Normalized Elimination Of The Small Component Research Articles

Transition from metal-ligand bonding to halogen bonding involving a metal as halogen acceptor a study of Cu, Ag, Au, Pt, and Hg complexes

Utilizing all-electron Dirac-exact relativistic calculations with the Normalized Elimination of the Small Component (NESC) method and the local vibrational mode approach, the transition from metal-halide to metal halogen bonding is determined for Au-complexes interacting with halogen-donors. The local stretching force constants of the metal-halogen interactions reveal a smooth transition from weak non-covalent halogen bonding to non-classical 3-center-4-electron bonding and finally covalent metal-halide bonding. The strongest halogen bonds are found for dialkylaurates interacting with Cl2 or FCl. Differing trends in the intrinsic halogen-metal bond strength, the binding energy, and the electrostatic potential are explained.

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).

Quantitative Assessment of the Multiplicity of Carbon–Halogen Bonds: Carbenium and Halonium Ions with F, Cl, Br, and I

CX (X = F, Cl, Br, I) and CE bonding (E = O, S, Se, Te) was investigated for a test set of 168 molecules using the local CX and CE stretching force constants k(a) calculated at the M06-2X/cc-pVTZ level of theory. The stretching force constants were used to derive a relative bond strength order (RBSO) parameter n. As alternative bond strength descriptors, bond dissociation energies (BDE) were calculated at the G3 level or at the two-component NESC (normalized elimination of the small component)/CCSD(T) level of theory for molecules with X = Br, I or E = Se, Te. RBSO values reveal that both bond lengths and BDE values are less useful when a quantification of the bond strength is needed. CX double bonds can be realized for Br- or I-substituted carbenium ions where as suitable reference the double bond of the corresponding formaldehyde homologue is used. A triple bond cannot be realized in this way as the diatomic CX(+) ions with a limited π-donor capacity for X are just double-bonded. The stability of halonium ions increases with the atomic number of X, which is reflected by a strengthening of the fractional (electron-deficient) CX bonds. An additional stability increase of up to 25 kcal/mol (X = I) is obtained when the X(+) ion can form a bridged halonium ion with ethene such that a more efficient 2-electron-3-center bonding situation is created.

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 of NESC and 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‐component NESC algorithm 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‐component NESC methods are discussed and their use for the calculation of spin‐orbit coupling (SOC) effects in connection with spin‐orbit splittings and SOC‐corrected energies are demonstrated. The impact of scalar relativistic and spin‐orbit effects on molecular properties is presented. WIREs Comput Mol Sci 2014, 4:436–467.This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods

Exploring Bonding in Heavy Atom Chemistry with Dirac-Exact Methods

The normalized elimination of the small component (NESC) method is a first principles 2-component relativistic approach that leads to the Dirac-exact description of one-electron systems. It is a powerful method to routinely investigate chemical and physical properties of molecules containing relativistic atoms. The vibrational modes of mercury halides are investigated to derive via the corresponding local HgX (X = H, F, Cl, Br, I, At) stretching modes an appropriate measure for the HgX bond strength. It is shown that HgF bonding in HgF4 is stronger than that in HgF2, which is a result of enhanced charge transfer from Hg to the four F atoms and the formation of electron deficient 2e-4c bonds with strong ionic character. A generally applicable bonding model for HgX molecules is outlined. The relativistic counterpart of the Schrodinger equation is the Dirac equation (14, 15), which provides the basis for an exact description of a single electron in an external field. The step from a single electron to a multiple electron problem requires relativistic calculations both for the one-electron and the two-electron part of the Hamiltonian. The quantum mechanical methodology to reliably calculate molecular properties of molecules containing relativistic atoms developed much slower than its non-relativistic counterpart. In the last 15 years, the Dirac-exact relativistic methods have been developed, which lead to accurate 4-component results with a 2-component or even 1-component approach. The first Dirac-exact relativistic 2-component method was developed by Dyall (16) in form of the normalized elimination of the small component (NESC) approach. Dyall developed the NESC method at a time when most quantum chemists followed an operator-driven strategy to convert Dirac's four-component description by a series of transformations into a two-component one. Contrary to these attempts, he attacked the problem by starting from a matrix formulation of the Dirac equation and then carried out all the pertinent transformations by adhering to the matrix

Relativistic Calculation of Hyperfine Parameters of Mercury Compounds

The normalized elimination of the small component (NESC) method is a Dirac-exact relativistic method that leads to reliable first order response properties such as contact densities, Mössbauer shifts, electric field gradients, quadrupole coupling constants, or hyperfine structure constants for heavy atoms. In this review, the calculation of these hyperfine parameters with a NESC analytical derivatives formalism is discussed and demonstrated for mercury containing molecules. There is a distinct need for accurate calculated hyperfine parameters because the possibilities of experiment are limited in a case such as mercury. This need can be fulfilled if, beside scalar relativistic effects, the influence of spin-orbit coupling, electron correlation and the finite dimension of the nucleus are accounted for.

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.

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.

Analytic Calculation of Contact Densities and Mössbauer Isomer Shifts Using the Normalized Elimination of the Small-Component Formalism.

The analytic linear response formalism for the calculation of the effective contact densities ρ̅ in the context of the normalized elimination of the small component (NESC) method is developed and implemented. The formalism is tested for the calculation of contact densities and contact density differences in a series of mercury cations and mercury-containing molecules. The calculations carried out at the NESC/SCF, NESC/MP2, and NESC/CCSD levels of theory demonstrate high sensitivity of the contact density to the local coordination environment and the oxidation state of mercury. The NESC/MP2 results are in a very good agreement with the NESC/CCSD ones, which suggests that the former method can be used as a cost-effective alternative to high-level ab initio calculations.

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.

No-Pair Bonding in Coinage Metal Dimers

High-level ab initio calculations at the coupled cluster with single and double substitutions and perturbative treatment of triple substitutions, CCSD(T), level of theory have been carried out for the dimers of coinage metal atoms Cu, Ag, and Au in the ground 1Sigma(g)+ state and in the excited 3Sigma(u)+ state. All of the calculations have been carried out with the inclusion of scalar-relativistic effects via the normalized elimination of the small component (NESC) method. For the dimers in the triplet state, nonzero bond dissociation energies are obtained which vary from 1.3 kcal/mol for 3Cu2 to 4.6 kcal/mol for 3Au2. Taking into account that, in bulky high-spin copper clusters, the bond dissociation energy per atom increases steeply to the value of ca. 19 kcal/mol, the results obtained in the present paper suggest that the bond dissociation energy per atom in high-spin gold clusters may reach extremely high values exceeding 20 kcal/mol thus becoming comparable to the usual bonding due to the spin-pairing mechanism.

Bonding in mercury-alkali molecules: Orbital-driven van der Waals complexes.

The bonding situation in mercury-alkali diatomics HgA (2Σ+) (A = Li, Na, K, Rb) has been investigated employing the relativistic all-electron method Normalized Elimination of the Small Component (NESC), CCSD(T), and augmented VTZ basis sets. Although Hg,A interactions are typical of van der Waals complexes, trends in calculated De values can be explained on the basis of a 3-electron 2-orbital model utilizing calculated ionization potentials and the De values of HgA+(1Σ+) diatomics. HgA molecules are identified as orbital-driven van der Waals complexes. The relevance of results for the understanding of the properties of liquid alkali metal amalgams is discussed.

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.

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.

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.

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.