Four-Component Damped Density Functional Response Theory Study

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Four-component damped density functional response theory study of UV/vis absorption spectra and phosphorescence parameters of group 12 metal-substituted porphyrins Thomas Fransson, Trond Saue, and Patrick Norman J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.6b00030 • Publication Date (Web): 15 Mar 2016 Downloaded from http://pubs.acs.org on March 29, 2016

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Journal of Chemical Theory and Computation

Four-component damped density functional response theory study of UV/vis absorption spectra and phosphorescence parameters of group 12 metal-substituted porphyrins Thomas Fransson,† Trond Saue,‡ and Patrick Norman∗,† †Department of Physics, Chemistry and Biology, Link¨ oping University, SE-581 83 Link¨ oping, Sweden ‡Laboratoire de Chimie et Physique Quantiques, UMR 5626 CNRS — Université Toulouse III-Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse, France E-mail: [email protected]

Abstract The influences of group 12 (Zn, Cd, Hg) metal-substitution on the valence spectra and phosphorescence parameters of porphyrins (P) have been investigated in a relativistic setting. In order to obtain valence spectra, this study reports the first application of the damped linear response function, or complex polarization propagator, in the four-component density functional theory framework [as formulated in J. Chem. Phys. 133, 064105 (2010)]. It is shown that the steep increase in the density of states as due to the inclusion of spin-orbit coupling yields only minor changes in overall computational costs involved with the solution of the set of linear response equations. Comparing single-frequency to multi-frequency spectral calculations, it is noted that the number of iterations in the iterative linear equation solver per frequency grid-point

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decreases monotonously from 30 to 0.74 as the number of frequency points goes from one to 19. The main heavy-atom effect on the UV/vis-absorption spectra is indirect and attributed to the change of point group symmetry due to metal-substitution, and it is noted that substitutions using heavier atoms yield small red-shifts of the intense Soret-band. Concerning phosphorescence parameters, the adoption of a fourcomponent relativistic setting enables the calculation of such properties at a linear order of response theory, and any higher-order response functions does not need to be considered—a real, conventional, form of linear response theory has been used for the calculation of these parameters. For the substituted porphyrins, electronic coupling between the lowest triplet states is strong and results in theoretical estimates of lifetimes that are sensitive to the wave function and electron density parametrization. With this in mind, we report our best estimates of the phosphorescence lifetimes to be 460, 13.8, 11.2, and 0.00155 s for H2 P, ZnP, CdP, and HgP, respectively, with the corresponding transition energies being equal to 1.46, 1.50, 1.38, and 0.89 eV.

Introduction For the purposes of achieving a fundamental understanding of molecular properties and chemical reactions, quantum chemical calculations have arisen as an indispensable tool, evolving from sophisticated state-of-the-art calculations accessible only to specialists, to more widely available black-box programs. This development is largely a result of the advancements in density functional theory (DFT), which is today applicable for investigations in both quantum chemistry and solid state physics. For molecular systems, the most cherished approximation is probably the Born–Oppenheimer approximation, which is typically imposed for all but very special systems or when considerating conical intersections. Next to this approximation, the second most common assumption is likely the neglect of effects of relativity, typically included only as scalar relativistic effects—if at all. However, for some molecular properties, spin-orbit coupling is not only highly influential, but even represents the origin

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of the desired observable, and it has been noted 1 that relativity adds a third dimension on the standard two-dimensional view of quantum chemical models (the other dimensions being the one- and N-particle descriptions). As such, further development in four-component wave function theory and DFT is important and we here report the first application of the damped linear response function in the four-component time-dependent DFT (TDDFT) framework, 2 investigating the influences of metal-substitutions on porphyrins as an illustrative example. The choice of porphyrins is due to the plethora of experimental and theoretical studies devoted to this category of compounds, owing in turn to their crucial importance for life on Earth—porphyrin derivatives are e.g. involved in photosynthesis and oxygen transport and storage—as well as their many technological applications. In terms of the latter, areas of application include uses in photodynamic therapy, 3,4 solar cells,, 5–7 photodetectors, 8 other photochemistry, 9,10 catalysis, 11,12 multicolor thermometers, 13 data storage, 14 molecular recognition, 15 non-linear optics, 16 triplet photosensitizers, 17 and other nanoarchitectures. 18–20 In addition to the many possible applications, the high degree of molecular symmetry makes porphyrins a popular test case for the assessment and evaluation of new theoretical methods. As a result, the electronic spectrum of free-based porphyrin (H2 P), as well as ZnP, have been studied extensively using, e.g., TDDFT, 21–25 many-body perturbation theory (MBPT), 26 field-induced surface hopping (FISH), 27 multi-configurational second-order perturbation theory (CASPT2), 28 restricted-active-space self-consistent field (RASSCF) theory, 29 the density matrix renormalization group (DMRG) method, 30 coupled cluster singles and approximate doubles (CC2), 31,32 configuration interaction singles (CIS), 33 symmetryadapted cluster configuration interaction (SAC-CI), 34 the improved virtual orbital-complete active space configuration interaction (IVO-CASCI) method, 35 quantum Monte Carlo, 36 and combinations of density functional theory and multi-reference configuration interaction methods (DFT/MRCI). 37,38 Also triplet states and phosphorescence parameters have been investigated for H2 P and ZnP, 24,25,32,35,39–41 whereas no prior studies have been found for

