Duo: A general program for calculating spectra of diatomic molecules

Abstract Duo is a general, user-friendly program for computing rotational, rovibrational and rovibronic spectra of diatomic molecules. Duo solves the Schrodinger 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 Program title: Duo Catalogue identifier: AEZJ_v1_0 rogram summary URL: http://cpc.cs.qub.ac.uk/summaries/AEZJ_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 187443 No. of bytes in distributed program, including test data, etc.: 6968371 Distribution format: tar.gz Programming language: Fortran 2003. Computer: Any personal computer. Operating system: Linux, Windows, Mac OS. Has the code been vectorized or parallelized?: Parallelized RAM: Case dependent, typically 10 MB Classification: 4.3, 4.9, 16.2, 16.3. Nature of problem: Solving the Schrodinger equation for the nuclear motion of a diatomic molecule with an arbitrary number and type of couplings between electronic states. Solution method: Solution of the uncoupled problem first, then basis set truncation and solution of the coupled problem. A line list can be computed if a dipole moment function is provided. The potential energy and other curves can be empirically refined by fitting to experimental energies or frequencies, when provided. Restrictions: The current version is restricted to bound states of the system. Unusual features: User supplied curves for all objects (potential energies, spin–orbit and other couplings, dipole moment etc.) as analytic functions or on a grid is a program requirement. Running time: Case dependent. The test runs provided take seconds or a few minutes on a normal PC.

[1]  António J. C. Varandas,et al.  Analytical potentials for triatomic molecules from spectroscopic data: IV. Application to linear molecules , 1978 .

[2]  B. Sutcliffe The separation of electronic and nuclear motion in the diatomic molecule , 2007 .

[3]  J. A. Coxon,et al.  Application of direct potential fitting to line position data for the X1Σg+ and A1Σu+ states of Li2 , 2006 .

[4]  Gabriel G. Balint-Kurti,et al.  The Fourier grid Hamiltonian method for bound state eigenvalues and eigenfunctions , 1989 .

[5]  Hirao,et al.  FTIR Emission Spectra, Molecular Constants, and Potential Curve of Ground State GeO. , 1999, Journal of molecular spectroscopy.

[6]  J. Huffaker Diatomic molecules as perturbed Morse oscillators. VI. High‐precision eigenfunctions. , 1981 .

[7]  B. R. Johnson The renormalized Numerov method applied to calculating bound states of the coupled‐channel Schroedinger equation , 1978 .

[8]  J. Tennyson,et al.  Hybrid variational–perturbation method for calculating ro-vibrational energy levels of polyatomic molecules , 2014, 1411.6098.

[9]  J. Tennyson,et al.  Towards efficient refinement of molecular potential energy surfaces: Ammonia as a case study , 2011 .

[10]  M. Child,et al.  Analytical approximations for adiabatic and non-adiabatic matrix elements of homonuclear diatomic Rydberg states. Application to the singlet p-complex of the hydrogen molecule , 1997 .

[11]  M. Hada,et al.  Ab initio study of ground and excited states of 6Li40Ca and 6Li88Sr molecules. , 2013, The Journal of chemical physics.

[12]  S. Hancocks Potential energy , 2016, BDJ.

[13]  J. Simons Resonance state lifetimes from stabilization graphs , 1981 .

[14]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[15]  J. Goodisman Dipole‐Moment Function for Diatomic Molecules , 1963 .

[16]  A. N. Tikhonov,et al.  Solutions of ill-posed problems , 1977 .

[17]  P. Dwivedi,et al.  Diatomic molecules as perturbed Morse oscillators. III. Perturbed eigenfunctions and Franck–Condon factors , 1977 .

[18]  A. Zaitsevskii,et al.  Direct deperturbation analysis of the AΠ2∼BΣ+2 complex of LiAr7,6 isotopomers , 2005 .

[19]  J. Ogilvie,et al.  An Effective Hamiltonian to Treat Adiabatic and Nonadiabatic Effects in the Rotational and Vibrational Spectra of Diatomic Molecules , 2007 .

[20]  John Lund,et al.  A Sine-Collocation Method for the Computation of the Eigenvalues of the Radial Schrödinger Equation , 1984 .

[21]  John M. Blatt,et al.  Practical points concerning the solution of the Schrödinger equation , 1967 .

