Zeroth-order Regular Approximation Hamiltonian Research Articles

Coordination of Mercury(II) in Water Promoted over Hydrolysis in Solvated Clusters [Hg(H2O)1-6](aq)2+: Insights from Relativistic Effects and Free Energy Analysis.

Understanding the nature of the interaction between mercury(II) ions, Hg2+, and water molecules is crucial to describe the stability and chemical behavior of structures formed during solvation, as well as the conditions that favor the Hg2+ coordination or inducing water hydrolysis. In our study, we explored exhaustively the potential energy surface of Hg2+ with up to six water molecules. We analyzed electronic and Gibbs free energies, binding, and nuclear magnetic resonance parameters. We used the zeroth-order regular approximation Hamiltonian, including scalar and spin-orbit relativistic corrections for free energy calculations and geometry optimizations to explore the interplay between electron correlation and relativistic effects. We analyzed intermolecular interactions with energy decomposition analysis, quantum theory of atoms in molecules, and natural bond orbital. Additionally, we used the four-component Dirac Hamiltonian to compute solvent effect on the magnetic shielding and J-coupling constants. Our results revealed that the water hydrolysis by Hg2+ requires a minimum of three water molecules. We found that the interaction between Hg2+ and water molecules is an orbital interaction due to relativistic effects and the most stable structures are opened-shape clusters, reducing the number of oxygen-mercury contacts and maximizing the formation of hydrogen bonds among water molecules. In these types of clusters, Hg2+ promotes the water hydrolysis over coordination with oxygen atoms. However, when we considered the change associated with the transfer of a cluster from the ideal gas to a solvated system, our solvation free energy analysis revealed that closed-shape clusters are more favorable, maximizing the number of oxygen-mercury contacts and reducing the formation of hydrogen bonds among water molecules. This finding suggests that, under room conditions, the coordination of Hg2+ is more favorable than hydrolysis. Our results have significant implications for understanding Hg2+ behavior in water, helping to develop targeted strategies for mercury remediation and management, and contributing to advancements in the broader field of environmental chemistry.

Spin-orbit ZORA and four-component Dirac-Coulomb estimation of relativistic corrections to isotropic nuclear shieldings and chemical shifts of noble gas dimers.

Hartree-Fock and density functional theory with the hybrid B3LYP and general gradient KT2 exchange-correlation functionals were used for nonrelativistic and relativistic nuclear magnetic shielding calculations of helium, neon, argon, krypton, and xenon dimers and free atoms. Relativistic corrections were calculated with the scalar and spin-orbit zeroth-order regular approximation Hamiltonian in combination with the large Slater-type basis set QZ4P as well as with the four-component Dirac-Coulomb Hamiltonian using Dyall's acv4z basis sets. The relativistic corrections to the nuclear magnetic shieldings and chemical shifts are combined with nonrelativistic coupled cluster singles and doubles with noniterative triple excitations [CCSD(T)] calculations using the very large polarization-consistent basis sets aug-pcSseg-4 for He, Ne and Ar, aug-pcSseg-3 for Kr, and the AQZP basis set for Xe. For the dimers also, zero-point vibrational (ZPV) corrections are obtained at the CCSD(T) level with the same basis sets were added. Best estimates of the dimer chemical shifts are generated from these nuclear magnetic shieldings and the relative importance of electron correlation, ZPV, and relativistic corrections for the shieldings and chemical shifts is analyzed.

What factors influence the metal-proton spin-spin coupling constants in mercury- and cadmium-substutited rubredoxin?

The indirect metal-proton spin-spin coupling constants between protons in cysteine groups and the mercury or cadmium nucleus have been calculated for a small model of Me-rubredoxin complex (Me = Cd, Hg) by means of density functional theory with zeroth-order regular approximation Hamiltonian (DFT-ZORA). The calculated spin-spin coupling constants, in spite of the moderate size of the model system, are in good agreement with the values measured in NMR experiment, which are in the 0.29-0.56 Hz range for the Cd complex and in the 0.57-2.20 Hz range for the Hg complex. The robustness of the chosen method has been verified by calculations with a number of different exchange-correlation functionals and basis sets. Additionally, it has been shown that the short- and long-distance metal-proton coupling constants are affected mainly by the values of the metal-proton distance and the H-N-C-C dihedral angle.

