2389:. Franck–Condon principles can be applied when the interactions between the chromophore and the surrounding solvent molecules are different in the ground and in the excited electronic state. This change in interaction can originate, for example, due to different dipole moments in these two states. If the chromophore starts in its ground state and is close to equilibrium with the surrounding solvent molecules and then absorbs a photon that takes it to the excited state, its interaction with the solvent will be far from equilibrium in the excited state. This effect is analogous to the original Franck–Condon principle: the electronic transition is very fast compared with the motion of nuclei—the rearrangement of solvent molecules in the case of solvation. We now speak of a vertical transition, but now the horizontal coordinate is solvent-solute interaction space. This coordinate axis is often labeled as "Solvation Coordinate" and represents, somewhat abstractly, all of the relevant dimensions of motion of all of the interacting solvent molecules.
1712:
2420:
equilibrium. The rearrangement of the solvent molecules according to the new potential energy curve is represented by the curved arrows in Figure 7. Note that while the electronic transitions are quantized, the chromophore-solvent interaction energy is treated as a classical continuum due to the large number of molecules involved. Although emission is depicted as taking place from the minimum of the excited state chromophore-solvent interaction potential, significant emission can take place before equilibrium is reached when the viscosity of the solvent is high, or the lifetime of the excited state is short. The energy difference between absorbed and emitted photons depicted in Figure 7 is the solvation contribution to the
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1707:{\displaystyle =\underbrace {\int \psi _{v}'^{*}\psi _{v}\,d\tau _{n}} _{\displaystyle {{\text{Franck–Condon}} \atop {\text{factor}}}}\underbrace {\int \psi _{e}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\,d\tau _{e}} _{\displaystyle {{\text{orbital}} \atop {\text{selection rule}}}}\underbrace {\int \psi _{s}'^{*}\psi _{s}\,d\tau _{s}} _{\displaystyle {{\text{spin}} \atop {\text{selection rule}}}}+\underbrace {\int \psi _{e}'^{*}\psi _{e}\,d\tau _{e}} _{\displaystyle 0}\int \psi _{v}'^{*}{\boldsymbol {\mu }}_{N}\psi _{v}\,d\tau _{v}\int \psi _{s}'^{*}\psi _{s}\,d\tau _{s}.}
190:. High-energy photon absorption leads to a transition to a higher electronic state instead of dissociation. In examining how much vibrational energy a molecule could acquire when it is excited to a higher electronic level, and whether this vibrational energy could be enough to immediately break apart the molecule, he drew three diagrams representing the possible changes in binding energy between the lowest electronic state and higher electronic states.
186:
photoproducts in a single step, the absorption of a photon, and without a collision. In order for a molecule to break apart, it must acquire from the photon a vibrational energy exceeding the dissociation energy, that is, the energy to break a chemical bond. However, as was known at the time, molecules will only absorb energy corresponding to allowed quantum transitions, and there are no vibrational levels above the dissociation energy level of the
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132:= 0 vibrational level of the ground electronic state and upon absorbing a photon of the necessary energy, makes a transition to the excited electronic state. The electron configuration of the new state may result in a shift of the equilibrium position of the nuclei constituting the molecule. In Figure 3 this shift in nuclear coordinates between the ground and the first excited state is labeled as
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2213:) and their environment. Franck–Condon metaphors are appropriate because molecules often interact strongly with surrounding molecules, particularly in liquids and solids, and these interactions modify the nuclear coordinates of the chromophore in ways closely analogous to the molecular vibrations considered by the Franck–Condon principle.
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Franck–Condon considerations can also be applied to the electronic transitions of chromophores dissolved in liquids. In this use of the Franck–Condon metaphor, the vibrational levels of the chromophores, as well as interactions of the chromophores with phonons in the liquid, continue to contribute to
2148:
modulates the intensity of transitions, i.e., it contributes with a factor on the order of 1 to the intensity of bands whose order of magnitude is determined by the other selection rules. The table below gives the range of extinction coefficients for the possible combinations of allowed and forbidden
168:
potential of simple harmonic oscillators, in more realistic potentials, such as those shown in Figure 1, energy spacing decreases with increasing vibrational energy. Electronic transitions to and from the lowest vibrational states are often referred to as 0–0 (zero zero) transitions and have the same
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level of the excited state to the zero-phonon level of the ground state or to higher phonon levels of the ground state. Just like in the Franck–Condon principle, the probability of transitions involving phonons is determined by the overlap of the phonon wavefunctions at the initial and final energy
159:
The applicability of the Franck–Condon principle in both absorption and fluorescence, along with Kasha's rule leads to an approximate mirror symmetry shown in Figure 2. The vibrational structure of molecules in a cold, sparse gas is most clearly visible due to the absence of inhomogeneous broadening
144:
proportional to the square of the (vertical) overlap of the vibrational wavefunctions of the original and final state (see
Quantum mechanical formulation section below). In the electronic excited state molecules quickly relax to the lowest vibrational level of the lowest electronic excitation state (
98:
Classically, the Franck–Condon principle is the approximation that an electronic transition is most likely to occur without changes in the positions of the nuclei in the molecular entity and its environment. The resulting state is called a Franck–Condon state, and the transition involved, a vertical
59:
Schematic representation of the absorption and fluorescence spectra corresponding to the energy diagram in Figure 1. The symmetry is due to the equal shape of the ground and excited state potential wells. The narrow lines can usually only be observed in the spectra of dilute gases. The darker curves
