In the following, we first investigate how slip displacement affects the couplings, where the slip displacement is shown in Fig. 3.1. Considering the center of the two molecules C1 = (x1, y1, z1) and C2 = (x2, y2, z2), we define ∆x = x2 − x1 ,∆y = y2 − y1 and
∆z = z2 − z1. And the slip displacement represent the displacement either along the x-axis (△x) or the y-axis (△y).
In this section, we apply our theory to three representative model systems. We start with an ethylene dimer system for which we compare the effect of slip displacement on overlap with the effect of slip displacement on the electronic couplings calculated from the RAS-SD-EOM approach to examine the validity of using overlap as an indicator of SF couplings. The dependence of slip displacement on SF couplings in tetracene and pyrene is also studied to verify our theory further.
3.2.1 Ethylene
To elucidate the validity of using certain overlap integrals to estimate SF couplings, we first study the SF coupling of the ethylene dimer at different slip displacement. Ethylene is the simplest conjugated system and the simple HOMO and LUMO of the Ethylene provide convenience for us to illustrate the behavior of the overlap in different slip displacement.
Figure 3.1 shows the geometry of the ethylene dimer system, where the face-to-face dis-tance between the two ethylene molecules is fixed at 4.5Å (△z = 4.5Å ), which is deter-mined by geometry optimization with MP2/6-31G calculation. Hereafter, all calculations are carried out using 6-31G basis set. We aim to show that effective couplings, two elec-tron integrals, and overlap integrals relevant in the SF process have the same behaviors when the slip displacement changes.
Figure 3.2 shows that the results of the ethylene dimer system. Figure 3.2(a) shows the HOMO and LUMO of an ethylene molecule. In Fig. 3.2(b), we plot the coupling between|ψT T⟩ and |ψCT⟩ derived from the RAS-SD-EOM method, ⟨ψT T|H|ψCT⟩, as a function of the long axis (x-axis) displacement. Clearly,⟨ψT T|H|ψCT⟩, as △y is zero , exhibits a maximum at△x = 1.5Å, which is in agreement with previous calculations [5].
In addition, we also plot⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩ as a function of △x, and both of them follow the dependence of△x on the diabatic coupling closely. And it indicates that couplings between the two most important CSFs in the four-electron-four-orbital model indeed dominate the diabatic couplings.
In Fig. 3.2(c), we show the dependence of △x on relevant two electron integrals, which exhibits△x dependence similar to ⟨ψT T|H|ψCT⟩. The behaviors of ⟨ψT T|H|ψCT⟩ can be estimated by the behaviors of these two electron integrals, which are approximated by the overlap integral, ⟨hA|lB⟩. Figure 3.2(d) also shows that ⟨hA|lB⟩ as a function of
△x, and the behavior of ⟨hA|lB⟩ shows excellent correspondence with the behavior of
⟨ψT T|H|ψCT⟩ ( Figure 3.2(b) ).
On the other hand, ⟨ψT T|H|ψCT⟩ is zero when △x is equal to zero regardless of the value of△y, which we didn’t show in the Fig. 3.2, and this phenomenon was also men-tioned before [5]. Calculations of⟨hA|lB⟩, which is a function of △y with △x = 0 shows
z
x y
∆z
∆x
∆y
Figure 3.1: Illustration of the slip displacement, where the yellow points are the centers of the ethylene monomers.
the same behavior with⟨ψT T|H|ψCT⟩. Apparently, in the ethylene system, ⟨hA|lB⟩ is an excellent indicator of the dependence of the slip displacement on⟨ψT T|H|ψCT⟩.
Besides, the dependence of the slip displacement on ⟨hA|lB⟩ can be explained by a mirror plane. This mirror plane is perpendicular to the long axis and in the middle of the face-to-face ethylene dimer such that HOMO is symmetric while LUMO is anti-symmetric to the mirror plane. This mirror plane also explain the zero if ⟨hA|lB⟩ in a face-to-face ethylene dimer system. Moreover,⟨hA|lB⟩ is zero at △x = 0 regardless of the value of
△y because the mirror plane remains in the dimer system.