[22]  Hajigeorgiou,et al.  Experimental Born-Oppenheimer Potential for the X1Sigma+ Ground State of HeH+: Comparison with the Ab Initio Potential. , 1999, Journal of molecular spectroscopy.

[23]  D. Tannor,et al.  Introduction to Quantum Mechanics: A Time-Dependent Perspective , 2006 .

[24]  Jacek Karwowski,et al.  Inverse problems in quantum chemistry , 2009 .

[25]  J. Tennyson,et al.  The ab initio calculation of spectra of open shell diatomic molecules , 2016, 1605.02301.

[26]  J. W. Cooley,et al.  An improved eigenvalue corrector formula for solving the Schrödinger equation for central fields , 1961 .

[27]  J. Huffaker Diatomic molecules as perturbed Morse oscillators. I. Energy levels , 1976 .

[28]  R. T. Pack,et al.  Separation of Rotational Coordinates from the N‐Electron Diatomic Schrödinger Equation , 1968 .

[29]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[30]  W. Kutzelnigg The adiabatic approximation I. The physical background of the Born-Handy ansatz , 1997 .

[31]  B. R. Johnson New numerical methods applied to solving the one‐dimensional eigenvalue problem , 1977 .

[32]  Markus Reiher,et al.  Inverse quantum chemistry: Concepts and strategies for rational compound design , 2014, 1401.1512.

[33]  Howard S. Taylor,et al.  Stabilization Method of Calculating Resonance Energies: Model Problem , 1970 .

[35]  J. Huffaker Diatomic molecules as perturbed Morse oscillators. V. Centrifugal distortion , 1980 .

[36]  M. Child,et al.  Analog of the Hellmann-Feynman theorem in multichannel quantum-defect theory , 2001 .

[37]  R. Herman,et al.  Theory of energy shifts associated with deviations from Born-Oppenheimer behavior in 1Σ-state diatomic molecules , 1966 .

[38]  E. Colbourn,et al.  The values of in diatomic molecules: implications for adiabatic and molecular fine structure calculations , 1979 .

[39]  D. Schwenke A unified derivation of Hamiltonian and optical transition matrix elements for open shell diatomic and polyatomic molecules using transformation tools of modern quantum mechanics. , 2015, The Journal of chemical physics.

[40]  Martin Schütz,et al.  Molpro: a general‐purpose quantum chemistry program package , 2012 .

[41]  R. Nesbet,et al.  Electronic Structure of C2 , 1966 .

[42]  Barry Simon,et al.  Schrödinger operators in the twentieth century , 2000 .

[43]  P. Bunker,et al.  The breakdown of the Born-Oppenheimer approximation: the effective vibration-rotation hamiltonian for a diatomic molecule , 1977 .

[44]  J. Tennyson Vibration–rotation transition dipoles from first principles , 2014 .

[45]  Peng Zhang,et al.  Structure and spectroscopy of ground and excited states of LiYb. , 2010, The Journal of chemical physics.

[46]  V. Mandelshtam,et al.  Calculation of the density of resonance states using the stabilization method. , 1993, Physical review letters.

[47]  F. Gilmore,et al.  Angular momentum coupling, potential curves and radiative selection rules for heavy diatomic molecules, with particular reference to Kr2 and Kr2+ , 1975 .

[48]  T. Yamabe,et al.  Electronic structure of C−70 , 1993 .

[49]  R. Field,et al.  The spectra and dynamics of diatomic molecules , 2004 .

[50]  J. K. Cashion,et al.  Testing of Diatomic Potential‐Energy Functions by Numerical Methods , 1963 .

[51]  H. Korsch,et al.  Milne's differential equation and numerical solutions of the Schrodinger equation. I. Bound-state energies for single- and double-minimum potentials , 1981 .

[52]  Nicholas J. Higham,et al.  The Accuracy of Floating Point Summation , 1993, SIAM J. Sci. Comput..

[54]  Warren M. Kosman,et al.  Inverse perturbation analysis: Improving the accuracy of potential energy curves☆ , 1975 .

[55]  B. Shore Solving the radial Schrödinger equation by using cubic‐spline basis functions , 1973 .

[56]  P. Julienne,et al.  Avoided crossings between bound states of ultracold Cesium dimers , 2008, 0806.2583.

