CALCULATION OF THE ELECTRONIC STRUCTURE OF SOLIDS IN THE SEVENTIES

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CALCULATION OF THE ELECTRONIC STRUCTURE OF SOLIDS IN THE SEVENTIES

F. Herman

To cite this version:

F. Herman. CALCULATION OF THE ELECTRONIC STRUCTURE OF SOLIDS IN THE SEV- ENTIES. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-13-C3-20. �10.1051/jphyscol:1972303�.

�jpa-00215037�

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CALCULATION OF THE ELECTRONIC STRUCTURE OF SOLIDS IN THE SEVENTIES

F. HERMAN IBM Research Laboratory San Josi, California 951 14 U. S. A.

RCsumC. - On esquisse differentes voies possibles que pourra prendre 1'Btude theorique de la structure electronique des solides dans les quelques annkes a venir. Cet aperCu comprend de courtes discussions concernant les cristaux simples, les surfaces cristallines et imperfections loca- lisks, les solides non cristallins, les cristaux complexes et la matiere condensk dans des conditions inhabituelles. Les progr6s dans toutes ces directions dependront des developpements dans les domaines de I'experience et de la technologie, des calculateurs et des methodes numiriques, de la theorie pure. On prevoit que le champ d'application de la theorie de 1'Ctat solide s'elargira considerablement dans la prochaine decade et qu'une collaboration croissante s'etablira entre theoriciens des solides, chimistes quantiques, m6tallurgistes, ceramistes, geophysiciens, astrophy- siciens, chimistes des polymeres et biologistes.

Abstract. - Some possible directions that theoretical studies of the electronic structure of solids may take during the next several years are sketched. This survey includes brief discussions of simple crystals, crystal surfaces and localized imperfections, non-crystalline solids, complex crystals, and condensed matter under unusual conditions. Progress in all of these areas will depend on experimental and technological developments, on advances in computers and computational methods, and on intrinsic theoretical advances. It is anticipated that the subject matter of solid state theory will broaden considerably during the coming decade, and that there will be increasing collaboration among solid state theorists, quantum chemists, metallurgists, ceramicists, geophysicists, astrophysicists, polymer chemists, and molecular biologists.

During the coming decade, theoretical studies of the electronic structure of solids can be expected t o follow five mutually complementary paths, as outlined below. Progress in each of these directions will depend on theoretical innovations, on parallel experimental advances [I], [2], on technological challenges and breakthroughs [3]-[5], and on advances in computers and computational methods [6]-[a]. There should be strong interplay between theory and experiment, as there has been in the past [I]-[2], with computers being used increasingly to control the gathering and reduction of experimental data in the light of advanced theoretical models and interpretational schemes.

Computers will be used by theoreticians more and more to test scientific ideas and to develop physical and chemical intuition through simulation studies.

Computer graphics [9], [lo] will be used by experi- mentalists and theoreticians alike to display complicated results and to gain insight into the essential nature of these results.

We anticipate a broadening of the subject matter of solid state theory, and of electron theory in particular, during the seventies. This broadening will involve the extension of solid state ideas, methods, and concepts to quantum, organic, and polymer chemistry, to theoretical metallurgy and ceramics, to

geophysics and astrophysics, and to molecular biology. Ten years from now, solid state theorists will be studying a much wider range of materials than they are today, using suitably modified versions of theories that were originally developed to handle relatively simple materials.

Of course, solid state physics has already made modest inroads into other disciplines (recall, for example, the application of the BCS theory of super- conductivity to nuclear physics [Ill). However, the coming decade should see a flowering of interdiscipli- nary studies on a wide scale, as many of us leave our narrow specialties and take a more universal view of our scientific opportunities.

I. Electronic structure and related properties of simple crystals. - The first path will be highly traditional : detailed calculations of the electronic structure and related physical properties of simple crystals (1 to 4 atoms per unit cell, say). However, there will be increasing emphasis on a unified theory of electrons and lattice vibrations, and on first-principles methods for determining lattice vibrational spectra and electron- lattice coupling parameters. There will also be increasing interest in improving the theory and the practical analysis of many-electron effects in solids. Recognition

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972303

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C3-14 F. HERMAN that a solid is not a free-electron gas, but rather a medium composed of alternating regions of high and low electron density will play a key role in future studies.

It has been remarked at a number of conferences that progress on electron correlation effects has been impeded by the lack of communication between quantum chemists (who have considerable experience with atoms and small molecules) and solid state many- body theorists (who have a profound knowledge of idealized solids but little patience with real atoms and molecules). Perhaps greater familiarity with recent developments in many-body theories of the electronic structure of atoms and molecules [12] will suggest to us suitable methods for dealing with electron correlation effects in real solids. In any event, it is clear that these effects are there and must be reckoned with.

