Vibrational response functions for multidimensional electronic spectroscopy in non-adiabatic models Filippo Troiani

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Vibrational response functions for multidimensional electronic spectroscopy

in non-adiabatic models

Filippo Troiani

Centro S3, CNR-Istituto di Nanoscienze, I-41125 Modena, Italy∗

(Dated: February 7, 2023)

The interplay of nuclear and electronic dynamics characterizes the multi-dimensional electronic

spectra of various molecular and solid-state systems. Theoretically, the observable eﬀect of such

interplay can be accounted for by response functions. Here, we report analytical expressions for

the response functions corresponding to a class of model systems. These are characterized by the

coupling between the diabatic electronic states and the vibrational degrees of freedom resulting

in linear displacements of the corresponding harmonic oscillators, and by non-adiabatic couplings

between pairs of diabatic states. In order to derive the linear response functions, we ﬁrst perform

the Dyson expansion of the relevant propagators with respect to the non-adiabatic component of

the Hamiltonian, then derive and expand with respect to the displacements the propagators at given

interaction times, and ﬁnally provide analytical expressions for the time integrals that lead to the

diﬀerent contributions to the linear response function. The approach is then applied to the derivation

of third-order response functions describing diﬀerent physical processes: ground state bleaching,

stimulated emission, excited state absorption and double quantum coherence. Comparisons between

the results obtained up to sixth order in the Dyson expansion and independent numerical calculation

of the response functions provide an evidence of the series convergence in a few representative cases.

I. INTRODUCTION

Multidimensional coherent spectroscopy represents a

powerful tool for investigating ultrafast dynamical pro-

cesses occurring in molecular and solid-state systems1–6.

In fact, the dependence of the nonlinear spectra on mul-

tiple frequencies allows one to separate diﬀerent and oth-

erwise overlapping contributions, and to establish corre-

lations between the observed excitation energies.

These processes often involve an interplay between

electronic and vibrational degrees of freedom, which

plays an important role in processes such as charge or

energy transfer and determines the observed coherent

beatings7–13. In a semiclassical representation of the sys-

tem dynamics, ultrashort laser pulses induce impulsive

transitions to diﬀerent electronic states. This triggers

the wave packet motion on the corresponding potential

energy surfaces, with features that depend on the speciﬁc

form of the electron-phonon coupling. In many cases of

interest, such coupling is represented in terms of the lin-

early displaced-oscillator model, where each vibrational

mode is represented as an independent harmonic oscilla-

tor, which undergoes an electronic-state dependent dis-

placement of the origin14–23. This adiabatic picture can

be integrated in a number of respects, including devia-

tions from harmonicity24,25, coupling between diﬀerent

modes26,27, dependence of the vibrational frequencies on

the electronic state28.

The interplay between electronic and nuclear degrees

of freedom is even closer in the presence of vibronic cou-

plings, which result in coherent population transfer be-

tween the diabatic states and hopping of the vibrational

wave packet between the corresponding potential energy

surfaces11. Its eﬀects have been observed in a variety

of physical systems, ranging from molecular crystals to

J-aggregates29, from polymeric ﬁlms to natural and arti-

ﬁcial light-harvesting systems. A detailed and quantita-

tive explanation of the observed multidimensional spec-

tra requires a detailed theoretical description of these

complex system, and possibly of its interaction with the

environment. The general understanding of the multidi-

mensional spectra, and speciﬁcally the capability of dis-

entangling the electronic and vibrational coherences, can

be possibly favored by the investigation of relatively sim-

ple systems, such as molecular dimers30,31. On the other

hand, a number of reduced models have been introduced

in order to allow the rationalization of the observed spec-

tra and to provide a semi-quantitative understanding of

the underlying dynamics in terms of a few electronic lev-

els and vibrational modes11,32–38.

Here we consider linear and nonlinear response func-

tions in a class of multilevel non-adiabatic model systems

deﬁned as follows. The vibrational degrees of freedom

are described by harmonic oscillators, which undergo a

diﬀerent displacement for each of the electronic diabatic

states. The Hamiltonian also includes terms that co-

herently couple pairs of diabatic states, thus introducing

non-adiabaticity.

The linear response functions are identiﬁed (up to a

prefactor) with speciﬁc propagators, which are computed

in three steps. First, the propagators are expanded in

a Dyson series with respect to the non-adiabatic com-

ponent of the Hamiltonian: each term in the series thus

corresponds to a given number of transitions between the

diabatic electronic states. For given number of transition

and for given values of these transition times, the propa-

gator can be formally (though not physically) identiﬁed

with the adiabatic response functions, whose analytical

expressions have been derived in Ref. 23 within a coher-

ent state approach. After performing the Taylor expan-

sion of such response function with respect to the relevant

displacement, we integrate with respect to the interaction

arXiv:2210.00786v3 [quant-ph] 5 Feb 2023

times, and obtain simple analytical expressions for each

of the contributions. Third-order response functions are

then derived, after decomposing them into the product

of three propagators.

