The Franck-Condon Principle

Derivation

The Franck-Condon Principle

D-313 Home PU-306 Threads light · matter · waves · energy Depends on The Born-Oppenheimer Approximation, Molecular Rotational and Vibrational Spectra, Einstein Coefficients and Spontaneous Emission

Statement

For an electric-dipole vibronic transition between an initial state \( |e', v'\rangle \) and a final state \( |e'', v''\rangle \), the transition dipole matrix element factorises, under the Born–Oppenheimer separation and the Condon approximation, into an electronic transition dipole \( \mathbf{M}(R_0) \) evaluated at a fixed reference geometry times a vibrational overlap integral. The line intensity is therefore \( I \propto |\mathbf{M}(R_0)|^{2}\,\big|\langle \chi_{v''}|\chi_{v'}\rangle\big|^{2} \), and the dimensionless nuclear factor \( q_{v''v'} = \big|\langle \chi_{v''}|\chi_{v'}\rangle\big|^{2} \) — the Franck–Condon factor — is the squared overlap of the vibrational wavefunctions belonging to the two different electronic potential-energy surfaces.

Why it matters

The Franck–Condon principle explains the vibrational structure ("progression") seen in electronic absorption and emission bands of molecules: why a single electronic transition splits into a comb of lines, and why the intensity envelope peaks at some non-trivial \( v'' \) rather than at \( v''=0 \). It is the quantitative expression of the intuitive statement that electronic rearrangement is fast compared with nuclear motion, so the nuclei are effectively frozen during the electronic jump — a "vertical" transition on a potential-energy diagram.

It supplies the nuclear weighting that multiplies the electronic transition moment in the Einstein coefficients, and hence fixes relative band intensities, fluorescence line shapes, non-radiative rates, and the mirror symmetry between absorption and emission spectra.

Assumptions

Born–Oppenheimer separability of each state.If dropped, the total state cannot be written as \( \psi_e(\mathbf{r};R)\,\chi_{ev}(R) \); the transition moment no longer splits into an electronic factor times a nuclear overlap, and vibronic intensity must be computed with fully coupled nuclear–electronic states. Electric-dipole approximation for the light–matter coupling.If dropped, magnetic-dipole and electric-quadrupole terms enter; the operator is no longer \( \hat{\boldsymbol{\mu}} \) and the selection/intensity analysis changes. Condon approximation: the electronic transition dipole \( \mathbf{M}(R) \) is essentially constant over the range of \( R \) spanned by the overlapping vibrational wavefunctions.If dropped, \( \mathbf{M}(R) \) must be expanded about \( R_0 \) (Herzberg–Teller coupling); the linear term lets nominally forbidden transitions "borrow" intensity, and the clean squared-overlap form fails. Orthogonality of the electronic states at every fixed nuclear geometry, \( \int \psi_{e''}^{*}(\mathbf{r};R)\,\psi_{e'}(\mathbf{r};R)\,d\mathbf{r} = \delta_{e''e'} \).If dropped (non-orthogonal or non-adiabatic electronic manifolds, e.g. near a conical intersection), the nuclear-dipole term no longer vanishes and the derivation's cancellation is lost.

Derivation

\[ |\Psi_{e v}\rangle = \psi_e(\mathbf{r};R)\,\chi_{ev}(R) \]

Write each vibronic state as an electronic wavefunction (parametric in the nuclear coordinates \( R \)) times a nuclear vibrational wavefunction, per the Born–Oppenheimer approximation. Rotation is suppressed for clarity. A

\[ \boldsymbol{\mu}_{fi} = \langle \Psi_{e'' v''}\,|\,\hat{\boldsymbol{\mu}}\,|\,\Psi_{e' v'}\rangle, \qquad I \propto \omega^{3}\,|\boldsymbol{\mu}_{fi}|^{2} \]

The electric-dipole transition rate (Einstein \( A \) for emission, \( B \) for absorption) is governed by the modulus-squared of the transition dipole matrix element; this is the quantity we must evaluate. A

\[ \hat{\boldsymbol{\mu}} = \hat{\boldsymbol{\mu}}_{\mathrm{el}}(\mathbf{r}) + \hat{\boldsymbol{\mu}}_{\mathrm{nuc}}(R), \qquad \hat{\boldsymbol{\mu}}_{\mathrm{el}} = -e\sum_i \mathbf{r}_i, \quad \hat{\boldsymbol{\mu}}_{\mathrm{nuc}} = +e\sum_\alpha Z_\alpha \mathbf{R}_\alpha \]

