Resonant-Convergent PCM Response Theory for the Calculation of

Article pubs.acs.org/JPCA

Resonant-Convergent PCM Response Theory for the Calculation of Second Harmonic Generation in Makaluvamines A−V: Pyrroloiminoquinone Marine Natural Products from Poriferans of Genus Zyzzya Bruce F. Milne*,†,‡ and Patrick Norman*,§ †

Nano-Bio Spectroscopy Group and ETSF Scientific Development Centre, Department of Materials Physics, University of the Basque Country, CFM CSIC-UPV/EHU-MPC and DIPC, Avenida de Tolosa 72, E-20018 Donostia, Spain ‡ Centre for Computational Physics, Department of Physics, University of Coimbra, Rua Larga, 3004-516 Coimbra, Portugal § Department of Physics, Chemistry and Biology, Linköping University, SE-581 83 Linköping, Sweden ABSTRACT: The first-order hyperpolarizability, β, has been calculated for a group of marine natural products, the makaluvamines. These compounds possess a common cationic pyrroloiminoquinone structure that is substituted to varying degrees. Calculations at the MP2 level indicate that makaluvamines possessing phenolic side chains conjugated with the pyrroloiminoquinone moiety display large β values, while breaking this conjugation leads to a dramatic decrease in the calculated hyperpolarizability. This is consistent with a charge-transfer donor-π-acceptor (D−π−A) structure type, characteristic of nonlinear optical chromophores. Dynamic hyperpolarizabilities calculated using resonance-convergent timedependent density functional theory coupled to polarizable continuum model (PCM) solvation suggest that significant resonance enhancement effects can be expected for incident radiation with wavelengths around 800 nm. The results of the current work suggest that the pyrroloiminoquinone moiety represents a potentially useful new chromophore subunit, in particular for the development of molecular probes for biological imaging. The introduction of solvent−solute interactions in the theory is conventionally made in a density matrix formalism, and the present work will provide detailed account of the approximations that need to be introduced in wave function theory and our program implementation. The program implementation as such is achieved by a mere combination of existing modules from previous developments, and it is here only briefly reviewed.

■

INTRODUCTION

technologies, microfabrication, (bio)imaging, and NLO microscopy.13−15 In the following sections, we present a computational study of the quadratic optical responses of a group of MNPs with a common cationic pyrroloiminoquinone (PIQ+) structural motif and called makaluvamines; see Figure 1. These compounds are secondary metabolites isolated from IndoPacific marine poriferans (sponges) of the genus Zyzzya and have been previously shown to inhibit topoisomerase II.18−20,20−24 The computational NLO evaluation of the makaluvamines undertaken shows that those compounds possessing a phenolic side chain conjugated to the PIQ+ subunit (thus forming a conjugated donor−acceptor (D−π−A) structure) have β values making these NPs interesting candidates as NLO chromophores in themselves. The PIQ+ moiety represents a potential new acceptor group for use in D−π−A chromophores.

Natural products (NPs) have played a vital role in organic chemistry since its beginnings as a true discipline in the 19th century. Until the last few decades, NPs were almost exclusively of terrestrial origin, but recently (as a result of the development of cheap and reliable diving techniques) the field of marine natural products (MNPs) has entered the scene and promises to yield unusual chemistries that may provide a whole new realm of possibilities and inspiration for chemists working in areas related to pharmacology and medicine.1,2 Recently, interest in (M)NPs has begun to be shown by scientists working on the chemistry and physics of organic materials. Of particular relevance to the present work has been the investigation of NPs for potential applications in photonics using both experimental and theoretical approaches.3−8 Photonics has been identified as one of the grand challenges in basic energy research and as one of the key enabling technologies (KETs) for the 21st century.9−12 Materials displaying strong nonlinear optical (NLO) properties have been investigated for some time due to their potential for use as ultrafast low energy-loss alternatives to conventional electronics in the field of optoelectronics and have applications in the transmission, manipulation, and storage of data. Further applications of NLO materials can be found in sensing © XXXX American Chemical Society

Special Issue: Jacopo Tomasi Festschrift Received: October 10, 2014 Revised: January 10, 2015

