Two-Photon Excited Fluorescence and Molecular Reorientations in Liquid Solutions

Linus Ryderfors; Emad Mukhtar; Lennart B.-Å. Johansson

doi:10.1007/s10895-007-0189-x

Journal of Fluorescence

September 2007, Volume 17, Issue 5, pp 466–480 | Cite as

Two-Photon Excited Fluorescence and Molecular Reorientations in Liquid Solutions

Authors
Authors and affiliations

Linus Ryderfors
Emad Mukhtar
Lennart B.-Å. JohanssonEmail author

Original Paper

First Online: 21 June 2007

Received: 19 February 2007

Accepted: 28 March 2007

86 Downloads
3 Citations

Abstract

Theoretical expressions are derived that relate the two-photon excited fluorescence depolarisation experiments to the molecular symmetry and the rotational motions of fluorescent molecules. Diffusive rotational motions in liquid solvents are considered, as well as the influence of fast unresolved motions (e.g. librations). The results obtained are compared with one-photon excited fluorescence depolarisation experiments. The derived theoretical expressions can be applied for detailed analyses of the molecular rotation in solvent. Several of the results are useful for determining and assigning the components of two-photon absorption tensors.

Keywords

Two-photon excited fluorescence Two-photon excited fluorescence depolarisation Two-photon excited anisotropy Theory of rotational motion Rotational diffusion Librations Perylene

Abbreviations

D: diffusion frame coordinate system
$ D^{{{\left( C \right)}}}_{{{\text{nm}}}} {\left( {\alpha ,\beta ,\gamma } \right)} $: a Wigner rotation matrix element
(D_X, D_Y, D_Z): the diagonal elements of the diffusion tensor $\tilde D$
D(t): difference curve constructed from depolarisation experiments
L: laboratory coordinate system
M: molecular coordinate system
OPE: one-photon excitation
$ \ifmmode\expandafter\vec\else\expandafter\vecabove\fi{\mu } $: electronic transition dipole moment
Ω_TP: two-photon polarisation ratio
Ω_AB: α _AB, β _AB, β _AB denote the Eulerian angles that transform from the A to the B frame
r(t): time-resolved fluorescence anisotropy
S(t): sum curve constructed from depolarisation experiments
ρ(t): fluorescence relaxation
$ \tilde T $: two-photon absorption transition tensor
TPE: two-photon excitation

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Notes

Acknowledgement

This work was financially supported by the Swedish Research Council.

Appendix A

Orientational transformations

Vectors and tensors represented by Cartesian components are likely to appear more transparent than their irreducible representations, while the latter are much more convenient to use for orientational transformations. This becomes especially evident for transformations that involve several coordinate systems. In this work the irreducible representation is used through-out in the derivations, but the important results are always transformed into the Cartesian representation. This Appendix focuses on elaborating Eqs. 1–4. The orientational averaging of these equations concerns the transformations between the D- and L-frame using Eq. 5, and applying the solutions to the conditional probability (cf. Eq. 6). The integrals, exemplified by Eq. 8, are easy to calculate by exploiting the orthogonality of the Wigner matrix elements [31].

In this work only one-colour two-photon absorption is considered where the intermediate state is far from resonance. Under these restrictions the TP-absorption tensor $ \tilde T $ is a symmetric real Cartesian tensor [30]. The relation between the symmetric real Cartesian (Q _XX, Q _XY,...,Q _ZZ) and irreducible (Q ₀, Q _±1, Q _±2) second rank tensor components are given by [24, 38]:

$$\begin{array}{*{20}c} {Q_{0} = \frac{1}{{{\sqrt 6 }}}{\left( {3Q_{{ZZ}} - TrQ} \right)}} \\ {Q_{{ \pm 1}} = \mp {\left( {Q_{{ZX}} \pm iQ_{{ZY}} } \right)}} \\ {Q_{{ \pm 2}} = \frac{1}{2}{\left( {Q_{{XX}} - Q_{{YY}} \pm 2iQ_{{XY}} } \right)}} \\ \end{array} $$

(23)

The orientational transformation of the irreducible tensor components of the rank K from a coordinate system A to another one B, by using the Eulerian angles (Ω _AB = α _AB, β _AB, γ _AB) is given by [31]:

$$Q^{B}_{{Kq}} = {\sum\limits_{q'} {D^{{{\left( 2 \right)}*}}_{{q{q}\ifmmode{'}\else$'$\fi}} {\left( {\Omega _{{AB}} } \right)}Q^{A}_{{K{q}\ifmmode{'}\else$'$\fi}} } }$$

(24)

The equations used to describe the sum and difference curves that correspond to TPE by linearly and circularly polarised light are derived from Eqs. 2 and 3. In the former case, the excitation polarisation with respect to the Cartesian laboratory coordinate systems is $ \hat \lambda _{ex} = \left( {0,0,1} \right) $, whereas the parallel and perpendicular settings of the emission polariser are $ \hat \lambda _{em} = \left( {0,0,1} \right)$ and $ \hat \lambda _{em} = \left( {1,0,0} \right)$, respectively.

$$S^{l} {\left( t \right)} = {\left\langle {{\left\{ {T^{L}_{{ZZ}} {\left( 0 \right)}} \right\}}^{2} {\left\{ {M^{L}_{{ZZ}} {\left( t \right)} + 2M^{L}_{{XX}} {\left( t \right)}} \right\}}} \right\rangle }\rho {\left( t \right)}$$

(25a)

$$D^{l} {\left( t \right)} = {\left\langle {{\left\{ {T^{L}_{{ZZ}} {\left( 0 \right)}} \right\}}^{2} {\left\{ {M^{L}_{{ZZ}} {\left( t \right)} - M^{L}_{{XX}} {\left( t \right)}} \right\}}} \right\rangle }\rho {\left( t \right)}$$

(25b)

