Vibrationally resolved optical spectra of modified diamondoids obtained from time-dependent correlation function methods

Shiladitya Banerjee , Tony Stüker and Peter Saalfrank *
Institut für Chemie, Universität Potsdam, Karl-Liebknecht-Straße 24-25, D-14476 Potsdam-Golm, Germany. E-mail: peter.saalfrank@uni-potsdam.de

First published on the web 1st July 2015

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Optical properties of modified diamondoids have been studied theoretically using vibrationally resolved electronic absorption, emission and resonance Raman spectra. A time-dependent correlation function approach has been used for electronic two-state models, comprising a ground state (g) and a bright, excited state (e), the latter determined from linear-response, time-dependent density functional theory (TD-DFT). The harmonic and Condon approximations were adopted. In most cases origin shifts, frequency alteration and Duschinsky rotation in excited states were considered. For other cases where no excited state geometry optimization and normal mode analysis were possible or desired, a short-time approximation was used. The optical properties and spectra have been computed for (i) a set of recently synthesized sp²/sp³ hybrid species with CC double-bond connected saturated diamondoid subunits, (ii) functionalized (mostly by thiol or thione groups) diamondoids and (iii) urotropine and other C-substituted diamondoids. The ultimate goal is to tailor optical and electronic features of diamondoids by electronic blending, functionalization and substitution, based on a molecular-level understanding of the ongoing photophysics.

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I. Introduction

Diamondoids form a family of hydrocarbons, consisting of repeated units of connected cyclohexane rings.^1–3 The first member of this series is adamantane, C₁₀H₁₆, a saturated diamondoid with sp³-hybridized carbon atoms. Higher homologues are known as diamantane, C₁₄H₂₀, triamantane C₁₈H₂₄, tetramantane C₂₂H₂₈, pentamantane and so on. Diamondoids are chemically quite inert, hard compounds.⁴ They are good (monochromatic) electron emitters with negative electron affinity,^5–7 transparent wide-gap materials, absorbing light significantly around and above about 6 eV,^8–10i.e., in the UV. Diamondoids are excellent fluorescing materials, known to fluoresce both in the solid state and in the gas phase. For absorption to and fluorescence from, the lowest electronic excited states, often pronounced vibrational finestructures in spectra have been observed.¹⁰ Vibrations involving excited states are also important for the interpretation of resonance Raman spectra, another important tool to characterize diamondoids.¹¹ In general, vibronic spectra are powerful tools to unravel details of optical properties of diamondoids and the focus of this paper.

Due to the diverseness of their shape and composition and the ability to be functionalized (see below), one hopes that diamondoids have tunable optical and electronic properties for possible applications. Ref. 12 reported the effects of C–H and interstitial substitution on the HOMO–LUMO energy and the band gaps of a range of diamondoids, using density functional theory computations. Along these lines, the synthesis of artificial diamondoids has greatly advanced in recent years,¹³ as well as their spectroscopic investigation.^14–18 The modified diamondoids studied in these works can roughly be classified according to three main categories:

(i) “Electronically blended diamondoids”, i.e., diamondoid subunits connected to each other by sp²-hybridized C atoms (or other unsaturated units). The subunits may be comprised of the same (DiaDia, two diamantanes; AdaAda, two adamantanes) or different (AdaDia, AdaDiaAda) diamondoid molecules. Several of these (and related) molecules have been synthesized by Schreiner and co-workers.^11,13 For example, DiaDia which exists as E and Z stereoisomers has been the subject of recent physico-chemical¹¹ and theoretical¹⁸ characterization. In DiaDia, a greatly reduced HOMO–LUMO gap is found compared to pristine diamondoids, as well as a strongly enhanced CC vibration in resonance Raman. Among other experimental studies, the valence photoelectron spectra of selected diamondoids, joined by single or double CC bonds have been measured in the recent past.¹⁹

To extend this work and identify trends in CC-blended diamondoids, other and also more complicated species such as AdaDiaAda (with two CC double bonds) will be considered here.

