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Multi-Photon Absorption of Molecules
Multi-Photon Absorption of Molecules Peter Cronstrand, Yi Luo and Hans Ågren Theoretical Chemistry, Royal Institute of Technology, AlbaNova University Center, SE-106 91 Stockholm, Sweden
Abstract Recent applications of response theory formulations of Olsen and Jørgensen on multi-photon absorption of molecules have been briefly reviewed. The connection between the calculated microscopic and experimentally measured macroscopic properties is derived. The performance of various computational approaches ranging from Hartree–Fock and Coupled Cluster to Density Functional theory for these spectral properties is analyzed. Using analytic response theory results as reference, the validity of the commonly applied few-states models for twoand three-photon absorption of molecules have been examined. A design strategy for three-dimensional chargetransfer multi-photon absorption systems is presented. Contents 1. Introduction 2. Multi-photon cross-sections 2.1. Macroscopic approach 2.2. Microscopic approach 2.3. Three-photon absorption 3. Response functions 4. Few-states models 4.1. Two-photon absorption 4.2. Three-photon absorption 5. Validity of few-states models 6. Three-dimensional systems 7. Comparison between DFT and ab initio results 8. Conclusion Acknowledgements References
1 2 2 4 6 6 9 9 10 10 13 16 20 20 20
1. INTRODUCTION The field of multi-photon absorption (MPA) can on a superficial level be described as having experienced three phases of activity, each separated with periods of about 30 years; initially a theoretical discovery, a phase of experimental confirmation and applications, and the present phase characterized by a dynamic interplay between experiment and theory. The pioneering prediction by Göppert-Mayer [1] already in 1930 pointed out a truly unique feature; the ability of matter to absorb more than one single photon at a time. In these early days the phenomenon was far from having a chance of being experimentally confirmed and remained therefore as a rather exotic, albeit interesting, aspect of light-matter interaction. The birth of the laser in 1960 released a floodgate of research in particular within the ADVANCES IN QUANTUM CHEMISTRY, VOLUME 50 ISSN: 0065-3276 DOI: 10.1016/S0065-3276(05)50001-7
© 2005 Elsevier Inc. All rights reserved
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evolving discipline of non-linear optics (NLO). So the phenomena of two-photon absorption (TPA) [2] and three-photon absorption (3PA) [3] could soon thereafter be confirmed, in 1961, respectively, 1963. On the theoretical side we consider the formulation of response theory of Olsen and Jørgensen in 1985 [4] as a large step towards and implementation of a practical scheme for large scale molecular calculations of multi-photon absorption (MPA). Traditionally, the physical origin of all MPA processes has been traced to various orders of the nonlinear susceptibilities, χ (1) , χ (3) , χ (5) , . . . . However, much more efficient computational schemes can be obtained by examining the resonance conditions in the sum-over-states (SOS) representation and by identifying products of first, second, third order of transition moments. A decisive simplification is therefore given by an evaluation the multiphoton transition moments as residues of (hyper)polarizabilities or, as encoded in response methodology, as residues of linear, quadratic or cubic response functions. These considerations enabled the MPA cross-sections to be extracted at the same order as the corresponding (hyper)polarizability; i.e., σ TPA from β and σ 3PA from γ . This is a considerable simplification from, for example, evaluating the 3PA cross-section from the fifth order polarizability. Other advantages of response theory for general molecular property calculations are well known, and will probably be reviewed in other contributions to this volume. Calculations using response theory was greatly promoted in the late 90s by the discovery of organic systems with large cross sections along with a growing attention motivated by a new generation of novel photonic technologies [5–15]. Although theoretical modelling has been available at an ab initio level since quite long time, it has not until recently reached a stage where it can match the increasing pool of experimental results. The original response theory work on Hartree–Fock and multi-configurational quadratic response functions have been extended to cubic response functions and a wider selection of wavefunctions, in particular to the coupled cluster hierarchies of electron correlated wave functions, and lately also to density functional theory (DFT). The present implementation of response theory in the program DALTON thus provides a flexible and powerful toolbox for theoretical modelling of multi-photon absorption of which only some aspects will be covered in this review. We start by deriving some basic relations in MPA which later is connected to computational schemes viable for direct response calculations. In the following section we recast the attained expressions into modified sum-over-states (SOS) expressions suitable for truncation into so called few-states models. Henceforth we review a sample of applications starting with examining the validity of the few-states models, followed by some results for explicit three-dimensional systems. Finally, we demonstrate the applicability of DFT by comparing with CC results and present DFT calculations of three-photon absorption for a set of chromophores.
2. MULTI-PHOTON CROSS-SECTIONS 2.1. Macroscopic approach On a macroscopic scale, the MPA processes can be elucidated by considering the rate of absorbed energy per volume unit when subjected to an external electric field, E d absorbed energy (1) = j · E. dt volume time
Multi-Photon Absorption of Molecules
3
The current density, j , induced in a non-magnetic medium with no free charge carriers, can be expanded as ∂ ∂P + c∇ × M − ∇ × Q + · · · (2) ∂t ∂t where the terms represent the electric dipole, magnetic dipole and electric quadrupole polarization. In the field of nonlinear optics the latter terms are neglected in general and the electric dipole term is expanded as j=
P = P(1) + P(2) + P(3) + P(4) + P(5) + · · ·
(3)
where it is sufficient to consider terms of odd order and the highlighted terms because of the time averaging procedure [16] (1)
Pi
(3) Pi Pi(5)
= χij (−ω; ω)Ej e−iωt + c.c. + · · · , (1)
= =
(3) 3χij kl (−ω; ω, −ω, ω)Ej e−iωt Ek∗ eiωt El e−iωt + c.c. + · · · , 10χij(5)klmn (−ω; ω, −ω, ω, −ω, ω)Ej e−iωt Ek∗ eiωt ∗ iωt × El e−iωt Em e En e−iωt + c.c. + · · · .
(4) (5)
(6)
In the expansion of the rate of absorbed energy we can determine the first order contribution of the rate of absorbed energy as (1) (1) ∂P · E = 2ω Im χij (−ω; ω) Ei Ej∗ . (7) ∂t This corresponds to one-photon transitions and the remaining terms can be ascertained as (3) (3) ∂P (8) · E = 6ω Im χij kl (−ω; ω, −ω, ω) Ei Ej∗ Ek El∗ , ∂t (5) (5) ∂P (9) · E = 20ω Im χij klmn (−ω; ω, −ω, ω, −ω, ω) Ei Ej∗ Ek El∗ Em En∗ ∂t which correspond to two- and three-photon absorption. The expressions can be abbreviated nc 2 E (in cgs units) [17] and identifying the further by introducing the intensity, I , as I = 2π one-, two- and three-photon absorption coefficients, α, β and γ , as (1) 2π hω ¯ Im χij (−ω; ω) , nc 24π 2 h¯ 2 ω (3) Im χij kl (−ω; ω, −ω, ω) , β= 2 2 n c 160π 3 h¯ 3 ω (5) γ = Im χij klmn (−ω; ω, −ω, ω, ω, −ω) 3 3 n c which can be related to the phenomenological equations describing the attenuation light beam experiencing OPA, TPA and 3PA as α=
dI = −αI − βI 2 − γ I 3 . dz
(10) (11) (12) of a
(13)
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Trough the relations σ TPA =
h¯ ωβ N
(14)
σ 3PA =
h¯ 2 ω2 γ N
(15)
and
one can finally define the OPA, TPA and 3PA cross-sections, σ OPA , σ TPA and σ 3PA .
2.2. Microscopic approach At a microscopic level, the susceptibilities correspond to (hyper)polarizabilities which in principle are attainable by quantum chemistry methods, but less feasible for larger systems. Instead of evaluating the imaginary part of (hyper)polarizabilities as such we can achieve a considerable downshift of the order of the property by only considering the resonant terms for γ (−ω; ω, −ω, ω) with ω = 12 ωf where ωf is the excitation energy to the final two-photon state |f . Under these conditions it is possible to rewrite γ as γαβγ δ (−ω; ω, −ω, ω) = h¯ −3 P1,3 0|µα |kk|µγ |f f |µβ |mm|µδ |0 × (ωk − ω)(−iΓf /2)(ωm − ω) km 0|µγ |kk|µα |f f |µβ |mm|µδ |0 + (ωk − ω)(−iΓf /2)(ωm − ω) =i
0|µα |kk|µγ |f f |µβ |mm|µδ |0 2h¯ −3 P−σ,2 P1,3 Γf (ωk − ω) (ωm − ω) m k
=i
2h−1
¯ ∗ Sαγ Sδβ . Γf
(16)
Hence, in the vicinity of two-photon resonance it is sufficient to evaluate the two-photon transition matrix elements Sαβ defined as 0|µα |kk|µβ |f 0|µβ |kk|µα |f −1 Sαβ = h¯ (17) + . ωk − ω ωk − ω k
Due to the slow convergence of these so called sum-over-states (SOS) expressions an explicit summation does not form a viable option for ab initio methods, except when they can be truncated to a few leading terms. These truncations have traditionally been considered legitimate for so-called charge transfer (CT) systems, where the excitation scheme is completely dominated by a few major excitation channels. In addition, they may also serve as a valuable tool for interpretation purposes because of their ability to display the relation of the TPA probability, δ TPA , to other somewhat more intuitive quantities. A more computationally efficient approach is offered by residue analysis of the (hyper)polarizabilities. The
Multi-Photon Absorption of Molecules
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single residue of the first order hyper-polarizability can be written as lim (ω2 − ωf )βij k (−ωσ ; ω1 , ω2 )
ω2 →−ωf
1 0|µi |nn|µj |f f |µk |0 0|µj |nn|µi |f f |µk |0 + = (ωn − ω1 − ωf ) (ωn + ω1 ) h¯ n 1 0|µi |nn|µj |f 0|µj |nn|µi |f + f |µk |0. = (ωnf − ω1 ) (ωn0 + ω1 ) h¯ n
(18)
It is clear that the term inside the brackets is connected to the two-photon absorption matrix element given by equation (17), when evaluated for ω1 = −ωf 0 /2. The final step in order to relate the microscopic origin to the macroscopic detection consists in relating the two coordinate systems; that of the laboratory and that of the molecules. Specifically we need to be able to relate the quantity that defines the macroscopic coordinates in the microscopic realm, that is the polarization of light, with the transition dipole moments evaluated at the quantum level. Since experiments rarely are made on single molecules, but an ensemble of molecules which for gases or liquids has no preferred direction, the relation must include a full orientation averaging. Thus, we want to establish a relation such as
0f S (λ, ν) 2 = (λA νB λ∗ ν ∗ )(lAa lBb lCc lDd ) S 0f S 0f ∗ . (19) C D AB DE In the beginning of 1970s, Monson and McClain [18,19] derived the following relations for accomplishing this task δ TPA = F δF + GδG + H δH
(20)
where F , G and H are defined as F = −|λ · ν ∗ |2 + 4|λ · ν|2 − 1, ∗ 2
2
∗ 2
2
(21)
G = −|λ · ν | − |λ · ν| + 4,
(22)
H = 4|λ · ν | − |λ · ν| − 1 and δF =
a,b
∗ Saa Sbb ,
δG =
(23)
∗ Sab Sab ,
a,b
δH =
∗ Sab Sba .
