Calculation of transient absorption - Excitation energy transfer in pheophorbide a complexes

3.7.2 Further approximations to the transfer rates

If theJ_mn(t)is assumed to be time-independent, the ratek_m→ncan be calculated in approxima-tion by utilizing the correlaapproxima-tion funcapproxima-tion [39]

C_m^app.1_→_n(t) = |J_mn|²

h² hS_DCL(t, 0)i_ther. (3.111) The respective rate calculated with this approximation will be denoted ask^app.1_m_→_n. The approx-imation is possible if the mutual orientation and distance between two chromophores mand ndoes not change dramatically during the simulation time. Since the CC structure does not change too much for a simulation time of 10 ps, the restriction to such short time scales permits this approximation.

A further approximation is possible if the energy gap fluctuations of both chromophoresmand nare uncorrelated. hS_DCL(t, 0)imay be factorized into two independent contributions. The respective rate using this second approximation (which will be denoted ask^app.2_m→n) is calculated via integrating the correlation functionC_m^app.2→n[39]:

C_m^app.2→n(t) = |J_mn|²

Obviously, the ratek^app.2_m→ncorresponds to the EET rate given in Eq. 3.101.

3.8 Calculation of transient absorption

In the following section, the formula for the computation of the transient absorption will be derived. A widespread approach is calculating the third order optical response via the respec-tive nonlinear response function (cf. [107, 108, 109]). Two dimensional spectra calculated via nonlinear response functions are shown in [110, 111].

To circumvent the computation of a function with a triple time-dependence [85] in this thesis, the transient absorption is calculated by solving the TDSE directly, including the electromag-netic field. With this approach the time-dependent polarization can be calculated directly. This polarization can then be utilized to calculate an absorption spectrum of a probe pulse, after an initial pump pulse has excited the system. The approach of directly calculating the total polar-ization has already been presented by the Geva group in [112] (for a two-level chromophore in a mono-atomic liquid). However, the P₄system in solution that will be the model system for transient absorption in this thesis is a much more complex system.

It is important to remind that the previously defined second excited state wave function of the CC always reckons two chromophores of the CC in the first excited electronic state. The double excitation of one chromophore in the complex is not allowed (additional comments have been made in Sec. 3.2).

In the current model the excitation of the CC from the electronic ground state to the first excited electronic state and the excitation from the first excited electronic state of the CC to the second excited electronic state of the CC is treated. An excitation of more than two Pheos within the complex is neglected. This approximation is reasonable, since for the utilized field strengths the total population of the second excited state of the CC is much smaller than the population of the first excited state. The population of a third excited electronic state of the CC, even if it was allowed, would be very small.

The respective TDSE is written

−i¯h∂

∂tΦ(t) = (H0+H₁+H2+H_field(t))_Φ(t). (3.113)

The CC wave function Φ(t) was defined in Eq. 3.2 and includes the electronic ground, first excited and second excited state. The respective matrix elementsH₀,H₁andH₂were explained in Sec. 3.3. The electromagnetic field Hamiltonian H_fieldwas already included in Eq. 3.24, but is given here again to improve the understandability of the following text:

H_field(t) =−

∑

superpo-sition over eleven states, one ground state, four different first excited and six different second excited electronic states. Together with the definitions of H0, H₁ and H2 (Eqs. 3.27, 3.28, 3.29) and the CC wave functionΦ(t)(Eq. 3.2), the TDSE in Eq. 3.113 can be rewritten in terms of the expansion coefficientsA_α. This gives the three equations

−i¯h∂ andAnm(t)of the same expansion coefficient.

