Sum-over-states Research Articles

A simple molecular orbital picture of RIXS distilled from many-body damped response theory.

Ab initio calculations of resonant inelastic x-ray scattering (RIXS) often rely on damped response theory, which prevents the divergence of response solutions in the resonant regime. Within the damped response theory formalism, RIXS moments are expressed as the sum over all electronic states of the system [sum-over-states (SOS) expressions]. By invoking resonance arguments, this expression can be reduced to a few terms, an approximation commonly exploited for the interpretation of computed cross sections. We present an alternative approach: a rigorous formalism for deriving a simple molecular orbital picture of the RIXS process from many-body calculations using the damped response theory. In practical implementations, the SOS expressions of RIXS moments are recast in terms of matrix elements between the zero-order wave functions and first-order frequency-dependent response wave functions of the initial and final states such that the RIXS moments can be evaluated using complex response one-particle transition density matrices (1PTDMs). Visualization of these 1PTDMs connects the RIXS process with the changes in electronic density. We demonstrate that the real and imaginary components of the response 1PTDMs can be interpreted as contributions of the undamped off-resonance and damped near-resonance SOS terms, respectively. By analyzing these 1PTDMs in terms of natural transition orbitals, we derive a rigorous, black-box mapping of the RIXS process into a molecular orbital picture. We illustrate the utility of the new tool by analyzing RIXS transitions in the OH radical, benzene, para-nitroaniline, and 4-amino-4'-nitrostilbene. These examples highlight the significance of both the near-resonance and off-resonance channels.

Hybrid calculation of the partition functions of prolate top molecules: Using HOCl as the test case

The partition function of a molecule can be obtained by a direct sum-over-states (SOS) calculation using the molecule’s quantum energy levels. These energies can be those obtained by a model fitted to experimental data. For molecules with “exotic” behaviour (such as large amplitude vibrational motion) physically motivated models can be required. Often, however, such models can be limited in the range of quantum levels that can be reliably calculated, making a converged SOS calculation impractical. Here we present a hybrid approach for prolate top molecules. Our hybrid approach allows the simple extension from a limited SOS calculation to a good approximation for the infinite sum limit. This is achieved by adding easily calculated closed-form corrections to the limited SOS result. We test our approach using the Generalised SemiRigid Bender (GSRB) program as the physically motivated model. HOCl serves as a good test case because for it the GSRB model can extend to high enough levels for convergence of the direct SOS calculation. Then, by limiting the range of GSRB calculated rotational levels included in the SOS, we confirm our hybrid approach. The technique we present here should be particularly applicable to quasilinear molecules, including those exhibiting quantum monodromy. The Appendix presents step-by-step instructions for implementing this hybrid procedure. Also presented there is a brief discussion of how to adapt the hybrid technique to approximately account for rotational or vibrational dissociation in the case that that is necessary.

Characterization of tetrakis(thiadiazole)porphyrazine metal complexes by magnetic circular dichroism and magnetic circularly polarized luminescence.

The magnetic circular dichroism (MCD) spectra of metal complexes of tetrakis(thiadiazole)porphyrazines ([TTDPzM] with M = 2HI , ZnII , MgII (H2 O), and CdII ) have been recorded in dimethyl formamide solution. Together with the UV-Vis spectra, the MCD spectra provide useful information about the structure and electronic properties of the complexes. The experimental UV-Vis and MCD spectra compare pretty well with DFT calculations of two sorts, based either on the sum-over-states (SOS) approach or on the complex polarization propagator approach. They further corroborate the findings and interpretation of MCD spectra of porphyrazines based on the model of Michl for peripheral molecular orbitals. Magnetic circularly polarized luminescence (MCPL) spectra, quite uncommon in the literature, have been recorded for [TTDPzM] (M = 2HI , ZnII , MgII (H2 O)).

A mononuclear cobalt(II) salophen-type complex: Synthesis, theoretical and experimental electronic absorption and infrared spectra, crystal structure, and predicting of second- and third-order nonlinear optical properties.