(ii) “Functionalized diamondoids”,^15–17i.e., species where one or multiple H atom(s) were substituted by particular functional groups like hydroxyl, cyano, amino, or thiol (–SH) groups. Sulfur-containing diamondoids are particularly interesting because of the ease of their attachment to metal surfaces; their synthesis dates back nearly to a decade.^20,21 It is possible to attach multiple functional groups, resulting in, e.g., dithiol, trithiol, and so on. Structural isomers also exist, depending on the position of the functional group (e.g., adamantane-1-thiol and adamantane-2-thiol) or the relative positions of two or more functional groups (e.g., adamantane-1,2-dithiol and adamantane-1,3-dithiol). Recently, Landt and co-workers showed that the incorporation of a thiol group in adamantane lowers its optical gap.¹⁵ This opens up the speculation on tunable optical properties of diamondoids by thiolization.

To identify trends in mono- vs. di-substitution as well as effects of the position of the functional group(s), on the optical gaps and the vibronic absorption, emission and resonance Raman spectra, various thiol- and dithiol-adamantanes will be studied in this work.

Also, the substitution (of two H atoms) by S groups, leading to thiones, e.g. adamantane thione, C₁₀H₁₄S, have been discussed in the literature as possible routes towards “tuned” materials. In a recent theoretical work,²² Vörös and co-workers predicted by PBE0/cc-pVTZ calculations for adamantane and [1(2,3)4]-pentamantane, that the optical gap is reduced gradually by the systematic substitution of thione groups. For adamantane-1,2,5,6-tetrathione and adamantane hexathione, for example, the optical gap was estimated to be in the visible range, between 2–3 eV.

No vibronic effects and real spectra were considered in that work, however, which we will do here for selected thionized diamondoids.

As a sideline, also several adamantanes functionalized with alcohol (–OH) and bromo (–Br) groups will be studied.

(iii) “Doping” or “C-substituting” diamondoids”, i.e., replacing one or more methine (CH) or methylene (CH₂) groups of pristine diamondoids by iso-electronic groups such as N and O, is the final route of tuning properties of diamondoids to be studied here. This can lead to the formation of molecules with entirely new electronic and optical properties. For instance, substitution of the four –CH groups of adamantane by the iso-electronic N atom results in urotropine or hexamethylene tetramine, (CH₂)₆N₄. In ref. 14, experiment has shown that the absorption and fluorescence of urotropine are very distinct from those of adamantane, with a “smooth”, redshifted absorption band with no pronounced vibrational finestructure.

In the present work, we shall study some of the photophysical properties of urotropine by electronic structure methods, as well as of several other N- or O-substituted adamantanes.

Our work focuses on comparing vibrationally resolved absorption, emission and resonance Raman (rR) spectra of representative diamondoids from each of the three categories mentioned, using a two-state model with the ground state (g, or S₀) and one bright, excited state (e, usually S₁). In certain cases also a wider range of excited states was considered, then, however, mostly only for vertical electronic excitation energies. The ground and excited states are calculated by hybrid density functional theory (DFT) and linear-response time-dependent DFT (TD-DFT), respectively. Pristine diamondoids will be used as a reference. We shall also compare to experiment where possible. Our ultimate goal is to understand the photophysics of modified diamondoids on a fundamental level, and – based on this understanding – to help developing criteria for the “tuning” of optoelectronic properties of these versatile materials.

In order to arrive at vibrationally resolved spectra, we use a time-dependent correlation function approach as pioneered in chemical physics by Heller and co-workers.^23,24 The time-dependent approach can offer computational advantages by avoiding the computation of Franck–Condon factors. In particular in the harmonic approximation which we use here, quasi-analytic expressions are available for auto- and cross-correlation functions. By combining (TD-)DFT with the correlation function approach in harmonic approximation, vibronically resolved spectra become available for medium-sized molecules with moderate computational effort and acceptable accuracy. The largest molecule treated here is AdaDiaAda, C₃₄H₄₄, with 228 normal modes. Other diamondoids have been studied elsewhere with the same methodology¹⁸ and bigger molecules of different type, e.g., β-carotene (C₄₀H₅₆), in ref. 25. In that reference, also the inclusion of Duschinsky rotation for resonance Raman spectra when calculated with time-dependent correlation functions has been put forward. (The further extension of the time-dependent approach to Herzberg–Teller corrections for resonance Raman was realized in ref. 26 and 27.) Duschinksy rotation, i.e., the rotation of normal modes in the electronically excited state relative to ground state modes, is an effect which will be considered here in many but not all examples.

The paper is organized as follows. The methods, models and approximations used for the computation of the spectra are described in Section II. Results are presented and discussed in Section III, for electronically blended diamondoids (Section III A), thiol- and thione-substituted diamondoids (Section III B) and urotropine (Section III C). Section IV summarizes the work and provides an outlook for possible future investigations in this field.

