The interplay between electronic coupling and vibrational motion in Charge and Energy Transfer

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University of Pisa

Doctoral Thesis

The interplay between electronic

coupling and vibrational motion in

Charge and Energy Transfer

Author:

Lorenzo Cupellini

Supervisor:

Prof. Benedetta Mennucci

A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy

in the

Chemistry and Materials Sciences

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Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.

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UNIVERSITY OF PISA

Abstract

Department of Chemistry and Industrial Chemistry Chemistry and Materials Sciences

Doctor of Philosophy

The interplay between electronic coupling and vibrational motion in Charge and Energy Transfer

by Lorenzo Cupellini

A faithful description of transfer processes in molecular systems requires the accurate determination of two interactions, namely, the electronic cou-pling and the coucou-pling to vibrational degrees of freedom. Both of these interactions are needed for the transfer to occur, and their relative magni-tude determines which mechanism the transfer dynamics will follow. In this work, we studied how the electronic coupling and the vibrational fluctua-tions contribute to charge and energy transfer processes. Particular attention was given to the reproduction of experimentally accessible quantities from first-principles simulations. We show that the employed quantum-chemical methods are a promising tool to describe energy and charge transfer in a wide range of systems.

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Introduction

Photoinduced charge and energy transfer processes characterize many of the physical and chemical phenomena that occur in condensed phase. Organic photovoltaics rely on photoinduced charge transport to transform electro-magnetic radiation into electrical energy. A much more complicated frame-work is present in nature, where photosynthetic organisms have developed a large variety of antenna complexes, that are able to absorb sunlight and funnel the energy to the reaction centers. There, charge transfer reactions ultimately transform in chemical energy every single photon that hit the an-tenna complexes.

Quantum mechanics rules the world of molecular systems, where charge and energy transfer take place. On one hand, the properties of molecules can be only understood with a quantum mechanical treatment. On the other hand, the very nature of transfer processes is modeled by quantum mechanics. Two important factors regulate electronic transfers across space. The first is the interaction, named electronic coupling, between electronic wavefunctions lo-calized in different regions of space, for example on different molecules. The second factor is the interaction between the electronic wavefunction and the large number of nuclear vibrations, which is commonly called vibronic cou-plingor electron-phonon coupling. Not only both these interactions are essen-tial for the transfer process to occur, but their relative magnitude also deter-mines its dynamics.

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The methods of quantum chemistry are nowadays able to accurately solve the Schr¨odinger equation for molecules, and predict their properties. How-ever, their application to transfer processes has gained importance only in the last years. The quantum chemical methods employed to model energy and charge transfer are in evolution. The complexity of the supramolecular systems where the transfer normally occurs, and the presence of additional interactions with the embedding environment, still represent a great chal-lenge to the computational methods.

This work aims at devising reliable computational methods and protocols to describe charge and energy transfer in molecular or supramolecular systems. The main objectives will be the accurate determination of electronic coupling and vibronic interactions, and the reproduction of observable that can be directly compared to experiments.

The Thesis is organized as follows:

Chapter 2 is focused on the theoretical characterization of electronic coupling in energy and charge transfer, and the computational strategies employed to evaluate this quantity. A special attention is reserved to a class of diabatiza-tion schemes, that allow the exact determinadiabatiza-tion of the electronic coupling, within the approximations of the quantum chemical method.

Chapter 3 presents the density matrix formalism in which the vibronic cou-pling can be understood as a spectral density function; the relation between spectral density and vibronic lineshape is then discussed. Finally I give an overview of the most commonly known methods to determine the spectral density.

Chapter 4 presents the various regimes of energy and charge transfer. A detailed description of the commonly employed weak coupling limit is given, as well as an overview of the other limits. A special attention is given to the

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modified Redfield theory and to the description of the vibronic lineshape in coupled excitations.

Chapter 5 presents the results obtained for the description of photoprotec-tion by triplet quenching in the CP29 minor light harvesting complex of higher plants.

Chapter 6 is focused on the description of the excitonic Hamiltonian of the LH2 antenna complex of purple bacteria. An explanation of the temperature dependence of the spectroscopic properties is also achieved.

Chapter 7 describes the combined experimental and computational efforts to characterize the mechanism of photoinduced charge separation in a molec-ular dyad composed by a BODIPY unit and a C60fullerene.

Finally, Chapter 8 presents the results obtained on the photophysical and spectroscopic properties of the PCP antenna. Here, the excitonic Hamilto-nian is combined with energy transfer calculations to obtain both spectral and dynamical properties.

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Chapter 2

The Electronic Coupling in Charge

and Energy Transfer

The electronic coupling is a key quantity in determining both the dynamics and the qualitative mechanism of energy and charge transfer. As we will see, the magnitude of the electronic coupling, compared to the exciton-phonon interactions, determines whether states localized on different sites are mixed to generate delocalized exciton states. An accurate computation of the elec-tronic coupling is mandatory, not only to predict the rates of energy and charge transfer processes, but also to determine their qualitative behavior. In this Chapter, I will present an overview on the definition of the electronic coupling in energy and charge transfer, as well as on the most commonly used methods to compute the coupling.

2.1 Singlet Electronic Coupling

The off-diagonal matrix element of the electronic Hamiltonian between sin-glet excited states is named sinsin-glet electronic coupling, and represents the electronic interaction between two excited singlet states localized on differ-ent sites, e.g. A and B. Such coupling is thus formally defined as VAB =

hA∗B| ˆHel|AB∗i, where ˆHelis the total electronic Hamiltonian of the system,

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Within a first-order perturbation expansion, it can be shown that the singlet electronic coupling VABbetween two sites A and B can be written as a sum

of three terms: [1,2]

VAB = VCoul+ Vexch+ Vovlp (2.1)

where VCoul and Vexch are respectively the Coulomb and exchange

interac-tions between the transition density matrices of A and B, whereas Vovlpis a

term proportional to the overlap of the transition densities.

V_Coul= Z dr1dr2 ρtr∗A,0i(r1) 1 r12 ρtr_B,0j(r2) (2.2) V_exch= − Z dr1dr2γA,0itr∗ (r1, r2) 1 r12 γ_B,0jtr (r1, r2) (2.3) V_ovlp= − Z dr1dr2ρtr∗A,0i(r1)ρtrB,0j(r2) (2.4)

Here, r12= |r1− r2|, and the transition density matrix between states 0 and

jis defined as:

γ_0jtr(r, r0) = N Z

Ψ∗_j(r, r2, . . . rN)Ψ0(r0, r2, . . . rN) dr2. . . drN (2.5)

and the transition density ρtr

0jis equal to the diagonal of γ0jtr: ρtr0j(r) = γtr0j(r, r).