A tetradentate Schiff base ligand, (4-nitro-N,N'-o-phenylene-bis(5-methoxysalicylideneimine)) (H2L), and its cobalt(II) complex (CoL•DMF) were synthesized. Elemental analysis, and FT-IR and 1H NMR spectra were used to confirm the molecular structure of H2L ligand. In addition, elemental analysis, molar conductance measurement, infrared and electronic absorption spectroscopies as well as single crystal X-ray diffraction were employed to distinguish the molecular structure of the complex that has a slightly distorted square-planar geometry around the cobalt(II) ion. The geometries of H2L and CoL were optimized by means of the DFT/(U)PBE0/Def2-TZVP level of theory. Good agreement of the optimized geometric parameters of the complex with the corresponding experimental ones and successfully reproduction of the FT-IR spectra of the ligand and complex validated the applied method. Based on the optimized structure of CoL, its electronic absorption spectrum was reproduced by using time-dependent density functional theory (TDDFT) at the same level. Its analysis was carried out to assign the observed electronic transitions and determine their characters. The computed nonlinear optical (NLO) parameters (β, γ and < γ > values) of H2L and CoL by using TDDFT at the same level combined with the sum-over-states (SOS) method are considerable larger values than reference compound i.e. urea. The ligand and the complex in the absence of nitro and methoxy moieties referred to as H2L' and CoL' respectively were optimized to elucidate the effect of these moieties on hyperpolarizabilities values.

Absolute Nonlinear Refractive Index Spectra Determination of Organic Molecules in Solutions.

It has been a great challenge to measure the spectrum of pure bound-electronic third-order nonlinear refraction ( n2) of organic chromophores in solutions because of the spurious contribution from the solvent and cuvette walls. In order to circumvent this problem, we present here a new method to obtain a highly accurate absolute n2 value of organic molecules in solutions with a self-referenced nonlinear ellipse rotation (NER) technique. As a proof of concept, we measured n2 spectra of two well-known chromophores, rhodamines B and 6G dissolved in methanol, in the range from ∼600 to 1200 nm. Our results pointed out that these two dyes present similar dispersion curves with strong negative nonlinearities near the one-photon absorption band and small positive values at long wavelengths. Furthermore, the negative signal of the dyes can be strong enough to cancel and even invert the positive nonlinear refraction of the solvent (methanol) as the solution's concentration increases. To understand the n2 spectrum and its connection to molecular properties of organic chromophores, we employed the sum-over-states (SOS) approach within the few-energy-level model and observed an excellent agreement between the experimental and theoretical spectra. In this way, we believe that, employing our NER technique and the SOS model, it is possible to determine both experimentally and theoretically the absolute magnitude and spectra of pure electronic n2 for a large variety of other organic molecules.

A new method to analyze and understand molecular linear and nonlinear optical responses via field-induced functions: a straightforward alternative to sum-over-states (SOS) analysis.

The sum-over-states (SOS) method allows the computation of polarizabilities and hyperpolarizabilities additively from the contributions of different electronic excited states in a given molecule or cluster. Subsequent analysis of the main excited configurations contributing to the relevant excited states allows characterizing the orbitals involved in the linear and nonlinear optical response. Unfortunately, the chemically relevant information that can be obtained by SOS is hindered by a series of methodological and computational drawbacks. Among these drawbacks, we can highlight the high computational cost, problems arising from nonconvergent series and errors caused by the inaccurate description of excitation energies and/or higher excited state matrix elements. For this reason, coupled-perturbed schemes are currently widely used to determine the NLO potential of molecules and materials. However, such a choice limits the amount of intuitive chemical information that, on the other hand, can be retrieved by a successful SOS computation. In this work, we present and discuss a novel computational strategy that offers the means to extract the useful chemical insights from a coupled-perturbed calculation at almost negligible extra computational cost providing a transparent picture about orbital contributions to the properties of interest. The proposed method is based on the generation and further analysis of field-induced orbitals, FIOs, from the analytic or numerical derivatives of the dipole moment. Orbital symmetry rules are derived using group theory and the method is tested for a series of small and medium size systems.

