to the sum of the zero- and first-order perturbational energy terms was decomposed into different physically meaningful components, like electrostatics, exchange and overlap effects. The first-order wave function obtained in the framework of this perturbation theory also consists of terms having clear physical significance: intramolecular correlation, polarization, charge transfer, dispersion and combined polarization–charge transfer excitations. The second-order energy, however, does not represent a simple sum of the respective contributions, owing to the intermolecular overlap. Here we propose an approximate energy decomposition scheme by defining some “partial Hylleraas functionals” corresponding to the different physically meaningful terms of the first-order wave functions. The sample calculations show that at large and intermediate intermolecular distances the total second-order intermolecular interaction energy contribution is practically equal to the sum of these “physical” terms, while at shorter distances the overlap-caused interferences become of increasing importance. Received: 18 June 2001 / Accepted: 28 August 2001 / Published online: 16 November 2001     

Perturbation expansion theory corrected from basis set superposition error II. Charge transfer,pair correlationand dispersion terms

The second-order perturbation theory based on the locally projected molecular orbitals is developed. A few test calculations with cc-pVDZ and aug-cc-pVDZ basis sets are carried out for the dimers, (H₂O)₂ and (HF)₂. The charge transfer terms remove the deficiency of the locally projected self-consistent field method for molecular interaction (LP SCF MO MI), and the potential energy curves calculated with aug-cc-pVDZ are very close to the corresponding curves of the counterpoise-corrected SCF energy. Only after adding the spin-exchanged dispersion type to the dispersion and intra-molecular pair correlation terms, the calculated potential energy curves become close to those of the counterpoise-corrected second-order Møller–Plesset (MP2). Pragmatic approaches for reducing the influence of the basis set superposition error are proposed.     

Analytic second-order energy derivatives in natural orbital functional theory

The analytic energy gradients in the atomic orbital representation have recently been published (Mitxelena and Piris in J Chem Phys 146:014102, 2017) within the framework of the natural orbital functional theory (NOFT). We provide here an alternative expression for them in terms of natural orbitals, and use it to derive the analytic second-order energy derivatives with respect to nuclear displacements in the NOFT. The computational burden is shifted to the calculation of perturbed natural orbitals and occupancies, since a set of linear coupled-perturbed equations obtained from the variational Euler equations must be solved to attain the analytic Hessian at the perturbed geometry. The linear response of both natural orbitals and occupation numbers to nuclear geometry displacements need only specify the reconstruction of the second-order reduced density matrix in terms of occupation numbers.     

The perturbation theory of the extended Hartree–Fock approximation for two-electron atoms

The first-order 1/Z perturbation theory of the extended Hartree–Fock approximation for two-electron atoms is described. A number of unexpected features emerge: (a) it is proved that the orbitals must be expanded in powers of Z^?1/2, rather than in Z^?1 as expected; (b) it is shown that the restricted Hartree–Fock and correlation parts of the orbitals can be uncoupled to first order, so that second-order energies are additive; (c) the equation describing the first-order correlation orbital has an infinite number of solutions of all angular symmetries in general, rather than only one of a single symmetry as expected; (d) the first-order correlation equation is a homogeneous linear eigenvalue-type equation with a non-local potential. It involves a parameter μ and an eigenvalue ω(μ) which may be interpreted as the probability amplitude and energy of a virtual correlation state. The second-order correlation energy is 2μ²ω. Numerical solutions for the first-order correlation orbitals, obtained variationally, are presented. The approximate second-order correlation energy is nearly 90% of the exact value. The first-order 1/Z perturbation theory of the natural-orbital expansion is described, and the coupled first-order integro-differential perturbation equations are obtained. The close relationship between the first-order extended Hartree–Fock correlation orbitals and the first-order natural correlation orbitals is discussed. A comparison of the numerical results with those of Kutzelnigg confirms the similarity.     

Separability of tight and roaming pathways to molecular decomposition

Recent studies have questioned the separability of the tight and roaming mechanisms to molecular decomposition. We explore this issue for a variety of reactions including MgH(2) → Mg + H(2), NCN → CNN, H(2)CO → H(2) + CO, CH(3)CHO → CH(4) + CO, and HNNOH → N(2) + H(2)O. Our analysis focuses on the role of second-order saddle points in defining global dividing surfaces that encompass both tight and roaming first-order saddle points. The second-order saddle points define an energetic criterion for separability of the two mechanisms. Furthermore, plots of the differential contribution to the reactive flux along paths connecting the first- and second-order saddle points provide a dynamic criterion for separability. The minimum in the differential reactive flux in the neighborhood of the second-order saddle point plays the role of a mechanism divider, with the presence of a strong minimum indicating that the roaming and tight mechanisms are dynamically distinct. We show that the mechanism divider is often, but not always, associated with a second-order saddle point. For the formaldehyde and acetaldehyde reactions, we find that the minimum energy geometry on a conical intersection is associated with the mechanism divider for the tight and roaming processes. For HNNOH, we again find that the roaming and tight processes are dynamically separable but we find no intrinsic feature of the potential energy surface associated with the mechanism divider. Overall, our calculations suggest that roaming and tight mechanisms are generally separable over broad ranges of energy covering most kinetically relevant regimes.     

Accurate Gaussian basis sets for the H2O molecule generated with the molecular improved generator coordinate Hartree–Fock method

The molecular improved generator coordinate Hartree–Fock (MIGCHF) method is used to generate accurate basis sets of primitive Gaussian-type functions for the H₂O molecule. Sequences of increasing size atom centered basis sets are employed to explore the accuracy that can be achieved with this method. Using the O(24s14p8d5f2g1h);H(22s9p5d2f1g) basis set, the HF and second-order electron correlation energies of the H₂O ground state at the experimental geometry are computed as −76.0674680 and −0.3491935 hartree, respectively. The HF energy is in error by 20 μhartree and the second-order correlation energy corresponds to 96.5% of an estimate of the limiting value. The relevance of the present calculations is to show the accuracy that can be achieved in studies of small polyatomic molecules with the MIGCHF method.     

