Lecture 9 Electronic Spectroscopy

Lecture 9 Electronic Spectroscopy Molecular Orbital Theory: A Review - LCAO approximaton & AO overlap - Variation Principle & Secular Determinant - Homonuclear Diatomic MOs - Energy Levels, Bond Order and Bonding vs. Anti-bonding - Heteronuclear Diatomics - Classification of Electronic States in Diatomics Electronic Spectroscopy: Diatomics - Potential Energy Curves and Electronic States - Vibrational Coarse Structure: Progressions & Sequences - Franck-Condon Principle - Rotational Fine Structure - Electronic & Vibronic Transitions - Fluorescence & Phosphorescence Electronic Spectroscopy: Polyatomics Different systems: - AH 2 triatomic molecules - Simple molecules: formaldehyde - Aromatic molecules: Hückel Theory - Transition metal complexes: crystal field theory Electronic transitions: - Selection rules - Summary of chromophores

Molecular Orbital Theory The simplest way to understand molecular electronic spectra is to consider molecules in the context of molecular orbital (MO) theory. In this theory, the MO s are constructed from linear combinations of atomic orbitals (LCAO). It should be emphasized that MO s are not literally LCAO s: this is just an approximation chosen for simplicity, and is adequate for this discussion. The mathematically unwieldy valence bond (VB) theory is often used to describe molecular structure as well, including the concepts of F and B bonds, hybridization and spin pairing - we will not discuss this here. Homonuclear diatomic molecules are the best starting point to consider MO theory: 1. Consider the two nuclei, without electrons, at equilibrium nuclear distance apart. 2. Construct MOs about these two nuclei in the same manner as AOs are added to a single bare nucleus. 3. Feed e- into the MO s in pairs (m S = ±½) in order of increasing energy using the aufbau principle - this yields the ground configuration of the molecule. Linear combinations of atomic orbitals are used to construct the MOs. Why is this possible? Close to the nucleus, the MO wave function will resemble the AO wave function for an atom of which the nucleus is part.

LCAO Approximation The MO wave function R is expressed as a linear combination of AO wave functions P i on both nuclei: R ' j i c i P i where c i is the coefficient (i.e., weighting factor) of the wavefunction P i. Not all LCAOs form MOs which are different from the AOs, but effective LCAO s arise when: 1. The AOs have comparable energies. 2. The AOs overlap as much as possible. 3. The AOs have same symmetry properties w.r.t. certain symmetry elements of the molecule. Consider a homonuclear diatomic molecule with nuclei labelled 1 and 2. The LCAO-MO wavefunction is: R ' c 1 P 1 % c 2 P 2 In order to consider points 2 and 3, it must be realized that each possible molecular orbital must form a basis for some irreducible representation of the point group of the molecule. In the same way that vibrational wavefunctions must form bases for irreducible representations, electronic wavefunctions must also form bases for irreducible representations, and can be assigned with symmetry species of the molecular point group.

Overlap of Atomic Orbitals Consider the N 2 molecule: (a) The N 1s AOs satisfy condition (1), with identical energies, but not (2), since the high nuclear charge causes the 1s AOs to be close to the nuclei, and there is little overlap. (b) The N 2s AOs satisfy (1) and (2), as well as (3): since they are spherically symmetrical, they are allowed by symmetry to overlap with one another. Negative overlap is shown here. (c) Positive overlap, same as (b). (d) The N 2s(1) and 2p x (2) satisfy (1) and (2), but not (3). The 2s is spherically symmetric, but 2p x is antisymmetric w.r.t. reflection through the plane, thus there are positive and negative overlaps which result in a total overlap of zero. (e) The N 2s(1) and 2p z (2) satisfy all three conditions, provided that the 2p z (2) is in the proper orientation.

