Shape of the Absorption and Fluorescence Spectra of Condensed

Article pubs.acs.org/JPCA

Shape of the Absorption and Fluorescence Spectra of Condensed Phases and Transition Energies Miguel Lagos*,† and Rodrigo Paredes*,‡ †

Departamento de Ciencias Aplicadas and ‡Departamento de Ciencias de la Computación, Facultad de Ingeniería, Universidad de Talca, Campus Los Niches, Curicó, Chile ABSTRACT: General integral expressions for the temperature-dependent profile of the spectral lines of photon absorption and emission by atomic or molecular species in a condensed environment are derived with no other hypothesis than: (a) The acoustic vibrational modes of the condensed host medium constitute the thermodynamic energy reservoir at a given constant temperature, and local electronic transitions modifying the equilibrium configuration of the surroundings are multiphonon events, regardless of the magnitude of the transition energy. (b) Electron−phonon coupling is linear in the variations of the bond length. The purpose is to develop a theoretical tool for the analysis of the spectra, allowing us to grasp highly accurate information from fitting the theoretical line shape function to experiment, including those spectra displaying wide features. The method is illustrated by applying it to two dyes, Lucifer Yellow CH and Coumarin 1, which display fluorescence maxima of 0.41 and 0.51 eV fwhm. Fitting the theoretical curves to the spectra indicates that the neat excitation energies are 2.58 eV ± 2.5% and 3.00 eV ± 2.0%, respectively.

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INTRODUCTION The analysis of electromagnetic absorption and emission spectra has been an invaluable source of knowledge in many fields of science and a standard tool in a variety of technical applications from long ago.1−3 However, there is an important technical point that remains still at issue. The interpretation of the spectra, in either quantitative chemical analysis or precise excitation energy measurements, demands curve fitting to the spectral features, particularly when they exhibit heavy overlapping or nonuniform background.4 When the peaks are broad, as those observed in many fluorescence spectra, little information on the target can be grasped from them. Currently, the selection of an analytical expression for fitting a specific spectral component is made on the basis of simple semi− empirical criteria.5 Gauss and Lorentz distributions,6,7 and a convolution of them, the Voigt distribution,8,9 are the most commonly used profiles to model the shape of the peaks. From an analytic view, Gauss and Lorentz curves are very different. While the tails of the former decay exponentially to zero and contribute weakly to the peak area, the area under the tails of the latter model constitutes a significant fraction of the total. Under these circumstances, simply integrating the line shape over the energy interval means that the measured intensity is sensitive to the decision of where to place the background limits. Hence the choice of the model for the lineshapes may have a substantial effect on the assigned peak intensities.10 Symmetric model distributions are used almost exclusively. This contradicts the empirical and theoretical evidence, which shows that the lineshapes of most spectral features given by condensed phases and large molecules are temperaturedependent and asymmetric11−13 when plotted against photon © XXXX American Chemical Society

energy. This applies to both narrow peaks, as those observed in infrared absorption spectroscopy, as well as wide spectral distributions, like the ones exhibited by the fluorescent photon absorption or emission by organic molecules. Besides the formal proof based on field-theory methods,12,13 peak asymmetry has a very fundamental explanation. Because of the second law of thermodynamics, any spontaneous transition between two well-defined states of the weakly coupled electron and electromagnetic radiation fields should increase the entropy. Then, the thermal bath, constituted by the acoustic vibrational modes of the host medium, must take a finite amount of the transferred energy. The entropy law makes more frequent the absorption of photons with energy higher than the energy ΔE of the electronic transition, while enhancing the emission of photons at energies lower than ΔE. Statistically, the energy difference is gained by the phonon bath. Determining the net excitation energy of the electronic bond or vibronic mode from an asymmetric peak requires an accurate model for the line shape, well founded on the physics of the target undergoing the photon absorption or emission process, because it hardly coincides with the maximum of the peak, even in narrow ones.12,13 Admitting the width of the peaks as the experimental error may be enough for some applications, but it does not take advantage of the full power of the method. Broad fluorescence maxima usually have Stokes shifts larger than, or as large as, the distribution breadth, and determining the precise energy of the bonding orbital involved in the transition Received: August 26, 2014 Revised: October 16, 2014

A

dx.doi.org/10.1021/jp508605c | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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wide bands but not necessarily on the limiting case of Gaussian ones.

