Large Spin Contamination Research Articles

Nature of partial sigma bond.

This study investigates the formation of partial sigma (σ) covalent bonds in experimentally synthesizable biradicals formed from hydrogenated and fluorinated C8, C20, and C60 cage structures, by assessing their stability, geometry, and bonding character in singlet and triplet states using restricted B3LYP-D3/6-31+G(d,p) theory, natural bond orbital (NBO) analysis, and complete active space self-consistent field (CASSCF) method. The results show that these partial σCC bonds have Wiberg bond orders of 0.38 to 0.48 and bond lengths ranging from 2.62 Å to 5.93 Å. Cage size influences the characteristics of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), with electrons favoring more antibonding orbitals in smaller cages where electrons reside more on the exterior of the cage and favoring bonding orbitals in larger ones where electrons are more in the interior. Fluorination enhances electron density on bonding orbitals. The analysis further clarified that the differentiation between antibonding and bonding features of HOMOs and LUMOs extends beyond merely electron transfer from s- to p-atomic orbitals, also noting possible interactions of the same symmetry repel. The study also introduces hyperconjugation from α-position CH bonds as a factor in stabilizing partial σ-bond formation. The results also caution against the use of broken symmetry methodology in unrestricted SCF wavefunctions for biradicals, such as those in this study as it may cause large spin contamination and thus errors in the calculated electronic properties results.

When do tripdoublet states fluoresce? A theoretical study of copper(II) porphyrin.

Open-shell molecules rarely fluoresce, due to their typically faster non-radiative relaxation rates compared to closed-shell ones. Even rarer is the fluorescence from states that have two more unpaired electrons than the open-shell ground state, since they involve excitations from closed-shell orbitals to vacant-shell orbitals, which are typically higher in energy compared to excitations from or out of open-shell orbitals. States that are dominated by the former type of excitations are known as tripdoublet states when they can be described as a triplet excitation antiferromagnetically coupled to a doublet state, and their description by unrestricted single-reference methods (e.g., U-TDDFT) is notoriously inaccurate due to large spin contamination. In this work, we applied our spin-adapted TDDFT method, X-TDDFT, and the efficient and accurate static-dynamic-static second order perturbation theory (SDSPT2), to the study of the excited states as well as their relaxation pathways of copper(II) porphyrin; previous experimental works suggested that the photoluminescence of some substituted copper(II) porphyrins originate from a tripdoublet state, formed by a triplet ligand π → π* excitation antiferromagnetically coupled with the unpaired d electron. Our results demonstrated favorable agreement between the X-TDDFT, SDSPT2 and experimental excitation energies, and revealed noticeable improvements of X-TDDFT compared to U-TDDFT, not only for vertical excitation energies but also for adiabatic energy differences. These suggest that X-TDDFT is a reliable tool for the study of tripdoublet state fluorescence. Intriguingly, we showed that the aforementioned tripdoublet state is only slightly above the lowest doublet excited state and lies only slightly higher than the lowest quartet state, which suggests that the tripdoublet of copper(II) porphyrin is long-lived enough to fluoresce due to a lack of efficient non-radiative relaxation pathways; an explanation for this unusual state ordering is given. Indeed, thermal vibration correlation function (TVCF)-based calculations of internal conversion, intersystem crossing, and radiative transition rates confirm that copper(II) porphyrin emits thermally activated delayed fluorescence (TADF) and a small amount of phosphorescence at low temperature (83K), in accordance with experiment. The present contribution is concluded by a few possible approaches of designing new molecules that fluoresce from tripdoublet states.

