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

196	J. M. ANDRÉ ET AL .

we obtain:

In all the other schemes (UHF, PHF, EHF), the dissociation limit is the correct one

corresponding to two neutral hydrogen atoms (2H-); each FSGO-hydrogen atom energy is thus obtained by the simple variational procedure:

dissociation limit for two hydrogens:

As a final comment, it is interesting to note that this FSGO study of the hydrogen molecule offers a new and simple illustration of the behavior of sophisticated Hartree-

Fock schemes like UHF, PHF and EHF. Furthermore, it provides a very efficient numerical example of instabilities in the standard Hartree-Fock method. It is important to see that the UHF, PHF and EHF schemes all correct the wrong RHF behavior and lead to the correct dissociation limit. However, the UHF and PHF schemes only correct the wave function for large enough interatomic distances and the effect of projection in the PHF scheme even results in a spurious minimum. The EHF scheme is thus the only one which shows a lowering of the energy with respect to RHF for all interatomic distances.

3.Subminimal basis set Hartree-Fock-type calculations of the hydrogen molecule in an external electric field.

When applying an external electrical field to the FSGO model of the hydrogen molecule, one expects that the floating gaussian will be moved in accordance with the polarity of the field, i.e., displaced towards the positive pole. Thus, near equilibrium internuclear distances, a minimum should be obtained close to the middle of the molecule. On the other hand, continuing to move the floating gaussian towards the positive pole, a barrier should appear close to the hydrogen atom the gaussian is floating towards. After having passed that barrier, the “energy catastrophe” of the unbound perturbing potential should produce an infinitely negatively stable position. This is the type of behaviour which is listed in Table 3 and illustrated in Figure 3. For the equilibrium internuclear distance (R =

1.474 a.u.) and the optimal exponent		we compute the energy as a
function of the orbital position	for various strength of the external electrical field (F
and 0.5 a.u.). The energy formulae can be obtained from the
Hamiltonian for the hydrogen molecule		in the electric field	(z being the axis

of the molecule):

FSGO HARTREE-FOCK INSTABILITIES OF HYDROGEN		197
where	is the Hamiltonian of the isolated molecule and	is the z component of the
dipole moment operator (in a.u.)

with the summation over nuclei and the i	summation over electrons.		stands for the
nuclear charges, while z denotes the z-coordinate.
In the RHF case with the doubly occupied	orbital	one obtains the mean value of the
energy:

Thus, we observe that when applying a finite electric field to a molecule, in addition to the instability with respect to the internuclear distance R, one obtains an instability connected to a change of the electric field strentgh F, as intuitively explained previously. In a more general way, the possible instabilities can be rationalized as follows: at some

fixed values of the internuclear distance R and the FSGO exponent a the energy may be viewed as a function, , of the FSGO orbital position only. In view of the cylindrical symmetry of the problem, the position is determined by the z coordinate of the FSGO

center,	_ The formula for the function	is:
where	is a constant equal to:

which contains the mean value of the kinetic energy, the electron-electron and the nuclear repulsions, as well as the nuclear dipole moment interaction with the electric field. As one can see, if the field is positive, the energy goes to when

When expressing the nuclear attraction integrals in the FSGO basis, one has explicitly:

where Fo is the standard error function:

198	J. M. ANDRÉ ET AL.

The optimal positions of the FSGO orbital correspond to the minima of the function They may be found by imposing the necessary condition:

The derivative of the error function is given by:

where:

By performing the derivative of the error function, one finds easily that the optimal position of the FSGO necessarily satisfies the following equation:

FSGO HARTREE-FOCK INSTABILITIES OF HYDROGEN	199
where:
It is convenient to define a further function,	for the discussion which follows, in

which

200	J. M. ANDRÉ ET AL .

At first sight, we are forced to solve this equation numerically, but its overall form allows a qualitative insight into the number of solutions and their approximate values. For example, one easily see that S represents a sum of two identical quasi-atomic (onedimensional) functions each centered on the corresponding hydrogen nucleus. The functions are quite similar to Gaussian functions, but they differ by their onedimensionality and by a different radial dependence. Indeed, instead of the usual

exponential behaviour, one has the		function that is positive, even with respect to
z, and, as it seen from the previous equation, has its maximum at			while it vanishes
for large z.
This information is sufficient to analyze the qualitative behaviour of			Indeed, two
limiting cases may be considered.
For one limiting case (small values of		i.e., close to equilibrium distances), the
function can be easily evaluated. By putting		it looks like a single one-dimensional
orbital centered in the symmetry center of the molecule. As one can see for
there is only one solution,	of the	equation and it corresponds obviously to a

single minimum of the energy (symmetric solution) as seen in Figure 3. For very large values of the equation cannot be longer satisfied and does not attain negative values. This corresponds to the energy changing monotonically qualitatively. The energy

minimum at	is clearly unstable when F increases. As one sees from Figure 4 for
moderate values of F, one should observe two		values satisfying the equation. One of

