Quasi-periodic and fractal polymers: Energy structure and carrier transfer

M. Mantela,1 K. Lambropoulos,1 M. Theodorakou,1 and C. Simserides1, ∗

1Department of Physics, National and Kapodistrian University of Athens,Panepistimiopolis, Zografos, GR-15784, Athens, Greece

(Dated: October 11, 2021)

We study the energy structure and the coherent transfer of an extra electron or hole along aperiodicpolymers made of N monomers, with fixed boundaries, using B-DNA as our prototype system. Weuse a Tight-Binding wire model, where a site is a monomer (e.g., in DNA, a base pair). We considerquasi-periodic (Fibonacci, Thue-Morse, Double-Period, Rudin-Shapiro) and fractal (Cantor Set,Asymmetric Cantor Set) polymers made of the same monomer (I polymers) or made of differentmonomers (D polymers). For all types of such polymers, we calculate the HOMO and LUMOeigenspectrum, the HOMO-LUMO gap and the density of states. We examine the mean over timeprobability to find the carrier at each monomer, the frequency content of carrier transfer (Fourierspectra, weighted mean frequency of each monomer, total weighted mean frequency of the polymer),and the pure mean transfer rate k. Our results reveal that there is a correspondence between thedegree of structural complexity and the transfer properties. I polymers are more favorable for chargetransfer than D polymers. We compare k(N) of quasi-periodic and fractal sequences with that ofperiodic sequences (including homopolymers) as well as with randomly shuffled sequences. Finally,we discuss aspects of experimental results on charge transfer rates in DNA with respect to ourcoherent pure mean transfer rates.

I. INTRODUCTION

Today, the electronic structure of biological molecules[e.g. proteins, enzymes, peptides and nucleic acids(DNA, RNA)] and their charge transfer and transportproperties attract considerable interest among the phys-ical, chemical, biological and medical communities, aswell as a broad spectrum of interdisciplinary scientistsand engineers1–6. DNA plays a fundamental role in ge-netics and molecular biology since its sequence of bases,adenine (A), guanine (G), cytosine (C), and thymine (T),contains the genetic code of living organisms. The base-pair stack of the DNA double helix creates a nearly one-dimensional π-pathway that favors charge transfer andtransport. The term transfer means that a carrier, cre-ated (e.g. by oxidation or reduction) or injected at aspecific place, moves to a more favorable location, while,the term transport implies the application of voltage be-tween electrodes. Charge transfer through DNA plays acentral role in DNA damage and repair2,7,8, so it maybe a critical issue in carcinogenesis and mutagenesis9,10.For example the rapid hole migration from other bases toguanine is connected to the fact that direct strand breaksoccur preferentially at guanines9. Furthermore, it mightbe an indicator of discrimination between pathogenic andnon-pathogenic mutations at an early stage11.

Charge movement is usually ascribed to two types ofmechanisms12,13: (i) incoherent or thermal hopping be-tween nearest neighboring or more distant sites and (ii)coherent hopping or tunneling or superexchange. Theterm tunneling implies quantum mechanical tunneling,between two sites, e.g., the carrier donor and the car-rier acceptor, through a bridge. The term superexchange,not to be confused with the similar term in magnetism,emanates from the distant interaction between the twosites, e.g. the donor and the acceptor, through a bridge.

However, we have shown systematically14–18 that, in thecoherent regime, all sites contribute with finite occupa-tion probabilities, although those with adequate on-siteenergies, for the initial placement of the carrier in thesequence, are more favored. This conclusion holds bothfor the wire model (where the site is a base pair) andfor the extended ladder model (where the site is a base)that we have used so far. The coherent mechanism isexpected to dominate carrier movement in the low tem-perature regime. In natural DNA, it is more likely thata hole will be created at a guanine which has the high-est HOMO of all bases19 and an electron will be createdat a thymine which has the lowest LUMO of all bases19.However, coherently, if e.g. the hole is initially created orinjected at an adenine, charge transfer will mainly be ac-complished through adenines and similarly for other ini-tial conditions16. Typically, in coherent transfer, chargeis never exactly localized but there is a mean over timeoccupation probability to find it at each site, the carrierdoes not exchange energy with the environment duringits transfer and this way it can travel short distances;strictly quantum mechanically, just a percentage of thecarrier reaches the last site.

Typically, in thermal hopping, charge is localized, thecarrier exchanges energy with the environment during itstransfer and this way it can travel far longer than via thecoherent mechanism. If d0 is a typical nearest neighbordistance, e.g., 3.4 A, and two sites stand off ∆r havingon-site energy difference ∆E, then, maybe one could pre-sume an equation k = k0 exp(−∆E/kBT ) exp(−∆r/d0),–or a similar one with other mathematical form– to qual-itatively describe thermal hopping.

In the present work, we take B-DNA as a prototypesystem, because, apart from its biological and nanoscien-tific importance, it has a rather long persistence lengthof around 50 nm or 150 base pairs20. However, there

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are several studies concerning charge and energy trans-fer in other aperiodic polymer systems 21–23. We studythe coherent regime, cf. Eq. (12), this time for aperiodicpolymers. Although unbiased coherent charge transfer inDNA nearly vanishes after 10 to 20 nm14–16,18, DNA stillremains a promising candidate as an electronic compo-nent in molecular electronics, e.g. as a short molecularwire or a nanocircuit element24,25. Favoring geometriesand base-pair sequences have still to be explored, e.g., in-corporation of sequences serving as molecular rectifiers,use of non-natural bases or using the triplet acceptor an-thraquinone for hole injection25.

Research has recently shown that carrier movementthrough B-DNA can be manipulated. Using variousnatural and artificial nucleobases (chemical modifica-tion) with different highest occupied molecular orbital(HOMO) levels, the hole transfer rate through DNA canbe tuned26. The carrier transfer rate strongly depends onthe difference between HOMO energies (for hole trans-fer) or LUMO energies (for electron transfer) and so itcan be increased by many orders of magnitude with ap-propriate sequence choice15–18. Furthermore, structuralfluctuations is another factor which influences quantumtransport through DNA molecular wires27.

We know that many factors (e.g. aqueousness, coun-terions, extraction process, electrodes, purity, substrate,structural fluctuations, geometry), influence carrier mo-tion along DNA. These factors are either intrinsic or ex-trinsic. Here we focus on the most important of the in-trinsic factors, i.e. the effect of alternating the base-pairsequence, which affects the overlaps across the π-stack.The aim of this work is a comparative examination ofthe influence of base-pair sequence on charge transfer, inaperiodic sequences. Ab initio calculations28–36, used toexplore experimental results and the underlying mecha-nisms, are currently limited to short segments for compu-tational reasons. Here we study rather long sequences, sowe employ the Tight-Binding (TB) model which allowsto address systems of realistic length14–16,37–53.

There are several works devoted to the study of trans-fer and transport in specific DNA structures using vari-ants of the Tight-Binding method12,15,16,38,39,51,54–57.Here, we employ a TB wire model, where the basepairs are the sites of the chain, to study the spectraland charge transfer properties of deterministic aperiodic[Thue-Morse (TM), Fibonacci (F), Double-Period (DP),Rudin-Shapiro (RS), Cantor Set (CS), Asymmetric Can-tor Set (ACS)] DNA segments. The relevant parametersare the on-site energies of base pairs and the hopping in-tegrals between successive base pairs. We have to solve asystem of N coupled equations for the time-independentproblem, and a system of N coupled first order differen-tial equations for the time-dependent problem. We studyHOMO and LUMO eigenspectra, HOMO-LUMO gapsand the relevant density of states (DOS) as well as themean over time probabilities to find the carrier at eachsite. We are also interested in the frequency content ofcarrier movement, hence, we analyze the Fourier spectra

of the time-dependent probability to find the carrier ateach site, the weighted mean frequency of each monomerand the total weighted mean frequency of the polymer.Finally, we study the pure mean transfer rate from acertain site to another, which describes the easiness ofcharge transfer; it gives us a measure of how much of thecarrier is transferred and also of how fast this process is.

The rest of the paper is organized as follows: InSec. II, we provide some details on the studied deter-ministic aperiodic sequences and we outline our nota-tion. In Sec. III we delineate the basic theory behind thetime-independent (Sec. III A) and the time-dependent(Sec. III B) problem. In Sec. IV, we discuss our resultsfor polymers made of the same monomer and polymersmade of different monomers. Here, for DNA, a monomeris a base pair. Finally, in Sec. V, we state our conclusions.

