Two-component Methods Research Articles

Making four- and two-component relativistic density functional methods fully equivalent based on the idea of “from atoms to molecule”

It is shown that four- and two-component relativistic Kohn-Sham methods of density functional theory can be made fully equivalent in all the aspects of simplicity, accuracy, and efficiency. In particular, this has been achieved based solely on physical arguments rather than on mathematical tricks. The central idea can be visualized as "from atoms to molecule," reflecting that the atomic information is employed to "synthesize" the molecular no-pair relativistic Hamiltonian. That is, the molecular relativistic Hamiltonian can, without loss of accuracy, be projected onto the positive energy states of the isolated Dirac atoms with the projector approximated simply by the superposition of the atomic ones. The dimension of the four-component Hamiltonian matrix then becomes the same as that of a two-component one. Another essential ingredient is to formulate quasirelativistic theory on matrix form rather than on operator form. The resultant quasi-four-component, normalized elimination of the small component, and symmetrized elimination of the small component approaches are critically examined by taking the molecules of MH and M(2) (M=At, E117) as examples.

Explaining the derivation and functioning of the exact infinite-order two-component Dirac theory in terms of the Hückel model

The transition from the four-component Dirac theory to the exact two-component formalism is considered by using a simple algebraic model of the Dirac hamiltonian. This model is found to correspond to the four-center Huckel matrix and permits to replace the complex operator algebra by very easy matrix operations. This mapping of operators onto number matrices shows how the exact two-component relativistic hamiltonians are derived. It also explains certain conceptual aspects of the relation between four-component and two-component hamiltonians and their eigenfunctions. The generalized Huckel model of a four-center heteroatomic π-electron system can be used to analytically analyze the essential features of a variety of different exact and approximate two-component methods of relativistic quantum chemistry.

Gradients for two-component quasirelativistic methods. Application to dihalogenides of element 116

The authors report the implementation of geometry gradients for quasirelativistic two-component Hartree-Fock and density functional methods using either the zero-order regular approximation Hamiltonian or spin-dependent effective core potentials. The computational effort of the resulting program is comparable to that of corresponding nonrelativistic calculations, as it is dominated by the evaluation of derivative two-electron integrals, which is the same for both types of calculations. Besides the implementation of derivatives of matrix elements of the one-particle Hamiltonian with respect to nuclear displacements, the calculation of the derivative exchange-correlation energy for the open shell case involves complicated expressions because of the noncollinear approach chosen to define the spin density. A pilot application to dihalogenides of element 116 shows how spin-orbit coupling strongly affects the chemistry of the superheavy p-block elements. While these molecules are bent at a scalar-relativistic level, spin-orbit coupling is so strong that only the 7p3/2 atomic orbitals of element 116 are involved in bonding, which favors linear molecular geometries for dihalogenides with heavy terminal halogen atoms.

Reliability and Detecting Change Following Short-Term Creatine Supplementation: Comparison of Two-Component Body Composition Methods

The purpose of the present study was twofold: firstly, to assess the reliability of various body composition methods, and secondly, to determine the ability of the methods to estimate changes in fat-free mass (FFM) following creatine (Cr) supplementation. Fifty-five healthy male athletes (weight 78.3 +/- 10.3 kg, age 21 +/- 1 years) gave informed consent to participate in this study. Subjects' FFM was estimated by hydrostatic weighing (HW), air-displacement plethysmography (ADP), bioelectrical impedance analysis (BIA), near-infrared spectroscopy (NIR), and anthropometric measurements (ANTHRO). Measurements were taken on 2 occasions separated by 7 days to assess the reliability of the methods. Following this, 30 subjects returned to the laboratory for an additional test day following 7 days of Cr supplementation (20 g.d(-1) Cr + 140 g.d(-1) dextrose) to assess each method's ability to detect acute changes in FFM. In terms of reliability, we found excellent test-retest correlations for all 5 methods, ranging from 0.983 to 0.998 (p < 0.001). The mean biases for the 5 methods were close to 0 (range -0.1 to 0.3 kg) and their 95% limits of agreement (LOAs) were within acceptable limits (HW = -1.1 to 1.7 kg; ADP = -1.1 to 1.2 kg; BIA = -1.0 to 1.0 kg; NIR = -1.4 to 1.4 kg); however, the 95% LOAs were slightly wider for ANTHRO (-2.4 to 2.6 kg). Following Cr supplementation there was a significant increase in body mass (from 77.9 +/- 10.1 kg to 78.9 +/- 10.3 kg, p = 0.000). In addition, all 5 body composition techniques detected the change in FFM to a similar degree (mean change: HW = 0.9 +/- 0.6 kg; ADP = 0.9 +/- 0.6 kg; BIA = 0.9 +/- 0.6 kg; NIR = 0.8 +/- 0.5 kg; ANTHRO = 1.0 +/- 0.7 kg; intraclass correlation coefficient = 0.962). We conclude that between-day differences in FFM estimation were within acceptable limits, with the possible exception of ANTHRO. In addition, all 5 methods provided similar measures of FFM change during acute Cr supplementation.

