Coherent Two-Quantum Two-Dimensional Electronic Spectroscopy

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Coherent Two-Quantum Two-Dimensional Electronic Spectroscopy Using Incoherent Light Darin James Ulness, and Daniel B. Turner J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.7b09443 • Publication Date (Web): 09 Nov 2017 Downloaded from http://pubs.acs.org on November 10, 2017

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The Journal of Physical Chemistry

Coherent Two-Quantum Two-Dimensional Electronic Spectroscopy Using Incoherent Light †

Darin J. Ulness and Daniel B. Turner

∗,‡

†Department of Chemistry, Concordia College, Moorhead MN 56562, USA ‡Department of Chemistry, New York University, New York NY 10003, USA E-mail: [email protected]

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ACS Paragon Plus Environment


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Abstract Two-quantum two-dimensional electronic spectroscopy (2Q 2D ES) may provide a measure of electron-correlation energies in molecules. Attempts to obtain this profound but elusive signal have relied on experimental implementations using femtosecond laser pulses, which induce an overwhelming background signal of nonresonant response. Here we explore theoretically the signatures of electron correlation in coherent 2Q 2D ES (4)

measurements that use spectrally incoherent light, I

2Q 2D ES. One can use such

fields to suppress nonresonant response, and therefore this method may better isolate the desired signature of electron correlation. Using an appropriate treatment of the multi-level Bloch electronic system, we find that I

(4)

2Q 2D ES present an opportunity

to measure electron-correlation energies in molecules.

Introduction An independent-electron model is a mean-field approximation of the Coulombic repulsions between electrons. Independent-electron models are widely used by chemists because the models afford tractable descriptions of bonding and reactivity. The Hartree–Fock approximation is the most accurate independent-electron model, and calculations using this approximation can converge to within 99% of the exact energy of most molecules. However, the residual energy of a few eV, the correlation energy, is the same magnitude as a chemical bond, and this makes qualitatively useful predictions difficult even for bond-dissociation 1

energies and other simple properties. In fact, chemical bonds are only formed under the Hartree–Fock approximation at the level of average energy, not eigenenergy.

2,3

In the Hartree–Fock approximation, the wavefunction for each electron is written as a linear combination of functions in the coordinate space of the i wavefunction Ψ(r1 , r2 , . . . ) is a product state,

th

electron, fn (ri ). The total

Ψ(r1 , r2 , . . . ) = (∑ fn (r1 )) × (∑ fm (r2 )) × . . . , n

m

2


(1)

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and each linear combination may be as large as necessary to converge under the constraint of self-consistency. This approach fails to produce wavefunctions that account for the electron– electron interactions arising from Coulombic repulsions. A variety of quantum-chemical computational methods transcend these separable Hartree–Fock states and can produce more accurate results, but the beloved ‘orbital’ picture of a molecule fails.

3

In a 2007 report, Mukamel and coworkers suggested that two-quantum two-dimensional electronic spectroscopy (2Q 2D ES), a coherent four-wave mixing technique, can provide a 4

measure of electron-correlation energy. This claim was surprising because the correlation energy is defined relative to the Hartree–Fock calculation and is perceived to be unmeasurable.

1,2

2Q 2D ES requires four femtosecond laser pulses in the sequence depicted in Fig.

(1). The 2Q pulse sequence directly probes two-quantum transitions between the ground state, ∣g⟩, and the doubly excited state, ∣f ⟩, during time interval τ2 . Through Fourier transformation, these coherent oscillations become peaks along the two-quantum axis, ω2 , and are correlated to emission frequencies ω3 . Two pathways contribute to the 2Q signal.

4

One pathway, R7 , describes emission at frequency ωf e through a coherent superposition of the doubly excited and singly excited states. The other pathway, R8 , describes emission at frequency ωeg through a coherent superposition of the ground and singly excited states. These pathways are of opposite sign and lead to the blue and red peaks, respectively, in the spectrum in Fig. (1). A 2Q spectrum should be sensitive to the correlation energy of the doubly excited electronic state, ∣f ⟩, meaning the degree to which it is a product state of two singly excited electronic states.

The ground, singly excited, and doubly excited spectroscopic states,

{∣g⟩ , ∣e⟩ , ∣f ⟩}, arise from the electronic occupancy of the HOMO, ∣H⟩ and LUMO, ∣L⟩. These molecular orbitals are the frontier orbitals relevant to spectroscopic measurements of the ground and excited electronic states of molecules. Their spatial components are given