II. SEQUENCES AND NOTATION

In our prototype system, B-DNA, we mention only thebase sequence of the 5′ − 3′ strand. For example, wedenote two successive monomers by YX, meaning thatthe base pair X-Xcompl is separated and twisted by 3.4

A and 36◦, respectively, relatively to the base pair Y-Ycompl, around the B-DNA growth axis. Xcompl (Ycompl)is the complementary base of X (Y).

The deterministic aperiodic sequences considered inthis work are either quasi-periodic or fractal. Such struc-tures are generally known as binary substitutional se-quences, i.e., based on a binary alphabet, like {0, 1} andgenerated using appropriate substitution rules.

A. Fibonacci

The Fibonacci (F) sequence is named after the Italianmathematician Leonardo Pisano (Fibonacci) who intro-duced it to Western European mathematics in his 1202book Liber Abaci, in a study of the population growthof rabbits58. However, this sequence appears many cen-turies before in Indian mathematics59. Fibonacci consid-ers the growth of an idealized rabbit population, assum-ing that a single newly born pair of rabbits (N) are putin a field, and rabbits are able to mate at the age of onemonth so that at the end of its second month a maturepair (M) can produce another pair of rabbits. Rabbitsnever die and a mating pair always produces one newpair every month from the second month on. The puzzlethat Fibonacci posed was: how many pairs will exist inone year? The collection of every month’s population is:F0 = N, F1 = M, F2 = MN, F3 = MNM, F4 = MN-MMN, etc. Using e.g. the two-letter alphabet {G, A},we can define the Fibonacci generation Fg by the sub-stitution rules A → G, G → GA, starting with F0 = A.Hence, F0 = A, F1 = G, F2 = GA, F3 = GAG, F4 =GAGGA, etc. If Ng is the Fibonacci number of genera-tion g, and we set N0 = N1 = 1, the recurrence relation

3

Ng = Ng−1 +Ng−2 produces the number sequence 1, 1,2, 3, 5, 8, 13, 21, 34, ... .

B. Thue-Morse

The Thue-Morse (TM) or Prouhet-Thue-Morse se-quence was first studied by Eugene Prouhet in 1851, whoapplied it to number theory60. The systematic study wasleft to Axel Thue who, in 1906, applied it on his studyof words combinatorics61. The most important contribu-tion to the sequence was made in 1921 by Marston Morsein the context of differential geometry and topologicaldynamics62, which brought the sequence to worldwideattention. In its simplest form, the TM sequence canbe defined by the recursive relations Sn = {Sn−1S+

n−1}and S+

n = {S+n−1Sn−1} (for n ≥ 1), with S0 = 0 and

S+0 = 163. Using e.g. the two-letter alphabet {G, A} we

can build up the sequence using the substitution rulesG→GA and A→AG. Hence, TM0 = G, TM1 = GA,TM2 = GAAG, TM3 = GAAGAGGA, etc.

C. Double-Period

The double-period (DP) sequence has its origin in thestudy of system dynamics and laser applications to non-linear optical fibers64. It is closely connected with theTM sequence: the n-th stage is Sn = {Sn−1S+

n−1} and

S+n = {Sn−1Sn−1} (for n ≥ 1), with S0 = 0 and S+

0 = 1.Using e.g. the two-letter alphabet {G, A}, we can definethe n-th generation by the substitution rules G→GA,A→GG. Hence, starting with DP0 = G, then DP1 =GA, DP2 = GAGG, DP3 = GAGGGAGA, etc.

D. Rudin-Shapiro

The Rudin-Shapiro (RS) aka Golay-Rudin-Shapirosequence is named after Marcel Golay, Walter Rudinand Harold S. Shapiro, who independently investigatedits properties65–67. It is generated starting with +1, +1and employing the rules:

+1,+1→ +1,+1,+1,−1+1,−1→ +1,+1,−1,+1−1,+1→ −1,−1,+1,−1−1,−1→ −1,−1,−1,+1 .

Using e.g. the two-letter alphabet {G, A} and employ-ing the inflation rule: GG→GGGA, GA→GGAG,AG→AAGA, AA→AAAG, the first generations areRS1 = GG, RS2 = GGGA, RS3 = GGGAGGAG, etc.

E. Cantor Set

The Cantor Set (CS), introduced by mathematicianGeorg Cantor, is one of the most well-known determinis-tic fractals68. It is built by splitting a straight line seg-ment in three, removing the middle third, then removingthe middle third of each of the two new straight line seg-ments and the process is repeated ad infinitum. Usinge.g. the two-letter alphabet {G, A} and the substitution

rules G→GAG, A→AAA, we can define the n-th gener-ation (n = 0, 1, 2, ...) as follows: CS0 = G, CS1 = GAG,CS2 = GAGAAAGAG, etc.

F. Asymmetric Cantor Set

The Asymmetric Cantor Set (ACS), is built by split-ting a straight line segment in four, removing the sec-ond quarter, then removing the second quarter of eachof the three new straight line segments and the processis repeated ad infinitum. Using e.g. the two-letter al-phabet {G, A} and the substitution rules G→GAGG,A→AAAA, we can define the n-th generation (n = 0, 1,2, ...) as follows: ACS0 = G, ACS1 = GAGG, ACS2 =GAGGAAAAGAGGGAGG, etc.

One could think of many types of aperiodic polymers,some of which are shown synoptically in Table I. We justgive an example of each type, e.g., for Fibonacci I se-quences we give the example G, C, CG, CGC, CGCCG,CGCCGCGC, ..., but there are obviously other similarsequences e.g. C, G, GC, GCG, GCGGC, GCGGCGCG,..., A, T, TA, TAT, TATTA, TATTATAT, ..., T, A, AT,ATA, ATAAT, ATAATATA, ....

TABLE I. Examples of the types of polymers studied in thiswork. I (D) denotes polymers made of identical (different)monomers. We only mention the 5′ − 3′ base sequence alongone of the two strands.

type sequence example notation

Fibonacci I G, C, CG, CGC, F G(C)

CGCCG, ...

Fibonacci D G, A, AG, AGA, F G(A)

AGAAG, ...

Thue-Morse I G, GC, GCCG, TM G(C)

GCCGCGGC, ...

Thue-Morse D A, AG, AGGA, TM A(G)

AGGAGAAG, ...

Double Period I T, TA, TATT, DP T(A)

TATTTATA, ...

Double Period D A, AG, AGAA, DP A(G)

AGAAAGAG, ...

Rudin-Shapiro I AA, AAAT, RS A(T)

AAATAATA, ...

Rudin-Shapiro D AA, AAAG, RS A(G)

AAAGAAGA, ...

Cantor Set I T, TAT, CS T(A)

TATAAATAT, ...

Cantor Set D A, AGA, CS A(G)

AGAGGGAGA, ...

Asymmetric C, CGCC, ACS C(G)

Cantor Set I CGCCGGGGCGCCCGCC, ...

Asymmetric A, AGAA, ACS A(G)

Cantor Set D AGAAGGGGAGAAAGAA, ...

4

III. THEORY

In this article, we use a simple wire model, where thesite is a monomer (e.g. in DNA, a base pair). We callµ the monomer index, µ = 1, 2, . . . , N . We assume thatthe state or movement of an extra hole or electron canbe expressed through the monomer HOMOs or LUMOs,respectively, cf. Eqs. (2) and (8) below.

A. Stationary States - Time-independent problem

The TB wire model Hamiltonian can be written as

HW =

N∑µ=1

Eµ |µ〉〈µ|+(N−1∑µ=1

tµ,µ+1 |µ〉〈µ+ 1|+ h.c.

).

