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12…The Absorption of Light

12…The Absorption of Light

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P. Douglas et al.

The energy match is given by:

E2 À E1 ẳ DE ẳ hm ẳ hc=k


where E1 is the energy of the lower energy state, E2 the energy of the higher

energy state, DE is the difference in energy between the two states, h is Planck’s

constant, c the speed of light, and m and k the frequency and wavelength of the

absorbed light. If this energy-matching criterion is not met, then photons of this

particular energy cannot be absorbed, and the material will transmit, reflect, or

scatter them.

The absorption process involves three waves: the two electronic waves of the

two energy states, plus the EMR wave of the incident light. A detailed understanding of the coupling mechanism between these requires quantum mechanics,

but the essence of the question, in mechanistic terms, is, ‘‘How efficiently can the

incident photon wave perturb the positions and momenta of the electron in the

lower state wave such that it can be pushed into positions and momenta corresponding to the higher energy wave?’’

The mechanism of interaction is via the oscillatory force an electromagnetic

wave exerts on the dipole formed between the nucleus and an electron. For a

‘picture’ of the process it is easiest to consider the case of the hydrogen atom. At

any time there is an instantaneous dipole between the electron and nucleus. The

wavelength of visible light is many times bigger than the atomic diameter, and as

the wave passes, both the electron and nucleus experience essentially the same

magnitude of electromagnetic oscillatory force, but, as they are oppositely

charged, the force acts on the two ends of the dipole, the nucleus and electron, in

opposite directions. This electromagnetic force causes an oscillation in the dipole,

and perturbs the electron wavefunction. If this perturbation is of the right frequency, the electron can be pushed from positions and momenta characteristic of

the lower state to those characteristic of the higher state; the electron wavefunction

is then that corresponding to the higher state rather than the lower state, energy is

absorbed and the transition made. If the perturbation cannot shift the electron from

the lower to the upper orbital, the electron remains in the lower orbital, no energy

is absorbed, although the oscillating electric field caused by the perturbed electron

oscillation can result in a scattered photon of the same frequency as the incident

radiation. (This latter process gives rise to Rayleigh scattering, scattering of light

at the same frequency of incident light, which occurs when any inhomogeneous

material is irradiated.)

In a similar way that the calculation of orbital overlap for the formation of MOs

depends on the integral of two wavefunctions, the transition probability is also

calculated from an integration involving two wavefunctions; this time those corresponding to the lower and upper states. But an additional term, to identify the

effect of the electromagnetic radiation in coupling these two states together is

^, which for a single electron is -er,

required, this is the dipole moment operator, l

where -e is the charge on an electron and r is position relative to the nucleus, i.e.

the vector position along x, y, z axes with the nucleus at zero. The ‘‘allowedness’’

1 Foundations of Photochemistry


of an electric dipole absorption from E1 to E2, is calculated from the transition

dipole moment, M12 (i.e. the electric dipole moment associated with the transition

between the two states):


^W2 ds

M12 ẳ W1 l


where W1 and W2 are the wavefunctions for the electronic ground and excited

states, 1 and 2, respectively, and the integral is across all space ($….ds). M12 is a

vector, and the transition probability for absorption is given by the square of the

magnitude of the transition dipole moment, i.e. |M12|2 (the vertical bars indicate it

is the magnitude of the vector, i.e. its length, which is squared, rather than squaring

the vector itself).

If the overall integral is large, then there is a high probability that an incident

photon will be absorbed, the transition takes place, and the molecule is excited into

the higher energy state, 2. If the integral is small, then there is a low probability of

absorption. If the integral is zero, there will be no absorption at all. From equation

(1.23) the two wavefunctions must have some overlap in space, but also the

symmetry of the wavefunctions, when combined with the symmetry properties of

the dipole moment operator, must be such as to give a non-zero resultant integral.

A consideration of those cases, where, for reasons of wavefunction symmetry, the

integral is zero, leads to what are called selection rules. The Parity Selection Rule: Atomic Transitions and Electron

Orbital Angular Momentum

For an atom the dipole moment operator has inversion symmetry across the

nucleus; therefore the absorption integral is zero if both the electron wavefunctions

are of the same parity, but non-zero if they are of different parity. Thus an

s ? s orbital transition is ‘forbidden’, but s ? p orbital transition is ‘allowed’.

