Nondynamical Correlation Effects Research Articles

Bonding in the ClOO(2A″) and BrOO(2A″) radical: Nonrelativistic single-reference versus relativistic multi-reference descriptions in density functional theory

The 2A″ state of ClOO and BrOO is investigated using single- and multi-reference DFT (density functional theory). Unrestricted DFT (UDFT) carried out with the BLYP functional exaggerates the ionic character of the X–OO bond (X=Cl, Br) and by this its bond dissociation energy, while UDFT with the B3LYP functional underestimates ionic character and bond dissociation energies. In previous investigations, this was overlooked and has led to a misleading interpretation of single determinant UDFT results for XOO(2A″). The correct description of the two radicals can only be achieved by an appropriate configurational state mixing between the doublet ground state (leading to ionic character) and the first excited state, a quartet state leading to covalent character of the X–OO bond. REKS (spin-restricted ensemble-referenced Kohn–Sham) theory provides a multireference description and is capable of describing bonding in XOO correctly provided it is carried out with the hybrid functional B3LYP rather than BLYP, which is the result of the self-interaction error of pure exchange functionals and the inclusion of non-specific non-dynamic electron correlation effects suppressing the specific ones introduced by REKS. The relevance of results for the DFT description of related compounds (HOOO radical, MOO with M =Cu, Ag or Au) or multi-reference problems in general is discussed.

Externally Corrected Coupled-Cluster Approaches: Energy versus Amplitude Corrected CCSD

The externally corrected coupled-cluster methods with singles and doubles (ecCCSD), which exploit some independently available wave function as a source of higher-than-pair clusters, are considered. The focus is on methods that employ a modest-size multireference (MR) configuration interaction (with singles and doubles, CISD) wave function as the external source. Both the amplitude- and energy-corrected CCSD methods are employed, the former correcting the standard single reference (SR) CCSD equations for triples and quadruples, while the latter accounts for the nondynamic correlation effects when evaluating the energy by employing the MR CISD wave function in lieu of the single determinantal (usually Hartree-Fock) reference in the asymmetric energy formula. The performance and relationship of both types of approaches is illustrated by computing the rotational and vibrational energy levels using the potential generated by these various methods and by comparing the calculated spectra with the experimental ones for the simplest first-row hydride, namely the LiH molecule. A special attention is paid to the role of core-correlation effects, in which case we also consider the HF molecule.

Singlet-triplet gaps in diradicals by the spin-flip approach: A benchmark study

The spin-flip approach has been applied to calculate vertical and adiabatic energy separations between low-lying singlet and triplet states in diradicals. The spin-flip model describes both closed- and open-shell singlet and (low-spin) triplet states within a single reference formalism as spin-flipping, e.g., α→β, excitations from a high-spin triplet (Ms=1) reference state. Since both dynamical and nondynamical correlation effects are much smaller for the high-spin triplet states than for the corresponding singlet states, the spin-flip models yield systematically more accurate results than their traditional (non-spin-flip) counterparts. For all the diradicals studied in this work, the spin-flip variant of the coupled-cluster model with double excitations yields energy separations which are within less than 3 kcal/mol of the experimental or the highly accurate multireference values. In most cases the errors are about 1 kcal/mol.

Energy- versus amplitude-corrected coupled-cluster approaches. III. Accurate computation of spectroscopic data exemplified on the HF molecule

The so-called energy-corrected coupled-cluster method with singles and doubles (CCSD), which is closely related to the recently proposed method of moments of Kowalski and Piecuch, accounts for the nondynamic correlation effects, which are missing in the standard single-reference CCSD, by projecting onto the modest-size multireference (MR) configuration-interaction SD (CISD) wave function when evaluating the energy, rather than onto the single-determinantal CCSD reference. The ability of this approach to generate reliable potential energy curves (PECs) or surfaces over a wide range of geometries is examined using a realistic ab initio model of the HF molecule. The PECs obtained with the correlation-consistent polarized valence double (triple, quadruple) zeta cc-pVXZ (T,Q) basis sets are used, together with the PEC extrapolated to the complete basis set limit, to compute the rotational and vibrational levels, which are then compared with the experimentally determined values as well as with the directly measured vibrational frequencies. A comparison is also made with the earlier results [X. Li, J. Mol. Struct.: THEOCHEM 547, 69 (2001)] that were obtained with the amplitude-corrected CCSD method, namely the so-called reduced MR CCSD that exploits the same MR CISD wave function, as well as with recently published renormalized CCSD(T) results [P. Piecuch et al., J. Chem. Phys. 115, 5796 (2001)]. It is shown that both the amplitude- and energy-corrected CCSD PECs may be used to generate highly accurate and reliable spectroscopic data once the basis set limit is achieved.

