Franck–Condon overlap factors

Two harmonic potential energy surfaces, a ground state g and an excited state e. Set the two vibrational frequencies, the displacement and the starting level; the page recalculates every overlap ⟨χ′v′|χ″v″⟩, the Franck–Condon factors and their share of the whole transition. Equations are at the bottom.

Inputs

cm⁻¹ cm⁻¹ Å k = μω² follows from itamu v′ = 0 … this − 1 (table rows)
Examples:

Potential energy surfaces and overlaps

Vertical arrows are the Franck–Condon (vertical) excitations from v″; each arrow's width follows its FCF. Every excited level is shaded with its overlap integrand χ′v′χ″v″ (blue where it adds, red where it cancels), all on one scale; the selected level (click a level, a table row, or use the menu below) is shaded darker. The excited state sits Te above the ground state; that gap is not to scale (axis break). Click a ground level to start from it (a hot state).

    Overlap integrand

    The overlap is the net area under χ′χ″ (blue minus red). Equal blue and red areas give zero: the two wavefunctions are orthogonal.

    Vibronic progression

    Each line sits at ν̃ − ν̃₀₀ = v′ωe − v″ωg with height FCF: the band shape of the electronic transition in the harmonic, Condon approximation. Click a line to select it.

    Equations

    Notation follows J. L. McHale, Molecular Spectroscopy, 2nd ed. (CRC Press, 2017): ″ marks the ground state (g), ′ the excited state (e); q = R − R₀ is the ground-state displacement coordinate.

    1. Harmonic potentials and force constants

      V_g(R) = \tfrac12 \mu\omega_g^2 (R-R_0)^2, \qquad V_e(R) = T_e + \tfrac12 \mu\omega_e^2 (R-R_0-d)^2, \qquad k = \mu\omega^2 = \mu\,(2\pi c\,\tilde\omega)^2

    2. Vibrational energy levels

      E_v = \hbar\omega\left(v+\tfrac12\right), \qquad G(v) = \tilde\omega\left(v+\tfrac12\right)\ \text{in cm}^{-1}

    3. Harmonic-oscillator wavefunctions

      \chi_v(q) = N_v\, H_v\!\left(\alpha^{1/2} q\right) e^{-\alpha q^2/2}, \qquad \alpha = \frac{\mu\omega}{\hbar}, \qquad N_v = \left(\frac{\alpha^{1/2}}{2^v\, v!\, \pi^{1/2}}\right)^{1/2}
      H_0 = 1,\quad H_1 = 2y,\quad H_2 = 4y^2-2,\quad H_3 = 8y^3-12y,\qquad H_{v+1}(y) = 2y\,H_v(y) - 2v\,H_{v-1}(y)

      In laboratory units, α [Å⁻²] = μ [amu] · ω̃ [cm⁻¹] / 33.715. The excited-state functions are centred at q = d.

    4. Franck–Condon factor (Condon approximation)

      \langle \psi_e \chi'_{v'} | \hat\mu | \psi_g \chi''_{v''} \rangle \approx \mu_{eg}\, \langle \chi'_{v'} | \chi''_{v''} \rangle, \qquad I_{v' \leftarrow v''} \propto |\mu_{eg}|^2\, \mathrm{FC}_{v'v''}, \qquad \mathrm{FC}_{v'v''} = \left| \int_{-\infty}^{\infty} \chi'_{v'}(q)\, \chi''_{v''}(q)\, dq \right|^2

    5. Sum rule, the "% of total" column

      \sum_{v'=0}^{\infty} \mathrm{FC}_{v'v''} = \sum_{v'} \langle \chi''_{v''} | \chi'_{v'} \rangle \langle \chi'_{v'} | \chi''_{v''} \rangle = \langle \chi''_{v''} | \chi''_{v''} \rangle = 1 \quad\Rightarrow\quad \%\ \text{of total} = 100\, \mathrm{FC}_{v'v''}

      The excited-state functions form a complete set, so the whole transition from v″ adds up to 100 %. What the shown rows miss goes to higher v′.

    6. Parity selection rule (d = 0)

      H_v(-y) = (-1)^v H_v(y) \;\Rightarrow\; \chi_v(-q) = (-1)^v \chi_v(q), \qquad d = 0:\ \ \chi'_{v'}\chi''_{v''}\ \text{has parity}\ (-1)^{v'+v''}
      v'+v''\ \text{odd}:\quad \int_{-\infty}^{\infty} (\text{even})\times(\text{odd})\, dq = 0 \quad\Rightarrow\quad \text{parity forbidden}

      With d ≠ 0 the integrand has no definite parity, so no transition is parity forbidden.

    7. Orthogonality (identical states)

      \omega_e = \omega_g,\ d = 0:\quad \langle \chi'_{v'} | \chi''_{v''} \rangle = \delta_{v'v''}

      Both sets are eigenfunctions of the same Hamiltonian, so different levels are orthogonal and only v′ = v″ survives. The table also marks an overlap "≈ orthogonal" when its FCF falls below 10⁻⁶ for any other reason (nearly identical states, or a node in the overlap at these parameters).