Split the total dipole operator into its electronic part (depending only on electron coordinates) and its nuclear part (depending only on nuclear coordinates). This is an exact algebraic decomposition. A

\[ \boldsymbol{\mu}_{fi} = \int d R\; \chi_{v''}^{*}(R) \left[ \int d\mathbf{r}\; \psi_{e''}^{*}\,\hat{\boldsymbol{\mu}}_{\mathrm{el}}\,\psi_{e'} \right] \chi_{v'}(R) \;+\; \int d R\; \chi_{v''}^{*}(R)\,\hat{\boldsymbol{\mu}}_{\mathrm{nuc}}(R) \left[ \int d\mathbf{r}\; \psi_{e''}^{*}\,\psi_{e'} \right] \chi_{v'}(R) \]

Substitute the factorised states of Step 1 and the operator of Step 3, then order the integrations: the electronic integral over \( \mathbf{r} \) is performed at fixed \( R \), the nuclear integral over \( R \) outside. The nuclear operator \( \hat{\boldsymbol{\mu}}_{\mathrm{nuc}}(R) \) commutes past the \( \mathbf{r} \)-integral because it does not act on electrons. B

\[ \int d\mathbf{r}\; \psi_{e''}^{*}(\mathbf{r};R)\,\psi_{e'}(\mathbf{r};R) = \delta_{e''e'} = 0 \quad (e'' \neq e') \;\;\Longrightarrow\;\; \text{nuclear term} = 0 \]

For a genuine electronic transition \( e'' \neq e' \), the electronic states are orthogonal at every fixed geometry, so the bracket multiplying the nuclear-dipole term vanishes identically. The permanent nuclear dipole cannot drive an electronic transition. B

\[ \mathbf{M}(R) \equiv \int d\mathbf{r}\; \psi_{e''}^{*}(\mathbf{r};R)\,\hat{\boldsymbol{\mu}}_{\mathrm{el}}\,\psi_{e'}(\mathbf{r};R), \qquad \boldsymbol{\mu}_{fi} = \int d R\; \chi_{v''}^{*}(R)\,\mathbf{M}(R)\,\chi_{v'}(R) \]

Define the electronic transition dipole moment \( \mathbf{M}(R) \), a function of the nuclear geometry through the parametric dependence of the electronic states. Only the electronic term of Step 4 survives. B

\[ \mathbf{M}(R) \approx \mathbf{M}(R_0) \quad\Longrightarrow\quad \boldsymbol{\mu}_{fi} \approx \mathbf{M}(R_0)\int d R\; \chi_{v''}^{*}(R)\,\chi_{v'}(R) = \mathbf{M}(R_0)\,\langle \chi_{v''}|\chi_{v'}\rangle \]

Condon approximation: \( \mathbf{M}(R) \) varies slowly on the scale of nuclear zero-point motion, so evaluate it at a representative geometry \( R_0 \) (typically the ground-state equilibrium) and take it outside the nuclear integral. What remains is a pure vibrational overlap. C

\[ I \propto \omega^{3}\,|\boldsymbol{\mu}_{fi}|^{2} = \omega^{3}\,|\mathbf{M}(R_0)|^{2}\,\big|\langle \chi_{v''}|\chi_{v'}\rangle\big|^{2}, \qquad q_{v''v'} \equiv \big|\langle \chi_{v''}|\chi_{v'}\rangle\big|^{2} \]

Insert into the intensity expression of Step 2. The electronic factor \( |\mathbf{M}(R_0)|^2 \) sets the overall electronic band strength; the vibrational structure is carried entirely by the Franck–Condon factor \( q_{v''v'} \). A

Result

\[ \boxed{\; q_{v''v'} = \big|\langle \chi_{v''}|\chi_{v'}\rangle\big|^{2} = \left| \int \chi_{v''}^{*}(R)\,\chi_{v'}(R)\,dR \right|^{2}, \qquad I_{v''v'} \propto \omega^{3}\,|\mathbf{M}(R_0)|^{2}\,q_{v''v'} \;} \]

Reading. The intensity of a vibronic line is the electronic transition strength times the squared overlap of the two vibrational wavefunctions. Crucially, \( \chi_{v'} \) and \( \chi_{v''} \) live on different electronic potential-energy surfaces — generally with different equilibrium geometries and force constants — so they are not orthogonal, and the overlap can be large for \( v'' \neq v' \). The most intense line corresponds to the upper vibrational state whose wavefunction has greatest amplitude directly above the lower state's equilibrium geometry: the "vertical" transition.