Theoretical advances during the coming decade will lead to a better understanding of superconducti- vity, magnetism and other cooperative phenomena, hot electron and other electronic transport processes, infrared and Raman spectra, optical and photoemission spectra, ultraviolet and X-ray spectra, and many other similar topics. There will be improved theories of excitons and polarons, and improved treatments of their interactions with phonons, magnons, and impurities. More sophisticated studies of transition metal compounds [13], [14] will be forthcoming, and these studies will lead to a better appreciation of magnetic semiconductors [I 51 and the metal-insulator transition [16].

Improved band structure calculations will undoubtedly play an important role in many of these investigations. First-principles methods [17]-[20] will be gradually refined, as will semi-empirical methods [20]- [22]. The fundamental assumptions and approxima- tions underlying both types of methods will be criti- cally examined in the light of new experimental evi- dence and increasing theoretical sophistication.

Before attempting to apply existing methods to more complex materials, we should have a very clear picture of the limits of validity of these methods. It would be hazardous to apply these methods uncriti- cally to unfamiliar situations.

During the past few years, there has been considerable interest in developing a physically realistic theory of exchange and correlation effects in solids [23]-[26].

Since all existing theories are based on a wide variety of simplifying assumptions, it is difficult to decide which theories are the most satisfactory. Up to now, there have been few conclusive demonstrations that one theory is significantly better than another. Usually, a new theory of exchange and/or correlation effects is tested by incorporating it into a band structure calculation and comparing the results of these calculations with experiment. More often than not, there are numerical uncertainties inherent in the band structure calculations that completely obscure the comparison

between theory and experiment, and hence invalidate the proposed test of the exchange-correlation theory.

If we are to make significant progress in our treatment of exchange and correlation effects in solids, we must first perfect our first-principles band structure calculations to the point where we can be quite confi- dent of the numerical accuracy of the energy levels, total energy, electronic wave functions, and other calculated quantities. As long as the errors associated with incomplete convergence, neglect of relativistic effects, use of the muffin-tin approximation, lack of self-consistency, and similar shortcomings can be presumed large, say as large as the differences between the predictions of competing exchange-correlation theories, we will have no way of checking the adequacy of these theories. Moreover, if we set out to check a theory, we should carry out calculations for a large number of substances, and compare theory and experiment for each of these, so that we can be sure that the agreement is meaningful and systematic, rather than circumstantial or spurious.

Up to now, there has been great emphasis on the calculated energy levels, and very little emphasis on the electronic wave functions. Now that we are entering a period where we will want to evaluate a wide variety of physical quantities, we will be well advised to determine the electronic wave functions with a high degree of accuracy, so that we can use these with confidence in calculations of matrix elements of various sorts.

11. Crystal surfaces, interfaces, dislocations, and localized imperfections in nearly perfect crystals.

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Earlier work on surfaces states [27], dislocations [28], and the defect solid state [29] relied heavily on simple physical models and phenomenological reasoning.

During the coming decade, there will be stronger emphasis on fundamentals, and much greater utiliza- tion of computer experiments 1301. An underlying theme will be the investigation of electron energy levels, vibrational modes, and atomic arrangements in the neighborhood of point, line, and plane defects. Ano- ther important theme will be the physics of tunneling [31]. Still another will be the study of chemi- sorption and catalysis. Computers will be used increasingly to test scientific theories and to develop physical and chemical intuition [6], [7]. It is likely that progress in this area will come about through increasing inter- action among solid state theorists, quantum chemists, crystallographers, and theoretical metallurgists.

The study of molecular clusters (spherical, rod-like, and disk-like) embedded in a crystalline medium will undoubtedly be a popular topic in the seventies, going far beyong traditional crystal field theory and ligand field theory in attempting to explain the optical and magnetic properties of impurity ions and other imperfections in solids.

One promising approach to the theory of molecular clusters in solids is the semi-empirical approach of

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Messmer and Watkins 1321. These authors and their collaborators have already demonstrated the effective- ness of an approximate molecular orbital treatment [33]

in dealing with vacancies, interstitials, and substitu- tional impurities in diamond-type crystals. In this work, the host (diamond) lattice is represented by a cluster of 35 carbon atoms and the electronic energy levels as well as the total energy of the cluster are determined with and without the imperfection. It is possible to determine all of these quantities, within the framework of the semi-empirical extended Hiickel method [34], as a function of the nuclear coordinates.

In this way, it is possible to decide which atomic arrangements are favored energetically, and what the corresponding energy level structures are.

