Book Review to appear in The International Journal of Quantum Chemistry
Quantal Density Functional Theory
Viraht Sahni, Author (Springer 2004)
For about 15 years Viraht Sahni, with students and collaborators, has developed
what first was called the “work formalism” of density functional theory (DFT) and now
goes by Quantal Density Functional Theory (Q-DFT). Neither name really conveys the
motivation and content of the work but Sahni’s new book does.
To paraphrase his perspective, the Kohn-Sham (K-S) implementation of the DFT
variational principle is an ingenious mathematical procedure that lacks physical content
and interpretability. The Hohenberg-Kohn theorems, in contrast, focus attention on a
physical observable, the electron number density, and on its relationship to a oneparticle potential. This centrality suggested to Sahni the value of reformulating DFT in
terms of fields and their associated quantum mechanical sources (number density,
current density, kinetic energy density, etc.) If the fields are conservative, then the
potentials are related to the sources by calculating the work. This, plus appeal to
various versions of Ehrenfest’s theorem, gives physical insight because it renders the
many-fermion problem in as close to a classical mechanical and electrodynamical form
as can be imagined.
Sahni’s speaking style is didactic and one can almost hear him speaking in the
writing. He is careful to prove theorems and not sweep aside details. A number of the
results are not narrowly tied to Q-DFT but are generally useful. Examples include the
survey of cusp conditions (Sect. 2.9.1), asymptotics of wave functions and their
relationship with ionization potentials (Sect. 2.9.2), portions of Chapter 6 on the onebody equation for the square root of the density, proof of the harmonic potential theorem
in Appendix B, etc. Sahni also makes good use of Hooke’s atom and related species,
another sign of his attention to didactic quality.
I found one of the more of the rewarding parts to begin at Section 5.2. It uses
the adiabatic coupling parameter approach to show how contributions to the K-S
potentials are related to the physical fields that arise in Q-DFT. The discussion then
goes (Sects. 5.3 - 5.5) to interpretation of the functional derivatives for the K-S exchange
and correlation potentials in general, and then for the density functional for the HartreeFock model. Those results have been published, but they seem much more accessible
and understandable in this form.
No book is without flaws. I found few. The use of “static” on p. 9 (after eq. 2.14)
is peculiar since the density under discussion is manifestly time-dependent. The phase
factor discussion below eq. (2.110) is a little blithe with respect to the quantal sources for
the kinetic energy and current density fields. On (p. 206) it is said that the Hartree-FockSlater total energy is the expectation value of the Coulomb Hamiltonian with respect to
the Slater determinant of HFS one-electron eigenfunctions. So far as I know, this
procedure was not called HFS nor is it what Slater emphasized. In the early days, the
computer costs were too high to compute a single-determinant expectation value total
energy.
Note the famous 1963 Herman and Skillman book “Atomic Structure
Calculations”. It uses the HFS one-electron Hamiltonian but has no total energies.
When the total energy expression corresponding to Slater local exchange was
recognized as having a different exchange-term coefficient from Kohn and Sham’s, the
X approach sprang up. Lindgren (Phys. Rev. Lett. 19, 382 (1965)) suggested that  be
chosen so that the total energy Sahni mentions equaled the Hartree-Fock energy atom
by atom. So far as I know, that was the only use of that expectation value. Usually the
term HFS means X, often but not always with  = 1. This leads to the second flaw in
Sect. 8.2. The discussion on p.213 of the optimal  is confusing at best. The topic is
anachronistic, but for the record, the standard choices were given by Schwarz (Phys.
Rev. B 5, 2466 (1972)). Those choices do not really correspond to  “... determined by
the minimization of the energy with respect to this parameter” as stated on p. 213.
Interesting discussions of that whole period are in Slater “The Calculation of Molecular
Orbitals” (Chapter 1) and “Solid State and Molecular Theory” (Chapter 33).
These modest deficiencies do not impair the overall value of this book. It is an
interesting, unconventional, and useful contribution to the DFT literature.