PY3P05 Bohn-Oppenheimer Approximation oBorn-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated. oThis leads to molecular wavefunctions that are given in terms of the electron positions (r i ) and the nuclear positions (R j ): oInvolves the following assumptions: oElectronic 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. oThe nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fast- moving electrons.
PY3P05 Molecular spectroscopy oElectronic transitions: UV-visible oVibrational transitions: IR oRotational transitions: Radio ElectronicVibrational Rotational E
PY3P05 Rotational motion oMust first consider molecular moment of inertia: oAt right, there are three identical atoms bonded to “B” atom and three different atoms attached to “C”. oGenerally specified about three axes: I a, I b, I c. oFor linear molecules, the moment of inertia about the internuclear axis is zero. oSee Physical Chemistry by Atkins.
PY3P05 Rotational motion oRotation of molecules are considered to be rigid rotors. oRigid rotors can be classified into four types: oSpherical rotors: have equal moments of intertia (e.g., CH 4, SF 6 ). oSymmetric rotors: have two equal moments of inertial (e.g., NH 3 ). oLinear rotors: have one moment of inertia equal to zero (e.g., CO 2, HCl). oAsymmetric rotors: have three different moments of inertia (e.g., H 2 O).
PY3P05 Quantized rotational energy levels oThe classical expression for the energy of a rotating body is: where a is the angular velocity in radians/sec. oFor rotation about three axes: oIn terms of angular momentum (J = I ): oWe know from QM that AM is quantized: oTherefore,, J = 0, 1, 2, …, J = 0, 1, 2, …
PY3P05 Quantized rotational energy levels oLast equation gives a ladder of energy levels. oNormally expressed in terms of the rotational constant, which is defined by: oTherefore, in terms of a rotational term: cm -1 oThe separation between adjacent levels is therefore F(J) - F(J-1) = 2BJ oAs B decreases with increasing I =>large molecules have closely spaced energy levels.
PY3P05 Rotational spectra selection rules oTransitions are only allowed according to selection rule for angular momentum: J = ±1 oFigure at right shows rotational energy levels transitions and the resulting spectrum for a linear rotor. oNote, the intensity of each line reflects the populations of the initial level in each case.
PY3P05 Molecular vibrations oConsider simple case of a vibrating diatomic molecule, where restoring force is proportional to displacement (F = -kx). Potential energy is therefore V = 1/2 kx 2 oCan write the corresponding Schrodinger equation as where oThe SE results in allowed energies v = 0, 1, 2, …
PY3P05 Molecular vibrations oThe vibrational terms of a molecule can therefore be given by oNote, the force constant is a measure of the curvature of the potential energy close to the equilibrium extension of the bond. oA strongly confining well (one with steep sides, a stiff bond) corresponds to high values of k.
PY3P05 Molecular vibrations oThe lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations. oTransition occur for v = ±1 oThis potential does not apply to energies close to dissociation energy. oIn fact, parabolic potential does not allow molecular dissociation. oTherefore more consider anharmonic oscillator.
PY3P05 Anharmonic oscillator oA molecular potential energy curve can be approximated by a parabola near the bottom of the well. The parabolic potential leads to harmonic oscillations. oAt high excitation energies the parabolic approximation is poor (the true potential is less confining), and does not apply near the dissociation limit. oMust therefore use a asymmetric potential. E.g., The Morse potential: where D e is the depth of the potential minimum and
PY3P05 Anharmonic oscillator oThe Schrödinger equation can be solved for the Morse potential, giving permitted energy levels: where x e is the anharmonicity constant: oThe second term in the expression for G increases with v => levels converge at high quantum numbers. oThe number of vibrational levels for a Morse oscillator is finite: v = 0, 1, 2, …, v max
PY3P05 Vibrational-rotational spectroscopy oMolecules vibrate and rotate at the same time => S(v,J) = G(v) + F(J) oSelection rules obtained by combining rotational selection rule ΔJ = ±1 with vibrational rule Δv = ±1. oWhen vibrational transitions of the form v + 1 v occurs, ΔJ = ±1. oTransitions with ΔJ = -1 are called the P branch: oTransitions with ΔJ = +1 are called the R branch: oQ branch are all transitions with ΔJ = 0
PY3P05 Vibrational-rotational spectroscopy oMolecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 – 4000cm -1 0.01 to 0.5 eV). oVibrational transitions accompanied by rotational transitions. Transition must produce a changing electric dipole moment (IR spectroscopy). P branch Q branch R branch
PY3P05 Electronic transitions oElectronic transitions occur between molecular orbitals. oMust adhere to angular momentum selection rules. oMolecular orbitals are labeled, , , , … (analogous to S, P, D, … for atoms) oFor atoms, L = 0 => S, L = 1 => P oFor molecules, = 0 => , = 1 => oSelection rules are thus = 0, 1, S = 0, =0, = 0, 1 oWhere = + is the total angular momentum (orbit and spin).
PY3P05 The End! oAll notes and tutorial set available from http://www.physics.tcd.ie/people/peter.gallagher/lectures/py3004/ oQuestions? Contact: email@example.com@tcd.ie oRoom 3.17A in SNIAM