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MOLSCAT: a program for non-reactive quantum scattering calculations on atomic and molecular collisions

Jeremy M. Hutson, C. Ruth Le Sueur

arXiv:1811.09584v2physics.chem-phcond-mat.quant-gasphysics.atom-ph

TL;DR

Nonreactive atom and molecule collisions require scattering observables across applications including cross sections, rate coefficients, spectroscopy, transport, and ultracold resonance properties. MOLSCAT constructs coupled-channel equations and propagates wavefunctions or log-derivative matrices from short range to asymptotic distances to extract S matrices. MOLSCAT provides low-energy scattering lengths and effective ranges, supports non-eigenfunction bases and user-specified couplings, and characterizes resonances as functions of energy or external fields.

  • Problem

    Nonreactive atom and molecule collisions require scattering observables across applications including cross sections, rate coefficients, spectroscopy, transport, and ultracold resonance properties.

    Source: §1 · p. 2

  • Method

    MOLSCAT constructs coupled-channel equations and propagates wavefunctions or log-derivative matrices from short range to asymptotic distances to extract S matrices.

    Source: PROGRAM SUMMARY · p. 2

  • Results

    MOLSCAT provides low-energy scattering lengths and effective ranges, supports non-eigenfunction bases and user-specified couplings, and characterizes resonances as functions of energy or external fields.

    Source: PROGRAM SUMMARY · p. 2

  • Takeaways & Limitations

    MOLSCAT supports scattering calculations for built-in and user-specified coupling cases, including low-energy properties and field-dependent resonances.

    Source: PROGRAM SUMMARY · p. 2

    The last version of MOLSCAT produced by Green and Hutson in collaboration was version 14 in 1994, and there has been no fully documented publication since.

    Source: §7 · p. 14

Abstract

from arXiv · show

MOLSCAT is a general-purpose program for quantum-mechanical calculations on nonreactive atom-atom, atom-molecule and molecule-molecule collisions. It constructs the coupled-channel equations of atomic and molecular scattering theory, and solves them by propagating the wavefunction or log-derivative matrix outwards from short range to the asymptotic region. It then applies scattering boundary conditions to extract the scattering matrix (S matrix). Built-in coupling cases include atom + rigid linear molecule, atom + vibrating diatom, atom + rigid symmetric top, atom + asymmetric or spherical top, rigid diatom + rigid diatom, rigid diatom + asymmetric top, and diffractive scattering of an atom from a crystal surface. Interaction potentials may be specified either in program input (for simple cases) or with user-supplied routines. For the built-in coupling cases, MOLSCAT can loop over partial wave (or total angular momentum) to calculate elastic and inelastic cross integral sections and spectroscopic line-shape cross sections. Post-processors are available to calculate differential cross sections, transport, relaxation and Senftleben-Beenakker cross sections, and to fit the parameters of scattering resonances. MOLSCAT also provides an interface for plug-in routines to specify coupling cases (Hamiltonians and basis sets) that are not built in; plug-in routines are supplied to handle collisions of a pair of alkali-metal atoms with hyperfine structure in an applied magnetic field. For low-energy scattering, MOLSCAT can calculate scattering lengths and effective ranges and can locate and characterize scattering resonances as a function of an external variable such as the magnetic field.

PROGRAM SUMMARY

MOLSCAT constructs and solves coupled-channel equations for nonreactive quantum scattering, propagating solutions from short range to asymptotic distances to obtain S matrices and scattering properties. It supports low-energy calculations, non-eigenfunction bases, user-specified couplings, external fields, and resonance characterization.