Of course, the results obtained depend upon the number of atoms included in the cluster, the semi- empirical parameters employed, and the treatment of the ⁽⁽surface ⁾⁾ atoms (boundary conditions). Even though the method is still in an early stage of deve- lopment, it appears highly promising. It already pro- vides a bridge between a localized picture of electronic structure, according to which the energy levels are supposed to depend primarily on the chemical identity of the constituent atoms and their short range order, and a delocalized picture, which in the present context reduces to an LCAO treatment of the band structure.

To illustrate this point, consider three polytypes of germanium : (a) ordinary diamond-type, or cubic, symbol 3 C ; (6) hypothetical 2 H, similar to wurtzite, except that all four atoms in the unit cell are identical ; and (c) hypothetical 4 H, with eight atoms per unit cell. In all three of these polytypes, every atom has the same four nearest neighbors and the same twelve next-nearest neighbors. Thus, every 17 (1

+

4

+

12) atom cluster is identical, whichever polytype it comes from. Differences in the crystal structures of 3 C, 2 H, and 4 H show up only in the third and higher order neighbors. To what extent is the band structure of germanium, and the related optical spectrum, determined by the short range order, as typified by the 17 atom clusters ?

Ortenburger, Rudge, and Herman [35] have recently calculated the band structure and optical spectrum of 3 C, 2 H, and 4 H germanium, using the empirical pseudopotential method. The optical spectra (really ^E,) are compared in figure 1. It will be seen that the gross features of the optical spectrum are the same for all three polytypes, while the fine details are quite different. It is reasonable to conclude that the gross features (location of fundamental absorption edge, average value of main peak, etc.) are determined by the short range order, while the fine details (sharpness and location of subsidiary peaks) are determined by the long range order. As the complexity (number of atoms per unit cell) of the crystal increases, from 2 to 4 to 8, the optical spectrum becomes smoother, and begins to approach in shape (though not precisely in location) the optical spectrum of amorphous germanium, which

FIG. 1 . - Comparison of theoretical optical spectra (actually c2) for three polytypes of germanium (cubic, 2 H, and 4 H), showing the changes produced by going from a tetrahedrally coordinated crystal containing 2 atoms per unit cell (cubic), to similar crystals having 4 (2 H) and 8 (4 H) atoms per unit cell. In all three polytypes, the nearest and next-nearest neighbors of any atom are identically located. For hexagonal 2 H and 4 H, the spectra for parallel and perpendicular polarization are shown, as well as the weighted averages of these spectra. Also shown for comparison is the experimental optical spectrum of amorphous germanium. (After Ortenburger, Rudge, and Herman,

reference [35]).

is also shown in figure 1. It is possible that a more realistic treatment of the atomic arrangements in amorphous germanium [36] will lead to an energy level structure and optical spectrum which is more nearly in agreement with the experimental spectrum.

Another promising approach to molecular clusters is the scattered wave method developed by Johnson and co-workers [37]. There are five principal features : (a) Using the muffin-tin approximation adapted to molecules, a non-relativistic self-consistent solution of the molecular wave equation is obtained. The method of solution is closely related to the KKR method, which has been used so successfully in band structure calculations, especially for close-packed structures. (b) The one-electron eigensolutions and the total electronic energy of the cluster are determined on the basis of the statistical exchange approximation [23], [38] ; the Xcc version of this approximation is recommended. (c) The values of cr within the ion core regions (atomic spheres) are determined from prior calculations for the corresponding free atoms. In the inter-atomic region, where the potential is flat, the value of a is determined by suitable averaging. ( d ) One- electron excitation energies are determined from Sla- ter's transition state theory [23], 1391, or by taking the differences between the total energies of the initial and final states of the cluster. (e) Open shell configurations are handled by taking an average over configurations - the hyper-Hartree-Fock approximation [23], [39], and multiplet structure can be obtained by subsequent calculations.

At the moment, the cluster is confined to a spherical region, and the scattered wave method is programmed to be most effective for symmetrical molecules, though it can also deal with non-symmetrical ones. It is likely

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C3-16 F. HERMAN

that future versions of this and related methods will be generalized to handle planar molecules (e. g., aromatic hydrocarbons) efficiently, as well as long chain molecules (polymers).

The scattered wave molecular cluster (SWMC) method has already been used to study a wide variety of polyatomic molecules and radicals such as SF,,

and (Mn0,)-. The computer programs for carrying out these self-consistent calculations are more than 100 times as fast as comparable programs using the H F MO LCAO SCF method [40], and, according to Slater and Johnson [41], the results appear to be in better agreement with experiment than such LCAO results. The SWMC and related methods will undoubtedly play a major role during the coming decade in advancing our knowledge and understanding of isolated large molecules and of molecular clusters embedded in solids.

