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Duo: a general program for calculating spectra of diatomic molecules

Sergei N. Yurchenko, Lorenzo Lodi, Jonathan Tennyson, Andrey V. Stolyarov

arXiv:1601.06531v2physics.comp-phphysics.chem-ph

TL;DR

Duo addresses the lack of a general program for diatomic molecules with coupled electronic states. It solves direct and inverse nuclear-motion problems, computes spectra and line lists, and supports arbitrary couplings between electronic states.

  • Problem

    Complex diatomic molecules require coupled-state treatments, but no general program was available for solving this problem.

  • Method

    Duo solves direct and inverse nuclear-motion problems using electronic-state couplings, including spin-orbit, electronic-rotational, spin-rotational, and spin-spin interactions.

  • Results

    Duo computes rotational, ro-vibrational, and rovibronic spectra, line positions, line intensities, and line lists for general diatomic molecules.

  • Takeaways & Limitations

    Duo provides a flexible framework used within ExoMol to study multiple diatomic species and is also applicable to ultracold diatomic molecules.

  • Takeaways & Limitations

    Duo does not currently provide explicit functionality for treating quasibound states.

Abstract

from arXiv · show

Duo is a general, user-friendly program for computing rotational, rovibrational and rovibronic spectra of diatomic molecules. Duo solves the Schrödinger equation for the motion of the nuclei not only for the simple case of uncoupled, isolated electronic states (typical for the ground state of closed-shell diatomics) but also for the general case of an arbitrary number and type of couplings between electronic states (typical for open-shell diatomics and excited states). Possible couplings include spin-orbit, angular momenta, spin-rotational and spin-spin. Corrections due to non-adiabatic effects can be accounted for by introducing the relevant couplings using so-called Born-Oppenheimer breakdown curves. Duo requires user-specified potential energy curves and, if relevant, dipole moment, coupling and correction curves. From these it computes energy levels, line positions and line intensities. Several analytic forms plus interpolation and extrapolation options are available for representation of the curves. Duo can refine potential energy and coupling curves to best reproduce reference data such as experimental energy levels or line positions. Duo is provided as a Fortran 2003 program and has been tested under a variety of operating systems.

Program summary

Duo solves diatomic nuclear-motion Schrödinger problems with arbitrary electronic-state couplings, producing energy levels and, when dipole data are supplied, line lists; it also supports empirical curve refinement.

  • Program capabilities: Duo solves the nuclear-motion Schrödinger equation for diatomic molecules with an arbitrary number and type of couplings between electronic states.The program first solves the uncoupled problem, then truncates the basis and solves the coupled problem.
  • Program capabilities: A line list can be computed when a dipole moment function is provided.The program summary identifies line-list generation as an available output.
  • Curve refinement: Duo can empirically refine potential-energy and other curves by fitting experimental energies or frequencies.The fitting capability applies when reference experimental data are provided.
  • Implementation: The program is written in Fortran 2003, is parallelized, and runs on Linux, Windows, and Mac OS.The typical memory requirement is below 10 MB, although the summary states that memory is case dependent.

1. Introduction

The paper presents Duo as a general solution to coupled electronic-state diatomic spectroscopy, extending the usual isolated-state treatment while also addressing direct and inverse spectral problems.

  • Background: For a 1Σ± diatomic state, rovibrational levels follow from a one-dimensional Schrödinger equation involving the potential curve, reduced mass, rotational quantum number J, and vibrational quantum number υ.The equation is described within the Born-Oppenheimer or adiabatic approximation.
  • Inverse problem: The inverse problem determines the potential V_state(r) that reproduces experimentally obtained energy levels E_υJ.The paper contrasts this task with the direct problem of solving for levels from a specified potential curve.
  • Motivation: Complex electronic structures require coupled-state treatment because interactions between electronic terms prevent treating each state in isolation.The authors motivate Duo by noting the lack of a general program for the coupled problem.
  • Contribution: Duo solves direct and inverse problems for general diatomics with arbitrary electronic-state couplings, including spin-orbit, electronic-rotational, spin-rotational, and spin-spin interactions.It also interpolates and extrapolates curves and calculates line positions and intensities.
  • Paper scope: The paper covers coupled nuclear-motion theory, line intensities and line lists, curve fitting, functional forms, and program structure.These topics correspond to the paper’s Sections 2 through 6.

