Spin exchange term

Here, Q is the electronic energy of a single determinant ab, K is the spin exchange term that will be dealt with later, and Sat is the overlap integral between the two AOs a and b. 

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. 

In the first term, Uc, usually called the Coulombic term, the initially excited electron on D returns to the ground state orbital while an electron on A is simultaneously promoted to the excited state. In the second term, called the exchange term, Liex, there is an exchange of two electrons on D and A. The exchange interaction is a quantum-mechanical effect arising from the symmetry properties of the wavefunctions with respect to exchange of spin and space coordinates of two electrons. 

In deahng with quantities whose associated operators do not ict on spin variables, we may use (4) and its two-electron analogue to derive parallel results for the spin-firee densities. It is necessary only to change lower-case letters to upper-case p — P and tt — II), to replace variables x by r, and (in the usual case where not more than one group is in a state of non-zero total spin), to put a factor before the exchange term in (20). 

Therefore, considering Bloch states, a positive exchange term coupling opposite spins is added to the Hamiltonian (11) ... 

Matrix elements for the valence functions were taken with the effective core potential the coulomb and exchange terms were handled exactly, numerically, without any parameterization and a Phillips-Kleinman projection operator term was also used. Spin-orbit coupling effects amongst the valence orbitals were treated semi-empirically using the operator... 

In this article we consider problems concerning the interpretation of unsaturated, steady-state NMR spectra of spin systems which are in a state of dynamic equilibrium. Spin exchange processes may occur with frequencies between a few sec-1 and several thousand sec-1 and thus modify the spectral lineshapes. In this case we use the terms dynamic NMR and dynamic spectra. The analysis of dynamic NMR lineshapes constitutes an important, and often unique, source of information about intra- and inter-molecular reaction rates. This is especially true for degenerate reactions where the products are chemically identical with the substrates. For this and similar reasons, dynamic NMR analysis has attracted considerable attention for about twenty years. 

For a pulse-type NMR experiment, the assumption has a straightforward interpretation, since the pulse applied at the moment zero breaks down the dynamic history of the spin system involved. The reasoning presented here, which leads to the equation of motion in the form of equation (72), bears some resemblance to Kaplan and Fraenkel s approach to the quantum-mechanical description of continuous-wave NMR. (39) The crucial point in our treatment is the introduction of the probabilities izUa which are expressed in terms of pseudo-first-order rate constants. This makes possible a definition of the mean density matrix pf of a molecule at the moment of its creation, even for complicated multi-reaction systems. The definition of the pf matrix makes unnecessary the distinction between intra- and inter-molecular spin exchange which has so far been employed in the literature. 

The magnitude of the exchange parameters is consistent with a rather high transition point temperature, TN = 30.8 K. The term //3 corresponds to four-spin exchange interactions... 

Assume that the spin-orbitals are well localized and that the spin-orbital pair A, B is well separated from the spin-orbital pair C, D. Then it may also be assumed that the virtual spin-orbitals are located in different spaces and the latter may be assigned to spaces of the occupied spin-orbitals. Thus, if (CD I v ITU) should be large then T and U must correspond to C and D. Under this assumption, at least one of the two X2 -matrix elements appearing in each of the expressions (167)—(169) can be expected to be small. Therefore, the contributions (167)—(169) may be expected to be much smaller than the contribution (166). If the former are neglected and the latter is completed to include its exchange terms, the expression (162), by making use of Eq. (147), becomes... 

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