demands an equally precise knowledge of the physical process taking place in the target. To stress the point, consider the large availability of very precise spectra of commercial fluorescent dyes displaying one or two broad asymmetric maxima. Although each maximum is associated with the excitation or de-excitation of a single bonding orbital, no analysis of the spectra is done to identify the bondings involved in the transitions and to determine their precise excitation energies. The primary objective of this paper is to introduce a theoretical scheme for obtaining accurate physical data of the sample from fitting the calculated spectral distributions to experiment, no matter how the width or asymmetric shape may be. The main hypotheses are: (a) The acoustic vibrational modes of the condensed host medium constitute the thermodynamic energy reservoir at a given, constant temperature. When the target is in the solid state or embedded in a solid, the main source for the temperature-dependent broadening, asymmetry, and shift of the spectral features are the acoustic modes of vibration of the crystal.12,13 When dissolved or dispersed in a liquid medium, the longitudinal acoustic excitations of the medium play the same role. Acoustic phonons obey Bose−Einstein statistics, and their energy spectra form a quasi-continuum with no gap at the ground level. Hence, any local electronic transition that modifies in any way the equilibrium configuration of the surrounding lattice will be a multiphonon event, regardless of the magnitude of the transition energy. Optical and impurity modes are a very different matter because their excitation energies have finite gap and produce recognizable spectral features, and they often constitute the object of study of farinfrared spectroscopy. (b) Electron−phonon coupling is linear in the bond length variations. The energy transfer with the phonon bath is activated by the sudden lattice distortion caused by the change in the state of the bonding electrons, and the distortions of the molecule, crystal, or hosting medium are governed mainly by the interaction energy terms linear in the ionic displacements. Hence linear electron−phonon interaction terms determine the peak profiles and shifts and are more than a high-order refinement. Higher order terms modify the frequency of the modes or the specific volume and may be important only for transitions involving substantial modification of bond configuration. No other simplifying assumption or approximation, additional to linear electron−phonon coupling, is made. When the electronic orbital undergoing the transition weakly affects the equilibrium configuration of the neighboring ions, the spectral feature is narrow, with breadth on the order of the Debye energy or smaller and manifestly asymmetric. Infrared absorption peaks are usually in this category. Conversely, the breadth of spectral peaks associated with electronic transitions producing substantial lattice distortions may be much larger than the Debye energy of the solid. This kind of transition has Gaussian shape as the asymptotic limit for large width. The inherent asymmetry takes the form of a Stokes shift of the whole distribution toward higher or lower photon energies in the absorption or emission spectral bands, respectively.13,14 The F center, that is, a crystallographic defect of ionic crystals in which an anionic vacancy is filled by one or more electrons, belong to this class. Because of the absence of a positive ionic core, the F-electron wave functions are very extended and cause strong lattice distortions upon excitation. The corresponding absorption and emission bands are particularly wide, almost identical and approximately Gaussian.14−17 The focus is now on

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LINESHAPE FUNCTION FOR LINEAR COUPLING The Hamiltonian. The derivation of a general expression for the line shape function, satisfying the hypotheses stated above, is available in the relatively recent literature.12 The normalized distribution of photon energies is obtained with the methods of quantum field theory, with no additional simplifying assumption or approximation that may compromise its applicability within the starting conditions. The resulting expression has the form of the integral of a rapidly oscillating function and hence is hard to work out by standard numerical methods. However, an asymptotic limit of it, holding for line widths on the order of the Debye energy of the solid or smaller, has been written in closed form and was shown to fit with high precision the asymmetric profiles of many experimentally observed infrared absorption bands.12,13 The general integral line shape function is studied in what follows for any strength of the electron−phonon coupling, that is, any line width, with the purpose of deriving explicit analytical expressions for the line shape, useful for the quantitative interpretation of the spectra. The Hamiltonian of the material medium interacting with the electromagnetic field can be separated as12 H = H0 + H1 + H2 + H3

(1)

where H0 =

∑ ℏωq⎛⎝aq†aq +

1 ⎞⎟ + 2⎠

gqS cS†cS(aq

aq†̅ )

⎜

q

+

∑ qS

H1 =

S

∑ ∑ gqS′ ScS†′cS(aq − aq†̅ ) S≠S′

H2 =

−

∑ εScS†cS

q

∑ ℏckην†k ⃗ηνk ⃗ νk ⃗

(2)

(3)

(4)

and H3 =

∑ Q S′ Sνkc⃗ S†′cS(ηνk ⃗ − ην†(−k ⃗)) S′S

(5)

Here H0 is the Hamiltonian of the condensed material medium and aq is the phonon operator of the mode q = (μ,q⃗), with wavevector, frequency, and polarization vector q⃗, ωq, and êq. The index μ characterizes the branch of the mode, or identifies localized modes associated with eventual impurities or crystal defects. It is denoted also q̅ ≡ (μ, − q⃗). The Fermion operator cS†, where S = (α,l),⃗ creates an electronic state ψα(r ⃗ − l),⃗ bound to the ion core at the site l,⃗ with α labeling the excitation state. The operator cS†cS accounts for the occupancy of the internal state of the ion located at l.⃗ By the hermiticity of H0, gqS* = − gq S . ̅

The system represented by H0 has a class of stationary states satisfying ⟨cS†cS⟩ = 0 for any S . They describe the unperturbed system with the ionic cores playing harmonic oscillations around the equilibrium configuration, with all of the electronic orbitals in ground state and responding adiabatically to the nuclear motions. The second and third terms of H0 open chances for the electronic orbitals to be excited, modifying in B



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general the equilibrium configuration the ionic cores by the term linear in the phonon operators. The term H1 expresses that the electronic states of each atomic constituent of the material medium are disturbed by the presence of their neighbors. The diagonal electron−phonon coefficients gqS are associated with local distortions of the

where

GqS ′ S =

(6)

are the eigenstates of H0 in which only one electronic quantum number is allowed to be excited. They represent well a solid in which the concentration of electrons in excited single−electron states is low enough to disregard their interaction. The energy eigenvalues13 ⎛

∑ ℏωq⎝nq + ⎜

q

1 ⎞⎟ + εS − 2⎠

∑ q

|gqS |2 ℏωq

FS ′ S(ℏck ; T ) =

q

q

μ ⎛ 1 ⎞ exp⎜− |GqS ′ S|2 ⎟|GqS ′ S| μq Lnqq(|GqS ′ S|2 ) ⎝ ⎠ (nq + μq )! 2

q

(12)

(13)

In eq 13, T is the absolute temperature of the sample and ES′S = ES′ − ES is the energy difference between the two phonondressed electronic states. The general expression for the line-shape function, eq 13, is as exact as the Hamiltonian defined by eqs 1−5 because no additional assumption or approximation was introduced in the derivations.

