Gauge-origin Independence Research Articles

NMR chemical shift computations at second-order Møller-Plesset perturbation theory using gauge-including atomic orbitals and Cholesky-decomposed two-electron integrals.

We report on a formulation and implementation of a scheme to compute nuclear magnetic resonance (NMR) shieldings at second-order Møller-Plesset (MP2) perturbation theory using gauge-including atomic orbitals (GIAOs) to ensure gauge-origin independence and Cholesky decomposition (CD) to handle unperturbed and perturbed two-electron integrals. We investigate the accuracy of the CD for the derivatives of the two-electron integrals with respect to an external magnetic field and for the computed NMR shieldings, before we illustrate the applicability of our CD-based GIAO-MP2 scheme in calculations involving up to about 100 atoms and more than 1000 basis functions.

Relativistic Effects in Magnetic Circular Dichroism: Restricted Magnetic Balance and Temperature Dependence

Magnetic circular dichroism of transition metal complexes and open-shell systems are challenging to simulate and analyze, mainly due to the interplay of spin-orbit couplings and finite-magnetic-field included Zeeman effects with the complex selection rules dictated by the circularly polarized light. In this work, we introduce an ab initio relativistic two-component formalism based on the restricted magnetic-balanced Hamiltonian for simulating MCD spectra. Both homogeneous finite magnetic field and relativistic effects are included variationally in the ground state reference. Finite-field London orbitals are used to enforce the constrained gauge-origin independence in the calculation using localized atomic orbitals. Through benchmark studies of AuCl4-, Pt(CN)42-, and Mo(CN)83-, we discuss how relativistic effects are manifested in MCD for both closed-shell and open-shell molecular complexes and how the interplay between spin-orbit coupling and magnetic field modulates the MCD selection rules. Finally, an investigation on temperature-dependent MCD is carried out and compared to experiment.

Linear and sublinear scaling computation of the electronic g-tensor at the density functional theory level.

We present an efficient and low-scaling implementation of a density functional theory based method for the computation of electronic g-tensors. It allows for an accurate description of spin-orbit coupling effects by employing the spin-orbit mean-field operator. Gauge-origin independence is ensured by the use of gauge-including atomic orbitals. Asymptotically linear scaling with molecule size is achieved with an atomic orbital based formulation, integral screening methods, and sparse linear algebra. In addition, we introduce an ansatz that exploits the locality of the contributions to the g-tensor for molecules with local spin density. For such systems, sublinear scaling is obtained by restricting the magnetic field perturbation to the relevant subspaces of the full atomic orbital space; several criteria for selecting these subspaces are discussed and compared. It is shown that the computational cost of g-tensor calculations with the local approach can fall below the cost of the self-consistent field calculation for large molecules. The presented methods thus enable efficient, accurate, and gauge-origin independent computations of electronic g-tensors of large molecular systems.

Aromatic Pathways in Porphycene Derivatives Based on Current-Density Calculations.

Magnetically induced current densities have been calculated for porphycenes at the density functional theory level using gauge-including atomic orbitals to ensure gauge-origin independence and a fast basis-set convergence of the current densities. The current densities have been analyzed by using the gauge-including magnetically induced current (GIMIC) method. The porphycenes are aromatic, sustaining strong diatropic ring currents. The ring-current pathways have been determined by integrating the strength of the current density passing selected bonds. The calculations show that the ring current of the porphycenes splits into an outer and inner branch at the pyrrolic rings implying that the ring current involves all 26 π electrons of the porphycenes, which is similar to the ring current of porphyrins. The pyrrolic rings of the aromatic porphycenes do not sustain any significant local ring currents. Dihydroporphycene with four inner hydrogens is antiaromatic with weakly aromatic pyrrolic rings. The annelated benzoic rings in benzoporphycene sustain local paratropic ring currents, whereas the global ring current of dibenzoporphycene splits into an outer and inner branch at the benzoic rings. Comparison of calculated1H NMR shieldings with ring-current strengths shows that interactions between the inner hydrogen and the neighbor nitrogen is more significant for differences in the 1H NMR shieldings than variations in global ring-current strengths. Calculated excitation energies show that the antiaromatic dihydroporphycene has a smaller optical gap than the aromatic porphycene, even though its HOMO-LUMO gap is larger.

Coupled-cluster theory for atoms and molecules in strong magnetic fields.

An implementation of coupled-cluster (CC) theory to treat atoms and molecules in finite magnetic fields is presented. The main challenges for the implementation stem from the magnetic-field dependence in the Hamiltonian, or, more precisely, the appearance of the angular momentum operator, due to which the wave function becomes complex and which introduces a gauge-origin dependence. For this reason, an implementation of a complex CC code is required together with the use of gauge-including atomic orbitals to ensure gauge-origin independence. Results of coupled-cluster singles-doubles-perturbative-triples (CCSD(T)) calculations are presented for atoms and molecules with a focus on the dependence of correlation and binding energies on the magnetic field.

