Justyna
Kozłowska
*,
Marta
Chołuj
,
Robert
Zaleśny
and
Wojciech
Bartkowiak
*
Department of Physical and Quantum Chemistry, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, PL-50370 Wrocław, Poland. E-mail: justyna.kozlowska@pwr.edu.pl; wojciech.bartkowiak@pwr.edu.pl
First published on 15th February 2017
In the present contribution we study the influence of spatial restriction on the two-photon dipole transitions between the X1Σ+ and A1Σ+ states of lithium hydride. The bond-length dependence of the two-photon absorption strength is also analyzed for the first time in the literature. The highly accurate multiconfiguration self-consistent field (MCSCF) method and response theory are used to characterize the electronic structure of the studied molecule. In order to render the effect of orbital compression we apply a two-dimensional harmonic oscillator potential, mimicking the topology of cylindrical confining environments (e.g. carbon nanotubes, quantum wires). Among others, the obtained results provide evidence that at large internuclear distances the TPA response of lithium hydride may be significantly enhanced and this effect is much more pronounced upon embedding of the LiH molecule in an external confining potential. To understand the origin of the observed variation in the two-photon absorption response a two-level approximation is employed.
Another area of research of increasing prominence concerns linear and nonlinear electric properties of spatially restricted atoms, ions and molecules. Basically, it is expected that embedding a quantum system in the confining cages will affect its electronic density distribution which, in turn, may be reflected through changes in linear and nonlinear optical (L&NLO) phenomena. Thus far, numerous theoretical results presented in the literature demonstrate that spatial restriction significantly modifies nonresonant electric dipole properties of atoms and molecules.9–14,17–19,23,24,31,32 In particular, it was reported that the values of effective linear polarizability (α) as well as second hyperpolarizability (γ) decrease together with the increasing strength of orbital compression.9,10,12,13,17–19,31,33 On the other hand, the behavior of dipole moment (μ) and first hyperpolarizability (β) differs depending on the topology of the confining environment and the system under consideration.9,12,32,33 It should be underscored that there are only a limited number of theoretical studies concerning the evaluation of the molecular quantities that govern the NLO processes in the resonant regime upon confinement.9 This work aims to fill the existing gap. In so doing, in the present contribution the focus is put on the exploration of the effect of spatial confinement on the two-photon absorption (TPA) response of a model molecular system.
TPA, which is a third-order NLO phenomenon, may be described as the electronic excitation of a quantum object induced by the simultaneous absorption of two photons of the same or different energy and, in general, is characterized by several attractive features. Besides the benefits of application of the TPA phenomenon in the field of spectroscopy (it enables the exploration of spectroscopic states which are one-photon forbidden due to symmetry), there are also a number of technological applications of this NLO process.34 These include high-resolution fluorescence microscopy,35,36 fabrication of optoelectronic logical circuits,37,38 three-dimensional optical data storage39 or nondestructive imaging of biological tissues,35,40 just to name a few. As the development of multiphoton based applications relies on the quest for chromophores with large TPA responses, considerable efforts are directed toward the design of appropriate molecular species.34,41,42 Although initially the attention was mainly focused on push–pull dipolar molecular structures,42,43 over the years it has been shifted to quadrupoles,44,45 multichromophoric dendrimeric systems46,47 or nanodots.48,49 Another promising route to accomplish large TPA responses involves alteration of bond lengths. Particularly, it has been shown that the molecular (hyper)polarizabilities as well as the TPA probability (δgf) exhibit nonmonotonic changes as a function of the bond-length alternation parameter.50,51 Moreover, the results of theoretical and experimental studies clearly indicate that environmental effects, especially solvent polarity, significantly influence the TPA strength of molecular systems.52–54
According to the results of some experimental studies, the spatial confinement effect may be considered as another important factor contributing to the changes of TPA strength.55–59 For example, it was demonstrated that exposing molecular systems to high pressure leads to the reduction of both one- and two-photon absorption responses.55 On the contrary, an enhancement of δgf was reported for different organic molecules confined between the interlayer spaces of clay minerals.58,59 Some important conclusions might be also found in recent work concerning the properties of different molecular species enclosed inside metal–organic framework (MOF) materials, which are emerging as unique structures due to their extraordinarily high porosity. Particularly interesting is the observation that systems containing chromophores incorporated into the MOF pores exhibit very strong TPA intensities.57,60 Thereby, in addition to the already known potential applications of MOFs (e.g. drug delivery, catalysis or hydrogen storage) they are also considered as an element of new two-photon-pumped microlasers.60 In some measure, these findings are in line with those emerging from our recent study performed on the HCCCN molecule embedded in a repulsive potential of cylindrical symmetry.9 Based on the conducted analysis it was found that the absolute value of the second-order transition moment (Sgfij) increases together with the increasing confinement strength. To the best of our knowledge the study in question provides still the only ab initio results quantifying the influence of spatial confinement on the resonant NLO properties.
