FranckâCondon Models for Simulating the Band Shape of Electronic

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Franck-Condon Models for Simulating the Band Shape of Electronic Absorption Spectra Shaohong L. Li, and Donald G. Truhlar J. Chem. Theory Comput., Just Accepted Manuscript • Publication Date (Web): 10 May 2017 Downloaded from http://pubs.acs.org on May 14, 2017

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Journal of Chemical Theory and Computation

May 8, 2017 revised for JCTC

Franck-Condon Models for Simulating the Band Shape of Electronic Absorption Spectra Shaohong L. Li and Donald G. Truhlar* Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA

Abstract. Band shape is an essential ingredient in the simulation of electronic absorption spectra. The excitation of multiple series of vibrational levels during an electronic excitation is a main contributor to band shapes. Here we present two simple models based on the Franck-Condon displaced-harmonic-oscillator model. The models are both derived from the time-dependent formulation of electronic spectroscopy. They assume that the transition dipoles do not depend on geometry and that the potential energy surfaces are locally quadratic; one model is second order in time and is called LQ2, and the other is third order in time and is called LQ3. These models are suitable for simulating the unresolved vibronic band shapes of electronic spectra that involve many vibrational modes. The models are straightforward and can be easily applied to simulate absorption spectra that are composed of many electronic transitions. As compared to carrying out molecular dynamics simulations, they require relatively few electronic structure calculations, and the additional cost for constructing the spectra is negligible. Therefore the models are suitable for simulating the spectra of complex systems such as transition metal complexes.

Keywords: electronic spectroscopy, Franck-Condon principle, spectroscopic band shape, UV/Vis spectroscopy

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2 1. INTRODUCTION Simulation of line shapes is an important ingredient needed for understanding and interpreting experimental UV/Vis spectra. With the recent improvements in electronic structure theories, especially equation-of-motion and linear-response methods and quasidegenerate and multi-state perturbation methods, it is now feasible to calculate excitation energies and transition probabilities for many types of electronic transitions of fairly large and complex molecules, with an error of only a few tenths of an electron volt. However, in order to compare to or interpret experimental spectra, one must consider the broadening of spectral lines. It is well known that spectral lines have a finite width caused by many broadening factors, such as the finite lifetime of excited states, collisions (pressure broadening), thermal Doppler effects, and resolution of the spectrometer. Furthermore, electronic excitation spectra are broadened by vibronic effects, as already recognized in the Franck-Condon principle and in Herzberg-Teller vibronic theory. 1,2 In particular, each electronic transition is always accompanied by vibrational transitions; as a result each electronic transition gives rise to a vibronic band.3,4,5,6 For unresolved spectra where the individual vibrational lines are not discernable, the spectrum simply appears as a broadened band. These broadening factors are often mimicked by convoluting the spectral lines with Gaussian, Lorentzian, or Voigt functions, whose width can be estimated a priori from experimental conditions or a posteriori by making a best fit to experimental spectra. In the present article, we attempt to account for vibronic broadening more directly, without, however, calculating all the individual vibronic-stateto-vibronic-state transition probabilities, which is impractical for complex systems. We are especially interested in an approach that is suitable for the electronic spectroscopy of complex systems, such as transition metal complexes, that are composed of a fairly large number of electronic excitations. Some approaches to simulate vibronic effects are already available; however, as we will show below, most of the existing methods are not suitable for this kind of system. First, many of the methods are too complicated to be applied to many electronic excitations of complex systems. Second, most experimental spectra of complex systems are not resolved, which makes the effort of accurately simulating the fine vibrational structure unnecessarily ambitious. In many situations, what we care about is just an unresolved spectrum, such as the ones usually obtained experimentally. In this work we will present two models suitable for this purpose, and we will demonstrate their usefulness by testing them on two multi-mode molecules, namely naphthalene and permanganate. First we briefly review some existing vibronic band shape models. The simplest is to employ the classical Franck-Condon principle and calculate only the vertical electronic transitions and convolute each transition with a broadening function, whose width is chosen empirically.7,8,9,10,11 This approach is convenient but has several flaws, especially the ambiguity in the choice of the width parameter, but also that this model does not recognize the asymmetry of the vibronic band shape, in particular that the band maximum


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3 is not located at the vertical excitation energy (an issue that has recently been studied by Lasorne et al.12). At the other extreme are sophisticated time-independent13,14,15,16,17 and timedependent18,19,20,21,22 approaches to explicitly simulate the vibronic excitations. These methods can generate highly accurate vibronic spectra, but they require special expertise to use, and they may be unsuitable for complex molecules. For example, transition metal complexes often have tens of low-lying electronic states contributing to the electronic spectra. The vibronic bands often overlap, obscuring fine vibrational details and leading to structureless spectra. In this case it would be a substantial amount of work, and it may not be necessary, to simulate the vibrational structures of all the relevant bands. A method with sophistication intermediate between the two types of methods above is the ensemble method.23,24,25,26,27 It simulates vibronic excitations by single-point excitations at an ensemble of geometries representing the initial nuclear wave functions of the molecule, which are sampled, for example, from a molecular dynamics simulation,23,24,25 a ground-state quantum harmonic oscillator wave function,26 or a harmonic-approximation Wigner distribution.27 This method is relatively inexpensive, and it can simulate the unresolved vibronic band shape, as well as going beyond the Condon approximation by taking into account the dependence of transition dipole moment on the geometry. However, it may require a large sample of geometries to represent the initial nuclear wave function of complex molecules with many vibrational degrees of freedom, and the large number of single-point excitations can be costly to compute. To overcome the limitations of the methods above, we seek models that are simple enough to be applicable to complex systems without adding a major load to the cost and effort, yet are able to capture the overall profile of unresolved bands. The models we investigate in this work are a symmetric Gaussian model and an asymmetric bell-shaped model derived from the time-dependent formulation of absorption spectroscopy developed by Heller and co-workers28,29,30,31 and that is implemented in the ORCA18,32 computer program. The Gaussian model was derived in ref 24 but was not tested. Here we correct some mistakes in the derivation, extend it to the asymmetric model, and show that the simple models are satisfactory for simulating unresolved vibronic spectra. One particular scenario in which these models will be useful is the evaluation of electronic structure methods for simulating an unresolved experimental spectrum. In many of such applications, by using the methods presented here for the width and shape of the bands, combined with the electronic structure method for the location and strength of the bands, we can ensure that the factor that limits the quality of comparison of the simulation to experiment is the electronic structure method itself rather than the treatment of the band shape.



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4 2. THEORY 2.1. Assumptions about the potential energy surfaces and the transition dipole moments. We will work under the Condon approximation, which assumes that the transition dipole moment does not vary with the geometry in the Franck-Condon region relevant to spectroscopy. We also assume that the potential energy surfaces (PESs) are locally quadratic in the Franck-Condon region. To carry out the mathematics, we will also use a model in which the PESs of the excited states have the same frequencies and normal modes (without Duschinsky rotation33,34) as the ground state and only the equilibrium geometry is shifted; however, the final result is more general, as will be discussed below. These approximations have been evaluated in previous studies, in conjunction with what are called the IMDHO (independent mode, displaced harmonic oscillator),35 LCM (linear coupling model),36 and VG (vertical gradient)37 models. Those studies showed that these approximations can often yield satisfactory results when combined with either time-independent or time-dependent methods. The approximations are made to make the calculations practical and straightforward for complex cases like transition metal complexes, since under these approximations the only quantities needed for simulating a spectrum using the models presented in this work are the vertical excitation energies, oscillator strengths, ground-state vibrational frequencies, and the gradients of excited state PESs with respect to nuclear coordinates at the ground state equilibrium geometry. We also assume zero temperature so that all the excitations start from the ground vibrational state of the ground electronic state. 2.2. The LQ2 model for the vibronic band shape. We present here a straightforward derivation of the model; readers are referred to the literature for more details of this kind of method.18,23,28,29,30,31 We start from the time-domain expression of the absorption cross section of the nth excited state under the Born-Oppenheimer (BO) and FC approximation, incorporating homogeneous and inhomogeneous broadening18,23 ∞

σ n (E) ∝ f n Re ∫ χ 00 χ n (t) e 0

i(E+E00 )t/!− Γnt/!−(1/2)Θn2t 2 /! 2

dt

(1)

where n labels an excited electronic state, E is the energy of the incident photon, fn is the oscillator strength of the electronic transition, Re is the operator that takes the real part, χ00 is the nuclear wave function of the ground vibrational state on the ground electronic state PES, χn is the time-dependent nuclear wave function on the excited electronic state −iH t/!

χ 00 where Hn is the nuclear-motion Hamiltonian of PES defined by χ n (t) = e n the excited electronic state, E00 is the energy of the ground vibrational state of the ground electronic state, Γn is the homogeneous broadening factor, and Θn is the inhomogeneous broadening factor. The Dirac bracket denotes integration over the nuclear coordinates. This is the expression for a single excited electronic state; if multiple excited electronic


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5 states are involved, one must sum their contributions. Figure 1 illustrates E00 and other energetic quantities to be used in following equations.

Figure 1. Illustration of the PESs of the electronic ground ( E0 (q) ) and excited ( En (q) ) a and states and of some important energies (E00 and En0 ) and energy differences ( Δ E0n v ). Δ E0n

Next we employ an approximation to the overlap integral of the nuclear wave functions, under the Franck-Condon displaced-harmonic-oscillator assumption,31 ⎡ ⎛ ⎤ ⎞ 2 ⎢ ⎜ ⎡Δ (n) ⎤ ⎥ ⎟ iω j t ⎢⎣ j ⎥⎦ −iω t ⎟ − i ΔE a t⎥ χ 00 χ n (t) = exp ⎢∑⎜ − (1− e j ) − ⎢ ⎜ 2 2 ⎟ ! 0n ⎥ ⎟ ⎢ j⎜ ⎥ ⎠ ⎣ ⎝ ⎦

(2)

where the sum runs over all normal modes j, Δ (n) j is the displacement of dimensionless normal coordinate j from the ground-state equilibrium geometry to the excited-state a is the electronic energy equilibrium geometry, ωj is the frequency of mode j, and Δ E0n difference between the excited electronic state and the ground electronic state at their respective equilibrium geometries (sometimes called the classical adiabatic excitation energy). Equations 1 and 2 can already be used for numerical simulation, and they are implemented in ORCA, where the model is called the “IMDHO model”. However, our goal is a simpler model, and next we derive a simple Gaussian model. This model was reported in ref 24 but here we present a more straightforward derivation, by keeping only



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6 the low-order terms in the Taylor expansion of the exponential term in eq 2. Such loworder terms will give a useful approximation to the vibronic band. Expanding the exponential term in eq 2 to second order in time, −iω t 1 (3) e j ≈ 1− iω j t − ω 2j t 2 2 it gives ⎡ ⎛ ⎞⎤ ⎤2 ⎢ ⎛ iω t ⎧ ⎜ ⎡Δ (n) ⎟⎥ ⎞ ⎫ 2 ⎢⎣ j ⎥⎦ 2 2 ⎟⎥ ⎪⎡ ⎪ i ⎤ a ⎜ ⎟ χ 00 χ n (t) = exp ⎢∑⎜ − j ⎨⎢Δ (n) +1 − Δ E t + − ω t (4) ⎬ j ⎟⎥ ⎟ ! 0n ∑⎜ ⎢ ⎜ 2 ⎪⎩⎣ j ⎦⎥ 4 ⎪ ⎭⎠ j ⎜ ⎟⎥ ⎢ j⎝ ⎝ ⎠⎦ ⎣ By using eq 4 and neglecting the broadening factors, eq 1 can be rewritten as

σ n (E) ∝ f n Re ∫

∞ −α t 2 +iβn (E)t dte n 0

≈ f ne

2 −⎡⎣βn (E)⎤⎦ /4α n

(5)

where

⎡ a ω j ⎧⎪⎡ (n) ⎤2 ⎫⎪⎤ E ⎢ Δ E0n − E00 βn (E) = − + ∑ ⎨⎢Δ j ⎥ +1⎬⎥ ⎦ ! ⎢ ! 2 ⎩⎪⎣ ⎭⎪⎥⎦ j ⎣

(6)

and ⎡ (n) ⎤2 2 ⎢⎣Δ j ⎥⎦ ω j αn = ∑ 4

(7)

j

With the harmonic approximation, we have

∑

ωj

j

E = 00 2 !

(8)

and a Δ E0n +∑ j

!ω j ⎡ (n) ⎤2 v Δ = Δ E0n 2 ⎢⎣ j ⎥⎦

(9)

v is the vertical excitation energy) so that eq 6 simplifies to (where Δ E0n

βn (E) =

1 v E − Δ E0n !

(

)

Combining eqs 5, 7, and 10 yields the LQ2 approximation:


(10)

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7

⎛ ⎞ 2 ⎟ ⎜ v E − Δ E0n ⎜ ⎟ σ n (E) ∝ f n exp ⎜ − ⎟ ⎜ 2 ⎡ (n) ⎤2 2 ⎟ ⎜⎜ ! ∑⎢⎣Δ j ⎥⎦ ω j ⎟⎟ j ⎝ ⎠

(

)

(11)

This is a Gaussian function centered at the vertical excitation energy, with a half width at

⎡ ⎤2 2 ω j , and with height proportional to the half maximum (HWHM) of ! (ln 2)∑ ⎢Δ (n) j ⎣ j ⎥⎦ oscillator strength. v (vertical To apply the above model, we need fn (oscillator strength), Δ E0n

excitation energy), ωj (ground-state vibrational frequencies), and Δ (n) j (displacements of the excited-state equilibrium geometry relative to the ground state equilibrium geometry in dimensionless normal coordinates) from electronic structure calculations. The first three can be readily obtained from electronic structure calculations. Under the displacedharmonic-oscillator approximation, the displacements can be calculated from the normal modes and frequencies of the ground-state PES and the gradient of the excited-state PES with respect to nuclear coordinates at the ground-state equilibrium geometry, as discussed in the next subsection. 2.3. Deriving the displacement of excited-state equilibrium geometry relative to the ground state equilibrium geometry in the displaced-harmonic-oscillator approximation. In this subsection, when we refer to “ground state” and “excited state” we mean ground and excited electronic state, respectively. Let q be the vector of the mass-weighted Cartesian displacements from the ground-state equilibrium geometry to a nonequlibrium geometry, defined by

qi = mi xi − xi0

(

)

(12)

where i labels one of the 3N Cartesian coordinates (N is the number of atoms), xi is the ith Cartesian coordinate of the nonequlibrium geometry, mi is the mass of the atom corresponding to xi, and xi0 is a Cartesian coordinate of the ground-state equilibrium geometry. The excited-state potential energy is expanded to second order about the ground-state equilibrium geometry as

1 ⎡ ⎤T En (q) = En0 + ⎢g(n) ⎥ q + q T Fq ⎣ ⎦ 2


(13)


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8 where En0 is the energy, g(n) is the gradient, and F is the Hessian of the excited-state PES at the ground-state equilibrium geometry. Notice that because of the mass weighting in eq 12, F has units of frequency squared. Because we made the displaced-harmonicoscillator approximation, F is the same as for the ground state. Therefore, the unitless matrix A whose rows are ground-state normal mode eigenvectors diagonalizes the Hessian: AFA T = Λ

(14)

where Λ is a diagonal matrix whose diagonal elements are squares of the ground-state harmonic frequencies. The equilibrium geometry of the excited state, q0(n) , is found by setting ∂En/∂q = 0,

∂En ∂q

q=q0(n)

= 0 = g(n) + Fq0(n)

(15)

Therefore q0(n) = −F−1g(n) = −A T Λ−1Ag(n)

(16)

Then the displacement in terms of dimensionless normal coordinates can be computed by its definition in terms of normal coordinates,38 1 1 4 (n) (17) Δ (n) = Λ q" 0 ! where q! 0(n) is the normal coordinate displacement vector defined by ⎡ (n) ⎤T (n) ⎡ ! (n) ⎤T ! (n) ⎢⎣q0 ⎥⎦ Fq0 = ⎢⎣q0 ⎥⎦ Λq0

(18)

which, combined with eq 14, gives q! 0(n) = Aq0(n)

(19)

(Notice that both sides of eq 17 are unitless.) Combining eqs 17 and 19, we can compute Δ(n) by 1 1 4 (n) (20) Δ (n) = Λ Aq0 ! (n) The components Δ (n) j of the vector Δ can be used for eq 11.


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9 2.4. The LQ3 model. The LQ2 model of eq 11 is obtained by expanding the exponential term in eq 2 to the second order. A natural extension is to expand it to a higher order. If we expand it to the third order, we have ∞

σ n (E) ∝ f n Re ∫ dte 0

−α nt 2 +i[ βn (E)t+γ nt 3 ]

∞

= f n ∫ dte 0

−α nt 2

(

cos βn (E)t + γ nt 3

)

(21)

where α and βn(E) are given by eqs 7 and 10, and

γn =

1 ⎡ (n) ⎤2 3 ∑Δ ω 12 ⎢⎣ j ⎥⎦ j

(22)

j

The integrand on the right-hand side of eq 21 involves a Gaussian centered at t = 0, whose amplitude is modulated by a cosine function. The integral will have large values if the cosine function oscillates slowly near t = 0 and have small values otherwise. The integral will have its maximum when βn(E)t + γnt3 is close to zero near t = 0, which occurs if βn(E) is a small negative value since γn > 0. Therefore, this model gives the maximum of the band being slightly red-shifted compared to the vertical excitation energies. This red shift of the band maximum is consistent with the observation that the band maximum is normally to the red of the vertical excitation energy.24 Moreover, if we numerically integrate eq 21, we will see, as shown in the Results and Discussion section, that it gives a bell-shaped band with a heavier tail on the high-energy side along with the red-shifted maximum. In the remainder of the text we will call this model the LQ3 model. 2.5. Anharmonic potentials. Although the displaced-harmonic-oscillator assumption was used to facilitate the derivation, the result is more general. As already pointed out by Condon1 in the case of a dissociative potential curve and as remarked by Tannor and Heller31 in the general case of anharmonic potential curve, the considerations that determine the bandwidth in the general case are similar to the ones that determine it for the case of a nearly harmonic excited-state wave function. In particular, the width of the band is determined by the interval of energy over which the intensity integral has an appreciable value.1 This is determined by the local gradient (slope) and Hessian (second derivatives) of the excited-state potential surface in the vicinity of the vertical excitation geometry, but (because the excitation is a short-time event, during which the timedependent nuclear wave packet upon excitation can explore only the vicinity of the Franck-Condon region) the absorption event may be over before the excited state nuclear motion actually reaches the equilibrium geometry of the excited state, if there is one. Thus one may think of the harmonic oscillator wave functions in the derivation as simply being a reasonable local basis set to derive the band shape rather than actually representing excited-state vibrational eigenstates. (And Δ(n) does not represent an actual displacement vector but just a shorthand notation for the right-hand side of eq 17.) The



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10 validity then requires not that the excited state be a displaced harmonic oscillator but rather that it be locally quadratic in the Franck-Condon region.31

3. COMPUTATIONAL DETAILS Following the choice of electronic structure method in ref 52, the ground-state equilibrium geometry of naphthalene was optimized, and the vibrational modes and harmonic frequencies computed, by BHandHLYP/TZVP39,40 as implemented in Gaussian 09.41 TDDFT calculations for the vertical excitation energies, oscillator strengths, and analytic energy gradients were done with the same exchange-correlation functional and basis set. Following the choice of electronic structure method in ref 53, the ground-state equilibrium geometry of permanganate anion was optimized, and the vibrational modes and harmonic frequencies computed, by BPW9142 and the Slater-type TZ2P basis set.43 Vertical excitation energies and oscillator strengths were computed by the statistical average of orbital potential (SAOP) method44,45 and the TZ2P basis set. The gradients of the excited states were computed numerically for this molecule. Gaussian 0941 was used for the DFT and TDDFT calculations for naphthalene. ADF46,47,48 was used for the DFT and SAOP calculations for permanganate. The ORCA_ASA module of ORCA32 was used for the simulation of spectra based on the IMDHO model (eqs 1 and 2). A Python package, FCBand,49 was used for the simulation of spectra based on the LQ2 (eq 11) and LQ3 (eq 21) models. 4. RESULTS AND DISCUSSION In this section we validate the LQ2 model against the IMDHO model and against the experimental spectra of two molecules, naphthalene and permanganate. These two molecules were chosen because their experimental vibronic spectra are available in the literature,50,51 and also because there have been previous attempts to simulate these spectra.52,53 Because we used the electronic structure methods that were previously shown to have good performance for the systems in those studies, we can reduce the error caused by the electronic structure calculations when simulating the spectra so as to better examine the error caused by the band shape assumptions in the models. We remind the readers that accurately reproducing the vibrational structures of the experimental spectra would require more sophisticated models beyond the IMDHO, LQ2, or LQ3 models, which is not the purpose of this work. The purpose of the present work is to show that the simple models requiring only a small amount of electronic structure calculation can simulate an unresolved spectrum. Because of the simplicity of the model, it can be applied with a low cost to spectra that are composed of a fairly large number of excited states, even for a fairly complex system like a transition metal complex. We will


