PAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29, MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES

Transcription

1 Subject Chemistry Paper No and Title Module No and Title Module Tag 8: Physical Spectroscopy 29: Molecular Term Symbols and Selection Rules for Diatomic Molecules. CHE_P8_M29

2 TLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. The Hydrogen Molecule Ion MO Method 4. Molecular Term Symbols 5. Selection Rules 6. Summary

3 1. Learning Outcomes In this module, we shall learn about (a) The electronic structure of diatomic molecules (b) Molecular term symbols (c) Selection rules for electronic transitions. 2. Introduction efore we discuss the electronic structure of diatomic molecules, let us recapitulate our study of the quantum mechanical treatment of the hydrogen molecule ion, H 2+. You will recall that this is the only one electron system and can be treated exactly using quantum mechanics. However, the treatment uses elliptical coordinates and is difficult to solve. Instead, we use one of the approximate treatments, the Valence ond (V) method or the Molecular Orbital (MO) method, which we also continue using for the other molecular systems. Here, we use the Molecular Orbital method, which is based on the Linear Combination of tomic Orbitals (LCO) to obtain the molecular orbitals. We first discuss briefly the results obtained for H 2 + and then extend to other molecules. 3. The Hydrogen Molecule Ion MO method In the MO treatment, we construct linear combinations of the atomic orbitals involved in bond formation. In this case, the 1s orbital of each hydrogen atom participates and thus only two orbitals are involved in forming the linear combination. Let us label the two hydrogens as and. The linear combinations are then c c (1) where c and c are the coefficients of the two molecular orbitals and ψ and ψ are the wave functions for the 1s orbitals on atom and, respectively. Since we started with two atomic orbitals, we will get two molecular orbitals. Figure 1: onding of two hydrogen atoms

4 We will use Group Theory to obtain these linear combinations. ll homonuclear diatomic molecules belong to the D h molecular point group. D h E 2C φ σ v i 2S φ C 2 χ We find that the two atoms remain unshifted under the first three operations of the point group, but exchange positions in the last three. Therefore, the character is two for the first three and zero for the last three. Inspection of the character table shows that this reduces to Σ g + Σ u+. This is an infinite group and usual methods do not work. The two linear combinations are obtained by using the method of projection operators: ( ) g u ( ) (2) We have used lower case symbols here because it is customary to label molecular orbitals with small Greek letters and molecular states with capital Greek symbols. The first is a bonding orbital, since when we square this wave function, we obtain which shows an accumulation of electron density in the region between the two nuclei, as the middle overlap term has a positive sign. This is called a bonding orbital, while the lower molecular orbital is an antibonding orbital (Fig. 2). The bonding molecular orbital is a bound state with an energy minimum, but the antibonding molecular orbital is an unbound state, and an electron in this orbital spontaneously dissociates.

5 Figure 2: Variation of energy with internuclear distance for the bonding and antibonding orbitals In a similar way, we can combine atomic orbitals of the same energy and symmetry to get the energy level diagram shown in Figure 3 for homonuclear diatomic molecules of the second period. s in the case of atomic states, we start filling electrons into orbitals using the ufbau principle. For H 2, both electrons occupy the 1σ g orbital, and the electron configuration is (1σ g)2. The electron configurations of the second row atoms are shown in Figure 3. esides the σ g and σ u orbitals, we also note that there are π g and π u orbitals. The orbitals are labelled according to their angular momentum, denoted by their λ quantum numbers. This is the analogous Greek character for the l quantum number, but unlike the 2l + 1 possible values of the components, there are only two components for each value of λ, except for λ = 0, which has only one component, as follows: λ Designation σ π δ φ Components 0 ±1 ±2 ±3 It can be noted in Figure 3 that there is a change in the energy order after dinitrogen (total number of electrons = 14). Upto N 2, s-p mixing occurs, raising the σ level above the π level. s in the

6 case of atomic spectroscopy, it is cumbersome to write the electron configuration. Rather, the molecular state symbol can provide all information Figure 3: Molecular orbital energy level diagrams for the diatomic molecules of the second row elements 4. Molecular Term Symbols s in the atomic term symbols, 2S+1 L J, we write the analogous term symbol for molecular states with the corresponding Greek symbols as 2Σ+1 Λ Ω. In order to avoid confusion with the molecular term symbol, we continue using S for the spin quantum number. lso, in the case of homonuclear diatomic molecules, instead of the total angular momentum quantum number Ω, we write the parity g or u, as the case may be. Homonuclear diatomic molecules Let us now take up some examples of homonuclear diatomic molecules. Just as for atomic orbitals, the filled molecular orbitals do not contribute to the angular momentum, and the molecular state with all filled molecular orbitals is designated as 1 Σ g+. This means that, except 2 and O 2, all atoms depicted in Figure 3 have this ground state.