A

DOI: 10.1021/jp5102362 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

polarizabilities are tensors containing spatial information about the system’s optical responses. Although a number of different optical effects are employed in the wide range of current NLO applications, probably the most widely used (at this time) is second harmonic generation (SHG).15 The SHG process is a special case of the general phenomenon of sum frequency generation, and by using the semiclassical dipole formalism from eq 2 it is seen to correspond to the mixing of two incident electromagnetic waves of frequency ω to produce a scattered wave with frequency 2ω. Alternatively, a fully quantized description leads to an event where two photons of frequency ω are annihilated with the creation of one photon of frequency 2ω, which is emitted in the scattering process. The quadratic dependence on the field E(t) means that the origin of this effect is the second term on the right-hand side of eq 2. It can therefore be seen that the magnitude of the quadratic hyperpolarizability tensor β must be enhanced to produce improved SHG characteristics in microscopic systems. Response Theory for a Coherent System. A molecular system that interacts with the environment by no other means than via external electromagnetic fields can be described by a wave function for which the time evolution is governed by an equation of motion that takes the form25 ∂ 1 ⟨ψ (t )|Ω̂n|ψ (t )⟩ = ⟨ψ (t )|[Ω̂n, Ĥ ]|ψ (t )⟩ ∂t iℏ − γ ⟨ψ (t )|Ω̂ |ψ (t )⟩ n

n0

(3)

where the relaxation parameter γn0 is equal to one-half of the inverse lifetime of excited state |n⟩ and it has been assumed that the state of thermal equilibrium is equal to the ground state |0⟩. In the case of an isolated system, the finite lifetime is due to spontaneous radiative and nonradiative relaxation mechanisms. The time dependence of the phase-isolated wave function in eq 3 is viewed as a rotation in Hilbert space of the timeindependent ground state according to

Figure 1. Makaluvamine structures.

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THEORY Nonlinear Optics. Optical properties of macroscopic materials can be understood as being governed by the timedependent induced polarization, P(t), arising when the material is subject to an external time-dependent electromagnetic field, E(t)16,17 1 1 P(t ) = P 0(t ) + χ (1) E(t ) + χ (2) E(t )2 + χ (3) E(t )3 + ... 2 6

̂

|ψ (t )⟩ = e−iP(t )|0⟩

(4)

where the generator of rotations is the Hermitian operator P(̂ t ) =

†

†

∑ Pn(t )Ω̂n + [Pn(t )]*Ω̂n; Ω̂n = |n⟩⟨0|, Ω̂n = |0⟩⟨n| n>0

(5)

In situations when the external fields are weak in comparison with atomic fields, it is a common practice to separate the Hamiltonian into two parts, one time-independent, internal, part Ĥ 0 that describes the molecule in isolation and the remainder V̂ (t) that is due to the interactions with the timedependent external field. By means of perturbation theory, one thereafter writes the expectation value of a general Hermitian operator Ω̂,which typically corresponds to an observable such as the electric or magnetic dipole moment, as an expansion in frequency-space in orders of the external field strengths. The hereby defined Fourier amplitudes define response functions corresponding to molecular properties, such as polarizabilities and magnetizabilities. The explicit sum-overstates expressions for the lowest-order response functions become equal to25,26

(1)

Here χ(i) are the electromagnetic susceptibilities of the system and nonlinear effects come about through the higher-order terms of the expansion with χ(1) determining the linear response, χ(2) the quadratic response, and so on. For microscopic systems such as atoms, molecules, or clusters this can be expressed as 1 1 μ(t ) = μ0 + αE(t ) + βE(t )2 + γE(t )3 + ... (2) 2 6 where μ(t) is the field-induced dipole polarization. Here the macroscopic susceptibilities are replaced by the microscopic polarizability α and the quadratic and cubic hyperpolarizabilities β and γ. As with the susceptibilities, χ(i), these (hyper)B


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The Journal of Physical Chemistry A 1 ω ⟨⟨Ω̂; V̂β ⟩⟩ = − ℏ

⎡ ⟨0|Ω̂|n⟩⟨n|V̂ ω|0⟩ β ∑⎢ ⎢ − − iγn0 ω ω n > 0 ⎣ n0

⟨0|V̂β |n⟩⟨n|Ω̂|0⟩ ⎤ ⎥ ωn0 + ω + iγn0 ⎥⎦

collision induced dephasing that leads to a state of incoherency. In absence of the latter, a pure state would remain in a pure state and could thus equally well be described by the wavefunction-based response theory presented in the previous section. In this stage, perturbation theory is invoked to solve eq 8 and the response functions that stand in direct comparison with those presented in the previous section will read as27