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pristine CdP and HgP. Most studies to date have utilized a conventional linear response formalism, resolving the excited states in a bottom-up approach. Alternatively, recent developments of real-time (RT) propagation methods could be used to obtain the electronic absorption spectrum, as exemplified by the four-component RT-TDDFT 42 calculations of Ledge X-ray absorption spectra where the spin-orbit coupling is pivotal for a physically realistic system treatment. 43 In this study, we instead apply a damped linear response formalism, also known as the complex polarization propagator (CPP) approach, 44–46 in a four-component DFT framework. 2 In connection to relativistic calculations, we note that the damped linear response theory has also been implemented for two-component density functional theory using the zeroth-order regular approximation (ZORA). 47 With account made for the finite lifetime of excited states, a resonant-convergent approach is formulated that allows for the direct determination of molecular response properties in any spectral region. This approach has been successfully applied for a plethora of different molecular properties, e.g., Raman scattering, 48,49 circular dichroism, 50–52 optical rotatory dispersion, 53 X-ray absorption spectroscopy, 54,55 and C6 dispersion coefficients. 55,56 As our main focus is to demonstrate the virtues of the four-component CPP/DFT approach as well as to investigate trends due to metal-substitution, we determine only the electronic responses and merely refer to studies dedicated to vibrational effects 27,57 and we also adopt established assignments of spectral features. 34 For the valence spectra, measurements on H2 P and related compounds were initially rationalized using Gouterman’s four-orbital model, 58–60 where the relative intensity of the weak, low-energy (500−650 nm) Q-band in comparison to the strong, high-energy (350−400 nm) B-band, or Soret band, is understood as arising from transitions from the two highest occupied (HOMO and HOMO–1) to the two lowest unoccupied molecular orbitals (LUMO and LUMO+1). The transition moments between the involved orbitals are of similar size but opposite sign, leading to an effect of cancellation for the weak Q-band and strengthening for the intense B-band. This model has been shown sufficiently accurate for the low energy

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transitions, 23 and has been used for pigment design by tuning the four-orbital energy levels. 34 However, for transitions at higher energy, such as the L-, N-, and M-bands in the UV region, additional configuration mixing needs to be accounted for. 26,38 For free-base porphyrin, H2 P, the experimental transition energies have been measured as 1.98–2.41, 3.13–3.33, 3.65, and 4.25–4.68 eV for the Q-, B-, N-, and L-band, respectively. 61–63 In terms of triplet state properties, porphyrins have been reported to possess a strong inter-system crossing, with a quantum yield of 0.9 for the non-radiative S1 →T1 inter-system crossing in H2 P 64 as coupled through geometrical distortions. 37 However, the radiative decay from the T1 state has a very low quantum yield, observable mainly for metallo-porphyrins or in noble gas matrices, 37,65 and the mismatch to theoretical values indicates that the main decay is non-radiative. 39 For the calculation of the coupling strengths between the singlet and triplet states, or the phosphorescence parameters, spin-orbit effects can be introduced in the zeroth-order Hamiltonian 66 or using a perturbative approach. 39–41 Here we advocate the former, as the properties of interest appears at the correct response order so that single photon absorption/emission events are obtained from linear response theory, whereas a perturbative approach requires the solution of a quadratic response functions involving oneand two-electron spin-orbit operators. The goals of the present study are thus twofold: (i) to report a first application and demonstrate some of the virtues of the damped linear response function in the four-component relativistic regime, 2 by (ii) studying the effects on the porphyrin UV/vis absorption and emission spectra from metal substitution.

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Methodology For linear absorption of radiation, the property of interest is the molecular polarizability, which for exact states can be written as a sum-over-states formula 1 X h0|ˆ µβ |nihn|ˆ µα |0i µα |nihn|ˆ µβ |0i h0|ˆ ααβ (ω) = − , + h ¯ n>0 ωn0 − ω − iγn ωn0 + ω + iγn

(1)

where µ ˆα denotes the electric-dipole operator along molecular axis α, h ¯ ωn0 = En − E0 is the transition energy between ground state |0i and excited state |ni, and γn corresponds to the inverse (finite) lifetime of excited state |ni. For approximate state methods this expression turns into different forms of matrix equations, and in the random phase approximation (RPA) the most demanding computational step becomes solving the linear response equation, [1] E[2] − (ω + iγ)S[2] X(ω) = −EB ,

(2)

where the generalized Hessian, metric, and property gradient are written as E[2] , S[2] , and [2]

EB , respectively, and the solution vector (or linear response vector) is given as X(ω). For convenience, a common damping term has here been adopted for all excited states, i.e., γn = γ. In the infinite lifetime approximation, corresponding to γ → 0, the linear response function exhibits divergences at resonances, and can thus not be solved by direct inversion, or similar techniques. Standard methods of solving the linear response equation instead becomes a matter of finding the generalized eigenvalues and eigenvectors of the RPA matrix— corresponding to resonance energies and resonant-specific transition moments—typically in a bottom-up manner using some variation of the Davidson algorithm. If the relevant response properties are not found among the lowest solutions, as is the case for core-excitations, a restriction on the excitation channels can be imposed to disregard the low-energy solutions. 67 In the case of γ > 0, the explicit reference to excited states can be avoided. A large

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number of frequencies (well over 100) can be handled in parallel at the cost of single-frequency calculations, 68 so spectral ”windows” of some 10 eV can be readily determined without additional constraints. This in turns means that e.g. core-excitation regions of the spectrum can be treated on equal footing with the UV/vis region and without a need of introducing limited excitation channels 67,69 or a core-valence separation. 70,71 The polarizability in Eq. (1) is complex, with the real and imaginary parts being connected by the Kramer–Kronig relation and translating into different molecular properties. For the linear absorption of radiation, the cross section as due to a randomly oriented molecular sample is equal to σ(ω) =

