Formal relations connecting different approaches to calculate relativistic effects on molecular magnetic properties

In the present work a set of formal relations connecting different approaches to calculate relativistic effects on magnetic molecular properties are proven. The linear response (LR) within the elimination of the small component (ESC), Breit Pauli, and minimal-coupling approaches are compared. To this end, the leading order ESC reduction of operators within the minimal-coupling four-component approach is carried out. The equivalence of all three approaches within the ESC approximation is proven. It is numerically verified for the NMR nuclear-magnetic shielding tensor taking HX and CH3X (X=Br,I) as model compounds. Formal relations proving the gauge origin invariance of the full relativistic effect on the NMR nuclear-magnetic shielding tensor within the LR-ESC approach are presented.     

Relativistic effects on the nuclear magnetic shieldings of rare-gas atoms and halogen in hydrogen halides within relativistic polarization propagator theory

In this work an analysis of the electronic origin of relativistic effects on the isotropic dia- and paramagnetic contributions to the nuclear magnetic shielding sigma(X) for noble gases and heavy atoms of hydrogen halides is presented. All results were obtained within the 4-component polarization propagator formalism at different level of approach [random-phase approximation (RPA) and pure zeroth-order approximation (PZOA)], by using a local version of the DIRAC code. From the fact that calculations of diamagnetic contributions to sigma within RPA and PZOA approaches for HX(X=Br,I,At) and rare-gas atoms are quite close each to other and the finding that the diamagnetic part of the principal propagator at the PZOA level can be developed as a series [S(Delta)], it was found that there is a branch of negative-energy "virtual" excitations that contribute with more than 98% of the total diamagnetic value even for the heavier elements, namely, Xe, Rn, I, and At. It contains virtual negative-energy molecular-orbital states with energies between -2 mc2 and -4 mc2. This fact can explain the excellent performance of the linear response elimination of small component (LR-ESC) scheme for elements up to the fifth row in the Periodic Table. An analysis of the convergency of S(Delta) and its physical implications is given. It is also shown that the total contribution to relativistic effects of the innermost orbital (1s1/2) is by far the largest. For the paramagnetic contributions results at the RPA and PZOA approximations are similar only for rare-gas atoms. On the other hand, if the mass-correction contributions to sigma(p) are expressed in terms of atomic orbitals, a different pattern is found for 1s1/2 orbital contributions compared with all other s-type orbitals when the whole set of rare-gas atoms is considered.     

Calculation of electric-field gradients based on higher-order generalized Douglas-Kroll transformations

In this paper, the calculation of electric-field-like properties based on higher-order Douglas-Kroll-Hess (DKH) transformations is discussed. The electric-field gradient calculated within the Hartree-Fock self-consistent field framework is used as a representative property. The properties are expressed as an analytic first derivative of the four-component Dirac energy and the nth-order DKH energy, respectively. The differences between a "forward" transformation of the relativistic energy or the "back transformation" of the wave function is discussed in some detail. Detailed test calculations were carried out on the electric-field gradient at the halogen nucleus in the series HX (X=F,Cl,Br,I,At) for which extensive reference data are available. The DKH method is shown to reproduce (spin-free) four-component Dirac-Fock results to an accuracy of better than 99% which is significantly closer than previous DKH studies. The calculations of both the Hamiltonian and the property operator are shown to be essentially converged after the second-order transformation, even for elements as heavy as At. In addition, we have obtained results within the density-functional framework using the DKHZ and zeroth-order regular approximation (ZORA) methods. The latter results included picture-change effects at the scalar relativistic variant of the ZORA-4 level and were shown to be in quantitative agreement with earlier results obtained by van Lenthe and Baerends. The picture-change effects are somewhat smaller for the ZORA method compared to DKH. For heavier elements significant differences in the field gradients predicted by the two methods were found. Based on comparison with four-component Dirac-Kohn-Sham calculations, the DKH results are more accurate. Compared to the spin-free Dirac-Kohn-Sham reference values, the ZORA-4 formalism did not improve the results of the ZORA calculations.     

