On the photophysical properties of IrIII, PtII, and PdII (phenylpyrazole) (phenyldipyrrin) complexes.

The absorption and emission characteristics of (ppz)2(dipy)IrIII, (ppz)(dipy)PtII and (ppz)(dipy)PdII, where ppz stands for phenylpyrazole and dipy for a phenyl meso-substituted dipyrrin ligand, have been investigated by means of combined density functional theory and multireference configuration interaction including scalar relativistic and spin-orbit coupling effects. These results were compared with experimental spectra. The complexes exhibit a high density of low-lying electronically excited states originating from ligand-centered (LC) and metal-to-ligand charge transfer (MLCT) states involving the dipyrrin ligand. In addition, metal-centered (MC) states are found to be low-lying in the Pd complex. In all three cases, the first strong absorption band and the phosphorescence emission band stem from LC excitations on the dipyrrin ligand with small MLCT contributions. The MLCT states show more pronounced relaxation effects than the LC states, with the consequence that the first excited state with predominant singlet multiplicity is of SMLCT/LC type in the heavier Ir and Pt complexes. Substantial spin-orbit coupling between SMLCT/LC and TLC enables fast and efficient intersystem crossing (ISC) and a high triplet quantum yield. Phosphorescence rate constants are rather small in accord with the dominant LC character of the transitions. Out-of-plane distortion promotes nonradiative decay of the excited state population via the MC states thus explaining the lower phosphorescence quantum yield of the Pt complex. The spectral properties of the Pd complex are different in many aspects. Optimization of the S1 state yields a dipyrrin intraligand charge transfer (ILCT) state with highly distorted nuclear arrangement in the butterfly conformers leading to nonradiative deactivation. In contrast, the primarily excited SLC state and the SMLCT/LC state of the twist conformer have nearly equal adiabatic excitation energies. The lack of a driving force toward the SMLCT/LC minimum, the high fluorescence rate constant of the bright SLC state and its moderately efficient ISC to the triplet manifold explain the experimentally observed dual emission of the Pd complex at room temperature.

Electron-Rich Dipyrrolonaphthyridinediones: Synthesis and Optical Properties.

This article describes the design rationale for highly electron-rich dipyrrolonaphthyridinedione (DPND) derivatives bearing substituted amino groups at the 3 and 9 positions, which exhibit absorption in the red and emission in the red/NIR region of the spectrum. These novel dyes are easily synthesized through a two-step protocol consisting of bromination of the DPND molecule followed by Buchwald-Hartwig amination. We demonstrated that the diamino-dipyrrolonaphthyridinediones have high ionization energies (∼4.7 eV) and that the spectroelectrochemical properties can be rationally tuned by altering the nature of the peripheral substituted amino groups. All amino-DPNDs exhibit solvatofluorochromism, which has not been previously reported for dyes possessing this core. Theoretical calculations reveal that in all cases, the strongest absorption is exhibited by the S1 states which clearly correlate with the HOMO-LUMO orbital transition. As all higher states have lower oscillator strengths, it is clear that fluorescence is completely dominated by the excitation/deexcitation sequence S0 → S1, S1 → S0 and that there are no contributions to the fluorescence from excitations to higher states.

Solvatofluorochromic, non-centrosymmetric π-expanded diketopyrrolopyrrole.

A novel non-centrosymmetric π-expanded diketopyrrolopyrrole was designed and synthesized. Strategic placement of tert-butyl groups at the periphery of a diketopyrrolopyrrole allowed us to selectively fuse one moiety via tandem Friedel-Crafts-dehydration reactions, resulting in a non-centrosymmetric dye. The structure of the dye was confirmed by X-ray crystallography, revealing that it contains a nearly flat arrangement of four fused rings. Extensive photophysical studies of this new functional dye revealed that the intensity of its emission strongly depends on solvent polarity, which is typical for dipolar chromophores. In non-polar solvents, the fluorescence quantum yield is high whereas in polar solvents such as MeOH, it is 12%. However, upon two-photon excitation the compound behaves like a centrosymmetric dye, showing a two-photon absorption maximum at significantly shorter wavelengths than twice the wavelength of the one-photon absorption maximum.