PROGRAM SUMMARY

  • MOLSCAT propagates coupled-channel solutions outward and applies scattering boundary conditions to obtain S matrices and scattering properties, including elastic, inelastic, and line-shape cross sections.The program handles nonreactive collisions between atoms and molecules.Source: PROGRAM SUMMARY · p. 2
  • For low-energy collisions, MOLSCAT propagates efficiently to very long range and calculates scattering lengths and effective ranges.Scattering lengths can be complex and energy-dependent when required.Source: PROGRAM SUMMARY · p. 2
  • MOLSCAT propagates in a primitive basis that need not diagonalize the long-range Hamiltonians, then transforms solutions to the asymptotic basis before extracting the S matrix.Long-range boundary conditions are applied after the transformation.Source: PROGRAM SUMMARY · p. 2
  • An interface lets users specify additional coupled equations arising from Hamiltonians and basis sets beyond the built-in cases.Source: PROGRAM SUMMARY · p. 2
  • The external-field interface supports multiple fields and scaling the interaction potential to investigate how scattering properties vary.Source: PROGRAM SUMMARY · p. 2
  • MOLSCAT locates and characterizes low-energy Feshbach resonances versus collision energy, external fields, or the potential scaling factor.Source: PROGRAM SUMMARY · p. 2

1. Introduction

Nonreactive collisions produce observables ranging from cross sections and rate coefficients to low-energy scattering lengths and resonance properties. MOLSCAT addresses these problems with coupled-channel equations, propagating radial channel solutions in a basis set while accounting for internal structure, interactions, and external fields.

1. Introduction

  • The Hamiltonian includes the particles' internal terms and their interaction, with spin and external-field dependence accommodated in both internal and interaction terms.The internal Hamiltonian can represent atoms, vibrating or rotating molecules, and particles with electron or nuclear spins.Source: §1 · pp. 2–3
  • Scattering calculations yield elastic and inelastic cross sections, thermal rate coefficients, relaxation and line-shape observables, transport properties, and low-energy scattering lengths and resonance properties.These applications span ordinary atomic and molecular collisions, spectroscopy, predissociating complexes, and ultracold collisions.Source: §1 · p. 2
  • MOLSCAT solves the coupled-channel Schrödinger equation by propagating the radial coordinate numerically and representing the other coordinates with a basis set.The wavefunction expansion is projected onto basis functions to obtain coupled differential equations for channel functions.Source: §1 · p. 3
  • The program propagates a wavefunction matrix and its derivative, or a log-derivative matrix, subject to short-range boundary conditions.The log-derivative matrix is Y(R) = Ψ′(R)[Ψ(R)]−1.Source: §1 · p. 3
  • Basis choices and interaction matrix elements depend on the physical system, while symmetry can divide the equations into blocks solved across total angular momentum and parity.In the absence of external fields, total angular momentum and total parity are conserved; the loops may be repurposed for other symmetries.Source: §1 · p. 3
  • MOLSCAT also handles external-field interactions, where total angular momentum is no longer a good quantum number.Source: §1 · p. 3

1.1. Matrix of the interaction potential

MOLSCAT builds interaction and internal-Hamiltonian matrices for propagation in a chosen basis. The following subsections describe the interaction-potential expansion and the treatment of non-diagonal internal or centrifugal Hamiltonians.

1.1. Matrix of the interaction potential

  • The interaction potential is expanded over internal coordinates into radial coefficients and coupling matrices, which are calculated once and reused during propagation.Radial coefficients may be supplied explicitly or generated by numerical integration over internal coordinates.Source: §1.1 · pp. 3–4

1.2. Matrices of the internal and centrifugal Hamiltonians

  • MOLSCAT permits bases in which the internal or centrifugal Hamiltonian is non-diagonal, expanding the internal Hamiltonian into terms with scalar coefficients that can represent external fields.At infinite separation, allowed energies are eigenvalues of the internal Hamiltonian, and separated-pair wavefunctions are simultaneous eigenvectors of the internal and centrifugal Hamiltonians.Source: §1.2 · p. 4

1.3. Boundary conditions at short range

MOLSCAT propagates coupled-channel solutions from short range and matches them to asymptotic channel functions to extract scattering matrices. It supports WKB-based short-range conditions, exceptions for special systems, and basis transformations when the propagated basis is not asymptotic.

