Molecular Spectroscopy Flashcards
absorption spectroscopy
molecule undergoes a transition from a state of lower energy, E1, to one of higher energy, E2
emission spectroscopy
molecule undergoes a transition from a state of higher energy, E2, to one of lower energy, E1
transition dipole moment
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
Born-Oppenheimer approximation
nuclei of a molecule, being much heavier than the electrons, moves relatively slowly and may be trates 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
Born-Oppenheimer approximation: approximate molecular wavefunction as a product of electronic, vibrational and rotational wavefunctions
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
rotational spectroscopy: rigid rotor energy levels (in Joules, J)
J - rotational angular momentum quantum number (0, 1, 2,…)
r - bond length
I - moment of inertia
moment of intertia, I
mi - masses of the atoms
ri - their distances from the centre of mass of molecule
molecule rotates about its centre of mass:
m1r1 = m2r2
reduced mass of a molecule
μ = m1m2/(m1 + m2)
energy levels of rotational states
expressed as a rotational term
in cm-1
B - rotation constant
depends on the moment of intertia of the molecule
has units of cm-1
F(J) ∝ J(J+1)
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
centrifugal distortion
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
effect of centrifugal distortion is taken into account empirically by writing the rotational energy term as:
selection rules for rotational spectroscopy
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
ΔJ = ±1
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
applying the selection rules to the expression for the energy levels of a rigid rotor
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
intensities of lines in rotational spectra
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
population of a rotational energy level, J
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
relative population of rotational level, J
harmonic oscillator
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
energy levels of harmonic oscillator (in J)
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
energy levels of the vibrational states are expressed as a vibrational term:
G(v) = Ev/hc (in cm-1)
ωe - harmonic vibration wavenumber (in cm-1)
ωe - harmonic vibration wavenumber
depends on the force constant and reduced mass of the diatomic molecule
anharmonic oscillator energy levels
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
potential energy curve for diatomic molecules can be expressed in terms of a Taylor’s expansion about the equilibrium bond length (x = 0)
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
Morse potential - approach to determining energy levels of an anharmonic oscillator
De - depth of the potential well
x = r - re
when r → ∞,
V(x) → hcDe
when r → 0
V(x) → very large (although not infinite)
solving the Schrodinger equation for the Morse potential
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
selection rules for vibrational spectroscopy
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.
for heteronuclear diatomics, μ varies with x
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
dipole transition moment for vibrational spectroscopy
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
applying the selection rules to the expression for the energy levels of an anharmonic oscillator
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
intensities of lines in vibrational spectra
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)]»_space; 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
dissociation energies, D0
D0 differs from the well depth, De on account of the zero-point energy
Birge-Sponer plot
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)
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)
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
calculating De
if we know D0, we can calculate De by adding zero-point energy
we can also estimate De if we know ωe and ωeχe
calculating De - from ZPE
if we know D0, we can calculate De by adding zero-point energy
vibration-rotation spectroscopy
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
rovibrational selection rules
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
energy levels of the vibration-rotation terms
sum of the vibrational and rotational terms
when a vibrational transition, v + 1 ← v occurs, J changes by ±1 and by 0 (if ΔJ = 0 is allowed)
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
Q branch consists of lines at the harmonic wavenumber ωe
R branch consists of a series of equally spaced lines at higher wavenumbers than the Q branch
ωe + 2B
ωe + 4B
P branch consists of a series of equally spaced lines at lower wavenumbers than the Q branch
ωe - 2B
ωe - 4B
P and R branch lines are not exactly equally spaced
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
for an anharmonic oscillator, the vibration-rotation terms are:
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
the vibrational dependence of the rotational constant
Be - hypothetical value of the rotation constant at the bottom of the potential well
αe - vibration-rotation interaction constant
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
for the R branch
for the P branch
wavenumber difference denoted as:
considering these expressions for the wavenumbers of the P and R branches
P branch lines appear at lower wavenumbers than the pure vibrational transition
R branch lines appear at higher wavenumbers than the pure vibrational transition
combination differences
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
combination differences method:
electronic spectroscopy
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
selection rules
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)
The Franck-Condon principle
provides a basis for explaining the vibrational structure associated with an electronic transition
term symbols of diatomic molecules
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 «_space;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
term symbols for diatomic molecules: notation
term symbols for diatomic molecules: selection rules
term symbols for diatomic molecules: electronic transitions in N2
vibrational structure of electronic transition
there are no restrictions on the change of vibrational quantum number associated with an electronic transition
Franck-Condon principle
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
bands
vibrational transitions accompanying an electronic transition are called VIBRONIC TRANSITIONS
vibronic transitions give rise to BANDS
band system
the set of bands associated with a particular electronic transition
progression
a group of transitions with a common lower or upper vibrational level
sequence
a group of transitions with the same Δν
B-X band system of I2
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
vibronic transition wavenumber
the total energy T of a molecule in a particular electronic, vibrational and rotational state may be written:
T = Te + G(v) + F(J)
there are 5 spectroscopic values to determine: this requires a minimum of 5 lines to be measured
Deslandres table
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
dissociation energies
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)
- determine the wavenumber of the v’’ = 0 2. progression limit
- determine ṽ(v’max,0)
- determine 0-0 transition from spectrum
- to determine D0’
determining D0’’ for the electronic ground state
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:
determine De’’