Gauge-origin independent calculations of Jones birefringence

We present the first gauge-origin independent formulation of Jones birefringence at the Hartree-Fock level of theory. Gauge-origin independence is achieved through the use of London atomic orbitals. The implementation is based on a recently proposed atomic orbital-based response theory formulation that allows for the use of both time- and perturbation-dependent basis sets [Thorvaldsen, Ruud, Kristensen, Jo̸rgensen, and Coriani, J. Chem. Phys. 129, 214108 (2008)]. We present the detailed expressions for the response functions entering the Jones birefringence when London atomic orbitals are used. The implementation is tested on a set of polar and dipolar molecules at the Hartree-Fock level of theory. It is demonstrated that London orbitals lead to much improved basis-set convergence, and that the use of small, conventional basis sets may lead to the wrong sign for the calculated birefringence. For large basis sets, London orbitals and conventional basis sets converge to the same results.

Quantitative prediction of gas-phase F19 nuclear magnetic shielding constants

Benchmark calculations of (19)F nuclear magnetic shielding constants are presented for a set of 28 molecules. Near-quantitative accuracy (ca. 2 ppm deviation from experiment) is achieved if (1) electron correlation is adequately treated by employing the coupled-cluster singles and doubles (CCSD) model augmented by a perturbative correction for triple excitations [CCSD(T)], (2) large (uncontracted) basis sets are used, (3) gauge-including atomic orbitals are used to ensure gauge-origin independence, (4) calculations are performed at accurate equilibrium geometries [obtained from CCSD(T)/cc-pVTZ calculations correlating all electrons], and (5) vibrational averaging and temperature corrections via second-order vibrational perturbation theory (VPT2) are included. For the CCSD(T)/13s9p4d3f calculations corrected for vibrational effects, mean and standard deviation from experiment are -1.9 and 1.6 ppm, respectively. Less elaborate theoretical treatments result in larger errors. Consideration of relative shifts can reduce the mean deviation (through an appropriately chosen reference compound), but does not change the standard deviation. Density-functional theory calculations of absolute and relative (19)F nuclear magnetic shielding constants are found to be, at best, as accurate as the corresponding Hartree-Fock self-consistent-field calculations and are not improved by consideration of vibrational effects. Molecular systems containing fluorine-oxygen, fluorine-nitrogen, and fluorine-fluorine bonds are found to be more challenging than the other investigated molecules for the considered theoretical methods.

Gauge-origin independent calculation of magnetizabilities and rotational g tensors at the coupled-cluster level

An implementation of the gauge-origin independent calculation of magnetizabilities and rotational g tensors at the coupled-cluster (CC) level is presented. The properties of interest are obtained as second derivatives of the energy with respect to the external magnetic field (in the case of the magnetizability) or with respect to magnetic field and rotational angular momentum (in the case of the rotational g tensor), while gauge-origin independence and fast basis-set convergence are ensured by using gauge-including atomic orbitals (London atomic orbitals) as well as their extension to treat rotational perturbations (rotational London atomic orbitals). The implementation within our existing CC analytic second-derivative code is described, focusing on the required modifications concerning integral evaluation and treatment of the unperturbed and perturbed two-particle density matrices. An extensive set of test calculations for LiH and BH (up to the full configuration-interaction limit), for a series of simple hydrides (HF, H(2)O, NH(3), and CH(4)) as well as the more challenging molecules CO, N(2), and O(3) [employing the CC singles and doubles (CCSD) and the CCSD approximation augmented by a perturbative treatment of triple excitations] demonstrates the importance of electron correlation for high-accuracy predictions of magnetizabilities and rotational g tensors.

Gauge-Origin-Independent Coupled Cluster Singles and Doubles Calculation of Magnetic Circular Dichroism of Azabenzenes and Phosphabenzene Using London Orbitals

A computational study of the Faraday B term of magnetic circular dichroism at the coupled cluster singles and doubles level is presented for pyridine, pyrazine, pyrimidine, and phosphabenzene. Gauge-origin independence is obtained by expressing the B term as a total derivative of the one-photon dipole transition strength and using London orbitals. The high quality of the coupled cluster singles and doubles (CCSD) B terms makes these useful for the assignment of experimental spectra. Previous assignments of the experimental spectra based on the qualitative perimeter model are confirmed by the CCSD results for the three azines, while a reassignment is proposed for phosphabenzene. For non-overlapping bands, the B terms calculated at the equilibrium geometries are in good agreement with the experimental values. For overlapping bands, large deviations occur. Attributing a line width to the calculated equilibrium B terms leads to a large cancellation of positive and negative contributions. This cancellation may result in a large displacement of the band center maximum, leading to a large uncertainty in the assignments of "vertical experimental excitation energies" (pyridine). Bands may also completely vanish due to such cancellation (phosphabenzene). Explicit consideration of the cancellation yields simulated theoretical spectra that are in good agreement with experiment once the theoretical spectra are parallel displaced. A major contribution for this parallel displacement is the shift in the excitation energies due to correlation beyond CCSD, as seen when comparing vertical CCSD and CC3 equilibrium-geometry excitation energies.