The recent experimental studies concerning two-photon absorption properties of molecular species enclosed inside metal–organic framework materials or confined between the interlayer spaces of clay minerals57–60 encouraged us to undertake the present investigations. In order to gain a fundamental understanding of various aspects of multiphoton absorption in the presence of spatial confinement, in this article we provide a comprehensive theoretical description of the confinement-induced changes in the two-photon dipole transitions between the X1Σ+ and A1Σ+ states of the LiH molecule using high-level electron correlation treatments and response theory. Owing to the simplicity of its electronic structure, lithium hydride is often considered as an ultimate benchmark that allows for a precise assessment of the accuracy and reliability of various theoretical methods. Thus, the number of papers reporting highly accurate reference data for various properties of this molecule is very substantial (see for example ref. 10, 12, 18, 31 and 61–76). Among others, the potential energy curves and spectroscopic properties of many electronic states of LiH have been already thoroughly investigated.61–65 It is also worth noticing that several high quality theoretical studies are available concerning the electronic and vibrational contributions to the dipole moment and (hyper)polarizability of lithium hydride.10,12,18,31,66–76 However, with the notable exception of the study devoted to simulations of NLO properties of LiH using damped cubic response theory within TDDFT,77 we are not aware of previous experimental or theoretical studies on the TPA response of the LiH molecule, particularly under usual external conditions. Thus, although the main focus of this work is to get a deeper insight into the influence of orbital compression on the two-photon absorption phenomena, the value of δgf reported here for the unconfined LiH molecule might be also of significance for subsequent analysis. A further interest of this study is to explore the bond-length dependence of the investigated molecular quantities, for both free and spatially restricted lithium hydride.
(1) |
In an attempt to assess the effect of spatial restriction on the TPA of the LiH molecule we analyze the values of the second-order transition moment (Sgfij), which constitutes the basic molecular quantity that describes the two-photon absorption process:80
(2) |
(3) |
Under the assumption that the presence of the CT state dominates the response of molecular systems to the external electric field it is possible to reduce the expressions defining the second-order transition moment (eqn (2)) within the well-known two-level approximation:34,42,83,84
(4) |
In order to characterize the electronic structure of lithium hydride in the X1Σ+ and A1Σ+ states the multiconfiguration self-consistent field (MCSCF) method together with response function formalism was used, as implemented in the Dalton package.85 Specifically, the MCSCF method was applied for the ground state wave function, whereas the one- and two-photon dipole transition properties were obtained from the multiconfiguration linear and quadratic response functions (MCLR and MCQR).86–89 The computations were performed in C2v symmetry, using the ANO-L basis set.90 All electrons were correlated and all orbitals included in the active space during the MCSCF calculations. It should be noted that the dependence of the analyzed quantities on the internuclear distance (R) was also investigated. Therefore, the one- and two-photon transition dipole moments and excitation energy values, as well as the dipole moment difference between the excited and the ground electronic state, were computed as a function of R, both under vacuum and in the presence of HO potential. Such analyses were carried out for the internuclear distances between 1.3 and 20 a.u. It should be noted that although the sign of Sgfzz is undetermined, we checked phases of wave functions and response vectors at each distance to obtain a smooth curve presented in Fig. 1. The same applies to Fig. 2.