Overlap of Atomic Orbitals, 2 For two nuclei A and B, the total normalized probability density to find an electron corresponding to real wavefunctions R + and R - is N 2 (A 2 + B 2 + 2AB) where N is the normalization constant and: 1. A 2 is the probability density if e- confined to AO on A 2. B 2 is the probability density if e- confined to AO on B 3. 2AB is an extra contribution to the electron density, known as the overlap density. A bonding orbital has a region of constructive interference resulting from overlap of the AOs, placing the e- in a position where they interact strongly with both nuclei; hence, the energy is lowered compared to the interaction of a single nucleus with a single electron. An anti-bonding orbital has a region of destructive inteference resulting from overlap. In this case, the electron is largely excluded from the internuclear region and not involved in bonding; this has the effect of pulling the two nuclei apart: i.e., anti-bonding, nuclear-nuclear repulsion dominates. a) bonding b) anti-bonding

Variation Principle The important properties of the each MO are therefore the energy, E, and the values of the coefficients, c i, which describe the contributions of each of the AOs to the MO. These properties can be obtained by solving:,r ' ER Multiplying both sides by R * (the complex conjugate of R), integrating over all space and solving for E gives: E ',R dj mr( m R( R dj In order to calculate E exactly, R must be known exactly (which it is not). The common procedure is to take an educated guess at some trial MO wavefunction, R n, and calculate the corresponding energy, E n. Then a second guess is made with the wavefunction R m, which has the energy E m. The variation principle states that if E m < E n, then R m is closer than R n to the true wavefunction, R. This principle applies to the ground state only. As a result, the true ground state wavefunction can never be calculated exactly, but can be approached as closely as possible by varying parameters in the trial wavefunctions (e.g., values of c i ). So, these calculations are iterative, and normally handled by fast computers.

Homonuclear Diatomic MOs If we combine the following equations: R ' c 1 P 1 % c 2 P 2 E ' mr(,r dj m R( R dj and assuming that P 1 and P 2 are not complex, we have: Ē ' m(c 2 1 P 1,P 1 % c 1 c 2 P 1,P 2 % c 1 c 2 P 2,P 1 % c 2 2 P 2,P 2 ) dj m (c 2 1 P2 1 % 2c 1 c 2 P 1 P 2 % c 2 2 P2 2 ) dj If the AO wavefunctions are normalized: m P2 1 dj ' m P2 2 dj ' 1 and, is a hermitian operator: m P 1,P 2 dj ' m P 2,P 1 dj ' H 12 then the quantity S is known as the overlap integral: m P 1 P 2 dj ' S and is a measure of the overlap between AOs P 1 and P 2. This means that the equation for energy is simplified to: Ē ' c 2 1 H 11 % 2c 1 c 2 H 12 % c 2 2 H 22 c 2 1 % 2c 1 c 2 S % c 2 2

Secular Determinant If the variation principle is used to optimize c 1 and c 2, we obtain and from the modified energy equation, and set them equal to zero: c 1 (H 11 & E) % c 2 (H 12 & ES) ' 0 c 1 (H 12 & ES) % c 2 (H 22 & E) ' 0 (now E is used instead of E, since we are approaching the true wavefunction). These equations are known as the secular equations (i.e., independent of time), and values of E which satisfy them can be obtained by solving two equations in two unknowns, or the secular determinant: /0 H 11 & E H 12 & ES H 12 & ES H 22 & E /0 ' 0 H 12 is the resonance integral, often symbolized as $ H 11 = H 22 = " is the Coulomb integral in a homonuclear diatomic. Thus the secular determinant is written: which yields: /0 " & E $ & ES $ & ES " & E /0 ' 0 (" & E) 2 & ($ & ES) 2 ' 0 from which two values of E are extracted: E + and E - : E ± ' (" ± $)/(1 ± S)

Homonuclear Diatomic MOs, 2 For very approximate MO wavefunctions and energies, we can assume S = 0 and a similar, as in the atom, giving the result that " = E A, the AO energy. These assumptions yield E ± E A ± $ This rough approximation gives two MOs symmetrically displaced from E A by ±$ (or separated by 2$). Since $ is negative, the orbital with energy E A + $ lies lowest: E A - $ E A 2$ E A E E A % $ With the approximation S = 0, the secular equations are: c 1 (" & E) % c 2 $ ' 0 c 1 $ % c 2 (" & E) ' 0 If E = E + or E -, c 1 /c 2 = 1 or -1, respectively, and MOs are: R % ' N % (P 1 % P 2 ) R & ' N & (P 1 & P 2 ) N + and N - are the normalization constants from: m R2 % dj ' m R2 & dj ' 1 and neglecting the overlap integral, N + = N - = 2 1/2 : R ± ' 2 &1/2 (P 1 ± P 2 )

