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

Using the inverse of (III.7) we can write the density of an arbitrary extended system as

which means that the Fourier component of the density can be written

This should be compared to (III. 18) where the role of k in (IV.5) is played by a reciprocal lattice vector K.

Mermin's conceptual starting point is a set of vectors k in reciprocal space which correspond to sharp Bragg peaks in the experimental diffraction pattern. The non vanishing Fourier components are then to be found for wave vectors which can be characterized as the set of all integral linear combinations of a certain finite set of D basis

of the 32 crystallographic point groups. If we have a non crystallographic point group the rank D of the lattice can be larger than three. Such a system is called a quasicrystal. A system with a crystallographic point group and a lattice with a rank D higher than three is called an incommensurately modulated crystal.

An important and interesting question is obviously whether for quasicrystals and incommensurately modulated crystals there is anything corresponding to the Bloch functions for crystals. Momentum space may be a better hunting ground in that connection than ordinary space, where we have no lattice. Not only is there no lattice, one cannot even specify the location of each atom yet [8].

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by

space will be non vanishing only when the momentum p equals k plus a vector of the lattice L.

The problem is to "translate" the fact that certain terms are absent in the expansion (IV.3) to symmetry properties of the density in the sense of transformation properties under certain operations. We have a density with non vanishing Fourier components only for such wave vectors k which belong to the lattice L:

Mermin [9, 18] has given a recipe for the construction of a set of Fourier components for a density characterised by a certain space group. The space group is then specified by a point group G, a lattice of wave vectors in the sense discussed above, and a set of phase

transform of a function with no symmetries whatever. That last condition is imposed in order to avoid that the density obtained from (IV.7) gets any symmetries which are not

associated with the point group G, and also to prevent from vanishing on a set of wave

vectors so large that the lattice is thinned out to a sublattice for which the space group would have a different character. The components (IV.7) transform under the elements of the point group according to the fundamental rule (II.7).

An effective one electron Schrödinger equation with a local potential V(r) in position space, (atomic units),

corresponds in momentum space to the following equation [19],

Wave functions in position and momentum spacce are related as in (III. 16), and the Fourier component of the potential is

In density functional theories the potential is determined by the density, and consequently its Fourier components are related to those of the density. One can therefore connect the symmetry properties of the momentum functions, in other words the transformation

components of the density, (11.7).

What has been sketched here is obviously just the bare framework of a general investigation of the symmetry properties of momentum space functions in quasicrystals. With all the information available in the papers by Mermin and collaborators it should however be a very tempting enterprise to go ahead along the lines sketched and learn about the details of the symmetry properties of those wave functions - both in momentum and in positition space - which will be needed in quasiperiodic extended systems.

References

1.D.Shechtman, I. Blech, D. Gratias and J.W. Cahn, Phys. Rev. Letters, 53, 1951 (1984).

2.J. F. Cornwell, Group Theory in Physics. Vol. 1, Academic Press, London (1989).

3.See e.g. Electrons in Disordered Metals and at Metallic Surfaces, P. Phariseau, B.L Györffy and L. Scheire Eds., NATO Advanced Study Institute Series, Series B: Physics, Volume 42, Plenum Press New York and London (1979).

4.M.E. Esclangon, C.R. Acad. Sci. (Paris) 135, 891 (1902).

5.a S. Tanisaki, J. Phys. Soc. Japan , 16, 579 (1961).

b Y. Yamada, S. Shibuya and S. Hoshino, J. Phys. Soc. Japan , 18, 1594 (1963).

Quantum Chemistry Computations in Momentum Space

M. DEFRANCESCHI (1), J. DELHALLE (2), L. DE WINDT (1), P. FISCHER (1, 3), J.G. FRIPIAT (2)

(1)Commissariat à l'Energie Atomique, CE-Saclay, DSM/DRECAM/SRSIM, F-91191 Gif-sur-Yvette Cedex, France

(2)Facultés Universitaires Notre-Dame de la Paix, Laboratoire de Chimie Théorique

Appliquée, Rue de Bruxelles, 61, B-5000 Namur, Belgium

(3)Université de Paris-Dauphine, Ceremade, Place Maréchal de Lattre de Tassigny,

F-750I6 Paris, France

1. Introduction

In quantum mechanics, the state of a physical system is described by a vector of an

Hilbert space, represented by a linear superposition of eigenvectors of Hermitian

operators which result from a particular choice of a maximal set of commuting observables [1,2]. The various representations obtained in this way are connected by a

generalized Fourier transformation. The so-called Schrödinger method, normally used for

systems of electrons and nuclei, starts in an Hilbert space by taking the components of particle coordinates as a maximal set; consequently, the state function of the system is

written in the coordinate representation, and this leads to the familiar Schrödinger equation for determining the possible energies of atoms and molecules as eigenvalues of the total Hamiltonian operator in position space. The Schrödinger equation can be expressed in

representation becomes the ordinary Fourier transform of the state function in position

space, with appropriate factor :

Taking the Fourier transform of the ordinary Schrodinger equation yields, in atomic units,

139

Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 139–158.

