One of the most significant developments in computational atomic and molecular physics in recent years has been the introduction of B-spline basis sets in calculations of atomic and molecular structure and dynamics. B-splines were introduced in applied mathematics more than 50 years ago, but it has been in the 1990s, with the advent of powerful computers, that the number of applications has grown exponentially. In this review we present the main properties of B-splines and discuss why they are useful to solve different problems in atomic and molecular physics. We provide an extensive reference list of theoretical works that have made use of B-spline basis sets up to 2000. Among these, we have focused on those applications that have led to the discovery of new interesting phenomena and pointed out the reasons behind the success of the approach.

In this work a new direct (noniterative) algorithm to solve the time-dependent density-functional theory equations for molecular photoionization has been proposed and implemented, using a multicentric basis set expansion of B-spline functions and complete exploiting of the molecular point-group symmetry. The method has been applied to study the photoionization dynamics of CS2 and C6H6: the results confirmed the expectation of large screening effects in CS2. For C6H6 the screening effects have been found to play a minor role than in CS2, however, also in this case the quality of the final results is definitely improved. The method has proven suitable to study with confidence molecules of medium size, and there is still room for further improvement working on more elaborate treatment of the exchange-correlation functional.

The valence shell electronic states of pyrimidine and pyrazine have been studied experimentally and theoretically. The absolute photoabsorption cross sections have been measured between 4 and 40 eV, using synchrotron radiation, and are dominated by prominent bands associated with intravalence transitions. In contrast, the structure due to Rydberg excitations is weak, but series have been observed converging onto the X 2 B 2 or D 2 B 1 limits in pyrimidine and the X 2 A g or D 2 B 3g limits in pyrazine. A comparison between the photoabsorption spectrum of pyrazine-h 4 and that for pyrazine-d 4 , together with calculated transition energies, has helped clarify the assignments of the 6a g →npb 1u and npb 2u Rydberg series. The vibrational progressions associated with these states have been assigned through analogy with those in the corresponding photoelectron band. The time-dependent version of density functional theory has been used to calculate oscillator strengths and excitation energies for the optically allowed singlet-singlet valence transitions, and also to obtain the excitation energies for electric-dipole-forbidden and/or spin-forbidden transitions. These theoretical results have allowed many of the experimentally observed bands to be assigned and provide a generally satisfactory description of the valence shell photoabsorption spectrum. Several of the prominent bands appearing above the ionization threshold can be correlated with predicted intense intravalence transitions.