The paper is organized as follows. In Section II we

deﬁne the model systems to which the approach is ap-

plied. Section III contains the main results, namely the

expressions of the single- and multiple-time propagators,

and the corresponding (linear and nonlinear) response

functions. Section IV contains the main steps in the for-

mal derivation of the above results. Finally, we draw the

conclusions in Section V.

II. THE MODEL

FIG. 1. First model system (A), which includes two, non-

adiabatically coupled excited electronic states |1iand |2i.

Optical transitions (green arrows) couple |1iwith the ground

state |0iand |2iwith the doubly excited state |3i. Each elec-

tronic state |kiimplies a displacement by −zkof the harmonic

oscillator corresponding to the vibrational mode.

The present approach allows the derivation of the re-

sponse function in the presence of non-adiabatic cou-

plings between electronic and vibrational degrees of free-

dom. More speciﬁcally, it applies to models where the

vibrational modes can be described by harmonic oscilla-

tors and the coupling between these and the electronic

degree of freedom results in an electronic-state depen-

dent displacement of the oscillators. The eigenstates of

such displaced harmonic oscillator Hamiltonian are char-

acterized by the factorization of the electronic and vibra-

tional components, as results from the crude adiabatic

approximation39. The non-adiaticity is introduced by a

direct coupling between two electronic states, with no

involvement of the vibrational degrees of freedom40.

Within such a class of models, we consider in the fol-

lowing those that are complex enough to display the pro-

cesses of interest, but otherwise as simple as possible.

Throughout the paper, we assume that the non-adiabatic

coupling only involves the ﬁrst two excited states. The

corresponding Hamiltonian reads

H=H0+V=

N−1

ξ=0

H0,ξ +~[η|1ih2|+η∗|2ih1|],(1)

where Vrepresents the non-adiabatic term, H0=

ξ=1 H0,ξ includes all the adiabatic ones, and its

electronic-state speciﬁc components are given by

H0,ξ =|ξihξ|

~¯ωξ+

ζ=1

~ωζ(a†

ζ+zζ,ξ)(aζ+zζ,ξ)

.

(2)

In the following, and for the rest of the paper, we set

~≡1.

The eigenstates of the Hamiltonian Hcoincide with

those of the adiabatic part H0for ξ= 0 or ξ≥3. In

these subspaces and for G= 1, the eigenstates of H

and H0are in fact given by |ξ;n, −zξi, where |n, −zξi=

D(−zξ)|niare the displaced Fock states. Instead, due to

the non-adiabatic term V, the eigenstates |ξ, −z1iand

|ξ, −z2iof H0,1and H0,2don’t coincide with those of

H, which in general don’t have a simple analytical ex-

pression. Interestingly, the form of the above Hamil-

tonian, and speciﬁcally that of the non-adiabatic term,

changes qualitatively if one replaces the basis {|1i,|2i}

with {|+i,|−i}, formed by the states that diagonalize

He=Pξ=1,2~¯ωξ|ξihξ|+V. In such a basis, the coupling

between electronic and vibrational degrees of freedom is

has a non-diagonal component in the electronic basis,

which can be identiﬁed with the non-adiabatic part of

the Hamiltonian (see Appendix A).

In the following, we refer to two simple and yet inter-

esting model systems, corresponding to particular cases

of the above Hamiltonian H. The ﬁrst one, referred to

as model A, is represented by a four-level system with a

single vibrational mode (N= 4, G= 1, Fig. 1). Within

such model, we derive the expressions of the third-order

response functions, which include contributions from pro-

cesses such as excited state absorption, involving the dou-

bly excited state |3i. The second model, referred to as

model B, is represented by a three-level system with two

vibrational modes (N= 3, G= 2, Fig. 2) and can be

referred to a pair of coupled monomers; each monomer

is coupled to its own (localized) vibrational mode. For

this model we compute the ﬁrst-order response function,

and show how this can formally reduced to a single-mode

response function in the case of a symmetric dimer. The

dimer would in principle include a doubly excited state

|3i, which however doesn’t play any role in the linear re-

sponse functions considered for this model, and is thus

disregarded. In fact, the present approach could in prin-

ciple be applied to a single, more general model, which

includes both a doubly excited state and two vibrational

modes. However, this would complicate the analytical

expressions and make their physical meaning less trans-

parent, without introducing signiﬁcantly new elements.