While the method of Messmer and Watkins [32] is manifestly semi-empirical, that of Johnson and co- workers [37] is very nearly a first-principles method.

If the choice of ^ahad a firmer theoretical basis, and if the geometrical assumptions connected with the muffin-tin approximation could be ruled out as (c semi-empirical parameters >>, the SWMC method would qualify as a first-principles method by any reasonable set of standards. (Of course, the numerical success of the method will not be influenced by what we call it, but the credibility and significance of the results will.)

In its present form, the SWMC method assumes that there is an optimum value of ^afor each atom which is independent (or very nearly so) of its electronic configuration and local environment. Thus, optimized values of a obtained on the basis of free atom ground state calculations are used for the ground and excited states of molecules incorporating these atoms. It will require considerable further study to establish a firm theoretical basis for this assumption of a transferability.

A further difficulty with the ^{X a}approximation is that the optimum value of a is strongly Z dependent, so that different values of ^ahave to be used in different ion core (atomic sphere) regions, and still another (suitably averaged) value of ^ahas to be used in the inter-atomic region. As we have emphasized in recent papers [42], [43], this Z dependence leads to a discon- tinuous exchange potential, an unphysical result which is obscured but not eliminated by the muffin- tin approximation. We have advocated the use of an improved statistical exchange approximation (modified Xap method) which is Z-independent [42], [43].

Since the muffin-tin approximation has been most successful in solid state band calculations in applications to close-packed structures [17]-[19], and is of doubtful validity when used for open structures, such as diamond-type crystals, it is somewhat surprising that the SWMC method works as well as it does for non-symmetrical molecules, such as diatomics and

methane. The inclusion of the non-muffin-tin terms, and of the non-spherical terms in the various ion core regions, will be a significant step forward. Similarly, the incorporation of relativistic effects will be important in paving the way for more accurate studies of molecules containing heavy atoms.

Many investigators, particularly theoretical chemists, are concerned with the use of the muffin-tin approximation, as well as the statistical exchange approximation, in applications to molecules. Therefore, it would be instructive to carry out LCAO MO SCF calculations for molecules using the statistical rather than the Hartree-Fock exchange approximation.

This could be done using suitably modified versions of existing H F LCAO MO SCF computer programs.

By avoiding the use of the muffin-tin approximation, one could compare directly the results of statistical and Hartree-Fock exchange calculations for molecules, and, for example, determine the degree to which the former can approximate the latter.

However, LCAO MO SCF statistical exchange calculations would still require some suitable choice of the âparameters, assuming the ^Xamethod were employed. One could use a universal value of âfor the molecule as a whole, and obtain it by some opti- mization procedure, but then the value of âwould become a function of the internuclear distances, and would reduce to an average of the a parameters of the isolated atoms at infinite separation. It would bs far more satisfactory to adopt the X@ method 1421, [43], which already incorporates universal (2-independent) parameters. The same parameters could be used for the molecule in its normal atomic configuration, for the molecule during dissociation, and for the separated atoms.

At this writing, it is not clear how much additional computer time would be required to carry out LCAO MO SCF rather than SWMC SCF statistical exchange molecular structure calculations, particularly if the latter included non-muffin-tin and non-spherical term corrections. However, all such calculations would be considerably faster to perform than exact Hartree-Fock molecular structure calculations.

It might be useful to consider an alternate form of LCAO MO SCF statistical exchange molecular structure calculations, in which the usual atomic orbitals employed in LCAO MO SCF H F molecular structure calculations are replaced by more localized atomic orbitals, and additional basis functions extending over the entire molecule and falling pro- perly to zero at large distances from the molecule are introduced. The extended basis functions would take the place of the tails of the original atomic orbitals.

From a computational point of view, there would be a tradeoff between the extra effort involved in introducing the extended basis functions, and the considerable saving of effort arising from the smaller number of overlap integrals that one has to deal with because the atomic orbitals used are more

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localized. This alternate form is the molecular ana- logue of mixed basis methods that have become popular recently in solid state band structure calculations (for a review, see [44]).

As we have already stated, we expect the statistical exchange approximation to play an important role in the study of the defect solid state during the coming decade. The propre treatment of exchange and correlation effects, within the context of the statistical field method, remains one of the major theoretical challenges ahead 1231, [26].

