19082
J. Phys. Chem. B 2005, 109, 19082-19089
ARTICLES Nonlinear Infrared and Optical Responses of a Holstein-Peirls-Hubbard Dimer R. Pilot and R. Bozio* Department of Chemical Sciences and INSTM Research Unit, UniVersity of Padua, Via Marzolo, 1, I-35131 PadoVa, Italy ReceiVed: January 4, 2005; In Final Form: August 17, 2005
A simple model based on the Holstein-Peierls-Hubbard Hamiltonian has been used to calculate the nonlinear responses at infrared and optical frequencies. The model is applied to a molecular ion radical dimer in order to account for the contribution to the nonlinear responses arising from the intermolecular charge transfer excitations and from their coupling to both intramolecular and intermolecular phonons. A similar calculation has been performed on a model quadrupolar conjugated molecule characterized by intramolecular charge transfer excitations. The calculations are performed according to a collective electronic oscillator scheme by solving the Liouville equation for the bielectronic density matrix. Such a choice allows us to retain the ability, inherent in the Hubbard type models, to fully account for the on-site electron correlation effects. Calculated spectra are reported for one-photon and two-photon absorption (the latter in the form of the imaginary part of the Kerr susceptibility) and for third harmonic generation. Narrow resonances are observed in the infrared, related mostly to the coupling with intramolecular modes. The off-resonant contribution to the nonlinear susceptibility arising from the electron-phonon interactions appears to be marginal (in the order of 1%) in all cases.
I. Introduction For the calculation of microscopic nonlinearities, there are two conceptually different quantum-chemistry approaches: the derivative method and the sum-over-state (SOS) method. The first one involves quantum calculations of energy or dipole moment, and the nth order polarizability is related to the nth order derivative of the dipole or to the (n + 1)th order derivative of energy with respect to the electric field. This approach is usually exploited to perform calculations of static polarizabilities.1,2 The second one treats the effect of the electric field as a perturbation on the system and requires the computation of all the eigenenergies and the transition dipoles; the polarizabilities are finally expressed as nominally infinite sums over the excited states.3,4 This approach is the most common for computing dynamic polarizabilities. However, the computation of the eigenenergies and the transition dipoles poses a very difficult many-body problem, particularly since electron correlation is very important in low dimensional systems such as conjugated polymers or push-pull molecules: large scale numerical, full configuration interaction calculations show that nonlinear optical polarizabilities are very sensitive to electron correlation.5 An additional difficulty with the SOS method is the need to perform tedious summations over the excited states. This forces one to work with small systems, or to truncate the summations, which limits the accuracy for large systems in particular because the response results from dramatic cancellation between very large positive and negative contributions.6-8 Some computational * To whom correspondence should be addressed. E-mail: renato.bozio@ unipd.it.
advantages can be achieved by exploiting an alternative approach to the optical responses proposed by Mukamel and coworkers,9-12 which consists of abandoning the eigenstate representation altogether and considering the material system as a collection of “collective electronic oscillators” (CEOs). This approach maps the calculation of the optical responses onto the dynamics of coupled electronic oscillators representing the electron-hole pair components of the single-electron reduced density matrix. It has two main advantages: the first is the ability to represent an ensemble of systems in a compact way through a mixed state,12 and the second is that it avoids the calculation of the complete information of a quantum system, that is, its set of many-electron eigenstates and eigenenergies. The most part of this information is rarely used in the calculation of common observables (energies, dipole moments, spectra) which only depend on the expectation value of a few, typically one- and two-electron quantities. The reduced description provided by the single-electron operator takes into account only a small amount of relevant information which is called for and thus is much less expensive in terms of computational effort. Moreover, only a few oscillators turn out to be relevant in terms of contributions to the optical response.13,14 The computational advantages of this method, with respect to the SOS one, make it very useful for computing the optical excitation of large conjugated systems including linear polyenes, donor/acceptor substituted oligomers, poly-phenylenevinylene (PPV) oligomers, chlorophylls, naphthalene and PPV dimers, phenyl-acetylene dendrimers, and photosynthetic light harvesting systems.11 The large conjugated systems just mentioned are good candidates for nonlinear optical (NLO) application because they
10.1021/jp0500384 CCC: $30.25 © 2005 American Chemical Society Published on Web 09/21/2005
Nonlinear Responses of a HPH Dimer
J. Phys. Chem. B, Vol. 109, No. 41, 2005 19083
Figure 1. (A) TTF monomer structure (left) and (TTF+)2 dimer structure (right). (B) Model for TTF dimer.
