The approximate coupled-cluster methods CC2 and CC3 in a finite magnetic field.

In this paper, we report on the implementation of CC2 and CC3 in the context of molecules in finite magnetic fields. The methods are applied to the investigation of atoms and molecules through spectroscopic predictions and geometry optimizations for the study of the atmosphere of highly magnetized White Dwarf stars. We show that ground-state finite-field (ff) CC2 is a reasonable alternative to CCSD for energies and, in particular, for geometrical properties. For excited states, ff-CC2 is shown to perform well for states with predominant single-excitation character. Yet, for cases in which the excited state wavefunction has double-excitation character with respect to the reference, ff-CC2 can easily lead to completely unphysical results. Ff-CC3, however, is shown to reproduce the CCSDT behavior very well and enables the treatment of larger systems at a high accuracy.

Theoretical prediction of closed-shell paramagnetism for scandium and yttrium hydride.

Following chemical intuition, one would expect that all closed-shell molecules are diamagnetic. However, it is known that this is not the case for some second-row hydrides with low-lying unoccupied π orbitals due to an unquenching of the total angular momentum in the presence of an external magnetic field. In this article, the transition-metal hydrides ScH and YH are investigated, assuming a similar unquenching effect involving low-lying unoccupied π and δ orbitals formed from the metal d orbitals rather than the p orbitals. We are comparing results obtained with various quantum-chemical methods (HF, CCSD, CCSD(T), CCSDT) and basis sets. The obtained positive values for the magnetizabilities clearly indicate paramagnetic behavior. Vibrational effects on the magnetizability tensor are also considered, but these effects are small and do not change the overall conclusion that both ScH and YH are further examples for closed-shell paramagnetism.

Cholesky Decomposition-Based Implementation of Relativistic Two-Component Coupled-Cluster Methods for Medium-Sized Molecules.

A Cholesky decomposition (CD)-based implementation of relativistic two-component coupled-cluster (CC) and equation-of-motion CC (EOM-CC) methods using an exact two-component Hamiltonian augmented with atomic-mean-field spin-orbit integrals (the X2CAMF scheme) is reported. The present CD-based implementation of X2CAMF-CC and EOM-CC methods employs atomic-orbital-based algorithms to avoid the construction of two-electron integrals and intermediates involving three and four virtual indices. Our CD-based implementation extends the applicability of X2CAMF-CC and EOM-CC methods to medium-sized molecules with the possibility to correlate around 1000 spinors. Benchmark calculations for uranium-containing small molecules were performed to assess the dependence of the CC results on the Cholesky threshold. A Cholesky threshold of 10-4 is shown to be sufficient to maintain chemical accuracy. Example calculations to illustrate the capability of the CD-based relativistic CC methods are reported for the bond-dissociation energy of the uranium hexafluoride molecule, UF6, with up to quadruple-ζ basis sets, and the lowest excitation energy in the solvated uranyl ion [UO22+(H2O)12].

Magnetic optical rotation from real-time simulations in finite magnetic fields.

We present a numerical approach to magnetic optical rotation based on real-time time-dependent electronic-structure theory. Not relying on perturbation expansions in the magnetic field strength, the formulation allows us to test the range of validity of the linear relation between the rotation angle per unit path length and the magnetic field strength that was established empirically by Verdet 160 years ago. Results obtained from time-dependent coupled-cluster and time-dependent current density-functional theory are presented for the closed-shell molecules H2, HF, and CO in magnetic fields up to 55 kT at standard temperature and pressure conditions. We find that Verdet's linearity remains valid up to roughly 10-20 kT, above which significant deviations from linearity are observed. Among the three current density-functional approximations tested in this work, the current-dependent Tao-Perdew-Staroverov-Scuseria hybrid functional performs the best in comparison with time-dependent coupled-cluster singles and doubles results for the magnetic optical rotation.

Computation of NMR shieldings at the CASSCF level using gauge-including atomic orbitals and Cholesky decomposition.

We present an implementation of coupled-perturbed complete active space self-consistent field (CP-CASSCF) theory for the computation of nuclear magnetic resonance chemical shifts using gauge-including atomic orbitals and Cholesky decomposed two-electron integrals. The CP-CASSCF equations are solved using a direct algorithm where the magnetic Hessian matrix-vector product is expressed in terms of one-index transformed quantities. Numerical tests on systems with up to about 1300 basis functions provide information regarding both the computational efficiency and limitations of our implementation.

Cholesky decomposition of complex two-electron integrals over GIAOs: Efficient MP2 computations for large molecules in strong magnetic fields.

