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Theoretical Analysis of the Anomalous Spectral Splitting of Tetracene in 4He Droplets Heather D. Whitley,†,‡ Jonathan L. DuBois,†,‡ and K. Birgitta Whaley*,† †
Department of Chemistry and Kenneth S. Pitzer Center for Theoretical Chemistry, University of California, Berkeley, California 94720, United States ‡ Lawrence Livermore National Laboratory, PO Box 808, Livermore, California 94550, United States ABSTRACT: We present a theoretical analysis of the electronic absorption spectra of tetracene in 4He droplets based on many-body quantum simulations. Using the path integral ground state approach, we calculate one- and two-body reduced density matrices of the most strongly localized He atoms near the molecule surface and use these to investigate the helium ground-state quantum coherence and correlations when tetracene is in its electronic ground and excited states. We identify a trio of quasi-one-dimensional, strongly localized atoms adsorbed along the long axis of the molecule that show some quantum coherence among themselves but far less with the remaining solvating helium. We evaluate the single-particle natural orbitals of the localized He atoms by diagonalization of the one-body density matrix and use these to construct single- and many-particle solvating helium basis states with which the zero-phonon spectral features of the tetracene4HeN absorption spectrum are then calculated. The absorption spectrum resulting from the three-body density matrix for the strongly bound trio of helium atoms is in very good agreement with the experimental data, accounting quantitatively for the anomalous splitting of the zero-phonon line [Hartmann, M.; Lindinger, A.; Toennies, J. P.; Vilesov, A. F. Chem. Phys. 1998, 239, 139; Krasnokutski, S.; Rouille, G.; Huisken, F. Chem. Phys. Lett. 2005, 406, 386]. Our results indicate that the combination of strong localization and the quasi-one-dimensional nature of trios of helium atoms adsorbed along the long axis of tetracene leads to a quantum coherent, yet highly correlated ground state for the helium density closest to the molecule. The spectroscopic analysis shows that this feature accounts quantitatively for the anomalous splittings and hitherto unexplained fine structure observed in the absorption spectra of tetracene and suggests that it may be responsible for the corresponding zero-phonon splittings in other quasione-dimensional planar aromatic molecules.
’ INTRODUCTION The tetracene molecule is one of the experimentally most wellstudied molecules in helium droplets.110 One motivating factor for this proliferation of experimental studies is the observation of an anomalous splitting of the zero-phonon line (ZPL) in its absorption spectrum in helium. This was first measured by laserinduced fluorescence (LIF) in 1998 by Hartmann et al., who observed that the purely vibronic or ZPL is split by ∼1 cm1, with a red shift of 103.1 cm1 relative to the bare molecule.1 This splitting is not observed in electronic spectra of tetracene embedded in or attached to clusters of other rare gas atoms. It has therefore been ascribed to be a unique feature of the He quantum solvation environment. The two components of the split ZPL were termed R and β lines, with the lower-energy absorption to the red being termed R and the higher-energy absorption termed β.8 Because, in this case, the β line showed additional fine structure while the R line was structureless, subsequent studies with tetracene complexes have assigned electronic spectral transitions to be R-type or β-type, according to whether they exhibit fine structure or are structureless. The fine structure of the β band has been tentatively assigned to rotational excitations of the molecule that are coupled to the r 2011 American Chemical Society
excitations of the first shell of helium.8 However, no adequate explanation has been advanced for the splitting of the ZPL, which is much larger than what would be expected for resolved rotational structure. This anomalous splitting has therefore been attributed to the unique nature of helium and various speculations offered as to its microscopic origin. These include the coexistence of different configurations of helium near the tetracene molecule or “tunneling” types of excitations of one or more atoms adsorbed in the double-well-like part of the Hetetracene potential energy surface that is located above the central aromatic rings of the molecule.6 Subsequent studies of ArNtetracene complexes in He droplets showed that the addition of a single Ar atom to the droplet does not suppress the ZPL splitting. Instead, the spectrum now consists of an Rβ pair split by a larger value, ∼3.4 cm1, and shifted by ∼114 cm1 from the bare molecule ZPL, together Special Issue: J. Peter Toennies Festschrift Received: January 11, 2011 Revised: May 16, 2011 Published: May 16, 2011 7220
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The Journal of Physical Chemistry A with an additional β-type (i.e., possessing fine structure) peak shifted by ∼140 cm1 from the bare molecule line.5 The isolated β-type absorption with the larger red shift was assigned to a Artetracene complex where the Ar atom is localized at the surface of the molecule, over the aromatic rings, disrupting the typical equilibrium configuration of the adsorbed helium, whereas the Rβ pair was assigned to an Artetracene complex in which the Ar atom is localized at a potential minimum in a plane parallel to the molecular plane that is not directly above the aromatic rings and consequently has less influence on the solvating helium configuration.5 More recently, additional studies have been carried out for Netetracene, H2Otetracene, and H2tetracene clusters in helium droplets, using various isotopes and spin states of hydrogen.8,10 The spectra of clusters with p-H2, o-H2, and HD exhibit the same features as those seen in the Artetracene experiments described above, that is, a single β-type line with a large spectral red shift relative to the bare molecule and additional Rβ pairs with smaller spectral red shifts.8 The splitting of the Rβ pairs, as well as their relative intensities, are found to be dependent on both the nuclear spin/rotational state and the mass of the hydrogen moiety, whereas the spectral shift of the single β-type line is relatively independent of the nuclear spin/rotational state and mass. However, clusters with D2, both para and ortho, show only the Rβ pairs and not the additional single β-type lines.8 In ref 8, the isolated β-type features were assigned to clusters in which the hydrogen molecule is localized above the tetracene molecule in the most tightly bound sites, disturbing the helium density to such an extent that the splitting is quenched (whatever its source), while the Rβ pairs were tentatively assigned to different configurations of helium near the molecule.8 The Netetracene and H2Otetracene clusters were found to exhibit similarly complex spectra.10 Comparison with the absorption spectra of other planar aromatic molecules is useful and revealing. Another such molecule that has been extensively studied in helium droplets is phthalocyanine.7,1118 For free-base phthalocyanine, a ZPL splitting is observed only in emission, in contrast to tetracene where the splitting of ZPL spectral lines is seen in both emission and absorption. In the case of phthalocyanine, a model for emission spectral splitting based on the existence of two different metastable helium configurations was proposed11 and is supported by a path integral Monte Carlo (PIMC) study of Hephthalocyanine clusters.15 One key point in this model is that interconversion between the two helium configurations is possible upon electronic excitation. For phthalocyanine, this was confirmed by pumpprobe experiments.14 This model is consistent with a relatively long (∼nanoseconds) lifetime of the aromatic molecule in the excited state. In a pumpprobe investigation of the ZPL splitting of tetracene in helium droplets, it was found that the two states responsible for the ZPL splitting have even longer lifetimes and do not interconvert at all for pulse delay times less than ∼100 ns.3,6,9 This lack of interconversion could indicate that there exists a very large energetic barrier between the two presumed configurations of helium in both ground (S0) and electronic excited (S1) states, so that interconversion does not occur on this time scale. However, it is also possible that the splitting of the ZPL of tetracene in helium droplets is not due to the existence of metastable helium configurations and that it has a different origin from the ZPL splitting for phthalocyanine in helium. This argument is supported by our recent PIMC study of structure, energetics, and