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MODELS FOR THE THERMAL BATH AND ITS COUPLING WITH THE ELECTRON FIELD Long-Wavelength Vibrational Modes. The main hypothesis of the present calculation of optical band shapes is that the thermal bath is constituted by the acoustic modes of the extended medium hosting the electronic orbital going through the transition. Acoustic modes have linear dispersion relations with no energy gap, and hence any local electronic transition involving configurational changes is expected to be accompanied by the creation and annihilation of many lowenergy phonons. These are the elementary excitations of long vibrational waves, with wavelengths much longer than the bond lengths, which extend all over the hosting medium. Therefore, the energy-dependent spectral distribution of the photons emitted or absorbed and the transition time rates depend on the properties of the condensed medium surrounding the molecule or crystal cell comprising the orbital undergoing the transition. Regarding eqs 11 and 12, the line shape function FS′S (ℏck,T) is entirely determined by the diagonal electron− phonon coupling coefficients gqS and the dispersion relations ωq of the vibrational modes of the acoustic branch.

(8)

(9)

⟨S′{nq + μq }|cS†′cS|S{nq}⟩

∏

⎩

∞

between two states with electronic quantum numbers S and S ′ depends on the dispersion relation ωq of the vibrational modes of the condensed embedding medium. Thus, line shifts are expected to occur in the spectra of the same molecule in different hosting media. General Expression for the Transition Rate. It has been shown that the matrix elements of cS†′cS between the eigenstates |S {nq}⟩ of H0 can be written as13

=

⎪

∫−∞ d(ℏck)FS′S(ℏck ; T ) = 1

is the phonon−dressed energy of the excited orbital. The transition energy

ES ′ S = ES ′ − ES

⎧ ⎪

which satisfies

|gqS |2 ℏωq

∞

∫−∞ dt exp⎨∑ |GqS′S|2

⎛ E − ℏck ⎞ exp⎜i S ′ S t⎟ ⎝ ⎠ ℏ

, nq = 0, 1, 2, ...

have a first term representing the energy of the ions vibrating around their new equilibrium positions, nq being the excitation number of the distorted mode q, a second term giving the energy of the one−electron orbital in the rigid undistorted environment, and a third term accounting for the energy released when the configurational degrees of freedom are allowed to accommodate to the new equilibrium positions, consistent with the excited local electronic state S . In the language of field theory, ϵS is the bare energy of the one− electron state S and

∑

1 2π ℏ

⎡ ⎤⎫ ⎛ ℏωq ⎞ ⎪ ⎟(1 − cos(ωqt )) + i sin(ωqt )⎥⎬ × ⎢− coth⎜ ⎪ ⎢⎣ ⎥⎦⎭ ⎝ 2kBT ⎠

(7)

ES = ϵS −

(11)

and is the generalized Laguerre polynomial of the variable x in standard notation. The experimental setup is able to select one-photon energy transfers between the material medium and the electromagnetic field; hence the rate of photon absorption or emission events is given exactly by Fermi’s golden rule applied to the interaction term H3 of the Hamiltonian. Replacing eqs 5 and 10 in Fermi’s rule, the thermal average over the initial vibrational states and the sum over all possible final ones can be done exactly by virtue of the summation properties of the generalized Laguerre and Bessel functions.12 After some mathematical steps, the probability per unit time and energy that a localized electronic state S will undergo a transition to S ′ with the transfer of one photon of energy ℏck, c denoting the speed of light, k = 2π/λ, and λ being the wavelength, turns out to be given by the integral expression12,13

of the electronic states, turning their energy levels into bands of finite width. In general, H1 is a comparatively small term and is usually neglected (Condon approximation). H2 is the Hamiltonian of the free electromagnetic field, and ην†k ⃗ creates a photon with well-defined momentum ℏk⃗ and polarization index ν. The term H3 governs the interaction of the electromagnetic and electron fields. Its coefficients Q S ′ Sνk ⃗ do not depend on the vibrational state of the material medium. Bare and Dressed Electronic States. The set of the states |S {nq}⟩ such that

ES{nq} =

ℏωq

Lμn (x)

lattice, whereas the terms in gqS′S contribute to the hybridization

H0|S{nq}⟩ = ES{nq}|S{nq}⟩

gqS ′ − gqS

nq!

(10) C



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can always be chosen to eliminate such linear terms. In effect, by just replacing

When the hosting medium is homogeneous and isotropic, as an amorphous or polycrystalline solid or a liquid solvent, and waves much longer than the interatomic distances or crystallite sizes are almost the only excited ones, the medium can be assumed to be a material continuum. The configurational degrees of freedom are the displacement u⃗(r)⃗ of each point r ⃗ due to the deformation, and the harmonic part H0′ of the Hamiltonian H0 takes the form H0′ =

∫

⎛ pp β ∂ui ∂ui ⎞ α ∂ui ∂uj ⎟ d3r ⃗⎜⎜ i i + + 2 ∂xi ∂xj 2 ∂xj ∂xj ⎟⎠ ⎝ 2ρ

ui( r ⃗) = ui′( r ⃗) + δi( r ⃗)

in the expression for the harmonic Hamiltonian H0′ , where δi(r)⃗ is a vector field such that [ui(r)⃗ , δj(r ′⃗ )] = 0, one can express H′0 in terms of the new displacements u′i while adding or removing eventual linear terms from it. The commutation relations in eq 15 hold for the new variables ui′, and hence their canonical conjugates remain the same. Physically, the procedure has a very important meaning because it involves the choice of the equilibrium configuration. Terms linear in the strains are indicative of a stress field causing a distortion δi(r)⃗ from the assumed equilibrium condition. Hence, if a linear term

