Small Component Method Research Articles

Calculations of atomic magnetic nuclear shielding constants based on the two-component normalized elimination of the small component method.

A new method for calculating nuclear magnetic resonance shielding constants of relativistic atoms based on the two-component (2c), spin-orbit coupling including Dirac-exact NESC (Normalized Elimination of the Small Component) approach is developed where each term of the diamagnetic and paramagnetic contribution to the isotropic shielding constant σiso is expressed in terms of analytical energy derivatives with regard to the magnetic field B and the nuclear magnetic moment 𝝁. The picture change caused by renormalization of the wave function is correctly described. 2c-NESC/HF (Hartree-Fock) results for the σiso values of 13 atoms with a closed shell ground state reveal a deviation from 4c-DHF (Dirac-HF) values by 0.01%-0.76%. Since the 2-electron part is effectively calculated using a modified screened nuclear shielding approach, the calculation is efficient and based on a series of matrix manipulations scaling with (2M)3 (M: number of basis functions).

Calculations of electric dipole moments and static dipole polarizabilities based on the two-component normalized elimination of the small component method.

The analytical energy gradient and Hessian of the two-component Normalized Elimination of the Small Component (2c-NESC) method with regard to the components of the electric field are derived and used to calculate spin-orbit coupling (SOC) corrected dipole moments and dipole polarizabilities of molecules, which contain elements with high atomic number. Calculated 2c-NESC dipole moments and isotropic polarizabilities agree well with the corresponding four-component-Dirac Hartree-Fock or density functional theory values. SOC corrections for the electrical properties are in general small, but become relevant for the accurate prediction of these properties when the molecules in question contain sixth and/or seventh period elements (e.g., the SO effect for At2 is about 10% of the 2c-NESC polarizability). The 2c-NESC changes in the electric molecular properties are rationalized in terms of spin-orbit splitting and SOC-induced mixing of frontier orbitals with the same j = l + s quantum numbers.

Analytical energy gradient for the two-component normalized elimination of the small component method.

The analytical gradient for the two-component Normalized Elimination of the Small Component (2c-NESC) method is presented. The 2c-NESC is a Dirac-exact method that employs the exact two-component one-electron Hamiltonian and thus leads to exact Dirac spin-orbit (SO) splittings for one-electron atoms. For many-electron atoms and molecules, the effect of the two-electron SO interaction is modeled by a screened nucleus potential using effective nuclear charges as proposed by Boettger [Phys. Rev. B 62, 7809 (2000)]. The effect of spin-orbit coupling (SOC) on molecular geometries is analyzed utilizing the properties of the frontier orbitals and calculated SO couplings. It is shown that bond lengths can either be lengthened or shortened under the impact of SOC where in the first case the influence of low lying excited states with occupied antibonding orbitals plays a role and in the second case the jj-coupling between occupied antibonding and unoccupied bonding orbitals dominates. In general, the effect of SOC on bond lengths is relatively small (≤5% of the scalar relativistic changes in the bond length). However, large effects are found for van der Waals complexes Hg2 and Cn2, which are due to the admixture of more bonding character to the highest occupied spinors.

Dirac‐exact relativistic methods: the normalized elimination of the small component method

Dirac‐exact relativistic methods, i.e., 2‐ or 1‐component methods which exactly reproduce the one‐electron energies of the original 4‐component Dirac method, have established a standard for reliable relativistic quantum chemical calculations targeting medium‐ and large‐sized molecules. Their development was initiated and facilitated in the late 1990s by Dyall's development of the normalized elimination of the small component (NESC). Dyall's work has fostered the conversion ofNESCand related (later developed) methods into routinely used, multipurpose Dirac‐exact methods by which energies, first‐order, and second‐order properties can be calculated at computational costs, which are only slightly higher than those of nonrelativistic methods. This review summarizes the development of a generally applicable 1‐componentNESCalgorithm leading to the calculation of reliable energies, geometries, electron density distributions, electric moments, electric field gradients, hyperfine structure constants, contact densities and Mössbauer isomer shifts, nuclear quadrupole coupling constants, vibrational frequencies, infrared intensities, and static electric dipole polarizabilities. In addition, the derivation and computational possibilities of 2‐componentNESCmethods are discussed and their use for the calculation of spin‐orbit coupling (SOC) effects in connection with spin‐orbit splittings andSOC‐corrected energies are demonstrated. The impact of scalar relativistic and spin‐orbit effects on molecular properties is presented.WIREs Comput Mol Sci2014, 4:436–467.This article is categorized under:Electronic Structure Theory > Ab Initio Electronic Structure Methods

Calculation of response properties with the normalized elimination of the small component method

The normalized elimination of the small component method is a first principles two‐component relativistic approach that leads to the Dirac‐exact description of one‐electron systems. Therefore, it is an ideal starting point for developing procedures, by which first‐ and second‐order response properties can be routinely calculated. We present algorithms and methods for the calculation of molecular response properties such as geometries, dipole moments, hyperfine structure constants, vibrational frequencies and force constants, electric polarizabilities, infrared intensities and so forth. The described formalisms are applied to molecules containing mercury and other heavy elements, which require a relativistic treatment. Perspectives for the future development and application of Dirac‐exact methods are outlined. © 2013 Wiley Periodicals, Inc.