Units check. The vibrational wavefunctions are normalised, \( \int |\chi|^2 dR = 1 \), so \( \langle \chi_{v''}|\chi_{v'}\rangle \) is dimensionless and \( q_{v''v'} \in [0,1] \) is a pure number. The completeness sum \( \sum_{v''} q_{v''v'} = 1 \) confirms it is a probability weighting. The dimensional content of the intensity resides entirely in \( \omega^3 |\mathbf{M}(R_0)|^2 \), with \( [\mathbf{M}] = \mathrm{C\,m} \).

Limiting cases

Identical surfaces (same equilibrium \( R_0 \), same \( \omega \)): \( \langle \chi_{v''}|\chi_{v'}\rangle = \delta_{v''v'} \), so only the \( \Delta v = 0 \) (0–0) line appears — no progression.
Small displacement, equal frequency (Huang–Rhys \( S \ll 1 \)): the 0–0 band dominates and \( q_{0v''} = e^{-S}S^{v''}/v''! \) — a Poisson progression of mean \( S \).
Large displacement (\( S \gg 1 \)): a broad progression peaking near \( v'' \approx S \), reproducing the classical vertical transition to the upper-state turning point.
Frequency change, no displacement: squeezing/stretching of the vibrational wavefunction alone gives a non-trivial 0–0 factor \( q_{00} = 2\sqrt{\alpha_1\alpha_2}/(\alpha_1+\alpha_2) < 1 \) with \( \alpha = \mu\omega/\hbar \).
Sum rule: \( \sum_{v''} q_{v''v'} = 1 \) fixes the total electronic band intensity independent of how it is distributed among vibrational lines.

Breaks when

The Condon approximation fails. If \( \mathbf{M}(R_0) \approx 0 \) (a symmetry-forbidden transition) or \( \mathbf{M}(R) \) varies rapidly, the leading term is inadequate; the linear expansion \( \mathbf{M}(R) \approx \mathbf{M}(R_0) + (\partial \mathbf{M}/\partial R)_{R_0}(R-R_0) \) (Herzberg–Teller) governs the intensity, and vibronic "intensity borrowing" activates non-totally-symmetric modes.
Born–Oppenheimer breaks down. Near a conical intersection or avoided crossing the electronic states mix through nuclear kinetic coupling; the state cannot be factorised and the electronic-orthogonality cancellation of Step 5 is invalid (non-adiabatic dynamics).
The upper surface is repulsive / dissociative. The final vibrational states form a continuum, so discrete Franck–Condon factors are replaced by a continuous reflection of \( |\chi_{v'}|^2 \) onto the repulsive wall (direct dissociation, predissociation), giving a broad structureless band.
Strong rotational–vibronic coupling. When rotation is not separable (large Coriolis/centrifugal effects, Renner–Teller), the neglected rotational wavefunctions reshape the line strengths and the pure vibrational overlap is no longer the whole nuclear factor.

Failure modes

Assuming orthogonality across surfaces: writing \( \langle \chi_{v''}|\chi_{v'}\rangle = \delta_{v''v'} \). The two vibrational functions belong to different potentials (shifted \( R_0 \), different \( \omega \)) and are not orthogonal — that non-orthogonality is the entire point.
Importing the \( \Delta v = \pm 1 \) rule: there is no harmonic \( \Delta v \) selection rule for vibronic transitions; every \( v'' \) is allowed, weighted only by \( q_{v''v'} \).
Forgetting to square: reporting \( \langle \chi_{v''}|\chi_{v'}\rangle \) as the intensity instead of its modulus squared, or dropping the modulus for complex/real-sign-changing overlaps.
Confusing the two factors: attributing vibrational intensity variation to \( \mathbf{M}(R_0) \), or attributing the overall electronic strength to \( q_{v''v'} \). They are distinct: electronic vs nuclear.
Peaking at 0–0 by default: assuming the strongest line is always \( v''=0 \); with displaced surfaces the maximum sits near \( v'' \approx S \).
Neglecting the \( \omega^3 \) (or \( \omega \)) frequency factor when comparing lines spread over a wide spectral range, especially in emission.

