Direct perturbation theory in terms of energy derivatives: fourth-order relativistic corrections at the Hartree-Fock level

In this work, the quantum-chemical treatment of relativistic effects by means of direct perturbation theory is extended from its lowest order, DPT2, to the next higher order, DPT4. The required theory is given in terms of energy derivatives with the DPT4 energy correction defined as the corresponding second derivative with respect to the relativistic perturbation parameter λ(rel) = c(2) and c as the speed of light. To facilitate the implementation in standard quantum-chemical program packages, a general formulation of DPT starting from a nonrelativistic Lagrangian is developed, thereby expanding both wave function and operators in terms of λ(rel). The corresponding expressions, which incorporate in an additive manner scalar-relativistic and spin-orbit contributions, are given at the Hartree-Fock level and have been implemented in the CFOUR program package using the available analytic second-derivative techniques. The accuracy of the DPT4 corrections at the HF level is investigated by comparison with rigorous four-component calculations. Scalar-relativistic and spin-orbit contributions are analyzed individually and the importance of the various terms to those corrections is discussed. Furthermore, the basis-set dependence of the computed DPT4 corrections is investigated.     

Direct perturbation theory in terms of energy derivatives: scalar-relativistic treatment up to sixth order

A formulation of sixth-order direct perturbation theory (DPT) to treat relativistic effects in quantum-chemical calculations is presented in the framework of derivative theory. Detailed expressions for DPT6 are given at the Hartree-Fock level in terms of the third derivative of the energy with respect to the relativistic perturbation parameter defined as λ(rel)=c(-2). They were implemented for the computation of scalar-relativistic energy corrections. The convergence of the scalar-relativistic DPT expansion is studied for energies and first-order properties such as dipole moment and electric-field gradient within the series of the hydrogen halides (HX, X = F, Cl, Br, I, and At). Comparison with spin-free Dirac-Coulomb calculations indicates that the DPT series exhibits a smooth and monotonic convergence. The rate of convergence, however, depends on the charge of the involved nuclei and significantly slows down for heavy-element compounds.     

Relativistic energy-consistent pseudopotentials--recent developments

The direct adjustment of two-component pseudopotentials (scalar-relativistic + spin-orbit potentials), to atomic total energy valence spectra derived from four-component multiconfiguration Dirac-Hartree-Fock all-electron calculations based on the Dirac-Coulomb-Breit Hamiltonian, has been made a routine tool for an efficient treatment of heavy main-group elements. Both large-core (nsp valence shell) and small-core ((n - 1)spd nsp valence shell) potentials have been generated for all the post-d elements of groups 13-17. At the example of lead and bismuth compounds (PbHal, BiH, BiO, BiHal (Hal = F, Cl, Br, I)), we show how small-core and large-core potentials can be combined in accurate, yet computationally economic, spin-free-state-shifted relativistic electronic structure calculations of molecular ground and excited states.     

Leading-order relativistic effects on nuclear magnetic resonance shielding tensors

We present perturbational ab initio calculations of the nuclear-spin-dependent relativistic corrections to the nuclear magnetic resonance shielding tensors that constitute, together with the other relativistic terms reported by us earlier, the full leading-order perturbational set of results for the one-electron relativistic contributions to this observable, based on the (Breit-)Pauli Hamiltonian. These contributions are considered for the H(2)X (X = O,S,Se,Te,Po) and HX (X = F,Cl,Br,I,At) molecules, as well as the noble gas (Ne, Ar, Kr, Xe, Rn) atoms. The corrections are evaluated using the relativistic and magnetic operators as perturbations on an equal footing, calculated using analytical linear and quadratic response theory applied on top of a nonrelativistic reference state provided by self-consistent field calculations. The (1)H and heavy-atom nuclear magnetic shielding tensors are compared with four component, nearly basis-set-limit Dirac-Hartree-Fock calculations that include positronic excitations, as well as available literature data. Besides the easy interpretability of the different contributions in terms of familiar nonrelativistic concepts, the accuracy of the present perturbational scheme is striking for the isotropic part of the shielding tensor, for systems including elements up to Xe.     