3



–EB

+EC

spectroscopic signal states pathways +EA

|f 〉

ELO

τ3

τ2

τ1

|e〉 |g〉

R7

2Q spectrum

R8

⎟f〉〈e⎜

⎟e〉〈g⎜

τ3

⎟f〉〈g⎜

⎟f〉〈g⎜

τ2

⎟e〉〈g⎜

⎟e〉〈g⎜

τ1

⎟g〉〈g⎜

⎟g〉〈g⎜

Δ2

ωfg ω2

pulse sequence

sample

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Δ1 ωfe ωeg ω3

Figure 1: 2Q 2D ES basics for femtosecond pulses. The pulse sequence generates a coherence between ∣g⟩ and ∣f ⟩ during time interval τ2 and then correlates the oscillations to emission frequencies along ω3 . Usually τ1 = 0. Red and blue peaks have opposite signs. We describe a method to generate an analogous 2Q 2D spectrum using incoherent light. by ∣g⟩ = ∣H, H⟩

(2a)

1 ∣e⟩ = √ (∣H, L⟩ + ∣L, H⟩) 2

∣f ⟩ = ∣L, L⟩ ,

(2b) (2c)

where the spin factor of (∣↑, ↓⟩ − ∣↓, ↑⟩) is suppressed. These singlet states have energies of Eg , Ee , and Ef , respectively, and the states belong to the independent-electron model,

∣i, j⟩ ≡ ∣i⟩ ⊗ ∣j⟩. The total energy for each state is a sum of the individual eigenenergies, î ⊕ H ˆ j ) ∣i, j⟩ = (Ei + Ej ) ∣i, j⟩ . (H

(3)

In the absence of electron correlation, ∣f ⟩ = ∣e⟩ ⊗ ∣e⟩, and the energy of the doubly excited state has an energy exactly twice that of the singly excited state, Ef = 2Ee . As a consequence, the emission frequencies of the signal pathways will be equal, ωf e = ωeg , and, because the response functions are identical except for the sign difference, the total 2Q signal will vanish. In contrast, when the electron correlation is nonzero, ∣f ⟩ ≠ ∣e⟩ ⊗ ∣e⟩, and the 2Q 2D spectrum will contain two distinct peaks having identical 2Q frequencies but distinct

4


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emission frequencies. One can compute the correlation energy by subtracting the emission frequencies, ∆1 = ωf e − ωeg , or by using ωf g and the absorption maximum, ωeg , yielding ∆2 = ωf g − 2ωeg . Although it seems that both ∆1 and ∆2 will equal Ef − 2Ee , the distinct many-body characters of the pathways means that the values ∆1 and ∆2 can be distinct.

5

Previous attempts to measure electron-correlation energies in molecules using 2Q 2D ES with femtosecond laser pulses were plagued by nonresonant response that overwhelmed the desired signal.

6–9

Nonresonant response arises from the extreme instantaneous power

of a laser pulse and its interactions with the cuvette and the solvent.

10

There is no known

method to suppress the nonresonant response that arises from femtosecond laser pulses. One remedy to this issue may be to use spectrally incoherent (so-called ‘noisy’) light, instead of femtosecond pulses, in the 2Q 2D measurement. Noisy light has been used to measure conventional coherent 2D ES.

11–13

While noisy light does produce nonresonant response when

all of the excitation fields are coincident, we demonstrated previously the ability manipulate the interferometric delays and exploit the persistent nature of the noisy fields to measure the desired molecular signal and simultaneously suppress the nonresonant response.

12

It

is not clear if noisy light will allow for the unambiguous extraction of electron-correlation signals, and therefore in this work we develop noisy-light based 2Q 2D ES from a theoretical perspective. Noisy light has been employed in nonlinear optical spectroscopy since 1983.

14–20

Like its

ultrashort pulse counterparts, the nonlinear optical signal is treated semiclassically whereby the classical electromagnetic fields act pertubatively on the quantum-mechanical material to n

th

order (third order for the technique considered in this work). For both noisy light

and short pulses, the classical electromagnetic field description is justified because of the large photon flux present in the sample. Further, both short pulses and noisy light are taken to be spatially coherent and spectrally broadband. The two light sources differ in how the electromagnetic field is made manifest from the broad spectrum. For short pulses, the spectral frequency components are phase locked to create a near transform-limited pulse in

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time. In contrast, the spectral frequency components of noisy light are completely phase incoherent (phase unlocked ). This produces a persistent or ‘always on’ electromagnetic field that has a stochastic function of time as its envelope. One refers to the field as being color locked because each the spectral frequency (color) is coherent only with itself. To make analytic progress in the description of noisy-light based nonlinear optical signals, one makes several additional assumptions about the nature of the light. Noisy light is taken to be cross-spectrally pure,

21

which separates its temporal and spatial behaviors. This

enables one to consider a spatially coherent but spectrally incoherent field. One also takes the noisy light to obey circular complex Gaussian statistics.

22

This allows higher-order time

correlation functions—which appear as a result of the temporal integration of the nonlinear optical signals by the detector—to be expressed as products of two-point time-correlation functions. Finally, the stochastic function representing the electromagnetic field is considered to be ergodic so that the time averaging process of the detector is equivalent to the ensemble averaged signal. This allows two-point time correlation functions to fall under the Wiener– Khintchine theorem,

23

which is the formal representation of the concept of color locking in

this context. Each stochastic function representing the electromagnetic field has an associated correlation (or coherence) time, which is analogous to the temporal duration of a short pulse originating from a phase-locked laser. This coherence time gives noisy light its time-resolving ability despite the persistent fields. As a matter of notation, noisy-light analogs of short pulse or conventional cw methods carry a I

(n)

preface to the initialism for the nonlinear process.