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spectral shifts of HeNtetracene clusters, in which, unlike our earlier study of Hephthalocyanine,15 only a single configuration of helium was found for both electronic states of the tetracene molecule in clusters containing up to N = 150 He atoms.19 As shown there, the first solvation shell is complete between N ≈ 20 and 50. Consequently, any metastable lowenergy configurations that could modify the absorption spectra would likely have appeared at smaller sizes. However, no metastable configurations were seen, implying that thermal population of metastable helium configurations does not produce the splitting of the ZPL in the electronic absorption spectrum of tetracene. The zero-phonon spectral line profile of tetracene was revisited in a recent study of the laser-induced fluorescence spectra of linear polyaromatics.2 In that work, a similar splitting of the ZPL was reported for anthracene, and the ZPL absorption profile of both molecules, tetracene and anthracene, were fit to the sum of four Lorentzian lines with constant width. This allowed fitting of additional smaller spectral features on the high-energy side of the split ZPL absorption peaks for tetracene, some of which were visible but not analyzed in the earlier experiments of ref 1. A fit to Lorentzians with width 0.38 cm1 gives relative peak positions of 1.06, 0.00, 1.15, and 2.07 cm1 and relative intensities of 0.23:1.04:0.10:0.05, where we are centering the spectrum at the most intense (second) peak, which corresponds to the β line of the ZPL. The first peak corresponds to the R line of the ZPL. The third and fourth peaks lie at somewhat higher energy but still within the absorption profile of the ZPL. The presence of the additional lines, which we shall refer to as γ and δ, provides valuable additional information for rationalization of the origin of this spectral splitting for tetracene and other polyaromatic hydrocarbons. Krasnokutski et al. suggested in ref 2 that the R and β peaks might be due to different helium configurations around the molecule, while the γ and δ peaks might be due to vibrational excitations of the first solvation shell.2 In the present study, we make an analysis of the electronic absorption spectrum of tetracene in helium clusters based on path integral ground-state Monte Carlo calculations of the properties of the full many-body ground state of the solvating helium in the S0 and S1 states. On the basis of the structure of the ground-state reduced density matrix, we establish first that the one-dimensional trios of helium atoms adsorbed along the long axis of tetracene that we have previously characterized in ref 19 occupy a quantum coherent but highly correlated ground state. We then use the eigenstates of the single-particle reduced density matrix, also known as the natural orbitals,20 to construct singleand many-particle solvating helium basis states with which the zero-phonon spectral features of the tetracene4HeN absorption spectrum are then calculated within a multimode mean field description. The theoretical spectrum obtained from a correlated three-particle basis shows remarkable agreement with the experimental data, accounting quantitatively for not only the anomalous splitting of the ZPL into R and β peaks but also for the additional peaks γ and δ that were assigned in the recent experiments of ref 2. This leads to the novel interpretation that the anomalous ZPL structures are the result of absorption from a tetracene molecule solvated by a strongly bound and strongly correlated coherent quantum adsorbate that gives rise to effective FranckCondon factors associated with transitions between different modes of the adsorbate. As a comparison with this ground-state analysis, we have also investigated whether thermal excitation of anisotropic vibrational excitations of the most 7221
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strongly adsorbed helium atoms in the clusters might be responsible for the anomalous splitting, using the path integral imaginary time correlation function (PICF) approach to calculate the helium vibrational excitation energies. The resulting modes are however too high in energy to be populated at the temperature of the experiments. We therefore conclude that the ZPL splitting for tetracene may well be a strictly ground-state phenomenon that reflects the unusual nature of the one-dimensional subsystem of the most strongly adsorbed He atoms on each side of the molecule. We suggest that this peculiar origin of the ZPL splitting is not unique to tetracene and may also play a role in the zerophonon spectral absorption profiles of other effective onedimensional planar aromatic molecules, such as anthracene,2,21 pentacene,7 naphthalene,22 and certain indoles.23
’ GROUND-STATE QUANTUM COHERENCE AND CORRELATIONS OF STRONGLY ADSORBED HELIUM ON TETRACENE Studies of the coherence and correlations in systems of quantum bosons have often focused on the computation of reduced density matrices (RDMs).24 In this study, we examine the single-particle RDM and the pair correlation functions for He atoms localized near the surface of the tetracene molecule. The pth-order RDM for the ground state, Ψ, of a system of N spinless bosons is evaluated by integrating out the degrees of freedom for N p particles from the probability density function |Ψ|2: 0
0
Fp ðr1 , ... , rp , r1 , ..., rp Þ Z N! 0 0 ¼ Ψðr1 , ... , rp , rpþ1 , ..., rN ÞΨðr1 , ... , rp , rpþ1 , ..., rN Þ drpþ1 ... drN ! ðN pÞ
ð1Þ The pth-order RDM is thus a measure of the probability of annhilating p particles at positions {r1, ..., rp} and creating those p particles at positions {r01, ..., r0p} given any possible configuration of the remaining N p particles. The pth-order RDM thus defined is a measure of the spatial coherence of p particles within the ground state of the N body system. The pth-order RDM can be diagonalized to yield a system of p-body wave functions, which are eigenfunctions of the pth-order density operator. These wave functions are called the natural functions. The eigenvalues of the diagonal RDM correspond to the extent to which those p-body states are occupied in the many-body system. The full manybody Hamiltonian eigenfunction can also be expressed as a superposition of these p-body states. The one-body density matrix (OBDM) is a primary quantity of interest in the study of BoseEinstein condensates (BECs).25,26 The diagonal OBDM gives a measure of how many effective single-particle states are occupied in a system of N bosons, while the off-diagonal OBDM describes the spatial coherences between these states.20 The single-particle states that diagonalize the OBDM are called the natural orbitals of the system. A perfect or “simple” BEC is a system of N bosons where all particles are in the same single-particle ground state.26 This situation yields a diagonal OBDM with a single, nonzero diagonal term n0, and the system is said to be fully condensed.20,25 When the dominant diagonal term n0 is less than the total number of bosons N, the system is said to be depleted, while if more than one diagonal term of the OBDM is nonzero and of order N, the system is said to be fragmented.27,28 As discussed extensively by
Leggett in ref 26, these definitions are valid for bosonic systems with any number of bosons N > 1. Thus, an ideal system of N noninteracting bosons constitutes a simple BEC, in which the ground state is always fully condensed and the single occupied natural orbital is the ground state of the N-body Hamiltonian, that is, n0 = N. A system of N weakly interacting bosons can form a fragmented BEC, in which several natural orbitals have similar occupancy, and hence, several ni are of order N. Detailed studies have shown that in the presence of an additional external trapping potential, both the interactions between particles and the geometry of the confining external potential determine the number of occupied natural orbitals24 and hence the extent of fragmentation of the system. For weakly interacting Bose systems such as cold atomic gases, fragmentation and depletion occur together. However, application of this terminology to bulk liquid helium is complicated by the higher density and consequent greater role of interactions than that in cold atomic gases. Bulk liquid helium has a significantly lower condensate fraction (n0 ≈ 0.1) and thus exhibits significant depletion, but the noncondensate density is spread over a very large number of single-particle states and is not concentrated in a few states. Thus, bulk helium is a depleted but not a fragmented Bose system. We will see below that the most strongly adsorbed component of helium solvating a tetracene molecule in a helium cluster shows some characteristics of each of these well-known bosonic reference situations, although the system does not directly map onto either of these. We perform ground-state simulations of 4HeNtetracene clusters in the S0 and S1 molecular states using the path integral ground-state method described together with the electronic state-specific interaction potentials in ref 19. In our groundand finite-temperature path integral calculations for this system described in that work, we found that the helium density is highly structured, consisting of several layers of “solid” helium near the molecular surface due to the strongly localizing Hemolecule interaction potential and the strong correlations between the helium atoms. This high level of local structure in the helium density is consistent with the solvating behavior of helium around other planar aromatic molecules.15,19,29,30 Our calculations for HeNtetracene clusters with 24 e N e 150 show that the most strongly bound atoms are also the most strongly localized and lie in three positions along the long (x) axis of the molecule, on either side of the molecular plane. We employ a coordinate convention of x along the long axis in the plane, y along the short (transverse) axis in the plane, and z along the perpendicular axis. We find that while each of the three atoms in these trios of highly localized atoms exchange weakly with each other, they do not appreciably exchange with other atoms in the cluster, implying a high degree of independence from the remaining less weakly bound helium density around the molecule. This enables us to examine the effect of these six most strongly bound helium atoms on the tetracene electronic absorption spectrum separate from that of the remaining helium. Before presenting our spectral analysis, we first analyze the quantum coherence and correlations exhibited by these trios of adsorbed helium atoms. The three-atom subsystems are thus studied with path integral ground-state calculations for He6 tetracene, in which each side of the molecular plane has three strongly localized atoms associated with it. Zero- and finitetemperature path integral calculations were also made for a larger cluster, with N = 24, in order to study the extended structure of the HeHe pair correlations with respect to the x, y, and z dimensions and to examine the influence of temperature on these correlations. 7222
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Figure 1. Single-particle reduced density matrix (OBDM) with respect to x for He6tetracene at T = 0 K in the S0 state. The coordinate x lies along the long axis of the molecule (see text).
