A scheme is presented for performing linear-combination-of-atomic-orbitals (LCAO) self-consistent-field (SCF) ab initio Hartree–Fock calculations of the electronic structure of periodic systems. The main aspects which characterize the present method are (i) a thorough discussion of both translational and local symmetry properties and the derivation of general formulas for the transformation of all the relevant monoelectronic and bielectronic terms under symmetry operators. (ii) The use of general yet powerful criteria for the truncation of infinite sums; in particular, the Coulomb electron–electron interactions are subdivided into terms corresponding to intersecting or nonintersecting charge distributions; the latter are grouped into shell contributions and the interaction is evaluated by multipolar expansions; the exchange interaction may be evaluated with great precision by retaining a relatively small number of two-electron integrals according to a truncation criterion which fully preserves its nonlocal character. (iii) The use of a procedure for performing integrals over k, as needed in the evaluation of the Fermi energy and in the reconstruction of the Fock matrix, which is particularly simple because it employs partially intersecting small spheres as integration subdomains where linear extrapolation is admitted. A comparison is finally made of our fundamental equations in the critical SCF stage with those obtainable by a recent proposal which uses Fourier transforms to express Coulomb and exchange integrals.

Exact-exchange Hartree–Fock calculations for periodic systems. I. Illustration of the method†

A central problem in one-electron band calculations is the proper inclusion of exchange and correlations. We have performed a first-principles calculation by utilizing the Green's-function method with an energy-dependent nonlocal self-energy operator obtained by replacing the Coulomb potential in the exchange operator by a dynamically screened interaction. Diamond was chosen as a prototype of covalent materials. To be consistent with a variety of experimental facts, we have taken the dielectric matrix of the medium within the time-dependent screened Hartree-Fock approximation, thereby including both local-field and electron-hole (excitonic) effects. Previous calculations along similar lines have been restricted either to a random-phase-approximation frequency-independent dielectric function or to a plasmon-pole approximation. We have investigated for the first time the role of a realistic frequency and wave-vector-dependent dielectric matrix, and examined the relative importance of the electron-hole excitations and of the plasma resonance across the range of the valence and conduction bands. The correlated band structure was calculated by diagonalizing the quasiparticle equation of motion in a local orbital basis in order to exploit the local character of both the self-energy operator and the orbitals spanning these bands. We have found that the plasma resonance does not contribute appreciably in the energy range about the band gap while it contributes significantly to the valence bandwidth. Our values of 7.4 eV (band gap) and 25.2 eV (valence band-width) are in good agreement with reflectivity and photoemission experiments. Implications for the local-density and the energy-independent Coulomb-hole plus screened-exchange approximations are discussed. In addition, our method, by utilizing an energy-dependent self-energy, has also enabled us to calculate quasiparticle damping times (specifically, intraband Auger decay rates) that are consistent with photoemission spectra.

Dynamical aspects of correlation corrections in a covalent crystal

The visible optical spectrum of two typical polydiacetylene (PDA) crystals, $\mathrm{poly}[2,4\ensuremath{-}\mathrm{h}\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{y}\mathrm{n}\mathrm{e}\ensuremath{-}1,6\ensuremath{-}\mathrm{d}\mathrm{i}\mathrm{o}\mathrm{l}\ensuremath{-}\mathrm{b}\mathrm{i}\mathrm{s}\ensuremath{-}(p\ensuremath{-}\mathrm{t}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{e}\phantom{\rule{0ex}{0ex}}\mathrm{s}\mathrm{u}\mathrm{l}\mathrm{f}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e})]$ (PTS) with an acetylene-like structure and poly[5,7-dodecadiyne-1,2-diol-bis-(phenylurethane)] (TCDU) with a butatriene-like structure, is investigated with the use of a first-principles Green's-function formalism of charge-transfer exciton theory The energy-band structures of the ground state are calculated first at the Hartree-Fock level with nonlocal exchange and they are corrected afterward for electron correlation effects using the electron-polaron model of Toyozawa The electron-hole interaction ${\stackrel{^}{H}}_{e,h}$ is treated by first-order perturbation theory in the framework of the Lax-Koster-Slater resolvent method as proposed by Takeuti The influence of screening on ${\stackrel{^}{H}}_{e,h}$ is also investigated with the help of an $\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}$-dependent static dielectric function The results show that correlation corrections play an important role in the calculation of the optical spectrum since they substantially modify the Green's function of the host crystal as well as ${\stackrel{^}{H}}_{e,h}$ Both the singlet and triplet excitons are found to be delocalized with a radius of 25-30 \AA{} in PDA's: the Frenkel exciton contributes only \ensuremath{\sim}30% of the binding energy The calculations predict the first-singlet absorption maximum for PTS at 21-22 eV and for TCDU at 17-18 eV, with an exciton bandwidth of \ensuremath{\sim}29 and \ensuremath{\sim}35 eV, respectively The exciton binding energy is found to be \ensuremath{\sim}04 eV for both systems Triplet levels are located at 08-09 eV below the singlets at $K=0$ and they form relatively flat bands The theoretical ionization potentials are 57 and 50 eV for PTS and TCDU crystals, respectively