(14)

which is quadratic in the strain components ∂ui/∂xj and invariant to an arbitrary rotation of axes. In eq 14, use is made of Einstein’s summation convention over indexes appearing twice in a single term, ρ is the density, xi, xj = x, y, z are the components of r,⃗ and pi, pj = px, py, pz stand for the canonically conjugated variables of ui, uj = ux, uy, uz in the sense

∂u

∫ d3r σ⃗ ij ∂xi

is added to H0′ , the transformation given by eq 22 takes it back to the purely quadratic form of eq 14 by choosing a deformation field δ⃗(r)⃗ such that

(15)

and both α and β are coefficients related to the elastic constants. If β = 0, the system supports just pressure waves because ∇·u⃗ = ∂ui/∂xi is the volume variation per unit volume of the material medium and α is the bulk elastic modulus. Assuming periodic boundary conditions in the volume V = LxLyLz, which encloses N = NxNyNz molecules or crystalline cells of the hosting medium, the transformation ℏ eq̂ (aq − aq†̅ ) exp(iq ⃗ · r ⃗) 2Vρωq

∑ q

ℏρωq

p ⃗ ( r ⃗) = i ∑

2V

q

eq̂ (aq† + aq ̅ ) exp( −iq ⃗ · r ⃗)

α(∇·δ ⃗)δij + β

(16)

(17)

N 2π ni , ni = 0, ± 1, ± 2, ... ± i , i = x , y , z Li 2 (18)

and the polarization unit vectors of the longitudinal branch μ = l and the two transversal ones, μ = t1 and t2, are elq̂ ⃗ = q ,̂ et̂ 1q ⃗ , and et̂ 2q ⃗ . The former is in the direction of q⃗, and the latter two are in the plane normal to q⃗, in general, êq = −êq.̅ Calling also ωq =

αδμL + β ρ

σij →

(19)

⎛ ⎜

q

1 ⎞⎟ 2⎠

(20)

The phonon commutation relations [aq , aq ′] = 0, [aq , aq†′] = δqq ′

∑ ⟨S′|σij|S⟩cS†′cS

(25)

where vectors |S ⟩ and |S ′⟩ are elements of an orthonormal basis of the space of one−electron states. Long Waves with Discrete Sources in the Molecular Scale. Photon emission and absorption processes, and hence the excitation and de-excitation of the long acoustic waves, occur at discrete sites of a crystalline or molecular structure, and the sources for the vibrational waves are transitions of electronic orbitals whose dimensions are on the atomic scale. Hence, writing an explicit expression for the electron−phonon coefficients gqS demands the introduction of the local structure of the nuclear sites l ⃗ in the interaction term eq 23 of the Hamiltonian. To this aim, notice that expression eq 23 can be expanded as

q

∑ ℏωq⎝aq†aq +

(24)

S′S

the procedure yields H0′ =

∂δi + σij = 0 ∂xj

The new variables u′i represent oscillations around the new equilibrium positions, modified by vectors δ⃗(r)⃗ . In the theory of elasticity the energy per unit volume of a distorted material medium is given by σijεij, where σij and εij = (∂ui/∂xj + ∂uj/∂xi)/ 2 are the stress and strain tensors. Equation 23 is consistent with this because, by the symmetry of σij, its products with the tensor ∂ui/∂xj and with the symmetric part εij of the latter are equivalent. If the undistorted situation is defined as the class of states for which the quantum numbers associated with the electronic degrees of freedom have their ground-state values, the excitation of one of these quantum numbers will add a linear term like the one of expression eq 23 to the purely quadratic Hamiltonian H0′ . Therefore, if the electronic degrees of freedom are to be taken into account, the Hamiltonian for the nuclear motions must incorporate a term of the general form given by eq 23. The coefficients σij do depend on the electronic variables and, in the second quantized formalism, this dependence is made explicit by making the substitution

is replaced in H′0. The index q = (μ,q⃗) characterizes the vibrational modes; the components of the wavevectors q⃗ are given by qi =

(23)

j

[ui( r ), ⃗ uj( r ′⃗ )] = [pi ( r ), ⃗ pj ( r ′⃗ )] = 0, [ui( r ), ⃗ pj ( r ′⃗ )] = iℏδijδ( r ⃗ − r ′⃗ )

u ⃗ ( r ⃗) =

(22)

(21)

ensure that the canonical commutators given by eq 15 are satisfied. The absence of terms linear in the strain components in the harmonic Hamiltonian eq 14 is because the displacements u(⃗ r)⃗

∂σ

∂u

∫ d3r σ⃗ ij ∂xi = ∫ d3r ⃗ ∂∂x (σijui) − ∫ d3r ⃗ ∂xij ui j

D

j

j

(26)



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and that the first volume integral on the right-hand side of the expression written above can be transformed into a surface integral by the divergence theorem of Gauss. The surface integral can be dropped because it corresponds to the external work made by forces applied on the surfaces of the sample, and in our problem the only energy source or sink is a molecular transition. Recalling now the theory of elasticity,18 in the absence of body forces, the i component f i of the force per unit volume acting in a material medium is related to the stress tensor σij by the equation f i = ∂σij/∂xj. To demonstrate this, observe that the total force acting in a portion of the material (the volume integral ∫ V f i dV) is exerted at the surface of that portion by the rest of the medium (the surface integral ∫ S σij dSj). The equation follows from identifying the two integrals and recalling the divergence theorem of Gauss. Hence ∂u