Helium One-Body Density Matrix. In our path integral ground-state calculations, the N-body helium Hamiltonian is given by " # N N p2 ^2 ðnÞ ^ ¼ V^ HeHe þ V^Hetet r þ ð2Þ H 2mHe i i¼1 j 1 all show finite amplitude in all three regions, with or without nodes. Helium Pair Correlation Functions. In order to gain further insight into helium correlations among the quasi-one-dimensional trio of atoms along the long molecular axis, we studied the pair correlation function g(x1,x2) for N = 6 at T = 0 and for N = 24 at both T = 0 and 0.625 K. At T = 0, pair correlation functions were calculated from the diagonal part of the two-body reduced density matrix, for example Z N! jΨðr1 , ... , rN Þj2 dr3 ... drN dy1 dy2 dz1 dz2 gðx1 , x2 Þ ¼ ðN 2Þ! ð4Þ At finite temperature, the pair correlation function is evaluated from the full N body finite temperature density matrix F(R,T) Z Z Z V 2 ðN 1Þ dN 2 x dN y dN zFðR, TÞ gðx1 , x2 Þ ¼ NZ ð5Þ with V the volume and Z the partition function. Both of these are diagonal operators that can be evaluated by standard sampling of the respective density matrices. In the N = 6 calculation, we performed the integration over z > 0 and z < 0 regions separately to allow analyis of correlations specifically among members of a single trio of three helium atoms on the same side of the molecular plane. The ground-state pair correlation function with respect to x for three atoms on one side of tetracene, g(x1,x2), is shown in Figure 3. The pair correlation hole along the diagonal, x1 = x2, results from the strong HeHe repulsions. By contrast, in the corresponding pair correlation functions g(y1,y2) and g(z1,z2) (not shown), all three helium atoms are seen to be localized in a single sharply peaked region, y1 = y2 and z1 = z2, respectively. Because of this strong localization with respect to y and z, which reflects the quasi-one-dimensional and surface-adsorbed nature of the trio of helium atoms, the heliumhelium interactions
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Figure 3. Two-body correlation function g(x1,x2) for He6tetracene at T = 0 K for the S0 electronic state of tetracene. The two-body correlation function for the S1 state is very similar. Only pairs of atoms that are located on the same side of the tetracene molecule are included in this average.
necessarily lead to a pronounced pair correlation hole with respect to the remaining coordinate x. Furthermore, elongation of the g(x1,x2) peaks localized at xi = 0 and xj = (3 that correspond to adjacent pairs of helium atoms indicates that the displacement of these atoms along x is strongly correlated. The remaining peaks localized around |xi| = |xj| ≈ 3, which correspond to correlations between the two outer atoms on the tetracene surface, show only a slight elongation, implying a weaker correlation between these more distant atoms. The pair correlation functions for the larger cluster with respect to motion in x, y, and z for N = 24 at T = 0.625 K for both the S0 and S1 states of tetracene are shown in Figures 46, respectively. The corresponding pair corrrelations at T = 0 K are virtually identical and thus are not shown here. Comparison of the right and left panels in each of these figures shows that when tetracene is in the S1 state, the pair correlations are qualitatively similar but more strongly peaked, indicating a higher degree of localization than that in S0. The following analysis of helium correlations for tetracene in S0 therefore applies also to tetracene in S1. In the y and z dimensions, we find solid-like correlations with the pair density well-localized and strongly peaked. g(z1,z2) is strongly localized above and below the tetracene molecule in the region of 2 e |z| e 5. The pair correlation function for the short (y) axis shows a more complex structure, reflecting the existence of layers of helium on both sides of the central trio of quasi-onedimensional helium atoms. The peak in Figure 5 at y1 = y2 = 0 corresponds to the trio of strongly localized atoms along the long (x) axis, and its intensity reflects the robustness of this structure even in the presence of more helium atoms. The set of four next most intense peaks in g(y1,y2) are elongated with respect to either y1 or y2 but not along the diagonal. This reflects the increasing delocalization of the He atoms that are located beyond the most strongly confining region of the Hetetracene interaction potential and that are not part of the core quasi-one-dimensional helium trio. In contrast to the strongly localized, solid-like pair correlations seen in the y and z coordinates, the x pair correlation function 7224
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Figure 4. Two-body correlation function g(z1,z2) for He24tetracene at T = 0.625 K for the S0 (left panel) and S1 (right panel) electronic states of tetracene.
Figure 5. Two-body correlation function g(y1,y2) for He24tetracene at T = 0.625 K for the S0 (left panel) and S1 (right panel) electronic states of tetracene.
