Two-Component Hamiltonians

In the preceding chapters, we set out from fundamental physical theory to arrive at a suitable theory for calculations on atoms and molecules, which still features four-component one-particle states. However, we noted that not only for small nuclear charges the contribution of the lower components of these spinors are small indeed. Hence, attempts were made to find Hamiltonians which do not require lower components for the one-particle states and which therefore are more convenient from a conceptual and — if possible —from a computational point of view. The principal options for such an elimination of small components are now introduced. 

The negative-energy solutions are a troublesome aspect of the Dirac Hamiltonian and clearly of limited interest in chemical applications. Over the years, much effort has therefore been spent on eliminating the positronic degrees of freedom of the Dirac Hamiltonian. Most such efforts start from the Dirac equation in the molecular field 

The identification of the operator R becomes clear when considering the effect of the unitary transformation on the orbitals 

For positive-energy solutions we want the lower components to be zero, which implies 

The Pauli Hamiltonian has no lower bound and is therefore not recommended for variational calculations. It is nowadays mosUy used in low-order perturbation calculations on light atoms (Z 40). 

The second term can be thought of as an effective kinetic energy operator that goes to the non-relativistic one when V 0. Proper renormalization gives the Infinite Order Regular Approximation (lORA) [17], often approximated by scaled ZORA [16], which improves on ZORA. 

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Kedziera, D. (2005) Convergence of approximate two-component Hamiltonians How far is the Dirac limit. Journal of Chemical Physics, 123, 074109-1-074109-5. 

Sadlej AJ, Snijders JG, van Lenthe E, Baerends EJ (1995) Relativistic regular two-component Hamiltonians, four component regular relativistic Hamiltonians and the perturbational treatment of Dirac s equation. J. Chem. Phys. 102 1758-1766... 

Once the matrix representation of the i -operator is known, it can immediately be used to determine the matrix representation of the two-component Hamiltonian h+, which is, of course, given by Eq. (44) and reads... 

In this Chapter, we will show how a whole family of one- and two-component quasi-relativistic Hamiltonians can conveniently be derived. The operator difference between the quasi-relativistic Hamiltonians and the Dirac equation can be explicitly identified and used in perturbation expansions. Expressions are derived for a direct perturbation theory scheme based on quasi-relativistic two-component Hamiltonians. The remaining difference between the variational energy obtained using quasi-relativistic Hamiltonians and the energy of the Dirac equation is estimated numerically by applying the direct perturbation theory ap-... 

The two-component Hamiltonian of Eq. (1) is invariant under the time reversal operation. In the one-electron case and for a special choice of phases, the time reversal operator T is given by... 

More accurate two-component Hamiltonians can be obtained by successive application of higher-order DK transformations [12,49],... 

Recently, the accuracy of two-component Hamiltonians was analyzed in terms of [90,92],... 

In recent years, higher orders of the DK transformation were formulated and explored in benchmark calculations on small molecules. Furthermore, it was shown that highly accurate transformed two-component Hamiltonians can be generated via the DK transformations of higher orders. These Hamiltonians converge quite well for the known elements of the periodic table limits of accuracy become noticeable only for elements with Z > 120. Higher orders of DK transformed Hamiltonians yield only small corrections for molecular observables thus, for most applications with normal demands of accuracy, DK2 is a reasonable, efficient, and well established choice. A valuable alternative is provided by the ZORA scheme, as comparison of available results shows. On the other hand, in the near future, accurate four-component approaches are expected to be essentially restricted to benchmark calculations due to their computational requirements. 

Van Lenthe, E., R. van Leeuwen, E.J. Baerends and J.G. Snijders, 1994, Relativistic regular two-component Hamiltonians, in New Challenges in Computational Quantum Chemistry, eds R. Broer, P.J.C. Aerts and PS. Bagus (Department of Chemical Sciences and Material Science Centre, University of Groningen, Netherlands). 

Because of the li (i) operators in this expression, this Hamiltonian is (less rigorously) called a four-component Hamiltonian in order to distinguish it from more approximate Hamiltonians that contain one-electron operators refering to 2-spinor representations, which are therefore called two-component Hamiltonians. In Eq. (8.66) the nucleus-nucleus repulsion operators are incorporated in the last term on the right-hand side of that equation abbreviated as... 

D. Kgdziera. Convergence of Approximate Two-Component Hamiltonians How far is the Dirac Limit J. Chem. Phys., 123 (2005) 074109. 

We can split the 1 2 2 term into scalar and spin-orbit operators, which makes it possible to define a spin-free one-electron Hamiltonian. The two-component Hamiltonian for the positive-energy states can then be written... 

Van Lenthe E, Baerends EJ, Snijders JG. Relativistic regular two-component Hamiltonians. J Chem Phys [Internet]. 1993 [cited 2014 Feb 24] 99(6) 4597. Available from http //link.aip.org/ link/JCPSA6/v99/i6/p4597/sl Agg=doi. 

At the time both LS-based methods (one component calculations followed by a separate calculation of the spin-orbit contributions), two- and four-component methods were developed. Despite the increase in computer power the high computational cost of four-component methods restricts their applicability to relatively small molecular systems. However, different flavors of two-component Hamiltonian have matured in the past years and are now approaching the computational efficiency of one-component methods (ZORA, X2C, etc.). (See for instance references [1,28-35]). As a result, for chemical reactions or spectroscopic studies, one-component approaches treating spin-orbit coupling a posteriori are preferred. 

Liu W, Peng D. Exact two-component Hamiltonians revisited. J Chem Phys. 2009 131 031104. Nakajima T, Hirao K. The Douglas-Kroll-Hess Approach. Chem Rev. 2011 112 385-402. Belpassi L, Storchi L, Quiney HM, Tarantelli F. Recent advances and perspectives in four-component Dirac-Kohn-Sham calculations. Phys Chem Chem Phys. 2011 13 12368-12394. Peng D, Reiher M. Exact decoupling of the relativistic Fock operator. Theor Chem Acc. 2012 131 1081. 

There has been much excitement in the relativistic quantum chemistry community regarding the possibility of constructing a formally exact two-component Hamiltonian for molecular calculations [50-56], as outlined in several review articles recently [56-59]. To be specific, an exact Hamiltonian can be constructed relatively straightforwardly at the one-electron level. Many-electron effects can be built into the approach in a pragmatic way with the help of model potentials [60-62]. For perspectives on a systematic incorporation of electron correlation into relativistic quantum chemical methods with many-electron wavefunctions, see Kutzelnigg [58] Liu [59] Saue [56] Saue and Visscher [63]. For a perspective on DFT, see van Wiillen [64]. 

X2C ( eXact 2-Component ) is an umbrella acronym [56] for a variety of methods that arrive at an exactly decoupled two-component Hamiltonian, with X2C referring to one-step approaches [65]. Related methods to arrive at formally exact two-component relativistic operators are, for example, infinite-order methods by Barysz and coworkers (BSS = Barysz Sadlej Snijders, lOTC = infinite-order two-component) [66-69] and normalized elimination of the small component (NESC) methods [70-77]. We discuss here an X2C approach as it has been implemented in a full two-component form with spin-orbit (SO) coupling and transformation of electric property operators to account for picture-change (PC) corrections [14],... 

Computational Tools for Predictive Modeling 315 The X2C two-component Hamiltonian matrix of equation (12.24) is then obtained via... 

Liu W and Peng D 2009 Exact two-component hamiltonians revisited. J. Chem. Phys. 131(3), 031104-4. 

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