In Silico Stark Effect: Determination of Excited-State Polarizabilities of

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In Silico Stark Effect: Determination of Excited-State Polarizabilities of Squaraine Dyes L. Orian,† R. Pilot,*,†,‡ and R. Bozio†,‡ †

Dipartimento di Scienze Chimiche, Università degli Studi di Padova, Via Marzolo 1, 35131 Padova, Italy Consorzio INSTM, Via G. Giusti 9, 50121 Firenze, Italy

‡

S Supporting Information *

ABSTRACT: The static electric polarizabilities (α) of a quadrupolar squaraine dye are investigated in silico, either as the excess polarizability, i.e., the change from the ground to the lowest excited state, Δα, or as those of the two states separately, depending on the approach. The polarizabilities are worked out by making use of the energy and dipole moment Taylor expansions as a function of the electric field (E), in which α is represented by the quadratic and linear terms, respectively, and also by means of the linear response approach. Dipoles and energies are computed at a few values of the electric field, with different strategies that consider the geometry of the molecule either frozen in the ground state or relaxed at each E value. From a physical standpoint, the most appropriate approach to describing the molecular polarizability depends on the processes in which the molecule is involved: for example, fluorescence methods provide information about relaxed excited states, and absorption methods are used to determine the polarizability changes of excited states frozen in the ground-state conformation. We show that the excited-state polarizability does not strongly differ from the ground-state polarizability when the field is applied along the main axis of the squaraine. In contrast, remarkable differences are found when the field is applied perpendicular to the molecular plane due to a large geometrical distortion of the molecular backbone.

1. INTRODUCTION Excited states of organic molecules act as mediators in the physical processes occurring in artificial devices (such as lightemitting diodes (LEDs)1 and photovoltaic cells2−4) and in biological systems (such as those involved in vision5). Electronic pump−probe techniques6−8 and, in particular, time-resolved vibrational spectroscopy9 can be used to probe the ultrafast structural changes associated with the temporal evolution of excited states. Importantly, as recently reviewed by Gruenke,10 time-resolved vibrational spectroscopy can also be combined with plasmonic substrates (as in surface-enhanced Raman scattering, SERS11−16) to address the interactions between molecular excitons of J-aggregates and metallic nanoparticles. An important parameter in the description of the properties of molecular ground and excited states is the polarizability. It is related, for example, to the absorption coefficient17 and to the Raman scattering cross section;18 moreover, it directly enters the expression of the dispersion forces,19 which control the assembly of organic molecules into aggregates. 20 The magnitude of the dispersion forces is expected to vary upon excitation, reflecting the changes in molecular polarizability from the ground to the excited state. An in-depth knowledge of the excited-state properties is therefore of considerable interest in order to gain better insight into the physical processes in which they are involved. To date, the change in polarizability between the excited state and © 2017 American Chemical Society

ground state (Δα), also called excess polarizability, has been reported in a limited number of papers in the literature, likely because of the fact that accurate calculations of excited-state geometries and the properties of sizable molecules in the presence of perturbations such as external fields have been tackled only in the past few years. For example, in pioneering work by Grozema et al., time-dependent density functional theory (TDDFT) calculations are used to work out the static Δα in several conjugated oligomers (diphenylpolyenes and oligothiophenes), showing a positive Δα for the transition to the lowest excited state; all calculations are referenced to ground-state geometries.21 Other theoretical studies were carried out by Improta et al. on tetra-phenyl-porphyrins,22 by van der Horst et al. on conjugated polymers,23 by Hinchliffe et al. on azulene,24 by Christiansen et al. on benzene,25 and by Ye et al. on diphenylpolyenes26 with different approaches. Data on the excited-state polarizability are also available from the experiment, typically recovered in two kinds of measurements: electronic absorption/emission27,28 and flash photolysis timeresolved microwave conductivity (FP-TMRC).29,30 The former technique exploits the Stark effect: the application of an electrical field to the sample induces a shift in the absorption or fluorescence spectrum that has a quadratic dependence on the Received: December 1, 2016 Revised: January 31, 2017 Published: February 1, 2017 1587

