Diamagnetic term

However, there are also higher-order diamagnetic corrections to Hmg as they result from the partitioning of the Dirac equation 

These are expected to be negligibly small because of the coefficient 1 /c.  

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 

After introducing the LCAO approximation, with the bond-order matrix P as a result of the SCF procedure, we get 

The integrals over atomic orbitals include second-order momenta for all combinations of the Cartesian coordinates, i.e. x, xy, xz, y2, yz and z2, which can be arranged into six symmetric matrices 

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The diamagnetic term, adia, corresponds to the opposing field resulting from the effect of the imposed field upon the electron cloud surrounding the nucleus. In this case, electrons closer to the nucleus give rise to greater shielding than distant ones. 

The idea of a constant diamagnetic term (32,33) has been challenged, and it was concluded from semiempirical calculations that oAA cannot be neglected (39). Mason (40,41) suggested that diamagnetic terms be calculated using Fly-gare s expression (42) ... 

A compromise between the opposing interpretations of Pople and Mason is offered by Webb and co-workers (46). These authors conclude that it is improper to equate the diamagnetic terms used by the two authors, since in contrast to Pople s o A, Flygare s contains significant and variable nonlocal contributions and is comparable rather to the gauge-dependent term in the original Ramsey expression (29). 

After this step one may already identify the paramagnetic and diamagnetic terms as arising from the pp and pn first order orbital corrections to the zeroth order wave function. The contribution is then further approximated by neglecting all off-diagonal terms in the pn-pn part of the Hessian... 

The atom-plus-ligand local-term approximation has been justified a posteriori for 30) and 57) shielding, since certain expected relationships of chemical shifts, such as additivity of substituent effects, were found to be absent for the observed shifts, but were fulfilled for i.e., after correction for the atom-plus-ligand diamagnetic term. Some examples are given in the following sections. 

Table II shows that the observed 29) shifts 8 (taken as positive downfield) show a degree of linear additivity in the methyl-substituted methanes, with a shortfall for neopentane. Table II shows also the division of the shielding into atom-plus-ligand diamagnetic and paramagnetic parts, relative to methane for which era is 295 ppm. The diamagnetic terms were calculated by Flygare s method 25). With each substitution of hydrogen by carbon, aa for the central carbon increases by 28 ppm but o-p increases by about 37 ppm, and the line moves 9 ppm downfield. Analogous relationships have been demonstrated for shielding in methyl-substituted NH3 and NH4+ 30). Table II shows that the shortfall at neopentane is in the paramagnetic term.

Fig. 5. Plot of shifts in substituted methanes against the number of substituents. The observed shifts are compared with these corrected for the atom-plus-ligand diamagnetic term. [Appeared in Mason (57).]...

When discussing the electrical conductivity of metals, we described them in terms of ionic cores and delocalised valence electrons. The core electrons contribute a diamagnetic term to the magnetic susceptibility, but the valence electrons can give rise to paramagnetism or one of the cooperative effects we have described. 

Chemical shifts, as treated by Ramsey (113), are comprised of separate diamagnetic and paramagnetic contributions. The diamagnetic term was considered to arise from the response of the extranuclear electrons to the external magnetic field and had been given earlier by Lamb (65) as... 

Xd is the electronic contribution to the paramagnetism, obtainable either from the ESR signal or by direct measurements with subtraction of the diamagnetic term RH is the Hall coefficient, the formula being due to Friedman (1971). K is the Knight shift, g is found to drop from 1 to about 0.3 in this range and a is then about 50 2 1 cm x. This then should mark the metal-insulator transition. A plot of g deduced from the various data is shown in Fig. 10.20. This we consider further evidence for the absence of quantum interference in liquids (Section 2). 

The diamagnetic term local electronic circulations around the nucleus induced by the applied field B0. According to the Lenz rule, the resultant intramolecular field opposes B0, so that the nucleus will be shielded. As known from the Lamb formula, eq. 3.3 [9],... 

As the diamagnetic term is set to zero, the procedure is named CTOCD-DZ. Total current becomes completely paramagnetic in form, and contains two terms that are expressed within the original coordinates system as a function of r, L e. 

Having applied equations (11-13) to more than one hundred different crystal structures, we have found empirically that reasonable diamagnetic terms are obtained when Pn = Nn - qn, difference between the total number of valence electrons Nn and the partial charge qn and when the summation is limited to rn distances less than 220 pm. This procedure allows to have Pn - Zn for light atoms (Z < 10) as originally suggested (50) and avoids to add diamagnetic terms over 1000 ppm when heavy elements are present. 

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