Improving the simulation of vibrationally resolved electronic spectra of phenanthrene: A computational Investigation

Chemical Physics Letters 628 (2015) 35–42

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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett

Improving the simulation of vibrationally resolved electronic spectra of phenanthrene: A computational Investigation Min Pang, Pan Yang, Wei Shen, Ming Li, Rongxing He ∗ Key Laboratory of Luminescence and Real-Time Analytical chemistry (Southwest University), Ministry of Education, College of Chemistry and Chemical Engineering, Southwest University, Chongqing 400715, China

a r t i c l e

i n f o

Article history: Received 23 August 2014 In final form 11 March 2015 Available online 20 March 2015

a b s t r a c t Based on the density functional theory and its time-dependent extension, the properties of the ground and the first excited states of phenanthrene were calculated. In harmonic and anharmonic approximations, the well-resolved absorption and emission spectra of phenanthrene were simulated using the Franck–Condon approximation combined with the Herzberg–Teller and Duschinsky effects, and the results reproduced the experimental spectra very well. The mirror symmetry breakdown between absorption and emission spectra is induced mainly from the Herzberg–Teller effect and Duschinsky mode mixing. Moreover, most of the vibrational modes were tentatively assigned and compared with the experiment. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Polycyclic aromatic hydrocarbons (PAHs) are the largest class of mutagens and carcinogens as far as we know [1,2]. Phenanthrene, as the prototypical molecule of non-linear PAHs, has been studied extensively [3–19]. Its X-ray structure was reported by Trotter and Mason et al. [3,4] and refined by Kay et al. [5]. The lowest excitation energy was investigated by Otokozawa et al. [6] who concluded that the S1 ← S0 (S0 is ground state and S1 is first excited state) absorption is weak and should be identified as 1 Lb and similar result could be found in Grimme’s work [7]. The Raman [8,9], IR [10], fluorescence and fluorescence excitation [11–14] spectra were also reported. The optical spectra enable us to have a further understanding of the excitation energies and equilibrium geometries. However, drawbacks were that only the main vibrational bands in the experimental spectra were assigned and the tentative assignments in the experimental spectra [12] needed to be confirmed based on the theoretical simulations. The reasons of mirror symmetry breakdown between the excitation and fluorescence spectra also needed to be analyzed in theory. As for quantum mechanical methods, if reliable vibrational structures were obtained, it may provide a direct link between the spectral property and geometrical parameters [15]. Therefore, a number of theoretical studies were performed to explore the properties of phenanthrene [16–19].

∗ Corresponding author at: School of Chemistry and Chemical Engineering, Southwest University, Chongqing 400715, China. E-mail address: [email protected] (R. He). http://dx.doi.org/10.1016/j.cplett.2015.03.020 0009-2614/© 2015 Elsevier B.V. All rights reserved.

Although the equilibrium geometries and vibrational frequencies were reported in many theoretical investigations, little effort was made in the theoretical simulation of the high-resolved absorption and emission spectra together with the intensity, symmetries and frequencies. Therefore, the simulations of well-resolved spectra and the detailed assignments of vibrational modes are helpful to understand the relationship between equilibrium geometries and vibrational structures of phenanthrene. It is well-known that the transition dipole moments of strong dipole-allowed transitions could be considered independent of the nuclear coordinates (Condon approximation) [20]. This approximation is usually satisfactory for transitions characterized by large oscillator strengths, giving rise to the so-called Franck–Condon (FC) spectrum. However, it is often inadequate for weak and forbidden transitions like the S1 ↔ S0 band of phenanthrene [21]. Therefore, it is necessary to consider the Herzberg–Teller effect, causing the phenomenon of intensity borrowing [22]. The Duschinsky mixing of normal coordinates in initial and final states should be included for a proper description of Herzberg–Teller effect [13,23–26]. In addition, although the harmonic approximation could provide a relatively effective general treatment of vibrational structure, the obtained harmonic frequencies usually have some systematic errors as the specific anharmonicity in the fundamental frequencies is ignored. Therefore, the anharmonic effect should also be taken into account for the spectra simulation and the assignments of vibrational modes [27,28]. In this Letter, the properties of the S0 and S1 states were obtained using the density functional theory (DFT) and its time-dependent extension (TD-DFT) with different basis sets. The excitation

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M. Pang et al. / Chemical Physics Letters 628 (2015) 35–42

energies were used to compare with the experimental values and the S1 state was identified as 1 Lb . The spectra simulated in harmonic and anharmonic approximation were compared with the experimental findings, and a good agreement between the simulated and experimental spectra could be found. The breakdown of mirror symmetry between the absorption and emission spectra was reproduced very well, and the possible reasons were given. Especially, the main vibrational modes in the G1,2,...,Nd (t) =

   ...