Discussion

The physical content is a separation of timescales. An electronic transition, driven by the fast-oscillating optical field and rearranging light electrons, is essentially instantaneous compared with the slow, heavy nuclei. On a potential-energy diagram the transition is therefore drawn as a vertical line at fixed \( R \): the nuclear coordinate and momentum are unchanged at the instant of the electronic jump. Quantum-mechanically this "verticality" is encoded not as a delta function but as the overlap \( \langle \chi_{v''}|\chi_{v'}\rangle \), which is largest when the upper vibrational wavefunction has maximum amplitude — classically, a turning point where the nuclei move slowly — directly above the lower state's equilibrium geometry.

The Franck–Condon factor is the nuclear weighting that the Born–Oppenheimer approximation inserts between the electronic transition moment and the observed intensity. It connects directly to the Einstein coefficients derived earlier: for spontaneous emission \( A_{v''v'} \propto \omega_{v''v'}^{3}\,|\mathbf{M}(R_0)|^{2}\,q_{v''v'} \), so the vibrational progression in a fluorescence spectrum is a direct readout of overlaps. Because emission from \( v'=0 \) and absorption to \( v''=0 \) sample the same set of overlaps in reverse, absorption and emission bands display an approximate mirror symmetry about the 0–0 line whenever the two surfaces have similar force constants.

Temperature enters through the population of the initial vibrational levels: at finite \( T \), lines originating from thermally populated \( v' > 0 \) ("hot bands") appear on the low-energy side, each carrying its own set of Franck–Condon factors weighted by the Boltzmann factor \( e^{-E_{v'}/k_B T} \). The displacement between surfaces is quantified by the dimensionless Huang–Rhys factor \( S = \mu\omega\,(\Delta R)^2/(2\hbar) \), which equals the mean number of vibrational quanta excited and controls the peak position and reorganisation energy \( \lambda = S\hbar\omega \) central to electron-transfer theory.

A deeper, equivalent formulation is time-dependent. Writing the absorption cross-section as the Fourier transform of the nuclear autocorrelation function \( C(t) = \langle \chi_{v'}\,|\,e^{-i\hat{H}_{e''}t/\hbar}\,|\,\chi_{v'}\rangle \) (with \( \mathbf{M}(R_0) \) factored out), the Franck–Condon factors are recovered as the discrete Fourier components at the vibronic frequencies. In this picture the initial vibrational wavepacket is placed on the upper surface at \( t=0 \) and its recurrences with the ground state generate the spectrum; the Condon approximation is precisely the statement that \( \mathbf{M} \) can be pulled outside the time evolution. This unifies bound-state progressions and continuum (dissociative) reflection spectra in a single expression.

Common misconceptions. The Franck–Condon principle does not say the transition itself takes literally zero time, nor that \( \Delta v \) is fixed; it says the nuclear coordinate and momentum are conserved through the transition, so intensity flows to whichever upper level best matches the frozen nuclear configuration. The overlap being non-zero for \( v'' \neq v' \) is not a violation of orthogonality — the states are orthogonal only within a single surface, and here they belong to different surfaces.

Worked examples

Displaced harmonic surfaces — a Franck–Condon progression. A diatomic has upper-state vibrational wavenumber \( \tilde{\nu} = 300\ \mathrm{cm^{-1}} \), reduced mass \( \mu = 1.0\times10^{-25}\ \mathrm{kg} \), and an equilibrium bond-length shift between the two electronic states of \( \Delta R = 0.10\ \text{Å} = 1.0\times10^{-11}\ \mathrm{m} \). The two surfaces have equal force constants. Find the Huang–Rhys factor and the Franck–Condon factors \( q_{0,v''} \) for the \( v'=0 \) progression, and identify the most intense line. B

\[ \omega = 2\pi c\,\tilde{\nu} = 2\pi (2.998\times10^{10}\ \mathrm{cm\,s^{-1}})(300\ \mathrm{cm^{-1}}) = 5.65\times10^{13}\ \mathrm{rad\,s^{-1}} \]

Convert the spectroscopic wavenumber to angular frequency. Symbolic form first, numbers second. A

\[ S = \frac{\mu\,\omega\,(\Delta R)^2}{2\hbar} = \frac{(1.0\times10^{-25})(5.65\times10^{13})(1.0\times10^{-11})^2}{2(1.055\times10^{-34})} = \frac{5.65\times10^{-34}}{2.11\times10^{-34}} = 2.68 \]

The Huang–Rhys factor for equal-frequency displaced oscillators; dimensionless (kg · s⁻¹ · m² / (J·s) = 1). B

\[ q_{0,v''} = e^{-S}\,\frac{S^{v''}}{v''!}, \qquad e^{-2.68} = 0.0686 \]