Decoupling of the Dirac equation correct to the third order for the magnetic perturbation

A two-component relativistic theory accurately decoupling the positive and negative states of the Dirac Hamiltonian that includes magnetic perturbations is derived. The derived theory eliminates all of the odd terms originating from the nuclear attraction potential V and the first-order odd terms originating from the magnetic vector potential A, which connect the positive states to the negative states. The electronic energy obtained by the decoupling is correct to the third order with respect to A due to the (2n+1) rule. The decoupling is exact for the magnetic shielding calculation. However, the calculation of the diamagnetic property requires both the positive and negative states of the unperturbed (A=0) Hamiltonian. The derived theory is applied to the relativistic calculation of nuclear magnetic shielding tensors of HX (X=F,Cl,Br,I) systems at the Hartree-Fock level. The results indicate that such a substantially exact decoupling calculation well reproduces the four-component Dirac-Hartree-Fock results.     

Near-thermal reactions of Au(+)(1S,3D) with CH3X (X = F,Cl)

Reactions of Au(+)((1)S) and Au(+)((3)D) with CH(3)F and CH(3)Cl have been carried out in a drift cell in He at a pressure of 3.5 Torr at both room temperature and reduced temperatures in order to explore the influence of the electronic state of the metal on reaction outcomes. State-specific product channels and overall two-body rate constants were identified using electronic state chromatography. These results indicate that Au(+)((1)S) reacts to yield an association product in addition to AuCH(2)(+) in parallel steps with both neutrals. Product distributions for association vs HX elimination were determined to be 79% association/21% HX elimination for X = F and 50% association/50% HX elimination when X = Cl. Reaction of Au(+)((3)D) with CH(3)F also results in HF elimination, which in this case is thought to produce (3)AuCH(2)(+). With CH(3)Cl, Au(+)((3)D) reacts to form AuCH(3)(+) and CH(3)Cl(+) in parallel steps. An additional product channel initiated by Au(+)((3)D) is also observed with both methyl halides, which yields CH(2)X(+) as a higher-order product. Kinetic measurements indicate that the reaction efficiency for both Au(+) states is significantly greater with CH(3)Cl than with CH(3)F. The observed two-body rate constant for depletion of Au(+)((1)S) by CH(3)F represents less than 5% of the limiting rate constant predicted by the average dipole orientation model (ADO) at room temperature and 226 K, whereas CH(3)Cl reacts with Au(+)((1)S) at the ADO limit at both room temperature and 218 K. Rate constants for depletion of Au(+)((3)D) by CH(3)F and CH(3)Cl were measured at 226 and 218 K respectively, and indicate that Au(+)((3)D) is consumed at approximately 2% of the ADO limit by CH(3)F and 69% of the ADO limit by CH(3)Cl. Product formation and overall efficiency for all four reactions are consistent with previous experimental results and available theoretical models.     

Mean field linear response within the elimination of the small component formalism to evaluate relativistic effects on magnetic properties

The linear response within the elimination of the small component formalism is aimed at obtaining the leading order relativistic corrections to magnetic molecular properties in the context of the elimination of the small component approximation. In the present work we extend the method in order to include two-body effects in the form of a mean field one-body operator. To this end we consider the four-component Dirac-Hartree-Fock operator as the starting point in the evaluation of the second order relativistic expression of magnetic properties. The approach thus obtained is the fully consistent leading order approximation of the random phase approximation four-component formalism. The mean field effect on the relativistic corrections to both the diamagnetic and paramagnetic terms of magnetic properties taking into account both the Coulomb and Breit two-body interactions is considered.     

A fully relativistic method for calculation of nuclear magnetic shielding tensors with a restricted magnetically balanced basis in the framework of the matrix Dirac-Kohn-Sham equation

A new relativistic four-component density functional approach for calculations of NMR shielding tensors has been developed and implemented. It is founded on the matrix formulation of the Dirac-Kohn-Sham (DKS) method. Initially, unperturbed equations are solved with the use of a restricted kinetically balanced basis set for the small component. The second-order coupled perturbed DKS method is then based on the use of restricted magnetically balanced basis sets for the small component. Benchmark relativistic calculations have been carried out for the (1)H and heavy-atom nuclear shielding tensors of the HX series (X=F,Cl,Br,I), where spin-orbit effects are known to be very pronounced. The restricted magnetically balanced basis set allows us to avoid additional approximations and/or strong basis set dependence which arises in some related approaches. The method provides an attractive alternative to existing approximate two-component methods with transformed Hamiltonians for relativistic calculations of chemical shifts and spin-spin coupling constants of heavy-atom systems. In particular, no picture-change effects arise in property calculations.     