possess easily polarizable electronic charge (delocalized π electrons). To date, however, the NLO parameters exhibited by π-conjugated molecular systems fall short of the values required for technological applications by 1 or 2 orders of magnitude. A possible solution, reported in the literature,15 is to exploit the high polarizabilities of molecules in an electronic excited state. The main difficulty with this idea is that a two-step process is required: in the first step, the molecules have to be pumped to an excited state, and in the second one, the NLO response is probed. Optical processes in molecular radical ions with unpaired electrons bear some analogies to those of neutral excited molecules: radical ion spectra are in fact characterized by transitions involving half-occupied molecular orbitals (highest occupied molecular orbital (HOMO) for radical cations and lowest unoccupied molecular orbital (LUMO) for radical anions). These transitions have low energies and, usually, high oscillator strengths. Similar transitions can also be observed in transient absorption spectra of electronic excited molecules, that is, spectra of molecules that have been initially brought to an electronic excited state. Adopting a single-configuration description of electronic states, these transitions correspond to the transfer of π electrons from the next-to-HOMO (HOMO-1) to the HOMO in a radical cation, or from the LUMO to the nextto-LUMO (LUMO+1) in a radical anion. It is then interesting to perform a study of the NLO properties of molecular radical ions, in particular if one considers that some of them are chemically and photochemically stable, which makes their study easier than that of neutral exited molecules. It is well-known that tetrathiafulvalene (TTF) produces stable radical cations. Moreover, solid background knowledge exists about this molecule based on experimental measurements, such as UV-vis-NIR, IR, and Raman spectra and analyses of these data based on quantum mechanical models.16-18 Another interesting aspect of these radicals is that they can form stable aggregates of various dimensions where the properties of the single radical, previously described, can be associated with those typical of molecular clusters. For example, in a linear cluster, the charge delocalization is high because electrons can be transferred among all of the molecules of the system: this translates into a high oscillator strength and then into potentially high hyperpolarizabilities. Even in the smallest of these clusters, that is, in a π radical dimer, a new, strong charge transfer (CT) transition appears at energies lower that those of the intramolecular π-π* transitions of the open-shell radicals and with a very high oscillator strength. The latter feature is related to the fact that the CT transition amounts to the transfer of a full electronic charge over intermolecular distances.
The purpose of this work is to characterize theoretically the NLO responses of the smallest radical cluster of TTF, that is, a dimer (see Figure 1A), making use of a Holstein-PeierlsHubbard (HPH) Hamiltonian to model its electronic properties and of the CEO method to calculate the nonlinear susceptibilities. Moreover, it is known from the literature that also vibrations can play an important role in determining the NLO properties: 22 on the basis of this background, the electron-phonon coupling (e-p) has been included in our model as well. The rationale underlying our computational approach is a semiempirical one. All of the relevant interaction parameters in our HPH Hamiltonian are deduced from comparison with an extensive set of experimental data.16-18 Their validation comes from being able to reproduce, with a single set of parameters, the energy and oscillator strengths of electronic and vibronic absorptions, Raman intensities, and resonance Raman excitation profiles. Nonetheless, they remain empirical quantities whose values are likely to be affected by the presence in the real system of additional interactions which are not explicitly accounted for in our model Hamiltonian. Keeping the modeling as simple and physically transparent as possible, on one hand, allows one to obtain insight into the interplay of the main interactions and, on the other hand, opens the way to exploring the evolution of the optical responses with increasing system size and dimensions. The paper is organized as follows. In section II, the model used to describe the dimer is outlined and the final expressions for the polarizability and the second hyperpolarizability are reported. Section III presents the simulated spectra of a radical dimer and of a model system exhibiting intramolecular charge transfer. A summary of the paper is presented in section IV. Appendixes 1-4 provide calculation details and are available in the Supporting Information. II. Theory A. The Dimer Model System. In our approach, a dimer like that shown in Figure 1A is represented by a two-electron, twosite model (Figure 1B). In practice, both TTF+• radicals are represented by one-electron sites: each site corresponds to just one molecular orbital, that is, the half-occupied HOMO of the TTF+• unit. It is known that the most important properties of the dimer related to the CT transition can be described by means of the Holstein-Peierls-Hubbard Hamiltonian:17
H ˆ HPH ) H ˆH + H ˆ EMV + H ˆ EIP + H ˆV
The meaning of the first term on the right-hand side of eq 1 is as follows:
H ˆ H ) -tTˆ + U(nˆ 1vnˆ 1V + nˆ 2vnˆ 2V)
base function + + + + |S0〉 ) x2(aˆ 1v aˆ 2V + aˆ 2v aˆ 1V)|0〉 + + + + 1 |CT(〉 ) /x2(aˆ 1vaˆ 1V ( aˆ 2v aˆ 2V)|0〉
(2)
H ˆ H is the Hubbard dimer Hamiltonian and accounts only for the electronic features of the system related to the CT process.