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.

Two-component relativistic methods for the heaviest elements.

Different generalized Douglas-Kroll transformed Hamiltonians (DKn, n=1, 2,...,5) proposed recently by Hess et al. are investigated with respect to their performance in calculations of the spin-orbit splittings. The results are compared with those obtained in the exact infinite-order two-component (IOTC) formalism which is fully equivalent to the four-component Dirac approach. This is a comprehensive investigation of the ability of approximate DKn methods to correctly predict the spin-orbit splittings. On comparing the DKn results with the IOTC (Dirac) data one finds that the calculated spin-orbit splittings are systematically improved with the increasing order of the DK approximation. However, even the highest-order approximate two-component DK5 scheme shows certain deficiencies with respect to the treatment of the spin-orbit coupling terms in very heavy systems. The meaning of the removal of the spin-dependent terms in the so-called spin-free (scalar) relativistic methods for many-electron systems is discussed and a computational investigation of the performance of the spin-free DKn and IOTC methods for many-electron Hamiltonians is carried out. It is argued that the spin-free IOTC rather than the Dirac-Coulomb results give the appropriate reference for other spin-free schemes which are based on approximate two-component Hamiltonians. This is illustrated by calculations of spin-free DKn and IOTC total energies, r(-1) expectation values, ionization potentials, and electron affinities of heavy atomic systems.

Spin–orbit interactions and supersymmetry in two-component relativistic methods

The spin–orbit coupling effects are investigated for the Douglas–Kroll and infinite-order two-component relativistic hamiltonians. The infinite-order theory reproduce all important features of the four-component Dirac theory, including the supersymmetric pattern of energy level. The Douglas–Kroll method can not give the right values of the spin orbit splitting.

Spin−Orbit and Relativistic Effects on Structures and Stabilities of Group 17 Fluorides EF3 (E = I, At, and Element 117): Relativity Induced Stability for the D3h Structure of (117)F3

Spin−orbit and scalar relativistic effects on geometries, vibrational frequencies, and energies for group 17 fluorides EF3 (E = I, At, and element 117) are evaluated with two-component methods using relativistic pseudopotentials and effective one-electron spin−orbit operators. The inclusion of relativistic effects makes the D3h structure of (117)F3 a stable local minimum, whereas IF3 and AtF3 retain C2v local minima even with relativistic effects. The valence shell electron pair repulsion model is not appropriate to explain the molecular structure of (117)F3. The geometries of EF3 (E = I, At, and element 117) molecules are optimized at the HF level with and without spin−orbit effects. Spin−orbit interactions elongate the bond lengths and decrease the harmonic vibrational frequencies. In the case of AtF3, spin−orbit interactions increase the bond lengths by 0.044 and 0.023 Å for and , respectively. Spin−orbit effects widen the bond angle of C2v structures of IF3 and AtF3, i.e., spin−orbit effects diminish the second-order Jahn−Teller term. The bond angle αe of AtF3 increases by 3.9° due to spin−orbit interactions in addition to the increase of 4.8° by scalar relativistic effects. For (117)F3, spin−orbit effects increase the bond length by 0.109 Å. The spin−orbit interactions stabilize (117)F3 by a significant margin (∼1.2 eV). This stabilization of the molecule compared with open p-shell atoms is quite unusual. Enhanced ionic bonding may be responsible for this stabilization because the electronegative F atom can effectively polarize or attract electrons from the destabilized 7p3/2 spinors of element 117 due to huge spin−orbit splitting of 7p.