This is the Laporte selection rule (also known as the parity selection rule), perhaps

the most commonly observed consequence of which is the low intensities of

d–d transitions in transition metal complexes. By itself, the parity selection rule

would suggest that an s ? f transition is allowed. However, consideration of

conservation of angular momentum, restricts changes to those transitions in which

Dl = ± 1. (The possibilities of an increase or decrease in l arise because of the

vector nature of momenta, which can oppose or reinforce one another).

Although the strongest interaction between electromagnetic radiation and

chemical structures is via the dipole interaction, there are weaker multipole and

magnetic interactions which may become important in certain circumstances,

notably when considering f–f transitions in lanthanide (III) ions.


P. Douglas et al. The Spin Selection Rule: Atomic Transitions and Spin–Orbit


The spin-angular momentum of an absorbed photon can be incorporated into an

absorbing atom as electron orbital angular momentum, i.e. a difference of ± 1 in

the l quantum number for atomic transitions. However, there is no electric-dipole

mechanism for the direct conversion of photon spin angular momentum into a

change in electron spin angular momentum. (ESR spectroscopy has its origin in

the interaction of the magnetic, rather than electric, field of the incident radiation).

Thus, the general electron spin selection rule is DS = 0 and absorption takes place

with no change in electron spin angular momentum, so that, notably, transitions

between singlet and triplet states are spin-forbidden.

However, as discussed in Sect. 1.4.3, atoms with high Z, i.e. ‘heavy atoms’, can

have electron spin angular momentum and electron orbital angular momentum

‘coupled’, such that, in optical transitions of heavy atoms, a change in electron

spin angular momentum, an electron ‘spin flip’, can be coupled to a change in

orbital angular momentum while total angular momentum is conserved. Thus,

when all three types of angular momentum involved in absorption: photon, electron spin, and electron orbital angular momentum, can be coupled together,

absorptions for which DS = ± 1 can become allowed. For example, the n = 6


P1 ? 1S0 transition of the Hg atom is very strongly allowed, even though it is a

triplet–singlet transition for which DS = 0. Selection Rules and Light Absorption in Molecules

The probabilities of absorption by structures larger than an atom are determined in

essentially the same way. For diatomic and linear polyatomic molecules the orbital

momentum selection rule becomes DK = 0, ± 1, where K is the symbol for

molecular total orbital angular momentum. For systems with appropriate symmetry, the rules are g $ u, ? $ +, - $ -. For non-linear polyatomic molecules

which have some symmetry, the selection rule for orbital angular momentum can

be given in terms of symmetry properties [45]. For centrosymmetric systems, the

rule g $ u still holds. In addition, the allowed transitions can be related to the

symmetry properties of a particular point group. For example, benzene has D6h

symmetry, and electric dipole transitions are symmetry forbidden from the A1

ground state to B1, B2 or other A1 states (see Fig. 1.15) [46]. However, the

symmetry arguments only hold for completely rigid structures. In practice,

vibrations can act to distort molecular structures sufficiently to allow some

absorption for transitions, which would be ‘forbidden’ if only the symmetry of the

electronic wavefunctions were to be considered. These transitions are usually

much weaker than fully allowed ones. For example, the lowest energy (longest

wavelength) absorption band in benzene corresponds to the symmetry forbidden


A1g ? 1B2u transition (see Fig. 1.15). Although this transition is forbidden for

pure electronic states, the structure of benzene is distorted from that of a pure

1 Foundations of Photochemistry


hexagon because of interaction with the various vibrational modes. This is termed

vibronic coupling (vibrational ? electronic). This reduces the symmetry and the

selection rule is partially relaxed. Nevertheless this band, which is seen around

256 nm, is 30 times weaker than the second band at 203 nm and 200 times weaker

than the third absorption band at 183 nm [21].

The spin selection rule, DS = 0, also applies to molecules, and transitions

between, for example, molecular singlet and triplet states are spin-forbidden.