Three-photon circular dichroism: towards a generalization of chiroptical non-linear light absorption.

We present the theory of three-photon circular dichroism (3PCD), a novel non-linear chiroptical property not yet described in the literature. We derive the observable absorption cross section including the orientational average of the necessary seventh-rank tensors and provide origin-independent expressions for 3PCD using either a velocity-gauge treatment of the electric dipole operator or a length-gauge formulation using London atomic orbitals. We present the first numerical results for hydrogen peroxide, 3-methylcyclopentanone (MCP) and 4-helicene, including also a study of the origin dependence and basis set convergence of 3PCD. We find that for the 3PCD-brightest low-lying Rydberg state of hydrogen peroxide, the dichroism is extremely basis set dependent, with basis set convergence not being reached before a sextuple-zeta basis is used, whereas for the MCP and 4-helicene molecules, the basis set dependence is more moderate and at the triple-zeta level the 3PCD contributions are more or less converged irrespective of whether the considered states are Rydberg states or not. The character of the 3PCD-brightest states in MCP is characterized by a fairly large charge-transfer character from the carbonyl group to the ring system. In general, the quadrupole contributions to 3PCD are found to be very small.

Origin-independent two-photon circular dichroism calculations in coupled cluster theory.

We present the first origin-independent approach for the treatment of two-photon circular dichroism (TPCD) using coupled cluster methods. The approach is assessed concerning its behavior on the choice of the basis set and different coupled cluster methods. We also provide a comparison of results from CC2 with those from density functional theory using the CAM-B3LYP functional. Concerning the basis set we note that in most cases an augmented triple zeta basis or a doubly augmented double zeta basis is needed for reasonably converged results. In the comparison of different coupled cluster methods results from CCSD, CC3 and CC2 have been found to be quite similar in most cases, while CCS results differ remarkably from the results at the higher levels. However, this proof-of-principle study also shows that further benchmarking of DFT and CC2 against accurate coupled cluster reference values (e.g. CCSD or CC3) is needed.

Pyrrolo[3,2-b]pyrroles-From Unprecedented Solvatofluorochromism to Two-Photon Absorption.

A combined experimental and theoretical study of the two-photon absorption (2PA) properties of a series of quadrupolar molecules possessing a highly electron-rich heterocyclic core, pyrrolo[3,2-b]pyrrole, is presented. In agreement with quantum-chemical calculations, large 2PA cross-section values, σ2PA ≈10(2) -10(3)  GM (1 GM=10(50)  cm(4) s photon(-1) ), are observed at wavelengths of 650-700 nm, which correspond to the two-photon allowed but one-photon forbidden transitions. The calculations also predict that increased planarity of this molecule through removal of two N-substituents leads to further increase in the σ2PA values. Surprisingly, the most quadrupolar pyrrolo[3,2-b]pyrrole derivative, containing two 4-nitrophenyl substituents at positions 2 and 5, demonstrates a very strong solvatofluorochromic effect, with a fluorescence quantum yield as high as 0.96 in cyclohexane, whereas the fluorescence vanishes in DMSO.

Benchmarking two-photon absorption cross sections: performance of CC2 and CAM-B3LYP.

We investigate the performance of CC2 and TDDFT/CAM-B3LYP for the calculation of two-photon absorption (TPA) strengths and cross sections and contrast our results to a recent coupled cluster equation-of-motion (EOM-EE-CCSD) benchmark study [K. D. Nanda and A. I. Krylov, J. Chem. Phys., 2015, 142, 064118]. In particular, we investigate whether CC2 TPA strengths are significantly overestimated compared to higher-level coupled-cluster calculations for fluorescent protein chromophores. Our conclusion is that CC2 TPA strengths are only slightly overestimated compared to the reference EOM-EE-CCSD results and that previously published overestimated cross sections are a result of inconsistencies in the conversion of the TPA strengths to macroscopic units. TDDFT/CAM-B3LYP TPA strengths, on the other hand, are found to be 1.5 to 3 times smaller than the coupled-cluster reference for the molecular systems considered. The unsatisfactory performance of TDDFT/CAM-B3LYP can be linked to an underestimation of excited-state dipole moments predicted by TDDFT/CAM-B3LYP.