In the evaluation of the correlation functions that appear in Eqs. 25a and 25b, it is recommended to transform these into the basis of irreducible tensors, which leads to:

$$S^{l} {\left( t \right)} = \frac{1}{9}{\left[ {{\left\{ {Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}} \right\}}^{2} + 6{\left\langle {{\left\{ {T^{L}_{0} {\left( 0 \right)}} \right\}}^{2} } \right\rangle }} \right]}Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{M}\rho {\left( t \right)}$$

(26a)

$$D^{l} {\left( t \right)} = \frac{1}{9}{\left[ {6Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}{\left\langle {T^{L}_{0} {\left( 0 \right)}M^{L}_{0} {\left( t \right)}} \right\rangle } + 3{\sqrt 6 }{\left\langle {{\left\{ {T^{L}_{0} {\left( 0 \right)}} \right\}}^{2} M^{L}_{0} {\left( t \right)}} \right\rangle }} \right]}\rho {\left( t \right)}$$

(26b)

The corresponding properties obtained when using the circularly polarised light for excitation, is derived in a similar manner. Then $ \hat \lambda _{ex} = \left( {{1 \mathord{\left/ {\vphantom {1 {\sqrt 2 ,}}} \right. \kern-\nulldelimiterspace} {\sqrt 2 ,}} \pm {i \mathord{\left/ {\vphantom {i {\sqrt 2 ,0}}} \right. \kern-\nulldelimiterspace} {\sqrt 2 ,0}}} \right) $ and the parallel and perpendicular settings of the emission polariser are $ \hat \lambda _{em} = \left( {0,0,1} \right) $ and $ \hat \lambda _{em} = \left( {1,0,0} \right)$, respectively. This leads to:

$$S^{c} {\left( t \right)} = {\left\langle {T^{L}_{2} {\left( 0 \right)}T^{L}_{{ - 2}} {\left( 0 \right)}} \right\rangle }Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{M}\rho {\left( t \right)}$$

(27a)

$$D^{c} {\left( t \right)} = {\sqrt {\frac{3}{2}} }{\left\langle {T^{L}_{2} {\left( 0 \right)}T^{L}_{{ - 2}} {\left( 0 \right)}M^{L}_{0} {\left( t \right)}} \right\rangle }\rho {\left( t \right)}$$

(27b)

The orientational averaging of Eqs. 26a–27b follows much the same route and therefore the details are only presented for Eqs. 26a and 26b, i.e. for the sum and difference decays obtained using the linearly polarised excitation. Since the trace of a tensor is invariant to orientational transformations, the averaging only involves the components of $ \tilde T $, which are of the first and second order. The first order term reads:

$$ {\left\langle {T^{L}_{0} {\left( 0 \right)}M^{L}_{0} {\left( t \right)}} \right\rangle } = \frac{1} {5}{\sum\limits_{i,m,q} {e^{{ - E_{i} t}} a^{{{\left( 2 \right)}}}_{{i,m}} } }a^{{{\left( 2 \right)}}}_{{i, - q}} T^{D}_{m} M^{D}_{q} {\left( { - 1} \right)}^{q} $$

(28)

For brevity in the notation used, the times of the absorption (t = 0) and emission (t = t) events are not explicitly indicated. When evaluating the expansion coefficients $ a_{i,l}^{(K)} $ the relation a ² + b ² = N ² was used [32] and the following normalisation of the tensor components was introduced: {μ _X}² + {μ _Y}² + {μ _Z}² = 1 and {T _XX}² + {T _YY}² + {T _ZZ}² + 2{T _XY}² + 2{T _XZ}² + 2{T _YZ}² = 1. By using these criteria Eq. 28 can be written as;

$$\begin{array}{*{20}l} {{{\left\langle {T^{L}_{0} {\left( 0 \right)}M^{L}_{0} {\left( t \right)}} \right\rangle } = \frac{1}{{10}}e^{{ - E_{2} t}} {\left( {\beta _{1} + \alpha _{1} } \right)} + \frac{1}{{10}}e^{{ - E_{0} t}} {\left( {\beta _{1} - \alpha _{1} } \right)} + } \hfill} \\ {{ - {\left( {\frac{1}{{10}}} \right)}{\left( {T^{D}_{1} + T^{D}_{{ - 1}} } \right)}{\left( {M^{D}_{1} + M^{D}_{{ - 1}} } \right)}e^{{ - E_{1} t}} + {\left( {\frac{1}{{10}}} \right)}{\left( {T^{D}_{1} - T^{D}_{{ - 1}} } \right)}{\left( {M^{D}_{1} - M^{D}_{{ - 1}} } \right)}e^{{ - E_{{ - 1}} t}} + } \hfill} \\ {{ - {\left( {\frac{1}{{10}}} \right)}{\left( {T^{D}_{2} - T^{D}_{{ - 2}} } \right)}{\left( {M^{D}_{2} - M^{D}_{{ - 2}} } \right)}e^{{ - E_{{ - 2}} t}} } \hfill} \\ \end{array} $$

(29a)

Here,

$$\beta _{1} = \frac{1}{2}{\left( {T^{D}_{2} + T^{D}_{{ - 2}} } \right)}{\left( {M^{D}_{2} + M^{D}_{{ - 2}} } \right)} + T^{D}_{0} M^{D}_{0} $$

(29b)

and

$$\begin{array}{*{20}c} {\alpha _1 = \frac{{b^2 - a^2 }}{{2N^2 }}\left[ {\left( {T_2^D + T_{ - 2}^D } \right)\left( {M_2^D + M_{ - 2}^D } \right) - 2T_0^D M_0^D } \right] + } \hfill \\ { + \frac{{\sqrt 2 ab}}{{N^2 }}\left[ {\left( {T_2^D + T_{ - 2}^D } \right)M_0^D + T_0^D \left( {M_2^D + M_{ - 2}^D } \right)} \right]} \hfill \\ \end{array} $$

(29c)