Figure 6. Two-body correlation function g(x1,x2) for He24tetracene at T = 0.625 K for the S0 (left panel) and S1 (right panel) electronic states of tetracene.
g(x1,x2) shown in Figure 6 exhibits a more delocalized, fluid-like character (Figure 6). In order to extract the correlations of the trios of most strongly localized He atoms and compare with the N = 6 cluster, we evaluate a restricted pair correlation g0 (x1,x2),
which is constrained to pairs of atoms satisfying the conditions z1/|z1| = z2/|z2| and |y1 y2| e 2.4 Å. The pair correlation hole, the enhanced localization in the S1 electronic state of tetracene compared to that in the S0 state, and the six peaks corresponding 7225
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Figure 7. Two-body correlation function g0 (x1,x2) between pairs of atoms located on the same side of the tetracene molecule in He24tetracene at T = 0.625 K for the S0 (left panel) and S1 (right panel) electronic states of tetracene. The correlation function has been constrained to allow direct comparison with Figure 6 (see text).
to the quasi-one-dimensional trio of most strongly localized atoms are readily apparent in Figure 7. Analysis of these pair correlation functions show that as more atoms are added to the cluster, the three atoms lying along the long axis of the molecule experience a stronger degree of localization with respect to y and z, due to the interactions and correlations between He atoms enhancing the one-dimensional character of the trio of strongly localized atoms. In addition, the marked differences between the pair correlations with respect to the x, y, and z coordinates provide further validation for the separation of these degrees of freedom in our evaluation of the OBDM and natural orbitals.
|Φæ, where the helium states are eigenfunctions of the N-body Hamiltonian consisting of the kinetic energies of all N atoms and the Hetetracene and HeHe interactions. We will only consider transitions between tetracene electronic states S0 and S1 in the absorption spectrum, and therefore there are only two relevant molecular states, |ψ0æ and |ψ1æ. The dipole correlation function eq 8 for the Hetetracene system is then
’ CALCULATION OF ABSORPTION SPECTRA Within the electric dipole approximation, the absorption spectrum of the Hetetracene cluster is obtained from the Fourier transform of the transition dipole correlation function34 Z 2π dt eiωt ÆμðtÞμð0Þæ IðωÞ ¼ ð6Þ p
IðpωÞ ¼
where the correlation function of the electronic transition dipole operator μ is defined by ^ ðtÞ^ Tr½^ Fμ μ ð0Þ kÆΨk j pi jΨi æÆΨi jμðtÞμð0ÞjΨk æ
ÆμðtÞμð0Þæ ¼
∑
∑i
ð7Þ
Here, {|Ψkæ} are the eigenvectors of the full cluster Hamiltonian and F^ = ∑i pi|ΨiæÆΨi| is the density matrix. When F^ is defined in the basis of eigenstates of the Hamiltonian, the time dependence can be expressed as ÆμðtÞμð0Þæ ¼
¼
∑i pi eiE t ÆΨi jμeiHt^ μjΨi æ i
∑if pi eiðE E Þt jÆΨi jμjΨf æj2 f
i
ð8Þ
where i and f denote the initial and final states, respectively. Due to the weak interaction of helium with a molecule, which implies that the presence of helium does not significantly modify the electronic structure of the molecule, we assume that the state of the system can be written as a product of a particular vibronic state of the molecule, |ψæ, and the state of the N helium atoms,
ÆμðtÞμð0Þæ ¼
∑if pi eitðE
1f
E0i Þ
jÆΦ1f jΦ0i æj2 jÆψ1 jμjψ0 æj2 ð9Þ
and the absorption spectrum is given by 2π p
∑if pi δðω ðE1f E0i ÞÞjÆΦ1f jΦ0i æj2jÆψ1 jμjψ0 æj2 ð10Þ
where the superscripts on the N-body eigenfunctions of the helium atoms indicate that these are the helium states that are associated with the S0 and S1 states of the tetracene molecule, respectively. For convenience, we shall refer to the former as helium states in S0 and the latter as helium states in S1. The transition intensity in the Hetetracene cluster is therefore proportional to both the statistical weight of the transition, pi, which is determined by the population of the initial state, and the helium FranckCondon factor, |ÆΦ1f |Φ0i æ|2, while the transition energy must be equal to the difference in the total energy of the cluster in the final and initial states. The S0 f S1 spectrum therefore includes contributions 6 0, where pi ¼ 6 0. The from all transitions with |ÆΦ1f |Φ0i æ|2 ¼ transition energy, E1f E0i = ΔES0fS1 þ ΔEfi, is shifted from the bare molecule transition energy ΔES0fS1 by ΔEfi, the difference in energy between the final and initial many-body states of the helium component. ΔEfi therefore corresponds to the observed spectral shift of the molecular transition relative to the gas phase. In ref 19, we made perturbative estimates of the spectral shift ΔE00 for small clusters with N e 20 helium atoms from zero- and finitetemperature path integral calculations, obtaining good agreement with available experimental data.4 Before embarking on quantitative analysis of eq 9, we first discuss qualitatively how this relates to the various features exhibited in the spectrum of tetracene in helium droplets.13,6,7,9 In general, interaction potentials of excited S1 state molecules with helium do not differ significantly from the corresponding 7226
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The Journal of Physical Chemistry A potentials for the ground S0 states.15,19,3537 Indeed, our calculations in ref 19 show that the helium configuration around tetracene in the S1 state is very similar to that in the S0 state. It is therefore reasonable to assume that the equilibrium positions of the helium atoms in the two states are approximately equal. This implies that the FranckCondon factors should be highest for helium vibrational states i = f, so that the initial states with maximal statistical weight pi will yield the most intense spectral lines. For a ground or thermal initial state distribution, these will correspond to the zero-phonon features of the molecular absorption spectrum. Additional features at higher energies will correspond to simultaneous excitation of the electronic degrees of freedom of the molecule and of the surrounding helium degrees of freedom. In general, one may expect that either or both collective and molecule-localized helium vibrations will contribute to the absorption spectrum. This is confirmed by experimental observations of both coupling to bulk-like collective helium modes, constituting extended phonon wings that are typically separated from the ZPL by a gap7,38 and additional sharp features.7 Note that for separate spectral lines to be resolved in an experimental measurement, the relevant shifts ΔEfi must be different for each line; transitions having the same shift will additively contribute to the total intensity of a single spectral line. We now address the calculation of absorption spectra. Equation 9 provides a recipe for computing the tetracenehelium electronic absorption spectrum at both zero and finite temperature if the exact helium cluster eigenstates in the presence of the molecule are known. At zero temperature, only the lowest helium eigenstate in S0 would contribute, with p0 = 1, while at finite temperatures (provided the cluster is in thermal equilibrium), multiple helium states in S0 would contribute with weights pi given by the Boltzmann factors pi = eE0i/kT. However, while the exact helium excitation energies, E0i may be obtained from path integral correlation function (PICF) calculations,39 the corresponding helium wave functions are generally not accessible from quantum Monte Carlo methods,; therefore, we cannot evaluate the helium FranckCondon factors. Nevertheless, one can still investigate the energetics of possible spectral transitions with evaluation of the exact helium excitation energies E0i and E1f with the molecule in the S0 and S1 states, respectively. To this end, we have performed PICF calculations for clusters with 1 e N e 24 that evaluated the collective excitations of helium with respect to motion along the x, y, and z axes.40 We focused our effort on these smaller clusters because we are primarily concerned with elucidating the influence of the helium close to the tetracene molecule on its absorption spectrum. From inversion of imaginary time correlation functions39,41,42 of collective coordinates of the form ^x ∑N i=1 xi, that is, the N-atom Bose symmetrized position operator of the helium density relative to tetracene, we determined that the frequencies for such collective helium vibrational motion with respect to the x axis lie between 1.5 and 7.0 cm1 in S0 and between ∼0.0 and 9.0 cm1 in S1, for these cluster sizes (Table 1). The magnitude of the resulting energy difference Δ1 Δ0, between the two lowest symmetryallowed energy spectral lines Δ1 = E11 E10 and Δ0 = E01 E00 is found to be consistent with the experimentally observed ∼1 cm1 splitting of the ZPL.40 However, given the magnitude of the excitation energies E01 E00, thermal population of these collective excited states at the droplet temperature of ∼0.4 K will be negligible. The corresponding excitations with respect to the short axis y and the perpendicular axis z were found to be an order