Within the time-dependent DFT (TD-DFT) formalism, V_exch is replaced by an exchange-correlation term Vxcdirectly dependent on the

exchange-cor-relation term of the chosen DFT functional: [1]

V =

Z

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In order to achieve a better understanding of the main aspects of the singlet electronic coupling, let us consider a simple four-orbital model. [2] Within this model, we consider two occupied orbitals, φAand φB, localized

respec-tively on sites A and B, and their virtual counterparts φ0

Aand φ0B. To make a

further simplification, let us assume that the orbitals located on different sites are orthogonal. The Slater determinant wavefunctions of the states |AB∗_i

and |A∗_Bi_{are thus:}

|A∗Bi =φAφ¯0AφBφ¯B

|AB∗i =φAφ¯AφBφ¯0B

where the bar over the orbital ( ¯φ) represents a β spin orbital, i.e. ¯φ(r, s) = ϕ(r)β(s), while the other orbitals are α orbitals, φ(r, s)=ϕ(r)α(s). The deter-minants |AB∗_i_{and |A}∗_Bi_{differ in the occupation of two spin orbitals; from}

Slater’s rules, only the two-electron terms of the Hamiltonian survive:

VAB = Z Z dr1dr2ds1ds2φ¯0∗A(r1, s1) ¯φA(r1, s1) 1 r12 ¯ φ∗_B(r2, s2) ¯φ0B(r2, s2)− Z Z dr1dr2ds1ds2φ¯0∗A(r1, s1) ¯φ0B(r1, s1) 1 r12 ¯ φ∗_A(r2, s2) ¯φB(r2, s2) (2.7)

After integration over the spin coordinates, one obtains the two terms of the singlet coupling:

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VAB = Z Z dr1dr2ϕ0∗A(r1)ϕA(r1) 1 r12 ϕ∗_B(r2)ϕ0B(r2)− Z Z dr1dr2ϕ0∗A(r1)ϕ0B(r1) 1 r12 ϕ∗_A(r2)ϕB(r2) (2.8)

The two terms of this equation indeed represent V_Couland V_exchof equations (2.2) and (2.3), where in this simplified model γtr

X(r1, r2) = ϕ∗X(r1)ϕ0X(r2)

(X = A, B). Obviously, since the orbitals of A and B were assumed to be or-thogonal to each other, the overlap term of the coupling is zero. In Figure 2.1 the two diagrams corresponding to the integrals of eq. (2.8) are represented.

D*

A

a.

D*

A

b.

Figure 2.1: Pictorial representation of the four-orbital model of the singlet electronic coupling between a donor state D∗_{and an acceptor state A. The}

Coulomb coupling (a) and the exchange coupling (b) are depicted, along with the relevant orbital superpositions and interactions.

It is worth noting that the results of equations (2.2) and (2.3) are completely general, and not limited to a single excitation formalism. [3, Chapter 14] To prove this point in our four-orbital model, let us consider a doubly excited state, e.g. |AB∗∗_{i =}

φ_Aφ¯_Aφ0_Bφ¯0_B

. Now, the determinants |A∗_Bi_{and |AB}∗∗_i

differ by three orbital substitutions. Therefore, all terms of the matrix ele-ment hA∗_{B| ˆ}_H

el|AB∗iare zero. Because the density matrix is a one-electron

operator, γtr_(r

1, r2)only contains the contributions from single excitations,

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The first-order expression allows the dissection of the coupling in three terms, of which Vexch and Vovlp are short-range contributions. [1] As VCoul

is the most important term in the singlet coupling, the remainder of this Sec-tion shall be focused on the calculaSec-tion of this term, leaving the discussion of short-range terms to Section 2.2.

2.1.1 The Coulomb coupling

The Coulomb coupling (2.2) is by far the most important contribution to the electronic coupling between singlet states. At sufficiently large distances be-tween sites, VCoulcan be approximated by the well-known multipolar

expan-sion; because the transition densities sum to zero, the first non-zero order is the dipole–dipole term (See eq. 2.9).

The calculation of VCoulcan be performed by expressing the transition

densi-ties in a basis of atomic orbitals and analytically calculating the Coulomb in-tegral. [4] However, the first calculation of VCoulfrom the integration of

tran-sition densities was developed by Krueger et al. by using a numerical inte-gration over a three-dimensional grid. [5] This numerical approach is called Transition Density Cube (TDC), and in principle gives the exact VCoulwith

an appropriately chosen grid, and has been extensively employed. [6–10] The analytical integration, however, offers much more stable results, and a faster calculation time. Moreover, it allows the inclusion of explicit screening of the Coulomb interaction by a dielectric medium. [4,11]

2.1.1.1 Approximate calculation of the Coulomb coupling

The first useful approximation of the Coulomb coupling consist in the simple dipolar expansion of the interaction (Point Dipole Approximation, or PDA).

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The Coulomb coupling is thus computed as the electrostatic interaction be-tween the transition dipole moments of the two sites A and B:

V_Coul(PDA)= µA· µB R3 AB −3(µA· RAB)(µB· RAB) R5 AB (2.9)

where RAB is the vector distance between A and B, and µA and µB are

the electronic transition dipole moments of A and B. The dipolar term in eq. (2.9) yields the well known R−3 _{asymptotic dependence of the singlet}

electronic coupling. Where applicable, the PDA has the clear advantage of only needing experimental data, namely the transition dipole moments and the distance between the chromophore centers, provided that the orientation of the transition dipole moments is known. For this reason, the PDA has been widely employed, even outside of its applicability range, to compute elec-tronic couplings in many situations. [12–16] Anyway, particular care should be taken when using the PDA, which can result in very large overestimates of the Coulomb coupling. [17, 18] As a rule of thumb, the PDA should be used only when RABis much larger than the spatial extent of the electronic

transition.

Another widely used method to compute V_Coulis based on the projection of the transition densities ρtr

Aand ρtrBonto atomic transition charges (TrCh). The

Coulomb coupling is then computed as the electrostatic interaction between those charges: V_Coul(TrCh)=X i∈A j∈B qiqj Rij (2.10)

where the indices i and j run over the atoms of A and B, respectively, qiand

qjare the transition charges of atoms i and j, and Rijis the distance between

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a long time to compute Coulomb couplings. [19, 20] Arguably, a definition of atomic charges that is physically accurate and adequate for electrostatic interaction is the one based on electrostatic potential (ESP) fitting. The cal-culation of the Coulomb coupling with ESP transition charges (TrEsp) was first presented in 2006, [21] and represents now a widely used method, par-ticularly in conjunction with molecular dynamics simulations. [17,22,23] The transition charges are usually computed once a relaxed chromophore struc-ture, [21] making the approximation that the transition charges are the same at every geometry. This makes the TrEsp method a fast alternative to cal-culate a large number of electronic couplings, for example on a molecular dnamics simulation. [22–24]

2.2 Triplet Electronic Coupling

The general definition of the electronic coupling given for singlet states in Section 2.1 is also applicable to triplet excited states, the only difference be-ing the spin symmetry, that nullifies the transition density. Resortbe-ing for simplicity to the same four-orbital model of Section 2.1, the interacting triplet states of two sites A and B can be written as:

|A∗Bi =φAφA0φBφ¯B

|AB∗i =φAφ¯AφBφB0

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VAB = Z Z dr1dr2ds1ds2φ0∗A(r1, s1) ¯φA(r1, s1) 1 r12 ¯ φ∗_B(r2, s2) ¯φ0B(r2, s2)− Z Z dr1dr2ds1ds2φ0∗A(r1, s1)φ0B(r1, s1) 1 r12 ¯ φ∗_A(r2, s2) ¯φB(r2, s2) (2.11)

Here, the transition density is identically zero:

ρtr(r1) =

Z

ds1ϕ0∗A(r1)α(s1)ϕA(r1)β(s1) = 0 (2.12)

and thus the integration over the spin coordinates only leaves the exchange term of eq (2.11): V_exch= Z Z dr1dr2ϕ0∗A(r1)ϕ0B(r1) 1 r12 ϕ∗_A(r2)ϕB(r2) (2.13)

The Coulomb coupling VCoulvanishes, and only the exchange and overlap

short-range contributions remain. The triplet energy transfer, thus, proceeds via the electron exchange first described by Dexter, who extended the F¨orster theory of energy transfer to interactions involving dark states. [25]

An analysis of Vexch (eq. 2.13) in the simple four-orbital model reveals that

the exchange integral is nothing but the electrostatic interaction between two charge distributions, ρx(r1) = ϕ0∗A(r1)ϕ0B(r1) and ρ0x(r2) = ϕ∗_A(r2)ϕB(r2),

which are the product of two occupied or two virtual orbitals that belong to different molecules. [2,25] Given that each ϕ orbital decays exponentially with the distance, ρx and ρ0x will both be negligible, unless A and B are in

close contact. If this is the case, ρxand ρ0x will be large enough in the same

region of van der Waals contact between A and B. Owing to the distance r12

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the overlaps R ρx(r)ρ0x(r)drare very small. [25] Nevertheless, the necessity

of a close separation between the two sites also means that the overlap con-tribution might not be negligible at all.