By converting the irreducible tensor components into Cartesian ones, the right-hand side of Eq. 29a reads,

$$\begin{array}{*{20}c} {\left\langle {T_0^L \left( 0 \right)M_0^L \left( t \right)} \right\rangle = \frac{1}{{10}}e^{ - E_2 t} \left( {\beta _1 + \alpha _1 } \right) + \frac{1}{{10}}e^{ - E_0 t} \left( {\beta _1 - \alpha _1 } \right) + } \hfill \\ { + \left( {\frac{2}{5}} \right)T_{XY}^D \mu _X^D \mu _Y^D e^{ - E_{ - 2} t} + \left( {\frac{2}{5}} \right)T_{YZ}^D \mu _Y^D \mu _Z^D e^{ - E_1 t} + \left( {\frac{2}{5}} \right)T_{XZ}^D \mu _X^D \mu _Z^D e^{ - E_{ - 1} t} } \hfill \\ \end{array} $$

(30a)

in which

$$\beta _{1} = T^{D}_{{XX}} {\left( {\mu ^{D}_{X} } \right)}^{2} + T^{D}_{{YY}} {\left( {\mu ^{D}_{Y} } \right)}^{2} + T^{D}_{{ZZ}} {\left( {\mu ^{D}_{Z} } \right)}^{2} - \frac{1}{3}Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}$$

(30b)

$$\begin{array}{*{20}l} {{\alpha _{1} = \frac{{D_{X} }}{\Delta }{\left( {T^{D}_{{YY}} {\left( {\mu ^{D}_{Y} } \right)}^{2} + T^{D}_{{ZZ}} {\left( {\mu ^{D}_{Z} } \right)}^{2} - 2T^{D}_{{XX}} {\left( {\mu ^{D}_{X} } \right)}^{2} + T^{D}_{{XX}} + {\left( {Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}} \right)}\mu ^{D}_{X} - \frac{2}{3}Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}} \right)} + } \hfill} \\ {{\frac{{D_{Y} }}{\Delta }{\left( {T^{D}_{{ZZ}} {\left( {\mu ^{D}_{Z} } \right)}^{2} + T^{D}_{{XX}} {\left( {\mu ^{D}_{X} } \right)}^{2} - 2T^{D}_{{YY}} {\left( {\mu ^{D}_{Y} } \right)}^{2} + T^{D}_{{YY}} + {\left( {Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}} \right)}\mu ^{D}_{Y} - \frac{2}{3}Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}} \right)} + } \hfill} \\ {{\frac{{D_{Z} }}{\Delta }{\left( {T^{D}_{{XX}} {\left( {\mu ^{D}_{X} } \right)}^{2} + T^{D}_{{YY}} {\left( {\mu ^{D}_{Y} } \right)}^{2} - 2T^{D}_{{ZZ}} {\left( {\mu ^{D}_{Z} } \right)}^{2} + T^{D}_{{ZZ}} + {\left( {Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}} \right)}\mu ^{D}_{Z} - \frac{2}{3}Tr \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}} \right)}} \hfill} \\ \end{array} $$

(30c)

By evaluating in Eq. 26b the term of the second order in components of $ \tilde T $, one obtains that

$$\begin{array}{*{20}l} {{{\left\langle {{\left\{ {T^{L}_{0} {\left( 0 \right)}} \right\}}^{2} M^{L}_{0} {\left( t \right)}} \right\rangle } = } \hfill} \\ {{{\sum\limits_{i,m,{m}\ifmmode{'}\else$'$\fi,q} {e^{{ - E_{i} t}} {\left( \begin{aligned} & \begin{array}{*{20}c} {2} & {2} & {2} \\ \end{array} \\ & \begin{array}{*{20}c} {0} & {0} & {0} \\ \end{array} \\ \end{aligned} \right)}{\left( {\begin{array}{*{20}c} {2} & {2} & {2} \\ {{ - m}} & {{ - {m}\ifmmode{'}\else$'$\fi}} & {{m + {m}\ifmmode{'}\else$'$\fi}} \\ \end{array} } \right)}a^{{{\left( 2 \right)}}}_{{i,m + {m}\ifmmode{'}\else$'$\fi}} a^{{{\left( 2 \right)}}}_{{i, - q}} T^{D}_{m} T^{D}_{{{m}\ifmmode{'}\else$'$\fi}} M^{D}_{q} {\left( - \right)}^{{m + {m}\ifmmode{'}\else$'$\fi + q}} } }} \hfill} \\ \end{array} $$

(31)

The matrices are 3-j symbols which are tabulated elsewhere [31]. By working out the sum in Eq. 31 this equation reads:

$$\begin{array}{*{20}l} {{{\left\langle {{\left\{ {T^{L}_{0} {\left( 0 \right)}} \right\}}^{2} M^{L}_{0} {\left( t \right)}} \right\rangle } = e^{{ - E_{2} t}} {\left( {\beta _{2} + \alpha _{2} } \right)} + e^{{ - E_{0} t}} {\left( {\beta _{2} - \alpha _{2} } \right)} + } \hfill} \\ {{ + {\left( {\frac{1}{{35}}} \right)}{\left[ {2T^{D}_{0} {\left( {T^{D}_{2} - T^{D}_{{ - 2}} } \right)} - {\sqrt {\frac{3}{2}} }{\left( {T^{D}_{1} T^{D}_{1} - T^{D}_{{ - 1}} T^{D}_{{ - 1}} } \right)}} \right]}{\left( {M^{D}_{2} - M^{D}_{{ - 2}} } \right)}e^{{ - E_{{ - 2}} t}} + } \hfill} \\ {{ - {\left( {\frac{1}{{35}}} \right)}{\left[ {T^{D}_{0} {\left( {T^{D}_{1} + T^{D}_{{ - 1}} } \right)} - {\sqrt 6 }{\left( {T^{D}_{2} T^{D}_{{ - 1}} + T^{D}_{1} T^{D}_{{ - 2}} } \right)}} \right]}{\left( {M^{D}_{1} + M^{D}_{{ - 1}} } \right)}e^{{ - E_{1} t}} + } \hfill} \\ {{ + {\left( {\frac{1}{{35}}} \right)}{\left[ {T^{D}_{0} {\left( {T^{D}_{1} - T^{D}_{{ - 1}} } \right)} - {\sqrt 6 }{\left( {T^{D}_{2} T^{D}_{{ - 1}} - T^{D}_{1} T^{D}_{{ - 2}} } \right)}} \right]}{\left( {M^{D}_{1} - M^{D}_{{ - 1}} } \right)}e^{{ - E_{{ - 1}} t}} } \hfill} \\ \end{array} $$

