First the salient features of the U(ν+1) algebraic approach associated to ν equivalent oscillators are presented. Then we introduce the 1D case through the connection of the U(2) algebra with the Morse/Pöshl-Teller potentials with the goal of describing the vibrational degrees of freedom of non linear polyatomic molecules. The coordinates and momenta are then identified and generalized to any potential, providing the possibility to solve the 1D Schrödinger equation for general potentials by purely algebraic means using the concept of transformation brackets. A new procedure to calculate of Franck-Condon factors is presented. Because of their importance in linear molecules the U(3) model is introduced, emphasizing its connection with configuration space. It is shown the application of the U(2) ×U(3) ×U(2) algebraic approach to describe the Raman spectroscopy of the CO₂ molecule. The U(3) model is applied to consider general potentials to describe linear-to-bend transition in triatomic molecules. Finally the U(4) model is introduced to describe 3D systems for general potentials. The Hydrogen atom as well as the 3D Morse systems are analyzed by purely algebraic means as a benchmark to show how to apply the algebraic method for potentials with spectroscopic interest.

r0pt Figure The first application of the quantum graph model to vibrational quantum dynamics of molecules is reported. The usefulness of the approach is demonstrated for the astructural molecular ion CH₅⁺, an enigmatic system of high-resolution molecular spectroscopy and molecular physics, challenging our traditional understanding of chemical structure and rovibrational quantum dynamics. The vertices of the quantum graph correspond to different versions of the molecule (120 in total for CH₅⁺), while the differently colored edges represent different collective nuclear motions transforming the distinct versions into one or another. These definitions allow the mapping of the complex low-energy vibrational quantum dynamics of CH₅⁺ onto the motion of a one-dimensional particle confined in a quantum graph. The quantum graph model provides a simple and intuitive qualitative understanding of the intriguing low-energy vibrational dynamics of CH₅⁺ and is able to reproduce, with just two adjustable parameters related to the two different motions (indicated by the red and blue lines in the figure), the lowest vibrational energy levels of CH₅⁺ (and CD₅⁺) with remarkable accuracy.

I shall present an improvement and applications of the Hierarchical Intertwined Reduced-Rank Block Power Method (J. Chem. Phys, 146, 204110 (2017)) for solving the vibrational Schroedinger equation. The improvement decreases the memory required to compute a spectrum. Variational calculations for molecules with a dozen atoms are now possible on a desktop computer. The memory cost scales linearly with the number of atoms in the molecule. We apply the HI-RRBPM to compute vibrational spectra of uracil and naphthalene, with 12 and 18 atoms, respectively. The HI-RRBPM uses a direct product basis but: 1) it is not necessary to store a direct-product-basis matrix representation of the Hamiltonian matrix (for naphthalene the size of the matrix would be  ∼ 10⁴⁸); 2) it is not necessary to store vectors whose length is equal to the size of the direct-product basis. This is accomplished by using sum-of-product (SOP) basis functions stored in a canonical polyadic tensor format and generated by evaluating matrix-vector products. The number of terms in the SOP basis functions is minimized by optimising the factors. Representing vibrational wavefunctions as optimised SOPs reveals the essential entangledness and provides new understanding. The method only works if the Hamiltonian is itself a SOP.

We present a new method for solving the vibrational Schroedinger equation for polyatomic molecules. It has the following advantages: 1) the size of the matrix eigenvalue problem is the size of the required pruned (nondirect product) polynomial-type basis; 2) it requires solving a regular, and not a generalized, symmetric matrix eigenvalue problem; 3) accurate results are obtained even if quadrature points and weights are not good enough to yield a nearly exact overlap matrix; 4) the potential matrix is diagonal; 5) the matrix-vector products required to compute eigenvalues and eigenvectors can be evaluated by doing sums sequentially, despite the fact that the basis is pruned. To achieve these advantages we use sets of nested Leja points and appropriate Leja quadrature weights and special hierarchical basis functions. Matrix-vector products are inexpensive because transformation matrices between the basis and the grid, and their inverses, are lower triangular. Vibrational energy levels of CH₂NH are calculated with the new method. For this purpose a simple harmonic oscillator kinetic energy operator and a quartic force field are used.

The first excited electronic state of the NO₃ radical is a dark state just less than 1 eV above the ground state, and its vibronic features have been observed optically by the Hirota, Okumura and Miller groups. In addition, it has been directly accessed by the Neumark group via photodetachment of the nitrate anion. This ²E^′′ state represents a case where the Jahn-Teller effect is neither weak enough to be treated with low-order (linear or quadratic) models nor is it profoundly distorted to the degree that it can be accurately represented by a quasistatic lower-symmetry geometry. While efforts to treat this state with polynomial Jahn-Teller model Hamiltonians have been carried out, none of these can really be regarded as being quantitatively successful. In this work, we use newly available methodology to treat this electronic state with a semi-global potential based on permutationally invariant polynomials and report results of the corresponding energy levels and simulated photodetachment spectrum.

