Optimal representation for semiclassical surface hopping methods

Diabatic and adiabatic representations: Electronic structure caveats

In this paper, we describe implementation of the semiclassical Liouville method for simulating molecular dynamics on coupled electronic surfaces in the electronic adiabatic representation. We cast the formalism in terms of semiclassical motion on Born–Oppenheimer potential energy surfaces with nonadiabatic coupling arising from the coordinate dependence of the adiabatic electronic eigenstates. Using perturbation theory and asymptotic evaluation of the resulting time integrals, we derive an expression for the probability of transition between adiabatic states which agrees with the result given previously by Miller and George [W. H. Miller and T. F. George, J. Chem. Phys. 56, 5637 (1972)]. We also demonstrate numerically the equivalence of semiclassical trajectory-based calculations in the adiabatic and diabatic representations by performing molecular dynamics simulations on a model two-state system and comparing with exact quantum mechanical results. Excellent agreement between the exact and semiclassical treatments is obtained in both representations.

Simulations of electronically nonadiabatic processes may employ either the adiabatic or diabatic representation. Direct dynamics calculations are usually carried out in the adiabatic basis because the energy, force, and state coupling can be evaluated directly by many electronic structure methods. However, although its straightforwardness is appealing, direct dynamics is expensive when combined with quantitatively accurate electronic structure theories. This generates interest in analytically fitted surfaces to cut the expense, but the cuspidal ridges of the potentials and the singularities and vector nature of the couplings at high-dimensional, nonsymmetry-determined intersections in the adiabatic representation make accurate fitting almost impossible. This motivates using diabatic representations, where the surfaces are smooth and the couplings are also smooth and-importantly-scalar. In a recent previous work, we have developed a method called diabatization by deep neural network (DDNN) that takes advantage of the smoothness and nonuniqueness of diabatic bases to obtain them by machine learning. The diabatic potential energy matrices (DPEMs) learned by the DDNN method yield not only diabatic potential energy surfaces (PESs) and couplings in an analytic form useful for dynamics calculations, but also adiabatic surfaces and couplings in the adiabatic representation can be calculated inexpensively from the transformation. In the present work, we show how to extend the DDNN method to produce good approximations to global permutationally invariant adiabatic PESs simultaneously with DPEMs. The extended method is called permutationally restrained DDNN.

The Born-Oppenheimer approximation assumes nuclear motion evolves on single, uncoupled potential energy surfaces, widely used to solve the time-independent Schrödinger equation for atomistic systems. However, for near-degenerate same-symmetry electronic states, avoided crossings in the potential energy curves occur and non-adiabatic couplings (NACs) become significant. In such cases, the adiabatic approximation is unsuitable for high-resolution spectroscopy. A unitary transformation to the diabatic representation can eliminate NACs, resulting in smooth molecular property curves that may cross. Computing this adiabatic-to-diabatic transformation (AtDT) is desirable but non-analytic for multi-state coupled systems, necessitating numerical solutions. It remains unclear if current methods yield numerically exact AtDTs ensuring rovibronic energy level equivalence between adiabatic and diabatic pictures. We demonstrate (for the first time) numerically exact equivalence of adiabatic and diabatic representations for -state diatomic molecules using abinitio data for , CH, and a model 10-state system. We show how the equivalence can be efficiently used to assess the importance of non-adiabatic effects and the impact of omitting them when computing rovibronic energies of diatomic molecules. The adiabatic and diabatic representations of the spectroscopic model, including all coupling terms, have been implemented in the diatomic code Duo.

The nonadiabatic transition probability 𝒫 between two electronic states is studied in the semiclassical approximation. Extensive use is made of the concept of global function introduced in the diabatic representation by Delos and Thorson (their t (s) function) and by ourselves in the adiabatic representation (the Massey function, T (S)). Consideration of the shapes of these global functions results in a classification of the different models which have been proposed earlier by Landau and Zener, by Rosen and Zener, by Delos and Thorson, by Nikitin, and by Bandrauk. The relationships which exist between all of these models are displayed. For all of them, the transition probability 𝒫 can be expressed as a simple analytic expression involving two parameters, ξ and ϑo. ξ is the Massey parameter. ϑo measures the initial decoupling, i.e., the component of the adiabatic states on the basis set of the diabatic states at the initial time (corresponding to the initial excitation in a photochemistry experiment). For a given value of ϑo, the different models predict relationships between 𝒫 and ξ that do not differ greatly, so that a kind of ’’universal function’’ relating 𝒫 and ξ can be defined. The previous considerations are applied to three specific examples of nonadiabatic coupling encountered in the case of H2, N2+ and CH2+. For each of them, the global functions t (s) and T (S) are calculated ab initio. The relationship between their shape and the nonadiabatic transition probability is discussed.

