Atomic Unitary Transformation

Since relativistic corrections for light atoms are usually small and may be neglected, and can be approximated by the nonrelativistic Hamiltonian matrices 

However, for the heavy-light off-diagonal blocks H ja, a relativistic transformation approximation should be considered. 

A straightforward idea for a local approximation to the heavy-heavy blocks is to transform only the heavy-atom block-diagonal blocks of the relativistic Hamiltonian matrix H, 

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. 

A priori relativistic corrections to off-diagonal blocks in the Hamiltonian cannot be neglected. This problem has already been addressed in Ref. [665] in the framework of the so-called two-center approximation, in which the DKH 

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The unitary transform does the same thing as a similarity transform, except that it operates in a complex space rather than a real space. Thinking in terms of an added imaginary dimension for each real dimension, the space of the unitary matrix is a 2m-dimensionaI space. The unitary transform is introduced here because atomic or molecular wave functions may be complex. 

However, as is well known, the trace is invariant with respect to choice of basis functions that are related by a unitary transformation. Thus, rather than working with the basis of eigenfunctions we may, following eqn (4.31), work with respect to the basis of atomic orbitals, to write... 

There is no need to pass from the basis set of the ATs L,Mi,S,Ms) to the basis set of the atomic multiplets (LS),J,Mj) since such a unitary transformation does not lead to a gain in the computational effort. In the basis set of the L,Mi, S, Ms) functions, the operator Vax is diagonal but the operator Hso has off-diagonal matrix elements. In contrast, in the basis set of the (LS),J,Mj) kets, the operator Hso is diagonal, but the operator Vax has off-diagonal matrix elements. Therefore, none of these basis sets is appropriate for considering the Zeeman term as a small perturbation. 

A unitary transformation between the various orbitals of an individual atom we shall refer to this as space invariance. 

The unitary transformation from the basis of the CMOs to the basis of the LMOs of the /-system does not change the covalent contribution to the effective crystal field. According to numerical estimates the resonance integrals (,/. between d-AO and LPs of the donor atoms by I0-H00 times overcomes the resonance integrals between d- AO and any other LMOs and thus dominates the resonance interaction of the d- and /-systems. So, as it has been shown in [71], restricting the summation in eq. (4.82) by the sum of diagonal elements (L = L) over only the LPs results in error in the estimated splitting of the (/-levels of 0.1 eV. This precision is comparable to that of the EHCF method itself. This estimate is described by the formula ... 

Whitten and Pakkanen proposed a so-called localization scheme in which the interaction of the adsorbate levels with the large number of metal levels can be described by those orbitals for which the interaction < Htf t> is large. A unitary transformation can be introduced such that <4>k I Heff I 4t> I is maximized. This procedure yields one set of functions which physically represent orbitals localized on the designated surface atoms, bonds between these atoms, as well as bonds linking the designated atoms with the remainder of the cluster. A second set of functions consists of so-called interior orbitals. They are treated as an invariant core in each configuration for the step of the configuration... 

The current work presented here must be necessarily considered as a new step towards an easy, comprehensive and cheap way to obtain atomic and molecular wavefunctions. In the same path we are attempting here to show that the Jacobi Rotation techniques may be considered viable alternative procedures to the usual SCF and Unitary Transformation algorithms. 

We could make a unitary transformation on the atomic displacements to obtain coordinates that correspond to the kinds of local modes envisaged by Kleinman and Spitzer, three for each kind of atom in the crystal. As in the treatment of... 

The actual sign ("phase") of the molecular orbital at any given point r of the 3D space has no direct physical significance in fact, any unitary transformation of the MO s of an LCAO (linear combination of atomic orbitals) wavefunction leads to an equivalent description. Consequently, in order to provide a valid basis for comparisons, additonal constraints and conventions are often used when comparing MO s. The orbitals are often selected according to some extremum condition, for example, by taking the most localized [256-260] or the most delocalized [259,260] orbitals. Localized orbitals are often used for the interpretation of local molecular properties and processes [256-260]. The shapes of contour surfaces of localized orbitals are often correlated with local molecular shape properties. On the other hand, the shapes of the contour surfaces of the most delocalized orbitals may provide information on reactivity and on various decomposition reaction channels of molecules [259,260]. 

As demonstrated in Chapter 6 for a stationary state, the atomic force law is obtained by setting the generator equal to the momentum conjugate to the coordinate r, the electronic coordinate that is integrated over the basin of the atom It follows immediately from eqn (8.35), which shows the effect of an infinitesimal unitary transformation on an operator, that the generator F = e-p induces an infinitesimal uniform translation of the electronic coordinate r by — e. Each component of the vector e is an arbitrary, infinitesimal real number, fixed for all r and, from eqn (8.35),... 

Proceeding as before in the field-free case, the variations in the state function are replaced by operators which act as generators of infinitesimal unitary transformations. That is, 5 P = ( — lh)F where F is an infinitesimal Hermitian operator (F = eG). Introducing the notion of generators into the result for the variation of the atomic action integral yields... 

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