=−

∫ d3rf⃗ ⃗ ·u⃗

j

V=

1 2 +

∑ v(l′⃗ − l ⃗) + ⃗ l ′≠ l⃗

1 2

2

⃗

∂u

=

⃗ ⃗ = − ∑ Fl⃗ ′− l ⃗ · u ⃗( l ) (29)

This equation may seem quite evident when thinking of the material medium as a discrete structure from the beginning. The interest of eq 29 is that it shows quite clearly the consistency between the continuous and the discrete schemes, represented by the Hamiltonian of eq 14 and a discrete system in which the potential energy of interaction between the ions, playing small oscillations u ⃗l ⃗ around the equilibrium positions l,⃗ is a sum of two-body terms V=

1 2

∂u

∫ d3r σ⃗ ij ∂xi j

=

1 2 ×

⃗

SS ′ l ⃗ ≠ l ⃗ ′

→ ⎯

→ ⎯

→ ⎯

⃗ v(l″ − l‴)|α′l′⃗d′⟩·eq̂ ∑ ∑ ∑ ∑ ∑ ⟨α l d⃗ |∇ ⎯→ ⎯ → ⎯ → ⎯ αα ′ l l⃗ ′ → d d′ l″≠l‴ ⃗

q

→ ⎯ → ⎯ ℏ † [exp(iq ⃗ ·l″) − exp(iq ⃗ ·l‴)]cα†l d⃗ c⃗ α ′ l ′→ ⃗ ⎯ (aq − a q ) d′ ̅ 2Vρωq

(33)

⎯r , functions ⟨→ ⎯r |α l d⃗ ⃗ ⟩ In the space of electron coordinates → e e → ⎯ → ⎯ → ⎯ ⃗ ⃗ ⃗ and ⟨ re|α′l′d′⟩ have negligible overlap unless l = l′ and d⃗ = d′. ⎯ → ⎯ → ⎯ ⎯ ⎯r − → ⎯r |∇v(→ re′ assuming Also ⟨→ l″ − l‴)|re′⟩ is a function of → e e → ⎯ ⎯ → ⎯ ⎯r − → re′ is in the vicinity of l″ − l‴. significant values only when→ e The dominant terms of the sum are then those with l ⃗ = l′⃗, d⃗ = → ⎯ → ⎯ → ⎯ d′, and l″ − l‴ = ±d , hence

∑ v(l′⃗ + ul⃗ ′ − l ⃗ − u ⃗l ⃗) ⃗

⃗ l ′≠ l⃗

∑ ∑ ⟨S|∇v( l ⃗ − l′⃗)|S′⟩·(u ⃗l ⃗ − ul⃗ ′)cS†cS′

The Fermion operators create states |S ⟩ constituting an orthonormal basis of the space of one-electron states. As usual, the basis (|S ⟩) is chosen as the eigenstates of a single electron moving in the attractive field of the ionic cores of positive ⃗ charge; the latter stay fixed to their equilibrium positions {l}. The important point is that the basis states may belong to two main groups: those trapping the electron at a single lattice site l,⃗ such as an atomic or ionic orbital, and those in which the electron is confined to the vicinity of two adjacent sites, l ⃗ and l ⃗ + d ⃗ , with similar probability of finding it in any of the two neighborhoods. The latter case corresponds to covalent orbitals and will be the ones which will be considered principally in what follows. Each basis state |S ⟩ = |α l d⃗ ⃗ ⟩ is characterized by a quantum number α determining the excitation state, a vector l ⃗ associated with a site of the crystalline or molecular structure, and a vector d⃗ connecting l ⃗ with a neighboring site. If no covalent bond exists at l ⃗ then d⃗ = 0. Hence

(28)

l ⃗ ′≠ l ⃗

1 2

(32)

Combining this with eq 27, the linear electron−phonon term becomes

j

(31)

cS†

⃗

∂u

⃗

∂li ∂l j

⃗ l ′≠ l⃗


(27)

∑ F l⃗ ⃗− l′δ( r ⃗ − l ⃗)


⃗

⃗ ⃗ ⃗ Recalling ∇v(−l)⃗ = −∇v(l)⃗ and replacing Fl⃗ ′− l ⃗ = −∇v(l′−l), the term linear in the displacements reduces to the right-hand side of eq 29. The interaction v of the nuclear degrees of freedom depends on the state of the electron field. In the adiabatic approximation, the effective energy of interaction between the core ions is the energy eigenvalue of the electronic system for a fixed ionic configuration { l ⃗ + u ⃗l ⃗}. In the scheme of second quantization, the internuclear potentials are held as operators in the space of one-electron states. In particular, the term linear in the displacements is written as

In a discrete lattice or molecular system, forces are localized at the nodes l ⃗ of the structure, while the displacements u⃗ vary slowly on the scale of the distances between nodes. Therefore, writing the total force exerted on the ion at l ⃗ by the rest of the ions as a sum of two−body interaction terms ∑l ′⃗ F l⃗ ⃗ − l ′⃗ , one sees that f ⃗ ( r ⃗) has the general form

l ⃗≠ l ⃗′

⃗

∑ ∇v(l′⃗ − l ⃗)·(ul⃗ ′ − u ⃗l ⃗) ⃗ l ′≠ l⃗

∑ uli′ ∂ v(l′ − l ) u lj⃗

j

f ⃗ ( r ⃗) =

1 2

(30)

