^{1,2,a)}, Jonathan L. DuBois

^{1,2}and K. Birgitta Whaley

^{1}

### Abstract

Spectral shifts of electronic transitions of tetracene in heliumdroplets are investigated in a theoretical study of –tetracene clusters with . Utilizing a pairwise interaction for the state of tetracene with helium that is extended by semiempirical terms to construct a potential for the state of tetracene with helium, the spectral shift is calculated from path integral Monte Carlo calculations of the helium equilibrium properties with tetracene in the and states at and at . The calculated spectral shifts are in quantitative agreement with available experimental measurements for small values of at and show qualitative agreement for larger (10–20). The extrapolated value of the spectral shift in large droplets is of the experimentally measured value. We find no evidence of multiple configurations of helium for any cluster size for either the or state of tetracene. These results suggest that the observed spectral splitting of electronic transitions of tetracene in large heliumdroplets is not due to the coexistence of static metastable helium densities, unlike the situation previously analyzed for the phthalocyanine molecule.

We are grateful to Professor J. P. Toennies for discussions which inspired us to pursue this project and to A. Slenzcka and R. Lehnig for discussion of their experimental data. H. D. Whitley acknowledges the support of the AFOSR through a NDSEG research fellowship. This work was performed, in part, under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.

I. INTRODUCTION

II. THEORY AND COMPUTATIONAL METHODS

A. Hamiltonian and interaction potentials

B. Computational methods

C. Spectral shift of electronic absorption spectra

1. Zero temperature shift

2. Perturbative estimates of zero temperature shift

3. Finite-temperature shift

III. RESULTS AND DISCUSSION

A. Helium configurations and clusterenergies

B. Spectral shift

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Fluid drops
- 31.0
- Ground states
- 20.0
- Liquid helium
- 19.0
- Wave functions
- 16.0
- Molecular spectra
- 15.0

## Figures

The (a) and (b) helium–tetracene interaction potential in wavenumbers near the global minimum at . The skeletal bond form of the tetracene molecule is also shown for reference. Contours are shown from the global minima up to about . The surface of the carbon skeleton of the tetracene molecule is restricted to the region and . (c) The difference in the and helium–tetracene interaction potentials, , shown in wavenumbers, near the global minimum at . The overall difference is repulsive near the carbons in the 9–12 positions (using the numbering systems of Ref. 38) due to the inclusion of . Contours are shown only for .

The (a) and (b) helium–tetracene interaction potential in wavenumbers near the global minimum at . The skeletal bond form of the tetracene molecule is also shown for reference. Contours are shown from the global minima up to about . The surface of the carbon skeleton of the tetracene molecule is restricted to the region and . (c) The difference in the and helium–tetracene interaction potentials, , shown in wavenumbers, near the global minimum at . The overall difference is repulsive near the carbons in the 9–12 positions (using the numbering systems of Ref. 38) due to the inclusion of . Contours are shown only for .

The average energy per He atom as a function of for –tetracene clusters in the and at . The finite-temperature estimate of the chemical potential [Eq. (23)] is also shown. The ground state chemical potential (blue triangles) for pure clusters are calculated from the data of Ref. 56.

The average energy per He atom as a function of for –tetracene clusters in the and at . The finite-temperature estimate of the chemical potential [Eq. (23)] is also shown. The ground state chemical potential (blue triangles) for pure clusters are calculated from the data of Ref. 56.

Parallel cuts of the helium densities seen in clusters at the maxima in the (left) and (right) states calculated via PIMC simulations at . The top panel shows one He atom localized at the molecule (seen for ), the center panel shows two atoms at the molecule (seen for ), while the lower panel shows three atoms localized near the molecule (seen for ).

Parallel cuts of the helium densities seen in clusters at the maxima in the (left) and (right) states calculated via PIMC simulations at . The top panel shows one He atom localized at the molecule (seen for ), the center panel shows two atoms at the molecule (seen for ), while the lower panel shows three atoms localized near the molecule (seen for ).

Cuts of the He–tetracene potentials along near the global minima for the and states.

Cuts of the He–tetracene potentials along near the global minima for the and states.

Cuts of the helium densities for atoms near tetracene at the density maxima in the (left) and (right) electronic states of tetracene calculated via PIMC simulations at . Contours are shown in the density range .

Cuts of the helium densities for atoms near tetracene at the density maxima in the (left) and (right) electronic states of tetracene calculated via PIMC simulations at . Contours are shown in the density range .

Density profile for He atoms on one side of the tetracene molecule calculated via PIMC simulations at . Profiles in the left (right) panels correspond to the state. Contours are shown for two planes parallel to the molecule at in the upper panels and at in the lower panels. The cuts shown are taken from a single PIMC simulation and do not necessarily reflect the full symmetry of the cluster that would result from averaging over many runs.

