Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200

126

G. DEL RE

case here considered eqn 39 is a reasonable approximation of the exact expression.

Two remarks may be added here. First, as was shown in a preceding paper [3], a correction must be added to the expression of the electron chemical potential whenever the given molecule is in the presence of another molecule or of a solid surface. Second, although we have referred to the scheme and to Löwdin's population analysis, no implication is made that the above analysis depends on either assumption. As has been mentioned, it has been designed for general all-valence SCF schemes. Also the introduction of Mulliken's population analysis is straightforward, since in that case

and the whole derivation above can be applied to the Mulliken population without any difficulty.

Acknowledgement. The Author thanks the Italian National Research Council (CNR) and the Italian Ministry of Universities (MURST) for support.

References

1.J. Hinze, M. A. Whitehead, and H. H. Jaffé, J. Am. Chem. Soc., 85, 184

(1963) .

2.G. Del Re, J. Chem. Soc. Far. Trans., 77, 2067-2076 (1981).

3.R. S. Mulliken, J. Chem. Phys., 2, 782 (1934).

4.G. Berthier, Ph. Millié, and A. Veillard, J. Chimie Phys. , 62, 8 (1965).

5.G. De Brouckère, Theor. Chim. Acta, 19, 310 (1970).

6.G. Del Re, P. Otto, J. Ladik, Isr. J. Chem., 19, 265 (1970).

7.R. S. Mulliken, J. Am. Chem. Soc., 72, 600 (1950); 74, 811 (1952).

8.G. Del Re, P. Otto, J. Ladik, Int. J. Quant. Chem., 37, 497 (1990).

9.R. T. Sanderson, Science, 114, 670 (1951).

10.A. Streitwieser jr., Molecular Orbital Theory for Organic Chemists, Wiley & Sons, New York, 1961.

11.G. Berthier, A. Daoudi, M. Suard, J. Mol. Struct. (Theochem), 179, 407

(1988).

12.G. Del Re, J. Mol. Struct. (Theochem), 169, 487 (1988).

13.J.-Y. Saillard, R. Hoffmann, J. Am. Chem. Soc., 106, 2006 (1984), 74, 811 (1952).

14.L. C. Cusachs, J. Chem. Phys., 43, S157 (1965).

Quasicrystals and Momentum Space

J.L. CALAIS

Quantum Chemistry Group, University of Uppsala

Box 518, S - 75120 - Uppsala, Sweden

1. Introduction

In November 1984 the world of crystallography was thoroughly shaken by the news that

"forbidden" peaks characteristic of icosahedral symmetry had been recorded in electron diffraction diagrams of an Al-Mn-alloy [1]. According to "classical" crystallography long

range order is compatible with rotations through multiples of			but not with
rotations through	. The point group of a	space group	must be one of the 32
crystallographic point groups and the icosahedral		group is certainly not one of them.

Scientific results of that nature are among the most interesting ones, since they open up qualitatively new perspectives.

A crystal is an extended system with (in principle) perfect long range order, which is invariant under all operations of a certain space group. At the other extreme we have disordered systems with a "completely" random arrangement of its constituent atoms. Intermediate cases with more or less short range order have been known for a long time [3]. What was unexpected in the paper by Shechtman et al. [1] was the combination of long range order and a non crystallographic point group. Already in 1902 the French mathematician Esclangon [4] pointed out, however, that arrangements which are aperiodic but non random are possible. And even though the paper by Shechtman et al. [1] must be regarded as the one which opened up this new field of crystallography, it seems that some Japanese results 20 years earlier [5] should also be interpreted as providing experimental evidence for the existence of quasi-periodic structures.

During the nearly ten years which have passed since the appearance of the "Shechtman paper" a large amount of both experimental and theoretical research has been carried out on quasiperiodic structures. For more material about quasicrystals we refer to a paper in La

Recherche by the French collaborator in the Shechtman team [6], to a thesis by Dulea [7J, and to a survey paper with a large number of references [8].

Last year a magnificent paper by Mermin appeared in the Reviews of Modern Physics [9], as the (so far) crowning contribution to a series of papers describing nothing less than a reformulation of crystallography [10 - 18]. Emphasising reciprocal space concepts Mermin and his collaborators have been able to treat both "classical" crystals and quasicrystals with the same method. As is often the case with truly original work this first of all throws new light on the theory of the "ordinary" space groups, which leads to a deeper understanding of notions and relationships believed to be well known. Then it provides a straightforward

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Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 127–138. © 1996 Kluwer Academic Publishers. Printed in the Netherlands.

128	J. L. CALAIS

classification of both crystals and quasicrystals, as well as incommensurately modulated crystals and quasicrystals. The procedure offers a simple explanation of how seemingly contradictory concepts can in fact be combined in a perfectly consistent manner. This major achievement definitely deserves to become better known by all physicists and chemists who work with extended systems. One of the aims of the present paper is to contribute towards that goal.

Over the last few years there has been an increasing interest in using momentum space concepts for both molecules and polymers and also to perform explicit calculations directly in momentum space rather than making the detour over position space [19, 20].