Calculation of static and dynamic linear magnetic response in approximate time-dependent density functional theory

We report implementations and results of time-dependent density functional calculations (i) of the frequency-dependent magnetic dipole-magnetic dipole polarizability, (ii) of the (observable) translationally invariant linear magnetic response, and (iii) of a linear intensity differential (LID) which includes the dynamic dipole magnetizability. The density functional calculations utilized density fitting. For achieving gauge-origin independence we have employed time-periodic magnetic-field-dependent basis functions as well as the dipole velocity gauge, and have included explicit density-fit related derivatives of the Coulomb potential. We present the results of calculations of static and dynamic magnetic dipole-magnetic dipole polarizabilities for a set of small molecules, the LID for the SF6 molecule, and dispersion curves for M-hexahelicene of the origin invariant linear magnetic response as well as of three dynamic polarizabilities: magnetic dipole-magnetic dipole, electric dipole-electric dipole, and electric dipole-magnetic dipole. We have also performed comparison of the linear magnetic response and magnetic dipole-magnetic dipole polarizability over a wide range of frequencies for H2O and SF6.

Calculation of origin-independent optical rotation tensor components in approximate time-dependent density functional theory

We outline an implementation of the origin-independent optical rotation tensor, which includes electric dipole-magnetic dipole and electric dipole-electric quadrupole polarizability. The method is based on approximate time-dependent density functional theory. We utilize time-periodic magnetic-field-dependent basis functions as well as a modified velocity-gauge formulation of dynamic polarizability tensors in order to obtain a gauge-origin independence. To ensure gauge-origin independence of the results within a given numerical accuracy, density fit coefficient derivatives are employed. A damping constant has been introduced into the linear response equations to treat both resonance and nonresonance regions of optical activity. We present calculations for trans-2,3-dimethyloxirane and derivatives thereof as well as calculations for androst-4,17-dien-3-one. In the Appendix, we derive the equivalence between the common-gauge origin and gauge-including atomic orbitals formulations for the optical rotation tensor in time-dependent DFT.

Solvent Effects on Raman Optical Activity Spectra Calculated Using the Polarizable Continuum Model

The integral equation formulation of the polarizable continuum model (IEFPCM) has been extended to the calculation of solvent effects on vibrational Raman optical activity spectra. Gauge-origin independence of the differential scattering intensities of right and left circularly polarized light is ensured through the use of London atomic orbitals. Density functional theory (DFT) calculations have been carried out for bromochlorofluoromethane, methyloxirane, and epichlorhydrin. The results indicate that solvent effects on the ROA differential scattering intensities can be substantial, and vary in sign and magnitude for different vibrational modes. It is demonstrated that both direct and indirect effects are important in determining the total solvent effects on the ROA differential scattering intensities. Local field effects are shown to be in general small, whereas electronic nonequilibrium solvation has a profound effect on the calculated solvent effects compared to an equilibrium solvation model. For molecules with several conformations, the changes in the relative stability of the different conformers also lead to noticeable changes in the ROA spectra.

Gauge-origin-independent magnetizabilities of solvated molecules using the polarizable continuum model

We present an implementation of the polarizable continuum model in its integral equation formulation for the calculation of the magnetizabilities of solvated molecules. The gauge-origin independence of the calculated magnetizabilities and the fast basis set convergence are ensured through the use of London atomic orbitals. Our implementation can use Hartree-Fock and multiconfigurational self-consistent-field (MCSCF) wave functions as well as density-functional theory including hybrid functionals such as B3LYP. We present the results of dielectric continuum effects on water and pyridine using MCSCF wave functions, as well as dielectric medium effects on the magnetizability of the aromatic amino acids as a model for how a surrounding protein environment affects the magnetizability of these molecules. It is demonstrated that the dielectric medium effects on the magnetizability anisotropies of the aromatic amino acids may be substantial, being as large as 25% in the case of tyrosine.

Calculation of optical rotation with time-periodic magnetic-field-dependent basis functions in approximate time-dependent density-functional theory

We report the implementation of a method for the calculation of optical rotation. This method is based on the time-dependent density-functional theory and utilizes time-periodic magnetic-field-dependent basis functions. The calculations are based on a density fit. It is demonstrated that additional terms in the analytical expression appearing from derivatives of the approximated Coulomb potential are necessary to provide the gauge-origin independence of the results within a given numerical accuracy. Contributions from these terms also restore the symmetry between the electric and magnetic perturbations in the optical rotation tensor.

