The Multi-Electron Problem in Molecular Physics and Quantum Chemistry
In this chapter, we shall meet up with some approaches to treating the multi- or many-electron problem in molecular physics and quantum chemistry. Among them are the Slater determinant approach and the Hartree-Fock equations to which it leads, which we wi
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In this chapter, we shall meet up with some approaches to treating the multi- or manyelectron problem in molecular physics and quantum chemistry. Among them are the Slater determinant approach and the Hartree-Fock equations to which it leads, which we will discuss for both closed and open electronic shells. An important concept is the correlation energy between electrons, and we will introduce several general methods for dealing with it.
7.1 Overview and Formulation of the Problem 7.1.1 The Hamiltonian and the Schrodinger Equation
In the following sections, we continue what was begun in Chaps. 4 and 5, where we already introduced some important methods using simple molecules as examples. Here, we deal with approaches to finding the electronic wavefunctions of molecules in general, including complex molecules. In the general case, N electrons with the coordinates r j, j = 1, ... , N move in the Coulomb field of the M nuclei with coordinates R K , K = 1, ... ,M and nuclear charge numbers ZK, and are also coupled to each other via the Coulomb interactions. The nuclei are taken to be fixed at their equilibrium positions R K , which they possess in the molecule under consideration. For an electron with the coordinate r j, we thus find an overall potential given by: V(rj) =
L VK(rj) ,
(7.1)
K
where the individual contributions consist of the Coulomb interaction energies between the electron j and the nucleus K: ZKe 2
VK(rj) = - - - - - 47rE'oIRK -rjl
(7.2)
The Hamiltonian for the electron with index j then contains the operators for the kinetic energy and the potential energy, i. e. it is given by: H(rj)
==
li,2
H(j)
= --VJ + V(rj) 2mo
.
H. Haken et al., Molecular Physics and Elements of Quantum Chemistry © Springer-Verlag Berlin Heidelberg 2004
(7.3)
148
7 The Multi-Electron Problem
(In a more exact treatment, the spin-orbit interaction would also have to be taken into account, but we shall neglect it here.) Between the electron with index j and an electron with index [ there is in addition a Coulomb interaction, whose potential energy is given by: (7.4) The interaction energy of all the electrons may then be written as: (7.5) The factor 1/2 guarantees that the Coulomb interactions between each pair of electrons are not counted twice in the sum, since the indices j and [ run over all electrons independently of one another, the only limitation being that an electron does not interact with itself, i. e. j f= [. After these preparatory definitions, we are ready to write down the Hamiltonian of the overall system; it has the form: N
H
=L
H(j)
+ Hint.
(7.6)
j=!
The Schr6dinger equation is then (7.7) where the wavefunction 1ft depends on all the electronic coordinates. Although the Hamiltonian H does not explicitly contain the electron spins, it is still important that the wavefunction 1ft also be a function of the spin coordinates, so that we can take the Pauli exclusion principle into account in a suitable manner, as we have already seen in Sect. 4.4. While it is in fact possible to solve the one-
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