Relativistic electric and magnetic property operators for two‐component transformed Hamiltonians

A general procedure is presented for the derivation of property operators for electric and magnetic perturbations for Hamiltonians derived from the Dirac Hamiltonian by a partially block‐diagonalizing unitary transformation. The procedure involves a regularized expansion in powers of p ²/m²c². Property operators are expressed in terms of the solid spherical harmonics. Expressions for the free‐particle Foldy–Wouthuysen, Douglas–Kroll, and Barysz–Sadlej–Snijders transformations are compared with the well‐known Pauli results. Explicit examples of a constant electric field and a constant magnetic field are given. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 412–421, 2000     

Picture change and calculations of expectation values in approximate relativistic theories

The effect of the so-called picture change on expectation values of one-electron operators in approximate two(one)-component relativistic theories is discussed. This effect is expected to be particularly large for operators which assume large values in the vicinity of heavy nuclei. The numerical results illustrating the picture change effect on electric field gradients at nuclei have been obtained in the spin-free Pauli and Douglas–Kroll approximations. It has been found that the picture change effect lowers the electric field gradient at I in HI by about 1 a.u. Very large picture change effect (−8 a.u.) has been calculated for HAt. It is concluded that in accurate calculations of expectation values of operators involving high inverse powers of the electron–nucleus distance the picture change, which accompanies the transformation of the Dirac (Dirac–Coulomb) equation to approximate two(one)-component relativistic Hamiltonians, must be taken into account. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 68: 159–174, 1998     

Perturbation energy expansions based on two-component relativistic Hamiltonians

The approximate elimination of the small-component approach provides ansätze for the relativistic wave function. The assumed form of the small component of the wave function in combination with the Dirac equation define transformed but exact Dirac equations. The present derivation yields a family of two-component relativistic Hamiltonians which can be used as zeroth-order approximation to the Dirac equation. The operator difference between the Dirac and the two-component relativistic Hamiltonians can be used as a perturbation operator. The first-order perturbation energy corrections have been obtained from a direct perturbation theory scheme based on these two-component relativistic Hamiltonians. At the two-component relativistic level, the errors of the relativistic correction to the energies are proportional to ⁴ Z ⁴, whereas for the relativistic energy corrections including the first-order perturbation theory contributions, the errors are of the order of ⁶ Z ⁶–⁸ Z ⁸ depending on the zeroth-order Hamiltonian.Contribution to the Björn Roos Honorary Issue     

All-electron scalar relativistic basis sets for the 6p elements

New segmented all-electron relativistically contracted (SARC) basis sets have been developed for the elements ₈₁Tl–₈₆Rn, thus extending the SARC family of all-electron basis sets to include the 6p block. The SARC basis sets are separately contracted for the second-order Douglas–Kroll–Hess and the zeroth-order regular approximation scalar relativistic Hamiltonians. Their compact size and segmented construction are best suited to the requirements of routine density functional theory (DFT) applications. Evaluation of the basis sets is performed in terms of incompleteness and contraction errors, orbital properties, ionization energies, electron affinities, and atomic polarizabilities. From these atomic metrics and from computed basis set superposition errors for a series of homonuclear dimers, it is shown that the SARC basis sets achieve a good balance between accuracy and size for efficient all-electron scalar relativistic DFT applications.     

Economical treatments of relativistic effects and electron correlation in WH6

The equilibrium geometries and relative stabilities of several structural isomers of tungsten hexahydride, WH₆, have been obtained at different levels of quantum chemical calculations. The performance of various strategies to (i) include electron correlation, viz. density functional theory based approaches, Møller/Plesset perturbation and coupled cluster theory, and to (ii) account for scalar relativistic effects, viz. various relativistic effective core potentials, first order perturbation theory, a quasi-relativistic treatment employing a Pauli Hamiltonian, and use of the Douglas/Kroll operator, are compared to the best theoretical data available. It is shown that relativistic and electron correlation effects are most important for the high-symmetry species, that these effects give rise to opposite trends in relative energies, and that overall the relativistic effects dominate. The most efficient way to incorporate relativistic effects appears to be via the use of relativistic effective core potentials, while the correlation energies are best taken account of using a conventional method such as CCSD(T). © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1604–1611, 1998     