TABLE 1: Exact Eigenvalues and Eigenvectors of the Hubbard Hamiltonian eigenvector
+ + |T1〉 ) aˆ 1v aˆ 2v|0〉 + + + + aˆ 2V - aˆ 2v aˆ 1V)|0〉 |T0〉 ) 1/x2(aˆ 1v + + |T-1〉 ) aˆ 1Vaˆ 2V|0〉
|1〉 ) b1|S0〉 + c1|CT+〉 ) |2〉 ) b2|S0〉 + c2|CT+〉 ) |3〉 ) |CT-〉 ) Aˆ + 3 |0〉 |4〉 ) |T1〉 ) Aˆ + 4 |0〉 |5〉 ) |T0〉 ) Aˆ + 5 |0〉 |6〉 ) |T-1〉 ) Aˆ + 6 |0〉
b1,2 ) [1 + (E1,2/2t)2]-(1/2)
b1,22 + c1,22 ) 1
1/
(1)
eigenvalue Aˆ + 1 |0〉 Aˆ + 2 |0〉
E 1 ) EE 2 ) E+ E3 ) U E4 ) 0 E5 ) 0 E6 ) 0 E( ) 1/2[U ( (U2 + 16t2)1/2]
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Figure 2. Energy levels and transition dipoles for the Hubbard dimer model. + aˆ iσ
The operators and aˆ iσ are, respectively, the creation and annihilation operators of an electron with spin σ ) v,V on site i; + nˆ iσ ) aˆ iσ aˆ iσ is the number operator for the electron with spin σ on site i. The operator
Tˆ )
∑σ (aˆ 1σ+ aˆ 2σ + aˆ 2σ+ aˆ 1σ)
(3)
transfers one electron from one site to the other. The parameter U is the correlation energy, and t is the charge transfer integral:
ˆ |ψ2〉 t ) 〈ψ1|H
(4)
where ψ1 and ψ2 are the half-occupied HOMOs of radicals 1 and 2, respectively. It should be emphasized that the Hubbard Hamiltonian provides a minimal representation of electronic correlations because the right term in eq 2 makes use of operators, not mean values. The price to pay for this is that the many-electron molecular wave function of each radical is replaced by a single unpaired electron orbital, the HOMO in our case. This is acceptable as long as we neglect the polarization of lower orbitals and the mixing between CT and π-π* transitions. By diagonalizing Hamiltonian (2), exact eigenvalues and eigenvectors are obtained: they are reported in Table 1. By introducing creation and annihilation operators of eigenstates (defined in Table 1), Aˆ + ˆ n, respectively, we get n and A
H ˆH )
∑n EnAˆ +n Aˆ n
(5)
and
1 R ˆ ) - ed(nˆ 2 - nˆ 1) ) R13(Aˆ + ˆ 3 + Aˆ + ˆ 1) + 1A 3A 2 ˆ 3 + Aˆ + ˆ 2) (6) R23(Aˆ + 2A 3A where R ˆ is the dipole operator in the point charge approximation (the point charges are located at the center of charge of the HOMO, which coincides by symmetry with the molecular center), e is the electron charge, and d is the vectorial distance between the sites. These formulas directly identify which transitions are optically (one-photon) allowed. In Figure 2, a level diagram for the system is reported. The fourth term on the right-hand side of eq 1, that is H ˆ v, accounts for vibrational modes, treated as harmonic oscillators. Its expression is
H ˆV )
pωi
pωe (Ve2 + ue2) 4 (7)
∑i 4 (P1i2 + P2i2 + Q1i2 + Q2i2) + ∑e
Qni, Pni, and ωi are the dimensionless coordinate, momentum,
Figure 3. The three intermolecular modes included in the model.