(32a)

in which,

$$\beta _{2} = - \frac{1}{{70}}{\left[ {{\left\{ {2{\left( {T^{D}_{2} + T^{D}_{{ - 2}} } \right)}T^{D}_{0} - {\sqrt {\frac{3}{2}} }{\left( {T^{D}_{1} T^{D}_{1} + T^{D}_{{ - 1}} T^{D}_{{ - 1}} } \right)}} \right\}}{\left( {M^{D}_{2} + M^{D}_{{ - 2}} } \right)} + 2{\left\{ {2T^{D}_{2} T^{D}_{{ - 2}} - {\left( {T^{D}_{0} } \right)}^{2} + T^{D}_{1} T^{D}_{{ - 1}} } \right\}}M^{D}_{0} } \right]}$$

(32b)

and

$$\begin{array}{*{20}l} {{\alpha _{2} = - \frac{1}{{35}}\left[ {\frac{{b^{2} - a^{2} }}{{2N^{2} }}{\left( {{\left\{ {2{\left( {T^{D}_{2} + T^{D}_{{ - 2}} } \right)}T^{D}_{0} - {\sqrt {\frac{3}{2}} }{\left( {T^{D}_{1} T^{D}_{1} + T^{D}_{{ - 1}} T^{D}_{{ - 1}} } \right)}} \right\}}{\left( {M^{D}_{2} + M^{D}_{{ - 2}} } \right)} - 2{\left\{ {2T^{D}_{2} T^{D}_{{ - 2}} - {\left( {T^{D}_{0} } \right)}^{2} + T^{D}_{1} T^{D}_{{ - 1}} } \right\}}M^{D}_{0} } \right)} + } \right.} \hfill} \\ {{\left. { + \frac{{{\sqrt 2 }ab}}{{N^{2} }}{\left( {{\left\{ {2{\left( {T^{D}_{2} + T^{D}_{{ - 2}} } \right)}T^{D}_{0} - {\sqrt {\frac{3}{2}} }{\left( {T^{D}_{1} T^{D}_{1} + T^{D}_{{ - 1}} T^{D}_{{ - 1}} } \right)}} \right\}}M^{D}_{0} + {\left\{ {2T^{D}_{2} T^{D}_{{ - 2}} - {\left( {T^{D}_{0} } \right)}^{2} + T^{D}_{1} T^{D}_{{ - 1}} } \right\}}{\left( {M^{D}_{2} + M^{D}_{{ - 2}} } \right)}} \right)}} \right]} \hfill} \\ \end{array} $$

(32c)

When applying these equations it might be more convenient have Eqs. 32b and 32c expressed in the Cartesian components. The results are then given by,

$$\begin{array}{*{20}l} {{\beta _{2} = - \frac{2}{{35}}\frac{1}{{3{\sqrt 6 }}}\left[ {{\left( {T^{D}_{{XX}} } \right)}^{2} {\left\{ {1 - 3{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{YY}} } \right)}^{2} {\left\{ {1 - 3{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} {\left\{ {1 - 3{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + } \right.} \hfill} \\ {{ + T^{D}_{{XX}} T^{D}_{{YY}} {\left\{ {2 - 6{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + T^{D}_{{XX}} T^{D}_{{ZZ}} {\left\{ {2 - 6{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + T^{D}_{{YY}} T^{D}_{{ZZ}} {\left\{ {2 - 6{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + } \hfill} \\ {{\left. { + {\left( {T^{D}_{{XY}} } \right)}^{2} {\left\{ { - 3 + 9{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{XZ}} } \right)}^{2} {\left\{ { - 3 + 9{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{YZ}} } \right)}^{2} {\left\{ { - 3 + 9{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}}} \right]} \hfill} \\ \end{array} $$

(33a)

$$\alpha _2 = - \frac{2}{{35}}\frac{1}{{3\sqrt 6 }}\left[ {\frac{{D_X }}{\Delta }\xi _X + \frac{{D_Y }}{\Delta }\xi _Y + \frac{{D_Z }}{\Delta }\xi _Z } \right]$$

(33b)

$$\begin{array}{*{20}l} {{\xi _{X} = {\left( {T^{D}_{{XX}} } \right)}^{2} {\left\{ {3{\left( {\mu ^{D}_{X} } \right)}^{2} - 1} \right\}} + {\left( {T^{D}_{{YY}} } \right)}^{2} {\left\{ {3{\left( {\mu ^{D}_{Z} } \right)}^{2} - 1} \right\}} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} {\left\{ {3{\left( {\mu ^{D}_{Y} } \right)}^{2} - 1} \right\}} + } \hfill} \\ {{ + T^{D}_{{XX}} T^{D}_{{YY}} {\left\{ {6{\left( {\mu ^{D}_{Y} } \right)}^{2} - 2} \right\}} + T^{D}_{{XX}} T^{D}_{{ZZ}} {\left\{ {6{\left( {\mu ^{D}_{Z} } \right)}^{2} - 2} \right\}} + T^{D}_{{YY}} T^{D}_{{ZZ}} {\left\{ {6{\left( {\mu ^{D}_{X} } \right)}^{2} - 2} \right\}} + } \hfill} \\ {{ + {\left( {T^{D}_{{XY}} } \right)}^{2} {\left\{ { - 9{\left( {\mu ^{D}_{Y} } \right)}^{2} + 3} \right\}} + {\left( {T^{D}_{{XZ}} } \right)}^{2} {\left\{ { - 9{\left( {\mu ^{D}_{Z} } \right)}^{2} + 3} \right\}} + {\left( {T^{D}_{{YZ}} } \right)}^{2} {\left\{ { - 9{\left( {\mu ^{D}_{X} } \right)}^{2} + 3} \right\}}} \hfill} \\ \end{array} $$