A method for fitting ab initio determined spin-orbit coupling interactions, in the Breit-Pauli approximation, based on quasidiabatic representations using neural network fits is reported. The algorithm generalizes our recently reported neural network approach for representing the dipole interaction. The S0, S1, and T1 states of formaldehyde are used as an example. First, the two singlet states S0 and S1 are diabatized with a modified Boys Localization diabatization method. Second, the spin-orbit coupling between singlet and triplet states is transformed to the diabatic representation. This removes the discontinuities in the adiabatic representation. The diabatized spin-orbit couplings are then fit with smooth neural network functions. The analytic representation of spin-orbit coupling interactions in a diabatic basis by neural networks will make accurate full-dimensional quantum dynamical treatment of both internal conversion and intersystem crossing possible, which will help us to gain better understanding of both processes.

An energy-based method is proposed for the diabatization of the OH(2Pi)+F(2P)-->O(3P)+HF(1Sigma+) reaction. It is demonstrated that the diabatic representation obtained is regularized, i.e., the residual derivative couplings do not present singularities at the conical intersections appearing along the reaction path. This method only requires the knowledge of the 1,2 3A" and 1 3A' eigenvalues and does not require any adjustable parameter. Thus, many convergence problems arising in other derivative-based diabatization methods are avoided, and the description of the configuration space along the reaction path is enormously simplified. Three-dimensional coupled diabatic energy surfaces are obtained by an interpolation procedure using approximately 4000 accurate ab initio points. The angular resolved photodetachment cross sections are obtained in the diabatic and adiabatic representations using a wave packet method. An excellent agreement is obtained with recent experimental data [D. M. Neumark, Phys. Chem. Chem. Phys. 7, 433 (2005)] for high electron kinetic energies where only the triplet electronic states contribute.

An analytical polynomial formulation of the exponential of fully symmetric matrices is proposed. This polynomial formula is derived using the Cayley - Hamilton theorem and the polynomial Euclidean division. The application field of these developments is time-dependent wavepacket calculations involving multiple potential energy surface crossing. The use of this polynomial formulation in time-dependent calculations is directly connected to the choice of the split operator, proposed by Feit and Fleck (1982 J. Comput. Phys. 47 412) as a numerical propagation scheme. We show in this paper that this analytical polynomial formula is numerically more efficient than the scheme proposed by Almeida and Metiu (1988 Chem. Phys. Lett. 146 47) where the exponential of the kinetic operator is evaluated in the momentum and diabatic representation and the exponential of the potential operator is evaluated in the coordinate and adiabatic representation. The use of an optical potential ( negative imaginary potential) to absorb the wavepacket near the edges of the grid point is quite simple to implement as it appears as a multiplicative factor in front of the polynomial expression. Our scheme is exemplified on the photodissociation of a diatomic molecule involving three potential energy curves.

Exact nonadiabatic quantum evolution preserves many geometric properties of the molecular Hilbert space. In the first paper of this series ["Paper I," S. Choi and J. Vaníček, J. Chem. Phys. 150, 204112 (2019)], we presented numerical integrators of arbitrary-order of accuracy that preserve these geometric properties exactly even in the adiabatic representation, in which the molecular Hamiltonian is not separable into kinetic and potential terms. Here, we focus on the separable Hamiltonian in diabatic representation, where the split-operator algorithm provides a popular alternative because it is explicit and easy to implement, while preserving most geometric invariants. Whereas the standard version has only second-order accuracy, we implemented, in an automated fashion, its recursive symmetric compositions, using the same schemes as in Paper I, and obtained integrators of arbitrary even order that still preserve the geometric properties exactly. Because the automatically generated splitting coefficients are redundant, we reduce the computational cost by pruning these coefficients and lower memory requirements by identifying unique coefficients. The order of convergence and preservation of geometric properties are justified analytically and confirmed numerically on a one-dimensional two-surface model of NaI and a three-dimensional three-surface model of pyrazine. As for efficiency, we find that to reach a convergence error of 10-10, a 600-fold speedup in the case of NaI and a 900-fold speedup in the case of pyrazine are obtained with the higher-order compositions instead of the second-order split-operator algorithm. The pyrazine results suggest that the efficiency gain survives in higher dimensions.