(1)Eµ is the on-site energy of the µ-th monomer, and tµ,λ =t∗λ,µ is the hopping integral between monomers µ and λ.The state of a polymer can be expressed as

|P〉 =

N∑µ=1

vµ |µ〉 . (2)

Substituting Eqs. (1) and (2) to the time-independentSchrodinger equation

H |P〉 = E |P〉 , (3)

we arrive to a system of N coupled equations

Eµvµ + tµ,µ+1vµ+1 + tµ,µ−1vµ−1 = Evµ, (4)

which is equivalent to the eigenvalue-eigenvector problem

H~v = E~v, (5)

where H is the hamiltonian matrix of order N , composedof the TB parameters Eµ and tµ,λ, and ~v is the vectormatrix composed of the coefficients vµ (which can be cho-sen to be real). The diagonalization of H leads to thedetermination of the eigenenergy spectrum (eigenspec-trum), {Ek}, k = 1, 2, . . . , N , for which we suppose thatE1 < E2 < · · · < EN , as well as to the determination ofthe occupation probabilities for each eigenstate, |vµk|2,where vµk is the µ-th component of the k-th eigenvector.{vµk} are normalized, and their linear independence ischecked in all cases.

Having determined the eigenspectrum, we can com-pute the density of states (DOS), generally given by

g(E) =

N∑k=1

δ(E − Ek). (6)

Changing the view of a polymer from one (e.g. top) to theother (e.g. bottom) side of the growth axis, reflects thehamiltonian matrix H of the polymer on its main antidi-agonal. This reflected Hamiltonian, Hequiv, describes the

equivalent polymer16. H and Hequiv are connected bythe similarity transformation Hequiv = L−1HL, whereL(= L−1) is the unit antidiagonal matrix of order N .Therefore, H and Hequiv have identical eigenspectra(hence the equivalent polymers’ DOS is identical) and

their eigenvectors are connected by vµk = vequiv(N−µ+1)k.

Generally,

equiv(YX. . . Z) = Zcompl . . .YcomplXcompl. (7)

B. Time-dependent problem

To describe the spatiotemporal evolution of an extracarrier (hole/electron), inserted or created (e.g. by oxida-tion/reduction) at a particular monomer of the polymer,we consider the state of the polymer as

|P(t)〉 =

N∑µ=1

Cµ(t) |µ〉 , (8)

where |Cµ(t)|2 is the probability to find the carrier at theµ-th monomer at time t. Substituting Eqs. (1) and (8)in the time-dependent Schrodinger equation

i~∂

∂t|P(t)〉 = H |P(t)〉 , (9)

we arrive at a system of N coupled differential equations

i~dCµdt

= EµCµ + tµ,µ+1Cµ+1 + tµ,µ−1Cµ−1. (10)

Eq. (10) is equivalent to a 1st order matrix differentialequation of the form

~C(t) = − i~H ~C(t), (11)

where ~C(t) is a vector matrix composed of the coefficientsCµ(t), µ = 1, 2, . . . , N . Eq. (11) can be solved with theeigenvalue method, i.e., by looking for solutions of the

form ~C(t) = ~ve−i~Et ⇒ ~C(t) = − i

~E~ve− i

~Et. Hence,Eq. (11) leads to the eigenvalue problem of Eq. (5), thatis, H~v = E~v. Having determined the eigenvalues andeigenvectors of H, the general solution of Eq. (11) is

~C(t) =

N∑k=1

ck~vke− i

~Ekt. (12)

In other words, the coefficients Cµ(t), µ = 1, 2, . . . , N ,are given by a superposition of the time evolution of thestationary states with time-independent coefficients ck.Hence, this is a coherent phenomenon. The coefficientsck are determined from the initial conditions. In partic-ular, if we define the N ×N eigenvector matrix V , withelements vµk, then it can be shown that the vector ma-trix ~c, composed of the coefficients ck, k = 1, 2, . . . , N ,is given by the expression

~c = V T ~C(0). (13)

5

Suppose that initially the extra carrier is placed at theλ-th monomer, i.e., Cλ(0) = 1, Cµ(0) = 0,∀µ 6= λ. Then,

~c =[vλ1 . . . vλk . . . vλN

]T. (14)

In other words, the coefficients ck are given by therow of the eigenvector matrix which corresponds to themonomer the carrier is initially placed at. In this work,we choose λ = 1, i.e., we initially place the carrier at thefirst monomer. From Eq. (12) it follows that the proba-bility to find the extra carrier at the µ-th monomer is

|Cµ(t)|2 =

N∑k=1

c2kv2µk+2

N∑k=1

N∑k′=1k′<k

ckck′vµkvµk′ cos(2πfkk′t).

(15)

fkk′ =1

Tkk′=Ek − Ek′

h, ∀k > k′, (16)

are the frequencies (fkk′) or periods (Tkk′) involved incharge transfer. If m is the number of discrete eigenener-gies, then, the number of different fkk′ or Tkk′ involved in

carrier transfer is S =(m2

)= m!

2!(m−2)! = m(m−1)2 . If there

are no degenerate eigenenergies (which holds for all casesstudied here, but e.g. does not hold for cyclic homopoly-mers15), then m = N . If eigenenergies are symmetricrelative to some central value, then, S decreases (thereexist degenerate fkk′ or Tkk′). Specifically, in that case,

S = m2

4 , for even m and S = m2−14 for odd m.

From Eq. (15), in the absence of degeneracy and forreal ck, vµk, it follows that the mean over time probabilityto find the extra carrier at the µ-th monomer is⟨

|Cµ(t)|2⟩

=

N∑k=1

c2kv2µk. (17)

Furthermore, from Eq. (15) it can be shown that theone-sided Fourier amplitude spectrum that correspondsto the probability |Cµ(t)|2 is given by

|Fµ(f)| =N∑k=1

c2kv2µkδ(f) + 2

N∑k=1

N∑k′=1k′<k

|ckck′vµkvµk′ |δ(f − fkk′).

(18)Hence, the Fourier amplitude of frequency fkk′ is2|ckvµkck′vµk′ |. We can further define the weighted meanfrequency (WMF) of monomer µ as

fµWM =

N∑k=1

N∑k′=1k′<k

|ckvµkck′vµk′ |fkk′

N∑k=1

N∑k′=1k′<k

|ckvµkck′vµk′ |

. (19)

WMF expresses the mean frequency content of the extracarrier oscillation at monomer µ. Having determined the

WMF for all monomers, we can now obtain a measureof the overall frequency content of carrier oscillations inthe polymer: Since fµWM is the weighted mean frequencyof monomer µ and

⟨|Cµ(t)|2

⟩is the mean probability of

finding the extra carrier at monomer µ, we define thetotal weighted mean frequency (TWMF) as

fTWM =

N∑µ=1

fµWM

⟨|Cµ(t)|2

⟩. (20)

A quantity that evaluates simultaneously the magni-tude of coherent charge transfer and the time scale ofthe phenomenon, is the pure mean transfer rate14

kλµ =

⟨|Cµ(t)|2

⟩tλµ

. (21)

tλµ is the mean transfer time, i.e., having placed the car-rier initially at monomer λ, the time it takes for the prob-ability to find the extra carrier at monomer µ, |Cµ(t)|2,

to become equal to its mean value,⟨|Cµ(t)|2

⟩, for the

first time. For the pure mean transfer rates,

kλµ = kµλ =

kequiv(N−λ+1)(N−µ+1) = kequiv(N−µ+1)(N−λ+1), (22)

where the superscript “equiv” refers to the equivalentpolymer in the sense of Eq.(7).

TABLE II. The HOMO/LUMO hopping integrals tµ,λ, inmeV, between successive base pairs µ, λ.

µ, λ tµ,λH tµ,λL µ, λ tµ,λH tµ,λL

Ref.19 Ref.19 Ref.19 Ref.19

AA ≡ TT −8 −29 AT 20 0.5

AG ≡ CT −5 3 AC ≡ GT 2 32

TA 47 2 TG ≡ CA −4 17

TC ≡ GA −79 −1 GG ≡ CC −62 20

GC 1 −10 CG −44 −8

IV. RESULTS

In this article, the TB parameters for B-DNA are takenfrom Ref.19. The HOMO/LUMO base-pair on-site ener-gies are19 EG-C = −8.0/ − 4.5 eV, EA-T = −8.3/ − 4.9eV. The hopping integrals are given in Table II.

At this point, we mention that any sign alteration ofthe hopping integrals does not affect the results presentedbelow, since the hamiltonian matrices we deal with inthe Wire Model are irreducible, symmetric and tridiag-onal. This can be shown as follows: Let us suppose aN ×N irreducible tridiagonal hermitian matrix T , withdiagonal elements Tk = ak and non-diagonal elementsT(k,k+1) = rk+1e

−iθk+1 , rk+1 > 0, ∀k = 1, . . . N − 1.