II. Methods

The vibrationally resolved absorption, emission and resonance Raman spectra have been calculated using the time-dependent correlation function approach, as popularized by Heller and co-workers.^23,24 Here we work in harmonic and Condon approximations, using two-state models. Normal-mode coordinates are used throughout (rather than curvilinear coordinates, which can lead to somewhat different results²⁸), the temperature is 0 K and possible effects of an environment are neglected.

In a first, more sophisticated model, called here the IMDHOFAD (independent mode displaced harmonic oscillator with frequency alteration and Duschinsky rotation) method, full geometry optimizations are carried out for the ground (g) and the selected bright, electronically excited state (e) and normal mode analyses are performed for both. Frequency alterations in the excited state and the Duschinsky rotation are taken fully into account. The method is described in more detail in ref. 18 and 25 and therefore only briefly reiterated below. In situations where optimization of excited states is not so trivial, unphysical, or simply unwanted (in order to save computational effort and allow for screening many molecules), the IMDHO (STA) approach will be used instead. In this approach,²⁹ frequency alteration and Duschinsky rotation are not accounted for, and in addition the “short-time approximation” (STA) is used. Excited-state displacements are obtained from a local extrapolation scheme.

The first approach of above which is based on two fully optimized harmonic potentials, belongs to a broader class of models called “adiabatic”. The latter approach, on the other hand, belongs to so-called “vertical” methods for which only information (on gradients and/or Hessians) at a Franck–Condon point for vertical transitions between two potentials enters. The vertical approach is not only more economic as it avoids excited-state optimizations, it can also be of advantage if the excited state optimization is difficult, for practical or principal reasons, and then it is sometimes physically also more sensible. This is particularly so when large-amplitude motions and/or geometry displacements in excited states take place.²⁸

For the adiabatic, IMDHOFAD model optimizations and normal mode analyses have been performed using density functional theory (DFT) and linear-response, time-dependent DFT (TD-DFT). Specifically, the B3LYP hybrid functional^30,31 together with a triple zeta valence polarized (TZVP,³²) basis set has been used throughout, if not explicitly stated otherwise. The validity of this method for diamondoids (good ratio of accuracy and computational effort), has been proven in ref. 18. The GAUSSIAN09³³ quantum chemistry package has been used for optimizations and normal mode analyses. A FORTRAN code developed earlier²⁵ was used to calculate the Duschinsky matrix and the dimensionless origin shifts between the normal modes of the two electronic states. The code is then used to calculate the auto-correlation functions and cross-correlation functions using the time-dependent approach. Spectra are obtained from Fourier transformed correlation functions by using the FFTW (Fastest Fourier Transform in the West) package.³⁴ The IMDHO (STA) calculations, on the other hand, have been carried out directly with the quantum chemical package ORCA,³⁵ on the (TD-)B3LYP/TZVP level of theory as well.

In the time-domain, the absorption cross-section is expressed as a Fourier transform of an autocorrelation function (we use atomic units in what follows)^18,23

	(1)

where ω_L is the frequency of the excitation radiation, c is the velocity of light in vacuum and E₀ is the adiabatic minima separation energy between the ground and excited states. E^g₀ is the zero point energy of the initial vibrational state of the ground electronic state, and the transition dipole moment between g and e states, which is taken constant and equal to the value at the equilibrium geometry (0_g) of the ground state, in the Condon approximation. Further, |ϕ^g₀(0)〉 is a product vibrational, initial wavefunction comprising 3N − 6 vibrational ground state normal modes of the electronic ground state (N is the number of atoms). Γ is the width (of the Lorentzian) used for damping the autocorrelation function, 〈ϕ^g₀(0)|e^−iĤ_etϕ^g₀(0)〉. Ĥ_e is the field-free, excited state nuclear Hamiltonian. In the harmonic approximation (even in the IMDHOFAD model), the expressions for multi-dimensional autocorrelation functions can be obtained analytically.^23,25 Their computation requires optimized geometries of ground and excited states, ground and excited state normal frequencies ω^g₁,ω^g₂,…,ω^g_3N−6;ω^e₁,ω^e₂,…,ω^e_3N−6 and corresponding normal modes and, derived from these quantities, the Duschinsky matrix and the so-called dimensionless origin shifts for modes.