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Table 1. Lowest Excitation Energies for Collective Vibrational Motion of Helium with Respect to the Long Axis of the the Tetracene Molecule in 4HeNTetracene Clusters, Extracted from PICF Calculationsa N
Δ0 (cm1)
Δ1 (cm1)
2
2.51 ( 0.07
2.64 ( 0.03
0.13 ( 0.07
4
4.11 ( 0.03
3.28 ( 0.03
0.841 ( 0.056 1.62 ( 0.1214
6
7.03 ( 0.04
8.65 ( 0.11
14
3.29 ( 0.70
5.05 ( 0.03
24
1.54 ( 0.01
0.00 ( 0.05
3.98 ( 0.02
3.40 ( 0.01
ZPL splitting (cm1)
1.76 ( 0.7024 1.54 ( 0.05
Δ0 = E01 E00 is the lowest helium excitation energy for the molecular electronic ground state (S0). Δ1 = E11 E10 is the lowest helium excitation energy for the molecular electronic excited state (S1). In the case of N = 24, two different frequencies are resolved from inversion of the imaginary time correlation function.40 Also shown are the estimated ZPL splittings that are calculated from the energy difference between the lowest vibrational excitations in S1 and in S0, that is, Δ1 Δ0, for each cluster. a
of magnitude larger. Consequently, we may conclude that the population of collective vibrational states of the solvating helium cannot account for the anomalous splitting of the ZPL in tetracenehelium clusters. Instead of such collective modes, we have found evidence that the modes responsible for the ZPL splitting arise due to strongly correlated single-particle-like excitations of the strongly adsorbed helium atoms. We calculate the zero-temperature absorption spectrum of the most strongly adsorbed and quasi-one-dimensional trios of helium atoms that lie on either side of the tetracene molecule along its long axis, using the natural orbitals as a basis for the helium states. We proceed by constructing single-particle and many-particle excitations using the natural orbitals as basis functions. We will focus on the splitting of the ZPL, that is, on the relative frequencies and intensities of the multiple components of this, and will not concern ourselves here with the absolute value of the spectral lines. This is consistent with restriction to the spectral effects of just the trios of helium atoms along the x axis of the molecule. The effect of the remaining solvating helium atoms will contribute only to the overall spectral shift of the ZPL relative to the bare molecule, which was estimated in ref 19 and which can be neglected here for the purpose of understanding the anomalous ZPL splitting. Spectral Absorption in the Natural Orbital Basis. The natural orbital calculation yields an expression for the He ground state in the S0 and S1 states of tetracene in terms of a superposition of single-particle states jΦ0 æ ¼
5
∑ i¼0
pffiffiffiffi ni jφx, i æjφy æjφz æ
ð11Þ
where the states {|φx,iæ}, |φyæ, and |φzæ are the natural orbitals with respect to x, y, and z, respectively. The T = 0 absorption spectrum would be determined by the overlap of this He ground state in S0 with the eigenstates in S1 according to eq 9. Because the electronic transition is fast relative to the time scale for motion of the nuclei, the positions of the He atoms can be assumed to be static during the transition from S0 to S1. The absorption of a photon by the tetracene molecule thus represents an instantaneous measurement of the coupled Hetetracene system. This is a key assumption of our model, as it allows the 7227
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system to exhibit characteristics consistent with the singleparticle states in the superposition. The single-particle natural orbitals computed from diagonalization of the OBDM are not necessarily eigenfunctions of the many-body cluster Hamiltonian, although they are derived from many-body simulations of this and thus explicitly incorporate the effects of all HeHe correlations. Because of this latter feature, the orthonormal single-particle basis provided by the natural orbitals constitutes a convenient basis for investigating the properties of the three atoms adsorbed on tetracene. We use the natural orbitals here as a basis in which to expand the manybody He states for these three atoms in order to calculate the spectral density function and the absorption spectrum. We do this here in two levels of approximation, first evaluating the absorption spectrum using the natural orbitals directly as a singleparticle basis and subsequently using these to construct a threeparticle basis that allows for correlated single-particle excitations. Expanding Æμ(t)μ(0)æ in the single-particle natural orbital basis φi and employing the corresponding effective single-particle Hamiltonians p2 ^2 ^ ð0Þ r þ V Hetet 2mHe 2 ^ 0ð1Þ ¼ p r^2 þ V^ ð1Þ H Hetet 2mHe
^ H
0ð0Þ
Table 2. Matrix Elements of the Hamiltonians in Equation 12 Evaluated in the Effective Single-Particle Basis state label i
energy in S0 (cm1)
energy in S1 (cm1)
0
66.36
69.05
1 2
60.28 61.18
63.39 63.81
3
62.60
63.19
4
51.04
54.69
5
51.77
53.74
¼
ð12Þ
leads to the single-particle analogue of eq 9 ÆμðtÞμð0Þæ ¼
∑ ni eitðΔE ijmk
S0 f S1 Þ
ð0Þ
ð1Þ†
0 Uim ðtÞUkj0
ðtÞÆφ0i jφ1k æÆφ1j jφ0m æjÆψ1 jμjψ0 æj2 ð13Þ
iH0 (R)t
0 (R)
where U (t) e , R = 0 and 1, are the effective time evolution operators for the He states in the presence of ground and excited electronic states of tetracene and ΔES0fS1 is the molecular electronic energy difference between the two states. Note that given the key assumption that the helium atoms in the strongly localized trio remain stationary during the transition from S0 to S1, it is the eigenstates and eigenvalues of an effective single-particle Hamiltonian like eq 12 that determine the dynamics of the transition rather than the full many-body Hamiltonian. We therefore evaluate all elements of the Hamiltonians in eq 12 in the natural orbital bases obtained from the path integral ground-state calculations in S0 and S1, using the Hetetracene interaction potentials employed in ref 19. The off-diagonal matrix elements i ¼ 6 m and j ¼ 6 k of H0 (0) and H0 (1) are found to be at least 2 orders of magnitude smaller than the diagonal elements. This indicates that for this trio of atoms, the natural orbitals are indeed good approximations to energy eigenfunctions as a result of the strong localization. We can therefore evaluate a spectrum by treating the natural orbitals as approximate eigenfunctions of the single-particle Hamiltonians in eq 12. This leads to ÆμðtÞμð0Þæ
∑ij ni eitðÆE æ ÆE æÞjÆφ0i jφ1j æj2 jÆψ1jμjψ0 æj2 1j
0i
ð14Þ
where ð0Þ
ÆE0i æ ¼ ES0 þ Hii0
and
ð1Þ
ÆE1j æ ¼ ES1 þ Hjj0
ð15Þ
are the diagonal matrix elements of the effective Hamiltonians, eq 12, and ES0 and ES1 are the molecular energies of the states ψ0 and ψ1 of the isolated tetracene molecule. The matrix elements