DOI: 10.1021/acs.jpca.6b12090 J. Phys. Chem. A 2017, 121, 1587−1596

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The Journal of Physical Chemistry A field itself. Shifts in the fluorescence spectrum provide information about the polarizability of a geometrically relaxed excited state. On the other hand, absorption shifts are related to the polarizability change of an excited state frozen in the ground-state geometry. In contrast, in FP-TMRC, the sample is excited with nanosecond pulses, and the polarizability change, which translates into a variation of the dielectric constant of the sample, is revealed as a phase shift of a microwave beam. In FPTMRC, the properties of a relaxed excited state are typically probed. Squaraines are a very promising class of compounds that have been widely investigated for applications in several technological fields such as photoconductivity,31 LEDs,32 and photovoltaics.33,34 Squaraines are characterized by a central squaralium unit acting as an acceptor, linked to two side donor units. When the two donors are identical, these molecules show only minor solvatochromism in fluorescence and in absorption because of the fact that the ground and excited states maintain nondipolar character upon excitation.35 Quadrupolar squaraines are also known to possess very strong nonlinear optical properties, in particular, very large two-photon absorption cross sections,36 as normally expected for donor−acceptor−donor systems.37−39 To the best of our knowledge, the excited-state polarizability of squaraine dyes has not been reported in the literature so far. In this article, we calculate ground- and excited-state static properties of a symmetric model squaraine (2,4-bis[4-(N,Ndimethylamino)-phenyl], CH3-Sq) in vacuo. The level of theory is validated for the analogous tert-butyl compound, the experimental absorption spectrum and crystallographic structure of which have been recently reported.33 Different approaches to the calculation of the excess polarizability are adopted and discussed and are aiming at providing a general tool for in silico Stark effect experiments.

and angles closely match. Solvent effects (chloroform) on the computed excitation energies of C4H9-Sq were accounted for using the conductor-like screening model (COSMO)48,49 as implemented in the ADF program.42 We have used a solventexcluding surface with an effective radius for chloroform of 3.17 Å derived from the macroscopic density, molecular mass, and a relative dielectric constant of 4.8 g/mol. The empirical parameter in the scaling function in the COSMO equation was chosen to be 0.0. The radii of the atoms were taken to be MM3 radii,50 divided by 1.2, giving 1.350 Å for H, 1.700 Å for C, 1.517 Å for O, and 1.608 Å for N. The experimental UV−vis spectrum in chloroform shows a maximum at 641 nm. The lowest excitation energy calculated at the COSMO-SAOP/ TZ2P all electron level is 633 nm and has the strongest oscillatory strength ( f = 2.128). The same transition in vacuo occurs at 542 nm (f = 1.710). It is worth noting that the lowest absorption of CH3-Sq in vacuo is 530 nm (f = 1.441). As expected, there is a negligible effect due to the length of the alkyl pendants. Functional LB9451 was also tested for the calculation of the excitation energies in chloroform of C4H9-Sq, but a worse agreement with the experimental data was obtained. In fact, at COSMO-LB94/TZ2P all electron, the lowest excitation energy is 671 nm ( f = 2.06) and overestimates by 30 nm the experimental value. Excited-state geometry optimizations were carried out at BLYP/TZ2P all electron level of theory using tight integration and convergence criteria. It has been reported that care must be taken when computing the absorption spectrum of squaraines using the TD-DFT approach because deviations of tens of nanometers may be obtained when compared to the experiment and to other ab initio approaches such as SAC/CI (symmetry adapted cluster/ configuration interaction).52,53 The very nice agreement with the experiment, when combined with the COSMO description of the solvent, enhances our confidence in the validity of our approach (SAOP/TZ2P all electron level), which was not specifically benchmarked in refs 52 and 53. Electric polarizability can be worked out in several approaches, as illustrated in the following. The total Hamiltonian can be expressed as