Vibrational auto-ionization resonance widths (γ) of v=1, n^* ≈ 14 Rydberg states of CaF are calculated in this work, based on results of a global multi-channel quantum defect fit. The calculation indicates that the n.36 pΠ eigen-channel has the shortest vibrational auto-ionization lifetime,  ∼ 10 ps, which is at least 4× shorter than the lifetime of all other CaF eigen-channels, in agreement with experimental observations. In addition, the calculation successfully reproduces the experimental observations that γ of the 14.36 pΠ⁻ rotational sequence (where the superscript `-' indicates negative Kronig symmetry) are nearly N-independent, while those of the 14.36 pΠ<sup>+</sup> rotational sequence (where the superscript `+' indicates positive Kronig symmetry) decrease quickly as a function of N, i.e. γ(N=10) ≈ \frac12 γ(N=1). By examining the eigen-channel composition of the two rotational sequences of state of opposite Kronig symmetry, we are able to show that the significantly faster decrease of γ for the 14.36 pΠ⁺ rotational sequence is caused by the stronger l-uncoupling interaction in the positive Kronig symmetry manifold. Based on a valence-precursor model (first suggested by Mulliken), the significantly faster vibrational auto-ionization rate of the n.36 pΠ eigen-channel is explained based on the electronic properties of its valance-precursor state, the C²Π state, for which the electron density is polarized toward the fluorine atom.

In the sub-picosecond singlet fission (SF) process, two coherent triplet excitons (¹(T₁T₁)) are generated from one singlet exciton (S₁) through a mysterious "multi-exciton" (¹ME) intermediate: S₁ ↔ ¹ME ↔ ¹(T₁T₁) → 2T₁.M. B. Smith and J. Michl, Chem. Rev. 110, 6891 (2010).rganic semiconducting materials that undergo exothermic SF double the efficiency associated with high-energy incident photons and allow the photovoltaics to exceed the Shockley-Queisser limit of 33.7%.W. Shockley and H. J. Queisser, J. Appl. Phys. 32, 510 (1961).nderstanding the character of ¹ME serves as the key to deciphering the ultrafast SF mechanism. Based on a popular hypothesis, the complete SF procedure involves two steps of charge transfer (CT) and ¹ME is a superposition of localized and charge-separated excited states.T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, J. Chem. Phys. 141, 074705 (2014).owever, the straightforward experimental and theoretical evidence to support this hypothesis is still in darkness due to the challenges to measure ultrafast photochemistry and to describe charge-separated and multiply-excited energy levels. Herein we modeled the local SF reactions occurring in hexacene, pentacene and bis(6,13-bis(triisopropylsilylethynyl)pentacene)benzene in the crystalline or solution phase. The efficient and accurate density functional theory (DFT) based approaches developed in the earlierS. R. Yost, J. Lee, M. W. B. Wilson, T. Wu, D. P. McMahon, R. R. Parkhurst, N. J. Thompson, D. N. Congreve, A. Rao, K. Johnson, M. Y. Sfeir, M. G. Bawendi, T. M. Swager, R. H. Friend, M. A. Baldo and T. Van Voorhis, Nat. Chem., 6, 492 (2014).nd present study were utilized to evaluate energies of S₁, ¹ME and ¹(T₁T₁), as well as the non-adiabatic coupling between each pair of them. Our results shed light on the roles of localized and charge-separated excited states in the complete SF mechanism and propose the strategy for the rational design of oligoacene-based SF materials.

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Footnotes:

M. B. Smith and J. Michl, Chem. Rev. 110, 6891 (2010).O W. Shockley and H. J. Queisser, J. Appl. Phys. 32, 510 (1961).U T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, J. Chem. Phys. 141, 074705 (2014).H S. R. Yost, J. Lee, M. W. B. Wilson, T. Wu, D. P. McMahon, R. R. Parkhurst, N. J. Thompson, D. N. Congreve, A. Rao, K. Johnson, M. Y. Sfeir, M. G. Bawendi, T. M. Swager, R. H. Friend, M. A. Baldo and T. Van Voorhis, Nat. Chem., 6, 492 (2014).a

X̃ ²Π NCS is a Renner-degenerate linear molecule whose rovibronic spectrum is greatly complicated by the Renner effect and all-pervading resonances. As an alternative avenue to understanding this spectrum, we have calculated values of the ro-vibronic energies, intensities, and rotational constants by direct numerical solution of the rovibronic Schrödinger equation with the RENNER program.J. Freund, S. C. Galleguillos Kempf, P. Jensen, U. Nagashima, T. Hirano, J. Mol. Spectrosc. 345, 31–38 (2018). DOI: 10.1016/j.jms.2017.11.010; T. Hirano, U. Nagashima, P. Jensen, J. Mol. Spectrosc., (2018), https://doi.org/10.1016/j.jms.2017.12.011.ll values obtained are in good agreement with the available experimental data. Ro-vibronic spectra are also simulated. The Renner calculations are based on three-dimensional potential energy surfaces and dipole moment surfaces computed ab initio for NCS in the X̃ ²Π electronic ground state at the core-valence, full-valence MR-SDCI+Q/[aug-cc-pCVQZ(N, C, S)] level of theory.