Molecular orbital investigation of the protonated H2X2AlNHn(CH3)3-n+ (X = F, Cl, and Br; n = 0-3) complexes

Structures of protonated alane-Lewis base donor-acceptor complexes H2X2AlNHn(CH3)(3-n)+ (X = F, Cl, and Br; n = 0-3) as well as their neutral parents were investigated. All the monocations H2X2AlNHn(CH3)(3-n)+ are Al-H protonated involving hypercoordinated alane with a three-center two-electron bond and adopt the C(s) symmetry arrangement. The energetic results show that the protonated alane-Lewis complexes are more stable than the neutral ones. They also show that this stability decreases on descending in the corresponding periodic table column from fluorine to bromine atoms. The calculated protonation energies of HX2AlNHn(CH3)(3-n) to form H2X2AlNHn(CH3)(3-n)+ were found to be highly exothermic. The possible dissociation of the cations H2X2AlNHn(CH3)(3-n)+ into X2AlNHn(CH3)(3-n)+ and molecular H2 is calculated to be endothermic.     

Parity nonconservation contribution to the nuclear magnetic resonance shielding constants of chiral molecules: a four-component relativistic study

A systematic four-component relativistic study of the parity nonconservation (PNC) contribution to the (isotropic) NMR shielding constants of chiral molecules is presented for the P enantiomers of the series H(2)X(2) (X=(17)O,(33)S,(77)Se,(125)Te,(209)Po). The PNC contributions are obtained within a linear response approach at the Hartree-Fock level. A careful design of the basis sets is necessary. The four-component relativistic results based on the Dirac-Coulomb Hamiltonian are compared with the nonrelativistic Levy-Leblond results and those obtained by the spin-free modified Dirac Hamiltonian. The calculations confirm the nonrelativistic scaling law Z(2.4) of the PNC contribution with respect to nuclear charge Z. However, the calculations also show that the overall scaling is significantly modified by relativistic effects. The scalar relativistic effect scales as Z(4.7) for the selected set of molecules, whereas the spin-orbit effect, of opposite sign, scales better than Z(6) and completely dominates the PNC contribution for the heaviest elements. This opens up the intriguing possibility of the experimental observation of PNC effects on NMR parameters of molecules containing heavy atoms. The presented formalism is expected to be valuable in assisting the search for suitable candidate molecules.     

Relativistic effects on the nuclear magnetic resonance shielding of FX (X = F, Cl, Br, I, and At) molecular systems

We present ab inito full four-component and spin-free calculations of the NMR shielding parameter, σ, in the FX (X = F, Cl, Br, I and At) molecular systems. A different expression that overcomes the traditional non-relativistic (NR) approximation used to calculate the relationship between spin-rotation constants and the paramagnetic terms of σ(p) are given. Large deviations from NR results are obtained for σ(X; X = I and At) and for σ(F; FAt). σ(∥)(p)(I; FI) is zero within the NR approach but -447.4 parts per million from our calculations. The electronic origin of relativistic corrections are analyzed. All passive SO contributions are obtained as a difference between full four-component calculations and spin-free ones. Considering relativistic effects on the anisotropy, we obtain a deviation of 10% for I and 25% for At. σ(∥)(SO)(X) is always negative and σ(∥)(SF)(X) is always positive; the passive SO becomes larger than the SF one for X = Br, I, and At. Both σ(∥)(SO)(X) and σ(⊥)(SO)(X) have a functional dependence such as a Z(X)(b) being the exponent 3.5 and 3.65, respectively. The passive SO contribution to the anisotropy has a similar functional dependence with an exponent of 3.60, meaning that its perpendicular component is larger than its corresponding parallel component.     

Unexpected geometrical effects on paramagnetic spin-orbit and spin-dipolar 2J(FF) couplings

The second-rank tensor character of the paramagnetic spin-orbit and spin-dipolar contributions to nuclear spin-spin coupling constants is usually ignored when NMR measurements are carried out in the isotropic phase. However, in this study it is shown that isotropic (2)J(FF) couplings strongly depend on the relative orientation of the C-F bonds containing the coupling nuclei and the eigenvectors of such tensors. Predictions about such effect are obtained using a qualitative approach based on the polarization propagator formalism at the RPA, and results are corroborated performing high-level ab initio spin-spin coupling calculations at the SOPPA(CCSD)/EPR-III//MP2/EPR-III level in a model system. It is highlighted that no calculations at the RPA level were carried out in this work. The quite promising results reported in this paper suggest that similar properties are expected to hold for the second-rank nuclear magnetic shielding tensor.     