6

0 20 40 60 80 100 120 140-8.12

-8.05

-7.98

-7.91

-7.84 F G(C) HOMO

0 20 40 60 80 100 120 140-4.54

-4.52

-4.50

-4.48

-4.46 F G(C) LUMO

0 5 10 15 20 25 30 35-8.12

-8.05

-7.98

-7.91

-7.84 TM G(C) HOMO

0 5 10 15 20 25 30 35-4.54

-4.52

-4.50

-4.48

-4.46 TM G(C) LUMO

0 10 20 30 40 50 60 70

-8.36

-8.32

-8.28

-8.24DP T(A) HOMO

Eige

nene

rgie

s (e

V)

0 10 20 30 40 50 60 70

-4.95

-4.92

-4.89

-4.86

-4.83 DP T(A) LUMO

0 10 20 30 40 50 60 70

-8.36

-8.32

-8.28

-8.24RS A(T) HOMO

0 10 20 30 40 50 60 70

-4.95

-4.92

-4.89

-4.86

-4.83 RS A(T) LUMO

0 10 20 30 40 50 60 70 80 90

-8.36

-8.32

-8.28

-8.24CS T(A) HOMO

0 10 20 30 40 50 60 70 80 90

-4.95

-4.92

-4.89

-4.86

-4.83 CS T(A) LUMO

0 40 80 120 160 200 240-8.12

-8.05

-7.98

-7.91

-7.84

N

ACS C(G) HOMO

0 40 80 120 160 200 240-4.54

-4.52

-4.50

-4.48

-4.46 ACS C(G) LUMO

FIG. 1. Eigenspectra of F G(C), TM G(C), DP T(A), RS A(T), CS T(A), ACS C(G) polymers, for the HOMO regime andthe LUMO regime, for a few generations. The horizontal axis shows the number of monomers in the polymer N .

0 20 40 60 80 100 120 140

-8.25

-8.10

-7.95

-7.80 F G(A) HOMO

0 20 40 60 80 100 120 140-4.95

-4.80

-4.65

-4.50

-4.35F G(A) LUMO

0 5 10 15 20 25 30 35

-8.25

-8.10

-7.95

-7.80TM A(G) HOMO

0 5 10 15 20 25 30 35-4.95

-4.80

-4.65

-4.50

-4.35TM A(G) LUMO

0 10 20 30 40 50 60 70

-8.25

-8.10

-7.95

-7.80 DP A(G) HOMO

Eige

nene

rgie

s (e

V)

0 10 20 30 40 50 60 70-4.95

-4.80

-4.65

-4.50

-4.35DP A(G) LUMO

0 10 20 30 40 50 60 70

-8.25

-8.10

-7.95

-7.80 RS A(G) HOMO

0 10 20 30 40 50 60 70-4.95

-4.80

-4.65

-4.50

-4.35RS A(G) LUMO

0 10 20 30 40 50 60 70 80 90

-8.25

-8.10

-7.95

-7.80 CS A(G) HOMO

0 10 20 30 40 50 60 70 80 90-4.95

-4.80

-4.65

-4.50

-4.35 CS A(G) LUMO

0 40 80 120 160 200 240

-8.25

-8.10

-7.95

-7.80ACS A(G) HOMO

N0 40 80 120 160 200 240

-4.95

-4.80

-4.65

-4.50

-4.35ACS A(G) LUMO

FIG. 2. Eigenspectra of F A(G), TM A(G), DP A(G), RS A(G), CS A(G), ACS A(G) polymers, for the HOMO regime andthe LUMO regime, for a few generations. The horizontal axis shows the number of monomers in the polymer N .

Since T is hermitian, T(k+1,k) = rk+1eiθk+1 . Now,

suppose a diagonal N × N matrix D, with elementsd1 = 1, dk = dk−1e

iθk ,∀k = 2, . . . , N . Then D is uni-

tary, and the similarity transformation T = D−1TDleads to the matrix T with diagonal elements Tk = akand non-diagonal elements T(k,k+1) = rk+1. Hence, thetridiagonal hermitian matrix T has the same eigenvalues

with the tridiagonal real symmetric matrix T , which haspositive non-diagonal entries69. Let us further supposethat T is real. Then, θk = 0 or θk = π depending onwhether T(k,k+1) > 0 or T(k,k+1) < 0. The elements of Dwill be dk = dk−1(±1),∀k = 2, . . . , n. Hence the matrix

T , which has positive entries, has the same eigenvalueswith T , which differs by T in that its off-diagonal ele-

7

-8.1 -8.05 -8 -7.95 -7.90

500

1000

-4.53 -4.5 -4.470

400

800

-8.1 -8.05 -8 -7.95 -7.90

200

400

-4.53 -4.5 -4.470

400

800

-8.36 -8.33 -8.3 -8.27 -8.240

300

600

-4.95 -4.9 -4.850

1000

2000

-8.36 -8.33 -8.3 -8.27 -8.240

200

400

-4.95 -4.9 -4.850

500

1000

-8.36 -8.33 -8.3 -8.27 -8.240

100

200

-4.95 -4.9 -4.850

70

140

-8.1 -8.05 -8 -7.95 -7.90

50

100

-4.53 -4.5 -4.470

100

200

FIG. 3. Density of states of F G(C), TM G(C), DP T(A), RS A(T), CS T(A), ACS C(G) polymers, for the HOMO and theLUMO regime, for a generation with large N .

-8.3 -8.2 -8.1 -8 -7.90

200

400

-4.9 -4.8 -4.7 -4.6 -4.50

500

1000

-8.3 -8.2 -8.1 -8 -7.90

200

400

-4.9 -4.8 -4.7 -4.6 -4.50

150

300

-8.3 -8.2 -8.1 -8 -7.90

250

500

-4.9 -4.8 -4.7 -4.6 -4.50

400

800

-8.3 -8.2 -8.1 -8 -7.90

250

500

-4.9 -4.8 -4.7 -4.6 -4.50

350

700

-8.3 -8.2 -8.1 -8 -7.90

20

40

60

-4.9 -4.8 -4.7 -4.6 -4.50

50

100

-8.3 -8.2 -8.1 -8 -7.90

50

100

-4.9 -4.8 -4.7 -4.6 -4.50

50

100

FIG. 4. Density of states of F G(A), TM A(G), DP A(G), RS A(G), CS A(G), ACS A(G) polymers, for the HOMO and theLUMO regime, for a generation with large N .

ments have negative signs in arbitrary positions. Finally,if ~v is an eigenvector of T , then D−1~v is an eigenvectorof T .

A. Eigenspectra, Density of States, Energy Gaps

In Figs. 1 and 2, we present the HOMO and LUMOeigenspectra, for increasing N , of I and D polymers, re-

spectively, and in Figs. 3 and 4 the corresponding DOS.For both I and D polymers, we notice that in quasi-periodic polymers the DOS has rather acute subbands,while in fractal polymers the DOS is fragmented andspiky. In Figs. 3 and 4, for illustration purposes, the DOShas been calculated for polymers made of a very big num-ber of monomers N . This value is shown in each panel.Of course, the persistence length of DNA is around 50nm or 150 base pairs20. On the other hand, if we stretch

8

and join the DNA of all chromosomes of a single cell,that would give us a length of the order of a meter andwould consist of billions of base pairs.

For I polymers, i.e., polymers made of identicalmonomers (cf. Figs. 1 and 3), we observe that all eigen-values are symmetric relative to the monomer’s on-siteenergy (this, obviously, also holds for the DOS). This ob-servation can be mathematically proven as follows: ForN even, the hamiltonian matrix of a generic I polymeris H = EµI + TGK , where Eµ is the (constant) on-siteenergy, I is the identity matrix and TGK is the Golub-Kahan matrix, containing only the non-diagonal elementsof H, i.e., the HOMO or LUMO hopping integrals tµ,λ.It can easily be shown that TGK = P TBP , where P isthe perfect shuffle matrix and

B =

(O A

AT O

), A =

t1,2 t2,3

t3,4 t4,5. . .

. . .

tN−1,N

.