Theoretical study on nonlinear optical properties of N-(6-hydroxyhexyl)-5-nitroazophenyl carbazole

The first-, second- and third-order nonlinear optical properties of N-(6-hydroxyhexyl)-5-nitroazophenyl carbazole in gas phase employing sum-over-states (SOS) method have been calculated for the first time. The ground state molecular structure of N-(6-hydroxyhexyl)-5-nitroazophenyl carbazole was obtained by the geometrical optimizations based on the B3LYP/6-31[Formula: see text]G(d) level. The energy of excited states and transition dipole moments between different excited states were obtained by using the time-dependent density functional theory (TDDFT) based on the CAM-B3LYP/Sadlej POL level. Charge transfer during electron excitation was analyzed based on hole and electron distributions. Our calculations showed that the N-(6-hydroxyhexyl)-5-nitroazophenyl carbazole has good nonlinear optical properties and its nonlinear optical properties arise from charge-transfer excitation and local excitation but charge transfer plays the leading role.

Twisted angle effects in the absorption spectra of carbon nanotube

The spectral characteristics of carbon nanotubes is essential for the development of optoelectronic components and sensors with application in many fields. The linear polarizability absorption spectra of the carbon nanotube with a twisted angle have been calculated by using the tight-binding model and sum-over-state method. First, a structural model of carbon nanotubes based on PPP(Pariser-Parr_Pople) was established, then a certain carbon atom was moved to a position where the carbon nanotubes are angled, and the torsional carbon nanotube structure model was calculated based on the change in bond length. Finally, the absorption spectrum at different twisted angles based on SOS(Sum-over-State) was calculated, and then the detailed analysis was made. It is found that the twisted angle has real impact on the absorption spectra of different types carbon nanotubes. The torsional angle leads to changes in the position and number of absorption peaks of the carbon nanotubes. Its influence on absorption spectra of different types of carbon nanotubes is different, but mainly concentrated in the low frequency region. Exactly, altering the twisted angle has an important effect on the absorption spectra of the carbon nanotube. This work offers a guidance of fabricating carbon nanotubes in the applications of photonic device in many fields.

Diagonal Born–Oppenheimer correction for coupled-cluster wave-functions

ABSTRACTWe examine how geometry-dependent normalisation freedom of electronic wave-functions affects extraction of a meaningful diagonal Born–Oppenheimer correction (DBOC) to the ground-state Born–Oppenheimer potential energy surface (PES). By viewing this freedom as a kind of gauge-freedom, it is shown that DBOC and the resulting associated mass-dependent adiabatic PES are gauge-invariant quantities. A sum-over-states (SOS) formula for DBOC which explicitly exhibits this invariance is derived. A biorthogonal formulation suitable for DBOC computations using standard unnormalised coupled-cluster (CC) wave-functions is presented. This is shown to lead to a biorthogonal version of SOS formula with similar properties. On this basis, different computational schemes for evaluating DBOC using approximate CC wave-functions are derived. One of this agrees with the formula used in the current literature. The connection to adiabatic-to-diabatic transformations in non-adiabatic dynamics is explored and complications arising from biorthogonal nature of CC theory are identified.

Oxazole Dyes with Potential for Photoluminescence Bioprobes: A Two-Photon Absorption Study

In this work, six π-conjugated oxazole compounds dissolved in dichloromethane were characterized with linear and nonlinear optical measurements. Z-scan with femtosecond laser pulses was employed to determine the two-photon absorption (TPA) spectra. Other photophysical parameters, such as absorbance, solvatochromism, lifetime fluorescence, and fluorescence anisotropy, were evaluated with linear optical techniques. The experimental TPA cross section spectra were adjusted by the sum-over-states (SOS) model, by which important parameters such as transition dipole moments and broadening parameters were determined. To better understand the TPA spectra of the oxazole compounds, quantum-chemical calculations using the response function formalism and the density functional theory level of theory were performed. Using the results provided by the quantum-chemical calculations and the broadening parameters estimated through the application of the SOS model, the TPA spectra were simulated by the superposition (summati...

Two-dimensional electronic spectroscopy as a tool for tracking molecular conformations in DNA/RNA aggregates.

