Experimental and theoretical studies are reported of the short-lived and delayed fluorescence of anthracene single crystals, excited by single- and double-photon absorption. A giant-pulse ruby laser provides the primary source of radiation of 14 400 cm−1 (up to 1027 photons/cm2·sec) and is also used to generate second-harmonic radiation from ADP, as well as stimulated Raman radiation of 12 800 and 17 500 cm−1 from liquid oxygen. The time dependence of the fluorescence intensity is studied as a function of laser intensity, crystal temperature and excitation wavelength. The very intense fast fluorescence with a half-life of 30 nsec at 300°K, characteristic of singlet exciton decay, and the relatively weak delayed fluorescence which involves intermediate triplet states, are separated using sectored disks. It is concluded that the triplet state at 14 750 cm−1 can be populated (i) by direct absorption of laser photons involving an activation energy of 350 cm−1; (ii) via two-photon absorption, presumably leading to a vibrationally excited state of the 1B2u exciton, followed by intersystem crossing; (iii) via one-photon (second-harmonic) excitation from levels≥700 cm−1 into the singlet absorption band, followed by conversion of the singlet exciton into a triplet pair. The latter process is suggested by the observed activation energy of 700 cm−1. In agreement with these interpretations, the delayed fluorescence intensity is found to vary with the second to fourth power of the laser intensity depending on the experimental conditions. Also, light of 17 500 cm−1 leads exclusively to Process (i), light of 12 800 cm−1 exclusively to (ii). Triplet lifetimes from 2–17 msec are obtained, depending on crystal purity, which indicates that unimolecular triplet decay is an extrinsic, radiationless process. A singlet—triplet intersystem crossing rate constant of about 3×10−5 sec−1 is estimated. The triplet—triplet annihilation rate constant is found to be about 5×10−11 cm3 sec−1. This value considered together with the triplet-pair creation process suggests a triplet exchange rate ≳ 1013 sec−1 and a triplet diffusion constant ≳o5×10−4cm2/sec.

Radiationless transitions between two electronic states are studied for a system consisting of a polyatomic molecule in a medium where vibrational relaxation is rapid. The transition rate is then governed by a vibronic matrix element and a vibrational overlap factor. Only the latter, known as the Franck—Condon factor and denoted by F, is investigated in detail. For harmonic oscillators F derives from shifts in equilibrium distance (displacements) and shifts in frequency (distortions). It is shown that for radiationless transitions involving large energy gaps (E), F is dominated by distortions, whereas for optical transitions it is dominated by displacements of the oscillators. These distortions lead to an approximately exponential decrease of F with increasing E. An isotope rule for F is derived which is valid for both displaced and distorted oscillators provided E is not too small. Since all formulas are of the form F (E), where E is the principal variable, comparison with experiment is only possible for a class of related molecules. Within this class E should vary a great deal but all other parameters should either be constant or vary in a simple, systematic manner. The dependence of some of these parameters on the number of oscillators involved in the transition is examined. Finally the theory is extended to molecular crystals and the corresponding liquids. It is concluded that in these systems the rate constant of an intramolecular radiationless transition must be equal to or smaller than that in dilute solutions, except when excited dimers are present in the initial state.

The theory of Part I is applied to nonradiative transitions from the lowest triplet state to the ground state of aromatic hydrocarbons. A previously communicated empirical relation between triplet energy, triplet lifetime, and the relative number of hydrogen atoms per molecule is substantiated and its physical implications are discussed. It is transformed into a relation between the Franck-Condon factor of the transitions and the triplet energy. In this form it can be compared with the theoretical expressions derived in Part I. No satisfactory theoretical representation of the empirical formula could be obtained on the basis of a harmonic-oscillator description of the normal modes of the molecules. However, introduction of anharmonicity leads to excellent agreement between theory and experiment. A one-parameter formula is derived which accounts with good accuracy for the dependence of the triplet lifetime on the triplet energy and the number of carbon and hydrogen atoms in the molecule. This formula shows that the Franck-Condon factors relevant to the radiationless triplet-ground-state transition are governed by CH or CD stretching modes, which behave in this respect as completely degenerate for a given molecule. The single adjustable parameter is related to the anharmonicities of these modes in the ground and triplet state. The analysis confirms that the purely radiative triplet lifetime of most if not all aromatic hydrocarbons is close to 30 sec and that the Franck-Condon factor is the only parameter in the expression for the nonradiative triplet decay constant which varies considerably between different aromatic hydrocarbons. Finally the temperature dependence of the Franck-Condon factors is considered. A theoretical treatment indicates a very small temperature dependence below 400°K. This result seems to be borne out by an analysis of recent experiments, leading to the conclusion that the observed temperature effects are associated with bimolecular processes. A notable exception is benzene, where the triplet lifetime is temperature-dependent down to very low temperatures, possibly due to a Jahn-Teller distortion.

