Molecular spectroscopy — rotational (rigid rotor), vibrational (harmonic oscillator, Morse potential), rotational-vibrat

5.1.4 · Chemistry › Physical Chemistry (Advanced)

1. Rotational spectroscopy — the rigid rotor

KIYA chahiye: quantised rotational energy levels.

Spectrum kaisa dikhta hai. Selection rule $Δ J = \pm 1$ (molecule mein permanent dipole hona chahiye). Absorption $J \to J + 1$ : $\tilde{ν} = F (J + 1) - F (J) = B [(J + 1) (J + 2) - J (J + 1)] = 2 B (J + 1)$ Toh lines $2 B, 4 B, 6 B, \dots$ par aati hain — equally spaced by $2 B$ . Spacing measure karo → $B$ milega → $I$ milega → $r$ milega. Yahi poora point hai.

2. Vibrational spectroscopy

2a. Harmonic oscillator

Selection rule $Δ v = \pm 1$ (aur dipole change honi chahiye). Saari transitions usi $\tilde{ν}_{e}$ par absorb karti hain → harmonic model ek hi line predict karta hai. Reality mein overtones aur convergence dikhti hai ⇒ hume Morse potential chahiye.

2b. Morse potential (anharmonicity)

Figure — Molecular spectroscopy — rotational (rigid rotor), vibrational (harmonic oscillator, Morse potential), rotational-vibrat

3. Rotational–vibrational coupling (ro-vibrational spectrum)

Selection rules: $Δ v = + 1$ , $Δ J = \pm 1$ .

$Δ J = + 1$ → R branch (higher $\tilde{ν}$ )
$Δ J = - 1$ → P branch (lower $\tilde{ν}$ )
$Δ J = 0$ (zyaadatar diatomics ke liye forbidden) → Q branch hota (band centre par gap).

4. Electronic spectroscopy — the Franck–Condon principle

Agar upper-state well displaced hai ( $r_{e}^{'} > r_{e}^{''}$ ), toh $v^{''} = 0$ se vertical line ek high $v^{'}$ par land karti hai → intensity $v^{'} > 0$ par peak karti hai, characteristic Franck–Condon intensity envelope deti hai.

Common mistake Classic errors ko steel-man karna

(1) " $I$ ke liye ek atom ki mass use karo." Sahi lagta hai kyunki rotation ek ball ke swinging jaisa dikhta hai. Fix: dono atoms centre of mass ke baare mein move karte hain → reduced mass $μ$ use karo. $I = μ r^{2}$ . (2) "Harmonic oscillator dissociate ho sakta hai." Sahi lagta hai kyunki real bonds toot jaate hain. Fix: ek parabola ki infinite walls hoti hain — ye kabhi dissociate nahi karta. $D_{e}$ ke liye Morse flattening chahiye. (3) "Ground vibrational energy zero hai." Sahi lagta hai (koi excitation nahi). Fix: zero-point energy $= \frac{1}{2} \tilde{ν}_{e} \neq = 0$ Heisenberg se. (4) "Rotational lines perfectly evenly spaced hoti hain." Sirf rigid rotor ke liye sahi; centrifugal distortion + $B_{v}$ change P/R branches ko asymmetric banata hai. (5) "Brightest electronic line hamesha $0 \to 0$ hoti hai." Sirf tab jab wells aligned hon. Agar displaced hain, vertical Franck–Condon overlap higher $v^{'}$ par peak karta hai.

Recall Feynman: 12-saal ke bacche ko samjhao

Socho ek molecule do balls on a spring jaisi hai. Ye spin kar sakti hai (rotation — thodi si energy chahiye, microwave jaisi), spring par andar-bahar wobble kar sakti hai (vibration — zyaada energy chahiye, heat/IR jaisi), aur iski electrons ek nayi arrangement mein jump kar sakti hain (bahut zyaada energy chahiye, UV light ki flash jaisi). Kyunki spinning, wobbling, aur electron-jumping mein bahut alag-alag energy lagti hai, hum bata sakte hain ki molecule kaun sa colour light "peeta" hai. Spinning lines ke beech ki gaps batati hain spring kitni lambi hai; wobble lines ka paas aana batata hai spring kitni strong hai aur kab tootegi. Jab electrons jump karte hain, itni tez jump karte hain ki balls ko hilne ka time nahi milta — jaise bina blur ki photo — aur yahi "frozen snapshot" rule (Franck–Condon) batata hai ki kaun si wobble mein land hoti hai.

