Abstract
The recent advances in the use of density functional theory (DFT) in quantum chemistry and in material sciences are considered from the perspective of the quantum theoretical description of the microscopic world. A viewpoint is presented on how to think about the quantum revolution and how to fit the DFT developments into it.
This paper is dedicated to Kostas Gavroglu as a token of my admiration, affection and respect. Sam Schweber.
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Notes
- 1.
See Bethe and Jackiw (1968).
- 2.
See for example Ren et al. (2012).
- 3.
See especially Kenneth Wilson’s plea to theoretical physicists making lattice gauge theory calculations to learn what quantum chemists using DFT had accomplished. Wilson (1990).
- 4.
- 5.
For the development of the RSPt code at Los Alamos see Wills et al (2010).
- 6.
See for example Höltje and Folkers (2003).
- 7.
We are using “modern“ and “postmodern” in the sense introduced by Forman (2012).
- 8.
Our presentation is based on Kohn’s Nobel lecture, Kohn (1999).
- 9.
- 10.
We do attach great importance to the notion of language. And in this we differ from Gavroglu and Simões. Thus, we do not consider the shift from atomic orbitals to molecular orbitals as a change in styles of reasoning, but rather a change in the language to address the problem of chemical structure. See Gavroglu and Simões (2012, 6–7).
- 11.
Conversely, the success of the approach of describing physical phenomena in terms of effective field theories is a reflection of the fact that appropriately isolated physical phenomena in a certain energy regime, probed and analyzed by instruments able to resolve effects only within a certain range of length scales, can be described most simply by a set of effective degrees of freedom appropriate to that scale.
Relativistic quantum field theories implicitly make statements about arbitrarily short space-time distances, and thus about arbitrarily high energies and momenta. Yet no conceivable experiment will be able to probe such distances. When calculating the predictions of a given theory the contributions stemming from these high-energy, short-distance components are divergent. The renormalization program to circumvent these divergences which was developed by Weisskopf, Bethe, Schwinger, Feynman, and by Dyson after World War II was given a new and deeper interpretation by Wilson and others in the early 1970s. Wilson was able to exhibit the effect of all possible modifications of a given theory beyond a certain cut-off in energy as a re-parametrization of all possible interactions between the entities that are assumed to populate the low-energy domain of that theory. Furthermore, he showed that starting from any set of interactions at the cut-off scale, a low energy physics description at a given level of accuracy could be formulated that depended only on a few relevant parameters (Wilson 1983). The great accomplishment by Wilson, Weinberg and others was demonstrating the universality of the low-energy physics, which resulted from the renormalization process thus justifying the use of effective field theories. See Weinberg (1995–2000) and in particular volume 2, Applications.
- 12.
Robert Laughlin won the Nobel laureate for work he did explaining the quantum Hall effect, and David Pines is one of the most distinguished solid-state theorists.
- 13.
Physicists believe that the standard model constitutes a “theory of everything” in the sense that it “explains” the existence of neutrons, protons and mesons, their properties and the interactions between them, which in turn account for the existence of nuclei, … . The quantum electrodynamics component of the electroweak theory in turn explains the electromagnetic interactions between electrons and nuclei and accounts for the existence of atoms and molecules, and the “weak” component explains the radioactivity of nuclei.
- 14.
Note that Eq. 5.2 only takes into account the Coulomb interactions between the charges and neglects spin and magnetic interactions, as well as the possibility of emission and absorption of photons. Systems for which this is a valid approximate Hamiltonian to calculate their level structure are called Coulomb systems.
- 15.
See Kadanoff (2011) and the references therein.
- 16.
The Born-Oppenheimer Anzatz consists in approximating the wave function for the electron-nuclear system as a product of a wave function for the nuclei and a wave function for the electrons in which the nuclei are considered clamped at the positions R 1, R 2, … R Nn It is justified on the basis that \( m<<{M}_z \). See for example Combes et al. (1981).
- 17.
A great deal can be said about “emergent” properties if one knows what symmetry group is broken, and what subgroups are left as a symmetry—and all this can be stated irrespective of the dynamics involved.
- 18.
We shall make some comments regarding this question in Sect. 5.1.2.
- 19.
As is well known, this issue was addressed by Phillip Anderson in 1972 in Science.
- 20.
Furthermore, the fact that the size of nuclei is very small compared to atomic dimensions and that their mass is very large compared to that of an electron justifies the approximation that in atoms and molecules the nuclei interact with the electrons only through electrostatic Coulomb forces.
- 21.
We are neglecting the effects of cosmic rays or the radioactivity of nuclei.
- 22.