Designation for MOs Every linear combination of two AOs gives two MOs, one higher and one lower in energy than the AOs. Formation of MOs are shown below for a homonuclear diatomic. The MOs are designated with symmetry labels from the D 4h point group, and includes the AO from which the MO was derived. F orbitals will be cylindrically symmetric about the internuclear axis, whereas the B orbitals will not. The g and u subscripts refer to symmetry or antisymmetry w.r.t. inversion through the centre of the molecule. Often an asterisk denotes the u type MO, and anti-bonding character w.r.t. the nodal plane z to the internuclear axis. Those without an asterisk (g) are bonding MO s.

Energy Levels MO s from the 1s, 2s and 2p AOs are arranged in order of increasing energy, holds for all first row diatomic molecules except O 2 and F 2. Because the expected resonance integral from the 2p z AOs is larger than that from the 2p x,y AOs, the expected order of MO energies is (at least for O 2 and F 2 ): For other first-row diatomics the F g 2s and F g 2p interact (same symmetry), and push each other apart such that F g 2p is above B u 2p:

Homonuclear Diatomic MOs, 3 Here is a correlation diagram for diatomic molecules, showing the limits for united (molecules) and separated atoms. Approximate positions of the AO s for each molecule are indicated by the dashed lines. Energy level orderings for first-row diatomics:

Bond Order The electronic structure of any first-row diatomic is obtained by feeding in the electrons pairwise into the MOs in order of increasing energy. B-orbitals are doubly degenerate, accomodating four electrons each. e.g., N 2 : (F g 1s) 2 (F ( u 1s)2 (F g 2s) 2 (F ( u 2s)2 (B u 2p) 4 (F g 2p) 2 The bonding character of an electron in a bonding orbital is approximately cancelled by the anti-bonding character of an electron in an anti-bonding orbital. Thus in N 2 : bonding of two electrons in F g 1s cancelled by F u* 1s, and same for F g 2s and F u* 2s. There are six electrons remaining in B u 2p and F g 2p. The measure of net bonding in a diatomic molecule is its bond order, b, given by: b ' ½(n & n * ) where n is the number of e- in bonding orbitals and n * is the number of e- in anti-bonding orbitals. So in N 2, n = 6 and n * = 3, so b = 3, consistent with a triple N/N bond. Exercise: calculate the bond order in H 2, He 2 and C 2 (1) The greater the bond order between two atoms of a given pair of elements, the shorter the bond. (2) The greater the bond order, the greater the bond strength.

More on bonding & anti-bonding The bond orders for some diatomic species are given below Bond Order r e / (pm) D e / (kj mol -1 ) HH 1 74.14 432.1 NN 3 109.76 941.7 HCl 1 127.45 427.7 CH 1 114 435 C-C 1 154 368 C=C 2 134 720 C/C 3 120 962 Numbers in italics are mean values for polyatomic molecules. Further data is available from Tables 14.2, 14.3, Atkins 6 th Edition. The ground configuration of oxygen: (F g 1s) 2 (F ( u 1s)2 (F g 2s) 2 (F ( u 2s)2 (F g 2p) 2 (B u 2p) 4 (B ( g 2p)2 is consistent with a double bond. States with variable multiplicity can arise from this configuration, and according to Hund s rule, the state with the higher multiplicity is lower in energy. In O 2, the two electrons in the unfilled B g 2p orbital may have their spins parallel, S = 1, or antiparallel, S = 0, giving multiplicites of 3 and 1. So the ground state of oxygen is the triplet state, with two parallel unpaired electrons. In F 2, the ground configuration is: (F g 1s) 2 (F ( u 1s)2 (F g 2s) 2 (F ( u 2s)2 (F g 2p) 2 (B u 2p) 4 (B ( g 2p)4 which is consistent with a single bond.