1.3. Boundary conditions at short range

  • WKB-based short-range boundary conditions are more efficient than requiring the wavefunction to vanish, and the default for locally closed channels converges faster with respect to Rmin.Separate conditions can be imposed for open and locally closed channels; systems with an energetically accessible origin or a Fermi pseudopotential may require other conditions.Source: §1.3 · p. 4

1.4. The scattering matrix

  • At long range, MOLSCAT matches propagated wavefunctions or log-derivative matrices to analytic free-channel functions, then converts the resulting K matrix to the S matrix.Open channels use Ricatti-Bessel functions and closed channels use modified spherical Bessel functions.Source: §1.4 · p. 5
  • When both the internal Hamiltonian and angular-momentum operator are diagonal, the propagation and asymptotic matching channels are the same.Source: §1.4 · p. 5
  • When the propagated basis is not asymptotic, MOLSCAT transforms the wavefunction or log-derivative matrix into a basis that diagonalizes the internal Hamiltonian and end-over-end angular momentum.For degenerate or near-degenerate subspaces, additional commuting operators can help resolve the basis.Source: §1.4 · p. 5

1.5. Cross sections

MOLSCAT uses S matrices to calculate cross sections and locate resonances, and provides tools for low-energy scattering properties. Its energy reference can be set directly or tied to a scattering threshold or pair of monomer states.

1.5. Cross sections

  • MOLSCAT accumulates degeneracy-averaged state-to-state integral cross sections from S matrices; it also calculates line-shape cross sections for spectroscopic-line broadening, shifting, and mixing.Line-shape calculations use the same kinetic energy for the upper and lower transition states.Source: §1.5 · pp. 5–6
  • The S-matrix eigenphase sum lets MOLSCAT converge on far-off scattering resonances as a function of energy.Such resonances may produce sharp energy-dependent cross-section features and may be interpreted as predissociating van der Waals-complex states.Source: §1.5 · p. 6

1.6. Reference energy

  • A reference energy can be supplied directly or defined by a scattering threshold or pair of monomer states, including field-dependent states.Source: §1.6 · p. 6

1.7.2. Characterisation of zero-energy Feshbach resonances

  • MOLSCAT can converge on and characterize zero-energy Feshbach resonances in scattering length versus external field, using different parameters for elastic and inelastic resonant behavior.Elastic resonances have a simple pole characterized by position, width, and background scattering length; inelastic behavior requires additional parameters.Source: §1.7.2 · p. 7

1.7.3. Effective range

  • MOLSCAT calculates the effective range using both low-energy expansions after the first energy, avoiding use of the expansion that is numerically unstable near a pole or zero crossing.Source: §1.7.3 · p. 7

1.8. Scattering wavefunctions

MOLSCAT can calculate multichannel scattering wavefunctions and retains an infinite-order sudden (IOS) implementation for cross sections. The IOS code uses fixed-orientation S matrices and does not provide low-energy features such as scattering lengths.

1.8. Scattering wavefunctions

  • MOLSCAT can calculate a multichannel wavefunction that is incoming in one channel and outgoing in all channels.The method generalizes that of Hutson and Thornley.Source: §1.8 · p. 7

1.9. Infinite-order sudden approximation

  • The IOS implementation computes degeneracy-averaged and line-shape cross sections from fixed-orientation S matrices, but does not provide low-energy features such as scattering lengths.Its dynamical factors are numerical integrals over those S matrices, while spectroscopic coefficients encode rotor levels and angular-momentum coupling; the code is retained for backwards compatibility.Source: §1.9 · p. 7

2. Systems handled

MOLSCAT supports close-coupling calculations for multiple atom and molecule pairings, plus plug-in coupling cases. It also implements dynamical approximations that trade accuracy for computational savings.

2. Systems handled

  • Built-in close-coupling cases cover atom–rotor and atom–diatom collisions, rotor pairs, asymmetric-top combinations, and atom scattering from a corrugated surface.The surface case is restricted to centrosymmetric lattices with real potential matrices.Source: §2 · p. 7
  • The built-in close-coupling calculations use a fully coupled space-fixed representation, performed separately for each total angular momentum and parity.Source: §2 · p. 8
  • Effective-potential, coupled-states, and decoupled L-dominant approximations can reduce computation time at the expense of accuracy, although support varies by collision type.Source: §2 · p. 8
  • A plug-in interface lets users specify coupled equations for additional collision pairs, with supplied routines for Mg + NH and hyperfine-coupled 85Rb2 in magnetic fields.Source: §2 · p. 8

3. Propagators

MOLSCAT offers several propagation methods with different efficiency and accuracy trade-offs. Hybrid short- and long-range propagation is useful when low-energy calculations must reach very large separations.

