Five-Photon Absorption and Selective Enhancement of Multiphoton Absorption Processes.

We study one-, two-, three-, four-, and five-photon absorption of three centrosymmetric molecules using density functional theory. These calculations are the first ab initio calculations of five-photon absorption. Even- and odd-order absorption processes show different trends in the absorption cross sections. The behavior of all even- and odd-photon absorption properties shows a semiquantitative similarity, which can be explained using few-state models. This analysis shows that odd-photon absorption processes are largely determined by the one-photon absorption strength, whereas all even-photon absorption strengths are largely dominated by the two-photon absorption strength, in both cases modulated by powers of the polarizability of the final excited state. We demonstrate how to selectively enhance a specific multiphoton absorption process.

The concept of multiphoton absorption (MPA) dates back to 1931, when it was predicted theoretically by Maria Göppert-Mayer.[1] After the introduction of high-intensity laser light sources, multiphoton absorption has gained a lot of attention during the past five decades, with the first experimental proof for two-photon absorption as the starting point in 1961.[2]

Two-photon absorption has by now found a wide range of applications.[3−6] Simultaneously, electronic excitations involving absorption of up to five photons have been observed experimentally.[7] The theoretical treatment of multiphoton absorption has also been pushed forward during the last decades, and computational methods for the calculation of two-photon transition strengths are available for SCF-based[8,9] as well as for correlated wave-function-based methods.[10−13] Implementations of three-photon absorption have been realized only for SCF-based methods,[14−16] but SCF-based density-functional theory studies give good agreement with experiment for TPA and 3PA processes for small and medium-sized molecules.[17,18] In our group, a recursive open-ended response theory[19] has been expanded to the treatment of single residues of response functions, enabling the calculation of multiphoton absorption cross sections to infinite order using SCF-based methods, with results having been reported for four-photon absorption.[20]

In order to design molecules with high multiphoton absorption cross sections, general and qualitative structure–property relations are important. However, very few general structure–property relations exist for higher-order multiphoton absorption properties, although some interesting observations have been made in the case of three-photon absorption.[21,22] The major scope of this work is both to present the first computational treatment of five-photon absorption and to study the relations between different multiphoton absorption processes. As experimental data for four- and five-photon absorption are very limited, in particular when it comes to complete spectra, we have to rely on a computational treatment of these properties to compare their behavior. In order to amplify the general observations presented here, we will focus on centrosymmetric molecules.

The multiphoton absorption cross sections calculated for the molecules in Figure 1 are shown in Table 1 for the p-dinitrobenzene (PDNB) molecule, which has D2 symmetry, and in Figure 2 for the diaminoazobenzene (C symmetry) and indigo molecules (C2 symmetry). Note that the units for the absorption cross sections differ and that the cross sections differ by orders of magnitude. The results reported have been calculated using the CAM-B3LYP functional[23] and the augmented correlation-consistent polarized valence double-ζ basis sets (aug-cc-pVDZ) of Dunning and Woon.[24,25] A recent study has shown that this level of theory is a good compromise between accuracy and computational cost.[20] The calculations have been performed using a recursive response theory code[19,20] interfaced to a development version of the Dalton program.[26]

Figure 1

Three centrosymmetric molecules that have been studied in this work: para-dinitrobenzene (1), diaminoazobenzene (2), and indigo (3).

Table 1

Calculated MPA Cross Sections for PDNB and Their Ratios

				jPA cross sections [cm2j sj–1/photonj–1]						cross section ratios
no.	energy [eV]		irrep	OPA (10–19)	TPA (10–54)	3PA (10–86)	4PA (10–118)	5PA (10–150)		OPA/3PA (1062)	TPA/4PA (1061)	3PA/5PA (1062)	OPA/5PA (10128)
1	3.90	1	b3g		3.44		0.08				43 860
2	3.95	1	au			0.04		0.02				269
3	4.39	1	b2g		9.26		0.96				9689
4	4.42	1	b1u	2.46		1.78		0.13		138 013		1338	18 466
5	4.62	1	b2u	89.4		803		65.5		11 132		1226	1365
6	5.03	1	b3u	1240		52 557		13 226		2359		397	94
7	6.06	2	b2u	278		1695		1911		16 390		89	145
8	6.06	1	b1g		29 100		263 357				110
9	6.15	1	ag		801 169		3 041 400				263
10	6.20	2	b1g		21 602		4 893 130				4
11	6.68	2	b2u	607		7395		42 997		8220		17	14
12	6.79	2	b3u	1626		60 375		412 524		2693		15	4
13	6.82	2	au			2.93		1.27				231
14	6.85	2	b3g		1.90		2.57				740
15	6.92	3	au			25.1		1.48				1694

Figure 2

Multiphoton absorption behavior of the two molecules under investigation. TPA cross sections have been scaled with 1032. 3PA cross sections are scaled with 1065, 4PA cross sections are scaled with 1094, and 5PA cross sections are scaled with 10126.