$$\begin{array}{*{20}l} {{\xi _{Y} = {\left( {T^{D}_{{YY}} } \right)}^{2} {\left\{ {3{\left( {\mu ^{D}_{Y} } \right)}^{2} - 1} \right\}} + {\left( {T^{D}_{{XX}} } \right)}^{2} {\left\{ {3{\left( {\mu ^{D}_{Z} } \right)}^{2} - 1} \right\}} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} {\left\{ {3{\left( {\mu ^{D}_{X} } \right)}^{2} - 1} \right\}} + } \hfill} \\ {{ + T^{D}_{{XX}} T^{D}_{{YY}} {\left\{ {6{\left( {\mu ^{D}_{X} } \right)}^{2} - 2} \right\}} + T^{D}_{{XX}} T^{D}_{{ZZ}} {\left\{ {6{\left( {\mu ^{D}_{Y} } \right)}^{2} - 2} \right\}} + T^{D}_{{YY}} T^{D}_{{ZZ}} {\left\{ {6{\left( {\mu ^{D}_{Z} } \right)}^{2} - 2} \right\}} + } \hfill} \\ {{ + {\left( {T^{D}_{{XY}} } \right)}^{2} {\left\{ { - 9{\left( {\mu ^{D}_{X} } \right)}^{2} + 3} \right\}} + {\left( {T^{D}_{{XZ}} } \right)}^{2} {\left\{ { - 9{\left( {\mu ^{D}_{Y} } \right)}^{2} + 3} \right\}} + {\left( {T^{D}_{{YZ}} } \right)}^{2} {\left\{ { - 9{\left( {\mu ^{D}_{Z} } \right)}^{2} + 3} \right\}}} \hfill} \\ \end{array} $$

(33c)

$$\begin{array}{*{20}c} {\xi _Z = \left( {T_{ZZ}^D } \right)^2 \left\{ {3\left( {\mu _Z^D } \right)^2 - 1} \right\} + \left( {T_{XX}^D } \right)^2 \left\{ {3\left( {\mu _Y^D } \right)^2 - 1} \right\} + \left( {T_{YY}^D } \right)^2 \left\{ {3\left( {\mu _X^D } \right)^2 - 1} \right\} + } \hfill \\ { + T_{XX}^D T_{YY}^D \left\{ {6\left( {\mu _Z^D } \right)^2 - 2} \right\} + T_{XX}^D T_{ZZ}^D \left\{ {6\left( {\mu _X^D } \right)^2 - 2} \right\} + T_{YY}^D T_{ZZ}^D \left\{ {6\left( {\mu _Y^D } \right)^2 - 2} \right\} + } \hfill \\ { + \left( {T_{XY}^D } \right)^2 \left\{ { - 9\left( {\mu _Z^D } \right)^2 + 3} \right\} + \left( {T_{XZ}^D } \right)^2 \left\{ { - 9\left( {\mu _X^D } \right)^2 + 3} \right\} + \left( {T_{YZ}^D } \right)^2 \left\{ { - 9\left( {\mu _Y^D } \right)^2 + 3} \right\}} \hfill \\ \end{array} $$

To evaluate Eq. 26a, one only needs to work out the following static average,

$$\begin{array}{*{20}l} {{{\left\langle {{\left\{ {T^{{_{L} }}_{0} {\left( 0 \right)}} \right\}}^{2} } \right\rangle } = \frac{1}{5}{\sum\limits_m {T^{D}_{m} T^{D}_{{ - m}} {\left( - \right)}^{m} = \frac{1}{5}{\left[ {2T^{D}_{2} T^{D}_{{ - 2}} - 2T^{D}_{1} T^{D}_{{ - 1}} + T^{D}_{0} T^{D}_{0} } \right]}} } = } \hfill} \\ {{ = \frac{1}{5}\frac{1}{6}{\left[ {4{\left\{ {{\left( {T^{D}_{{XX}} } \right)}^{2} + {\left( {T^{D}_{{YY}} } \right)}^{2} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} - T^{D}_{{XX}} T^{D}_{{YY}} - T^{D}_{{XX}} T^{D}_{{ZZ}} - T^{D}_{{YY}} T^{D}_{{ZZ}} } \right\}} + 12{\left\{ {{\left( {T^{D}_{{XY}} } \right)}^{2} + {\left( {T^{D}_{{XZ}} } \right)}^{2} + {\left( {T^{D}_{{YZ}} } \right)}^{2} } \right\}}} \right]}} \hfill} \\ \end{array} $$

(34)

Finally, we comment on the case of using the circular polarised excitation and evaluate the corresponding equations, i.e. Eqs. 27a and 27b. Considering the averages involved one observes that

$${\left\langle {T^{L}_{2} {\left( 0 \right)}T^{L}_{{ - 2}} {\left( 0 \right)}M^{L}_{0} {\left( t \right)}} \right\rangle } = - {\left\langle {{\left\{ {T^{L}_{0} {\left( 0 \right)}} \right\}}^{2} M^{L}_{0} {\left( t \right)}} \right\rangle }$$

(35a)

and

$${\left\langle {T^{L}_{2} {\left( 0 \right)}T^{L}_{{ - 2}} {\left( 0 \right)}} \right\rangle } = {\left\langle {{\left\{ {T^{L}_{0} {\left( 0 \right)}} \right\}}^{2} } \right\rangle }$$

(35b)

Now making use of the Eqs. 30a–30c, and 33a–35b in the sum and difference curves given by Eqs. 26a–27b, as well as the definition of the TPE-anisotropy, one obtains the final and complete expressions for r ^l(t) and r ^c(t), which are presented in Appendix B.