For the sake of clarity, these two features are kept sep-

arate, and investigated independently from one another

in the two models.

Finally, in deriving the response functions, we as-

sume that the system dynamics is triggered by a Franck-

Condon transition between the electronic states, induced

by the interaction with the external electric ﬁeld. This is

followed by a free evolution of the system, resulting from

the interplay between the electronic and vibrational de-

grees of freedom.

FIG. 2. Second model system (B), which includes two, non-

adiabatically coupled excited electronic states |1iand |2i.

These correspond to the excitation respectively of the ﬁrst

and second monomer that form a dimer. Transitions between

the ground (g) and excited state (e) of each monomer can

be induced optically (green arrows). The excitation of each

monomer results in a displacement by −zeof the correspond-

ing vibrational mode (harmonic oscillator).

III. MAIN RESULTS

The central result of the present article is represented

by the time propagators between the excited states be-

longing to the subspace Se. This, result is then used

to derive the expressions of the linear and nonlinear re-

sponse functions. In the following, we present a brief

discursive description of the method (Subsec. III A), fol-

lowed by the presentation of the ﬁnal expressions (Sub-

secs. III B–III E). The formal derivations of the results

are presented in Section IV.

A. Brief description of the method

The relevant time propagator is the matrix element

of the time evolution operator between the states |σ; 0i

and |σ0; 0i: these are given by the product of the diabatic

electronic states that are coupled by the non-adiabatic in-

teraction V(σ, σ0= 1,2), and of the vibrational ground

state of the undisplaced harmonic oscillator. The time-

evolution operator is computed by performing a Dyson

expansion with respect to V: each term in the expan-

sion corresponds to an electronic pathway,i.e. to a given

sequence of electronic states e1, . . . , eM(an alternating

sequence of |1iand |2i), being M−1 the order of the

expansion. The overall time evolution of the system that

one can associate to each electronic pathway consists of

a sequence of sudden transitions between the two dia-

batic states, interleaved by time intervals during which

the system remains in the same electronic state (Fig. 3).

For the vibrational state, each transition between the

states |1iand |2iimplies a hopping of the coherent state

from one potential energy surface to the other, being

these two relatively displaced parabolas. The resulting

time evolution resembles that induced by sequences of

delta-like laser pulses within the linearly displaced har-

monic oscillator model23. This formal analogy allows us

to use in the present case the analytical expressions that

have recently been derived for the vibrational component

of the response function in the adiabatic case (R).

The following step consists in the integration over all

the possible values of the non-adiabatic interaction times.

In order to perform such integration analytically, we per-

form a Taylor expansion of R, which can be written as

the product of M(M+1)/2 double exponential functions.

Each term in the Dyson expansion (order M−1 in the

non-adiabatic coupling constant η) thus gives rise to a

number of inﬁnite terms, one for each set of orders ki

(i= 1, . . . , M(M+ 1)/2) of the Taylor expansions. For-

mally, each of these terms can be written as a product of

exponential functions, that oscillate during the intervals

of duration τj(j= 1, . . . , M) with a frequency Ωj. This

is given by the sum of an electronic and a vibrational

contributions. The former corresponds to the energy ¯ωj

(~≡1) of the electronic state for the relevant time in-

terval (speciﬁed by j) and electronic pathway (speciﬁed

by M−1); the latter one is the energy qjωof the qj-th

eigenstate of the undisplaced harmonic oscillator. The

values of qjresult from those of klin a one-to-many cor-

respondence. Physically, one can thus associate to each

term of the Taylor expansion a vibronic pathway, deﬁned

by a sequence of electronic and vibrational states |ej;qji,

with j= 1, . . . , M. Besides, each of these term is propor-

tional to the displacements (z1or z2) or their diﬀerence

to the power of kT=PM(M+1)/2

i=1 ki. Being the modulus

of the displacement typically smaller than one, this se-

ries is expected to converge, even though the number of

terms increases rapidly with kT.

These functions can be analytically integrated, and

give a formally simple result, consisting - for each vi-

bronic pathway - in the sum of Mterms, each one os-

cillating at a frequency Ωj. If all these frequencies diﬀer

from one another, the oscillating terms e−iΩjtare multi-

plied by constants Aj. If kof those frequencies coincide,

then each of the multiple e−iΩjtis multiplied by a mono-

mial ajtrj, with rj= 0, . . . , k −1 (it follows from the

calculations that the number of identical frequencies for

each vibronic pathway cannot exceed (M−1)/2). Being

this feature common to all the terms that result from the

Taylor expansion, the entire contribution of order up to

M−1 in ηis given by the sum of terms that oscillate at

the frequencies ¯ω1and ¯ω2(diabatic state energies) and of

their vibrational replicas, multiplied by polynomial func-

tions of t, of order (M−2)/2 for even Mand (M−1)/2

for odd M.