ω

+

(6)

and ω ω 1 ⟨⟨Ω̂; V̂β 1 , Vγ̂ 2⟩⟩ = 2 ℏ

∑ 71,2

1 ω ⟨⟨Ω̂; V̂β ⟩⟩ = − ℏ

ω ω ⎡ ⟨0|Ω̂|n⟩⟨n| V̂β 1|k⟩⟨k|Vγ̂ 2|0⟩ ⎢ ⎢ n , k > 0 ⎣ [ωn0 − (ω1 + ω2) − iγn0][ωk 0 − ω2 − iγk 0]

∑

+

+

ω ⟨0|V̂β |n⟩⟨n|Ω̂|0⟩ ⎤ ⎥ + ωn0 + ω + iγn0 ⎥⎦

ω ω ⟨0|Vγ̂ 2|n⟩⟨n| V̂β 1|k⟩⟨k|Ω̂|0⟩

[ωn0 + ω2 + iγn0][ωk 0 + (ω1 + ω2) + iγk 0] ⎤ ⎥ − ω2 − iγk 0] ⎥⎦

ω ω 1 ⟨⟨Ω̂; V̂β 1 , Vγ̂ 2⟩⟩ = 2 ℏ

(7)

ω ω ⟨0|V̂β 1|m⟩⟨m|Ω̂|n⟩⟨n|Vγ̂ 2|0⟩

∑×⎢ mn

⎢⎣ (ωm0 + ω1 + iγm0)(ωnm − ωσ − iγnm) ω

+ +

+

ω

⟨0|V̂β 1|m⟩⟨m|Ω̂|n⟩⟨n|Vγ̂ 2|0⟩ (ωn0 − ω2 − iγn0)(ωmn + ωσ + iγmn) ω ω ⟨0|Ω̂|m⟩⟨m|V̂β 1|n⟩⟨n|Vγ̂ 2|0⟩

(ωn0 − ω2 − iγn0)(ωm0 − ωσ − iγm0) ⎤ ⎥ + ω1 + iγm0)(ωn0 + ωσ + iγn0) ⎥⎦

ω ω ⟨0|V̂β 1|m⟩⟨m|Vγ̂ 2|n⟩⟨n|Ω̂|0⟩

(ωm0

(11)

where ℏωmn denotes the energy separation of states |m⟩ and |n⟩. Comparison of Response Functions for Coherent and Incoherent Systems. If we compare the two sets of response functions presented in the two previous sections, there are notable differences. The response functions derived in wave function theory contain summations that exclude the ground state; even in the limit of infinite excited state lifetimes (γn0 → 0), they are free of secular divergencies, that is, singularities of response functions in the limit of zero frequencies, or static external fields. The response functions derived in density matrix theory contain summations that include the ground state, and secular divergencies are therefore present in these cases. An even more striking difference is that response functions of the same order have a varying number of terms in the summations. In coherent and incoherent state response theory the number of terms in the expression for the first-order nonlinear response function equals six and eight, respectively. This issue was recognized by Bloembergen and coworkers in 1977,28 and they started an experimental search to find collision induced spectral resonances for sodium vapor. To date, however, no experimental evidence has been produced of such resonances. We will here provide a quantitative determination of what these differences in response expressions amount to, and we will focus on the response functions pertinent to the coupling to an external electric field, namely, the first-order hyperpolarizabilities. Expressions for these molecular properties are obtained from the general formulas by substituting Ω̂ = μ̂ α, V̂ ωβ = −μ̂β, and so on, as to arrive at

(8)

It has again been assumed that the ground state represents the state of thermal equilibrium as expressed by the use of the Kronecker delta functions and which is a reasonable assumption in electronic structure theory of molecular materials where gap energies are substantially larger than kBT. The damping parameters apply to individual elements of the density matrix and contain two separate contributions according to col γmn = (Γm + Γn)/2 + γmn