The derivation of the overlap contribution (2.4) in Ref. [1] is only possible within the TD-DFT framework. Indeed, the corresponding CI expression is somewhat different, even at the first order. [26] Moreover, it is important to notice that the Pauli repulsion can affect the orbitals of A and B, thus indirectly affecting the whole coupling. In fact, in many cases, the exchange integral alone accounts for only half of the total triplet coupling. [27]

2.3 Charge Transfer Electronic Coupling

The charge transfer coupling is responsible for the migration of an electron or a positive charge (a hole) from a donor to an acceptor molecule, as well as the mixing of charge-separated states with neutral states. The definition of the charge transfer coupling varies depending on the type of charge transfer process. For example, in a ground state hole transfer,

A++ B → A + B+

the relevant coupling is the matrix element hA+_{B| ˆ}_H

el|AB+i.

Assuming that the orbitals of A and B are orthogonal and frozen upon io-nization, we can express the states |A+_Bi_{and |AB}+_i _{with respect to the}

ground Hartree-Fock state of |ABi:

A+B = ˆa_i|ABi

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where ˆai and ˆaj are the destruction operators that remove an electron

re-spectively from the orbitals φi of A and φj of B. The electronic coupling

between orbitals φiand φj of the isolated fragments A and B.

In analogy, one can obtain the coupling for a ground state electron transfer process,

A−+ B → A + B− as hA−_{B| ˆ}_H

el|AB−i, where now the relevant states are expressed through the

operators that create an electron on the orbitals φaof A and φbof B:

A−B = ˆa†_a|ABi

AB− = ˆa†_b|ABi

obtaining the electronic coupling as [29]

be-tween virtual orbitals φaand φb.

This frozen-orbital approach has been used to compute hole and electron transport in organic molecular crystals, [30–33] and to simulate scanning

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tunnelling microscopy images. [29, 34] The schemes for hole and electron transfer in the ground state are represented in Figure 2.2.

B

A

−

a.

B

A

+

b.

Figure 2.2: Pictorial representation of the four-orbital model of the charge-transfer coupling in ground-state electron (a) and hole (b) transfer. The processes mentioned above involve only charged ground states. Other charge transfer processes involve, instead, excited states. The couplings between locally excited states and charge-separated states are important to describe photoinduced charge separation processes, depicted in Figure 2.3. This type of coupling may be expressed through the direct, frozen-orbital scheme seen above; however, more sophisticated methods are gen-erally needed to compute these couplings. A particular class of diabatiza-tion methods, suitable for any type of charge-transfer coupling, is detailed in Section 2.4.

B

A*

a.

B

A*

b.

Figure 2.3: Pictorial representation of the four-orbital model of the charge-transfer coupling in photoinduced electron (a) and hole (b) transfer.

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2.4 The fragment difference schemes

Among the methods that are usually employed in the calculation of the elec-tronic coupling between two sites, the schemes based on a calculation on the entire dimer provide a way to automatically include all the coupling terms. These schemes are based on the diabatization of the electronic Hamiltonian of the dimer, and in principle yield the exact coupling within the electronic structure method used. [2,35]

Let us consider a molecular system composed of two fragments, that can be either two separate molecules or two fragments of the same molecule. An electronic structure calculation on the entire system necessarily yields the adiabatic states, which are the eigenstates of the electronic Hamiltonian at a specific nuclear geometry. However, a diabatic picture better describes the states involved in energy transfer and charge transfer, for example in the Forster theory of energy transfer and in the Marcus theory of charge transfer. Within this picture, the electronic Hamiltonian is written in a basis of local-ized states, and it is not diagonal. For example, considering only two states, the Hamiltonian matrix reads:

Hel=

Ei Vij

Vij Ej

!

(2.16)

where Eiand Ejare the energies of the diabatic states, and Vijis the electronic

coupling between those states.

At the avoided crossing, the condition Ei = Ej means that the energy gap

between the adiabatic states is 2Vif, therefore Vif may be computed as half

of the energy gap. This condition holds either in a symmetric system, in which the diabatic states must have the same energy, or at a specific geom-etry. In the other cases, a localization scheme is needed to (i) define the di-abatic states, and (ii) find the transformation matrix between adidi-abatic and

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diabatic bases (adiabatic to diabatic or ATD transformation). There is no unique choice for the diabatic states, and several schemes have been pro-posed. The Fragment Charge Difference (FCD) scheme was introduced for the calculation of charge transfer couplings, [36] after which the analog Frag-ment Excitation Difference (FED) scheme was proposed for Energy Trans-fer. [37] Finally, the Fragment Spin Difference (FSD) scheme was introduced to deal with the Triplet Energy Transfer coupling. [27] Very recently, the Frag-ment Transition Difference (FTD) scheme was proposed as an alternative to FED for the energy transfer coupling. [38]

The Fragment Difference schemes rely on the definition of an additional op-erator (which for now we shall call ˆY), representing an observable which has its extrema in the diabatic states; ˆY can be defined in such a way that it has eigenvalues 0 and ±1. The localized states are therefore those states that di-agonalize the additional operator ˆY. For example, it is possible to express the matrix elements of ˆY in a basis formed by 2 adiabatic states: ˆY is a real symmetric 2 by 2 matrix. Y(adiabatic)= Y1 Y12 Y12 Y2 ! ; Y(localized)= 1 0 0 −1 ! (2.17)

The eigenvectors of the matrix Y form the unitary matrix transformation U (i.e., U†

YU is diagonal), which is the ATD transformation, within the as-sumption that the two adiabatic states are a linear combination of the local-ized states of interest. By applying the same transformation to the diagonal energy matrix, one can obtain the Hamiltonian in the diabatic basis, and in particular the electronic coupling Vif.

U† E1 0 0 E2 ! U = Ei Vif Vif Ef ! , U = cos θ − sin θ sin θ cos θ ! (2.18)

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The transformation angle θ satisfies

tan 2θ = Y12

Y11− Y22 (2.19)

And the coupling Vif can be directly calculated by equation (2.20)

Vif = (E2− E1) ·

Y12

p(Y1− Y2)2+ 4Y122

(2.20)

In the following sections, I will discuss in detail the definition of the addi-tional operator ˆY in the different Fragment Difference schemes.

2.4.1 Fragment Charge Difference

When dealing with charge transfer processes, the electronic states of interest, the coupling among which should be calculated, are states in which an elec-tron is moved from a donor to an acceptor moiety, or vice-versa (See Section 2.3). The diabatic states are those where the electron is completely localized on one of the two fragments, whereas the adiabatic states will be mixed to some degree. Within the FCD scheme, the diabatic states are defined as the states that either maximize or minimize the charge difference between donor and acceptor fragments. [36] Therefore, the ATD transformation matrix can be found by diagonalizing the ∆q matrix defined as follows:

∆qnm= Z r∈D ρnm(r)dr − Z r∈A ρnm(r)dr (2.21)

where ρnm(r)is the transition density between states m and n, if m 6= n, and

the state density if m = n.