The emission cross-section can also be expressed as the Fourier transform of a time-dependent correlation function

	(2)

where ω_E is the frequency of the emitted radiation, E^e₀ the zero point energy of the excited electronic state, |ϕ^e₀(0)〉 the corresponding product vibrational state, from which the emission takes place. |e^−iĤ_gtϕ^e₀(0)〉 is the time-dependent wave-packet obtained by propagation of |ϕ^e₀〉 under the influence of the field-free, ground state nuclear Hamiltonian Ĥ_g. Further, is the transition dipole moment, evaluated now at the optimized geometry of the excited state (0_e). Like absorption, expressions for multi-dimensional autocorrelation functions can be obtained quasi-analytically.^23,25

The resonance Raman (rR) cross-section is calculated using

	(3)

where ω_L is the frequency of the exciting light, ω_S the frequency of the scattered radiation and is the Raman polarizability for rR scattering from vibrational level i to f, in the ground electronic state. For 0 → 1 Raman scattering as considered in this work, i = 0, f = 1. q and q′ refer to the x, y or z directions, along which the components of the transition dipole moment are considered. In the time-dependent regime, the Raman polarizability can be calculated as a Fourier transform of a cross-correlation function^18,24

	(4)

The quantity 〈ϕ^g_f(0)|e^−iĤ_etϕ^g_i(0)〉 is the cross-correlation function. Similar to the autocorrelation functions, analytical expressions also exist for the evaluation of cross-correlation functions in multi-dimensions.^24,25 Note that a different broadening factor is usually adopted for rR spectra. Here we compute rR spectra by fixing the excitation wavelengths ω_L at certain values close to electronic excitation energies, as a function of the Raman shift. For 0 → 1 Raman scattering, we then obtain a stick spectrum as a function of the ground state vibrational frequencies, which is broadened here by Lorentzians of appropriate FWHM.

As outlined above, the time-dependent approach offers a variant, the short-time approximation (STA) which can be applied for absorption, emission and rR intensities.^24,29 As stated, the STA is not always an approximation but may, as a representative of the “vertical” models, in certain special situations be even more accurate than the full IMDHOFAD model. In the STA approach to absorption as implemented in ORCA,²⁹ excited-state energies are calculated for a particular set of geometries at and around the ground state geometry and quadratic fits to these energies are used to estimate a harmonic excited surface without explicitly optimizing its geometry and performing a normal mode analysis. Rather, this method neglects frequency alterations and mode-mixing in the excited state and will therefore be referred to as the IMDHO (STA) approach in what follows. Specifically, mode displacements are calculated using the relation, for absorption²⁹

	(5)

Here, is the gradient of the excited state potential energy V_e along the mth mass weighted normal coordinate of a ground state mode, at the equilibrium geometry of the ground state (Franck–Condon point) and ω^e_m is the excited state frequency of the mth normal mode. In the IMDHO model, excited state frequencies are assumed to be the same as the ground state frequencies ω^g_m. The dimensionless shift of the mth normal mode, Δ_m is then obtained from the relation

	(6)

The values obtained are used to calculate the absorption spectra, and similar procedures exist for emission and resonance Raman spectra.²⁹

III. Results

A. Electronically blended diamondoids

Let us first study molecules containing one or two CC double bonds which connect simple diamondoids, such as adamantane (Ada) or diamantane (Dia). Specifically, we consider AdaAda, AdaDia, DiaDia and AdaDiaAda, which are experimentally known.¹¹ DiaDia exists as two geometric isomers, E and Z, depending on the geometry around the CC bond connecting the two diamantane units and has been studied with analogous theoretical models in ref. 18. Most optical and electronic properties of E and Z are very similar and so are their energies. For AdaAda and AdaDia no analogous isomers exist. Fig. 1 shows the optimized geometries in their ground electronic states for the two other diamondoids containing a single CC bond as studied here, namely AdaAda and AdaDia.