∑ 71,2 ⎡

where ℏωn0 denotes the energy separation of states |n⟩ and |0⟩, ωσ is the sum of optical frequencies ω1 and ω2, and V̂ ω is the quantum mechanical operator (overbars denote fluctuation operators) responsible for the coupling of the quantum system to the external classical electromagnetic field oscillating with an angular frequency ω (and likewise for ω1 and ω2). Response Theory for an Incoherent System. The key difference in the formulation of a response theory for a system in the liquid phase as compared with the above treatment of the gas pase is that it is no longer reasonable to neglect intermolecular interactions. These interactions take several forms ranging from a physical dielectric screening of external fields to solute−solvent chemical bonding. There is an element of time dependence in these interactions due to solute and solvent molecular dynamics (MD) and which can be accounted for by a statistical phase space sampling, be it by means of Monte Carlo or MD simulations, but there is also a more subtle effect caused by the solvent−solute interactions and which points toward the very core of response theory, namely, the equation of motion itself. In a liquid, there are frequent weak intermolecular collisions that do not alter state populations of the involved systems but that change the overall phases of wave functions in uncontrollable ways. For this reason, it becomes unreasonable to describe the ensemble of molecules as coherent and it can thus not be represented as a pure state. Incoherent ensembles of molecules are best treated in a density matrix formalism, where the equation of motion reads as27 ∂ 1 ρmn = [Ĥ , ρ ]̂ mn − γmn(ρmn − δn0δm0) ∂t iℏ

(10)

and

ω ω ⟨0|Vβ̂ 1|n⟩⟨n|Ω̂|k⟩⟨k|Vγ̂ 2|0⟩

[ωn0 + ω1 + iγn0][ωk 0

⎡ ⟨0|Ω̂|n⟩⟨n|V̂ ω|0⟩ β ∑⎢ ⎢ iγn0 ω ω − − n ⎣ n0

βαβγ ( −ωσ ; ω1 , ω2) = ⟨⟨μα̂ ; μβ̂ , μγ̂ ⟩⟩ω1, ω2

(9)

where the first term represents the inverse lifetimes (Γn = 1/τn) associated with spontaneous relaxation and the second term, which applies to off-diagonal elements only, represents the

(12)

For the first-order hyperpolarizability, we note that the second term in the right-hand side of eq 7 is replaced by terms number two and three in the right-hand side of eq 11. Let us C


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The Journal of Physical Chemistry A rewrite the second and third terms of eq 11 to identify what separates the two formulas; the numerators are identical and can therefore be left aside

NnA(ωσ ) = Al[1][E[2] − ℏ(ωσ + iγ )S[2]]−ln1 NnB(ω1) = [E[2] − ℏ(ω1 + iγ )S[2]]−nl1 Bl[1]

1 (ωnm − ωσ − iγnm)(ωm0 + ω1 + iγm0) 1 + (ωmn + ωσ + iγmn)(ωn0 − ω2 − iγn0) =

=

=

NnC(ω2) = [E[2] − ℏ(ω2 + iγ )S[2]]−nl1 Cl[1]

(ωnm

⎡ ⎤ 1 1 1 ⎢ ⎥ − − ωσ − iγnm) ⎢⎣ (ωm0 + ω1 + iγm0) (ωn0 − ω2 − iγn0) ⎥⎦

(ωnm

⎡ ⎤ ωnm − ωσ − i(γn0 + γm0) 1 ⎢ ⎥ − ωσ − iγnm) ⎢⎣ (ωm0 + ω1 + iγm0)(ωn0 − ω2 − iγn0) ⎥⎦

(ωm0

⎡ i(γnm − γn0 − γm0) ⎤ 1 ⎥ ⎢1 + + ω1 + iγm0)(ωn0 − ω2 − iγn0) ⎢⎣ (ωnm − ωσ − iγnm) ⎥⎦

For convenience, we have here adopted a common broadening parameter for all excited states, that is, γn0 = γ, so that the relaxation matrix R̃ [2] becomes equal to γ times the overlap matrix S[2]; see eq 71 of ref 25 for further details of the handling of the relaxation matrix and also other matrix definitions. It would be trivial to introduce different parameters in the program implementation for the separate electronic excitations, for example, based on orbital energy differences, but we have restrained from doing so in the present work. We note that relaxation and overlap matrices of third order vanish in the single determinant approximation so that, apart from the last term involving the third-order Hessian E[3] in eq 15, the expression for the quadratic response functions parallels the exact-state expression for the hyperpolarizability in eq 7. In fact, the exact-state expression could be written on the same matrix form as the SCF expression with a mere replacement of electron excitation operators â†a âi with state transfer operators | n⟩⟨0|, and it would be seen that in the exact state basis the third-order Hessian vanishes. The interactions between the solute and the dielectric medium enters into the system Hamiltonian (as opposed to relaxation that enters as a phenomenological, separate, term in the equation of motion). From an implementation point of view, therefore, the modifications needed from a vacuum CPP approach to the PCM-CPP counterpart are isolated to terms E[2] and E[3] in eq 15 and which are already dealt with in detail in the corresponding developments of resonant-divergent quadratic response functions.37,38 As long as we are not considering collision-induced resonances and remain in a purestate representation of the solute, the implementation of the PCM-CPP model merely requires the interfacing of existing routines in the program. This is the approach we have taken in the present work, and it thus corresponds to the approximation of setting the expression in eq 14 to zero.