Note on the atomic correlation energy

In the introductory section, we compare the total, kinetic, nuclear-electron, Coulomb, exchange, and correlation energies of ground-state atoms. From the analyses of the data, one can conclude that the Hartree-Fock (HF) model is notably good and might require only a small perturbation to become essentially an “accurate” model. For this reason and considering past literature, we present a semiempirical extension of the HF model. We start with a calibration of three independent models, each one with an effective Hamiltonian, which introduces a small perturbation on the kinetic, the nuclear-electron, or the Coulomb HF operators. The perturbations are expressed as very simple functions of products of orbital probability density. The three perturbations yield very equivalent results and the computed ground-state energies are reasonably near to the accurate nonrelativistic energies recently provided by E. Davidson and his collaborators for the 2–18 electron systems and the estimates by Clementi and his collaborators for the 19–54 electron systems. The first ionization potentials from He to Cs, the second ionization potentials from Li to Zn, and excitation energies for npⁿ, 3dⁿ, and 4s¹3dⁿ configurations are used as additional verification and validation. The above three effective Hamiltonians are then combined in order to redistribute the correlation energy correction in a way which exactly satisfies the virial theorem and maintains the HF energy ratios between kinetic, nuclear-electron, and electron-electron interaction energies; the resulting effective Hamiltonian, named “virial constrained,” yields good quality data comparable to those obtained from the three independent effective operators. Concerning excitation energies, these effective Hamiltonians yield values only in modest agreement with experimental data, even if definitively superior to HF computations. To further improve the computed excitation energies, we applied an empirical scaling in the vector coupling coefficient; this correction yields very reasonable excitations for all the configurations that we have considered. We conclude that the use of effective potentials to introduce small perturbations density-dependent onto the HF model constitutes a broad class of practical and reliable semiempirical solutions to atomic many-electron problems, can provide an alternative to popular proposals from density functional theory, and should prepare the ground for “generalized HF models.” © 1997 John Wiley & Sons, Inc. Int J Quant Chem 62: 571–591, 1997     

Derivation and pilot application of a scheme for intermediate Hamiltonians in Fock space

A Fock space formulation of the intermediate Hamiltonian approach is derived by introducing shift operators in the equations determining effective Hamiltonians in Fock space. A non-hermitian intermediate Hamiltonian is constructed from the Fock space Bloch equation. An alternative derivation, based on a similarity transformation expression, is presented providing access to hermitian intermediate Hamiltonians. In a pilot application, the potential curves of the two lowest¹ _g ⁺ states of the H₂ molecule are calculated demonstrating the applicability of the scheme.     

An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation

The authors report the implementation of a simple one-step method for obtaining an infinite-order two-component (IOTC) relativistic Hamiltonian using matrix algebra. They apply the IOTC Hamiltonian to calculations of excitation and ionization energies as well as electric and magnetic properties of the radon atom. The results are compared to corresponding calculations using identical basis sets and based on the four-component Dirac-Coulomb Hamiltonian as well as Douglas-Kroll-Hess and zeroth-order regular approximation Hamiltonians, all implemented in the DIRAC program package, thus allowing a comprehensive comparison of relativistic Hamiltonians within the finite basis approximation.     

Symmetric group approach to relativistic CI. IV. Representations of one-electron spin operators and their products in a symmetric group-adapted basis of N-electron spin functions

A detailed study on representations of one-electron spin operators and of their products in an N-electron S_N-adapted spin space is presented. Some conclusions relevant for their evaluation and implementation in relativistic two-component SGA-CI calculations are derived. © 1997 John Wiley & Sons, Inc.     

Expectation Values in Spin-Averaged Douglas–Kroll and Infinite-Order Relativistic Methods

The change of picture for the r ^–1 operator which occurs on passing from the four component relativistic schemes to two component theories is investigated for the spin-averaged Douglas–Kroll approximation and the recently proposed infinite-order approach. For nuclei already with moderately large values of the nuclear charge the change of picture contribution is found to be relatively important. Its neglect significantly affects the calculated values of the total relativistic contribution to the expectation value of r ^–1. A numerical method for the calculation of the total relativistic contribution to the expectation values of r ^–1, which avoids the explicit use of the appropriately transformed r ^–1 operator, is devised and tested. Also the differences between the Douglas–Kroll approximation and the infinite-order scheme are investigated.     

Symmetric group approach to relativistic CI. I. General formalism

General formulas for matrix elements of spin-dependent operators in a basis of spin-adapted antisymmetrized products of orthonormal orbitals are derived. The resulting formalism may be applied to construction of the Hamiltonian matrices both for Pauli and for projected no-pair relativistic configuration interaction methods. From a formal point of view, it is a generalization of the symmetric group approach to the CI method for the case of spin-dependent Hamiltonians. © 1997 John Wiley & Sons, Inc.     

