One-Electron Case

The fcth component of the current density may be rewritten in a way similar 

In order to resolve the two terms on the right-hand side, we rearrange the eigenvalue equation of the one-electron stationary Dirac equation so that we obtain an expression for the 4-spinor tp, 

By taking the transpose and complex conjugate, we obtain the analogous expression for tp (exploiting aj = aj.). 

To clarify the notation, we write the momentum-dependent term on the right-hand side explicitly component-wise 

Relativistic Quantum Chemistry. Markus Reiher and Alexander Wolf 

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The peak potential is equal to the standard potential of the A/B couple as in the one-electron case. The peak is thinner that a one-electron Nernstian peak by a factor of 2. Thus, the peak width, counted from the half-peak to the peak, is 0.882(TZT/F) (i.e., 22.7 mV at 25°C). 

The curves in Figure 1.25a may thus be used to represent the variations in the convoluted current with the standard potential separation. Similarly, the curves in Figure 1.25b may be viewed as representing the slopes of the convoluted current responses. The cyclic voltammetric current responses themselves can be derived from the integral equation (1.58) in the same way as described earlier in the one-electron case. Curves such as those shown in Figure 1.26a are obtained. 

In this chapter, the primary focus is on the one-electron case. Dropping the q = I labels, we can write the r-space, first-order, density matrix as... 

The simplest way to gain a better appreciation for tlie hole function is to consider the case of a one-electron system. Obviously, the Lh.s. of Eq. (8.6) must be zero in that case. However, just as obviously, the first term on the r.h.s. of Eq. (8.6) is not zero, since p must be greater than or equal to zero throughout space. In die one-electron case, it should be clear that h is simply the negative of the density, but in die many-electron case, the exact form of the hole function can rarely be established. Besides die self-interaction error, hole functions in many-electron systems account for exchange and correlation energy as well. 

This formula gives the integral F°A F B F y F D) in terms of non-redundant AO integrals only, at least where all four AOs are distinct. Just as for the one-electron case, we can expand some stabilizers and reduce the dimension of some sets of DCRs when some AOs are the same. 

Let us find expressions for the matrix elements of these operators for a subshell of equivalent electrons with respect to the relativistic wave functions, in a one-electron case defined by (2.15), and for a subshell, by (9.8). 

It is convenient to define the zero of energy as in the one-electron case to correspond to the electron at infinity. Then orbital energies for bound electrons are negative, and the ionization energy for an electron in orbital i is... 

The total orbital angular momentum and the total spin, described by L and 5, may be coupled exactly as in the one-electron case (p. 56). Their resultant total angular momentum is described by a quantum number /, with possible values L + S, L + — 5 and for... 

The next section shows that these general statements are important for determining the electronic structure of many-electron atoms even though they are deduced from the one-electron case. 

These two criteria appear to be generalizations of the localization criteria named after Edmiston-Ruedenberg and Boys respectively [1, 2, 3]. Nevertheless the pair functions ipp, constructed by either of these criteria, are generally symmetry-adapted (in spite of having localized features as well) unlike for the one-electron case there is no strict alternative symmetry-adapted vs. localized [5],... 

We have used the same symbol L for the transformed one-electron part of the Hamiltonian as in section 6, although there is a slight difference. In the pure one-electron case of sec. 6, the operator L was determined by the condition that its nondiagonal part vanishes, i.e. that it does not couple states within the model space with states outside of it. Now we cannot require, a priori, that the nondiagonal part of L vanishes. However, we decompose L into a contribution L that corresponds to its diagonal part, and a non-diagonal remainder, and make a similar decomposition of the two-electron operator G. We shall see that the non-diagonal remainders do not contribute to expectation values. 

For the quasireversible one-step, one-electron case, we can evaluate A s) by applying the condition ... 

Matsuda and Ayabe (5) coined the term quasireversible for reactions that show electron-transfer kinetic limitations where the reverse reaction has to be considered, and they provided the first treatment of such systems. For the one-step, one-electron case,... 

The two-component Hamiltonian of Eq. (1) is invariant under the time reversal operation. In the one-electron case and for a special choice of phases, the time reversal operator T is given by... 

This pseudopotential has some points in common with our earlier one-electron case ... 

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