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11 show that the models give a reasonable shape for the vibronic bands and thus strike a balance between accuracy and computational cost. 4.1. Naphthalene. The S0→S2 transition is the first dipole-allowed transition of naphthalene that has π→π* (HOMO→LUMO) character. It is considered a prototypical system of the bright excitations of aromatic systems and has been extensively studied.50,52,54,55 Figure 2 compares two models to experiment. The IMDHO model, despite the simplicity of the approximations, reproduces the vibrational features of the experimental spectrum quite well, especially in the low-energy region. The LQ2 model, which is an approximation to the IMDHO model, gives a satisfactory overall shape of the band, although, as expected from the symmetry of the Gaussian function, the LQ2 model gives a symmetric band, while the experimental band is heavier on the right tail than on the left tail.

Figure 2. Absorption spectrum of the S0→S2 transition of naphthalene as given by the IMDHO model (red solid line), the LQ2 model (black dash line), and the experiment 50 (blue dotted-dash line). (The spectra are horizontally translated to align the 0-0 excitation peak at the zero of energy so as to reduce the error in the position of the bands caused by the electronic structure method. The absorption strengths are scaled arbitrarily for good visual comparison.) The LQ3 model does a better job of simulating the asymmetry of the band. Figure 3 shows three simulations for the same excitation of naphthalene. The LQ3 model has a maximum shifted to lower energy by ~650 cm-1 (0.08 eV) compared to the LQ2 model, whose maximum is located at the vertical excitation energy. The LQ3 model also has an asymmetric shape with a heavier right tail; although it has a non-physical negative left tail, one can simply replace the negative values with zeros for the purpose of simulation. Another interesting comparison can be made between the LQ3 model and the IMDHO model convoluted with a large empirical inhomogeneous broadening of Θn = 800 cm-1;



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12 such a large broadening is adopted to simulate an unresolved spectrum. The LQ3 model and the IMDHO model have an excellent match (except for the negative left tail of the LQ3 model). Although the match would not be as good if a different broadening factor were used for the IMDHO model, the important point here is that the LQ3 model (like the very close LQ2 model) is capable of simulating the overall shape of an unresolved spectrum.

Figure 3. Absorption spectrum of the S0→S2 transition of naphthalene given by the LQ2 model (black dash line), the LQ3 model (red solid line), and the IMDHO model with a large empirical broadening (blue dotted-dash line). (Although it is not necessary in this case, for consistency with the treatment in Figure 2, the spectra are horizontally translated to align the 0-0 excitation at zero of energy. The absorption strengths are scaled to have a maximum of unity.) -

4.2. Permanganate. The permanganate ion, MnO4 , has a closed-shell ground state belonging to the A1 irreducible representation (irrep) of the Td point group,56 and only A1→T2 singlet-singlet transitions are dipole and spin allowed. The absorption spectrum of the four lowest A1→T2 transitions has been studied experimentally50 and theoretically.53 We followed ref 53 for the electronic structure calculations and the empirical adjustment of vertical excitations, and combined them with our models to simulate the spectrum and to compare with the experimental one in ref 50. Tables 1 and 2 list the computed vibrational frequencies of the ground state and the vertical excitation energies respectively. Table 1. Vibrational frequencies of the ground state of permanganate. a mode 1–2

irrep e

frequency (cm-1) 349


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13 3–5 6 7–9

t2 399 a1 875 t2 940 a Computed by BPW91/TZ2P.

Table 2. Oscillator strengths and vertical excitation energies of permanganate. fa Evert (eV) a ΔEadj (eV) b Eadj-vert (eV) c 1 T2 6.89E-03 3.00 -0.54 2.46 2 T2 2.10E-03 4.03 -0.44 3.59 3 T2 9.58E-03 4.92 -0.79 4.13 4 T2 2.30E-03 5.95 -0.52 5.43 a Oscillator strengths (f) and vertical excitation energies (Evert) computed by SAOP/TZVP. b

Empirical adjustment of vertical excitation energies following ref 53.

c

Adjusted vertical excitation energies, Eadj-vert = Evert + ΔEadj

Figure 4 shows that the IMDHO model is less successful at reproducing the vibrational features of the experiment for permanganate than it was for naphthalene; as shown in ref 53, one needs more sophisticated methods to simulate the vibrational features of permanganate. The simulated 2 T2 and 3 T2 bands are stronger than experiment; such stronger bands may be partly due to the electronic structure as well as to the band shape model. Nonetheless, the LQ2 model is able to produce the overall unresolved profile of the IMDHO model. For this case consisting of many bands across a wide energy range, the difference between the LQ2 model and the LQ3 model (not shown here) is inconsequential.



