Hamiltonian other-orbit interaction

We are now in a position to present the total electronic Hamiltonian by summing over all possible electrons i. We must be very careful, however, not to count the various interactions twice. Thus on summing over i, we modify all terms which are symmetric in i and j by a factor of 1 /2. These terms are the electron-electron Coulomb interaction (3.141), the orbit-orbit interaction (3.145), the spin-spin interaction (3.151) and the spin-other-orbit interaction from (3.144) and (3.153). 

In this Hamiltonian (5) corresponds to the orbital angular momentum interacting with the external magnetic field, (6) represents the diamagnetic (second-order) response of the electrons to the magnetic field, (7) represents the interaction of the nuclear dipole with the electronic orbital motion, (8) is the electronic-nuclear Zeeman correction, the two terms in (9) represent direct nuclear dipole-dipole and electron coupled nuclear spin-spin interactions. The terms in (10) are responsible for spin-orbit and spin-other-orbit interactions and the terms in (11) are spin-orbit Zeeman gauge corrections. Finally, the terms in (12) correspond to Fermi contact and dipole-dipole interactions between the spin magnetic moments of nucleus N and an electron. Since... 

Judd, Crosswhite, and Crosswhite (10) added relativistic effects to the scheme by considering the Breit operator and thereby produced effective spin-spin and spin-other-orbit interaction Hamiltonians. The reduced matrix elements may be expressed as a linear combination of the Marvin integrals,... 

It can be shown (Veseth, 1970) that all electron-nuclear distances, r ) can be referred to a common origin, and, neglecting only the contribution of spin-other-orbit interactions between unpaired electrons, the two-electron part of the spin-orbit Hamiltonian can be incorporated into the first one-electron part as a screening effect. The spin-orbit Hamiltonian of Eq. (3.4.2) can then be written as... 

Racah parameters for f configurations = electronic charge = single-electron operators = direct Slater parameters = two-electron operators = exchange Slater parameters = one of the special Cartan s groups = Planck s constant /2ir = hamiltonian = electrostatic interaction = spin-orbit interaction = spin-spin interaction = spin-other-orbit interaction... 

The consequence is that we must treat the spin-orbit and the spin-other-orbit interactions separately we cannot combine them as in the Breit-Pauli Hamiltonian. The reason is that the functions on which the momentum operators operate are derived from the small component, and only in the nonrelativistic limit where the large and small components are related by kinetic balance can we rewrite the spatial part of the spin-other-orbit interaction in the same form as the spin-orbit interaction. The reader... 

Although the electrostatic and spin-orbit interactions are by far the most important terms in the Hamiltonian, other smaller interactions have to be considered in order to get a good agreement between experimental and calculated energy levels. Diagonalization of the energy matrix which incorporates only the electrostatic and spin-orbit interaction, often results in discrepancies between experimental and calculated levels of several hundred cm (Wyboume 1965). 

An indirect mode of anisotropic hyperfine interaction arises as a result of strong spin-orbit interaction (174)- Nuclear and electron spin magnetic moments are coupled to each other because both are coupled to the orbital magnetic moment. The Hamiltonian is... 

Quantum chemical methods may be divided into two classes wave function-based techniques and functionals of the density and its derivatives. In the former, a simple Hamiltonian describes the interactions while a hierarchy of wave functions of increasing complexity is used to improve the calculation. With this approach it is in principle possible to come arbitrarily close to the correct solution, but at the expense of interpretability of the wave function the molecular orbital concept loses meaning for correlated wave functions. In DFT on the other hand, the complexity is built into the energy expression, rather than in the wave function which can still be written similar to a simple single-determinant Hartree-Fock wave function. We can thus still interpret our results in terms of a simple molecular orbital picture when using a cluster model of the metal substrate, i.e., the surface represented by a suitable number of metal atoms. 

The Born- Oppenheimer approximation. With spin-orbit and other relativistic interactions omitted, the Hamiltonian of a polyatomic molecule is... 

The first term in eq. (1) Ho represents the spherical part of a free ion Hamiltonian and can be omitted without lack of generality. F s are the Slater parameters and ff is the spin-orbit interaction constant /<- and A so are the angular parts of electrostatic and spin-orbit interactions, respectively. Two-body correction terms (including Trees correction) are described by the fourth, fifth and sixth terms, correspondingly, whereas three-particle interactions (for ions with three or more equivalent f electrons) are represented by the seventh term. Finally, magnetic interactions (spin-spin and spin-other orbit corrections) are described by the terms with operators m and p/. Matrix elements of all operators entering eq. (1) can be taken from the book by Nielsen and Koster (1963) or from the Argonne National Laboratory s web site (Hannah Crosswhite s datafiles) http //chemistry.anl.gov/downloads/index.html. In what follows, the Hamiltonian (1) without Hcf will be referred to as the free ion Hamiltonian. 

Thus, the main relativistic effects are (1) the radical contraction and energetic stabilization of the s and p orbitals which in turn induce the radial expansion and energetic destabilization of the outer d and f orbitals, and (2) the well-known spin-orbit splitting. These effects will be pronounced upon going from As to Sb to Bi. Associated with effect (1), it is interesting to note that the Bi atom has a tendency to form compounds in which Bi is trivalent with the 6s 6p valence configuration. For this tendency of the 6s electron pair to remain formally unoxidized in bismuth compounds (i.e. core-like nature of the 6s electrons), the term inert pair effect or nonhybridization effect has been often used for a reasonable explanation. In this context, the relatively inert 4s pair of the As atom (compared with the 5s pair of Sb) may be ascribed to the stabilization due to the d-block contraction , rather than effect (1) . On the other hand, effect (2) plays an important role in the electronic and spectroscopic properties of atoms and molecules especially in the open-shell states. It not only splits the electronic states but also mixes the states which would not mix in the absence of spin-orbit interaction. As an example, it was calculated that even the ground state ( 2 " ) of Bij is 25% contaminated by Hg. In the Pauli Hamiltonian approximation there is one more relativistic effect called the Dawin term. This will tend to counteract partially the mass-velocity effect. 

Dealing with electrons we know that the dominant interaction between them is the Coulomb repulsion corrected, because electrons are fermions, by interactions induced by their spin. The spin-orbit interaction is already included in the one-electron Dirac Hamiltonian but the two-electron interaction should also include interactions classically known as spin-other-orbit, spin-spin etc... Furthermore a relativistic theory should incorporate the fact that the speed of light being finite there is no instantaneous interaction between particles. The most common way of deriving an effective Hamiltonian for a many electron system is to start from the Furry [11] bound interaction picture. A more detailed discussion is given in chapter 8 emd we just concentrate on some practical considerations. 

A common feature of all CI/SO methods, is to couple only the wave functions taken as basis functions to represent the Hamiltonian. No other configurations than the one already present in the considered wave functions are coupled by spin-orbit interaction. This means that some external configurations, close in energy to the ones of interest, may have large spin-orbit interactions with the states of interest. Neglecting them leads obviously to an imcontrolled loss of... 

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