them corresponds to a minimum and the other to a maximum of the energy. One example of a single minimum is given in Figure 3 for the cases, and 0.25 a.u. Thus, there is a certain critical electric field value for which the energy curve changes qualitatively from the one having a single minimum to that with no minimum at all. From Figure 3, one easily see the existence of one and no minima in the curve

according to the strength of the field. For large values of F, there is no root in the S- function and no minimum is found for the This is the case for in Figure 3. Since the coordinate origin has been placed at the center of the molecule, the contribution

of the nuclei to the dipole moment is 0 and the total dipole moment is equal to					(in
a.u.). At large values of		the energy of the molecule mainly comes from the interaction
of the dipole moment with the electric field and therefore has the asymptotic form
From Figure 3, it is also seen that the energy changes linearly with				for large	For
small	values, one has a strong influence of the nuclei (through the first term of the
	equation). Also, it is easily proved that the asymptotic behaviour of the energy
comes	from its linear dependence of		for large values of this variable and that its

asymptotic slope is 2F.
The other limiting case corresponds to the limit of large values of			In order to remain
concise, this has not been illustrated. For		one finds three values of	for which
The first one is _	and the remaining two are close to		and

(the two positions of the nuclei). They correspond to the broken symmetry behaviour

analyzed in section 2. For very large values of	no value of	satisfies the equation,
and the function does not attain negative values. This means that the energy			has
no extremum.

FSGO HARTREE-FOCK INSTABILITIES OF HYDROGEN

201

For intermediate values of F, either four or two	values satisfy the equation. This
corresponds to one or two energy minima, respectively,	both of which are unstable when

F increases. When broken symmetry solutions are observed in the absence of external

electric field	one might expect a single minimum appearing either at nucleus a
or b, by turning on the electric field. In fact, for			the minimum eventually appears
only at the nucleus a,	because the	values for its	left hand side (at the nucleus a)

negative branch are slightly smaller than for its right hand side (at the nucleus b) positive branch. Both instabilities (with respect to F) should however be extremely close one to

another. A more precise numerical analysis confirms also the possibility of two minima

202 J. M. ANDRÉ ET AL .

at low electric field and at a sufficiently large distance to allow for broken symmetry solutions.

Acknowledgements

The authors would like to acknowledge the support of the convention		9.4593.92 from
the Belgian National Science Foundation	Fonds National de la Recherche
Scientific and of its associated Foundation	Fonds de	la Recherche

Fondamentale et Collective). One of us (L.P.) acknowledges also the support by the BST-

439/23/93 project of the Department of Chemistry, University of Warsaw, Poland.

D.H.M. thanks the Belgian Ministry of Science (SPPS) for financial support.

References

1.C.C.J. Roothaan, Rev. Mod. Phys. 32, 179 (1960).

2.P.-O. Löwdin, J. Appl. Phys. Suppl. 33, 251 (1962).

3.A.A. Frost, J. Chem. Phys. 47, 3707 (1967).

4.G. Berthier, Folia Chimica Acta, 15, 67 (1987)

Electronic Charge Density of Quantum Systems in the Presence of an Electric Field: a Search for Alternative Approaches

G.P. ARRIGHINI and C. GUIDOTTI

Dipartimento di Chimica e Chimica Industriale. Università di Pisa

Via Risorgimento 35, 56126 Pisa, Italy

1. Introduction

Many fundamental properties of atoms and molecules could come within our reach through the "simple" knowledge of the electronic distribution density Among these

properties we limit ourselves to quote as particularly significant the various multipole moments of the distribution itself, electric potential and field generated by it in the surrounding space. The list of properties that we could master grows longer if we were in a position to establish how the electronic distribution density is polarized under the action of external electric and magnetic fields, inasmuch as one might evaluate also various kinds of (generally nonlinear) response parameters of matter, a piece of information that is nowadays of utmost importance for a vast series of research programs endowed with prominent technological significance (for instance, oriented toward the very ambitious goal of "designing" molecularly-thought materials in such strategic fields as photonics,

optoelectronics, etc. [1-3]). Although the electronic density is physically defined in

a space of lowdimensionality according to the proper modeling adopted for the system under investigation), the canonical approach to the computation of such fundamental q u a n t i t y involves the preliminary obtainment of the electronic wavefunction, a solution to the Schrodinger equation depending on the totality of the

DNe space coordinates associated with the Ne present electrons. Without going into the

slightest detail, we simply restrict our comments on this point to re-emphasize what is well known even to quantum chemistry students, the fact that the usually accepted descriptions of the quantum behaviour of many-electron systems correspond to approximate solutions to the Schrödinger equation, most frequently built up in terms of one-electron wavefunctions, i.e. orbitals. Hartree-Fock (HF) orbitals constitute almost invariably the output of ab initio molecular calculations carried out by today's computer

program packages up to

and lead obviously to an orbital picture of the

electronic distribution as a sum of contributions from each of the occupied orbitals. Overcoming the HF accuracy so as to take into account effects beyond the mean-field approximation (electron correlation) is permitted at the cost of handling much smaller molecules, while for systems containing a very large number of electrons even the HF level of description becomes untenable and one has to turn to empirical or semiempirical models.