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Footnotes:

J. Freund, S. C. Galleguillos Kempf, P. Jensen, U. Nagashima, T. Hirano, J. Mol. Spectrosc. 345, 31–38 (2018). DOI: 10.1016/j.jms.2017.11.010; T. Hirano, U. Nagashima, P. Jensen, J. Mol. Spectrosc., (2018), https://doi.org/10.1016/j.jms.2017.12.011.A

We have constructed ab initio 3D potential energy surfaces (PESs) for X̃  ²Π NCS in core-valence SDCI+Q/[aCVQZ(N,C,S)] calculations. The B₀ value predicted from these PESs deviates only 0.05%  from the corresponding experimental values for NC³²S and NC³⁴S. Since we have quite accurate 3D PESs, we can determine both the equilibrium structure and the r₀ structure accurately: r_e(N-C) = 1.1778 Å, r_e(C-S) = 1.6335 Å, and ∠_e(N-C-S) = 180^°. The ro-vibrationally averaged structure, determined as expectation values over DVR3D wavefunctions, has 〈r(N-C)〉₀ = 1.1836 Å, 〈r(C-S)〉₀ = 1.6356 Å, and 〈∠(N-C-S) 〉₀ = 172.5^°. The 3D PESs show that the X̃ ²Π NCS has its potential energy minimum at a linear configuration, and hence it is a "linear molecule." Experimentally, B₀ values are reported for two isotopologues only.A. Maeda, H. Habara, T. Amano, Mol. Phys., 105, 477-495 (2007).sing the expectation values given above as the initial guess, a bent r₀ structure having an 〈∠(N-C-S) 〉₀ of 172.2^° is deduced from the experimentally reported B₀ values for NC³²S and NC³⁴S. It shows that the linear molecule NCS has a "bent" ro-vibrationally averaged structure, confirming our previous predictions:T. Hirano, U. Nagashima, J. Mol. Spectrosc., 314, 35-47 (2015); T. Hirano, U. Nagashima, P. Jensen, J. Mol. Spectrosc. 343, 54-61 (2018).ny linear molecule is observed as being bent on ro-vibrational average. See Ref. c T. Hirano, U. Nagashima, P. Jensen, J. Mol. Spectrosc. (2018), https://doi.org/10.1016/j.jms.2017.12.011.or further discussion of this molecule. ²Π NCS is a typical Renner molecule. The Renner spectroscopy of this molecule will be presented in a separate talk.J. Freund et al, "Computational spectroscopy of NCS in the Renner-degenerate Electronic state X̃ ²Π."html:<hr /><h3>Footnotes: A. Maeda, H. Habara, T. Amano, Mol. Phys., 105, 477-495 (2007).U T. Hirano, U. Nagashima, J. Mol. Spectrosc., 314, 35-47 (2015); T. Hirano, U. Nagashima, P. Jensen, J. Mol. Spectrosc. 343, 54-61 (2018).a T. Hirano, U. Nagashima, P. Jensen, J. Mol. Spectrosc. (2018), https://doi.org/10.1016/j.jms.2017.12.011.f J. Freund et al, "Computational spectroscopy of NCS in the Renner-degenerate Electronic state X̃ ²Π."

It is possible to compute vibrational spectra using the Heidelberg implementation of the multi-configuration time-dependent Hartree method. However, the Heidelberg program can only be used if the potential energy surface is a sum of products (SOP). I shall present a new collocation-based MCTDH approach that can be used with general potential energy surfaces. This is imperative if one wishes to compute very accurate spectra. Collocation obviates the need for quadrature and facilitates using complicated kinetic energy operators. When the basis is good, the accuracy of collocation solutions to the Schroedinger equation is not sensitive to the choice of the collocation points. We test the collocation MCTDH equations by showing that they can be used to compute accurate vibrational energy levels of CH₃. MCTDH, with or without collocation, uses a direct-product basis. I shall demonstrate that by using so-called hierarchical basis functions, it is possible to both benefit from the advantages of collocation and prune the MCTDH basis. These new computational tools will make it possible to use MCTDH-type methods to compute very accurate spectra.