Nondynamic Electron Correlation 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.

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.

Influence of the self-interaction error on the structure of the DFT exchange hole

Approximate density functional theory (DFT) covers long-range non-dynamic electron correlation via the exchange functional while the correlation functional includes just the short-range dynamic electron correlation effects. We show that the self-interaction error of approximate exchange functionals (local density approximation, LDA and others) mimics the long-range correlation effects. For this purpose the exchange hole is investigated at the Hartree–Fock, the LDA, and the self-interaction corrected (SIC)-LDA levels of theory.

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.

Extracting polarized atomic orbitals from molecular orbital calculations

We present a class of methods for extracting a polarized atomic orbital (EPAO) minimal basis set from a converged molecular orbital (MO) calculation. Unlike minimal basis sets obtained from previous approaches, EPAOs rigorously contain the occupied molecular orbital space. EPAOs achieve this exactness because their spatial extent is not restricted. Nonetheless, EPAOs are optimally localized with respect to a localization criterion and are essentially single-centered. EPAOs provide an alternative scheme for partitioning the electron density into atomic subspaces. Therefore, they can be used to determine atomic and chemical group properties such as charge populations. Since EPAOs provide a compact description to the occupied space, they may have other computational applications such as in local correlation methods. Additionally, the EPAOs give a description of valence antibonding orbitals that may be appropriate for nondynamical electron correlation. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 169–184, 2000

Density functional theory with fractionally occupied frontier orbitals and the instabilities of the Kohn–Sham solutions for defining diradical transition states: Ring-opening reactions

Density functional theory with fractionally occupied frontier orbitals was combined with an analysis of the instability of the Kohn–Sham solutions and applied to the study of the ring-opening reactions of 1,2-dioxetene, 1,2-dithiete, 1,2-diselenete, and 1,2-ditellurete. These molecules possess transition structures and reactive intermediates for the ring-opening reactions which exhibit strong nondynamical electron correlation. All restricted density functional theory (DFT) solutions for the closed shell transition state structures for all these species are unstable. The solutions with the hybrid Hartree–Fock DFT functionals, B3LYP and B3PW91, are triplet unstable, while for the pure DFT functional BLYP the instability is due to a violation of the Aufbau principle. The same types of instabilities were found for the 1,2-diselenete and 1,2-ditellurete intermediates. Lower energy stable solutions for the diradical transition structures were found with unrestricted DFT methods allowing fractionally occupied orbitals. This DFT approach indicates a decrease in the active orbital space from four fractionally occupied natural orbitals in earlier multireference predictions to two fractionally occupied Kohn–Sham orbitals.

Accurate calculations on excited states: new theories applied to the –OX, –XO, and –XO 2 (X=Cl and Br) chromophores and implications for stratospheric bromine chemistry

Electronic excitation energies are determined using single-reference based theories derived from response equations involving perturbation theory and coupled-cluster theory. These methods are applied to the singlet manifold of excited electronic states of the HClO, HBrO, HOClO, HOBrO, HClO 2, and HBrO 2 molecules. Vertical excitation energies (VEEs) for the lowest symmetry unique triplet states are also computed using coupled-cluster theory. Both the singlet and triplet coupled-cluster VEEs are found to be in excellent agreement with experiment for HOBr and HOCl. The reliability of the various perturbation theory approaches is assessed by comparison to the linear response coupled-cluster method, and the CIS(D3) approach is found to perform the best, on average. In order to obtain reliable VEEs for these molecules, CIS(D3) is the recommended approach, unless excited state non-dynamical electron correlation is significant, in which case the recently proposed CIS(D 0) technique may perform better, on average. For both the HClO 2 and HBrO 2 molecules, it is found that the lowest singlet excited electronic state occurs at too high an energy for it to be accessible from solar radiation in the lower stratosphere. The stratospheric chemistry implications of this, especially for bromine chemistry are discussed. It is suggested that YBrO 2 molecules may in fact be the only stable stratospheric bromine reservoir species under light conditions.

Energies and analytic gradients for a coupled-cluster doubles model using variational Brueckner orbitals: Application to symmetry breaking in O4+

We describe an alternative procedure for obtaining approximate Brueckner orbitals in ab initio electronic structure theory. Whereas approximate Brueckner orbitals have traditionally been obtained by mixing the orbitals until the coefficients of singly substituted determinants in the many-electron wave function become zero, we remove singly substituted determinants at the outset and obtain orbitals which minimize the total electronic energy. Such orbitals may be described as variational Brueckner orbitals. These two procedures yield the same set of exact Brueckner orbitals in the full configuration interaction limit but differ for truncated wave functions. We consider the simplest variant of this approach in the context of coupled-cluster theory, optimizing orbitals for the coupled-cluster doubles (CCD) model. An efficient new method is presented for solving the coupled equations defining the energy, doubles amplitudes, and orbital mixing parameters. Results for several small molecules indicate nearly identical performance between the traditional Brueckner CCD method and the variational Brueckner orbital CCD approach. However, variational Brueckner orbitals offer certain advantages: they simplify analytic gradients by removing the need to solve the coupled-perturbed Brueckner coupled-cluster equations for the orbital response, and their straightforward extensions for inactive orbitals suggests possible uses in size-extensive models of nondynamical electron correlation. Application to O4+ demonstrates the utility of variational Brueckner orbitals in symmetry breaking cases.

