Molecular Spectroscopy

Question 1

absorption spectroscopy

Answer 1

molecule undergoes a transition from a state of lower energy, E1, to one of higher energy, E2

Question 2

emission spectroscopy

Answer 2

molecule undergoes a transition from a state of higher energy, E2, to one of lower energy, E1

Question 3

transition dipole moment

Answer 3

intensity of spectral line - arising from a molecular transition between initial state (Ψi) and final state (Ψf) depends on the electric dipole moment

for a transition to be dipole allowed, the expectation value of the dipole moment operator (the transition dipole moment, R) must be non-zero

Question 4

the (electric) dipole moment

Answer 4

Question 5

Born-Oppenheimer approximation

Answer 5

nuclei of a molecule, being much heavier than the electrons, moves relatively slowly and may be treated as stationary while the electrons move in their field

approximation allows us to select an internuclear separation in a diatomic molecule and solve the Schrodinger equation for electrons at that nuclear separation
- this calculation can be repeated for different nuclear separations
- allowing us to explore how the energy of a molecule varies with bond length
- obtain a molecular potential energy surface

Question 6

Born-Oppenheimer approximation: approximate molecular wavefunction as a product of electronic, vibrational and rotational wavefunctions

Answer 6

electron motion much faster that molecular vibrations which are much faster than molecular rotations

all of the different types of molecular energy can be treated separately

Question 7

rotational spectroscopy: rigid rotor energy levels (in Joules, J)

Answer 7

J - rotational angular momentum quantum number (0, 1, 2,…)
r - bond length
I - moment of inertia

Question 8

moment of intertia, I

Answer 8

mi - masses of the atoms
ri - their distances from the centre of mass of molecule

molecule rotates about its centre of mass:
m1r1 = m2r2

Question 9

reduced mass of a molecule

Answer 9

μ = m1m2/(m1 + m2)

Question 10

energy levels of rotational states

Answer 10

expressed as a rotational term

in cm-1

Question 11

B - rotation constant

Answer 11

depends on the moment of intertia of the molecule

has units of cm-1

Question 12

F(J) ∝ J(J+1)

Answer 12

energy separation between adjacent rotational energy levels increases as J increases

B for small molecules is typically in the region 0.1 - 10 cm-1

B ∝ 1/I
- therefore larger molecules have more closely spaces rotational energy levels than smaller molecules

Question 13

centrifugal distortion

Answer 13

reduces the rotation constant and the energy levels are slightly closer together than the rigid rotor expression predicts

D - centrifugal distortion ~ related to the vibrational wavenumber of a diatomic molecule, ωe

Question 14

effect of centrifugal distortion is taken into account empirically by writing the rotational energy term as:

Answer 14

Question 15

selection rules for rotational spectroscopy

Answer 15

selection rules are found by evaluating the transition dipole moment
Ψr - rotation wavefunction

double prime ‘’ - initial state
single prime ‘ - final state

this integral is only non-zero if the molecule has a permanent dipole moment, μ and ΔJ = ±1
- these are the rotational selection rules for linear rotors

Question 16

ΔJ = ±1

Answer 16

when a photon is absorbed by a molecule, angular momentum of the system must be conserved

photon has a spin angular momentum of 1, when molecule absorbs photon, its rotational angular momentum J must increase by 1

Question 17

applying the selection rules to the expression for the energy levels of a rigid rotor

Answer 17

the wavenumbers of the allowed J + 1 ← 1 transitions are:

spectrum consists of a series of equally spaced lines with wavenumbers: 2B, 4B, 6B….
- separated by 2B

Question 18

intensities of lines in rotational spectra

Answer 18

intensities increase with increasing J and pass through a maximum before tailing off as J becomes large

existence of maximum arises because of a maximim in the population of rotational energy levels

Question 19

population of a rotational energy level, J

Answer 19

given by the Boltzmann expression

N - total number of molecules in the sample
gJ = (2J +1) - degeneracy of the level J
k - Boltzmann constant
T - rotational temperature of the sample