26:
Franck–Condon principle energy diagram. Since electronic transitions are very fast compared with nuclear motions, the vibrational states to and from which absorption and emission occur are those that correspond to a minimal change in the nuclear coordinates. As a result, both absorption and emission
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241:
article titled "A Theory of
Intensity Distribution in Band Systems". Here he formulates the semiclassical formulation in a manner quite similar to its modern form. The first joint reference to both Franck and Condon in regard to the new principle appears in the same 1926 issue of Physical Review in
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vibrations) with the electronic transitions of chromophores embedded as impurities in the lattice. In this situation, transitions to higher electronic levels can take place when the energy of the photon corresponds to the purely electronic transition energy or to the purely electronic transition
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was concerned with the mechanisms of photon-induced chemical reactions. The presumed mechanism was the excitation of a molecule by a photon, followed by a collision with another molecule during the short period of excitation. The question was whether it was possible for a molecule to break into
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In the original Franck–Condon principle, after the electronic transition, the molecules which end up in higher vibrational states immediately begin to relax to the lowest vibrational state. In the case of solvation, the solvent molecules will immediately try to rearrange themselves in order to
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electronic transitions along with the incomplete validity of the factorization of the total wavefunction into nuclear, electronic spatial and spin wavefunctions means that the selection rules, including the Franck–Condon factor, are not strictly observed. For any given transition, the value of
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molecule the nuclear coordinates axis refers to the internuclear separation. The vibronic transition is indicated by a vertical arrow due to the assumption of constant nuclear coordinates during the transition. The probability that the molecule can end up in any particular vibrational level is
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function. The solvent-chromophore interaction is drawn as a parabolic potential in both electronic states. Since the electronic transition is essentially instantaneous on the time scale of solvent motion (vertical arrow), the collection of excited state chromophores is immediately far from
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coordinates of the atoms constituting the molecule do not have time to change during the very brief amount of time involved in an electronic transition. However, this physical intuition can be, and is indeed, routinely extended to interactions between light-absorbing or emitting molecules (
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The Franck–Condon principle, in its canonical form, applies only to changes in the vibrational levels of a molecule in the course of a change in electronic levels by either absorption or emission of a photon. The physical intuition of this principle is anchored by the idea that the nuclear
43:
The Franck-Condon
Principle describes the intensities of vibronic transitions, or the absorption or emission of a photon. It states that when a molecule is undergoing an electronic transition, such as ionization, the nuclear configuration of the molecule experiences no significant change.
71:
Semiclassical pendulum analogy of the Franck–Condon principle. Vibronic transitions are allowed at the classical turning points because both the momentum and the nuclear coordinates correspond in the two represented energy levels. In this illustration, the 0–2 vibrational transitions are
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is illuminated by light corresponding to the electronic transition energy, some of the chromophores will move to the excited state. Within this group of chromophores there will be a statistical distribution of solvent-chromophore interaction energies, represented in the figure by a
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Energy diagram illustrating the Franck–Condon principle applied to the solvation of chromophores. The parabolic potential curves symbolize the interaction energy between the chromophores and the solvent. The
Gaussian curves represent the distribution of this interaction
1021:{\displaystyle P=\left\langle \psi _{e}'\psi _{v}'\psi _{s}'\right|{\boldsymbol {\mu }}\left|\psi _{e}\psi _{v}\psi _{s}\right\rangle =\int \psi _{e}'^{*}\psi _{v}'^{*}\psi _{s}'^{*}({\boldsymbol {\mu }}_{e}+{\boldsymbol {\mu }}_{N})\psi _{e}\psi _{v}\psi _{s}\,d\tau }
2410:. Immediately after the transition to the ground electronic state, the solvent molecules must also rearrange themselves to accommodate the new electronic configuration of the chromophore. Figure 7 illustrates the Franck–Condon principle applied to solvation. When the
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It should be clear that the quantum mechanical formulation of the Franck–Condon principle is the result of a series of approximations, principally the electrical dipole transition assumption and the Born–Oppenheimer approximation. Weaker magnetic dipole and electric
60:
represent the inhomogeneous broadening of the same transitions as occurs in liquids and solids. Electronic transitions between the lowest vibrational levels of the electronic states (the 0–0 transition) have the same energy in both absorption and fluorescence.
99:
transition. The quantum mechanical formulation of this principle is that the intensity of a vibronic transition is proportional to the square of the overlap integral between the vibrational wavefunctions of the two states that are involved in the transition.
2124:
may lower the probability of a transition or prohibit it altogether. Rotational selection rules have been neglected in the above derivation. Rotational contributions can be observed in the spectra of gases but are strongly suppressed in liquids and solids.
93:
of the vibrational level of the molecule in the originating electronic state. In the semiclassical picture of vibrations (oscillations) of a simple harmonic oscillator, the necessary conditions can occur at the turning points, where the momentum is zero.