(13)

The first term above is identified as the second term in eq 7, and the second term thereby constitutes the difference between, on the one hand, coherent and, on the other hand, incoherent state response theory. The correction term will read as Δ=

1 ℏ2

∑ m,n

∑ 71,2 i(γnm − γn0 − γm0)⟨0|μβ̂ |m⟩⟨m|μα̂ |n⟩⟨n|μγ̂ |0⟩ (ωnm − ωσ − iγnm)(ωm0 + ω1 + iγm0)(ωn0 − ω2 − iγn0) (14)

We note that, in absence of collision-induced dephasing, this correction term will vanish because the γ factor in the numerator is then equal to zero. Approximate State Implementation. Since relaxation was introduced in wave function mechanics by Norman and coworkers under the name of the complex polarization propagator (CPP) approach,25 it has been implemented in several approximate electronic structure theory methods such as Hartree−Fock, Kohn−Sham density functional theory (DFT), multiconfiguration self-consistent field, and coupled cluster. In some cases, the implementations encompass nonrelativistic and two- and four-component relativistic descriptions of the system as well as linear and nonlinear optical response processes.29 Recently, we also reported the extension of the CPP approach to a hybrid quantum mechanical−molecular mechanical (QM/MM) model to study chromophores in a polarizable embedding such as a protein or a snapshot of a liquid.30 In the present work, we demonstrate the combination of the CPP approach at the DFT level of theory and a polarizable continuum model (PCM) in which a cavity is formed around the solute and the outside is represented by a dielectric medium.31−36 The expression for the resonant-convergent quadratic response function in the single-determinant self-consistent field (SCF) framework takes the form25 ⟨⟨Â ; B̂, Ĉ⟩⟩ω1, ω2 =

■

COMPUTATIONAL DETAILS The geometries of the makaluvamines were optimized using the GAMESS-US program39 with a PCM representation of the acetonitrile solvent. The choice of solvent was made because it is suitable for the dissolution of the polar PIQ+ chromophores and is a commonly used solvent in experimental studies of hyperpolarizability that has been employed in several previous theoretical studies of NLO chromophores.40 Geometries were optimized at the B3LYP41,42 level with the polarized splitvalence Def2-SVP basis set.43 Because the makaluvamines have been obtained as salts in the case of both natural and synthetic compounds, the cationic form of the pyrroloiminoquinone core was the only one considered even where deprotonation of the iminium moiety could, in principle, yield a neutral compound. With the use of the GAMESS-US program, static hyperpolarizabilities were calculated at the MP2 and DFT levels of theory using the finite-field method based on numerical differentiation of dipole moments and employing the diffuseaugmented Def2-SVPD basis set designed for response calculations.44 In the case of DFT, the functionals BLYP, B3LYP, and CAM-B3LYP were considered, out of which the range-separated hybrid functional CAM-B3LYP can be

[2] C ∑ 71,2[NnA(ωσ )Bnm N m (ω 2 ) [2] C + NnB(ω1)A nm N m (ω 2 ) [2] A +NnB(ω1)Cnm Nm (ωσ )] [3] [3] + NnA(ωσ )[Enml + Enlm ]NmB(ω1)NlC(ω2)

(15)