Some thoughts about the stability and reliability of commonly used exchange?correlation functionals ? coverage of dynamic and nondynamic correlation effects

The self-interaction error (SIE) of commonly used density functional theory (DFT) exchange functionals mimics long-range (nondynamic) pair correlation effects in an unspecified way. Slater exchange suffers from a larger SIE and, therefore, covers more nondynamic correlation effects than Becke exchange, which is the reason why exchange–correlation (XC) functionals based on Slater exchange lead to stabler restricted DFT solutions than those based on Becke exchange. However, the stability of an XC functional does not guarantee higher accuracy. On the contrary, if system-specific nondynamic correlation effects have to be introduced via the form of the wave function, these will be suppressed by nondynamic correlation effects already covered by the exchange functional. Hybrid functionals suffer less from the SIE and, therefore, cover a smaller number of nondynamic electron correlation effects. Accordingly, they are better suited when nondynamic electron correlation has to be introduced by the form of the wave function. It is shown that, for example, broken-symmetry unrestricted DFT calculations are more accurate when carried out with B3LYP than BLYP contrary to claims made in the literature.

Perturbative corrections to the equation-of-motion spin–flip self-consistent field model: Application to bond-breaking and equilibrium properties of diradicals

We present perturbative corrections to a recently introduced spin–flip self-consistent field (SF-SCF) wave function. The spin–flip model describes both closed and open shell singlet states within a single reference formalism as spin–flipping, e.g., α→β, excitations from a triplet (Ms=1) reference state for which both dynamical and nondynamical correlation effects are much smaller than for the corresponding singlet state. The simplest spin–flip model employs a SCF wave function for the reference state, and the resulting equations for target states are therefore identical to configuration interaction singles (in spin–orbital form). While being a qualitatively correct zero-order wave function, SF-SCF should be augmented by dynamical correlation corrections to achieve a quantitative accuracy. The results demonstrate that the second-order approximation to the more theoretically complete spin–flip coupled-cluster model (truncated at double substitutions) represents a systematic improvement over the SF-SCF model.

Application and development of multiconfigurational localized perturbation theory

Generalization of localized perturbation theory, which results with a method able to span the spin space correctly, is presented. This generalization is achieved by using a multiconfigurational (MC) wave function as the reference. This is the most comprehensive expansion used within MC–LMP2 approach to date, with, however, low computational cost [computational scaling with system size (N) of the new method is O(N3)]. Recently, we have reported the successful Jaguar2 (J2) model for calculating atomization energies. Within the MC–LMP2 framework, the J2 model for calculating heats of formation is based on the generalized valence bond–perfect pairing (GVB–PP) wave function. The J2 model was applied only to closed shell cases because of the perfect pairing (PP) restriction in the reference function. In order to describe other systems, the PP restriction needs to be lifted. This work describes efforts in that direction. The PP restriction can be lifted by a restricted configuration interaction (RCI) procedure applied to the GVB–PP wave function. In this paper, the equations describing the application of LMP2 theory to self-consistent RCI wave function are derived and explained. The RCI wave function is a “true” MC expansion as opposed to the GVB–PP, which uses only a single spin eigenfunction (SEF). We also present the self-consistent (SC) optimization of the RCI wave function. The SC–RCI–LMP2 is the first MC–LMP2 method where the spin space is spanned in the reference. This is important for describing the nondynamical correlation (near degeneracy) effects associated, for example, with bond breaking processes. The SC–RCI–LMP2 is an efficient method applicable to large systems; it is shown to reproduce the potential energy surfaces calculated by the complete active space–second order perturbation (CAS–SCF–PT2) method. This is demonstrated, for the first time, on some widely used test cases.