Spin-orbit coupling calculations with the two-component normalized elimination of the small component method

A new algorithm for the two-component Normalized Elimination of the Small Component (2cNESC) method is presented and tested in the calculation of spin-orbit (SO) splittings for a series of heavy atoms and their molecules. The 2cNESC is a Dirac-exact method that employs the exact two-component one-electron Hamiltonian and thus leads to exact Dirac SO splittings for one-electron atoms. For many-electron atoms and molecules, the effect of the two-electron SO interaction is modeled by a screened nucleus potential using effective nuclear charges as proposed by Boettger [Phys. Rev. B 62, 7809 (2000)]. The use of the screened nucleus potential for the two-electron SO interaction leads to accurate spinor energy splittings, for which the deviations from the accurate Dirac Fock-Coulomb values are on the average far below the deviations observed for other effective one-electron SO operators. For hydrogen halides HX (X = F, Cl, Br, I, At, and Uus) and mercury dihalides HgX2 (X = F, Cl, Br, I) trends in spinor energies and SO splittings as obtained with the 2cNESC method are analyzed and discussed on the basis of coupling schemes and the electronegativity of X.

Removal of Mercury from the Environment: A Quantum-Chemical Study with the Normalized Elimination of the Small Component Method

1,3-Benzenediamidoethanethiolatemercury [BDT-Hg or BD(S)-Hg] and its derivatives are investigated utilizing the Dirac exact relativistic normalized elimination of the small component method in connection with B3LYP, CCSD(T), and polarizable continuum calculations. It is shown that the chelating energy of BDT-Hg can be significantly increased by replacing sulfur with selenium or tellurium, thus leading to BD(Se)-Hg or BD(Te)-Hg. In this particular case, the chalcogenophilicity of mercury increases from S to Te because increasing the E-Hg bond lengths leads to a reduction of ring strain. Various possibilities of increasing the metal (M) chelating strength in BDT-M complexes are investigated, and suggestions for new chelating agents based on the BDT-M template are made.

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.

Development, Implementation, and Application of an Analytic Second Derivative Formalism for the Normalized Elimination of the Small Component Method.

Analytical second derivatives for the normalized elimination of the small component (NESC) method are derived for the first time and implemented for the routine calculation of NESC vibrational frequencies and other second order molecular properties using the scalar relativistic form of NESC. Using response theory, the second derivatives of the transformation matrix U connecting the large and the pseudolarge components of the relativistic wave function are correctly derived. The 24 derivative terms involving the NESC Hamiltonian and the NESC renormalization matrix are individually tested, and their contributions to the energy Hessian are calculated. 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 second derivatives are tested. It is shown that second order properties can routinely be calculated in combination with Hartree-Fock, density functional theory, Moller-Plesset perturbation theory, and any other electron correlation corrected quantum chemical method provided analytic second derivatives are available in the nonrelativistic case. The general applicability of the analytic NESC Hessian is demonstrated by benchmark calculations for NESC/DFT calculations involving up to 1500 basis functions.

An improved algorithm for the normalized elimination of the small-component method

A new algorithm for the iterative solution of the normalized elimination of the small component (NESC) method is presented that is less costly than previous algorithms and that is based on (1) solving the NESC equations for the uncontracted rather than contracted basis (“First-Diagonalize-then-Contract”), (2) a new iterative procedure for obtaining the NESC Hamiltonian (“iterative TU algorithm”), (3) the renormalization scheme connected to the picture change, and (4) a finite nucleus model with a Gaussian charge distribution. The accuracy of NESC energies, which match those of 4-component Dirac calculations, is demonstrated. Test calculations with CCSD(T), DFT, and large basis sets including high angular momentum basis functions (f,g,h,i) are presented to prove the general applicability of the new NESC algorithm. Comparison with other algorithms of solving the NESC equations are shortly discussed and time savings are presented.

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.

Calculation of X-ray photoelectron spectra with the use of the normalized elimination of the small component method

A method for the calculation of X-ray photoelectron spectra (XPS) based on the use of the normalized elimination of the small component (NESC) formalism combined with the restricted active space state interaction (RASSI) approach with atomic mean field integrals (AMFI) is developed. Benchmark calculations carried out for the 4f XPS of U5+show that the NESC/RASSI/AMFI method is capable of reproducing the results of the full 4-component relativistic calculations with excellent accuracy. The NESC/RASSI/AMFI method is applied to study the 4p and 5p XPS of ytterbium phosphide YbP. The results of the calculations suggest an alternative interpretation of the satellite peaks in the 4p XPS of YbP.

Resolution of identity Dirac-Kohn-Sham method using the large component only: Calculations of g-tensor and hyperfine tensor

A new relativistic two-component density functional approach, based on the Dirac-Kohn-Sham method and an extensive use of the technique of resolution of identity (RI), has been developed and is termed the DKS2-RI method. It has been applied to relativistic calculations of g and hyperfine tensors of coinage-metal atoms and some mercury complexes. The DKS2-RI method solves the Dirac-Kohn-Sham equations in a two-component framework using explicitly a basis for the large component only, but it retains all contributions coming from the small component. The DKS2-RI results converge to those of the four-component Dirac-Kohn-Sham with an increasing basis set since the error associated with the use of RI will approach zero. The RI approximation provides a basis for a very efficient implementation by avoiding problems associated with complicated integrals otherwise arising from the elimination of the small component. The approach has been implemented in an unrestricted noncollinear two-component density functional framework. DKS2-RI is related to Dyall's [J. Chem. Phys. 106, 9618 (1997)] unnormalized elimination of the small component method (which was formulated at the Hartree-Fock level and applied to one-electron systems only), but it takes advantage of the local Kohn-Sham exchange-correlation operators (as, e.g., arising from local or gradient-corrected functionals). The DKS2-RI method provides an attractive alternative to existing approximate two-component methods with transformed Hamiltonians (such as Douglas-Kroll-Hess [Ann. Phys. 82, 89 (1974); Phys. Rev. A 33, 3742 (1986)] method, zero-order regular approximation, or related approaches) for relativistic calculations of the structure and properties of heavy-atom systems. In particular, no picture-change effects arise in the property calculations.