Note: Derivation of two-photon circular dichroism--Addendum to "Two-photon circular dichroism" [J. Chem. Phys. 62, 1006 (1975)].

This addendum shows the detailed derivation of the fundamental equations for two-photon circular dichroism which are given in a very condensed form in the original publication [I. Tinoco, J. Chem. Phys. 62, 1006 (1975)]. In addition, some minor errors are corrected and some of the derivations in the original publication are commented.

Excited state polarizabilities for CC2 using the resolution-of-the-identity approximation.

We report an implementation of static and frequency-dependent excited state polarizabilities for the approximate coupled cluster single and doubles model CC2 as analytic second derivatives of an excited state quasienergy Lagrangian. By including appropriate conditions for the normalization and the phase of the eigenvectors, divergent secular terms are avoided. This leads to response equations in a subspace orthogonal to the unperturbed eigenvectors. It is shown how these projected equations can be solved without storage of the double excitation part of the eigenvectors. By exploiting the resolution-of-the-identity approximation and a numerical Laplace transformation, the quadratic scaling of the main memory demands of RI-CC2 with the system size could be preserved. This enables calculations of excited state polarizabilities for large molecules, e.g., linear polyacenes up to decacene with almost 2500 basis functions on a single compute node within a few days. For a test set of molecules where measurements are available as reference data, we compare the orbital-relaxed and unrelaxed CC2 approaches with experiment to validate its accuracy. The approach can be easily extended to other response methods, in particular CIS(D∞). The latter gives results which, in the orbital-relaxed case, are within a few percent of the CC2 values, while coupled cluster singles results deviate typically by about 20% from orbital-relaxed CC2 and experimental reference data.

Optical rotation calculations on large molecules using the approximate coupled cluster model CC2 and the resolution-of-the-identity approximation.

We investigate the performance of the approximate coupled cluster singles- and doubles model CC2 in the prediction of optical rotations of organic molecules. For this purpose we employ a combination of two test sets from the literature which include small and medium-sized rigid organic molecules and a series of helicenes. CC2 calculations on molecules as large as 11-helicene became possible through a recent implementation of frequency-dependent second-order properties for CC2 which makes use of the resolution-of-the-identity approximation for the electron repulsion integrals. The results are assessed with respect to the accuracy of the absolute values of the optical rotation and the prediction of the correct sign, which is crucial for the determination of absolute configurations. The performance of CC2 is compared with that of density functional theory at the B3LYP and CAM-B3LYP levels. Furthermore we investigated the influence of the molecular geometry and the one-electron basis set and tested to which extent spin-component scaling changes the results.

Intermolecular charge transfer enhances two-photon absorption in yellow fluorescent protein.

We present a quantum chemical study of the two-photon absorption (TPA) properties of yellow fluorescent protein (YFP), a mutant of the extensively studied green fluorescent protein. The aromatic chromophore of YFP has a π-stacking interaction with the aromatic ring of a tyrosine residue (Tyr203) in a parallel-displaced structure with a distance of about 3.4 Å. We study the TPA spectrum of the π-stacking system of YFP using the well-established Coulomb-attenuated B3LYP density functional (CAM-B3LYP) and the second-order approximate coupled-cluster model CC2. This work presents both the first comprehensive study of the two-photon absorption spectrum of YFP and the largest-scale coupled-cluster calculation of two-photon absorption that has ever been performed. We analyze the intermolecular charge-transfer (ICT) transitions in this stacked system and show that the ICT transitions are an important mechanism for enhancing the TPA cross sections in YFP. We investigate the distance dependence of the ICT transitions and show that their TPA cross sections are strongly dependent on the separation of the aromatic moieties. This provides a means for tuning the TPA properties of YFP and other structurally related fluorescent proteins through molecular engineering.

Rotational averaging of multiphoton absorption cross sections.