Density functional theory: coverage of dynamic and non-dynamic electron correlation effects

The electron correlation effects covered by density functional theory (DFT) can be assessed qualitatively by comparing DFT densities ρ(r) with suitable reference densities obtained with wavefunction theory (WFT) methods that cover typical electron correlation effects. The analysis of difference densities ρ(DFT)-ρ(WFT) reveals that LDA and GGA exchange (X) functionals mimic non-dynamic correlation effects in an unspecified way. It is shown that these long range correlation effects are caused by the self-interaction error (SIE) of standard X functionals. Self-interaction corrected (SIC) DFT exchange gives, similar to exact exchange, for the bonding region a delocalized exchange hole, and does not cover any correlation effects. Hence, the exchange SIE is responsible for the fact that DFT densities often resemble MP4 or MP2 densities. The correlation functional changes X-only DFT densities in a manner observed when higher order coupling effects between lower order N-electron correlation effects are included. Hybrid functionals lead to changes in the density similar to those caused by SIC-DFT, which simply reflects the fact that hybrid functionals have been developed to cover part of the SIE and its long range correlation effects in a balanced manner. In the case of spin-unrestricted DFT (UDFT), non-dynamic electron correlation effects enter the calculation both via the X functional and via the wavefunction, which may cause a double-counting of correlation effects. The use of UDFT in the form of permuted orbital and broken-symmetry DFT (PO-UDFT, BS-UDFT) can lead to reasonable descriptions of multireference systems provided certain conditions are fulfilled. More reliable, however, is a combination of DFT and WFT methods, which makes the routine description of multireference systems possible. The development of such methods implies a separation of dynamic and non-dynamic correlation effects. Strategies for accomplishing this goal are discussed in general and tested in practice for CAS (complete active space)-DFT.

Electron–electron counterbalance density for molecules: Exchange and correlation effects

Numerical results for the electron–electron counterbalance density in a series of molecules are reported. A compact density-dependent expression for the electron–electron counterbalance density in symmetric systems is used in the discussion of the importance of the exchange and correlation effects in electron-pair distributions. In addition, the behavior of the electron–electron counterbalance density in the dissociation process of the H2 molecule is analyzed in the light of the occurrence of strong nondynamical correlation effects. Finally, the advanced density-dependent expressions allow a direct comparison with the charge concentration index and the electron–electron coalescence density, thus revealing the decisive role of exchange and correlation effects in describing the chemical bonding in molecules.

Size-consistent wave functions for bond-breaking: the equation-of-motion spin-flip model

A new approach to the bond-breaking problem is proposed. Both closed and open shell singlet states are described within a single reference formalism as spin-flipping, e.g., α → β , excitations from a triplet ( M s=1) reference state for which both dynamical and non-dynamical correlation effects are much smaller than for the corresponding singlet state. Formally, the new theory can be viewed as an equation-of-motion (EOM) model where excited states are sought in the basis of determinants conserving the total number of electrons but changing the number of α and β electrons. The results for two simplest members of the proposed hierarchy of approximations are presented.