PtF6(2-) dianion and its detachment spectrum: a fully relativistic study

In this work we calculate the photoelectron spectrum of the PtF(6)2- dianion by application of the third-order Dirac-Hartree-Fock one-particle propagator technique. Relativistic effects and electron correlation are hereby treated on a consistent theoretical basis which is mandatory for systems containing heavy elements. A PtF6(2-) gas phase photoelectron spectrum is not yet available and our calculations therefore have predictive character. As it is characteristic for dianionic systems a strong dependence on basis set size and molecular geometry is observed. In contrast to the already calculated PtCl(6)2- photoelectron spectrum no valence orbital inversion due to strong interplay of spin-orbit coupling and electron correlation is observed. Furthermore an unusually strong spin-orbit splitting was found for the sigma-type subvalence 1t1u molecular spinor despite its very small platinum p population. The double ionization threshold is strongly lowered by relativistic effects now enabling an interatomic Coulombic decay process after ionization from the sigma-bonding orbitals. The results stress the importance of spin-orbit coupling for the understanding of the spectral structure which cannot be reproduced by a scalar-relativistic treatment only.     

Remarkable interplay of electron correlation and relativity in the photodetachment spectrum of PtCl(6)2-

In this work we calculate the photoelectron spectrum of the PtCl(6) (2-) dianion by application of the recently developed third-order Dirac-Hartree-Fock implementation of the one-particle propagator technique allowing for a consistent treatment of spin-orbit and scalar relativistic effects together with electron correlation. For PtCl(6) (2-) a gas phase photoelectron spectrum is available showing clearly discernible structures not reproducible by a nonrelativistic or purely scalar-relativistic computation. A population analysis of the valence orbitals allows for an assignment of the photoelectron peaks and reveals the strong influence of relativity in combination with electron correlation.     

Relativistic calculation of indirect NMR spin-spin couplings using the Douglas-Kroll-Hess approximation

We have employed the Douglas-Kroll-Hess approximation to derive the perturbative Hamiltonians involved in the calculation of NMR spin-spin couplings in molecules containing heavy elements. We have applied this two-component quasirelativistic approach using finite perturbation theory in combination with a generalized Kohn-Sham code that includes the spin-orbit interaction self-consistently and works with Hartree-Fock and both pure and hybrid density functionals. We present numerical results for one-bond spin-spin couplings in the series of tetrahydrides CH(4), SiH(4), GeH(4), and SnH(4). Our two-component Hartree-Fock results are in good agreement with four-component Dirac-Hartree-Fock calculations, although a density-functional treatment better reproduces the available experimental data.     

Importance of spin-orbit coupling for the assignment of the photodetachment spectra of AuX2- (X=Cl, Br, and I).

Recently, photodetachment spectra of AuX2-, X=Cl, Br, and I, have been reported [D. Schr?der et al., Angew. Chem. Int. Ed. 2003, 42, 311] followed by a scalar-relativistic theoretical study of the assignment of these spectra [B. Dai, J. Yang, Chem. Phys. Lett. 2003, 379, 512]. Herein, the photodetachment spectra of the title molecules are reassigned, taking spin-orbit coupling into account and employing relativistic-effective core potentials for gold and the halogen atoms. The composition of the AuX2 electronic states are further analyzed in terms of scalar-relativistic electronic states. The relevance of spin-orbit coupling in the spectroscopy of heavy elements is emphasized by comparing the results of this work with experiment and with other scalar-relativistic theoretical studies.     

Through-space spin-spin coupling in van der Waals dimers and CH/pi interacting systems. An ab initio and DFT study

The through-space J(HH) and J(CH) spin-spin coupling constants of model van der Waals dimers (involving methane, ethylene, and benzene), and of selected compounds showing the CH/pi interaction, have been investigated by means of DFT and ab initio calculations. In the range of intermolecular separations for which the interaction is stabilizing, weak couplings (0.1-0.3 Hz) are predicted for J(CH), while the corresponding J(HH) couplings are much smaller. The relative contributions (Fermi-contact, spin-orbit, and spin-dipole) are strongly dependent on the geometry of the dimers and on the distance; the non-negligible values of J(CH) for pi systems stem largely from an incomplete cancellation of spin-orbit terms. The results obtained for the larger molecules, that is, acetonitrile@calix[4]arene 5, the imine 6, and the aryl ester 7 are consistent with those on the model dimers. For 7, the occurrence of a through-space mechanism for the transmission of coupling is established by examining trends in the magnitude of couplings as a function of the number of intervening covalent bonds.     

