25: Many-Electron Atoms

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When two or more electrons are present in a system, the TISEq equation cannot be solved analytically. Thus for the vast majority of chemical applications, we must rely on approximate methods. We will explore some of these approximations in this and further chapter, starting from the many-electron atoms (all atoms other than hydrogen). It is important to stress that because of the nature of approximations, this is still a very active field of scientific research, and improved methods are developed every year.

The electronic Hamiltonian for a many-electron atom can be written as:

\[ \hat{H}({\bf r}_1,{\bf r}_2,\ldots,{\bf r}_N)=\sum_{i=1}^N \left(-{\dfrac {\hbar ^{2}}{2m_e }}\nabla_{i}^{2}-{\dfrac {Ze^{2}}{4\pi \varepsilon _{0}r_{i}}} \right)+{\dfrac {e^{2}}{4\pi \epsilon _{0}}}\sum_{i<j}\dfrac{1}{r_{i j}}, \tag{10.1} \]

where \(Z\) is the nuclear charge, \(m_e\) and \(e\) are respectively the mass and charge of an electron, \({\bf r}_i\) and \(\nabla_i^2\) are the spatial coordinates and the Laplacian of each electron, \(r_{i}=|{\bf r}_i|\), and \(r_{ij}=|{\bf r}_i-{\bf r}_j|\) is the distance between two electrons (all other symbols have been explained in previous chapters). The TISEq is easily written using Equation 22.3.1.

25.1: Many-Electron Wave Functions
When we have more than one electron, the sixth postulate that we discussed in chapter 24 comes into place. In other words, we need to account for the spin of the electrons and we need the wave function to be antisymmetric with respect to exchange of the coordinates of any two electrons.
25.2: Approximated Hamiltonians
In order to solve the TISEq for a many-electron atom we also need to approximate the Hamiltonian, since analytic solution using the full Hamiltonian as in Equation 26.1 are impossible to find. The most significant approximation used in chemistry is called the variational method.