Appendix B

Coefficients of r ^l(t)

The relation between the pre-exponential factors and the absorption and emission tensor components, in the five-exponential general expression for the anisotropy in Eq. 9, is given by:

$$\begin{array}{*{20}l} {{\beta ^{l} = {\left( {T^{D}_{{XX}} } \right)}^{2} {\left\{ {9{\left( {\mu ^{D}_{X} } \right)}^{2} - 3} \right\}} + {\left( {T^{D}_{{YY}} } \right)}^{2} {\left\{ {9{\left( {\mu ^{D}_{Y} } \right)}^{2} - 3} \right\}} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} {\left\{ {9{\left( {\mu ^{D}_{Z} } \right)}^{2} - 3} \right\}} + } \hfill} \\ {{ + T^{D}_{{XX}} T^{D}_{{YY}} {\left\{ {1 - 3{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + T^{D}_{{XX}} T^{D}_{{ZZ}} {\left\{ {1 - 3{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + T^{D}_{{YY}} T^{D}_{{ZZ}} {\left\{ {1 - 3{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + } \hfill} \\ {{ + {\left( {T^{D}_{{XY}} } \right)}^{2} {\left\{ {2 - 6{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{XZ}} } \right)}^{2} {\left\{ {2 - 6{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{YZ}} } \right)}^{2} {\left\{ {2 - 6{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}}} \hfill} \\ \end{array} $$

(36)

$$\alpha ^{l} = \frac{{D_{X} }}{\Delta }\xi ^{l}_{X} + \frac{{D_{Y} }}{\Delta }\xi ^{l}_{Y} + \frac{{D_{Z} }}{\Delta }\xi ^{l}_{Z} $$

(37)

$$\begin{array}{*{20}l} {{\xi ^{l}_{X} = {\left( {T^{D}_{{XX}} } \right)}^{2} {\left\{ {3 - 9{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{YY}} } \right)}^{2} {\left\{ {3 - 9{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} {\left\{ {3 - 9{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + } \hfill} \\ {{ + T^{D}_{{XX}} T^{D}_{{YY}} {\left\{ { - 1 + 3{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + T^{D}_{{XX}} T^{D}_{{ZZ}} {\left\{ { - 1 + 3{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + T^{D}_{{YY}} T^{D}_{{ZZ}} {\left\{ { - 1 + 3{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + } \hfill} \\ {{ + {\left( {T^{D}_{{XY}} } \right)}^{2} {\left\{ {6{\left( {\mu ^{D}_{Y} } \right)}^{2} - 2} \right\}} + {\left( {T^{D}_{{XZ}} } \right)}^{2} {\left\{ {6{\left( {\mu ^{D}_{Z} } \right)}^{2} - 2} \right\}} + {\left( {T^{D}_{{YZ}} } \right)}^{2} {\left\{ {6{\left( {\mu ^{D}_{X} } \right)}^{2} - 2} \right\}}} \hfill} \\ \end{array} $$

(38a)

$$\begin{array}{*{20}l} {{\xi ^{l}_{Y} = {\left( {T^{D}_{{YY}} } \right)}^{2} {\left\{ {3 - 9{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{XX}} } \right)}^{2} {\left\{ {3 - 9{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} {\left\{ {3 - 9{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + } \hfill} \\ {{ + T^{D}_{{XX}} T^{D}_{{YY}} {\left\{ { - 1 + 3{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + T^{D}_{{XX}} T^{D}_{{ZZ}} {\left\{ { - 1 + 3{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + T^{D}_{{YY}} T^{D}_{{ZZ}} {\left\{ { - 1 + 3{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + } \hfill} \\ {{\; + {\left( {T^{D}_{{XY}} } \right)}^{2} {\left\{ {6{\left( {\mu ^{D}_{X} } \right)}^{2} - 2} \right\}} + {\left( {T^{D}_{{XZ}} } \right)}^{2} {\left\{ {6{\left( {\mu ^{D}_{Y} } \right)}^{2} - 2} \right\}} + {\left( {T^{D}_{{YZ}} } \right)}^{2} {\left\{ {6{\left( {\mu ^{D}_{Z} } \right)}^{2} - 2} \right\}}} \hfill} \\ \end{array} $$

(38b)

$$\begin{array}{*{20}l} {{\xi ^{l}_{Z} = {\left( {T^{D}_{{ZZ}} } \right)}^{2} {\left\{ {3 - 9{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{XX}} } \right)}^{2} {\left\{ {3 - 9{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{YY}} } \right)}^{2} {\left\{ {3 - 9{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + } \hfill} \\ {{ + T^{D}_{{XX}} T^{D}_{{YY}} {\left\{ { - 1 + 3{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + T^{D}_{{XX}} T^{D}_{{ZZ}} {\left\{ { - 1 + 3{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + T^{D}_{{YY}} T^{D}_{{ZZ}} {\left\{ { - 1 + 3{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + } \hfill} \\ {{ + {\left( {T^{D}_{{XY}} } \right)}^{2} {\left\{ {6{\left( {\mu ^{D}_{Z} } \right)}^{2} - 2} \right\}} + {\left( {T^{D}_{{XZ}} } \right)}^{2} {\left\{ {6{\left( {\mu ^{D}_{X} } \right)}^{2} - 2} \right\}} + {\left( {T^{D}_{{YZ}} } \right)}^{2} {\left\{ {6{\left( {\mu ^{D}_{Y} } \right)}^{2} - 2} \right\}}} \hfill} \\ \end{array} $$

(38c)

$$\delta ^{l} = 3{\left\{ {{\left( {T^{D}_{{XX}} } \right)}^{2} + {\left( {T^{D}_{{YY}} } \right)}^{2} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} } \right\}} + 2{\left\{ {T^{D}_{{XX}} T^{D}_{{YY}} + T^{D}_{{XX}} T^{D}_{{ZZ}} + T^{D}_{{YY}} T^{D}_{{ZZ}} } \right\}} + 4{\left\{ {{\left( {T^{D}_{{XY}} } \right)}^{2} + {\left( {T^{D}_{{XZ}} } \right)}^{2} + {\left( {T^{D}_{{YZ}} } \right)}^{2} } \right\}}$$

