Relativistic calculation of nuclear magnetic shielding using normalized elimination of the small component

The normalized elimination of the small component (NESC) theory, recently proposed by Filatov and Cremer, is extended to include magnetic interactions and applied to the calculation of the nuclear magnetic shielding in HX (X=F, Cl, Br, I) systems. The NESC calculations are performed at the levels of the zeroth-order regular approximation (ZORA) and the second-order regular approximation (SORA). The calculations show that the NESC-ZORA results are very close to the NESC-SORA results, except for the shielding of the I nucleus. Both the NESC-ZORA and NESC-SORA calculations yield very similar results to the previously reported values obtained using the relativistic infinite-order two-component coupled Hartree-Fock method. The difference between NESC-ZORA and NESC-SORA results is significant for the shieldings of iodine.     

Analytic calculation of second-order electric response properties with the normalized elimination of the small component (NESC) method

Analytic second derivatives of the relativistic energy for the calculation of electric response properties are derived utilizing the normalized elimination of the small component (NESC) method. Explicit formulas are given for electric static dipole polarizabilities and infrared intensities by starting at the NESC representation of electric dipole moments. The analytic derivatives are implemented in an existing NESC program and applied to calculate dipole moments, polarizabilities, and the infrared spectra of gold- and mercury-containing molecules as well as some actinide molecules. Comparison with experiment reveals the accuracy of NESC second order electric response properties.     

Relativistically corrected electric field gradients calculated with the normalized elimination of the small component formalism

Based on the analytic derivatives formalism for the spin-free normalized elimination of the small component method, a new computational scheme for the calculation of the electric field gradient at the atomic nuclei was developed and presented. The new computational scheme was tested by the calculation of the electric field gradient at the mercury nucleus in a series of Hg-containing inorganic and organometallic compounds. The benchmark calculations demonstrate that the new formalism is capable of reproducing experimental and theoretical reference data with high accuracy. The method developed can be routinely applied to the calculation of large and very large molecules and holds considerable promise for the interpretation of the experimental data of biologically relevant compounds containing heavy elements.     

On convergence of the normalized elimination of the small component (NESC) method

The convergence behavior of the iterative solution of the normalized elimination of the small component (NESC) method is investigated. A simple and efficient computational protocol for obtaining the exact positive-energy eigenvalues of the relativistic Hamiltonian starting from the energies obtained within the regular approximation is suggested. The protocol is based on the analysis of the relationship between the eigenvalues of the quasi-relativistic Hamiltonian in the regular approximation and the positive-energy eigenvalues of the exact relativistic Hamiltonian which was derived in the course of this work. This article is dedicated to Wim Nieuwpoort on the occasion of his 75th birthday.     

Calculations of the isotropic hyperfine coupling constants in free radicals

For a number of free radicals the results of non-empirical (ab initio) and semi-empirical (INDO, DEPAC, CNDO/SP) calculations of the isotropic hyperfine coupling constants are compared.     

Relativistically corrected geometries obtained with analytical gradients: normalized elimination of the small component using an effective potential

For the quasi-relativistic normalized elimination of small component using an effective potential (NESC-EP) method, analytical energy gradients were developed, programmed, and implemented in a standard quantum chemical program package. NESC-EP with analytical gradients was applied to determine geometry, vibrational frequencies, and dissociation enthalpies of ferrocene, tungsten hexafluoride, and tungsten hexacarbonyle. Contrary to non-relativistic calculations and calculations carried out with RECPs for the same compounds, NESC-EP provided reliable molecular properties in good agreement with experiment. The computational power of NESC-EP results from the fact that reliable relativistic corrections are obtained at a cost level only slightly larger than that of a non-relativistic calculation.     

Density functional theory predictions of isotropic hyperfine coupling constants

The reliability of density functional theory (DFT) in the determination of the isotropic hyperfine coupling constants (hfccs) of the ground electronic states of organic and inorganic radicals is examined. Predictions using several DFT methods and 6-31G, TZVP, EPR-III and cc-pVQZ basis sets are made and compared to experimental values. The set of 75 radicals here studied was selected using a wide range of criteria. The systems studied are neutral, cationic, anionic; doublet, triplet, quartet; localized, and conjugated radicals, containing 1H, 9Be, 11B, 13C, 14N, 17O, 19F, 23Na, 25Mg, 27Al, 29Si, 31P, 33S, and 35Cl nuclei. The considered radicals provide 241 theoretical hfcc values, which are compared with 174 available experimental ones. The geometries of the studied systems are obtained by theoretical optimization using the same functional and basis set with which the hfccs were calculated. Regression analysis is used as a basic and appropriate methodology for this kind of comparative study. From this analysis, we conclude that DFT predictions of the hfccs are reliable for B3LYP/TZVP and B3LYP/EPR-III combinations. Both functional/basis set scheme are the more useful theoretical tools for predicting hfccs if compared to other much more expensive methods.     