A computational strategy to simulate two-dimensional electronic spectra (2DES) is introduced, which allows us to analyse ground state dynamics and to sample and measure different conformations attained by flexible molecular systems in solution. An explicit mixed quantum mechanics/molecular mechanics (QM/MM) approach is employed for the evaluation of the necessary electronic excited state energies and transition dipole moments. The method is applied towards a study of the highly flexible water-solvated adenine-adenine monophosphate (ApA), a system featuring two interacting adenine moieties that display various intermolecular arrangements, known to deeply affect their photochemical outcome. Molecular dynamics simulations and cluster analysis have been used to select the molecular conformations, reducing the complexity of the flexible ApA conformational space. By using our sum-over-states (SOS) approach to obtain the 2DES spectra for each of these selected conformations, we can discern spectral changes and relate them to specific nuclear arrangements: close lying π-stacked bases exhibit a splitting of their respective 1La signal traces; T-stacked bases exhibit the appearance of charge transfer states in the low-energy Vis probing window while displaying no 1La splitting, being particularly favoured when promoting amino to 5-ring interactions; unstacked and distant adenine moieties exhibit signals similar to those of the adenine monomer, as is expected for non-interacting nucleobases. 2DES maps reveal the spectral fingerprints associated with specific molecular conformations, and are thus a promising option to enable their quantitative spectroscopic detection beyond standard 1D pump-probe techniques. This is expected to aid the understanding of how nucleobase aggregation controls and modulates the photostability and photo-damage of extended DNA/RNA systems.

Stabilizing g-States in Centrosymmetric Tetrapyrroles: Two-Photon-Absorbing Porphyrins with Bright Phosphorescence.

Using time-dependent density functional theory (TDDFT) and sum-overstates (SOS) formalism, we predicted significant stabilization of 2P-active g-states in a compact fully symmetric porphyrin, in which all four pyrrolic fragments are fused with phathalimide residues via the β-carbon positions. The synthesis of a soluble, nonaggregating meso-unsubstituted tetraarylphthalimidoporphyrin (TAPIP) was then developed, and the spectroscopic measurements confirmed that a strongly 2P-active state in this porphyrin is stabilized below the B (Soret) state level. Single-crystal X-ray analysis revealed near-ideally planar geometry of the TAPIP macrocycle, while its tetra-meso-arylated analogue (meso-Ar4TAPIP) was found to be highly saddled. Consistent with these structural features, Pt meso-Ar4TAPIP phosphoresces rather weakly (ϕphos = 0.05 in DMF at 22 °C), while both Pt and Pd complexes of TAPIP are highly phosphorescent (ϕphos = 0.45 and 0.23, respectively). In addition PdTAPIP exhibits non-negligible thermally activated (E-type) delayed fluorescence (ϕfl(d) ∼ 0.012). Taken together, these photophysical properties make metal complexes of meso-unsubstituted tetaarylphthalimidoporphyrins the brightest 2P-absorbing phosphorescent chromophores known to date.

Molecular Design Principles for Magneto-Electric Materials: All-Electric Susceptibilities Relevant to Optimal Molecular Chromophores

Magneto-electric (M-E) response at the molecular level arises from the interaction of matter with the electric and magnetic fields of light, and can manifest itself as nonlinear M-E magnetization (MNL) or M-E rectification (PNL). However, there is presently a limited understanding of how molecular material properties impact M-E response. Here we investigate the relationship between M-E nonlinear coefficients and the third-order electric susceptibility, χ(3), finding that MNL is proportional to χxxxx(3) while PNL scales with χzzxx(3) due to a cascaded nonlinearity. By applying a sum-over-states (SOS) expression for the elements of χ(3) to valence-bond charge-transfer (VB-CT) models, we formulate practical guidelines for the design of materials expected to exhibit enhanced M-E properties. On this basis, we predict that many conventional nonlinear optical chromophores with large values of χxxxx(3) may be suitable for generating optical magnetism at low intensities. In the case of M-E rectification, analysis ...

Theoretical and experimental research on the third-order nonlinear optical properties of the N-6-hydroxy-hexyl-4-azophenyl-carbazol

The third-order nonlinear optical properties of the N-6-hydroxy-hexyl-4-azophenyl-carbazol have been investigated both theoretically and experimentally. Using Z-scan method with a femtosecond laser (440[Formula: see text]nm, 190[Formula: see text]fs), the third-order nonlinear optical properties of the N-6-hydroxy-hexyl-4-azophenyl-carbazol were determined. The second-order hyperpolarizability [Formula: see text] of the experiment result was found to be [Formula: see text][Formula: see text]esu. We have also calculated the second-order hyperpolarizability based on the sum-over-states (SOS) method. The theoretical value was in accord with the experiment result. The result showed that the N-6-hydroxy-hexyl-4-azophenyl-carbazol was one of the promising third-order nonlinear optical materials.