The effects of electron correlation on the degree of bond alternation and electronic structure of oligomers of polyacetylene

Full geometry optimizations on oligoenes have been performed with Hartree–Fock and density functional theory in combination with double zeta and triple zeta quality basis sets with primary focus on the degree of bond length alternation and on the energy gap. Monitoring the dependence of the computed properties on the oligomer size provides new insights into the reliability of the calculations, which are analyzed in terms of dynamical and nondynamical electron correlation. Our theoretical bond length alternation values for the oligomers of polyacetylene extrapolate to significantly smaller values than what has been established by experiments and earlier theoretical predictions. The exact exchange mixing to the exchange-correlation functional not only improves the agreement of the theoretical gap of oligoenes with experimental excitation energies but also increases the computed bond length alternations. Based on a newly proposed one parameter functional of Becke, the effect of the exact exchange mixing has been further identified, and a practical measure of nondynamical correlation energy has been suggested. The popular Becke–Lee–Yang–Parr density functional produces bond alternation values that are unacceptably small.

Electron correlation on torsional and inversion barrier heights of acetaldehyde and acetone in the S 1(n,π ∗) state: CASSCF and CAS-MP2 study

The torsional and inversion barrier heights of acetaldehyde and acetone in the S 1 state were studied using CASSCF and CAS-MP2 wavefunctions, where the geometries of the conformers were optimized at the CASSCF level and the dynamical electron correlation energies were evaluated by the multiconfigurational second-order perturbation theory (CAS-MP2). The CASSCF wavefunction largely overestimates the inversion barrier height, while it slightly underestimates the torsional barrier height. The second-order correction to the torsional barrier height is positive, but that to the inversion barrier height is negative. The results of this study suggest that the nondynamical electron correlation, especially the correlation of π and 3p x (O) electrons for the torsional barrier and the dynamical electron correlation for the inversion barrier, is important. Calculated torsional and inversion barrier heights agreed reasonably with experimental results.

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

Ab Initio Characterization of the Isomerism between the μ-η2:η2-Peroxo- and Bis(μ-oxo)dicopper Cores

The interconversion of model compounds {[(NH3)3Cu]2(μ-η2:η2-O2)}2+ (1) and {[(NH3)3Cu]2(μ-O)2}2+ (2) has been examined using multireference second-order perturbation theory with an 8-electron/8-orbital active space. At this level of theory, 1 and 2 are separated by only 0.3 kcal/mol, and the barrier to isomerization is predicted to be very low based on single-point energy calculations for intermediate structures. The flat nature of the potential energy surface along the interconversion coordinate derives from a balancing of Coulomb forces and nondynamic electron correlation. The latter effect depends critically on the significant energy change experienced by the 13au virtual orbital on passing from one isomer to the other. In addition, solvation electrostatics favor 2 over 1.

Geometry and electronic structure of model small nitrones and their tautomers

Ab initio MO studies have been performed on the parent nitrone [CH 2N(H)O], its tautomers and methyl derivatives. The effect of non-dynamical electron correlation has been taken into account using the CASSCF wavefunction. Different kinds of excitation patterns have been applied in the (2,2) to (12,12) range in order to appreciate the effect of proper orbital selection. Not only the π (a″) orbitals were found to be essential for a reliable description of nitrone molecular properties but also certain σ (a′) ones have also been shown to play an important role. Calculating the Mayer's bond indices and performing localization studies revealed four fully formed covalent bond and a fifth one which could be regarded either as a covalent bond polarised into the N → O bond direction or as a partially back-donated lone pair into the O → N direction. Comparing the calculated geometrical parameters of the parent nitrone to those of its oxime and nitroso tautomers (using several methods from HF/6–31G to CCSD(T)/TZP and MR-CISD/6–31IG ∗ levels of theory) also points to this direction. Since from CASSCF studies and natural orbital analyses the biradical contribution was found to be small, the electronic structure of the nitrone can be regarded as intermediate between the hypervalent and zwitterionic forms.

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.

Exploring the Boundary between Aromatic and Olefinic Character: Bad News for Second-Order Perturbation Theory and Density Functional Schemes

The question whether [10]annulene prefers olefinic structures with alternate single and double bonds or aromatic structures like all other small to medium sized uncharged (4n + 2){pi} electron homologs (e.g. benzene, [14]annulene) has been controversial for more than 20 years. Our new results suggest that only the high-order correlated methods will be able to correctly predict the [10]annulene potential energy surface. The UNO-CAS results and the strong oscillation of the MP series show that nondynamical electron correlation is important. Consequently, reliable results can only be expected at the highest correlated levels like CCSD(T) method, which predicts the olefinic twist structure to be lower in energy by 3-7 kcal/mol. This prediction that the twist structure is lower in energy is supported by (a) the MP2-R12 method, which shows that large basis sets favor the olefinic structure relative to the aromatic, and (b) the fact that both structures are about equally affected by nondynamical electron correlation. We conclude that [10]annulene is a system which cannot be described adequately by either second-order Moller-Plesset perturbation theory or density functional methods. 13 refs., 3 tabs.