(39)

$$\gamma ^{l}_{{ - 2}} = \frac{{T^{D}_{{XY}} {\left( {3T^{D}_{{XX}} + 3T^{D}_{{YY}} + T^{D}_{{ZZ}} } \right)} + 2T^{D}_{{XZ}} T^{D}_{{YZ}} }}{{\delta ^{l} }}\mu ^{D}_{X} \mu ^{D}_{Y} $$

(40a)

$$\gamma ^{l}_{1} = \frac{{T^{D}_{{YZ}} {\left( {3T^{D}_{{YY}} + 3T^{D}_{{ZZ}} + T^{D}_{{XX}} } \right)} + 2T^{D}_{{XY}} T^{D}_{{XZ}} }}{{\delta ^{l} }}\mu ^{D}_{Y} \mu ^{D}_{Z} $$

(40b)

$$\gamma ^{l}_{{ - 1}} = \frac{{T^{D}_{{XZ}} {\left( {3T^{D}_{{XX}} + 3T^{D}_{{ZZ}} + T^{D}_{{YY}} } \right)} + 2T^{D}_{{XY}} T^{D}_{{YZ}} }}{{\delta ^{l} }}\mu ^{D}_{X} \mu ^{D}_{Z} $$

(40c)

Coefficients of r ^c(t)

The relation between the pre-exponential factors in Eq. 14 and the absorption and emission tensor components are given by:

$$\begin{array}{*{20}l} {{\beta ^{c} = {\left( {T^{D}_{{XX}} } \right)}^{2} {\left\{ {1 - 3{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{YY}} } \right)}^{2} {\left\{ {1 - 3{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} {\left\{ {1 - 3{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}}} \hfill} \\ {{\quad \;\,T^{D}_{{XX}} T^{D}_{{YY}} {\left\{ {2 - 6{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + T^{D}_{{XX}} T^{D}_{{ZZ}} {\left\{ {2 - 6{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + T^{D}_{{YY}} T^{D}_{{ZZ}} {\left\{ {2 - 6{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}} + } \hfill} \\ {{\quad {\left( {T^{D}_{{XY}} } \right)}^{2} {\left\{ { - 3 + 9{\left( {\mu ^{D}_{Z} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{XZ}} } \right)}^{2} {\left\{ { - 3 + 9{\left( {\mu ^{D}_{Y} } \right)}^{2} } \right\}} + {\left( {T^{D}_{{YZ}} } \right)}^{2} {\left\{ { - 3 + 9{\left( {\mu ^{D}_{X} } \right)}^{2} } \right\}}} \hfill} \\ \end{array} $$

(41)

$$\alpha ^{c} = \frac{{D_{X} }}{\Delta }\xi ^{c}_{X} + \frac{{D_{Y} }}{\Delta }\xi ^{c}_{Y} + \frac{{D_{Z} }}{\Delta }\xi ^{c}_{Z} $$

(42)

with the $ \xi ^{c}_{j} = \xi _{j} $ for j = X, Y, Z given by Eq. 33c. Furthermore

$$\delta ^{c} = {\left( {T^{D}_{{XX}} } \right)}^{2} + {\left( {T^{D}_{{YY}} } \right)}^{2} + {\left( {T^{D}_{{ZZ}} } \right)}^{2} - {\left\{ {T^{D}_{{XX}} T^{D}_{{YY}} + T^{D}_{{XX}} T^{D}_{{ZZ}} + T^{D}_{{YY}} T^{D}_{{ZZ}} } \right\}} + 3{\left\{ {{\left( {T^{D}_{{XY}} } \right)}^{2} + {\left( {T^{D}_{{XZ}} } \right)}^{2} + {\left( {T^{D}_{{YZ}} } \right)}^{2} } \right\}}$$

(43)

$$\gamma ^{c}_{{ - 2}} = \frac{{T^{D}_{{XY}} {\left( {2T^{D}_{{ZZ}} - T^{D}_{{XX}} - T^{D}_{{YY}} } \right)} - 3T^{D}_{{XZ}} T^{D}_{{YZ}} }}{{\delta ^{c} }}\mu ^{D}_{X} \mu ^{D}_{Y} $$

(44a)

$$\gamma ^{c}_{1} = \frac{{T^{D}_{{YZ}} {\left( {2T^{D}_{{XX}} - T^{D}_{{YY}} - T^{D}_{{ZZ}} } \right)} - 3T^{D}_{{XY}} T^{D}_{{XZ}} }}{{\delta ^{c} }}\mu ^{D}_{Y} \mu ^{D}_{Z} $$

(44b)

$$\gamma ^{c}_{{ - 1}} = \frac{{T^{D}_{{XZ}} {\left( {2T^{D}_{{YY}} - T^{D}_{{XX}} - T^{D}_{{ZZ}} } \right)} - 3T^{D}_{{XY}} T^{D}_{{YZ}} }}{{\delta ^{c} }}\mu ^{D}_{X} \mu ^{D}_{Z} $$

(44c)