This is an important issue because the source of the acoustic waves dissipating part of the energy available for the electronic transition always has a discrete structure. However, the medium of the waves may be a crystalline or amorphous solid, or a liquid, and may be eventually better described by the continuous model. Equation 29 tells us that the Hamiltonian eq 14 will be suited for the three situations, no matter the discrete character of the source, provided the waves have long wavelengths. Expanding in powers of the displacements and retaining terms up to the second order, the interaction energy can be written as

∂u

∫ d3r σ⃗ ij ∂xi j

⃗ v(d )⃗ |α′ l d⃗ ⃗⟩·eq̂ = 2i ∑ ∑ ⟨α l d⃗ |∇ ⎯ αα ′ ql→ d⃗

ℏ 2Vρωq

⎡ ⎛ d ⃗ ⎞⎤ ⎛ d ⃗ ⎞ × exp⎢iq ⃗ ·⎜ l ⃗ + ⎟⎥ sin⎜ ·q ⃗⎟cα†l d⃗ c⃗ α l d⃗ (⃗ aq − aq†̅ ) ′ ⎢⎣ ⎝ 2 ⎠⎥⎦ ⎝ 2 ⎠

(34)

Therefore, the coefficients of the diagonal part of the linear electron−phonon coupling term E



∑ gqScS†cS(aq − aq†̅ )

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condition that there is a total number N of different wavevectors q⃗ in the sample, that is

(35)

Sq

V (2π )3

of the Hamiltonian H0 are gqS = −2iFα⃗ l d⃗ ⃗ ·eq̂

⎡ ⎛ ℏ d ⃗ ⎞⎤ ⎛ d ⃗ ⎞ exp⎢iq ⃗ ·⎜ l ⃗ + ⎟⎥ sin⎜ ·q ⃗⎟ , S ≡ (α l d⃗ ⃗) 2Vρωq 2 ⎠⎥⎦ ⎝ 2 ⎠ ⎣⎢ ⎝

μ

∑ (eμ̂ q ⃗· i )̂ 2 sin 2(dqx) μ

μ

3(ΔF )2 π 2ℏρvs3 ℏvs β= 2akBT v τ= st 2a

(37)

J (τ ; T ) =

When the orbital going through the transition is an hybridized state with several lobes then

I (τ ) =

⎡ ⎛ ℏ d ⃗ ⎞⎤ ⎛ d ⃗ ⎞ exp⎢iq ⃗ ·⎜ l ⃗ + ⎟⎥ sin⎜ ·q ⃗⎟ ⎢⎣ ⎝ 2Vρωq 2 ⎠⎥⎦ ⎝ 2 ⎠

q

(40)

(41)

Long wavelength acoustic modes have the main role in the energy exchange with the thermal bath; then, a Debye model should provide a precise enough description of the frequencies ωμq ⃗ . They can be written accordingly as ⎪

⎪

a π ℏvs

∞

⎧ ⎪

∫−∞ dτ exp⎨⎩−αJ(τ ; T ) ⎪

(48)

Written in terms of the Debye temperature ΘD = ℏvsqD/kB, the definitions in eq 45 read β = 8.081 × 10−2ΘD/T and τ = 1.058 × 1010ΘDt, with t in seconds, for photon absorption or emission by an electronic orbital of octahedral symmetry placed inside a face-centered cubic cell. In this particular geometry, which is quite illustrative of the general case, d = a/2 and aqD = 2(3π2)1/3, and the auxiliary functions J(τ;T) and I(τ) appear as shown in Figure 1 for T = ΘD/2. It is seen in Figure 1 that J(τ;ΘD/2) vanishes and has a minimum at τ = 0, exhibits an inflection point close to τ = 0.2 and a maximum near τ = 0.43, and then goes asymptotically to a constant value J(∞;ΘD/2) = 2.0644. There are two essential points to be noted in the comparison of the two curves: (a) Function I(τ) is practically linear in the neighborhood of τ = 0 in which the other function has a minimum and can be represented by a parabola. (b) Function I(τ) has a maximum at τ = 0.2, very close to the point in which the other function has the inflection point. Therefore, J(τ;T) varies linearly with τ in a neighborhood of the point in which I(τ) is stationary. Properties (a) and (b) of the auxiliary functions J(τ;T) and I(τ) hold at any temperature and are expected to be important in the evaluation of the line shape function Fl′l(ℏck;T)

∫ d3q ⃗ ∑

⎧ vsq q = |q ⃗| ≤ q D ωμq ⃗ = ⎨ ⎩0 otherwise

(47)

⎪

2

μ

dx ⎛⎜ sin x ⎞⎟ 1− sin(2τx) ⎝ x x ⎠

(46)

⎪

Model for the Acoustic Modes and Explicit Expressions for the Line-Shape Function. As the wavevectors q⃗ of the vibrational modes q = (μ, q⃗) form a quasi-continuum of density V/(2π)3 in the first Brillouin zone, defined in eqs 18, the sum over q can be carried out by making the substitution V (2π )3

aqD

dx ⎛⎜ sin x ⎞⎟ 1− coth(βx) sin 2(τx) x ⎝ x ⎠

⎡ ⎤⎫ 2a + i ⎢αI (τ ) − (ℏck − El ′ l)τ ⎥⎬ ℏvs ⎣ ⎦⎭

∑ |GqS′ S| = 2 |ΔF |3 [sin 2(dqx) + sin 2(dqy) + sin 2(dqz)] Vρℏωq μ

∑→

∫0

aqD

Fl ′ l(ℏck ; T ) =

where vectors d⃗ point along the symmetry directions of the lobes. If the orbital has octahedral symmetry with six orthogonal lobes