1

  ˛N d 

× exp it

˛=˛1

...

N

d



 ˛Nd



ω˛ −

Nd  

1 j + 2

j=1

∞

dt exp[it(ωnm − ω)] · G(t)

(1)

−∞

 nm denotes the where  is Planck’s constant, c is the speed of light,  electronic transition dipole moment. Using the relation of Slater sum, Eq. (1) can be rewritten as

2 2ω   nm   3 ⎡



× exp ⎣−



∞

⎤

sj {2¯ j + 1 − (¯ j + 1)eitωj − ¯ j e−itωj }⎦

(2)

j

Considering the general case, a molecular system consists of Nd modes exhibiting the Duschinsky effect and N modes without mode-mixing. In this case, G(t) can be written as



l= / 1,2,...,Nd

G(t) = G1,2,...,Nd (t)

l

Gl (t)



l +

exp it

1 2



ωl −



l +

1 2

2  

  ωl

(4)

and

˛Nd



⎫⎤ ⎬ ωj ⎦ ⎭

(5)

That is, G1,2,...,Nd (t) represents the correlation function of the mixing modes (i.e., the Duschinsky effect).  nm depends on the The electronic transition dipole moment  nuclear coordinates,  nm =   nm (0) + 

 nm ∂ ∂Q



Q + ···

(6)

0

The zero-order term of this expansion is generally referred as the Franck–Condon approximation for strongly allowed transitions, while the first-order term is the so-called Herzberg–Teller effect for weakly allowed or forbidden transitions. The absorption coefficient for the electronic transition m → n in the anharmonic correction can be expressed as [27] ˛(ω) =

2 2ω   nm   3 ⎡



× exp ⎣−



∞

dteit(ωnm +˝0 −ω)−nm |t|

−∞

⎤

sj (1 − 3 j ){2¯ j + 1 − (¯ j + 1)eitωj − ¯ j e−itωj }⎦

(7)

j

The quantities ˝0 and j stand for the first-order anharmonic correction given by



˝0 = −2

j Sj ωj

(8)

j

and j =

aj3 dj aj2

=

aj3 dj 0.5ωj2

(9)

where ωj is harmonic vibrational normal-mode frequency, Sj is Huang–Rhys factor, dj denotes the displacement. 3. Results and discussion 3.1. Excitation energies and equilibrium geometries

dteit(ωnm −ω)−nm |t|

−∞



l

 l



Geometry optimizations and frequency calculations: The excitation energies, equilibrium geometries and frequencies of vibrational modes in both S0 and S1 states were calculated using the density functional theory and its time-dependent extension with different hybrid exchange functionals (B3LYP, PBE0 and BHandHLYP containing 20%, 25%, and 50% of exact HF exchange) with TZVP, QZVP and cc-pVDZ basis sets in this work. As a comparison, the geometrical structure and frequencies of the ground state were also calculated by MP2 and the first excited state was carried out using the meta-GGA (M062X) and long-range corrected (CAM-B3LYP) functionals. The result calculated by B3LYP/cc-pVDZ was adopted to compare with the previous theoretical and experimental data. Other results were provided in the Supplementary material as a reference. All computations above were carried out using the gaussian09 software package [29]. Simulation of well-resolved spectra: The absorption and emission spectra were simulated using the FCclasses program [30]. The Huang–Rhys factor and the first-order anharmonic corrected spectra were calculated using our own code and the values were given in Table S7 (see Supplementary material). With the harmonic oscillator approximation, the absorption or emission coefficient for the electronic transition m → n in Condon approximation can be expressed as [22]

˛(ω) =





Pml  n (Ql ) ml (Ql )



l

˛1

2. Computational methods and theoretical details

2 42 ω   nm  a(ω) =  3c





2    Pm1 ...Nd  n (Q˛ 1 )...n (Q˛ Nd ) × m1 (Q1 )...mNd (QNd ) 

absorption and emission spectra were assigned, which is meaningful to understand the photophysical properties of phenanthrene.



Gl (t) =



˛1

1  ˛ + 2

G1,2,...,Nd (t) and Gl (t) denote the correlation functions defined by

(3)

Generally for PAHs, the low-energy absorption spectrum is mainly characterized by two low-lying excited states, the so-called 1 L and 1 L states, with different absorption intensity that will a b determine the relative position of these two states. The 1 La has a strong absorption in ultraviolet region while the 1 Lb has a weak transition [14,31]. For phenanthrene, our calculations display that S1 state has a very weak absorption with a small oscillator strength (f = 0.0011). This indicates that the characteristic of S1 could be identified as 1 Lb , which was consistent with the published work from Otokozawa et al. [6]. The calculated vertical excitation energies ( E) and oscillator strengths (f) from S0 to S1 together with the experimental data [12] were listed in Table S1.