Conceptually it is very valuable to work so to speak in parallel in position and momentum space, since corresponding concepts often help to "clarify each other". In the present paper we want to confront - in a preliminary way - the procedures proposed by Mermin and collaborators with certain momentum space notions. We expect first of all to get a better understanding of these procedures. And more specifically we want to use Mermin's results for investigating the symmetry properties of momentum wave functions for quasiperiodic systems. In this connection it is important to distinguish the closely related by still different notions of "reciprocal space" and "momentum space". In a certain sense these terms denote the same object. Momentum space and position space refer to different representations of wave functions related by Fourier transforms for any types of systems. The term

"reciprocal space", on the other hand is normally used only in connection with solids. Until relatively recently this concept has been used only about crystals: solids with long range order which are invariant under one of the 230 space groups that can exist in three dimensions.

Mermin's "generalised crystallography" works primarily with reciprocal space notions centered around the density and its Fourier transform. Behind the density there is however a wave function which can be represented in position or momentum space. The wave functions needed for quasicrystals of different kinds have symmetry properties - so far to a large extent unknown. Mermin's reformulation of crystallography makes it attractive to attempt to characterise the symmetry of wave functions for such systems primarily in momentum space.

In the next setion we review some key concepts in Mermin's approach. After that we summarise in section III some aspects of the theory of (ordinary) crystals, which would seem to lead on to corresponding results for quasicrystals. A very preliminary sketch of a study of the symmetry properties of momentum space wave functions for quasicrystals is then presented in section IV.

2. Indistinguishability and Identity

As stressed by Mermin and collaborators [9 - 18] it is far too restrictive to define the structural indistinguishability of two mesoscopically homogeneous materials with reference to identical densities. Instead of the densities themselves one should study the properties of the correlation functions,

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Here V is the volume of the Born-von Kármán region, i.e. that part of position space which is repeated as a result of the fundamental periodic boundary conditions. The integration in (II. 1) is carried out over that region, which we denote by BK.

For n = 1 we have for example,

i.e. the average density of the system.

Two densities		and	are said to be indistinguishable if all their correlation functions
and	are identical. As		shown by Mermin and collaborators their Fourier transforms
then have some very interesting properties. We can always expand a density				in plane
waves,

The wave vectors k can be expressed in terms of any basis vectors we choose. At the moment there is neither a direct nor a reciprocal lattice. Using (II.3a) in (II. 1) we see that the Fourier components of two indistinguishable densities can differ only by a phase factor:

The gauge function		is linear in its argument:
A related concept is that of phase function			which relates the Fourier components of
a density	and those of a transformed density obtained by letting a point group
operation g work on r:

If g is an element of the point group of the material meaning that and are indistinguishable for all elements g in that group, corresponding Fourier components can differ only by a phase factor:

A "generalised" space group is specified by a point group and the associated phase functions The ordinary space groups constitute special cases of these generalised

space groups.

Since (gh)k = g(hk) we get with (II.7)

which implies

130	J. L. CALAIS

From this group compatibility condition Mermin and his collaborators have derived both

all the "ordinary" crystallographic and the quasicrystallographic space groups.

If gk = k,	(II.7) implies that either the Fourier component		vanishes or the phase
function	is an integer or zero. Another way of expressing that important result is to
say, that given a phase function		those wave vectors k, for which that function is not
equal to an integer or zero, determine a set of vanishing Fourier components				The

number of vanishing terms in the Fourier expansion (II.3a) of the density is a kind of measure of the degree of symmetry in the system.

3. Momentum space characteristics of crystals

The traditional characterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for such an operation is where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get

The details of the operation Rr can be further specified by the matrix which represents the operation R in a suitably chosen coordinate system [2], in which also the vector r is expressed. For the operation on a function of r we need the inverse of the space group operation,

We thus have for an arbitrary function f(r),

A crystal characterised by a space group G has an electron density p(r) which is invariant

under all elements	of G:
The electron density is the diagonal element of the number density matrix		, i.e the
first order reduced density matrix after integration over the spin coordinates,:

A transformation of the number density matrix N under a space group operation means that both variables are transformed:

The following relation and its inverse hold between the elements of the number density matrices in momentum and position space [21]:

The momentum space counterpart of (III.6) can therefore be written,

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Thus the point group part of the operation works on the momentum coordinates and the translation part gives rise to a phase factor. We notice that this phase factor reduces to 1 in the diagonal elements, or in general when the difference between the the two arguments of

is a reciprocal lattice vector.

If the elements of the number density matrix in position space are invariant under all operations of the space group, i.e. if

we get with (III.8), that their momentum space counterparts satisfy

The momentum distribution, i.e. the diagonal element of (III. 10) then satisfies

The reciprocal form factor [22] is the Fourier transform of the momentum distribution,

Using (III. 11) we see that the reciprocal form factor of a crystal which is invariant under a space group, satisfies the relations,

for all point group elements R of the space group.

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly.