Furthermore, systems whose characteristic time T, mass M, and length L, are such that \( M{L}^2/T>>h \) are “macroscopic” and described by classical mechanics, those for which \( M{L}^2/T\approx h \) are “microscopic” and described by quantum mechanics.
- 23.
This equation links the atomistic dynamics with thermodynamics through statistical mechanics in a derivation of the equation of state for a very dilute gas.
- 24.
Eg. the Navier-Stokes description of fluid flow in terms of a local velocity field as compared to the Newton’s equations describing the motion of the individual molecules.
- 25.
Or through more localized, specific atomic or molecular forces and are also coupled to the electromagnetic field.
- 26.
See Gavroglu and Simões (2012) for the details and subsequent developments.
- 27.
For a thorough account of these developments see Gavroglu (1995).
- 28.
Upon receiving the 1993 Orsted medal for his contributions to the teaching of physics, Bethe in his acceptance speech stated “that there is a certainty principle in quantum theory and that the certainty principle is far more important for the world and us than the uncertainty principle. That doesn’t say that the uncertainty principle is wrong. It says that the uncertainty principle just tells you that the concepts of classical physics, position, and velocity, are not applicable to atomic structure.” Bethe (1993).
- 29.
It did so first in terms of phenomenological inter-nucleonic potentials; thereafter in attempts to determine these potentials on the basis of meson theories; and more recently in terms of the standard model.
- 30.
See for example Fermi (1932). In the article Fermi stated that he considered the stability and other properties of the electron open questions due to the infinite self-energy problem in quantum electrodynamics. He thought the problem was to be resolved by a future theory. Subsequently, further infinities were encountered in QED. Divergences due to self energy and vacuum polarization effects plagued quantum field theories throughout the 1930s. These problems were partially resolved by the post World War II renormalization program.
- 31.
See Brown and Rechenberg (1996).
- 32.
One might add to these the cosmological—consisting of galaxies and their constituents, their evolution and dynamics. These hierarchies were not considered to be independent: accurate measurements of atomic energy levels reveal nuclear and subnuclear properties. Similarly, the recent startling discovery of the necessity of the presence of cold dark matter—consisting of as yet undiscovered subnuclear entities—in order to make sense of new cosmological observational data is proof of the linkage between the various levels. But it must also be noted that these observations have not destabilized our amazingly accurate representations of the atomic world. And needless to say, the linkage of the levels is made explicit as soon as one tries to answer evolutionary questions.
- 33.
- 34.
Our presentation is based on Kohn’s Nobel lecture (Kohn 1999).
- 35.
In fact, it yields an exact result for the energy of an atom in the limit \( Z\to \infty \). This was proved by Lieb and Simon (1977). Elliott Lieb and Walter Thirring are responsible for extensive and generative contributions to the rigorous mathematical proofs of the quantum mechanical accounts of the stability of atoms, molecules and solids. See Thirring (2001).
- 36.
More generally when 1/r potentials are involved TF theory can be made to yield important results using analytical methods. Thus TF methods can also make statements about the stability of systems where gravitational forces are also involved. See the informative and insightful article by Spruch (1991) and the references therein to the papers by Lieb and by Thirring. See also Lieb (1981) and Thirring (1981).
- 37.
- 38.
See Lieb (1981).
- 39.
- 40.
- 41.
More precisely, they proved that \( {E}^{(N)}\ge - AN \), with A a finite, N-independent constant.
- 42.
- 43.
See Kohn’s Nobel lecture (Kohn 1999).
- 44.
For example, the approximate agreement between X-ray scattering by neutral atoms and cross-sections calculated using TF density distributions.
- 45.
j includes the spin quantum number.
- 46.
- 47.
- 48.
As can be seen in Engel and Dreizler’s comprehensive account.
- 49.
The proof of the four-color map problem using computers was undoubtedly one of the factors responsible for the change.
- 50.
- 51.
- 52.
See for example Wills et al. (2010).
- 53.
See Chap. 1 of Wills et al. (2010).
- 54.
Quantum chemists can now calculate many properties—such as structure, binding energies, vibrational frequencies, …—for systems containing hundreds of atoms and more. Similarly activation energies of chemical reactions involving fairly complex molecules can now be computed.
- 55.
As noted, the most stable crystal structure for a particular combination of atoms can now be calculated and determined even when the structure is not known experimentally. See note 77.
- 56.
See Weinberg (1995–2000), in particular volume 2, Applications.
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Schweber, S., BenPorat, G. (2015). Quantum Chemistry and the Quantum Revolution. In: Arabatzis, T., Renn, J., Simões, A. (eds) Relocating the History of Science. Boston Studies in the Philosophy and History of Science, vol 312. Springer, Cham. https://doi.org/10.1007/978-3-319-14553-2_5
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