^{1}, Chao-Ping Hsu

^{2,a)}and Graham R. Fleming

^{3}

### Abstract

Triplet-triplet (TT) energy transfer requires two molecular fragments to exchange electrons that carry different spin and energy. In this paper, we analyze and report values of the electronic coupling strengths for TT energy transfer. Two different methods were proposed and tested: (1) Directly calculating the off-diagonal Hamiltonian matrix element. This direct coupling scheme was generalized from the one used for electron transfer coupling, where two spin-localized unrestricted Hartree-Fock wave functions are used as the zero-order reactant and product states, and the off-diagonal Hamiltonian matrix elements are calculated directly. (2) From energy gaps derived from configuration-interaction-singles (CIS) scheme. Both methods yielded very similar results for the systems tested. For TT coupling between a pair of face-to-face ethylene molecules, the exponential attenuation factor is , which is about twice as large as typical values for electron transfer. With a series of fully stacked polyene pairs, we found that the TT coupling magnitudes and attenuation rates are very similar irrespective of their molecular size. If the polyenes were partially stacked, TT couplings were much reduced, and they decay more rapidly with distance than those of full-stacked systems. Our results showed that the TT coupling arises mainly from the region of close contact between the donor and acceptor frontier orbitals, and the exponential decay of the coupling with separation depends on the details of the molecular contacts. With our calculated results, nanosecond or picosecond time scales for TT energy-transfer rates are possible.

Two of the authors (C.P.H. and Z.Q.Y.) acknowledge the National Science Council of Taiwan (Grant No. NSC93-2113-M-001-012). One of the authors (Z.Q.Y.) wishes to acknowledge financial support from Academia Sinica. The work at Berkeley was supported by the Director, Office of Science, Office of Basic Energy Sciences, Chemical Sciences Division, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.

I. INTRODUCTION

II. THEORY AND METHODS

A. Direct coupling

B. Energy-gap-based method: Configuration-interaction singles

C. Computational details

III. RESULTS AND DISCUSSION

A. Direct coupling

B. Couplings derived from CIS

C. Moving along the reaction coordinate

D. Effects of size and intermolecular contacts

E. Nature of TT couplings

F. TT energy-transfer rates

IV. CONCLUSIONS

### Key Topics

- Cancer
- 41.0
- Electron transfer
- 30.0
- Energy transfer
- 18.0
- Band gap
- 8.0
- Basis sets
- 7.0

## Figures

A schematic picture of TT energy transfer as two simultaneous electron transfers between the donor and acceptor .

A schematic picture of TT energy transfer as two simultaneous electron transfers between the donor and acceptor .

Face-to-face arrangements of (A) two ethylenes, and (B) an ethylene and a methaniminium cation. denotes the intermolecular distance.

Face-to-face arrangements of (A) two ethylenes, and (B) an ethylene and a methaniminium cation. denotes the intermolecular distance.

Distance and basis set dependence of TT coupling, calculated from the DC scheme for the two-ethylene system. Bases sets used are as follows: 3-21G (filled squares), (open squares), DZP (filled circles), and (open circles).

Distance and basis set dependence of TT coupling, calculated from the DC scheme for the two-ethylene system. Bases sets used are as follows: 3-21G (filled squares), (open squares), DZP (filled circles), and (open circles).

Distance and basis set dependence of TT coupling. Shown are half of the CIS energy gaps between the two lowest triplet states for the two-ethylene system. Basis sets used are as follows: 3-21G (filled squares), (open squares), DZP (filled circles), (open circles), and aug-cc-pVTZ (open triangles).

Distance and basis set dependence of TT coupling. Shown are half of the CIS energy gaps between the two lowest triplet states for the two-ethylene system. Basis sets used are as follows: 3-21G (filled squares), (open squares), DZP (filled circles), (open circles), and aug-cc-pVTZ (open triangles).

