# Molecular Spectroscopy and Energy Levels

Contains basic facts to know, and an introduction to approximations and selection rules for molecular energy levels.

Molecular Spectroscopy & Energy Levels Notes

Basic Spectroscopic Facts to Know

Energy Levels for Rotation –

- Linear molecules, diatomics: EJ = BJ(J+1) – DJ2(J+1)2
- Spherical tops (SF6, etc): EJ = BJ(J+1)
- Symmetric tops: EJK = BJ(J+1) + (A-B)K2

Energy Levels for Vibration –

- Normal modes of polyatomic molecules (SHM): Ev = ωe(v+½)
- Local modes, real diatomics, etc: Ev = ωe(v+½) - ωexe(v+½)2

Vibrations in Electronic Spectra –

- Franck-Condon Principle: (Franck-Condon Factor).
- If re changes a lot, many vibrations excited. If re doesn’t change then Δv = 0.
- Only totally symmetric modes can be excited with many v’ in absorption.

Gross Selection Rules –

- Pure rotation in IR, Microwave: molecules with dipole moments.
- Pure rotational Raman: all molecules except spherical tops.
- Vib-Rot in IR: a dipole must change or appear in the vibrational motion.
- Vib-Rot Raman: polarisability must change in the motion.
- Electronic Spectra: all molecules.

Selection Rules for rotational level changes –

- One-photon transitions (microwave, IR, UV), no other coupled angular momentum present: ΔJ = ±1.
- One-photon, other angular moment present: ΔJ = 0, ±1.
- Two-photon (Raman): ΔJ = 0, ±2.

Selection Rules for Vibration level changes –

- IR, Raman Spectra, harmonic oscillator: Δv = ±1.
- Real molecules: Δv = ±1, ±2, etc (overtones much weaker than fundamental).
- Electronic Spectra: Δv = any, symmetric modes only. Intensities from Franck-Condon factors.
- Exclusion Rule: in centro-symmetric molecules IR and Raman activity are mutually exclusive.

Selection Rules for Electronic Transitions in Diatomics –

- ΔΛ = 0, ±1. ΔS = 0. g ←→u. +←→+, -←→-.

Branches –

- P branch ΔJ = -1. R branch ΔJ = +1. Q branch ΔJ = 0.
- IR Spectra in diatomics show P and R branches.
- IR Spectra in parallel bands of linear molecules: P and R.
- IR Spectra in perpendicular bands of linear molecules: P,Q,R.
- IR Spectra in parallel bands of symmetric tops: P,Q,R.

(others are more complex).

Electronic Spectra of diatomics –

- P and R branches in Σ−Σ transitions.
- P,Q and R in Σ−Π or vice-versa, and most others.

Constants –

B = h/(8π2cI) I = μr2

Energy Levels and Concepts

Units and Nomenclature –

Molecular Spectroscopy uses several non-SI units. Should usually stick to the units given. Often there is a ratio of hv/kT.

Energy: cm-1, Hz, eV. cm-1 x c (in cm s-1) = Hz, i.e. cm-1 x 3x108 x 100 = Hz.

Molecules:

cm-1, eV, Hz x h, kT are all per molecule.

kJ mol-1, RT imply per N molecules.

Also useful: at 298K, kT ≈ 207cm-1.

Nomenclature:

Upper state – single prime (J’, B’, v’, etc).

Lower state – double prime (J’’, B’’, etc.)

What is Spectroscopy?

Change of one quantum state to another → spectroscopic transition. Energy for this provided by EM radiation. Exchange of energy between radiation and matter.

Time-dependent Schrodinger Equation is important. In a weak oscillating EM field → t-dependent. Perturbation Theory to describe the transition.

hv = Eupp – Elow [ variation of μ over time ]

Electric dipole transitions most common in Molecular Spectroscopy, but should be aware that light waves have oscillating electric and magnetic fields, so transfer could occur magnetically:

Therefore magnetically induced transition 104 times less likely than electrically.

Systems with no oscillating electric dipole moment → ESR and NMR.

Radiative Relaxation –

Spontaneous Emission:

Level 2 → Level 1 by relaxation, giving out a photon of energy hv21.

Occurs in the absence of external radiation. Still involves transition dipole moment – quantum basis (interaction with radiation vacuum state).

If there are N2 molecules in level 2, energy is emitted at the rate:

I21 = N2 hv12 A

Comparing this with the probability, A, of emission shows that intensity depends on v4.

Hence, spontaneous emission irrelevant in microwave, but often dominant relaxation mechanism in UV (sets time limit for molecule in upper state).

Induced Emission / Absorption –

Coupling between oscillating transition moment and oscillating electromagnetic field.

Probability, P12 = ρ(v12) B12

Where ρ is the radiation density.