(24)
a,b
For linearly polarized light this implies that F = G = H = 2. The microscopically determined two-photon probability, δ TPA , is subsequently related to the macroscopic cross-section, σ TPA , as 4π 2 αa05 ω2 TPA 24π 2 hω ¯ 2 (3) 4π 2 hω ¯ 2 4 4 −3 TPA = Im χ e a E δ = (25) δ . 0 h c c2 c2 Provided the Bohr radius, a0 and the speed of light are given in cgs units and the frequency, ω and the TPA probability, δ TPA in atomic units the resulting unit will be cm4 s photon−1 . The result can be generalized further by introducing the finite lifetime broadening as σ TPA =
σ TPA =
4π 3 αa05 ω2 TPA δ (ωf − 2ω, Γf ) c
(26)
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through the normalized Lorentzian
(ωf − 2ω, Γf ) =
Γf 1 . π (ωf − 2ω)2 + Γf2
(27)
2.3. Three-photon absorption In complete analogy with TPA, 3PA cross-sections determined by third order transition moments which in turn can be evaluated through a single residue of the second order hyperpolarizability. Again, orientational averaging is required in order to relate the intrinsic coordinates of a single molecule to the ensemble of freely moving particles as measured in the laboratory coordinate system. As for TPA one therefore seeks to evaluate: 2 0f 0f ∗ δ 3PA = T 0f (λ, ν, ξ ) = λA νB ξC TABC λD νE ξF TDEF . (28) According to McClain [20,21] the orientation averaged values for the 3PA probability δ3PA for linearly (L) and circularly (C) polarized light can be written as: 1 (2δG + 3δF ), 35 1 = (5δG − 3δF ), 35
δL3PA =
(29)
δC3PA
(30)
where δF =
Tiij Tkkj ,
(31)
Tij k Tij k .
(32)
i,j,k
δG =
i,j,k
Similar to TPA, we can finally relate the orientationally averaged three-photon absorption probabilities, δ3P , to the three-photon cross-section, σ 3PA , as σ 3PA =
4π 4 a08 α ω3 δ 3PA (ωf − 3ω, Γf )(ω) . 3c Γf
(33)
With the same convention concerning the units as for TPA the final cross-sections will be obtained in units of cm6 s2 photon−1 .
3. RESPONSE FUNCTIONS To some extent response theory [4,22] can be seen as an elaborate scheme of timedependent perturbation theory. By means of this theory we can avoid the explicit summation of the expressions attained in the previous section by solving algebraic equations. Secondly, the formalism is analytically transferable, i.e., the same technique may subsequently be applied to retrieve properties from a wave function irrespective of the actual parameterization of the wave function.
Multi-Photon Absorption of Molecules
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For exact states these can be given in their spectral representation with −A; Bω1 =
P
0|A|pp|B|0 ω p − ω1
p
(34)
as the linear response function and −A; B, Cω1 ,ω2 =
P
0|A|pp|B|qq|C|0 ¯ (ωp + ω0 )(ωq − ω2 )
(35)
p,q=0
¯ as the quadratic response function, where p|B|q = p|B − 0|B|0|q and −ω0 = ω1 + ω2 +· · · . P is the permutation operator. If the chosen operator is the dipole operator, µ, the response functions, µi ; µj , µk , . . .ω1 ,ω2 ,... will correspond to the (hyper)polarizabilities α, β and γ . The response functions contain inherently information about the excited states. The poles determine the location of the excitation energies and further information can be retrieved by examining resonance conditions through a residue analysis. From the expression for the linear response function for an exact wave function, where the unperturbed Hamiltonian is diagonal, we can easily evaluate transition dipole moments between the ground state, |0, and an excited state, |f as lim (ω1 − ωf )µi ; µj ω1 = 0|µi |f f |µj |0.
ω1 →−ωf
(36)
The single residue of the quadratic response function provides information on the twophoton transition matrix elements lim (ω2 − ωf )µi ; µj , µk −ω1 ,ω2 0|µi |nn|(µj − 0|µj |0)|f =− ω n − ω2 n 0|µj |nn|(µi − 0|µi |0)|f + f |µk |0 ω n − ω1
ω2 →−ωf
(37)
where ω1 + ω2 = ωm . From the double residue of the same response function one can deduce the transition dipole moments between excited states lim (ω2 − ωm )µi ; µj , µk −ω1 ,ω2 lim (ω1 − ωf ) ω1 →ωf ω2 →−ωm
= −0|µi |f f | µi − 0|µi |0 |ii|µk |0. (38) We emphasize that this is done from the reference state, |0, preferably the ground state, with no further reference to any excited state. For approximate wavefunctions the two-photon transition matrix element, Sαβ , can within the response terminology be evaluated directly as [4] [2] F B F SAB = −NjA (ωf /2)Bj[2] k Nk (ωf ) − Nj (−ωf /2)A(j k) Nk (ωf ) 1 [3] [3] B F + S − ω S − NjA (ωf /2) Ej[3] ω f j lk f j kl Nj (−ωf /2)Nk (ωf ). (kl) 2
(39)
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In analog, the three-photon tensor elements are evaluated as B C F Tabc = NjA (ωf /3)Tj[4] klm (ωf /3, ωf /3, ωf /3)Nk (−ωf /3)Nl (−ωf /3)Nm (ωf ) B CF − NjA (ωf /3) Tj[3] kl (−ωf /3, ωf /3 − ωf /3)Nk (ωf /3)Nl (−ωf /3, ωf /3) C B + Tj[3] kl (−ωf /3, 2ωf /3)Nk (−ωf /3)Nl (ωf /3)
F BC + Tj[3] kl (ωf , −2ωf /3)Nk (ωf )Nl (ωf /3, −ωf /3) [3] B C F − NjA (ωf /3) Bj[3] kl Nk (−ωf /3)Nl (ωf ) + Cj kl Nk (−ωf /3, ωf ) [2] BF CF + NjA (ωf /3) Bj[2] k Nk (−ωf /3, ωf ) + Cj k Nk (−ωf /3, ωf ) B CF C BF + A[2] j k Nj (−ωf /3)Nk (−ωf /3, ωf ) + Nj (−ωf /3)Nk (−ωf /3, ωf )
B C F + NjF (ωf )NkBC (−ωf /3, −ωf /3) − A[3] j kl Nj (ωf /3)Nk (−ωf /3)Nl (ωf ) (40)
where −1
NjX (ωa ) = E [2] − ωa S [2] j k Xk[1] ,
[2] E − ωa S [2] j k NkF (ωf ) = 0
X ∈ {A, B, C},
(41) (42)
and
[3] (ω1 , ω2 )NlB (ω1 )NmC (ω2 ) E [2] − (ω1 + ω2 )S [2] NjBC (ω1 , ω2 ) = Tklm [2] B [2] C − Ckl Nl (ω1 ) − Bkl Nl (ω2 ),
[2] [3] (−ωa , ωf )NlX (−ωa )NlF (ωf ) E − (ωf − ωa )S [2] NjXF (ωa , ωf ) = Tklm [2] F Nl (ωf ), − Xkl
X ∈ {B, C}.
(43)
[4] The terms Tj[3] kl and Tj klm , which are separate from the three-photon transition matrix elements, are short-hand notation for
[3] [3] [3] Tj[3] (44) kl (ω1 , ω2 ) = Ej (kl) − ω1 Sj kl − ω2 Sj kl ,
[4] [4] [4] [4] [4] Tj klm (ω1 , ω2 , ω3 ) = Ej (klm) − ω1 Sj k(lm) − ω2 Sj l(km) − ω3 Sj m(kl) . (45)
The many parameters not defined here can be found in the paper of Olsen and Jørgensen [4]. Again it is noteworthy that the same approach can be applied for a wide selection of wavefunctions, from low scaling methods as Hartree–Fock (HF) and density functional theory (DFT) to highly correlated schemes as coupled cluster (CC). Response theory at the self-consistent field (SCF) level has often been applied for the calculations of multi-photon absorption of large organic molecules. Considering the size of these systems and computational tractability the method of choice for incorporating electron correlation is DFT. Recently, response theory up to the fourth property order has been implemented in the framework of DFT, which enables improvements of the predictions with a high degree of correlation.
Multi-Photon Absorption of Molecules
9
4. FEW-STATES MODELS 4.1. Two-photon absorption The explicit formulas for the transition matrix and tensor elements are normally given as sum-over-states (SOS) expressions. An option is therefore to enforce a truncation of the SOS-expression and only include a few dominating states and excitation channels. This may be motivated by the increasing energy term in the denominator or the assumption that only a few excitation paths actually will contribute in a full summation. The convergence rates with respect to the inclusion of states in the summation are known to be slow, except for charge-transfer (CT) systems which fortunately—but not surprisingly—coincides with a class of systems proposed for TPA and 3PA applications. These so-called few-states models where only a limited set of excited states and accompanying transition moments are addressed to represent the full excitation scheme is clearly inferior to response theory as a methodology, but the decomposition into simple properties such as excitation energies and transition dipole moments can enable valuable interpretation and promote an enhanced intuitive understanding through so-called structure-to-property relations. Following the orientationally averaging procedure devised by Monson and McClain [18], but rearranging the terms in a slightly more intuitive fashion we can rewrite the two-photon probability for linear polarized light δLTPA , and circular polarized light, δCTPA , as δLTPA = 24δ1 + 8δ2 − 16δ3 ,
(46)
= 16δ1 + 12δ2 − 24δ3
(47)
(µ0i · µif )(µ0j · µjf ) , ωi ωj
(48)
(µ0i × µif ) · (µ0j × µjf ) , ωi ωj
(49)
(µ0i × µ0j ) · (µif × µjf ) . ωi ωj
(50)
δCTPA where δ1 =
ij
δ2 =
ij
δ3 =
ij
This reformulation emphasizes the vector nature of the transition dipole moments. The terms δ1 and δ2 thus describe the alignments—or absence of alignments—of the channels leading to the actual two-photon state, i.e., indirectly the symmetry of the excited state. Obviously, a perfect alignment is preferable, however not possible in each molecular point group. Far more intriguing is the task of controlling the interference term, δ3 . In order to avoid negative contributions, the summation would ultimately consist of terms entangled in arrangements such as µ00 , µ11 ⊥ µ01 and where µ00 is directed in the opposite direction to µ11 . These conditions can easily be fulfilled by arbitrary vectors, but are not likely to occur for transition dipole moments for a real system. From inspection it is also clear that the D2h molecular point group is less appropriate, since all excitation paths unavoidably will be orthogonal. The distribution between the different terms δ1 , δ2 and δ3 thus defines a
