Lecture 3 Many electron atoms The orbital approximation

Lecture 3 Many - electron atoms

The orbital approximation Putting electrons into orbitals similar to those in the hydrogen atom gives a useful way of approximating the wavefunction of a manyelectron atom.

Electron spin Electrons have an intrinsic rotation called spin, which may point in only two possible directions, specified by a quantum number ms. Two electrons in the same orbital with the opposite spin are paired. Unpaired electrons give rise to paramagnetism.

ms can take only one of two values: +1/2 or -1/2 Pauli exclusion principle When the spin quantum number ms is included, no two electrons in an atom may have the same set of quantum numbers. Thus a maximum of two electrons can occupy any orbital.

Effective nuclear charge The electrostatic repulsion between electrons weakens their binding in an atom; this is known as screening or shielding. The combined effect of attraction to the nucleus and repulsion from other electrons is incorporated into an effective nuclear charge.

The electrostatic repulsion between negatively charged electrons has a large influence on the energies of orbitals. Thus the ionization energy of a neutral helium atom (two electrons) is 24. 58 e. V compared with 54. 40 e. V for that of He+ (one electron). The effect of repulsion is described as screening or shielding. The combined effect of attraction to the nucleus and repulsion from other electrons gives an effective nuclear charge Zeff, which is less than that (Z) of the ‘bare’ nucleus.

ε(ionization energy) = - Z 2 eff. R/n 2 σ = Z –Zeff. Where σ is the screening constant Zeff = Zactual - (electron/electron repulsion effect)

Orbital Penetration and Shielding Effects The radial distribution function of s orbitals is said to penetrate the core electron density. The probability of finding s orbital electrons from the valence shell is significant in the core. Because of this, the effective nuclear charge felt is higher, and thus the energy level of the orbital is lower. This makes the filling order: 3 s, then 3 p, then 3 d. This affects the order of filling between the 4 s and 3 d orbitals as well. Since the 4 s orbital allows more penetration of the electron close to the nucleus, it is favoured over the 3 d orbital for filling. The filling order becomes: 1 s, 2 p, 3 s, 3 p, 4 s, 3 d, 4 p, 5 s, 4 d, 5 p, 6 s, 4 f, 5 d, 6 p, 7 s, 5 f, 6 d, 7 p, 8 s

The Aufbau Principle The Aufbau principle states (quite simply) that electrons will preferential occupy the lowest energy orbital possible, and therefore when assigning the electrons of a given atom to subshells, one starts at the lowest empty orbital (1 s) and continues to fill in order of increasing energy until all of the electrons for that atom are accommodated. Because not all electrons feel the same “pull” from the nucleus (Zeff), the order in which the subshells are filled rely on both the principal quantum number n, and the angular quantum number l.

Hund’s Rule The Aufbau principle describes how subshells must be filled with respect to energy, but it doesn’t address the order of filling of the individual orbitals. Hund’s Rule of Maximum Multiplicity states that orbitals should be filled to give the maximum number of unpaired electrons:

E. g. What is the orbital filling pattern for carbon? Given the Aufbau Principle, carbon has: 1 s 2 2 p 2. How are the p orbitals filled? Two possible solutions arise: