Modeling Optical Spectra of Large Organic Systems Using Real-Time

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Modeling Optical Spectra of Large Organic Systems using RealTime Propagation of Semiempirical Effective Hamiltonians Soumen Ghosh, Amity Andersen, Laura Gagliardi, Christopher J. Cramer, and Niranjan Govind J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00618 • Publication Date (Web): 16 Aug 2017 Downloaded from http://pubs.acs.org on August 22, 2017

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Journal of Chemical Theory and Computation

Draft of August 12, 2017 Prepared for Journal of Chemical Theory and Computation 1 2 3

Modeling Optical Spectra of Large Organic Systems using Real-Time Propagation of Semiempirical Effective Hamiltonians 7

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Soumen Ghosh†, Amity Andersen§, Laura Gagliardi*†, Christopher J. Cramer*†, and Niranjan Govind*§ 10

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Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, MN 55455-0431, USA. 14

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Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99338, USA 16

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Abstract: 21

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We present an implementation of a time-dependent semiempirical method (INDO/S) in NWChem using real-time (RT) propagation to address, in principle, the entire spectrum of valence electronic excitations. Adopting this model, we study the UV/Vis spectra of medium-sized systems like P3B2, f-coronene, and in addition much larger systems like ubiquitin in the gas phase and the betanin chromophore in the presence of two explicit solvents (water and methanol). RT-INDO/S provides qualitatively and often quantitatively accurate results when compared with RT- TDDFT or experimental spectra. Even though we only consider the INDO/S Hamiltonian in this paper, our implementation provides a framework for performing electron dynamics in large systems using semiempirical Hartree-Fock (HF) Hamiltonians in general. 3

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Introduction 35

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In the last decade, performing large-scale static and dynamic calculations has become essential in different areas of chemical research,1 including, e.g., modeling the explicit solvation of chromophores,2-5 the active sites of enzymes,6,7 homogeneous clusters,8-11 proteins,12-16 photochemistry,17,18 and the properties of crystals,19,20 polymers,21-23 and nanomaterials.24,25. Performing ab initio electronic structure calculation on large systems has become significantly more practical, thanks to advances in computational hardware and new efficient algorithms. Density functional theory (DFT) calculations are now regularly performed for systems of up to O(103) atoms (using local functionals), although more stringent limitations are associated with post-Hartree-Fock (HF) methods that take increasing account of electron correlation. Even in favorable instances, however, such large-scale calculations tend to be restricted to molecular (or supermolecular) geometries that are static. In many instances, it is important not only to model interactions of the system with its environment but also its dynamical properties. When explicit solvation is included, many relevant chemical systems require from 103 to 104 atoms for a physical treatment, and such systems are not yet trivially accessible with ab initio DFT and post-HF methods. 5

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Combined quantum mechanical/molecular mechanical (QM/MM) methods have often proven effective for the study of large systems,26 but the accuracy of QM/MM methods is strongly 57

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dependent on the chosen size of the QM region,5,27-29 the accuracy of the electronic structure methods used, and the minimization of boundary effects.30 Treating an entire system at a single level of theory avoids such complications, and approximate electronic structure methods employing effective Hamiltonians have long been the subject of extensive study owing to their computational efficiency. Such methods were first developed by neglecting selected electronelectron repulsion integrals, and computing the remaining terms in the Hamiltonian by correlating them with different physical properties of relevant chemical systems. The first semiempirical methods to appear in the literature were restricted to π orbitals in organic systems, e.g., the Hückel31 and Pariser-Parr-Pople (PPP)32-34 methods. Subsequently, Hoffmann and co-workers extended the Hückel method to describe both π and σ electronic bonding.35 Pople and coworkers then developed all-valence-electron semiempircal methods based on HF theory, including three semiempirical approximations: complete neglect of differential overlap (CNDO),36,37 intermediate neglect of differential overlap (INDO)38 and neglect of diatomic differential overlap (NDDO)39. Among these approximate methods, INDO received special attention from Zerner and co-workers, who parametrized the INDO (INDO/S) method specifically to compute accurately the UV/Vis spectra of organic molecules at the configuration interaction including single excitations (CIS) as well as within the random phase approximation.40-44 INDO/S parameters are now available for transition metals,45,46 actinides,47 and lanthanides48. Modifications to the INDO/S method, INDO/X, have also been reported.49 An alternative effective Hamiltonian model formally derived from density functional theory, namely the density functional tight binding (DFTB) model,50 has also been extensively developed by a number of groups51-57. 29

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Considering electron dynamics, several quantum mechanical approaches are available. Within the Born-Oppenheimer (BO) approximation, these include, e.g., direct integration of the Schrodinger equation for very small systems (e.g., H2)58, real-time configuration interaction singles (CIS),59,60 real-time time-dependent density functional theory (RT-TDDFT),61-65 and real-time timedependent Hartree-Fock (RT-TDHF)66,67. In the framework of semiempirical methods, wave packet dynamics with an extended Hückel Hamiltonian68 has been applied to gas phase systems,69 molecules in solution,70 and dye-sensitized semiconductor interfaces71-76 (recently, this latter model has been combined with nuclear dynamics as well77,78). Real-time dynamics approaches formulated for DFTB have also been used to study electron dynamics79 as have analogs formulated for semiempirical HF methods,80-83 with both applied to the study of both ground and excited states in small- to medium-sized systems. In the specific case of DFTB, interesting recent applications include the study of non-adiabatic electron-hole dynamics in nanoscale solar cell materials.84-86 45

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Regarding the prediction of UV/Vis spectra, the CIS method has been extensively explored with semiempirical methods,87,88 with the INDO/S Hamiltonian having been used in particular to study excited states in organic electronics,89-91 conjugated polymers,92-95 biological molecules,96-99 and nanoparticles.100,101 Other approaches applied to study very large systems include combining CIS with PPP to study the UV/Vis spectra of organic dyes102,103 and combining linear-response timedependent (LR-TD) theory with DFTB104,105. Since traditional implementations of CIS and LRTD approaches are typically “bottom-up” approaches, they are mostly used to study the lowest excited states and can become prohibitively expensive for large systems and systems with high densities of states. Real-time approaches mitigate this issue as they can be used to capture broad 58

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spectral regions depending on the size of the time step and the length of the simulation. 65,106-108 More recently real-time approaches have also been used to study excited-state absorption.109-111 In this context, we should also mention other viable approaches that can address wide absorption spectra, the complex polarization,112-114 damped response approaches115,116 and multishift linear solvers117. Other approaches include the self-consistent constricted variational DFT approach by Ziegler and co-workers118, the simplified Tamm-Dancoff119 and simplified TDDFT reported by Grimme and co-workers120. and the recent symmetric Lanczos algorithm and kernel polynomial method (KPM) due to Brabec and co-workers121. 14

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In this paper, we present an implementation of RT-INDO/S, in the NWChem,122 computational chemistry program for the modeling UV/Vis spectra of large molecular complexes. We first outline the general methodology for real-time density matrix propagation with a semiempirical Hamiltonian. Next, we apply the approach to medium size molecular complexes like P3B2 (C72H38N20), and f-coronene (C108H42N12) in the gas phase. Finally, to highlight our approach, we calculate the UV/Vis absorption spectra of tyrosine isolated in the gas phase and present in the protein ubiquitin, and also of the betanin red pigment molecular complex in explicit aqueous and alcohol (methanol) solutions. We show that, without any modification of the existing parameters, RT-INDO/S provides qualitatively and often quantitatively accurate results when compared with experiment or RT- TDDFT results and it is thus applicable to a large class of systems. 26

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Theory: 31

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Before discussing the INDO/S approximation, we briefly discuss ab-initio HF theory to highlight changes in the equations due to the approximations. 3

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Ab-initio Hartree-Fock (HF) Theory: 36

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In Hartree-Fock theory123, for a closed-shell system, the molecular orbitals (MO), 𝜓𝑖 are considered as linear combination of atomic orbitals (LCAO) 𝜙𝜇 , 39

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𝜓𝑖 = ∑ 𝐶𝑖𝜇 𝜙𝜇 42

𝜇

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(1)

where 𝐶𝑖𝜇 is the linear expansion coefficient. Molecular orbitals are optimized by solving the matrix equation, 46

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𝐹𝐶 = 𝑆𝐶𝐸 49

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where F and S are the Fock and overlap matrices, C is the matrix of linear expansion coefficients and E is the diagonal matrix containing the MO eigenvalues, respectively. Matrix elements of S are given by, 52

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𝑆𝜇𝜐 = ∫ 𝜙𝜇∗ (𝑟1 )𝜙𝜐 (𝑟1 )𝑑𝑟1 5

(3)

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and those of F are given by, 57 58 59 60 ACS Paragon Plus Environment


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1 𝐹𝜇𝜐 = 𝐻𝜇𝜐 + ∑ 𝑃𝜆𝜎 [(𝜇𝜐|𝜆𝜎) − (𝜇𝜆|𝜐𝜎)] 2 5

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𝜆𝜎

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where the one-electron contribution is given by 8

atoms

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𝐻𝜇𝜐 = 1

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∫ 𝜙𝜇∗ (𝑟1 ) [−

1 2 𝑍𝐴 ∇ + ∑ ] 𝜙 (𝑟 )𝑑𝑟 (5) 2 𝑟1 − 𝑅𝐴 𝜐 1 1 𝐴

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where ZA is the nuclear charge of atom A, 𝑟1 is the electronic position and RA is the atomic position and the two-electron electron-repulsion integrals are computed as 15

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(𝜇𝜐|𝜆𝜎) = ∬ 𝜙𝜇∗ (𝑟1 )𝜙𝜐 (𝑟1 ) 19

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1 ∗ 𝜙 (𝑟 )𝜙 (𝑟 )𝑑𝑟 𝑑𝑟 𝑟12 𝜆 2 𝜎 2 1 2

(6)

where 𝑟12 is the interelectronic distance. The one electron reduced density matrix is given by 20 21

𝑀𝑂

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∗ 𝑃𝜆𝜎 = ∑ 𝐶𝜇𝑖 𝐶𝑖𝜎

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(7)

𝑖

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where the index i runs over only the occupied molecular orbitals. 27 28 30

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INDO/S Approximation: 31

The INDO/S method is based on the INDO/1 method38 and can be summarized as follows: 3

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(i) Overlap integrals (eq. 3) are neglected unless 𝜇 = 𝜐. Given S = 1, this reduces Eq. 2 to 35

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𝐹𝐶 = 𝐶𝐸 37

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(8)

(ii) Integrals of type (𝜇𝜐|𝜆𝜎) are set equal to zero unless 𝜇 = 𝜐 and 𝜆 = 𝜎, in which case, if the two pairs of basis functions are located on different atoms 40

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(𝜇𝜐|𝜆𝜎) = 𝛿𝜇𝜐 𝛿𝜆𝜎 𝛾𝐴𝐵 (μ on atom A and  on atom B) (9) 43

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where the Mataga-Nishimoto-Weiss formula40 takes 45

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𝛾𝐴𝐵 = 48

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𝑓𝛾 𝑅𝐴𝐵 +

2𝑓𝛾 (𝛾𝐴𝐴 +𝛾𝐵𝐵)

(10)

and 𝛾𝐴𝐴 = 𝐼𝐴 − 𝐴𝐴 , where IA and AA are the ionization potential and electron affinity of atom A,40,41 respectively, and Weiss suggested a value of 1.2 bohrs∙hartree for 𝑓𝛾 parameter. 𝑅𝐴𝐵 is the distance between atom A and B. Single-center electron-repulsion integrals in the INDO model are expressed through Slater-Condon parameters40 54

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(𝑠𝑠|𝑠𝑠) = (𝑠𝑠|𝑝𝑝) = 𝐹 0 5 56 57 58 59 60 ACS Paragon Plus Environment

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1 (𝑠𝑝|𝑠𝑝) = 𝐺 1 (𝑠𝑝) 3 4 5 6

(𝑝𝑥 𝑝𝑥 |𝑝𝑥 𝑝𝑥 ) = 𝐹 0 (𝑝𝑝) + 7 8 9

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𝐹 2 (𝑝𝑝) (11)

(𝑝𝑥 𝑝𝑥 |𝑝𝑦 𝑝𝑦 ) = 𝐹 0 (𝑝𝑝) − 10 1

2 2 𝐹 (𝑝𝑝) 25

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(𝑝𝑥 𝑝𝑦 |𝑝𝑥 𝑝𝑦 ) = 25 𝐹 2 (𝑝𝑝) 12 13 14 16

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(iii) Diagonal one-electron matrix elements for atomic orbitals ϕμ centered on atom A are computed as 18

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𝑐𝑜𝑟𝑒 𝐻𝜇𝜇 = 𝑈𝜇𝜇 − ∑ 𝑍𝐵 𝛾𝐴𝐵

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𝐵≠𝐴

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where 𝑈𝜇𝜇 is the one-center core integrals. 25

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(iv) Two-center off-diagonal one-electron matrix elements are calculated using the weighted overlap approach 27

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𝑐𝑜𝑟𝑒 𝐻𝜇𝜐 = 𝑓𝜇𝜐 (𝛽𝜇0 + 𝛽𝜐0 )𝑆𝜇𝜐

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where the parameter fμυ is equal to 1.267 and 0.585 for σ and π overlap41, respectively, 𝑆𝜇𝜐 is computed using Slater orbitals (i.e., atomic orbitals are not assumed to be orthonormal for this step), and a single βA is used for each element and semiempirically adjusted to yield best agreement with reference spectroscopic data. 35

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Using the above approximations, F can be defined for closed-shell system as, 38

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𝐵≠𝐴

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𝐵≠𝐴 𝜆∈𝐵

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(𝜇, 𝜐 ∈ 𝐴, 𝜆 ∈ 𝐵)

(14)

𝐹𝜇𝜆 = 𝑓𝜇𝜐 (𝛽𝜇0 + 𝛽𝜐0 )𝑆𝜇𝜐 − 𝑃𝜇𝜆 𝛾𝐴𝐵 (𝜇 ∈ 𝐴, 𝜆 ∈ 𝐵 ) 47

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The INDO/S approximation uses a Slater-type minimal basis set. All parameters used in this paper are reported in the supporting information (Table S1). Equation 14 can be easily extended for openshell systems and our implementation can perform such calculations. However, all the systems reported in this paper are closed-shell. 51

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Time-Dependent Fock Matrix: Unlike ground state DFT calculations where the Fock and density matrices are purely real (with real basis functions), both become complex quantities when propagated in time as we shall see shortly. Using Eq. 4, the time-dependent Fock matrix in the presence of an applied time-dependent field is given by, 57

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𝑐𝑜𝑟𝑒 𝐹𝜇𝜐 (𝑡) = 𝐻𝜇𝜐 + 𝐺𝜇𝜐 [𝑃(𝑡)] − 𝐷𝜇𝜐 𝐸(𝑡)

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where 𝐺𝜇𝜐 is the two-electron component of the Fock matrix element, 𝐷𝜇𝜐 is an element of the dipole matrix, 7

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𝑥 𝐷𝜗𝜇 = ∫ 𝜑𝜇∗ (𝑟)𝑥𝜑𝜗 (𝑟) 𝑑𝑟

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and E is the external electric field. 13

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𝐸(𝑡) = 𝛿(𝑡)𝐸0 16

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where 𝐸0 is the strength of the delta electric field. Under the influence of the weak perturbation, the difference between the instantaneous (𝑃(𝑡)) and initial density (𝑃0 ) matrices can be written as, 19

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𝑊(𝑡) = 𝑃(𝑡) − 𝑃0 = −𝑖[𝐸 ∙ 𝐷, 𝑃0 ] (18) 20 2

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Time Propagation: Time propagation of the density matrix is governed by the von Neumann equation, 24

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𝜕𝑃 = [𝐹(𝑡), 𝑃(𝑡)] 𝜕𝑡

(19)

During time propagation, the density matrix is in general complex. In the case of the Fock matrix, it can be shown that the contribution to the imaginary part comes exclusively from the exact exchange part of the Fock matrix.65 32

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In semiempirical HF methods, the commutator [𝐹0 , 𝑃0 ] (where 𝐹0 and 𝑃0 are the Fock and density matrices before perturbation, respectively) is small but nonzero owing to the zero differential overlap approximation. In such cases, the evolution equation can be re-written as80,82, 36

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𝑖 39 41

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𝜕(𝑃0 + 𝑊(𝑡)) = [𝐹(𝑃0 + 𝑊(𝑡)), (𝑃0 + 𝑊(𝑡))] 𝜕𝑡

𝜕𝑊(𝑡) = [𝐹(𝑃0 + 𝑊(𝑡)), (𝑃0 + 𝑊(𝑡))] − [𝐹0 , 𝑃0 ] 𝜕𝑡

(20) (21)

Using a small scaling factor 𝑔 (equal to 10-5 times the norm of W), the von Neumann operator, within linear-response, is constructed as, 46

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𝐿𝑊(𝑡) =

[𝐹(𝑃0 + 𝑔𝑊(𝑡)), (𝑃0 + 𝑔𝑊(𝑡))] − [𝐹0 , 𝑃0 ] 𝜕𝑊(𝑡) = −𝑖 𝜕𝑡 𝑔

(22)

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where L is called the Liouville superoperator. This way the response of the system can be studied under the influence of a weak delta function pulse.59 54

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Chebyshev Propagator: A general overview of different time propagation schemes can be found in reviews by Kosloff124 and by Castro and coworkers.125 Here we propagate Eq. 22 using the iterative Chebyshev algorithm126 57

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𝐿 𝑊(𝑡) = 𝑒 𝐿∆𝑡 𝑊 𝑠𝑡𝑎𝑟𝑡 = ∑(2 − 𝛿𝑛0 )𝐽𝑛 (𝛼∆𝑡)𝑇𝑛 ( ) 𝑊 𝑠𝑡𝑎𝑟𝑡 = ∑(2 − 𝛿𝑛0 )𝐽𝑛 (𝛼∆𝑡)𝜁𝑛 𝛼 5

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𝑛

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𝑛

where 𝐽𝑛 are Bessel functions of the first kind of order n, ∆𝑡 is the time step, and α is a parameter required to scale eigenvalues of the Fock matrix into the range [−1,1]. In our implementation 10

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where 𝜀𝑚𝑎𝑥 and 𝜀𝑚𝑖𝑛 are the maximum and minimum eigenvalues of the Fock matrix, respectively. Usually, α needs to be slightly bigger than the spectral range of the Fock matrix for efficient propagation. 𝜁𝑛 is the modified Chebyshev series80, 17

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𝐿 𝜁𝑛 = 𝑇𝑛 ( ) 𝑊 𝑠𝑡𝑎𝑟𝑡 𝛼 21

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(25)

The modified Chebyshev polynomials can be evaluated recursively using following equations, 2

𝜁0 = 𝑊 𝑠𝑡𝑎𝑟𝑡 23 25

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𝐿𝑊 𝑠𝑡𝑎𝑟𝑡

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𝜁𝑛 = 2 29 31

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𝛼

(26)

𝐿𝜁𝑛−1 + 𝜁𝑛−2 𝛼

From the above equations, it is clear that the Chebyshev propagator only involves matrix multiplications. However, in principle, determination of the parameter α requires on-the-fly diagonalization of the Fock matrix. For weak or no external perturbations, where fluctuations in the eigenspectrum are small, only a single diagonalization step is required to obtain the scaling factor. In our implementation the Chebyshev expansion is truncated to an accuracy of 10-10. 38

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Dipole Moment: The time-dependent dipole moment is computed in the atomic orbital (AO) basis according to 41

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𝜇(𝑡) = 𝑇𝑟[𝐷𝑃(𝑡)] 43

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(27)

Real-time calculations are performed for each polarization of the external field pulse and the complex on-diagonal polarizability tensor is computed as 45

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𝛼𝑖𝑖 = 49

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𝜇𝑖 (𝜔) 𝑖 = 𝑥, 𝑦, 𝑧 𝐸𝑖 (𝜔)

(28)

where 𝜇𝑖 (𝜔) and 𝐸𝑖 (𝜔) are the Fourier transforms of the dipole moment and electric fields in the i direction. The absorption cross section formula is obtained from127,62, 65 52

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𝜎(𝜔) = 56

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4𝜋𝜔 𝐼𝑚[𝛼(𝜔)] (29) 𝑐

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1 𝑆(𝜔) = 𝑇𝑟[𝜎(𝜔)] 3 4 6

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(30)

Computational Details: 8

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All calculations were performed with a development version of NWChem. Geometries for P3B2 and f-coronene were obtained from reference 108 and the geometry for ubiquitin was obtained from reference 104, respectively. RT-TDDFT and LR-TDDFT results with B3LYP/6-31G(d) method for P3B2 and f-coronene were obtained from reference 108. We have also performed INDO/S-CIS calculations within a (20e,20o) active space using the ZINDO-MN128 code to compare with RT-INDO/S results for P3B2 and f-coronene. INDO/S-CIS results are also reported in the supporting information for comparison. We note that the trends of our RT-INDO/S results compared to INDO/S-CIS are consistent with INDO/S-RPA results of Baker and Zerner and coworkers42,43 when compared with INDO/S-CIS, suggesting that RT and RPA calculations at the INDO/S level would show agreement similar to that we have previously observed with DFT (where RT-TDDFT is formally equivalent to RPA, but we do not have access to an INDO/S-RPA implementation). The gas-phase geometry of isolated tyrosine was optimized at the B3LYP129/ccpVTZ130 level. QM/MM calculations were performed to prepare the solvated betanin dye system in two solvents, water and methanol. Solvent partitions of 3080-water and 1263-methanol molecules were assigned as the MM regions, and the betanin molecule was treated as the QM region, respectively. The water molecules were assigned SPC/E force field parameters131, while the methanol molecules were assigned AMBER ff99 parameters132. For the betanin molecule, the Def2-SVP basis set133 and hybrid PBE0134 functional was used. Initially, both the water-solvated and methanol-solvated betanin system geometries were optimized. After a 1 ps of equilibration with the Berendsen thermostat (NVT)135 at 298 K and a time step of 0.25 fs, QM/MM molecular dynamics simulations were performed for 10 ps. 10 snapshots of the trajectory were extracted to generate cluster models having a 7 Å thick shell of solvent molecules. Cluster generation and visualization were done with VMD136. 38

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For all systems considered, starting from the ground state, RT-INDO/S calculations were carried out under the influence of a weak delta function electric field pulse with a field strength of 0.0001 a.u. A time step of 0.1 a.u. (0.0024fs) was used and the system was propagated for 10000 time steps (24 fs). Since infinite time propagation is not practical, we multiply the time-dependent dipole moment by a damping factor 𝑒 −𝑡/𝜏 with τ = 200 a.u. before taking the Fourier transform in order to produce the absorption spectra. This has the effect of introducing artificial and identical lifetimes for all of the excited states. 46

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First, we have performed calculations on two medium size organic molecules, P3B2 and fcoronene, composed of 130 and 162 atoms, respectively. We compare our results with previous RT-TDDFT and LR-TDDFT calculations.108 As mentioned before, performing LR-TDDFT or INDO/S-CIS (see, e.g., Figures S1 and S2) calculations for the same energy range for which realtime calculations are performed would require the calculation of several thousands of roots. Realtime methods, on the other hand, are able to obtain spectra for a broad energy range because the 58

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delta function electric field pulse samples all frequencies and the resolution of a spectral window depends only on the time step and length of propagation. For instance, to resolve an excitation of energy ω, one requires a time step < π / ω, and typically a tenth of that for accuracy. In both cases, RT-INDO/S spectra show qualitative agreement with both RT-TDDFT and LR-TDDFT (Figures 2 and 3). However, the intensities of different peaks predicted by RT-INDO/S are different from RT-TDDFT, which is not unexpected for approximate Hamiltonians and the limitations of the Slater-type minimal basis sets. 1

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Figure 1. Molecular structures of (a) P3B2, (b) f-coronene. Blue, light blue and white colors represent nitrogen, carbon and hydrogen, respectively. 23

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Figure 2. RT-INDO/S, LR-TDDFT and RT-TDDFT spectra of P3B2. LR-TDDFT and RT-TDDFT calculations were performed using the B3LYP/6-31G(d)137 method. Only 200 roots were calculated with LR-TDDFT. LR-TDDFT and RT-TDDFT spectra were adapted with permission from reference 108. Copyright 2015 American Chemical Society. 4