(23)By performing the Singular Value Decomposition of theupper bidiagonal matrix A, i.e., by writing it as A =USW T , we obtain

B = J

(−S 0

0 D

)JT , J =

1√2

(U U

−W W

). (24)

So, finally,

TGK = P TJ

(−S 0

0 S

)JTP . (25)

Hence, the eigenvalues of TGK are given by the positiveand negative values of the diagonal matrix S, i.e., theyare symmetric around zero70. Hence, since, H = EµI +TGK , the eigenvalues of H are symmetric around Eµ.For N odd, we can add a zero row and a zero columnto TGK so that it is again of even order and follow theaforementioned procedure. Then, two degenerate trivialeigenvalues will appear apart from the symmetric ones71.So, the eigenvalues of H occur by omitting the zero rowand column, hence they are symmetric around Eµ, whichis also an eigenvalue.

For D polymers, i.e., polymers made of differentmonomers (cf. Figs. 2 and 4), the eigenenergies and theDOS gather around the two monomer’s on-site energies.

The energy gap of a monomer is the difference be-tween its LUMO and HOMO levels. The energy gapof a polymer is the difference between the lowest level ofthe LUMO regime and the highest level of the HOMOregime, because we assume that the orbitals - one persite - which contribute to the HOMO (LUMO) band areoccupied (empty), since in both possible monomers thereis an even number of pz electrons contributing to the πstack. In Fig. 5 we present the energy gaps (calculated forlarge N ; cf. Figs. 3 and 4) and the HOMO and LUMO

band limits of all aperiodic polymers examined in thiswork. The G-C (A-T) monomer gap is always greaterthan the gaps of I polymers made of G and C or A andT. D polymers have smaller HOMO-LUMO gaps than Ipolymers (cf. upper panel of Fig. 5). Furthermore, thelower HOMO (LUMO) band limit of D polymers is al-ways between the lower and upper HOMO (LUMO) bandlimit of I polymers consisted of A and T, while the upperHOMO (LUMO) band limit of D polymers is always be-tween the lower and upper HOMO (LUMO) band limitof I polymers consisted of G and C (cf. lower panel ofFig. 5).

B. Mean over time Probabilities

The main aspects of our results for the mean over timeprobabilities for I and D polymers are summarized inFigs. 6 and 7 (where we show only two consecutive gener-ations) and in Figs. A.1 and A.2 in Appendix A (where weshow many consecutive generations), for some examplecases. We suppose that the extra carrier is initially placedat the first monomer. A general observation is that usu-ally these probabilities are distributed to monomers closeto the one the carrier was initially placed at.

The mean over time probabilities of finding the extracarrier at each monomer of a polymer depends on thesequence on-site energies and magnitude of hopping pa-rameters between successive monomers. This can moreeasily be seen in I polymers (cf. Fig. 6), where onlythe hopping integrals affect the energy structure. Forthe Thue-Morse G(C) polymers, the probabilities arepalindromic for odd generation numbers. This is dueto the fact that the Hamiltonian matrices of these poly-mers are palindromic, i.e., reading them from top leftto bottom right and vice versa gives the same result18.This property stems directly from the sequence struc-ture. For Cantor Set A(T) polymers, the mean overtime probability for an extra hole is almost totally dis-tributed at the four (or three for generation 1) start-ing monomers, regardless of N , while for an extra elec-tron the probabilities are almost semi-palindromic, i.e.⟨|Cµ(t)|2

⟩=⟨|CN−µ+1(t)|2

⟩, µ = 2, 4, ..., N − 1. In

this case, even if the sequence structure is the same forHOMO and LUMO, the magnitude of hopping integralshas a stronger effect on the results. Another example isthe Rudin-Shapiro A(T) sequence where the mean overtime probability for an extra electron is almost totallydistributed at the four starting monomers, regardless ofN , while for holes it is basically distributed at monomers1, 2, 3 and 6. Regarding the extra hole in AsymmetricCantor C(G) polymers, the probability is much higherfor monomers 1, 2, 9, 10 of every 32-monomer period.Generally, for I polymers, the mean over time probabili-ties are significant only rather close to the first monomer,although in some cases we observe non-negligible proba-bilities at more distant monomers.

9

F G

(C)

F C

(G)

TM G

(C)

TM C

(G)

DP G

(C)

DP C

(G)

RS G

(C)

RS C

(G)

CS G

(C)

CS C

(G)

ACS

G(C

)AC

S C

(G) --

F A

(T)

F T

(A)

TM A

(T)

TM T

(A)

DP A

(T)

DP T

(A)

RS A

(T)

RS T

(A)

CS A

(T)

CS T

(A)

ACS

A(T

)AC

S T

(A) --

F G

(A)

TM A

(G)

DP A

(G)

RS A

(G)

CS A

(G)

ACS

A(G

)

2,9

3,0

3,1

3,2

3,3

3,4

3,5

3,6

A-T monomer gap

G-C monomer gap

HO

MO

-LU

MO

gap

(eV)

F G

(C)

F C

(G)

TM G

(C)

TM C

(G)

DP G

(C)

DP C

(G)

RS G

(C)

RS C

(G)

CS G

(C)

CS C

(G)

ACS

G(C

)AC

S C

(G) --

F A

(T)

F T

(A)

TM A

(T)

TM T

(A)

DP A

(T)

DP T

(A)

RS A

(T)

RS T

(A)

CS A

(T)

CS T

(A)

ACS

A(T

)AC

S T

(A) --

F G

(A)

TM A

(G)

DP A

(G)

RS A

(G)

CS A

(G)

ACS

A(G

)

-8,5

-8,0

-7,5

-7,0

-6,5

-6,0

-5,5

-5,0

-4,5

HOMOhigh HOMOlow LUMOlow LUMOhigh

HOMO band limits

LUMO band limits

HO

MO

, LU

MO

ban

d lim

its

FIG. 5. Energy gaps (left) as well as HOMO and LUMO band limits (right), at the large N limit, for all aperiodic polymersconsidered in this work. Squares: I Polymers, i.e., made of the same monomer. Blue stars: D Polymers, i.e., made of differentmonomers. The green (purple) dashed line shows the energy gap of the G-C (A-T) base pair.

0 4 8 12 16 2010-8

10-6

10-4

10-2

100F G(C) HOMO

6 7

0 4 8 12 16 2010-3

10-2

10-1

100

F G(C) LUMO 6 7

0 2 4 6 8 10 12 14 1610-1010-810-610-410-2100

TM G(C) HOMO 3 4

0 2 4 6 8 10 12 14 16

10-2

10-1

100TM G(C) LUMO 3

4

0 2 4 6 8 10 12 14 1610-4

10-3

10-2

10-1

100

DP T(A) HOMO

Mea

n Pr

obab

ilitie

s

3 4

0 2 4 6 8 10 12 14 1610-8

10-6

10-4

10-2

100

DP T(A) LUMO

3 4

0 2 4 6 8 10 12 14 1610-4

10-3

10-2

10-1

100

RS A(T) HOMO 3 4

0 2 4 6 8 10 12 14 16

10-8

10-6

10-4

10-2

100RS A(T) LUMO 3

4

0 3 6 9 12 15 18 21 24 27

10-6

10-4

10-2

100

CS T(A) HOMO 2 3

0 3 6 9 12 15 18 21 24 2710-5

10-4

10-3

10-2

10-1

100CS T(A) LUMO 2

3

0 2 4 6 8 10 12 14 16

10-8

10-6

10-4

10-2

100

ACS C(G)HOMO

1 2

0 2 4 6 8 10 12 14 1610-3

10-2

10-1

100ACS C(G) LUMO 1

2

FIG. 6. Mean over time probabilities to find the extra carrier at each monomer µ = 1, . . . , N , having placed it initially at thefirst monomer, for two consecutive generations (the number of which is denoted at each panel’s legend,) for G(C), TM G(C),DP T(A), RS A(T), CS T(A), ACS C(G) polymers, for HOMO and LUMO.

Generally, for D polymers, the mean over time prob-abilities are almost negligible further than the firstmonomer. An exception is the Rudin-Shapiro A(G)sequence where the probabilities for both HOMO andLUMO are almost totally distributed at the three start-ing monomers of each polymer, regardless its length.Likewise, the mean over time probability for the extraelectron in Cantor Set A(G) polymers is almost totallydistributed at the first and third monomer of each poly-mer, regardless its length. An extra electron in Double-Period A(G) reaches somehow more distant monomers.