P. Cronstrand et al.
10
clarifying signature of the processes underlying a particular TPA cross-section. Apparently, it is useful to distinguish between states where the dominating contributions are from parallel (P) or orthogonal (O) sub-paths. This may be especially relevant for non-symmetrical species where the character of the two-photon state cannot be determined directly by referring to symmetry rules. Few-states models are obtained by truncating the summation in equation (17) to include a finite number of excited states. For instance, by truncating equation (46) to three states and restrict to two dimensions one obtains if
δ TPA = 8
(µ0i µif )2 (2 cos2 (θ0i ) + 1) . (ωi )2
(51)
1f
The angle, θ01 between the relevant transition dipole moments, µ01 and µ1f , is unknown from an experimentally point a view and have to be assumed to be zero, which may be motivated for purely one-dimensional systems. Including yet a state will yield a four state model and the possibility of constructive or destructive interference between the excitation channels.
4.2. Three-photon absorption The monochromatic three-photon transition tensor element Tabc is defined as 0|µa |mm|µb |nn|µc |f Tabc = Pa,b,c . (ωm − 2ωf /3)(ωn − ωf /3) n,m
(52)
Confining to two states this can be rewritten as 0f
Tzzz = 27 ×
ff
0f
2 3 2µz (µ00 z − µz ) − (µz )
2ωf2
0f
= 27 ×
0f
µz [2( µz )2 − (µz )2 ] 2ωf2
(53)
and, for a one-dimensional system, the total three-photon absorption probability δ 3PA for linearly polarized light will become δL3PA =
(Tzzz )2 . 7
(54)
5. VALIDITY OF FEW-STATES MODELS The matrix equations describing multi-photon excitations given in the response section are indeed not very illuminating, but by inspecting the E and S terms under certain conditions we can identify a connection with the conventional SOS expressions. If H0 On |0 = En |n, as for instance is the case for FCI, the second-order Hessians and overlap matrices, will be diagonal and the third-order matrices will vanish, with the final consequence that the equation for the TP transition moment [equation (39)] will equal the SOS expression in equation (17). Thus, when approaching FCI we can expect a convergence for the absorption cross-sections between the two formalisms. At other levels of theory, the two approaches will be different even when performing a complete summation of the excited states.
Multi-Photon Absorption of Molecules
11
For a small diatomic such as LiH and a small basis set (here we confine to STO-3G) we can include all excited states in the SOS summations, both at the Hartree–Fock, and the full-configuration interaction (FCI) level. In Table 1 we present the results for twoand three-photon transition matrix elements between the ground state X 1 Σ + and the two lowest singlet excited states of Σ + symmetry. At the FCI level, the SOS and response approaches agree on the final property values, whereas, at the SCF level, this is no longer Table 1. Two-photon Szz and three-photon matrix elements Tzzz for LiH at the SCF and FCI levels with the STO-3G basis set Property
Szz Tzzz
State 11 Σ + 21 Σ + 11 Σ + 21 Σ +
SCF SOSa 42.417 10.680 5939.8 −663.65
Response
FCI SOSa
Response
42.406 10.344 5857.6 −1134.7
76.490 −8.2558 13355.1 −39.695
76.490 −8.2558 13355.1 −39.695
All quantities are given in a.u. Taken from [15]. a The SOS calculations include all states in the given representation.
Table 2. A comparison of truncated sum-over-states values with response results for twophoton Szz and three-photon Tzzz matrix elements States
Szz
Tzzz
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
−62.867 −62.836 −62.774 −61.970 −80.226 −75.316 −71.047 −70.996 −71.030 −71.165 −70.458 −70.416 −70.946 −70.548 −70.528
−1767.6 −1763.8 −1746.3 −1596.7 −3210.7 −2214.0 −3032.2 −3017.9 −3012.1 −3021.9 −2942.4 −2954.8 −3106.7 −3236.4 −3236.6
Response
−70.142
−5442.7
Results are obtained for the 21 A1 charge-transfer state of pNA at the SCF/6-31G level. All quantities are given in a.u. Taken from [15].
P. Cronstrand et al.
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so due to the contributions from the third-order Hessian matrix. The discrepancy is more pronounced for the higher order property Tzzz . We also note a rapid convergence for all properties with respect to the number of states included in the SOS summations at the FCI level. Already with inclusion of five of the total 29 excited states, the properties are converged to within 1%. At the SCF level, on the other hand, the values predicted by SOS (all states included) and response agree to within a few percent for Szz and for Tzzz of the first excited state. For the second excited state, however, the results predicted with the two methods differ by a factor of two. This shows that the quality of truncated SOS models depends not only on the property at hand but that it is also state specific. A similar observation can be made for para-nitro-aniline (pNA) as shown in Table 2. For TPA we note a mutual convergence between SOS and direct response after inclusion of 7
Fig. 1. Molecular structures.
Multi-Photon Absorption of Molecules
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Table 3. A comparison of truncated sum-over-states values with response results for twophoton Szz (a.u.) and three-photon Tzzz (×104 a.u.) matrix elements Molecule
AF-240 AF-260 AF-370 AF-385 AF-386 AF-390 DTT DTT-DD(101) DTT-AD(102) DTT-DD(103) DTT-AD(104) DTT-AA
ωa
4.08 4.11 4.30 4.13 4.27 4.16 3.17 (4.34) 3.10 (4.19) 2.97 3.04 (4.07) 2.97 3.01 (4.04)
Osc. str.
1.328 1.222 0.365 1.477 1.159 1.430 1.886 2.153 2.219 2.545 2.390 2.275
Szz
Tzzz
SOSb
Response
SOSb
Response
129.2 156.4 79.5 15.7 65.0 123.2 99.0 107.3 436.4 206.5 424.9 622.0
129.4 148.2 81.9 14.0 60.9 118.2 94.0 190.2 430.0 256.5 416.7 674.4
2.31 1.61 0.22 3.20 1.84 2.36 11.54 15.15 13.51 22.06 17.44 18.79
4.05 4.30 1.65 3.78 2.61 3.86 7.48 9.78 18.31 14.68 20.15 13.35
Results are obtained for a series of chromophores at the SCF/6-31G level. The excitation energies ω are given in eV. Taken from [15]. a The state with strongest TPA is given in parenthesis for symmetrical molecules. b A two-state model is employed for all AF compounds except AF-370 where a three-states model was used. A two-state model is employed for all DTT compounds except symmetrical DTT where a three-states model is used for Szz .
states, whereas for 3PA it appears as if the value is converged, while still deviating from the direct response value by more than 50%. The majority of systems proposed for TPA applications are various types of π-conjugated structures modified by the attachments of electron accepting (A) and donating (D) groups, as for instance those shown in Fig. 1. As depicted in Table 3 and Fig. 2, the agreement between the few-states models (FSM) and response methods for Szz is striking for all compounds. The deviation for TPA is occasionally 10%, but mostly below 5% if DTT-DD is disregarded. This is in sharp contrast to the performance of FSMs for Tzzz which show substantial discrepancies when compared against response results. The mean value of deviation is close to 40%, and the error can occasionally exceed 200%. Perhaps even more crucial is the fact that few-states models fail to predict the same ordering of the compounds as the response values with respect to the strength of Tzzz .
6. THREE-DIMENSIONAL SYSTEMS The two principal strategies for optimizing multi-photon cross sections have been to either propose new types of π-conjugated structures or to modify structures by attaching electron accepting and/or electron donating groups at certain locations. Introducing electron accepting (A) and electron donating (D) groups to conjugated systems has the well
14
P. Cronstrand et al.
Fig. 2. Correlation between results obtained with few-states models and the response method for two-photon Szz and three-photon Tzzz matrix elements. The systems included are derivatives of stilbene (triangle), AF (square), and DTT (ring). known effect of localizing the otherwise de-localized highest occupied and lowest unoccupied orbitals (HOMO-LUMO) and thereby establishing an effective charge-transfer path across the molecule. Due to the increase of transition dipole moment guiding this transition and an overall alignment involving all transition dipole moments, this technique leads to enhancements of several orders of magnitude for TPA [10,11]. However, the attachment of various functional groups also affect the symmetry of the molecule, which, as seen from the generalized few-states model formula, may have a significant influence by imposing restrictions among the excitation paths leading to the final multi-photon absorption state. Para-cyclophane (PCP), see Fig. 3, constitutes an interesting example in this context since it is explicitly three-dimensional and also offers the possibility of through-space delocalization as a means of gaining contributions to the cross-sections. From the agreements between the direct response and the few-states models results for TPA, displayed in Table 4 or Fig. 4, it is evident that it is sufficient to consider three states in order to incorporate the major sources to the cross sections for all PCP compounds. The distributions among the terms, δ1TPA , δ2TPA and δ3TPA , are displayed in the histograms in Fig. 5 for compounds PCP0, PCP1, PCP2 and PCP8. In general the first excited state can be classified as having a clear P-character and it is also far more intense than the second state. The dipolar structures, PCP0 and PCP1, show similar features. Both are, as expected, completely dominated by the δ1TPA term, which indicates an alignment between the relevant transition dipole moments and a pronounced P-character. PCP2 and PCP8 belong to the D2 point group, which is not fortunate in terms of the TPA probability, because of the restrictions of the transition dipole operators impelled by symmetry rules. A typical path to
Multi-Photon Absorption of Molecules
15
Fig. 3. Molecular structures.
Fig. 4. The two-photon probability, δ TP , in 105 a.u. as determined from response and few-states models for the first (1) and second (2) excited state for the systems PCP0-PCP8.