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Figure 3. RT-INDO/S, LR-TDDFT and RT-TDDFT spectra of f-coronene. LR-TDDFT and RTTDDFT calculations were performed using B3LYP/6-31G(d) method. Only 40 roots were calculated with LR-TDDFT. LR-TDDFT and RT-TDDFT spectra were adapted with permission from reference 108. Copyright 2015 American Chemical Society. 25

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While P3B2 and f-coronene are fairly large systems for RT-TDDFT calculations, performed with the B3LYP hybrid exchange-correlation functional, for INDO/S these systems are relatively small in size. To evaluate the performance of the RT-INDO/S method for considerably larger organic systems, we have also calculated the UV/Vis absorption spectrum of a small protein ubiquitin. Ubiquitin is a regulatory protein consisting of 76 amino acids. UV/Vis spectra for the entire protein in the gas phase and in aqueous solution are available and the low energy part of the spectrum includes a distinct feature associated with the single tyrosine chromophore in ubiquitin. QM/MM138 and TD-DFTB104,105 calculations have been performed for the entire protein and the tyrosine molecule in the gas phase. QM/MM calculation predicted a redshift of ~4 nm (~0.06 eV) upon embedding the chromophore in the protein environment. However, TD-DFTB calculations predict that low energy part of the spectra is not dominated only by tyrosine spectra. Other excitations observed in the low energy part of the spectrum appear due to the underestimation of charge transfer excitations,105 a well-known issue in DFTB method139,140. Ubiquitin has 1231 atoms, so that even in a small energy range, there can be thousands of single orbital transitions, making linear response and CI approaches prohibitively expensive to capture all relevant roots. However, RT-INDO/S calculations for tyrosine in the gas phase and in the ubiquitin environment are practical and predict a blueshift of ~0.04 eV in the spectra (Figure 4) in near quantitative agreement with experiment. That suggests that this method is potentially suitable for studying environmental effects in large protein molecules. In addition, it is important to note that unlike TD-DFTB calculation, low energy part of the RT-INDO/S spectrum is only dominated by tyrosine. This is expected, since, semiempirical HF methods do not severely underestimate charge transfer excitations like DFTB methods.88,141,142 52

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Figure 4. RT-INDO/S spectra of tyrosine in gas phase and in ubiquitin. Experimentally observed red shift of 0.06 eV in tyrosine spectra at 4.49 eV when embedded in ubiquitin protein environment is qualitatively (~0.04 eV) reproduced with the RT-INDO/S method. Red and black vertical lines indicate the peaks in tyrosine spectra in gas phase and ubiquitin spectra respectively. 30

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Another area where the RT-INDO/S method may be particularly useful is for the computation of the UV/Vis spectra of organic dyes in explicit solvents. Betanin is an organic dye, well known for its application in the food, cosmetic, and pharmaceutical industries.143,144 The photophysical properties of betanin in aqueous and alcoholic solutions have been studied.145 Gas phase UV/Vis spectra of betanin computed with RT-INDO/S shows a maximum at 2.24 eV which agrees fairly well with an LR-TDDFT (PBE0/6-31+G(d,p)146 ) excitation energy of 2.35 eV (Figure S3). In both water and methanol solutions, the experimental UV/Vis absorption spectra of betanin exhibit a maximum at 2.32 eV.145 It has been suggested recently that large numbers of solvent molecules can be required to get convergence of excitation energy in a truncated model.27,29 To model the UV/Vis spectra of betanin in aqueous and methanol solutions, we have treated the complex and the surrounding solvent explicitly. We have extracted large solvated clusters, 837-861 atoms for water and 816-882 atoms for methanol [Figures 5b, 5d], from initial QM/MM simulations described earlier, respectively. 46

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RT-INDO/S calculations were performed on clusters extracted from 10 snapshots for each of the two solvents and averaged to generate a composite spectrum. Results obtained from individual spectra are reported in the supporting information. Our calculations predict maxima at 2.31 eV and 2.34 eV in methanol and water (Figure 6) in near quantitative agreement with experiment.145 These results indicate that the RT-INDO/S method, with existing parameters, can be used to accurately calculate solvatochromic shifts in the spectra of organic dyes in explicit solvents. 5

