Selection Rules for One-Photon Transitions

Heuristic selection rules for one-photon transitions may be obtained by using Eq. 1.9 or 1.10 for the perturbation W in the expression for c (t), Eq. 1.97. This procedure yields the matrix element 

While this discussion based on electrostatics ignores the time dependence in PF(t) and omits the effects of magnetic fields associated with the light wave, it does anticipate some of our final results in this section regarding electric dipole and electric quadrupole contributions to the matrix elements m fF(t) fc . It yields no insight into magnetic multipole transitions or into the nature of the time-ordered integrals in the Dyson series expansion of Eq. 1.96. 

The quantity k r is equivalent to k r cos 0 = 27r r cos d/X, where 6 is the angle formed between the vectors k and r. The matrix elements ( m W t) k limit r to the molecular dimensions over which the wave functions k and w are appreciable, i.e., r 10 A in typical cases. The shortest wavelengths X used in molecular spectroscopy are on the order of 10 A for vacuum-ultraviolet light, and are of course much longer for visible, IR, and microwave spectroscopy. Hence k r is typically much less than 1, and the series expansion of exp(ik r) converges rapidly. In the special geometry we have assumed for our vector potential, 

The first term in m W t) k requires the matrix element w 5/5x k . This can be obtained by evaluating the commutator 

Collecting these results for the first and second terms in ( m W t) k, we summarize that 

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The selection rules will be mentioned briefly here. In general, the process of multiphoton absorption is similar to that of single-photon absorption. The multiple photons are absorbed simultaneously to a real excited state in the same quantum event, where the energy of the transition corresponds to the sum of the energies of the incident photons. Thus selection rules for these transitions may be derived from the selection rules for one-photon transitions as they can be considered multiple one-photon transitions [20]. 

It can be shown that the angular part of this integral vanishes [1] unless A/ = / — / = 1 not zero), and unless Am = m — m = 0 or 1. Evaluation of the radial part is difficult this factor is nonzero regardless of An = n — n, provided A/ = 1. Hence we can summarize the El selection rules for one-photon transitions in hydrogenlike atoms. 

For centrosymmetric molecules, however, A/ige = 0 and accordingly <52 state = 0. Moreover, 2 PA into one-photon-allowed states is forbidden according to the parity selection rule whereas one-photon transitions in centrosymmetric systems are accompanied by a change in state parity (g—>u or u—>g), 2PA is only possible between states of the same parity (g—>g or u—>u). Indeed, this feature has long been exploited in spectroscopy to obtain information complementary to that accessible from 1PA for example, see Refs. [122] and [123]. 

The selection rules for multiphoton transitions are clearly different from the usual dipole selection rules, since each photon carries an angular momentum 1 Thus, for two-photon transitions, one rule is A J = 0 2, but further control can be exercised by selecting the polarisation of the light the A J = 0 transitions are only possible if the laser light is linearly polarised (i.e. contains both circular polarisation), while the choice of either circular polarisation results in an increase or a decrease of J. Detailed discussion of the selection rules for two-photon transitions can be found in several papers [453, 455, 459]. For multiphoton transitions, the same principles apply, and the role of polarisation is still more significant. A general reference is [460], in which selection rules are derived from first principles, and a list of selection rules for two-photon transitions is given in table 9.1. 

The selection rules for two-photon transitions are different and many dyes that have a large coefficient for one-photon absorption exhibit two-photon cross sections in the order of at most a few GM [14]. Correspondingly, a two-photon emission (excited state two photons) is observed as a weak emission, since only the total energy is conserved, not the energy of each individual photon. Thus, emission is a weak continuum of interest more of astrophysicists than of chemistry practitioners, but advancement in the field has been remarkable [15]. 

The simplest example of multiphoton absorption is two-photon absorption. Two-photon transitions have selection rules, which differ from the selection rules for one-photon absorption. It is required for one-photon transitions that the levelsy and / have the opposite parity, and for two-photon transitions the levels j and / should have the same parity. 

A complex 7THG can result from one-, two-, or three-photon resonances. One-photon resonance occurs when the fundamental frequency co is close to an allowed electronic transition. Two-photon resonance occurs when 2co is close to a two-photon allowed electronic transition. For centrosymmetric molecules the two-photon selection rule couples states of like inversion symmetry, e.g. g <- g. For acentric molecules one-photon transitions can also be two-photon allowed. Three-photon resonance occurs when 3co is close to the energy of an electronic transition the same symmetry rules apply as for one-photon transitions. 