and angular frequency of the ith intramolecular normal mode of the nth molecule; ue, Ve, and ωe are the corresponding terms for intermolecular vibrational modes. By intramolecular (or internal) modes, we mean the vibrational normal modes of individual molecules in the dimer, and by intermolecular (or external) modes, we mean the displacements of one molecule with respect to the other. Figure 3 shows the three symmetric (ag) intermolecular modes included in the model: they are the antiphase combinations of the rigid traslational modes of the radicals in the dimer. The second and third terms on the right-hand side of eq 1 account for the electron-vibration interaction at the linear coupling approximation. It is assumed that intramolecular modes modulate the HOMO site energy while intermolecular modes modulate the charge transfer integral. In fact, they modify the distance and then the overlap between the functions |ψ1〉 and |ψ2〉. The expressions for H ˆ EMV and H ˆ EIP are then
H ˆ EMV ) -
∑i
gi
qiN ˆ
x2
(8)
and
H ˆ EIP ) -
∑e geueTˆ
(9)
where
N ˆ ) nˆ 2 - nˆ 1 and qi )
1 (Q1i - Q2i) x2
gi ) (∂/∂Qi)0 and ge ) (∂t/∂ue)0 are, respectively, the coupling constants for intramolecular (EMV) and intermolecular (EIP) modes. The Hellman-Feymann theorem guarantees that only totally symmetric modes have a nonzero EMV coupling constant. The last point which should be stressed is that eqs 8 and 9 show that the physical difference between intra- and intermolecular vibrations is mathematically translated into a different kind of coupling: intramolecular vibrations can couple only states with different symmetry (operator N ˆ is antisymmetric with respect to the exchange of sites 1 and 2), while intermolecular vibrations can couple only states with the same symmetry (operator Tˆ is symmetric). B. Equation of Motion for the Bielectronic Density Matrix. In our treatment, we aim at retaining the most advantageous feature of the Hubbard model, that is, considering explicitly the electron correlation. For this purpose, we introduce a variant in the CEO method by making use of a bielectronic density matrix defined, in second quantization, as σσ′ + + ) aˆ i,σ aˆ j,σ′aˆ n,σ′aˆ m,σ Fˆ nm,ij
(10)
where the spin indexes must be equal in pairs because there is no correlation between states with different spin functions. Its
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J. Phys. Chem. B, Vol. 109, No. 41, 2005 19085
expectation value is denoted by
eq 17 is rewritten as
σσ′ σσ′ Fnm,ij ) 〈Ψ(t)|Fˆ nm,ij |Ψ(t)〉
ipF˘(t) ) [h(t) + f(t),F(t)]
(19)
h(t) ) hH + hEMV(t) - u -1{F(ω)〈T(ω)〉}T
(20)
f(t) ) hEXT(t) - u -1{G(ω)u{E(t)〈N(t)〉}}T
(21)
(11) where
The Liouville equation is σσ′ σσ′ ipF˘nm,ij ) 〈Ψ(t)|Fˆ nm,ij ,H ˆ |Ψ(t)〉
(12) and
where
ˆ EXT H ˆ )H ˆ HPH + H
(13)
and
1 ˆ ‚E(t) ) ed‚E(t)N H ˆ EXT ) -R ˆ 2
(14)
uz 1 N ˆz R ˆ ) ed0 1 + 2 d0kz
(15)
[
]
d is the distance between the sites, d0 is the equilibrium distance, kz ) x(2Mωz/p), M is the reduced mass of the dimer, and E is the electric field which has been chosen to be polarized along z. In HEXT, the dependence of intermolecular distance on intermolecular vibrational modes is taken into account as expansion series at first order. The dimer is assumed to be in a perfectly eclipsed configuration along the z axis. With this assumption, the only nonzero coupling constant is gz, and then, only uz is coupled.18 (In real structures, such as (TFF+•)2(W6O12), TTF dimers are not perfectly eclipsed but they present a small slippage of 0.5 Å along the y direction: this allows the system to be polarized along the y direction too and allows the constants gx and gy to be non zero. However, the largest component of polarization is by far along z and gz is remarkably larger than the other two:18 this justifies the assumption of a perfectly eclipsed dimer.) Since R and E have only the z component, from now on, Rz and Ez will be indicated simply as R and E. C. Solution of the Equation of Motion in the Liouville Space. If the expressions for the vibrational operators are considered explicitly, it can be realized that eq 12 is not in closed form because the right term contains three-electron terms of F. The equation can be recast and solved in the random phase approximation (RPA),19 that is, by substituting vibrational operators by their expectation values. The time dependent total Hamiltonian H ˆ (t), rewritten in the RPA, is reported in eq 16; Appendix 1 (Supporting Information) shows explicit calculations and the definitions for all of the terms on the right-hand side of eq 16:
hˆ T(t) ) hˆ H(t) + hˆ EMV(t) + hˆ EIP(t) + hˆ EXT(t)
(16)
F(ω) and G(ω) are defined in eqs A1-13 and A1-14; u {f(t)} is the Fourier transform of f(t) and is defined in eq A2-9. h(t) collects all of the terms that do not show explicit dependence on the external electric field and f(t) the other ones. The eigenvalues and eigenvectors of h(0) T coincide with the ones of hH, shown in Table 1, provided the charge transfer integral is redefined at the new equilibrium distance induced by electron-phonon coupling (see Appendix 3 in the Supporting Information). Equation 19 is solved in a perturbative way20 by expanding the density matrix in a power series of E:
F(t) ) F(0) + F(1)(t) + F(2)(t) + ...