Structural and spectroscopic study of the excited electronic states of silver dihalides by quantum chemical methods

The structural and electronic properties of the excited electronic states of AgX(2) (X = F, Cl, Br, and I), have been calculated, taking electron correlation and spin-orbit coupling into account and employing improved relativistic-effective-core potentials for silver and the halogen atoms. The relative ordering of the excited states of these molecules has been discussed via molecular-orbital arguments. The spin-orbit splittings of three degenerate electronic states ((2)Pi(g), (2)Pi(u), and (2)Delta(g)) have been calculated and the spin-orbit induced inter-state (Sigma - Pi) coupling has been discussed. The composition of the spin-orbit eigenstates is analyzed in terms of scalar-relativistic electronic states. Finally, a theoretical prediction of the photodetachment bands of the title molecules has been accomplished.     

The barrier height of the F+H2 reaction revisited: coupled-cluster and multireference configuration-interaction benchmark calculations

Large scale coupled-cluster benchmark calculations have been carried out to determine the barrier height of the F+H2 reaction as accurately as possible. The best estimates for the barrier height of the linear and bent transition states amount to 2.16 and 1.63 kcal/mol, respectively. These values include corrections for core correlation, scalar-relativistic effects, spin-orbit effects, as well as the diagonal Born-Oppenheimer correction. The CCSD(T) basis-set limits are estimated using extrapolation techniques with augmented quintuple and sextuple-zeta basis sets, and remaining N-electron errors are determined using coupled-cluster singles, doubles, triples, quadruples calculations with up to augmented quintuple-zeta basis sets. The remaining uncertainty is estimated to be less than 0.1 kcal/mol. The coupled-cluster results are used to calibrate multireference configuration-interaction calculations with empirical scaling of the correlation energy.     

Density functional theory study of indirect nuclear spin-spin coupling constants with spin-orbit corrections

This work outlines the calculation of indirect nuclear spin-spin coupling constants with spin-orbit corrections using density functional response theory. The nonrelativistic indirect nuclear spin-spin couplings are evaluated using the linear response method, whereas the relativistic spin-orbit corrections are computed using quadratic response theory. The formalism is applied to the homologous systems H2X (X=O,S,Se,Te) and XH4 (X=C,Si,Ge,Sn,Pb) to calculate the indirect nuclear spin-spin coupling constants between the protons. The results confirm that spin-orbit corrections are important for compounds of the H2X series, for which the electronic structure allows for an efficient coupling between the nuclei mediated by the spin-orbit interaction, whereas in the case of the XH4 series the opposite situation is encountered and the spin-orbit corrections are negligible for all compounds of this series. In addition we analyze the performance of the density functional theory in the calculations of nonrelativistic indirect nuclear spin-spin coupling constants.     

Analytical evaluation of first-order electrical properties based on the spin-free Dirac-Coulomb Hamiltonian

We report an analytical scheme for the calculation of first-order electrical properties using the spin-free Dirac-Coulomb (SFDC) Hamiltonian, thereby exploiting the well-developed density-matrix formulations in nonrelativistic coupled-cluster (CC) derivative theory. Orbital relaxation effects are fully accounted for by including the relaxation of the correlated orbitals with respect to orbitals of all types, viz., frozen-core, occupied, virtual, and negative energy state orbitals. To demonstrate the applicability of the presented scheme, we report benchmark calculations for first-order electrical properties of the hydrogen halides, HX with X = F, Cl, Br, I, At, and a first application to the iodo(fluoro)methanes, CH(n)F(3 - n)I, n = 0-3. The results obtained from the SFDC calculations are compared to those from nonrelativistic calculations, those obtained via leading-order direct perturbation theory as well as those from full Dirac-Coulomb calculations. It is shown that the full inclusion of spin-free (SF) relativistic effects is necessary to obtain accurate first-order electrical properties in the presence of fifth-row elements. The SFDC scheme is also recommended for applications to systems containing lighter elements because it introduces no extra cost in the rate-determining steps of a CC calculation in comparison to the nonrelativistic case. On the other hand, spin-orbit contributions are generally small for first-order electrical properties of closed-shell molecules and may be handled efficiently by means of perturbation theory.     