Perturbation theory of relativistic corrections

Methods for perturbation theory of relativistic corrections for an electron in a Coulomb field are divided into three categories: (1) in terms of 4-component spinors; (2) in terms of the ‘large components’ of the Dirac spinor; (3) involving a Foldy-Wouthuysen type transformation, where one attempts to obtain a two-component spinor different from the ‘large component’. In methods of category 1 (the ‘direct perturbation theory’ of paper I of this series, the related approaches by Rutkowski as well as by Gesteszy, Grosse, and Thaller and a somewhat different one by Moore) the wave function, the energy and the Hamiltonian are analytic inc ^?2. No divergent terms arise. In methods of category 2 (that of the elemination of the small component as well as a similarity transformation in intermediate normalization) wave function and energy are still analytic inc ^?2, but the effective Hamiltonian no longer is. Regularized results can be obtained by controlled cancellation of divergent terms. In category 3 both the effective Hamiltonian and the wave function are highly singular and non-analytic inc ^?1. A controlled cancellation of divergent terms is at least very difficult. These pathologic feature survive in the non-relativistic limit and have hence little to do with relativistic effects. They are related to the fact that forr → 0 the sign of the quantum number κ rather than that of the energy determines which component of the Dirac spinor is large and which is small. In the limitr → 0 andc → ∞ the Foldy-Wouthuysen wave function of a 2p _1/2 state is a 1p wave function. Hierarchies of transformations of the Dirac equation and its non-relativistic limit are presented and discussed. Finally the problem of the regularization of effective Hamiltonians on 2-component level ‘for electrons only’ is addressed.     

Energy expressions for atomic configurations in the L–S coupling scheme

The matrix elements of the spin-free Hamiltonian between two atomic configuration state functions (CSF 'S ) in the L–S coupling scheme are expressed in terms of the atomic integrals F^k's and G^k's. Using these general expressions, the matrix elements have been obtained for all the atomic configurations with three valence electrons that have not been solved so far by earlier methods. The scope for applying this new approach to obtain the Auger line energies and the promotion energies of metals that involve more than two partially filled shells is indicated. The energy expressions for some of the relevant configurations are tabulated.     

NMR shielding constants of CuX,AgX, and AuX (X = F,Cl, Br,and I) investigated by density functional theory based on the Douglas–Kroll–Hess Hamiltonian

Two‐component relativistic density functional theory (DFT) with the second‐order Douglas–Kroll–Hess (DKH2) one‐electron Hamiltonian was applied to the calculation of nuclear magnetic resonance (NMR) shielding constant. Large basis set dependence was observed in the shielding constant of Xe atom. The DKH2‐DFT‐calculated shielding constants of I and Xe in HI, I₂, CuI, AgI, and XeF₂ agree well with those obtained by the four‐component relativistic theory and experiments. The Au NMR shielding constant in AuF is extremely more positive than in AuCl, AuBr, and AuI, as reported recently. This extremely positive shielding constant arises from the much larger Fermi contact (FC) term of AuF than in others. Interestingly, the absolute values of the paramagnetic and the FC terms are considerably larger in CuF and AuF than in others. The large paramagnetic term of AuF arises from the large d‐components in the Au d_π –F p_π and Au sd_σ–F p_σ molecular orbitals (MOs). The large FC term in AuF arises from the small energy difference between the Au sd_σ + F p_σ and Au sd_σ–F p_σ MOs. The second‐order magnetically relativistic effect, which is the effect of DKH2 magnetic operator, is important even in CuF. This effect considerably improves the overestimation of the spin‐orbit effect calculated by the Breit–Pauli magnetic operator. © 2013 Wiley Periodicals, Inc.     

Recent Development of Relativistic Molecular Theory

Summary. Today it is common knowledge that relativistic effects are important in the heavy-element chemistry. The continuing development of the relativistic molecular theory is opening up rows of the periodic table that are impossible to treat with the non-relativistic approach. The most straightforward way to treat relativistic effects on heavy-element systems is to use the four-component Dirac-Hartree-Fock approach and its electron-correlation methods based on the Dirac-Coulomb(-Breit) Hamiltonian. The Dirac-Hartree-Fock (DHF) or Dirac-Kohn-Sham (DKS) equation with the four-component spinors composed of the large- and small-components demands severe computational efforts to solve, and its applications to molecules including heavy elements have been limited to small- to medium-size systems. Recently, we have developed a very efficient algorithm for the four-component DHF and DKS approaches. As an alternative approach, several quasi-relativistic approximations have also been proposed instead of explicitly solving the four-component relativistic equation. We have developed the relativistic elimination of small components (RESC) and higher-order Douglas-Kroll (DK) Hamiltonians within the framework of the two-component quasi-relativistic approach. The developing four-component relativistic and approximate quasi-relativistic methods have been implemented into a program suite named REL4D.In this article, we will introduce the efficient relativistic molecular theories to treat heavy-atomic molecular systems accurately via the four-component relativistic and the two-component quasi-relativistic approaches. We will also show several chemical applications including heavy-element systems with our relativistic molecular approaches.     