Net Emission / Absorption –

Must consider populations:

↓: Iem = N2 hv12 [A + Bρ(v12)]

↑: Iabs = N1 hv12Bρ(v12)

i.e. net depends on different in N1 and N2. Thermal Equilibrium,

Absorption Coefficients –

Linear (weak field):

dI = - Iαcdx

I = intensity, c = molecular concentration, α = absorption coefficient.

Integrate over path:

I = Ioe-αlc [ Beer-Lambert Law ]

Orbital Approximation –

e.g. H2 = 1σg2

CO = 1σ22σ23σ24σ21π45σ2

Pauli: ψtot is antisymmetric wrt exchange of electrons.

i.e. H2 (1σg2) → ψ = 1σg(α)(1) 1σg(β)(2) – 1σg(α)(2) 1σg(β)(1)

Born-Oppenheimer Approximation –

Electrons and nuclei experience same forces of magnitude but nuclei ~ 4 times more massive, so electrons move much more rapidly.

Energy of a molecule (but not atom) depends on the relative positions of nuclei. Need to consider this. Born-Oppenheimer Approximation → nuclear and electronic motion are separable and independent (reasonable as nucleus moves much slower).

Good approximation → nuclei fixed.

Hence,

ψtot = ψel(qel) ψnucl(qnucl)

Similarly, can separate nuclear motion into vibration and rotation:

ψtot ≈ ψel(qel) ψvib(qvib) ψrot(qrot)

Etot = Eel + Evib + Erot

ΔEel >> ΔEvib >> ΔErot

Approximation is very bad when:

Jahn-Teller Effect: direct coupling of electronic and vibrational motions.

Lambda-type Doubling: electronic and rotational coupling.

Translational –

Separated from other degrees of freedom so reference to Centre of Mass.

Also, usually only concerned with changes within molecule – only effect of translation is small (Doppler Effect).

Not usually necessary to consider since:

ΔEtrans << kT

Can be separated rigorously (no approximation).

(Xo, Yo, Zo) = coordinates of centre of mass.

, and similar for Yo, Zo.

Particle i at (Xi, Yi, Zi), where Xi = Xo + xi, etc.

Thus coordinates relative to centre of mass.

Xo, Yo, Zo variation = translational motion. Completely separately measurably from “internal” coordinations xi, yi, zi.

LCAO Approximation –

From Orbital Approximation, electronically:

ψ = φ1φ2 … φn [ each φ represents 1 electron ]

LCAO → Linear Combination of Atomic Orbitals,

φ = where, cm = mixing coefficient, χm = atomic orbital.

Variation Principle – adjust cm so that energy is as low as possible (more accurate).

LCAO Approach works for small molecules, but computationally very difficult.

Bonding & Antibonding Molecular Orbitals –

Consider H2. H• → χ = N e-r (1 electron). Label each H 1sA and 1sB.

Consider symmetry,

Molecule is symmetrical, so match e-density, → ψ identical round each nucleus or same except for change of sign.

Hence,

φ = cA1sA + cB1sB

cA = cB,

φ = 1sA + 1sB [ Normalisation not included ]

σ → cylindrically symmetric about A-B.

g → gerade, symmetric wrt centre of symmetry.

However, can also have:

cA = -cB [ e-density still the same ]

φ = 1sA - 1sB

Selection Rules –

Intensity (N’’ – N’) < ψ’ | μz | ψ’’ > 2

μz is the key factor.

μz =

Pure rotational spectrum,

ψel’ = ψel’’ & ψvib’ = ψvib’’

Symmetry tells us that observation of rotational transition requires the molecule to have a non-zero electric dipole moment.

First integral (λα)gives rotational selection rules:

ΔJ = 0, ±1

ΔΛ (or ΔK) = 0, ±1

Second integral (μ) cannot be factorised into only vibrational / electronic parts.

Expand μα as power series in vibrational coordinate Q.

Vibrational transition → first term vanishes.

2nd term → use orthogonality and factorises.

Hence, for allowed vibrational transition, dipole moment must change as vibration occurs.

Also, Δv = ±1 for simple harmonic oscillator.

Allowed electronic transition, must contain totally symmetric molecular point group (Σ+ for diatomics).

Symmetric vibration (e.g. for diatomics) = and Δvα .

Non-symmetric vibration - Δvα = 0,2,4… (odd changes the symmetry).

Inversion and Parity Rule –

E* f(X,Y,Z) = f(-X,-Y,-Z) where E* = parity.

E*ψ = ±ψ + → even parity, - → odd parity.

< ψ’ | μz | ψ’’ > ≠ 0 for allowed, → integrand symmetric wrt E*. μz is antisymmetric, therefore ψ’ψ’’ must be antisymmetric.

Hence,

+ ↔ - is allowed, + ↔ + and - ↔ - are forbidden.

Parity Rule different for magnetic dipole (even). Gives opposite result.

Centrosymmetrically,

Electric g ↔ u allowed.

Magnetic g ↔ g / u ↔ u are allowed.

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