o Rotational transitions

o Vibrational transitions

o Electronic transitions

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o Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in can be separated.

o This leads to molecular wavefunctions that are given in terms of the electron positions (ri) and the nuclear positions (Rj):

ˆ ˆ ˆ " (rˆi ,R j ) ="electrons(rˆi ,R j )"nuclei (R j ) o Involves the following assumptions: ! o 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.

o The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the fast- moving electrons.

PY3P05 o Electronic transitions: UV-visible o Vibrational transitions: IR o Rotational transitions: Radio

E

Electronic Vibrational Rotational

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o Must first consider molecular moment of inertia:

2 I = "miri i o At right, there are three identical bonded to “B” and three different atoms attached to “C”. ! o Generally specified about three axes: Ia, Ib, Ic. o For linear molecules, the moment of inertia about the internuclear axis is zero. o See Physical Chemistry by Atkins.

PY3P05 o Rotation of molecules are considered to be rigid rotors. o Rigid rotors can be classified into four types:

o Spherical rotors: have equal moments of inertia (e.g., CH4, SF6).

o Symmetric rotors: have two equal moments of inertial (e.g., NH3).

o Linear rotors: have one moment of inertia equal to zero (e.g., CO2, HCl).

o Asymmetric rotors: have three different moments of inertia (e.g., H2O).

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o The classical expression for the energy of a rotating body is:

2 E a =1/2Ia"a

where !a is the angular velocity in radians/sec.

! 2 2 2 o For rotation about three axes: E =1/2Ia"a +1/2Ib"b +1/2Ic"c o In terms of angular momentum (J = I!): ! J 2 J 2 J 2 E = a + b + c 2Ia 2Ib 2Ic o We know from QM that AM is quantized:

J = J(J +1)!2 , J = 0, 1, 2, … ! J(J +1)! o Therefore, E J = , J = 0, 1, 2, … 2I !

PY3P05 ! o Last equation gives a ladder of energy levels. o Normally expressed in terms of the rotational constant, which is defined by: !2 ! hcB = => B = 2I 4"cI o Therefore, in terms of a rotational term:

! F(J) = BJ(J +1) cm-1 o The separation between adjacent levels is therefore ! F(J) - F(J-1) = 2BJ o As B decreases with increasing I =>large molecules have closely spaced energy levels.

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o Transitions are only allowed according to for angular momentum:

"J = ±1 o Figure at right shows rotational energy levels transitions and the resulting spectrum for a linear rotor. o Note, the intensity of each line reflects the populations of the initial level in each case.

PY3P05 o Consider simple case of a vibrating , where restoring force is proportional to displacement (F = -kx). Potential energy is therefore

V = 1/2 kx2 o Can write the corresponding Schrodinger equation as

!2 d 2" + [E #V]" = 0 2µ dx 2 2 2 ! d " 2 2 + [E #1/2kx ]" = 0 2 µ dx m m where µ = 1 2 m1 + m2 ! o The SE results in allowed energies 1/ 2 # k & E = (v +1/2)!" " = % ( v = 0, 1, 2, … ! v $ µ'

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! !

o The vibrational terms of a molecule can therefore be given by G(v) = (v +1/2)v˜

1/ 2 1 # k & v˜ = % ( 2"c $ µ' ! o Note, the force constant is a measure of the curvature of the potential energy close to the equilibrium! extension of the bond. o A strongly confining well (one with steep sides, a stiff bond) corresponds to high values of k.

PY3P05 o 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. o Transition occur for "v = ±1 o This potential does not apply to energies close to dissociation energy. o In fact, parabolic potential does not allow molecular dissociation. o Therefore more consider anharmonic oscillator.

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o A molecular potential energy curve can be approximated by a parabola near the bottom of the well. The parabolic potential leads to harmonic oscillations. o At high excitation energies the parabolic approximation is poor (the true potential is less confining), and does not apply near the dissociation limit. o Must therefore use a asymmetric potential. E.g., The :

2 "a(R"Re ) V = hcDe (1" e )

where De is the depth of the potential minimum and 1/ 2 ! # µ" 2 & a = % ( $ 2hcDe '

PY3P05 ! o The Schrödinger equation can be solved for the Morse potential, giving permitted energy levels: 2 G(v) = (v +1/2)v˜ " (v˜ +1/2) xev˜

2 where x is the constant: a ! e x = e 2µ" ! o The second term in the expression for G increases with v => levels converge at high quantum numbers. ! o The number of vibrational levels for a Morse oscillator is finite:

v = 0, 1, 2, …, vmax

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o Molecules vibrate and rotate at the same time => S(v,J) = G(v) + F(J)

S(v,J) = (v +1/2)v˜ + BJ(J +1) o Selection rules obtained by combining rotational selection rule !J = ±1 with vibrational rule !v = ±1. ! o When vibrational transitions of the form v + 1 ! v occurs, !J = ±1. o Transitions with !J = -1 are called the P branch:

v˜ P (J) = S(v +1,J "1) " S(v,J) = v˜ " 2BJ o Transitions with !J = +1 are called the R branch:

! v˜ R (J) = S(v +1,J +1) " S(v,J) = v˜ + 2B(J +1) o Q branch are all transitions with !J = 0

! PY3P05 o Molecular vibration spectra consist of bands of lines in IR region of EM spectrum (100 – 4000cm-1 0.01 to 0.5 eV). o Vibrational transitions accompanied by rotational transitions. Transition must produce a changing electric dipole moment (IR ).

Q branch

P branch R branch

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o Electronic transitions occur between molecular orbitals. o Must adhere to angular momentum selection rules. o Molecular orbitals are labeled, ", #, $, … (analogous to S, P, D, … for atoms) o For atoms, L = 0 => S, L = 1 => P o For molecules, % = 0 => ", % = 1 => # o Selection rules are thus

$% = 0, ±1, $S = 0, $"=0, $& = 0, ±1 o Where & = % + " is the total angular momentum (orbit and spin).

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