The electronic matrix elements governing intersystem crossing in aromatic hydrocarbons are derived and evaluated. The derivation is equivalent to Fano's treatment of resonance scattering. It is shown that the best first-order description for crossings that show simple unimolecular behavior involves molecular states that are both spin contaminated and vibronically contaminated, but can be adequately represented by an expansion to second order in pure-spin adiabatic Born-Oppenheimer states. The corresponding first- and second-order matrix elements are expanded about a nuclear equilibrium configuration and the expansions are terminated through the application of a symmetry argument. This yields five different types of matrix elements with small or vanishing cross terms; these matrix elements are associated with five experimentally distinguishable mechanisms, namely (1) direct spin-orbit coupling, (2) vibronically induced spin-orbit coupling, (3) mixed vibronic and spin-orbit coupling, (4) resonant spin-orbit coupling, and (5) vibronically induced resonant spin-orbit coupling. To distinguish these mechanisms use is made of isotope effects, spin polarization and vibrational selection. The available experimental data on singlet-to-triplet crossing in naphthalene and anthracene are analyzed in detail and compared with qualitative and quantitative theoretical predictions. It is concluded that at low temperature singlet-to-triplet crossing in naphthalene is dominated by crossing to the third-lowest triplet state via the third mechanism. At high temperature the dominant process involves either resonant crossing to the fourth-lowest triplet state via the fifth mechanism or resonant crossing to the fifth-lowest triplet state via the fourth mechanism. In an appendix spin-rotational and orbital-rotational coupling are shown to be too small to contribute measurably to these crossings.

The rate constant of the reaction catalyzed by the enzyme carbonic anhydrase II, which removes carbon dioxide from body fluids, is calculated for a model of the active site. The rate-determining step is proton transfer from a zinc-bound water molecule to a histidine residue via a bridge of two or more water molecules. The structure of the active site is known from X-ray studies except for the number and location of the water molecules. Model calculations are reported for a system of 58 atoms including a four-coordinated zinc ion connected to a methylimidazole molecule by a chain of two waters, constrained to reproduce the size of the active site. The structure and vibrational force field are calculated by an approximate density functional treatment of the proton-transfer step at the Self-Consistent-Charge Density Functional Tight Binding (SCC-DFTB) level. A single transition state is found indicating concerted triple proton transfer. Direct-dynamics calculations for proton and deuteron transfer and combinations thereof, based on the Approximate Instanton Method and on Variational Transition State Theory with Tunneling Corrections, are in fair agreement and yield rates that are considerably higher and kinetic isotope effects (KIEs) that are somewhat higher than experiment. Classical rate constants obtained from Transition State Theory are smaller than the quantum values but the corresponding KIEs are five times larger. For multiple proton transfer along water bridges classical KIEs are shown to be generally larger than quantum KIEs, which invalidates the standard method to distinguish tunneling and over-barrier transfer. In the present case, a three-way comparison of classical and quantum results with the observed data is necessary to conclude that proton transfer along the bridge proceeds by tunneling. The results suggest that the two-water bridge is present in low concentrations but makes a substantial contribution to proton transport because of its high efficiency. Bridging structures containing more water molecules may have lower energies but are expected to be less efficient. The observed exponential dependence of the KIEs on the deuterium concentration in H(2)O/D(2)O mixtures implies concerted transfer and thus rules out substantial contributions from structures that lead to stepwise transfer via solvated hydronium ions, which presumably dominate proton transfer in less efficient carbonic anhydrase isozymes.