ω Im [α] , ε0 c

(3)

where α ¯ denotes the isotropic average of the complex polarizability. It is to be noted that the pre-factor is occasionally written with ε−1 0 → 4π in the literature, which is numerically correct as long as atomic units are used. For the construction of a computationally tractable linear response function in fourcomponent theory, we adopt the methodology described in Ref. 2, where certain symmetries are exploited in order to obtain an efficient algorithm. For a nonzero complex frequency z = ω + iγ, the solution vector X(z) will lack well-defined Hermiticity and time-reversal symmetry, and in order to get real equations we decompose the solution vector into four unique symmetry-adapted components according to X = X++ + X−+ + X+− + X−− ,

(4)

where the signs corresponding to Hermiticity and time-reversal symmetry, respectively (explicit reference to the complex frequency is avoided, for brevity). We further convert the time-reversal antisymmetric elements into a symmetric form by extracting an imaginary phase according to Uh,t = iU

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−h,−t

,


(5)


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reversing the sign of both Hermiticity and time-reversal symmetry. Applied to the solution vector, this decomposition yields X = X++ + X−+ + i(X

−+

+X

++

).

(6)

For the coupling to an external electric field, the property gradient is associated to a Hermitian (h = +1) and time-reversal symmetric (t = +1) operator, and is thus designated as [1]++

EB

. The components of the solution vectors are now expanded in two sets of orthonormal

vectors of different Hermiticity,

X =

p X k=1

p X

X− =

m X

m X

+

+

+ a+ k bk ,

X =

+ a+ k bk ,

− a− k bk ,

k=1

−

− a− k bk ,

X =

k=1

k=1

where the reference to time-reversal symmetry is removed, as all properties are now timereversal symmetric. The RPA linear response function in Eq. (2) is solved in a reduced space, where the reduced equation takes the form 

˜ ++ E

˜ +− γ S ˜ +− −ω S

0    −ω S ˜ −+ ˜ −− ˜ −+ E 0 γS    −γ S ˜ −+ ˜ −− −ω S ˜ −+ 0 E   ˜ +− ω S ˜ +− ˜ ++ 0 −γ S E



+

 a    a−    a   ˜−  ˜+ a



 [1]+ ˜   − EB        0    = ,    0       0 

(7)

˜ ++ and E ˜ −− are square sub-matrices of dimension p and m, respectively, and S ˜ +− where E ˜ −+ are rectangular sub-matrices of dimensions (p × m) and (m × p), respectively. It and S is to be noted that this reduced equation is real, such that no complex equations needs to be explicitly calculated. From this the linear response function can now be computed, and from the imaginary part of this function the absorption cross section is then obtained. For the calculation of phosphorescence lifetimes, or the strength of the coupling between 8


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singlet ground state and the lowest triplet excited state, we calculate the oscillator strengths as f 0←f =

x,y,z 2mωf 0 X |h0 |rα | f i|2 , 3¯h α

(8)

using standard bottom-up routines for the calculation of the explicit excited states. From this, the transition rates of spontaneous emission are given according to k 0←f =

2ωf20 e2 0←f f . mc3 4πε0

(9)

Finally, if all spin sub-levels of the triplet state are assumed to be equally populated, the phosphorescence lifetime is given as, summing over triplet sub-levels i

τ=P

1 i

ki0←f

.

(10)

Computational details Molecular structures have been optimized with use of the Gaussian03 program, 72 and the ground and lowest triplet state geometries have been optimized using restricted and unrestricted KS-DFT, respectively, using the B3LYP 73,74 functional and Dunning’s all-electron cc-pVDZ basis set 75 for the non-metals. For the metals a Stuttgart ECP was utilized, adopting ECP28MWB 76 with the double-ζ basis set as provided in the Dirac program distribution for Zn and Cd, and ECP60MWB 77 as implemented in Gaussian03 for Hg. All calculations of response properties have been performed with the Dirac program 78 into which Eq. (2) has been implemented in the four-component DFT approximation. Here an uncontracted versions of the all-electron DK3 basis sets 79 have been adopted for metal atoms. For hydrogen, carbon, and nitrogen contracted 6-31G 80 basis sets were chosen for all spectrum calculations, while the phosphorescence parameter calculations also utilized Dunning’s basis sets augmented with diffuse functions. 81 The Coulomb-attenuated B3LYP

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(CAM-B3LYP) functional 82 has been used for most UV/vis spectrum calculations, with B3LYP results discussed in connection to spectrum effects due to metal-substitution. Additional test calculations utilizing alternative functionals were performed for reasons of comparison and with results presented in Supplementary Materials. The phosphorescence parameters have been obtained using the B3LYP and PBE0 83 functionals. Relativistic calculations have been performed using the Dirac–Coulomb Hamiltonian with (SS|SS) integrals neglected and replaced by interatomic SS energy corrections, 84 with comparisons made to results obtained using the non-relativistic Lévy-Leblond 85 and Dyall’s spin-free 86 Hamiltonians. 87 For the relativistic calculations the small component basis sets were generated using the condition of restricted kinetic balance, and rotations to negativeenergy orbitals have been excluded. For substitutions with the heavier metal atoms as well as for calculations using large basis sets, an atomic start approach 88 has been utilized in order to obtain a reasonable starting wave function for the ground state calculations. In order to obtain explicit excited states a standard bottom-up eigenvalue solver has been utilized for the phosphorescence parameter calculations, as well as calculations illustrating the density-of-states for the systems. All remaining calculations have utilized the CPP algorithm summarized above and described in more detail in Ref. 2. A value for the damping parameter γ of 1000 cm−1 (0.124 eV) has been adopted, yielding spectral features with a resolution expected to match that of experiment.