P. Cronstrand et al.
16
Table 4. The two-photon probability, δ TP , in a.u. as calculated by the 3-states model, the 5-states model and by response theory at the SCF level with the 6-31G basis set response
Molecule
ωexp
ωtheo
3state δTPA
5state δTPA
δTPA
0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8
416
345 270 306 297 341 337 361 344 363 360 361 358 363 342 361 356 355 352
9.549e+05 7.748e+04 3.183e+05 1.086e+04 3.059e+02 9.588e+02 1.366e+04 4.232e+03 2.647e+05 9.117e+05 5.552e+05 1.734e+05 6.258e+05 1.323e+04 3.894e+05 1.633e+05 1.504e+03 6.447e+03
7.383e+05 3.376e+04 3.299e+05 3.707e+03 3.524e+01 2.751e+03 1.368e+04 4.229e+03 2.647e+05 9.117e+05 5.552e+05 1.734e+05 6.222e+05 1.451e+04 3.894e+05 1.633e+05 1.504e+03 6.447e+03
8.15E+05 7.24E+04 2.81E+05 3.15E+03 7.38E+02 2.44E+03 2.00E+04 3.61E+03 2.63E+05 8.84E+05 7.04E+05 3.76E+05 5.91E+05 1.50E+04 3.99E+05 1.76E+05 3.75E+03 1.80E+04
– – 446 433 479 491 441 444 – –
the final excited state will include orthogonal dipole operators which will not only reduce δ1TP at the cost of δ2TPA , but most likely also lead to destructive interference by adding a substantial negative contribution from the δ3TPA term. It is noteworthy that the distinction between P- and O-character still applies even when the system has an undefined symmetry. As demonstrated by Fig. 5 and as indicated by equation (48) an explicit dipolar structure seems to be preferable because other configurations will then inevitably introduce nonparallel sub-paths in the excitation scheme with destructive interference as a consequence. Thus, the supposed flexibility when going from 1D to higher dimensions appears to be somewhat fictitious in the sense that an unambiguous dipolar structure still will be the most efficient TPA property of this system.
7. COMPARISON BETWEEN DFT AND AB INITIO RESULTS Quantum chemical modelling of non-linear absorption has until the implementation of density functional response theory been limited to Hartree–Fock or semi-empirical methods for extensive systems. DFT improves the description offered by HF, by supplying fractions of correlation energy obtained at a moderate computational cost. Because of the possibility within the response framework to go through the CC hierarchy and so apply a convergence scheme in the n-electron space for any available property, the low-scaling methods HF and DFT can be thoroughly benchmarked. The convergence and accuracy of results is well documented for energies [23–26], excited state energies, transition dipole moments, from
Multi-Photon Absorption of Molecules
17
Fig. 5. The distribution between δ1TP , δ2TP and δ3TP in a.u. as calculated by the three-states TP and δ TP from response. model compared with the total value δtot resp
ground-to-excited-state as well as excited-to-excited-state and (hyper)polarizabilities. The general trend is that the initial CCS-value, which is of similar quality as HF, is somewhat over-corrected by the CC2-method, and stabilized by CCSD somewhere in between the predictions of CCS and CC2. The refinement when moving to CC3 is often modest, which reflects that the most important contribution relates to the inclusion of singles and doubles. This roughly sketched oscillatory pattern for the sequence HF–CC2–CCSD is reproduced for excitation energies, δ TPA and δ 3PA as depicted in Table 5. The moderate contributions from triples, as estimated from the insignificant differences between excitation energies
P. Cronstrand et al.
18
Table 5. H2 O. Excitation energies ω (in eV), two-photon probabilities, δ TPA (in 103 a.u.) and three-photon probabilities, δ 3PA (in 104 a.u.) for linear polarized light as calculated by response theory at the CC, DFT and HF levels State
1B2
1B1
ω
δ TPA
δ 3PA
ω
δ TPA
δ 3PA
HF CC2 CCSD CC3 B3LYP BLYP LDA
8.60 7.20 7.57 7.58 6.87 6.24 6.54
0.07 0.24 0.16 – 0.18 0.25 0.24
3.46 13.96 8.35 – 11.9 26.1 19.5
10.27 8.86 9.33 9.35 8.28 7.50 7.87
0.64 2.72 1.64 – 1.93 2.82 2.63
0.49 3.72 2.06 – 2.96 5.51 4.72
State
2A1 ω
δ TPA
δ 3PA
ω
δ TPA
δ 3PA
10.87 9.53 9.91 9.91 9.01 8.35 8.61
0.24 0.48 0.33 – 0.39 0.49 0.49
2.98 8.56 5.26 – 6.60 8.82 10.6
11.75 10.35 10.79 10.82 9.80 8.98 9.31
0.72 2.46 1.83 – 1.91 2.94 2.70
3.45 54.66 26.69 – 17.9 39.8 26.5
HF CC2 CCSD CC3 B3LYP BLYP LDA
2B2
The aug-cc-pVTZ basis set was employed for ω and δ 3PA , whereas δ TPA was obtained by the Sadlej basis set.
predicted by CCSD and CC3, strengthen the predictive credibility of the δ TPA and δ 3PA values obtained at the CCSD level. While the excitation energies predicted by CCSD neatly is bracketed between the HF and B3LYP results, the orderings between the estimates for δ TPA and δ 3PA . are more irregular. The HF results are uniformly the lowest, but CCSD and B3LYP interchangeably predict the largest value. The overall mutual agreement between CCSD and B3LYP seems to support the use of B3LYP for exploring the δ TPA and δ 3PA for larger structures. Optionally in conjunction with another low-scaling method as HF, in order to attain a balanced description. We should note that performance of B3LYP for larger structures are hampered by “overpolarization”, due to incorrect asymptotic behaviour of the functional. This might lead to overestimation of the cross-sections. In Table 6 and in Fig. 6 we display the three-photon absorption probabilities, δ 3PA , for the first excited state for the series of modified trans-stilbene and DTT molecules, see Fig. 1. As seen in Fig. 6 the qualitative agreement concerning trends between HF and DFT is comforting, though the enhancement when attaching substituents is in general predicted to be more dramatic with DFT than for HF. A homologous (AA or DD) substitution will raise δ 3PA approximately by a factor between 2 and 18. Indisputably, AD substituted compounds give the largest responses and supersedes the non-substituted systems with at least one
Multi-Photon Absorption of Molecules
19
Table 6. Excitation energies in eV, three-photon probabilities, δ 3PA , for linear polarized light in 106 a.u. and three-photon cross sections, σ 3PA in 10−82 cm6 s2 as calculated by response at the HF and DFT levels with 6-31G basis set Molecule
TS TS-DD TS-AA TS-AD DTT DTT-DD(101) DTT-AA DTT-AD(102)
Exp. ω
HF ω
δ 3PA
≈ 4.0a 3.32b
4.59 4.35 4.22 4.05 3.17 3.10 3.01 2.97
12.3 0.121 32.9 0.277 77.8 0.598 289 1.96 806.08 2.63 1382.17 4.20 2590.45 7.25 4808.37 12.9
3.06c 2.88d 2.67e 2.92e 2.85d
DFT σ 3PA
ω
δ 3PA
4.08 76.6 3.67 261 3.34 138 2.78 5990 2.66 11574 2.50 31842.6 2.25 114951 1.92 649761
σ 3PA 0.533 1.32 0.525 13.1 22.2 50.8 134 472
a From Refs. [27,28]. b From Ref. [29]. c From Ref. [30]. d From Ref. [31]. e From Ref. [32].
Fig. 6. Comparison between HF and B3LYP results obtained for TS and DTT.
order of magnitude and often close to two. As demonstrated by the substantial difference between TS- and DTT-based systems, the electron richness of the basic building block, interpreted as the strength of the π-center, also strongly influences the σ 3PA .
P. Cronstrand et al.
20
8. CONCLUSION Different computational approaches for calculations of multi-photon absorption cross sections of molecules have been discussed. It is clearly shown that the response theory formulation of Olsen and Jørgensen [4] has great advantages. For instance, using the residue of response functions, one can calculate the MPA cross-sections at the same order as the corresponding (hyper)polarizability, which is a significant simplification compared to computing the MPA cross section from the (2M − 1)th order polarizability. The importance of response theory is further illustrated by the fact that the 3PA cross section of molecules converges very slowly with respect to the number of states included. Over the years, response theory has been implemented at different computational levels, such as Hartree– Fock, MCSCF, Coupled Cluster and Density Functional Theory. We believe that the recent development of response theory at the DFT level opens new opportunities for a variety of applications.
ACKNOWLEDGEMENTS This work was supported by the Swedish Research Council (VR), the Carl Trygger Foundation (CTS) and by a grant from the photonics project run jointly by the Swedish Materiel Administration (FMV) and the Swedish Defense Research Establishment (FOI). The computing time provided by National Supercomputer Center in Linköping (NSC) is gratefully acknowledged.
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M. Göppert-Mayer, Ann. Phys. 9 (1930) 273. W. Kaiser, C.G. Garret, Phys. Rev. Lett. 7 (1961) 229. S. Singh, L.T. Bradley, Phys. Rev. Lett. 12 (1964) 612. J. Olsen, P. Jørgensen, J. Chem. Phys. 82 (1985) 3235. M. Albota, D. Beljonne, J.L. Brédas, J.E. Ehrlich, J. Fu, A.A. Heikal, S.E. Hess, T. Kogej, M.D. Levin, S.R. Marder, D. McCord-Maughon, J.W. Perry, H. Röckel, M. Rumi, G. Subramaniam, W.W. Webb, X. Wu, C. Xu, Science 281 (1998) 1653. B.A. Reinhardt, L.L. Brott, S.J. Clarson, A.G. Dillard, J.C. Bhatt, R. Kannan, L. Yuan, G.S. He, P.N. Prasad, Chem. Mater. 10 (1998) 1863. G.S. He, P.P. Markowicz, T. Lin, P.N. Prasad, Nature 415 (2002) 767. P. Norman, Y. Luo, H. Ågren, Chem Phys. Lett. 286 (1998) 8. C.-K. Wang, O. Macak, Y. Luo, H. Ågren, J. Chem. Phys. 114 (2001) 9813. P. Norman, Y. Luo, H. Ågren, J. Chem. Phys. 111 (1999) 7759. P. Cronstrand, Y. Luo, H. Ågren, J. Chem. Phys. 117 (2002) 11102. P. Macak, Y. Luo, P. Norman, H. Ågren, J. Chem. Phys. 113 (2002) 7055. P. Macak, Y. Luo, H. Ågren, Chem. Phys. Lett. 330 (2000) 447. Y. Luo, P. Norman, P. Macak, H. Ågren, J. Phys. Chem. A 104 (2000) 4718. P. Cronstrand, P. Norman, Y. Luo, H. Ågren, J. Chem. Phys. 121 (2004) 2020. H. Mahr, in: H. Rabin, C.L. Tang (Eds.), Quantum Electronics, vol. IA, Academic Press, New York, 1975, p. 225. R.W. Boyd, Nonlinear Optics, Academic Press, London, 2003. P.R. Monson, W.M. McClain, J. Chem. Phys. 53 (1970) 29. W.M. McClain, J. Chem. Phys. 55 (1971) 2789. W.M. McClain, J. Chem. Phys. 58 (1973) 324.