Figure 8. Calculated spectrum from the single-particle model in the natural orbital basis. Red lines correspond to the individual transitions with relative intensities indicated by their height; the black line shows the continuous spectrum resulting from convolution with a Lorentzian line shape function of width 0.38 cm1. (1) H(0) ii and Hjj are evaluated in the bases of the diagonal OBDM in each electronic state, where the total OBDM is constructed as a product of the individual diagonal density matrices with respect to each coordinate x, y, and z. The diagonal elements of the Hamiltonians for each of the basis states are listed in Table 2. A key implication of this property of the natural orbitals for the heliumtetracene system is that the Fourier transform of the resulting dipole correlation function can then be readily made, resulting in an absorption spectrum
I1 ðpωÞ ¼
2π p
∑ij ni δðω ðΔES ð0Þ
0
f S1
ð1Þ
þ ÆHjj æ
ÆHii æÞjjÆφ0i jφ1j æj2 jÆψ1 jμjψ0 æj2
ð16Þ
Note that the intensities of these spectral components are given by the product of the eigenvalues of the OBDM in S0 and the differential overlap of the natural orbitals in the two electronic states. We see that the differential evolution of the natural orbitals under the single-particle Hamiltonians in eq 12 results in several zero- temperature spectral components at slightly different energies whenever more than one nonzero ni value exists. This is a true zero-temperature spectrum and does not imply any thermal population of the single-particle states. The resulting “singleparticle” spectrum is shown in Figure 8. Because, as noted above, we are not interested in the overall spectral shift here (which was studied in detail in ref 19), we show the spectrum here as a function of frequency with respect to the most intense transition and with 7228
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spectral components scaled relative to this (i.e., we do not incorporate the molecular transition dipole moment). The red lines in Figure 8 indicate the transitions calculated from the singleparticle model, with the height of each line indicating its relative intensity. The black line is the result of convolution of those data with a Lorentzian line shape function of width 0.38 cm1, the same width as that used in the spectral fitting of ref 2. The dominant feature of the spectrum is the β-type feature, which includes contributions from transitions i = j = 0 and i = j = 2 as well as a contribution from the transition i = 1, j = 3. The single-particle spectrum also shows a secondary transition at about 0.6 cm1 with a weak shoulder at about 1.4 cm1 and several weaker transitions at positive relative energies. These secondary transitions to the red of the β-type feature are close to the observed R line of tetracene and correspond to transitions i = j = 1 and i = j = 4. The spectrum also shows features at about 1.0 and 3.0 cm1, close to the γ and δ lines of the measured tetracene spectrum,2 corresponding to i = j = 5 and i = j = 3, respectively. The remaining very weak features at around (7.5 cm1 correspond to transitions between i = 0(2) and j = 2(0). To include the possibility that the coherent quantum correlations of the ground state may give combination modes of excitation in the natural orbital modes and to take this into account in the zero-temperature absorption spectrum, we have constructed an effective three-particle ground-state wave function, |Φ0,3æ, as a tensor product of the ground-state wave function in the single-particle basis
Table 3. Matrix Elements of the Hamiltonians in Equation 18 Evaluated in the Effective Three-Particle Basis three-particle
single-particle
energy in S0
energy in S1
state index p
labels ijk
(cm1)
(cm1)
1 2
012 013
187.82 189.24
196.25 195.63
3
014
177.68
187.13
4
015
178.41
186.19
5
023
190.14
196.05
6
024
178.59
187.55
7
025
179.32
186.61
8
034
180.00
186.93
9 10
035 045
180.73 169.18
185.98 177.48
11
123
184.06
190.39
12
124
172.50
181.89
13
125
173.23
180.95
14
134
173.92
181.27
15
135
174.65
180.32
16
145
163.10
171.82
17 18
234 235
174.82 175.55
181.69 180.74
19
245
164.00
172.25
20
345
165.41
171.62
jΦ0, 3 æ ¼ jΦ0 æ X jΦ0 æ X jΦ0 æ pffiffiffiffiffiffiffiffiffiffiffi ni nj nk 1 2 3 jφx, i φx, j φx, k æjφ1y φ2y φ3y æjφ1z φ2z φ3z æ ¼P C i¼0 j¼i þ 1 k¼j þ 1 5
5
∑ ∑
5
∑
ð17Þ where the subscripts are the labels of the natural orbitals calculated from our path integral ground-state simulations, the superscripts label the particles, P is the permutation operator, and C is a normalization constant. To maintain consistency with the total occupation numbers calculated from diagonalization of the single-particle OBDM, we assume that two or more atoms cannot simultaneously occupy the same natural orbital. The basis functions for evaluation of the Hamiltonian matrix elements are then Fock states of three unique functions in the natural orbital basis, weighted by the value of (ni)1/2 for Reach x orbital. The resulting wave function is normalized so that dR |Φ0,3(R)|2 = 1. The spectrum is then evaluated from the analogue of eq 14 with φi replaced by the three-particle Fock states and the effective Hamiltonians in eq 12 replaced by " # 3 p2 ^2 ð0Þ 0ð0Þ ^ ¼ r þ V^ Hei tet H 2mHe i i¼1 " # 2 3 p ð1Þ 0ð1Þ 2 ^ ^ H r þ V^ Hei tet ¼ ð18Þ 2mHe i i¼1
∑
∑
Again, we neglect the heliumhelium interactions in evaluation of the Hamiltonian matrix elements because the positions of these three strongly localized He atoms are essentially constant during the molecular electronic transition from S0 to S1 and the effects of heliumhelium interactions are very similar in the two electronic states. They will therefore effectively cancel out in energy differences E1q E0p, where p,q denote the three-particle
states, and thus do not appear in the exponential factors of the corresponding dipole correlation function. As in the singleparticle analysis above, we evaluated the matrix elements of the Hamiltonians in eq 18 in the three-particle basis and verified that the off-diagonal elements are again much smaller than the diagonal elements. The resulting absorption spectrum is I3 ðpωÞ ¼
2π p
∑pq np δðω ðΔES
0
f S1
0 ð1Þ 0 ð0Þ þ ÆHqq æ ÆHpp æÞjjÆφ0p jφ1q æj2 jÆψ1 jμjψ0 æjj2
ð19Þ Thus, for the three-particle formulation, the absorption components are given by the differences in the diagonal matrix elements of the three-particle Hamiltonian, eq 18, while their intensities are given by the overlaps of the tensor product states in S0 and S1 and can be reduced to combinations of FranckCondon overlaps of the natural orbitals. The diagonal elements of the Hamiltonians for each of the basis states are listed in Table 3, and the resulting “three-particle” spectrum is shown in Figure 9. As in Figure 8, the individual transitions are shown as red lines, and the black line shows the spectrum resulting from convolution with a Lorentzian line shape of width 0.38 cm1. To ease comparison with the single-particle spectrum, we refer to the three-particle transitions by indices p,q, where p and q are pointers to the triples ijk of single-particle indices in Table 3. For greater clarity, the transitions shown in Figure 9 are listed explicitly in Table 4. Similar to the single-particle spectrum, we find that the β-type feature in the three-particle spectrum contains contributions from multiple transitions (p = q = 1, 10, 16, and 19.) The R-type feature at about 1.0 cm1 corresponds to p = q = 3 and 12. The features in the region of the γ peak at about 1.0 cm1 correspond to transitions with p = q = 4, 7, 8, 14, 7229
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Figure 9. Calculated spectrum from the three-particle model based on a tensor product of single-particle states in the natural orbital basis. Red lines correspond to the individual transitions with relative intensities indicated by their height; the black line shows the continuous spectrum resulting from convolution with a Lorentzian line shape function of width 0.38 cm1.