On the Correlation Energy of π-Electrons in Planar Hydrocarbons

We considered the nondynamical E(ND)π and dynamical E(D)π correlation energies of π electrons in a wide variety of planar hydrocarbons. The former could be conveniently calculated within the CASSCF formalism by using modest basis sets. The dynamical part of the correlation energy was studied with the CASPT2 method. It appeared that E(D)π was sensitive to the basis set. It is also found that the ab initio E(ND)π and E(D)π values follow very simple additivity rules, which allow fairly good estimates of the nondynamical and dynamical correlation effects of π electrons simply by counting the carbon and hydrogen atoms. Small deviations from the additivity of E(D)π are found in benzene (2.1 kcal/mol), naphthalene (3.3 kcal/mol), and cyclobutadiene (−3.0 kcal/mol), indicating that some care has to be exercised in applying the additivity rules to (anti)aromatic molecules. Nondynamical correlation E(ND)π exhibits even more pronounced deviations from the additivity in the systems characterized by a π-electron delocalization larger than that in linear polyenes. A novel electrostatics + correlation interpretation of (anti)aromaticity is introduced which sheds new light on an old but central problem of chemistry. It is also suggested that endo- and exoaromaticity should be distinguished. An interesting result of the present calculations is that the Hartree−Fock electron−electron (Vee) interactions and the nondynamical correlation are much more favorable in cyclobutadiene's (CBD's) transition structure (TS) than in its ground state (GS). It appears, however, that the overwhelming effect in the CBD(TS) is an increase in the nuclear repulsion (Vnn), which is higher by 86.8 kcal/mol than in the GS. Consequently, the propensity of CBD to assume a rectangular geometry in the GS occurs inter alia because of a dramatic relief in the nuclear repulsion. The opposite is the case in the GS of benzene, where the dominating Vne in the regular hexagon prevails over an increase in Vee and Vnn repulsions caused by the D6h formation. Intriguing and counterintutitive results are obtained by comparing the E(ND)π of the CBD(GS) and benzene with those of corresponding linear polyenes. The E(ND)π of the CBD(GS) is higher by 8 kcal/mol than that of the 1,3-butadiene, whereas the E(ND)π of benzene is lower by 5.7 than that of hexatriene (in kcal/mol). The (anti)aromatic (de)stabilization of CBD and benzene relative to 1,3-butadiene is 40.7 and 28.4 kcal/mol, respectively. The Vne attraction in both compounds is appreciably higher (i.e., less favorable) than that in the reference molecule, 1,3-butadiene. However, this is overcompensated in benzene by more advantageous Vee and Vnn terms, but it is not the case for CBD. This difference makes benzene exoaromatic and CBD exoantiaromatic.

New approach to the state-specific multireference coupled-cluster formalism

A new development is presented in the framework of the state-specific multireference (MR) coupled-cluster (CC) theory (MRCC). The method is based on the CASSCF (complete active space self-consistent field) wave function and it is designed specifically for calculating excited electronic states. In the proposed approach, the cluster structure of the CC wave operator and the method to determine this operator are the key features. Since the general formulation of the CASCC method is uncontracted, i.e., allows the interaction between the nondynamic and dynamic correlation effects to affect both the CAS reference function and the CC correlation wave operator, the method is expected to perform better than contracted perturbative approaches such as the CASPT2 (second-order perturbation theory based on the CAS wave function) method. Also, the CASCC method is not a perturbative approach and is not based on selection of an unperturbed Hamiltonian, which in the case of the CASPT2 method often leads to the “intruder state” problem. CASCC calculations of the lowest totally symmetric excited state of the H8 model system using the internally contracted and uncontracted approaches reveal some interesting features of the methodology.

Relativistic MCSCF by means of quasidegenerate direct perturbation theory. II. Preliminary applications

The relativistic multiconfiguration self-consistent field (MCSCF) theory by means of (quasidegenerate) direct perturbation theory (DPT) is formulated and implemented through the leading order. Systematic investigations of nondynamical correlation and relativistic effects in berylliumlike systems are carried out. For this sequence, it appears that the single-state first-order DPT is valid only for the nuclear charge up to 30 or so. Calculations for the spectroscopic constants of the molecules of homonuclear halogen dimers X2 (X=F, Cl, Br, I, At) and TlX (X=F, Cl, Br, I) are performed at all-electron SCF (self-consistent field) and complete active space SCF levels, with the aim to demonstrate the efficiency of DPT by comparing with other relativistic calculations. Spin–orbit coupling affects the energies of the open-shell atoms and is taken into account for the binding energies.