To better understand the definition of ∆q, let us consider the case of an electron transfer from a donor D to an acceptor A: the diabatic states are

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|ii = |D−Aiand |fi = |DA−i. In the diabatic basis, following by the defini-tion, ∆q is diagonal, with ∆qiand ∆qf being respectively +1 and -1. If there

is some coupling Vif between the states, the adiabatic states (|mi and |ni)

will be a combination of |ii and |fi. Therefore, in the adiabatic basis, ∆qmn

will be non-zero, and ∆qmand ∆qnwill have values closer to zero.

In a “real world” calculation, the eigenvalues of ∆q will not be exactly ±1; eigenvalues far from the ideal values indicate that the adiabatic states are not combinations of only two localized states |D−_Ai_{and |DA}−_i_{. The magnitude}

of the eigenvalues of ∆q, therefore, is arguably a test for the applicability of this scheme. I will return later on this point, when presenting the multi-state approach.

2.4.2 Fragment Excitation Difference

In the case of EET, the number of electrons on each moiety remains the same in the two diabatic states of interest, while the difference between the two states is the number of excited electrons. In order to define, as in the FCD approach, the localized states, the FED scheme defines the “excitation num-ber” difference between donor and acceptor. [37] The excitation number is the integral of the excitation density, which in turn is defined as the sum of attachment and detachment densities:

ρex(r) = ρAtt(r) + ρDet(r) (2.22)

In for single-excitation methods, the attachment and detachment densities integrate to1

2, so the excitation density integrates to 1. The difference density

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For a localized excitation, both attachment and detachment densities are lo-calized either on the donor or the acceptor, and the difference in the excita-tion number is +1 or -1.

In order to define an additional operator, the notion of excitation number should be extended to off-diagonal elements. In CI-singles, the transition density ρnmbetween two excited states m and n reads:

ρmn(r) = − X ij X a t(m)_ia t(n)∗_ja φi(r)φ∗j(r) + X i X ab t(m)_ia t(n)∗_ib φa(r)φ∗b(r) (2.23)

where indices i, j and a, b indicate occupied and virtual MOs, respectively, and tiaare the CIS amplitudes. When m = n eq. 2.23 reduces to the

differ-ence density: we can therefore identify the first and second terms in the r.h.s. of eq. 2.23 with the transition attachment and detachment densities. In this way, the elements of the excitation density can be defined as:

ρex_mn(r) =X ij X a t(m)_ia t(n)∗_ja φi(r)φ∗j(r) + X i X ab t(m)_ia t(n)∗_ib φa(r)φ∗b(r) (2.24)

We also note that R ρex

mn(r)dr = δmn.

In analogy to the FCD method, the additional “excitation difference” opera-tor ∆x is finally defined as

∆xmn= Z r∈D ρex_mn(r)dr − Z r∈A ρex_mn(r)dr (2.25)

Equations (2.25) and (2.21) are similar, but ∆x contains the excitation densi-ties instead of the state (transition) densidensi-ties.

For an EET process between a donor D and an acceptor A, the initial and final diabatic states are |ii = |D∗_Ai_{and |fi = |DA}∗_i_{. In the basis defined by}

|iiand |fi, ∆x is diagonal. Following the definition in equation (2.25) ∆x_i and ∆x will be respectively 1 and -1. Also in this case, in a real calculation

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the eigenvalues ∆xiand ∆xfwill be only approximately equal to ±1; also in

this case, a large deviation from the ideal values indicates that two adiabatic states are not sufficient to reconstruct perfectly localized states.

Finally, one can note, from eq. (2.22), that a charge-transfer state, where attachment and detachment densities are localized on different moieties, would have ∆x equal to zero.

2.4.3 Limits of the 2-state formulation

Applying the transformation U in eq. (2.18) to the adiabatic states yields states that are as much as possible localized, and similar to the initial and final states. [36] However, in many cases, a 2-state adiabatic basis is not sufficient to retrieve completely localized states: in fact, an adiabatic state could be the combination of many diabatic states of both donor and accep-tor. Moreover, charge-transfer states can mix with excitonic states, and vice-versa: the diabatic states obtained from a 2-state FCD analysis will contain contributions from locally excited states, and those obtained from a 2-state FED analysis will contain CT components [2, 39,40].

The suitability of the adiabatic 2-state basis to reproduce localized states can be assessed by a test on the eigenvalues of the additional operator matrix Y. Since the eigenvalues are variational, and, by definition, in a localized state l, Yll = ±1, the more localized the obtained diabatic states, the closer to ±1

the eigenvalues of Y will be. In our calculations, we set a threshold of 0.95 on the absolute value of the eigenvalues of Y to decide whether the obtained coupling is meaningful. If the eigenvalue test fails, the 2-state model will not yield accurate couplings.

Another issue with the 2-state model is its inability to compute the couplings between states that are more weakly coupled. Let us consider, as an example,

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D

DA

A

V

₁₁

V

₂₂

V

₂₁

V

12

Figure 2.4: 4-states scheme for a donor-acceptor pair. The first and sec-ond states of the two molecules are strongly mixed in the adiabatic picture. The couplings V11and V22are easily computed from a 2-states FED

calcu-lation on the first or second two states, respectively, of the DA dimer. On the contrary, the couplings V12and V21can be computed only with a

multi-state scheme.

the FED method applied to a DA pair, as depicted in Figure 2.4. Assuming that the first states of D and A have similar energies, the adiabatic states 1 and 2 will be approximately the combination of the first state of D with the first state of A, while the adiabatic states 3 and 4 will be the combination of the second state of D with the second state of A. The 2-state FED model applied to the first two adiabatic states would therefore yield the coupling between the first excited states of A and D, V11, and the coupling V22among

the second states will be given by a 2-state FED calculation on the second pair of adiabatic states. Within this model, however, it is impossible to compute the cross-couplings V12and V21, because there are no adiabatic states that are

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2.4.4 Multi-State variants

To overcome the limits outlined in Section 2.4.3, one can resort to a multi-state formulation. The generalization of the 2-multi-state model to a multi-multi-state model is not straightforward, as the additional operators defined in Section 2.4 only have three different eigenvalues (0 or ±1). This means that the diag-onalization of these operators will only separate the adiabatic basis in three subspaces, and therefore there is no unique choice for the transformation U from the adiabatic to the diabatic basis.

2.4.4.1 Multi-State Fragment Charge Difference

The FCD scheme was generalized to a multi-state formulation by Yang and Hsu. [39] Here I report a brief summary of the formulation.

Without loss of generality, we consider the total dimer to be neutral, and the ground state to be close to a neutral structure. The excited states of the dimer will be combinations of donor-to-acceptor CT states (such as |D+_A−_i_,

|D+_A−∗_i_,|D+∗_A−_i_{, ...), locally excited (LE) states (such as |DA}∗_i_{and |D}∗_Ai_),

and possibly higher energy acceptor-to-donor CT states (such as |D−_A+_i_,

|D−A+∗i,|D−∗A+i, ...). Donor-to-acceptor CT states will have ∆q = +2, acceptor-to-donor CT states will have ∆q = −2; finally, LE states will have ∆q = 0. We note that the LE subspace also includes the ground state. In the adiabatic n-state basis, the Hamiltonian is diagonal, and ∆q is not: by diagonalizing ∆q we obtain the unitary transformation matrix T to the basis

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in which ∆q is diagonal: T†        ∆q11 ∆q12 · · · ∆q1n ∆q21 ∆q22 · · · ∆q2n ... ... ... ... ∆qn1 ∆qn2 · · · ∆qnn        T =        ∆qa 0 · · · 0 0 ∆qb · · · 0 ... ... ... ... 0 0 · · · ∆qz        (2.26)

We shall call this basis the T -basis. Ideally, ∆q should have eigenvalues only 0 and ±2, so that there is a choice on the columns of matrix T. In real applica-tions, however, the eigenvalues of ∆q are not exactly 0 and ±2, so that a diag-onalization will yield a particular transformation matrix T. In any case, the transformation to the T -basis separates three subspaces: donor-to-acceptor CT (or CT1), acceptor-to-donor CT (CT2) and LE subspaces.