M. Pang et al. / Chemical Physics Letters 628 (2015) 35–42

37

Table 1 Calculated bond lengths (Å) and angles (degree) for the ground and first singlet excited states of phenanthrene using the B3LYP/cc-pVDZ level. Parameter

S0 Babaa

Bonds (Å) C1 –C2 C1 –C3 C3 –C5 C5 –C6 C1 –C7 C7 –C9 C9 –C11 C11 –C13 C13 –C3 C5 –H15 C13 –H23 C11 –H21 C9 –H19 C7 –H17 Angle (deg) C2 –C1 –C3 C1 –C3 –C5 C3 –C5 –C6 C3 –C1 –C7 C1 –C7 –C9 C7 –C9 –C11 C9 –C11 –C13 C11 –C13 –C3 C1 –C3 –C13 C3 –C5 –H15 C11 –C13 –H23 C9 –C11 –H21 C7 –C9 –H19 C1 –C7 –H17 a b c d e

1.453 1.427 1.433 1.367 1.416 1.387 1.409 1.385 1.416

S1 Manogaranb 1.461 1.404 1.439 1.337 1.409 1.367 1.400 1.365 1.406 1.073 1.073 1.072 1.072 1.069 119.1 119.8 121.1 118.1 121.3 120.4 119.6 121.0 119.8 118.1 120.4 120.0 119.9 120.0

Exp1.c 1.457 1.408 1.442 1.335 1.404 1.393 1.378 1.366 1.422

118.8 120.5 120.8 118.5 119.9 121.4 120.6 120.0 120.1

Exp2.d 1.460 1.416 1.447 1.352 1.399 1.395 1.393 1.391 1.421 1.050 1.110 1.060 1.100 1.080 118.9 120.5 120.7 118.8 120.9 121.3 119.7 119.7 120.5 118.4 121.3 118.0 119.0 122.0

Calce 1.459 1.429 1.437 1.362 1.417 1.385 1.410 1.383 1.417 1.093 1.093 1.092 1.092 1.090

Babaa 1.455 1.479 1.413 1.380 1.437 1.426 1.420 1.432 1.431

119.1 119.7 121.2 117.9 121.5 120.4 119.5 121.1 119.6 118.3 120.3 120.1 119.7 119.9

Calce 1.432 1.465 1.415 1.400 1.427 1.399 1.408 1.405 1.412 1.093 1.093 1.091 1.092 1.090 119.7 118.7 121.6 116.9 121.7 121.0 119.2 121.5 119.8 118.8 120.1 120.5 119.2 119.8

Calculated data with the CAM-B3LYP/6-311++G(d,p) level scaled by 0.983 (Ref. [14]). Optimized value at HF/4-21G level by Manogaran et al. [19]. X-ray data reported by Kay et al. [5]. Neutron diffraction reported by Kay et al. [5]. Calculated values with the B3LYP/cc-pVDZ level.

The equilibrium geometrical parameters of S0 and S1 states optimized by B3LYP/cc-pVDZ were used to compare with the available theoretical and experimental data (Table 1). The atom number˚ is ing was shown in Figure 1. For S0 state, the C9 –H19 (1.092 A) underestimated by 0.008 A˚ compared with the neutron diffrac˚ but it is better than the result calculated with tion value (1.100 A), ˚ The C5 –C6 (1.362 A) ˚ and C9 –C11 (1.410 A) ˚ are HF/4-21G (1.072 A). overestimated respectively by 0.027 and 0.032 A˚ when compared ˚ while are similar to the with the X-ray values (1.335 and 1.378 A), ˚ The differences of neutron diffraction results (1.352 and 1.393 A). bond angles are almost within 1◦ except the overestimation of

C9 –C11 –H21 and underestimation of C1 –C7 –H17 by 2.1◦ in comparison with the neutron diffraction values. The analysis of bond lengths and angles showed that the hydrogen atoms labeled as H17 and H18 were forced to approach each other closely and therefore lead to a change in the bond strength of C7 –H17 and C8 –H18 as caused by ␲-delocalization [5,32]. The calculated geometrical parameters of S0 state are in accordance with the available theoretical [14,19] and experimental data [5], indicating that the present theoretical methods are reasonable. For S1 state, its geometrical parameters were calculated and compared with the theoretical data reported by Baba [14]. One can see that the present results are consistent with Baba’ data [14]. To the best of our knowledge, the molecular structure of S1 state has not been investigated experimentally, thus the calculated geometry of S1 state is only a reasonable prediction. 3.2. State symmetries and vibrational frequencies

Figure 1. Molecular structure, coordinate axes, and numbering of carbon and hydrogen atoms in phenanthrene.

Phenanthrene is expected to possess C2v symmetry by our calculation which is coincident with those previous studies [12,14]. It has 66 normal modes including ten out of plane modes of b1 , twenty-two planar modes of b2 symmetry, twenty-three Condon allowed a1 modes and eleven a2 modes [12]. The symmetries and vibrational frequencies of phenanthrene calculated by B3LYP/cc-pVDZ were listed in Table 2 in comparison with the available experimental and theoretical data [12,14]. Obviously, the calculated frequencies of S0 and S1 states are consistent with the experimental and previous calculated results. The maximum deviation is just 54.6 cm−1 in S0 state and 30.1 cm−1 in S1 . What’s more, the value of mode 47 (1430 cm−1 ) is close to

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M. Pang et al. / Chemical Physics Letters 628 (2015) 35–42

Table 2 Calculated vibrational frequencies (cm−1 ) in the S0 and S1 states of phenanthrene together with the experimental data (only the a1 and b2 modes were listed). Symmetry

a1 modes

b2 modes

a b c d

State S0

Mode

Mode

Expa

B3LYPb

5 7 13 17 23 34 35 37 39 40 42 44 45 48 49 52 54 56 58 61 62 64 66

246 408 544 714 833 1063

247 411 555 725 841 1065 1111 1165 1178 1229 1260 1325 1388 1454 1471 1567 1652 1671 3169 3180 3186 3197 3216

5 8 13 17 23 34 35 37 38 41 42 44 45 48 50 52 53 56 58 61 62 64 66

446 504 629 723 886 1015 1061 1158 1177 1237 1293 1388 1437 1486 1538 1618 1666 3165 3169 3179 3194 3205

9 12 15 20 26 32 33 36 39 40 43 46 47 49 51 54 55 57 59 60 63 65

9 10 15 16 24 32 33 36 38 41 43 46 47 50 51 53 55 57 59 60 63 65

1165 1199 1257 1355 1423 1450 1526 1580 1613

471 500 620 731 875 1019

1434

State S1 Exp.a

Exp.d

Cal.c (scaled)

B3LYPb

238 395 517 673 817 1021 1038

238 392 516 672 819 1036 1065

1207 1246 1337 1377 1430 1487 1529 1602

1155 1243 1260 1342 1374 1449 1483 1534 1578

231 395 520 688 813 1057 1067 1116 1138 1217 1236 1305 1371 1439 1491 1510 1544 1680 3257 3278 3282 3296 3319

238 401 519 690 828 1048 1087 1141 1153 1242 1256 1348 1397 1446 1469 1520 1548 1643 3173 3182 3188 3206 3222

435 486 600 706 853 965 1008 1110 1154 1191 1266 1384 1403 1460 1493 1596 1645 3252 3257 3276 3284 3299

437 495 598 710 858 985 1041 1124 1171 1230 1277 1413 1430 1459 1516 1557 1573 3170 3175 3181 3203 3210

429 486 591 702 833 957 1016

1373 1383

425 484 589 701 829 954 1013 1119 1188 1221 1265 1408 1428 1524 1597

Experimental frequencies reported by Warren [12]. Calculated values at the B3LYP/cc-pVDZ level. Calculated values in CAM-B3LYP/6-311++G(d,p) taken from Ref. [14]. Experimental frequencies of the first singlet state reported by Baba [14].

both the experimental (1428 cm−1 ) and theoretical (1403 cm−1 ) data reported by Baba et al. [14], but has a deviation of 57 cm−1 when compared with the experimental value (1373 cm−1 ) given by Warren et al. [12]. So, we considered that the experimental data (1428 cm−1 ) given by Baba et al. [14] is reliable (see the red marks in Table 2). Differences between vibrational frequencies of S0 and S1 were just dozens of wave numbers. Therefore, the accurate frequencies and equilibrium geometries of S0 and S1 states make it possible to correctly simulate the electronic spectra of phenanthrene. 3.3. Simulated optical spectra in harmonic and anharmonic approximations 3.3.1. Harmonic approximation In this work, the well-resolved spectra of phenanthrene were simulated using a model of displaced oscillator considering Franck–Condon approximation incorporated with Herzberg–Teller effect [33] (named FH). Due to the interferential features, the

Duschinsky mixing [34] of normal coordinates in the initial and final states was included for a proper description of the Herzberg–Teller term. Based on the frequencies and symmetries of experimental results, we tentatively assigned the vibrational bands in FH absorption spectrum. Relevant normal modes of the electronic states were listed in Table 3. In comparison with the experimental data, the main transitions 51 (238 cm−1 , a1 ), 171 (690 cm−1 , a1 ), 201 (710 cm−1 , b2 ), 231 (828 cm−1 , a1 ) and 501 (1469 cm−1 , a1 ) were coincident with peaks located at 238 cm−1 (a1 ), 673 cm−1 (a1 ), 702 cm−1 (b2 ), 817 cm−1 (a1 ) and 1487 cm−1 (a1 ) in experimental absorption spectrum reported by Warren et al. [12]. For transitions 81 , 91 and 151 , the symmetries and positions were simulated accurately, but the intensities were overestimated. While for band 451 , the intensity was underestimated seriously. The relative intensity difference between theory and experiment may be caused by the systematic error in harmonic approximation and the displacement between optimized geometries of S0 and S1 . To clarify the contribution of Herzberg–Teller effect on the vibrational intensities of different bands, spectra labeled as FC and

M. Pang et al. / Chemical Physics Letters 628 (2015) 35–42

39

Table 3 Relevant normal modes of the electronic states in the FH absorption spectrum of Phenanthrene, calculated with B3LYP/cc-pVDZ. na

51 81 91 121 131 151 171 201 231 341 351 361 371 381 401 431 441 451 461 471 481 491 501 511 531 551

State S1

State S0 b

sym

ω

a1 a1 b2 b2 a1 b2 a1 b2 a1 a1 a1 b2 a1 a1 b2 b2 a1 a1 b2 b2 a1 b2 a1 b2 a1 b2

238 401 437 495 519 598 690 710 828 1048 1087 1124 1141 1153 1230 1277 1348 1397 1413 1430 1446 1459 1469 1516 1548 1573

c

d

 e

 e

 e

f

I

ı

m1

m2

m3

ω

9.6 18.9 17.2 7.0 14.3 27.1 33.7 3.1 8.4 2.2 1.5 7.4 0.5 2.5 3.8 5.6 7.5 2.4 2.7 2.9 24.5 3.2 14.2 6.5 26.3 20.9

−1.77249 −8.00753 0.00000 0.00000 −1.64695 0.00000 −5.21775 0.00000 1.90890 −2.47486 0.24656 0.00000 −0.39242 0.57829 0.00000 0.00000 1.83350 −5.31238 0.00000 0.00000 −5.95045 0.00000 −1.16237 0.00000 −4.29448 0.00000

5 [1.00] 7 [0.98] 9 [0.99] 10 [0.99] 13 [0.98] 15 [0.99] 17 [0.99] 18 [0.99] 23 [0.99] 34 [0.96] 35 [0.97] 36 [0.99] 39 [0.98] 37 [0.44] 41 [0.96] 43 [0.98] 44 [0.79] 45 [0.70] 46 [0.87] 47 [0.83] 49 [0.75] 50 [0.91] 52 [0.23] 51 [0.78] 54 [0.67] 53 [0.81]

7 [0.00] 13 [0.03] 10 [0.01] 9 [0.01] 7 [0.02] 17 [0.01] 13 [0.00] 15 [0.01] 13 [0.01] 37 [0.02] 39 [0.01] 55 [0.00] 37 [0.02] 39 [0.32] 45 [0.01] 55 [0.01] 48 [0.17] 48 [0.23] 47 [0.07] 45 [0.09] 46 [0.10] 47 [0.05] 40 [0.17] 55 [0.21] 48 [0.13] 55 [0.11]

13 [0.00] 5 [0.00] 15 [0.00] 15 [0.00] 23 [0.00] 10 [0.00] 7 [0.00] 10 [0.00] 48 [0.00] 35 [0.01] 34 [0.01] 33 [0.00] 42 [0.00] 40 [0.20] 38 [0.01] 45 [0.00] 46 [0.03] 49 [0.01] 53 [0.04] 50 [0.07] 48 [0.08] 45 [0.02] 48 [0.16] 45 [0.01] 56 [0.12] 45 [0.03]

−9.26 −10.42 −8.96 −9.27 −36.23 −30.64 −35.28 −27.44 −12.24 −16.56 −23.76 −33.98 −37.55 −11.63 −6.74 −16.58 23.18 8.35 24.79 −7.63 −24.97 −27.53 −97.54 −21.42 103.30 −44.97

ωexp

g

238 395 429 486 517 591 673 702 817 1021 1038

1337 1377 1373 1430 1383 1487 1602

ωexp

i y (1)

i z (1)

FCj

0.000000 0.000000 0.004159 0.002833 0.000000 −0.006163 0.000000 −0.002111 0.000000 0.000000 0.000000 0.004425 0.000000 0.000000 −0.003019 0.004067 0.000000 0.000000 0.002749 −0.003067 0.000000 −0.003059 0.000000 −0.004878 0.000000 0.008372