Interactions between relativistic subsystems: consequences of the two-component approximation

The operator for interactions between two relativistic subsystems is derived in the framework of the two-component ‘‘for electrons only’’ approximation to the Dirac–Coulomb (DC) theory. The consequences of the transition from the Dirac picture to the two-component scheme are analysed. It is shown that upon this transition the interaction operator undergoes the so-called change of picture transformation. This analysis shows that the two-component methods require that all terms of the DC hamiltonian are transformed in the same way. The same conclusion applies to all relativistic hamiltonians used in two-component many-electron theories. The long-range limit of the interaction operator in the two-component formalism is analysed and shown to convert into products of multipole operators in the changed picture. The presented formalism can be used in both perturbation and supermolecular (variational) approaches to the theory of intermolecular interactions. In comparison with non-relativistic theories, the only difference amounts to the evaluation of matrix elements with wave functions of the two-component formalism. Whenever the spin–orbit coupling effects are negligible, the present theory can be further reduced to the one-component (scalar, spin-averaged) form.

Two-component methods of relativistic quantum chemistry: from the Douglas–Kroll approximation to the exact two-component formalism

The two-component methods of relativistic quantum chemistry based on the Foldy–Wouthuysen (FW) transformations of the Dirac hamiltonian are reviewed. Following the strategy designed by Douglas and Kroll, the FW transformation is carried out in two steps. The first amounts to performing the exact free-particle FW transformation. At variance with other approaches, the second step is written in the form, which results in a nonlinear operator equation. This equation can be solved iteratively, leading to two-component hamiltonians of arbitrarily high accuracy in even powers of the fine structure constant. All these hamiltonians can be classified according to their completeness with respect to the leading order in the fine structure constant. On passing to the basis set representation one obtains the usual Douglas–Kroll hamiltonian and all possible higher-order approximations. By a simple modification of the operator equation which determines the block-diagonalizing transformation one can obtain numerical infinite-order solutions, i.e. one can obtain the exact numerical solution for the separation of the pure electronic part of the Dirac spectrum. This gives the exact two-component method for the use in relativistic quantum chemistry. The computational aspects of this approach are discussed as well. The transition from the Dirac formalism to any two-component approximation is accompanied by the change of all operators, including those which correspond to external perturbations and lead to properties of different orders. This so-called change of picture problem is given particular attention and its importance for certain operators is identified.

The change of picture of the Hellmann–Feynman force operator in approximate relativistic methods

The calculation of energy gradients in one- and two-component approximate methods of relativistic quantum chemistry which follow from the block-diagonalisation of the Dirac hamiltonian, is considered. It is indicated that the transition to these approximate methods involves the so-called change of picture for all operators. In particular this applies to the Hellmann–Feynman force operator. To avoid explicit transformation of this operator to the new picture two schemes for its modelling in terms of the usual Coulomb attraction operators are proposed. One of them is based on the point charge nuclear dipole moment model. The other one, the so-called shifted nucleus model, involves a parametrised shift of the nucleus. Both these models lead to a semianalytical method for the evaluation of energy gradients in approximate relativistic calculations, in which the relaxation terms is obtained analytically whereas the matrix elements of the Hellmann–Feynman force are calculated by using the finite difference method. The accuracy and relative merits of the two models are analysed. The two models are tested within the one-component Douglas–Kroll method. A large change of picture contribution is found in calculations of intramolecular forces in the coinage metal hydrides. This shows that any approximate relativistic technique for calculations of the energy gradients and for the relativistic geometry optimisation must take into account the change of picture contribution to the evaluated Hellmann–Feynman force.

Application of orthogonal functions to differential pulse voltammetric analysis: Simultaneous determination of tin and lead in soft drinks

The application of two-component analysis methods, differentiation of signals and orthogonal function, to the resolution of partially overlapping differential pulse voltammetric (DPV) peaks is demonstrated. The study was extended to differential pulse cathodic stripping voltammetry (DPCSV). A binary system of tin(II) and lead(II), having 72 mV of DPV peaks separation in 0.04 M acetic/ o-phosphoric/boric acids’ mixture, was used as a model throughout the work. The stripping voltammetric analysis data processed by orthogonal functions and the first-derivative ( 1D) methods, were successfully applied to the simultaneous determination of both metals in canned soft drinks. Moreover, the applicability of the methods were demonstrated by the recovery of lead in a drinking water sample.