3. Propagators

  • LDMA offers better accuracy than LDMD for very strongly coupled problems, but its higher cost makes it primarily useful for those production runs or system setup.For setup, it can output interaction-matrix eigenvalues and nonadiabatic couplings at specified distances.Source: §3 · p. 8
  • LDMG offers fourth- or fifth-order symplectic propagation that costs 1.5–3 times more per step than LDMD or LDJ but can be efficient when high precision is required.Its errors can be smaller for a given step size.Source: §3 · p. 8
  • WKB is limited to single-channel problems, while DV and RMAT are not recommended except for special purposes and VIVS has largely been superseded by AIRY.LDJ is very stable but has largely been superseded by LDMD.Source: §3 · pp. 8–9
  • AIRY’s increasing step size makes it efficient at long range, while pairing it with fixed-step LDMD at shorter range suits low-energy calculations requiring very large separations.AIRY is less efficient for fast-varying potentials; LDMD is efficient and stable at short and medium range.Source: §3 · pp. 8–9

4. Computer time

MOLSCAT’s computer time depends on the number of coupled equations and the radial steps needed for the desired accuracy. The required step size is tied to the minimum local wavelength.

4. Computer time

  • Computer time is approximately proportional to N^3 for N coupled equations, with a practical limit of a few hundred to several thousand equations depending on computer speed and memory.The number of radial steps also affects computation time.Source: §4 · p. 9
  • Computer time scales linearly with the number of radial steps, and the step size is typically proportional to the minimum local wavelength.The step count depends on the steps needed to reach the desired accuracy.Source: §4 · p. 9
  • For scattering calculations, the maximum local kinetic energy may be approximated by the collision energy plus the interaction-potential well depth.Source: §4 · p. 9

5. Plug-in functionality

MOLSCAT’s plug-in interfaces let users provide potential representations and define additional basis sets and coupling matrices. Its internal field and scaling structures also support calculations and resonance searches across external variables.

5.1. Potential or potential expansion coefficients

  • Potential expansion coefficients can be supplied directly, calculated by the program from a user-supplied potential, or returned together by a replacement routine.For simple exponential and inverse-power potentials, parameters can be entered in the input file.Source: §5.1 · p. 9

5.2. Basis sets and coupling matrices

  • User routines can define additional basis sets, select calculation elements, and supply coupling matrices and internal Hamiltonian and angular-momentum matrices, including field-dependent Hamiltonian terms.Source: §5.2 · p. 9

5.3. External fields and potential scaling

  • MOLSCAT supports multiple electric, magnetic, or photon fields at arbitrary relative orientations and can scan an external-field variable or potential scaling factor, including for low-energy resonance characterization.Potential scaling can apply to the whole interaction potential or a subset of its expansion coefficients.Source: §5.3 · p. 10

6. Distributed files and example calculations

The distribution includes source code, documentation, build and example files, auxiliary data, and licensing information. Its examples exercise the program’s propagation, scattering, line-shape, field-dependent, and resonance capabilities across varied collision systems.