The Hamiltonian hT(t) contains F(t) in the expectation values of the vibrational operators which are consequently expanded in a power series:
〈O〉 ) 〈O〉(0) + 〈O〉(1) + 〈O〉(2) + ...
(17)
Physically, the system can interact with the field E only if it has a dipole moment R; thus, if the two electrons occupy the same site, this consideration suggests that only singlet states evolve in time. This is mathematically confirmed by the term (1 - δσσ′) which is nonzero only if two electrons have different spins. Including only the singlet eigenvectors of h(0) T and making use of the new definition
hT(t) ) h(t) + f(t)
(18)
(23)
where 〈O〉(n) ) Tr[OF(n)] and O ) uz,qi. The essential feature of our calculation is that the vibronic perturbation to the electronic states and to the intermolecular distance is only considered to first order but the perturbation by the electric field is carried out to all appropriate orders for the calculation of the expectation values of vibronic operators. Appendix 2 (Supporting Information) provides all calculation details for the solution of eq 19. D. Polarizabilities and Hyperpolarizabilities. In the frequency domain, that is, assuming a monochromatic electric field E(t) ) 1/2[Eω1e-iω1t + E-ω1eiω1t], response theory provides the following expressions that relate the response (R) to the perturbation (E):
R(1)(t) ) R(2)(t) )
Equation 12 may then be rewritten within the RPA and in terms of matrices instead of bielectronic operators:
ipF˘σσ′(t) ) [hT(t),Fσσ′(t)](1 - δσσ′)
(22)
R(3)(t) )
0 [R(-ω1;ω1) exp(-iω1t) + c.c.]E1 2
(24)
0 [β(-2ω1;ω1,ω1) exp(-i2ω1t) + 2β(0;ω1,4 ω1) + β(2ω1;-ω1,-ω1) exp(+i2ω1t)]E12 (25) 0 [γ(-3ω1;ω1,ω1,ω1) exp(-3iω1t) + 8 3γ(-ω1;ω1,-ω1,ω1) exp(-iω1t) + c.c.]E13 (26)
R is the polarizability, β the first hyperpolarizability, and γ is the second hyperpolarizability. The two different Fourier components of the second hyperpolarizability, namely, γ((3ω1;-ω1,-ω1,-ω1) and γ((ω1;-ω1,-ω1,(ω1), will be named the third harmonic generation (THG) and Kerr components, respectively, according to the physical meaning associated with them.
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The expectation value for R, calculated from eq A1-15, is
[
]
〈uz(t)〉 1 〈Rˆ (t)〉 ) ed0 1 + 〈N ˆ (t)〉 2 d0kz
2 0x2πE1
〈R((ω1;-ω1)〉(1)
2gzωz
T11 and
p(ωe2 + γe2)
(27)
By introducing the expansion of F(t) in eq 27 and comparing 〈R(t)〉(n) with equations derived from response theory, the following expressions for R and γ are achieved
R((ω1;-ω1) )
〈uz〉(0) )
(28)
γ((3ω1;-ω1,-ω1,-ω1) ) 8 〈R((3ω1;-ω1,-ω1,-ω1)〉(3) (29) 0x2πE13
Y(-ω1) )
8c12E2 E22 - (-ω1 + iΓ12)2
The expression achieved for R was obtained working in Liouville space: Rice21 got the same expression working in the usual Hilbert space. As could be predicted from symmetric considerations, only internal vibrations affect the linear dynamic properties. In fact, in the previous expression, only the propagator for intramolecular vibrations D(ω) appears. Intermolecular vibrations only modify the equilibrium distance (which is a static property) of the system by the term 〈uz〉(0). III. Numerical Calculations
and
γ((ω1;-ω1,-ω1,(ω1) ) 8 〈R((ω1;-ω1,-ω1,(ω1)〉(3) (30) 3 30x2πE1 Appendix 4 (Supporting Information) gives the explicit form of the Fourier components and some calculation details. In a centrosymmetric system, β ) 0 by symmetry. To write the expressions for the polarizabilities, more than one convention is found in the literature. We followed the same convention as in the textbook by Butcher and Cotter.20 At first order, the equations can be solved analytically and the explicit expression for R is