C. Frequency Content

The frequencies involved in charge transfer are givenby Eq. (16). Hence, the maximum frequency is deter-mined by the maximum difference of eigenenergies, i.e.,by the upper and lower limits of the HOMO or LUMOband (calculated for large N ; cf. Figs. 3 and 4). Thesemaximum frequencies for all studied polymers are shownin Fig. 8.

The Fourier spectra of the time-dependent probabilityto find an extra electron or hole at each monomer aregenerally in the THz regime, mainly in the FIR and MIRpart of the electromagnetic spectrum. When the domi-

10

0 4 8 12 16 2010-8

10-6

10-4

10-2

100

F G(A) HOMO 6 7

0 4 8 12 16 2010-15

10-10

10-5

100

F G(A) LUMO 6 7

0 2 4 6 8 10 12 14 1610-9

10-6

10-3

100

TM A(G) HOMO 3 4

0 2 4 6 8 10 12 14 1610-18

10-12

10-6

100

TM A(G) LUMO 3 4

0 2 4 6 8 10 12 14 1610-8

10-6

10-4

10-2

100

Mea

n Pr

obab

ilitie

s

DP A(G) HOMO 3 4

0 2 4 6 8 10 12 14 1610-16

10-12

10-8

10-4

100DP A(G) LUMO

3 4

0 2 4 6 8 10 12 14 1610-12

10-9

10-6

10-3

100

RS A(G) HOMO 3 4

0 2 4 6 8 10 12 14 1610-20

10-15

10-10

10-5

100

RS A(G) LUMO 3 4

0 3 6 9 12 15 18 21 24 2710-8

10-6

10-4

10-2

100

CS A(G) HOMO 2 3

0 3 6 9 12 15 18 21 24 2710-12

10-8

10-4

100

CS A(G) LUMO 2 3

0 2 4 6 8 10 12 14 1610-9

10-6

10-3

100

ACS A(G) HOMO 1 2

0 2 4 6 8 10 12 14 1610-18

10-12

10-6

100

ACS A(G) LUMO 1 2

FIG. 7. Mean over time probabilities to find the extra carrier at each monomer µ = 1, . . . , N , having placed it initially at thefirst monomer, for two consecutive generations (the number of which is denoted at each panel’s legend,) for F G(A), TM A(G),DP A(G), RS A(G), CS A(G), ACS A(G) polymers, for HOMO and LUMO.

nant frequencies, i.e. those with greater Fourier ampli-tudes, are smaller (bigger), the carrier transfer is slower(faster). Extensive examples of the Fourier spectra ofthe probability to find an extra carrier at the first and atthe last monomer, having placed it initially at the firstmonomer, for I and D aperiodic polymers, for the HOMOand the LUMO regime, can be found in Refs.72,73. Re-cently, we have also analyzed18,74 the frequency contentof periodic polymers, using the TB wire model or the TBextended ladder model, including the Fourier spectra, theWMFs and the TWMF as a function of N , with detailsin Refs.75,76.

In Fig. 9 we depict the TWMF as a function of Nfor the various types of aperiodic polymers. We noticethat the TWMF generally stabilizes as the generationnumber increases. In all cases, TWMF are in the region≈ 10−2 − 102 THz.

D. Pure Mean Transfer Rates

Next, we study the pure mean transfer rates from thefirst to the last monomer, k1,N , or from now on, justk. We depict k(N) either for HOMO or for LUMO, forI and D polymers in Fig. 10. In all cases, k(N) is adecreasing function. Generally, the degree of coherenttransfer difficulty is greater for D polymers. Overall, ourresults suggest that I polymers, which are simpler casesin terms of energy intricacy, are more efficient regardingcoherent hole and electron transfer.

We include in each panel of Fig. 10, k(N) of homopoly-

F G

(C)

F C

(G)

F A

(T)

F T

(A)

F G

(A) --

TM G

(C)

TM C

(G)

TM A

(T)

TM T

(A)

TM A

(G) --

DP

G(C

)D

P C

(G)

DP

A(T

)D

P T

(A)

DP

A(G

) --R

S G

(C)

RS

C(G

)R

S A

(T)

RS

T(A

)R

S A(

G) --

CS

G(C

)C

S C

(G)

CS

A(T

)C

S T

(A)

CS

A(G

) --AC

S G

(C)

ACS

C(G

)AC

S A

(T)

ACS

T(A

)AC

S A(

G)

0

20

40

60

80

100

120

HOMOLUMO

max

freq

uenc

y

FIG. 8. The maximum frequency of the Fourier spectrum, forthe HOMO and the LUMO regime of Fibonacci, Thue-Morse,Double Period, Rudin-Shapiro, Cantor Set, Asymmetric Can-tor Set polymers, at the large N limit.

mers (e.g., A...) which are the “champions” among pe-riodic polymers in terms of efficiency of coherent carriertransfer18, i.e., in terms of magnitude of k and of slowerdecrease of k(N). It seems that k(N) of homopolymers isan unreachable limit for aperiodic polymers. Comparingperiodic polymers18 with aperiodic polymers in terms ofk(N), we realize that although generally periodic poly-mers are more efficient, specific aperiodic polymers canbe better than specific periodic ones.

11

0 20 40 60 80 100 120 14010-2

10-1

100

101

102

G(C) C(G) A(T) T(A) G(A)

HOMO F

0 5 10 15 20 25 30 3510-3

10-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

HOMO TM

0 10 20 30 40 50 60 7010-3

10-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

HOMO DP

0 10 20 30 40 50 60 7010-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

HOMO RS

(THz)

0 10 20 30 40 50 60 70 80 9010-3

10-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

HOMO CS

0 40 80 120 160 200 24010-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

HOMO ACS

0 20 40 60 80 100 120 14010-2

10-1

100

101

102

G(C) C(G) A(T) T(A) G(A)

LUMO F

TWMF

0 5 10 15 20 25 30 3510-3

10-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

LUMO TM

0 10 20 30 40 50 60 7010-3

10-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

LUMO DP

0 10 20 30 40 50 60 7010-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

LUMO RS

0 10 20 30 40 50 60 70 80 9010-3

10-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

LUMO CS

N

0 40 80 120 160 200 24010-2

10-1

100

101

102

G(C) C(G) A(T) T(A) A(G)

LUMO ACS

FIG. 9. Total Weighted Mean Frequency (TWMF) as a function of the number of monomers N in the polymer, having placedthe carrier initially at the first monomer, for Fibonacci, Double Period, Rudin-Shapiro, Cantor Set, Asymmetric Cantor Setpolymers, for the HOMO (upper half) and the LUMO (bottom half) regime. D Polymers, i.e., made of different monomers,are denoted by blue stars.

In each panel of Fig. 10, we also take the best ofaperiodic polymers in terms of k(N) and shuffle ran-domly the sequence of its monomers. In all cases, ex-cept for Cantor Set HOMO, this random shuffle de-teriorates severely k(N). For Cantor Set, A(T) andT(A) have identical k(N) because the Cantor Set rulesfor A(T) and T(A) produce equivalent polymers, cf.Eq. (7). For equivalent polymers, k(N) from the firstto the last monomer are identical, cf. Eq. (22). Forexample, TAT ≡ ATA, TATAAATAT ≡ ATATTTATA,TATAAATATAAAAAAAAATATAAATAT ≡ ATATT-TATATTTTTTTTTATATTTATA and so on. Similarly,the Cantor Set rules for G(C) and C(G) produce equiv-alent polymers, which have identical k(N). In CantorSet HOMO, the best sequences in terms of k(N) are

A(T) and T(A), where the hopping integrals involvedare tAA = tTT = − 8 meV, tAT = 20 meV, tTA = 47meV, and we have just one on-site energy, that of A-T. From these hopping integrals, tAA has the smallestabsolute value. Given the structure of the Cantor Set se-quences, making the random shuffle, the number of tAA

decreases, while the numbers of the bigger hopping inte-grals, tAT and tTA increase. For this reason, in CantorSet HOMO, the random shuffle increases k(N). In Can-tor Set LUMO, this argument is inverted because nowthe best sequences in terms of k(N) are G(C) and C(G),where the hopping integrals involved are tGG = tCC =20 meV, tGC = − 10 meV, tCG = − 8 meV, and wehave just one on-site energy, that of G-C. In this case,the random shuffle decreases the number of the bigger

