The electrons in an atom occupy distinct orbitals that are defined by the quantum numbers n, I and m, and the electron spin, s. All the relevant information about the energies and arrangements of the electrons within atoms are governed by the laws o f quantum mechanics. Each orbital is a representation o f the three dimensional wavefunction if/ which is substituted into the Schrodinger equation to obtain the required information. The wavefiinctions represent the probability distribution, given by o f the electron in space about the nucleus. The way in which electrons fill the orbitals and the transitions that are allowed between them are governed by a definite set o f selection rules that are set by the allowed values o f the quantum numbers. Table 2-1 summarises the quantum numbers and their properties.
Table 2-1: The quantum numbers and their properties
Q u a n t u m n u m b e r
S y m b o l A llo w e d valu es D e s c r ip t io n
Principal n 1, 2, 3, ... 00 D efines the energy and the size o f the orbital
Orbital I / < ( « - ! ) Determines the electron angular momentum and the shape o f the orbital
M agnetic mi or U m < ± ( / - 1) Describes the behaviour o f electrons in a m agnetic field and specifies the direction o f a particular orbital
Spin ms 0 1 s ±'/2 Associated with the spin angular m omentum o f the
electron
The shapes o f the orbitals are also defined by the quantum numbers. The shapes of some o f the orbitals that can be occupied by an electron o f the hydrogen atom are shown in Figure 2-1. For example, for integer values of n and with / = m/ = 0 the orbitals are
2 Mo l e c u l a r St r u c t u r e, Sp e c t r o s c o p ya n d Ra d i at i o n Ch e m i s t r y 3 2
spherical and are referred to as s orbitals. The orbitals are labelled according to the value o f n as ns, e.g. 1 s, 2s, etc.
For / = 1 (n > 2) twin-lobed p-orbitals are produced. From Table 2-1 it can be seen that for / = 1 there are three possible values o f m/ (-1 ,0 , 1). Hence there are three p-orbitals
labelled «px, «Py and where by convention «pz is associated with mi = 0.
It thus follows that there are five d-orbitals produced for / = 2 (/? > 3; w/ = -2, -1, 0, 1, 2), there are seven f-orbitals for / = 3 (« > 4; w/ = -3, -2 ,-1 , 0, 1,2, 3), and so on.
The orbitals are thus labelled as /zs, /zp, né, nï, ng, «h, etc. for / = 0, 1, 2, 3, 4, etc.
respectively where n is used as a prefix to define the relative distance from the nucleus
and / defines the shape o f the orbital.
I = 0. m = 0: y, 1s * 2s 3s yj. n = 1 n = 2 . 1 = 1: n = 2 n = 3 m = 0
2
px y m = +1, -1Figure 2-1 : Examples of some of the orbitals of the hydrogen atom
The energy o f each orbital depends primarily on two factors: (i) the attraction between
the electrons and the nucleus, and (ii) the repulsion between the electrons. Thus the
energies o f different orbitals are different for different atoms with varying sizes of nuclei and numbers o f electrons. For a single electron such as in a hydrogen atom, there is no contribution to the energy due to the interactions between the electrons, so the
orbitals with the same n are degenerate, i.e. they all have the same energy and orbital
energies increase with increasing n. This is not the case for many-electron atoms in
2 Mo l e c u l a r St r u c t u r e, Sp e c t r o s c o p ya n d Ra d i a t i o n Ch e m i s t r y 3 3
lie at a lower energy than the d orbitals. This arises due to the electron-electron
repulsion in the many electron atom systems, and the shielding o f nuclear charge o f the
outer electrons by the sphere o f the inner electrons. The effect o f shielding on s, p and d
electrons differ according to their penetration which is defined by their probability
distributions. Hence it is more probable that an s electron may be found near the nucleus than a p or a d electron, and so the effect due to shielding will be less, which means that it is more tightly bound to the nucleus.
There are three basic rules that are followed in the treatment o f many-electron atoms which gives rise to the electronic configuration:
(1) No two electrons in the same orbital can have the same set o f quantum numbers n, /,
m or 5. This imposes a limit to the number o f electrons that can occupy any one orbital
(defined by n, /, and m) to two electrons having opposite spin ( 5 = +K and -V2). This rule
is known as the Pauli Principle.
(2) Electrons occupy different orbitals starting at the lowest energy first (i.e. n = 1, / =
0). The orbital energies increase with increasing n and /. The order o f the energy levels
is: Is, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, etc. {See Figure 2-2).
6d
Figure 2-2: The order of the orbital energy levels
(3) Electrons will occupy degenerate orbitals unpaired, and with their spins parallel.
This is known as H und’s Rule. For example, in the ground state nitrogen atom
containing 7 electrons the first two s-orbitals are filled and there are 3 electrons in the 2p orbitals, one in each p%, py and p% orbital. All the p-electrons will also have parallel
2 M o le c u la r S t r u c tu r e , S p e c tr o s c o p y a n d R a d ia tio n C h e m is tr y_________________________3 4 spins, such that the configuration will be Is^ ( tl) , 2s^ (t I), 2px,y,z (tD, tD, tD). If another electron is added, however, it will then have to pair up with one o f the p-electrons, with its spin anti-parallel.
Thus the electronic configurations of atoms may be obtained by placing the electrons into the appropriate orbitals one by one using the above rules. This procedure is referred to as the building-up principle. A complete set o f electrons for a given n is referred to as a closed shell. For example the Is^ electrons o f helium form a complete K-shell, the 2s^, 2p^ electrons form the L-shell, etc.
2.2.2 Ions
Ions are atoms that have more or less electrons than the number o f protons in the nucleus, thus giving them a net negative or positive charge. Thus ionic configurations are determined by either adding or removing electrons from its ground state neutral atom counterpart until the required charge state is obtained. The configurations of positive ions (cations) are determined by removing the electrons in a specific order. The
ionisation energy of an atom is the minimum energy required to remove an electron
from the atom. Thus the first ionisation energy is the energy required to remove one electron from a neutral atom, the second ionisation energy is the energy required to remove one electron from a positive ion with a single charge, etc. The ionisation potentials o f selected ions that are relevant to the work related in this thesis can be found in Appendix A. The electronic configuration o f negative ions (anions) is determined by adding electrons to the ground state neutral atom configuration using the building up procedure outlined above. The electron affinity o f an atom is the energy released from a gas phase atom when an electron attaches to it.