Relativistic diffusion Monte Carlo method: Zeroth-order regular approximation-diffusion Monte Carlo method in a spin-free formalism

A diffusion Monte Carlo (DMC) method for the relativistic zeroth-order regular approximation (ZORA) is proposed. In this scheme, a novel approximate Green's function is derived for the spin-free ZORA Hamiltonian. Several numerical tests on atoms and small molecules showed that by combining with the relativistic cusp-correction scheme, the present approach can include both relativistic and electron-correlation effects simultaneously. The correlation energies recovered by the ZORA-DMC method are comparable with the nonrelativistic DMC results and superior to the coupled cluster singles and doubles with perturbative triples correction results when the correlation-consistent polarized valence triple-zeta Douglas-Kroll basis set is used. For the heavier CuH molecule, the ZORA-DMC estimation of its dissociation energy agrees with the experimental value within the error bar.

Spin–Spin Artificial DNA Intercalated with Silver Cations: Theoretical Prediction

The indirect nuclear spin-spin coupling constants of Ag(+) cation intercalated between imidazole rings in DNA chains are calculated by means of DFT with relativistic effects taken into account by the use of the zeroth-order regular approximation Hamiltonian (DFT-ZORA). The calculations model how the (1)J((15)N,(109)Ag) coupling constant is affected by different types of geometry deformations and by the presence of water, which is simulated by means of the polarizable continuum model and explicitly present water molecules. Calculations for systems containing two and three imidazole pairs are also carried out to model the influence of stacking interactions. The computed (1)J((15)N,(109)Ag) spin-spin coupling constant is in the range of 85-105 Hz (depending on the computational model) and is in good agreement with the experimental value (ca. 92 Hz). This coupling constant is very little affected by the presence of solvent, stacking interactions, and geometry deformations. Such behavior is explained by visualization of the coupling path by means of coupling energy density (CED). Bigger models allow the coupling constant between two adjacent silver ions to be computed, and give a value of approximately 1 Hz, which is probably too small to be of practical interest. The (2)J((15)N,(15)N) value is calculated to be about 2.5 Hz, and is therefore of measurable magnitude.

Spin-orbit relativistic long-range corrected time-dependent density functional theory for investigating spin-forbidden transitions in photochemical reactions

A long-range corrected (LC) time-dependent density functional theory (TDDFT) incorporating relativistic effects with spin-orbit couplings is presented. The relativistic effects are based on the two-component zeroth-order regular approximation Hamiltonian. Before calculating the electronic excitations, we calculated the ionization potentials (IPs) of alkaline metal, alkaline-earth metal, group 12 transition metal, and rare gas atoms as the minus orbital (spinor) energies on the basis of Koopmans' theorem. We found that both long-range exchange and spin-orbit coupling effects are required to obtain Koopmans' IPs, i.e., the orbital (spinor) energies, quantitatively in DFT calculations even for first-row transition metals and systems containing large short-range exchange effects. We then calculated the valence excitations of group 12 transition metal atoms and the Rydberg excitations of rare gas atoms using spin-orbit relativistic LC-TDDFT. We found that the long-range exchange and spin-orbit coupling effects significantly contribute to the electronic spectra of even light atoms if the atoms have low-lying excitations between orbital spinors of quite different electron distributions.

Improving upon the ZORA Hamiltonian

AbstractThe metric perturbation method is used to derive what is called the normalized zeroth‐order regular approximation (ZORA) Hamiltonian. This Hamiltonian, although derived in a different way, turns out to be equivalent to the infinite‐order regular approximation (IORA) operator of Dyall and van Lenthe. The normalized ZORA Hamiltonian is analyzed in terms of its expansion with respect to the leading order of the fine structure constant. Through the leading second‐order in the fine structure constant, the normalized ZORA Hamiltonian recovers all terms of what is known as the first‐order regular approximation (FORA). The relation of the regular approximation to methods based on the Douglas–Kroll transformation is discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Time-dependent quasirelativistic density-functional theory based on the zeroth-order regular approximation

A time-dependent quasirelativistic density-functional theory for excitation energies of systems containing heavy elements is developed, which is based on the zeroth-order regular approximation (ZORA) for the relativistic Hamiltonian and a noncollinear form for the adiabatic exchange-correlation kernel. To avoid the gauge dependence of the ZORA Hamiltonian a model atomic potential, instead of the full molecular potential, is used to construct the ZORA kinetic operator in ground-state calculations. As such, the ZORA kinetic operator no longer responds to changes in the density in response calculations. In addition, it is shown that, for closed-shell ground states, time-reversal symmetry can be employed to simplify the eigenvalue equation into an approximate form that is similar to that of time-dependent nonrelativistic density-functional theory. This is achieved by invoking an independent-particle approximation for the induced density matrix. The resulting theory is applied to investigate the global potential-energy curves of low-lying LambdaS- and omega omega-coupled electronic states of the AuH molecule. The derived spectroscopic parameters, including the adiabatic and vertical excitation energies, equilibrium bond lengths, harmonic and anharmonic vibrational constants, fundamental frequencies, and dissociation energies, are in good agreement with those of time-dependent four-component relativistic density-functional theory and ab initio multireference second-order perturbation theory. Nonetheless, this two-component relativistic version of time-dependent density-functional theory is only moderately advantageous over the four-component one as far as computational efforts are concerned.

On the physical meaning of the ZORA Hamiltonian

By expanding the Foldy-Wouthuysen representation of the Dirac equation near the free-particle solution it is shown that the Hamiltonian of the zeroth-order regular approximation (ZORA) leads to an infinite summation of the leading relativistic corrections to the free-particle, non-relativistic energy. The analysis of the perturbation expansion of the ZORA Hamiltonian reveals that the ZORA Hamiltonian recovers all terms of the Breit-Pauli theory to second order. This result is general and applies not only to hydrogen-like atomic ions (as was demonstrated before) but also to a wide variety of physical problems. ZORA is analogous to the random phase approximation in many-body theory in the sense that both methods include an infinite-order summation of the asymptotically non-vanishing terms. This highlights the difference between ZORA and the Douglas-Kroll method, with the latter being analogous to finite-order many-body perturbation theory. On the basis of this analysis the performance of ZORA when calculating various molecular properties is discussed.

Relativistic regular approximations revisited: An infinite-order relativistic approximation

The concept of the regular approximation is presented as the neglect of the energy dependence of the exact Foldy–Wouthuysen transformation of the Dirac Hamiltonian. Expansion of the normalization terms leads immediately to the zeroth-order regular approximation (ZORA) and first-order regular approximation (FORA) Hamiltonians as the zeroth- and first-order terms of the expansion. The expansion may be taken to infinite order by using an un-normalized Foldy–Wouthuysen transformation, which results in the ZORA Hamiltonian and a nonunit metric. This infinite-order regular approximation, IORA, has eigenvalues which differ from the Dirac eigenvalues by order E3/c4 for a hydrogen-like system, which is a considerable improvement over the ZORA eigenvalues, and similar to the nonvariational FORA energies. A further perturbation analysis yields a third-order correction to the IORA energies, TIORA. Results are presented for several systems including the neutral U atom. The IORA eigenvalues for all but the 1s spinor of the neutral system are superior even to the scaled ZORA energies, which are exact for the hydrogenic system. The third-order correction reduces the IORA error for the inner orbitals to a very small fraction of the Dirac eigenvalue.