Exact 2-component Hamiltonian/method

As seen in equation (26), the quasi-relativistic Hamiltonian and the operators describing the difference between the exact Dirac Hamiltonian and the quasi-relativistic one are now explicitly separated and the direct perturbation theory method can be applied. In the direct perturbation theory approach, the metric is also affected by the perturbation [12]. Note that the interaction matrix is block diagonal at the lORA level of theory, whereas the coupling between the upper and the lower components still appears in the metric. 

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],... 

In a second step, in order to determine the influence of the anharmonicity in the exact potential we will expand the term up to higher powers of the components of r and treat them as small perturbations to the harmonic approximation of the Hamiltonian by means of first order perturbation theory. These perturbative calculations offer insight into the effects of the anharmonic parts of the potential onto the energies and the form of the wave functions. For a discussion of the basis set method and the computational techniques used for the numerical calculation of the exact eigenenergies and eigenfunctions in the outer potential well we refer the reader to [7]. In the following we discuss the results of these numerical calculations of the exact eigenenergies and wave functions and... 

These terms include the ft)-component of M 1 and M 2 thus, applying the perpendicular field Hamiltonian leads to a mixture of different Ms for the wave function of the next FC order. Thus, the dimensions in the perpendicular case are larger than those in the parallel case see Tables 13.1 and 13.2. Note that the M expansion is naturally introduced by the Hamiltonian, and we did not take any consideration of the initial or g functions. This feature comes from the exact structure of the FC wave function and shows an important merit of the FC method the correct wave function expansion is always achieved, even when we start from a simple initial function. 

The operator Eq. (1.4) is not easy to solve since it involves terms which are linear in ap [7]. One of the possible way to solve the Eq. (1.4) is by means of some iterative scheme. It can be made through some odd powers of a, say, k = 2, 3,... (with a denoting the fine structure constant, a = 1/c). Then, the unitary transformation U will be exact through the same order in a. Simultaneously, this will lead to the approximate form h2], k = 2, 3,... oih. Thus the method leads to a series of two-component relativistic Hamiltonians whose accmacy is determined by the accuracy of the iterative solution for R. In each step of the iteration the analytical form of the R operator (Eq. (1.4)) and the Hamiltonian (Eq. (1.5)) have to be derived. 

Molecular level computer simulations based on molecular dynamics and Monte Carlo methods have become widely used techniques in the study and modeling of aqueous systems. These simulations of water involve a few hundred to a few thousand water molecules at liquid density. Because one can form statistical mechanical averages with arbitrary precision from the generated coordinates, it is possible to calculate an exact answer. The value of a given simulation depends on the potential functions contained in the Hamiltonian for the model. The potential describing the interaction between water molecules is thus an essential component of all molecular level models of aqueous systems. 

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