7 For 2, the electron configuration is (1π u) 2. We need not consider the inner filled orbitals for finding the state because these contribute zero orbital and zero spin angular momenta. Let us consider the possible arrangements of the two electrons, keeping the Pauli Exclusion Principle in mind Row M λ M S The first row has M L = 2 and M S = 0; therefore, it belongs to a single Δ state. Its other component is in row 6. The second row has M L = 0 and M S = 1 and therefore belongs to a triplet Σ state. Its two other components are in rows 3 and 5, leaving row 6 with both M L and M S equal to zero, and hence this is also a Σ state, but is a singlet. We also know that triplet states have symmetric spin components and antisymmetric spatial components, so the triplet state is 3 Σ - and the singlet state is 1 Σ +. Finally, both electrons are in u orbitals, so the final state is u u = g. We finally have the three states arising from this configuration, i.e. 1 Δ g, 3 Σ g - and 1 Σ g +. rranged in order of increasing energy according to Hund s rules, the states are 3Σ g - < 1Δ g < 1 Σ g + The only difference in the case of O 2 is that here both unpaired electrons are in π g orbitals, but since the product of two g states is also g, the overall states are the same as for 2.

8 Heteronuclear diatomic molecules For heteronuclear diatomic molecules, the procedure is the same, but the following points must be kept in mind: The two participating orbitals do not have equal energies. The more electronegative atom has lower energy levels. When two orbitals of the same symmetry of the two atoms combine, the lower of the two goes down in energy, and the upper one gets raised in energy. In other words, the lower energy level (bonding) will have more character of the more electronegative element, and the antibonding orbital will have more of the higher energy orbital. 2. The energy scheme is the same as for the corresponding isoelectronic diatomic molecule, i.e. CO will have a similar energy level diagram as N Upto 14 electrons, the scheme of nitrogen is followed, but beyond that, the scheme of O 2 is followed, i.e. NO will have a similar energy level diagram as O Since the centre of symmetry is lost, there is no g or u notation. Instead, we may write the value of Ω as a subscript. Example 1 Deduce the term symbol for NO. Solution The MO diagram for NO is shown below:

9 esides the 4 core electrons, there are 11 more electrons. The levels up to π2p are completely filled and do not contribute to the angular momentum, leaving (π2p*) 1. Therefore, the term symbol is 2 Π. The possible values of Ω are 1+½ and 1-½, i.e. 3/2 and ½. So the possible states are 2 Π½ and 2 Π3/2 with the former as the ground state because the level is less than half-filled Selection Rules There are five quantities to be considered, and so five selection rules. 1. The selection rule for change in the spin quantum number is the same as in atomic spectroscopy, i.e. ΔS = The orbital angular momentum selection rule is ΔΛ = 0, ±1. Conservation of angular momentum requires that the angular momentum must change by one unit, so for ΔΛ = 0, the rotational quantum number changes; ΔJ = ±1 in order to conserve the angular momentum. For ΔΛ = ±1, the allowed rotational transitions are ΔJ = 0, ±1. 3. The total angular momentum must change as ΔΩ = 0, ±1, but Ω = 0 0 is not permitted in order to conserve angular momentum, as in atomic spectroscopy. 4. The reflection symmetry must not change, i.e. + + and - -, but + -. This only applies to Σ states. 5. The parity must change, i.e. g u, but g g and u u. 6. Summary The electronic configuration of molecules can be written similar to that of atomic configurations. In a similar way, molecular term symbols can also be derived. Upto nitrogen, N 2, s-p interaction causes the energy level ordering to be different compared to the rest of diatomics. The molecular term symbols give a concise representation of the various angular momenta in the molecule. The selection rules do not allow transitions between states of the same parity and different multiplicity or reflection symmetry. The orbital angular momentum selection rule is ΔΛ = 0, ±1.