2. COMPUTATIONAL METHODS The geometry of 2,4-bis[4-(N,N-dimethylamino)-phenyl] squaraine (CH3-Sq) was fully optimized without any symmetry constraint. Calculations were carried out with the BLYP functional40,41 as implemented in the Amsterdam density functional (ADF) program,42 combined with a TZ2P basis set for all of the atoms taken from the ADF library. The TZ2P basis set is a large uncontracted set of Slater-type orbitals (STOs), is of triple-ζ quality, and has been augmented with two sets of polarization functions on each atom: 2p and 3d in the case of H and 3d and 4f in the case of C, N, and O. The frozencore approximation was employed for ground-state geometry optimizations: up to 1s for C, N, and O. A very accurate Becke integration grid and ZLMfit fitting function were used.43 The level of theory is defined in the text BLYP/TZ2P sc. The SAOP (statistical averaging of (model) orbital potentials) potential,44,45 combined with an all electron TZ2P basis set, has been used for TD-DFT calculations, i.e., for the calculation of the excitation energies and polarizabilities, because it is well suited for excited states and response properties.46,47 The level of theory was validated by optimizing the geometry of 2,4-bis[4-(N,N-dibutylamino)-phenyl] squaraine (C4H9-Sq) and calculating its absorption spectrum; the single-crystal X-ray structure and UV−vis absorption spectrum in chloroform of this compound have been recently reported.33 The BLYP/ TZ2P sc-optimized geometry of C4H9-Sq in vacuo is in very nice agreement with the experimental geometry: bond lengths

H = H0 + H1

(1)

where H0 describes the unperturbed molecule and H1 = −μ·E represents its interaction with an external electric field. μ and E are the electric dipole moment and the static electric field, respectively. The energy (E) of the molecule can be expressed, in the presence of the perturbation E, as a Taylor expansion around the energy at E = 0 ⎛ dE ⎞ 1 ⎛ d2E ⎞ E(Ei) = E(0) + ⎜ ⎟ Ei + ⎜ 2 ⎟ Ei2 + ... 2 ⎝ dEi ⎠ ⎝ dEi ⎠ E = 0 E =0 i

i

(2)

where i = X, Y, Z. This expression is simplified for the case in which E is polarized along one of the Cartesian components. E(Ei) represents the energy of the molecule, either in the ground state or in one of the excited states, in the presence of an external electric field applied along direction i. Because the Hellmann−Feynman theorem guarantees that dE dEi

=

dH dEi

= −⟨μi ⟩, by the derivation of eq 2 we can also

write 1588

DOI: 10.1021/acs.jpca.6b12090 J. Phys. Chem. A 2017, 121, 1587−1596

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Figure 1. 2,4-Bis[4-(N,N-dimethylamino)-phenyl] squaraine (CH3-Sq): fully optimized molecular structure at the BLYP/TZ2P sc level of theory (A) and frontier Kohn−Sham molecular orbitals (level: SAOP/TZ2P all electron) (B and C). Different color codes are used for HOMO (red/blue) and LUMO (orange/cyan).

Table 1. HOMO−LUMO Excitation Energy (in eV) in the Presence of an Applied Electric Field along Different Directionsa field direction field

X

0 0.0001 0.0005 0.001 0.005

2.34 2.34 2.34 2.34

(1.441, (1.441, (1.441, (1.441,

Y 2.34 2.34 2.34 2.34 2.34

98.3%) 98.3%) 98.3%) 98.3%)

(1.441, (1.441, (1.441, (1.441, (1.437,

Z 98.3%) 98.3%) 98.3%) 98.3%) 98.3%)

2.34 2.34 2.34 2.35

(1.441, (1.441, (1.442, (1.455,

98.3%) 98.3%) 98.3%) 97.1%)

The electric field is applied only in TD-DFT calculations. The oscillator strength and the monoelectronic character percentage are indicated in parentheses. a

⎛ d2E ⎞ ⎛ dE ⎞ ⟨μi ⟩ = −⎜ − ⎜ 2 ⎟ Ei + ... ⎟ ⎝ dEi ⎠ E = 0 ⎝ dEi ⎠ E = 0 i

Finally, the polarizability can be calculated as the linear coefficient in the dipole moment vs the electric field or as the quadratic coefficient in the energy vs the electric field relations, respectively.