The extension of this approach to the multimode case

is rather straightforward, because the dynamics of the

Gvibrational modes are independent from one another.

The Dyson expansion is of the propagator is not modiﬁed

by the presence of multiple modes. On the other hand,

the Taylor expansion has to be performed for each of

the adiabatic response functions R, resulting in a larger

number of vibronic pathways. Each of these is given

by a sequence of states |ej;qji, where qjdeﬁnes a G-

dimensional vibrational (Fock) state. The ﬁnal expres-

sion of the response function is thus identical to that

discussed above, apart from the replacement - in the fre-

quencies Ωj- of the single-mode energies qjωwith their

multimode counterparts PG

ζ=1 ωζqj,ζ .

In the multitime propagators of interest, the overall

evolution of the system is divided in three time inter-

vals (TL,TC, and TR), delimited by optically-induced

transitions between the subspace Se, and the ground or

doubly excited states. The generalization of the above

procedure thus requires two independent Dyson expan-

sions, one for each of the time evolutions that take place

in Se, during the waiting times TLand TR(the evolution

in TCalways takes place outside from the subspace Se,

and therefore does not require a further expansion). The

overall function at deﬁned interaction times can be writ-

ten as a product of ML(ML+ 1)/2 + 1 + MR(MR+ 1)/2

double exponential functions, being ML(MR) the order

in the Dyson expansion for the ﬁrst (third) time interval.

In the ﬁnal step, the integration is performed indepen-

dently with respect to the interaction times belonging to

the intervals TLand TR. This gives rise to the functions

of order ML−1 and MR−1 in η, and that depend respec-

tively on TLand TR, in the same way as the single-time

propagators depend on t.

B. Time propagators

We consider the case where the system modeled by the

Hamiltonian H[Eqs. (1-2)] undergoes a Franck-Condon

transition from the ground state |0; 0ito |1; 0i, corre-

sponding to a generic linear superposition of Hamiltonian

eigenstates. This will evolve in time, under the com-

bined eﬀect of the adiabatic (H0,e) and non-adiabatic

(V) terms.

More speciﬁcally, the propagators are written as the

sum of diﬀerent terms, each one corresponding order

(M−1) in the non-adiabatic interactions. The nona-

diabatic interactions take place at times tk, with tM−1<

tM−2<· · · < t1and result in transitions between, e.g.,

FIG. 3. (a) Main steps for computing the single- and multi-

time propagators: Dyson expansion with respect to the non-

adiabatic coupling; Taylor expansion with respect to the dis-

placements; integration over the interaction times. (b) Elec-

tronic and vibronic pathways (and related frequencies Ωj) as-

sociated to the terms obtained after the Dyson and Taylor

expansions. The diagram refers to the case where the initial

and ﬁnal states concide with |1i.

states |1; qk+1iand |2; qki(where the qkspecify the Fock

states of the undisplaced harmonic oscillator). Between

two consecutive nonadiabatic interactions, for time inter-

vals τk=tk−1−tk, the system evolves freely under the

eﬀect of the Hamiltonian H0and accumulates the phase

Ωkτk. The response functions are eventually derived by

integrating over the interaction times tk. Between two

consecutive non-adiabatic interactions, for time intervals

of duration τk=tk−1−tk, the system evolves freely, un-

der the eﬀect of the Hamiltonian H0, and accumulates

the phase Ωkτk. The response functions are eventually

derived by integrating over the interaction times.

1. Oﬀ-diagonal elements

If the number of transitions that has taken place in

the time tis odd, M−1 = 2n+ 1, the initial and ﬁnal

excited states diﬀer. The resulting time propagator, i.e.

the matrix element of the time-evolution operator US=

e−iHt, can be written in the form:

h2; 0|US|1; 0i=

∞

n=0

η∗|η|2nF2n+1(t)

∞

n=0

η∗|η|2nX

k





2n+2

j=1

Aj(t)e−iΩjt



k

,(3)

where it is intended that all the functions and parame-

ters in the curly brackets depend on k(see below). The

function F2n+1(t), corresponding to the order 2n+ 1 in

the Dyson expansion, is given by the sum of monomials

Aj(t) = ajtrj, with rj≤n, multiplied by terms that

oscillate at the frequencies:

Ωj=ω qj+ ¯ω1,for even j

ω qj+ ¯ω2,for odd j,(4)

with qjnon-negative integers. These frequencies are thus

given by the sum of two terms: the energy of the diabatic

states (¯ω1or ¯ω2), and an integer multiple of the vibra-

tional frequency ω.