6.1. Distributed files

  • The distribution provides documentation, Fortran source, a GNU makefile, example inputs and outputs, auxiliary potential data, a README, and the GNU General Public License, Version 3.Source: §6.1 · p. 10

6.2.1–6.2.2. Propagator and decoupling examples

  • The propagator example compares combinations of short- and long-range methods in close-coupling calculations for an atom and rigid rotor, while the decoupling example compares available approximations for a vibrating rotor.The latter uses LDMD/AIRY propagation and first runs close-coupling calculations.Source: §6.2.1 · p. 10, §6.2.2 · p. 11

6.2.3. Location of quasibound state (Feshbach resonance) for Ar + HF

  • The Ar + HF example estimates a narrow resonance from eigenphase sums, then demonstrates that its far-from-resonance formula cannot safely locate the resonance using points spanning it.The example uses two sets of five energy points.Source: §6.2.3 · p. 11

6.2.4–6.2.5. Line-shape cross sections

  • The Ar + CO2 and Ar + H2 examples calculate Raman line-shape cross sections, including CO2 S(10) line properties and H2 rotational Raman widths and shifts across a shape resonance.The H2 example evaluates S matrices at the same kinetic energy for different rotational states.Source: §6.2.4 · p. 11, §6.2.5 · p. 11

6.2.6–6.2.7. Rotor collision cross sections

  • The para-H2 + ortho-H2 example compares close-coupling with coupled-states calculations and direct potential coefficients with program expansion; the He + ortho-NH3 example tests convergence before full cross sections.The H2 calculations illustrate equivalence between the two potential-input approaches.Source: §6.2.6 · p. 12, §6.2.7 · p. 12

6.2.8–6.2.9. Spherical-top and atom-surface scattering

  • The Ar + CH4 example targets converged inelastic cross sections while allowing the elastic partial-wave sum to terminate before convergence; the He + LiF example demonstrates diffractive atom-surface scattering.Source: §6.2.8 · p. 12, §6.2.9 · p. 12

6.2.10. Cross sections for Mg + NH in a magnetic field

  • The Mg + NH magnetic-field example uses a plug-in basis suite and accumulates cross sections across total-projection and parity values; excluding off-diagonal spin-spin terms does not save time in this case.The test run uses a smaller basis set than the cited Wallis et al. calculations.Source: §6.2.10 · pp. 12–13

6.2.11. Characterisation of magnetically tunable Feshbach resonances and calculation of effective range for 85Rb + 85Rb

  • The 85Rb + 85Rb example characterizes four field-dependent low-energy resonances with distinct elastic and inelastic behavior and calculates effective range across the strong aa-channel resonance near 850 G.The four cases include a scattering-length pole, two oscillatory features at excited thresholds, and one with strong background inelasticity.Source: §6.2.11 · p. 13

7. Program history

MOLSCAT grew from a broad close-coupling scattering program into a version with expanded basis-set plug-ins and additional low-energy and external-field capabilities. The 2019 changes extend both the systems it can model and the scattering properties it can calculate.

7. Program history

  • MOLSCAT’s history runs from 1970s close-coupling and cross-section capabilities through added coupling cases, resonance fitting, and post-processors.Earlier versions included decoupling approximations and outputs for elastic, inelastic, and line-shape cross sections.Source: §7 · p. 13
  • Version 14 (1994) was the last version produced by Green and Hutson in collaboration, and no fully documented publication followed it.A parallel version, PMP MOLSCAT, was based on version 14.Source: §7 · p. 14

7.1. Principal changes in version 2019.0 •

  • The extended plug-in mechanism permits propagation in non-eigenfunction basis sets, with transformation to a basis diagonal in both operators before long-range matching.This makes implementing systems with complicated interaction partners simpler.Source: §7.1 · p. 14
  • The plug-in functionality supports calculations and loops over values of electric, magnetic, and/or photon fields.Source: §7.1 · p. 14
  • MOLSCAT processes the S matrix to calculate low-energy scattering lengths, volumes, or hypervolumes, and can extrapolate a_0(k) to obtain the effective range.The scattering lengths or volumes may be complex and energy-dependent.Source: §7.1 · p. 14
  • MOLSCAT can locate and characterize external-field Feshbach resonances, including elastic poles in scattering length or volume and more complicated inelastic signatures.Source: §7.1 · p. 14
  • MOLSCAT can calculate a multichannel wavefunction incoming in one channel and outgoing in all open channels.Source: §7.1 · p. 14
  • The update permits sensible combinations of short- and long-range propagators, more general starting log-derivative boundary conditions, and a symplectic propagation approach.Source: §7.1 · p. 14
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