Supplementary information for \Unraveling Electronic Absorption Spectra using Nuclear Quantum Effects: Photoactive Yellow Protein and Green Fluorescent Protein Chromophores in Water" Tim J. Zuehlsdorff,1, a) Joseph A. Napoli,2 Joel M. Milanese,1 Thomas E. Markland,2, b) and Christine M. Isborn1, c) 1)Chemistry and Chemical Biology, University of California Merced, Merced, California 95343, USA 2)Department of Chemistry, Stanford University, Stanford, California 94305, USA (Dated: 15 June 2018) a)Electronic mail: tzuehlsdorff@ucmerced.edu b)Electronic mail: [email protected] c)Electronic mail: [email protected] 1 CONTENTS I. Double-counting in the E-ZTFC approach 2 A. Updated model systems 3 B. A double-counting free approach: an average vibronic shape function applied to vertical excitation energies obtained from configurations where the chromophore geometry is optimized in a fixed solvent environment 12 C. Application to the GFP chromophore anion 16 D. Concluding remarks 20 II. Simulation cell sizes and QM regions 20 III. Convergence tests: QM region and MM box 23 IV. Convergence tests: AIMD and AI-PIMD sampling 26 V. Computing generalized moments of a distribution 28 VI. Quantifying non-Gaussian features in the ensemble absorption spectra 29 VII. Structural analysis 31 A. Collective motion 31 B. Nuclear quantum effects on specific bond lengths 34 VIII. Analysis of solvent effects 36 IX. Franck-Condon Shape Functions 39 X. Absorption spectra for the Ensemble and E-ZTFC approaches 47 References 47 I. DOUBLE-COUNTING IN THE E-ZTFC APPROACH The E-ZTFC approach includes chromophore nuclear degrees of freedom both from sam- pling the chromophore motion in the ensemble of vertical excitation energies and from the 2 ground state vibrational wave function used to construct the zero temperature Franck- Condon shape function. To characterize how this double counting of the chromophore nuclear motion affects the computed spectra, we introduced a simple model system in Ref.1, consisting of an ensemble of identical displaced harmonic oscillators coupled to a uniform classical solvent environment described by the solvent reorganization energy. We found that when the ensemble spectrum was sampled from a Boltzmann distribution the double- counting in the E-ZTFC approach led to systematic overestimation of the spectral width compared to the exact solution of the model system, but that this overestimation was small in the limit of large solvent reorganization energies. In the present work, the ensemble of vertical excitation energies is sampled from both AIMD and AI-PIMD trajectories, with the AI-PIMD ensemble corresponding to a quantum Wigner distribution. This quantum distribution includes nuclear quantum effects, including zero-point energy, allowing the system to sample more anharmonic regions of the potential energy surface. To test the validity of combining the E-ZTFC approach with an AI-PIMD ensemble that accounts for nuclear quantum effects on the ground state potential energy surface, we here again explore the effects of double counting the nuclear motion of the chromophore. We first extend our simple model system to include a Wigner distribution of the nuclear degree of freedom. Furthermore, to assess the behavior of the approach for anharmonic degrees of freedom, we extend the model system to that of an ensemble of identical displaced Morse oscillators. We later compare computed spectral results for a real system with and without double counting for a small subset of configurations. A. Updated model systems The model system of an ensemble of identical displaced harmonic oscillators coupled to a classical uniform solvent environment has been described in detail in Ref.1. Here we summarize the main expressions for the E-ZTFC approach based on an ensemble of classical and a quantum configurations, as well as the exact solution to the model system based on the finite temperature Franck-Condon spectrum with Gaussian solvent broadening. The exact solution to the model system of the displaced harmonic oscillators can be 3 written as 1 1 " 2 # exp [−D(2n + 1)] X X Di+j (! − (i − j)) σexact (! −∆ ; T ) / p (n+1)injexp − (1) harmonic eg 4πλk T i!j! 4λk T B i=0 j=0 B −1 h 1 i where n = exp − 1 , ∆eg is the adiabatic energy gap between the two potential kB T energy surfaces, and D is the Huang-Rhys factor, which, in the natural units of the model system is given by D = d2=2 (where d is the displacement of between the two minima of the potential energy surfaces). The parameter λ specifies the solvent reorganization energy measured in units of ~!0, where !0 is the frequency of the harmonic oscillator. The E-ZTFC absorption spectra for the model system, both based on a classical Boltz- mann ensemble distribution and the fully quantum mechanical Wigner distribution are given by: −D 1 i 2 E(cl)-ZTFC e X D (! − i) σ (! − ∆ ; T ) / exp − (2) harmonic eg p i! 4k T (D + λ) 4πkBT (D + λ) i=0 B e−D σE(qm)-ZTFC(! − ∆ ; T ) / harmonic eg s D 2π 2λkBT + h i tanh 1 2kBT 2 3 1 X Di 6 (! − i)2 7 × exp 6− 7 (3) i! 4 D 5 i=0 2 2λkBT + h i tanh 1 2kBT D The main difference between the two expressions is a factor of h i for the quantum tanh 1 2kBT ensemble, which approaches D as T ! 