Question 20

relative population of rotational level, J

Answer 20

Question 21

harmonic oscillator

Answer 21

for small molecular vibrations, molecular bond obeys Hooke’s law and the restoring force, dV(x)/dx = -kx

k - force constant of bond
x = r - re
re - equilibrium bond length

integrating expression for restoring force gives the potential energy:

V(x) = 1/2 kx^2

Question 22

energy levels of harmonic oscillator (in J)

Answer 22

vibrational energy levels are quantised

v - vibrational quantum number (v = 0, 1, 2,…)
ω - vibration angular frequency in rad s-1
ω = (k/μ)^1/2

Question 23

energy levels of the vibrational states are expressed as a vibrational term:

G(v) = Ev/hc (in cm-1)

Answer 23

ωe - harmonic vibration wavenumber (in cm-1)

Question 24

ωe - harmonic vibration wavenumber

Answer 24

depends on the force constant and reduced mass of the diatomic molecule

Question 25

anharmonic oscillator energy levels

Answer 25

a real diatomic molecule is not a harmonic oscillator when r → ∞, the molecule dissociates into two neutral atoms - at this point, dV/dx = 0 ~ curve flattens out when r → 0, the positive charges on the nuclei cause mutual repulsion - dV/dx increases ~ curve becomes steeper

Question 26

potential energy curve for diatomic molecules can be expressed in terms of a Taylor's expansion about the equilibrium bond length (x = 0)

Answer 26

for small displacements: molecule behaves as a harmonic oscillator - the Taylor's series is truncated at the x^2 term for larger displacements, the higher order (anharmonic) terms become more important

Question 27

Morse potential - approach to determining energy levels of an anharmonic oscillator

Answer 27

De - depth of the potential well x = r - re when r → ∞, V(x) → hcDe when r → 0 V(x) → very large (although not infinite)

Question 28

solving the Schrodinger equation for the Morse potential

Answer 28

ωeχe - the first anharmonic wavenumber as the vibrational quantum number increases, the anharmonic term becomes more significant - vibrational energy levels become more closely spaces until they converge at the dissociation limit

Question 29

selection rules for vibrational spectroscopy

Answer 29

selection rules are found by evaluating the transition dipole moment Ψv - vibration wavefunction this integral is only non-zero if the molecule has a permanent dipole moment, μ.

Question 30

for heteronuclear diatomics, μ varies with x

Answer 30

this variation can be expressed as a Taylor's series subscript e - refers to equilibrium bond length substitue μ into the expression for the dipole transition moment to give: Ψ'v and Ψ''v are eigenfunctions of the same Hamiltonian - therefore orthogonal ~ the first term = 0

Question 31

dipole transition moment for vibrational spectroscopy

Answer 31

first-term of this expression is non-zero if Δv = ±1 - this is the vibrational selection rule for a harmonic oscillator higher order terms in the expansion (arising from anharmonicity), modify the vibrational selection rule to Δv = ±1, ±2, ±3,... transitions with Δv = ±1 are known as fundamental vibrational transitions transitions with Δv = ±2, ±3,... are known as first, second... vibrational overtones the effects of anharmonicity are relatively small, therefore overtones are much weaker than fundamental transitions

Question 32

applying the selection rules to the expression for the energy levels of an anharmonic oscillator

Answer 32

the wavenumbers of v + 1 ← v transitions are: thus, the spacing between adjacent vibrational energy levels decreases as v increases in order to determine ωe and ωeχe, at least two transition wavenumbers are required

Question 33

intensities of lines in vibrational spectra

Answer 33

intensities of absorption lines in vibrational spectrum govered by the population of the lower vibrational state the relative population of vibration level v is: for most diatomic molecules: hc[G(v) - G(0)] >> kT - so only the ground vibrational state (v'' = 0) is significantly populated at normal temperatures bands with v'' ≠ 0 referred to as hot bands - the intensities of these bands increase with temperature