Green's-function study of optical properties of polymers: Charge-transfer exciton spectra of polydiacetylenes

The use of self-consistent band calculations to understand the electronic structure of solids is reviewed. The formal basis for the approach is first examined to emphasise that not all band calculations are applications of the same theory. Similarly the critical points, both fundamental and mundane, of the self-consistency process are examined. Again, the central theme is the idea that perhaps some aspects of the self-consistent calculation are not completely established. On the other hand, band structure technique, while still developing, has been rather well explored. Thus one can examine it from the general point of view of seeing what the various techniques imply about the nature of the band structure problem. Finally, some applications of self-consistent band structure calculations are examined.

http://iopscience.iop.org/article/10.1088/0034-4885/44/2/002/pdf

Self-consistent energy band calculations

Publisher Summary Spectroscopy, in one form or another, plays a key role in the evolution of modern physics-optical spectroscopy and atomic physics being the canonical example. One studies the energy dependence of the emission or absorption of some kind of particle or field by a selected sample material. This energy dependence reveals dynamical information about the sample. The most successful kinds of spectroscopy are those in which the particle and sample interact only weakly, because then the dynamical information is characteristic of the unperturbed sample. The chapter provides a comparison between four spectroscopies of solids revealing the range of energies and wave vectors accessible to each. The diagonal line represents optical absorption. It is impossible to independently vary the energy and momentum of a photon, as Maxwell's equations fix one once the other is known. Therefore, any excitation in a solid that is created by absorption of a photon must lie on the line shown. A beam of particles is incident on the sample and scatters through some angle, losing certain energy. If multiple scattering is not a problem, then the energy lost and momentum transferred in the sample are ascribed to the creation of a single excitation. By varying the scattering angle and the energy loss, an entire region in energy-momentum space is studied.

Inelastic Electron Scattering Spectroscopy

In this paper, we examine the optical properties of LiF. The fundamental calculations performed are a self-consistent Hartree-Fock band-structure calculation. As has been found previously, such a calculation has substantial errors when compared with experiment. Correlation corrections are needed and are included by means of the electronic-polaron method. The usual electronic polaron is extended to include short-range polarization corrections which the usual model neglects but which are peculiarly large for LiF. The energy bands are evaluated directly by the linear-combination-of-local-basis-functions method along with the crystal wave functions for 89 points in the irreducible wedge of the first Brillouin zone. Using these results and the Lemann-Taut interpolation scheme the density of states, the joint density of states, and the imaginary part of the dielectric function are computed. These results which neglect the formation of excitons and "plasmons" are in poor agreement with experiment. The effect of the exciton formation is included by a Koster-Slater one-band---one-site calculation. We find that the inclusion of exciton effects substantially improves the level of agreement with experiment. This is in direct contrast with recent work of Menzel et al. in which energy-band theory alone is found to account for the optical properties of LiF. This disagreement is discussed in detail.

Optical properties of LiF

A recent ab initio band-structure calculation by Piela et al. for solid methane is reviewed critically. The analysis used by Piela et al. to determine which contributions to the Fock matrix are large is shown to be in error. This error has the effect that Piela et al. neglect terms in the two-center matrix element which are large and easy to evaluate and retain terms in the one-center matrix element which are small and difficult to evaluate.

A. Barry Kunz

Papers

Optical properties of LiF

Comments on "Electron band structure of solid methane: Ab initio calculations"