Rotational averaging of tensors is a crucial step in the calculation of molecular properties in isotropic media. We present a scheme for the rotational averaging of multiphoton absorption cross sections. We extend existing literature on rotational averaging to even-rank tensors of arbitrary order and derive equations that require only the number of photons as input. In particular, we derive the first explicit expressions for the rotational average of five-, six-, and seven-photon absorption cross sections. This work is one of the required steps in making the calculation of these higher-order absorption properties possible. The results can be applied to any even-rank tensor provided linearly polarized light is used.

Vibrational frequency scaling factors for correlation consistent basis sets and the methods CC2 and MP2 and their spin-scaled SCS and SOS variants.

We present scaling factors for vibrational frequencies calculated within the harmonic approximation and the correlated wave-function methods coupled cluster singles and doubles model (CC2) and Møller-Plesset perturbation theory (MP2) with and without a spin-component scaling (SCS or spin-opposite scaling (SOS)). Frequency scaling factors and the remaining deviations from the reference data are evaluated for several non-augmented basis sets of the cc-pVXZ family of generally contracted correlation-consistent basis sets as well as for the segmented contracted TZVPP basis. We find that the SCS and SOS variants of CC2 and MP2 lead to a slightly better accuracy for the scaled vibrational frequencies. The determined frequency scaling factors can also be used for vibrational frequencies calculated for excited states through response theory with CC2 and the algebraic diagrammatic construction through second order and their spin-component scaled variants.

Calculation of two-photon absorption strengths with the approximate coupled cluster singles and doubles model CC2 using the resolution-of-identity approximation.

An implementation of two-photon absorption matrix elements using the approximate second-order coupled-cluster singles and doubles model CC2 is presented. In this implementation we use the resolution-of-the-identity approximation for the two-electron repulsion integrals to reduce the computational cost. To avoid storage of large arrays we introduce in addition a numerical Laplace transformation of orbital energy denominators for the response of the doubles amplitudes. The error due to the numerical Laplace transformation is found to be negligible. Using this new implementation, we performed a series of benchmark calculations on substituted benzene and azobenzene derivatives to get reference values for TD-DFT results. We show that results obtained with the Coulomb-attenuated B3LYP functional are in reasonable agreement with the coupled-cluster results, whereas other density functionals which do not have a long-range correction give considerably less accurate results. Applications to the AF240 dye molecule and a weakly bound molecular tweezer complex demonstrate that this new RI-CC2 implementation allows for the first time to compute two-photon absorption cross sections with a correlated wave function method for molecules with more than 70 atoms and to apply this method for benchmarking TD-DFT calculations on molecules which are of particular relevance for experimental studies of two-photon absorption.

Large scale polarizability calculations using the approximate coupled cluster model CC2 and MP2 combined with the resolution-of-the-identity approximation.

We present an implementation of static and frequency-dependent polarizabilities for the approximate coupled cluster singles and doubles model CC2 and static polarizabilities for second-order Mo̸ller-Plesset perturbation theory. Both are combined with the resolution-of-the-identity approximation for electron repulsion integrals to achieve unprecedented low operation counts, input-output, and disc space demands. To avoid the storage of double excitation amplitudes during the calculation of derivatives of density matrices, we employ in addition a numerical Laplace transformation for orbital energy denominators. It is shown that the error introduced by this approximation is negligible already with a small number of sampling points. Thereby an implementation of second-order one-particle properties is realized, which avoids completely the storage of quantities scaling with the fourth power of the system size. The implementation is tested on a set of organic molecules including large fused aromatic ring systems and the C(60) fullerene. It is demonstrated that exploiting symmetry and shared memory parallelization, second-order properties for such systems can be evaluated at the CC2 and MP2 level within a few hours of calculation time. As large scale applications, we present results for the 7-, 9-, and 11-ring helicenes.

Trithiatetrazocine cations, [RCN4S3]+; planar sulfur-nitrogen 10pi aromatics.