We can transform the Hamiltonian in the T -basis:

T†        E1 0 · · · 0 0 E2 · · · 0 ... ... ... ... 0 0 · · · E_n        T =        Haa Hab · · · Hac Hba Hbb · · · Hbc ... ... ... ... Hca Hcb · · · Hzz        =     HCT1 HCT1,LE HCT1,CT2 HLE,CT1 HLE HLE,CT2 HCT2,CT1 HCT2,LE HCT2     (2.27)

However, as there are multiple choices for the T -basis, this Hamiltonian is not well defined. Therefore, we introduce a constraint on the Hamiltonian to define the diabatic basis without uncertainty. The diabatic states that belong to the same eigenspace of ∆q should not be coupled, namely, the correspond-ing blocks of the Hamiltonian (CT1, CT2 and LE) should be diagonal. [39,40]

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Let us define a block transformation T0 _{that diagonalizes the blocks of the} Hamiltonian: T0†     HCT1 HCT1,LE HCT1,CT2 HLE,CT1 HLE HLE,CT2 HCT2,CT1 HCT2,LE HCT2    T 0 ₌     E_CT1 _H˜_CT1,LE _H˜_CT1,CT2 ˜ HLE,CT1 ELE H˜LE,CT2 ˜ HCT2,CT1 H˜CT2,LE ECT2     (2.28) T0is block diagonal: T0=     T0CT1 0 0 0 T0LE 0 0 0 T0CT2     (2.29) so that T0† αHαT0α= Eα ; α =CT1,CT2,LE.

The new basis is the final diabatic basis. The transformation U from the adi-abatic to the diadi-abatic basis is therefore simply TT0_.

In practice, as the eigenvalues of the ∆q matrix are not exactly 0 or ±1, we need a threshold value to assign the states in the T -basis to the CT1,CT2, and LE subspaces. Yang and Hsu used a value of 1 in their work, so that if |∆q| < 1the state is assigned to the LE subspace, otherwise it is assigned to CT1 or CT2. [39]

A formulation with three selected states was used by Voityuk to compute couplings for photoinduced charge separation. [40] On the contrary, the pro-tocol by Yang and Hsu heads toward an automation of the method, where many excited states are included in the adiabatic space until the value of the selected coupling converges to a stable value. [39] However, their work high-lighted that this automatic procedure is rather unstable in some cases, and many excited states are needed to reach convergence on the coupling value.

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2.4.4.2 Multi-State Fragment Excitation Difference

The approach used for multi-state FCD can be used also in the FED scheme. In this case, the eigenspaces of the operator ∆x are the states localized on the acceptor (such as |DA∗_i_{, |DA}∗0

i...), with eigenvalue +1, the donor localized states (such as |D∗_Ai_,|D∗0

Ai, ...), with eigenvalue -1, and the CT states (both |D+_A−_i_{and |D}−_A+_i_{), which have eigenvalue 0.}

In analogy to equation (2.26), the first step is the diagonalization of ∆x in the adiabatic basis, to obtain the unitary transformation matrix T:

T†        ∆x11 ∆x12 · · · ∆x1n ∆x21 ∆x22 · · · ∆x2n ... ... ... ... ∆xn1 ∆xn2 · · · ∆xnn        T =        ∆xa 0 · · · 0 0 ∆xb · · · 0 ... ... ... ... 0 0 · · · ∆x_z        (2.30)

The second transformation T0 _{is, similarly to eq. (2.27), the diagonalization}

of the blocks of the Hamiltonian written in the T -basis, corresponding to acceptor localized states (AL), donor localized states (DL), and CT states.

T0†     HAL HAL,CT HAL,DL HCT,AL HCT HCT,DL HDL,AL HDL,CT HDL    T 0₌     E_AL _H˜_AL,CT _H˜_AL,DL ˜ HCT,AL ECT H˜CT,DL ˜ HDL,AL H˜DL,CT EDL     (2.31)

The FED couplings are found in the ˜HAL,DLblock.

As in the multi-state FCD scheme, the actual eigenvalues of ∆x are not ex-actly 0 and ±1. In analogy with ref. [39], we can set a threshold of 0.5 to assign the states to the AL, DL or CT eigenspaces.

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2.4.5 Other diabatization schemes

In the previous sections, I focused on the diabatization schemes that rely on an additional operator, defined as the difference in some quantity between two molecular fragments. All these schemes are based on a specific defini-tion of diabatic states, and a specificadefini-tion of the molecular fragments. The Generalized Mulliken Hush (GMH) method [41] is based on the dipole oper-ator as an additional operoper-ator, but it does not need a specification of molecu-lar fragments. Subotnik et al. showed that GMH is an approximation of the Boys orbital localization scheme applied to state localization. [42] Notably, the GMH scheme can be generalized to a multi-state formulation, but it has been found less numerically stable than the equivalent multi-state FCD for-mulation. [39] Later, a modification of the Boys scheme was devised, to treat the EET coupling. [43]

Among the other diabatization schemes, the method by Arag´o and Troisi is based on the calculation of atomic transition charges of the target dia-batic states, and the reconstruction of the same properties from the adiadia-batic states. [44] In principle, this method is capable of calculating both EET and CT couplings for an arbitrary number of states. However, it requires the cal-culation of the target diabatic states, in addition to the excited states of the whole system.

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Chapter 3

The coupling to Vibrations

3.1 Density operator formalism

When dealing with a large number of degrees of freedom, it is often useful to use a reduced picture, where the important properties are calculated by explicitly describing only part of the degrees of freedom. For example, one could focus on the electronic part of a molecular wavefunction, implicitly taking into account the nuclear part. To do so, it is useful to characterize the electronic state as a statistically mixed state, described by its reduced density operator ˆρ. The density operator formalism is able to treat a part of a larger quantum system, such as a spin that is entangled to another, or an electronic system that interacts with a bath. For a pure state, namely a wavefunction ψ(t), the density operator is defined as: [45]

ˆ

ρ(t) = |ψ(t)i hψ(t)| (3.1)

conversely, for a purely statistical mixture of states with probabilities pk, the

density operator reads:

ˆ

ρ(t) =X

k

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From these definitions, it is easy to see that the density operator is Hermitian, and the trace of its matrix projection on a complete (arbitrary) basis is equal to 1. The diagonal matrix elements of ˆρ are named populations, whereas off-diagonal element are named coherences. Importantly, the trace of ρ2_{is less}

than or equal to the unity, and T rρ2 _{= 1}_{only for a pure state. [45]}

The expectation value of a generic operator ˆAcan be obtained by tracing its corresponding matrix with the density matrix:

¯

A(t) = T r(Aρ(t)) =X

kl

hk| ˆA|li hl| ˆρ|ki (3.3)

