Exact decoupling of the Dirac Hamiltonian. I. General theory

Exact decoupling of positive- and negative-energy states in relativistic quantum chemistry is discussed in the framework of unitary transformation techniques. The obscure situation that each scheme of decoupling transformations relies on different, but very special parametrizations of the employed unitary matrices is critically analyzed. By applying the most general power series ansatz for the parametrization of the unitary matrices it is shown that all transformation protocols for decoupling the Dirac Hamiltonian have necessarily to start with an initial free-particle Foldy-Wouthuysen step. The purely numerical iteration scheme applying X-operator techniques to the Barysz-Sadlej-Snijders (BSS) Hamiltonian is compared to the analytical schemes of the Foldy-Wouthuysen (FW) and Douglas-Kroll-Hess (DKH) approaches. Relying on an illegal 1/c expansion of the Dirac Hamiltonian around the nonrelativistic limit, any higher-order FW transformation is in principle ill defined and doomed to fail, irrespective of the specific features of the external potential. It is shown that the DKH method is the only valid analytic unitary transformation scheme for the Dirac Hamiltonian. Its exact infinite-order version can be realized purely numerically by the BSS scheme, which is only able to yield matrix representations of the decoupled Hamiltonian but no analytic expressions for this operator. It is explained why a straightforward numerical iterative extension of the DKH procedure to arbitrary order employing matrix representations is not feasible within standard one-component electronic structure programs. A more sophisticated ansatz based on a symbolical evaluation of the DKH operators via a suitable parser routine is needed instead and introduced in Part II of this work.     

Exact decoupling of the Dirac Hamiltonian. II. The generalized Douglas-Kroll-Hess transformation up to arbitrary order

In order to achieve exact decoupling of the Dirac Hamiltonian within a unitary transformation scheme, we have discussed in part I of this series that either a purely numerical iterative technique (the Barysz-Sadlej-Snijders method) or a stepwise analytic approach (the Douglas-Kroll-Hess method) are possible. For the evaluation of Douglas-Kroll-Hess Hamiltonians up to a pre-defined order it was shown that a symbolic scheme has to be employed. In this work, an algorithm for this analytic derivation of Douglas-Kroll-Hess Hamiltonians up to any arbitrary order in the external potential is presented. We discuss how an estimate for the necessary order for exact decoupling (within machine precision) for a given system can be determined from the convergence behavior of the Douglas-Kroll-Hess expansion prior to a quantum chemical calculation. Once this maximum order has been accomplished, the spectrum of the positive-energy part of the decoupled Hamiltonian, e.g., for electronic bound states, cannot be distinguished from the corresponding part of the spectrum of the Dirac operator. An efficient scalar-relativistic implementation of the symbolic operations for the evaluation of the positive-energy part of the block-diagonal Hamiltonian is presented, and its accuracy is tested for ground-state energies of one-electron ions over the whole periodic table. Furthermore, the first many-electron calculations employing sixth up to fourteenth order DKH Hamiltonians are presented.     

Analytic energy gradients for the spin-free exact two-component theory using an exact block diagonalization for the one-electron Dirac Hamiltonian

We report the implementation of analytic energy gradients for the evaluation of first-order electrical properties and nuclear forces within the framework of the spin-free (SF) exact two-component (X2c) theory. In the scheme presented here, referred to in the following as SFX2c-1e, the decoupling of electronic and positronic solutions is performed for the one-electron Dirac Hamiltonian in its matrix representation using a single unitary transformation. The resulting two-component one-electron matrix Hamiltonian is combined with untransformed two-electron interactions for subsequent self-consistent-field and electron-correlated calculations. The "picture-change" effect in the calculation of properties is taken into account by considering the full derivative of the two-component Hamiltonian matrix with respect to the external perturbation. The applicability of the analytic-gradient scheme presented here is demonstrated in benchmark calculations. SFX2c-1e results for the dipole moments and electric-field gradients of the hydrogen halides are compared with those obtained from nonrelativistic, SF high-order Douglas-Kroll-Hess, and SF Dirac-Coulomb calculations. It is shown that the use of untransformed two-electron interactions introduces rather small errors for these properties. As a first application of the analytic geometrical gradient, we report the equilibrium geometry of methylcopper (CuCH(3)) determined at various levels of theory.     