0.002036 0.001936 0.000000 0.000000 −0.004997 0.000000 0.007223 0.000000 −0.002751 0.000488 −0.001779 0.000000 0.000832 −0.002163 0.000000 0.000000 −0.003020 −0.002845 0.000000 0.000000 0.002501 0.000000 −0.007821 0.000000 0.004777 0.000000

0.001206 0.038279 0.000000 0.000000 0.002176 0.000000 0.028124 0.000000 0.004910 0.008895 0.087343 0.000000 0.000364 0.000581 0.000000 0.000000 0.006287 0.055388 0.000000 0.000000 0.073837 0.000000 0.002190 0.000000 0.041281 0.000000

h

238 392 425 484 516 589 672 701 819 1036 1065 1019 1155 1221 1265 1342 1374 1408 1428 1449 1483 1524 1578 1597

a Fundamental vibrations, assigned as nq , where n is the excited normal mode, and q is quantum number. These correspond to totally symmetric modes or vibronically induced nonsymmertric modes. b Frequencies relative to the electronic origin band. c Relative peak intensity, the 0–0 origin band intensity is 100. d Dimensionless displacement. e In square brackets the J(m x , n)2 , values = 0.00 are actually values <0.005. f

ω = ω − ωm 1 g The values in laser jet spectrum published by Warren et al. [12]. h The values reported by Baba et al. [14]. i First derivatives of the electric dipole transition moment (au/(Å amu1/2 )) for the relatively active normal modes. j Franck–Condon factors smaller than 0.000001 are neglected.

HT were simulated (see Figure 2). The FC spectrum was simulated in Franck–Condon approximation without the Duschinsky rotation, Herzberg–Teller effect as well as the change of frequencies between the ground and excited states. It is assumed that the excited-state normal modes are identical to those of the ground state, apart from a displacement between them. S0 normal mode frequencies were used for absorption and S1 frequencies were used for emission. Spectrum HT was simulated in Herzberg–Teller approximation including the Duschinsky mixing by setting the transition dipoles at the geometry of the ground and excited states to zero (i.e. setting the FC term to zero). As shown in Figure 2, the intensities of FC spectrum are much weaker than those in FH one, while the intensities of HT spectrum are almost the same as the FH one except the intensity of 0–0 transition. It indicated that the Herzberg–Teller effect is essential in obtaining a full and rich vibrational structure. This can also be found by investigating the derivatives of the electric dipole transition moments with respect to the normal coordinates. The dipole moment derivatives along Z axis (as plotted in Figure 1) have nonzero values for a1 vibrations and that the values of modes 17 and 50 are more intense which exhibit the higher HT activity. The dipole moment derivatives along Y axis (see Figure 1) have relatively large values for b2 modes which indicated that the Herzberg–Teller approximation also has an effect on the b2 -type modes. In order to give a visualized sight of Duschinsky contribution in Herzberg–Teller approximation, spectrum HTNO-D was simulated. The only difference between HT and HTNO-D was that the Duschinsky couplings were switched off. Compared the HTNO-D spectrum with HT one, we can find that the intensities of the fundamental transition of several modes changed greatly

especially for mode 50. It indicated that Duschinsky mixing affected this mode seriously and it can be identified further by the calculation of Duschinsky matrix elements. Mode 50 is mainly projected on three modes 52 , 40 and 48 , the values of Duschinsky matrix elements are J[52 , 50]2 = 0.23, J[40 , 50]2 = 0.17 and J[48 , 50]2 = 0.16, respectively. Inspection of the maximal Duschinsky matrix values of all the relevant normal modes in Table 3, it is interesting to find that the values of maximal Duschinsky matrix elements of modes located in the range 0–1000 cm−1 (modes 8, 9, 12, 13, 15, 17 and 23) are very large (greater than 0.98), while the values were relatively small for modes within the high frequency region (modes 45, 48, 50 and 53). It indicated that the Duschinsky effect is weak for the low frequency modes and relatively evident when it comes to the high frequency modes. Similar result was given by Hohlneicher [35]. The emission spectrum was also simulated and compared with the experimental result reported by Warren et al. [12]. Relevant normal modes of the electronic states were listed in Table S5. The simulated spectrum agreed with the experimental emission spectrum well. Transitions assigned as 71 (411 cm−1 , a1 ), 171 (725 cm−1 , a1 ), 491 (1471 cm−1 , a1 ) are in consistent with the previous experimental peaks (407 cm−1 (a1 ), 714 cm−1 (a1 ) and 1450 cm−1 (a1 ), respectively). However, the intensities of transitions assigned as 451 (1388 cm−1 , a1 ), 491 (1471 cm−1 , a1 ) and 541 (1652 cm−1 , b2 ) (they correspond respectively to the peaks located at 1355, 1450 and 1580 cm−1 in the experimental emission spectrum) were underestimated. This may be caused by the systematic error under harmonic approximation. Therefore, the first-order anharmonic effect should be taken into account and it will be discussed in Section 3.3.2. The FC, HT, HTNO-D and FH