Results and discussion Geometry effects of metal-substitution For substituted porphyrins, the π-conjugation favors a planar structure and distortions typically only occur due to the substituents, 89 with the size of the metal atom determining whether it can fit in the cavity or if a non-planar structure is imposed. Depending on the intended substitution, the porphyrin cavity has been reported to be able to accommodate 10


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metal atoms with ionic radii in the interval of 75–90 pm 89,90 while remaining planar. As such, zinc can easily fit into the cavity while mercury is too large to retain a planar structure. By comparison, cadmium is a limiting case with, for instance, the Cd(II) ionic radius being equal to 95 pm. 90 In the present case, we find a planar ground state equilibrium structure for CdP, so the S0 symmetries of H2 P, ZnP, CdP, and HgP are found to be D2h , D4h , D4h , and C4v , respectively. These results are consistent with earlier studies for H2 P, ZnP, and CdP, 35,39,57,91 whereas structures for HgP have not been found in the literature. For the lowest triplet state, H2 P retains a planar D2h symmetry, whereas the other species yields structures of lower symmetry as compared to that of the ground state. For ZnP the triplet state remains planar but loose some rotational symmetry. CdP becomes saddle-like in the triplet state, and, finally, HgP has a dome-like S0 state and a saddle-like T1 state. Accordingly, excited triplet state equilibrium structures are found to span the point groups D2h , D2h , C2v , and C2v for H2 P, ZnP, CdP, and HgP, respectively. This is again in agreement with previous findings. 39 The full structures are visualized and tabulated in Supplementary Materials. A note on the choice of Cartesian axes: we have placed molecules in the yz-plane (with z along the NH · · · HN axis for H2 P). In the original Gouterman model, the x- and z-axis are interchanged, and the designation of the excited states is thus different—what is here given as singlet states of B1u symmetry is in studies adopting the Gouterman choice of axes given as states of B3u symmetry, and vice versa. This difference is most easily elucidated by studying the intensity of the spectral peaks as out-of-plane transitions (B3u in our work) are significantly weaker than the in-plane transitions. We also note that a non-relativistic labeling of states as belonging to the singlet or triplet manifolds will be used also in the present work even though we are well aware that spin is no longer a good quantum number due to the spin-orbit coupling. This is done in order to facilitate the comparison to previous studies, but caution is called for as this can at times be misleading if the electronic coupling between states of different spin multiplicity is significant.

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Convergence of the damped linear response function General convergence properties of the adopted algorithm to solve the damped linear response function have been discussed in earlier studies. At the four-component relativistic level of theory, a proof-of-principle implementation was presented in the original work and the convergence was studied for a very small system—lithium hydride (LiH). 2 At the non-relativistic level, it was for the first time demonstrated that highly efficient calculations of spectra could be performed with the adaption of the algorithm to handle hundreds of optical frequencies in parallel. 68 In the non-relativistic framework, time-reversal symmetry is not exploited in the same manner so the reduced-space algorithm is in this case also modified but there is of course reason to believe that spectral calculations should also be more or less equally efficient in the relativistic framework. Now, with the implementation completed and ready for use, we can perform the verification of this presumption by the calculation UV/vis-absorption spectra for systems of real interest. In Fig. 1, we present the convergence behavior for the calculation of the complex polarizability at n optical frequencies simultaneously, covering an increasingly larger spectral region centered about 3.63 eV and expanding by steps of 0.07 eV to each side. This calculation is performed for HgP at the four-component relativistic CAM-B3LYP level of theory, a level at which the strong dipole-allowed transition (the B-band) is found at 3.61 eV. The number of iterations required for converging all frequencies is reported, as resulting from treating an increasing number of frequencies simultaneously. Due to an increasing number of trial vectors in each iteration that can (potentially) be added to the reduced space, the total number of required iterations decrease steadily with the number of frequencies. The dimension of the reduced space after convergence is reached grows with an increasing number of frequencies, although the number of iterations decreases. However, the increase in the dimension of the reduced space is slow, going from 120 for a single frequency to 262 for a set of 19 frequencies. It is the reduced space equation that is explicitly solved by means of matrix inversion and we can easily afford reduced space dimensions that are an order of magnitude larger before it is 12


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anywhere near becoming a computational bottleneck, memory-wise or CPU-wise. A treatment of over a hundred frequencies in parallel is clearly manageable from this point of view, just as noted in the non-relativistic case. 68 If we consider the number of solver iterations per frequency, we observe a monotonic decrease, amounting to 0.74 for n = 19. It is clear that the CPP approach benefits from treating several frequencies simultaneously, and, since the overhead is small to construct multiple Fock matrices in parallel, the overall wall-time for the calculation of an entire spectrum is not significantly exceeding that of the calculation of a single excited state or a static polarizability.

UV/vis-absorption spectra Relativistic effects Before discussing to the results of the response calculations, we note a few technical details: By default in the Dirac program, the starting self-consistent field (SCF) density is obtained by mean of diagonalizing an empirically corrected bare nucleus Hamiltonian. This choice is too poor to reach convergence for systems as complex as those studied in the present work. For this reason, we have adopted an atomic guess start. 88 Concerning more extensive basis set investigations, we refer to previous studies on UV/vis-absorption that find only modest dependencies of these linear response properties on the addition of polarization and diffuse functions. 23,28,92,93 As most standard basis sets are optimized for non-relativistic calculations, it can be expected that they perform worse for relativistic calculations where the inner core orbitals are typically contracted. For the metal atoms, we therefore employ uncontracted basis sets and the issue is only a concern for the description of the core orbitals of C and N. Performed tests of decontraction showed, however, that the use of non-relativistic, contracted, basis sets for light elements does not deteriorate the accuracy in results for UV/vis-absorption properties of porphyrins. This result is not very surprising since the small relativistic contraction of the 1s-orbitals will have a negligible effect on the valence electron density involved in the electronic transitions under study. 13