(octahedral symmetry)

1 2

∫0

the line shape function 12 turns into

(39)

2

(45)

and the auxiliary functions (38)

→ ⎯ d

(44)

α=

2 |ΔF |2 sin 2(dqx) (simple bonding between neighbors) Vρℏωq3

gqS = −2i ∑ Fα⃗ l d⃗ ⃗ ·eq̂

V N

The Debye model is usually employed when dealing with crystalline solids. However, amorphous solids and liquids also exhibit linear variation of the frequency with the magnitude |q ⃗| = 2π /λ of the wavevector. The latter has a cut-off qD = π/a because the wavelength λ cannot be smaller than twice the molecular dimension a. Introducing the adimensional magnitudes

where i,̂ j,̂ and k̂ are the unit vectors along the coordinate axes. As the polarization vectors eμ̂ q ⃗ (μ = l, t1, t2, longitudinal and two transversal modes) form an orthonormal basis, the sum over μ on the right-hand side gives unity, and then

∑ |GqS ′ S|2 =

(43)

D

aqD = (6π 2)1/3 , a3 ≡

⃗ v(d ⃗)|α l d⃗ ⃗⟩ is the expectation value of where Fα⃗ l d⃗ ⃗ = −⟨α l d⃗ |∇ the force between the ionic cores at l ⃗ and l ⃗ + d⃗, for an excitation state α of the bonding orbital. This way, for a transition between two states |α l d⃗ ⃗⟩ and |α′ l d⃗ ⃗⟩ of a bonding orbital connecting two neighboring sites situated along the x axis, in which only the excitation index varies, the magnitudes |GqS ′ S|2 defined by eq 11 satisfy

2 |ΔF |2 Vρℏωq3

d3q ⃗ = N

which gives

(36)

∑ |GqS′ S|2 =

∫|q ⃗|≤q

(42)

where the speed of sound vs does not depend on the branch index μ. The cut-off wavenumber qD is determined by the F



Article

The parameter α may vary over a wide range. The excitation or de-excitation of an ionic core inner orbital, exchanging a high-energy photon with the electromagnetic field, has little effect on the cohesive forces of the crystal lattice or molecular structure. Thus, ΔF, and hence α, are expected to be very small in such kind of transitions, and the line shape becomes simply Fl′l(ℏck;T) = δ(ℏck − El′l) (α → 0). However, notice that the zero-phonon intensity decays rapidly with α. For α = 1 and T = ΘD/2, which is generally close to room temperature, the intensity of the zero-phonon line is exp(−2.0644) = 0.127. As the line shape function is normalized, the contribution of the integral is 0.873.

■

EXAMPLES OF APPLICATIONS Emission Spectra. The open circles in Figure 3 represent the measured fluorescence spectrum of an aqueous solution of

Figure 1. Auxiliary functions J(τ;T) and I(τ) for temperature T = ΘD/ 2. In general, J(τ;T) = J(−τ;T) and I(τ) = −I(−τ).

involving transitions of orbitals that couple strongly to the acoustic modes. Figure 2 depicts J(τ;T) for four very different

Figure 3. Comparison of the experimental fluorescence spectrum of the commercial dye Lucifer Yellow CH in water with the curve given by eq 50 for the three parameters shown in the inset. Notice the high precision of the dressed energy E = El′l of the electronic transition given by the fit between theory and experiment. The maximum of the spectral distribution is 0.322 eV below the energy released by the transition.

Figure 2. Auxiliary function J(τ;T) for T varying over a wide range. The function conserves its general shape, and the inflection point always remains near τ = 0.2. At T ≲ ΘD/10, function J(τ;T) has negligible variation with T and is proportional to T for T ≳ ΘD/2.

temperatures. At temperatures T ≪ ΘD/2, function J(τ;T) is almost independent of T, and for T ≳ ΘD/2, the temperature dependence goes asymptotically to a scale factor linear in T. The inflection point keeps close to τ = 0.2 at any temperature. When τ → ±∞ function, J(τ;T) converges rapidly to the constant J( ±∞ ; T ) =

1 2

∫0

aqD

dx ⎛⎜ sin x ⎞⎟ 1− coth(βx) ⎝ x x ⎠

Lucifer Yellow CH, a commercial fluorescent dye used in cell biology,19 provided by Invitrogen molecular probes of the company Life Technologies (www.fluorophores.tugraz.at/ substance/410). The solid line depicts the calculated energydependent photon intensity Fl′l(ℏck;T), as given by eq 50, with the parameters α, β and E = El′l indicated inside the Figure, and taking the same normalization for the two curves. The parameter E = 2.580 eV, which follows from the fitting, is the dressed energy of the electronic transition, that is, the full amount of energy released by the orbital undergoing the emission process. The maximum of the spectral distribution is at Emax = 2.258 eV, which is 0.322 eV below the available energy E; the latter figure provides an estimation of the energy taken by the heat reservoir and yielding the Stokes shift. The procedure constitutes a method for measuring E = El′l with a precision that, according to Figure 3, is of about ±1% of the fwhm of the peak. In the next subsection, a more precise way to estimate the error is explained. Figure 4 shows similar data on the fluorescence of the fragrant organic chemical compound Coumarin 1, which has a number of applications (www.fluorophores.tugraz.at/ substance/567). The fit of the theoretical curve to the experimental data gives E = 3.00 eV for the dressed energy