However, in the presence of heavy atoms, either in the molecular structure itself,

or in the solvent, the rule may be relaxed enough for singlet to triplet absorptions

to be detected, although usually they are still weak absorption bands. Selection Rules for Vibrational Transitions, Rotational

Transitions, and Raman Scattering

The vibrational transitions of a molecule can be probed directly using infrared

spectroscopy and are subject to the gross selection rule that for a change in a

vibrational state brought about by the absorption or emission of a photon, there

must be an accompanying change in the dipole moment of the molecule. Homonuclear diatomics are an important group of molecules which do not absorb IR

radiation because of this selection rule.

Pure rotational spectra can be observed in the gas phase; however the selection

rule for rotational transitions requires the molecule to have a permanent electric

dipole. Homonuclear diatomics are, again, an important group of molecules which

do not show microwave absorption because of this selection rule.

Although these selection rules limit IR and microwave absorption, electronic

transitions can be, and usually are, simultaneously accompanied by both vibrational and rotational transitions, and for an electronic transition there is no

restriction on the associated change in vibrational state. Vibrational transitions

accompanying an electronic transition are called vibronic transitions. In high

resolution gas phase work, these, along with their accompanying rotational transitions give rise to what is called an electronic band system. In solution the

vibrational structure is usually not well resolved, but there is often some structure,

a vibrational progression, corresponding to transitions into a number of electronic

levels in the upper electronic state (see Fig. 1.20).

Raman spectroscopy is a complementary technique used to probe vibrational

and rotational modes. It is based on the inelastic scattering of light. Upon irradiation of a sample with a monochromatic light source (typically a laser), some of

the incident photons collide with the molecules and lose energy to vibrational/

rotational modes, and will emerge from the sample with lower energy (Stokes

radiation); other photons may gain energy from vibrationally hot states and will

emerge with a higher energy (anti-Stokes radiation); finally some photons will be

directly scattered without a change in frequency (Rayleigh radiation). The

observed energy ‘losses’ or ‘gains’ provide information about the vibrational and

rotational states of molecules. The gross selection rule for rotational Raman


P. Douglas et al.

Fig. 1.20 UV/Vis absorption spectrum of anthracene in cyclohexane for the S0 ? S1 transition.

The vibrational progression in the absorption spectrum corresponds to transitions to excited-state

vibrations. The hatched area under the peak corresponds to the integrated absorption coefficient

(IAC) as defined in Eq. 1.26

transitions is that the molecule must be anisotropically polarisable, while for

vibrational Raman transitions the polarisability should change as the molecule

vibrates. Absorbance, Transmittance, Molar Absorption (Extinction)

Coefficient, Beer–Lambert Law and Deviations from Beer–

Lambert Law

For a parallel monochromatic radiation beam, where the proportion of radiation

absorbed by a substance is independent of the intensity of the incident irradiation,

i.e. the probability of absorption is linearly dependent on incident intensity, which

is the usual case for single photon transitions, each successive layer of thickness

dx absorbs an equal fraction -dI/I of radiant intensity I, and integration across a

finite thickness, x, with an initial irradiation intensity I0 gives:

lnI0 =It ị ẳ bx


where It is the transmitted intensity and b is a constant dependent upon the sample.

This is Lambert’s law. Beer showed that b is proportional to concentration (strictly

limited to low concentrations), and, as is most commonly the case, using log10, this


log10 I0 =It ị ẳ log10 I0 =Io Iabs ịị ẳ ecx


1 Foundations of Photochemistry


Table 1.6 Typical values of molar absorption coefficients (e) for some electronic transitions in

organic molecules and metal complexes


Transition type

e/mol-1 dm3


Highly conjugated organic


Small aromatic compounds

Spin-allowed p ? p*

Spin-allowed, symmetry forbidden lowest

energy p ? p*

Carbonyl compounds

Spin-allowed n ? p*

Metal complexes

Spin- and Laporte-allowed charge transfer

Tetrahedral metal complexes

Spin-allowed, partially Laporte-forbidden


Octahedral and square planar metal Spin-allowed, Laporte-forbidden d ? d


All systems

Spin forbiddena









These values increase with systems containing high atomic number atoms due to increased

spin–orbit coupling

where Iabs is the absorbed intensity, e is the molar absorption (formerly called

extinction) coefficient, and c is the concentration of absorbing material. If c is

given in mol dm-3, e is the decadic molar absorption coefficient (although usually

the term decadic is omitted); x, the pathlength, is usually give in cm (a 1 cm

pathlength cell being the most commonly used in solution phase experimental

photochemistry; note pathlength is often given the symbol l), so the units of e are

usually mol-1 dm3 cm-1. e is wavelength dependent. log10(I0/It) is called absorbance, or optical density (usually given the symbol A, Abs, or OD), and it varies

linearly with concentration and path length. For a solution made up of a mixture of

absorbers, i, the total absorbance at a given wavelength is the sum of P

the absorbances of the individual components at that wavelength, i.e. Atotal = x eici.