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In Fig. 2, the effects of relativity—as well as of an imposed planarity in panel (d) to change point group from C4v to D4h —are reported for HgP. Included in the figure are black bars to indicate the excitation energies and oscillator strengths of the intense transitions, and colored circles to indicate the positions of all excited states. A color code of the circles is adopted to show the characteristics of the respective transitions: In panels (a) and (b), blue circles mark states of triplet spin symmetry, and, for all figure panels, colors of black and red are used to separate transitions that are electric dipole-allowed and forbidden, respectively. For the nonrelativistic calculation presented in panel (a), the degeneracy of the excited states are also included for all transitions up to the intense B-band and, here again, red color designates dipole-forbidden excitations. The intense B-band is labeled by ”2” in black, indicating a dipole- and spin-allowed transition to a doubly degenerate state spanning the irreducible representation E. Likewise, the lowest state in the spectrum is a doubly degenerate triplet state that, in a non-relativistic notation, is written as 3 E and thus contain six underlying states. With spin-orbit interactions taken into account for these six degenerate excited states, four of them belong to the irreducible representation E and two to A2 —the latter two are thus electric-dipole forbidden. For HgP, it is seen that the relativistic effects of the spectral features are well accounted for using a scalar relativistic Hamiltonian, with discrepancies in obtained energies amounting to less than 0.01 eV, as compared to results obtained using the relativistic Dirac–Coulomb Hamiltonian. In comparison, the fully non-relativistic calculation displays discrepancies of up to 0.6 eV, but the larger errors are mainly found for the triplet and low-intensity singlet states. In terms of oscillator strengths, we note that the spin-mixing of close-lying states can be substantial, with intensity up to some 10% being transfered from singlet to triplet states (using the non-relativistic designation). The imposition of planarity results in a spectrum shown in panel (d). It is seen that this strained molecular structure displays an absorption spectrum that, for most parts, is very much like that obtained for the relaxed molecular structure. Upon closer inspection, however, it is noted that quite large differences can be

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found in the energetics and transition moments of the triplet states, e.g., the near degeneracy seen in panel (c) of the six components of the second excited 3 E state is broken. From a computational perspective, the spectrum calculation for HgP reveals an important aspect, namely the large differences in the number of transitions, or density-of-states (DOS), to deal with depending on whether a relativistic or non-relativistic Hamiltonian is employed. If a bottom-up linear response approach is adopted, a total of 43 excited states needs to be converged in the calculation before the intense B-band is reached. Admittedly, some of these states (10) are dipole-forbidden and others belong to different irreducible representations, which are facts that can be exploited in the calculation to improve convergence and lessen the requirements on computational resources. But this is besides the point, because, in a general perspective, biochemical molecular systems are rarely ever as symmetric as the metal-substituted porphyrins and, even in cases when the pristine chromophores are highly symmetric, there is typically a need to include a description of the symmetrybreaking environment—as, for example, a molecular mechanics (MM) description of a liquid environment. In such a situation, it becomes a highly demanding calculation to determine UV/vis-spectra at the relativistic level of theory for systems of biochemical interest. However, it is precisely in this situation, when the DOS is high, that the CPP response theory approach has its main advantages. Since a direct address is made of an arbitrary frequency ”window”, the CPP approach is more or less insensitive to the DOS—mainly affecting the convergence rate, but also this aspect turns out not all too severe in practice. So it is clear that relativistic spectrum calculations benefit strongly from being performed in conjunction with the CPP methodology. Metal-substitution effects The UV/vis-absorption spectra of the free-based and metal-substituted porphyrins obtained at the relativistic CAM-B3LYP level of theory are illustrated in Fig. 3, with excitation energies and intensities reported in Table 1. The position of the electronic excited states are

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indicated by circles that are color coded as in Fig. 2, with black and red colors denoting electric dipole-allowed and forbidden states, respectively. In addition, we have also determined spectra with use of the B3LYP functional and results can be found in Table 1 and illustrated in Supplementary Materials. Focusing on the CAM-B3LYP results, it is clear that the most significant effect in terms of spectral features is due to the metal-substitution, with all three metal-porphyrins showing very similar spectra. As compared to H2 P, the main difference shown in the spectra of the metal-porphyrins are the degeneracy in the B-band, significant differences in the intensities of the N-band, and, to a smaller extent, the intensity differences in the L-band. There are also clear changes to be observed in the low-energy triplets, but we postpone a discussion around this observation until later when we discuss the phosphorescence parameters. We further note that the Q-band in the theoretical spectra, situated at approximately 2.4 eV, are almost invisible, as opposed to experimental measurements. This is due to a lack of vibronic effects in the present study. 23,39,57,94 As stated in the introduction, the valence absorption spectrum of especially H2 P has been extensively investigated in the literature and we refer to the compilation of theoretical data, including oscillator strengths, found in Ref. 22 as well as the TDDFT review on the subject. 24 We adopt the designations of the different bands from Ref. 34 and note that all singlet in-plane transitions of B1u and B2u symmetry are of π → π ∗ character (as is the lowest triplet transition 39 ), while the out-of-plane singlet transitions of B3u symmetry are of n → π ∗ character. We recall that for free-based porphyrin, the experimental spectrum is characterized by a weak Q-band at 1.98–2.41 eV, a very intense B-band at 3.13–3.33 eV, followed by relatively weak N- and L-bands at 3.65 and 4.25–4.68 eV, respectively. 61–63 Further, the lowest triplet transition (B2u 41 ) is found experimentally at 1.56 eV. 61 The CAM-B3LYP results reported in Fig. 3 displays too high excitation energies for all singlet bands, the overestimation amounting to about 0.2, 0.4, 0.9, and 0.9 eV for the Q-, B-, N-, and L-band, respectively. Results using a B3LYP functional yields a similar trend but with smaller energy discrepancies as compared