Poisson progression for the \( v'=0 \) initial level of displaced harmonic surfaces (standard displaced-oscillator overlap). B

\[ q_{0,0}=0.069,\; q_{0,1}=0.184,\; q_{0,2}=0.246,\; q_{0,3}=0.220,\; q_{0,4}=0.147 \]

Evaluate the Poisson weights; the maximum is at \( v'' = \lfloor S \rfloor = 2 \). Partial sum \( \approx 0.87 \), the remainder in higher \( v'' \). A

\[ S = 2.68, \qquad q_{0,v''} = e^{-2.68}\,\frac{2.68^{\,v''}}{v''!}, \qquad v''_{\max} = 2 \]

Reading. The absorption band is not a single line but a comb of five main lines; the strongest is the 0→2 transition, not 0→0, a direct signature of the geometry change on electronic excitation. Units: all \( q \) dimensionless, summing toward 1.

Frequency change, no displacement — the 0–0 factor. Two electronic surfaces share the same equilibrium geometry, but the upper-state vibrational frequency is twice the lower, \( \omega_2 = 2\omega_1 \). Compute the 0–0 Franck–Condon factor from the overlap of the two ground-state Gaussians. B

\[ \chi_0^{(i)}(R) = \left(\frac{\alpha_i}{\pi}\right)^{1/4} e^{-\alpha_i R^2/2}, \qquad \alpha_i = \frac{\mu\,\omega_i}{\hbar} \]

Normalised harmonic ground states centred at the common origin; the width parameter \( \alpha_i \) scales with frequency. A

\[ \langle \chi_0^{(2)}|\chi_0^{(1)}\rangle = \left(\frac{\alpha_1\alpha_2}{\pi^2}\right)^{1/4}\!\!\int_{-\infty}^{\infty}\! e^{-(\alpha_1+\alpha_2)R^2/2}\,dR = \left(\frac{\alpha_1\alpha_2}{\pi^2}\right)^{1/4}\sqrt{\frac{2\pi}{\alpha_1+\alpha_2}} \]

Gaussian integral \( \int e^{-a R^2/2}dR = \sqrt{2\pi/a} \) with \( a = \alpha_1+\alpha_2 \). B

\[ q_{00} = \big|\langle \chi_0^{(2)}|\chi_0^{(1)}\rangle\big|^{2} = \frac{2\sqrt{\alpha_1\alpha_2}}{\alpha_1+\alpha_2} \]

Square and simplify; a symmetric function of the two widths, equal to 1 only when \( \alpha_1=\alpha_2 \). B

\[ \alpha_2 = 2\alpha_1 \;\Rightarrow\; q_{00} = \frac{2\sqrt{\alpha_1\cdot 2\alpha_1}}{\alpha_1+2\alpha_1} = \frac{2\sqrt{2}\,\alpha_1}{3\alpha_1} = \frac{2\sqrt{2}}{3} = 0.943 \]

Insert the frequency ratio; \( \alpha_1 \) cancels, leaving a pure number. A

\[ q_{00} = \frac{2\sqrt{\alpha_1\alpha_2}}{\alpha_1+\alpha_2} \xrightarrow{\ \omega_2 = 2\omega_1\ } \frac{2\sqrt{2}}{3} \approx 0.94 \]

Reading. Even with no bond-length change, a change of vibrational frequency alone reduces the 0–0 overlap below unity (here to 0.94) and lends a little intensity to \( \Delta v \) even quanta. Units: dimensionless, as required for a Franck–Condon factor.