A combination of Kohn–Sham density functional theory and multi-reference configuration interaction methods

An effective Hamiltonian in a basis of spin- and space-symmetry adapted configuration state functions (CSF), which includes information from Kohn–Sham density functional theory (DFT), is used to calculate configuration interaction (CI) wave functions for the electronic states of molecules. The method emphasizes on states of multiconfigurational character which cannot be represented by conventional DFT. The CI matrix elements are constructed empirically by using the exact operator and corrections from DFT. Both the optimized KS orbitals from the parent determinant and the corresponding KS potential from the parent state density are used. Depending on their energy gap the CI off-diagonal elements between CSF are exponentially scaled to zero to avoid double counting of electron correlation. The selection of the most important CSF describing nondynamical correlation effects and the use of an approximate resolution of the identity (RI) for the evaluation of the two-electron integrals allows a very efficient DFT/MRCI treatment of molecules with several hundreds of electrons. As applications, the prediction of excitation energies for singlet and triplet states of organic molecules and transition metal complexes, the calculation of electronic circular dichroism spectra and investigations of the energetics of diradicals are presented. It is found, that the new DFT/MRCI approach gives results of high accuracy (rms errors for relative energies <0.2 eV) comparable to those from sophisticated ab initio treatments.

Coupled-cluster methods with internal and semi-internal triply and quadruply excited clusters: CCSDt and CCSDtq approaches

Extension of the closed-shell coupled-cluster (CC) theory to studies of bond breaking and general quasidegenerate situations requires the inclusion of the connected triply and quadruply excited clusters, T3 and T4, respectively. Since the complete inclusion of these clusters is expensive, we explore the possibility of incorporating dominant T3 and T4 contributions by limiting them to active orbitals. We restrict T3 and T4 clusters to internal or internal and semi-internal components using arguments originating from the multireference formalism. A hierarchy of approximations to standard CCSDT (CC singles, doubles, and triples) and CCSDTQ (CC singles, doubles, triples, and quadruples) schemes, designated as the CCSDt and CCSDtq approaches, is proposed and tested using the H2O and HF molecules at displaced nuclear geometries and C2 at the equilibrium geometry. It is demonstrated that the CCSDt and CCSDtq methods provide an excellent description of bond breaking and nondynamic correlation effects. Unlike perturbative CCSDT and CCSDTQ approaches, the CCSDt and CCSDtq approaches do not fail at large internuclear separations, in spite of using the restricted Hartree–Fock reference. All CCSDt and CCSDtq approaches are essentially n6 procedures and yet they are shown to provide reliable information about T3 and T4 components, whose standard evaluation requires expensive n8 and n10 steps.

Simultaneous handling of dynamical and nondynamical correlation via reduced multireference coupled cluster method: Geometry and harmonic force field of ozone

The geometry and vibrational frequencies of the ozone molecule are studied using the reduced multireference coupled cluster method with singles and doubles (RMR CCSD) relying on a two-electron/two-orbital (2,2) active space. The role played by the choice of the reference space is analyzed in detail by considering both the spin adapted, two-configuration reference space and the complete three-configuration reference space based on the (2,2) active space. It is shown that while the two-reference RMR CCSD approach, using the C2v symmetry-adapted (2,2) active space, provides a better description of the geometry and harmonic frequencies of symmetric vibrational modes than the standard single-reference (SR) CCSD method, the three-reference approach is essential to achieve an accurate description of the harmonic frequency of the asymmetric stretching mode. A unique behavior of ozone in this regard is highlighted by a comparison with other symmetric triatomics. The comparison with results obtained by other methods, including the SR configuration interaction with singles and doubles (CISD), the standard SR CCSD, as well as two-reference and large reference space MR CISD methods, clearly demonstrates the superiority of the RMR CCSD approach. In fact, the performance of a small reference space RMR CCSD method is similar to that of a large reference space MR CISD, thus enabling a compact description of both dynamical and nondynamical correlation effects in states having a multireference character.