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M. Pang et al. / Chemical Physics Letters 628 (2015) 35–42

Figure 2. Simulated well-resolved absorption (left) and emission (right) spectra of phenanthrene, calculated with B3LYP/cc-pVDZ, within a range of about 2210 cm−1 from the 0–0 transition frequency (set to zero). The intensities of modes in the emission HT and HTNO-D spectra were five times stronger than those in FH and FC ones.

emission spectrum of phenanthrene were simulated and listed in Figure 2. Results showed that the Herzberg–Teller effect was noticeable in providing the full and rich emission spectrum. 3.3.2. Anharmonic approximation The above discussion showed that the vibrational intensities of several bands simulated with harmonic approximation were underestimated when compared with corresponding experimental results. Therefore, the first-order anharmonic correction was used to simulate the spectra of phenanthrene and the result was expected to improve the spectral profile in comparison with the experiment. The present method has been performed successfully in previous reports [27,28]. In order to exhibit separately the contribution of anharmonicity on the vibrational intensity, the spectrum was simulated in Franck–Condon approximation (FC spectrum). It is known that the frequencies of excited state cannot be calculated by TD-DFT methods in anharmonic approximation, thus the S0 anharmonic frequencies were used to evaluate roughly the anharmonic effect of emission spectrum. The frequencies of S0 in harmonic and anharmonic approximations were calculated with B3LYP/cc-pVDZ level, and the results were listed in Table S6. The simulated FC emission spectra with harmonic and anharmonic effects together with the experimental result were listed in Figure 4. For the fundamental transitions, only the totally symmetric vibrational normal modes (a1 -type) may appear in the FC spectrum in harmonic approximation. Therefore, in the experimental emission spectrum, bands 151 and 241 with b2 -type vibration were

not included in following discussion. Compared the harmonic FC (abbreviated as fc-har) emission spectrum with the experimental one, the most noticeable difference is that the vibrational intensities of transitions 451 , 491 and 541 were underestimated extremely. In addition, band 51 does not appear in fc-har spectrum. However, in anharmonic FC (shorten as fc-anhar) emission spectrum these two problems were improved markedly. Firstly, band 51 appears in the fc-anhar spectrum although with a quite weak vibrational intensity. Secondly, the intensities of fundamental transitions 451 , 491 and 541 were improved which are almost comparable with the experimental intensities of corresponding modes. Excluding the non-totally symmetric vibrations, we found that fc-anhar emission spectrum simulated correctly the most of main vibrational bands except for band 131 , and reproduced the experimental spectrum profile very well. According to Eq. (7), the anharmonic quantity ˝0 only shifts the location of harmonic 0–0 excitation energy, but has no influence on vibrational intensity. This is the so-called dynamic correction to the main peak of 0–0 line in spectra [27]. Contrarily, another anharmonic quantity j from the first-order contribution will provide a possible contribution to the change of vibrational intensity as shown in Eq. (7). Therefore, we calculated the values of j for all totally symmetric vibrational modes based on Eq. (9), and the results were given in Table S7. Compared Eq. (2) with Eq. (7), one can find that the anharmonic effect just corrects the Franck–Condon factors, which finally leads to the change of vibrational intensity. As for the absorption spectrum, although the anharmonic effect corrects the relative intensities of transitions 451 and 481 in FC

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Figure 3. Theoretical absorption (left) and emission (right) spectra of phenanthrene simulated with the B3LYP/cc-pVDZ level in harmonic approximation together with the experimental result [12].

Figure 4. Theoretical FC emission spectra of phenanthrene simulated at the B3LYP/cc-pVDZ level in harmonic and anharmonic approximations together with the experimental results [12].