[57]  N. Kuz'menko,et al.  Solution of the radial Schrödinger equation by a modified “shooting” method , 1987 .

[58]  R. Bernstein,et al.  DISSOCIATION ENERGY AND VIBRATIONAL TERMS OF GROUND-STATE (X ¹$Sigma$/ sub g/$sup +$) HYDROGEN. , 1968 .

[59]  J. Tennyson,et al.  Radiative lifetimes and cooling functions for astrophysically important molecules , 2016, 1601.07997.

[60]  Calvin Jary,et al.  An accurate analytic potential function for ground-state N2 from a direct-potential-fit analysis of spectroscopic data. , 2006, The Journal of chemical physics.

[61]  G. Herzberg Molecular spectra and molecular structure. Vol.1: Spectra of diatomic molecules , 1950 .

[62]  Lloyd N. Trefethen,et al.  The Exponentially Convergent Trapezoidal Rule , 2014, SIAM Rev..

[63]  E. Tiemann,et al.  The X(1)Σg(+) ground state of Mg2 studied by Fourier-transform spectroscopy. , 2013, The Journal of chemical physics.

[64]  P. Dwivedi,et al.  Diatomic molecules as perturbed Morse oscillators. IV. Franck-Condon factors for very high J , 1978 .

[65]  Carl C. Haugen,et al.  Long-range damping functions improve the short-range behaviour of ‘MLR’ potential energy functions , 2011 .

[66]  A. J. Merer,et al.  Λ-Type doubling parameters for molecules in Δ electronic states , 1987 .

[67]  Vargas,et al.  Strongly convergent method to solve one-dimensional quantum problems. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[68]  G. Johnson The Schrödinger equation , 1998 .

[69]  Zimei Rong,et al.  Calculation of Displacement Matrix Elements for Morse Oscillators , 2003, International Conference on Computational Science.

[70]  A. D. Smith,et al.  Uniform semiclassical calculation of resonance energies and widths near a barrier maximum , 1981 .

[71]  J. Bowman,et al.  Perturbative inversion of the HOCl potential energy surface via singular value decomposition , 1999 .

[72]  Deyin Zhao,et al.  A Mathematica program for the two-step twelfth-order method with multi-derivative for the numerical solution of a one-dimensional Schrödinger equation , 2004, Comput. Phys. Commun..

[73]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[74]  C. Hill,et al.  ExoMol molecular line lists - XIII: The spectrum of CaO , 2015, 1512.08987.

[75]  P. Hajigeorgiou,et al.  Born–Oppenheimer breakdown in the ground state of carbon monoxide: A direct reduction of spectroscopic line positions to analytical radial Hamiltonian operators , 1993 .

[76]  N. Handy,et al.  The adiabatic approximation , 1996 .

[77]  Masatoshi Sekine,et al.  Accurate basis set by the CIP method for the solutions of the Schrödinger equation , 2004 .

[78]  B. Lindberg A new efficient method for calculation of energy eigenvalues and eigenstates of the one‐dimensional Schrödinger equation , 1988 .

[79]  R. L. Roy,et al.  Representing Born -Oppenheimer breakdown radial correction functions for diatomic molecules , 2002 .

[80]  Jenning Y. Seto,et al.  A Computer Program for Fitting Diatomic Molecule Spectral Data to Potential Energy Functions , 2006 .

[81]  I. Røeggen The inversion eigenvalues of non-σ states of diatomic molecules, expressed in terms of quantum numbers , 1971 .

[82]  Laurence S. Rothman,et al.  Einstein A-coefficients and statistical weights for molecular absorption transitions in the HITRAN database , 2006 .

[84]  S. Sauer,et al.  The Rotational g Factor of Diatomic Molecules in State 1Σ+ or 0+ , 2007 .

[85]  J. Hutson,et al.  A new approach to perturbation theory for breakdown of the Born-Oppenheimer approximation , 1980 .

[86]  R. L. Roy,et al.  LEVEL: A computer program for solving the radial Schrödinger equation for bound and quasibound levels , 2017 .

[87]  W. Liu,et al.  Energies and widths of quasibound levels (orbiting resonances) for spherical potentials , 1978 .

[88]  R. Guardiola,et al.  On the numerical integration of the Schrödinger equation in the finite-difference schemes , 1982 .