MkMRCC, APUCC and APUBD approaches to 1,n-didehydropolyene diradicals: the nature of through-bond exchange interactions

Mukherjee-type (Mk) state specific (SS) multi-reference (MR) coupled-cluster (CC) calculations of 1,n-didehydropolyene diradicals were carried out to elucidate singlet-triplet energy gaps via through-bond coupling between terminal radicals. Spin-unrestricted Hartree–Fock (UHF) based coupled-cluster (CC) computations of these diradicals were also performed. Comparison between symmetry-adapted MkMRCC and broken-symmetry (BS) UHF-CC computational results indicated that spin-contamination error of UHF-CC solutions was left at the SD level, although it had been thought that this error was negligible for the CC scheme in general. In order to eliminate the spin contamination error, approximate spin-projection (AP) scheme was applied for UCC, and the AP procedure indeed eliminated the error to yield good agreement with MRCC in energy. The CCD with spin-unrestricted Brueckner's orbital (UB) was also employed for these polyene diradicals, showing that large spin-contamination errors at UHF solutions are dramatically improved, and therefore AP scheme for UBD removed easily the rest of spin-contaminations. Pure- and hybrid-density functional theory (DFT) calculations of the species were also performed. Three different computational schemes for total spin angular momentums were examined for the AP correction of the hybrid DFT. The AP DFT calculations yielded the singlet-triplet energy gaps that were in good agreement with those of MRCC, AP UHF-CC and AP UB-CC. Chemical indices such as the diradical character were calculated with all these methods. Implications of the present computational results are discussed in relation to previous RMRCC calculations of diradical species and BS calculations of large exchange coupled systems.

A benchmark study of the vertical electronic spectra of the linear chain radicals C2H and C4H

The ability of coupled-cluster models to predict vertical excitation energies is tested on the electronic states of carbon-chain radicals of particular relevance to interstellar chemistry. Using spin-unrestricted and -restricted reference wave functions, the coupled-cluster singles and doubles (CCSD) model and a triples-including model (CC3) are tested on the sigma radicals C(2)H and C(4)H. Both molecules exhibit low-lying excited states with significant double-excitation character (as well as states of quartet multiplicity) and are thus challenging cases for excited-state approaches. In addition, we employ two diagnostics for the reliability of the CC results: the approximate excitation level (AEL) relative to the ground state and the difference between excitation energies obtained with spin-unrestricted and spin-restricted reference wave functions (the U-R difference). We find that CCSD yields poor excitation energies for states with AEL significantly larger than ca. 1.1 and/or large U-R differences, as well as for certain states exhibiting large spin contamination or other inadequacies in the reference determinant. In such cases, connected triple excitations can be included in the model and generally provide improved results. Furthermore, we find that large discrepancies exist between CC and multireference (MR) results for certain states. These disagreements are not related to basis-set effects, but likely arise from the lack of spin adaptation in conventional spin-orbital CC implementations and active space selection in the MR models.

Reactions of F+(3P) and F+(1D) with Silicon Oxide. Possibility of Spin-Forbidden Processes

High level ab initio and density functional theory calculations have been carried out to study the potential energy surfaces associated with the reactions of F(+) in its (3)P ground state and in its (1)D first excited state with silicon dioxide. The structures and vibrational frequencies of the stationary points of both potential energy surfaces were obtained at the B3LYP/6-31G(d) level. Final energies were calculated at the B3LYP/6-311+G(3df,2p) and at the G3X levels of theory. [Si, O(2), F](+) singlet and triplet state cations present very different bonding characteristics. The most favorable reactions path in F(+)((3)P) + SiO(2) reactions should yield O(2) + SiF(+), while in the reactions in the first excited state, only a charge exchange process, yielding F((2)P) + SiO(2)(+)((2)A), should be observed. However, both potential energy surfaces cross each other, because although the entrance F(+)((3)P) + SiO(2) lies 34.5 kcal/mol below F(+)((1)D) + SiO(2), the global minimum of the singlet PES lies 10.3 kcal/mol below the global minimum of the triplet. The minimum energy crossing point between them is close to the global minimum, and the spin-orbit coupling is not zero, suggesting that very likely some of the products will be formed in the singlet hypersurface. The existence of instabilities and large spin-contamination in the description of some of the systems render the DFT calculations unreliable.

