1.3 Polyelectronic Atoms and Molecules

The solution of Schrodinger equation for atoms with more than one electron is not easy. The first problem comes due to the fact that the solution for even Helium atom is not exact. The helium atom can be termed as an example of three body problem i.e with two electrons and one nucleus. So it can be concluded that there exists no exact solutions for three or more interacting particles. Thus any solution obtained can be defined as approximate solution or representing closely to the real and true solutions. It should be noted that the wave function can have more than one functional form which thus makes the choice of the function difficult.

A complication with multi-electron species is the inclusion of the electron spin. In Quantum Chemistry, spin is defined using the quantum number s. This can only take the value of ½.It is quantified in such a manner so that its projection on the Z-axis is either +h or -h. It is denoted by the quantum number ms.This can also be termed as ‘up spin’ or ‘down spin’, which refers to the values of +1/2  and -1/2 respectively. In the Schordingler equation the electron spin is written as a one-electron intergral. This integral is a product of a spatial function which is a function of coordinates of electron and a spin function that depends on the spin.A solution arising out of it are called as spin orbitals denoted by . The resulting spatial part describes the distribution of electron density in space.From now on the spatial part will be referred to as an orbital and represented using Φ for atomic orbitals  and Ψfor molecular orbitals.

The spin part defines the electron spin  and is labelled α or β. These spin functions have the value 0 or 1 depending on the quantum  number ms of the electron. Thus ,,, . Each spatial  orbital can accommodate two electrons, with paired spins. In order to predict the electronic  structure of a polyelectronic atom or a molecule, the Aufbau principle is employed, in which  electrons are assigned to the orbitals, two electrons per orbital. We need to remember that  electrons occupy degenerate states with a maximum number of unpaired electrons  (Hund's rules), and that there are certain situations where it is energetically more favourable  to place an unpaired electron in a higher-energy spatial orbital rather than pair it with  another electron. For most of the situations that we shall be interested in the number of electrons, N, will be an even number that occupy the N/2 lowest-energy orbitals.  Electrons are indistinguishable. If we exchange any pair of electrons, then the distribution of  electron density remains the same. According to the Born interpretation, the electron density  is equal to the square of the wavefunction. It therefore follows that the wavefunction must  either remain unchanged when two electrons are exchanged, or else it must change sign. For electrons the wavefunction is required to change sign which is called the antisymmetry  principle.

1.4 The Born-Oppenheimer Approximation

The Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular  species, H₂+ , when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Born-Oppenheimer  approximation. The masses of the nuclei are much greater than the masses of the electrons (the  resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of  the electron). This means that the electrons can adjust almost instantaneously to any changes  in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Born-Oppenheimer approximation the  total wavefunction for the molecule can be written in the following form:

(1.10)

The total energy equals the sum of the nuclear energy (the electrostatic repulsion between the positively charged nuclei) and the electronic energy. The electronic energy comprises the kinetic and potential energy of the electrons moving in the electrostatic field of the nuclei,  together with electron-electron repulsion: ETOT = E(electrons) + E(nuclei).  When the Born-Oppenheimer approximation is used we concentrate on the electronic  motions; the nuclei are considered to be fixed. For each arrangement of the nuclei the  Schrodinger equation is solved for the electrons alone in the field of the nuclei. If it is desired  to change the nuclear positions then it is necessary to add the nuclear repulsion to the  electronic energy in order to calculate the total energy of the configuration.