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of the solvent. Assuming the solvent relaxation time is short compared with the lifetime of the electronic excited state, emission will be from the lowest solvent energy state of the excited electronic state. For small-molecule solvents such as water or
128:. Figure 1 illustrates the Franck–Condon principle for vibronic transitions in a molecule with Morse-like potential energy functions in both the ground and excited electronic states. In the low temperature approximation, the molecule starts out in the
84:
are relatively instantaneous compared with the time scale of nuclear motions, therefore if the molecule is to move to a new vibrational level during the electronic transition, this new vibrational level must be instantaneously compatible with the
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https://chem.libretexts.org/Courses/Pacific_Union_College/Quantum_Chemistry/13%3A_Molecular_Spectroscopy/13.07%3A_The_Franck-Condon_Principle#:~:text=The%20Franck%2DCondon%20Principle%20describes,molecule%20experiences%20no%20significant%20change
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The first integral after the plus sign is equal to zero because electronic wavefunctions of different states are orthogonal. Remaining is the product of three integrals. The first integral is the vibrational overlap integral, also called the
1269:{\displaystyle =\int \psi _{e}'^{*}\psi _{v}'^{*}\psi _{s}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\psi _{v}\psi _{s}\,d\tau +\int \psi _{e}'^{*}\psi _{v}'^{*}\psi _{s}'^{*}{\boldsymbol {\mu }}_{N}\psi _{e}\psi _{v}\psi _{s}\,d\tau }
232:
James Franck recognized that changes in vibrational levels could be a consequence of the instantaneous nature of excitation to higher electronic energy levels and a new equilibrium position for the nuclear interaction potential.
2034:
1950:{\displaystyle \iint \psi _{v}'^{*}\psi _{e}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\psi _{v}\,d\tau _{e}d\tau _{n}\approx \int \psi _{v}'^{*}\psi _{v}\,d\tau _{n}\int \psi _{e}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\,d\tau _{e}.}
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and is the fundamental assumption of the Franck–Condon principle. Combining these equations leads to an expression for the probability amplitude in terms of separate electronic space, spin and vibrational contributions:
556:
2097:) is a function of nuclear coordinates. Since the dependence is usually rather smooth it is neglected (i.e., the assumption that the transition dipole surface is independent of nuclear coordinates, called the
260:
Figure 1 in Edward Condon's first publication on what is now the Franck–Condon principle . Condon chose to superimpose the potential curves to illustrate the method of estimating vibrational transitions.
221:
upwards to the a curves in
Diagram I. the particles will have a potential energy greater than D' and will fly apart. In this case we have a very great change in the oscillation energy on excitation by
762:
359:
2324:) is determined by lattice parameters. Because the energy of single phonons is generally quite small, zero- or few-phonon transitions can only be observed at temperatures below about 40
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levels. For the Franck–Condon principle applied to phonon transitions, the label of the horizontal axis of Figure 1 is replaced in Figure 6 with the configurational coordinate for a
701:
of the initial and final state. The overall wavefunctions are the product of the individual vibrational (depending on spatial coordinates of the nuclei) and electronic space and
310:
2322:
2091:
533:{\displaystyle {\boldsymbol {\mu }}={\boldsymbol {\mu }}_{e}+{\boldsymbol {\mu }}_{N}=-e\sum \limits _{i}{\boldsymbol {r}}_{i}+e\sum \limits _{j}Z_{j}{\boldsymbol {R}}_{j}.}
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and is a stabilizing interaction, that is, the solvent molecules can move and rotate until the energy of the interaction is minimized. The interaction itself involves
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of the lattice. The upwards arrows represent absorption without phonons and with three phonons. The downwards arrows represent the symmetric process in emission.
1963:
209:. Here we have D > D' and D' > D". At the same time the equilibrium position of the nuclei moves with the excitation to greater values of
642:{\displaystyle P=\left\langle \psi '\right|{\boldsymbol {\mu }}\left|\psi \right\rangle =\int {\psi '^{*}}{\boldsymbol {\mu }}\psi \,d\tau ,}
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is determined by all of the selection rules, however spin selection is the largest contributor, followed by electronic selection rules. The
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over the spatial coordinates of the electrons would not depend on the nuclear coordinates. However, in the Born–Oppenheimer approximation
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2109:. The remaining two integrals contributing to the probability amplitude determine the electronic spatial and spin selection rules.
148:), and from there can decay to the electronic ground state via photon emission. The Franck–Condon principle is applied equally to
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The Franck–Condon principle has a well-established semiclassical interpretation based on the original contributions of
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molecules. These surrounding molecules may interact with the chromophores, particularly if the solvent molecules are
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the structure of the absorption and emission spectra, but these effects are considered separately and independently.
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potential energy in Figure 6 is represented as that of a harmonic oscillator, and the spacing between phonon levels (
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energy plus the energy of one or more lattice phonons. In the low-temperature approximation, emission is from the
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Energy diagram of an electronic transition with phonon coupling along the configurational coordinate
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of the individual transitions. Vibronic transitions are drawn in Figure 2 as narrow, equally spaced
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Coolidge, A. S, James, H. M. and
Present, R. D. (1936). "A study of the Franck–Condon Principle".
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Condon, E. (1928). "Nuclear motions associated with electron transitions in diatomic molecules".
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Condon, E. (1926). "A theory of intensity distribution in band systems (Meeting abstract)".