Two-Photon Absorbing Phosphorescent Metalloporphyrins: Effects of π-Extension and Peripheral Substitution.

The ability to form triplet excited states upon two-photon excitation is important for several applications of metalloporphyrins, including two-photon phosphorescence lifetime microscopy (2PLM) and two-photon photodynamic therapy (PDT). Here we analyzed one-photon (1P) and degenerate two-photon (2P) absorption properties of several phosphorescent Pt (II) porphyrins, focusing on the effects of aromatic π-extension and peripheral substitution on triplet emissivity and two-photon absorption (2PA). Our 2PA measurements for the first time made use of direct time-resolved detection of phosphorescence, having the ability to efficiently reject laser background through microsecond time gating. π-Extension of the porphyrin macrocycle by way of syn-fusion with two external aromatic fragments, such as in syn-dibenzo- (DBP) and syn-dinaphthoporphyrins (DNP), lowers the symmetry of the porphyrin skeleton. As a result, DBPs and DNPs exhibit stronger 2PA into the one-photon-allowed B (Soret) and Q states than fully symmetric (D4h) nonextended porphyrins. However, much more 2P-active states lie above the B state and cannot be accessed due to the interfering linear absorption. Alkoxycarbonyl groups (CO2R) in the benzo-rings dramatically enhance 2PA near the B state level. Time-dependent density functional theory (TDDFT) calculations in combinations with the sum-over-states (SOS) formalism revealed that the enhancement is due to the stabilization of higher-lying 2P-active states, which are dominated by the excitations involving orbitals extending onto the carbonyl groups. Furthermore, calculations predicted even stronger stabilization of the 2P-allowed gerade-states in symmetric Pt octaalkoxycarbonyl-tetrabenzoporphyrins. Experiments confirmed that the 2PA cross-section of PtTBP(CO2Bu)8 near 810 nm reaches above 500 GM in spite of its completely centrosymmetric structure. Combined with exceptionally bright phosphorescence (ϕphos = 0.45), strong 2PA makes Pt(II) complexes of π-extended porphyrins a valuable class of chromophores for 2P applications. Another important advantage of these porphyrinoids is their compact size and easily scalable synthesis.

Resolving Electronic Transitions in Synthetic Fluorescent Protein Chromophores by Magnetic Circular Dichroism.

The detailed electronic structures of fluorescent chromophores are important for their use in imaging of living cells. A series of green fluorescent protein chromophore derivatives is examined by magnetic circular dichroism (MCD) spectroscopy, which allows the resolution of more bands than plain absorption and fluorescence. Observed spectral patterns are rationalized with the aid of time-dependent density functional theory (TDDFT) computations and the sum-over-state (SOS) formalism, which also reveals a significant dependence of MCD intensities on chromophore conformation. The combination of organic and theoretical chemistry with spectroscopic techniques also appears useful in the rational design of fluorescence labels and understanding of the chromophore's properties. For example, the absorption threshold can be heavily affected by substitution on the phenyl ring but not much on the five-member ring, and methoxy groups can be used to further tune the electronic levels.

The effects of exact exchange of density functionals on the evaluation of second hyperpolarizabilities of streptocyanines using sum-over-states method

TDDFT calculations are used to discuss the effects of the exact exchange (Ex) on the evaluation of second hyperpolarizabilities (γyyyy) of streptocyanines (3–9 carbons). The results confirm that TDDFT calculations are reliable on the reproduction of the first excitation energies (E01) compared with experiments (within 0.3eV deviations). The sum-over-states (SOS) calculations based on TDDFT suggest that the percentages of Ex determines the γyyyy. The nonlinear optical (NLO) active region of streptocyanines is the π-bridge conjugation where the bond length is between 2 and 3au. Within this area, the higher the percentage of Ex is, the lower the γyyyy is (γyyyy is normally negative for streptocyanines). The γyyyy can be decomposed into T and N contributions according to the three-level SOS method. The decomposition shows that the T term significantly decreases with the increasing percentages of Ex, while N term gradually increases. Therefore, hybrid generalized gradient approximation (hGGA), hybrid meta-generalized gradient approximation (hybrid meta-GGA) and long-range corrected (LRC) functionals tend to overestimate γyyyy. The functional tuning process can improve the results due to the decreasing of ω. This means the deviation of γyyyy do NOT corresponds to the delocalization error (DE). Consequently, the pure GGA, pure meta-GGA, and LRC with small ω can give qualitative correct results.