Exact decoupling of the Dirac Hamiltonian. IV. Automated evaluation of molecular properties within the Douglas-Kroll-Hess theory up to arbitrary order

In Part III [J. Chem. Phys. 124, 064102 (2005)] of this series of papers on exact decoupling of the Dirac Hamiltonian within transformation theory, we developed the most general account on how to treat magnetic and electric properties in a unitary transformation theory on the same footing. In this paper we present an implementation of a general algorithm for the calculation of magnetic as well as electric properties within the framework of Douglas-Kroll-Hess theory. The formal and practical principles of this algorithm are described. We present the first high-order Douglas-Kroll-Hess results for property operators. As for model properties we propose to use the well-defined radial moments, i.e., expectation values of r(k), which can be understood as terms of the Taylor-series expansion of any property operator. Such moments facilitate a rigorous comparison of methods free of uncertainties which may arise in a direct comparison with experiment. This is important in view of the fact that various approaches to two-component molecular properties may yield numerically very small terms whose approximate or inaccurate treatment would not be visible in a direct comparison to experimental data or to another approximate computational reference. Results are presented for various degrees of decoupling of the model properties within the Douglas-Kroll-Hess scheme.     

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.     

Douglas–Kroll–Hess Theory: a relativistic electrons-only theory for chemistry

A unitary transformation allows to separate (block-diagonalize) the Dirac Hamiltonian into two parts one part: solely describes electrons, while the other gives rise to negative-energy states, which are the so-called positronic states. The block-diagonal form of the Hamiltonian no longer accounts for the coupling of both kinds of states. The positive-energy (‘electrons-only’) part can serve as a ‘fully’ relativistic electrons-only theory, which can be understood as a rigorous basis for chemistry. Recent developments of the Douglas–Kroll–Hess (DKH) method allowed to derive a sequence of expressions, which approximate this electrons-only Hamiltonian up to arbitrary-order. While all previous work focused on the numerical stability and accuracy of these arbitrary-order DKH Hamiltonians, conceptual issues and paradoxa of the method were mostly left aside. In this work, the conceptual side of DKH theory is revisited in order to identify essential aspects of the theory to be distinguished from purely computational consideration.     

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.     

Convergence of approximate two-component Hamiltonians: how far is the Dirac limit

A systematic elimination of the off-diagonal parts of the Dirac Hamiltonian is carried out in the spirit of the Douglas-Kroll [Ann. Phys. 82, 87 1974] approach and the recently proposed infinite-order two-component method. The present approach leads to a series of approximate two-component Hamiltonians which are exact through a certain order in the external potential. These Hamiltonians are used to study the convergence pattern of approximate two-component theories. It is shown that to achieve an acceptably high accuracy for low-lying one-electron levels in heavy and superheavy systems one needs to use approximate Hamiltonians of prohibitively high order in the external potential. One can conclude that the finite-order two-component Hamiltonians are of limited usefulness in accurate relativistic calculations for heavy and superheavy systems.     

Analytic second derivatives for the spin-free exact two-component theory

The formulation and implementation of the spin-free (SF) exact two-component (X2c) theory at the one-electron level (SFX2c-1e) is extended in the present work to the analytic evaluation of second derivatives of the energy. In the X2c-1e scheme, the four-component one-electron Dirac Hamiltonian is block diagonalized in its matrix representation and the resulting "electrons-only" two-component Hamiltonian is then used together with untransformed two-electron interactions. The derivatives of the two-component Hamiltonian can thus be obtained by means of simple manipulations of the parent four-component Hamiltonian integrals and derivative integrals. The SF version of X2c-1e can furthermore exploit available nonrelativistic quantum-chemical codes in a straightforward manner. As a first application of analytic SFX2c-1e second derivatives, we report a systematic study of the equilibrium geometry and vibrational frequencies for the bent ground state of the copper hydroxide (CuOH) molecule. Scalar-relativistic, electron-correlation, and basis-set effects on these properties are carefully assessed.     