The reaction of bicyclic sulfur-nitrogen heterocyles RCN 5S 3 (R = F 3C, Ph, Me 2N, 2-FC 6H 4, 2,6-F 2C 6H 3) with [Hg(SO 2) 2][AsF 6] 2 in liquid SO 2 yielded the corresponding trithiatetrazocinium-hexafluoroarsenates [RCN 4S 3][AsF 6] as yellow solids and a red-brown insoluble byproduct with the approximate composition Hg 3N 2. Single crystal structure determinations of the salts and theoretical calculations on the trithiatetrazocine cations [RCN 4S 3] (+) revealed the cations to be planar eight-membered heterocycles. The [RCN 4S 3] (+) cations are the missing link in the series of known valence isoelectronic eight-membered 10pi azocines, that is, [S 4N 4] (2+) and RC(NSN) 2CR. In contrast to neutral trithiatetrazocines and dithiatetrazocines RC(NSN) 2CR carrying donor substituents, which have a folded butterfly structure with a transannular S-S bond, the positive charge on the [RCN 4S 3] (+) cations always stabilizes the planar structure independent of the nature of the substituent. Experimentally obtained structures and theoretical calculations (geometries, frequencies, charges, nucleus-independent chemical shifts (NICS)) are in agreement with the description of [RCN 4S 3] (+) as planar eight-membered 10pi Huckel aromatic. The single crystal structure determinations of some 3,5-R-1,2,4,6-thiatriazinyl hexafluoroarsenates (R = Ph, 2-FC 6H 4, 4-FC 6H 4), which were isolated from the above reaction as well, are also included.

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.

Open-ended recursive calculation of single residues of response functions for perturbation-dependent basis sets.

We present theory, implementation, and applications of a recursive scheme for the calculation of single residues of response functions that can treat perturbations that affect the basis set. This scheme enables the calculation of nonlinear light absorption properties to arbitrary order for other perturbations than an electric field. We apply this scheme for the first treatment of two-photon circular dichroism (TPCD) using London orbitals at the Hartree-Fock level of theory. In general, TPCD calculations suffer from the problem of origin dependence, which has so far been solved by using the velocity gauge for the electric dipole operator. This work now enables comparison of results from London orbital and velocity gauge based TPCD calculations. We find that the results from the two approaches both exhibit strong basis set dependence but that they are very similar with respect to their basis set convergence.

Open-Ended Recursive Approach for the Calculation of Multiphoton Absorption Matrix Elements.

We present an implementation of single residues for response functions to arbitrary order using a recursive approach. Explicit expressions in terms of density-matrix-based response theory for the single residues of the linear, quadratic, cubic, and quartic response functions are also presented. These residues correspond to one-, two-, three- and four-photon transition matrix elements. The newly developed code is used to calculate the one-, two-, three- and four-photon absorption cross sections of para-nitroaniline and para-nitroaminostilbene, making this the first treatment of four-photon absorption in the framework of response theory. We find that the calculated multiphoton absorption cross sections are not very sensitive to the size of the basis set as long as a reasonably large basis set with diffuse functions is used. The choice of exchange-correlation functional, however, significantly affects the calculated cross sections of both charge-transfer transitions and other transitions, in particular, for the larger para-nitroaminostilbene molecule. We therefore recommend the use of a range-separated exchange-correlation functional in combination with the augmented correlation-consistent double-ζ basis set aug-cc-pVDZ for the calculation of multiphoton absorption properties.

Analytic Molecular Hessian Calculations for CC2 and MP2 Combined with the Resolution of Identity Approximation.

An implementation of analytic second derivatives for the approximate coupled cluster singles and doubles model CC2 and for second-order Møller-Plesset perturbation theory (MP2) will be presented. The RI approximation for the two-electron repulsion integrals is used to reduce memory demands, operation count, and I/O requirements. During the calculation, the storage of [Formula: see text] quantities (where [Formula: see text] is a measure for the system size) can completely be avoided. It is shown that with the MP2 method and an appropriate scaling of the harmonic frequencies, especially C-F stretch frequencies are reproduced much better in comparison to experiments than with the B3LYP density functional. Similar advantages are observed for molecules with strong, internal van der Waals interactions. Spin scaling offers additional improvements in these cases. The implementation has been tested for molecules with up to 81 atoms and 684 basis functions.