Spin-other-orbit interaction matrix elements

The frozen-core (fc) approach is not restricted to spin-independent electronic interactions the spin-orbit (SO) interaction between core and valence electrons can be expressed by a sum of Coulomb- and exchange-type operators. The matrix element formulas can be derived in a similar way as the Sla-ter-Condon rules.27 Here, it is not important whether the Breit-Pauli spin-orbit operators or their no-pair analogs are employed as these are structurally equivalent. Differences with respect to the Slater-Condon rules occur due to the symmetry properties of the angular momentum operators and because of the presence of the spin-other-orbit interaction. It is easily shown by partial integration that the linear momentum operator p is antisymmetric with respect to orbital exchange, and the same applies to t = r x p. Therefore, spin-orbit... 

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,... 

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. 

In the molecular orbital description of the states of formaldehyde the active nonbonding orbital is mainly a pi(0) orbital. The inr states involve p (0) and Py C) orbitals. To have a significant value, the spin-orbit coupling matrix elements must include orbitals that have significant overlap. The effective matrix element for this particular interaction is of the type (py(0), (Rp (0)). The operator (R, rotates the p, orbital on the o.xygen into strong overlap with the Py orbital on oxygen and this is the portion of (R that can effectively mix the As and Mi states. The other components of (R are not so effective. 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

Second-order effects arising from the product of matrix elements involving J+ L and L+ S operators have the same form as 7J+S. In the case of H2, the second-order effect seems to be smaller than the first-order effect, but in other molecules this second-order effect will be more important than the first-order contribution to the spin-rotation constant. These second-order contributions can be shown to increase in proportion with spin-orbit effects, namely roughly as Z2, but the direct spin-rotation interaction is proportional to the rotational constant. For 2n states, 7 is strongly correlated with Ap, the spin-orbit centrifugal distortion constant [see definition, Eq. (5.6.6)], and direct evaluation from experimental data is difficult. On the other hand, the main second-order contribution to 7 is often due to a neighboring 2E+ state. Table 3.7 compares calculated with deperturbed values of 7 7eff of a 2II state may be deperturbed with respect to 2E+ by... 

From A L = 0, 1 we conclude that S may have non-vanishing matrix elements with other S states and P states. In (3d) there are no other S states and there is only one P state, namely P the selection rule on the spin quantum number, S, is satisfied. Since S has only one value of J, namely, 5/2, the selection rule on J restricts the spin-orbit interaction to Ss/2 with P6/2. Finally we must connect states of the same Mj, but the matrix element is independent of the particular value of Mj chosen. Thus, the interaction between 85/2 and P5/2 will be described by the matrix element... 

A model hamiltonian should have the structure of the full hamiltonian, but could in principle have terms consisting of higher order products of annihilation and creation operators. Here we limit considerations to such operators that contain a one-electron part and an electron-electron interaction part. The number of independent matrix elements can be considerably reduced by symmetry considerations and by requiring compatibility with other operator representatives. It is clear that the form of the spectral density requires that the hamiltonian commutes with the total orbital angular momentum and with various spin operators. These are given in the limited basis as... 

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