R((ω1;-ω1) )
fosc 0[Ω(-ω1)2 - (-pω1 + iΓ)2]
(31)
where
[(
fosc ) 2 e d0 +
〈uz〉 kz
)]
(0)
c1 E2 and
fosc and Ω are the oscillator strength and the frequency, respectively, of the only oscillator necessary to describe R: due to the simplicity of the model and to the use of the bielectronic density matrix, potential advantages of the CEO method are not exploited in this case. On the other hand, the CEO method allows us to account for the coupling with the vibrational dynamics in a simple way by using the random phase approximation. Note that accounting for this coupling in a SOS scheme would have implied extending the sum to vibrational states. With simple algebraic manipulation, eq 31 may be recast in the form
(
)
〈uz〉(0) e d0 + kz Y(-ω1) R((ω1;-ω1) ) 40 1 + D(-ω1) Y(-ω1) where
TABLE 2: Parameters Introduced in the Model (They Are Deduced from Analysis of Experimental Raman and IR Spectra of the Monomer and of the Dimer16-18) Γi1 ) Γ1i ) 750 cm-1 (i ) 2,3); Γ11 ) 100 cm-1 Γij ) 1500 cm-1 (i,j ) 2,3) t ) 0.30 eV, U ) 1.13 eV, γi ) 20 cm-1 d0,z ) 3.42 Å, d0,x ) d0,y ) 0 Å intramolecular mode
ωi (cm-1)
gi (eV)
q1 q2 q3 q4
512 761 1420 1506
0.067 0.049 0.133 0.042
intermolecular mode
ωz (cm-1)
gz‚kz (eV/Å)
uz
116.5
-0.171
2
Ω(-ω1) ) xE22 + 8c12E2D(-ω1) (32)
2
In this section, numerical simulations of the dimer properties are presented. The simulations were carried out using the parameters shown in Table 2. They were obtained from analysis of experimental IR and Raman spectra of (TTF)2(W6O19).18 Only the ground state is assumed to be populated in the absence of interaction with the field.
A. First Order Spectra. Figure 4 shows the imaginary part of R. In the calculated linear absorption spectrum, the peak labeled “e” corresponds to the charge transfer transition, that is, the |1〉 f |3〉 one (cf. Figure 2); the peaks labeled from “a” to “d” are vibrational resonances of the four intramolecular modes, q1 to q4, introduced in the simulations. Table 3 summarizes the results.
2
(33) Figure 4. Imaginary part of R calculated for the dimer model. The inset shows the infrared region of the spectrum in an enlarged scale.
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TABLE 3: Numerical Results for the Linear Absorption Spectrum of the Dimer (ωi Are the Calculated Frequencies, and Ωi Are the Input (Unperturbed) Frequencies) ωi (cm-1)
Ωi (cm-1)
492 751 1351 1501 11211
512 761 1420 1506
peak a b c d e
It is worth noting that in a centrosymmetric system antisymmetric vibrational modes (i.e., intramolecular vibrations) are allowed but symmetric ones (i.e., intermolecular vibrations) are forbidden. As pointed out in previous work,18 the calculated vibrational frequencies (ωi) are downshifted with respect to the Raman unperturbed frequencies (Ωi). This is a well-known phenomenon observed in the experimental infrared spectra of CT crystals and ion radical salts.17 It is worth noting that also vibronically induced transitions from |S1〉 to the symmetry forbidden |S2〉 excited state are predicted in the framework of Herzberg-Teller coupling but they do not show up in the linear electron-phonon coupling scheme we adopted. This fact can be associated with the approximation we have used to close the equation of motion of the density matrix (RPA). B. Third Order Spectra. Figure 5 shows the imaginary part of the Kerr component. The peaks labeled from “a” to “d” are
Figure 6. THG component of γ calculated for the dimer model. Peak m is very weak and is not appreciable in the scale of the figure.
Figure 7. Imaginary part of the THG component for γ: enlargement in the vibrational region.