We present a survey of our work about applications of symbol calculus to quantum mechanics. Symbol calculus embodies the rules of algebra for the linear partial differential operators with variable coefficients, when such operators are expressed as functions of the non-commuting quantities xj and ~/~xj (j=l ..... n). In quantum mechanics the symbol calculus generates S~-C~sica£ expansions, i.e. perturbative series in powers of Planck's constant ~, that are both numerically useful and mathematically interesting. As the complexity of such expansions increases very rapidly with the order, algebraic computer systems have proven most useful in their study [l] . We present here a few of our results, obtained with the extensive use of the formula manipulation language REDUCE (except when specified) : in particular we have found new algorithms to solve the stationary SchrBdinger equation formally in powers of ~. Basic facts about quantum mechanics and semi-classical methods are found in [3] any textbook such as . More precise references, about symbol calculus for instance, are listed in our earlier articles [4j.~ ~

We present a comprehensive comparison of the following mixed quantum-classical methods for calculating electronic transition rates: (1) nonequilibrium Fermi's golden rule, (2) mixed quantum-classical Liouville method, (3) mean-field (Ehrenfest) mixed quantum-classical method, and (4) fewest switches surface-hopping method (in diabatic and adiabatic representations). The comparison is performed on the Garg-Onuchic-Ambegaokar benchmark charge-transfer model, over a broad range of temperatures and electronic coupling strengths, with different nonequilibrium initial states, in the normal and inverted regimes. Under weak to moderate electronic coupling, the nonequilibrium Fermi's golden rule rates are found to be in good agreement with the rates obtained via the mixed quantum-classical Liouville method that coincides with the fully quantum-mechanically exact results for the model system under study. Our results suggest that the nonequilibrium Fermi's golden rule can serve as an inexpensive yet accurate alternative to Ehrenfest and the fewest switches surface-hopping methods.

The semiclassical theory developed in the preceding papers of this series is applied to the calculation of cross sections for the reaction ${\mathrm{C}}^{6+}\ensuremath{-}\mathrm{H}(1s)\ensuremath{\rightarrow}{\mathrm{C}}^{5+}(n)+{\mathrm{H}}^{+}$, $n=3\ensuremath{-}6$, over the energy range 0.05-30.0 keV/amu. Approximate variationally optimized translational factors, which generalize those of Bates and McCarroll, are employed in both adiabatic and diabatic representations. Cross sections obtained for a given basis, using the method of Piacentini and Salin or that of Bates and McCarroll, are compared with those from the new theory. The sensitivity of the cross sections to the choice of translational factor is explored and problems connected with the high-energy optimization of the translational factors are discussed. It is also shown that at the lowest energy the use of straight-line trajectory is incorrect. A new method to generate and store molecular-basis matrix elements with translational factors was developed to provide the essential input to the scattering program.

Exciton dissociation at donor–acceptor heterojunctions is one of the key processes that determine the energy conversion efficiency of organic solar cells. Here, we theoretically investigate the exciton dissociations at oligothiophene/C60 donor–acceptor heterojunctions by using the long-range corrected time-dependent density functional theory and quantum dynamics calculations. We analyze the absorption spectra, electronic structures in adiabatic and diabatic representations, vibronic coupling, and dynamics of the exciton dissociations. The coupling strength and relative energy between the exciton and charge transfer (CT) states depend sensitively on the donor–acceptor distance and the stacking structure, which in turn affects the absorption spectra and the dynamics of exciton dissociations. Our quantum dynamics calculations exhibit ultrafast exciton dissociations owing to the strong coupling between the exciton and CT states, as observed in the time-resolved spectroscopies of the donor–acceptor interfaces of conjugated polymers and fullerenes.

A theoretical description of multi-channel fragmentation of the Na 3+ cluster-ion in collision with a He target

Semiempirical potential energy surfaces for Br–HBr are constructed, based on empirical Kr–HBr, Kr–Br and Kr–Kr potentials. The electrostatic forces due to the interaction of the quadrupole of the Br atom with the multipoles of the HBr molecule are included, and make a substantial contribution to the well depths. The potentials are presented in diabatic and adiabatic representations, with and without spin–orbit coupling. When spin–orbit coupling is included, the adiabatic well depth is 342 cm−1, at a linear Br–H–Br configuration. Fully coupled bound-state calculations are carried out, and the ground state is found to be bound by 269 cm−1. If the complex can be observed spectroscopically, it will provide a sensitive probe of the potential energy surfaces in the entrance and exit valleys of the Br+HBr reaction. In addition, the complex is likely to be important as a product of photodissociation of HBr in HBr dimer.