In large-scale quantum-chemical calculations, the electron-repulsion integral (ERI) tensor rapidly becomes the bottleneck in terms of memory and disk space. When an external finite magnetic field is employed, this problem becomes even more pronounced because of the reduced permutational symmetry and the need to work with complex integrals and wave function parameters. One way to alleviate the problem is to employ a Cholesky decomposition (CD) to the complex ERIs over gauge-including atomic orbitals. The CD scheme establishes favorable compression rates by selectively discarding linearly dependent product densities from the chosen basis set while maintaining a rigorous and robust error control. This error control constitutes the main advantage over conceptually similar methods such as density fitting, which relies on employing pre-defined auxiliary basis sets. We implemented the use of the CD in the framework of finite-field (ff) Hartree-Fock and ff second-order Møller-Plesset perturbation theory (MP2). Our work demonstrates that the CD compression rates are particularly beneficial in calculations in the presence of a finite magnetic field. The ff-CD-MP2 scheme enables the correlated treatment of systems with more than 2000 basis functions in strong magnetic fields within a reasonable time span.

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.

Spin contamination in MP2 and CC2, a surprising issue.

When calculating the spin multiplicity at either the second-order Møller-Plesset (MP2) or the iterative second-order approximate coupled-cluster singles and doubles (CC2) levels of theory using the same strategy for the calculation of the expectation value as in regular CC theory together with the usual definitions of the MP2 and CC2 density matrices, artificial spin contamination occurs in closed-shell molecules. Non-intuitively, for open-shell systems, results at the MP2 or CC2 levels of theory based on this procedure even suggest stronger contamination at the correlated level than for the Hartree-Fock reference, although treatment of electron correlation should lower spin contamination. In this Communication, the reasons behind this inconsistency are investigated and a solution is proposed, which removes spin contamination for closed-shell molecules and leads to physically meaningful results for open-shell cases. Additionally, we show that CC2 significantly outperforms MP2 in describing systems with a strongly spin-contaminated reference with a performance similar to that of full coupled-cluster with singles and doubles substitutions (CCSD).

Full triples contribution in coupled-cluster and equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fields.

Coupled-cluster as well as equation-of-motion coupled-cluster methods play an important role whenever high accuracy is warranted. Concerning excitation energies, consideration of triple excitations is typically required to reach an accuracy better than 0.1-0.3 eV. In the context of strong magnetic fields such accuracy is needed for the prediction of spectra of strongly magnetized White Dwarfs. In addition it turns out that in order to correctly model the behavior of energies with respect to the magnetic field strength, triple excitations are required. Due to avoided crossings which are extremely often encountered in the context of strong magnetic fields, double-excitation character can be transferred between electronic states of the same symmetry. We report an implementation of the full finite-field coupled-cluster with single, double, and triple substitutions (CCSDT) and the equation-of-motion-CCSDT models and apply them to the prediction of field-dependent transition wavelengths for sodium as well as to the four lowest singlet states of the CH+ molecule in a strong magnetic field.

Coupled-cluster techniques for computational chemistry: The CFOUR program package.

An up-to-date overview of the CFOUR program system is given. After providing a brief outline of the evolution of the program since its inception in 1989, a comprehensive presentation is given of its well-known capabilities for high-level coupled-cluster theory and its application to molecular properties. Subsequent to this generally well-known background information, much of the remaining content focuses on lesser-known capabilities of CFOUR, most of which have become available to the public only recently or will become available in the near future. Each of these new features is illustrated by a representative example, with additional discussion targeted to educating users as to classes of applications that are now enabled by these capabilities. Finally, some speculation about future directions is given, and the mode of distribution and support for CFOUR are outlined.

GW quasiparticle energies of atoms in strong magnetic fields.

Quasiparticle energies of the atoms H-Ne have been computed in the GW approximation in the presence of strong magnetic fields with field strengths varying from 0 to 0.25 atomic units (0.25 B0=0.25 ℏe-1a0 -2≈58 763 T). The GW quasiparticle energies are compared with equation-of-motion ionization-potential (EOM-IP) coupled-cluster singles-and-doubles (CCSD) calculations of the first ionization energies. The best results are obtained with the evGW@PBE0 method, which agrees with the EOM-IP-CCSD model to within about 0.20 eV. Ionization potentials have been calculated for all atoms in the series, representing the first systematic study of ionization potentials for the first-row atoms at field strengths characteristic of magnetic white dwarf stars. Under these conditions, the ionization potentials increase in a near-linear fashion with the field strength, reflecting the linear field dependence of the Landau energy of the ionized electron. The calculated ionization potentials agree well with the best available literature data for He, Li, and Be.

Transition-Dipole Moments for Electronic Excitations in Strong Magnetic Fields Using Equation-of-Motion and Linear Response Coupled-Cluster Theory.

