Many-Electron Case

For an N-electron system, we start from the definition of the many-electron current density [14], 

The wave function Y is an eigenfunction of the many-electron Dirac-Coulomb 

Considering only multiplicative operators for ggg, the last term is zero because 

In order to resolve the remaining terms, first we consider only one of the terms. If we choose h 2), we find 

As a first-order differential equation, the radial Dirac equation, which is the central equation to be solved in the case of atoms, requires a discretization scheme, and several options are at hand of which some should be presented here for the sake of completeness. The first one is analogous to the Nu-merov procedure for second-order differential equations without first derivatives [491,492]. The derivation in terms of Taylor series expansions provides a derivation which is easier to understand. However, using operator techniques is the most elegant way for this particular task. 

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The variational procedure in a many-electron space may be considered as several consecutively executed variational procedures in one-electron spaces (the procedure may be iterative if a self-consistency is required). This means that fulfilment of the one-electron minimax principle is a necessary (but, in general, not sufficient) condition for the fulfilment of a similar principle in a many-electron case. Therefore one should not expect a many-electron generalization of Eq. (7) being valid when, say, parameters L and S are varied. 

We assume for simplicity that the two-electron atom is described by a Hamiltonian (29) in which //(I) and //(2) are the hydrogen-like Dirac Hamiltonians and h, 2) = 0. Apparently, after this simplification the problem is trivial since it becomes separable to two one-electron problems. Nevertheless we present this example because it sheds some light upon formulations of the minimax principle in a many-electron case. 

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. 

The theory of upper and lower bounds is used in order to get estimates of the resolvent in the one-electron as well as in the many-electron case. The issue of completeness of the basis sets employed is central. 

It is shown in Chapter 11 that eq. (8) holds (with / replaced by j) for any operator J with components Jx, Jy, Jz that obey the angular momentum CRs. (The quantum number j determines the eigenvalues of J2, from eq. (11.4.40).) Consequently, eq. (8) applies also to the many-electron case with / replaced by L,... 

A symmetry-adapted perturbation theory approach for the calculation of the Hartree-Fock interaction energies has been proposed by Jeziorska et al.105 for the helium dimer, and generalized to the many-electron case in Ref. (106). The authors of Refs. (105-106) developed a basis-set independent perturbation scheme to solve the Hartree-Fock equations for the dimer, and analyzed the Hartree-Fock interaction energy in terms of contributions related to many-electron SAPT reviewed in Section 7. Specifically, they proposed to replace the Hartree-Fock equations for the... 

The extension of Schrodinger s energy functional to the many-electron case, including the Lagrange multiplier 1 is (in analogy with eqn (5.58))... 

Equation (6.14) has a classical analogue which states that the force exerted on the matter contained in a region 2 is equal to the negative of the pressure acting on each element of the surface bounding the region. A local form of the force law is readily obtained in the same manner as used in the derivation of the time derivative of the current density in eqn (5.29). No problems arise in extending the expression to the many-electron case and for a stationary state the result is... 

Eq. 20 can be solved iteratively and yields the same one-particle energies as the corresponding Dirac-equation. The radial functions P K r) correspond to the large components. In the many-electron case the correct nonlocal Hartree-Fock potential is used in Eq. 21, but a local approximation to it in Eqs. 22. Averaging over the relativistic quantum number k leads to a scalar-relativistic scheme. 

Since the quantum numbers l or L are seldom well-defined in a one-electron valence orbital or a many-electron case (a) basis set, and since, for a linear... 

Spatial symmetry constraints of which the above example is the simplest. In a many-electron case, the constraint may be expressed as a product of orbital symmetries. 

For the many electron case Eq. (28) would simply become... 

Moving to the many-electron case involves the use of analogous quantum numbers for which the usual one-electron-atom labels are retained. 

From section 6.6 we understand that the nonrelativistic energy is not the first term of the series expansion, but the rest energy mgC is. Therefore, the energy eigenvalues will differ by meC. We will later see that this result transfers to the many-electron case. Since a constant W can be added to the time-independent Dirac equation, Eq. (6.7),... 

The result of the different origin behavior of the radial functions transfers to the many-electron case and creates substantial drawbacks for numerical methods applied to solve the radial equation. Numerical solution methods... 

The relation to the spin density can be made more explicit by invoking a Gordon decomposition of the current density to produce expressions for charge- and spin-related currents [392,397]. Although we have already encountered the Gordon decomposition for the 4-current in section 8.8.1, Appendix F considers explicitly the decomposition of its spatial components, that is, of the current density, in standard notation. From Appendix F, we take the result for the many-electron case,... 

In the Dirac many-electron case, Eqs. (9.151) and (9.152) allow us to determine the ratio of the first coefficients in the series expansion of Pj(r) and Qi r) in the point-like nucleus case ... 

In section 6.9 we already introduced finite-size models of the atomic nucleus and analyzed their effect on the eigenstates of the Dirac hydrogen atom. This analysis has been extended in the previous sections to the many-electron case. It turned out that neither the electron-electron interaction potential functions nor the inhomogeneities affect the short-range behavior of the shell functions already obtained for the one-electron case. Table 9.5 now provides the total electronic energies calculated for the hydrogen atom and some neutral many-electron atoms obtained for different nuclear potentials provided by Visscher and Dyall [439], who also provided a list of recommended finite-nucleus model parameters recommended for use in calculations in order to make computed results comparable. 

This monograph presents atomic structure theory from the nonrelativistic perspective with an emphasis on calculations. Relativistic effects are considered by quasi-relativistic HamUtonians, and the Dirac many-electron case is only addressed in the appendix. However, the book provides a good presentation of tire general philosophy and strategy in numerical atomic structure theory. Prior to this monograph, Froese Fischer published a now classic book on numerical nonrelativistic Hartree-Fock theory in the 1970s [475]. Another classic text on this subject was delivered by Hartree in the 1950s [493]. 

Nevertheless, we adopt the restriction of a block-diagonal V for the sake of clarity. The more general case considering also off-diagonal potential contributions is then discussed in chapter 15. We now study decoupling of large and small components for one-electron equations only, which can then be generalized to the many-electron case. 

A generalization to the many-electron case and the relation to response theory is straighforward along the principles highlighted in the preceding sections. 

This one-electron energy e, (A) may be generalized to the many-electron case to yield E (A). According to the basic philosophy of the generalized DKH theory [611], the global unitary transformation U A) is constructed step by step as the product... 

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