Similar to Eq. 3.59, an ansatz to the expansion coefficients was made that eliminates the phase factor caused by the transition energy∆E_g→e, the electronic ground state interactions as well as the kinetic energy operatorsTm, all included inH₀. The ansatz is different for the electronic ground, first and second excited state expansion coefficient:

A˜₀= A₀exp

Inserting Eqs. 3.118 and 3.119 into Eq. 3.115 gives i¯h∂

3.8 Calculation of transient absorption

Inserting Eqs. 3.118, 3.119 and 3.120 into Eq. 3.116 yields i¯h ∂

Inserting Eqs. 3.119 and 3.120 into Eq. 3.117 gives

−i¯h∂

The electromagnetic field E(t), which is already included in the formulas given above, is written as

E(t) =E_pu(t)e^iΨ^pu+E_pr(t)e^iΨ^pr+c.c., (3.124) with

E_p(t) =epEp(t)e⁻^iω^p^t,p∈ {pu,pr}, (3.125) with the pump and probe beam frequencies ω_pu, ω_pr, the respective phases Ψpu, Ψpr, with the polarization vectors of the pump and the probe beam e_pu ande_prand the respective field envelopes E_pu and E_pr. The formulas for the electromagnetic field (Eqs. 3.124 and 3.125) are inserted into the Eqs. 3.121, 3.122 ,3.123 for the expansion coefficients ˜Aα(t). At this point the so-called rotating wave approximation [113] is applied. All terms that include an exponential function of the form exp(2iω_pt)_{or exp}(−2iω_pt)(those terms oscillating with high frequency) are neglected [114].

3.8.1 Time-dependent polarization

After solving Eq. 3.113, the calculated time-dependent wave function Φ(t)can be utilized to calculate the time-dependent dipole moment expectation value defined as [85]

d(t,E) =hh_Φ(R(t),t,E)|µˆ|_Φ(R(t),t,E)ii_ther =hh_Φ(t)|µˆ|_Φ(t)ii_ther. (3.126) Thet- andE-dependence inΦ(t) =_Φ(R(t),t,E)denotes that the calculated expansion coeffi-cients ˜Aα(τ)(cf. Eq. 3.2 for the CC wave function) at timet = τdepend directly on the actual MD trajectory and the field envelope betweent = 0 andt = τ. The thermal averageh...i_ther is carried out with respect to the equilibrium distribution of nuclear coordinates. In terms of the expansion coefficients ˜Aα(t), the dipole moment expectation value reads [85]

d(t,E) =h

∑

α,β

dαβA˜^∗_α(t,E)A^˜_β(t,E)i_ther. (3.127) The polarization (dipole density) is then calculated as

P(t,E) =n_CCd(t,E), (3.128) with the CC density n_CC. However, to calculate transient absorption spectra, the probe pulse polarization in signal direction has to be derived from the overall polarization d(t,E)_{. In the} next step, two different methods will be presented, which allow to carry out this calculation.

3.8.2 Signal polarization, method I

Dohmke and co-workers proposed a method [113] that will be mainly utilized within this the-sis. According to the two beams in Eq. 3.124, a power expansion with respect to the polarization at some pointris carried out. This power expansion can be written as [113, 115]

P(r,t) =

∑

m,n

eⁱ⁽^mk^pu^r⁺^nk^pr^r⁾P⁽^mn⁾(t). (3.129) Here, for once,mandndo not count chromophores, they run over all integers and theP⁽^mn⁾(t) are the polarization amplitudes of the different spectroscopic signals in the direction mk_pu+ nkpr. If the phasesΨpare defined asΨp =kpr, withp ∈ {pu,pr}, Eq. 3.129 can be interpreted as a Fourier transform [115, 116]:

P(t;Ψpu,Ψpr) =

∑

m,n

eⁱ⁽^mΨ^pu⁺^nΨ^pr⁾P⁽^mn⁾(t). (3.130) Such an interpretation would yield the following conclusion: ifP(t;Ψpu,Ψpr)is known for the whole Ψpu,Ψpr configuration space, the respective inverse Fourier transform would generate all amplitudes P⁽^mn⁾[115]. In practice, it is only possible to computeP(t;Ψpu,Ψpr)for a finite

, N different values of the polarization amplitudes P⁽^mn⁾ may be calculated via the following equation [115]:

[m,n]indicates a finite sum overmandn, according to theN different pairs of

Ψ_pu^l ,Ψ^l_pr⁰ . It was shown in [113] that (if the rotating wave approximation is used) only a signal into the directionmk_pu+ (1−m)k_prappears. If higher order terms (m,n > 2) are neglected, Eq. 3.131