Flashcards

"Rigid rotor" kaun sa model hai aur iska key assumption kya hai?

Diatomic as two masses on a fixed-length massless rod; rotation ke dauran bond length constant rehti hai.

Rotational energy wavenumbers mein

F (J) = B J (J + 1)

with

B = h /8 π^{2} c I

I = μ r^{2}

Adjacent rotational absorption lines ke beech spacing

2 B

(lines at

2 B, 4 B, 6 B, \dots

), toh

Δ J = + 1

deta hai

\tilde{ν} = 2 B (J + 1)

I

mein reduced mass kyun use karte hain?

Dono atoms centre of mass ke baare mein orbit karte hain; two-body problem ek effective mass

μ = m_{1} m_{2} / (m_{1} + m_{2})

mein reduce ho jaata hai.

Harmonic oscillator levels aur

\tilde{ν}_{e}

G (v) = \tilde{ν}_{e} (v + \frac{1}{2})

\tilde{ν}_{e} = \frac{1}{2 π c} k / μ

Zero-point energy kya hai aur nonzero kyun?

\frac{1}{2} \tilde{ν}_{e}

; Heisenberg potential minimum par exactly rest karne se rokta hai.

Morse potential expression

V (r) = D_{e} [1 - e^{- a (r - r_{e})}]^{2}

r \to \infty

par

D_{e}

tak flatten ho jaata hai.

Morse (anharmonic) energy levels

G (v) = \tilde{ν}_{e} (v + \frac{1}{2}) - \tilde{ν}_{e} x_{e} (v + \frac{1}{2})^{2}

; levels converge karte hain.

Harmonic model dissociation ke liye kyun fail karta hai?

Parabola ki infinite walls aur equal spacing hoti hai; toot nahi sakta ya converge nahi kar sakta — Morse ye fix karta hai.

P vs R branch

Δ J = + 1

, higher

\tilde{ν}

; P:

Δ J = - 1

, lower

\tilde{ν}

; Q (

Δ J = 0

) usually forbidden, band origin par gap rehta hai.

R-branch line positions

\tilde{ν}_{R} = \tilde{ν}_{0} + 2 B (J + 1)

P-branch line positions

\tilde{ν}_{P} = \tilde{ν}_{0} - 2 B J

P/R branch asymmetry kaun karata hai?

Centrifugal distortion aur

B_{v} = B_{e} - α_{e} (v + \frac{1}{2})

(rotation–vibration coupling).

Franck–Condon principle batao

Electronic transitions vertical hoti hain (nuclei frozen); intensity sabse zyaada maximum vibrational-wavefunction overlap ke liye hoti hai.

Franck–Condon factor

∣ ⟨ ψ_{v^{'}}^{'} ∣ ψ_{v^{''}}^{''} ⟩ ∣^{2}

, squared vibrational overlap integral.

Energy scales ka order

Electronic

≫

Vibrational

≫

Rotational (eV

≫

0.1 eV

≫

10⁻³ eV).

Pure rotational absorption ke liye selection rule

Δ J = \pm 1

aur molecule mein permanent dipole hona chahiye.

Connections

Born-Oppenheimer Approximation — electronic/nuclear motion ko alag karna justify karta hai.
Reduced Mass and Two-Body Problem — $I = μ r^{2}$ aur $\tilde{ν}_{e}$ ka basis.
Quantum Harmonic Oscillator — vibrational ladder & zero-point energy.
Angular Momentum in QM — $J (J + 1) ℏ^{2}$ eigenvalues.
Heisenberg Uncertainty Principle — zero-point energy kyun exist karta hai.
Dissociation Energy and Birge-Sponer Plot — Morse spacings se $D_{0}$ nikalna.
Boltzmann Distribution — populations $\Rightarrow$ rotational line intensities.