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14 Figure 4. Absorption spectrum of the lowest four A1→T2 singlet-singlet transitions of permanganate given by the IMDHO model, the LQ2 model, and the experiment.50 (The absorption strengths are scaled arbitrarily for comparison.) 5. CONCLDUING REMARKS In this work we presented the derivation of a simple Gaussian model (i.e., the second-order LQ2 model) and an asymmetric bell-shaped model (i.e., the third-order LQ3 model) for calculating the vibronic band shape of the electronic spectroscopy of molecules under the Condon approximation (transition dipole independent of geometry) and the locally quadratic approximation. These models are approximations to a more sophisticated time-dependent model (the IMDHO model). By testing the new models on the low-lying excitations of two molecules, naphthalene and permanganate, we showed that the models are capable of capturing the unresolved band shape predicted by the IMDHO model. Among the two models, the LQ2 model is simpler and has the advantage of being analytic, but it does not describe the asymmetry of the bands, whereas the LQ3 model introduces an asymmetry shape and a red-shifted maximum but needs numerical integration. (The LQ3 model also gives a region of small negative values in the lowenergy tail of the spectrum, but in practice these negative values can simply be set to zero.) The LQ3 model requires the same input data as the LQ2 model but it requires one to carry out a one-dimensional integration numerically rather than evaluating an analytic formula. As an additional merit, the fact that the LQ2 and LQ3 models are independent of the electronic structure method provides flexibility to treat solvent effects. Solvent effects can be incorporated by using the vertical excitation model (VEM) employing an implicit solvation model.57,58 The ground-state frequencies can be evaluated by a self-consistent reaction-field calculation,59,60 the excited state energies by the VEM model, and the excited state gradient as the gradient of the VEM excitation energy. There are systems for which the Condon approximation or locally quadratic approximation is not expected to be valid. The Condon approximation is not suitable for a dark band whose excitation is weak or forbidden by spatial symmetry at the equilibrium geometry but becomes allowed due to vibronic interactions. An example would be the 1 B2u – 1A1g transition of benzene.61 The present method is also not suitable for cases where the excited-state PES differs significantly from quadratic in the Franck-Condon region, such as in photoionization of butatriene where the excited-state surface has a conical intersection (and therefore is not smooth) in the Franck-Condon region.62 A case where both approximations break down is excitation to the S1 (1A2) state of formaldehyde.63 In summary, the LQ2 and LQ3 models are suitable for simulating the unresolved electronic spectra of many complex molecules, for the following reasons. First, most


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15 experimental spectra of complex systems do not have resolved vibrational fine structure. Second, the LQ2 and LQ3 models do not require decision making or human intervention; therefore they can be easily streamlined to simulate spectra of complex systems such as transition-metal complexes that involve tens or even hundreds of electronic excitations. (This is already available in the FCBand package.)49 Third, the required electronic structure calculations such as ground-state vibrational frequencies and excited-state gradients are relatively inexpensive, and no complicated calculations like excited-state geometry optimization are needed. The models are independent of the electronic structure method and can be used along with any electronic structure method that is good for the problem at hand, or even with the use of different methods for ground-state vibrational frequencies and electronic excitation energies. Given the quantities provided by the electronic structure calculations, the added cost for simulating the spectrum is negligible. ! AUTHOR

INFORMATION

Corresponding Author

*E-mail: [email protected] ORCID Shaohong L. Li: 0000-0003-3875-6199 Donald G. Truhlar: 0000-0002-7742-7294 Notes

The authors declare no competing financial interest. ! ACKNOWLEDGMENTS

This work was supported in part by the U. S. Department of Energy, Office of Basic Energy Sciences, under grant no. DE-SC0008666. S.L.L was also supported by the Frieda Martha Kunze Fellowship and a Doctoral Dissertation Fellowship at the University of Minnesota. Computational resources were provided by the University of Minnesota Supercomputing Institute and by the DOE Office of Science User Facilities at Molecular Science Computing Facility in the William R. Wiley Environmental Molecular Sciences Laboratory of Pacific Northwest National Laboratory and at the National Energy Research Scientific Computing Center (Contract No. DE-AC02-05CH11231).

!

REFERENCES



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