Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 115

Computing the energy levels of a system of N electrons, interacting with each other through the Coulomb force, and placed in an external potential V₁(r) is a very important problem in physics and chemistry. It is encountered in the determination of the energy levels of atoms (in which case the external potential for the electrons¹ is the Coulomb potential created by the nucleus –Zq²/4πε₀r), or of molecules as well, or of electrons in a solid (submitted to a periodic potential), or in an aggregate or a nanocristal, etc. It is a problem where two ingredients simultaneously play an essential role: the fermionic character of the electrons, which forbids them to occupy the same individual state, and the effects of their mutual interactions. Ignoring the Coulomb repulsion between electrons would make the calculation fairly simple, and similar to that of § 1 in Complement C_XIV, concerning free fermions in a box; the free plane wave individual states would have to be replaced by the energy eigenstates of a single particle placed in the potential V₁(r). This would lead to a 3-dimensional Schrodinger equation, which can be solved with very good precision, although not necessarily analytically.

However, be it in atoms or in solids, the repulsion between electrons plays an essential role. Neglecting it would lead us to conclude, for example, that, as Z increases, the size of atoms decreases due to the attractive effect of the nucleus, whereas the opposite occurs²! For N interacting particles, even without taking the spin into account, an exact computation would require solving a Schrödinger equation in a 3N-dimensional space; this is clearly impossible when N becomes large, even with the most powerful computer. Hence, approximation methods are needed, and the most common one is the Hartree-Fock method, which reduces the problem to solving a series of 3-dimensional equations. It will be explained in this complement for fermionic particles.

The Hartree-Fock method is based on the variational approximation (Complement E_XI), where we choose a trial family of state vectors, and look for the one that minimizes the average energy. The chosen family is the set of all possible Fock states describing the system of N fermions. We will introduce and compute the “self-consistent” mean field in which each electron moves; this mean field takes into account the repulsion due to the other electrons, hence justifying the central field method discussed in Complement A_XIV. This method applies not only to the atom’s ground state but also to all its stationary states. It can also be generalized to many other systems such as molecules, for example, or to the study of the ground level and excited states of nuclei, which are protons and neutrons in bound systems.

This complement presents the Hartree-Fock method in two steps, starting in § 1 with a simple approach in terms of wave functions, which is then generalized in § 2 by using Dirac notation and projector operators. The reader may choose to go through both steps or go directly to the second. In § 1, we deal with spinless particles, which allows discussing the basic physical ideas and introducing the mean field concept keeping the formalism simple. A more general point of view is exposed in § 2, to clarify a number of points and to introduce the concept of a one-particle (with or without spin) effective Hartree-Fock Hamiltonian. This Hamiltonian reduces the interactions with all the other particles to a mean field operator. More details on the Hartree-Fock methods, and in particular their relations with the Wick theorem, can be found in Chapters 7 and 8 of reference [5].