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do depend (parametrically) on the nuclear coordinates, so that the integral (a so-called
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This separation of the electronic and vibrational wavefunctions is an expression of the
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Diagram I. shows a great weakening of the binding on a transition from the normal state
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213:. If we go from the equilibrium position (the minimum of potential energy) of the
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whereas chromophore excited state lifetimes range from a few picoseconds to a few
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at ambient temperature, solvent relaxation time is on the order of some tens of
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2029:{\displaystyle \int \psi _{e}'^{*}{\boldsymbol {\mu }}_{e}\psi _{e}\,d\tau _{e}}
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minimize the interaction energy. The rate of solvent relaxation depends on the
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line shapes. Equal spacing between vibrational levels is only the case for the
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Noyes, W. A. (1933). "The correlation of spectroscopy and photochemistry".
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picture, the vibrational levels and vibrational wavefunctions are those of
2931:"Light absorption and fate of excitation energy: Franck-Condon principle"
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Condon, E. (1926). "A theory of intensity distribution in band systems".
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Franck, J. (1926). "Elementary processes of photochemical reactions".
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Franck, J. (1926). "Elementary processes of photochemical reactions".
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Life
Sciences at the University of Illinois at Urbana-Champaign
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The closest Franck–Condon analogy is due to the interaction of
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Spectra of Atoms and
Molecules (Topics in Physical Chemistry)
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Birge, R. T. (1926). "The band spectra of carbon monoxide".
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The spin-independent part of the initial integral is here
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for the transition between these two states is given by
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The Franck–Condon principle is a statement on allowed
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extended this insight beyond photoreactions in a 1926
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Compendium of
Chemical Terminology, 2nd Edition (1997)
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Quantum chemistry rule regarding vibronic transitions
2529:"A Theory of Intensity Distribution in Band Systems"
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1960:This factorization would be exact if the integral
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757:{\displaystyle \psi =\psi _{e}\psi _{v}\psi _{s}.}
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2854:Harris, Daniel C.; Michael D. Bertolucci (1978).
273:transition from the initial vibrational state (
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35:are shown favoring transitions with changes in
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2812:: CS1 maint: multiple names: authors list (
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2194:Spin forbidden but orbitally allowed
2186:Spin allowed but orbitally forbidden
2154:Intensities of electronic transitions
354:{\displaystyle |\epsilon 'v'\rangle }
7:
2464:Zero-phonon line and phonon sideband
2361:Consider chromophores surrounded by
2341:Franck–Condon principle in solvation
2334:Zero-phonon line and phonon sideband
242:an article on the band structure of
2907:. Oxford: Oxford University Press.
2572:Transactions of the Faraday Society
2502:Transactions of the Faraday Society
2336:for further details and references.
2241:Franck–Condon principle for phonons
496:
468:
305:{\displaystyle |\epsilon v\rangle }
179:Transactions of the Faraday Society
27:produce molecules in vibrationally
2929:Rabinowitch E.; Govindjee (1969).
2305:
2149:spin and orbital selection rules.
1511:
1435:
1347:
277:) of the ground electronic level (
14:
2317:{\displaystyle \hbar \Omega _{i}}
365:). The molecular dipole operator
177:In a report published in 1926 in
2444:Ultraviolet-visible spectroscopy
2439:Molecular electronic transition
1723:as a product of two integrals:
697:are, respectively, the overall
2434:Born–Oppenheimer approximation
978:
948:
769:Born–Oppenheimer approximation
371:is determined by the charge (−
328:
289:
265:Quantum mechanical formulation
1:
2527:Condon, Edward (1926-12-01).
312:, to some vibrational state (
173:Development of the principle
139:. In the simplest case of a
118:quantum harmonic oscillators
2905:Molecular Quantum Mechanics
2779:Journal of Chemical Physics
2449:Quantum harmonic oscillator
2178:Spin and orbitally allowed
2984:
2873:Bernath, Peter F. (1995).
2086:{\displaystyle \psi '_{e}}
124:of molecules, such as the
2903:; R. S. Friedman (1999).
2856:Symmetry and spectroscopy
2745:Reviews of Modern Physics
2095:transition dipole surface
2056:{\displaystyle \psi _{e}}
388:as well as the charges (+
2766:10.1103/RevModPhys.5.280
2116:transitions between two
2883:Oxford University Press
2725:10.1103/PhysRev.28.1157
2647:10.1103/PhysRev.28.1182
2553:10.1103/PhysRev.28.1182
269:Consider an electrical
2686:10.1103/PhysRev.32.858
2618:10.1103/PhysRev.27.637
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2164:extinction coefficient
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690:{\displaystyle \psi '}
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198:to the excited states
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82:Electronic transitions
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2417:Gaussian distribution
2385:and can also include
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2287:{\displaystyle q_{i}}
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665:{\displaystyle \psi }
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545:probability amplitude
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2514:10.1039/tf9262100536
2469:Sudden approximation
2383:van der Waals forces
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2271:
2142:Franck–Condon factor
2107:Franck–Condon factor
2101:is often allowed).
2099:Condon approximation
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1964:
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1350:Franck–Condon
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2858:. New York: Dover.
2792:1936JChPh...4..193C
2758:1933RvMP....5..280N
2717:1926PhRv...28.1157B
2678:1928PhRv...32..858C
2639:1926PhRv...28.1182C
2610:1926PhRv...27..637.