TABLE 4: Interpretation of the Resonances for the Spectrum of Figure 5 (ωi Are the Calculated Frequencies, and Ωi Are the Input (Unperturbed) Frequencies) peak
ωi (cm-1)
a b c d e f
482 742 1358 1497 6641 11301
Ωi (cm-1) 512 761 1420 1506 2 ph. |1〉 f |2〉 1 ph. |1〉 f |3〉
q1 q2 q3 q4
TABLE 5: Interpretation for the Resonances of Figure 6 Figure 5. Imaginary part of the Kerr component of γ calculated for the dimer model.
vibrational resonances associated with the four intramolecular modes introduced in the model calculation. Peak d is very weak and can hardly be distinguished in the enlarged spectrum shown in the inset of the figure. As in the linear spectrum, the modes appear downshifted with respect to the input frequencies. Peak e (Figure 5) corresponds to the two-photon absorption (TPA), and peak f corresponds to the saturation of the CT transition. Thus, the system behaves like a saturable absorber when irradiated in the CT absorption region. The negative sign of the vibrational resonances in Figure 5 can be interpreted analogously to the saturation of the CT transition. Table 4 summarizes the results. Figure 6 shows the square of the absolute value for the THG component of γ. The interpretation of the electronic peaks is given in Table 5. To analyze the vibrational region, it is more appropriate to use the imaginary part and not the absolute value. Figure 7 shows the enlargement in the vibrational region for the imaginary part, and Table 6 summarizes the results. All of the
peak
ωi (cm-1)
i l m
3751 6611 10964
3 ph. |1〉 f |3〉 2 ph. |1〉 f |2〉 1 ph. |1〉 f |3〉
TABLE 6: Interpretation of the Resonances for the Spectrum of Figure 7 peak
ωi (cm-1)
a b c d e f g h
166 252 450 485 520 745 1344 1499
3 ph. q1 3 ph. q2 3 ph. q3 1 ph. q1 3 ph. q4 1 ph. q2 1 ph. q3 1 ph. q4
intramolecular vibrations appear both as one- and three-photon resonances; their frequencies are downshifted with respect to the input ones. The three-photon resonance of the q4 vibration is an exception in the sense that it is not downshifted. This is probably due to the simultaneous presence in the same spectral region, near 500 cm-1, of some vibrational resonances which can give rise to interference phenomena. The vibrational
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Pilot and Bozio
Figure 8. Example of quadrupolar structure.
resonances shown in Figure 7 are particularly intense with respect to the electronic ones; this is due to the fact that their bandwidth is very narrow (γ ) 20 cm-1). It should be pointed out that the contribution of vibrations to the NLO response is limited to the region where they are resonant. In fact, far from the IR region, at optical frequencies, the vibrational contribution, estimated by setting the coupling constants to zero, is about 1%. This consideration holds both for the Kerr component and for the THG component. Studies of the vibronic contribution to the hyperpolarizability β of pushpull polyenes are reported in the literature. On the basis of site models, Painelli found a strong contribution to the first hyperpolarizability (β), but in the static limit,22,23 on the other hand, by a quantum-chemistry approach, Agren found for the dynamic second hyperpolarizabilty (γ) (in particular for TPA) just a weak effect of the vibrations on the NLO response.24 By now, we have found, in both components of γ, only one kind of vibrations, that is, the intramolecular ones. From symmetry considerations, however, also intermolecular phonons are expected to appear as two-photon resonances in Kerr and THG spectra; a stronger electron-phonon coupling may make them appreciable. It is reasonable to expect that, in a conjugated symmetric structure, like that shown in Figure 8, the electronphonon coupling is stronger than in the dimer. These quadrupolar structures, including both push-push and pull-pull elongated chromophores, are drawing considerable attention because of their high TPA coefficients which make them good candidates for applications such as optical power limiting and two-photon florescence imagining.25,26 Our aim is then to describe the quadrupolar structure in Figure 8 using the same model adopted for the dimer but with a different set of input parameters. The analogy between the systems can be seen if we think that the donors D (in Figure 8) correspond to the TTF•+ molecule in the symmetric dimer. The π chain plays two roles: the first is to make the electron transfer between the donors in the quadrupolar structure more efficient than that in the dimer; this effect can be translated into an increased value for the parameter t. The second role is to change the relative position of the terminal units by means of its bending and stretching modes: this effect can be introduced in the model replacing intermolecular frequencies with intramolecular ones that are typical of conjugated bonds. Therefore, we can simulate the quadrupolar structure by introducing in the model the set of parameters of Table 7. The spectrum shown in Figure 9 is the enlargement in the region of intermolecular vibrations of the imaginary part of the Kerr component: the intermolecular mode appears as a twophoton resonance, but its intensity is however very low. IV. Conclusions By making use of a simple chemical model, the goal of our investigation has been to study the NLO responses of a symmetric radical dimer and the effects of the vibronic coupling to both intramolecular and intermolecular modes. Since our focus has been on the role of the CT interactions, from the
Figure 9. Enlargement in the intermolecular vibrational region of the Kerr component calculated with the parameters of Table 7.