Question 34

dissociation energies, D0

Answer 34

D0 differs from the well depth, De on account of the zero-point energy

Question 35

Birge-Sponer plot

Answer 35

when several vibrational transitions are observed, a graphical technique (Birge-Sponer plot) used to determine dissociation energy, D0 Birge-Sponer plot: sum of successive intervals ΔG(v) = G(v +1) - G(v) from v = 0 to the dissociation limit is D0 (just as the height of a ladder is the sum of the separation of its rungs)

Question 36

since ΔG = ωe - 2ωeχe(v + 1), the area under the plot of ΔG against v + 1/2 is equal to the sum of the separations between the energy levels (and thus equal to D0)

Answer 36

this method usually overestimates D0 - higher order anharmonic terms result in a deviation from linearity as v increases if we know D0, we can calculate De by adding zero-point energy we can also estimate De if we know ωe and ωeχe

Question 37

calculating De

Answer 37

if we know D0, we can calculate De by adding zero-point energy we can also estimate De if we know ωe and ωeχe

Question 38

calculating De - from ZPE

Answer 38

if we know D0, we can calculate De by adding zero-point energy

Question 39

vibration-rotation spectroscopy

Answer 39

each line in a high resolution vibrational spectrum of a gas-phase heteronuclear diatomic molecule consists of many closely spaces lines - arise from many rotational transitions accompanying each vibrational transition (rovibrational transitions) - these spectra referred to as band specta

Question 40

rovibrational selection rules

Answer 40

Δv = ±1, ±2,... ΔJ = ±1 ΔJ = 0 is forbidden - pure vibrational transitions are not observed - position at which it would occur is known as the band centre - exceptions for molecules having orbital angular momentum about their internuclear axis ~ in this case, rovibrational selection rules are: Δv = ±1, ±2,... ΔJ = 0, ±1

Question 41

energy levels of the vibration-rotation terms

Answer 41

sum of the vibrational and rotational terms

Question 42

when a vibrational transition, v + 1 ← v occurs, J changes by ±1 and by 0 (if ΔJ = 0 is allowed)

Answer 42

the absorptions fall into groups called branches the transition wavenumbers of the lines in each branch can be determined: - apply the appropriate ΔJ selection rule - calculate difference between the vibration-rotation terms

Question 43

Q branch consists of lines at the harmonic wavenumber ωe

Answer 43

Question 44

R branch consists of a series of equally spaced lines at higher wavenumbers than the Q branch ωe + 2B ωe + 4B

Answer 44

Question 45

P branch consists of a series of equally spaced lines at lower wavenumbers than the Q branch ωe - 2B ωe - 4B

Answer 45

Question 46

P and R branch lines are not exactly equally spaced

Answer 46

R branch lines converge and the P branch lines diverge away from the band centre - this can be explained if we take anharmonicity into account

Question 47

for an anharmonic oscillator, the vibration-rotation terms are:

Answer 47

rotation constant, B written with a subscript 'v' - it depends on the vibrational state as the vibrational quantum number, v increases, internuclear separation, r increases - moment of inertial increases (I = μr^2) B ∝ 1/r^2 B therefore decreases as v increases

Question 48

the vibrational dependence of the rotational constant

Answer 48

Be - hypothetical value of the rotation constant at the bottom of the potential well αe - vibration-rotation interaction constant

Question 49

transition wavenumbers of the lines in each branch of a vibration-rotation spectrum of an anharmonic oscillator determined: - applying appropriate ΔJ selection rule - calculate difference between the vibration-rotation terms

Answer 49

Question 50

for the R branch

Answer 50

Question 51

for the P branch

Answer 51

Question 52

wavenumber difference denoted as:

Answer 52

Question 53

considering these expressions for the wavenumbers of the P and R branches

Answer 53

P branch lines appear at lower wavenumbers than the pure vibrational transition R branch lines appear at higher wavenumbers than the pure vibrational transition

Question 54

combination differences

Answer 54

every transition depends on two rotation constants B'' and B' - for the molecules in its lower vibrational state (v'') and upper vibrational state (v') to detemine B'' and B': use combination differences method - differencs in wavenumber between two transitions with a COMMON LOWER vibration-rotation level gives information about energy differences between rotational levels in the UPPER vibrational state - differences in wavenumber between two transitions with a COMMON UPPER vibration-rotation level gives information about energy differences between rotational levels in the LOWER vibrational state