2545:1926PhRv...28.1182C
2267:. The lattice mode
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1081:
1061:
947:
927:
907:
831:
818:
805:
2735:2011-12-28 at the
2696:2011-12-28 at the
2657:2011-12-28 at the
2373:is referred to as
2355:
2314:
2284:
2238:
2152:
2083:
2070:
2053:
2026:
1970:
1947:
1888:
1841:
1756:
1736:
1704:
1657:
1598:
1594:
1592:
1585:
1537:
1526:
1524:
1506:
1458:
1450:
1448:
1430:
1370:
1362:
1360:
1342:
1294:
1266:
1197:
1177:
1157:
1082:
1062:
1042:
1018:
928:
908:
888:
819:
806:
793:
754:
687:
662:
639:
530:
504:
476:
351:
302:
262:
227:James Franck, 1926
114:quantum mechanical
74:
62:
41:
2968:Molecular physics
2958:Quantum chemistry
2825:Herzberg, Gerhard
2800:10.1063/1.1749818
2459:Vibronic coupling
2201:
2200:
1532:
1530:
1522:
1520:
1515:
1453:
1451:
1446:
1444:
1439:
1365:
1363:
1358:
1356:
1351:
1289:
1287:
495:
467:
400:) and locations (
375:) and locations (
2975:
2944:
2942:
2941:
2918:
2896:
2880:
2869:
2850:
2834:
2817:
2811:
2803:
2769:
2728:
2711:(6): 1157–1181.
2689:
2650:
2633:(6): 1182–1201.
2621:
2587:
2557:
2556:
2539:(6): 1182–1201.
2524:
2518:
2517:
2497:
2491:
2485:
2323:
2321:
2320:
2315:
2313:
2312:
2293:
2291:
2290:
2285:
2283:
2282:
2157:
2092:
2090:
2089:
2084:
2078:
2062:
2060:
2059:
2054:
2052:
2051:
2035:
2033:
2032:
2027:
2025:
2024:
2011:
2010:
2001:
2000:
1995:
1988:
1987:
1978:
1956:
1954:
1953:
1948:
1943:
1942:
1929:
1928:
1919:
1918:
1913:
1906:
1905:
1896:
1884:
1883:
1870:
1869:
1859:
1858:
1849:
1834:
1833:
1821:
1820:
1807:
1806:
1797:
1796:
1787:
1786:
1781:
1774:
1773:
1764:
1754:
1753:
1744:
1713:
1711:
1710:
1705:
1700:
1699:
1686:
1685:
1675:
1674:
1665:
1653:
1652:
1639:
1638:
1629:
1628:
1623:
1616:
1615:
1606:
1593:
1586:
1581:
1580:
1579:
1566:
1565:
1555:
1554:
1545:
1525:
1523:
1521:
1518:
1516:
1513:
1507:
1502:
1501:
1500:
1487:
1486:
1476:
1475:
1466:
1449:
1447:
1445:
1442:
1440:
1437:
1431:
1426:
1425:
1424:
1411:
1410:
1401:
1400:
1395:
1388:
1387:
1378:
1361:
1359:
1357:
1354:
1352:
1349:
1343:
1338:
1337:
1336:
1323:
1322:
1312:
1311:
1302:
1275:
1273:
1272:
1267:
1258:
1257:
1248:
1247:
1238:
1237:
1228:
1227:
1222:
1215:
1214:
1205:
1195:
1194:
1185:
1175:
1174:
1165:
1143:
1142:
1133:
1132:
1123:
1122:
1113:
1112:
1107:
1100:
1099:
1090:
1080:
1079:
1070:
1060:
1059:
1050:
1027:
1025:
1024:
1019:
1010:
1009:
1000:
999:
990:
989:
977:
976:
971:
962:
961:
956:
946:
945:
936:
926:
925:
916:
906:
905:
896:
881:
877:
876:
875:
866:
865:
856:
855:
841:
836:
832:
827:
814:
801:
763:
761:
760:
755:
750:
749:
740:
739:
730:
729:
696:
694:
693:
688:
686:
671:
669:
668:
663:
648:
646:
645:
640:
625:
620:
619:
618:
617:
597:
586:
581:
577:
539:
537:
536:
531:
526:
525:
520:
514:
513:
503:
488:
487:
482:
475:
457:
456:
451:
442:
441:
436:
427:
363:bra–ket notation
360:
358:
357:
352:
347:
339:
331:
311:
309:
308:
303:
292:
228:
207:
122:potential energy
108:
2983:
2982:
2978:
2977:
2976:
2974:
2973:
2972:
2948:
2947:
2939:
2937:
2928:
2925:
2915:
2899:
2893:
2872:
2866:
2853:
2847:
2823:
2804:
2775:
2741:
2737:Wayback Machine
2705:Physical Review
2702:
2698:Wayback Machine
2666:Physical Review
2663:
2659:Wayback Machine
2627:Physical Review
2624:
2597:Physical Review
2593:
2569:
2566:
2564:Further reading
2561:
2560:
2533:Physical Review
2526:
2525:
2521:
2499:
2498:
2494:
2486:
2482:
2477:
2454:Morse potential
2430:
2343:
2304:
2296:
2295:
2274:
2269:
2268:
2243:
2231:
2206:
2122:selection rules
2065:
2064:
2043:
2038:
2037:
2016:
2002:
1990:
1979:
1962:
1961:
1934:
1920:
1908:
1897:
1875:
1861:
1850:
1825:
1812:
1798:
1788:
1776:
1765:
1745:
1728:
1727:
1691:
1677:
1666:
1644:
1630:
1618:
1607:
1571:
1557:
1546:
1533:
1492:
1478:
1467:
1454:
1416:
1402:
1390:
1379:
1366:
1328:
1314:
1303:
1290:
1279:
1278:
1249:
1239:
1229:
1217:
1206:
1186:
1166:
1134:
1124:
1114:
1102:
1091:
1071:
1051:
1031:
1030:
1001:
991:
981:
966:
951:
937:
917:
897:
867:
857:
847:
846:
842:
792:
788:
777:
776:
741:
731:
721:
710:
709:
705:wavefunctions:
679:
674:
673:
654:
653:
609:
605:
587:
570:
566:
555:
554:
515:
505:
477:
446:
431:
418:
417:
408:
396:
383:
340:
332:
322:
321:
283:
282:
267:
244:carbon monoxide
239:Physical Review
229:
226:
205:
175:
138:
126:Morse potential
109:
103:
50:
33:potential wells
17:
12:
11:
5:
2981:
2979:
2971:
2970:
2965:
2960:
2950:
2949:
2946:
2945:
2924:
2923:External links
2921:
2920:
2919:
2913:
2897:
2891:
2870:
2864:
2851:
2845:
2821:
2786:(3): 193–211.