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W.M. McClain, J. Chem. Phys. 57 (1972) 2264. O. Christiansen, P. Jørgensen, C. Hättig, Int. J. Quantum Chem. 68 (1998) 1. O. Christiansen, A. Halkier, H. Koch, P. Jørgensen, T. Helgaker, J. Chem. Phys. 108 (1998) 2801. O. Christiansen, H. Koch, P. Jørgensen, J. Chem. Phys. 103 (1995) 7429. P. Cronstrand, O. Christiansen, P. Norman, H. Ågren, Phys. Chem. Chem. Phys. 3 (2001) 2567. P. Cronstrand, O. Christiansen, P. Norman, H. Ågren, Phys. Chem. Chem. Phys. 2 (2000) 5357. G. Hohlneicher, B. Dick, J. Photochem. 27 (1984) 215. M.S. Gudipati, M. Mauds, J. Daverkausen, G. Hohlneicher, Chem. Phys. 192 (1995) 37. J.W. Robinson, Handbook of Spectroscopy, vol. II, CRC Press, Florida, 1974. M. Rumi, J.E. Ehrlich, A.A. Heikal, J.W. Perry, S. Barlow, Z. Hu, D. McCord-Maughon, T.C. Parker, H. Röckel, S. Thayumanavan, S.R. Marder, D. Beljonne, J.L. Brédas, J. Am. Soc. 122 (2000) 9500. [31] O.-K. Kim, K.-S. Lee, H.Y. Woo, K.-S. Kim, G.S. He, J. Swiatkiewicz, P.N. Prasad, Chem. Mater. 12 (2000) 284. [32] L. Ventelon, L. Moreaux, J. Mertz, M. Blanchard-Desce, Chem. Commun. (London) 1999 (1999) 2055.
Abstract Recent applications of response theory formulations of Olsen and Jørgensen on multi-photon absorption of molecules have been briefly reviewed. The connection between the calculated microscopic and experimentally measured macroscopic properties is derived. The performance of various computational approaches ranging from Hartree–Fock and Coupled Cluster to Density Functional theory for these spectral properties is analyzed. Using analytic response theory results as reference, the validity of the commonly applied few-states models for twoand three-photon absorption of molecules have been examined. A design strategy for three-dimensional chargetransfer multi-photon absorption systems is presented. Contents 1. Introduction 2. Multi-photon cross-sections 2.1. Macroscopic approach 2.2. Microscopic approach 2.3. Three-photon absorption 3. Response functions 4. Few-states models 4.1. Two-photon absorption 4.2. Three-photon absorption 5. Validity of few-states models 6. Three-dimensional systems 7. Comparison between DFT and ab initio results 8. Conclusion Acknowledgements References
1 2 2 4 6 6 9 9 10 10 13 16 20 20 20
1. INTRODUCTION The field of multi-photon absorption (MPA) can on a superficial level be described as having experienced three phases of activity, each separated with periods of about 30 years; initially a theoretical discovery, a phase of experimental confirmation and applications, and the present phase characterized by a dynamic interplay between experiment and theory. The pioneering prediction by Göppert-Mayer [1] already in 1930 pointed out a truly unique feature; the ability of matter to absorb more than one single photon at a time. In these early days the phenomenon was far from having a chance of being experimentally confirmed and remained therefore as a rather exotic, albeit interesting, aspect of light-matter interaction. The birth of the laser in 1960 released a floodgate of research in particular within the ADVANCES IN QUANTUM CHEMISTRY, VOLUME 50 ISSN: 0065-3276 DOI: 10.1016/S0065-3276(05)50001-7
© 2005 Elsevier Inc. All rights reserved
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P. Cronstrand et al.
evolving discipline of non-linear optics (NLO). So the phenomena of two-photon absorption (TPA) [2] and three-photon absorption (3PA) [3] could soon thereafter be confirmed, in 1961, respectively, 1963. On the theoretical side we consider the formulation of response theory of Olsen and Jørgensen in 1985 [4] as a large step towards and implementation of a practical scheme for large scale molecular calculations of multi-photon absorption (MPA). Traditionally, the physical origin of all MPA processes has been traced to various orders of the nonlinear susceptibilities, χ (1) , χ (3) , χ (5) , . . . . However, much more efficient computational schemes can be obtained by examining the resonance conditions in the sum-over-states (SOS) representation and by identifying products of first, second, third order of transition moments. A decisive simplification is therefore given by an evaluation the multiphoton transition moments as residues of (hyper)polarizabilities or, as encoded in response methodology, as residues of linear, quadratic or cubic response functions. These considerations enabled the MPA cross-sections to be extracted at the same order as the corresponding (hyper)polarizability; i.e., σ TPA from β and σ 3PA from γ . This is a considerable simplification from, for example, evaluating the 3PA cross-section from the fifth order polarizability. Other advantages of response theory for general molecular property calculations are well known, and will probably be reviewed in other contributions to this volume. Calculations using response theory was greatly promoted in the late 90s by the discovery of organic systems with large cross sections along with a growing attention motivated by a new generation of novel photonic technologies [5–15]. Although theoretical modelling has been available at an ab initio level since quite long time, it has not until recently reached a stage where it can match the increasing pool of experimental results. The original response theory work on Hartree–Fock and multi-configurational quadratic response functions have been extended to cubic response functions and a wider selection of wavefunctions, in particular to the coupled cluster hierarchies of electron correlated wave functions, and lately also to density functional theory (DFT). The present implementation of response theory in the program DALTON thus provides a flexible and powerful toolbox for theoretical modelling of multi-photon absorption of which only some aspects will be covered in this review. We start by deriving some basic relations in MPA which later is connected to computational schemes viable for direct response calculations. In the following section we recast the attained expressions into modified sum-over-states (SOS) expressions suitable for truncation into so called few-states models. Henceforth we review a sample of applications starting with examining the validity of the few-states models, followed by some results for explicit three-dimensional systems. Finally, we demonstrate the applicability of DFT by comparing with CC results and present DFT calculations of three-photon absorption for a set of chromophores.
2. MULTI-PHOTON CROSS-SECTIONS 2.1. Macroscopic approach On a macroscopic scale, the MPA processes can be elucidated by considering the rate of absorbed energy per volume unit when subjected to an external electric field, E d absorbed energy (1) = j · E. dt volume time
Multi-Photon Absorption of Molecules
3
The current density, j , induced in a non-magnetic medium with no free charge carriers, can be expanded as ∂ ∂P + c∇ × M − ∇ × Q + · · · (2) ∂t ∂t where the terms represent the electric dipole, magnetic dipole and electric quadrupole polarization. In the field of nonlinear optics the latter terms are neglected in general and the electric dipole term is expanded as j=
P = P(1) + P(2) + P(3) + P(4) + P(5) + · · ·
(3)
where it is sufficient to consider terms of odd order and the highlighted terms because of the time averaging procedure [16] (1)
Pi
(3) Pi Pi(5)
= χij (−ω; ω)Ej e−iωt + c.c. + · · · , (1)
= =
(3) 3χij kl (−ω; ω, −ω, ω)Ej e−iωt Ek∗ eiωt El e−iωt + c.c. + · · · , 10χij(5)klmn (−ω; ω, −ω, ω, −ω, ω)Ej e−iωt Ek∗ eiωt ∗ iωt × El e−iωt Em e En e−iωt + c.c. + · · · .
(4) (5)
(6)
In the expansion of the rate of absorbed energy we can determine the first order contribution of the rate of absorbed energy as (1) (1) ∂P · E = 2ω Im χij (−ω; ω) Ei Ej∗ . (7) ∂t This corresponds to one-photon transitions and the remaining terms can be ascertained as (3) (3) ∂P (8) · E = 6ω Im χij kl (−ω; ω, −ω, ω) Ei Ej∗ Ek El∗ , ∂t (5) (5) ∂P (9) · E = 20ω Im χij klmn (−ω; ω, −ω, ω, −ω, ω) Ei Ej∗ Ek El∗ Em En∗ ∂t which correspond to two- and three-photon absorption. The expressions can be abbreviated nc 2 E (in cgs units) [17] and identifying the further by introducing the intensity, I , as I = 2π one-, two- and three-photon absorption coefficients, α, β and γ , as (1) 2π hω ¯ Im χij (−ω; ω) , nc 24π 2 h¯ 2 ω (3) Im χij kl (−ω; ω, −ω, ω) , β= 2 2 n c 160π 3 h¯ 3 ω (5) γ = Im χij klmn (−ω; ω, −ω, ω, ω, −ω) 3 3 n c which can be related to the phenomenological equations describing the attenuation light beam experiencing OPA, TPA and 3PA as α=
dI = −αI − βI 2 − γ I 3 . dz
(10) (11) (12) of a
(13)
P. Cronstrand et al.
4
Trough the relations σ TPA =
h¯ ωβ N
(14)
σ 3PA =
h¯ 2 ω2 γ N
(15)
and
one can finally define the OPA, TPA and 3PA cross-sections, σ OPA , σ TPA and σ 3PA .