P⁽¹⁰⁾is the polarization due to the pump pulse,P⁽⁰¹⁾is the polarization due to the probe pulse and the P⁽²⁻¹⁾ and P⁽⁻¹⁺²⁾ represent combined contributions in the directions 2kpu −kpr as well as −kpu+2kpr. If for the pairs

Ψ^l_pu,Ψ^l_pr⁰

the values (0, 0), (π/2, 0), (π, 0), (3π/2, 0) are chosen, the following result can be derived for the polarization in probe pulse direction [113, 115]:

P⁽⁰¹⁾= ¹

4(P(t; 0, 0) +P(t;π/2, 0) +P(t;π, 0) +P(t; 3π/2, 0)). (3.133) If the additional approximation that both pulses are not overlapping in time can be made, Eq. 3.133 can further be simplified. In this case only two contributions have to be calculated [113]. However, in this work always Eq. 3.133 will be utilized. This rendered it possible to cal-culate the polarization in probe pulse direction for overlapping as well as for non-overlapping pulses.

3.8 Calculation of transient absorption

3.8.3 Signal polarization, method II

Another very intuitive approach for the calculation of the signal polarization was suggested by Pullerits and co-workers in [117, 118]. It is assumed that a lot off CCs are distributed in space, each numbered by the index j. The polarization in probe pulse directionkpris then calculated via

P(k_pr,t) =

∑

e^ik^pr^r^jP(r_j,t;E(r_j,t)). (3.134) At first, the local polarization at a lot of pointsrj is calculated. Of course, at each point rj the local electromagnetic fieldE(r_j,t)has a different phase. The summation over all the different local polarizations with the local phase prefactorse^ik^pr^r^jgives the polarization in probe pulse di-rection. However, since thee^ik^pr^r^j depend very sensitively onr_jand fluctuate between negative and positive values, the convergence of this method is extremely bad. Millions of local polar-izations have to be taken into account to achieve a convergence of the formula (cf. Sec. 5.2.3).

3.8.4 Differential transient absorption

The frequency dependent transient absorption after the delay timet_delaycan be written as [85, 113]

∆S(_ω,t_delay) =_2ωIm^hE_pr^∗ (ω)_∆P(_k_pr_,ω)ⁱ_, _(3.135) with the frequencyω, the Fourier transformed electromagnetic field

E_pr(ω) =

Z _∞

−_∞dte^iωtE_pr(t), (3.136) and the Fourier transformed differential polarization in probe pulse direction

∆P(k_pr,ω) =

Z _∞

−_∞dte^iωt∆P(k_pr,t). (3.137) The differential polarization in probe pulse direction can be written as

∆P(_k_pr,t) =_P(_k_pr,t;Epu 6=0)−_P(_k_pr,t;Epu =0). (3.138) The differential polarization is simply the polarization of the system when a pump pulse was present, minus the polarization of the system, when the pump pulse was absent.

It should be mentioned here that Eq. 3.135 follows directly (via Fourier transform) from the formula for the total energy that is dissipated or gained by the probe pulse in a medium [85, 113]:

Z _∞

−_∞dtE˙^∗_pr(t)P(k_pr,t). (3.139) 3.8.5 Transient anisotropy

In the last section, the differential transient absorption∆S(ω,t_delay)(Eq. 3.135) was introduced.

Next, it will be distinguished between the case when pump and prope pulse polarization are parallel (e_pu ke_pr) and the case when they are orthogonal (e_pu ⊥ e_pr). The respective differen-tial transient absorptions are noted∆S^kand∆S^⊥. From∆S^k and∆S^⊥the transient anisotropy r can be calculated. The computation ofr gives a quantity that may be compared directly to experimental measurements. It is written as (cf. [9])

r= ^∆S

k−_∆S^⊥

∆S^k+2∆S^⊥. (3.140)

Im Dokument Excitation energy transfer in pheophorbide a complexes (Seite 51-56)