Water-catalyzed O-H insertion/HI elimination reactions of isodihalomethanes (CH2X-I, where X = Cl, Br, I) with water and the dehalogenation of dihalomethanes in water-solvated environments

A combined experimental and theoretical investigation of the ultraviolet photolysis of CH2XI (where X = Cl, Br, I) dihalomethanes in water is presented. Ultraviolet photolysis of low concentrations of CH2XI (where X = Cl, Br, I) in water appears to lead to almost complete conversion into CH2(OH)2 and HX and HI products. Picosecond time-resolved resonance Raman (ps-TR3) spectroscopy experiments revealed that noticeable amounts of CH2X-I isodihalomethane intermediates were formed within several picoseconds after photolysis of the CH2XI parent compound in mixed aqueous solutions. The ps-TR3 experiments in mixed aqueous solutions revealed that the decay of the CH2X-I isodihalomethane intermediates become significantly shorter as the water concentration increases, indicating that the CH2X-I intermediates may be reacting with water. Ab initio calculations found that the CH2X-I intermediates are able to react relatively easily with water via a water-catalyzed O-H insertion/HI elimination reaction to produce CH2X(OH) and HI products, with the barrier for these reactions increasing as X changes from Cl to Br to I. The ab initio calculations also found that the CH2X(OH) product can undergo a water-catalyzed HX elimination reaction to make H2C=O and HX products, with the barrier to reaction decreasing as X changes from Cl to Br to I. The preceding two water-catalyzed reactions produce the HI and HX leaving groups observed experimentally, and the H2C=O product further reacts with water to make the other CH2(OH)2 product observed in the photochemistry experiments. This suggests that that the CH2X-I intermediates react with water to form the CH2(OH)2 and HI and HX products observed in the photochemistry experiments. Ultraviolet photolysis of CH2XI (where X = Cl, Br, I) at low concentrations in water-solvated environments appears to lead to efficient dehalogenation and release of two strong acid leaving groups. We very briefly discuss the potential influence of this photochemistry in water on the decomposition of polyhalomethanes and halomethanols in aqueous environments.     

An SCF and CI Study of the 1,3 Shift in the HX?CHY⇌XCH?YH Isoelectronic Series: X,YCH2, NH,and O

Ab initio molecular orbital calculations at SCF level with the 3-21G, 6-31G, and 6-31G** basis sets and CI level with the 6-31G basis set have been carried out for an isoelectronic series HX? CH?Y and X?CH? YH, where X, Y can be CH₂, NH, and O. Optimized structures (3-21G and 6-31G**) for both tautomers and the 1,3 hydrogen shift transition states are reported. The relative stabilities of the isomers and the barriers of the 1,3 shift are discussed in terms of proton affinities and bond orders. It is shown that both the relative stabilities of the tautomers and the relative barrier heights can be explained qualitatively using simple proton affinity arguments and that the barrier heights are quantitatively related to bond orders.     

Quantum-chemical calculations of NMR chemical shifts of organic molecules: XV. Relativistic calculations of <Superscript>29</Superscript>Si NMR chemical shifts of silanes

Calculations of ²⁹Si NMR chemical shifts of 68 silanes possessing various substituents, in particular, with heavy halogens attached to silicon atom, were carried out applying an efficient calculation scheme of locally dense basis set in the framework of the electron density functional theory utilizing the Keal–Tozer functional combined with relativistic Dyall basis sets on a four-component relativistic level. The main factors of calculation accuracy of silicon chemical shifts were analyzed including the relativistic effects, environmental impact, and vibrational corrections. The mean absolute calculation error for the studied compounds series accounting for all mentioned factors was 14.0 ppm for the nonrelativistic calculation and 6.7 ppm for the four-component relativistic calculation at the range of silicon chemical shifts variation of ~250 ppm.     

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.     

Normal halogen dependence of 13C NMR chemical shifts of halogenomethanes revisited at the four‐component relativistic level

The ‘Normal Halogen Dependence’ of ¹³C NMR chemical shifts in the series of halogenomethanes is revisited at the four‐component relativistic level. Calculations of ¹³C NMR chemical shifts of 70 halogenomethanes have been carried out at the density functional theory (DFT) and MP2 levels with taking into account relativistic effects using the four‐component relativistic theory of Dirac‐Coulomb within the different computational methods (4RPA, 4OPW91) and hybrid computational schemes (MP2 + 4RPA, MP2 + 4OPW91). The most efficient computational protocols are derived for practical purposes. Relativistic shielding effect reaches as much as several hundreds of ppm for heavy halogenomethanes, and to account for this effect in comparison with experiment at the qualitative level, relativistic Dyall's basis sets of triple‐zeta quality or higher are to be used within the framework of the four‐component relativistic theory taking into account solvent effects. Relativistic geometrical optimization (as compared with the non‐relativistic level) is essential for the molecules containing at least two iodines at one carbon atom. Copyright © 2016 John Wiley & Sons, Ltd.     