If a transition is forbidden by the spin-selection rule, the molar absorption

coefficient is typically 10-5–10-3 mol-1 dm3 cm-1, irrespective of whether the

transition is Laporte- or vibrationally-allowed. If a transition is spin-allowed but

parity forbidden, e is typically of the order of 100–103 mol-1 dm3 cm-1. If the

transition is both spin- and Laporte-allowed, e is large (103–105 mol-1 dm3 cm-1)

and the absorption is said to be ‘‘fully-allowed’’. Typical values for ‘‘allowed’’ and

‘‘forbidden’’ transitions in some common organic and inorganic complexes are

given in Table 1.6.

Apparent deviations from the Beer–Lambert law arise mainly because of

instrumental factors such as: stray light, sample fluorescence, and use of a wide

radiation bandwidth. Real deviations arise because of high concentrations which

introduce solute–solute interactions and changes in e with solution refractive

index, and concentration dependent chemical equilibria.


P. Douglas et al. The Strength or Probability of Absorption

We saw earlier that the probability of electric dipole absorption is related to the

transition dipole moment. However, there are a variety of terms commonly used to

describe the strength or probability of absorption. The ‘allowedness’ or ‘forbiddenness’ of the transition, and oscillator strength, f, are useful ideas where the

relative, rather than absolute value, of the strength of coupling is required. These

terms are factors used to describe how likely absorption is by reference to the

‘ideal oscillator’ of a free electron where the transition is ‘fully allowed’ and both

the ‘allowedness’ and oscillator strength are unity.

The molar absorption (extinction) coefficient, Einstein coefficient and absorption cross-section are commonly used measures of transition probability. The first

three are used for atomic and molecular species in the gas or solution phase, while

the latter is commonly used in solid-state studies. These are absolute measures of

absorption probability and are ultimately derived from the transition dipole

moment, and are therefore all related. They can be measured experimentally from

the absorption spectrum and can, in some cases, be calculated using molecular

orbital theory programs. (Note that generally MO calculations will give the

oscillator strength for any ‘forbidden’ transition as zero).

The method of calculation of these parameters from the absorption spectrum is

illustrated in Fig. 1.20, where the spectrum is plotted as e vs wavenumber (~m in

units of cm-1), instead of the conventional wavelength units [47]. The integrated

absorption coefficient, (IAC), is the area of the absorption peak, which is given by:






The IAC is proportional to the square of the transition dipole moment, i.e.

|M12|2. The related oscillator strength, f, of the transition is given by:

f ¼ 4:33 Â 10






The maximum value of f for a fully-allowed transition is 1 and it is a unitless


The IAC is also related to the B12 Einstein coefficient for spontaneous

absorption by:




eð~mÞd~m ¼ B12 h~mNA = ln 10


1 Foundations of Photochemistry


where NA is Avogadro’s number, and the term ln10 arises from the use of the

decadic molar absorption coefficient, i.e. one based on log10. Finally, the Einstein

coefficient for spontaneous emission, A21, can be related to B12 by:

B12 ẳ A21 =8phc~m3 ị


and, thus to the IAC by:

A21 ẳ

8pc~m2 ln 10IACị



1:30ị Types of Transitions

While bearing in mind the limitations of our approximations to the Schrödinger

equation for complex molecules, it is still often found that transitions arise predominantly from the movement of an electron from one molecular or atomic

orbital to another. The nature of the transition is then described by the two orbitals

involved; the most common types of transition are given below.

1. Transitions between orbitals localised on atoms; e.g. d–d transitions of

transition metal salts, f–f transitions of lanthanide ions. Such metal-centred

(MC) transitions are ubiquitous in transition metal and lanthanide complexes.

They are relatively weak because they are symmetry (Laporte) forbidden.