2773:
2752:(4): 280–287.
2739:
2700:
2672:(6): 858–872.
2661:
2622:
2591:
2565:
2562:
2559:
2558:
2519:
2492:
2479:
2478:
2476:
2473:
2472:
2471:
2466:
2461:
2456:
2451:
2446:
2441:
2436:
2429:
2426:
2387:hydrogen bonds
2342:
2339:
2338:
2337:
2311:
2307:
2303:
2281:
2277:
2242:
2239:
2227:
2205:
2202:
2199:
2198:
2195:
2191:
2190:
2187:
2183:
2182:
2179:
2175:
2174:
2160:
2081:
2077:
2073:
2050:
2046:
2023:
2019:
2015:
2009:
2005:
1999:
1994:
1986:
1982:
1977:
1973:
1969:
1958:
1957:
1946:
1941:
1937:
1933:
1927:
1923:
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1912:
1904:
1900:
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1887:
1882:
1878:
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1868:
1864:
1857:
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1848:
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1828:
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1811:
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1673:
1669:
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1660:
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1651:
1647:
1643:
1637:
1633:
1627:
1622:
1614:
1610:
1605:
1601:
1597:
1591:
1584:
1578:
1574:
1570:
1564:
1560:
1553:
1549:
1544:
1540:
1536:
1529:
1519:selection rule
1505:
1499:
1495:
1491:
1485:
1481:
1474:
1470:
1465:
1461:
1457:
1443:selection rule
1429:
1423:
1419:
1415:
1409:
1405:
1399:
1394:
1386:
1382:
1377:
1373:
1369:
1341:
1335:
1331:
1327:
1321:
1317:
1310:
1306:
1301:
1297:
1293:
1286:
1276:
1265:
1262:
1256:
1252:
1246:
1242:
1236:
1232:
1226:
1221:
1213:
1209:
1204:
1200:
1193:
1189:
1184:
1180:
1173:
1169:
1164:
1160:
1156:
1153:
1150:
1147:
1141:
1137:
1131:
1127:
1121:
1117:
1111:
1106:
1098:
1094:
1089:
1085:
1078:
1074:
1069:
1065:
1058:
1054:
1049:
1045:
1041:
1038:
1017:
1014:
1008:
1004:
998:
994:
988:
984:
980:
975:
970:
965:
960:
955:
950:
944:
940:
935:
931:
924:
920:
915:
911:
904:
900:
895:
891:
887:
884:
880:
874:
870:
864:
860:
854:
850:
845:
840:
835:
830:
826:
822:
817:
813:
809:
804:
800:
796:
791:
787:
784:
765:
764:
753:
748:
744:
738:
734:
728:
724:
720:
717:
685:
682:
661:
650:
649:
638:
635:
632:
628:
624:
616:
612:
608:
603:
600:
596:
593:
590:
585:
580:
576:
573:
569:
565:
562:
541:
540:
529:
524:
519:
512:
508:
502:
498:
494:
491:
486:
481:
474:
470:
466:
463:
460:
455:
450:
445:
440:
435:
430:
426:
404:
392:
379:
350:
346:
343:
338:
335:
330:
301:
298:
295:
291:
266:
263:
224:
188:potential well
174:
171:
136:
101:
89:positions and
49:
46:
29:excited states
15:
13:
10:
9:
6:
4:
3:
2:
2980:
2969:
2966:
2964:
2961:
2959:
2956:
2955:
2953:
2936:
2932:
2927:
2926:
2922:
2916:
2914:0-19-855947-X
2910:
2906:
2902:
2901:Atkins, P. W.