2. Method of solution

Duo constructs and diagonalizes a coupled rovibronic Hamiltonian in a symmetry-adapted basis, starting from electronic-state curves and rotationless vibrational solutions.

  • Hamiltonian: The nuclear Hamiltonian combines electronic, vibrational, rotational, mass-polarization, and potential-energy contributions.The rotational angular momentum is R = J − L − S, linking total, orbital, and spin angular momenta.
  • Basis construction: Duo begins with clamped-nuclei electronic states described by spin S, orbital projection Λ, spin projection Σ, and an electronic-state label.The electronic wave functions depend parametrically on internuclear distance r.
  • Vibrational basis: For each electronic potential curve, Duo solves the J = 0 one-dimensional Schrödinger equation to obtain vibrational energies and wave functions.Vibrational states are indexed by energy ordering.
  • Basis construction: The rovibronic basis combines electronic, vibrational, and symmetric-top rotational functions, retaining states subject to angular-momentum constraints and user-selected vibrational cutoffs.The vibrational basis can be limited by an energy threshold or maximum vibrational quantum number υ_max.
  • Couplings: Rotational ladder operators generate S-uncoupling, L-uncoupling, and spin-electronic couplings between basis states with the corresponding selection rules.These terms connect states differing in spin and angular-momentum projections as specified by the coupling rules.
  • Curves and corrections: Electronic-structure matrix elements produce r-dependent curves for angular-momentum operators, while adiabatic corrections may be added to the Born-Oppenheimer potential.Duo can include supplementary operators such as spin-orbit coupling, dipole moments, and other corrections.
  • Diagonalization: Duo transforms the basis to states of defined parity, diagonalizes the two parity blocks, and obtains final rovibronic eigenvalues and eigenvectors.In the general case, J and parity are the only exact quantum-number labels; other assignments are approximate.
  • Output: The sample AlO output reports energies in cm−1 alongside exact and approximate quantum numbers and user-defined electronic-state labels.The exact labels include J, n, and parity; approximate labels include state, υ, Λ, S, Σ, and Ω.

2.1. Solution of the uncoupled vibrational problem

Duo primarily solves the uncoupled vibrational problem with sinc DVR, converting the Schrödinger equation into a matrix eigenvalue problem on a radial grid. The method converges rapidly, although grid truncation and floating-point summation limit accuracy in some cases.

  • sinc DVR: Duo’s sinc DVR discretizes the radial coordinate on a uniform grid and solves the resulting real symmetric matrix eigenvalue problem.The grid spans [rmin, rmax], with Δr=(rmax−rmin)/(Np−1), and the Hamiltonian has dimension Np×Np.
  • sinc DVR: Sinc DVR energies and wave functions converge approximately exponentially with the number of grid points.Figure 1 demonstrates this behavior for several Morse-potential energy levels.
  • Matrix elements: Duo evaluates vibrational-coordinate integrals by summing function values over the grid using the rectangle rule.For rapidly decaying nonsingular functions, this integration is exponentially accurate.
  • Matrix elements: Floating-point cancellation can prevent further improvement when matrix elements become very small, because significant digits are lost.Kahan compensated summation could reduce this error but is not implemented.
  • Finite differences: Finite-difference alternatives are available, but their eigenvalue error decreases as (Δr)^4 rather than exponentially with inverse grid spacing.The five-point finite-difference scheme produces a symmetric pentadiagonal matrix.

2.2. Levels lying close to dissociation

Near dissociation, loosely bound vibrational states extend to large internuclear distances, making uniform grids expensive to converge. Duo offers adaptive non-uniform mappings, but these are experimental and unavailable with sinc DVR.

  • Grid truncation: Grid endpoints must be chosen so that both potential values are much larger than the target vibrational energy.This ensures the wavefunction is negligible near the truncation boundaries.
  • Grid truncation: E14 requires rmax>90 Å and 1500 uniform-grid points to converge, compared with 6 Å and 120 points for E12 and 9 Å and 180 points for E13 at 10^-6 cm−1.The required rmax increases sharply as the state approaches dissociation.
  • Applications: Alkali-diatom excited states may require grids extending several tens or hundreds of Angstroms.Li2, Na2, and K2 are cited as examples of systems needing large rmax values.
  • Non-uniform grids: Duo implements adaptive analytical mappings for non-uniform grids, but they are experimental and can be used only with the less efficient five-point finite-difference method.They cannot currently be combined with sinc DVR.