A rapid generalized valence‐bond algorithm for semiempirical NDDO calculations

AbstractA recently developed algorithm based on the coupled cluster ansatz for the generalized valence‐bond perfect‐pairing model is adapted for the NDDO class of semiempirical methods. It allows for the calculation of the nondynamic electron correlation and executes as rapidly as the semiempirical molecular orbital method. To assess the interplay of the nondynamic and dynamic electron correlation vs. parameterization, preliminary results are presented for rotation about a double bond, dissociations about single bonds, and the relative stabilities of biradical isomers. © 1995 John Wiley & Sons, Inc.

Investigation of a diagnostic for perturbation theory. Comparison to the [formula omitted]1 diagnostic of coupled-cluster theory

A diagnostic for perturbation theory calculations, L 2, is defined and numerical results compared to the established L 1 diagnostic from coupled-cluster theory. L 2 is the lowest-order non-zero contribution to a Møller-Plesset perturbation expansion of T 1. L 2 is a reasonable estimate of the importance of non-dynamical electron correlation, although not as reliable as T 1. L 2 values ⩽0.012 suggest that low orders of perturbation theory should yield reasonable results; L 2 values between 0.012 and 0.015 suggest that caution is required in interpreting results from low orders of perturbation theory; L 2 values ⩾0.015 indicate that low orders of perturbation theory are not reliable for accurate results. Although not required mathematically, L 2 is always less than T 1 for the examples studied here.

Ab initio MO and approximate density functional theory studies on the lowest singlet and triplet states of s and as‐indacene

AbstractThe lowest singlet and triplet states of s‐ and as‐indacene have been studied by means of high‐level ab initio MO and approximate density functional methods. Among the geometrical and energetical details discussed are the equilibrium structure of s‐indacene (C2h or D2h), the structures and energies of the low‐lying s‐indacene triplet states, and the stability and geometry of the singlet and triplet states of as‐indacene. It is shown that single‐determinant‐based methods, such as Hartree‐Fock or MP2, are not suited to properly describe these molecules. Instead, methods are required which explicitly take into account nondynamical and dynamical electron correlation. The results obtained by density functional theory‐based methods compare very well with the most elaborate ab initio MO data and seem to provide an economical alternative even for molecules with a complicated electronic structure such as s‐ and as‐indacene. © 1995 John Wiley & Sons, Inc.

Vibrational analysis of the squarate ion based on ab initio molecular orbital calculations. A practical method to calculate vibrational force fields of non-bond-alternating conjugated molecules

Vibrational analysis of the squarate ion is performed on the basis of ab initio molecular orbital calculations. The vibrational force field is calculated with the Hartree-Fock (HF), the second- and third-order Møller-Plesset perturbation (MP2 and MP3), and the complete active space self-consistent field (CASSCF) methods, in order to study the effects of electron correlation. The 6G–31G ∗, 6–31 + G ∗, and 6–31 + G (df) basis sets are used for the purpose of examining the basis-set dependence of the results of calculations. It is found that the calculations at the MP3/6G–31G ∗ and CASSCF/ 6G–31G ∗ levels provide reasonable vibrational force fields for the squarate ion. The 6–31 + G ∗ and 6–31 + G(df) basis sets are not suitable for calculating the out-of-plane force field. The effects of nondynamic electron correlation are large on the vibrational frequencies of the e u CO stretching and b 1 g CC stretching modes. The origin of the large effects of electron correlation on these vibrational frequencies is studied by examining the calculated force constants in detail. It is found that a reasonable vibrational force field can also be obtained by taking an average of those at the HF/6G–31G ∗ and MP2/6G–31G ∗ levels. This method is also applied to the in-plane vibrational force field of benzene. The relation between the definition of molecular symmetry coordinates and the calculated force constants is discussed, in terms of the atomic motions corresponding to the unit displacements along the defined coordinates and the origin of the potential energy gained by those vibrational motions.

Analytic energy derivatives for ionized states described by the equation-of-motion coupled cluster method

The theory for analytic energy derivatives of excited electronic states described by the equation-of-motion coupled cluster (EOM-CC) method has been generalized to treat cases in which reference and final states differ in the number of electrons. While this work specializes to the sector of Fock space that corresponds to ionization of the reference, the approach can be trivially modified for electron attached final states. Unlike traditional coupled cluster methods that are based on single determinant reference functions, several electronic configurations are treated in a balanced way by EOM-CC. Therefore, this quantum chemical approach is appropriate for problems that involve important nondynamic electron correlation effects. Furthermore, a fully spin adapted treatment of doublet electronic states is guaranteed when a spin restricted closed shell reference state is used—a desirable feature that is not easily achieved in standard coupled cluster approaches. The efficient implementation of analytic gradients reported here allows this variant of EOM-CC theory to be routinely applied to multidimensional potential energy surfaces for the first time. Use of the method is illustrated by an investigation of the formyloxyl radical (HCOO), which suffers from notorious symmetry breaking effects.