Question 55

combination differences method:

Answer 55

Question 56

electronic spectroscopy

Answer 56

in the lowest vibrational state of the ground electronic state of the molecule - nuclei are at their equilibrium positions - nuclei are stationary when electronic transition occurs (i.e. HOMO-LUMO transition), the electronic distribution is changes - PES shifted to higher energy (molecule has been excited) - electrons no longer in their ground electronic configuration - less stable - PES shifts along x-axis (along internuclear separation) - if electron in bonding orbital shifts to higher energy antibonding orbital, average bond length increases, bottom of PES shifts to larger internuclear separation

Question 57

selection rules

Answer 57

allow us to determine whether a particular electronic transition is allowed or not the molar absorption coefficient is used to quantify the intensity of an electronic transition (Beer-Lambert law)

Question 58

The Franck-Condon principle

Answer 58

provides a basis for explaining the vibrational structure associated with an electronic transition

Question 59

term symbols of diatomic molecules

Answer 59

classify the projections of electronic angular momenta along the molecular axis in a molecule: - orbital angular momenta of electrons are coupled - gives L - spins of electrons are coupled - gives S resultant angular momentum in H atom, j orbital angular momentum, l spin angular momentum, s j = I l-s I ... I l+s I in molecule, coupling between L and S << coupling between L and internuclear axis and between S and internuclear axis - thus, only components of L and S along the internuclear axis (Λħ and Σħ) are well-defined Λ and Σ couple to give a total angular momentum projection along internuclar axis Ω = I Λ + Σ I

Question 60

term symbols for diatomic molecules: notation

Answer 60

Question 61

term symbols for diatomic molecules: selection rules

Answer 61

Question 62

term symbols for diatomic molecules: electronic transitions in N2

Answer 62

Question 63

vibrational structure of electronic transition

Answer 63

there are no restrictions on the change of vibrational quantum number associated with an electronic transition

Question 64

Franck-Condon principle

Answer 64

electronic transitions occur much more rapidly thn vibrational transitions - nuclei need to be in the same position before and after an electronic transition: this means the transiton takes place between points on the potential energy surfaces that lie on vertical line the intensity distribution of the vibrational components of an electronic component depend on the relative positions of the two electronic potential energy curves

Question 65

bands

Answer 65

vibrational transitions accompanying an electronic transition are called VIBRONIC TRANSITIONS vibronic transitions give rise to BANDS

Question 66

band system

Answer 66

the set of bands associated with a particular electronic transition

Question 67

progression

Answer 67

a group of transitions with a common lower or upper vibrational level

Question 68

sequence

Answer 68

a group of transitions with the same Δν

Question 69

B-X band system of I2

Answer 69

progressions from v'' > 0 are only visible inthe absorption spectrum when the harmonic vibration wavenumber is small such as in the b-X band system of I2

Question 70

vibronic transition wavenumber

Answer 70

the total energy T of a molecule in a particular electronic, vibrational and rotational state may be written: T = Te + G(v) + F(J)

Question 71

Deslandres table

Answer 71

organises the vibration transition energies differences between wavenumbers in adjacent columns: - correspond to vibrational separations in the LOWER electronic state differences between wavenumbers in adjacent rows: - correspond to vibrational separations in the UPPER electronic state

Question 72

dissociation energies

Answer 72

can be derived from electronic spectra zero-point dissociation energy, D0 - energy required to dissociate the molecule in a given electronic state from the lowest vibrational level of that state well-depth, De - energy from the equilibrium geometry (i.e. bottom of the potential well, to the dissociation limit) 1. determine the wavenumber of the v'' = 0 2. progression limit 3. determine ṽ(v'max,0) 4. determine 0-0 transition from spectrum 5. to determine D0'

Question 73

determining D0'' for the electronic ground state

Answer 73

almost impossible using the same method as for D'0 - this is because few vibrational levels are populated however, if the wavenumber difference between the atomic fragments is used, we can use:

Question 74

determine De''