Ab initio treatment of the chemical reaction precursor complex Br(2P)-HCN. 2. Bound-state calculations and infrared spectra

Rovibronic energy levels and properties of the Br(2P)-HCN complex were obtained from three-dimensional calculations, with HCN kept linear and the CN bond frozen. All diabatic states that correlate to the 2P3/2 and 2P1/2 states of the Br atom were included and spin-orbit coupling was taken into account. The 3 x 3 matrix of diabatic potential surfaces was taken from the preceding paper (paper 1). In agreement with experiment, we found two linear isomers, Br-NCH and Br-HCN. The calculated binding energies are very similar: D0 = 352.4 cm(-1) and D0 = 349.1 cm(-1), respectively. We established, also in agreement with experiment, that the ground electronic state of Br-NCH has |Omega| = (1/2) and that Br-HCN has a ground state with |Omega| = (3/2), where the quantum number, Omega, is the projection of the total angular momentum, J, of the complex on the intermolecular axis R. This picture can be understood as being caused by the electrostatic interaction between the quadrupole of the Br(2P) atom and the dipole of HCN, combined with the very strong spin-orbit coupling in Br. We predicted the frequencies of the van der Waals modes of both isomers and found a direct Renner-Teller splitting of the bend mode in Br-HCN and a smaller, indirect, splitting in Br-NCH. The red shift of the CH stretch frequency in the complex, relative to free HCN, was calculated to be 1.98 cm(-1) for Br-NCH and 23.11 cm(-1) for Br-HCN, in good agreement with the values measured in helium nanodroplets. Finally, with the use of the same potential surfaces, we modeled the Cl(2P)-HCN complex and found that the experimentally observed linear Cl-NCH isomer is considerably more stable than the (not observed) Cl-HCN isomer. This was explained mainly as an effect of the substantially smaller spin-orbit coupling in Cl, relative to Br.     

Relativistic heavy-atom effects on heavy-atom nuclear shieldings

The principal relativistic heavy-atom effects on the nuclear magnetic resonance (NMR) shielding tensor of the heavy atom itself (HAHA effects) are calculated using ab initio methods at the level of the Breit-Pauli Hamiltonian. This is the first systematic study of the main HAHA effects on nuclear shielding and chemical shift by perturbational relativistic approach. The dependence of the HAHA effects on the chemical environment of the heavy atom is investigated for the closed-shell X(2+), X(4+), XH(2), and XH(3) (-) (X=Si-Pb) as well as X(3+), XH(3), and XF(3) (X=P-Bi) systems. Fully relativistic Dirac-Hartree-Fock calculations are carried out for comparison. It is necessary in the Breit-Pauli approach to include the second-order magnetic-field-dependent spin-orbit (SO) shielding contribution as it is the larger SO term in XH(3) (-), XH(3), and XF(3), and is equally large in XH(2) as the conventional, third-order field-independent spin-orbit contribution. Considering the chemical shift, the third-order SO mechanism contributes two-thirds of the difference of approximately 1500 ppm between BiH(3) and BiF(3). The second-order SO mechanism and the numerically largest relativistic effect, which arises from the cross-term contribution of the Fermi contact hyperfine interaction and the relativistically modified spin-Zeeman interaction (FC/SZ-KE), are isotropic and practically independent of electron correlation effects as well as the chemical environment of the heavy atom. The third-order SO terms depend on these factors and contribute both to heavy-atom shielding anisotropy and NMR chemical shifts. While a qualitative picture of heavy-atom chemical shifts is already obtained at the nonrelativistic level of theory, reliable shifts may be expected after including the third-order SO contributions only, especially when calculations are carried out at correlated level. The FC/SZ-KE contribution to shielding is almost completely produced in the s orbitals of the heavy atom, with values diminishing with the principal quantum number. The relative contributions converge to universal fractions for the core and subvalence ns shells. The valence shell contribution is negligible, which explains the HAHA characteristics of the FC/SZ-KE term. Although the nonrelativistic theory gives correct chemical shift trends in present systems, the third-order SO-I terms are necessary for more reliable predictions. All of the presently considered relativistic corrections provide significant HAHA contributions to absolute shielding in heavy atoms.     

Spin-orbit Ab initio investigation of the photodissociation of dibromomethane in the gas and solution phases

A clear and reliable theoretical investigation on dibromomethane (CH(2)Br(2)) photodissociation is desired. The calculation must consider: (i) relativistic effects; (ii) the potential energy curves (PECs) of spin-orbit coupling states; (iii) geometry optimization by the method with both static and dynamic electron correlations; (iv) solvent effects on the photodissociation in the solution. All these have been considered in this study by state-of-the-art quantum chemical calculations. The experimentally observed photodissociation in the gas phase with products of spin-orbit-coupled states, Br((2)P(3/2)) and Br*((2)P(1/2)), was assigned by multi-state second order multiconfigurational perturbation theory in conjunction with spin-orbit interaction through complete active space state interaction (MS-CASPT2/CASSI-SO) PECs. The mechanisms of the experimentally observed photodissociation and photoisomerization in solvent were elucidated by the MS-CASPT2/CASSI-SO method combined with polarized continuum model of the solvent.     