A variationally stable quasi-relativistic method: low-order approximation to the normalized elimination of the small component using an effective potential

A simple and variationally stable quasi-relativistic method based on a modified low-order (LO) approximation to the normalized elimination of the small component (NESC) method is presented. The modification of the original LO-NESC scheme implies the use of an energy-independent factor in the relativistic correction to the potential energy. This factor cuts off the potential energy at short distances from the nucleus and in this way restores the variational stability of LO-NESC. The new method, dubbed LO-NESC-effective potential (EP) was tested in calculations on one-, two- and many-electron atoms. The LO-NESC-EP can be easily implemented into the existing nonrelativistic quantum-chemical program codes because its Hamiltonian matrix can be expressed entirely in terms of the integrals appearing in a nonrelativistic calculation. Received: 1 April 2002 / Accepted: 23 June 2002 / Published online: 30 August 2002     

Relativistic calculation of nuclear magnetic shielding tensor using the regular approximation to the normalized elimination of the small component. II. Consideration of perturbations in the metric operator

A previous relativistic shielding calculation theory based on the regular approximation to the normalized elimination of the small component approach is improved by the inclusion of the magnetic interaction term contained in the metric operator. In order to consider effects of the metric perturbation, the self-consistent perturbation theory is used for the case of perturbation-dependent overlap integrals. The calculation results show that the second-order regular approximation results obtained for the isotropic shielding constants of halogen nuclei are well improved by the inclusion of the metric perturbation to reproduce the fully relativistic four-component Dirac-Hartree-Fock results. However, it is shown that the metric perturbation hardly or does not affect the anisotropy of the halogen shielding tensors and the proton magnetic shieldings.     

Development and application of the analytical energy gradient for the normalized elimination of the small component method

The analytical energy gradient of the normalized elimination of the small component (NESC) method is derived for the first time and implemented for the routine calculation of NESC geometries and other first order molecular properties. Essential for the derivation is the correct calculation of the transformation matrix U relating the small component to the pseudolarge component of the wavefunction. The exact form of ?U/?λ is derived and its contribution to the analytical energy gradient is investigated. The influence of a finite nucleus model and that of the picture change is determined. Different ways of speeding up the calculation of the NESC gradient are tested. It is shown that first order properties can routinely be calculated in combination with Hartree-Fock, density functional theory (DFT), coupled cluster theory, or any electron correlation corrected quantum chemical method, provided the NESC Hamiltonian is determined in an efficient, but nevertheless accurate way. The general applicability of the analytical NESC gradient is demonstrated by benchmark calculations for NESC/CCSD (coupled cluster with all single and double excitation) and NESC/DFT involving up to 800 basis functions.     

Connection between the regular approximation and the normalized elimination of the small component in relativistic quantum theory

The regular approximation to the normalized elimination of the small component (NESC) in the modified Dirac equation has been developed and presented in matrix form. The matrix form of the infinite-order regular approximation (IORA) expressions, obtained in [Filatov and Cremer, J. Chem. Phys. 118, 6741 (2003)] using the resolution of the identity, is the exact matrix representation and corresponds to the zeroth-order regular approximation to NESC (NESC-ZORA). Because IORA (=NESC-ZORA) is a variationally stable method, it was used as a suitable starting point for the development of the second-order regular approximation to NESC (NESC-SORA). As shown for hydrogenlike ions, NESC-SORA energies are closer to the exact Dirac energies than the energies from the fifth-order Douglas-Kroll approximation, which is much more computationally demanding than NESC-SORA. For the application of IORA (=NESC-ZORA) and NESC-SORA to many-electron systems, the number of the two-electron integrals that need to be evaluated (identical to the number of the two-electron integrals of a full Dirac-Hartree-Fock calculation) was drastically reduced by using the resolution of the identity technique. An approximation was derived, which requires only the two-electron integrals of a nonrelativistic calculation. The accuracy of this approach was demonstrated for heliumlike ions. The total energy based on the approximate integrals deviates from the energy calculated with the exact integrals by less than 5 x 10(-9) hartree units. NESC-ZORA and NESC-SORA can easily be implemented in any nonrelativistic quantum chemical program. Their application is comparable in cost with that of nonrelativistic methods. The methods can be run with density functional theory and any wave function method. NESC-SORA has the advantage that it does not imply a picture change.     

Mean field linear response within the elimination of the small component formalism to evaluate relativistic effects on magnetic properties

The linear response within the elimination of the small component formalism is aimed at obtaining the leading order relativistic corrections to magnetic molecular properties in the context of the elimination of the small component approximation. In the present work we extend the method in order to include two-body effects in the form of a mean field one-body operator. To this end we consider the four-component Dirac-Hartree-Fock operator as the starting point in the evaluation of the second order relativistic expression of magnetic properties. The approach thus obtained is the fully consistent leading order approximation of the random phase approximation four-component formalism. The mean field effect on the relativistic corrections to both the diamagnetic and paramagnetic terms of magnetic properties taking into account both the Coulomb and Breit two-body interactions is considered.     

