Matrix element spin-other-orbit

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. 

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... 

It appears, therefore, that it is possible to obtain accurate expectation values of the spin-orbit operators for diatomic molecules. Matcha et a/.112-115 have provided general expressions for the integrals involved and from their work Hall, Walker, and Richards116 derived the diagonal one-centre matrix elements of the spin-other-orbit operator for linear molecules. Provided good Hartree-Fock wavefunctions are available, these should be sufficient for most calculations involving diatomic molecules. 

The first two terms represent spin-orbit coupling, whilst the second two are normally described as spin other-orbit terms. Following Fontana [39], Chiu pointed out that for matrix elements diagonal in the total electron spin S(S = S + S2), (8.208) is contracted to the sum of two terms,... 

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,... 

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... 

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. 

As a first step in applying these rules, one must examine > and > and determine by how many (if any) spin-orbitals > and > differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to aehieve maximal eoineidenee with those in the other determinant it is essential to keep traek of the number of permutations ( Np) that one makes in aehieving maximal eoineidenee. The results of the Slater-Condon rules given below are then multiplied by (-l) p to obtain the matrix elements between the original > and >. The final result does not depend on whether one ehooses to permute ... 

Yij is the sum over pairs of CSFs which differ by a single spin-orbital occupancy (i.e., one having (f>i occupied where the other has ( )j occupied after the two are placed into maximal coincidence-the sign factor (sign) arising from bringing the two to maximal coincidence is attached to the final density matrix element) ... 

Other two-electron operators are the mass-polarization and the spin-orbit coupling operator. A two-electron operator gives non-vanishing matrix elements between two Slater determinants if the determinants contain at least two electrons and if they differ in the occupation of at most two pairs of electrons. The second quantization representation of a two-electron operator must thus have the structure... 

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

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