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Figure 5. Betanin in explicit (a) water during QM/MM simulation, (b) water for the truncated model, (c) methanol during QM/MM simulation, (d) methanol for the truncated model. Blue, light blue, red and white colors represent nitrogen, carbon, oxygen and hydrogen respectively. 26

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Figure 6. RT-INDO/S spectra of betanin in methanol and water solution. 10 different snapshots are considered for each solvent. The averaged intensity is plotted for the final spectra. 57

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While we have demonstrated the ability of the RT-INDO/S method to predict UV/Vis spectrum for medium and large organic molecular complexes and also to compute solvatochromatic shifts, we note that the INDO/S model was developed essentially to compute vertical excitation energies and ionization potentials and is not suitable for optimizing geometries or dynamics, whether on the ground or excited state. Thus, prior to application of the RT-INDO/S model, appropriate geometries/trajectories must be obtained using other standard quantum or molecular mechanical techniques (or taken from experiment). In addition, our current implementation of RT-INDO/S is only appropriate for single excitations. Doubly and other higher excited states are not currently accessible. In addition, since minimal basis sets are used, Rydberg excited states are not accessible with this model. This problem may be overcome by including diffuse basis functions and further parametrization.147 Even with the limitations mentioned above, we anticipate that RT-INDO/S will be a useful technique for the study of the electronic excitations in large-scale systems. In addition, the real-time approach described here combined with semiempirical Hamiltonians may prove useful for the study of charge and exciton dynamics in large systems. 68-78,84-86 The INDO/S Hamiltonian has been previously used to calculate parameters in different model Hamiltonians to perform exciton dynamics successfully in different chemical systems.97,148,149 INDO/S has also been found to be accurate for calculating parameters to model charge transfer energies and electronic coupling, related to charge transfer rates in chemical systems. 83,150,151 Despite these successes, the INDO/S Hamiltonian has not been used in dynamical studies in combination with real-time approaches. We are currently investigating this as well as the possibility of using other effective Hamiltonians for dynamical applications. 29

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We have reported an implementation of a real-time INDO/S method for the computation of UV/Vis spectra for medium-sized systems like P3B2 and f-coronene, and large systems like gasphase ubiquitin and the betanin chromophore in explicit water and methanol solvents. It is encouraging that, with no adjustment to the underlying INDO/S parameters, UV/Vis spectra predicted at the RT-INDO/S level are in qualitative, and in the largest systems quantitative, agreement with either more accurate RT-TDDFT or experimental spectra. The low computational cost of this method makes this method quite general and opens up the possibility to study electronic excitations in large-scale systems in a fully quantum mechanical framework, including evaluating UV/Vis solvatochromic shifts for chromophores in explicit solvents and assessing environmental effects on the absorption spectra of chromophores embedded in macromolecules. Given the robust parameterization of INDO/S for transition-metal containing compounds, there is reason to expect similar utility of our RT-INDO/S approach when applied to such cases as well, although this remains to be assessed in further detail. 50

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The authors thank Dr. David Bowman for helpful discussions. This work was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences and the Office of Advanced Scientific Computing Research through the Scientific Discovery through Advanced Computing (SciDAC) program under Award Numbers DE-SC0008666 (S.G., C.J.C., L.G.) and KC-030106062653 58

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(A.A., N.G.). The research was performed using The Minnesota Supercomputing Institute (MSI) at the University of Minnesota and EMSL, a DOE Office of Science User Facility sponsored by the Office of Biological and Environmental Research and located at the Pacific Northwest National Laboratory (PNNL). PNNL is operated by Battelle Memorial Institute for the United States Department of Energy under DOE contract number DE-AC05-76RL1830. 10

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INDO/S parameters, INDOS-CIS spectrum of P3B2 and f-coronene, RT-INDO/S and LR-TDDFT spectrum of betanin, details of RT-INDO/S spectra for different snapshots of solvated betanin molecule and molecular geometries for all of the systems discussed herein are available in the supporting information. 18

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