Table 4. Spectral Transitions in the Effective Three-Particle Basis Evaluated from Equation 14a
a
relative transition energy (cm1)
index S0 i
index S1 f
1
1
0.00
5.36 101
2
2
2.04
6.93 102
intensity
3.21 10
3
2
4
11.48
3 4
3 2
1.01 8.78
4.05 102 1.49 103
4
4
0.66
3.34 102
5
5
2.52
6.83 102
5
7
11.96
3.16 103
6
6
0.53
3.99 102
7
5
8.30
1.47 103
7
7
1.14
3.30 102
8 9
8 9
1.51 3.18
5.17 103 4.26 103
10
10
0.13
2.49 103
11
11
2.10
6.35 102
11
13
11.54
2.94 103
12
12
0.96
3.71 102
13
11
8.72
1.37 103
13
13
0.72
3.06 102
14 15
14 15
1.09 2.76
4.80 103 3.96 103
16
16
0.30
2.32 103
17
17
1.57
4.73 103
18
18
3.24
3.90 103
19
19
0.19
2.28 103
20
20
2.23 4
Transitions with an intensity greater than 10
2.95 104
are listed.
and 17, while the features in the region of the δ peak arise from transitions with p = q = 2, 5, 9, 11, 18, and 20. In Figure 10, we compare the spectra from both models presented here with the fit to the experimental spectrum that
Figure 10. Calculated absorption spectra of tetracene in 4He droplets from the ground-state wave function of the trio of the three most strongly adsorbed helium atoms, compared with experimental data. The shaded curve shows the fit to the experimental absorption spectrum in ref 2. The dashed line is the model spectrum computed directly from the single-particle natural orbitals (see Figure 8). The filled circles connected by a solid line shows the model spectrum for the three-particle system, calculated from a tensor product of single-particle natural orbitals (see Figure 9). For both of these model spectra, a phonon wing in the form of ref 2 is added explicitly to the zero-temperature spectrum calculated from the trio of three adsorbed helium atoms.
was extracted in ref 2 (shaded) for tetracene in helium droplets at T ≈ 0.4 K. For the sake of comparison, the phonon wing has been added to the model spectra, computed here using the same functional form as that employed in ref 2. The spectra here have been scaled so that the intensity at the β lines are equal. Figure 10 shows remarkable agreement between the calculated spectrum and the fit to the experimental data. We see that in the threeparticle model, the absorption at 1.0 cm1 has a relative intensity of approximately 0.19 that of the primary peak at the zero-frequency reference. This feature is in very close agreement with the position and intensity of the R line in the fit to the experimental data.2 We also find that the additional higherenergy transitions, at þ1.0 and þ2.0 cm1, agree well with the γ and δ lines in the fit of the experimental data, although the intensity of the transition at þ2.0 cm1 is significantly greater than that seen in the experiment. This remarkable agreement between the measured absorption spectrum and the spectra calculated in the three-atom Fock state representation composed of natural orbitals of the helium atoms that are most strongly localized at tetracene suggest that this trio of strongly localized yet coherently correlated helium atoms is responsible for the ZPL splitting of tetracene in He droplets. The results imply that transitions involved in the ZPL are a property of the helium ground state in this system and reflect the coherent quantum correlations in the ground state of the three atoms due to the strong interparticle correlations induced by the heliumhelium interaction and their quasi-one-dimensional nature on the surface of tetracene.
’ DISCUSSION We have presented a new theoretical approach to compute the electronic absorption spectrum in a many-body quantum system, namely, to use the natural orbitals computed from the one-body density matrix that thereby take heliumhelium correlations 7230
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The Journal of Physical Chemistry A into account at zeroth order, to construct tensor product bases for higher-order density matrices that are used to calculate the zero-temperature absorption spectrum from dipole correlation functions. Formally, the approach constitutes a multimode mean field description. We have seen above that this multimode approach gives excellent agreement with the experimental ZPL spectrum for tetracene in helium clusters, thus clarifying the hitherto anomalous origin of the splitting observed in this system. From a theoretical perspective, it is important to understand how this natural-orbital-based calculation of the zero-temperature spectrum is related to the exact expression in terms of energy eigenstates in eq 10. There are several aspects to this. First, as noted earlier, it is not possible to calculate the excited-state wave functions for the full tetracenehelium cluster. The natural orbitals provide an accurate representation of the effects of coherent quantum correlations between single-particle components of the helium ground state. A spectral calculation in this natural orbital basis will therefore show several spectral components as a result of the multimode nature of the helium ground states in S0 and S1. In our derivation of the absorption spectrum in this basis, we noted that because of the strong localization of these trios of helium atoms on tetracene, the natural orbitals constitute good approximations to energy eigenfunctions and that this allowed the spectrum to be obtained in the usual form as a sum over delta functions of energy. The theory correctly satisfies energy conservation. However, the spectral weights pi accompanying these delta functions are not thermal but zerotemperature weights expressing the weighted distribution of the natural orbitals in the ground state. This means that even a zerotemperature spectrum will display some fine structure due to the ground-state quantum correlated motion of the helium at the tetracene molecule. Of course, the temperature of the experiments is not zero and is typically T = 0.4 K. However, all theoretical calculations indicate that the structure of helium droplets is very close to the ground state at the temperature T = 0.4 K of the experiments, validating the quantitative comparison of the experimental spectra with a zero-temperature analysis. A second important aspect concerns the restriction of the calculation of the zero-phonon absorption spectrum to just the quasi-one-dimensional trio of strongly bound helium atoms located along the long (x) axis of the tetracene molecule. The tensor product states used here constitute an approximation to the full three-particle density matrix of this trio. Our analysis showed that this results in a very accurate description of the dynamical evolution of the dipole moment operator that enters the calculation of the absorption spectrum. Consequently, we may take the three-particle spectrum as an accurate representation of the zero-temperature spectrum. It remains to assess the effect of the remaining helium atoms on the spectrum. As noted earlier, this trio of atoms is well-isolated from the surrounding helium environment, including the rest of the first solvation shell. We have monitored the permutation exchanges of these atoms both within the trio and with the other helium atoms in the first solvation shell. While exchange does occur within the trio, these atoms essentially do not exchange with the remaining helium density. This is consistent with observations in our earlier simulations of benzene.43 This ensures that the density matrix of this trio of helium atoms and the quantum coherence described by this are, to a good approximation, uncoupled from the wave function of the surrounding helium density. Consequently, the surrounding helium density contributes only to an
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overall spectral shift, as described in our earlier study,19 and does not affect the detailed structure of the ZPL, which is determined only by the ground-state quantum coherences and correlation. A further issue concerns the question as to whether the anomalous zero-phonon splitting might be alternately derived from thermal population of some subtly correlated local helium vibrations with low energy. We determined that collective helium vibrational modes are too high in energy for this explanation to be applicable to T = 0.4 K. One might then ask whether the energies of the natural orbitals in the effective single-particle Hamiltonians might be interpreted as some kind of correlated local helium vibration. While this is an interesting question for further study, the zero-temperature analysis in this work shows that in the presence of strongly correlated quantum coherence between adsorbed helium atoms, it is possible to obtain multiple spectral components for the ZPL without any thermal population of excited vibrational states. We cannot, however, completely rule out the possibility of thermal contributions in the form of He vibrations consistent with the natural orbital states to the experimentally measured spectrum. We now compare the behavior of tetracene with other planar aromatic molecules in helium. As noted earlier, all experimental measurements for tetracene indicate that the lifetimes of the electronically excited states in the droplets that are responsible for the ZPL features are relatively