2.2. Microscopic approach At a microscopic level, the susceptibilities correspond to (hyper)polarizabilities which in principle are attainable by quantum chemistry methods, but less feasible for larger systems. Instead of evaluating the imaginary part of (hyper)polarizabilities as such we can achieve a considerable downshift of the order of the property by only considering the resonant terms for γ (−ω; ω, −ω, ω) with ω = 12 ωf where ωf is the excitation energy to the final two-photon state |f . Under these conditions it is possible to rewrite γ as γαβγ δ (−ω; ω, −ω, ω) = h¯ −3 P1,3 0|µα |kk|µγ |f f |µβ |mm|µδ |0 × (ωk − ω)(−iΓf /2)(ωm − ω) km 0|µγ |kk|µα |f f |µβ |mm|µδ |0 + (ωk − ω)(−iΓf /2)(ωm − ω) =i
0|µα |kk|µγ |f f |µβ |mm|µδ |0 2h¯ −3 P−σ,2 P1,3 Γf (ωk − ω) (ωm − ω) m k
=i
2h−1
¯ ∗ Sαγ Sδβ . Γf
(16)
Hence, in the vicinity of two-photon resonance it is sufficient to evaluate the two-photon transition matrix elements Sαβ defined as 0|µα |kk|µβ |f 0|µβ |kk|µα |f −1 Sαβ = h¯ (17) + . ωk − ω ωk − ω k
Due to the slow convergence of these so called sum-over-states (SOS) expressions an explicit summation does not form a viable option for ab initio methods, except when they can be truncated to a few leading terms. These truncations have traditionally been considered legitimate for so-called charge transfer (CT) systems, where the excitation scheme is completely dominated by a few major excitation channels. In addition, they may also serve as a valuable tool for interpretation purposes because of their ability to display the relation of the TPA probability, δ TPA , to other somewhat more intuitive quantities. A more computationally efficient approach is offered by residue analysis of the (hyper)polarizabilities. The
Multi-Photon Absorption of Molecules
5
single residue of the first order hyper-polarizability can be written as lim (ω2 − ωf )βij k (−ωσ ; ω1 , ω2 )
ω2 →−ωf
1 0|µi |nn|µj |f f |µk |0 0|µj |nn|µi |f f |µk |0 + = (ωn − ω1 − ωf ) (ωn + ω1 ) h¯ n 1 0|µi |nn|µj |f 0|µj |nn|µi |f + f |µk |0. = (ωnf − ω1 ) (ωn0 + ω1 ) h¯ n
(18)
It is clear that the term inside the brackets is connected to the two-photon absorption matrix element given by equation (17), when evaluated for ω1 = −ωf 0 /2. The final step in order to relate the microscopic origin to the macroscopic detection consists in relating the two coordinate systems; that of the laboratory and that of the molecules. Specifically we need to be able to relate the quantity that defines the macroscopic coordinates in the microscopic realm, that is the polarization of light, with the transition dipole moments evaluated at the quantum level. Since experiments rarely are made on single molecules, but an ensemble of molecules which for gases or liquids has no preferred direction, the relation must include a full orientation averaging. Thus, we want to establish a relation such as
0f S (λ, ν) 2 = (λA νB λ∗ ν ∗ )(lAa lBb lCc lDd ) S 0f S 0f ∗ . (19) C D AB DE In the beginning of 1970s, Monson and McClain [18,19] derived the following relations for accomplishing this task δ TPA = F δF + GδG + H δH
(20)
where F , G and H are defined as F = −|λ · ν ∗ |2 + 4|λ · ν|2 − 1, ∗ 2
2
∗ 2
2
(21)
G = −|λ · ν | − |λ · ν| + 4,
(22)
H = 4|λ · ν | − |λ · ν| − 1 and δF =
a,b
∗ Saa Sbb ,
δG =
(23)
∗ Sab Sab ,
a,b
δH =
∗ Sab Sba .
(24)
a,b
For linearly polarized light this implies that F = G = H = 2. The microscopically determined two-photon probability, δ TPA , is subsequently related to the macroscopic cross-section, σ TPA , as 4π 2 αa05 ω2 TPA 24π 2 hω ¯ 2 (3) 4π 2 hω ¯ 2 4 4 −3 TPA = Im χ e a E δ = (25) δ . 0 h c c2 c2 Provided the Bohr radius, a0 and the speed of light are given in cgs units and the frequency, ω and the TPA probability, δ TPA in atomic units the resulting unit will be cm4 s photon−1 . The result can be generalized further by introducing the finite lifetime broadening as σ TPA =
σ TPA =
4π 3 αa05 ω2 TPA δ (ωf − 2ω, Γf ) c
(26)
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6
through the normalized Lorentzian
(ωf − 2ω, Γf ) =
Γf 1 . π (ωf − 2ω)2 + Γf2
(27)
2.3. Three-photon absorption In complete analogy with TPA, 3PA cross-sections determined by third order transition moments which in turn can be evaluated through a single residue of the second order hyperpolarizability. Again, orientational averaging is required in order to relate the intrinsic coordinates of a single molecule to the ensemble of freely moving particles as measured in the laboratory coordinate system. As for TPA one therefore seeks to evaluate: 2 0f 0f ∗ δ 3PA = T 0f (λ, ν, ξ ) = λA νB ξC TABC λD νE ξF TDEF . (28) According to McClain [20,21] the orientation averaged values for the 3PA probability δ3PA for linearly (L) and circularly (C) polarized light can be written as: 1 (2δG + 3δF ), 35 1 = (5δG − 3δF ), 35
δL3PA =
(29)
δC3PA
(30)
where δF =
Tiij Tkkj ,
(31)
Tij k Tij k .
(32)
i,j,k
δG =
i,j,k
Similar to TPA, we can finally relate the orientationally averaged three-photon absorption probabilities, δ3P , to the three-photon cross-section, σ 3PA , as σ 3PA =
4π 4 a08 α ω3 δ 3PA (ωf − 3ω, Γf )(ω) . 3c Γf
(33)
With the same convention concerning the units as for TPA the final cross-sections will be obtained in units of cm6 s2 photon−1 .
3. RESPONSE FUNCTIONS To some extent response theory [4,22] can be seen as an elaborate scheme of timedependent perturbation theory. By means of this theory we can avoid the explicit summation of the expressions attained in the previous section by solving algebraic equations. Secondly, the formalism is analytically transferable, i.e., the same technique may subsequently be applied to retrieve properties from a wave function irrespective of the actual parameterization of the wave function.
Multi-Photon Absorption of Molecules
7
For exact states these can be given in their spectral representation with −A; Bω1 =
P
0|A|pp|B|0 ω p − ω1
p
(34)
as the linear response function and −A; B, Cω1 ,ω2 =
P
0|A|pp|B|qq|C|0 ¯ (ωp + ω0 )(ωq − ω2 )
(35)
p,q=0
¯ as the quadratic response function, where p|B|q = p|B − 0|B|0|q and −ω0 = ω1 + ω2 +· · · . P is the permutation operator. If the chosen operator is the dipole operator, µ, the response functions, µi ; µj , µk , . . .ω1 ,ω2 ,... will correspond to the (hyper)polarizabilities α, β and γ . The response functions contain inherently information about the excited states. The poles determine the location of the excitation energies and further information can be retrieved by examining resonance conditions through a residue analysis. From the expression for the linear response function for an exact wave function, where the unperturbed Hamiltonian is diagonal, we can easily evaluate transition dipole moments between the ground state, |0, and an excited state, |f as lim (ω1 − ωf )µi ; µj ω1 = 0|µi |f f |µj |0.
ω1 →−ωf
(36)
The single residue of the quadratic response function provides information on the twophoton transition matrix elements lim (ω2 − ωf )µi ; µj , µk −ω1 ,ω2 0|µi |nn|(µj − 0|µj |0)|f =− ω n − ω2 n 0|µj |nn|(µi − 0|µi |0)|f + f |µk |0 ω n − ω1
ω2 →−ωf
(37)
where ω1 + ω2 = ωm . From the double residue of the same response function one can deduce the transition dipole moments between excited states lim (ω2 − ωm )µi ; µj , µk −ω1 ,ω2 lim (ω1 − ωf ) ω1 →ωf ω2 →−ωm
= −0|µi |f f | µi − 0|µi |0 |ii|µk |0. (38) We emphasize that this is done from the reference state, |0, preferably the ground state, with no further reference to any excited state. For approximate wavefunctions the two-photon transition matrix element, Sαβ , can within the response terminology be evaluated directly as [4] [2] F B F SAB = −NjA (ωf /2)Bj[2] k Nk (ωf ) − Nj (−ωf /2)A(j k) Nk (ωf ) 1 [3] [3] B F + S − ω S − NjA (ωf /2) Ej[3] ω f j lk f j kl Nj (−ωf /2)Nk (ωf ). (kl) 2
(39)
P. Cronstrand et al.
8
In analog, the three-photon tensor elements are evaluated as B C F Tabc = NjA (ωf /3)Tj[4] klm (ωf /3, ωf /3, ωf /3)Nk (−ωf /3)Nl (−ωf /3)Nm (ωf ) B CF − NjA (ωf /3) Tj[3] kl (−ωf /3, ωf /3 − ωf /3)Nk (ωf /3)Nl (−ωf /3, ωf /3) C B + Tj[3] kl (−ωf /3, 2ωf /3)Nk (−ωf /3)Nl (ωf /3)
F BC + Tj[3] kl (ωf , −2ωf /3)Nk (ωf )Nl (ωf /3, −ωf /3) [3] B C F − NjA (ωf /3) Bj[3] kl Nk (−ωf /3)Nl (ωf ) + Cj kl Nk (−ωf /3, ωf ) [2] BF CF + NjA (ωf /3) Bj[2] k Nk (−ωf /3, ωf ) + Cj k Nk (−ωf /3, ωf ) B CF C BF + A[2] j k Nj (−ωf /3)Nk (−ωf /3, ωf ) + Nj (−ωf /3)Nk (−ωf /3, ωf )
B C F + NjF (ωf )NkBC (−ωf /3, −ωf /3) − A[3] j kl Nj (ωf /3)Nk (−ωf /3)Nl (ωf ) (40)
where −1
NjX (ωa ) = E [2] − ωa S [2] j k Xk[1] ,
[2] E − ωa S [2] j k NkF (ωf ) = 0
X ∈ {A, B, C},
(41) (42)
and
[3] (ω1 , ω2 )NlB (ω1 )NmC (ω2 ) E [2] − (ω1 + ω2 )S [2] NjBC (ω1 , ω2 ) = Tklm [2] B [2] C − Ckl Nl (ω1 ) − Bkl Nl (ω2 ),
[2] [3] (−ωa , ωf )NlX (−ωa )NlF (ωf ) E − (ωf − ωa )S [2] NjXF (ωa , ωf ) = Tklm [2] F Nl (ωf ), − Xkl
X ∈ {B, C}.
(43)
[4] The terms Tj[3] kl and Tj klm , which are separate from the three-photon transition matrix elements, are short-hand notation for
[3] [3] [3] Tj[3] (44) kl (ω1 , ω2 ) = Ej (kl) − ω1 Sj kl − ω2 Sj kl ,
[4] [4] [4] [4] [4] Tj klm (ω1 , ω2 , ω3 ) = Ej (klm) − ω1 Sj k(lm) − ω2 Sj l(km) − ω3 Sj m(kl) . (45)
The many parameters not defined here can be found in the paper of Olsen and Jørgensen [4]. Again it is noteworthy that the same approach can be applied for a wide selection of wavefunctions, from low scaling methods as Hartree–Fock (HF) and density functional theory (DFT) to highly correlated schemes as coupled cluster (CC). Response theory at the self-consistent field (SCF) level has often been applied for the calculations of multi-photon absorption of large organic molecules. Considering the size of these systems and computational tractability the method of choice for incorporating electron correlation is DFT. Recently, response theory up to the fourth property order has been implemented in the framework of DFT, which enables improvements of the predictions with a high degree of correlation.