The time-dependent Schr¨odinger equation can be recast in the following form for the density operator:

∂ ∂tρ(t) = −ˆ i ~ h ˆ_{H(t), ˆ}_ρ(t)i (3.4)

Equation (3.4) is the quantum Liouville equation, and describes the evolu-tion of the density operator under the influence of the Hamiltonian H(t). This equation can be projected on a complete basis set, to give

∂ ∂tρjk(t) = − i ~ X m,n L_jk,mnρmn (3.5)

where the Liouville superoperator L is a 4-indices N2_{× N}2 _{matrix that}

op-erates on the density matrix ρ

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In the case of time-independent Hamiltonians, the solution of eq. (3.4) can be readily obtained. The diagonal terms, or populations, remain constant, while the coherences oscillate at the characteristic Bohr frequencies:

ρmn(t) = ρmn(0)e−i

(Em−En)

~ t (3.7)

Let us now operate the separation between nuclear and electronic degrees of freedom. The nuclear coordinates will be treated as a “bath” B that is in the canonical ensemble, while the electronic degrees of freedom will be our quantum “system” S. The total Hamiltonian can be thus partitioned as:

ˆ

H = ˆHS(q) + ˆHB(Q) + ˆHSB(q, Q) (3.8)

where ˆHS and ˆHB are the system and bath Hamiltonian, ˆHSB is the

inter-action between the two, and q and Q are respectively the system and bath coordinates. The total density matrix ρtotcan be reduced, by taking the

par-tial trace only on the bath coordinates Q:

ρ(t) = T rB[ρtot(t)] (3.9)

Now, ρ only contains the electronic degrees of freedom of the molecule of interest, whereas all the other coordinates make up the bath. If an operator

ˆ

Adepends only on the electronic degrees of freedom, its expectation value may still be calculated as

¯

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3.2 Linear absorption of a single molecule

In the density operator formalism, the linear, or first-order, response to an optical field E(t) = E0_{(t) cos(ωt)}_{can be calculated as: [45]}

χ(1)(t) = θ(t)T r(µρ(1)(t)) = i

~θ(t)(J (t) − J

∗_(t)) _(3.10)

where θ(t) is the Heaviside step function, which ensures that the causality principle is respected, µ is the dipole operator, and J(t) is the molecular dipole quantum correlation function, calculated on the equilibrium density matrix:

J (t) = hˆµ(t)ˆµ(0) ˆρeqi (3.11)

Notably, J(−t) = J∗_(t)_{. One can then obtain the absorption spectrum as: [45]}

A(ω) ∝ ω Z ∞ −∞ dtJ (t)eiωt = 2ω< Z ∞ 0 dtJ (t)eiωt (3.12)

The above expressions are completely general. For electronic spectroscopy, we must consider both electronic and nuclear degrees of freedom. For sim-plicity, let us consider only two electronic states, namely the ground state |giand the excited state |ei, which are coupled to the nuclear coordinates Q. The adiabatic Hamiltonian of the complete system thus reads:

H = |gi Hg(Q) hg| + |ei He(Q) he| (3.13)

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Hg(q) = T (q) + Ug(q) (3.14)

He(q) = ~ωeg0 + T (q) + Ue(q) (3.15)

Here, T (q) is the nuclear kinetic energy, Ug and Ueare the adiabatic

poten-tial energy surfaces (PESs) of ground and excited states, respectively, and ~ωeg0 is the adiabatic transition energy, namely the energy gap between the

minima of both PESs. Following the Born-Oppenheimer approximation, the off-diagonal system-bath coupling, or derivative coupling, is neglected in the adiabatic Hamiltonian. This term is responsible for non-radiative tran-sitions between electronic states, that is, transport terms in the Liouville su-peroperator. For the moment, we only notice that the transport terms can be phenomenologically added to the Liouville superoperator.

We can simplify the expression of J(t) in the case of electronic spectroscopy, making some approximations. First of all, we can safely assume that the equilibrium density matrix is the equilibrium ground electronic state |gi:

ρeq= |gi ρghg| (3.16)

Here, ρg is the nuclear density operator that is in thermal equilibrium with

the ground state:

ρg =

exp(−βHg)

T rBexp(−βHg) (3.17)

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ˆ

µ = µge(Q)|gihe| + µeg(Q)|eihg| (3.18)

As can be seen, µeg(Q) is a matrix element over the electronic degrees of

freedom, but is still an operator in the vibrational subspace, as it depends on the nuclear coordinates. The Condon approximation consists in neglecting the nuclear coordinate dependence of µeg, so that it is no longer an operator

on the nuclear degrees of freedom.

Now the dipole quantum correlation function J(t) reads:

J (t) = exp i ~Hgt µgeexp −i ~Het µegρg (3.19)

From now on, the angular brackets h. . .i will denote the trace of the operator argument calculated on a complete basis of the vibrational subspace, namely, the quantum mechanical average of the operator. In equation (3.19) we see that J(t) depends on the evolution of µegin both the ground and excited state

PESs. In the Condon approximation, µegcan be factored out of the quantum

mechanical average. By expanding the trace h. . .i on the vibrational basis of the ground state {|vi}, we obtain:

J (t) = |µeg|2 X v p(εv)e i ~εvt D v e −i ~Het v E = |µeg|2 X v p(εv)e i ~εvthv | v(t)i (3.20) Where εv is the energy of the vibrational state |vi, p(εv) is the Boltzmann

probability for the molecule to be in state |vi, and |v(t)i is the time evolu-tion of the ground-state nuclear wavepacket |vi under the influence of the excited-state Hamiltonian He(See Figure 3.1). Equation (3.20) is equivalent

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Hg He

v(0)

v(t)

Figure 3.1: Representation of the evolution of the wavepacket |vi on the excited-state PES. At t = 0, the wavepacket |v(0)i (blue) is pushed by the optical field from the ground-state PES Hg to the excited-state PES

He; |v(t)i is the evolution determined by He. In the Condon

approxima-tion, the vibronic linear response is determined by the time evolution of hv(0) | v(t)i

to equation (1) of ref. [46], except here we are considering a thermal equilib-rium of initial states. From this equation, J(t) can be calculated by propa-gating the initial ground-state wavepacket |ii on the excited-state PES, and computing the overlap hi|i(t)i at every time step. An alternative is to refer all the time evolution on the excited-state PES to ground-state properties. This method is based on the cumulant expansion that will be described in the next section.

3.2.1 The second-order cumulant expansion

Equation (3.19) can be referred directly to the ground-state evolution of the nuclear degrees of freedom, by taking the ground-state PES as the energy reference, and defining the “energy gap Hamiltonian” as:

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U = He− Hg− ~ωeg (3.21)

For the moment, ωeg is a completely arbitrary scaling parameter and it can

be chosen in such a way that hUi = 0. The time evolution of the energy gap Hamiltonian under the ground-state dynamics is then: [45]

U (t) = exp i ~Hgt U exp −i ~Hgt (3.22)

We can thus rewrite the time-evolution operator for the excited state on the basis of Hgand U(t), noting that now U(t) is a time-dependent Hamiltonian,

for which the time-ordered quantum propagator is

exp₊ −i ~ Z t t0 dτ U (τ ) ≡ 1 +P∞ n=1 −~i nRt t0dτn Rτn t0 dτn−1· · · Rτ2 t0 dτ1U (τn)U (τn−1) · · · U (τ1) (3.23)

and the evolution on the excited-state PES can be replaced by:

exp −i ~Het = exp(−iωegt) exp −i ~Hgt exp₊ −i ~ Z t 0 dτ U (τ ) (3.24)