φ | S gf zz | S gf,TLM zz | 〈δgf〉 | ω f | 〈g|z|f〉 | Δμz |
---|---|---|---|---|---|---|
0.0 | 103.9 | 60.4 | 3760 | 1.3 × 10−1 | 9.3 × 10−1 | 4.3 |
0.1 | 69.7 | 41.9 | 1360 | 1.5 × 10−1 | 7.0 × 10−1 | 4.5 |
0.2 | 24.9 | 16.4 | 178 | 1.8 × 10−1 | 3.1 × 10−1 | 4.7 |
The dependence of the diagonal component of the second-order transition moment of LiH on the internuclear distance, evaluated using the multiconfiguration quadratic response functions and two-level approximation, is depicted in Fig. 1. At this stage it should be mentioned that the R-dependence brings forth important features of the static electric properties (μ, α, β and γ) of molecular systems, as it has been already disclosed in many valuable scientific works.10,92–108 Although it is difficult to draw one general conclusion emerging from these studies, there are several important observations worth underscoring. Among others, the sign inversion of the dipole moment with the change in the internuclear separation is characteristic for various molecular systems (e.g. AlCl, AlF, AlH, BCl, BF, CO, CS, HBr, HCl, HF, MgHe, NaLi, SiO, SiS, YBr).97–100 Such observation is of relevance as it reflects the process of electron charge transfer inside the molecule. Moreover, on the basis of extensive theoretical studies Maroulis and co-workers have found that the variation of bond length results in substantial changes of (hyper)polarizability, which are quite distinct for different molecular systems.101–104 A thorough consideration of the connection of polarizability and hyperpolarizability derivatives to Raman and hyper-Raman spectra was also reported by Quinet and Champagne.109 From the theoretical point of view an important finding concerns also the fact that changes in the intermolecular distance may have a substantial influence on the electron correlation contribution to the studied electric properties.102,104
Quite recently, Lo and Klobukowski discussed the electronic structure as well as the response of μ and α of lithium hydride to the confining potential and also the dependence of the computed quantities on the internuclear distance.10 As it was found by the authors, the changes of μ and α as a function of bond length are substantial and slightly dependent on the external potential. The curves displayed in Fig. 1(a) clearly demonstrate that the value of the second-order transition moment of lithium hydride varies largely with the internuclear separation as well. For the sake of discussion performed herein, it should be noted that only the absolute values of Sgfzz are of significance for the magnitude of the TPA response. As one can notice the Sgfzz function exhibits a nonmonotonic behaviour even for the unconfined lithium hydride. Particularly, for R < Re the values of Sgfzz computed for the free LiH molecule are smaller than those determined at the experimental equilibrium distance. However, stretching the LiH bond length leads to an increase in the magnitude of the second-order transition moment and the Sgfzz(R) function exhibits two extrema equal to 499 and −723 a.u. at R = 5.8 a.u. and R = 8.5 a.u., respectively. Note that in the intermediate internuclear separation range the inversion of the Sgfzz sign occurs, while for large R (R > 16 a.u.) its value converges to zero.
Turning the attention to the results obtained for the spatially limited LiH molecule several interesting conclusions can be also drawn. As it turns out, upon embedding in the harmonic potential the second-order transition moment of LiH follows, in general, the same patterns of changes as presented by the unconfined molecule. Yet, the internuclear distances at which Sgfzz reaches its maxima (extrema) are shifted toward larger values of R. Likewise, the bond lengths for which inversion of the second-order transition moment sign is observed (so called “crossing point”) are also noticeably larger. Obviously, the shifts in the Sgfzz(R) function can be considered as a natural consequence of the fact that the employed confining potential causes an increase of the energy gap between the X1Σ+ and A1Σ+ states of LiH. Moreover, it follows from Fig. 1 that the influence of spatial restriction on the Sgfzz value is much more pronounced far from the equilibrium bond length of lithium hydride. In contrast to what was observed when R = Re, at larger internuclear distances the presence of an external potential results in a significant enhancement of the second-order transition moment with respect to the value obtained for the unconfined LiH molecule. It is notable that there is a three-fold increase of the maximum values of Sgfzz due to confinement. The above observations can be easily understood by the analysis of key parameters for the maximum, crossing point and minimum on the Sgfzz curve (for all confinement strengths), which are assembled in Table 2. In particular, for the internuclear distances under consideration, the excitation energy decreases by an order of magnitude, while 〈g|z|f〉 significantly increases, with respect to the values obtained at the experimental equilibrium distance (cf.Table 1). Moreover, for both free and spatially confined LiH, the crossing points are characterized by smaller ωf and Δμ values and larger OPA transition moments when compared to the data obtained for Smaxzz and Sminzz. According to the TLM, changes in the above mentioned spectroscopic parameters, and their mutual correlation, have a decisive impact on the second-order transition moment values (see the discussion below).
