Approximations Pauli-Schrodinger

The second aim concerns a presentation of the theory of one-electron atoms starting from its relativisitic foundation, the Dirac equation. The nonrelativis-tic Pauli and Schrodinger theories are introduced as approximations of this equation. One of the major purpose, about these approximations, has been to display, on the one side, the enough good concordance between the Dirac and the Pauli-Schrodinger theories for the bound states of the electron furthermore, but to a weaker extent, for the states of the continuum close to the freedom energy and, on the other side, the considerable discordances for... 

For anything but the most trivial systems, it is not possible to solve the electronic Schrodinger equation exactly, and approximate techniques must instead be used. There exist a variety of approximate methods, including Hartree-Fock (HF) theory, single- and multireference correlated ab initio methods, semiempirical methods, and density functional theory. We discuss each of these in turn. In Hartree-Fock theory, the many-electron wavefunction vF(r1, r2,..., r ) is approximated as an antisymmetrized product of one-electron wavefunctions, ifijfi) x Pauli principle. This antisymmetrized product is known as a Slater determinant. 

Though the Pauli potential Fpauii(r) = Fp(r) entering the Schrodinger equation at Eq. (45) is also only known approximately (for example, the von Weizsacker study of March and Murray [32] yielded Fp (r) = (5/3)c p(r) where the kinetic constant = (3h l0m) (3/Sn) ), the work of Lieb et al. [36] (see also [37]), who did not utilize the Pauli potential in their original paper, proves to be a splendid example of this approach, where important analytical progress proves possible. 

Abstract. This chapter concerns a presentation of the Darwin solutions of the Dirac equation, in the Hestenes form of this equation, for the central potential problem. The passage from this presentation to that of complex spinor is entirely explicited. The nonrelativistic Pauli and Schrodinger theories are deduced as approximations of the Dirac theory. 

However, from a theoretical and also a practical point of view, several teachings may be deduced. In particular, with the retardation, the Pauli approximation of the Dirac theory is no longer in strict agreement with the Schrodinger theory, as it is the case with the dipole approximation, when for example two states pl/2 and p3/2 are considered as unified in a single state p. Such a feature has a nonnegligible incidence. 

We can deduce from relations established in Sect. 9.2 that a direct passage of the vectors T- -(k) of the transitions sl/2 — pl/2 and sl/2 — pl/2 to a vector T- -(k) of a transition s —p is not possible. In other words, one of the effect of the retardation is to break the possibility to find an equivalence between the Pauli approximation and the Schrodinger theory, and the reason lies on the incidence of the retardation on the spherical parts of the Dirac wave functions, related to the presence of the spin. The incidence is already sensible, in the transitions of the discrete spectrum (see (9.38), (9.39), (9.40)) and this incidence may be amplified in the contribution of the continuum, independently of the incidence of the chosen values for the radial functions. 

Our task is to find approximate solutions to the time-independent Schrodinger equation (Eq. (2)) subject to the Pauli antisymmetry constraints of many-electron wave functions. Once such an approximate solution has been obtained, we may extract from it information about the electronic system and go on to compute different molecular properties related to experimental observations. Usually, we must explore a range of nuclear configurations in our calculations to determine critical points of the potential energy surface, or to include the effects of vibrational and rotational motions on the calculated properties. For properties related to time-dependent perturbations (e.g., all interactions with radiation), we must determine the time development of the... 

Although the full four-component treatment with the Dirac Hamiltonian is ideal, the computation of four-component wave functions is expensive. Thus, since small components have little importance in most chemically interesting problems, various two- or one-component approximations to the Dirac Hamiltonian have been proposed. From Eq. 10.32, the Schrodinger-Pauli equation composed of only the large component is obtained as... 

Note that no approximation has been made so far. The Breit-Pauli (BP) approximation [49] is introduced by expanding the inverse operators in the Schrodinger-Pauli equation in powers of (V — E) jlc and ignoring the higher-order terms. Instead, the BP approximation can be obtained truncating the Taylor expansion of the FW transformed Dirac Hamiltonian up to the (ptcf term. The one-electron BP Hamiltonian for the Coulomb potential V = Zr /r is represented by... 

In Section 2.1, the electronic problem is formulated, i.e., the problem of describing the motion of electrons in the field of fixed nuclear point charges. This is one of the central problems of quantum chemistry and our sole concern in this book. We begin with the full nonrelativistic time-independent Schrodinger equation and introduce the Born-Oppenheimer approximation. We then discuss a general statement of the Pauli exclusion principle called the antisymmetry principle, which requires that many-electron wave functions must be antisymmetric with respect to the interchange of any two electrons. 

---