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to experimental results, in this case being equal to about 0.3, 0.2, 0.2, and 0.3 eV with the same ordering of bands. It is to be noted that the discrepancies are in most cases larger for the B1u than for the B2u transitions. In terms of intensities, the B3LYP spectrum shows By - and Ny -bands of comparable intensities, while the experimental measurements report a difference of one order of magnitude. As a result of the poorly predicted intensities at the B3LYP level of theory, we choose to focus on the CAM-B3LYP spectra in the main article. In Supplementary Materials, we also present spectra for H2 P obtained with functionals where the parameters are tailored in order to gradually go from either CAM-B3LYP or BHandHLYP toward B3LYP, and it is demonstrated that changing the amount of exact exchange in the functional—either at the asymptotic limit or as a range-independent parameter—has quite different effects on the separate bands in the absorption spectrum. Following spectral changes due to altering the amount of exact exchange in the functional, it becomes apparent that the N-band in fact has contributions from two underlying states coupled to the ground state with the two different in-plane (yz) dipole operators. The Cartesian z-component, directed along the NH · · · HN axis in H2 P, has very low intensity at the CAM-B3LYP level as compared to the y-component (the other in-plane component), whereas the two transitions (referred to as band components Nz and Ny ) have comparable intensities at the B3LYP level. The transition energies for both Nz and Ny are strongly dependent on the amount of exchange, which is a clear indication of a charge-transfer character in these transitions—for Ny this effect is particularly strong, reducing the excitation energy by about 0.75 eV in going from CAM-B3LYP to B3LYP. In contrast, the effects on the B-band from altering the amount of exchange are much smaller, reducing transition energies of the Bz and By components by some 0.2 eV in going from CAM-B3LYP to B3LYP and showing a modest functional dependence for intensities. Our results are similar to what has previously been reported for these functionals 23,25 (cf. Table 1). Comparing our excitation energies to calculations using more computationally demanding methods, we note that our CAM-B3LYP energies are similar to results obtained using SAC-CI 34 or STEOM-CC, 95 while

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our B3LYP results are more in line with reported DFT/MRCI 38 results. In the framework of damped linear response theory, a partial explicit inclusion of nuclear motion can be addressed by a combination of MD simulations and CPP spectrum calculations, 96,97 but this is outside the scope of the present work. Here, vibrational effects are implicitly included in the damping parameter, as it phenomenologically accounts for broadening due to the lifetime of excited states, finite resolution in experimental measurements, and (symmetric) broadening effects. Explicit considerations of vibrational effects can be found in the literature for both H2 P 57 and ZnP, 94 where it has been concluded that the vibronically induced transitions via the Herzberg–Teller mechanism provide important corrections, particularly for the Q-band, whereas Franck–Condon corrections are less prominent. For the metal-porphyrins, experimental measurements have been found only for ZnP, and in this case the experimental excitation energies are reported as equal to 2.18–2.33, 3.13, 4.07, and 5.18 eV for the Q-, B-, N-, and L-bands. 98 As compared to free base porphyrin, the higher degree of symmetry in the metal-porphyrins imply that the two Cartesian components to each absorption band are fully degenerate and equally intense. In comparing results obtained with the CAM-B3LYP and B3LYP functionals, we note that, intensity-wise, spectra are quite similar and showing B-bands that are much stronger than the corresponding N-bands. But the energy separations between B- and N-bands is somewhat smaller at the B3LYP level as compared to the CAM-B3LYP results, again due to the better description of charge-transfer character of the N-band in latter case. The B3LYP results obtained here agrees well with previous theoretical considerations. 40,99 In comparing spectra for the separate metal-porphyrins, we note that the strong B-band becomes slightly more red-shifted the heavier the metal atom—this is consistent with measurements on substituted phthalocyanine derivatives, 100 where it was attributed to the increase in electron density. In addition, there are significant shifts of the triplet excitation energies that will be discussed in the next section concerned with the phosphorescence. For metal-substituted porphyrins, previous studies have shown that a loss of planarity of systems is the primary factor for spectral

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changes, whereas the electronic structure of the metal center is less important in this respect. 89 An effect of non-planarity has been observed to be a red-shift of ππ ∗ -bands, with a size proportional (in a non-linear fashion) to the distortion. In this study, we see that the small distortion to a dome-like structure induced by the Hg-substitution is not sufficient to change the electronic structure of the porphyrin to any significant degree, as is evident from the comparison of spectra for non-planar and planar HgP made in Fig. 2. The small changes imposed by the different substitutions are further supported by experimental studies of metal-substituted tetraphenylporphyrin. 61,101

Phosphorescence parameters For the calculation of phosphorescence parameters, an approach which incorporates the spinorbit effects in the zeroth-order Hamiltonian has the advantage of yielding the necessary transition moments and energies at the linear response theory level. By comparison, a perturbative approach requires the calculation of a quadratic response function in order to obtain such properties. In determining phosphorescence parameters, we have adopted a conventional bottom-up approach to converge the lowest triplet states. Since, according to Kasha’s rule, 102 light emission from organic molecules occur exclusively from the lowest electronically excited state of a given spin multiplicity, one needs only to converge a single root of the response matrix in connection with phosphorescence studies. In this case, it would be counter-productive to employ the CPP approach since, although transition energies and oscillator strengths are available also in this framework, it does require an integration of the absorption band (similar to what needs to be done to extract oscillator strengths from experimental absorption spectra). The resulting phosphorescence parameters for the porphyrins under study are found in Table 2, as obtained using two different functionals and three basis sets. Comparisons are made to experimental and theoretical studies of H2 P and ZnP, while we are not aware of any previous work for for CdP and HgP. In the spectrum calculations, we deemed CAM-B3LYP results as superior to those obtained with 19