Relativistic all-electron density functional calculations

The current status of relativistic density functional theory is reviewed. For most applications relevant to chemistry, relativistic corrections to the electron interaction and radiative corrections are not important, and the (four-component) Dirac–Kohn–Sham model can be viewed as a reference. More approximate (two-component) schemes are much more popular not only because of the lower computational effort, but also because, in chemistry, one is only interested in the electronic states of a system, and the two-component methods in one way or another project out the positronic states. It is even possible to arrive at one-component relativistic methods if one can neglect spin-orbit coupling, which is often done for closed-shell compounds containing heavy atoms. Various such schemes are available today, based on first-order perturbation theory or even including higher order corrections, based on the Douglas–Kroll–Hess transformation, or on the so-called regular approximation. In recent years, analytical geometry gradients have been implemented for all these methods, the gradient for the zeroth-order regular approximation (ZORA) being presented in this work for the first time. The availability of such geometry gradients is quite important for the applicability of these methods to “real chemistry”; that is, polyatomic molecules. Results of relativistic density functional methods are collected for some benchmark molecules. Then some results for various tungsten pentacarbonyl phosphines, platinum tricarbonyl phosphines, and molybdenum dinitrogen phosphine carbonyls, as obtained with first-order relativistic density functional calculations by means of direct perturbation theory (DPT), are presented. In another application, the force field of molybdenum, tungsten, and uranium hexafluoride in the metal–fluorine stretch region is calculated at both the DPT and ZORA levels. First-order relativistic calculations are reasonable for second- and third-row transition metal compounds (mainly 4d and 5d electrons involved in bonding), whereas, for uranium and gold compounds (mainly 5f, 6d, and 6s electrons involved), higher order relativistic corrections must be considered. © 1999 John Wiley & Sons, Inc. J Comput Chem 20: 51–62, 1999

Spin-orbit effects calculated by two-component coupled-cluster methods: test calculations on AuH, Au 2, TlH and Tl 2

We have modified the MOLFDIR code by Visscher and coworkers to use one-electron integrals of relativistic effective core potentials with spin–orbit operators and two-component molecular spinors for polyatomic molecules. The present methods incorporate spin–orbit interactions from the Hartree–Fock step through the use of two-component molecular spinors confirming Kramers' degeneracy and are particularly useful to estimate spin–orbit effects. Test calculations for the ground states of AuH, Au 2, TlH and Tl 2 are performed to show that spin–orbit effects vary depending upon the level of theory employed.

Measuring the body composition of elderly subjects: a comparison of methods

There is a paucity of data on differences between methods for the assessment of body composition in elderly subjects. Studies on younger adults suggest that such differences are of some practical significance at the individual level. In the present study the following methods of estimating percentage body fatness (BF%) were compared in healthy elderly men and women (mean age 70 (SD 6) years: densitometry; skinfold thickness; total body water; bioelectrical impedance (BIA) using an age-specific predictive equation and the manufacturers' equation; body mass index (BMI). Though BF% estimates from the various methods tended to be highly correlated with those from densitometry and with each other, differences between methods at the individual level were marked. In particular, the age-specific equations based on BMI and BIA systematically overestimated BF% relative to the other methods. Biases between BF% estimates derived from densitometry, skinfolds, BIA (manufacturers' equation) and total body water were less marked, indicating little evidence of systematic differences between these methods in elderly subjects. Individual differences between methods were slightly greater than those reported in some studies of younger adults, but this may be of little practical significance, and may be considered inevitable in view of variability between and within subjects in the extent to which the underlying assumptions of these two-component methods are met in elderly subjects.

"Demasking" the Temperature Rhythm after Simulated Time Zone Transitions

Simulated time zone transitions were performed in an isolation unit upon groups of one to four human subjects. In the first series of experiments, the adjustment of the circadian rhythm of body temperature, measured in the presence of sleep and other masking factors, was assessed by cosinor analysis and by cross-correlation methods. These methods modeled the circadian timing system either as a single component or as the sum of two components, those due to exogenous and endogenous influences. The one-component models described a more rapid adjustment of the temperature rhythm to the time zone transition than did the two-component models; we attribute this difference to the masking effects of the exogenous component. In a second series of experiments, we showed that the shift of the endogenous component, as assessed by the two-component models, was not significantly different from that measured during constant routines. The results also showed that, if the zeitgebers were phased in advance of the endogenous component, then advances of the endogenous component were produced only if this mismatch was less than about 10 hr. Mismatches greater than this, and cases where the zeitgebers were delayed with respect to the endogenous component, both produced delays of the endogenous component. We conclude that the two-component cross-correlation methods can be used to estimate shifts of the endogenous component of a circadian rhythm in the presence of masking factors. They are therefore an alternative to constant routines when these latter are impracticable to carry out.