    8. Same R₀, different frequencies, from v″ = 0 (McHale's problem)

      \langle \chi'_0 | \chi''_0 \rangle = N'_0 N''_0 \int_{-\infty}^{\infty} e^{-\bar\alpha q^2} dq = N'_0 N''_0 \sqrt{\frac{\pi}{\bar\alpha}}, \qquad \bar\alpha = \frac{\alpha_g+\alpha_e}{2}
      = \left(\frac{\alpha_g^{1/2}\alpha_e^{1/2}}{\pi}\right)^{1/2} \left(\frac{2\pi}{\alpha_g+\alpha_e}\right)^{1/2} = \left(\frac{2(\omega_g\omega_e)^{1/2}}{\omega_g+\omega_e}\right)^{1/2} \quad\Rightarrow\quad \mathrm{FC}_{00} = \frac{2(\omega_g\omega_e)^{1/2}}{\omega_g+\omega_e}
      \mathrm{FC}_{10} = 0 \quad (\text{parity: } \textstyle\int \text{even}\times\text{odd} = 0)
      \langle \chi'_2 | \chi''_0 \rangle = N'_2 N''_0 \int_{-\infty}^{\infty} \left(4\alpha_e q^2 - 2\right) e^{-\bar\alpha q^2} dq, \qquad \int_{-\infty}^{\infty} q^2 e^{-\bar\alpha q^2} dq = \frac{1}{2\bar\alpha}\sqrt{\frac{\pi}{\bar\alpha}}
      = \left(\frac{(\omega_g\omega_e)^{1/2}}{\omega_e+\omega_g}\right)^{1/2} \frac{\omega_e-\omega_g}{\omega_e+\omega_g} \quad\Rightarrow\quad \mathrm{FC}_{20} = \frac{(\omega_g\omega_e)^{1/2}}{\omega_e+\omega_g} \left(\frac{\omega_e-\omega_g}{\omega_e+\omega_g}\right)^2
      \text{in general:}\quad \langle \chi'_{2n} | \chi''_0 \rangle = \left(\frac{2(\omega_g\omega_e)^{1/2}}{\omega_g+\omega_e}\right)^{1/2} \frac{\sqrt{(2n)!}}{2^n\, n!} \left(\frac{\omega_e-\omega_g}{\omega_e+\omega_g}\right)^{n}, \qquad \langle \chi'_{2n+1} | \chi''_0 \rangle = 0

      FC₀₀ is 1 at ωe = ωg (the two functions overlap completely) and falls as the frequencies separate. FC₂₀ is zero at ωe = ωg (orthogonality) and grows with the frequency difference: only even v′ appear, a progression in steps of 2ωe.

    9. Displaced, same frequency, from v″ = 0 (Poisson progression)

      \mathrm{FC}_{v'0} = e^{-S}\, \frac{S^{v'}}{v'!}, \qquad S = \frac{\alpha d^2}{2} = \frac{\Delta^2}{2}\ \ (\text{Huang--Rhys factor}), \qquad \Delta = \alpha^{1/2} d

    10. General case: displaced and different frequencies, any v″ (the table's analytical column)

      \langle \chi'_{m} | \chi''_{n} \rangle = I_{00} \sqrt{\frac{m!\, n!}{2^{m+n}}} \sum_{k=0}^{\min(m,n)} \frac{\gamma^k}{k!}\, P_{m-k}\, Q_{n-k}
      P_j = \sum_{i=0}^{\lfloor j/2 \rfloor} \frac{\rho^{\,i}\, b'^{\,j-2i}}{i!\,(j-2i)!}, \qquad Q_j = \sum_{i=0}^{\lfloor j/2 \rfloor} \frac{(-\rho)^{i}\, b''^{\,j-2i}}{i!\,(j-2i)!}
      \rho = \frac{\omega_e-\omega_g}{\omega_e+\omega_g}, \quad \gamma = \frac{4(\omega_g\omega_e)^{1/2}}{\omega_g+\omega_e} = 2\sqrt{1-\rho^2}, \quad b' = -\frac{2\alpha_e^{1/2}\alpha_g\, d}{\alpha_g+\alpha_e}, \quad b'' = \frac{2\alpha_g^{1/2}\alpha_e\, d}{\alpha_g+\alpha_e}
      I_{00} = \langle \chi'_0 | \chi''_0 \rangle = \left(\frac{2(\omega_g\omega_e)^{1/2}}{\omega_g+\omega_e}\right)^{1/2} \exp\!\left(-\frac{\alpha_g\alpha_e\, d^2}{2(\alpha_g+\alpha_e)}\right)

      Derivation: insert the Hermite generating function Σv Hv(y) sv/v! = e2ys − s² for both states and do the Gaussian integral over q:

      \int_{-\infty}^{\infty} e^{2\alpha_e^{1/2}(q-d)s - s^2}\, e^{2\alpha_g^{1/2} q t - t^2}\, e^{-\alpha_e (q-d)^2/2 - \alpha_g q^2/2}\, dq = \sqrt{\frac{\pi}{\bar\alpha}}\; e^{-\alpha_g\alpha_e d^2/2(\alpha_g+\alpha_e)}\; e^{\rho s^2 - \rho t^2 + b' s + b'' t + \gamma s t}

      The coefficient of smtn, times N′mN″n m! n!, is the overlap above. With d = 0 (b′ = b″ = 0) it reduces to eq. 8; with ωe = ωg (ρ = 0, γ = 2) and n = 0 to eq. 9.

    11. Numerical check and transition wavenumbers

      \langle \chi'_{v'} | \chi''_{v''} \rangle \approx h \sum_i{}' \chi'_{v'}(q_i)\, \chi''_{v''}(q_i) \quad (\text{trapezoid rule, 6001 points}), \qquad \tilde\nu_{v'v''} - \tilde\nu_{00} = v'\tilde\omega_e - v''\tilde\omega_g

      The page evaluates eq. 10 for every row and also integrates the two wavefunctions numerically; the note under the table gives the largest difference.