Although they may not be the important transitions for any particular application of transition metal photochemistry, they will almost always be present.

These are the transitions that give many transition metal salts their characteristic colour and are found in some gemstones and minerals. For example, the

red colour in ruby is due to the d–d transitions in chromium (III) present at

certain sites in an aluminium oxide (corundum) crystal.

2. Transitions between atomic orbitals in mixed oxidation state transition

metal complexes. These transitions can be relatively strong and are known as

metal to metal charge transfer (MMCT) transitions. Two common examples

are those involving the Fe2+/Fe3+ centres in the pigment prussian blue and the

Fe2+/Ti4+ ions in sapphire.

3. Transitions between ligand and metal orbitals in transition metal complexes. These are called metal to ligand, and ligand to metal charge transfer

(MLCT and LMCT) transitions, respectively. These can be fully-allowed

transitions and are usually the source of colour in intensely coloured transition

metal complexes, e.g. the LMCT transitions in chromate, CrO42-, and permanganate, MnO-,

4 ions, and the intense absorptions in ferroin (tris(phenanthroline) iron(II)) and tris(2,20 -bipyridyl)ruthenium(II) (see Chap. 4).

4. Transitions between molecular orbitals in organic compounds: e.g. polyaromatics, organic dyes, biological colorants such as chlorophylls and carotenes

(see Chap. 4). The lowest energy transitions are those between the HOMO and


P. Douglas et al.

Fig. 1.21 Structures of some organic molecules that show charge transfer: a naphthazarin;

b quinhydrone (a complex between quinone and 1,4-dihydroxybenzene); c TMPD (electron

donor) and d TCNE (electron acceptor)

LUMO. The two most important types of transitions in organic molecules are

n ? p* and p ? p* transitions; r ? p* and r ? r* transitions; are usually

of such high energy that they occur in the vacuum UV region. Most molecules

are not of particularly high symmetry and for these, p ? p* transitions are

generally highly allowed and have high molar absorption coefficients. For those

molecules which do have high symmetry, some p ? p* transitions may be

symmetry forbidden and show significantly reduced molar absorption coefficients (e.g. some transitions of pyrene). Pure n ? p* transitions are symmetry

forbidden when the n orbital has r symmetry, such as carbonyls; but vibrations

are effective in reducing the degree of forbiddenness; and a phenomenon known

as intensity borrowing, by which the intensity of a forbidden transition lying

close to an allowed transition can be increased, allows many n ? p* transitions

to have significant molar absorption coefficients. (The need to introduce the

concept of ‘intensity borrowing’ is an example of the failure of the simple MO

approach to match the solutions of the Schrödinger equation for complex

systems.) For many organic molecules containing heteroatoms, such as O and

N, n ? p* and p ? p* transitions are energetically close and in some cases

the relative positions of the absorptions can be reversed in solvents of differing

polarity. In certain cases, transitions can take place between electron rich and

electron poor regions in complex organic molecules. These charge transfer

transitions usually have very high molar absorption coefficients and are

responsible for many of the intense colours found in typical organic dyes (see

Chap. 4), such as naphthazarin (Fig. 1.21a).

5. Transitions between the valence and conduction bands in semiconductors.

These are the origin of colour in semiconductor pigments and quantum dots. In

addition to colour from bandgap transitions, introduction of dopants or impurities

into semiconductors leads to localised energy levels on the dopant or impurity

atoms/molecules (so-called colour centres) and transitions between these levels

and those of the semiconductor conduction and valence bands become possible;

the colour in blue and yellow diamonds arises from these types of transitions.

6. Transitions between electronic energy levels in imperfect crystals. The

colours of some minerals and semi-precious gems, notably: Cairngorms or

smoky quartz where colour comes from defects in the quartz lattice caused by

1 Foundations of Photochemistry


radiation from nearby radioactive minerals, and blue topaz where similar

defects are introduced into otherwise colourless stones deliberately in the


7. Transitions of the hydrated electron. One of the effects of radiation on

aqueous systems is the ejection of electrons, which are then hydrated by surrounding water molecules. The blue colour of the hydrated electron arises

because of transitions between the electronic energy levels of the electron held

in the potential well created by these water molecules. Similar intense colours

due to solvated electrons are seen upon dissolving sodium or other alkali metals

in liquid ammonia or aliphatic amines.