111. Electronic structure and related properties of non-crystalline solids.

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Current efforts are directed primarily at uniformly distributed structural and/or chemical disorder, as in amorphous semiconductors and random binary alloys [45], 1521. Some of the topics of principal interest include atomic arrangements and radial distribution functions for disordered materials, effects of disorder on the electronic energy levels and the lattice vibrational spectrum, locali- zation of electrons and lattice vibrational modes, electronic transport mechanisms, percolation theory, optical and photoemission properties, and the effects of electron correlation on the electronic structure.

The emerging theory of non-crystalline solids has many points of contact with the theory of liquids 1531, the theory of liquid metals [54], the theory of liquid crystals [55J, and, more generally, with the theory of random functions [56].

Future efforts might involve detailed studies of polycrystalline materials and powders, in which orde- red crystallites or disordered grains, having random sizes and shapes, are separated by twin or grain boun- daries. There might also be studies of inhomogeneous or anisotropic disorder, as in stacking fault disorder.

The overall challenge here is to adapt the methods of solid state physics particularly the modern theory of disordered solids, to the broad and relatively unex- plored realms of theoretical metallurgy and ceramics, and to introduce a more fundamental approach into these technologically important areas [57].

The theory of random functions [56] and computer simulations [6], [7], [30] will undoubtedly play an increasingly important role in the theory of disordered solids. There is also considerable room for imaginative analogies. For example, it is conceivable that fruitful analogies can be drawn between individual dislocations and individual chain molecules (polymers), and that methods developed for dealing with disloca- tion networks 1281 can be applied or adapted to polymer networks 1581, and vice versa.

Most authors attempt to treat three-dimensional structural and/or chemical disorder in terms of formal theories, such as the coherent potential approximation [59]-[61]. We are rapidly reaching the stage when the fundamental assumptions of such theories will be tested by detailed numerical calculations. In the meanwhile, some authors are concentrating on less

sophisticated but more direct methods. For example, Polk [62], Shevchik and Paul [63], and Henderson and Herman [36] have recently constructed physically plausible atomic models of amorphous silicon and germanium, using direct inspection in some cases and computer simulation in others. Polk, Shevchik, and Paul concentrate on large isolated clusters of randomly arranged atoms having tetrahedral coordination, while Henderson and Herman work with smaller clusters of this type, but subject them to cyclic boundary conditions (as is commonly done in molecular dyna- mics 1641 and in Monte Carlo calculations [65].

All the authors just mentioned have already suc- ceeded in obtaining random tetrahedrally coordinated networks whose' radial distribution functions closely resemble their experimental counterparts. Work now in progress is aimed at determining the electron energy levels and lattice vibrational modes [36]. It will be interesting to see how the results obtained on the basis of such random atomic models compare with theoretical predictions based on other idealized models, for example, those recently discussed by Weaire and Thorpe [66], Ziman [52], and Gubanov 1671.

Much current thinking about electronic states in disordered systems emphasizes electron localiza- tion [50]-[52], a subject that has already received considerable attention in solid state physics [13], [14], [68] and quantum chemistry [69]. As solid state physicists learn more about quantum chemistry [70], and as we gain practical experience with random atomic models and networks we may be able to take advantage of the large body of chemical insight and knowledge concerning localized electronic states 1691, and apply this body of information to disordered systems.

IV. Electronic structure and related properties of complex crystals. - Most of the methods that have already been developed for treating the electronic structure of crystals 1171-[22] become progressively more difficult to employ as we move from simple crystals (1 to 4 atoms per unit cell) to more complex ones. In the case of semi-empirical methods, the number of adjustable parameters eventually grows too large to be manageable ; the same can be said of lattice vibrational calculations. Severe convergence problems usually arise if we attempt to deal with complex crystals by direct methods. Of course, there are always tricks, such as treating a complex crystal as a simple crystal plus a perturbation [71], but eventually we run out of tricks, and we are faced with the problem of dealing with a crystal containing 10 or 100 or even 1 000 atoms per unit cell.

There is already considerable theoretical interest in complex crystalline materials in such widely different fields as quantum optics [72] and geophysics. Quartz, garnet, and spinel are typical complex crystals having geophysical importance [73], [74]. There is also grow- ing interest in molecular crystals [75]. To illustrate

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C3-18 F. HERMAN the diversity of molecular crystals, we note that some,

like hexamine, C,H,,N,, are formed from almost spherical molecules, while others, like naphthalene, C,,H,, are formed from almost independent, anisotropic molecules. In these two examples, the individual molecules are held together by van der Waals forces.

In molecular crystals where hydrogen bonds determine the type of structure, we may have crystals with chains of molecules, crystals whose molecules are joined in layers, crystals whose molecules are joined in three- dimensional frameworks, and crystals whose molecules join to form a box in whose interior another molecule is trapped.