Longitudinal and transverse hyperpolarizabilities of carbon nanotubes: a computational investigation through the coupled-perturbed Hartree–Fock/Kohn–Sham scheme

Static electronic polarizability $$\alpha$$ and second hyperpolarizability $$\gamma$$ of semiconducting and conducting carbon nanotubes with radius up to 7.5 A are evaluated using the coupled-perturbed Hartree–Fock/Kohn–Sham scheme, as implemented in the periodic CRYSTAL14 code, and a split-valence basis set. Two density functionals, namely LDA (pure local) and B3LYP (hybrid), and the Hartree–Fock Hamiltonian are compared. A few PBE (gradient corrected) density functional data are also produced for comparison with previous calculations. Convergence of both longitudinal (L) and transverse (T) components is documented. It is shown how the second hyperpolarizability depends critically on the computational conditions, the more so the larger the radius of the nanotube (and thus the smaller the energy gap). The longitudinal component is sensibly affected by the truncation of the exact exchange series (HF and B3LYP), which must include electron–electron interactions at a distance up to 100 A in order to have $$\gamma _L$$ converged to better than 1 %. The transverse $$\gamma _T$$ component of conducting tubes critically depends on the number of k points in reciprocal space: at least 900 k points are required to converge better than 1 % at the LDA level. Coupled-perturbed results are compared to uncoupled values obtained from a sum-over-states (SOS) approach. The difference between the two is particularly important along the transverse direction and when pure DFT functionals are used: the coupled-perturbed correction can shrink the SOS value by several hundreds times. The ratio LDA/HF is roughly constant around 2 for $$\alpha _L$$ ; it ranges between 25 and 60 for $$\gamma _L$$ . As regards the convergence with the nanotube radius, the $$R^2$$ law is confirmed for $$\alpha _L$$ and $$\alpha _T$$ (normalized for the cell parameter) at all levels of theory. For the second hyperpolarizabilities $$\gamma _L$$ and $$\gamma _T$$ , a clear $$R^5$$ dependence is observed.

Comparison of simplified sum-over-state expressions to calculate resonance Raman intensities including Franck-Condon and Herzberg-Teller effects.

Sum-over-state (SOS) expressions to simulate absorption spectroscopy and resonance Raman (RR) scattering including Franck-Condon (FC) and Herzberg-Teller (HT) effects are described. Starting from the general SOS method, several simplified SOS formulae are derived. In particular, within the so-called independent mode displaced harmonic oscillator model, it is shown that including the vibronic structure in the absorption and RR spectra only requires the calculation of FC overlap integrals of the type θg0|θev, where g, e, and v stand for the electronic ground state, excited state, and vibrational quantum number, respectively. Additionally, an approximation of the latter approach is introduced, referred as the simplified Φe method, in which the FC factors are neglected. This method is advantageous from the computational point of view and it is demonstrated that it reproduces the main characteristics of the more involved approaches. The merits and drawbacks of the different methods are discussed by applying them to the prototypical compound of Rhodamine 6G. Overall, this work intends to unravel and clarify some differences in the SOS theories of RR scattering.

Magnetic circular dichroism of chlorofullerenes: Experimental and computational study

Magnetic circular dichroism (MCD) spectra of C60Cl6, C70Cl10 and C60Cl24 were measured and interpreted using a sum-over-state (SOS) protocol exploiting time dependent density functional theory (TDDFT). Unlike for plain absorption, the MCD spectra exhibited easily recognizable features specific for each chlorinated molecule and appear as a useful tool for chlorofullerene identification. MCD spectrum of C60Cl24 was below 400nm partially obscured due to scattering and low solubility. In all cases a finer vibrational structure of the electronic bands was observed at longer wavelengths. The TDDFT simulations provided a reasonable basis for interpretation of the most prominent spectral features.