Relativistic spin-orbit effects on hyperfine coupling tensors by density-functional theory

A second-order perturbation theory treatment of spin-orbit corrections to hyperfine coupling tensors has been implemented within a density-functional framework. The method uses the all-electron atomic mean-field approximation and/or spin-orbit pseudopotentials in incorporating one- and two-electron spin-orbit interaction within a first-principles framework. Validation of the approach on a set of main-group radicals and transition metal complexes indicates good agreement between all-electron and pseudopotential results for hyperfine coupling constants of the lighter nuclei in the system, except for cases in which scalar relativistic effects become important. The nonrelativistic Fermi contact part of the isotropic hyperfine coupling constants is not always accurately reproduced by the exchange-correlation functionals employed, particularly for the triplet and pi-type doublet radicals in the present work. For this reason, ab initio coupled-cluster singles and doubles with perturbative triples results for the first-order contributions have been combined in the validation calculations with the density-functional results for the second-order spin-orbit contributions. In the cases where spin-orbit corrections are of significant magnitude relative to the nonrelativistic first-order terms, they improve the agreement with experiment. Antisymmetric contributions to the hyperfine tensor arise from the spin-orbit contributions and are discussed for the IO2 radical, whereas rovibrational effects have been evaluated for RhC, NBr, and NI.     

Periodic trends in indirect nuclear spin-spin coupling tensors: relativistic density functional calculations for interhalogen diatomics

There have been significant advances in the calculation and interpretation of indirect nuclear spin-spin coupling (J) tensors during the past few years; however, much work remains to be done, especially for molecules containing heavy atoms where relativistic effects may play an important role. Many J tensors cannot be explained based solely on a nonrelativistic Fermi-contact mechanism. In the present work, the relativistic zeroth-order regular approximation density-functional (ZORA-DFT) implementation for the calculation of J has been applied to the complete series of homonuclear and heteronuclear diatomic halogen molecules: F(2), Cl(2), Br(2), I(2), At(2), ClF, BrF, IF, ClBr, ClI, and BrI. For all of these compounds, the reduced isotropic coupling constant (K(iso)) is positive and the reduced anisotropic coupling constant (DeltaK) is negative. With the exception of molecular fluorine, the magnitudes of K(iso) and DeltaK are shown to increase linearly with the product of the atomic numbers of the coupled nuclei. ZORA-DFT calculations of J for F(2) and ClF are in excellent agreement with the results obtained from multiconfigurational self-consistent-field calculations. The relative importance of the various coupling mechanisms is approximately constant for all of the compounds, with the paramagnetic spin-orbit term being the dominant contributor to K(iso), at approximately 70-80%. Available experimental stimulated resonant Raman spectroscopy data are exploited to extract the complete J((127)I,(127)I) tensor for iodine in two rotational states. The dependence of K(iso) and DeltaK on bond length and rovibrational state is investigated by using calculated results in combination with available experimental data. In addition to providing new insights into periodic trends for J coupling tensors, this work further demonstrates the utility of the ZORA-DFT method and emphasizes the necessity of spin-orbit relativistic corrections for J calculations involving heavy nuclei.     

Perturbative treatment of scalar-relativistic effects in coupled-cluster calculations of equilibrium geometries and harmonic vibrational frequencies using analytic second-derivative techniques

An analytic scheme for the computation of scalar-relativistic corrections to nuclear forces is presented. Relativistic corrections are included via a perturbative treatment involving the mass-velocity and the one-electron and two-electron Darwin terms. Such a scheme requires mixed second derivatives of the nonrelativistic energy with respect to the relativistic perturbation and the nuclear coordinates and can be implemented using available second-derivative techniques. Our implementation for Hartree-Fock self-consistent field, second-order Moller-Plesset perturbation theory, as well as the coupled-cluster level is used to investigate the relativistic effects on the geometrical parameters and harmonic vibrational frequencies for a set of molecules containing light elements (HX, X=F, Cl, Br; H2X, X=O, S; HXY, X=O, S and Y=F, Cl, Br). The focus of our calculations is the basis-set dependence of the corresponding relativistic effects, additivity of electron correlation and relativistic effects, and the importance of core correlation on relativistic effects.