long, on the order of tens of nanoseconds. The fluorescence lifetimes of R and β spectral lines of tetracene in He droplets are about 23 and 35 ns, respectively.6 In pumpprobe experiments with a delay of about 20 ns between the applied laser pulses and in measurements of the emission spectra of tetracene, no interconversion between the states corresponding to the R and β lines is observed,3,9 whereas in experiments where the delay is >100 ns, which is substantially longer than the fluorescence lifetime of the R and β spectral lines, there is evidence for interconversion.6 This indicates that the lifetime of the He states associated with the S1 excited state of the molecule responsible for the ZPL splitting must be at least ∼20 ns. Such a long lifetime of the helium states is in strong contrast with the situation for phthalocyanine11 and porphin7 and implies that the helium dynamics must be nondissipative on this time scale. This is consistent with the proposed zero-temperature origin of the anomalous ZPL splitting, that is, as derived from the ground-state correlated quantum coherence of the mostly strongly adsorbed component of the solvating helium in the S0 and S1 molecular states. In contrast, the zero-phonon splitting in phthalocyanine emission spectra is derived from the presence of metastable helium configurations that may decay via coupling to phonon and ripplon modes of the droplet. When analyzing why tetracene exhibits this zero-temperature split spectral absorption in helium whereas the phthalocyanine and porphin molecules do not, a key factor to take into consideration is the geometry of the aromatic system. The phthalocyanine and porphin molecules localize their most strongly bound solvating He atoms in effective two-dimensional geometries, in contrast to the one-dimensional geometry of the trio of most strongly bound helium atoms at the tetracene molecule. This greater dimensionality of the most strongly bound helium atoms at the phthalocyanine and porphin molecules leads to lower barriers for interconversion between different states of the adsorbed helium. It is also likely to result in a greater extent of coupling of these adsorbed atoms to the helium droplet vibrational modes. This is suggested by the marked difference between the phonon wings of the two-dimensional 7231
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The Journal of Physical Chemistry A phthalocyanine and porphin molecules and the phonon wings of linear molecules such as tetracene and pentacene.7 This offers additional evidence that, in addition to possessing very different ground-state densities, both the helium excitations and dynamics are quite different in the case of the linear molecules than in the two-dimensional molecular systems. For example, the two configurations of He near the surface of phthalocyanine in our previous PIMC study were seen to be related to each other by a rotation of the He density by 45.15 The energy scale for this rotational motion can be roughly estimated by assuming that the He adsorbed on the phthalocyanine rotates as a rigid disk without friction. Such an estimate yields a value of e0.1 K for the rotational energy of the He disk; the corrugation of the Hephthalocyanine interaction potential would increase this value in the S0 state. However, the lowering of the barriers in the Hephthalocyanine interaction potential that is seen upon excitation to S1 could facilitate this rotational motion and hence also the interconversion between the two metastable, effective 2D states of He in that system.13 There is no similar rearrangement mechanism for interconversion between the effective onedimensional states in the Helinear molecule systems. A further question is whether tetracene in helium is unique compared to the other linear aromatic molecules. While the specific nature of the quantum correlated helium coherence along the long axis of the molecule that was established in this work is specific to tetracene, the analysis given here suggests that related phenomena may be expected to occur also for other linear molecules in helium clusters. We therefore suggest that the features here observed for tetracene might be present in some form for any system with a similar geometry and highly anisotropic localization. With regard to linear molecules, we note that the absorption spectra of naphthalene, anthracene, and pentacene all exhibit multiple features in the ZPL region.2,7,21,22 In the case of naphthalene, the spectrum was assumed to consist of a single, very broad ZPL.22 It is possible that this broad feature may consist instead of multiple superimposed sharp features, such as has been proposed empirically to fit the spectrum of anthracene.2 The spectrum of pentacene also displays additional features between the ZPL and the phonon wing.7,21 Whether such features are due to excitation of vibrations of localized helium atoms or derive from the zero-temperature correlated quantum coherence effects demonstrated here for tetracene will depend on the details of the He density near each of these molecules. Microscopic theoretical studies of these systems will be necessary to determine the contribution of correlated coherent helium dynamics to the ZPL structure in the absorption spectra of other molecules. The interesting variations seen when the aromatic molecules are complexed with molecular hydrogen and its isotopomers8,10 can be expected to result from sensitivity of the quantum coherence of quasi-one-dimensional structures of helium atoms on the molecular surfaces but will similarly require detailed microscopic calculations for a full explanation. It would also be interesting to investigate to what extent these effects contribute to the zero-temperature ZPL splitting seen for nonlinear molecules such as the indoles.23
’ SUMMARY We have studied the dynamical features and absorption spectroscopy of strongly correlated He atoms localized near the surface of tetracene and used these to account for the anomalous splitting of the ZPL in the electronic absorption
ARTICLE
spectrum in helium droplets. Ground-state calculations show that the helium ground state for both ground and excited electronic states of tetracene contains a quasi-one-dimensional trio of helium atoms adsorbed along the long axis of the molecule and that this trio of helium atoms is characterized by strong and coherent heliumhelium correlations, as evidenced by analysis of the one-body density matrix. While there are permutation exchanges within the trio, exchanges with surrounding helium atoms are strongly suppressed because of the quasi-one-dimensional confinement of the helium density deriving from these three atoms. We have calculated natural orbitals from the onebody density matrix and used these to construct approximate three-body density matrices for this trio of strongly adsorbed helium atoms. This description of the most strongly bound helium density around the molecule was used to derive an expression for the zero-temperature absorption spectrum employing the dipole correlation function approach within a multimode mean field description that shows multiple spectral frequencies that may result from the correlated quantum coherence of the adsorbed helium ground states in the presence of ground and excited electronic states of the molecule. Explicit calculation of the spectrum due to these trios of atoms results in excellent agreement with the structure of the anomalous ZPL of tetracene, accounting for the relative positions and intensities of the well-established R and β lines, as well as for two additional smaller-intensity peaks at higher energy. The absolute frequencies of all transitions may be obtained from the relative positions by adding the bare molecule transition energy and the spectral shift resulting from the remaining solvating helium in the droplet. This analysis suggests that the zero-temperature electronic absorption spectrum of linear polyatomic molecules can show a novel splitting of the zero-phonon absorption line into multiple spectral lines as a result of a quantum coherent, but strongly correlated, set of helium atoms adsorbed on the quasi-planar molecular surface.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We are grateful to Prof. J. P. Toennies for initial discussions that inspired us to investigate the anomalous zero-phonon splitting of tetracene in helium droplets and to A. Slenzcka and R. Lehnig for discussion of their experimental data. This work was performed in part under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. ’ REFERENCES (1) Hartmann, M.; Lindinger, A.; Toennies, J. P.; Vilesov, A. F. Chem. Phys. 1998, 239, 139. (2) Krasnokutski, S.; Rouille, G.; Huisken, F. Chem. Phys. Lett. 2005, 406, 386. (3) Lindinger, A.; Toennies, J. P.; Vilesov, A. F. Phys. Chem. Chem. Phys. 2001, 3, 2581. (4) Even, U.; Al-Hroub, I.; Jortner, J. J. Chem. Phys. 2001, 115, 2069. (5) P€ ortner, N.; Vilesov, A. F.; Havenith, M. Chem. Phys. Lett. 2001, 343, 281. (6) Hartmann, M.; Lindinger, A.; Toennies, J. P.; Vilesov, A. F. J. Phys. Chem. A 2001, 105, 6369. 7232
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