Light absorption and emission by atoms or molecules are quantum mechanical pro-cesses. The allowed energy states of the system are described by Schr¨odinger’s equation:
HΨ = EΨ , (4.1)
where E is the energy. The wave function Ψ can in principle be understood as a rep-resentative of the location of the particle. H is the Hamilton operator which is the ob-servable corresponding to the total energy of the system. In general, no simple solutions of Schr¨odinger’s equation exist. It is, therefore, far from trivial to derive the absorption spectrum of a molecule from its molecular structure and to calculate consistently the effect of the environment on the spectrum [198–204].
For polyenes, however, the absorption maximum can be qualitatively understood from a model of a quantum mechanical particle located in a box [104, 205]. Since the
π
-electrons of polyenes are located in a diffuse cloud around the conjugated system, they can be assumed to perform a one-dimensional translational motion along the
π
-system.4.6. Qualitative Model of the Inter-Protein Shift 71
4.6.1 M
ODEL OF AP
AR TICLE IN AB
OXFigure 4.11 shows a one-dimensional box with impenetrable borders at x= 0 and x=L.
The potential energy of the particle is infinitely high beyond the borders. The probability of the particle to be beyond these borders is zero. Thus, the wave functionΨ must be equal to zero forx≤0 andx≥L. In the interval [0,L], the wave function takes the form:
Ψ = Asin (k x) , (4.2)
wherekis the wave number. Atx= 0, Eq. (4.2) is obviously zero, atx=Lit becomes zero if
sin
( k L )= 0, which holds true ifk L is a whole-numbered multiple ofπ
. It follows that the allowed values forkare those for whichk L=nπ
holds withn= 1, 2,. . .. The energy is consequently quantized. The connection betweenkand the energyEis given by:E = k2~2
2m , (4.3)
where m is the mass of the particle and ~ denotes Planck’s constant h divided by 2
π
.Substitutingkbynπ/L, the allowed energy values of the particle are obtained:
E = n2~2π2
2m L2 . (4.4)
In the standard model of a particle in a box, the allowed energy levels of a particle depend on its massm and on the length of the box L. The larger the value of L, the smaller the difference between the allowed energy levels.
For polyenes, the box extension L corresponds to the length of the
π
-system and the mass of the particle is the electron massme. The excitation energy is given by the energy difference between the first two energy levels that correspond to the ground stateS0and to the first excited stateS1, respectively. According to this model, the excitation energy of a polyene can be calculated in dependence on the extension of theπ
-system. In analogy, a generalized model of a particle in a box is formulated that qualitatively relates the electrostatic potential of the archaeal rhodopsins to their absorption maxima.4.6.2 M
ODEL OF AP
AR TICLE IN AB
OX WITHS
TEPP
OTENTIALIn comparison to the standard model of a particle in a box, the environment, here the protein potential, is included as an additional parameter. The electrostatic contribution of the environment is approximated by a step potential. The allowed energy levels of a particle depend on two additional parameters, the height of the potential step and its position. A schematic picture of the generalized model is shown in Figure 4.12. The box lengthL represents the extension of the
π
-system. The potential step is located at positiona, where 0<a<L. The height of the potential step is given by ∆Φ. For a particle of chargeqlocated in this potential, the stationary Schr¨odinger equation is given by:EΨ(x) = − ~2 2me
∂2
∂ x2 Ψ(x) + V(x) Ψ(x) , (4.5)
region II
Figure 4.12. Generalized model of a particle in a box with potential step. L denotes the length of the box, a the position of the step potential, ΦI and ΦII are the additional potentials in regionIand regionII, respectively, and∆Φdenotes the height of the potential step.
where E is the energy of the particle, Ψ its wave function, ~ denotes Planck’s constant divided by 2
π
and me is the mass of the particle, i.e., here the electron mass. The potential energyVis given by:V(x) =
where ΦI and ΦII denote the potential of the environment in region I and region II, re-spectively. The boundary conditionsΨ(0) = Ψ(L) = 0together with the continuity of the wave function and its derivative atx=a lead to a quantization condition for the allowed energy levels: wherek1andk2are the wave numbers in region I and region II, respectively, given by:
kj = q
2me/~2(E −qΦj) . (4.8)
The term (E – qΦj) can become negative for one of the two regions. In that case, the corre-sponding wave number kj is a complex number: kj=iρ, withρ = p
2me/~2|E −qΦj|.
The corresponding solution decays exponentially, reflecting the fact that the region is classically forbidden.
4.6. Qualitative Model of the Inter-Protein Shift 73
kal
mole
a [ ˚ A]
E
kal
mol e
BR x HR x SRII x
Figure 4.13. Dependence of the excitation energy ∆E on the position of the stepa and the potential difference∆Φ. The length of the boxLis set to 14.5 ˚A. The excitation energy∆E, given in kcal mol−1e−1, is indicated by the color code. The dashed line marks the position of the step in the archaeal rhodopsins ata= 7.25 ˚A. The crosses indicate the height of the potential step∆Φfor BR,HR and SRII.
Eq. (4.7) can be solved numerically leading to a discrete energy spectrum of the system.
Two limiting cases can be distinguished:
(i) When the step is located close to one of the potential walls of the box,i.e.,ais close to 0 or L, the model approaches the standard model of a particle in a box. Con-sequently, the dependence of ∆E, i.e., the energy difference between two allowed energy levels, ona and∆Φis low in this limit.
(ii) When the height of the potential step ∆Φ approaches zero, a similar effect is seen and the behavior of Eq. (4.7) resembles again the standard model of a particle in a box.
Figure 4.13 shows the excitation energy∆Efor a positively-charged particle. The positive charge corresponds to the charge of the retinal Schiff base, which delocalizes over the
π
-system upon excitation. The box lengthLwas set to 14.5 ˚A, representing the extension of the retinalπ
-system in the archaeal rhodopsins. The step positiona is varied from 3 to 11 ˚A and the height of the potential step ∆Φ varies from 0 to 20 kcal mol−1e−1. The qualitative picture offered by this model reveals an interesting aspect of spectral tuning.As can be seen, spectral tuning is most effective, if the controlling potential changes close to the center of the polyene
π
-system: then, even small changes in the height of the potential step∆Φhave a pronounced influence on the excitation energy∆E.APPLICATION TO BR, HRAND SRII
For the electrostatic potential of the three archaeal rhodopsins, two plateaus can be distinguished (cf. Figure 4.4): one in the β-ionone ring region and another one in the Schiff base region. The separation between these two plateaus is located approximately in the middle of the retinal
π
-system. The position of the step for the archaeal rhodopsins is indicated by the dashed line in Figure 4.13. The height of the potential step ∆Φ was defined as the difference between the potential at theβ-ionone ring and the potential at the Schiff base and is indicated by the crosses for BR, HR and SRII. The box lengthLwas set to 14.5 ˚A, corresponding to the extension of the retinalπ
-system.As can be seen in Figure 4.13, the excitation energy ∆E is significantly higher for SRII than for BR and HR, which have similar excitation energies. Using the generalized model of a particle in a box with step potential the observed differences in electrostatic potential in BR, HR and SRII are related to differences in the absorption maxima. The general trend of the experimentally measured absorption maxima could be reproduced by grouping BR and HR together and showing a significantly higher excitation energy for SRII (cf.
Figure 4.1).