Local relativistic exact decoupling

We present a systematic hierarchy of approximations for local exact decoupling of four-component quantum chemical Hamiltonians based on the Dirac equation. Our ansatz reaches beyond the trivial local approximation that is based on a unitary transformation of only the atomic block-diagonal part of the Hamiltonian. Systematically, off-diagonal Hamiltonian matrix blocks can be subjected to a unitary transformation to yield relativistically corrected matrix elements. The full hierarchy is investigated with respect to the accuracy reached for the electronic energy and for selected molecular properties on a balanced test molecule set that comprises molecules with heavy elements in different bonding situations. Our atomic (local) assembly of the unitary exact-decoupling transformation--called local approximation to the unitary decoupling transformation (DLU)--provides an excellent local approximation for any relativistic exact-decoupling approach. Its order-N(2) scaling can be further reduced to linear scaling by employing a neighboring-atomic-blocks approximation. Therefore, DLU is an efficient relativistic method well suited for relativistic calculations on large molecules. If a large molecule contains many light atoms (typically hydrogen atoms), the computational costs can be further reduced by employing a well-defined nonrelativistic approximation for these light atoms without significant loss of accuracy. We also demonstrate that the standard and straightforward transformation of only the atomic block-diagonal entries in the Hamiltonian--denoted diagonal local approximation to the Hamiltonian (DLH) in this paper--introduces an error that is on the order of the error of second-order Douglas-Kroll-Hess (i.e., DKH2) when compared with exact-decoupling results. Hence, the local DLH approximation would be pointless in an exact-decoupling framework, but can be efficiently employed in combination with the fast to evaluate DKH2 Hamiltonian in order to speed up calculations for which ultimate accuracy is not the major concern.     

Quasirelativistic theory. II. Theory at matrix level

The Dirac operator in a matrix representation in a kinetically balanced basis is transformed to the matrix representation of a quasirelativistic Hamiltonian that has the same electronic eigenstates as the original Dirac matrix (but no positronic eigenstates). This transformation involves a matrix X, for which an exact identity is derived and which can be constructed either in a noniterative way or by various iteration schemes, not requiring an expansion parameter. Both linearly convergent and quadratically convergent iteration schemes are discussed and compared numerically. The authors present three rather different schemes, for each of which even in unfavorable cases convergence is reached within three or four iterations, for all electronic eigenstates of the Dirac operator. The authors present the theory both in terms of a non-Hermitian and a Hermitian quasirelativistic Hamiltonian. Quasirelativistic approaches at the matrix level known from the literature are critically analyzed in the frame of the general theory.     

Relation between different variants of the generalized Douglas-Kroll transformation through sixth order

Wolf et al. have recently investigated a generalized Douglas-Kroll transformation. From a general class of unitary transformations that can be used in the Douglas-Kroll transformation, they pick one which is supposed to give, at a given order, an optimal transformed Dirac Hamiltonian. Results were presented through the fifth order. However, no data were given to demonstrate to which extent the so-called "optimal" Douglas-Kroll transformation is superior to other choices. In this work, the Douglas-Kroll transformation is extended to the sixth order for the first time, using computer algebra algorithms to obtain the working equations. It is shown how, at a given order, different variants of the Douglas-Kroll Hamiltonians are related. Various choices of the generalized transformation are examined numerically for the ground states of the one-electron atomic ions with nuclear charges Z=20, 40, 60, 80, 100, and 120. It is shown that compared to the improvement obtained by including the next order, the differences between various choices for the generalized Douglas-Kroll transformation are almost negligible. Results closest to the Dirac eigenvalues are not obtained with the optimal Douglas-Kroll transformation given by Wolf et al., but with the parametrization originally suggested by Douglas and Kroll.     

Nonvariational time-dependent multiconfiguration self-consistent field equations for electronic dynamics in laser-driven molecules

A time-dependent multiconfiguration self-consistent field (TDMCSCF) scheme is developed to describe the time-resolved electron dynamics of a laser-driven many-electron atomic or molecular system, starting directly from the time-dependent Schrodinger equation for the system. This nonvariational formulation aims at the full exploitations of concepts, tools, and facilities of existing, well-developed quantum chemical MCSCF codes. The theory uses, in particular, a unitary representation of time-dependent configuration mixings and orbital transformations. Within a short-time, or adiabatic approximation, the TDMCSCF scheme amounts to a second-order split-operator algorithm involving generically the two noncommuting one-electron and two-electron parts of the time-dependent electronic Hamiltonian. We implement the scheme to calculate the laser-induced dynamics of the two-electron H2 molecule described within a minimal basis, and show how electron correlation is affected by the interaction of the molecule with a strong laser field.     