Excited States and Orbital Shapes The singlet state of oxygen has the same configuration as the ground state, but is an excited state where the electron spins are anti-parallel (usually a low-lying excited state). Excited configurations of molecules, as for atoms, can give rise to more than one state. For example, the short-lived C 2 molecule with the excited configuration: (F g 1s) 2 (F ( u 1s)2 (F g 2s) 2 (F ( u 2s)2 (B u 2p) 1 (F ( g p)1 results in both triplet and singlet states, since the two electrons in the partially filled orbitals may have parallel or antiparallel spins. In the linear homonuclear molecules, the F and B bonding orbitals are simple to visualize. The contributing AOs and electron density of the bonding orbitals is shown below: (A) (A) Contributing AOs to bonding and anti-bonding F and F * MOs. (B) Resultant boundary surface of a F orbital in a linear diatomic. (C) Contributing AOs to bonding and anti-bonding B and B * MOs. (D) Cylindrical electron density in the B MO in a linear molecule. (B) (C) (D)

Heteronuclear Diatomics Heteronuclear diatomic molecules that have atoms that are sufficiently similar can be treated in a similar manner to the homonuclear diatomics (e.g., NO, CO, CN). In NO, 15 e-can be fed into the MOs to give the ground configuration: (F1s) 2 (F ( 1s) 2 (F2s) 2 (F ( 2s) 2 (F2p) 2 (B2p) 4 (B ( 2p) 1 (note the g and u subscripts are gone, since the molecules no longer have inversion centres). This configuration gives rise to a doublet state, since there is one unpaired electron. Molecules like SO and PO can be treated in this way as well, since they are still rather like homonuclear diatomics, since the 2s and 2p AO s on O are similar to the 3s and 3p AO s on P or S. Linear combinations can be made between AOs with the same symmetry on different atoms (these AOs would have very different energies). However, the resonance integral $ would be very small in these cases (e.g., overlap 1s from H with 1s from P), and the MOs are virtually unchanged in energy from the AOs (such LCAOs are not meaningful). For molecules like HCl, the MO diagram looks nothing like the homonuclear MO diagram. For instance, the KL3s 2 3p 5 configuration of the Cl atom has 3p e- (12.967 ev) which are similar in energy to the H 1s electron (13.598 ev). Of the 3p orbitals, only the 3p z AO has the proper symmetry to overlap with the H 1s AO, and c 1 /c 2 is no longer ±1. 3p x and 3p y AO s of Cl cannot overlap with the 1s AO of H (remain as lone pairs in MOs very similar to the AOs)

Heteronuclear Diatomics, 2 MO scheme for heteronuclear diatomics with similar atoms: Generally speaking, the AOs of the more electronegative atom will be lower in energy than the corresponding AOs of the less electronegative atom. Symmetry labels are appropriate for C 4v The MO scheme and orbitals involved in bonding are pictured below for molecules with dissimilar atoms like HF, HCl, etc.: Note the energies of the 2p lone pair and 2p AOs are very similar. The 2p z orbital has the right symmetry to form the 3F MO with the 1s AO on H.. The 2p x and 2p y AOs do not have the correct symmetry, and constitute the 2p lone pair (non-bonding)

Properties of Diatomics Properties of Homonuclear Diatomics Properties of Heteronuclear Diatomics

Classification of Electronic States We now concern ourselves with angular momenta due to orbital and spin motions of the electrons (for now, neglect the overall rotation of the molecule). The orbital and spin motions of electrons create magnetic moments which interact with one another in an analogous manner to the various types of couplings in atoms. For all diatomic molecules, the coupling approximation which best describes electronic states is analogous to the Russell- Saunders approximation for atoms: (1) All e- orbital angular momenta are coupled to give a resultant vector L (2) All e- spin angular momenta are coupled to give S If there is no highly charged nucleus in the molecule, the spinorbit coupling between L and S is weak enough that instead of coupling to one another they couple to the electrostatic field produced by the nuclear charges (this is known as Hund s case (a)): L S 7 E S