The subject of polymeric crystals [76] is of considerable importance in its own rigEt. Such crystals contain staggering numbers of atoms per unit cell. If we also allow for random atomic and molecular arrangements, and consider amorphous as well as crystalline polymeric materials, we encounter proteins, nucleic acids, fibers, elastomers, and plastics, in short, a wide range of natural and synthetic polymers. Pro- gress in understanding the electronic structure and related physical properties of such diverse materials will depend on collaborations involving solid state physicists, geophysicists, polymer chemists, and molecular biologists [77].

How do we even make a start in attacking such complicated materials ? Of course, even the most complex biological molecules are composed of a relatively small number of components [77], and this already greatly simplifies the analysis. Moreover, we do not have to treat a complex material as a whole in order to understand some of its properties. It may be sufficient to consider certain key portions, first in isolation, and then in a more natural environment.

This is all reminiscent of the molecular clusters we discussed earlier.

In dealing with a complex crystal, we might find it advantageous to partition the unit cell into a suitable number of sub-cells, each of which could be dealt with individually in the spirit of a molecular cluster.

We wouId have to impose appropriate boundary conditions on each cluster, and we would attempt to determine these boundary conditions by self-consistent iteration or relaxation. In the limit of sub-cells containing a very small number of atoms, we come back to the Wigner-Seitz-Slater polyhedral cellular method [78].

One can't help wondering whether this method was abandoned too early, that is to say, before modern electronic computers were being used to full advantage in band structure calculations. It is conceivable that the cellular method, or some improved version of it, could be made competitive with - or superior to

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the APW and KKR methods with the non-muffin- tin and non-spherical terms taken into account.

In any event, the first step toward dealing with complex crystals is clearly to perfect our methods for dealing with simple crystals [44]. It is possible that two or more of the traditional first-principles methods can be combined to yield a method that is considerably more accurate and efficient than any of its individual constituents. Such ideas have been discussed recently by Dalton [44].

V. Condensed matter under unusualconditions [79].

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We are already familiar with studies of band structure under pressure, with the occurrence of super- conductivity in high pressure modifications of materials, with the enormous complexity of some high pressure modifications, and with the intricacies of electronic transitions and phase transitions at high pressure 1791-1827.

The study of the properties of materials under high pressure is particularly suitable for computational investigations, since such study involves the analysis of conditions that are not easily attainable in the laboratory or in an accessible portion of nature. Of particular importance is the study of the materials of the earth's mantle and core under pressure [83]-[85].

Our knowledge of the interior of the earth would be advanced considerably if we could perform reliable calculations of the elastic, magnetic, and thermal properties of known or assumed constituents under pressure.

Solid state ideas are being used increasingly in geomagnetism and seismology [83]-[85]. During the past few years, solid state methods and concepts were applied with considerable success to the study of neutron stars and white dwarfs, which represent extreme states of matter [86]-[88]. During the coming decade, collaborations involving solid state physicists, geophysicists, and astrophysicists will greatly advance our understanding of pulsars, interstellar dust grains, comets, asteroids, meteors, the planets, the moon, and the earth itself. By the end of the seventies, solid state astrophysics and solid state geophysics will be major fields in their own right.

During the seventies, we expect computers to play an ever more important role in such scientific studies, as the situations we deal with become more complex, and as our theoretical ideas become more sophisticated and require more incisive tests and verifica- tions [77], [89]. Computer graphics 191, [lo] will be used ever more widely as the potentials of this tech- nique become more generally appreciated. Finally, the imaginative use of computers will tie experimental and theoretical activities more closely together, leading to more rapid advances on both sides.

It will be an exciting decade.

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1281 NABARRO (F. R. N.), Theory of Crystal Dislocations (Oxford University Press, New York, 1967) ; HIRTH (J. P.) and LOTHE (J.), Theory of Dislo- cations (McGraw-Hill, New York, 1968).

[29] LIDIARD (A. B.), in reference [8] p. 363 ; LIDIARD (A. B.) and NORGETT (M. J.), in reference [8] p. 385.

[30] BULLOUGH (R.), in reference [8] p. 413.

1311 DUKE (C. B.), Tunneling in Solids (Academic Press, New York, 1969).

1321 MESSMER (R. P.) and WATKINS (G. D.), Phys. Rev.

Letters, 1970, 25, 656 ; WATKINS (G. D.), MESS-

MER (R. P.), WEIGEL (C.), PEAK (D.) and COR-

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[33] POPLE (J. A.) and BEVERIDGE (D. L.), Approximate Molecular Orbital Theory (McGraw-Hill, New York, 1970).