In contrast, when△y = 0 and △x become non-zero in the dimer system, ⟨hA|lB⟩ has non-zero value since the mirror plane disappears. For a good correlation between
⟨ψT T|H|ψCT⟩ and HOMO-LUMO overlap, the behavior of ⟨ψT T|H|ψCT⟩ in our results can also be elucidated by this mirror plane.
The most important is that comparing Fig. 3.2(b) and (d) shows that the relative mag-nitude of⟨ψT T|H|ψCT⟩ can also be elucidated by ⟨hA|lB⟩. For example, ⟨ψT T|H|ψCT⟩ as△x = 1Å is smaller than ⟨ψT T|H|ψCT⟩ as △x = 2Å and ⟨hA|lB⟩ reveals the same relationship between as△x = 1Å and as △x = 2Å. This indicates that it is possible to calculate⟨ψT T|H|ψCT⟩ from ⟨hA|lB⟩.
Figure 3.2: (a) Visualizations of HOMO and LUMO in ethylene monomer. The depen-dence of△x on (b) couplings, (c) two electron integrals and (d) overlap. Other parameters are△y = 0Å and △z = 4.5Å.
3.2.2 Tetracene
The SF process was observed in tetracene crytals before, [9] hereafter SF process was observed in bistetracene solutions as well [31] , which reveals that the structure of the dimer can be manipulated. In order to apply our theory to bistetracene for molecular design, we aim to verify that our theory can be applied to the tetracene dimer system.
Figure 3.3(a) shows the HOMO and LUMO of a tetracene molecule. In a face-to-face tetracene dimer system, there is a mirror plane perpendicular to the short axis and in the middle of the dimer system such that HOMO is anti-symmetric while LUMO is symmetric to the mirror plane. Different from the ethylene dimer system, SF couplings,⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩, are expected to be zero at △y = 0 regardless of the value of △x because the mirror plane is perpendicular to the short axis and disappears only when△y becomes non-zero.
Hence, we plot the couplings as a function of△y (shown in Fig. 3.3(b)) to demonstrate the disappearence of the mirror plane in the tetracene dimer system. Figure 3.3(c) and (d)
show that the△y dependence of relevant two electron integrals and the HOMO-LUMO overlap agree very well with the△y dependence of ⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩. Even though the patterns of HOMO and LUMO in tetracene molecule are more complicated than the patterns of orbitals in ethylene molecule, our theory still can be applied to the tetracene dimer system.
In addition, from the results of ethylene and tetracene, we can find that the direction of the mirror plane in the ethylene face-to-face dimer is different from the direction of the mirror plane in the tetracene face-to-face dimer. At the same time, in the ethylene dimers,⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩ are zero at △x = 0 regardless of the value of △y in the ethylene dimers, while in the tetracene dimers, the couplings are zero at△x = 0 regardless of the value of△y. This implies that the direction of the mirror plane in the face-to-face dimer determines the appearance of the non-zero value of⟨T T |H|CT ⟩ and
⟨S1S1|H|CT ⟩, which is useful to predict the behaviors of couplings in an unknown dimer system.
In Fig. 3.3(b), the oscillatory behavior is observed. ⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩ exhibit a maximum at△y = 3.5Å, which also can be clearly illustrated by the HOMO-LUMO overlap. In Fig. 3.3(a), the mirror plane divides HOMO into two parts. When△y increases, the overlap between the first part of HOMO, or the postitive part, and LUMO is denoted as Spfor the overlap is always positive. While the overlap between the second part, or the negative part, and LUMO is always negative, hence Sn.