2.3. States beyond the dissociation limit

Duo can represent rotationally induced quasibound states and identify narrow resonances through stabilization, but it does not yet provide explicit quasibound-state treatment. Rotational calculations use a J=0 vibrational basis whose size scales modestly with the target vibrational range.

  • Quasibound states: For J>0, the rotational potential can create a barrier above dissociation that supports orbiting or rotationally predissociating resonances.These are quasibound states trapped temporarily before dissociation.
  • Quasibound states: Duo lacks explicit quasibound-state functionality, although long-lived resonances can be identified with the stabilization method.The method tracks levels as rmax increases, where resonances remain relatively stable and undergo avoided crossings.
  • Rotational convergence: For rotationally excited levels through vmax, a J=0 vibrational basis of about 1.25×vmax+2 functions is recommended.For example, v=30 requires a basis of size 40.
  • Hamiltonian terms: Duo includes spin-orbit, spin-rotational, spin-spin, and lambda-doubling terms in the Hamiltonian.These additional interactions extend the model beyond the non-relativistic Hamiltonian.
  • Non-adiabatic corrections: Born-Oppenheimer breakdown curves modify vibrational and rotational operators to model small non-adiabatic shifts.They can also be interpreted as position-dependent vibrational and rotational masses.

2.7. Representation of the couplings

Duo represents couplings in a Hund’s case (a) Λ-representation, while accepting Cartesian input and transforming it by a unitary change of basis. Correct phase handling and matrix dimensions are important for reliable coupled-state calculations.

  • Λ-representation: Duo’s coupling matrix elements and transition dipoles are defined in the Hund’s case (a) Λ-representation, where Lz is diagonal with signed projection Λ.Appropriate electronic phase choices make coupling matrix elements real, although the electronic wavefunctions remain complex.
  • Representation transformation: Duo accepts Cartesian input and transforms it into the Λ-representation by diagonalizing the Cartesian Lz matrix.The transformation is expressed tensorially as A_Duo=C†AC.
  • Matrix elements: Coupling rules allow all non-zero Cartesian matrix elements to be reconstructed from a single non-zero reference element in suitable cases.The example uses a non-zero spin-orbit matrix element between selected electronic components.
  • Phase consistency: Ab initio off-diagonal couplings can acquire arbitrary sign changes with geometry, so their phases must be post-processed consistently.Relative phases between different matrix elements must be preserved, and transition dipole phase changes require the same convention.
  • Computational scale: The final Hamiltonian dimension scales with electronic states, retained vibrational functions, spin multiplicity, and J, while typical energy calculations take a small fraction of a second.Blocks reach roughly a thousand dimensions only in rather complicated cases.

3. Line intensities and line lists

Duo computes transition probabilities, Einstein coefficients, absorption intensities, and line lists from molecular-state and dipole-moment information. Its ExoMol-compatible files support temperature-dependent absorption and emission spectra.

  • Line intensities: Einstein A coefficients provide transition probabilities and can be used to compute radiative lifetimes and cooling functions.The coefficients are reported in s^-1.
  • Line list format: A line list catalogues transition frequencies and intensities in two ExoMol files: States and Transitions.The States file stores energies, degeneracies, rotational quantum numbers, and molecular-state labels; the Transitions file stores upper and lower state indexes and Einstein A coefficients, with wavenumbers supplied additionally.
  • Line list format: The States file includes energy term values, degeneracies, J, vibrational and electronic labels, parity, and angular-momentum projections.The listed projections include Λ, Σ, and Ω.
  • Line list format: The Transitions file uses upper and lower state counting numbers and an Einstein-A coefficient, enabling general-temperature absorption or emission simulations.The output examples use 27Al16O State and Transition files.

4. Inverse problem

Duo refines potential-energy and coupling curves by fitting calculated energies or frequencies to experimental reference data. The procedure supports weighted least squares, parameter selection, curve constraints, quantum-number matching, and morphing toward reference curves.