Since photons have angular momentum of +1 or -1, an electronic state absorbing two photons simultaneously may change angular momentum by +2, 0. Two L = +1 photons cause a change of +2 a photon of L = +1 and one of I = -1 cause a change of 0 (A1 = 0, 2, AJ = 0, 2, AL = 0, 2, AS = 0). Thus the selection rules for two-photon absorption allow the excited electron to be either in an s or a d state, states which are of even-to-even parity or odd-to-odd parity such as f-f transitions, which now become allowed. An electron therefore cannot go from an s state... 

One very important aspect of two-photon absorption is that the selection rules for atoms or S5munetrical molecules are different from one-photon selection rules. In particular, for molecules with a centre of symmetry, two-photon absorption is allowed only for gog or uou transitions, while one-photon absorption requires g-o-u transitions. Therefore, a whole different set of electronic states becomes allowed for two-photon spectroscopy. The group-theoretical selection rules for two-photon spectra are obtained Ifom the symmetries... 

Thus it was not observed until lasers were invented. In principal, one-photon and two-photon excitation follow different selection rules. For example, the inner shell one-photon transitions in transition metal, rare earth, and actinide ions are formally forbidden by the parity selection rule. These ions have d- or/-shells and transitions within them are either even to even (d d) or odd to odd (f /). The electric dipole transition operator is equal to zero. 

The selection rules governing transitions between electronic energy levels are the spin rule (AS = 0), according to which allowed transitions must involve the promotion of electrons without a change in their spin, and the Laporte rule (AL = 1 for one photon). This parity selection rule specifies whether or not a change in parity occurs during a given type of transition. It states that one-photon electric dipole transitions are only allowed between states of different parity [45],... 

Relaxation of the rules can occur, especially since the selection rules apply strongly only to atoms that have pure Russell-Saunders (I-S) coupling. In heavy atoms such as lanthanides, the Russell-Saunders coupling is not entirely valid as there is the effect of the spin-orbit interactions, or so called j mixing, which will cause a breakdown of the spin selection rule. In lanthanides, the f-f transitions, which are parity-forbidden, can become weakly allowed as electric dipole transitions by admixture of configurations of opposite parity, for example d states, or charge transfer. These f-f transitions become parity-allowed in two-photon absorptions that are g g and u u. These even-parity transitions are forbidden for one photon but not for two photons, and vice versa for g u transitions [46],... 

Allowed transitions in the harmonic approximation are those for which the vibrational quantum number changes by one unit. Overtones - that is, the absorption of light at a whole number times the fundamental frequency - would not be possible. A general selection rule for the absorption of a photon is that the dipole... 

One other feature is worth noting at this poiit, and it concerns the case where the synergistic pair has a fixed mutual crientation, even if the pair itself rotates freely. While the local symmetry of each of the absorbers A and B determines the selection rules for the transitions they undergo, the symmetry of their relative juxtaposition also plays a role in determining the polarization characteristics of their synergistic photoabsoiption. This is principally manifest in the occurrence of two-photon circular dichroism where the A—B pair has definite handedness, as will be demonstrated in Section IX. Thus it transpires that not only the local symmetry, but also the global symmetry... 

So far we have concentrated attention on the vibrational manifold, without considering the rotational structure within this manifold. It should be noted at the outset that the rotational selection rules, AJ = 0, +1, valid for one-photon processes, completely break down for multiphoton processes, although the selection rules AM = 0, AK = 0 for a parallel transition remain valid. Thus, a wide range of J states associated with a particular vibrational state may become populated. Felker et al.22 have recently reported the observation of rotational coherence in large molecules. They observe a coherent superposition of precisely three J states, arising from AJ = 0, +1 in a one-photon process. The multiphoton process prepares a similar coherent population of J states, capable of exhibiting quantum interference phenomena, but many more J levels may be involved. 

The selection rules for parity are <-> for N even and for AT odd. Equation (6.1.30) will be explicitly discussed below for the examples of one-, two-, and three-photon transitions. 

Since the electronic selection rule is AA = 0, 1, 2, 3, three-photon excitations from, for example, a + state, can terminate in 1 An and 1 u states as well as the + and 1n,1 states accessible via one-photon transitions. 

The total parity of the final state is equal to the parity of the ionic level times the parity of the electron partial wave (which is even for even l and odd for odd l). For example, for a transition from a 1E+ molecular state to a 2II ion state, starting from J" = 4 (e-level, + parity), the J+ = 7/2 rotational level of the ion has two components one e-level (— parity), one /-level (+ parity) (see Fig. 8.16). The selection rule for allowed one-photon transitions is H— —. Consequently, for the transition into the ion e-level, the partial wave of the ejected electron is l = 0 (s) for the transition into the ionic /-level, the partial wave of the ejected electron is l = 1 (p). Equation (8.1.8a), with S -1- = 1/2, is satisfied for these l values. 

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