Hamiltonian, atomic spin-orbit

The formalism for treating light atom systems begins with the Breit equation. The atomic spin-orbit Hamiltonian is given by (5)... 

A prime on the summation sign as usual indicates the omission of diagonal terms. In the PPP-hamiltonian, this means the omission of terms where the operators would correspond to the same spin orbital. Thus, there is neither a term with Prr nor with UruTirv, but there is one with rirvUr-v The parameters Qr and Prs are matrix elements of Hcore between atomic spin orbitals Urv i) and and jrs is the electron repulsion integral between spin orbital densities Wr[Pg.177]

Although the individual orbitd-angular-momentum operators L, do not commute with the atomic Hamiltonian (11.1), one can show (Bethe and Jackiw, pp. 102-103) that L does commute with the atomic Hamiltonian [provided spin-orbit interaction (Section 11.6) is neglected]. We can therefore characterize an atomic state by a quantum number L, where L(L -I- 1) is the square of the magnitude of the toted electronic orbital angular momentum. The electronic wave function il/ of an atom satisfies L tfr = L(L -I- The total-electronic-orbital-angular-momentum quantum number L of an atom is specified by a code letter, as follows ... 

For polyatomic molecules the operator 5 for the square of the total electronic spin angular momentum commutes with the electronic Hamiltonian, and, as for diatomic molecules, the electronic terms of polyatomic molecules are classified as singlets, doublets, triplets, and so on, according to the value of 25 + 1. (The commutation of 5 and H holds provided spin-orbit interaction is omitted from the Hamiltonian for molecules containing heavy atoms, spin-orbit interaction is considerable, and 5 is not a good quantum number.)... 

In Chapter 15 we asserted that a set of commuting operators can have a set of common eigenfunctions. For example, the hydrogen atom spin orbitals can be chosen to be eigenfunctions of H, L, L, and S. The operator does not commute with the electronic Hamiltonian of the hJ ion. The physical reason for this is that all directions are not equivalent as they are with atoms, because of the presence of the two fixed nuclei. The operator does commute with the electronic Hamiltonian operator if the internuclear axis is chosen as the z axis. The molecular orbitals can be eigenfunctions ofLji... 

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

The spin magnetic moment Ms of an electron interacts with its orbital magnetic moment to produce an additional term in the Hamiltonian operator and, therefore, in the energy. In this section, we derive the mathematical expression for this spin-orbit interaction and apply it to the hydrogen atom. 

Thus, the total Hamiltonian operator H for a hydrogen atom including spin-orbit coupling is... 

While the Hamiltonian operator Hq for the hydrogen atom in the absence of the spin-orbit coupling term commutes with L and with S, the total Hamiltonian operator H in equation (7.33) does not commute with either L or S because of the presence of the scalar product L S. To illustrate this feature, we consider the commutators [L, L S] and [S, L S],... 

Fig. 1. BLYP/uncDZ mean dipole polarizability of the mercury atom as a function of frequency. All values in atomic units. SR+SO refers to calculations based on the Dirac-Coulomb Hamiltonians, whereas SR refers to calculations in which all spin-orbit interaction has been eliminated.

We extend the method over all three rows of TMs. No systematic study is available for the heavier atoms, where relativistic effects are more prominent. Here, we use the Douglas-Kroll-Hess (DKH) Hamiltonian [14,15] to account for scalar relativistic effects. No systematic study of spin-orbit coupling has been performed but we show in a few examples how it will affect the results. A new basis set is used in these studies, which has been devised to be used with the DKH Hamiltonian. 

Neglecting spin-orbit contributions (smaller than other relativistic corrections for the ground state of atoms, and zero for closed-shell ones), the Breit hamiltonian in the Pauli approximation [25] (weak relativistic systems) can be written for a many electron system as ... 

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