where the three response vectors equal D


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core structure. However, in cases where these are not connected by an easily delocalizable π-electron system the two groups remain essentially independent as in makaluvamines D, F, J, K, P, and V, all of which display β̃0 values that are of similar magnitude to those of the small unsubstituted makaluvamines. In systems such as these where charge transfer plays an important part in determining the optical response, the use of methods (such as MP2) that provide a reliable description of long-range electronic effects is clearly important. For this reason, we take the current MP2 β̃0 data as a benchmark against which to assess the performance of several density functional theory (DFT) methods. Although it is easy enough to obtain static hyperpolarizabilities with the MP2 method, its extension to the dynamic case is more complicated and hampered by the existence of uncorrelated poles in the response function and so it is common practice to employ methods based on timedependent density functional theory (TDDFT).58−60 In Table 1, we also present β̃0 values for the makaluvamines as obtained with use of the BLYP, B3LYP, and CAM-B3LYP functionals. These represent the generalized gradient approximation (BLYP) and Hartree−Fock (HF) hybrid variations. B3LYP contains a constant 20% HF exchange component while the Coulomb-attenuated CAM-B3LYP contains a variable HF component that rises from 19% (short-range interactions) to 65% (long-range). CAM-B3LYP was designed with the intention of providing an improved description of long-range electronic interactions such as charge transfer.61 This functional has been found to yield lower average errors than either GGA or conventional hybrid functionals in calculations of electronic excitations displaying charge transfer or even Rydberg character.51,52 The pure DFT functional BLYP was in most cases found to overestimate the values of β̃0 considerably. This is most likely due to the fact that the GGA approximation, like the original local density approximation from which it was developed, suffers from an incorrect description of the charge density far from the atomic nuclei. The GGA charge density drops to zero too rapidly and so it cannot be expected to perform well for properties such as hyperpolarizabilities that are largely determined by the more diffuse outer regions of a molecule. The addition of exact long-range exchange in the hybrid functionals leads in many cases to a significant improvement in the results for the hyperpolarizabilities, but it is also clear that there are still systems for which no clear improvement is found. The standard B3LYP functional generally performs well for the β̃0 trends, but it sometimes displays errors of some ±100% as compared with MP2 results. The worst cases appear to be the conjugated makaluvamines, where large overestimation of the hyperpolarizability occurs, presumably because of incomplete cancellation of the charge-transfer errors in the underlying BLYP functional. The most faithful replication of the magnitudes and trends provided by the MP2 method comes from the Coulombattenuated functional CAM-B3LYP and which performs well for all three classes of makaluvamine (PIQ+, PIQ+ with unconjugated side chain, and PIQ+ with conjugated side chain). Hyper-Rayleigh Scattering Susceptibilities. HyperRayleigh scattering (HRS) is a convenient method for SHG studies of chromophores in solution, particularly in the case of nonpolar molecules where the electric-field induced SHG

expected to perform well for response calculations on systems where charge-transfer is important such as in D−π−A chromophores.48−52 On the basis of the results for static hyperpolarisabilities, makaluvamines E, G, L, and M were selected for further evaluation of their NLO responses in the range of incident laser frequencies commonly used in SHG. Because these all possessed the longer conjugated side chain (see Figure 1), the smaller makaluvamine A was also included for comparison. Dynamic first hyperpolarizabilities were calculated at the PCMCPP/DFT level of theory, as outlined above and with use of a local version of the Dalton program.53,54 On the basis of the results obtained in the present work for the static hyperpolarisabilities and results of previous studies, the CAM-B3LYP functional was employed for all CPP calculations. The singly augmented double-ζ basis set of Dunning and coworkers (augcc-pVDZ) was used in these calculations because it provides a more complete and flexible set of diffuse functions.55,56 Linear response calculations were performed at the same level of theory to obtain information on the absorption spectra of the makaluvamine derivatives.

■

RESULTS AND DISCUSSION Static Hyperpolarizabilities. The results of the static hyperpolarizability calculations are given in Table 1. The values presented correspond to the orientationally averaged first hyperpolarizability in accordance with57 β0̃ =

∑

βi2 ; βi =

i=x ,y,z

3 5

∑

βijj (16)

j=x ,y,z

Table 1. Makaluvamine Static First Hyperpolarizabilities (au) β̃0 makaluvamine

MP2

BLYP

B3LYP

CAM-B3LYP

A B C D E G H I J K L M N P V

838 1325 1257 649 48742 32640 1103 962 988 451 53694 52124 1046 738 634

901 293 1358 1156 68697 71061 1193 939 970 1067 70038 68844 1014 706 1379

911 585 1321 410 68017 60785 1194 962 600 363 71278 70599 1033 315 460

970 1129 1344 376 46789 34034 1236 1027 756 163 50781 50232 1095 535 349

A clear division can be seen in the β̃0 values between the makaluvamines possessing an aromatic side chain that is conjugated with the PIQ+ moiety and those that either do not have this conjugation or in fact lack the amino substitution altogether. At the MP2 level, the conjugated makaluvamines (E, G, L, and M) display β̃0 values that are about 30 to 50 times larger than what is found for the other structures. The addition of a conjugated aromatic charge donor to the formally positively charged acceptor group would be expected to significantly increase the (hyper)polarizability of the PIQ+ E


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Figure 2. Hyper-Rayleigh scattering hyperpolarizabilities βHRS for makaluvamine derivatives in vacuum (green squares) and acetonitrile solution (red triangles). Two-photon resonances are indicated by vertical dashed lines in respective colors.