In a face-to-face tetracene dimer, where both△x and △y are equal to zero, the nearest distance between the positive part and the center of LUMO is equal to the nearest distance between the negative part and the center of LUMO. Hence, the absolute value of Sp and Snare equal at△y = 0Å, which results in ⟨hA|lB⟩ = 0.
At △y = 3.5Å with △x fixed to zero, the nearest distance between the negative part and the center of LUMO become even larger so the absolute value of Sn become much smaller than Sn at △y = 0. On the contrary, Sp at both △y = 0Å and △y = 3.5Å are similar because the nearest distance between the positive part and the center of LUMO at△y = 3.5Å is the same as at △y = 0Å. As a result, the HOMO-LUMO
over-HOMO
Two electron integral (meV) Coupling (meV)Overlap
(b)
Figure 3.3: (a) Visualizations of HOMO and LUMO in tetracene monomer. The depen-dence of△y on (b) couplings, (c) two electron integrals and (d) overlap. Other parameters are△x = 0Å and △z = 4Å.
lap (⟨hB|lA⟩ = Sp + Sn) reach the maxima at △y = 3.5Å, which also elucidates that
⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩ reach the maxima at △y = 3.5Å.
Note that other polyacenes, such as anthracene and pentacene, are in the same point group as tetracene. These polyacenes should also present the same mirror plane as tetracene [67].
Although we only do the calculations in tetracene, we expect that the similar dependence of slip displacement on SF couplings in pentacene and anthracene as in tetracene.
3.2.3 Pyrene
The dependence of slip displacement on the SF coupling in the pyrene, another four-ring system, is also investigated because the mirror plane presented in the face-to-face pyrene dimer system is different from in the face-to-face tetracene dimer system. And the mirror plane can be determined by Fig. 3.3(a), which shows the HOMO and LUMO in a pyrene molecule.
The results in pyrene also confirm the validity of our theory. The mirror plane to which
HOMO is symmetric while LUMO is anti-symmetric is perpendicular to the long axis, and in the middle of the face-to-face pyrene dimer system. Based on our theory,⟨T T |H|CT ⟩ and⟨S1S1|H|CT ⟩ are expected to be zero at △x = 0 regardless of the value of △y. And our results of⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩ shown in Fig. 3.3(b), which are functions of△x show the same geometric dependence with the overlap shown in Fig. 3.3(d).
A oscillatory behavior is also observed in Fig. 3.4(b) and a discrepancy is found at △x = 3Å in ⟨S1S1|H|CT ⟩. This phenomenon results from the special behavior of (hAhB|hAlA) as a function of△x. In Fig. 3.4(c), the value of (hAhB|hAlA) at any△x is always equal to or smaller than zero. Though⟨hA|lB⟩ which is shown in Fig. 3.4(d) can-not fully describe (hAhB|hAlA) as a function of△x, HOMO-LUMO overlap still exhibit a similar△x dependence with ⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩. Hence, we believe that the overlap is a good indicator of the dependence of the slip displacement on the effective SF couplings in the pyrene system despite the discrepancy.
From the results in ethylene, tetracene and pyrene, we find that ⟨T T |H|CT ⟩ and
⟨S1S1|H|CT ⟩ are zero in all three different face-to-face dimer systems. As a result, the zero value of⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩ must be a general property for all conjugated dimer systems. It is important because the property confirms that a face-to-face dimer can not be a good design of the SF material. The face-to-face dimer means⟨ψT T|H|ψCT⟩ must be zero in this dimer system, where SF process can not be observed.
Also, ⟨T T |H|CT ⟩ and ⟨S1S1|H|CT ⟩ in these three simer systems, exhibit one or more peaks instead of a broad band as the slip displacement along the specific direc-tion increases, which means ⟨ψT T|H|ψCT⟩ is sensitive to the slip displacements. This property of⟨ψT T|H|ψCT⟩ indicates that even though two dimers’ structure are similar,
⟨ψT T|H|ψCT⟩ in two dimer systems may be totally different.