Multi-Photon Absorption of Molecules
9
4. FEW-STATES MODELS 4.1. Two-photon absorption The explicit formulas for the transition matrix and tensor elements are normally given as sum-over-states (SOS) expressions. An option is therefore to enforce a truncation of the SOS-expression and only include a few dominating states and excitation channels. This may be motivated by the increasing energy term in the denominator or the assumption that only a few excitation paths actually will contribute in a full summation. The convergence rates with respect to the inclusion of states in the summation are known to be slow, except for charge-transfer (CT) systems which fortunately—but not surprisingly—coincides with a class of systems proposed for TPA and 3PA applications. These so-called few-states models where only a limited set of excited states and accompanying transition moments are addressed to represent the full excitation scheme is clearly inferior to response theory as a methodology, but the decomposition into simple properties such as excitation energies and transition dipole moments can enable valuable interpretation and promote an enhanced intuitive understanding through so-called structure-to-property relations. Following the orientationally averaging procedure devised by Monson and McClain [18], but rearranging the terms in a slightly more intuitive fashion we can rewrite the two-photon probability for linear polarized light δLTPA , and circular polarized light, δCTPA , as δLTPA = 24δ1 + 8δ2 − 16δ3 ,
(46)
= 16δ1 + 12δ2 − 24δ3
(47)
(µ0i · µif )(µ0j · µjf ) , ωi ωj
(48)
(µ0i × µif ) · (µ0j × µjf ) , ωi ωj
(49)
(µ0i × µ0j ) · (µif × µjf ) . ωi ωj
(50)
δCTPA where δ1 =
ij
δ2 =
ij
δ3 =
ij
This reformulation emphasizes the vector nature of the transition dipole moments. The terms δ1 and δ2 thus describe the alignments—or absence of alignments—of the channels leading to the actual two-photon state, i.e., indirectly the symmetry of the excited state. Obviously, a perfect alignment is preferable, however not possible in each molecular point group. Far more intriguing is the task of controlling the interference term, δ3 . In order to avoid negative contributions, the summation would ultimately consist of terms entangled in arrangements such as µ00 , µ11 ⊥ µ01 and where µ00 is directed in the opposite direction to µ11 . These conditions can easily be fulfilled by arbitrary vectors, but are not likely to occur for transition dipole moments for a real system. From inspection it is also clear that the D2h molecular point group is less appropriate, since all excitation paths unavoidably will be orthogonal. The distribution between the different terms δ1 , δ2 and δ3 thus defines a
P. Cronstrand et al.
10
clarifying signature of the processes underlying a particular TPA cross-section. Apparently, it is useful to distinguish between states where the dominating contributions are from parallel (P) or orthogonal (O) sub-paths. This may be especially relevant for non-symmetrical species where the character of the two-photon state cannot be determined directly by referring to symmetry rules. Few-states models are obtained by truncating the summation in equation (17) to include a finite number of excited states. For instance, by truncating equation (46) to three states and restrict to two dimensions one obtains if
δ TPA = 8
(µ0i µif )2 (2 cos2 (θ0i ) + 1) . (ωi )2
(51)
1f
The angle, θ01 between the relevant transition dipole moments, µ01 and µ1f , is unknown from an experimentally point a view and have to be assumed to be zero, which may be motivated for purely one-dimensional systems. Including yet a state will yield a four state model and the possibility of constructive or destructive interference between the excitation channels.
4.2. Three-photon absorption The monochromatic three-photon transition tensor element Tabc is defined as 0|µa |mm|µb |nn|µc |f Tabc = Pa,b,c . (ωm − 2ωf /3)(ωn − ωf /3) n,m
(52)
Confining to two states this can be rewritten as 0f
Tzzz = 27 ×
ff
0f
2 3 2µz (µ00 z − µz ) − (µz )
2ωf2
0f
= 27 ×
0f
µz [2( µz )2 − (µz )2 ] 2ωf2
(53)
and, for a one-dimensional system, the total three-photon absorption probability δ 3PA for linearly polarized light will become δL3PA =
(Tzzz )2 . 7
(54)
5. VALIDITY OF FEW-STATES MODELS The matrix equations describing multi-photon excitations given in the response section are indeed not very illuminating, but by inspecting the E and S terms under certain conditions we can identify a connection with the conventional SOS expressions. If H0 On |0 = En |n, as for instance is the case for FCI, the second-order Hessians and overlap matrices, will be diagonal and the third-order matrices will vanish, with the final consequence that the equation for the TP transition moment [equation (39)] will equal the SOS expression in equation (17). Thus, when approaching FCI we can expect a convergence for the absorption cross-sections between the two formalisms. At other levels of theory, the two approaches will be different even when performing a complete summation of the excited states.
Multi-Photon Absorption of Molecules
11
For a small diatomic such as LiH and a small basis set (here we confine to STO-3G) we can include all excited states in the SOS summations, both at the Hartree–Fock, and the full-configuration interaction (FCI) level. In Table 1 we present the results for twoand three-photon transition matrix elements between the ground state X 1 Σ + and the two lowest singlet excited states of Σ + symmetry. At the FCI level, the SOS and response approaches agree on the final property values, whereas, at the SCF level, this is no longer Table 1. Two-photon Szz and three-photon matrix elements Tzzz for LiH at the SCF and FCI levels with the STO-3G basis set Property
Szz Tzzz
State 11 Σ + 21 Σ + 11 Σ + 21 Σ +
SCF SOSa 42.417 10.680 5939.8 −663.65
Response
FCI SOSa
Response
42.406 10.344 5857.6 −1134.7
76.490 −8.2558 13355.1 −39.695
76.490 −8.2558 13355.1 −39.695
All quantities are given in a.u. Taken from [15]. a The SOS calculations include all states in the given representation.
Table 2. A comparison of truncated sum-over-states values with response results for twophoton Szz and three-photon Tzzz matrix elements States
Szz
Tzzz
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
−62.867 −62.836 −62.774 −61.970 −80.226 −75.316 −71.047 −70.996 −71.030 −71.165 −70.458 −70.416 −70.946 −70.548 −70.528
−1767.6 −1763.8 −1746.3 −1596.7 −3210.7 −2214.0 −3032.2 −3017.9 −3012.1 −3021.9 −2942.4 −2954.8 −3106.7 −3236.4 −3236.6
Response
−70.142
−5442.7
Results are obtained for the 21 A1 charge-transfer state of pNA at the SCF/6-31G level. All quantities are given in a.u. Taken from [15].
P. Cronstrand et al.
12
so due to the contributions from the third-order Hessian matrix. The discrepancy is more pronounced for the higher order property Tzzz . We also note a rapid convergence for all properties with respect to the number of states included in the SOS summations at the FCI level. Already with inclusion of five of the total 29 excited states, the properties are converged to within 1%. At the SCF level, on the other hand, the values predicted by SOS (all states included) and response agree to within a few percent for Szz and for Tzzz of the first excited state. For the second excited state, however, the results predicted with the two methods differ by a factor of two. This shows that the quality of truncated SOS models depends not only on the property at hand but that it is also state specific. A similar observation can be made for para-nitro-aniline (pNA) as shown in Table 2. For TPA we note a mutual convergence between SOS and direct response after inclusion of 7
Fig. 1. Molecular structures.
Multi-Photon Absorption of Molecules
13
Table 3. A comparison of truncated sum-over-states values with response results for twophoton Szz (a.u.) and three-photon Tzzz (×104 a.u.) matrix elements Molecule
AF-240 AF-260 AF-370 AF-385 AF-386 AF-390 DTT DTT-DD(101) DTT-AD(102) DTT-DD(103) DTT-AD(104) DTT-AA
ωa
4.08 4.11 4.30 4.13 4.27 4.16 3.17 (4.34) 3.10 (4.19) 2.97 3.04 (4.07) 2.97 3.01 (4.04)
Osc. str.
1.328 1.222 0.365 1.477 1.159 1.430 1.886 2.153 2.219 2.545 2.390 2.275
Szz
Tzzz
SOSb
Response
SOSb
Response
129.2 156.4 79.5 15.7 65.0 123.2 99.0 107.3 436.4 206.5 424.9 622.0
129.4 148.2 81.9 14.0 60.9 118.2 94.0 190.2 430.0 256.5 416.7 674.4
2.31 1.61 0.22 3.20 1.84 2.36 11.54 15.15 13.51 22.06 17.44 18.79
4.05 4.30 1.65 3.78 2.61 3.86 7.48 9.78 18.31 14.68 20.15 13.35
Results are obtained for a series of chromophores at the SCF/6-31G level. The excitation energies ω are given in eV. Taken from [15]. a The state with strongest TPA is given in parenthesis for symmetrical molecules. b A two-state model is employed for all AF compounds except AF-370 where a three-states model was used. A two-state model is employed for all DTT compounds except symmetrical DTT where a three-states model is used for Szz .
states, whereas for 3PA it appears as if the value is converged, while still deviating from the direct response value by more than 50%. The majority of systems proposed for TPA applications are various types of π-conjugated structures modified by the attachments of electron accepting (A) and donating (D) groups, as for instance those shown in Fig. 1. As depicted in Table 3 and Fig. 2, the agreement between the few-states models (FSM) and response methods for Szz is striking for all compounds. The deviation for TPA is occasionally 10%, but mostly below 5% if DTT-DD is disregarded. This is in sharp contrast to the performance of FSMs for Tzzz which show substantial discrepancies when compared against response results. The mean value of deviation is close to 40%, and the error can occasionally exceed 200%. Perhaps even more crucial is the fact that few-states models fail to predict the same ordering of the compounds as the response values with respect to the strength of Tzzz .
