The Pauli Principle and Determinantal Wavefunctions

Electrons are fermions—that is, indistinguishable particles. Thus, no observable physical property of a system can change if we simply rename or renumber the electrons. The manner in which this statement manifests itself mathematically is that the total wavefunction of an atom or molecule ( ) must be antisymmetric with respect to an exchange of the coordinates of any two electrons. This is a rule called the Pauli principle (you must just accept this rule, unless you want to delve into a textbook on advanced quantum mechanics). Antisymmetric means that the exchange of two electrons creates the negative of the original function. Consider an operator Pi,2 that permutes the position of any two electrons, arbitrarily denoted as electron 1 and 2, in a multi-electron system. P is an acceptable wavefunction if Pi,2 operating on produces - 4 (Eq. 14.16). 

Let s start checking for adherence to the Pauli principle by considering the simplest wavefunction for a two-electron system (Eq. 14.17), where a(l) is read as electron 1 in orbital Here, we use molecular orbitals (y/ s) to analyze molecules, but we could just as easily have used atomic orbitals (0 s) for atoms. 

Why is it that the exchange of two electrons creating the negative of the wavefunction does not change any physical observables Remember that any physical observable relates to which is completely invariant to electron interchange if P is antisymmetric, because (- P) = ( P). Note that P is in the form of a determinant  

A spin-orbit wavefunction of this form is called a Slater determinant, after the pioneering physicist who developed the concept. Any can be written as a Slater determinant using the general form shown, creating a total wavefunction that obeys the Pauli principle. Note that we have now collapsed the spatial and spin parts into a single symbolism, such that y/ (m) contains both space and spin components. 

Note also that if any two columns of a determinant are identical, the value of the determinant is zero. Thus, in order to have a non-zero wavefunction, no two electrons can occupy the same spin orbital, meaning that they cannot have the same spin when in the same spatial orbital—a statement of the Pauli principle in a more familiar form. 

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