Density profile for He atoms on one side of the tetracene molecule calculated via PIMC simulations at . Profiles in the left (right) panels correspond to the state. Contours are shown for two planes parallel to the molecule at in the upper panels and at in the lower panels. The cuts shown are taken from a single PIMC simulation and do not necessarily reflect the full symmetry of the cluster that would result from averaging over many runs.

Density profile for He atoms on one side of the tetracene molecule calculated via PIMC simulations at . Profiles in the left (right) panels correspond to the state. Contours are shown for two planes parallel to the molecule at in the upper panels and at in the lower panels. The cuts shown are taken from a single PIMC simulation and do not necessarily reflect the full symmetry of the cluster that would result from averaging over many runs.

Density profile for He atoms on one side of the tetracene molecule. Profiles in the left (right) panels correspond to the state. Contours are shown for two planes parallel to the molecule at in the upper panels and at in the lower panels. The cuts shown are taken from a single PIMC simulation and do not necessarily reflect the full symmetry of the cluster that would result from averaging over many runs.

Density profile for He atoms on one side of the tetracene molecule. Profiles in the left (right) panels correspond to the state. Contours are shown for two planes parallel to the molecule at in the upper panels and at in the lower panels. The cuts shown are taken from a single PIMC simulation and do not necessarily reflect the full symmetry of the cluster that would result from averaging over many runs.

Cuts of the helium density profile at the maximum density for at (upper panels) and (lower panels) calculated via PIMC simulations at . The left panels show results from calculations in the electronic state of tetracene. Results from calculations in the state are shown in the panel on the right.

Cuts of the helium density profile at the maximum density for at (upper panels) and (lower panels) calculated via PIMC simulations at . The left panels show results from calculations in the electronic state of tetracene. Results from calculations in the state are shown in the panel on the right.

Cuts of the helium density profile at the maximum density for at (upper panels) and (lower panels) calculated via PIMC simulations at . The left panels show results from calculations in the electronic state of tetracene. Results from calculations in the state are shown in the panel on the right.

The integrated difference in the helium density profiles for the first layer of He near tetracene in the and for clusters with , , and helium atoms calculated via PIMC simulations at . For and the difference has been integrated for so that only the first layer of helium near the molecule is considered. The plots shown are derived from a single PIMC simulation in each electronic state and do not necessarily reflect the full symmetry of the cluster that would result from averaging over many runs.

The integrated difference in the helium density profiles for the first layer of He near tetracene in the and for clusters with , , and helium atoms calculated via PIMC simulations at . For and the difference has been integrated for so that only the first layer of helium near the molecule is considered. The plots shown are derived from a single PIMC simulation in each electronic state and do not necessarily reflect the full symmetry of the cluster that would result from averaging over many runs.

Spectral shift of the tetracene transition as a function of the number of helium atoms in a He cluster with . Experimental data (triangles) are taken from Ref. 6. The inset shows the same data in the region of . The standard deviations of the plotted data are all , and are much smaller than the symbols shown on the plots. The experimentally measured point for has been shown with an error bar of since this value was not reported with certainty.

Spectral shift of the tetracene transition as a function of the number of helium atoms in a He cluster with . Experimental data (triangles) are taken from Ref. 6. The inset shows the same data in the region of . The standard deviations of the plotted data are all , and are much smaller than the symbols shown on the plots. The experimentally measured point for has been shown with an error bar of since this value was not reported with certainty.

Spectral shift of the tetracene transition as a function of the number of helium atoms in a He cluster from PIMC simulations at . Data for are fitted by an exponential function, (dotted lines). Standard deviations of the perturbative estimates of the spectral shift for are .

Spectral shift of the tetracene transition as a function of the number of helium atoms in a He cluster from PIMC simulations at . Data for are fitted by an exponential function, (dotted lines). Standard deviations of the perturbative estimates of the spectral shift for are .

## Tables

Interaction potential parameters for helium with tetracene.

Interaction potential parameters for helium with tetracene.

Parameters for the interaction potential.

Parameters for the interaction potential.

Comparison of the energy per particle in –tetracene clusters in the state for ground state VMC, DMC, and VPI calculations. The variance of the average (shown in parentheses) is in the last digit. The energy per particle for PIMC calculations at is listed in the last column. VMC and DMC results were taken from Ref. 41.

Comparison of the energy per particle in –tetracene clusters in the state for ground state VMC, DMC, and VPI calculations. The variance of the average (shown in parentheses) is in the last digit. The energy per particle for PIMC calculations at is listed in the last column. VMC and DMC results were taken from Ref. 41.

Average total energy for each of the clusters in this study calculated from PIMC calculations at . The variance of the mean, shown in parentheses, is in the last digit.

Average total energy for each of the clusters in this study calculated from PIMC calculations at . The variance of the mean, shown in parentheses, is in the last digit.

Configurations for clusters with .

Configurations for clusters with .

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