Polarizable continuum model study of solvent effects on electronic circular dichroism parameters

We present an implementation of the polarizable continuum model for the calculation of solvent effects on electronic circular dichroism spectra. The computational model used is density functional theory in the length-gauge formulation, and gauge-origin independence is ensured through the use of London atomic orbitals. Results of calculations carried out for methyloxirane and bicyclic ketones, camphor, norcamphor, norbornenone, and fenchone are presented, and the theoretically obtained solvent effects are compared with experimental observations.

Density functional theory calculation of electronic circular dichroism using London orbitals

We present the first implementation of gauge-origin independent electronic circular dichroism (ECD) at the density functional theory level in the length-gauge formulation, where gauge origin independence has been ensured through the use of London atomic orbitals. We investigate the dependence of the rotatory strengths on the choice of gauge formulation, the size of the basis set, and on the choice of density functional for several selected small molecules. The ECD parameters are also calculated for a few larger molecules, and the results are compared with experiment where available, as well as with previously published theoretical numbers. Using the length-gauge formulation instead of the velocity-gauge gives in general better basis set convergence. However, large basis sets are still needed in order to get reliable ECD results, and a double set of diffuse functions are in general required.

Density-functional theory calculations of optical rotatory dispersion in the nonresonant and resonant frequency regions.

The complex linear response function, which can be employed for calculations of second-order molecular properties in regions of strong absorption, is here extended to encompass the mixed electric-dipole-magnetic-dipole polarizability. The mixed electric-dipole-magnetic-dipole polarizability determines the optical rotation and, when absorption is taken into account, the full anomalous optical rotatory dispersion (ORD) spectra of chiral molecules can be calculated using first-principle quantum-chemical methods. Gauge-origin independence of the results is ensured through the use of London atomic orbitals. To illustrate the importance of taking the absorption process properly into account, we here apply this methodology to the study of the anomalous ORD of hydrogen peroxide, 3R-methylcyclohexanone, 4R-1,1-dimethyl-[3]-(1,2)-ferrocenophan-2-on, and the D(2) isomer of the C(84) fullerene.

Gauge-Origin Independent Density-Functional Theory Calculations of Vibrational Raman Optical Activity

We present density-functional theory calculations of vibrational Raman optical activity. Gauge-origin independence of the results is ensured by using London atomic orbitals, and the frequency dependence is included by using analytical response theory. The Raman optical activity circular intensity differences are obtained by combining analytical calculations of linear response functions with numerical differentiation with respect to nuclear distortions. The results are compared with experiment, previous Hartree -Fock calculations, and a more recent approximate sum-over-states approach.

Solvent effects on the NMR parameters of H2S and HCN

A recently developed method to calculate singlet and triplet gauge-origin–independent magnetic properties of solvated molecules is applied to the study of those parameters that determine an observed nuclear magnetic resonance signal: the magnetizability, the nuclear shieldings, and the indirect spin–spin coupling constants. The solvent is represented by a dielectric medium and the electronic structure of the solvated molecule by Hartree–Fock and multiconfigurational Hartree–Fock wave functions. For the properties that depend on the external magnetic field, we use London atomic orbitals to ensure gauge-origin independence and a rapid basis-set convergence. We find that the dielectric-medium effects on these molecules are substantial, being of the same order as rovibrational and electron correlation effects, and thus cannot be neglected if accurate comparisons with liquid-phase measurements are wanted. However, the present model is incapable of describing the close-range interactions that may occur in solution. It, therefore, represents the electrostatic effects of the bulk solvent, and the model is an initial approach towards a complete ab initio model for the study of magnetic properties of solvated molecules. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1281–1291, 1999

Solvent effects on nuclear shieldings and spin–spin couplings of hydrogen selenide

Solvent effects on the nuclear shielding and indirect spin–spin coupling constants of H2Se have been calculated by modeling the surroundings as a continuous dielectric medium. Gauge-origin independence of the nuclear shieldings is ensured by using London atomic orbitals in combination with linear response theory. We present the linear response function of a solvated molecule subject to triplet perturbations and use a new implementation of this theory to evaluate the Fermi-contact and spin–dipole contributions to the indirect spin–spin coupling constants. We present high-level calculations of the nuclear shielding and indirect spin–spin coupling constants of H2Se in vacuum and different solvents. Our results represent the first ab initio calculations of the spin–spin coupling constants in H2Se as well as the first investigation of medium effects on these properties. It is demonstrated that the solvent shifts of the spin–spin couplings are caused by a polarization of the molecular electronic structure as well as by changes in the geometry upon solvation.