Exact-decoupling methods

All exact-decoupling approaches can be related to the modified Dirac equation and we closely follow here the work presented in Refs. [16,647]. Two-component electrons-only Hamiltonians can be obtained from block-diagonalizing the four-component (one-electron) modified Dirac equation in matrix representation. As we have discussed in chapters 8 and 10 for four-component Dirac-Hartree-Fock-Roothaan calculations, basis functions for the small component must fulfill certain constraints as otherwise variational instability and a wrong nonrelativistic limit [547] would result. The correct nonrelativistic limit will be obtained if the kinetic-balance condition, p E c / holds 

For introducing the kinetic-balance restriction on the small component s basis functions into the one-electron equation, we define an operator s 

KB-transformed operators are crucial for the implementation of exact-decoup-ling methods. The free-particle Foldy-Wouthuysen transformation (cf. chapter 11) 

The two-component spinor functions cpfi can be chosen as spin orbitals which are the direct product of real scalar functions with spin functions, A (x) a,jS. Such a choice reduces the cost of basis orthogonalization since only real matrices are involved and different spins are decoupled. 

We apply the following notation for operator matrices in the discussion to come matrices written as M indicate real matrices in the basis of a set of spin-free (one-component) basis functions A. The notation M denotes matrices in two-component spinor space cpfi and M refers to the complete basis function space in which the 4-spinors (tpf, q ) are expanded. We therefore write the ( two-component ) standard matrices 

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The procedure not to apply the (a p)-operator onto the basis functions but to consider it explicitly and rewrite the one-electron equation is a major trick which has been dicsussed by many authors and which is intimately connected with the modified Dirac equation and the so-called exact-decoupling methods discussed in detail in section 14.1. 

The development of exact-decoupling methods has led to computational approaches that make relativistic all-electron quantum calculations for large molecules feasible. Because of their importance for practical calculations, it is advisable to compare all of them (the comparison to follow is based on Refs. [16,647]). Exact-decoupling methods have been given many names by different authors, but there exist only three different types. 

Thus, the evaluation of the X-operator in matrix form requires the diago-nalization of the modified Dirac-Roothaan equation, but its diagonalization posed the pitfalls that had led to the derivation of exact-decoupling methods in order to avoid pathologies originating from the negative-energy states in the first place. However, if this step can be accomplished with an approximate potential V —> V, which does not include the full electron-electron interaction, then it can be a very efficient procedure. 

So far, we have discussed the three variants of exact-decoupling methods. In principle, they are all exact two-component methods when employing the full electrostatic potential V. However, in practice, approximations are introduced in order to increase the efficiency (ideally without compromising the accuracy). The discussion to follow continues according to Ref. [16] and is independent of how the exact-decoupling transformation U is obtained. It therefore holds for all exact-decoupling methods. 

It should be emphasized that the first possibility is the standard choice and makes the scalar-relativistic versions of all three exact-decoupling methods identical when it comes to a comparison of expectation values. If the second possibility is adopted, clearly different terms emerge, as has been discussed in the literature [754,755]. 

This section collects results obtained with the three exact-decoupling methods within the same implementation and follows the discussion in Ref. [647]. The number of matrix operations necessary for the implementation of different two-component approaches has been collected in Table 14.2. The multiplication of a general matrix with a diagonal matrix requires O(m ) multiplications of floating-point numbers, where m is the dimension of the matrix identical to the number of (scalar) basis functions in this context. The multiplication of two general matrices scales formally as If m is large, the cost of the... 

M. Reiher. An efficient implementation of two-component relativistic exact-decoupling methods for large molecules. 

J. Autschbach, D. Peng, M. Reiher. Two-component relativistic calculations of electric-field gradients using exact decoupling methods Spin-orbit and picture-change effects. J. Chem. Theory Comput., 8 (2012)4239-4248. 

This approach was employed by Peralta et al. [649,661] and explored in detail by Thar and Kirchner [668] for the low-order DKH method. We refer to it as the diagonal local approximation to the Hamiltonian (DLH). It is clear that the DLH approximation can be applied to all relativistic exact-decoupling approaches. Obviously, the DLH approximation will work best at large interatomic distances. For example, it is a good approximation for heavy-atom molecules in solution for which it was conceived in Ref. [668]. Then, A represents the group of atoms that form one of the solute and/or solvent molecules. 

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],... 

For samples with well-resolved lines, the selective-decoupling method involves placing the rf field at the exact resonance frequency of the chosen resonance. Low rf power is used under these conditions so that only the selected resonance is affected. All other lines are far enough from resonance that their excitation is negligible. Be-... 

This e qnession for the propagators is still exact, as long as, the principal sub-manifold h and its complement sub-manifold h arc complete, and the characteristics of the propagator is reflected in the construction of these submanifolds (47,48). It should be noted that a different (asymmetric) metric for the superoperator space, Eq. (2.5), could be invoked so that another decoupling of the equations of motion is obtained (62,63,82-84). Such a metric will not be explored here, but it just shows the versatility of the propagator methods. 

We also remark that Eq. (5.44) may be decomposed into separate sets of equations for the odd and even ap(t) which are decoupled from each other. Essentially similar differential recurrence relations for a variety of relaxation problems may be derived as described in Refs. 4, 36, and 73-76, where the frequency response and correlation times were determined exactly using scalar or matrix continued fraction methods. Our purpose now is to demonstrate how such differential recurrence relations may be used to calculate mean first passage times by referring to the particular case of Eq. (5.44). 

Although the full Navier Stokes equations are nonlinear, we have studied a number of problems in Chap. 3 in which the flow was either unidirectional so that the nonlinear terms u Vu were identically equal to zero or else appeared only in an equation for the crossstream pressure gradient, which was decoupled from the primary linear flow equation, as in the ID analog of circular Couette flow. This class of flow problems is unusual in the sense that exact solutions could be obtained by use of standard methods of analysis for linear PDEs. In virtually all circumstances besides the special class of flows described in Chap. 3, we must utilize the original, nonlinear Navier Stokes equations. In such cases, the analytic methods of the preceding chapter do not apply because they rely explicitly on the so-called superposition principle, according to which a sum of solutions of linear equations is still a solution. In fact, no generally applicable analytic method exists for the exact solution of nonlinear PDEs. 

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