Problems

Show explicitly that \( \sum_{v''} q_{v''v'} = 1 \) for any fixed \( v' \).
Solution
Using completeness of the upper-surface vibrational basis, \( \sum_{v''} |\chi_{v''}\rangle\langle \chi_{v''}| = \hat{1} \). Then \( \sum_{v''} q_{v''v'} = \sum_{v''} \langle \chi_{v'}|\chi_{v''}\rangle\langle \chi_{v''}|\chi_{v'}\rangle = \langle \chi_{v'}|\Big(\sum_{v''}|\chi_{v''}\rangle\langle \chi_{v''}|\Big)|\chi_{v'}\rangle = \langle \chi_{v'}|\chi_{v'}\rangle = 1 \), since \( \chi_{v'} \) is normalised. The total electronic band intensity is therefore fixed regardless of how it is distributed over vibrational lines.
For the displaced-oscillator progression \( q_{0,v''} = e^{-S}S^{v''}/v''! \), find the value of \( v'' \) that maximises the intensity, and confirm your rule for the case \( S = 2.68 \) of Worked Example 1.
Solution
Take the ratio of successive terms: \( q_{0,v''+1}/q_{0,v''} = S/(v''+1) \). This exceeds 1 (intensity still rising) while \( v'' + 1 < S \), i.e. \( v'' < S-1 \), and drops below 1 for \( v'' > S-1 \). The maximum is at \( v''_{\max} = \lfloor S \rfloor \) (and both \( S-1 \) and \( S \) tie when \( S \) is an integer). For \( S = 2.68 \), \( v''_{\max} = 2 \), matching the computed \( q_{0,2} = 0.246 \) being the largest.
A transition has electronic transition dipole \( |\mathbf{M}(R_0)| = 2.5\ \mathrm{D} \) (1 D \( = 3.336\times10^{-30}\ \mathrm{C\,m} \)) and 0–0 Franck–Condon factor \( q_{00} = 0.30 \). Taking the \( \omega^3 \) factor as common to neighbouring lines, find the ratio of the 0→1 to the 0→0 line intensities if \( q_{01} = 0.36 \). Does the electronic dipole enter the ratio?
Solution
Within a narrow progression \( \omega^3 \) and \( |\mathbf{M}(R_0)|^2 \) are common to both lines, so \( I_{01}/I_{00} = q_{01}/q_{00} = 0.36/0.30 = 1.2 \). The electronic transition dipole cancels in the ratio and does not appear — it sets the absolute band strength, not the relative vibrational intensities. (Its value 2.5 D is a distractor for the ratio.)
Estimate the Huang–Rhys factor \( S \) for a molecule whose electronic excitation shifts the equilibrium bond length by \( \Delta R = 0.05\ \text{Å} \), with reduced mass \( \mu = 1.14\times10^{-26}\ \mathrm{kg} \) (roughly CO) and vibrational wavenumber \( \tilde{\nu} = 1740\ \mathrm{cm^{-1}} \) (an excited-state value). Is the 0–0 or a higher line strongest?
Solution
\( \omega = 2\pi c\tilde{\nu} = 2\pi(2.998\times10^{10})(1740) = 3.28\times10^{14}\ \mathrm{rad\,s^{-1}} \). With \( \Delta R = 5.0\times10^{-12}\ \mathrm{m} \): \( S = \mu\omega(\Delta R)^2/(2\hbar) = (1.14\times10^{-26})(3.28\times10^{14})(2.5\times10^{-23})/(2.11\times10^{-34}) \). Numerator \( = 1.14\times10^{-26}\times3.28\times10^{14} = 3.74\times10^{-12} \); \( \times 2.5\times10^{-23} = 9.35\times10^{-35} \). So \( S = 9.35\times10^{-35}/2.11\times10^{-34} = 0.44 \). Since \( S < 1 \), \( v''_{\max} = 0 \): the 0–0 band is strongest, with \( q_{00} = e^{-0.44} = 0.64 \) and \( q_{01} = 0.44\,e^{-0.44} = 0.28 \).
Explain, using the structure of the derivation, why a symmetry-forbidden electronic transition (\( \mathbf{M}(R_0) = 0 \)) can nevertheless show weak but non-zero vibronic bands, and which assumption must be relaxed. What vibrational modes become active?
Solution
The clean result relies on the Condon approximation of Step 7, \( \mathbf{M}(R)\approx\mathbf{M}(R_0) \). If \( \mathbf{M}(R_0)=0 \) by symmetry, the leading term vanishes and one must keep the next term of the Taylor expansion, \( \mathbf{M}(R) \approx \mathbf{M}(R_0) + (\partial\mathbf{M}/\partial R)_{R_0}(R-R_0) \) — the Herzberg–Teller correction. The transition moment then becomes \( (\partial\mathbf{M}/\partial R)_{R_0}\langle\chi_{v''}|(R-R_0)|\chi_{v'}\rangle \), which is non-zero for modes that lower the molecular symmetry so as to make the derivative symmetry-allowed. These non-totally-symmetric ("promoting" or "inducing") modes borrow intensity from nearby allowed electronic states, producing weak false-origin progressions built on \( \Delta v = 1 \) of the active mode rather than on the true 0–0. Thus the Born–Oppenheimer factorisation is retained, but the Condon approximation is relaxed.