An implementation of transition-dipole moments at the equation-of-motion coupled-cluster singles-doubles (EOM-CCSD) and CCSD linear response (LR) levels of theory for the treatment of atoms and molecules in strong magnetic fields is presented. The presence of a finite magnetic field leads, in general, to a complex wave function and a gauge-origin dependence, necessitating a complex computer code together with the use of gauge-including atomic orbitals. As in the field-free case, for EOM-CC, the evaluation of transition-dipole moments consists of setting up the one-electron transition-density matrix (TDM) which is then contracted with dipole-moment integrals. In the case of CC-LR, the evaluation proceeds with a modified TDM but additionally requires a second contribution accounting for the amplitude response which is missing in EOM-CC theory for properties. We present a selected set of transitions for the sodium atom and investigate the LiH molecule in both a parallel as well as a perpendicular magnetic field. The dependence of excited-state energies and transition moments on the magnetic-field strength is discussed with a focus on magnetic-field-induced avoided crossings. Additionally, the differences between field-dependent EOM-CCSD and CCSD-LR transition moments are investigated.

Equation-of-motion coupled-cluster methods for atoms and molecules in strong magnetic fields.

A program for the direct calculation of excitation energies of atoms and molecules in strong magnetic fields is presented. The implementation includes the equation-of-motion coupled-cluster singles-doubles (EOM-CCSD) method for electronically excited states as well as its spin-flip variant. Differences to regular EOM-CCSD implementations are due to the appearance of the canonical angular-momentum operator in the Hamiltonian causing the wave function to become complex. The gauge-origin problem is treated by the use of gauge-including atomic orbitals. Therefore, a modified Davidson method for diagonalizing complex non-Hermitian matrices is used. Excitation energies for selected atoms and molecules that are of importance in the astrochemical context are presented and their dependence on the magnetic field is discussed.

Current Density Functional Theory Using Meta-Generalized Gradient Exchange-Correlation Functionals.

We present the self-consistent implementation of current-dependent (hybrid) meta-generalized gradient approximation (mGGA) density functionals using London atomic orbitals. A previously proposed generalized kinetic energy density is utilized to implement mGGAs in the framework of Kohn-Sham current density functional theory (KS-CDFT). A unique feature of the nonperturbative implementation of these functionals is the ability to seamlessly explore a wide range of magnetic fields up to 1 au (∼235 kT) in strength. CDFT functionals based on the TPSS and B98 forms are investigated, and their performance is assessed by comparison with accurate coupled-cluster singles, doubles, and perturbative triples (CCSD(T)) data. In the weak field regime, magnetic properties such as magnetizabilities and nuclear magnetic resonance shielding constants show modest but systematic improvements over generalized gradient approximations (GGA). However, in the strong field regime, the mGGA-based forms lead to a significantly improved description of the recently proposed perpendicular paramagnetic bonding mechanism, comparing well with CCSD(T) data. In contrast to functionals based on the vorticity, these forms are found to be numerically stable, and their accuracy at high field suggests that the extension of mGGAs to CDFT via the generalized kinetic energy density should provide a useful starting point for further development of CDFT approximations.

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.

Spin-orbit couplings within the equation-of-motion coupled-cluster framework: Theory, implementation, and benchmark calculations.

We present a formalism and an implementation for calculating spin-orbit couplings (SOCs) within the EOM-CCSD (equation-of-motion coupled-cluster with single and double substitutions) approach. The following variants of EOM-CCSD are considered: EOM-CCSD for excitation energies (EOM-EE-CCSD), EOM-CCSD with spin-flip (EOM-SF-CCSD), EOM-CCSD for ionization potentials (EOM-IP-CCSD) and electron attachment (EOM-EA-CCSD). We employ a perturbative approach in which the SOCs are computed as matrix elements of the respective part of the Breit-Pauli Hamiltonian using zeroth-order non-relativistic wave functions. We follow the expectation-value approach rather than the response-theory formulation for property calculations. Both the full two-electron treatment and the mean-field approximation (a partial account of the two-electron contributions) have been implemented and benchmarked using several small molecules containing elements up to the fourth row of the periodic table. The benchmark results show the excellent performance of the perturbative treatment and the mean-field approximation. When used with an appropriate basis set, the errors with respect to experiment are below 5% for the considered examples. The findings regarding basis-set requirements are in agreement with previous studies. The impact of different correlation treatment in zeroth-order wave functions is analyzed. Overall, the EOM-IP-CCSD, EOM-EA-CCSD, EOM-EE-CCSD, and EOM-SF-CCSD wave functions yield SOCs that agree well with each other (and with the experimental values when available). Using an EOM-CCSD approach that provides a more balanced description of the target states yields more accurate results.