(3)

i

This expression corresponds to the Taylor expansion of the dipole moment as a function of the electrical field and can be recast by making use of the polarizability definition, leading to

⟨μi ⟩ = μi0 + αiiEi + ... where μi0 = −

( ) dE dEi

Ei = 0

3. RESULTS AND DISCUSSION The model squaraine is the 2,4-bis[4-(N,N-dimethylamino)phenyl] squaraine (CH3-Sq) (the optimized molecular structure, the HOMO and LUMO Kohn−Sham molecular orbitals are shown in Figures 1A, 1B and 1C, respectively). Its geometry, fully relaxed at the BLYP/TZ2P sc level of theory, has almost perfect D2h symmetry, although no symmetry constraint was imposed. Bond lengths and angles closely match the crystallographic corresponding ones of analog C4H9-Sq.33 The central C4O2 cyclobutadione bridge and the two electrondonating aniline units are consistent with a resonance-stabilized

(4)

and αii = −

d2E dEi2

( )

αii is the

Ei = 0

diagonal element ii of the rank two polarizability tensor α. Analogously, the energy expression can be recast as E(Ei) = E(0) − μi0 Ei −

1 αiiEi2 + ... 2

(5) 1589

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Figure 2. Molecular structures of CH3-Sq fully optimized in vacuo in the presence of an external electric field E = 0.005 au applied along X (A, view along the Y direction), Y (B, view along the X direction), Z (C, view along the X direction); bond lengths of CH3-Sq are fully optimized in vacuo in the presence of an external electric field E = 0.005 au applied along the Z direction (D). Level of theory: BLYP/TZ2P sc.

to a U-bent shape. When the field is applied along Y, only slight bending occurs. Finally, when the field is applied along Z, CH3Sq is stretched along the molecular axis. Interestingly, when inspecting this last geometry (Figure 2D), the two halves of CH3-Sq are not equivalent anymore. In Tables 2 and 3, the ground-state energies and the dipole moments of CH3-Sq calculated through optimization in the

structure.54 The C−O and C−N crystallographic bond lengths correspond to 1.23 and 1.36 Å, respectively, and nicely agree with the computed values (Figure 1C). A comparison with the C−O distance in ketones, i.e., 1.24 Å, and in particular with the C−N bond in aniline, 1.40 Å, reveals a certain extent of electron delocalization in the ground state. 3.1. Calculation of the Energy and Dipole Moment of the Ground and Excited States. The calculated excitations at the SAOP/TZ2P all electron level (the level of theory was validated against an experimental absorption spectrum, as discussed in the computational methods) reveal a strong absorption at 530 nm with an oscillator strength of 1.441, which has neat monoelectronic character (98.3%) and is ascribed to the HOMO−LUMO transition (Figure 1B,C). Neither the energy nor the nature of this excitation is significantly affected by the presence of an external electric field applied along the X and Y directions; as shown in Figure 1A, X, Y, and Z indicate the direction orthogonal to the molecular plane, the direction along the short molecular axis, and the direction along the long molecular axis of the squaraine, respectively. The only significant change is computed along the Z direction (Table 1). The HOMO−LUMO excitation energy increases with increasing electric field intensity along Z. Then, we applied the electric field selectively in three different directions, and the molecular structure was optimized at the BLYP/TZ2P sc level. The geometries of CH3-Sq optimized in the presence of the strongest field considered here (0.005 au) applied along X, Y, and Z are shown in Figure 2A− C, respectively. The relevant effect of the applied field is lowering the molecular symmetry to pseudo-C2v. The largest deformation occurs when the field is applied along X, i.e., orthogonal to the molecular plane: CH3-Sq planarity deviates

