Diamagnetic expectation value

The second-order terms give the magnetizability. The first term is known as the diamagnetic part and it is particularly easy to calculate since it is just the expectation value of the second moment operators. The second term is called the paramagnetic part. 

From the above discussion it becomes clear that in order to eliminate the spin-orbit interaction in four-component relativistic calculations of magnetic properties one must delete the quaternion imaginary parts from the regular Fock matrix and not from other quantities appearing in the response function (35). It is also possible to delete all spin interactions from magnetic properties, but this requires the use of the Sternheim approximation [57,73], that is calculating the diamagnetic contribution as an expectation value. 

The second term (e.g. the diamagnetic susceptibility) is simply the expectation value of and may be predicted by the Hartree-Fock theory safely 50). On the other hand, electron correlation may alter the SCF value of the first term (e.g. the paramagnetic susceptibility) to a certain extent, so that the whole correlation effect depends on the relative vdues of the two parts. 

A particularity of non-relativistic theory is that the introduction of vector potentials gives rise to second-order magnetic operators as well. These diamagnetic terms contribute as expectation values to second-order properties and are... 

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... 

Then the first derivative of the expectation value of Hx over the perturbed wave function lP(k2) is evaluated by a numerical method for k2 = 0. The set of magnetic parameters is expressed as the sum of the first-order (diamagnetic) and second-order (paramagnetic) contributions [1]... 

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... 

Besides the two anisotropies of x given below, a third piece of information to determine the principal components and to partition them into diamagnetic and paramagnetic contributions was an assumed value for the expectation value (c ) of the charge distribution (electronic + nuclear the electronic part was estimated from free-atom values). This is directly related to 2Xcc Xbb Xaa- The paramagnetic and diamagnetic contributions, and Xm. to these components were obtained as follows ... 

Here, e and m refer to the charge and mass of the electron, c is the velocity of light, N the Avogadro number and the expectation value (mean value) of the square distance of the electron from the nucleus. According to Eq. (11), diamagnetic susceptibilities should be independent of temperature which is observed in practice to a good approximation. Since all substances contain electrons, diamagnetism is an inherent property of all atomic and molecular systems. 

The first contribution is an expectation value of the second-order perturbation operator with the unperturbed wavefunction of the system. It is negative and is called the diamagnetic contribution The second contribution involves either the first derivative of the perturbed wavefunction, the first-order correction to the wavefunction, a sum over all the other unperturbed states or a linear response... 

The induced contribution consists therefore of a paramagnetic or sum-over-states contribution and a diamagnetic or ground-state expectation value term. Combining these with the contribution from the rigid charges, Eq. (6.15), yields... 

For all the magnetic linear response properties derived in Chapters 5 and 6 one would obtain expressions similar to the electronic-vibrational polarizability, Eq. (8.7). On the other hand, the diamagnetic contributions to the magnetic properties as well as all first-order properties, i.e. properties defined as first derivatives of the energy, will take the following simple expectation value form... 

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 ... 

---