Computer Physics Communications 14 (1978) 447—448 © North-Holland Publishing Company
BOOK REVIEW The Hartree—Fock Method for Atoms. A Numerical Approach Charlotte Froese Fischer, John Wiley and Sons, Inc., 1977. pp. xi + 308. Price: £17.20/$29.00
It is just 50 years since D.R. Hartree first described a numerical method for solving the Schrodinger equation with a non-Coulombic potential, along with the method of “the self-consistent field”. V. Fock introduced the “self-consistent field with exchange” in 1930, which he derived from a variational principle that provides the preferred formulation today. With M.J. Petrashen, he published a numerical solution for Na in 1934 using a method that turned out to be unsuitable for practical computation. It was Hartree’s numerical methods that proved to be the most satisfactory for solving the “Hartree—Fock equations”, as they came to be called, in those days of desk computation. When Hartree came to survey the field in his book, The Calculation ofAtomic Structures just 20 years ago, the era of electronic computation had dawned. This work described the current state of the art, emphasizing singleconfiguration approximations and configurations contaming at most one open shell, with applications mainly restricted to ground states. A large section was devoted to a discussion of numerical methods for integrating self-consistent field equations, some of them more suited to desk computation along with others still used on today’s electronic computers. The variation of solutions of the equations as a function of atomic number was also studied primarily as a means of generating good initial estimates for the iterative process needed to solve the Hartree—Fockequations. Professor Fischer, a former student of Hartree, brings the story up to date in her new book. The state of the art has been transformed by the rapid expansion of cornputing power and its widespread diffusion over the last 20 years, accompanied by a development of numerical and computational techniques, much of which will be familiar to readers of Computer Physics Communications. The theoretical foundation is still the variational method. 447
At the single configuration level, the Hartree—Fock equations have been solved for the ground states and some excited states of neutral atoms throughout the Periodic Table, as well as for some ionic states. This type of calculation has be~comealmost a matter of routine. The limitations of the single configuration approximation are clearly set out, leading up to an illuminating exposition of the Multiconfiguration Hartree—Fock (MCHF) method as a tool for studying electron correlation in many-electron atoms. Results obtained by the author and co-workers with her program, MCHF72, in the “separated.pair approximation”, for systems with up to 4 electrons are used by way of illustration. A short chapter, describing the ab initio calculation of certaia atomic properties such as ionization potentials and electron affinities, transition probabilities, hyperfine structure constants and isomer shift calibration constants, concludes the first part of the book. The residue deals with aumerical aspects of the solution of Hartree—Fock equations at a practical level. Methods of approximating boundary and eigenvalue problems by finite differeflce equations are surveyed and both direct and shooting methods for their solution are described. There is a short discussion of generalized eigenvalue problems. No formal error analysis is attempted, but some practical test problems for which analytic solutions are known serve to illustrate the magnitude of errors likely in Hartree—Fock problems. The book concludes with a short account of iterative methods for solving the MCHF equations. Each part of the book is supplemented by a short bibliography, and there is a useful table listing relevant programs in the CPC Program Library. The book gathers much information hitherto available only in research papers. As a clear and coherent exposition of the state of the art of numerical approxi-
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mation of the Hartree—Fock equations, this study will be warmly welcomed, as much by beginning graduate students as by mature research workers wishing to apply the Hartree—Fockmethod to practical problems. However, the author’s strict limitation of the subject matter has some drawbacks. Thus Roothaan’s “analytic” Hartree—Fock method, which approximates radial wave-functions by a linear combination of suitable basis functions rather than by using finite differences, is discussed in about half of page 222. The Roothaan method has been used extensively, despite uncertain errors, especially for molecular calculations, and a critical comparison of the results for atoms with those of the numerical approach is badly needed. Another, more fundamental, criticism is that students, who will rightly wish to read the book, will seek in vain for any indication of the physical limitations of the nonrelativistic, pure LS
coupling, Hartree—Fockmethod. In fact the author does not relate the single-configuration results for He to Rn, which are reported in tables 2—3, with experiment, though some comparisons of specific properties are made in chapter 5. Whilst it would be unreasonable to demand an extensi~iediscussion of the breakdown of the nonrelativistic description for neutral atoms with Z 40 or for highly ionized species of lower atomic number, some warning would not be out of place. But these are relatively minor criticisms. This work will surely take its place as a valuable and worthy addition to the literature of Computational Physics. I.P. GRANT Pembroke College Oxford, UK