There are remarkable parallels in the absorption behavior of the different states depending on whether an even or an odd number of photons is absorbed; see Table 1. The PDNB molecule deviates from this observation with its Au states, which cannot be reached by one-photon processes for symmetry reasons.

Investigating the results, we note that the qualitative agreement between the absorption cross sections is not as obvious for the even-order processes as for odd-photon absorption processes. The best correlation between the different even-photon absorption processes has been found in the diaminoazobenzene molecule.

The similarity in the multiphoton absorption cross sections observed in Figure 2 can be explained by considering the sum-over-states expressions for the multiphoton absorption cross sections. For a complete discussion of the basic theory of multiphoton absorption we refer to the literature,[20,27−30] and we restrict ourselves here only to provide the general expression for the multiphoton absorption cross sections that we will use in the analysis and to show the approximation we do in order to explain our observations.

The general j-photon absorption cross section σ is proportional to the square of the j-photon transition matrix element M for j ≥ 2.[20,27]with M being defined as (atomic units)[29]where ω is the frequency of perturbation X and where we have already used the fact that the ground-state dipole moment vanishes for centrosymmetric molecules.[30] μ is the dipole operator with respect to perturbation X, and ω is the excitation energy between the ground state 0 and the excited state l. f is the final state of the excitation, and the operator generates all permutations of the perturbations. j is the number of photons absorbed in the described process. The calculations we present in this Letter have been performed using this equation in a slightly different formulation. To interpret them, we now simplify this expression using symmetry considerations.

For centrosymmetric molecules we can easily identify the symmetry of the accessible final states f as well as the intermediate states l in eq 2. Because all electric dipole perturbations are ungerade and the ground state always gerade, the excited state li that appears in the numerator of eq 2 must be ungerade. All final states for odd-photon absorption processes must be ungerade. In contrast, TPA and all higher-order even-photon absorption processes are allowed only for excitations with gerade final states.

However, from Figure 2 and Table 1, we note that there is even a semiquantitative agreement between the absorption cross sections of different excited states. The orders of magnitude that are covered by the different cross section values are very similar in all cases, and the order of the states with respect to their MPA behavior is similar within the even- and odd-photon absorption groups. This is illustrated in particular by the ratios between the cross sections that have been calculated for the PDNB molecule. Apart from the one for the 1B1u state, which shows a rather small cross section, we find that all ratios between OPA and 3PA are within the range of about an order of magnitude. The behavior of the TPA/4PA ratios is rather similar with the exception of the one for the 1B1g state. We will return to this anomaly.

In order to explain these observations, we first give the explicit expressions for the one- and two-photon absorption matrix elements:where the individual perturbations are marked by lowercase letters. and represent ungerade and gerade states, respectively, and the index f marks the final state. is the number of ungerade excited states that are symmetry allowed. The operator generates all permutations of the perturbations and their individual frequencies.

Let us now consider the 3PA transition matrix elementFor a molecule with a strong, isolated OPA state, the dominating intermediate state of ungerade symmetry is the final state of interest; in the spirit of few-state models for multiphoton absorption cross sections, the 3PA matrix element then reduces toNote that in this expression the summation over the ungerade states has been dropped, but the summation over gerade states is not affected. The validity of this approach has been shown, for example, in work by Vivas and co-workers, where a three-level model for TPA yielded impressive results compared to experiment.[31] Using this expression, we can write the 3PA matrix element asThis quasi-few-state model now explains the similarity of the OPA and 3PA absorption cross sections, the 3PA cross section being proportional to the OPA cross section, and the proportionality factor being the excited-state polarizability divided by an energy denominator. The permutation operator that is still present in the approximated equation takes into account that several spatial components of MOPA can contribute to the transition dipole moment.

This is also an explanation for the low 3PA and 5PA cross sections of the Au states in PDNB since these states cannot be dominated by an OPA excitation according to eq 6. We can therefore separate 3PA excitations into OPA-dominated and pure-3PA absorptions.

We can extend this analysis to 4PA. The matrix element for four-photon absorption iswhere both the summations over k and m cover the same manifold of ungerade states. In the same manner, we can now assume a dominant contribution from as the intermediate gerade state and write eq 8 asAnalogous to the results for 3PA, we find that the 4PA cross sections are proportional to the TPA cross section, the proportionality factor being the excited-state polarizability divided by an energy term.