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other functionals. When it comes to phosphorescence parameters, the situation becomes different since it targets states of triplet spin multiplicity. It is well-known that the treatment of triplets with the CAM-B3LYP functional can be problematic, 103 a fact that is attributed to triplet instability due to the large amount of asymptotic exact exchange 104 or arising from the adiabatic approximation errors. 105 For this reason, we choose to present phosphorescence parameters obtained with the B3LYP and PBE0 functionals in Table 2. Concerning phosphorescence, PBE0 yields lifetime parameters that are some 30–80% larger than what is found with use of the B3LYP functional (for a given basis set) and energetics that are lower by about 0.1 eV. Adding polarization and diffuse functions to the basis sets, changes the B3LYP excitation energies by at most 0.09 eV, but there are differences in phosphorescence lifetimes by up to a factor of 6.0. This sensitivity in results for lifetimes is by far largest for free base porphyrin, where it changes from 167.2 s at the B3LYP/6-31G level to 968.7 s at the B3LYP/aug-cc-pVDZ level. By employing a triplet-ζ basis set, the lifetime is reduced to an intermediate value of 459.9 s. The strong variations in lifetimes with respect to changes in the basis set is predominantly due to a sensitiveness in the calculation of transition moments, which are amplified since it is the square of this property that enters in the evaluation of the corresponding lifetimes. For singlet–triplet transitions, the transition moments are obviously weak and therefore numerically sensitive for this reason alone. But in the case of porphyrin, there is also a more subtle reason as to why results vary so strongly with basis sets. There is an effective coupling of the lowest electronic states resulting in a ”borrowing” of intensity from one state to another. In such a situation, it can be wise to study the integrated intensity in a narrow frequency ”window” in order to get numerically more stable results. The summed oscillator strength of the first two triplet states of H2 P is much more stable towards basis set alterations, reducing the largest observed difference in lifetimes to a factor of 2.5 instead of 6.0. Doing the same comparison for results obtained with the two Dunning’s basis sets shows that, when the summed value for the two lowest states is considered, discrepancies in oscillator strengths decrease from 2.1

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to 1.2. The latter value is in much better line with the general understanding that, for DFT response calculations performed on medium- and large-size molecules, the aug-cc-pVDZ basis set should be quite accurate. The energy separation of the lowest triplets is larger for ZnP, CdP, and HgP and we note that the basis set sensitiveness is correspondingly much smaller than for H2 P. The strong dependence of the phosphorescence lifetime on the parametrization has also been observed in previous studies, 39 and is further illustrated by making a comparison to the results for porphyrins published by Minaev. 41 For free base porphyrin, lifetimes are reportedly varying by a factor of ten for different basis sets, 41 and the best results obtained in this work are comparable to our aug-cc-pVDZ results. Phosphorescence parameters for geometries that are not relaxed in the excited triplet state have also been obtained at the B3LYP and CC2 levels of theory. Not taking geometry relaxation into account can cause large errors in the calculation of phosphorescence lifetimes—the reported B3LYP lifetime of H2 P at the ground state geometry is an order of magnitude larger than what is obtained for the relaxed structure. 39 At the wave function correlated CC2 level of theory, on the other hand, a much shorter lifetime of 167 s is reported for free base porphyrin. 32 A contributing factor to this low value is clearly the overestimated transition energy of 2.04 eV, which is an error that is amplified due to the inverse cube dependence of the lifetime on the transition energy as is seen when combining Eqs. (8)–(10). It is thus clear that obtaining reliable benchmarks of phosphorescence lifetimes is difficult from a numerical point of view. We note, however, that the trends that we obtain are very clear: Comparing H2 P with ZnP, the metal-substitution yields a very large decrease in lifetime, amounting to 1–2 orders of magnitude depending on the adopted functional and basis set. Going from Zn to the heavier Cd atom only results in a minor decrease in lifetimes by some 20–40%. Finally, turning from Cd to the very heavy Hg atom, the lifetime decreases by two orders of magnitude, and the excitation energies decrease by 0.4–0.5 eV. We note here that the T1 optimized geometries of CdP and HgP are both of a C2v symmetry, with

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HgP being more distorted. Comparing to experimental measurements, we note that phosphorescence lifetimes of the porphyrins under study are overestimated by most theoretical considerations. However, the phosphorescence of H2 P is very difficult to observe experimentally, usually requiring an enhancement by internal or external heavy atom effects. Previous analyses of the phosphorescence parameters have shown that the short experimental lifetime is due not only to the radiative decay but also the non-radiative decay. Comparison to the decay rates reported in Ref. 106, for which estimates of the pure radiative decays are given, shows improved agreement to the theoretical values. Utilizing experimental excitation energies in combination with the most reliable transition moments obtained in this study, the best estimates for H2 P and ZnP becomes 380 and 7.7 s, respectively, as compared to 460 and 13.8 s, as reported in Table 2. Part of the discrepancy can thus be attributed to inaccuracies in excitation energies. We further note that the measurement showing the shortest lifetime was conducted in ethyl iodide solvent, 107 which influence the lifetime parameters by the external heavy atom effect, as well as the fact that the vibronic activity may increase the decay rate. 39 For more details on the comparison to experimental values, as well as a consideration of vibronic effects, we refer to the comprehensive studies of Minaev and co-workers. 39,41,65 In conclusion, theoretical benchmark values of the phosphorescence lifetimes of porphyrins are difficult to obtain for a large number of reason, but reasonably accurate estimates can be provided for H2 P, ZnP, CdP, and HgP amounting to 460, 13.8, 11.2, and 0.00155 s, respectively.