A sequential product of geminals and its applications

In this paper, an SPG function, which is associated with an extreme point of the set of N-representable second-order reduced density matrices, is used to perform the calculation of the ground state energy of LiH with the variation of internuclear separation. The result of our calculation essentially is in accordance with that of AGP function.     

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.     

First-order gradient correction for the exchange-energy density functional for atoms

Summary Spurred by earlier discoveries by Deb and others that a first-order correction in the kinetic energy functional leads to an improved kinetic energy, a first-order gradient term is studied as a component of the gradient-corrected functional for atomic exchange energy. This term is shown to improve the local density approximation to the exchange energy more than does the usual second-order gradient correction. The imperative of systematically deriving this gradient correction is discussed but not resolved.     

Second-order correlation potential in the Kohn–Sham approximation for atoms

A new type of correlation functional derived from the second-order expression for the correlation energy of an atom is proposed. The derived correlation potential contains one free parameter, which is determined by fitting the known pair correlation energy. The calculations with this potential in the Kohn–Sham approximation give rather accurate values for the matrix elements of different operators.     

Semirigorous bounds for the overlap of approximate wavefunctions

Alexander's probable upper and lower bounds to the overlap S between an approximate and the true wavefunctions are based on second-order perturbation theory and a special ordering of configurations. These weak points are removed in the present treatment and replaced by an exact expression for the energy lowering, plus a very reasonable postulate. The resulting lower bound to the overlap S is illustrated with examples taken from the literature.     

On the role of higher-order correlation effects on the induction interactions between closed-shell molecules

High-order correlation contributions to the second-order induction energy were studied for various representative van der Waals complexes. It was found that the induction energy obtained by the truncation of the relaxed M?ller-Plesset expansion in the second or third order is in most cases quite close to the induction energy computed with the coupled-cluster method (restricted to single and double excitations). Also, the effect of triples excitations on this perturbation term is usually small. However, given an oscillatory behaviour of the M?ller-Plesset induction corrections, the coupled-cluster method seems to be better suited to a reliable calculation of the induction energy. The sources of the remaining differences between the interaction energies computed by symmetry-adapted perturbation theory and those computed by the supermolecule coupled-cluster method (restricted to single, double, and noniterative triple excitations) are examined. It has been found that they can be attributed to the higher-order correlation terms in the second-order dispersion and exchange-induction corrections.     

Additive density functional correlation corrections to single particle theories

The correlation energy functional E_c of the Hartree-Fock density is investigated. It was previously established that E_c produces the exact ground-state energy when added to the Hartree-Fock energy. Except when certain degeneracies occur, it is here shown that E_c is bounded from below when the coordinates of the Hartree-Fock density are scaled uniformly by λ, as λ → ∞. Consequently, approximations to E_c should display this bounded property, which is a second-order perturbation energy. It is also shown that a corresponding result applies to that correlation energy functional, Ë_c, which is to be added to a completed exact exchange-only (in the OPM sense) density functional calculation. Scaling requirements are presented for each order in the perturbation expansion for Ë_c. For instance, the second-order term is dimensionless. © 1997 John Wiley & Sons, Inc.     

Possible artifacts occurring in the calculation of intermolecular energies from delocalized pictures

The intermolecular energy between two identical subsystems may be calculated from symmetrydelocalized MO's resulting for instance from a preliminary SCF calculation of the supersystem. Then each second-order energy correction mixes intramolecular correlation,R ^–6 intermolecular dispersion energy, andR ^–3 components. TheR ^–3 components disappear through subtle cancellations. The shifted Epstein-Nesbet energy denominators introduce an artificial second-order intermolecularR ^–1 component, which would be cancelled by off-diagonal third-order terms, as well as a bad asymptotic limit at infinite distances. TheR ^–1 artifact will also occur in strong symmetrical chemical bonds calculated in the Epstein-Nesbet perturbation scheme from delocalized MO's. These defects will occur in all variational approximate CI techniques which neglect off-diagonal elements between delocalized doubly excited determinants. These artifacts are avoided when using the Moller-Plesset definition of the zeroth order Hamiltonian or when starting from (SCF)localized MO's (even in the Epstein-Nesbet perturbation). The discussion is exemplified on an accurateab initio calculation of the Ar₂ molecule.     

Studies using the CASSCF wavefunction

The CASSCF method of Roos, Taylor and Siegbahn has been implemented and extended. In addition to the super-Cl and two-step convergence method, a second-order convergence method is implemented. Further, CASSCF energy gradients are available. The CASSCF method is extended by second-order Epstein-Nesbet perturbation theory. Applications of these methods to the single-triplet splitting of CH₂ and also the determination of the fundamental frequencies of H₂O are reported.     

Some remarks on the perturbation and variation–perturbation calculations of the free energy for quantum systems

A generalization of the Gibbs–Bogoliubov inequality F ? F₀ + 〈H ? H₀〉₀ for the free energy F is studied which leads to a variation principle for this quantity that may be of importance in certain computational applications to quantum systems. This approach is coupled with a study of the perturbation expansion of the free energy for a canonical ensemble with H = H₀ + λV in the general case when H₀ and V do not commute. The second- and high-order derivatives of the free energy with respect to the perturbation parameter λ are calculated. From the second-order term is finally obtained a second-order correction to the previous variational minimum for the free energy.