  • Inverse problem: Empirical refinement solves an inverse problem: finding potential-energy and coupling curves that reproduce observed energy levels or transition frequencies.The target data are typically extracted from experiment.
  • Optimization: The refinement is formulated as a nonlinear weighted least-squares problem over parameters defining the potential and coupling curves.Calculated energies or frequencies depend implicitly on the fitted parameters, while positive weights can reflect measurement uncertainties.
  • Optimization: Duo renormalizes input weights so that their sum is one and uses nonlinear conjugate gradients with LAPACK DGELSS by default.The built-in LINUR routine is available as an alternative, and individual curve parameters may be fixed or refined.
  • Constrained minimization: Reference-curve penalties constrain refined curves toward supplied curves, helping avoid unphysical behavior and underdetermined fits.Reference curves are typically obtained from ab initio calculations, with point and curve weights controlling their influence.
  • Level matching: Quantum numbers identify matching observed and calculated levels when refinement changes their order within a close-lying (J, τ) block.Duo uses J, parity, state, υ, and absolute values of Λ, Σ, and Ω for matching.
  • Curve morphing: Curve morphing scales an initial function by H(r) and is especially useful when experimental information is sparse.The initial function may be supplied analytically or as a spline interpolant.

5. Types of functional forms

Duo represents r-dependent potentials, dipole moments, and couplings with multiple analytical forms or numerical interpolation and extrapolation schemes. These choices include physically motivated long-range behavior and practical safeguards for sparse or limited-range data.

  • 5.1. Analytical representations: Duo provides parametrized analytical forms for r-dependent curves, including potentials and dipole moments, with equilibrium and long-range quantities available in the representations.The analytical-representation section defines T_e at the equilibrium geometry and includes long-range coefficients C_n.
  • 5.1. Analytical representations: Available analytical forms include Dunham, Taylor, Simons-Parr-Finlan, Murrell-Sorbie, Chebyshev, perturbed Morse, extended Morse, Morse long-range, and Šurkus-polynomial expansions.Damping functions can be added to Šurkus-polynomial expansions.
  • 5.1. Analytical representations: The perturbed and extended Morse forms generalize Morse potentials through additional coefficients, exponential tails, or distance-dependent exponents.The perturbed Morse asymptote is A_e + Σ_i a_i, while the extended form uses a distance-dependent exponent coefficient.
  • 5.1. Analytical representations: The Morse long-range form and Šurkus expansions provide reduced-variable constructions designed to represent asymptotic potential behavior.The Morse long-range form uses long-range potential and exponent-coefficient functions; Šurkus forms use a variable with a specified asymptote.
  • 5.2. Numerical representations: Numerical curves can be supplied as data points and automatically interpolated or extrapolated, using natural cubic or quintic splines within the specified range.Transformed-set interpolation, such as (r_i, r^2F_i) or (1/r_i, F_i), is not implemented.
  • 5.2. Numerical representations: Short-range extrapolation uses A + B/r, Ar + Br^2, or A + Br, selected by curve type; long-range extrapolation uses A + B/r^6, A + Br, or A/r^2 + B/r^3.The default forms differ for potentials, transition dipoles, angular-momentum curves, and other curves.
  • 5.2. Numerical representations: Extrapolation can introduce first-derivative discontinuities and incorrect dipole asymptotes, so analytical long-range forms are recommended for sensitive or loosely bound states.The dipole limits cited require powers m = 3 or 5 at short range and m = 4 or 7 at long range, unlike the implemented forms.

6. Program inputs and structure

Duo is configured through a plain-text input file specifying curves, Hamiltonian terms, dipoles, solution methods, and convergence settings. Its modular sections activate couplings, corrections, or tasks without code changes, and the distribution includes runnable examples.

  • Program inputs: Duo uses a plain-text input file to specify Hamiltonian terms, potential and coupling curves, dipole moments, solution methods, and convergence thresholds.Keywords and options are handled by Stone’s input parser.
  • Program structure: Adding an input-file section switches on different couplings, corrections, or tasks without altering the program code.This provides a modular way to configure calculations.
  • Program structure: The distribution includes source code, a manual, compiler makefiles, and four examples with sample inputs and outputs.The examples cover a Morse potential, PS and PH fitting cases, and a six-electronic-state ScH spectrum.
  • Program structure: The examples restrict rotational excitation to low J values so they run quickly.This condition applies to the supplied demonstration cases.

7. Conclusion

Duo is a flexible program for solving nuclear motion in diatomic molecules with non-adiabatically coupled electronic states. It computes multiple types of spectra from ab initio or semi-empirical inputs, which can also be fitted to experimental data.

  • Duo solves the nuclear motion problem for diatomic molecules with non-adiabatically coupled electronic states.
  • The code simulates pure rotational, rovibrational, and rovibronic spectra using ab initio or semi-empirical data.
  • Semi-empirical inputs can be obtained within Duo by fitting to experimental data.
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