© 1996 Kluwer Academic Publishers. Printed in the Netherlands.

Tsoucaris, decided to treat by Fourier transformation, not the Schrödinger equation itself, but one of its most popular approximate forms for electron systems, namely the Hartree-

Fock equations. The form of these equations was known before, in connection with electron-scattering problems [13], but their advantage for Quantum Chemistry calculations was not yet recognized.

numerical molecular orbitals computations, i.e. without atomic basis functions contrary to what happens in r-space where it is difficult to obtain accurate Hartree-Fock solutions for atoms, molecules and solids due to the need of representing the solutions in terms of a finite basis of known functions, e.g. the linear combination of atomic orbitals (LCAO)

approximation. For chemists interested in polyatomic molecules, the momentum method

is quite attractive because it is not limited to systems whose geometry determines the

coordinates to be used for integrating the position space equations, as for example polar

and/or spheroidal coordinates for diatomic molecules, see e.g. Ref. [15]. During the last

years, we have contributed to demonstrate that direct momentum space calculations are in

principle feasible for any molecule by studying hydrogen systems of increasing

complexity : the ground state at the SCF and MC-SCF level [16], an open-shell

system [17] and a chain of H atoms including an infinite number of electrons and nuclei

[18,19]. More complex systems have also been studied : atoms up to neon [20-28], cations [22,23, 28-31], anions [22, 23,27, 28], symmetric molecules [16, 17,32-36] as

well as asymmetric molecules such as or HF [38].

The advantages of the momentum approach are not only limited to the opportunity for direct numerical calculations for chemical systems, but it also offers the prospect of

selecting better bases of atomic functions on which rely almost all first principle quantum

mechanical calculations.

2. MOMENTUM SPACE EQUATIONS FOR A CLOSED-SHELL SYSTEM

The Fourier transformation method enables us to immediately write the momentum space equations as soon as the SCF theory used to describe the system under consideration

allows us to build one or several effective Fock Hamiltonians for the orbitals to be

determined. This includes a rather large variety of situations:

Closed-shell systems as defined in the standard Hartree-Fock theory [39-40].

Unrestricted monodeterminantal treatments using different orbitals for different spins

for open-shell systems (free radicals, triplet states, etc.) [41,42].

Roothaan open-shell treatments involving a closed-shell subsystem and outer unpaired

electrons interacting through two-index integrals of Coulomb and exchange type only

[43].

MC-SCF treatments written in terms of coupled Fock equations [44]. The simplest

mixing [46], or bonding-antibonding molecular problems [47], and more generally the

Clementi-Veillard electron-pair MC-SCF theory [48].

SCF treatments for infinite chains having translational symmetry [49,50],

In the recent past, we have investigated and published examples illustrating the different

Refs. [18, 19] a SCF treatment for infinite chains and finally in Ref [16] a MC-SCF

treatment were proposed.

In this contribution our purpose is to review the principles and the results of the

momentum space approach for quantum chemistry calculations of molecules and

polymers. To avoid unnecessary complications, but without loss of generality, we shall

consider in details the case of closed-shell systems.

2.1. RESTRICTED HARTREE-FOCK EQUATIONS

Since both position and momentum formulations contain exactly the same information, it

is convenient to start from the familiar position space expression and express it in

Hamiltonian including a kinetic term and an effective potential in which the electron-

nucleus attraction is balanced by the Coulomb-exchange potential approximating the real

electron-electron interaction. In atomic units, we have :

and by applying the Fourier transform to Eq. 3, we get the momentum space RHF equation. The linearity of the Fourier transformation allows a separate treatment of each of the terms occurring in the Hartree Fock equation.

Kinetic energy term.

The integral in Eq. 4 is readily evaluated if is replaced by its inverse Fourier

transform. After rearrangement of the terms, one finds that the integral over r yields the delta function Carrying out the remaining integral yields the final expression.

p will be used to denote the length of vector p.

Nuclear attraction, electron-electron repulsion, and exchange terms.

and

final expression of the nuclear term and the electron-electron repulsion term in p- representation are summarized below :

Using the convolution theorem the content of the square brackets in Eq. 9 is rewritten as :

Defining a quantity Wij(q),

and introducing it in Eq. 9 leads to the expression for the two-electron term :

With the above results, it is possible to write the expanded momentum space form of the

Hartree-Fock equations :

The equations to be fulfilled by momentum space orbitals contain convolution integrals

form factor F and the interaction terms Wij defined in terms of current momentum

coordinates are the momentum space counterparts of the core potentials and Coulomb

and/or exchange operators in position space. The nuclear field potential transfers a

momentum to electron i, while the interelectronic interaction produces a momentum

transfer between each pair of electrons in turn. Nevertheless, the total momentum of the

whole molecule remains invariant thanks to the contribution of the nuclear momenta [7].

2.2ORBITAL AND TOTAL ENERGIES

The calculation of in momentum space is analogous to that in position space. Starting

position space where the different contributions can be expressed as follows :

Kinetic energy term. Its expression is straightforward to write :