The number density matrix for a crystal with translation symmetry can be written in terms of its natural orbitals [23, 24], as

This is the most general expression obtained from a set of natural spin orbitals written in spinor form as

The orbitals are Bloch functions labeled by a wave vector k in the first Brillouin

zone (BZ), a band index µ , and a subscript i indicating the spinor component. The

combination of k and can be thought of as a label of an irreducible representation of the space group of the crystal. The quantity is the occupation function which measures

the degree of occupation at wave vector k in band

132	J. L. CALAIS

The momentum space counterpart of the Bloch orbital

vanishes unless k-p is a reciprocal lattice vector K [25]. In other words this function of the momentum variable p labeled by the wave vector k, vanishes except when p = k, and at equivalent points p = k+K in the other Brillouin zones.

We expand the density (III.5) in a Fourier series,

Here BK stands for "Born-von and denotes the basic region of periodicity associated with the periodic boundary conditions. That "large period" must be carefully distinguished from the "small period" associated with the crystal lattice. BK contains N cells of volume and thus has the volume Wave functions have the "large

period", but quantities like the density and the crystal potential have the "small period".

We first notice the following connection between the Fourier component (III. 17b) and the density matrix in momentum space, obtained from the inverse of (III.7):

Combining the inverses of (III. 14) and (III. 16) we get the natural expansion for a general element of the number density matrix in momentum space:

Here the component of the number density matrix associated with the wave vector k is thus

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133

Substituting (III. 19) in (III. 18) and using the special properties of (III. 14) we can then write the Fourier component of the density as

A more condensed expression is obtained using (III. 14) and (III.17b):

Here	is the Fourier component of the square of the absolute value of the Bloch
orbital	which can be written as a product of a plane wave and a function

having the periodicity of the lattice:

Using (III. 16) we can also write this Fourier component in terms of the momentum space orbitals as

If the density is invariant under the space group operation {R|m} we have with (III.4) and (III.17b),

134	J.L. CALAIS

It is important to distinguish between symmetry properties of wave functions on one hand and those of density matrices and densities on the other. The symmetry properties of wave functions are derived from those of the Hamiltonian. The "normal" situation is that the

Hamiltonian commutes with a set of symmetry operations which form a group. The eigenfunctions of that Hamiltonian must then transform according to the irreducible representations of the group. Approximate wave functions with the same symmetry properties can be constructed, and they make it possible to simplify the calculations.

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations according to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows

This implies that a density built up from such Bloch functions [cf (III.5) and (III.14)] is invariant under all such translations [the "little" period]:

Corresponding	relations for arbitrary space group elements ' ~ '	show that if the
orbitals	which make up the density transform asthe irreducible representations of

the space group, the density is invariant under all the operations of that group.

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions? In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space group, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetry.

The symmetry properties of the density show up experimentally as properties of its

Fourier components If those components vanish except when the wave vector k

equals one of the lattice vectors K of a certain reciprocal lattice, the general plane wave expansion of the density,

reduces to (III.17a). Since times an integer, we then have

If (III.24) holds we get the corresponding result for arbitrary space group elements

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135

The symmetry properties of the momentum space wave functions can be obtained either from their position space counterparts or more directly from the counterpart of the Hamiltonian in momentum space.

4. Momentum space characteristics of quasicrystals

One of the main points in the papers by Mermin and his collaborators [9 - 18] is the insistence on the primacy of reciprocal space. The properties of the Fourier transform of the density rather than the density itself determine those properties which are of importance for "generalized" crystallography. As pointed out by Mermin that point view was stressed in a paper by Bienenstock and Ewald already in 1962 [26].

Irrespective of the type of extended system we are interested in we impose periodic boundary conditions in position space - "the large period": BK. Such conditions imply a discretisation of momentum and reciprocal space |27] which means that integrations are replaced by summations:

The discrete momenta can be written as

where the are positive or negative integers or zero, and the very large even integer G

characterizes the BK region The reciprocal basis vectors do not require any

actual physical lattice, but can be seen as just providing a suitable framework. We have used (IV. 1) several times in the previous section, but there we had lattices both in direct and in reciprocal space, and then this procedure may have seemed more natural. In the present section there is definitely no lattice in direct space and the "lattice" in reciprocal space may be of a different nature from the ordinary ones. Because of the periodic boundary conditions, (IV. 1) should still be used, however.

The Fourier expansion of the density in an extended system which does not have any particular symmetry is

This sum over all reciprocal space vectors of the form (IV.2) should be carefully distinguished from the expansion (III.4) of the density of a periodic crystal. If the density has the "little period", the expansion (IV.3) reduces to a sum over all reciprocal lattice vectors. The general case (IV.3) and the periodic case (III.4) actually represent two extreme cases. The presence of "more and more symmetry" in the density can be gauged

by the disappearance of more and more Fourier components

in (IV.3). If some of the

Fourier components in (IV.3) vanish, but not necessarily all which do not correspond to a set of reciprocal lattice vectors, we have a Fourier expansion of a density with another type of long range order than the one known from traditional crystals. There are quasicrystals, incommensurately modulated crystals or incommensurately modulated

quasicrystals [9].