6. THREE-DIMENSIONAL SYSTEMS The two principal strategies for optimizing multi-photon cross sections have been to either propose new types of π-conjugated structures or to modify structures by attaching electron accepting and/or electron donating groups at certain locations. Introducing electron accepting (A) and electron donating (D) groups to conjugated systems has the well
14
P. Cronstrand et al.
Fig. 2. Correlation between results obtained with few-states models and the response method for two-photon Szz and three-photon Tzzz matrix elements. The systems included are derivatives of stilbene (triangle), AF (square), and DTT (ring). known effect of localizing the otherwise de-localized highest occupied and lowest unoccupied orbitals (HOMO-LUMO) and thereby establishing an effective charge-transfer path across the molecule. Due to the increase of transition dipole moment guiding this transition and an overall alignment involving all transition dipole moments, this technique leads to enhancements of several orders of magnitude for TPA [10,11]. However, the attachment of various functional groups also affect the symmetry of the molecule, which, as seen from the generalized few-states model formula, may have a significant influence by imposing restrictions among the excitation paths leading to the final multi-photon absorption state. Para-cyclophane (PCP), see Fig. 3, constitutes an interesting example in this context since it is explicitly three-dimensional and also offers the possibility of through-space delocalization as a means of gaining contributions to the cross-sections. From the agreements between the direct response and the few-states models results for TPA, displayed in Table 4 or Fig. 4, it is evident that it is sufficient to consider three states in order to incorporate the major sources to the cross sections for all PCP compounds. The distributions among the terms, δ1TPA , δ2TPA and δ3TPA , are displayed in the histograms in Fig. 5 for compounds PCP0, PCP1, PCP2 and PCP8. In general the first excited state can be classified as having a clear P-character and it is also far more intense than the second state. The dipolar structures, PCP0 and PCP1, show similar features. Both are, as expected, completely dominated by the δ1TPA term, which indicates an alignment between the relevant transition dipole moments and a pronounced P-character. PCP2 and PCP8 belong to the D2 point group, which is not fortunate in terms of the TPA probability, because of the restrictions of the transition dipole operators impelled by symmetry rules. A typical path to
Multi-Photon Absorption of Molecules
15
Fig. 3. Molecular structures.
Fig. 4. The two-photon probability, δ TP , in 105 a.u. as determined from response and few-states models for the first (1) and second (2) excited state for the systems PCP0-PCP8.
P. Cronstrand et al.
16
Table 4. The two-photon probability, δ TP , in a.u. as calculated by the 3-states model, the 5-states model and by response theory at the SCF level with the 6-31G basis set response
Molecule
ωexp
ωtheo
3state δTPA
5state δTPA
δTPA
0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8
416
345 270 306 297 341 337 361 344 363 360 361 358 363 342 361 356 355 352
9.549e+05 7.748e+04 3.183e+05 1.086e+04 3.059e+02 9.588e+02 1.366e+04 4.232e+03 2.647e+05 9.117e+05 5.552e+05 1.734e+05 6.258e+05 1.323e+04 3.894e+05 1.633e+05 1.504e+03 6.447e+03
7.383e+05 3.376e+04 3.299e+05 3.707e+03 3.524e+01 2.751e+03 1.368e+04 4.229e+03 2.647e+05 9.117e+05 5.552e+05 1.734e+05 6.222e+05 1.451e+04 3.894e+05 1.633e+05 1.504e+03 6.447e+03
8.15E+05 7.24E+04 2.81E+05 3.15E+03 7.38E+02 2.44E+03 2.00E+04 3.61E+03 2.63E+05 8.84E+05 7.04E+05 3.76E+05 5.91E+05 1.50E+04 3.99E+05 1.76E+05 3.75E+03 1.80E+04
– – 446 433 479 491 441 444 – –
the final excited state will include orthogonal dipole operators which will not only reduce δ1TP at the cost of δ2TPA , but most likely also lead to destructive interference by adding a substantial negative contribution from the δ3TPA term. It is noteworthy that the distinction between P- and O-character still applies even when the system has an undefined symmetry. As demonstrated by Fig. 5 and as indicated by equation (48) an explicit dipolar structure seems to be preferable because other configurations will then inevitably introduce nonparallel sub-paths in the excitation scheme with destructive interference as a consequence. Thus, the supposed flexibility when going from 1D to higher dimensions appears to be somewhat fictitious in the sense that an unambiguous dipolar structure still will be the most efficient TPA property of this system.
7. COMPARISON BETWEEN DFT AND AB INITIO RESULTS Quantum chemical modelling of non-linear absorption has until the implementation of density functional response theory been limited to Hartree–Fock or semi-empirical methods for extensive systems. DFT improves the description offered by HF, by supplying fractions of correlation energy obtained at a moderate computational cost. Because of the possibility within the response framework to go through the CC hierarchy and so apply a convergence scheme in the n-electron space for any available property, the low-scaling methods HF and DFT can be thoroughly benchmarked. The convergence and accuracy of results is well documented for energies [23–26], excited state energies, transition dipole moments, from
Multi-Photon Absorption of Molecules
17
Fig. 5. The distribution between δ1TP , δ2TP and δ3TP in a.u. as calculated by the three-states TP and δ TP from response. model compared with the total value δtot resp
ground-to-excited-state as well as excited-to-excited-state and (hyper)polarizabilities. The general trend is that the initial CCS-value, which is of similar quality as HF, is somewhat over-corrected by the CC2-method, and stabilized by CCSD somewhere in between the predictions of CCS and CC2. The refinement when moving to CC3 is often modest, which reflects that the most important contribution relates to the inclusion of singles and doubles. This roughly sketched oscillatory pattern for the sequence HF–CC2–CCSD is reproduced for excitation energies, δ TPA and δ 3PA as depicted in Table 5. The moderate contributions from triples, as estimated from the insignificant differences between excitation energies
P. Cronstrand et al.
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Table 5. H2 O. Excitation energies ω (in eV), two-photon probabilities, δ TPA (in 103 a.u.) and three-photon probabilities, δ 3PA (in 104 a.u.) for linear polarized light as calculated by response theory at the CC, DFT and HF levels State
1B2
1B1
ω
δ TPA
δ 3PA
ω
δ TPA
δ 3PA
HF CC2 CCSD CC3 B3LYP BLYP LDA
8.60 7.20 7.57 7.58 6.87 6.24 6.54
0.07 0.24 0.16 – 0.18 0.25 0.24
3.46 13.96 8.35 – 11.9 26.1 19.5
10.27 8.86 9.33 9.35 8.28 7.50 7.87
0.64 2.72 1.64 – 1.93 2.82 2.63
0.49 3.72 2.06 – 2.96 5.51 4.72
State
2A1 ω
δ TPA
δ 3PA
ω
δ TPA
δ 3PA
10.87 9.53 9.91 9.91 9.01 8.35 8.61
0.24 0.48 0.33 – 0.39 0.49 0.49
2.98 8.56 5.26 – 6.60 8.82 10.6
11.75 10.35 10.79 10.82 9.80 8.98 9.31
0.72 2.46 1.83 – 1.91 2.94 2.70
3.45 54.66 26.69 – 17.9 39.8 26.5
HF CC2 CCSD CC3 B3LYP BLYP LDA
2B2
The aug-cc-pVTZ basis set was employed for ω and δ 3PA , whereas δ TPA was obtained by the Sadlej basis set.
predicted by CCSD and CC3, strengthen the predictive credibility of the δ TPA and δ 3PA values obtained at the CCSD level. While the excitation energies predicted by CCSD neatly is bracketed between the HF and B3LYP results, the orderings between the estimates for δ TPA and δ 3PA . are more irregular. The HF results are uniformly the lowest, but CCSD and B3LYP interchangeably predict the largest value. The overall mutual agreement between CCSD and B3LYP seems to support the use of B3LYP for exploring the δ TPA and δ 3PA for larger structures. Optionally in conjunction with another low-scaling method as HF, in order to attain a balanced description. We should note that performance of B3LYP for larger structures are hampered by “overpolarization”, due to incorrect asymptotic behaviour of the functional. This might lead to overestimation of the cross-sections. In Table 6 and in Fig. 6 we display the three-photon absorption probabilities, δ 3PA , for the first excited state for the series of modified trans-stilbene and DTT molecules, see Fig. 1. As seen in Fig. 6 the qualitative agreement concerning trends between HF and DFT is comforting, though the enhancement when attaching substituents is in general predicted to be more dramatic with DFT than for HF. A homologous (AA or DD) substitution will raise δ 3PA approximately by a factor between 2 and 18. Indisputably, AD substituted compounds give the largest responses and supersedes the non-substituted systems with at least one
Multi-Photon Absorption of Molecules
19
Table 6. Excitation energies in eV, three-photon probabilities, δ 3PA , for linear polarized light in 106 a.u. and three-photon cross sections, σ 3PA in 10−82 cm6 s2 as calculated by response at the HF and DFT levels with 6-31G basis set Molecule
TS TS-DD TS-AA TS-AD DTT DTT-DD(101) DTT-AA DTT-AD(102)
Exp. ω
HF ω
δ 3PA
≈ 4.0a 3.32b
4.59 4.35 4.22 4.05 3.17 3.10 3.01 2.97
12.3 0.121 32.9 0.277 77.8 0.598 289 1.96 806.08 2.63 1382.17 4.20 2590.45 7.25 4808.37 12.9
3.06c 2.88d 2.67e 2.92e 2.85d
DFT σ 3PA
ω
δ 3PA
4.08 76.6 3.67 261 3.34 138 2.78 5990 2.66 11574 2.50 31842.6 2.25 114951 1.92 649761
σ 3PA 0.533 1.32 0.525 13.1 22.2 50.8 134 472
a From Refs. [27,28]. b From Ref. [29]. c From Ref. [30]. d From Ref. [31]. e From Ref. [32].
Fig. 6. Comparison between HF and B3LYP results obtained for TS and DTT.
order of magnitude and often close to two. As demonstrated by the substantial difference between TS- and DTT-based systems, the electron richness of the basic building block, interpreted as the strength of the π-center, also strongly influences the σ 3PA .
P. Cronstrand et al.
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8. CONCLUSION Different computational approaches for calculations of multi-photon absorption cross sections of molecules have been discussed. It is clearly shown that the response theory formulation of Olsen and Jørgensen [4] has great advantages. For instance, using the residue of response functions, one can calculate the MPA cross-sections at the same order as the corresponding (hyper)polarizability, which is a significant simplification compared to computing the MPA cross section from the (2M − 1)th order polarizability. The importance of response theory is further illustrated by the fact that the 3PA cross section of molecules converges very slowly with respect to the number of states included. Over the years, response theory has been implemented at different computational levels, such as Hartree– Fock, MCSCF, Coupled Cluster and Density Functional Theory. We believe that the recent development of response theory at the DFT level opens new opportunities for a variety of applications.
ACKNOWLEDGEMENTS This work was supported by the Swedish Research Council (VR), the Carl Trygger Foundation (CTS) and by a grant from the photonics project run jointly by the Swedish Materiel Administration (FMV) and the Swedish Defense Research Establishment (FOI). The computing time provided by National Supercomputer Center in Linköping (NSC) is gratefully acknowledged.
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