Brillouin–Wigner based multi-reference perturbation theory for electronic correlation effects

We investigate basis set reduction (BSR), a configuration-based multi-reference perturbation theory using an implicit effective Hamiltonian motivated by Brillouin-Wigner perturbation theory. This approach avoids the intruder-state and level-crossing problems by construction and yields a rapidly converging perturbation expansion. Formulated to systematically approximate multi-reference configuration interaction, BSR yields accurate results in second order, because it includes relaxation effects of the primary space wavefunction in the presence of the perturbation. We benchmark the method for molecules (CH2,O3) in which both dynamical and non-dynamical correlation effects are known to be important, obtaining accuracies of the order of 1 kcal/mol across the potential energy surface in second-order perturbation theory. We address the critical issues of perturbative orbital optimization for the primary orbital space, the choice of the secondary orbital space and the effects of single excitations.

Singlet-Triplet Splitting in Methylene: An Accurate Description of Dynamic and Nondynamic Correlation by Reduced Multireference Coupled Cluster Method

A classical multireference problem - the singlet-triplet separation in methylene - is examined by the recently introduced reduced multireference (RMR) singles and doubles coupled cluster (CCSD) method, using both double zeta plus polarization (DZP) and large atomic natural orbital (ANO) basis sets. In the former case, the performance of the RMR CCSD as well as of other approaches is assessed by a comparison with the full configuration interaction (FCI) result that represents the exact solution for this basis, while in the latter case a comparison is made with the experiment. It is shown that using a minimal two-configuration reference space, the RMR CCSD result compares well with either FCI or experiment; and is of the same quality as that provided by the two-reference state universal MR CCSD theory. Both MR CCSD approaches give a balanced description for the singlet and triplet states involved and correct the shortcomings of the single reference CCSD approach that is lacking in the presence of nondynamical correlation effects.

CASSCF, MP2, and CASMP2 studies on addition reaction of singlet molecular oxygen to ethylene molecule

The reaction mechanism and relative stabilities of the intermediates and transition states in the reaction of 1O2 plus ethylene molecule using ab initio molecular orbital (MO) theories at several levels of theory with the correction of the nondynamic and dynamic electron correlation effects were systematically investigated. Full geometry optimizations of the corresponding biradical (BR) intermediates, perepoxide (PE) and 1,2-dioxetane (DO) were performed by complete-active-space CASSCF{2,2}/6-31G* method with nondynamic electron correlation effect and Moller-Plesset MP2/6-31G* method with dynamic electron correlation effect. For the 1,4-biradical intermediates, new gauche-type 1BR state was found at both CASSCF and MP2 levels, corresponding to the transition state of the rotation motion of the O2 moiety. It was found from the intrinsic-reaction-coordinate (IRC) study that another gauche-type 1BR transition state connects smoothly to the reactant system 1O2+H2C(DOUBLE BOND)CH2 and the gauche minimum 1BR state, showing that the reaction through the 1,4-biradical intermediates initially proceeds through the gauche transition state to form the gauche minimum 1BR state, following that the free rotation of O2 moiety occurs due to the energy barrier less than 4.0 kcal/mol. The stability of the perepoxide is surprisingly sensitive to the levels of the theory and the basis sets employed. The coupled-cluster methods, CCSD and CCSD(T), gave the reasonable stabilities of 1,4-biradical intermediates, perepoxide, and dioxetane as a reaction product. From the results of the CCSD and CCSD(T) methods, the reaction of 1O2+H2C(DOUBLE BOND)CH2 proceeds by a two-step mechanism through the 1,4-biradical intermediates rather than through the perepoxide. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65: 787–801, 1997

Electron correlation effects on the d-d excitations in NiO

The partly filled 3d shell in solid transition metal compounds is quite localized on the transition metal ion and gives rise to large electron correlation effects. With the recently developed CASSCF/CASPT2 approach electron correlation effects can be accounted for efficiently. The CASSCF step accounts for the non-dynamical correlation and part of the dynamical correlation, the following CASPT2 step takes largely care of the remaining dynamical correlation in a perturbative way. This approach is applied to the d-d excitations in NiO for which both non-dynamical and dynamical electron correlation effects have substantial influence on the energy differences. Excitation energies that compare well to the experimental data are obtained and the importance of the different electron correlation effects can be assessed.