Table 2. E(S0) Values (in eV) in the Presence of an Electric Field Applied along Different Directionsa field direction field 0 0.0001 0.0005 0.001 0.005

X −1.23 −3.18 −1.25 −2.84

× × × ×

Y 10−4 10−3 10−2 10−1

0 −4.98 −1.23 −4.93 −1.23

× × × ×

Z 10−5 10−3 10−3 10−1

−1.52 −3.79 −1.51 −3.95

× × × ×

10−4 10−3 10−2 10−1

a

Level of theory BLYP/TZ2P sc. All values are relative to the groundstate energy value without an applied field.

presence of an electric field are shown. Energy values decrease with increasing field strength, and again the strongest stabilization of CH3-Sq is found when the field is applied along the main molecular axis. The lowest excitation energy, corresponding to the HOMO− LUMO transition, was then calculated for these geometries optimized in the presence of the electric field (Table 4). The lowest strong HOMO−LUMO excitation is blue-shifted when the field is applied along X and Z, with the largest effect predicted in the former case; it is almost unaffected when the field is applied along Y. The trend with the field along X is the 1590

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of the electric field and independently for the ground and excited states. A4: αii(S0) and αii(S1) are calculated by fitting μi (S0 or S1) vs electric field Ei with eq 4. The dipole moments are those reported in Tables 3 and 6 and refer to geometries relaxed for every value of the electric field and independently for the ground and excited states. A5: αii(S0) and αii(S1) are computed using the response theory as implemented in two different codes. In particular, αii(S0) are calculated with the ADF code, referring to a groundstate geometry optimized with E = 0. αzz(S0) and αzz(S1) are calculated, testing different levels of theory (BLYP, CAMB3LYP, and B3LYP combined with different basis sets), with the DALTON code,55−57 using a ground-state geometry optimized under a D2h symmetry constraint. These data are reported in Table S3; Table 7 shows the range into which αzz(S0) and αzz(S1) fall. All of the results are in satisfactory agreement, i.e., the different functional/basis set combinations provide similar values, with the largest deviation obtained as expected with the pure GGA method. In Figure 4A−D, the fits of the energy dependence on the electric field are shown for the zz component, as obtained following the A1−A3 approaches. The coefficient of the quadratic term corresponds to the static polarizability; the linear coefficient provides the permanent dipole moment. The latter turns out to be very small for both S0 and S1, consistent with the expected null value for a D2h molecule at E = 0. The fits of the dipole moment dependence on the zz component of the electric field (A4) are shown in Figure 5A,B. In Table 7, the results for all diagonal components of the polarizability are summarized; in the last column on the right, the orientationally averaged excess polarizability is reported for approaches A1−A4 1 and has been calculated as Δα = 3 (Δαxx + Δαyy + Δαzz) In Table 8, the comparison between experimental and calculated values of the excess polarizability is reported for several conjugated oligomers from ref 21. By inspecting the data of Table 7, one can notice the following: (1) In comparing methods A1 and A2, only Δαyy is unchanged. This is expected because upon relaxation of the ground-state geometry in the presence of the electric field, no significant deformation occurs only when the field is applied along Y (Figure 2B). (2) Methods A3 and A4 are expected to provide the same value of polarizability because they account for the geometric relaxation in the same way: a discrepancy between them arises from using the energy or the dipole moment. αii(S0) are in excellent agreement; in fact, they are consistent within 4% for all Cartesian components. Concerning the excited state, αzz(S1) is still in excellent agreement, αyy(S1) shows a slightly higher variation, and αxx(S1) differs quite a lot

Table 3. Modulus of the Ground-State Dipole Moments (in Debye) in the Presence of an Electric Field Applied along Different Directionsa field direction field 0 0.0001 0.0005 0.001 0.005

X

Y

Z

0.298 1.10 2.27 10.1