Recent Development of Relativistic Molecular Theory

Today it is common knowledge that relativistic effects are important in the heavy-element chemistry. The continuing development of the relativistic molecular theory is opening up rows of the periodic table that are impossible to treat with the non-relativistic approach. The most straightforward way to treat relativistic effects on heavy-element systems is to use the four-component Dirac-Hartree-Fock approach and its electron-correlation methods based on the Dirac-Coulomb(-Breit) Hamiltonian. The Dirac-Hartree-Fock (DHF) or Dirac-Kohn-Sham (DKS) equation with the four-component spinors composed of the large- and small-components demands severe computational efforts to solve, and its applications to molecules including heavy elements have been limited to small- to medium-size systems. Recently, we have developed a very efficient algorithm for the four-component DHF and DKS approaches. As an alternative approach, several quasi-relativistic approximations have also been proposed instead of explicitly solving the four-component relativistic equation. We have developed the relativistic elimination of small components (RESC) and higher-order Douglas-Kroll (DK) Hamiltonians within the framework of the two-component quasi-relativistic approach. The developing four-component relativistic and approximate quasi-relativistic methods have been implemented into a program suite named REL4D.In this article, we will introduce the efficient relativistic molecular theories to treat heavy-atomic molecular systems accurately via the four-component relativistic and the two-component quasi-relativistic approaches. We will also show several chemical applications including heavy-element systems with our relativistic molecular approaches.     

Relativistic theory for indirect nuclear spin–spin couplings within the polarization propagator approach

We present a relativistic theory for the nuclear spin–spin coupling tensor within the polarization propagator approach using the particle-hole Dirac–Coulomb–Breit Hamiltonian and the full four-component wave function. We give explicit expressions for the coupling tensor in the random-phase approximation, neglecting the Breit interaction. A purely relativistic perturbative electron–nuclear Hamiltonian is used and it is shown how the single relativistic contribution to the coupling tensor reduces to Ramsey's three second-order terms (Fermi contact, spin–dipole, and paramagnetic spin–orbit) in the nonrelativistic limit. The principal propagator becomes complex and the leading property integrals mix atomic orbitals of different parity. The well-known propagator expressions for the coupling tensor in the nonrelativistic limit is obtained neglecting terms of the order c^?n (n ? 1). © 1993 John Wiley & Sons, Inc.     

Connections between perturbation theory and the unitary transformation methods of deriving effective Hamiltonians

The connections between open shell Brillouin–Wigner perturbation theory and the Van Vleck unitary transformation formalisms for generating effective Hamiltonians are explored. An explicit expression is obtained relating the generator ? of the unitary transformation e^i? with the amplitudes to be found from perturbation theory. The “renormalization effects” needed to produce the explicit “orthogonal-Hermitian” form of the effective Hamiltonian in perturbation theory are related directly to the generator of the unitary transformation. The conclusions reached previously by Jørgensen and Brandow regarding the identity of the effective Hamiltonians of the formalisms are explicitly verified for the case that the generator ? satisfied the Kemble condition. The procedure suggests how the powerful techniques of perturbation theory can be used within the unitary transformation framework to guarantee properly renormalized wave functions.     

Spectroscopic constants and molecular properties of the X2Π and a4Σ− electronic states of the SiF radical

The potential energy curves (PECs) of the X²Π and a⁴Σ^? electronic states of the SiF radical have been studied by an ab initio quantum chemical method. The calculations have been made using the complete active space self‐consistent field (CASSCF) method, which is followed by the valence internally contracted multireference configuration interaction (MRCI) approach in combination with several correlation‐consistent basis sets. The effects on the PECs by the core‐valence correlation and relativistic corrections are included. The way to consider the relativistic correction is to use the third‐order Douglas–Kroll Hamiltonian approximation. The relativistic corrections are made at the level of cc‐pV5Z basis set. The core‐valence correlation corrections are performed using the cc‐pCV5Z basis set. To obtain more reliable results, the PECs determined by the MRCI calculations are also corrected for size‐extensivity errors by means of the Davidson modification (MRCI+Q). These PECs are extrapolated to the complete basis set limit by the total‐energy extrapolation scheme. Using these PECs, the spectroscopic parameters are determined and compared with those reported in the literature. With these PECs obtained by the MRCI+Q/CV+DK+56 calculations, the vibrational levels, inertial rotation, and centrifugal distortion constants of the first 20 vibrational state of each electronic state are calculated when the rotational quantum number J equals zero. Comparison with the Rydberg‐Klein‐Rees (RKR) data shows that the present results are reliable and accurate. The molecular constants of the X²Π and a⁴Σ^? electronic states determined by the MRCI+Q/CV+DK+56 calculations should be good prediction for future laboratory experiment. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011