Numerical methods every atomic and molecular theorist should know


This Technical Review encapsulates the methods and numerical techniques that have been so successfully used over the years to study the electron scattering of atoms and molecules. In the past few decades, these approaches have also proven effective in treating the time-dependent interaction of strong electromagnetic fields with atoms and molecules. There are clear synergies between the two that can be exploited computationally. The ideas discussed in this Technical Review have played an important role in shaping modern atomic and molecular physics, and we expect that future developments will build heavily on these foundations.

Key points

  • We overview the best-of-breed numerical methods being used to compute electron–atom and electron–molecule scattering cross-sections and to propagate the Schrödinger equation in time.

  • We describe the close-coupling, R-matrix and Kohn variational methods and briefly discuss the importance of complex scaling to practical scattering theory.

  • We outline how these techniques may be extended to treat the time-dependent Schrödinger equation.

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Change history

  • 21 February 2020

    An amendment to this paper has been published and can be accessed via a link at the top of the paper.


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This work was supported by the National Institute of Standards and Technology. H.G. acknowledges support from the National Research Council Fellowship Program.

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Correspondence to Barry I. Schneider.

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Self-consistent field

It is often synonymous with the Hartree–Fock method and mean-field method. It is based on Douglas Hartree’s assumption that each particle in an N-body system could be treated in the average field of the other N − 1 particles of the system.

Variational principle

A principle used in the calculus of variations concerned with finding functions that minimize, maximize or make stationary quantities defined by differential of integral equations.

Configuration interaction

A quantum chemical variational method for solving the Schrödinger equation of many-body systems. It relies on constructing linear combinations of Slater determinants of fixed spin orbitals. The unknown linear coefficients are computed using the Rayleigh–Ritz variational principle. In theory, the method is exact, but in practice, the computational limitations arising from a large number of configurations force the truncation of the expansions.


The R-matrix relates the channel wavefunction Fc to its derivative \({F}_{c}^{{\prime} }\) at the R-matrix boundary r = a0 by the equation: \({\bf{F}}={\mathscr{R}}\,{{\bf{F}}}^{{\prime} }\), where the labels on the elements of the matrix refer to channels, c.

Re-arrangement collisions

These are collisions in which the initial and final states are eigenstates of different unperturbed Hamiltonians.


In this context, it refers to a matrix transformation of a set of linear equations that are more suitable for finding a solution by iterative techniques.

Born–Oppenheimer approximation

An approximation in molecular systems that treats electronic and nuclear motion adiabatically. This assumption is based on the difference in timescales of electronic and nuclear motions. Consequently, the electronic wavefunction can be computed by freezing the positions of the nuclei. Repeating this calculation as a parametric function of nuclear positions produces a potential in which the nuclei move.

Quantum defect

In this context, it refers to an expression for the energy of a Rydberg state in an atom that replaces the hydrogenic integer quantum number n, by a non-integer value. The deviation from the integer is called the quantum defect. The correction accounts for the fact that the inner electrons partially screen the bare nuclear charge.

S-, T- and K-matrices

These are a family of scattering matrices that relate the initial states in a scattering process to the possible final states. All three matrices are related to each other mathematically. The S-matrix expresses the states in terms of exponential free waves, whereas the K-matrix expresses the states in terms of sine and cosine functions. The T-matrix uses a mixture of exponential and trigonometric functions.

Padé approximate

A numerical method that approximates a function by a rational function of two polynomials.

Coupled cluster

This is a numerical quantum chemical method used for describing many-body systems. Coupled cluster takes an assumed reference basic configuration (such as the Hartree–Fock wavefunction) and constructs multi-electron wavefunctions using an exponential operator ansatz to add electron correlation. The cluster expansion must be truncated in practice due to the complexity of the resulting nonlinear equations.

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Schneider, B.I., Gharibnejad, H. Numerical methods every atomic and molecular theorist should know. Nat Rev Phys 2, 89–102 (2020).

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