Thermochemistry of disputed soot formation intermediates C4H3 and C4H5

Accurate isomeric energy differences and standard enthalpies of formation for disputed intermediates in soot formation, C(4)H(3) and C(4)H(5), have been determined through systematic extrapolations of ab initio energies. Electron correlation has been included through second-order Z-averaged perturbation theory (ZAPT2), and spin-restricted, open-shell coupled-cluster methods through triple excitations [ROCCSD, ROCCSD(T), and ROCCSDT] utilizing the correlation-consistent hierarchy of basis sets, cc-pVXZ (X = D, T, Q, 5, and 6), followed by extrapolations to the complete basis set limit via the focal point method of Allen and co-workers. Reference geometries were fully optimized at the ROCCSD(T) level with a TZ(2d1f,2p1d) basis set. Our analysis finds that the resonance-stabilized i-C(4)H(3) and i-C(4)H(5) isomers lie 11.8 and 10.7 kcal mol(-1) below E-n-C(4)H(3) and E-n-C(4)H(5), respectively, several kcal mol(-1) (more, less) than reported in recent (diffusion Monte Carlo, B3LYP density-functional) studies. Moreover, in these systems Gaussian-3 (G3) theory suffers from large spin contamination in electronic wave functions, poor reference geometries, and anomalous vibrational frequencies, but fortuitous cancellation of these sizable errors leads to isomerization energies apparently accurate to 1 kcal mol(-1). Using focal-point extrapolations for isodesmic reactions, we determine the enthalpies of formation (delta(f)H(0) (composite function)) for i-C(4)H(3), Z-n-C(4)H(3), E-n-C(4)H(3), i-C(4)H(5), Z-n-C(4)H(5), and E-n-C(4)H(5) to be 119.0, 130.8, 130.8, 78.4, 89.7, and 89.1 kcal mol(-1), respectively. These definitive values remove any remaining uncertainty surrounding the thermochemistry of these isomers in combustion models, allowing for better assessment of whether even-carbon pathways contribute to soot formation.

CCSDT study of the fluoroperoxyl radical, FOO

The FOO radical was studied at the coupled-cluster (CC) theoretical level, employing both the perturbative calculation of the contribution of triple excitations, CCSD(T), and the exact evaluation, CCSDT. Two solutions were found, one of them with a large spin contamination. It is shown that spin contamination is a problem at the CCSD(T) level, while it is largely inmaterial at the complete CCSDT level. The full CCSDT level affords an FO distance of 1.632 ± 0.005 Å in reasonable agreement to experiment. The enthalpy of formation at the uncontaminated CCSDT level, Δ f H 0 298(FOO)=6.5±1 kcal/mol, is also in very good agreement with the experimental value of 6.24 ± 0.5 kcal/mol. Contrary to previous studies, CCSDT performs better than CCSD(T) in the calculation of the properties of this radical.

Validation study of meta-GGA functionals and of a model exchange–correlation potential in density functional calculations of EPR parameters

We report the first implementation of the DFT calculation of electronic g-tensors and hyperfine coupling constants using meta-GGA exchange–correlation functionals (both Laplacian and kinetic-energy-density dependent) and statistical average of orbital model exchange–correlation potentials (SAOP). The g-tensors for a set of eleven small main group radicals and for a series of ten 3d and two 4d transition metal complexes have been compared using two meta-GGA functionals (FT98 and PKZB), SAOP, as well as the BP86 GGA, and B3PW91 and BHPW91 hybrid functionals. The same functionals have been compared in the calculation of hyperfine coupling constants for thirteen small main group radicals and twelve 3d transition metal compounds. The results for main group radicals are satisfactory, both with the newly validated and previously tested functionals. In contrast, the accurate evaluation of the EPR parameters for 3d transition metal complexes remains a challenge for “pure” density functional theory: the new functionals (potentials) do not improve the results compared to GGA functionals, whereas hybrid functionals tend to provide superior results. SAOP leads to large spin contamination for a number of transition metal complexes. This has been traced back to the orbital-energy dependence of the potential.