A semi-empirical approach to the estimation of ESR isotropic hyperfine coupling constants in aromatic radicals

Using a generalised product wave function, an expression is given for the isotropic hyperfine coupling constant at a given atom in an aromatic -radical. By a consistent scheme of approximation the expression is cast in a form in which the coupling constant at a given atom in the radical can be evaluated from the results of a Hückel calculation, provided that certain integrals are known. A scheme for assigning and relating these integrals is given, coupling constants are calculated for ¹³C, ¹⁴N, ¹⁷O and ¹⁹F atoms, and the calculations compared with experiment.

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Zusammenfassung Bei Verwendung einer Produktwellenfunktion wird ein Ausdruck für die isotrope Hyperfeinkopplungskonstante eines Atomes in einem aromatischen -Radikal angegeben. Durch ein konsistentes Näherungsschema wird der Ausdruck in eine Form gebracht, bei der die Kopplungskonstante eines Atoms im Radikal aus Resultaten einer Hückelrechnung ausgewertet werden kann, vorausgesetzt, daß gewisse Integrale bekannt sind. Ein Schema, um diese Integrale zu kennzeichnen und miteinander in Beziehung zu setzen, wird angegeben, Kopplungskonstanten werden für ¹³C, ¹⁴N, ¹⁷O und ¹⁹F-Atome ausgerechnet und die Rechnungen mit dem Experiment verglichen.

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Résumé Calcul d'une expression pour la constante de couplage hyperfin isotrope sur un atome dans un radical aromatique à l'aide d'une fonction d'onde produit généralisé. A l'aide d'un schéma d'approximation cohérent cette expression est mise sous une forme telle que la constante de couplage sur un atome du radical peut être évaluée à partir des résultats d'un calcul de type Hückel, pourvu que certaines intégrales soient connues. On fournit un procédé pour déterminer et relier entre elles ces intégrales; les constantes de couplage pour ¹³C, ¹⁴N, ¹⁷O et ¹⁹F sont calculées et comparées aux données expérimentales.

     

On quantum-chemical estimates of constants of isotropic hyperfine coupling with protons in free radicals

A scheme of consitent quantum-chemical calculations of constants of isotropic hyperfine coupling (IHFC) with protons (a _H ^iso ) in free radicals is considered for the case where the spin populations ρ_s ^H are determined in the basic set of symmetrically orthogonalized atomic orbitals taking model σ-and π-electron fragments as an example. The competence of using two coefficients of proportionalityK(H) when estimating the proton IHFC constants by semiempirical quantum-chemical methods was demonstrated. Theoretical substantiation of empirical values of the above coefficients previously used is revealed. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 2, pp. 245–249, February, 1999.     

Density functional theory study of 14N isotropic hyperfine coupling constants of organic radicals

Nitrogen hyperfine coupling constants (hfccs) of organic radicals have been calculated by density functional theory (DFT) methodology. The capability of the B3LYP functional, combined with 6-31G*, TZVP and EPR-III basis sets, to reproduce experimental nitrogen coupling constant data has been analyzed for 109 neutral, cationic and anionic radicals, all of them containing at least one nitrogen atom. The results indicate that the selection of the basis set plays an important role in the accuracy of DFT calculations of hfccs, mainly in relation with the composition of the primitive functions and the quantum number of those functions. The main conclusion obtained is the high reliability of the scheme B3LYP/6-31G* for the prediction of nitrogen hfccs with very low computational cost.     

Study of influences of various excitation classes onab initio calculated isotropic hyperfine coupling constants

Summary Reliable prediction of the isotropic hyperfine coupling constantA _iso is still a difficult task forab initio calculations. Strong dependence on the method employed for its calculation has been found. Within a CI ansatzA _iso is considerably affected by the excitation classes taken into account within the CI calculation. In the present work the influence of various excitation classes onA _iso is examined. Calculations including all single, double, triple and a large part of the quadruple excitations are performed and the individual effects of the excitation classes are studied. It is found that the surprisingly good agreement found for S-CI treatments is due to large error cancellations. The importance of higher than double excitations arises from their indirect influence on the single excitations.     

Effects of static and dynamic perturbations on isotropic hyperfine coupling constants in some quinone radicals

The effects of solvent dielectric response on the isotropic hyperfine coupling constants of the 1,4-benzoquinone, 1,4-naphthoquinone and 9,10-anthraquinone anions and 1,4-naphthalenediol cation radicals were studied by electron spin resonance (ESR) spectroscopy and by the theoretical density functional method within the polarizable continuum model. Experimental results demonstrate that the isotropic hyperfine coupling constants can be obtained with high accuracy and that the effects of solvent impurities can be minimized by careful sample preparation. The results obtained correlate well with theoretical predictions from density functional theory calculations. For 1,4-naphthalenediol both the solvent dielectric response as well as rotational averaging of the hydroxy groups were calculated. The overall results highlight the importance of static and dynamic perturbations to the couplings and aid in the assignation process of the couplings to specific magnetic nuclei.