Potential-energy curves of the two lowest triplet states for the ethylene-methaniminium cation system [as in Fig. 2(B)]. The reaction coordinate is as described in Eq. (25). Calculations were performed at level, with an intermolecular distance fixed at 4.5 Å. The inset is a magnified plot of the region near the curve crossing, with grids representing 0.01 (abscissa) and 0.002 hartrees (ordinate).

Potential-energy curves of the two lowest triplet states for the ethylene-methaniminium cation system [as in Fig. 2(B)]. The reaction coordinate is as described in Eq. (25). Calculations were performed at level, with an intermolecular distance fixed at 4.5 Å. The inset is a magnified plot of the region near the curve crossing, with grids representing 0.01 (abscissa) and 0.002 hartrees (ordinate).

TT energy-transfer coupling for the asymmetric ethylene-methaniminium ion system, face to face stacked, as shown in Fig. 2(B). The coupling calculated by HF-CIS (open symbols with dashed lines) and DC (closed symbols and solid lines) with (squares), DZP (circles), and (triangles) basis sets. DC results were obtained at the transition-state geometry, i.e., where the minimum CIS energy gap was found.

TT energy-transfer coupling for the asymmetric ethylene-methaniminium ion system, face to face stacked, as shown in Fig. 2(B). The coupling calculated by HF-CIS (open symbols with dashed lines) and DC (closed symbols and solid lines) with (squares), DZP (circles), and (triangles) basis sets. DC results were obtained at the transition-state geometry, i.e., where the minimum CIS energy gap was found.

Stacked pairs of hexatrienes in three different arrangements. (A) A maximum contact is allowed. Panels (B) and (C) show two configurations where the close-contact area exists only in the terminal bonds. Pairs of *all-trans* butadienes and octatetraenes in similar arrangements were also studied.

Stacked pairs of hexatrienes in three different arrangements. (A) A maximum contact is allowed. Panels (B) and (C) show two configurations where the close-contact area exists only in the terminal bonds. Pairs of *all-trans* butadienes and octatetraenes in similar arrangements were also studied.

Dependence of molecular size and intermolecular contacts for TT couplings. (A) Couplings are for a pair of butadienes (triangles), hexatrienes (inverse triangles), and octatetraenes (squares) at a number of intermolecular distances. The open symbols are for the results from a fully stacked configuration [Fig. 7(A)] and the closed symbols represent the results from the flipped configuration [Fig. 7(B)]. Data for a pair of ethylenes (open circles) were also included for comparison. (B) For a pair of hexatrienes, we show the coupling of fully stacked (open squares), partially stacked as in Fig. 7(B) (filled squares), and as in Fig. 7(C) (open diamonds) configurations. All the results are from calculations.

Dependence of molecular size and intermolecular contacts for TT couplings. (A) Couplings are for a pair of butadienes (triangles), hexatrienes (inverse triangles), and octatetraenes (squares) at a number of intermolecular distances. The open symbols are for the results from a fully stacked configuration [Fig. 7(A)] and the closed symbols represent the results from the flipped configuration [Fig. 7(B)]. Data for a pair of ethylenes (open circles) were also included for comparison. (B) For a pair of hexatrienes, we show the coupling of fully stacked (open squares), partially stacked as in Fig. 7(B) (filled squares), and as in Fig. 7(C) (open diamonds) configurations. All the results are from calculations.

Effects of orientations in TT couplings. Couplings are for a pair of side-by-side ethylenes intermolecular distances obtained by (circles) and (triangles). For comparison, the results for face-to-face ethylenes are included (crosses).

Effects of orientations in TT couplings. Couplings are for a pair of side-by-side ethylenes intermolecular distances obtained by (circles) and (triangles). For comparison, the results for face-to-face ethylenes are included (crosses).

## Tables

TT energy transfer coupling (in meV) for a pair of ethylenes [Fig. 2(A)] calculated by direct coupling (DC) or configuration-interaction singles (CIS).

TT energy transfer coupling (in meV) for a pair of ethylenes [Fig. 2(A)] calculated by direct coupling (DC) or configuration-interaction singles (CIS).

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