Table 2. Visible Transition Wavelengths λ (nm) and Oscillator Strengths f for Makaluvamine Derivatives in Vacuum and Acetronitrile Solution S0 → S1a

S0 → S2b

vacuum

a

solution

vacuum

solution

makaluvamine

λ

f

λ

f

λ

f

λ

f

A E G L M

474 608 612 615 614

0.004 0.635 0.639 0.664 0.655

487 604 609 614 612

0.007 0.521 0.542 0.550 0.524

363 434 436 433 434

0.005 0.285 0.283 0.281 0.273

353 424 429 429 427

0.136 0.531 0.526 0.549 0.542

Dominating electronic excitation is from HOMO to LUMO. bDominating electronic excitation is from HOMO−1 to LUMO.

chromophore involved through its depolarization ratio (DR) value defined as

(EFISHG) cannot be applied. The NLO susceptibility corresponding to HRS is given by45 βHRS( −2ω; ω , ω) =

2 2 {⟨βZZZ ⟩ + ⟨βXZZ ⟩}

(17)

DR =

The full expressions used to calculate the β tensor orientational averages ⟨β2ZZZ⟩ and ⟨β2XZZ⟩ were given in previous works.8,45−47 In addition to measuring the first hyperpolarizability this approach can give information on the nature of the

2 ⟨βZZZ ⟩ 2 ⟨βXZZ ⟩

(18)

A value of DR around 1.5 corresponds to octupolar chromophore symmetry, while a value around 5 is indicative of a linear donor−acceptor type chromophore.8,45 F


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Table 3. Hyper-Rayleigh Scattering Hyperpolarizabilities ΒHRS and Depolarization Rations (DR) for Makaluvamine Derivatives in Vacuum and Acetonitrile Solution λ=∞

λ = 1907 nm

vacuum

solution

vacuum

solution

makaluvamine

βHRS

DR

βHRS

DR

βHRS

DR

βHRS

DR

A E G L M

206 15854 16324 16470 16086

2.8 4.9 4.9 4.9 5.0

302 15892 16598 17067 16459

2.8 4.8 4.8 4.8 4.9

241 28946 30200 30562 29767

2.8 5.0 4.9 4.9 5.0

360 28716 30420 31533 30220

2.7 4.9 4.8 4.9 4.9

Table 3 shows values of the HRS susceptibilities, βHRS, for makaluvamines A, E, G, L, and M calculated at λ = ∞ and λ = 1907 nm. The inclusion of PCM solvation on the values obtained from the TDDFT response calculations in the static (λ = ∞) limit causes an increase in βHRS, as was seen in previous work.8 Moving away from the hypothetical static limit to wavelengths that have more relevance to real-world applications, it is found that a very similar solvent enhancement of βHRS occurred at λ = 1907 nm. A small increase relative to the (λ = ∞) values was observed, but these finite wavelength values essentially suffer from no resonance enhancement effects that might obscure the solvent effects observed here. Of most interest in the present work was how the chosen compounds might behave in terms of HRS response in regions of the spectrum relating to experimental fundamental laser wavelengths commonly used in laboratory applications. For this reason a scan of wavelengths was performed from 1400 to 750 nm, and the resulting βHRS spectra are shown in Figure 2. In this Figure, triangles and squares mark actual data points from the calculations and the solid lines represent data point interpolations. Makaluvamines E, G, L, and M were seen to display two clear resonances at ∼850 and ∼1225 nm (variable depending on inclusion of solvent effects). In contrast, makaluvamine A was found to have only one resonance in this spectral region at ∼975 nm. Linear response TDDFT calculations presented in Table 2 showed that in E, G, L, and M these were due to twophoton resonances with the S0 → S1 excitation at ∼610 nm and the S0 → S2 excitation at ∼430 nm. In the case of makaluvamine A the optical absorption spectrum is blueshifted so that only the S0 → S1 at 474 (vacuum) and 487 nm (acetonitrile) has a two-photon resonance falling within the spectral window investigated here. The effect of acetonitrile solvation was found to behave in a λ-independent fashion. In the case of makaluvamine A, an increase in βHRS relative to vacuum was seen at all fundamental wavelengths when solvent effects were included. However, as λ becomes smaller the change in βHRS caused by the solvent increases with even greater changes seen immediately before and after the two-photon resonance at ∼975 nm. For the extended side-chain makaluvamines E, G, L, and M, the behavior was more complex with higher βHRS values observed in vacuum than with solvent at λ values greater than 1000 nm. This situation was reversed as the spectra approached the second resonance at ∼850 nm with the highest βHRS values now belonging to the solvated system. The largest resonance enhancements were observed on the short-wavelength side of the second resonance in makaluvamines E, G, L, and M with values of βHRS reaching approximately nine times the magnitude of those calculated at λ = 1907 nm or at the static level. This is particularly

