Diamagnetic electronic operator

Computation of the spin-orbit contribution to the electronic g-tensor shift can in principle be carried out using linear density functional response theory, however, one needs to introduce an efficient approximation of the two-electron spin-orbit operator, which formally can not be described in density functional theory. One way to solve this problem is to introduce the atomic mean-field (AMEI) approximation of the spin-orbit operator, which is well known for its accurate description of the spin-orbit interaction in molecules containing heavy atoms. Another two-electron operator appears in the first order diamagnetic two-electron contribution to the g-tensor shift, but in most molecules the contribution of this operator is negligible and can be safely omitted from actual calculations. These approximations have effectively resolved the DET dilemma of dealing with two-electron operators and have so allowed to take a practical approach to evaluate electronic g-tensors in DET. Conventionally, DET calculations of this kind are based on the unrestricted... 

The dominant diamagnetic Hamiltonian term is a simple one-electron operator and its expectation value, when the ground-state determinantal wave function is constructed from the set of occupied molecular spin-orbitals, is... 

This electronic Diamagnetic Shielding operator contributes to the ESR g-tensor. 

Most of the properties of molecules that one might evaluate from a molecular wave function, such as the dipole moment, quadrupole moment, field gradient at a nucleus, diamagnetic susceptibility, etc., are described by sums of one-electron operators of the general form... 

In Ramsey expressions, the diamagnetic electronic and spm-orbit operators are given by... 

The diamagnetic spin-orbit operator describes the direct interaction between the two orbital magnetic moments induced in the electron density by the presence of the magnetic moments of nuclei K and L. As for the diamagnetic shielding operator, this is a 3 x 3 nonsymmetric tensor operator, with in general nine independent elements. 

Two contributions to the magnetizability appear in the nonrelativistic electronic Hamiltonian in the presence of a magnetic vector potential O Eq. 11.40. One arises as an expectation value of the diamagnetic magnetizability operator, see O Eq. 11.47. The second involves a linear response contribution arising from the interaction of the magnetic dipole operator O Eq. 11.44 with itself We can, therefore, calculate the magnetizability from the expression ... 

The total molecular susceptibility has now been expressed as a sum over operators localized on the various atomic nuclei. But they operate on wave functions that extend over the whole molecule. If the average values of these atomic operators are not greatly dependent on parts of the wave function far removed from the nucleus in question and if the relevant properties of the electron distribution around each nucleus are not much different for a given type of atom in different molecules, the terms within each sum over n in Eq. (41) will be independent and constant. They will, in short, be additive atomic susceptibilities that can be evaluated from measured molecules and used to predict the susceptibility of any desired molecule. We have already demonstrated the additivity of the diamagnetic susceptibilities [Eq. (37)]. 

Spin Conservation. In elementary chemical acts, such as bond cleavage, bond formation, or electron transfer, magnitude and direction of spin are conserved. This means that both the expectation value of total electron spin (operator S2) and that of its z projection (operator Sz) remain constant the same holds for the nuclear spin states. Hence, for example, fragmentation of a molecule M in a singlet state S>, which is characterized by zero total spin and zero z projection of spin, into two particles 1 and 2 yields either two singlet (i.e., diamagnetic) species, or two radicals, the spins of which are paired in a specific manner such that spin conservation is fulfilled (in a way, they are... 

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