Hamiltonian many-electron atoms

Now consider a d ion as an example of a so-called many-electron atom. Here, each electron possesses kinetic energy, is attracted to the (shielded) nucleus and is repelled by the other electron. We write the Hamiltonian operator for this as follows ... 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

Because of interelectronic repulsions, the Schrodinger equation for many-electron atoms and molecules cannot be solved exactly. The two main approximation methods used are the variation method and perturbation theory. The variation method is based on the following theorem. Given a system with time-independent Hamiltonian //, then if

well-behaved function that satisfies the boundary conditions of the problem, one can show (by expanding

[Pg.271]

With internal nuclear motion neglected, the Hamiltonian for a many-electron atom with atomic number Z is... 

While using (4.14) and exact wave functions. This supports the conclusion of Drake [83] that for electric dipole transitions, by considering the commutator of with the atomic Hamiltonian in the Pauli approximation, we obtain Qwith relativistic corrections of order v2/c2 (see (4.18)-(4.20)). However, for many-electron atoms and ions, one has to use approximate (e.g., Hartree-Fock) wave functions, and then this term gives non-zero contribution, conditioned by the inaccuracy of the model adopted. 

Thus, we have expressed the non-relativistic Hamiltonian of a many-electron atom with relativistic corrections of order a2 in the framework of the Breit operator (formulas (1.15), (1.18)—(1.22)) in terms of the irreducible tensorial operators (second term in (1.15), formulas (19.5)—(19.8), (19.10)— (19.14), (19.20), respectively). 

Dependence of the main terms of the Hamiltonian of a many-electron atom on nuclear charge Z may be easily revealed in the following fashion. Let us consider the Hamiltonian of the kind (1.15)... 

We now move to the many-electron atom or molecule. Within the Bom-Oppenheimer38 approximation (i.e., neglect of nuclear motion) the Hamiltonian H becomes... 

TERMS OF THE HAMILTONIAN OPERATOR FOR A MANY-ELECTRON ATOM OR MOLECULE [14]... 

We have already dealt with the calculation of the wave functions of the hydrogen atom. We now proceed to consider many-electron atoms, first dealing with the simplest such example, the helium atom which possesses two electrons. The Hamiltonian for a helium-like atom with an infinitely heavy nucleus can be obtained by selecting the appropriate terms from the master equation in chapter 3. The Hamiltonian we use is... 

Until this point, the consideration of electron-electron repulsion terms has been neglected in the molecular Hamiltonian. Of course, an accurate molecular Hamiltonian must account for these forces, even though an explicit term of this type renders exact solution of the Schrddinger equation impossible. The way around this obstacle is the same Hartree-Fock technique that is used for the solution of the Schrddinger equation in many-electron atoms. A Hamiltonian is constructed in which an effective potential of the other electrons substitutes for a true electron-electron reg sion term. The new operator is called the Lock operator, F. The orbital approximation is still used so that F can be separated into i (the total number of electrons) one-electron operators, Fi (19). 

For heavy elements, all of the above non-relativistic methods become increasingly in error with increasing nuclear charge. Dirac 47) developed a relativistic Hamiltonian that is exact for a one-electron atom. It includes relativistic mass-velocity effects, an effect named after Darwin, and the very important interaction that arises between the magnetic moments of spin and orbital motion of the electron (called spin-orbit interaction). A completely correct form of the relativistic Hamiltonian for a many-electron atom has not yet been found. However, excellent results can be obtained by simply adding an electrostatic interaction potential of the form used in the non-relativistic method. This relativistic Hamiltonian has the form... 

Formulation of the methods of computational chemistry is reasonably straightforward. The hard work comes in its implementation. We begin with the electronic Hamiltonian for a molecule, a generalization of that for a many-electron atom given in Eq (9.2) ... 

A many electron atom in an intense, pulsed laser field is a formidable computational problem. This is because the multi-dimensional Hamiltonian... 