interesting from the point of view of bioimaging and biospectroscopy applications where wavelengths in the region of 800 nm are optimal for achieving deep tissue penetration. Although the resonance enhancement had decreased slightly for all of these compounds by 800 nm, large βHRS values of ∼150 × 103 au were found at this wavelength. The depolarization ratios (DRs) for makaluvamines A, E, G, L, and M are shown in Table 3. It is clear from these results that there is a fundamental difference in the nature of the chromophores involved in the optical responses, separating makaluvamine A from those with extended conjugated side chain. Makaluvamine A, which represents essentially the isolated PIQ+ core moiety, was found to have a DR value of 2.8, indicating that the chromophore involved in the HRS response is intermediate in nature between the octupolar extreme of DR = 1.5 and the linear donor−acceptor type chromophore corresponding to DR = 5. In PIQ+, at the most basic level, the cationic quaternary nitrogen and keto oxygen coupled through the iminoquinone ring system would be expected to form a simple linear donor−acceptor chromophore with DR of ∼5, but the various substitutions attached to this system clearly play an important role in perturbing the symmetry of this basic chromophore leading to the reduced DR value. These distortions of chromophore symmetry within the PIQ+ core, however, are completely overwhelmed in the case of the makaluvamines possessing the conjugated phenolcontaining side chains. The DR ratios found for makaluvamines E, G, L, and M were all 4.8 to 4.9, showing that in these molecules the chromophores are indeed of linear donor− acceptor symmetry. The well-separated neutral side chain and cationic PIQ+ core serve as good donor and acceptor subunits, and when linked by the conjugated π−system comprising the C−C double bond and nitrogen lone pair the whole creates an integral extended D−π−A chromophore. The lack of the double bond in makaluvamines D, J, K, P, and V breaks this conjugated bridge and reduces their NLO responses to the level of the small makaluvamines despite the fact that they possess the phenolic side chain, as can be seen from the results in Table 1.

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CONCLUSIONS

Resonant-convergent calculations of first-order nonlinear response properties of liquids have been enabled by a combination of the polarizable embedding method with the complex polarization propagator approach. A detailed analysis of second-order response functions shows that collisioninduced resonances are not included in our wave functionbased approach, and a presentation is made of the difference G


Article


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term in between wave function and density-matrix-based response theory. As an example application of the PCM-CPP approach, we are presenting hyper-Rayleigh scattering parameters for a series of conjugated makaluvamine derivatives. Evaluation of the resulting data suggests that the those makaluvamines possessing an extended phenol-containing side chain conjugated to the pyrroloiminiquinone core structure have a linear D−π−A type structure and display the large HRS susceptibilities expected for this type of chromophore. Furthermore, resonance effects produce greatly enhanced HRS responses at fundamental wavelengths of ∼800 nm, indicating that these compounds could be profitably used in the development of new HRS labels for applications in optical bioimaging.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: bruce@teor.fis.uc.pt. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from the Swedish Research Council (Grant No. 621-2010-5014) as well as a grant for computing time at the Swedish National Supercomputer Centre (NSC) are acknowledged. B.F.M. thanks the Portuguese Foundation for Science and Technology (project PTDC/FIS/103587/2008), the Donostia International Physics centre, and the Centre de Fisica de Materiales, University of the Basque Country, for financial support. B.F.M. acknowledges the provision of computer resources, technical support, and assistance from the Laboratory for Advanced Computing of the University of Coimbra, Portugal.

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