Considering the lowest-energy, bright (S₀ → S₁) absorption transitions of DiaDia, AdaAda and AdaDia, we show in Table 1 the HOMO–LUMO gaps ΔE_HL (calculated for Kohn–Sham orbital energies) and vertical excitation energies ΔE_vert (calculated from TD-DFT) for these compounds. We also compare to the lowest-energy, bright transitions for adamantane.¹⁸

Molecule	ΔEHL	ΔE0–0	ΔEvert	ΔEvibro
Adamantane (Ada)	8.15	6.54	7.32	6.84
AdaAda	6.34	5.20	5.60	5.48
AdaDia	6.28	5.16	5.55	5.44
DiaDia(E)	6.23	5.14	5.52	5.41
DiaDia(Z)	6.22	5.11	5.50	5.38

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Considering Ada as a reference first, experimentally it is known that this parent compound has an optical gap (the onset of absorption) of 6.49 eV,¹⁰ which is close to the experimental 0–0 transition in this case (see also Table 6 below). (The optical gap of Dia is found to be 6.40 eV according to that ref. 10.) From Table 1 we see that the theoretical ΔE_0–0 value is close to the experimental one. The vertical excitation energy is 7.32 eV for adamantane and unknown experimentally. From the table it is seen that the energy gaps decrease considerably when Ada/Dia subunits are connected by CC bonds. Taking ΔE_0–0 values as a measure for optical gap reduction, the latter is in the order of 1.3 eV and close to up to about 2 eV if ΔE_vert are considered (relative to Ada). Both the ΔE_0–0 and ΔE_vert values of AdaAda, DiaDia and AdaDia are all very similar (within ∼0.1 eV). The lowest allowed electronic transition to S₁ for the electronically blended diamondoids involves a dominant HOMO → LUMO transition, which is relatively weak for AdaAda, AdaDia and also the previously studied DiaDia(E). For DiaDia(E) the vertical transition energy to S₁ at ω₁ = 5.52 eV has an oscillator strength of f₁ = 0.00039.¹⁸ The HOMO is the CC π-orbital, and the LUMO is a diffuse orbital delocalized over the periphery of the molecule. A similar behavior is found for the species studied here, AdaAda and AdaDia. It should be noted that, due to relatively low oscillator strengths, the S₁ state does not dominate the spectrum, however. Rather, the most intense transition is due to a CC π → π* excitation at higher energies. For DiaDia(E), for example, π* is the LUMO+2, which, like π, is strongly localized around the CC bond. The corresponding excited state is S₄ in case of DiaDia(E), at a vertical transition energy of ω₄ = 6.34 eV and an oscillator strength of f₄ = 0.8327.¹⁸ (States S₂ and S₃ are dark states in case of DiaDia(E).)

Table 1 also demonstrates that ΔE_HL values should not be considered as reliable estimates of optical gaps, being about 0.8 eV larger than ΔE_vert. Another measure for optical excitations is the vibronic energy difference ΔE_vibro, i.e., the maximum of the vibronically resolved absorption spectra. ΔE_vibro is also listed in the table, showing similar (but slightly lower) values than ΔE_vert. For the unsaturated species, ΔE_vert is reduced by about 1.8 eV compared to a saturated diamondoid such as Ada. The strong reduction of the optical gap in CC-connected dimers by the here observed order of magnitude is very consistent with experimental findings.¹¹

To illustrate effects of the vibronic finestructure on spectra in detail, we show in Fig. 2 absorption, emission and resonance Raman spectra for AdaAda and AdaDia, obtained by using (TD-)B3LYP/TZVP/IMDHOFAD and the S₀ and S₁ states. Let us consider the absorption and emission spectra (upper panels) first. For both molecules it is seen that indeed, the 0–0 transitions are not clearly visible either in absorption or in emission. This is because some modes exhibit very high values of adimensional shifts, up to 4 (compared to the usual values which lie around 1). All absorption and emission spectra show a pronounced vibrational finestructure. For both molecules, the maximal absorption peak is close to the vertical excitation energies, i.e., ΔE_vert ∼ ΔE_vibr as already demonstrated in Table 1 (both ΔE_0–0 and ΔE_vert are shown as dashed, vertical lines). As a fine detail, we note that the absorption and emission spectra of AdaDia (b) are slightly more structured than those of AdaAda (a), but are similar otherwise. Also the corresponding spectra of DiaDia (see Fig. 10 of ref. 18) look similar.

