Subject Chemistry Paper No and Title 8: Physical Spectroscopy Module No and Title 29: Molecular Term Symbols and Selection Rules for Diatomic Molecules. Module Tag CHE_P8_M29 CHEMISTRY PAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29 , MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. The Hydrogen Molecule Ion – MO Method 4. Molecular Term Symbols 5. Selection Rules 6. Summary CHEMISTRY PAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29 , MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES 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 Before we discuss the electronic structure of diatomic molecules, let us recapitulate our study of + the quantum mechanical treatment of the hydrogen molecule ion, H2 . 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 Bond (VB) 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 Atomic Orbitals (LCAO) to obtain the molecular orbitals. + We first discuss briefly the results obtained for H2 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 A and B. The linear combinations are then cA A cB B (1) where cA and cB are the coefficients of the two molecular orbitals and ψA and ψB are the wave functions for the 1s orbitals on atom A and B, respectively. Since we started with two atomic orbitals, we will get two molecular orbitals. Figure 1: Bonding of two hydrogen atoms CHEMISTRY PAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29 , MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES We will use Group Theory to obtain these linear combinations. All homonuclear diatomic molecules belong to the D∞h molecular point group. φ φ D∞h E 2C∞ … ∞σv i 2S∞ … ∞C2 A A A A B B B B B B B A A A χ 2 2 2 0 0 0 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 Σg Σu . 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 ) A B (2) ( u ) A B 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 2 2 2 2 A B A 2 A B B 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. CHEMISTRY PAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29 , MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES 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. As in the case of atomic states, we start filling electrons into orbitals using the Aufbau principle. 2 For H2, both electrons occupy the 1σg orbital, and the electron configuration is (1σg) . The electron configurations of the second row atoms are shown in Figure 3. Besides the σg and σu orbitals, we also note that there are πg and πu 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 0 1 2 3 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 N2, s-p mixing occurs, raising the σ level above the π level. As in the CHEMISTRY PAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29 , MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES 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 2S+1 As in the atomic term symbols, LJ, we write the analogous term symbol for molecular states 2Σ+1 with the corresponding Greek symbols as ΛΩ. In order to avoid confusion with the molecular term symbol, we continue using S for the spin quantum number. Also, 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 1 + molecular state with all filled molecular orbitals is designated as Σg . This means that, except B2 and O2, all atoms depicted in Figure 3 have this ground state. CHEMISTRY PAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29 , MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES 2 For B2, the electron configuration is …(1πu) . 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. +1 -1 ↑↓ ↑↓ ↓↓ ↓↑↓ ↑ ↑ ↓ ↓ ↑ ↓ ↓ ↑↓ Row Mλ MS 1 +2 0 2 0 1 3 0 0 4 0 0 5 0 -1 6 -2 0 The first row has ML = 2 and MS = 0; therefore, it belongs to a single Δ state. Its other component is in row 6. The second row has ML = 0 and MS = 1 and therefore belongs to a triplet Σ state. Its two other components are in rows 3 and 5, leaving row 6 with both ML and MS 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 3Σ- and the singlet state is 1Σ+. 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. 1Δg, 3Σg- and 1Σg+. Arranged in order of increasing energy according to Hund’s rules, the states are 3Σg- < 1Δg < 1Σg+ The only difference in the case of O2 is that here both unpaired electrons are in πg orbitals, but since the product of two g states is also g, the overall states are the same as for B2. CHEMISTRY PAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29 , MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES Heteronuclear diatomic molecules For heteronuclear diatomic molecules, the procedure is the same, but the following points must be kept in mind: 1. 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 N2. 3. Upto 14 electrons, the scheme of nitrogen is followed, but beyond that, the scheme of O2 is followed, i.e. NO will have a similar energy level diagram as O2. 4. 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: CHEMISTRY PAPER :8, PHYSICAL SPECTROSCOPY MODULE: 29 , MOLECULAR TERM SYMBOLS AND SELECTION RULES FOR DIATOMIC MOLECULES Besides the 4 core electrons, there are 11 more electrons. The levels up to π2p are completely filled and do not contribute to 1 2 the angular momentum, leaving (π2p*) . Therefore, the term symbol is Π. The possible values of 2 2 Ω are 1+½ and 1-½, i.e.