Spin-contamination of coupled-cluster wave functions

The propensity of approximate solutions of the electronic Schrödinger equation to break spin-symmetry is directly related to the quality of the approximate wave function, and thus can be used as a diagnostic tool. The quasi-variational nature of the (valence) optimized orbitals coupled-cluster doubles methods, (V)OO-CCD, enables a discussion of the stability of coupled-cluster wave functions in terms of both spin-contamination and a corresponding energy lowering relative to the pure spin solutions. The spin-contamination of (V)OO-CCD models has been studied for bond-breaking processes and diradicals. The main findings are: (i) The OO-CCD method is stable for a relatively large range of nuclear distortions and is capable of eliminating even very large spin-contamination of the unrestricted Hartree–Fock solution given that the molecular electronic configuration remains essentially single-reference. When a spin-contaminated solution arises, the energy splitting rapidly becomes large and 〈Ŝ2〉 approaches the Hartree–Fock value; (ii) The VOO-CCD method, which is designed to approximate a multi-reference model, remains stable over broader ranges; however, for pure diradicals it becomes unstable. In these cases, spin-contamination is also very large, but the energy lowering for the spin-unrestricted solutions is negligible; (iii) Higher order corrections described by perturbation theory lead to smaller energy splittings between restricted and unrestricted (V)OO-CCD energies. However, in case of spin-contaminated (V)OO-CCD solutions, these corrections may lead to unphysical shapes of the potential energy surfaces. Thus, in order to quantitatively characterize the quality of the wave functions, both spin-contamination and energy lowering due to the breaking of spin-symmetry must be considered.

Spin Contamination in Hartree−Fock and Density Functional Theory Wavefunctions in Modeling of Adsorption on Graphite

We have examined the effect of spin contamination in single determinant wave function at the Hartree−Fock and density functional theory in studying adsorption on graphite. Particularly, we examined NO adsorption on a single-layer graphene where some carbon atoms were left unsaturated to simulate the active sites in char. Upon comparison with the results from the restricted open-shell Hartree−Fock (ROHF) method, we found that UHF has a large spin contamination. Consequently, this yields large errors in the Hartree−Fock ground-state wave function, energy-level splitting, and adsorption energy. Interestingly, the spin contamination using unrestricted density functional theory was small, and the energetic information was almost unaffected.

Modification of the g a u s s i a n − 2 theoretical model: The use of coupled-cluster energies, density-functional geometries, and frequencies

A family of modified GAUSSIAN−2 (G2M) calculational schemes have been proposed, based on geometry optimization and vibrational frequency calculations using the hybrid density-functional approach, and electron correlation evaluation using the coupled-cluster methods. The most accurate model, called G2M(RCC), gives the average absolute deviation of calculated atomization energies from experiment for 32 first-row compounds of 0.88 kcal/mol. The other two methods, called G2M(RCC,MP2) and G2M(rcc,MP2), exhibit the average absolute deviations of 1.15 and 1.28 kcal/mol, respectively, and can be used for the calculations of molecules and radicals of larger sizes containing up to six to seven heavy atoms. The G2M(rcc,MP2) model demonstrates an accuracy comparable to that of G2(MP2) and requires less intensive computations than the latter. The preference of the G2M(RCC) methods over the original G2 is expected to be particularly significant for the open shell systems with large spin contamination.

Restricted open-shell Hartree–Fock-based many-body perturbation theory: Theory and application of energy and gradient calculations

A new approach for many-body perturbation theory (MBPT) built upon a restricted open-shell Hartree–Fock (ROHF) reference function is presented. ROHF-MBPT is shown to give much improved results compared to unrestricted Hartree–Fock (UHF) MBPT in cases where there is large spin contamination of the UHF reference function, and to converge much more rapidly to the infinite-order coupled-cluster result. Equations for analytical gradients at the MBPT(2) level are described and implemented. ROHF-MBPT and restricted open-shell Hartree–Fock single- and double-excitation coupled cluster (ROHF-CCSD) applications are presented for several difficult cases. These include the structure and electron affinity of the CN radical; structure, binding energy, and vibrational frequencies of Li3; the structure and vibrational frequencies for the unobserved FCS molecule; and the multiplet structure of the Ni atom.