Appendix C

The influence of ultrafast depolarisations

The influence of ultrafast depolarisations is effectively treated as an uncertainty in the orientation of the transition vector/tensor components with respect to the D-frame. For any anisotropic molecule the orientational uncertainty can be described by the Eulerian angles Ω _MD = (α _MD, β _MD, γ _MD) and a model for the orientational distribution density f(Ω _MD). A simple model of f(Ω _MD) would be the Heaviside’s step function with $ - \alpha ^{{\max }}_{{MD}} \leqslant \alpha _{{MD}} \leqslant \alpha ^{{\max }}_{{MD}} $, $ \pi - \beta ^{{\max }}_{{MD}} \leqslant \beta _{{MD}} \leqslant \beta ^{{\max }}_{{MD}} $ and $ - \gamma ^{{\max }}_{{MD}} \leqslant \gamma _{{MD}} \leqslant \gamma ^{{\max }}_{{MD}} $, which was assumed in this work. In order to account for the depolarisation, all the equations that depend on the T ^D- and M ^D-components (cf. Eqs. 21 and 22) must be averaged with respect to f(Ω _MD). The sum-curves S ^j(t) (j = l or c) are invariant to the orientation of the coordinate system, which is to be expected since they are proportional to the orientation averaged emission. Correspondingly, it is possible to compute that Eq. 34 is invariant under the transformation from the M to the D frame. The difference curves D ^j(t) (j = l or c) are composed of two terms that are given by Eqs. 28 and 31. One of the terms (cf. Eqs. 28) is first order in $ \tilde T $-components. For arbitrary irreducible tensor components, $ T^{D}_{p} $, of an oblate ellipsoid, the superscript p = ±m symmetry implies that m = ±2 or m = 0. The other term is of second order in $ \tilde T $, generally $q':T_q^D T_{q'}^D $ with q + q′ = ±m. In the transformation from D to M, the $ \alpha _{MD}$-dependence of the Wigner rotation matrix elements is simply always $ e^{{ \pm im\alpha _{{MD}} }} $, the average of which is $ {\left| {\frac{{\sin m\alpha _{{{\text{MD}}}} }} {{m\alpha _{{{\text{MD}}}} }}} \right|} $. A second $ {\left| {\frac{{\sin m\alpha _{{{\text{MD}}}} }} {{m\alpha _{{{\text{MD}}}} }}} \right|} $-factor originates from averaging the second rank emission tensor components, $ M^{{\text{D}}}_{{\text{p}}} $. The net result is that the pre-exponentials $ r^{{\text{\kern1ptj}}}_{{\text{m}}} $ with j = l or c in the anisotropy decay r ^j(t) should be reduced by the factor $ {\left( {\frac{{\sin m\alpha _{{{\text{MD}}}} }} {{m\alpha _{{{\text{MD}}}} }}} \right)}^{2} $ in the presence of α _MD-depolarisations. To work out the β _MD-dependence of the anisotropy is more cumbersome. Considering an oblate molecule with its emission transition dipole polarised along the X ^M-axis, for which M–D orientational transformations (Eqs. 21 and 22) are carried out, the Eq. 29a–29c can then be written:

$$\begin{array}{*{20}l} {{{\left\langle {T^{L}_{0} {\left( 0 \right)}M^{L}_{0} {\left( t \right)}} \right\rangle } = \frac{1}{5}e^{{ - E_{2} t}} {\left\{ {{\left( {2 - x} \right)}{\left( {T^{M}_{2} + T^{M}_{{ - 2}} } \right)} + {\sqrt 6 }xT^{M}_{0} } \right\}}{\left\{ {1 - x \mathord{\left/ {\vphantom {x 4}} \right. \kern-\nulldelimiterspace} 4} \right\}}} \hfill} \\ {{ + \frac{1}{5}e^{{ - E_{0} t}} {\left\{ {9x{\left( {T^{M}_{2} + T^{M}_{{ - 2}} } \right)} + 3{\sqrt 6 }{\left( {1 - {3x} \mathord{\left/ {\vphantom {{3x} 2}} \right. \kern-\nulldelimiterspace} 2} \right)}T^{M}_{0} } \right\}}{\left\{ {x \mathord{\left/ {\vphantom {x 2}} \right. \kern-\nulldelimiterspace} 2 - 1} \right\}}} \hfill} \\ \end{array} $$

(45)

In Eq. 45, x denotes the average of sin² β _MD over the relevant distribution of the angle β _MD. The corresponding influence of β _MD-depolarisation on from Eq. 32a–32c leads to that;

$$\begin{array}{*{20}l} {{{\left\langle {{\left\{ {T^{L}_{0} {\left( 0 \right)}} \right\}}^{2} M^{L}_{0} {\left( t \right)}} \right\rangle } = } \hfill} \\ {{ = - \frac{1}{{35}}e^{{ - E_{2} t}} {\left\{ {{\left( {2 - x} \right)}T^{M}_{0} {\left( {T^{M}_{2} + T^{M}_{{ - 2}} } \right)} + {\sqrt 6 }x{\left\{ {2T^{M}_{2} T^{M}_{{ - 2}} - T^{M}_{0} T^{M}_{0} } \right\}}} \right\}}{\left\{ {1 - x \mathord{\left/ {\vphantom {x 4}} \right. \kern-\nulldelimiterspace} 4} \right\}} + } \hfill} \\ {{ - \frac{1}{{35}}e^{{ - E_{0} t}} {\left\{ {9xT^{M}_{0} {\left( {T^{M}_{2} + T^{M}_{{ - 2}} } \right)} + 3{\sqrt 6 }{\left( {1 - {3x} \mathord{\left/ {\vphantom {{3x} 2}} \right. \kern-\nulldelimiterspace} 2} \right)}{\left\{ {2T^{M}_{2} T^{M}_{{ - 2}} - T^{M}_{0} T^{M}_{0} } \right\}}} \right\}}{\left\{ {x \mathord{\left/ {\vphantom {x 2}} \right. \kern-\nulldelimiterspace} 2 - 1} \right\}}} \hfill} \\ \end{array} $$

(46)

The difference curve D ^l(t) for the linearly excited experiment Eq. 26b combines the terms of first (Eq. 45) and second order (Eq. 46) in $ \ifmmode\expandafter\tilde\else\expandafter\~\fi{T}$, whereas the difference curve D ^c(t) (cf. Eq. 27b) only depends on the term of second order, i.e. Eq. 46. These considerations mean that the linearly and circularly polarised anisotropy exhibit the same α _MD-dependence, but have a somewhat different dependence with respect to the β _MD-depolarisation. This is outlined in the Figs. 2, 3, 4, and 5. For an arbitrary molecular symmetry the resulting expressions for D ^j(t) (j = l or c) are too extensive to be explicitly given here.

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Authors and Affiliations

Linus Ryderfors
- 1
Emad Mukhtar
- 1
Lennart B.-Å. Johansson
- 2
Email author

1.Department of Photochemistry and Molecular ScienceUppsala UniversityUppsalaSweden
2.Department of Chemistry Biophysical ChemistryUmeå UniversityUmeåSweden

Two-Photon Excited Fluorescence and Molecular Reorientations in Liquid Solutions

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