Picture-change Effects Research Articles

Evaluation of picture change effects on fractional occupation number states in noble gas atoms

In the two-component relativistic density functional theory, the picture change error (PCE), which originates from insufficient transformation of operators, should be corrected. In this study, we examine the PCE in the fractional occupation number (FON) state based on the spin-free infinite-order two-component Hamiltonian. The PCE for the total and orbital energy changes is estimated with respect to the FON electrons of the highest occupied molecular orbital and 1s core orbital in noble gas atoms. PCE is significant in core orbitals in heavy elements but relatively small in light elements and valence orbitals. The delocalization error, which can be represented by the total energy deviation from the behavior of the exact energy, is overestimated by the lack of picture change correction of the two-electron operator and underestimated by that of the density operator. Corresponding to these results, the PCE influences the value of orbital energies and slope of orbital energy change to FON.

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple.

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnFn, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Quasi-Relativistic Calculation of EPR g Tensors with Derivatives of the Decoupling Transformation, Gauge-Including Atomic Orbitals, and Magnetic Balance

We present an exact two-component (X2C) ansatz for the EPR g tensor using gauge-including atomic orbitals (GIAOs) and a magnetically balanced basis set expansion. In contrast to previous X2C and four-component relativistic ansätze for the g tensor, this implementation results in a gauge-origin-invariant formalism. Furthermore, the derivatives of the relativistic decoupling matrix are incorporated to form the complete analytical derivative of the X2C Hamiltonian. To reduce the associated computational costs, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The quasi-relativistic X2C and DLU-X2C Hamiltonians accurately reproduce the results of the parent four-component relativistic theory when accounting for two-electron picture-change effects with the modified screened nuclear spin-orbit approximation in the respective one-electron integrals and integral derivatives. According to our benchmark studies, the uncontracted Dyall and segmented-contracted Karlsruhe x2c-type basis sets perform well when compared to large even-tempered basis sets. Moreover, (range-separated) hybrid density functional approximations such as LC-ωPBE and ωB97X-D are needed to match the experimental findings. The impact of the GIAOs depends on the distribution of the spin density, and their use may change the Δg shifts by 10-50% as shown for [(C5Me5)2Y(μ-S)2Mo(μ-S)2Y(C5Me5)2]-. Routine calculations of large molecules are possible with widely available and comparably low-cost hardware as demonstrated for [Pt(C6Cl5)4]- with 3003 basis functions and three spin-(1/2) La(II) and Lu(II) compounds, for which we observe good agreement with the experimental findings.

Hyperfine Coupling Constants in Local Exact Two-Component Theory

We present a highly efficient implementation of the electron-nucleus hyperfine coupling matrix within the one-electron exact two-component (X2C) theory. The complete derivative of the X2C Hamiltonian is formed, that is, the derivatives of the unitary decoupling transformation are considered. This requires the solution of the response and Sylvester equations, consequently increasing the computational costs. Therefore, we apply the diagonal local approximation to the unitary decoupling transformation (DLU). The finite nucleus model is employed for both the scalar potential and the vector potential. Two-electron picture-change effects are modeled with the (modified) screened nuclear spin-orbit approach. Our implementation is fully integral direct and OpenMP-parallelized. An extensive benchmark study regarding the Hamiltonian, the basis set, and the density functional approximation is carried out for a set of 12-17 transition-metal compounds. The error introduced by DLU is negligible, and the DLU-X2C Hamiltonian accurately reproduces its four-component "fully" relativistic parent results. Functionals with a large amount of Hartree-Fock exchange such as CAM-QTP-02 and ωB97X-D are generally favorable. The pure density functional r2SCAN performs remarkably and even outperforms the common hybrid functionals TPSSh and CAM-B3LYP. Fully uncontracted basis sets or contracted quadruple-ζ bases are required for accurate results. The capability of our implementation is demonstrated for [Pt(C6Cl5)4]- with more than 4700 primitive basis functions and four rare-earth single-molecule magnets: [La(OAr*)3]-, [Lu(NR2)3]-, [Lu(OAr*)3]-, and [TbPc2]-. Here, the results with the spin-orbit DLU-X2C Hamiltonian are in an excellent agreement with the experimental findings of all Pt, La, Lu, and Tb molecules.

Challenges in Multireference Perturbation Theory for the Calculations of the g-Tensor of First-Row Transition-Metal Complexes.