8. Charge transfer transitions between molecules in association. Here, the

transition occurs between the molecules themselves, with an electron transferred from one molecule to the other. The various colours observed with

molecular iodine (I2) with aromatic molecules are due to charge-transfer [48].

Intermolecular charge transfer is also responsible for the yellow colour in

quinhydrone, a complex formed between quinone and 1,4-dihydroxybenzene

(Fig. 1.21b). Naphthazarin can be considered as an intramolecular equivalent of

quinhydrone. Organic charge transfer complexes having strong absorption

bands can also be readily formed between aromatic molecules and strong

electron donors, such as N,N,N’,N’-tetramethyl-p-phenylenediamine (TMPD or

Wurster’s blue), Fig. 1.21c or acceptors, such as tetracyanoethylene ((TCNE),

Fig. 1.21d).

1.12.2 Absorption Spectra Absorption Spectra in the Gas Phase

Atoms lack vibrational and rotational energy levels. Low pressure atomic gases

show absorption lines which are extremely narrow, limited in width by Doppler

broadening due to the range of molecular velocities in a thermally equilibrated gas.

(The Doppler effect is a change in absorption/emission frequency due to relative

motion of the absorber/emitter and the source/detector. The reader will almost

certainly have experienced the effect while standing at a railway station as a nonstopping train passes sounding its horn. The sound of the horn shifts from a high

frequency to low frequency tone as it first approaches and then leaves the station).

At high pressures and temperatures interatomic interactions, i.e. collisions, and the

perturbing effects of neighbouring molecules, also cause lines to broaden.

For molecular species, gas-phase absorption spectra show a series of electronic

transitions upon which vibrational and rotational structure are superimposed. For

simple molecules in the gas phase this structure is simple enough for individual

electronic-vibrational–rotational transitions to be seen, particularly at low temperatures where only the very lowest vibrational and rotational energy states are


P. Douglas et al.

populated in the ground state. However, for molecules of any complexity there are

many vibrational levels, and since rotational energy spacings decrease with

molecular mass [14] rotational levels are much closer together, so that even in the

gas phase, where molecules are isolated from one another, the absorption spectrum

can become so complex that it resembles more a series of overlapping bands than

groups of discrete lines. Absorption Spectra in Solution

Absorption bands in solution are less structured and broader than those in the gas

phase. In solution, molecular rotation is hindered through collision with solvent

molecules such that rotational quantisation is lost and solvent–solute interactions

broaden vibrational bands further. The magnitude of the latter effect depends on

the strength of the solute–solvent interaction and it is therefore most pronounced in

polar solvents, so that spectra in a non-polar solvent such as hexane will generally

show more distinct vibrational structure than those recorded in a more polar solvent such as acetone or an alcohol. While molecular absorption bands in solution

may be very broad, there is almost always some electronic/vibrational structure

and usually a number of electronic absorption bands can be identified. Absorption Spectra in Solution: Effect of Aggregation

Consider the relatively simple case of two aromatic molecules. If they are close

enough, they may interact, or couple, leading to splitting of the excited-state into

two levels [49]. The effect of this coupling depends upon the way the transition

dipole moments of the two molecules are arranged. If they are parallel, the transition to the upper excited state becomes more probable, and this leads to a blue or

hypsochromic shift in the absorption spectrum. If, in contrast, the transition dipoles

are in a head-to-tail arrangement, the transition to the lower excited state becomes

more probable, and there is a red or bathochromic shift in the absorption spectrum.

(There are corresponding shifts in the fluorescence spectra and changes in the

emission quantum yield.) A third possibility is that the transition dipoles are

arranged in an oblique fashion. In this case the absorption band is split into two.

These cases are shown in Fig. 1.22. Similar behaviour exists with larger aggregates. For historical reasons, if the shift is to longer wavelengths, these are termed

J-aggregates. (After the scientist Edwin Jelley, who observed the effect with

photographic dyes [50]. It was also reported independently by G. Scheibe [51].) Jaggregates frequently have very sharp absorption bands and show considerable

potential in various functional dye systems [52]. The aggregates where there is a

shift to shorter wavelengths are termed H-aggregates (after the hypsochromic or

blue shift in the spectrum).

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12…The Absorption of Light

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