14 May 2026

Franck–Condon Principle

Fixed-nuclear-coordinate electronic transitions and the vibrational overlap factors that shape a molecular band system.

msc semester-ii molecular-spectra franck-condon-principle vibrational-overlap electronic-transitions

Electrons move on a much shorter time scale than nuclei. During an electronic absorption or emission event, the electronic distribution can change while the internuclear separation $R$ has essentially no time to change. This is the Franck–Condon principle:

An electronic transition occurs at fixed nuclear coordinates.

In a potential-energy diagram whose horizontal axis is $R$, such a transition is drawn vertically. “Vertical” means constant $R$; it does not mean that the nuclei move vertically or that energy is unconstrained.

When the two electronic states have different equilibrium bond lengths, a vertical transition from \(v''=0\) can overlap most strongly with an excited vibrational level \(v'>0\).

Quantum-mechanical factorization

In the Born–Oppenheimer approximation, neglecting rotation for the moment,

\[\Psi_i(\mathbf r,R) =\phi_i(\mathbf r;R)\chi_{v''}(R), \qquad \Psi_f(\mathbf r,R) =\phi_f(\mathbf r;R)\chi_{v'}(R).\]

The electric-dipole transition amplitude is

\[\mathbf M_{fi} =\int dR\int d\mathbf r\, \chi_{v'}^*(R)\phi_f^*(\mathbf r;R) \widehat{\boldsymbol\mu} \phi_i(\mathbf r;R)\chi_{v''}(R).\]

Perform the electronic integral first:

\[\mathbf M_{\mathrm e}(R) =\int \phi_f^*(\mathbf r;R) \widehat{\boldsymbol\mu} \phi_i(\mathbf r;R)\,d\mathbf r.\]

Then

\[\mathbf M_{fi} =\int \chi_{v'}^*(R)\, \mathbf M_{\mathrm e}(R)\, \chi_{v''}(R)\,dR.\]

The Condon approximation assumes that the electronic transition moment changes slowly over the range in which the vibrational wavefunctions are appreciable:

\[\mathbf M_{\mathrm e}(R)\simeq\mathbf M_{\mathrm e}(R_0).\]

Here $R_0$ is a representative internuclear separation within that overlap region; it is not a new dynamical coordinate.

It may therefore be taken outside the nuclear integral:

\[\mathbf M_{fi} \simeq \mathbf M_{\mathrm e}(R_0) \left\langle\chi_{v'}\middle|\chi_{v''}\right\rangle.\]

The Franck–Condon factor is

\[\boxed{ q_{v'v''} =\left| \int\chi_{v'}^*(R)\chi_{v''}(R)\,dR \right|^2. }\]

It is a probability-like overlap factor, not an additional energy-selection rule. Energy conservation still fixes the photon frequency.

Exactly displaced harmonic potentials

The origin of a vibrational progression can be seen analytically. Suppose the lower and upper potentials have the same curvature, but their equilibrium positions differ by $d$. Put

\[\mu_{\mathrm r}=\frac{m_1m_2}{m_1+m_2}, \qquad \alpha=\frac{\mu_{\mathrm r}\omega}{\hbar}, \qquad x=R-R_e''.\]

Here $\mu_{\mathrm r}$ is the nuclear reduced mass and $\omega$ is the common angular vibrational frequency of the two harmonic potentials.

Take the lower state to be $v’‘=0$. The normalized functions are

\[\chi_0''(x) =\left(\frac{\alpha}{\pi}\right)^{1/4} e^{-\alpha x^2/2},\]

and

\[\chi_n'(x) =\left(\frac{\alpha}{\pi}\right)^{1/4} \frac{H_n\!\left(\sqrt\alpha(x-d)\right)} {\sqrt{2^n n!}}\, e^{-\alpha(x-d)^2/2}.\]

Let $z=\sqrt\alpha x$ and $\delta=\sqrt\alpha d$. To evaluate the overlaps, use the Hermite generating function

\[\sum_{n=0}^{\infty}\frac{H_n(z-\delta)}{n!}t^n =e^{2(z-\delta)t-t^2}.\]

The Gaussian integral that generates the unnormalized overlap is

\[\begin{aligned} K(t) &=\frac1{\sqrt\pi}\int_{-\infty}^{\infty} e^{-[z^2+(z-\delta)^2]/2} e^{2(z-\delta)t-t^2}\,dz\\ &=\frac1{\sqrt\pi} e^{-\delta^2/2-2\delta t-t^2} \int_{-\infty}^{\infty} e^{-z^2+(\delta+2t)z}\,dz. \end{aligned}\]

Completing the square,

\[\int_{-\infty}^{\infty}e^{-z^2+bz}\,dz =\sqrt\pi\,e^{b^2/4}.\]

With $b=\delta+2t$,

\[\begin{aligned} K(t) &=e^{-\delta^2/2-2\delta t-t^2} e^{(\delta+2t)^2/4}\\ &=e^{-\delta^2/4}e^{-\delta t}. \end{aligned}\]

Comparing equal powers of $t$ and restoring the normalization $1/\sqrt{2^n n!}$ gives

\[\left\langle\chi_n'\middle|\chi_0''\right\rangle =e^{-\delta^2/4} \frac{(-\delta)^n}{\sqrt{2^n n!}}.\]

Define the dimensionless displacement

\[S=\frac{\delta^2}{2} =\frac{\mu_{\mathrm r}\omega d^2}{2\hbar}.\]

The overlap amplitude and factor become

\[\left\langle\chi_n'\middle|\chi_0''\right\rangle =e^{-S/2}\frac{(-\sqrt S)^n}{\sqrt{n!}},\] \[\boxed{ q_{n0}=e^{-S}\frac{S^n}{n!}. }\]

The factors are normalized:

\[\sum_{n=0}^{\infty}q_{n0} =e^{-S}\sum_{n=0}^{\infty}\frac{S^n}{n!} =e^{-S}e^S=1.\]

If $d=0$, then $S=0$ and only $q_{00}=1$. If the minima are displaced, the largest factor occurs near $n\simeq S$, so the strongest band need not end at $v’=0$. For unequal curvatures, the overlaps are no longer this simple Poisson distribution, but the defining integral remains valid.

What determines a measured band intensity

The Franck–Condon factor is only the vibrational part. A measured line or band intensity also contains

\[I_{v'v''J'J''} \propto N_{v''J''}\, \mathcal F(\nu)\, \left|\mathbf M_{\mathrm e}\right|^2 q_{v'v''}\, S_{J'J''}.\]

Here $N_{v’‘J’’}$ is the initial population, $\mathcal F(\nu)$ is the frequency-dependent radiation factor appropriate to absorption or emission, and $S_{J’J’’}$ is the rotational line-strength factor. Thus the Franck–Condon principle explains the vibrational envelope only after the electronic transition is allowed; it does not by itself determine the full observed intensity.

The Gaussian completion, Poisson recurrence, normalization, and zero-displacement limit are checked in the Maxima worksheet.