Each of the (2n+ 1)-th order terms in the Dyson ex-

pansion [Eq. 10] is given by the sum of diﬀerent contribu-

tions, one for each vector k= (k1, . . . .kM(M+1)/2). These

contributions result from the Taylor expansion of the adi-

abatic propagator, and are of order kT=PM(M+1)/2

i=1 ki

in the displacements zζ,ξ [see Eq. (2)]. The explicit de-

pendence of the contributions in the sum on kand on

the displacements can be expressed as follows:

h2; 0|US|1; 0i=

∞

n=0

η∗|η|2n

k



h(−i)2n+1ehM(z)χM(z,k)i2n+2

j=1

Aj(t)e−iΩjt



k

(5)

where M= 2n+ 1.

The frequencies Ωjand the functions Aj(t) depend on

konly through the integers qj, which specify the sequence

of vibrational states in the related pathway. These inte-

gers are given by the expression:

qj=

x=1

min(j,M+1−x)

y=max(1,j−x+1)

k(x−1)M−1

2(x−1)(x−2)+y.(6)

We note that the relation between kand qis not one-

to-one, for diﬀerent vectors kcan correspond to a same

The constant prefactor, denoted with Cin Eq. (10),

depends both on kand on the vector z= (z1,...,zM),

whose components coincide with the displacements of the

oscillator (here, these are given by zk=z2for odd kand

zk=z1for even k). Such dependence is expressed by the

functions hMand χM. The former one, whose general

expression is reported in Section IV, is here given by

hM(z) = −1

2Mz2

12 −z1z2,(7)

where zij ≡zi−zj. The latter one χM, which depends

both on zand on k, in the present case reads

χM(k,z) = QM−1

p=1 [(−1)pz1z21]k1+w[(−1)pz2z12]kM−p+1+w

QM(M+1)/2

l=1 kl!

×(z1z2)kM(M+1)/2

M−p

q=2

[(−1)p+1z2

12]kq+w,(8)

where w= (p−1)M−(p−1)(p−2)/2. The zero-phonon

line corresponds to q=0and χM= 1.

The expansion in Eq. (3) includes in principle an in-

ﬁnite number of terms, resulting from both the Dyson

and the Taylor expansions. However the relative impor-

tance in the former expansion is expected to decrease for

increasing values of the order 2n+ 1, especially in the

short-time limit (|η|t.1). As to the second expansion,

being in general |z1|,|z2|<1, the value of the constant

prefactor Cis also expected to rapidly decrease for in-

creasing values of the order kT, which deﬁnes the power

in the displacements.

2. Diagonal elements

If the number of transitions that has taken place in the

time tis even, the initial and ﬁnal excited states coincide.

The resulting propagators read:

hσ; 0|US|σ; 0i=

∞

n=0

|η|2nF2n(t)

∞

n=0

|η|2nX

k





2n+1

j=1

Aj(t)e−iΩjt



k

,(9)

where σ= 1,2. In the 0-th order contribution (n= 0),

the propagator is reduced to that derived for the adia-

batic case23. Analogously to the case of the oﬀ-diagonal

elements, the functions F2nare given by the sum of mono-

mial functions Aj(t) = ajtnj(with nj≤n), multiplied

by terms that oscillate at the frequencies

Ωj=ω qj+ ¯ω3−σ,for even j

ω qj+ ¯ωσ,for odd j.(10)

As to the dependence of the diﬀerent contributions on

k, resulting from the Taylor expansion, this is given by:

hσ; 0|US|σ; 0i=

∞

n=0

|η|2n

k



h(−i)2nehM(z)χM(z,k)i2n+2

j=1

Aj(t)e−iΩjt



k

(11)

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Vibrationalresponsefunctionsformultidimensionalelectronicspectroscopyinnon-adiabaticmodelsFilippoTroianiCentroS3,CNR-IstitutodiNanoscienze,I-41125Modena,Italy(Dated:February7,2023)Theinterplayofnuclearandelectronicdynamicscharacterizesthemulti-dimensionalelectronicspectraofvariousmolecularandsolid-...

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Vibrational response functions for multidimensional electronic spectroscopy in non-adiabatic models Filippo Troiani

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