[34] HOFFMANN (R.), J . Chem. Phys., 1963, 39, 1397.

[35] ORTENBURGER (1. B.), RUDGE (W. E.) and HERMAN (F.), J. Non-Crystalline Solids (in press).

[36] HENDERSON (D.) and HERMAN (F.), J. Non-Crystal- line Solids (in press) ; HENDERSON (D.), also reference [8] p. 175.

[37] JOHNSON (K. H,), J. Chem. Phys., 1966, 45, 3085 ; JOHNSON (K. H.) and SMITH (F. C.. Jr). in C o m ~ u - tational ~ e t h o d s in Band T'heor; (Plenum PI&, New York, 1971), p. 377 ; JOHNSON (K. H.), in Advances in Quantum Chemistry, Vol. 7, edited by P.-0. Lowdin (Academic Press, New York, 1972) ; JOHNSON (K. H.), this volume.

[38] SLATER (J. C.), Phys. Rev., 1951, 81,385 ; KOHN (W.) and SHAM (L. J.), Phys. Rev., 1965, 140, A1 133 ; HERMAN (F.) and SKILLMAN, Atomic Structure Calculations (Prentice-Hall, Englewood Cliffs, New Jersey, 1963).

[39] SLATER (J. C.), in Computational Methods in Band Theory (Plenum Press, New York, 1971), p. 447.

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[41] SLATER (J. C.) and JOHNSON (K. H.), Phys. Rev.

(in press). These authors take the view that the SCF-Xa method, unlike most other SCF methods, automatically includes long-range correlation, so that the energy of a system as a function of internuclear positions automatically reduces to the proper values at infinite internuclear distances.

See also : CONNOLLY (J. W. D and JOHNSON (K. H.), Chem. Phys. Letters, 1971, 10, 616.

Here, the SCF-Xa cluster method is regarded as an approximate version of the H F SCF method which has considerable advantages for determining molecular properties. In particular, one can get results nearly as accurate as those given by exact H F SCF calculations, but with the expen diture of at least two orders of magni- tude less computer time. The SCF-Xa cluster method is competitive, so far as computer time is concerned, with semi-empirical methods such as CNDO, and hence may be regarded as an alternative to such semi-empirical methods in studies of large molecules.

[42] HERMAN (F.) and SCHWARZ (K. H.), in reference [8]

p. 245 ; SCHWARZ (K. H.) and HERMAN (F.), this volume ; for earlier work on the XaB method, see reference [43].

[43] HERMAN (F.), VAN DYKE (3. P.) and ORTENBURGER (I. B.), Phys. Rev. Letters, 1969, 22, 807 ; HER-

MAN (F.), ORTENBURGER (I. B.) and VAN DYKE (J. P.), Intern. J. Quantum Chem., 1970, 3S, 827 ; ORTENBURGER (I. B.) and HERMAN (F.), in Computational Methods in Band Theory (Plenum Press, New York, 1971), p. 469.

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C3-20 F. HERMAN

[44] DALTON (N. W.), in reference 181 p. 113.

[45] MOTT (N. F.), Adv. Phys., 1967, 16, 49 ; Phil. Mag., 1968, 17, 1259 ; Comments Solid State Phys., 1970-71, 3, 123.

[46] MOTT (N. F.), editor, Amorphous and Liquid Semi- conductors (North-Holland, Amsterdam, 1970).

1471 TAUC (J.), editor, Amorphous and Liquid Semicon- ductors (Plenum Press, New York, 1971).

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[49] HORI (J.), Spectral Properties of Disordered Chains and Lattices (Pergamon Press, Oxford, 1968).

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[51] ZIMAN (J. M.), J. Phys. C (Pvoc. Phys. Soc.), 1968, 1, 1532 ; J. Phys. C (Solid St. Phys.), 1969, 2, 1230, 1704.

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[53] EGELSTAFF (P. A.), An Introduction to the Liquid State (Academic Press, London, 1967).

[54] MARCH (N. H.), Liquid Metals (Pergamon Press, Oxford, 1968).

1551 DE GENNES (P. G.), Comments Solid State Phys., 1969, 1, 213.

[56] PANCHEV (S.), Random Functions and Turbulence (Pergamon Press, Oxford, 1971). See also references, p. 425 ff.

[57] PASK (J. A.), editor, An Atomistic Approach to the Nature and Properties of Materials (Wiley, New York, 1967).

[58] FLORY (P. J.), Statistical Mechanics of Chain Mole- cules (Wiley-Interscience, New York, 1969) ; FERRY (J. D.), Viscoelastic Properties of Poly- mers, Second Edition (Wiley, New York, 1970).