Infinite-order quasirelativistic density functional method based on the exact matrix quasirelativistic theory

The exact one-electron matrix quasirelativistic theory [Kutzelnigg and Liu, J. Chem. Phys. 123, 241102 (2005)] is extended to the effective one-particle Kohn-Sham scheme of density functional theory. Several variants of the resultant theory are discussed. Although they are in principle equivalent, consideration of computational efficiency strongly favors the one (F(+)) in which the effective potential remains untransformed. Further combined with the atomic approximation for the matrix X relating the small and large components of the Dirac spinors as well as a simple ansatz for correcting the two-electron picture change errors, a very elegant, accurate, and efficient infinite-order quasirelativistic approach is obtained, which is far simpler than all existing quasirelativistic theories and must hence be regarded as a breakthrough in relativistic quantum chemistry. In passing, it is also shown that the Dirac-Kohn-Sham scheme can be made as efficient as two-component approaches without compromising the accuracy. To demonstrate the performance of the new methods, atomic calculations on Hg and E117 are first carried out. The spectroscopic constants (bond length, vibrational frequency, and dissociation energy) of E117(2) are then reported. All the results are in excellent agreement with those of the Dirac-Kohn-Sham calculations.     

A second-quantization framework for the unified treatment of relativistic and nonrelativistic molecular perturbations by response theory

A formalism is presented for the calculation of relativistic corrections to molecular electronic energies and properties. After a discussion of the Dirac and Breit equations and their first-order Foldy-Wouthuysen [Phys. Rev. 78, 29 (1950)] transformation, we construct a second-quantization electronic Hamiltonian, valid for all values of the fine-structure constant alpha. The resulting alpha-dependent Hamiltonian is then used to set up a perturbation theory in orders of alpha(2), using the general framework of time-independent response theory, in the same manner as for geometrical and magnetic perturbations. Explicit expressions are given to second order in alpha(2) for the Hartree-Fock model. However, since all relativistic considerations are contained in the alpha-dependent Hamiltonian operator rather than in the wave function, the same approach may be used for other wave-function models, following the general procedure of response theory. In particular, by constructing a variational Lagrangian using the alpha-dependent electronic Hamiltonian, relativistic corrections can be calculated for nonvariational methods as well.     

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.     

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.     

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.     

Retarded boson-fermion interaction in atomic systems

The retarded interaction between an electron and a spin-0 nucleus is derived from electrodynamical perturbation theory. The contribution of retardation at order v(2)c(2) mimics the Breit interaction [Phys. Rev. 34, 553 (1929); 36, 388 (1930); 39, 616 (1932)] with the Dirac matrix alpha(2) being replaced by p(2)m(2)c where p(2) is the linear momentum operator for the nucleus. An effective one-electron retardation operator is obtained in relative coordinates, and this can be used through all orders in perturbation theory without any problem of infinite degeneracy. A few steps of unitary transformation lead to the nonrelativistic limit. The leading terms in retardation corrections to energy are of order (m(e)m(n))alpha(2)Z(4)(alpha(2)m(e)c(2)). The implications for atomic systems are discussed.     

Local unitary transformation method for large-scale two-component relativistic calculations: case for a one-electron Dirac Hamiltonian

An accurate and efficient scheme for two-component relativistic calculations at the spin-free infinite-order Douglas-Kroll-Hess (IODKH) level is presented. The present scheme, termed local unitary transformation (LUT), is based on the locality of the relativistic effect. Numerical assessments of the LUT scheme were performed in diatomic molecules such as HX and X(2) (X = F, Cl, Br, I, and At) and hydrogen halide clusters, (HX)(n) (X = F, Cl, Br, and I). Total energies obtained by the LUT method agree well with conventional IODKH results. The computational costs of the LUT method are drastically lower than those of conventional methods since in the former there is linear-scaling with respect to the system size and a small prefactor.