An ab initio study of the reaction pathways for OH + C2H4 .fwdarw. HOCH2CH2 .fwdarw. products

The energetically favorable reaction paths for the unimolecular decomposition of the primary addition product of OH + C/sub 2/H/sub 4/ have been studied with ab initio techniques. Equilibrium geometries and transition structures were fully optimized with 3-21G and 6-31G* basis sets at the Hartree-Fock level. Heats of reaction and barrier heights have been computed with Moeller-Plesset perturbation theory up to fourth order, with and without annihilation of spin contamination. At the MP4 level barrier heights are lowered by 2-7 kcal/mol when the largest spin contaminant is removed. After the addition of OH + C/sub 2/H/sub 4/ to form the 2-hydroxyethyl radical, the most favorable reaction path (other than decomposition to reactants) is the (1,3)-hydrogen shift to form ethoxy radical followed by a dissociation into CH/sub 3/ + CH/sub 2/O. Other slightly higher energy paths include dissociation of ethoxy into H + CH/sub 3/CHO and decomposition of the 2-hydroxyethyl radical into H + HOCHCH/sub 2/.

Ab initio study of the carbon dioxide dimer and the carbon dioxide ion complexes [(CO2)2+ and (CO2)3+

Ab initio calculations were performed at several levels for neutral CO/sub 2/, CO/sub 2/ dimer, and the cations (CO/sub 2/)/sub 2//sup +/ and (CO/sub 2/)/sub 3//sup +/. At the highest level, a CO/sub 2/ neutral dimer of C/sub 2h/ symmetry (staggered side-by-side) is bound by 1.3 kcal/mol, which is 0.2 kcal/mol more stable than a T-shaped dimer (C/sub 2v/ symmetry). The dissociation energy of a (CO/sub 2/)/sub 2//sup +/ complex of C/sub 2h/ symmetry is calculated to be 16.2 kcal/mol which is 4.4 kcal/mol greater than the dissociation energy of the T-shaped (CO/sub 2/)/sub 2//sup +/ complex. Due to a smaller amount of spin contamination (spin polarization), the stability of this complex which is characterized by a partial bond between oxygens is overestimated at the (MP4SDQ/6-31G*) level relative to the T-shaped ion complex and CO/sub 2//sup +/ (plus CO/sub 2/) which have significant spin contamination. Projecting out the largest spin contaminant at the PMP2/6-31G* level and at the PMP4/3-21G level leads to an estimated decrease in the energy separation between the C/sub 2h/ symmetry complex and the T-shaped complex of 6.3 kcal/mol (relative to the (MP4SDQ/6-31G*) method). Two trimers of (CO/sub 2/)/sub 3//sup +/ were considered, a cyclic complex (C/submore » 3h/) and a cross-shaped complex (D/sub 2h/). The more stable (CO/sub 2/)/sub 3//sup +/ complex (D/sub 2h/) is bound by 16.0 kcal/mol with respect to monomers. Vibrational frequencies were used to compute entropies of reaction for (CO/sub 2/)/sub 2//sup +/ ..-->.. CO/sub 2/ + CO/sub 2//sup +/ and (CO/sub 2/)/sub 3//sup +/ ..-->.. 2CO/sub 2/ + CO/sub 2//sup +/.« less

Theoretical approach to fluorine substitution in X2NO and X2CN free radicals. Comparison between ab initio UHF and RHF + perturbation treatments

Non-empirical calculations have been performed to analyze the effects of fluorine substitution on the geometry and electronic properties of two series of π and σ radicals. Both UHF and RHF + perturbation methods have been used and the results are compared as a function of the basis set quality. As concerns geometry and spin-free electronic properties the results are independent of the UHF or RHF formalism, but highly sensitive to the basis set. The STO-3G basis is unable to reproduce even general trends. Polarization functions always play a relevant role and correlation effects seem not negligible at least for fluorine-containing radicals. The molecular shape of π radicals changes from a planar to a pyramidal geometry when increasing the electronegativity of the substituents. On the contrary, σ radicals always remain planar. Unprojected UHF spin densities are closer to the RHF + perturbation results for small spin contamination (X2NO). On the contrary, it is the projected UHF spin density which is in better agreement with the RHF + perturbation value for large spin contamination (X2CN). No simple correlation can be found between spin densities and gross atomic spin populations. For H2NO the spin density at nitrogen is smaller than at the oxygen nucleus, but substitution may enhance or reverse this trend.