In this article, we have studied 34 S = 1/2 complexes of first-row transition-metal complexes in d1, d5, d7, and d9 configurations in an attempt to determine the intrinsic accuracy of the scalar-relativistic complete active space self-consistent field (CASSCF) and N-electron valence perturbation theory (NEVPT2) methods, with respect to predicting molecular g-values. CASSCF calculations based on active spaces that contain only metal-based orbitals largely overestimate the g-values, compared to experiment and often fail to provide chemically meaningful results. Incorporation of dynamic correlation by means of the NEVPT2 method significantly improves the transition energies, with a typical error, relative to the experiment, of 2000-3000 cm-1. As a result, a lowering in the g-shift by almost an order of magnitude is obtained, relative to the CASSCF results. However, the g-shifts are still overestimated, compared to the experiment, since CASSCF leads to an overly ionic description of the metal-ligand bond and, hence, to spin-orbit coupling matrix elements that are too large. Inclusion of the double d-shell, along with appropriate bonding counterparts to the antibonding d-orbitals in the active space, led to the correct trends in the g-values for all studied complexes, with the linear regression coefficient ( R) equal to 0.93 over the entire dataset. Various technical aspects of the calculations such as the influence of relativity, importance of picture change effects, solvation effects, and comparison between second-order perturbation and effective Hamiltonian-based theories have also been systematically studied. In addition, g-tensor calculations were performed with five popular density functional theory (DFT) methods (B3LYP, M06L, M06, TPSSh, and PBE0) to compare with wave function (WF) methods. Our results suggest that WF-based methods are remarkably better than DFT methods. However, despite the fact that WF theory has come a long way in computing the properties of large, open-shell transition-metal complexes, methodological work is still necessary for truly high accuracies to be reached.

Two-Component Relativistic Calculations of Electric-Field Gradients Using Exact Decoupling Methods: Spin-orbit and Picture-Change Effects.

Electric field gradients (EFGs) for heavy nuclei in diatomic molecules (hydrogen halides and group-13 iodides), uranyl (UO22+), and a uranyl carbonate complex, were computed with different formally exactly (or nearly exactly) decoupled two-component relativistic Hamiltonians: the “exact two-component” (X2C) method, the Barysz–Sadlej–Snijders (BSS) approach, and up to 35th order in the Douglas–Kroll–Hess (DKH) expansion, utilizing a new implementation in the open-source NWChem quantum chemistry package. Results from two-component Hartree–Fock and density functional theory (DFT) calculations at the scalar relativistic approximation as well as including spin–orbit coupling are reported. Picture-change corrected EFGs obtained with X2C, BSS, and high-order DKH are shown to be numerically equivalent. Effects from spin–orbit coupling on the EFGs tend to be moderate but should not be excluded for reliable predictions. Picture-change effects on the EFG of a heavy atom can be as large as the correct picture-change ...

One-electron contributions to the g-tensor for second-order Douglas–Kroll–Hess theory

The electric g-tensor is a central quantity for the interpretation of electron paramagnetic resonance spectra. In this paper, a detailed derivation of the 1-electron contributions to the g-tensor is presented in the framework of linear response theory and the second-order Douglas-Kroll-Hess (DKH) transformation. Importantly, the DKH transformation in the presence of a magnetic field is not unique. Whether or not the magnetic field is included in the required Foldy-Wouthuysen transformation, different transformation matrices and, consequently, Hamiltonians result. In this paper, a detailed comparison of both approaches is presented, paying particular attention to the mathematical properties of the resulting Hamiltonians. In contrast to previous studies that address the g-tensor in the framework of DKH theory, the resulting terms are compared to those of the conventional Pauli theory and are given a physical interpretation. Based on these mathematical and physical arguments, we establish that the proper DKH transformation for systems with constant magnetic fields is based on a gauge-invariant Foldy-Wouthuysen transformation, i.e., a Foldy-Wouthuysen transformation including the magnetic field. Calculations using density functional theory (DFT) are carried out on a set of heavy, diatomic molecules, and a set of transition-metal complexes. Based on these calculations, the performance of the relativistic calculation with and without inclusion of picture-change effects is compared. Additionally, the g-tensor is calculated for the Lanthanide dihydrides. Together with the results from the other two molecular test sets, these calculations serve to quantify the magnitude of picture-change effects and elucidate trends across the periodic table.

Relativistic four-component calculations of electronic g-tensors in the matrix Dirac–Kohn–Sham framework

A relativistic four-component density functional approach based on the use of a restricted kinetically balanced basis (mDKS-RKB) is applied for calculations of electronic g-tensors. The reported approach includes spin-polarization using non-collinear spin-density exchange–correlation functionals in the unrestricted fashion. Benchmark relativistic calculations have been carried out for a few Mo and W complexes demonstrating reliability and efficiency of the implemented approach. The method provides an attractive alternative to existing approximate two-component methods with transformed Hamiltonians for relativistic calculations of electronic g-tensors of heavy-atom systems. In particular, no picture-change effects arise for property calculations.

X-ray constrained unrestricted Hartree–Fock and Douglas–Kroll–Hess wavefunctions

The extension of the X-ray constrained (XC) wavefunction approach to open-shell systems using the unrestricted Hartree-Fock formalism is reported. The XC method is also extended to include relativistic effects using the scalar second-order Douglas-Kroll-Hess approach. The relativistic effects on the charge and spin density on two model compounds containing the copper and iron atom are reported. The size of the relativistic effects is investigated in real and reciprocal space; in addition, picture-change effects are investigated and discussed for the isolated Cu atom. It is found that the relativistic terms lead to changes in the densities that are much smaller than those from the X-ray constraint. Nevertheless, the use of the relativistic corrections in the ab initio model always leads to an improvement in the agreement statistics. An interesting result of the unrestricted XC technique is the possibility of obtaining experimentally derived spin densities from X-ray data.