[89]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[90]  Robert A. van de Geijn,et al.  Restructuring the Tridiagonal and Bidiagonal QR Algorithms for Performance , 2014, ACM Trans. Math. Softw..

[91]  Sergei N. Yurchenko,et al.  ExoMol: molecular line lists for exoplanet and other atmospheres , 2012 .

[92]  Markus Meuwly,et al.  Morphing ab initio potentials: A systematic study of Ne–HF , 1999 .

[93]  R. Islampour,et al.  An extensive study of transformation of the diatomics Hamiltonian operator from laboratory- to body-fixed frame , 2015 .

[94]  J. Michael Finlan,et al.  New alternative to the Dunham potential for diatomic molecules , 1973 .

[95]  C. Amiot,et al.  Analysis of strongly coupled electronic states in diatomic molecules: Low-lying excited states of RbCs , 2003 .

[96]  Uwe V. Riss,et al.  Calculation of resonance energies and widths using the complex absorbing potential method , 1993 .

[97]  H. Katô Energy Levels and Line Intensities of Diatomic Molecules. Application to Alkali Metal Molecules , 1993 .

[98]  J. Huffaker Diatomic molecules as perturbed Morse oscillators. II. Extension to higher‐order parameters , 1976 .

[99]  Michael W. Schmidt,et al.  Spin-orbit coupling in molecules: Chemistry beyond the adiabatic approximation , 2003 .

[100]  J. Tennyson,et al.  The calculated rovibronic spectrum of scandium hydride, ScH , 2015, 1504.04051.

[101]  R. Bernstein,et al.  Dissociation Energy and Long‐Range Potential of Diatomic Molecules from Vibrational Spacings of Higher Levels , 1970 .

[102]  E. Sidky,et al.  Phase-amplitude method for calculating resonance energies and widths for one-dimensional potentials , 1999 .

[103]  R. Lefebvre,et al.  The use of the equivalence between boxing and scaling to determine resonance energies , 1986 .

[104]  J. Gallas Some matrix elements for morse oscillators , 1980 .

[105]  D. L. Cooper,et al.  Spin-orbit coupling in molecules , 1981 .

[106]  J. Watson The inversion of diatomic Born–Oppenheimer-breakdown corrections , 2004 .

[107]  F. Martín Ionization and dissociation using B-splines: photoionization of the hydrogen molecule , 1999 .

[108]  P. Bunker The electronic isotope shift in diatomic molecules and the partial breakdown of the Born-Oppenheimer approximation , 1968 .

[109]  H. Ishikawa,et al.  An accurate method for numerical calculations in quantum mechanics , 2002 .

[110]  D. Colbert,et al.  A novel discrete variable representation for quantum mechanical reactive scattering via the S-matrix Kohn method , 1992 .

[111]  Sandalphon,et al.  Improved molecular parameters for the Ballik-Ramsay system of diatomic carbon (b 3 Σg − → a 3 Πu) , 1988 .

[112]  John G. Herriot,et al.  Algorithm 507: Procedures for Quintic Natural Spline Interpolation [E1] , 1976, TOMS.

[113]  A. Császár,et al.  Grid-based empirical improvement of molecular potential energy surfaces. , 2014, The journal of physical chemistry. A.

[114]  W. Press,et al.  Numerical Recipes: The Art of Scientific Computing , 1987 .

[115]  C. Hill,et al.  Study of the electronic and rovibronic structure of the X ²Σ⁺, A ²Π, and B ²Σ⁺ states of AlO. , 2014, The Journal of chemical physics.

[116]  A. Bolotin,et al.  The generalized potential energy function for diatomic molecules , 1984 .

[117]  Ahmed F. Al-Refaie,et al.  The ExoMol database: Molecular line lists for exoplanet and other hot atmospheres , 2016, 1603.05890.

[118]  J. Tennyson,et al.  ExoMol: molecular line lists for exoplanet and other atmospheres , 2012, 1204.0124.

[119]  R. L. Roy,et al.  Adaptive analytical mapping procedure for efficiently solving the radial Schrödinger equation , 2008 .

[120]  A. Pashov,et al.  Construction of potential curves for diatomic molecular states by the IPA method , 2000 .

[121]  John A. Coxon,et al.  Direct potential fit analysis of the X 1Σ+ ground state of CO , 2004 .