For five-photon absorption, we can in a similar manner writenoting that we now have two ungerade intermediate states that can be substituted by the state. If we only introduce the final state for the second ungerade state ( in the equation) we obtainwhich is an expression for 3PA-dominated 5PA, which is representative for the Au states of the PDNB molecule. (In principle there should also be a term arising from introducing the final state for the first ungerade intermediate state. However, this would lead to a term that would correspond to the OPA transition moment scaled by the excited-state second hyperpolarizability and thus can be expected to be smaller.) In most cases, we can simplify eq 12 further by substituting all ungerade intermediate states by the final state, givingwhich shows that the 5PA absorption cross section can be expected to be proportional to the OPA transition moment scaled by the excited-state polarizability squared.

These expressions explain why we observe the orthogonality of the odd- and even-order multiphoton absorption cross sections. Furthermore, they also explain the strong parallelism of the odd- and even-order multiphoton absorption cross sections, as they are in all cases related to the lowest-order one- or two-photon absorption cross section, scaled by the energy difference between the excitation energy to the final state and the incoming laser frequencies, and scaled by (products of) excited-state polarizabilities. It is important to point out that this model does not allow quantitative predictions of higher-order MPA from OPA or TPA data, but it gives a qualitative to semiquantitative description of the behavior of the different properties.

Multiphoton absorption cross sections can be enhanced through resonances, occurring when the frequency (or sums of frequencies) of the incoming laser light matches that of intermediate excited states (e.g., when ω =/2 ≈ in the case of TPA). We note from the equations above that the frequency prefactor in the few-state models has the same poles as that of the TPA or 3PA processes, respectively. Thus, boosting TPA and 3PA processes through near-resonance conditions will also boost the higher-order multiphoton absorption cross sections.

Boosting a higher-order multiphoton absorption cross section selectively without a simultaneous increase in the lower-order multiphoton absorption cross section will thus be possible only if we can make the excited-state polarizability approach a pole for one of the frequency combinations of the incoming laser light. Consider for example the 4PA transition moment in eq 10. Whereas the TPA cross section, and thus also the 4PA cross section, will have a pole if there are intermediate states at half the energy of the excited state, the 4PA will be selectively enhanced if there is an excited state of ungerade symmetry at 3 /4. Similar considerations can be made for the 5PA process, and we can selectively enhance the 5PA process if there are intermediate states of gerade symmetry with energy 2 /5 and 4 /5.

If we now return to the unexpected enhancement of the 4PA cross section of the 2B1 (and the close-lying 1B1g and 1Ag states) in PDNB, we note that there exists an intermediate B2u state of ungerade symmetry at approximately 3/4 of the energy of these final states, thus creating a resonance in the excited-state polarizability and enhancement of the 4PA process relative to the TPA process.

To summarize our findings, we have presented the first 5PA cross section calculations at the density-functional level of theory using the CAM-B3LYP exchange–correlation functional for three centrosymmetric molecules. These calculations have shown that for molecules with an inversion center the selection rules for dipole-allowed transitions make the relative magnitude of all the even (odd) multiphoton absorption cross sections similar for different excited states. Assuming the validity of a few-state model for the states with the same symmetry as the final state of the multiphoton absorption process, we have shown that higher-order multiphoton absorption cross sections are proportional to the TPA (even-order absorption processes) or OPA (odd-order absorption processes) cross sections, scaled by an energy denominator and powers of the polarizability of the final excited state.

Our analysis shows that for centrosymmetric molecules excited states that display strong one- or two-photon absorption cross sections will also display strong multiphoton absorption cross sections. We have also shown how to selectively enhance the higher-order multiphoton absorption cross section by designing molecules with intermediate states that match selected frequency combinations for the frequency of the incoming light. We predict, for instance, a strong enhancement of 5PA processes by designing molecules with intermediate states of energy 2 /5 and 4 /5, as well as for 4PA, as illustrated by our results calculated for the PDNB molecule. These results will be valuable guidelines for designing molecules with strong higher-order multiphoton absorption cross sections.

Computational Details

All multiphoton absorption calculations have been performed using a recursive open-ended response theory code,[19] which has recently been expanded to the treatment of single residues of response functions.[20] The response code has been interfaced with a development version of the Dalton program.[26] All calculations have been performed using the aug-cc-pVDZ basis set from the Dunning family of basis sets[24] and the CAM-B3LYP density functional.[23] In all calculations of n-photon absorption the energy of the involved photons was the excitation energy divided by n.

The geometry of the para-dinitrobenzene molecule has been optimized using the cc-pVQZ basis set and the B3LYP functional[32,33] in Gaussian.[34] The geometries of the diaminoazobenzene and indigo molecules have been optimized using TURBOMOLE[35] with the B3LYP functional and the TZVP basis set.[36]

The rotational averaging of the calculated results has been performed following available literature,[37−39] and the calculation of the cross sections was done using an approach that has recently been generalized by two of us[20] based on work by Peticolas.[27] The detailed equations for this are given in the Supporting Information.