Restricted magnetically balanced basis applied for relativistic calculations of indirect nuclear spin–spin coupling tensors in the matrix Dirac–Kohn–Sham framework

The relativistic four-component density functional approach based on the use of restricted magnetically balanced basis (mDKS-RMB), applied recently for calculations of NMR shielding, was extended for calculations of NMR indirect nuclear spin–spin coupling constants. The unperturbed equations are solved with the use of a restricted kinetically balanced basis set for the small component while to solve the second-order coupled perturbed DKS equations a restricted magnetically balanced basis set for the small component was applied. Benchmark relativistic calculations have been carried out for the X–H and H–H spin–spin coupling constants in the XH4 series (X=C, Si, Ge, Sn and Pb). The method provides an attractive alternative to existing approximate two-component methods with transformed Hamiltonians for relativistic calculations of spin–spin coupling constants of heavy-atom systems. In particular, no picture-change effects arise in our method for property calculations.

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.

Analytic high-order Douglas-Kroll-Hess electric field gradients

In this work we present a comprehensive study of analytical electric field gradients in hydrogen halides calculated within the high-order Douglas-Kroll-Hess (DKH) scalar-relativistic approach taking picture-change effects analytically into account. We demonstrate the technical feasibility and reliability of a high-order DKH unitary transformation for the property integrals. The convergence behavior of the DKH property expansion is discussed close to the basis set limit and conditions ensuring picture-change-corrected results are determined. Numerical results are presented, which show that the DKH property expansion converges rapidly toward the reference values provided by four-component methods. This shows that in closed-shell cases, the scalar-relativistic DKH(2,2) approach which is of second order in the external potential for both orbitals and property operator yields a remarkable accuracy. As a parameter-dependence-free high-order DKH model, we recommend DKH(4,3). Moreover, the effect of a finite-nucleus model, different parametrization schemes for the unitary matrices, and the reliability of standard basis sets are investigated.

Resolution of identity Dirac-Kohn-Sham method using the large component only: Calculations of g-tensor and hyperfine tensor

A new relativistic two-component density functional approach, based on the Dirac-Kohn-Sham method and an extensive use of the technique of resolution of identity (RI), has been developed and is termed the DKS2-RI method. It has been applied to relativistic calculations of g and hyperfine tensors of coinage-metal atoms and some mercury complexes. The DKS2-RI method solves the Dirac-Kohn-Sham equations in a two-component framework using explicitly a basis for the large component only, but it retains all contributions coming from the small component. The DKS2-RI results converge to those of the four-component Dirac-Kohn-Sham with an increasing basis set since the error associated with the use of RI will approach zero. The RI approximation provides a basis for a very efficient implementation by avoiding problems associated with complicated integrals otherwise arising from the elimination of the small component. The approach has been implemented in an unrestricted noncollinear two-component density functional framework. DKS2-RI is related to Dyall's [J. Chem. Phys. 106, 9618 (1997)] unnormalized elimination of the small component method (which was formulated at the Hartree-Fock level and applied to one-electron systems only), but it takes advantage of the local Kohn-Sham exchange-correlation operators (as, e.g., arising from local or gradient-corrected functionals). The DKS2-RI method provides an attractive alternative to existing approximate two-component methods with transformed Hamiltonians (such as Douglas-Kroll-Hess [Ann. Phys. 82, 89 (1974); Phys. Rev. A 33, 3742 (1986)] method, zero-order regular approximation, or related approaches) for relativistic calculations of the structure and properties of heavy-atom systems. In particular, no picture-change effects arise in the property calculations.

Accurate and efficient treatment of two-electron contributions in quasirelativistic high-order Douglas-Kroll density-functional calculations

Two-component quasirelativistic approaches are in principle capable of reproducing results from fully relativistic calculations based on the four-component Dirac equation (with fixed particle number). For one-electron systems, this also holds in practice, but in many-electron systems one has to transform the two-electron interaction, which is necessary because a picture change occurs when going from the Dirac equation to a two-component method. For one-electron properties, one can take full account of picture change in a manageable way, but for the electron interaction, this would spoil the computational advantages which are the main reason to perform quasirelativistic calculations. Exploiting those picture change effects are largest in the atomic cores, which in molecular applications do not differ too much from the cores of isolated neutral atoms, we propose an elegant, efficient, and accurate approximation to the two-electron picture change problem. The new approach, called the "model potential" approach because it makes use of atomic (four- and two-component) data to estimate picture change effects in molecules, shares with the nuclear-only approach that the Douglas-Kroll operator needs to be constructed only once (not in each self-consistent-field iteration) and that no time-consuming multicenter relativistic two-electron integrals need to be calculated. The new approach correctly describes the screening of both the nearest nucleus and distant nuclei, for the scalar-relativistic as well as the spin-orbit parts of the Hamiltonian. The approach is tested on atomic and molecular-orbital energies as well as spectroscopic constants of the lead dimer.

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.