[122]  R. Field,et al.  New spectroscopic data, spin-orbit functions, and global analysis of data on the AΣu+1 and bΠu3 states of Na2 , 2007 .

[123]  R. L. Roy,et al.  Shape Resonances and Rotationally Predissociating Levels: The Atomic Collision Time‐Delay Functions and Quasibound Level Properties of H2(X 1Σg+) , 1971 .

[124]  A. Stolyarov,et al.  The phase formalism for the one-dimensional eigenvalue problem and its relation with the quantum Bohr-Sommerfeld rule , 1990 .

[125]  S. Walji,et al.  Dissociation energies and potential energy functions for the ground X (1)Σ(+) and "avoided-crossing" A (1)Σ(+) states of NaH. , 2015, The Journal of chemical physics.

[126]  R. Lefebvre Box quantization and resonance determination: the multichannel case , 1985 .

[127]  I. Gordon,et al.  Peculiarities of high-overtone transition probabilities in carbon monoxide revealed by high-precision calculation. , 2015, The Journal of chemical physics.

[128]  R. L. Roy,et al.  A new potential function form incorporating extended long-range behaviour: application to ground-state Ca2 , 2007 .

[129]  A. M. Shaw,et al.  Microwave spectroscopy and interaction potential of the long‐range He⋯Kr+ ion: An example of Hund’s case (e) , 1996 .

[130]  S. Falke,et al.  The AΣu+1 state of K2 up to the dissociation limit , 2006 .

[131]  James Demmel,et al.  Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers , 2008, SIAM J. Sci. Comput..

[132]  J. Komasa,et al.  Nonadiabatic corrections to the wave function and energy. , 2007, The Journal of chemical physics.

[133]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[134]  J. L. Dunham The Energy Levels of a Rotating Vibrator , 1932 .

[135]  R. Ferber,et al.  Fourier transform spectroscopy and direct potential fit of a shelflike state: application to E(4)1Σ(+) KCs. , 2011, The Journal of chemical physics.

[136]  Gang Li,et al.  EINSTEIN A COEFFICIENTS AND OSCILLATOR STRENGTHS FOR THE A2Π–X2Σ+ (RED) AND B2Σ+–X2Σ+ (VIOLET) SYSTEMS AND ROVIBRATIONAL TRANSITIONS IN THE X2Σ+ STATE OF CN , 2014 .

[137]  A. Zaitsevskii,et al.  High resolution spectroscopy and channel-coupling treatment of the A 1Σ+–b 3Π complex of NaRb , 2002 .

[138]  P. Jensen,et al.  Potential parameters of PH3 obtained by simultaneous fitting of ab initio data and experimental vibrational band origins , 2003 .

[139]  R. L. Roy,et al.  Potential energy, Λ doubling and Born–Oppenheimer breakdown functions for the B 1Πu “barrier” state of Li2 , 2003 .

[140]  Anthony J. Merer,et al.  Lambda-type doubling parameters for molecules in Π electronic states of triplet and higher multiplicity , 1979 .

[141]  Howard A. Levine Review: A. N. Tikhonov and V. Y. Arsenin, Solutions of ill posed problems , 1979 .

[142]  M. Viant,et al.  Microwave spectroscopy and interaction potential of the long‐range He...Ar+ ion , 1995 .

[143]  J. Poll,et al.  ON THE VIBRATIONAL FREQUENCIES OF THE HYDROGEN MOLECULE , 1966 .

[144]  Carl C. Haugen,et al.  Direct-potential-fit analyses yield improved empirical potentials for the ground X (1)Σ(+)(g) state of Be2. , 2014, The Journal of chemical physics.

[145]  L. Veseth Spin-orbit and spin-other-orbit interaction in diatomic molecules , 1970 .

[146]  J. Hutson Coupled channel methods for solving the bound-state Schrödinger equation , 1994 .

[147]  R. L. Roy,et al.  Algebraic vs. numerical methods for analysing diatomic spectral data: a resolution of discrepancies , 2004 .

[148]  R. L. Roy,et al.  Improved Parameterization for Combined Isotopomer Analysis of Diatomic Spectra and Its Application to HF and DF. , 1999 .

[149]  B. V. Noumerov A Method of Extrapolation of Perturbations , 1924 .