[59] SOVEN (P.), Phys. Rev., 1967, 156, 809 ; 1969, 178, 11 36.

[60] VELICKY (B.), KIRKPATRICK (S.) and EHRENREICH (H.), Phys. Rev., 1968, 175, 747 ; see also : SCHWARTZ (L.) and EHRENREICH (H.), Annals Phys., 1971, 64., 100.

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[62] POLK (D.), J. Non-Crystalline Solids, 1971, 5 , 365.

1631 SHEVCHIK (N. J.) and PAUL (W.), J. Non-Cvystalline Solids (in press).

[64] BERNE (B. J.) and FOR~TER (D.), Ann. Rev. Phys.

Chem., 1971,22,563.

[65] HAMMERSLEY (J. M.) and HANDSCOMB (D. C.), Monte Carlo Methods (Methuen, London, 1967).

[66] WEAIRE (D.), Phys. Rev. Letters, 1971, 26, 1541 ; WEAIRE (D.) and THORPE (M. F.), Phys. Rev., 1971, B 4,2508 ; THORPE (M. F.) andWEA1RE (D.), Phys. Rev., 1971, B 4 , 3518, Phys. Rev. Letters, 1971,27, 1581.

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[68] MARCH (N. H.) and STODDART (J. C.), Repts. Prog.

Phys., 1968, 36. Part 11, 533.

1691 EDMISTON (C.) and RUEDENBERG (K.), Rev. Mod.

Phys., 1963, 35, 457 ; J. Chem. Phys., 1965, 43, S97 ; in Quantum Theory of Atoms, Molecules and the Solid State (Academic Press, New York,

1966), p. 263 ; ENGLAND (W.) and RUEDENBERG (K.), Theoret. Chim. Acta (Berlin), 1971, 22, 196.

For other formulations, see reference [40], pp. 138-141.

[70] PHILLIPS (J. C.), Covalent Bonding in Crystals, Mole- cules and Polymers (University of Chicago Press,

1969).

[71] HERMAN (F.), J. Electronics, 1955, 1, 103. The perturbation method introduced here to relate diamond- and zinc-blende-type crystal band structures has been used recently by several authors to relate zinc-blende- and chalcopyrite-type crystal band structures.

[72] AKHMANOV (S. A.) and KHOKHLOV (R. V.), Soviet Phys. Uspekhi, 1968,11,394 ; BLOEMBERGEN (N.), Comments Solid State Phys., 1969/70,2,119, 161 ; WEMPLE (S. H.) and DIDOMENICO (M., Jr), Phys. Rev., 1971, B 3, 1338.

[73] AHRENS (L.), Distribution of the Elements in Our Planet (McGraw-Hill, New York, 1965).

1741 RINGWOOD (A. E.), Phys. Earth Planetary Znteviovs, 1970, 3, 109.

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[76] KELLER (A.), Repts. Pvog. Phys., 1968, 36. Part 11, 623.

[77] WATSON (J. D.), Molecular Biology of the Gene, Second Edition (W. A. Benjamin, New York, 1970) ; HANDLER (P.), editor, Biology and the Future of Man (Oxford University Press, New York, 1970). BRESLER (S. E.), Soviet Phys. Uspekhi, 1970,12,534.

[78] SLATER (J. C.), Phys. Rev., 1934, 45, 794 ; Rev. Mod.

Phys., 1934, 6, 209.

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[SO] PAUL (W.) and WARSCHAUER (D.), editors, Solids Under Pressure (McGraw-Hill, New York, 1963) ; DRICKAMER (H. G.), Solid State Phys., 1965, 17, 1 ; Comments Solid State Phys., 1970, 3, 53.

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1821 VOROPINOV (A. I.), GANDEL'MAN (G. M.) and POD-

VAL'NYI (V. G.), Soviet Phys. Uspekhi, 1970, 13, 56.

[83] STACEY (F. D.), Physics of the Earth (Wiley, New York, 1969) ; KAULA (W. M.), An Introduction to Planetary Physics : The Terrestrial Planets (Wiley, New York, 1968).

[84] STRAHLER (A. N.), The Earth Sciences, Second Edi- tion (Harper and Row, New York, 1971).

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[87] BAYM (G.), in reference [8] p. 267.

[88] CAMERON (A. G. W.), Ann. Rev. Astvon. Astrophys., 1970, 8, 170 ; OSTRIKER (J.), ibid, 1971, 9, 353 ; GINZBURG (V. L.), Soviet Phys. Uspekhi, 1971, 14, 83.

[89] GINZBURG (V. L.), Soviet Phys. Uspekhi, 1971,14,21.