12

0 30 60 90 120 15010-32

10-24

10-16

10-8

100

A...G(C)C(G)A(T)T(A)G(A)r A(T)

HOMO F

0 5 10 15 20 25 30 3510-18

10-12

10-6

100

HOMO TMA...G(C)C(G)A(T)T(A)A(G)r A(T)

0 10 20 30 40 50 60 7010-30

10-24

10-18

10-12

10-6

100

A...G(C)C(G)A(T)T(A)A(G)r A(T)

HOMO DP

0 10 20 30 40 50 60 7010-40

10-30

10-20

10-10

100

A...G(C)C(G)A(T)T(A)A(G)r A(T)

Hz)

HOMO RS

0 10 20 30 40 50 60 70 80 9010-12

10-9

10-6

10-3

100

A...G(C)C(G)A(T)T(A)A(G)r A(T)

HOMO CS

0 40 80 120 160 200 24010-40

10-30

10-20

10-10

100

A...G(C)C(G)A(T)T(A)A(G)r A(T)

HOMO ACS

0 30 60 90 120 15010-32

10-24

10-16

10-8

100

k (P

C...G(C)C(G)A(T)T(A)G(A)r C(G)

LUMO F

0 5 10 15 20 25 30 3510-45

10-36

10-27

10-18

10-9

100

C...G(C)C(G)A(T)T(A)A(G)r C(G)

LUMO TM

0 10 20 30 40 50 60 7010-21

10-14

10-7

100

C...G(C)C(G)A(T)T(A)A(G)r C(G)

LUMO DP

0 10 20 30 40 50 60 7010-5410-4510-3610-2710-1810-9100

C...G(C)C(G)A(T)T(A)A(G)r C(G)

LUMO RS

0 10 20 30 40 50 60 70 80 9010-18

10-12

10-6

100

C...G(C)C(G)A(T)T(A)A(G)r C(G)

LUMO CS

N

0 40 80 120 160 200 24010-50

10-40

10-30

10-20

10-10

100

C...G(C)C(G)A(T)T(A)A(G)r C(G)

LUMO ACS

FIG. 10. Pure mean transfer rates k of Fibonacci, Thue-Morse, Double Period, Rudin-Shapiro, Cantor Set, Asymmetric CantorSet polymers, homopolymers and randomly shuffled aperiodic polymers as a function of the number of monomers N in thepolymer, for the HOMO (upper half) and the LUMO (bottom half) regime. By blue stars we denote D Polymers, i.e., madeof different monomers.

hopping integrals tGG = tCC and decreases the numbersof the smaller hopping integrals tGC and tCG. However,apart from the exception of the Cantor set HOMO, gen-erally speaking, the conclusion is that aperiodic polymersposses some kind of order, i.e., a well-defined construc-tion rule that makes them more efficient than randompolymers in terms of k(N); therefore, when this rule isdestroyed, the transfer efficiency diminishes.

E. Transfer rates in experiments

Comparison of the coherent pure mean transfer ratesk of our prototype system, B-DNA, with experimentallyobtained transfer rates is a rather complicated issue. In

the past, the experimental transfer rates in donor - bridge(DNA) - acceptor systems were obtained using the con-centrations of different products generated e.g. when ahole is (PY) or is not (PN) transferred. The concentra-tions of PY and PN were indirectly measured by meth-ods like polyacrylamide gel electrophoresis and piperi-dine treatment77,78. Although these methods revealedsome aspects of hole transfer like the sequence depen-dence and the ability of transfer, they do not provide thekinetics of hole transfer in DNA79. Although, generally,greater concentration of PY implies greater charge trans-fer, there is no proof that the concentrations of PN andPY are proportional to the degree of transfer.

Quantum mechanically, only a fraction of the carrierreaches the acceptor through the bridge. For the same

13

reason, the definition of transfer time is problematic. Thetransfer rate should depend both on the amount and thespeed of transfer. However, the concentration of PY isnot strictly proportional to the amount of carrier trans-fer and not strictly inversely proportional to the time oftransfer. A more direct experimental approach is time-resolved spectroscopy, e.g. transient absorption, to ob-serve the products of charge transfer79–81.

Our point of view is different, since the quantity weuse, the pure mean transfer rate14, given by Eq. 21, usessimultaneously the magnitude of coherent charge trans-fer and the time scale of the phenomenon. However,our method applies to coherent transfer only and can-not cover incoherent mechanisms like thermal hopping.

It is a common assertion in the literature that whenthe fall of the transfer rate with respect to the lengthof a given DNA segment is described by an exponentialfit, the mechanism of transfer is superexchange, whereaswhen it is described by a power law fit, the mechanismof transfer is multi-step hopping. However, we stressthat the fitted parameters produced this way should betreated with care, especially when it comes to attribut-ing them to specific mechanisms. For example, in Ref.79,where the hole transfer kinetics of various short DNAsegments were experimentally investigated with time-resolved spectroscopy, the authors present an exponen-tial decay length β = 1.6 A−1 by fitting the experi-mental hole transfer rates of G(A)nG DNA oligomers(n = 0, 1, 2) to the exponential law K = K0e−βd, whered is the charge transfer distance, i.e., d = 3.4 × (N − 1)A. Using the transfer rate values of Ref.79, we observedthat, although β, determined as the slope of the linear fitln(K) = ln(K0)− βd is indeed ∼= 1.6 A−1, a direct expo-nential fit gives β ∼= 1.3 A−1, suggesting that the law ofdecay is not exactly exponential. On the contrary, the fitsof our theoretically obtained pure mean transfer rates, k,for the same system, give β ∼= 1.84 A−1 for β determinedas the slope of the linear fit ln(k) = ln(k0) − βd, andβ ∼= 1.79 A−1 for a direct exponential fit k = k0e−βd,suggesting closer convergence to an exponential decay.Similarly, in Ref.82, the authors experimentally study,with time-resolved spectroscopy, hole transfer through(GA)n and (GT)n sequences, where n = 2-12 is thenumber of repetition units. The authors fitted the ob-tained transfer rates to the power law K = K

′

0N−η,where N is the number of hopping steps between gua-nines (in our notation, N = N

2 − 1), reported the sameexponent for both sequences, i.e. η = 2, and suggestedthat this value provides evidence that the long-distancehole transfer occurs by multi-step hopping between gua-nines. From the rate values provided in Table I of Ref.82,we observed that, although η as a slope of the linear

fit ln(K) = ln(K

′

0

)− η ln(N ) is indeed 2 for both se-

quences, a direct power law fit yields η ∼= 1.4 for (GA)nand η ∼= 1.3 for (GT)n, suggesting that the rate decaydoes not follow exactly a power law. On the contrary,the fits of our theoretically obtained pure mean trans-fer rates, k, for (GA)n, give η ∼= 1.40 for η determined

as the slope of the linear fit ln(k) = ln(k

′

0

)− η ln(N ),

and η ∼= 1.56 A−1 for a direct power law fit k = k′

0N−η.The respective values for (GT)n are η ∼= 2 for both fits.Hence, our theoretical results suggest that the fall of k,as the length of the bridge increases, convergences to apower law and that the fall of the transfer rate is lesssteep when purines are on the same strand compared tothe case when purines are crosswise.

DNA is a dynamical structure, i.e., the geometry isnot fixed. Large variations of the TB parameters areexpected in real situations and also, large variations ofthe TB parameters have been obtained by different the-oretical methods by different authors, cf. e.g. Ref.14

and references therein. Hence, the parameters any TBmodel uses have to be utilized with care. In Ref.83, theauthors report experimentally deduced (by transient ab-sorption spectroscopy) charge separation rates, in cappedAn (n =1-7) and A3Gn (n =1-19) DNA hairpins with astilbenedicarboxamide hole donor and a stilbenedietherhole acceptor. We computed our theoretical coherentpure mean transfer rates, k, for the same systems with amodified parametrization: tAA → 1.6tAA, tAG → 2.1tAG,tGG → 2.25tAG (cf. Table II). In order to mimic thedonor and the acceptor, we added two sites at the endsof the TB chain, with on-site energies Edon = EA−T −0.1eV, Eac = EG−C + 0.1 eV. We used for the hopping in-tegral from the donor (last base pair) to the first basepair (acceptor) 100 meV (250 meV). Our results, alongwith the experimental ones, are depicted in Fig. 11.Apart from the A1 and A2 systems, for which we findmuch larger rates, the pure mean transfer rates k are ofthe same order of magnitude, in good quantitative agree-ment with the experimental transfer rates K. Actually,the same sequences An (n =1-7) and A3Gn (n =1-19)analyzed in Ref.83 had also been analyzed by the samegroup in Ref.84. In Ref.84, the authors mention a timeresolution of ca. 180 fs. Hence, roughly, only transferrates K < (1/180) PHz ≈ (1/200) PHz = 5 × 10−3 PHzcan be detected by this technique.