The importance of current contributions to shielding constants in density-functional theory.

The sources of error in the calculation of nuclear-magnetic-resonance shielding constants determined by density-functional theory are examined. Highly accurate Kohn-Sham wave functions are obtained from coupled-cluster electron density functions and used to define accurate-but current independent-density-functional shielding constants. These new reference values, in tandem with high-accuracy coupled-cluster shielding constants, provide a benchmark for the assessment of errors in common density-functional approximations. In particular the role of errors arising in the diamagnetic and paramagnetic terms is investigated, with particular emphasis on the role of current-dependence in the latter. For carbon and nitrogen the current correction is found to be, in some cases, larger than 10 ppm. This indicates that the absence of this correction in general purpose exchange-correlation functionals is one of the main sources of error in shielding calculations using density functional theory. It is shown that the current correction improves the shielding performance of many popular approximate DFT functionals.

Spin-free Dirac-Coulomb calculations augmented with a perturbative treatment of spin-orbit effects at the Hartree-Fock level.

A perturbative approach to compute second-order spin-orbit (SO) corrections to a spin-free Dirac-Coulomb Hartree-Fock (SFDC-HF) calculation is suggested. The proposed scheme treats the difference between the DC and SFDC Hamiltonian as perturbation and exploits analytic second-derivative techniques. In addition, a cost-effective scheme for incorporating relativistic effects in high-accuracy calculations is suggested consisting of a SFDC coupled-cluster treatment augmented by perturbative SO corrections obtained at the HF level. Benchmark calculations for the hydrogen halides HX, X = F-At as well as the coinage-metal fluorides CuF, AgF, and AuF demonstrate the accuracy of the proposed perturbative treatment of SO effects on energies and electrical properties in comparison with the more rigorous full DC treatment. Furthermore, we present, as an application of our scheme, results for the electrical properties of AuF and XeAuF.

The route to high accuracy in ab initio calculations of Cu quadrupole-coupling constants.

We report nonrelativistic and scalar-relativistic coupled-cluster calculations of the copper quadrupole-coupling constants for eleven small copper-containing compounds. It is shown to be necessary to treat both electron-correlation and scalar-relativistic effects on the same footing even for a qualitatively correct description, because both effects are significant and are strongly coupled in the case of Cu electric-field gradients. We show that the three scalar-relativistic schemes employed in the present study--the leading order of direct perturbation theory, the spin-free exact two-component theory in its one-electron variant, and the spin-free Dirac-Coulomb approach--provide accurate treatments of scalar-relativistic effects for the copper compounds under study. Furthermore, we demonstrate that results close to the basis-set limit can be obtained by augmenting large uncontracted standard basis sets for copper with additional steep functions. It is also shown that high-level correlation effects (those beyond the perturbative treatment of triple excitations) make important contributions in the present case.

Rotational spectra of rare isotopic species of fluoroiodomethane: determination of the equilibrium structure from rotational spectroscopy and quantum-chemical calculations.

Supported by accurate quantum-chemical calculations, the rotational spectra of the mono- and bi-deuterated species of fluoroiodomethane, CHDFI and CD(2)FI, as well as of the (13)C-containing species, (13)CH(2)FI, were recorded for the first time. Three different spectrometers were employed, a Fourier-transform microwave spectrometer, a millimeter/submillimter-wave spectrometer, and a THz spectrometer, thus allowing to record a huge portion of the rotational spectrum, from 5 GHz up to 1.05 THz, and to accurately determine the ground-state rotational and centrifugal-distortion constants. Sub-Doppler measurements allowed to resolve the hyperfine structure of the rotational spectrum and to determine the complete iodine quadrupole-coupling tensor as well as the diagonal elements of the iodine spin-rotation tensor. The present investigation of rare isotopic species of CH(2)FI together with the results previously obtained for the main isotopologue [C. Puzzarini, G. Cazzoli, J. C. López, J. L. Alonso, A. Baldacci, A. Baldan, S. Stopkowicz, L. Cheng, and J. Gauss, J. Chem. Phys. 134, 174312 (2011); G. Cazzoli, A. Baldacci, A. Baldan, and C. Puzzarini, Mol. Phys. 109, 2245 (2011)] enabled us to derive a semi-experimental equilibrium structure for fluoroiodomethane by means of a least-squares fit procedure using the available experimental ground-state rotational constants together with computed vibrational corrections. Problems related to the missing isotopic substitution of fluorine and iodine were overcome thanks to the availability of an accurate theoretical equilibrium geometry (computed at the coupled-cluster singles and doubles level augmented by a perturbative treatment of triple excitations).