Towards Accurate Simulation of Two-Dimensional Electronic Spectroscopy

Abstract

We introduce the basic concepts of two-dimensional electronic spectroscopy (2DES) and a general theoretical framework adopted to calculate, from first principles, the nonlinear response of multi-chromophoric systems in realistic environments. Specifically, we focus on UV-active chromophores representing the building blocks of biological systems, from proteins to nucleic acids, describing our progress in developing computational tools and protocols for accurate simulation of their 2DUV spectra. The roadmap for accurate 2DUV spectroscopy simulations is illustrated starting with benchmarking of the excited-state manifold of the chromophoric units in a vacuum, which can be used for building exciton Hamiltonians for large-scale applications or as a reference for first-principles simulations with reduced computational cost, enabling treatment of minimal (still realistic) multi-chromophoric model systems. By adopting a static approximation that neglects dynamic processes such as spectral diffusion and population transfer, we show how 2DUV is able to characterize the ground-state conformational space of dinucleosides and small peptides comprising dimeric chromophoric units (in their native environment) by tracking inter-chromophoric electronic couplings. Recovering the excited-state coherent vibrational dynamics and population transfers, we observe a remarkable agreement between the predicted 2DUV spectra of the pyrene molecule and the experimental results. These results further led to theoretical studies of the excited-state dynamics in a solvated dinucleoside system, showing that spectroscopic fingerprints of long-lived excited-state minima along the complex photoinduced decay pathways of DNA/RNA model systems can be simulated at a reasonable computational cost. Our results exemplify the impact of accurate simulation of 2DES spectra in revealing complex physicochemical properties of fundamental biological systems and should trigger further theoretical developments as well as new experiments.

Appendices

Appendix: Retarded Green’s function and third-order density matrix

In the absence of an external field ($\hat{H} = \hat{H}_{0}$), the free evolution of an unperturbed density matrix, $\hat{\rho }^{(0)} (t)$, is stated as

$$\hat{\rho }^{(0)} (t) = G(t)\hat{\rho }(0) = \varTheta (t)e^{{ - \left( {\frac{i}{\hbar }} \right)\hat{H}_{0} t}} \hat{\rho }(0)^{{\left( {\frac{i}{\hbar }} \right)\hat{H}_{0} t}}$$

(A.1)

where $\hat{\rho }(0) = \left| g \right\rangle \left\langle g \right|$ is the density matrix of the system in the GS equilibrium (g), and $\varTheta (t) = \int_{ - \infty }^{t} {d\tau \delta (\tau )}$ is the Heaviside step-function ensuring causality.

In the perturbation scheme described in Sect. 2.1 (Eq. 5), the third-order density matrix is stated as

$$\hat{\rho }^{(3)} (t) = G(t)\hat{\rho }(0) + \left( {\frac{i}{\hbar }} \right)^{3} \int\limits_{0}^{t} {d\tau_{3} } \int\limits_{0}^{{\tau_{3} }} {d\tau_{2} \cdots } \int\limits_{0}^{{\tau_{2} }} {d\tau_{1} } \,G(t - \tau_{3} )[{\text{H}}'(\tau_{3} )G(\tau_{3} - \tau_{2} )[{\text{H}}'(\tau_{2} )G(\tau_{2} - \tau_{1} )[{\text{H}}'(\tau_{1} )G(\tau_{1} )\rho (0)]]]$$

(A.2)

Lindblad equation and population transfer

In the CGF approach, population transfer is assumed to arise from fast (and thus memoryless) bath fluctuations, i.e. characterized by rapidly decaying correlation functions. Population transfers can be included phenomenologically by adding fluctuation and dissipation terms to the Liouville–von Neumann equation (Eq. 4), leading to the Lindblad equation

$$\dot{\hat{\rho }} = \frac{i}{\hbar }[\hat{H},\hat{\rho }] + \sum\limits_{\alpha } {\left( {\hat{V}_{\alpha } \hat{\rho }\hat{V}_{\alpha } \mathop - \limits^{\dag } \frac{1}{1}\hat{V}_{\alpha } \mathop V\limits^{{\dag {\mathbf{ \wedge }}\,}}_{\alpha } \hat{\rho } - \frac{1}{2}\hat{\rho }\hat{V}_{\alpha } \mathop V\limits^{{\dag {\mathbf{ \wedge }}\,}}_{\alpha } } \right)}$$

(A.3)

where $\hat{V}_{\alpha }$ are operators describing system–bath couplings in the most general form. Applying the secular approximation to the Green’s function, i.e. discarding the fast oscillating coherence terms that decay before population transfer is activated, results in the Pauli master equation describing the population relaxation

$$\frac{{{\text{d}}\rho_{\text{ee}} ( {\text{t)}}}}{{{\text{d}}t}} = - \sum\limits_{e'} {K_{ee,e'e'} \rho_{e'e'} (t)}$$

(A.4)

where K is the rate matrix, with elements $K_{ee,e'e'}$ depicting the rate of population transfer from state e into state e’. The solution of the differential equation is formally given by the population Green’s function

$$\rho_{e'e'} (t) = - \sum\limits_{e} {G_{e'e',ee} (t)\rho_{ee} (0)}$$

(A.5)

, and the elements of the matrix $G_{e'e',ee} (t)$ act as time-dependent weighting factors in Liouville pathways, where populations evolve in the excited state, like those of ESAs and SEs (in Eq. 9).

Phase functions and line shape

The phase functions can assume different forms depending on the level of sophistication applied to describe the vibrational dynamics of the system (an overview is given in Refs. [2, 47]). The main building block is the line shape function g _ij (t), which is the integral transformation of the autocorrelation function of bath fluctuations

$$g_{ij} (t) = \frac{1}{2\pi }\int {\frac{{C_{ij} (\omega )}}{{\omega^{2} }}} \left[\coth \left( {\frac{\hbar \omega }{{2k_{B} T}}} \right)(1 - \cos \,\,\omega t) + i\sin \,\,\omega t - i\omega t\right]d\omega$$

(A.6)

It can be obtained from MD simulations or in closed-form expressions derived from different models. For example, the homogeneous (anti-diagonal) broadening of the spectral signals arising due to coupling to a continuum of fast-decaying low-frequency modes can be expressed by the line shape function of the semi-classical Brownian oscillator (OBO) [124]:

$$g_{ij}^{OBO} (t) = \frac{{\lambda_{ij} }}{\varLambda }\left( {\frac{{2k_{B} T}}{\hbar \varLambda } - i} \right)(e^{ - \varLambda t} + \varLambda t - 1)$$

(A.7)

where λ _ij and Λ are the system–bath coupling strength and fluctuation timescale, respectively.