0. This factor guarantees that the quantum E-ZTFC spectrum retains finite broadening at low temperatures as a result of zero-point motion. The harmonic oscillator model has closed expressions for the E-ZTFC spectra based on both quantum and classical ensembles, as well as the exact solution. However, it does not contain any anharmonicity, meaning that a harmonic finite-temperature Franck-Condon (FTFC) spectrum forms an exact solution. Realistic systems like the ones considered in this work possess a significant degree of anharmonicity. It is therefore interesting to use an anharmonic Morse oscillator model to determine whether the E-ZTFC approach based on a harmonic zero temperature vibronic shape function and a fully anharmonic ensemble spectrum provides a better approximation to the exact solution than a pure FTFC spectrum in the harmonic approximation. For this reason we consider a model system of an ensemble 4 of identical displaced Morse oscillators coupled to the same classical solvent environment as the harmonic system. The Hamiltonians HGS and HES for the ground and excited state potential energy surfaces of the Morse oscillator are given by 1 d2 HMorse = − + A 1 − e−αx2 (4) GS 2 dx2 2 1 d 2 HMorse = − + A 1 − e−α(x−d) + ∆ (5) ES 2 dx2 eg For the purpose of this work, we fix the value of α such that α = p1 , which guarantees that 2A 1 2 the harmonic part of the potential is equal to 2 x , in agreement with the harmonic oscillator model system. The parameter A controls the degree of anharmonicity in the model system, with A ! 1 corresponding to the harmonic oscillator. The vertical excitation energy of the Morse oscillator system for a given position x can then be written as: Morse −α(x−d)2 −αx2 !vert (x) = ∆eg + A 1 − e − A 1 − e (6) The classical and quantum probability distribution function of position x on the ground state potential energy surface can be written as " 2 # Morse A (1 − exp [−αx]) ρcl (x; T ) / exp − (7) kBT 1 " EMorse # Morse X fgg;i Morse 2 ρ (x; T ) = exp − (x) (8) qm k T fgg;i i B Morse Morse th where EfGSg;i and fGSg;i(x) are the i energy eigenvalue and wavefunction of the Hamilto- Morse nian HGS . We can then construct the classical and quantum ensemble spectra of vertical excitations for the anharmonic model system. For the anharmonic model system, we again consider the influence of the solvent envi- p ronment in terms of a Gaussian broadening with standard deviation σ = 2kBT λ, where λ is the solvent reorganization energy. Because the Morse potential chosen has a quadratic term that agrees with the potential of the harmonic oscillator, the ZTFC shape function in the harmonic approximation including solvent broadening can be written as 1 X Di (! − i)2 σharmonic(! − ∆ ; T ! 0) / e−D exp − (9) ZTFC eg i! 4k T λ i=0 B The E-ZTFC approach based on the quantum and the classical Morse ensemble and the harmonic approximation for the vibronic shape function is then computed by convoluting Eqn. 9 with the respective ensemble spectra of vertical excitations. 5 The exact solution for the model anharmonic system is given by " EMorse # Morse X fGSg;i X Morse Morse 2 σ (! − ∆eg; T ) / exp − j exact k T fGSg;i fESg;j i B j 2 2 3 Morse Morse ! − EfGSg;i + EfESg;j × exp 6− 7 (10) 4 4kBT λ 5 where the sums run over all bound states of the ground and excited state Morse oscillator. Unlike for the harmonic oscillator, Eqn. 10 cannot be simplified to a closed expression. In this work we evaluate the Franck-Condon integrals between the vibrational wavefunctions of the displaced Morse oscillator numerically, considering a total of 30 eigenstates on each potential energy surface. Simulated E-ZTFC spectra based on a quantum and a classical ensemble distribution in comparison with the exact solution for a harmonic oscillator and two Morse oscillator systems with varying degrees of anharmonicity can be found in Fig. 1. The first column corresponds to a harmonic system with kBT = 0:175, which approximately corresponds to ~!0 the frequency of a C-C single bond oscillation at room temperature. The second column contains results for a Morse oscillator at the same temperature and moderate anharmonicity of A = 40.