The “I” carries dual meaning: “Incoherent” to reflect the use of spectrally incoherent light and “Interferometric” to reflect the nature of how the time resolution is achieved. The value of n indicates the number of noisy fields involved in the creation of the signal. In this case n = 4 because three noisy fields create the third-order polarization and one noisy field acts as the local oscillator in the spectrally resolved heterodyne-detection scheme. (4)

this work we develop I

2Q 2D ES.

6


11,12

Thus in

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The outline of this paper is as follows. First we review the coherent four-wave mixing signals for I

(4)

2Q 2D ES that arise under the Bloch model with incoherent light. We then

describe how the 2Q scan is implemented with noisy-light fields, and how this constrains the pathways that contribute to the detected signal. Finally, we describe and demonstrate a procedure to isolate the desired 2Q signal pathways and extract the correlation energy.

Theoretical The theoretical development begins by stating the input fields, then incorporating the material response, and finally producing the detected signal field. We detailed many of these aspects previously for I

(4)

2D ES,

11–13

and thus only present a summary below.

Input Fields We consider four input fields arranged in the conventional BOX geometry as indicated in Fig. (2). A local oscillator, ELO , brings the third-order signal, Esig , to quadrature at the detector for heterodyne detection and spectral interferometry. The temporal characteristics of the fields are EA (t) = E0 p(t)e

−iωt

EB (t, σ) = E0 p (t − σ)e ∗

∗ ∗

(4a) +iω(t−σ)

EC (t, τ, σ) = E0 p(t + τ − σ)e ELO (s, κ) = E0 p (s − κ)e ∗

∗ ∗

(4b)

−iω(t+τ −σ)

+iω(s−κ)

,

(4c) (4d)

where E0 is the field strength, p(t) is a complex stochastic envelope function, and ω is the carrier frequency of the light. The interferometric delays under experimental control are σ, τ , and ζ. We define the interferometric delay variable κ as κ = σ − τ + ζ for symmetry in the analysis. In noisy-light spectroscopies, the light persistently illuminates the material at

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all times, and in the mathematical analysis we label the specific interaction times τj and then integrate over them. In femtosecond-pulse measurements, there is a direct relationship between the interaction times and the interferometric delays, but in noisy-light measurements we must account for all of the potential interaction times. In the femtosecond 2Q 2D ES measurement, the two conjugate fields EA and EC interact ∗

with the sample before the conjugate field EB . This initiates the two-quantum coherence during time period τ2 , which is stepped in time. In this work we exploit the persistent nature of the noisy-light fields to capture the desired 2Q signal pathways using a nonrephasing-like scan of the interferometric variables.

Frequency Domain Detection and Spectral Interferometry We analyze the case of frequency-domain detection using a local oscillator and spectral interferometry.

12,13

The signal detected by the CCD is

˜ σ, τ, κ) = π −1 Re[ ∫ ∆I(ω,

+∞

dxe

+iωx

−∞

⟨Esig (σ, τ, t)ELO (κ, t − x)⟩ ], (3)

∗

t

(5)

where the integration associated with the x variable represents a white detector response (3) ∗ function, E˜LO (ωs , κ) is the local oscillator on the s timeline, and E˜sig (ωt , σ, τ ) is the third-

order signal on the t timeline. There are two timelines under the “bichromophoric” model to allow for appropriate stochastic averaging of the noisy fields responsible for generating the nonlinear signal (t timeline) and that attendant as the local oscillator (s timeline). The quotations on bichromophoric are used to indicate this is a formal extension of the bichromophoric model

24–26

needed in homodyne experiments

20,26,27

experiment. In the current case, the second generic chromophore

to the current heterodyne

26,28

in the bichromophoric

model represents the local oscillator. The bichromophoric model has a trivial presence in experiments using coherent ultrashort pulses and is almost never mentioned. For experiments involving noisy light, the bichromophoric model is not trivial.

8


17,18

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(a)

κ

EA

Esig E*LO

σ

ζ

* EB

(b)

Esig ELO

ϕ

〈

=

σ

〈〈

*

〉

ζ

–τ

* *

* *

–τ EC EB

〉

*

σ

〈

*

*

EA EB EC ELO

EA EB

EC

τ

* *

=

〉〈

ζ * *

+

〉〈

〉

EC ELO

(i)

〉

EA ELO

(ii)

ζ+ο–τ * *

(4)

Figure 2: Parameters for I 2Q 2D ES. (a) The BOX geometry for the spatially coherent beams. The three excitation fields generate signal in the phase-matched direction, Esig . Symbols

Coherent Two-Quantum Two-Dimensional Electronic Spectroscopy

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