TABLE 7: Set of Parameter Values Adopted to Simulate a Conjugated Quadrupolar Structure Γi1 ) Γ1i ) 750 cm-1 (i ) 2,3); Γ11 ) 100 cm-1 Γij ) 1500 cm-1 (i,j ) 2,3), d0,z ) 3.5 Å t ) 1 eV, U ) 1.5 eV, γi ) 20 cm-1 intermolecular mode
ωz (cm-1)
gz‚kz (eV/Å)
uz
1500
0.5
standpoint of describing its electronic properties, the dimer has been treated as a two-electron, two-site system where the electronic structure of each TTF radical is replaced by a single site corresponding to the half-occupied HOMO of the radical itself. This amounted to neglecting any possible mixing between CT transitions and localized (say, π*-π type) electronic transitions. To derive the linear and nonlinear responses, we have made use of the density matrix formalism: in particular, we have carried out the calculation by using a bielectronic density matrix. In a two-electron system, it allows one to take into account the correlation effect which has been recognized to be very important in determining the nonlinear response. Moreover, we have made use of the CEO picture, which requires one to work in the Liouville space instead of the Hilbert space and describes the optical response in terms of “electronic oscillators” instead of sums of terms containing dipoles and transition energies as in the usual sum-over-state method. The coupling with the vibrations has been introduced at the linear level; that is, the energies have been expanded up to the first order in the vibrational mode. The novel characteristics of our calculation, with respect to the literature, consist of the choice of the bielectronic density matrix and its association with the CEO picture. From the simulations, we have recognized the appearance of the CT transition and of the antisymmetric vibrational modes in the linear spectrum: their frequencies resulted to be downshifted with respect to the input ones. This is in agreement with a theoretical prediction regarding the case of more than one mode coupled to the same pair of electronic states and finds experimental confirmation in the IR spectra of these systems. The same results were obtained in the literature but by making use of linear response theory.21 The calculation of the third order spectra, THG and Kerr components, represents an original result. The Kerr spectrum showed, for example, that in correspondence with the CT transition the system behaves as a saturable absorber. From the vibration point of view, we have recognized multiphoton resonances for both kinds of vibrations and have found that, with the parameters determined from independent experiments for the TTF radical dimer, the coupling with the intermolecular modes produces much weaker resonances than
Nonlinear Responses of a HPH Dimer that with the intramolecular ones. Moreover, vibrations have shown a very small contribution (about 1%) to the off-resonant NLO response in the THG component as well as in the Kerr component. No evidence of the appearance of vibronic transitions (that is, electronic ones with a quantum of vibration) has been obtained: this is probably due to the scheme of approximation we used to close the equation of motion of the density matrix (random phase approximation). Recently, experimental investigations of the NLO response of polymer films containing (TTF+)2 dimers have been carried out in our laboratory.27 In particular, single-color pump-probe experiments have shown a strong, ultrafast saturation of the intermolecular CT absorption at 800 nm and have allowed us to estimate an excited state lifetime of the CT state of 400 ( 50 fs. We are now undertaking a more thorough investigation of the NLO responses, including resonant and off-resonant measurements in the infrared spectral range. Finally, it is worth noting that the use of Hubbard-type Hamiltonians turned out to be useful for theoretical modeling of a variety of nonconventional condensed matter systems characterized by strong electron correlations. We are planning to explore the applicability and usefulness of our approach for calculating the NLO responses of representative examples of such systems, like 1-D and 2-D conducting organic charge transfer crystals and mixed valence metal chain compounds. In view of this, the study of the (TTF+)2 radical dimer served also as a benchmark system to check that our CEO approach based on the bielectronic density matrix yields results for the linear optical properties in full agreement with those obtained by using the linear response theory. Acknowledgment. We gratefully acknowledge financial support from the Italian Ministry of Education and Research (MIUR) through the FIRB research project RBNE01P4JF “Molecular and organic/inorganic hybrid nanostructures for photonics”. Supporting Information Available: Appendixes 1-4, providing calculation details. This material is available free of charge via the Internet at http://pubs.acs.org.
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