V. CONCLUSION

We systematically studied the energy structure andthe coherent transfer of an extra carrier, electron orhole, along various categories of binary quasi-periodic(Fibonacci, Thue-Morse, Double-Period, Rudin-Shapiro)and fractal (Cantor Set, Asymmetric Cantor Set) poly-mers consisting of either the same monomer (I polymers)or different monomers (D polymers), using the TB wiremodel and B-DNA as a prototype system.

Regarding the energy structure of the polymers, wecalculated HOMO and LUMO eigenspectra and the den-sity of states, as well as the HOMO-LUMO gap. Theeigenenergies lie around the monomers’ on-site energies.We demonstrated that for I polymers, the eigenenergiesare always symmetric relative to the (constant) monomer

14

0 5 10 15 2010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

An expAn

A3Gn expA3Gn

k (P

Hz)

N

FIG. 11. Comparison of experimental hole transfer rates Kfor An and A3Gn segments83 (full circles) with our theoreti-cal coherent pure mean transfer rates k (empty circles), as afunction of the number of monomers in the polymer N . TheTB parametrization is described in the main text.

on-site energy. For both I and D polymers, in quasi-periodic cases the DOS has rather acute subbands, whilein fractal cases it is fragmented and spiky. D polymersposses smaller HOMO-LUMO gaps than I polymers andtheir band limits lie within the energy regions defined bythe respective limits of I polymers.

Next, we studied the mean over time probabilities tofind an extra hole or electron at each monomer of thepolymer, having it initially placed at the first monomer.For I polymers, the mean over time probabilities are sig-nificant only rather close to the first monomer, althoughin some cases we observe non-negligible probabilities atmore distant monomers. For D polymers, the mean overtime probabilities are generally negligible further thanthe first monomer.

Furthermore, we determined the frequency content ofcoherent extra carrier transfer via the total weightedmean frequency of the polymer, using the weighted mean

frequencies of the Fourier spectra that correspond to theprobabilities to find the carrier at each monomer. Weshowed that, in all cases, the TWMF lies in the THzregime, ≈ 10−2 − 102 THz, and generally stabilizes aftera few generations.

The study of the pure mean transfer rates, k(N), showsthat I polymers, which are simpler cases in terms of en-ergy intricacy, are more efficient than D polymers re-garding coherent hole and electron transfer. Comparingperiodic18 and aperiodic polymers reveals that althoughgenerally periodic polymers are more efficient, particularaperiodic polymers can be better than particular periodicones. However, the structurally simplest periodic poly-mers, i.e., the homopolymers18, represent an unreachablelimit for all aperiodic polymers. Furthermore, a randomshuffle of a quasi-periodic or fractal monomer sequencedestroys the deterministic character of its constructionrules, thus leading to vanishing transfer rates. As far ascomparison with experiments is concerned, large varia-tions of the TB parameters are expected in real situa-tions, hence modifications are necessary. Using a modi-fied parametrization, we were able to find hole pure meantransfer rates k of similar magnitude with experimentaltransfer rates K obtained by time-resolved spectroscopy.

ACKNOWLEDGEMENTS

M. Mantela wishes to thank the State ScholarshipsFoundation (IKY): This research is co-financed byGreece and the European Union (European Social Fund-ESF) through the Operational Programme “Human Re-sources Development, Education and Lifelong Learning”in the context of the project “Strengthening HumanResources Research Potential via Doctorate Research”(MIS-5000432), implemented by the State ScholarshipsFoundation (IKY). K. Lambropoulos wishes to acknowl-edge support by the Hellenic Foundation for Researchand Innovation (HFRI) and the General Secretariat forResearch and Technology (GSRT), under the HFRI PhDFellowship grant (GA no 260).

Appendix A: Appendix

15

0.0 0.2 0.4 0.6 0.8 1.010-12

10-9

10-6

10-3

100

F G(C) HOMO

123456789

0.0 0.2 0.4 0.6 0.8 1.010-12

10-9

10-6

10-3

100

TM G(C) HOMO

12345

0.0 0.2 0.4 0.6 0.8 1.010-5

10-4

10-3

10-2

10-1

100

DP T(A) HOMO

123456

0.0 0.2 0.4 0.6 0.8 1.0

10-6

10-4

10-2

100

123456

RS A(T) HOMO

Prob

abilit

ies

0.0 0.2 0.4 0.6 0.8 1.010-8

10-6

10-4

10-2

100

CS T(A) HOMO

1234

0.0 0.2 0.4 0.6 0.8 1.010-28

10-21

10-14

10-7

100

ACS C(G) HOMO

1234

0.0 0.2 0.4 0.6 0.8 1.010-4

10-3

10-2

10-1

100

Mean

F G(C) LUMO

123456789

0.0 0.2 0.4 0.6 0.8 1.010-3

10-2

10-1

100

TM G(C) LUMO

12345

0.0 0.2 0.4 0.6 0.8 1.010-9

10-6

10-3

100

DP T(A) LUMO

123456

0.0 0.2 0.4 0.6 0.8 1.010-16

10-12

10-8

10-4

100

RS A(T) LUMO

123456

0.0 0.2 0.4 0.6 0.8 1.010-6

10-4

10-2

100

( -1)/(N -1)

CS T(A) LUMO

1234

0.0 0.2 0.4 0.6 0.8 1.0

10-6

10-4

10-2

100

ACS C(G) LUMO

1234

FIG. A.1. Mean over time probabilities to find the extra carrier at each monomer µ = 1, . . . , N , having placed it initially atthe first monomer, for some consecutive generations, for G(C), TM G(C), DP T(A), RS A(T), CS T(A), ACS C(G) polymers,for the HOMO (upper half) and the LUMO (bottom half).

16

0.0 0.2 0.4 0.6 0.8 1.010-10

10-8

10-6

10-4

10-2

100

F G(A) HOMO

123456789

0.0 0.2 0.4 0.6 0.8 1.010-12

10-8

10-4

100

TM A(G) HOMO

12345

0.0 0.2 0.4 0.6 0.8 1.010-12

10-8

10-4

100

DP A(G) HOMO

123456

0.0 0.2 0.4 0.6 0.8 1.010-35

10-28

10-21

10-14

10-7

100

RS A(G) HOMO

123456

Prob

abilit

ies

0.0 0.2 0.4 0.6 0.8 1.010-9

10-6

10-3

100

CS A(G) HOMO

1234

0.0 0.2 0.4 0.6 0.8 1.010-24

10-18

10-12

10-6

100

ACS A(G) HOMO

1234

0.0 0.2 0.4 0.6 0.8 1.010-16

10-12

10-8

10-4

100

F G(A) LUMO

123456789

Mean

0.0 0.2 0.4 0.6 0.8 1.010-40

10-30

10-20

10-10

100

TM A(G) LUMO

12345

0.0 0.2 0.4 0.6 0.8 1.010-20

10-15

10-10

10-5

100

DP A(G) LUMO

123456

0.0 0.2 0.4 0.6 0.8 1.010-50

10-40

10-30

10-20

10-10

100

RS A(G) LUMO

123456

0.0 0.2 0.4 0.6 0.8 1.010-15

10-12

10-9

10-6

10-3

100

( -1)/(N -1)

CS A(G) LUMO

1234

0.0 0.2 0.4 0.6 0.8 1.010-35

10-28

10-21

10-14

10-7

100

1234

ACS A(G) LUMO

FIG. A.2. Mean over time probabilities to find the extra carrier at each monomer µ = 1, . . . , N , having placed it initially at thefirst monomer, for some consecutive generations, for F G(A), TM A(G), DP A(G), RS A(G), CS A(G), ACS A(G) polymers,for the HOMO (upper half) and the LUMO (bottom half).

17

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