Orbital and Spin Angular Momentum Orbital angular momentum, 7 L is very strongly coupled to the internuclear electrostatic field and precesses about this field - so that magnitude of L is not defined, and L is not a good quantum number. The 7 component of orbital angular momentum is defined such that the quantum number 7 = 0, 1, 2, 3,... All electronic states with 7 > 0 are doubly degenerate (electrons orbiting clockwise or counterclockwise, but have the same energy). If 7 = 0, there is no orbital motion and no degeneracy. 7 for a molecule is akin to L (S, P, D, F, G,...) for an atom: 7 = 0, 1, 2, 3, 4,... are assigned the letters E, A, ), M, ',... Spin angular momentum, E The coupling of S to the internuclear axis is not caused by the electrostatic field (no effect) but by the magnetic field along the axis due to the motion of the electrons. The S component along the internuclear axis is described by the quantum number E, which is analogous to m S in atoms, and can take the values E = S, S - 1,..., -S. In this case, S is a good quantum number, and for states with 7 > 0, there are 2S + 1 components corresponding to the number of values that E can take: so multiplicity is calculated as usual: 2S + 1, and indicated by a pre-superscript as for atoms: e.g., 3 A

Spin-Orbit Coupling The component of the total angular momentum (i.e., orbital plus spin) along the internuclear axis is given by S, where the quantum number S = *7 + E*. The spin-orbit interaction splits the components so that the energy level is shifted by )E = A7E, where A is the spin-orbit coupling constant: normal multiplet: component with lowest S has the lowest energy (A +ve) inverted multiplet: component with lowest S has the highest energy (A -ve) For E states there is no orbital angular momentum and no resulting magnetic field to couple S to the electrostatic field along the internuclear axis (the E state has one component, no matter what the multiplicity is). If there is a highly charged nucleus in the molecule, the spinorbit coupling may be large enough that L and S are not uncoupled by the electrostatic field of the nuclei. In this case, L and S couple to give J, as in an atom: L J S S This is Hund s case (c), where J precess about the internuclear axis, and has a component along this axis S (7 is no longer a good quantum number, S used as label).

Labels and Selection Rules The quantum numbers 7, S and S are not sufficient to label all of the states. We also need: # g or u which indicates that R e is symmetric or antisymmetric w.r.t. inversion through the molecule centre (only applied to homonuclear diatomics) # the symmetry of R e w.r.t. reflection across any F v plane containing the molecular axis. + and - correspond to R e that are symmetric and antisymmetric w.r.t. these reflections (e.g., 2 E g - and 3 E u+ ). The +/- symbolism is not often used with higher 7 states (A ±, ) ± not often seen). # the ground state is labelled as X, higher states with same multiplicity are A, B, C,... in order of increasing energy. For states with different multiplicity from that of the ground state, labels are a, b, c,... (sometimes used) Selection Rules (for Hund s case (a)): (1) )7 = 0, ±1 E-E, A-E, )-A allowed; )-E, M-) forbidden (2) )S = 0 This selection rule breaks down as nuclear charge increases, as in atoms. In H 2, triplet-singlet transitions are strictly forbidden, but in CO, the a 3 A-X 1 E + transition is weakly observed. (3) )E = 0; )S = 0, ±1 for transitions between multiplet components (4) +:-; +:+; -:- Relevant only for E-E transitions. (5) g:u; g:g; u:u E u+ -E g + & A u -E g + allowed; E g+ -E g + & A u -E u - forbidden

Hund s Cases All of Hund s cases are shown diagrammatically below, we are predominant concerned with (a). L, 7 = orbital angular momentum S, E = spin angular momentum J, S = L + S K = second or intermediate angular momentum N = molecular rotation

Key Concepts 1. The simplest way to understand molecular electronic spectra is to consider molecules in the context of molecular orbital (MO) theory. In this theory, the MO s are constructed from linear combinations of atomic orbitals (LCAO). 2. The variation principle is used to calculate (as closely as possible) the energy of the ground state, by calculating energies of a series of trial wavefunctions, and using the wavefunction with the resulting lowest energy as the best approximation to the overall electronic wavefunction. 3. The measure of net bonding in a diatomic molecule is its bond order: (1) The greater the bond order between two atoms of a given pair of elements, the shorter the bond. (2) The greater the bond order, the greater the bond strength. 4. Total orbital and spin angular momenta couple in diatomic molecules as well. If there is no highly charged nucleus in the molecule, the spin-orbit coupling between L and S is weak enough that instead of coupling to one another they couple to the electrostatic field produced by the nuclear charges (Hund s case (a)). If there is a highly charged nucleus in the molecule, the spin-orbit coupling may be large enough that L and S are not uncoupled by the electrostatic field of the nuclei. In this case, L and S couple to give J, as in an atom (Hund s case (c)).