Conclusions We report the implementation and a first application of the damped linear response function (or complex polarization propagator, CPP) in the four-component DFT framework, based on the formulation and proof-of-principle implementation presented by Villaume et al. 2 In

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our previous work, response matrices for a very small system were exported to an external program script with which the algorithm could be studied, and the present study therefore marks the first general implementation and use of the damped linear response function. We have studied the effects of group 12 metal-substitution on porphyrins—both in terms of UV/vis-absorption spectra and phosphorescence parameters. The advantages of using a damped linear response formalism are accentuated in the relativistic realm where the density of valence excited states is high due to the loss of spin symmetry. A standard, bottom-up, calculation using, e.g., the Davidson algorithm, requires the convergence of over 40 excited states at the four-component relativistic level of theory in order to address the lowest intense Soret band in porphyrins. For larger or less symmetric systems, the resolution of high-energy states would be hampered by such a high density of low-lying states. As such, while the studied molecules do not exhibit large relativistic effects for the singlet states, they provide a good illustration of the present method by allowing for the comparison of CPP and conventional response theory approaches. Regarding absorption spectra, good agreement to previous theoretical results and, to a lesser extent, experiments are obtained for H2 P and ZnP, while for CdP and HgP earlier work is not found in the literature. The largest influence of metal-substitution is due to changes in molecular symmetry, whereas the differences in spectra for ZnP, CdP, and HgP are small and amounting to a red-shift in the strong B-band that increases for the heavier metal atoms. The spectrum of especially H2 P is observed to exhibit a strong dependence on the amount of exact exchange in the functional, which allows the characterization of the N-band (but not the B-band) to be associated with charge-transfer transitions. Our relativistic approach with spin-orbit interaction included variationally furthermore simplifies the calculation of phosphorescence parameters in that they can be obtained at the linear response level, thus avoiding the calculation of a quadratic response function as in the perturbative approach. The calculation of lifetimes is shown to be strongly dependent on the parametrization in general and on the basis set in particular. It is demonstrated

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that, to a large extent, this is due to a strong electronic coupling among the lowest triplet states, resulting in a ”borrowing” of intensity. The weak spin-orbit coupling in H2 P is associated with a long phosphorescence lifetime that decreases by 1–2 orders of magnitude with the substitution of Zn. Whereas ZnP and CdP have similar lifetimes, the substitution with Hg (as compared to Cd) imply a decrease by another 2 orders of magnitude, as a result of a significant increase in the spin-orbit coupling. The best theoretical estimates of phosphorescence lifetimes are 460, 13.8, 11.2, and 0.0155 s for H2 P, ZnP, CdP, and HgP, respectively, and the corresponding transition energies are 1.46, 1.50, 1.38, and 0.89 eV.

Acknowledgments Financial support from the Swedish Research Council (Grant No. 621-2014-4646) is acknowledged. The calculations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at National Supercomputer Centre (NSC), Sweden.

Supporting Information Molecular structures for H2 P, ZnP, CdP, and HgP in the S0 and T1 states as well as UV/vis spectra obtained with other choices of exchange-correlation functionals are available as supporting information. This information is available free of charge via the Internet at http://pubs.acs.org.

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Table 1: Excitation energies (eV) and oscillator strengths for the first four in-plane singlet excited states of porphyrin derivatives. Four-fold rotational symmetry of ZnP, CdP, and HgP ensures that B1u and B2u are identical for ZnP and CdP, and B1 and B2 states are identical for HgP. 11 B1u /Q ∆E H2 P CAM-B3LYP/6-31G B3LYP/6-31G CAM-B3LYP/6-31G* 22 B3LYP/6-31G* 22 B3LYP/cc-pVTZ 65 SAC-CI 34 DFT/MRCI 38 STEOM-CC 95 PT2 28 Expt 61

H2 P CAM-B3LYP/6-31G B3LYP/6-31G CAM-B3LYP/6-31G* 22 B3LYP/6-31G* 22 B3LYP/cc-pVTZ 65 SAC-CI 34 DFT/MRCI 38 STEOM-CC 95 PT2 28 Expt 61 ZnP CAM-B3LYP/6-31G B3LYP/6-31G B3LYP/6-31G** 39 DFT/MRCI 38 Expt 98 CdP CAM-B3LYP/6-31G B3LYP/6-31G

HgP CAM-B3LYP/6-31G B3LYP/6-31G

f

21 B1u /B ∆E f

31 B1u /N ∆E f

41 B1u /L ∆E f

2.21 3(-4) 2.30 2(-3) 2.19 1(-3) 2.28 1(-6) 2.25 3(-3) 1.96 8(-3) 1.94 7(-3) 1.75 1(-3) 1.63 4(-3) 1.98 0.01 11 B2u /Q ∆E f

3.60 0.83 3.38 0.41 3.55 0.80 3.33 0.41 3.32 0.64 3.69 1.3 3.07 0.49 3.47 0.69 3.12 0.91 3.33 1.15 21 B2u /B ∆E f

4.30 0.60 3.89 0.81 4.27 0.59 3.86 0.78 3.96 0.55 4.41 0.82 3.56 0.98 4.06 0.93 3.42 0.83 3.65

Four-Component Damped Density Functional Response Theory Study

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