spectrum (see Figure S3), its influence on the final spectrum is almost invisible. 3.3.3. Mirror symmetry breakdown The most notable difference between the excitation and emission spectral profiles was the enormous changes in intensity of the corresponding vibrational bands. For experimental absorption spectrum, the intensities of several bands are very strong within the range 0–1000 cm−1 , especially the intensity of band 171 , which is almost comparable to that of the 0–0 line. While in the experimental fluorescence, the spectral profile is only dominated by the 0–0 transition and the intensity of band 171 is about one-sixth relative to that of the 0–0 line. This indicated that there was a remarkable asymmetry between the excitation and fluorescence and the mirror symmetry was broken down. Inspection of Figure 3 showed that the simulated absorption and fluorescence spectra reproduced perfectly the experimental mirror symmetry breakdown (MSB). However, what are the factors that lead to the MSB? Exploring these factors is crucial for understanding the photophysical properties of phenanthrene. Firstly, the differences between the vibrational frequencies of S0 and S1 might be responsible for MSB, even though their deviations were small as just discussed above. For example, the frequency difference of mode 17 in S0 and S1 is 21 cm−1 in experiment (35 cm−1 in calculation, see Table 2 and Figure 3), indicating that the band 171 in absorption and emission spectra is not symmetrically located at the two sides of the 0–0 line. For bands 51 , 131 , 151 and 451 , the same situation is found. However, the frequency difference should

not be the main reason that led to MSB. As pointed out by Warren et al. [12], the origin of MSB should involve interference of the Condon ‘allowed’ and vibronically induced moments for transitions involving totally symmetric vibrations (that is Herzberg–Teller effect) and mode mixing (Duschinsky contribution) of b2 -type modes. Inspection of Figure 2 showed that both the simulated FC absorption and emission spectra of phenanthrene are weak transition. However, the FH absorption spectrum is very strong (relative to the FC absorption) while the FH emission is still very weak. This suggests that the Herzberg–Teller contribution plays a significant role in the MSB, which is in good agreement with the results reported by Warren [12] and Hohlneicher [35]. Further, by inspecting the FH absorption spectra, one can find that the vibrational intensities mainly arose from the fundamental transition of modes 8, 9, 12, 13, 15, 17, 23, 43, 48, 53 and 55. In these modes, the transitions 8, 13, 17, 23, 48 and 53 are totally symmetric normal vibrational modes (a1 -type), while the others are of b2 -symmetry. The above discussion reveals that the Herzberg–Teller effect not only changes seriously the vibrational intensities of the fundamental transition of totally symmetric vibrational modes (such as modes 8, 13, 17, 23, 48 and 53), but also changes greatly the intensities of the fundamental transition of non-totally symmetric b2 -type modes (9, 12, 15 and 43). In addition, the values of Duschinsky matrix elements of modes 48, 53 and 55 are relative small (the maximal value, J[53 , 55]2 , is only 0.81). Among these three modes, mode 55 is b2 -type while modes 48 and 53 are a1 -type, indicating that the Duschinsky effect also contributes to the vibrational intensity change of totally symmetric vibrational modes (a1 -type). The

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present results not only confirmed the experimental findings (that is, the Herzberg–Teller effect of totally symmetric vibrations and the Duschinsky contribution of b2 -type modes should be responsible for MSB), but also proposed that both Herzberg–Teller and Duschinsky effects can change the intensities of totally and nontotally symmetric vibrational modes. Compared the FC and FH emission spectra (Figure 2), we found that the intensity changes between them are very small. Moreover, the intensity changes are mainly caused by the a2 -type modes (modes 1, 4 and 27), b1 -type modes (modes 22 and 31), and a1 type modes (modes 40 and 49). The similarity of HTNO-D and HT shows that the Duschinsky effect is negligible in emission spectrum. Comparison of absorption and emission spectra indicated that the Herzberg–Teller effect on the fundamentals can be constructive in absorption and destructive in emission by changing the vibrational intensities of different modes. Similar result was given by Hohlneicher [35]. This leads to the differences in the absorption and emission spectra, thus the mirror symmetry is broken down. 4. Conclusions In this work, the characteristic of the first singlet state was identified as 1 Lb by investigating the excitation energies of phenanthrene. In the approximations of harmonic and anharmonic oscillators, the well-resolved absorption and emission spectra were simulated using Frank-Condon approximation incorporated with the Herzberg–Teller and Duschinsky effects. The Herzberg–Teller effect and Duschinsky mode mixing were analyzed individually, and the good coincidence with experimental spectra indicated that the Herzberg–Teller and Duschinsky contributions were essential to weak vibrational transitions like S0 ↔ S1 of phenanthrene. In addition, we found that the first-order anharmonic effect can improve the electronic spectra of phenanthrene by modifying the positions and intensities of vibrational bands. The factors leading to MSB were also explored, and the results suggested that the MSB should be induced mainly by the Duschinsky and Herzberg–Teller effects. Further, most of the vibrational bands were tentatively assigned and compared with the experimental data. The theoretical insights presented in this work are expected to help us understand the photophysical properties of other non-linear polycyclic aromatic hydrocarbons. Acknowledgments This work was supported by the Natural Science Foundation of China (21173169, 20803059), Chongqing Municipal Natural

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