(49)

2

which follows from replacing sin (τx) by its average 1/2 in eq 46, and I(τ) drops to zero with the same speed. Then, the absolute value of the subintegral function in eq 48 for the line shape does not vanish at τ = ±∞, and it is much better to express it as ∞ a dτ {exp[−αJ(τ ; T )] Fl ′ l(ℏck ; T ) = π ℏvs −∞

∫

− exp[−αJ(∞ ; T )]} ⎧⎡ ⎤⎫ 2a (ℏck − El ′ l)τ ⎥⎬ × exp⎨i⎢αI(τ ) − ℏvs ⎦⎭ ⎩⎣ ⎪

⎪

⎪

⎪

+ exp[−αJ(∞ ; T )]δ(ℏck − El ′ l)

(50)

to separate explicitly the zero-phonon line, described by the δfunction, and deal with a converging integral. G



Article

Figure 5. Open circles are representative of the time rate of photon absorption by Lucifer Yellow CH dissolved in water. The spectrum is less clean than the corresponding emission spectrum. The contribution of transitions alternate to the one evidenced in Figure 3 is apparent. Solid line represents eq 50 for absorption, with the same values for α and β of Figure 3. The transition energy E = 2.515 eV shows a small shift of 2.5% from the value that fits the emission spectrum.

Figure 4. Fluorescence spectrum of the commercial dye Coumarin 1 in ethanol, as given by experiment and by eq 50 with the three parameters shown in the Figure. The dressed energy E = ES ′ S of the electronic transition is given by the fit between theory and experiment with a precision better than 1% of the fwhm. The most probable photon energy is 0.238 eV smaller than the energy released by the electronic transition. The steeper side of the theoretical curve shows wavy sectors evidencing the convergence of the integral in eq 50 reduces there, particularly for small values of α.

of the electronic transition causing the photon emission. The accuracy of this Figure is again on the order ±1% of the fwhm. The spectral distribution has its maximum at Emax = 2.762, 0.238 eV less than the total energy released. Notice that the steeper side of the theoretical distribution shows two wavy sectors near 2.90 and 3.15 eV, produced by the slow convergency of the integral of eq 50 for lower values of α. Absorption Spectra. Fluorescence spectra allow for more precise conclusions than the absorption spectra because photon-emitting transitions between different pairs of electronic states do not decay in general at the same rate and hence can be distinguished one another by the time they take to occur. Transitions with larger half life can be neatly observed by waiting enough time after finishing the irradiation burst to allow for the decay of the faster ones. On the contrary, photon absorption takes place during irradiation, and all of the allowed excitation processes occur in the same time interval. Absorption spectra are then more complex than the emission ones, as is observed in Figures 5 and 6, which display the absorption spectra of Lucifer Yellow CH in water and Coumarin 1 in ethanol. The electronic transition causing the emission features of Figures 3 and 4 has an important role in the absorption spectra described by open circles in Figures 5 and 6, but some mixing with other processes is also apparent. The solid lines in Figures 5 and 6 represent Fl′l(ℏck;T), as given by eq 50, for absorption processes and with the same numerical values for the parameters α and β obtained previously from the fit to the fluorescence data of the two compounds, and just E was slightly varied in both cases. The maximum of the Lucifer Yellow absorption spectrum is at Emax = 2.872 eV, while the energy released by the transition of the electronic orbital is E = 2.515 eV, that is, 0.357 eV less. The absorption spectrum of Coumarin 1 has its maximum at Emax = 3.304 eV, that is 0.244 eV above the corresponding energy released by the electronic transition. The electromagnetic field is the primary source of energy in photon absorption processes. The interesting point to be noted here is that the magnitude El′l is a characteristic of the two dressed electronic states of the

Figure 6. Open circles represent the time rate of photon absorption by Coumarin 1 in ethanol. A wider energy domain is shown to illustrate the contribution of alternate absorption processes, additional to the transition evidenced in Figure 4. The solid line represents eq 50 for absorption, with the same values for α and β of Figure 4. The transition energy E = 3.06 eV shows a small shift of 2.0% from the value that fits the emission data.

orbital involved in the transition and must be independent of any circumstance of the process, as is its direction, excitation, or de-excitation. Disregarding the fact that the data on photon absorption seem to be less clean than the fluorescence spectra, one can estimate the errors in the neat transition energies El′l as |ΔES ′ S| = 2.580 − 2.515 eV = 0.065 eV (Lucifer yellow CH) |ΔES ′ S| = 3.06 − 3.00 eV = 0.06 eV (Coumarin 1)

(51)

that is ES ′ S = 2.58 eV ± 2.5% (Lucifer yellow CH)

■

ES ′ S = 3.00 eV ± 2.0% (Coumarin 1)

(52)

AUTHOR INFORMATION

Corresponding Authors

*M.L.: E-mail: [email protected]. *R.P.: E-mail: [email protected]. H



Article

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was partially supported by the research fund Enlace− Fondecyt, Dirección de Investigación, Universidad de Talca, Chile (M.L. and R.P.) and by CONICYT, FONDECYT/ Regular 1131044, Chile (R.P.)

■

REFERENCES

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Shape of the Absorption and Fluorescence Spectra of Condensed

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