Our main concern in the study of many-electron atoms (and many-electron molecules) continues to be the energy eigenproblem. Accordingly, the wave-functions of interest are those that are eigenfunctions of the hamiltonian. 

The independent electron model serves as the reference basis. Fano s theory of autoionisation consists in describing the consequence of turning on an interaction between a sharp state and the underlying continuum, which are presumed initially to be devoid of correlations. Of course, the perturbation is a hypothetical one, since it cannot really be turned off. The independent electron atom, as such, does not exist. Hypothetical interactions are familiar in perturbation theory. They carry with them the implication that, if they could be removed, the zero-order Hamiltonian which would result can be solved exactly, providing the basis for a perturbative expansion. For a many-electron atom, this is clearly not so, but the idea is nevertheless convenient. It is a case of pretending that,... 

Although the proper point of departure for relativistic atomic structure calculations is quantum electrodynamics (QED), very few atomic structure calculations have been carried out entirely within the QED framework. Indeed, almost all relativistic calculations of the structure of many-electron atoms are based on some variant of the Hamiltonian introduced a half century ago by Brown and Ravenhall [1] to understand the helium fine structure. By decoupling the electron and radiation fields in QED to order a (the fine-structure constant) using a contact transformation. Brown and Ravenhall obtained a relativistic momentum-space Hamiltonian in which the electron-electron Coulomb interaction was surrounded by positive-energy projection operators. Owing to the fact that contributions from virtual electron-positron pairs are automatically projected out of... 

In this chapter, we turn to problems of quantum chemistry and of many-electron atomic and molecular physics for which fhe desideratum is the quantitative knowledge and easy conceptual understanding of dynamical processes and phenomena thaf depend explicifly on time. We focus on a theoretical and computational approach which computes q>(q,t) by solving nonperturbatively the many-electron TDSE for unstable states of atoms and small molecules. The time evolution of fhese states is caused either by the time-independent Hamiltonian, Ham ( -g-/ case of time-resolved autoionization—see below) or by the time-dependent Hamiltonian, H t) = Ham + Vext(f), where Vext(f) is the sum of the identical one-electron operators that couple the field of a strong pulse of radiation to the electronic and nuclear moments of N-electron atomic or molecular states of inferest, thereby producing, during and at the end of the interaction, final stafes in the ionization or the dissociation continua. 

The spin-orbit Hamiltonian of Eq. (31) is correct only for a bare nucleus. In the case of many-electron atoms where the nucleus is surrounded by a "core of electrons, the electrostatic potential, U (r), changes more rapidly with r because of the rapid change in shielding by the core as we... 

When the discussion is limited to a single open subshell and to perturbations within such a shell, an interesting formulation of the many-electron atomic problem can be achieved that exhibits useful particle-hole symmetry. The summations in the perturbation term of the hamiltonian then run only over electron states nlmu) of the open subshell (nQ. This means that the electron repulsion integrals can be expressed as... 

The Schrddinger equation for the one-electron atom (Chapter 6) is exactly solvable. However, because of the interelectronic repulsion terms in the Hamiltonian, the Schrbdinger equation for many-electron atoms and molecules is not separable in any coordinate system and cannot be solved exactly. Hence we must seek approximate methods of solution. The two main approximation methods, the variation method and perturbation theory, will be presented in Chapters 8 and 9. To derive these methods, we must develop further the theory of quantum mechanics, which is what is done in this chapter. 

For a many-electron atom, the operators for individual angular momenta of the electrons do not commute with the Hamiltonian operator, but their sum does. Hence we want to learn how to add angular momenta. 

The primary purpose of this chapter is to show how the D— oo limit of the hamiltonian for a many-electron atom can be altered so that it accounts approximately for the effects of finite D. The resulting hamiltonians, which we will call subhamiltonians, are almost as simple eis the D— cx) limit hamiltonians from which they are constructed. They are therefore in many respects quite crude. On the other h2ind, they incorporate no approximations which would destroy many-body effects, and so turn out to be quite useful for studying electron correlation. 

---