S max zz | S cp zz | S min zz | |
---|---|---|---|
φ = 0.0 | |||
R | 5.80 | 7.00 | 8.50 |
ω f | 5.7 × 10−2 | 4.4 × 10−2 | 4.9 × 10−2 |
〈g|z|g〉 | −2.9 | −1.8 | −0.5 |
〈f|z|f〉 | 0.2 | −2.0 | −4.1 |
Δμz | 3.1 | −0.2 | −3.6 |
〈g|z|f〉 | 2.5 | 3.2 | 2.8 |
φ = 0.1 | |||
R | 6.40 | 7.50 | 8.80 |
ω f | 4.8 × 10−2 | 3.8 × 10−2 | 4.3 × 10−2 |
〈g|z|g〉 | −3.3 | −2.0 | −0.6 |
〈f|z|f〉 | 0.03 | −2.2 | −4.1 |
Δμz | 3.33 | −0.2 | −3.5 |
〈g|z|f〉 | 2.6 | 3.3 | 2.8 |
φ = 0.2 | |||
R | 7.60 | 8.50 | 9.20 |
ω f | 3.0 × 10−2 | 2.0 × 10−2 | 3.0 × 10−2 |
〈g|z|g〉 | −3.9 | −2.1 | −0.9 |
〈f|z|f〉 | −0.1 | −2.2 | −3.1 |
Δμz | 3.8 | −0.1 | 2.2 |
〈g|z|f〉 | 2.8 | 3.4 | 3.0 |
As it follows from the data depicted in Fig. 1(b) the estimated Sgf,TLMzz values reproduce reasonably well those computed using the MCQR approach. Thus, the two-level approximation is sufficient to qualitatively explain the changes in the second-order transition moment as a function of internuclear separation. The dependence of the spectroscopic parameters contributing to Sgf,TLMzz on the bond length is illustrated in Fig. 2. A close inspection of the presented plots allows one to conclude that the change in Sgfzz is mostly governed by the variation of Δμz. Of particular importance are changes occurring for the excited state dipole moment. In this case a maximum and a minimum of the μz(R) function appear at internuclear distances close to those for which the peaks of Sgfzz functions are also located. On the other hand, in the ground electronic state of LiH the dipole moment reaches its maximum at bond length, where the potential energy curve crosses the Li+H− ionic potential curve,10 and yields zero values at larger internuclear distances. The enhancement of the second-order transition moment value at R > Re is also due to the decrease of the excitation energy value, accompanied by an increase of the one photon transition dipole moment. Nevertheless, these two quantities have considerably less impact on the nature of Sgfzz changes in the function of internuclear separation. Noteworthy, this observation applies to both free and spatially limited lithium hydride.
In Fig. 3, the variation of the one- and two-photon absorption strength of LiH with the internuclear separation is presented. Unsurprisingly, the R-dependence of 〈δgf〉 bears a strong resemblance to that of the diagonal component of the second-order transition moment of LiH. The total effect is even more pronounced as the maximum value of TPA strength under double perturbation, that is the confining potential with φ = 0.2 a.u. and the bond stretched to 7.6 a.u., is almost ninety times larger than 〈δgf〉 computed for unconfined LiH at the experimental equilibrium distance. However, one may find fundamental differences between the changes in the one- and two-photon absorption strength occurring due to the variation in the internuclear separation in the presence of confining HO potential (cf.Fig. 3(a)). Although the R-dependence of oscillator strength (f) demonstrates nonmonotonic behavior, for internuclear distances larger than Re the value of f is always greater than the one computed at the experimental bond distance. This applies to the result obtained under vacuum as well as upon embedding LiH in HO potential. In contrast to 〈δgf〉 the spatial confinement diminishes the OPA strength virtually in the whole range of R. The present findings remain in agreement with the observations made in a theoretical study concerning the absorption spectra of the p-nitroaniline (pNA) molecule embedded in different confining cages.22 In particular, it was demonstrated that under the influence of chemical pressure, imposed by the helium tube, the absorption maximum of pNA is shifted to larger wavelengths, losing some of its intensity.
Fig. 3 One- (a) and two-photon (b) absorption strength function of the free and spatially confined LiH molecule. Calculations were performed using the MCSCF wave function and the ANO-L basis set. |
The results of the performed calculations indicate a significant reduction of the two-photon absorption response of lithium hydride at its experimental equilibrium bond length upon confinement. On the other hand, large and nonmonotonic changes of the second-order transition moment value, and consequently TPA strength, were observed due to the variation in the internuclear separation. As it has been found that at distances larger than the equilibrium bond length a substantial enhancement of Sgfzz and 〈δgf〉 might occur. Moreover, it has been disclosed that the importance of the orbital compression effect increases in the case of highly distorted geometries of lithium hydride. According to the obtained results under double perturbation, i.e. when the bond length of the LiH molecule embedded in an external potential is strongly stretched, the TPA strength could increase by several orders of magnitude. Analysis of the results in terms of the two-level model leads to the conclusion that the observed changes in the TPA response are mostly governed by the variation in the difference between the ground- and excited electronic state dipole moment of LiH.
Summing up, the obtained results provide evidence that the “pure” spatial confinement effect might have a significant influence on the magnitude of the two-photon absorption response of molecular systems. According to our knowledge, some of the studied topics, including comprehensive analysis related to bond-length dependence of the investigated molecular quantities, have never been considered in the literature before. Moreover, it should not be overlooked that the highly accurate ab initio values of the second-order transition moment and TPA probability of the unconfined LiH molecule were reported herein for the first time in the literature.
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