2898:
2894:
2892:0-19-507598-6
2888:
2884:
2879:
2878:
2871:
2867:
2865:0-486-66144-X
2861:
2857:
2852:
2848:
2846:0-486-65821-X
2842:
2838:
2833:
2832:
2826:
2822:
2820:
2815:
2809:
2801:
2797:
2793:
2789:
2785:
2781:
2780:
2774:
2772:
2767:
2763:
2759:
2755:
2751:
2747:
2746:
2740:
2738:
2734:
2731:
2726:
2722:
2718:
2714:
2710:
2706:
2701:
2699:
2695:
2692:
2687:
2683:
2679:
2675:
2671:
2667:
2662:
2660:
2656:
2653:
2648:
2644:
2640:
2636:
2632:
2628:
2623:
2619:
2615:
2611:
2607:
2603:
2599:
2598:
2592:
2590:
2585:
2581:
2577:
2573:
2568:
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2554:
2550:
2546:
2542:
2538:
2534:
2530:
2523:
2520:
2515:
2511:
2507:
2503:
2496:
2493:
2489:
2484:
2481:
2474:
2470:
2467:
2465:
2462:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2431:
2427:
2425:
2423:
2418:
2413:
2409:
2405:
2401:
2396:
2390:
2388:
2384:
2380:
2379:electrostatic
2376:
2372:
2368:
2364:
2359:
2351:
2347:
2340:
2335:
2331:
2330:
2329:
2327:
2309:
2279:
2275:
2266:
2261:
2256:
2252:
2248:
2240:
2235:
2230:
2226:
2222:
2218:
2214:
2212:
2203:
2196:
2193:
2192:
2188:
2185:
2184:
2180:
2177:
2176:
2172:
2169:
2165:
2161:
2159:
2158:
2155:
2150:
2147:
2143:
2139:
2138:
2132:
2126:
2123:
2119:
2115:
2110:
2108:
2102:
2100:
2096:
2079:
2075:
2071:
2048:
2044:
2021:
2017:
2013:
2007:
2003:
1997:
1984:
1975:
1971:
1967:
1944:
1939:
1935:
1931:
1925:
1921:
1915:
1902:
1893:
1889:
1885:
1880:
1876:
1872:
1866:
1862:
1855:
1846:
1842:
1838:
1835:
1830:
1826:
1822:
1817:
1813:
1809:
1803:
1799:
1793:
1789:
1783:
1770:
1761:
1757:
1750:
1741:
1737:
1733:
1726:
1725:
1724:
1722:
1701:
1696:
1692:
1688:
1682:
1678:
1671:
1662:
1658:
1654:
1649:
1645:
1641:
1635:
1631:
1625:
1612:
1603:
1599:
1595:
1589:
1582:
1576:
1572:
1568:
1562:
1558:
1551:
1542:
1538:
1534:
1527:
1503:
1497:
1493:
1489:
1483:
1479:
1472:
1463:
1459:
1455:
1427:
1421:
1417:
1413:
1407:
1403:
1397:
1384:
1375:
1371:
1367:
1339:
1333:
1329:
1325:
1319:
1315:
1308:
1299:
1295:
1291:
1284:
1277:
1263:
1260:
1254:
1250:
1244:
1240:
1234:
1230:
1224:
1211:
1202:
1198:
1191:
1182:
1178:
1171:
1162:
1158:
1154:
1151:
1148:
1145:
1139:
1135:
1129:
1125:
1119:
1115:
1109:
1096:
1087:
1083:
1076:
1067:
1063:
1056:
1047:
1043:
1039:
1036:
1029:
1028:
1015:
1012:
1006:
1002:
996:
992:
986:
982:
973:
963:
958:
942:
933:
929:
922:
913:
909:
902:
893:
889:
885:
882:
878:
872:
868:
862:
858:
852:
848:
843:
833:
828:
824:
820:
815:
811:
807:
802:
798:
794:
789:
785:
782:
775:
774:
773:
770:
751:
746:
742:
736:
732:
726:
722:
718:
715:
708:
707:
706:
704:
700:
699:wavefunctions
683:
680:
659:
636:
633:
630:
626:
614:
606:
601:
598:
594:
591:
588:
578:
574:
571:
567:
563:
560:
553:
552:
551:
549:
546:
527:
522:
510:
506:
500:
492:
489:
484:
472:
464:
461:
458:
453:
443:
438:
428:
416:
415:
414:
412:
407:
403:
399:
395:
391:
387:
382:
378:
374:
370:
369:
364:
344:
341:
336:
333:
319:
315:
296:
293:
280:
276:
272:
264:
259:
255:
251:
249:
248:Raymond Birge
245:
240:
236:
235:Edward Condon
223:
220:
216:
212:
208:
201:
197:
191:
189:
184:
180:
172:
170:
167:
163:
157:
155:
151:
147:
142:
135:
131:
127:
123:
119:
115:
106:
100:
95:
92:
88:
83:
79:
70:
66:
58:
54:
47:
45:
38:
34:
30:
25:
21:
2963:Spectroscopy
2938:. Retrieved
2934:
2904:
2876:
2855:
2835:. New York:
2830:
2808:cite journal
2783:
2777:
2749:
2743:
2708:
2704:
2669:
2665:
2630:
2626:
2601:
2595:
2575:
2571:
2536:
2532:
2522:
2505:
2501:
2495:
2483:
2422:Stokes shift
2391:
2360:
2356:
2349:
2244:
2228:
2224:
2220:
2211:chromophores
2207:
2166:(ε) values (
2153:
2145:
2141:
2136:
2135:
2127:
2117:
2113:
2111:
2106:
2103:
2098:
2094:
1959:
1721:approximated
1720:
1718:
766:
651:
547:
542:
405:
401:
397:
393:
389:
380:
376:
372:
367:
366:
317:
313:
278:
274:
268:
257:
231:
218:
214:
210:
203:
199:
195:
193:
183:James Franck
176:
158:
154:fluorescence
146:Kasha's rule
133:
129:
111:
97:
78:James Franck
75:
68:
56:
42:
36:
23:
2578:: 536–542.