203

Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 203–218.

© 1996 Kluwer Academic Publishers. Printed in the Netherlands.

204	G. P. ARRIGHINI AND C. GUIDOTTI

The concerns we have expressed are bound to get even more acute if the problem under study demands that we are able to adequately describe distortion effects induced in the electron distribution by external fields. The evaluation of linear (and, still more, non linear) response functions [1] by perturbation theory then forces one to take care also of the nonoccupied portion of the complete orbital spectrum, which is entrusted with the role of representing the polarization caused by the external fields in the unperturbed electron distribution [4].

A still more outstanding role in quantum many-particle systems is assigned to the electron density by the Hohenberg and Kohn theorem [5], a not obvious statement

affirming the existence of a rigorous theoretical framework where one is allowed to obtain ground state properties of the system in terms of the ground state density alone. Unfortunately, although the electron kinetic and exchange-correlation energy

contributions are shown to be universal functionals of the density the theorem does

not offer any practical guide to their actual construction. In view of the extremely attracting perspective of treating many-electron systems at an accuracy level beyond the

HF one, without making recourse to wavefunction approaches, it is quite understandable that many efforts have been addressed to the development of density functional theories (DFT's) [6-8], There exists possibly general agreement that the most satisfactory DFT approach presently implemented, suggested by Kohn and Sham [9], actually fails the original program, because it involves a return to an orbital picture (Kohn-Sham orbitals) as a rescue from the difficulties posed by our insufficient knowledge of the basic universal functionals inherent of the procedure, particularly the kinetic energy one. As a consequence, troubles met with large molecules, that we presumed to be able to leave outdoors thanks to the novel approach, again enter home from the windows, thus challenging to a substantial extent applications concerning most of the chemically and technologically interesting problems.

The present (very preliminary) investigation follows a research line closer to the true spirit of the DFT's, moving in the same direction as some recent papers where the attention is focused on the development of a formalism able to lead to the electron

density

without invoking wavefunctions, orbitals in particular [10-15]. It is right to

recall that the seminal ideas of this approach are anything but new, their origin dating back to the atomic statistical model put forward more than sixty years ago by Thomas and Fermi. Without pretending to review the concerned literature during such a long period of time (but a very complete bibliography is collected in ref. [7]), we limit ourselves to point out as particularly relevant to the present work some additional papers

[16-26] where the manifest intent of revitalizing an old subject proceeds through the development of a general formalism that contemplates the Thomas-Fermi theory as a low-order level of approximation.

By the present paper we intend to start to explore the possibility of generating explicit,

approximate ways for calculating the electron density

of a quantum system

subjected to an external homogeneous and static electric field, without invoking, in the construction, orbitals as basic ingredients. Although the electronic distribution of the system is at the outset assumed to be describable in terms of (unspecified) occupied orbitals, we immediately shirk the orbital approach in favor of an integral representation

of the electronic density involving the knowledge of the quantum mechanical

propagator (QMP) [27-30]. A drastic ansatz for the latter quantity based on the known

QMP of a particle moving in a linear potential field is the key-step of the whole

procedure, by which we attain, without any further approximations, an explicit final

ELECTRONIC CHARGE DENSITY OF QUANTUM SYSTEMS		205
result for	expressed in terms of Airy function	and its first derivative. The only

applications of the result thus deduced will be restricted to the case of a model of independent particles moving in a quadratic potential while simultaneously acted by a static electric field: the predicted electric dipole moment induced by the field in the system is shown to be the exact value, despite the fact that the electronic charge density resulting from the approach is only an approximation to the correct one.

2. An approximate approach to the electronic density

The system under study is assumed to consist of electrons, possibly in the presence

of a nuclear framework. An orbital picture of the quantum behaviour of the system is then introduced on accepting the validity of an independent-particle model where each

electron moves in the field of an effective potential					which afterwards is left
substantially unspecified. We emphasize, however, that the choice of						is an essential
step of any modeling. Besides semiempirical forms, effective potentials
functionally dependent on the electron numeral density					are intuitively bound to play
a prominent role in applications.
The one-electron Hamiltonian operator					kinetic energy operator,
generates a complete spectrum of orbitals				according to the Schrödinger equation
The ground state of the system corresponds to				electrons occupying the			lowest-
energy	levels, so that the electron numeral density				is
To attain an expression of			which does not make explicit reference to the occupied
orbitals, we rewrite eq. (2.2) in the form
where we have introduced the Heavisidefunction					and the Fermi level energy			If
we make use of the following standard representation of
from eq.	(2.1) and the completeness of the spectrum of orbitals supported by							the
electron density n		(2.3), can be expressed as
(atomic units with			are used throughout this paper). Eq.			(2.5)	is a well

known result [12,14-17,20,24,25], that makes evident the key-role played by the diagonal