The quantum dipole correlation function then becomes

J (t) = e−iωegt µge(t) exp+ −i ~ Z t 0 dτ U (τ ) µegρg (3.25)

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Where now both µeg(t)and U(t) are calculated from the evolution driven by

the ground-state Hamiltonian. Since only the expectation values of µeg(t)

and U(t) are needed, one can in principle use eq. (3.25) on a classical nuclear trajectory, replacing the quantum expression (3.23) with a simple exponen-tial. [45] We take a step further by invoking the Condon approximation, we assume that µge is a constant with respect to time and nuclear coordinates,

and thus we can factorize it out of the quantum mechanical average:

J (t) = e−iωegt_|µ eg|2 exp₊ −i ~ Z t 0 dτ U (τ ) ρg = e−iωegt_|µ eg|2D(t) (3.26)

Now, all the information on the quantum dynamics of the excited state is contained in the “quantum dephasing function” D(t). At this stage, D(t) is still represented by the time-ordered infinite summation (3.23), but it can be expanded perturbatively in orders of U, in the so-called cumulant expan-sion. [47] Here a perturbation expansion is possible because U(t) is small if compared to the ground-state Hamiltonian Hg.

D(t) ' exp −i ~ Z t 0 dτ1hU (τ1)i − 1 ~2 Z t 0 dτ2 Z τ2 0 dτ1hU (τ2)U (τ1)i + . . . (3.27) This expansion is usually truncated at the second order. [45] If we take ωegso

that U(t) averages to zero, the first order also vanishes, and only the second order remains. Notably, the quantum mechanical average hU(τ2)U (τ1)idoes

not depend on the time origin τ1, and can be rewritten as hU(t)U(0)i. Up to

this second order, all the information on the dephasing is contained in the two-time quantum correlation function of U, which we shall call C(t):

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C(t) = 1

~2 hU (t)U (0)i (3.28)

The quantum correlation function is a complex quantity, and satisfies the condition

C(−t) = C∗(t) (3.29)

namely, the real part C0_(t) _{is even, and the imaginary part C}00_(t) _{is odd.}

It is considerably easier working in the frequency domain, introducing the Fourier transform of C(t): ˜ C(ω) = Z ∞ −∞ dt eiωtC(t) = ˜C0(ω) + ˜C00(ω) (3.30)

Following from the properties of the Fourier transform, ˜C(ω)is a real-valued function; moreover, ˜C0(ω)and ˜C00(ω)are even and odd functions of the fre-quency, respectively. Finally, ˜C(ω)satisfies the detailed balance condition,

˜

C(−ω) = e−β~ωC(ω)˜ (3.31)

and its even and odd part are related by

˜ C0(ω) = coth β~ω 2 ˜ C00(ω) ; ω > 0 (3.32)

that is, given either the real or imaginary part of the energy gap correlation spectrum, we can predict the other part. This relationship is one manifes-tation of the fluctuation-dissipation theorem. Due to its independence from temperature, the odd part of the spectral density ˜C00(ω) is the commonly

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used representation. From now on, we shall refer to ˜C00(ω)as the spectral densityof the exciton-phonon coupling. [45] The spectral density (3.30) thus is sufficient to compute the quantum dephasing function D(t) and the optical linear response, under the following approximations:

1. The Born-Oppenheimer approximation, that let us neglect off-diagonal derivative couplings in (3.13).

2. The Condon approximation, on which basis the dipole operator µge,

acting on the nuclear wavefunctions, is treated as a constant

3. The second-order truncation of the cumulant expansion, following which all terms such as hU(τ3)U (τ2)U (τ1)iand higher orders have been

neglected

Finally, J(t) can be written as:

J (t) = |µeg|2e−iωegt−g(t) (3.33)

where g(t) is the lineshape function:

g(t) = Z t 0 dτ2 Z τ2 0 dτ1C(τ1) (3.34) = − Z ∞ 0 ˜ C00(ω) πω2 coth β~ω 2 · cos(ωt) − 1 − i (sin(ωt) − ωt) (3.35)

Passing to eq. (3.35) we made use of the properties of C(t) (3.29) and the detailed balance on ˜C(ω)(3.31).

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Finally, the absorption spectrum is obtained directly from the lineshape func-tion, using (3.33) and (3.12):

A(ω) ∝ 2ω< Z ∞

0

dt ei(ω−ωeg)t−g(t) (3.36)

The formalism applied here is not limited to the calculation of absorption properties of a molecule. A lineshape function gab(t)can be defined between

any two potential energy surfaces a and b, and will allow to calculate how the vibrational Hamiltonian modulates the transition between a and b.

3.3 The Displaced Harmonic Oscillator model

The expressions for the linear response can be greatly simplified by employ-ing the most basic model of couplemploy-ing between electronic and vibrational degrees of freedom. This model is called Displaced Harmonic Oscillator (DHO), and consists in several approximations. First of all, the ground-state potential energy surface (PES) Hg, on which the nuclei move, is assumed to

be harmonic. The second assumption is that the second derivatives of both ground and excited-state PESs are equal. Thus, the only difference between the PESs of ground and excited state is the position of the minimum and the vertical offset.

Let us initially consider, for simplicity, a molecule with a single vibrational normal mode. Within the DHO model, this mode (with coordinate q and reduced mass m) is associated to the same harmonic frequency ω0in both the

ground and the electronically excited state, but with a different equilibrium geometry (Fig. 3.2). The two PESs read:

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0 ω0_eg ω_eg 0 d λ q E H_e H_g

Figure 3.2: Representation of the Displaced Harmonic Oscillator model. Here, Hgand Herepresent the ground and excited-state PESs, ωeg is the

vertical excitation energy, ω0

egis the energy difference between the two

min-ima, or adiabatic excitation energy, and λ is the reorganization energy.

Hg = p2 2m+ 1 2mω 2 0q2 (3.37) He= ~ω0eg+ p2 2m+ 1 2mω 2 0(q − d)2 (3.38)

where d is the displacement between ground and excited state equilibrium geometries. The energy gap Hamiltonian (3.21) can thus be defined by using Hgas a reference: Ueg(q) = He− Hg = ~ωeg0 − mω20dq + 1 2mω 2 0d2 (3.39)

The last term in the left member of eq. (3.39) is often referred to as the “re-organization energy” λ:

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λ = 1 2mω

2

0d2 (3.40)

The vertical transition energy ~ωegcan be defined as the difference between

the two PESs at the equilibrium geometry of the ground state, thus ωeg =

ω_eg0 + λ. It is useful to redefine the energy gap Hamiltonian Ueg in order to

exclude all constant terms:

U0eg(q) = −mω20dq = αq (3.41)

In this way, the average of U0

egcalculated on the ground-state nuclear

den-sity matrix ρgis zero. The parameters of the DHO, namely the displacement

dand the reorganization energy λ, can be obtained from the slope of the energy gap Hamiltonian α:

d = − α mω2 0 λ = 1 2 α2 mω2₀ (3.42)

Finally, we can introduce a dimensionless displacement parameter S, called the Huang-Rhys factor, and defined as:

S = λ ~ω0

= mω0d

2

2~ (3.43)

We will now apply the second-order cumulant expansion in order to cal-culate the vibronic lineshape within the DHO model, starting by the quan-tum correlation function of the energy gap C(t). From eq. (3.41), C(t) = α2hq(t)q(0)i, and, for a single harmonic oscillator,

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hq(t)q(0)i = ~ 2mω0

(¯n + 1)e−iω0t_{+ ¯}_ne+iω0t

(3.44)

where ¯n is the thermally averaged occupation number for the harmonic os-cillator:

¯

n(ω0, β) = (eβ~ω0− 1)−1 (3.45)

By putting (3.41) in (3.44), we obtain

C(t) = (~ω0)2S(¯n + 1)e−iω0t+ ¯ne+iω0t

(3.46)

Here, the only parameters needed to define C(t) are the mode frequency ω0

and the Huang-Rhys factor S defined in eq. (3.43). A double intergration of C(t)leads to the lineshape function:

g(t) = −S(¯n + 1)(e−iω0t_{− 1) + ¯}_n(e+iω0t_{− 1) − iSω}

0t (3.47)

In the low temperature limit, the absorption spectrum is then given by eq. (3.36):

A(ω) ∝ ω|µeg|2<

Z ∞

0

dt eiωegt_exp_S(e−iω0t_{− 1) + iSω}

0t

(3.48)

By including the Sω0 shift into ωeg, and taking the Taylor expansion of the

exponential term in S, we get the absorption spectrum with the expected vibronic progression: [45]

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A(ω) ∝ ω|µeg|2e−S X k Sk k!δ(ω − ω 0 eg− nω0) (3.49) where ~ω0

eg= ~ωeg− λis the adiabatic excitation energy. The intensity of the

infinitely narrow vibronic bands is proportional to Sk_{/k!: if S < 1, then the}

first peak corresponding to the 0 − 0 adiabatic transition is the most intense, otherwise the distribution will be peaked on some higher energy band (see Fig. 3.3 ). ω0_eg ω₀ ω I S = 0 S < 1 S > 1

Figure 3.3: Vibronic progression at T = 0K with different Huang-Rhys factors S. The larger the value of S, the longer the related vibronic pro-gression.

The limit of high temperatures (that is, β~ω0 1) gives for the dephasing

function: D(t) = exp 2SkBT ~ω0 cos(ω0t) =X n 1 n! 2SkBT ~ω0 n cosn(ω0t) (3.50)

which leads to an absorption spectrum that is a series of sidebands equally spaced on either side of ω0

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The total spectral density of a single-mode DHO can be obtained from eqs. (3.30) and (3.46):

˜

C(ω) = 2πω₀2S [(¯n + 1)δ(ω − ω0) + ¯nδ(ω + ω0)] (3.51)

This function quantifies how much a mode at frequency ωk is coupled to

the electronic excitation under study. The first and second terms in (3.51) describe upward and downward energy shifts of the system, respectively. Coupling to a vibration typically leads to an upshift of the energy gap tran-sition energy since energy must be put into the system and bath. However, as with hot bands, when there is thermal energy available in the bath, it also allows for down-shifts in the energy gap. The net balance of upward and downward shifts averaged over the bath follows the detailed balance expression (3.31).

3.3.1 Spectral density for multiple oscillators

In real systems, the electronic degrees of freedom will interact with a large number of nuclear modes that may reflect molecular vibrations, phonons, or intermolecular interactions. We can describe these modes as a continuous distribution of harmonic oscillators of varying mode frequency and coupling strength. The energy gap Hamiltonian is readily generalized to the case of a continuous distribution of motions if we statistically characterize the density of states and the strength of interaction between the system and this bath. This method is also referred to as the Spin-Boson Model used in NMR for treating a two-level spin-1/2 system interacting with a quantum harmonic bath.

The DHO model can be easily extended to a system with M harmonic modes by expressing the ground-state PES as

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Hg= M X k=1 p2 k 2mk +1 2mkω 2 kqk2 (3.52)

and the energy gap Hamiltonian as

U0(q) = −

M

X

k=1

mkdkωk2qk= α†q (3.53)

Finally, the reorganization energy is

λ = M X k=1 mkω2_kd2_k 2 = M X k=1 Sk~ωk (3.54)

Since the harmonic modes are independent of one another, the total spectral density ˜C(ω)is readily generalized from eq. (3.51):

˜ C(ω) = 2π M X k=1 ω_k2Sk[(¯nk+ 1)δ(ω − ωk) + ¯nkδ(ω + ωk)] (3.55) with ˜ C0(ω) = πX k ω_k2Sk coth(β~ωk/2) [δ(ω − ωk) + δ(ω + ωk)] ˜ C00(ω) = πX k ω_k2Sk[δ(ω − ωk) − δ(ω + ωk)] (3.56)

Passing to a continuum of vibrational modes, the spectral density ˜C00(ω)for ω > 0can be seen as the product of a density of states ξ(ω) and their coupling strength to the electronic excitation given by the factor πω2_S(ω)_{: [48]}

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˜

C00(ω) = πω2S(ω)ξ(ω) (3.57)

The total reorganization energy can be obtained from ˜C00(ω)by:

λ = ~ π Z ∞ 0 dωC˜ 00_(ω) ω (3.58)

Note that the spectral density in eq. (3.51) consists of δ-functions since the DHO model does not provide any relaxation channel for excited state vibra-tions. This would lead to coherent (undamped) vibrational motion. Once anharmonicities in the potential energy surfaces are taken into account, vi-brations are no longer fully independent and, as a result, we have vibrational relaxation. Further channels for vibrational relaxation are provided by a complex environment, such as solvent molecules or a protein moiety sur-rounding the solute. The finite lifetime of vibrationally excited states can be taken into account by multiplying expression (3.44) for the coordinate auto-correlation function by an exponential function e−γt_{that causes Lorentzian}

broadening of the peaks in the spectral density.

In real molecular systems, the modulation of the energy gap comes from both the internal vibrations of a molecule and the motion of the molecule in the environment. The internal vibrations are often high-frequency under-damped intramolecular modes, which reflect in sharp peaks of the spectral density. The low frequency intermolecular modes, induced by the surround-ing environment, result in a continuous contribution to the spectral density. The vibronic structure in high-resolution spectra can thus be interpreted as the sum of a low energy continuous band (often referred as phonon side-band, PSB) and a series of sharp contributions at higher frequency attributed to discrete vibrations. [49]

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On a final note, we remind that, in the DHO model, the second-order cu-mulant expansion is exact, and gives the same result as the total expansion. [CIT] This is generally not true when the vibration are anharmonic, or when the second derivatives of the excited-state PES are different from those of the ground state. In this last case, the energy gap Hamiltonian is not anymore a linear function of the nuclear coordinates, and the modes of the excited state are a combination of those of the ground state.

3.4 Computational evaluation of spectral densities

As seen in the previous sections, the spectral density contains, up to the sec-ond order in the energy gap, all the information on the dephasing between ground and excited state. The spectral density can be used to compute the absorption spectra, in addition to raman and nonlinear spectroscopies. [45]

3.4.1 The mixed quantum/classical approach

The calculation of spectral densities often involves the computation of the energy gap autocorrelation function hU(t)U(0)icl with a mixed

quantum/-classical approach, computing the energy gap with a quantum chemistry method along a classical molecular dynamics (MD) trajectory. [50–56] How-ever, quantum correlation functions are complex valued and do not have an exact classical equivalent, because the classical autocorrelation function is real and even in time. The classical spectral density only contains the even part ˜C0(ω); the odd part ˜C00(ω)may be obtained by exploiting the detailed balance relation (3.32). However, a classical autocorrelation function has a physical sense only in the high-temperature limit, where ˜C0 ˜C00namely, β~ω 1 for all ω where ˜C(ω)is non-negligible.

The interplay between electronic coupling and vibrational motion in Charge and Energy Transfer

University of Pisa

Doctoral Thesis