2508:: 536–542.
2408:nanoseconds
2404:picoseconds
2265:normal mode
2260:zero-phonon
2234:normal mode
2114:vibrational
2952:Categories
2940:2024-05-18
2881:. Oxford:
2604:(5): 640.
2475:References
2131:quadrupole
320:′),
219:vertically
162:Lorentzian
150:absorption
2395:viscosity
2375:solvation
2350:Figure 7.
2306:Ω
2302:ℏ
2221:Figure 6.
2197:10 to 10
2189:10 to 10
2181:10 to 10
2162:Range of
2118:different
2072:ψ
2045:ψ
2018:τ
2004:ψ
1993:μ
1985:∗
1972:ψ
1968:∫
1936:τ
1922:ψ
1911:μ
1903:∗
1890:ψ
1886:∫
1877:τ
1863:ψ
1856:∗
1843:ψ
1839:∫
1836:≈
1827:τ
1814:τ
1800:ψ
1790:ψ
1779:μ
1771:∗
1758:ψ
1751:∗
1738:ψ
1734:∬
1693:τ
1679:ψ
1672:∗
1659:ψ
1655:∫
1646:τ
1632:ψ
1621:μ
1613:∗
1600:ψ
1596:∫
1583:⏟
1573:τ
1559:ψ
1552:∗
1539:ψ
1535:∫
1504:⏟
1494:τ
1480:ψ
1473:∗
1460:ψ
1456:∫
1428:⏟
1418:τ
1404:ψ
1393:μ
1385:∗
1372:ψ
1368:∫
1340:⏟
1330:τ
1316:ψ
1309:∗
1296:ψ
1292:∫
1264:τ
1251:ψ
1241:ψ
1231:ψ
1220:μ
1212:∗
1199:ψ
1192:∗
1179:ψ
1172:∗
1159:ψ
1155:∫
1149:τ
1136:ψ
1126:ψ
1116:ψ
1105:μ
1097:∗
1084:ψ
1077:∗
1064:ψ
1057:∗
1044:ψ
1040:∫
1016:τ
1003:ψ
993:ψ
983:ψ
969:μ
954:μ
943:∗
930:ψ
923:∗
910:ψ
903:∗
890:ψ
886:∫
869:ψ
859:ψ
849:ψ
839:μ
821:ψ
808:ψ
795:ψ
743:ψ
733:ψ
723:ψ
716:ψ
681:ψ
660:ψ
634:τ
627:ψ
623:μ
615:∗
607:ψ
602:∫
592:ψ
584:μ
572:ψ
497:∑
469:∑
462:−
449:μ
434:μ
425:μ
409:) of the
386:electrons
384:) of the
349:⟩
334:ϵ
300:⟩
294:ϵ
258:Figure 5.
166:parabolic
69:Figure 3.
57:Figure 2.
24:Figure 1.
2827:(1971).
2733:Archived
2694:Archived
2655:Archived
2428:See also
2412:solution
2400:methanol
2080:′
1981:′
1899:′
1852:′
1767:′
1747:′
1668:′
1609:′
1548:′
1469:′
1381:′
1305:′
1208:′
1188:′
1168:′
1093:′
1073:′
1053:′
939:′
919:′
899:′
879:⟩
829:′
816:′
803:′
790:⟨
684:′
611:′
595:⟩
575:′
568:⟨
345:′
337:′
225:—
222:light...
141:diatomic
102:—
72:favored.
48:Overview
2788:Bibcode
2754:Bibcode
2713:Bibcode
2674:Bibcode
2635:Bibcode
2606:Bibcode
2541:Bibcode
2363:solvent
2353:energy.
2326:kelvins
2255:lattice
2247:phonons
1438:orbital
152:and to
112:In the
91:momenta
87:nuclear
2911:
2889:
2862:
2843:
2371:solute
2251:quanta
2146:weakly
1355:factor
652:where
411:nuclei
271:dipole
217:curve
31:. The
2837:Dover
2367:polar
2144:only
361:(see
206:'
105:IUPAC
2909:ISBN
2887:ISBN
2860:ISBN
2841:ISBN
2819:Link
2814:link
2771:Link
2730:Link
2691:Link
2652:Link
2589:Link
2381:and
2332:See
2232:, a
2063:and
1514:spin
703:spin
672:and
543:The
202:and
2796:doi
2762:doi
2721:doi
2682:doi
2643:doi
2614:doi
2580:doi
2549:doi
2510:doi
2253:of
2168:mol
281:),
246:by
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2328:.
2173:)
2171:cm
413::
250:.
181:,
156:.
137:01
80:.
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2022:e
2014:d
2008:e
1998:e
1976:e
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1940:e
1932:d
1926:e
1916:e
1894:e
1881:n
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1702:.
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825:s
812:v
799:e
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783:P
752:.
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589:|
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490:+
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454:N
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318:ε
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204:a
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