Vibrational peak spacings in absorption spectra approximately correspond to excited state vibrational frequencies of normal modes which are dominantly excited upon electronic excitation. (Similarly, peak spacings in emission spectra reflect dominant modes in the electronic ground state, after de-excitation.) For AdaAda and AdaDia, it is the CC stretching mode which is the principal contributor to the vibrational finestructure in absorption. This is a similar feature as observed previously for DiaDia¹⁸ and can be attributed to the location of the HOMO on the CC double bond connecting the two diamondoid units to each other. As reported for DiaDia in ref. 18, the CC bond lengths of AdaAda and AdaDia also elongate by 0.06–0.07 Å in the S₁ excited state involving a HOMO → LUMO transition. At the same time, the vibrational frequency of the CC vibrations soften from about 1700 cm⁻¹, to about 1530 cm⁻¹.¹⁸Fig. 2(a) and (b), middle panel, show the absorption autocorrelation functions for AdaAda and AdaDia, respectively. The most striking features are recurrences after about 22 fs in both cases, which translate into a vibrational level spacing of about 1520 cm⁻¹ (or about 0.19 eV) – in good correspondence to the softened CC vibration, also known from experiment.¹¹ The slightly more pronounced recurrence features in AdaDia compared to AdaAda, explain the slightly more pronounced vibrational features in the spectra of the former.

To put our calculations in a wider context, we note that theoretically^10,18 and also experimentally¹⁰ for Ada, as a reference, the lowest absorption band is highly vibrationally structured. Experimentally, the intensity of the first (S₁) absorption band increases at least up to about 7 eV.¹⁰ Experimentally, also the emission spectrum is highly finestructured, peaking at around 5.9 eV.¹⁰ In ref. 18, we found a highly structured fluorescence spectrum for Ada as well, with a maximum at around 6 eV. This and the absorption behavior found in ref. 11 gives us some confidence that the present trends in vibronic spectra for unsaturated diamondoids are reliable.

Concerning the rR spectra in Fig. 2, lowest panels, reported at excitation energies of 5.29 eV (a) and 5.26 eV (b), respectively, we note that also for these the CC stretching mode (now for the ground state with a frequency around 1700 cm⁻¹) is the dominant scatterer. The excitation energies were chosen slightly above the corresponding ΔE_0–0 values (cf.Table 1) to establish a resonance effect. Some C–H and –CH₂ bending modes around 1300–1400 cm⁻¹ and some low frequency C–C–C torsional modes are also intense. Again, the general behavior is similar to DiaDia¹⁸ and in good agreement with experimental data for CC-connected diamondoids.¹¹

Another system which will consider is AdaDiaAda. This molecule is particularly interesting because of the presence of two CC double bonds and also because of methodological interest. An attempt to obtain an optimized equilibrium geometry (on the B3LYP/TZVP level) for the S₁ state of AdaDiaAda was not successful. However, on changing the basis set to a less accurate 6-31G* basis set, again using the B3LYP functional, an optimized S₁ state was obtained, whose equilibrium geometry was confirmed by only real frequencies in the subsequent normal mode analysis. Fig. 3 shows the optimized geometries of the S₀ and S₁ states at the B3LYP/6-31G* level of theory. It can be seen that during the S₀ → S₁ transition, a torsion of one of the rings around one of the CC bonds occurs. This CC bond is elongated to 1.45 Å in the S₁ state, compared to its value of 1.35 Å in the ground state, hence facilitating the rotation of one of the diamondoid units around itself. There were slight rotations of some of the other cyclohexane rings as well.

One of the main effects of these rotations is that, many of the normal modes have very high values of dimensionless origin shifts, resulting in unphysically low vibronic overlap and a broad unphysical absorption spectrum. This is a case where the IMDHO (STA) approach as a method based on a “vertical” model is expected to be more realistic. Consequently, this method was used to obtain vibronic spectra.

The absorption and rR spectra obtained on the B3LYP/6-31G* level, using the IMDHO (STA) model are shown in Fig. 4(a) and (b) respectively.

The general characteristics of the absorption and rR spectra are similar to those of the electronically blended diamondoids described earlier. The contributors of the vibrational spacings in the absorption spectrum and the principal Raman scatterer in the rR spectrum are mainly the nearly degenerate stretching modes of the two CC bonds, with ground state vibrational frequencies around 1700 cm⁻¹ according to our B3LYP/TZVP calculations.