Relativistic effects on group-12 metal nuclear shieldings

The leading-order perturbation theory approach to relativistic effects on the nuclear magnetic shielding provides an economic method for obtaining the chemical shifts in heavy-element containing systems. The method features detailed analysis potential in terms of the different physical mechanisms affecting the shielding tensors of heavy nuclei. The perturbative nature, however, results in an increasing error with increasingly heavy elements in the system. In this work, we investigate the performance of the Breit-Pauli perturbation theory (BPPT) against fully relativistic four-component theory in computing the nuclear shielding constants as well as the chemical shifts with respect to corresponding atomic ions of group-12 metals, M = Zn, Cd, and Hg, in dimethyl M(CH(3))(2) and aqueous M(H(2)O)(6)(2+) complexes. It is shown that five out of the total of sixteen BPPT correction terms are responsible for most of the relativistic corrections for the chemical shift of studied metals. The relativity is important already for Cd and BPPT is proven to work well up to Hg for the chemical shift, as calibrated with the fully relativistic method.     

Nuclear charge-distribution effects on the NMR spectroscopy parameters

We present here a systematic study about the influence of the size and type of nuclear charge-distribution models (Gaussian and point-like) on the NMR spectroscopic parameters, the nuclear magnetic shielding σ and the indirect nuclear spin J-coupling. We found that relativistic effects largely enhance the nuclear charge-distribution effects (NChDE) on those parameters being them quite sensitive to the nuclear model adopted for calculations. Results for two rare gas atoms (Kr, Rn) and few molecular systems like HX, (X = Br, I, At), CH(4), SnH(4), SnIH(3), SnI(2)H(2), and PbIH(3) are presented. J-couplings are more sensitive than shieldings in both, relativistic and non-relativistic (NR) regimes. The highest effect (close to 11% of variation in relativistic calculations with that two different nuclear models) is observed for J(Pb-I) in PbIH(3). A similar effect is found for J(Pb-H) in the same molecule, close to 9%. The NChDE for σ(Sn) in SnI(4-n)H(n) with n = 1, 2 is as large as few ppm (between 3 and 8.56 ppm). For J(Sn-H) in this set of molecules, it goes from 37 Hz for SnH(4) to 54 Hz for SnI(2)H(2). Furthermore, we found that the vicinal NChDE is very small though not zero. For (1)J(Sn-H) in SnIH(3), the NChDE of iodine is close to 2 Hz (0.1%). We also studied the NChDE on the ground state electronic energies of atoms and molecules. We found that these effects are only important within the relativistic regime but not within the NR one. They are in good agreement with previous works.     

Fully relativistic coupled cluster and DFT study of electric field gradients at Hg in 199Hg compounds

We investigate the magnitude and interplay of relativistic and electron correlation effects on the electric field gradient (EFG) at the position of Hg in linear and bent HgL(2) (L = CH(3), Cl, Br, I) and trigonal planar [HgCl(3)](-) compounds using four-component relativistic Dirac-Coulomb (DC) and non-relativistic (NR) calculations at the Hartree-Fock (HF), DFT, MP2 and coupled cluster (CC) levels. The relativistic and electron correlation contributions to EFG have opposite signs and are not additive, demonstrating the importance of taking into account relativistic and electron correlation contributions on an equal footing. DC-MP2 overestimates the electron correlation correction by 0.48-0.56 a.u. for Hg-halides and by 0.8 a.u. for Hg(CH(3))(2), respectively, while DC-CCSD underestimates the correlation correction by 0.57-0.66 a.u. compared to the reference DC-CCSD-T data. EFGs obtained at the DC-DFT level vary considerably with the functional; DC-CAMB3LYP and DC-BH&H reproduce DC-CCSD-T results within 0.08-0.24 a.u. (1%-3%) for Hg(CH(3))(2) and Hg-halides, respectively. An updated value of the nuclear quadrupole moment of the I = 5/2 excited state of (199)Hg, Q((199)Hg) = 0.675(12) b is derived from the literature. This value compares well with that derived from our calculated EFG at the DC-CCSD-T level and the experimental data for Hg(CH(3))(2); Q((199)Hg) = 0.650 b.