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## Lecture 14: Molecular structure

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**Lecture 14: Molecular structure**• Rotational transitions • Vibrational transitions • Electronic transitions PY3P05**Bohn-Oppenheimer Approximation**• Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated. • This leads to molecular wavefunctions that are given in terms of the electron positions (ri) and the nuclear positions (Rj): • Involves the following assumptions: • Electronic wavefunction depends on nuclear positions but not upon their velocities, i.e., the nuclear motion is so much slower than electron motion that they can be considered to be fixed. • The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fast-moving electrons. PY3P05**Molecular spectroscopy**• Electronic transitions: UV-visible • Vibrational transitions: IR • Rotational transitions: Radio E Rotational Electronic Vibrational PY3P05**Rotational motion**• Must first consider molecular moment of inertia: • At right, there are three identical atoms bonded to “B” atom and three different atoms attached to “C”. • Generally specified about three axes: Ia, Ib, Ic. • For linear molecules, the moment of inertia about the internuclear axis is zero. • See Physical Chemistry by Atkins. PY3P05**Rotational motion**• Rotation of molecules are considered to be rigid rotors. • Rigid rotors can be classified into four types: • Spherical rotors: have equal moments of intertia (e.g., CH4, SF6). • Symmetric rotors: have two equal moments of inertial (e.g., NH3). • Linear rotors: have one moment of inertia equal to zero (e.g., CO2, HCl). • Asymmetric rotors: have three different moments of inertia (e.g., H2O). PY3P05**Quantized rotational energy levels**• The classical expression for the energy of a rotating body is: where ais the angular velocity in radians/sec. • For rotation about three axes: • In terms of angular momentum (J = I): • We know from QM that AM is quantized: • Therefore, , J = 0, 1, 2, … , J = 0, 1, 2, … PY3P05**Quantized rotational energy levels**• Last equation gives a ladder of energy levels. • Normally expressed in terms of the rotational constant, which is defined by: • Therefore, in terms of a rotational term: cm-1 • The separation between adjacent levels is therefore F(J) - F(J-1) = 2BJ • As B decreases with increasing I =>large molecules have closely spaced energy levels. PY3P05**Rotational spectra selection rules**• Transitions are only allowed according to selection rule for angular momentum: J = ±1 • Figure at right shows rotational energy levels transitions and the resulting spectrum for a linear rotor. • Note, the intensity of each line reflects the populations of the initial level in each case. PY3P05**Molecular vibrations**• Consider simple case of a vibrating diatomic molecule, where restoring force is proportional to displacement (F = -kx). Potential energy is therefore V = 1/2 kx2 • Can write the corresponding Schrodinger equation as where • The SE results in allowed energies v = 0, 1, 2, … PY3P05**Molecular vibrations**• The vibrational terms of a molecule can therefore be given by • Note, the force constant is a measure of the curvature of the potential energy close to the equilibrium extension of the bond. • A strongly confining well (one with steep sides, a stiff bond) corresponds to high values of k. PY3P05**Molecular vibrations**• The lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations. • Transition occur for v = ±1 • This potential does not apply to energies close to dissociation energy. • In fact, parabolic potential does not allow molecular dissociation. • Therefore more consider anharmonic oscillator. PY3P05**Anharmonic oscillator**• A molecular potential energy curve can be approximated by a parabola near the bottom of the well. The parabolic potential leads to harmonic oscillations. • At high excitation energies the parabolic approximation is poor (the true potential is less confining), and does not apply near the dissociation limit. • Must therefore use a asymmetric potential. E.g., The Morse potential: where Deis the depth of the potential minimum and PY3P05**Anharmonic oscillator**• The Schrödinger equation can be solved for the Morse potential, giving permitted energy levels: where xeis the anharmonicity constant: • The second term in the expression for G increases with v => levels converge at high quantum numbers. • The number of vibrational levels for a Morse oscillator is finite: v = 0, 1, 2, …, vmax PY3P05**Vibrational-rotational spectroscopy**• Molecules vibrate and rotate at the same time => S(v,J) = G(v) + F(J) • Selection rules obtained by combining rotational selection rule ΔJ = ±1 with vibrational rule Δv = ±1. • When vibrational transitions of the form v + 1 v occurs, ΔJ = ±1. • Transitions with ΔJ = -1 are called the P branch: • Transitions with ΔJ = +1 are called the R branch: • Q branch are all transitions with ΔJ = 0 PY3P05**Vibrational-rotational spectroscopy**• Molecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 – 4000cm-1 0.01 to 0.5 eV). • Vibrational transitions accompanied by rotational transitions. Transition must produce a changing electric dipole moment (IR spectroscopy). Q branch R branch P branch PY3P05**Electronic transitions**• Electronic transitions occur between molecular orbitals. • Must adhere to angular momentum selection rules. • Molecular orbitals are labeled, , , , … (analogous to S, P, D, … for atoms) • For atoms, L = 0 => S, L = 1 => P • For molecules, = 0 => , = 1 => • Selection rules are thus = 0, 1, S = 0, =0, = 0, 1 • Where = + is the total angular momentum (orbit and spin). PY3P05**The End!**• All notes and tutorial set available from http://www.physics.tcd.ie/people/peter.gallagher/lectures/py3004/ • Questions? Contact: • peter.gallagher@tcd.ie • Room 3.17A in SNIAM PY3P05