Compared to AdaAda and AdaDia (Fig. 2), for AdaDiaAda the absorption range is blue-shifted again, by about 0.5–0.6 eV. In the calculation this comes about by the fact that the adiabatic minima separation energy between S₀ and S₁ in the IMDHO (STA) approach does not involve a full optimization of the S₁ state. The “Frank–Condon-extrapolated” S₁ state is 0.4 eV higher in energy than what one finds for comparable systems, for which a full excited state geometry optimization was carried out. This is demonstrated in Table 2, where we compare adiabatic minima separation energy values for DiaDia (B3LYP/TZVP), AdaDiaAda (B3LYP/6-31G*) and AdaDiaAda (B3LYP/6-31G*, STA). Note that for AdaDiaAda (B3LYP/6-31G*/IMDHOFAD) and DiaDia(E) (B3LYP/TZVP/IMDHOFAD) the fully optimized adiabatic energy differences are very similar, in contrast to AdaDiaAda (B3LYP/TZVP/STA), which gives the mentioned higher value. The question arises then if indeed, the blue-shifted absorption spectrum (and also the rR spectrum) shown in Fig. 4 are realistic, or numerical artefacts. An experiment could be of great value here.

Molecule	Method	E 0 (eV)
DiaDia(E)	B3LYP/TZVP/IMDHOFAD	5.27
AdaDiaAda	B3LYP/TZVP/IMDHOFAD	5.28
AdaDiaAda	B3LYP/TZVP/STA	5.70

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B. Functionalized diamondoids: thiols, thiones and related species

We now turn to the second way of modifying diamondoids, by introducing ligands. We shall mostly focus on sulfur-containing materials, thiols and thiones, but also touch diamondoids functionalized with alcohol (–OH) and bromine (–Br) groups. For simplicity, as a parent compound only adamantane will be considered.

C. C-substituted diamondoids: urotropine and related compounds

By replacing entire methylene and methine groups by hetero-atoms we can realize another class of diamondoids, called “C-substituted” in this paper.

IV. Summary and outlook

In this work, a time-dependent approach to vibronically resolved spectroscopy based on autocorrelation and cross-correlation functions was applied, along with time-dependent (linear-response) density functional theory to understand certain optical properties of electronically blended, functionalized and “doped” (or “C-substituted”) diamondoids. In particular, absorption, emission and resonance Raman spectra were computed. The Condon and harmonic approximations were used for this. The IMDHO (STA)²⁹ and IMDHOFAD models^18,25 have been used for the calculation of time-dependent correlation functions. In cases when large-amplitude motions and large displacements in excited states take place, the IMDHOFAD may be difficult to realize. Fortunately in these cases a “vertical” approach like IMDHO (STA) could also be more realistic. If one or the other approach is more appropriate, is generally still an open question.

The absorption and emission spectra of the modified diamondoids are (almost all) redshifted with respect to the pristine diamondoids. The magnitude and trends of the redshift depends on the nature of the frontier orbitals which are involved in the electronic transition. This, again, depends on the nature, number and position of functional groups or substituents attached to the diamondoids. With adequate number of thione groups, for example, the absorption can be shifted to the visible region. Geometrical distortions during the electronic excitation also affect the resolution of the spectra. Important information about the vibrational normal modes playing dominant roles in the electronic excitation were extracted from the resonance Raman spectra, particularly in situations where well resolved absorption spectra could not be obtained due to significant geometrical distortions. Where comparison to experiment and previous theoretical results was possible, the present theory gave reasonable agreement. We made, however, also a large number of predictions which await experimental proof. It must be said, though, that simple explanations for observed trends are not always available. In these cases simulations are needed and therefore sufficiently accurate and at the same time sufficiently economic theoretical methods, to allow for systematic investigations of a large class of candidate materials. We believe that the present methodology is a suitable starting point.

Of course, the present methods could and should be improved. The application of more detailed basis sets and/or more accurate electronic structure methods is a worthwhile direction. Effects of anharmonicity and finite temperature might also be studied. Further, the inclusion of non-radiative transitions such as intersystem crossing by spin–orbit coupling, or internal conversion due to non-Born–Oppenheimer couplings, will be interesting to include. In this context it should be noted that the time-dependent correlation function approach may offer advantages over traditional, time-independent (Golden Rule type) approaches, too.^41,42

Acknowledgements

We sincerely thank Dr Dominik Kröner, Tao Xiong (Potsdam) as well Thomas Möller and Janina Maultzsch (TU Berlin) for fruitful discussions. Financial support by the “DFG-Forschergruppe 1282” (project Sa 547/11-1) is gratefully acknowledged.

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