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

96		W. KUTZELNIGG
The essential message is that the error goes as	and the optimum h as
This means very fast convergence with the number n of intervals, very
different from the example of appendix B where the	error only	decreased as
In this appendix we have argued that (C.5b) is valid for ’sufficiently
small’ h. That meant that h should not be significantly larger than		which

is not very restrictive. However (C.2) only holds for h satisfying (C.6), which limits its validity to extremely small h, in the limit (C.2) becomes even invalid. The two references to ‘small’ h must be clearly distinguished.

D.THE EXPONENTIAL FUNCTION WITH A LOGARITHMICALLY EQUIDISTANT GRID

We consider again (B.1), but with the transformation

The lower integration limit is now changed from 0 to If we want to discretize, we must also introduce a lower cut-off. I.e. rather than (D.2) we must consider

The integrand in (D.3) falls	off	rapidly for	but more slowly for
Therefore the ‘lower’ cut-off	is	more critical than the ‘upper’ cut-off		We have

We divide the domain

into n intervals, hence

We minimize the error with respect to

for hn fixed, ignoring terms of

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS

97

and we get for the discretization error

We want to take the limit	and	in order to obtain	There
is the difficulty that in this limit		becomes an indefinite phase.
Pictorially it is clear what this means.

The intervals near the maximum of the integrand F(z)give the largest contributions. If one changes both integration limits, the intervals close to the maximum are not only changed in length, but also their positions with respect to the maximum are shifted. It makes, especially for large h, a lot of a difference if the ’innermost’ interval has its center or a border at the maximum. The limit for the integration from

depends somewhat on the position of the innermost interval, especially for large h.

Since the limit and is not unique, we can either choose a procedure to make it unique, e.g. fix that there is always a border of an interval at z = 0, or

– what is more realistic – we accept the non-uniqueness and hence an incomplete information and average over the indefinite phase in some consistent way. Leaving the phase unspecified we get

Since [25,26]

98	W. KUTZELNIGG

the term with l = 1 dominates for sufficiently small h. If we take only this term in

(D.8) and form the mean square average over the phase

we get

This estimate is independent of a as is the estimate (D.6c) of the cut-off error.

Note	that	is a monotonically increasing function of h, while both
Re		and Im	oscillate between		and
The	discretization	error	for finite integration limits	and	contains in ad-

dition to (D.8) two extra terms (under the sum) that contain incomplete Gamma functions. We don't need their explicit form for the estimation of the dominating part of the overall error. Of course, expanding these extra terms in powers of h would lead to the error estimation (A.4), that holds for extremely small h (and sufficiently small l) which is rather irrelevant in the present context.

Somewhat similar to appendix C we have a discretization error that goes as

and a cut-off error

The minimum as function of h is

achieved (for large n) if

If and	happen to have opposite sign, the optimum error vanishes, while close
to its zero ε (h) has an inflection point.
The optimum interval length goes as		and the error as exp	This is

certainly a much faster convergence than for the choice of an equidistant grid for the exponential function as studied in appendix B.

We have not considered the next term in an 1 /n expansion of

which

would be needed to get the prefactor of

E. A GAUSSIAN WITH A LOGARITHMICALLY EQUIDISTANT GRID

We consider now (C.1) but with the transformation

Everything is similar to appendix D.

Now (D.3) is replaced by

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS

99

We further get

We minimize

with respect to

For the discretization error we get

The argument concerning the indefinite phase in the limit	is similar as in
appendix D. The counterpart of (D.8) is

100	W. KUTZELNIGG

Limitation to the term with l = 1 (which dominates for sufficiently small h and the same phase averaging as in appendix D leads to [24,25]

For small h this goes as

The condition analogous to (D.11a) is

Like for the last example the optimum h goes as

and the error as

The convergence is slower than for the same function with an equidistant grid, but both h and ε are (on this level of approximation) independent of i.e. essentially the same grid can be used for a very steep or a very flat Gaussian.

there is only a shift via the a-dependence of

and

References

1.S.F. Boys, Proc. Roy. Soc. A200, 542 (1950)

2.R.N. Hill, J. Chem. Phys. 83, 1173 (1985)

3.S. Huzinaga, J. Chem. Phys. 42, 1293 (1965)

4.F.B. v. Duijneveldt, IBM Tech. Res. Rep. RJ 945 (1971)

5.W . J . Hehre, R. Stewart and J.A. Pople, J. Chem. Phys 51, 2657 (1969)

6.C.M. Reeves, J. Chem. Phys. 39, 1 (1963)

K. Rudenberg, R.C. Raffinetti, R.D. Bardo in

Energy, Structure and Reactivity, Wiley, New York (1973)

R.C. Raffinetti, Int. J. Quant. Chem. Sym. 9, 289 (1975)

7.M.W. Schmidt and K. Ruedenberg, J. Chem. Phys. 71, 3951 (1979)

8.S. Huzinaga, M. Klobukwoski and H. Tatewaki, Can. J. Chem. 63, 1812 (1985)

9.J.D. Morgan and S. Haywood, unpublished, quoted in ref. 19

10.V. Mühlenkamp, Thesis, Bochum (1992)

11.D. Feller and E.R. Davidson, in Reviews in Computational Chemistry 1, K.B.

Lipkowitz and D.B. Boyd Eds., VCH, Weinheim (1990) p.1

12.H. Preuß, Z. Naturforsch. A11, 823 (1956), Mol. Phys. 8, 157 (1964)

13.J. L. Whitten, J. Chem. Phys. 39, 349 (1963)

14.B. Klahn and W.A. Bingel, Theor. Chim. Acta 44, 2 (1977)

15.B. Klahn and J.D. Morgan, ,7. Chem. Phys. 81, 410 (1984)

16.W. Klopper and W. Kutzelnigg, J. Mol. Struct. THEOCHEM. 135, 339 (1986)

17.W. Kutzelnigg, Theoret. Chim. Acta 68, 445 (1985)

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS

101

18.R. Franke and W. Kutzelnigg, Chem. Phys. Letters 199, 561 (1992)

19.J.D. Morgan III, in: Numerical determination of the electronic structure of atoms, diatomic and polyatomic molecules, M. Defranceschi and J . Delhalle

Eds., (Kluwer, Dordrecht (1989) p. 49

20.J.R. Mohallem and M. Trsic, Int. J. Quant. Chem. 33, 555 (1988)

21.W. Kutzelnigg and St. Vogtner, to be published

22.K. Szalewicz, B. Jeziorski, H..J. Monkhorst, J.G. Zabolitzky, J. Chem. Phys.

78, 1420 (1983)

23.See e.g. J Stoer, Einführung in die Numerische Mathematik I, Springer, Berlin

(1972)

24.This estimate of the discretization error ought to be known in numerical mathematics. Usually it is easier to derive formulas like this than too look them up

in the literature.

25.M. Abramowicz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York (1965)

26.I.S. Gradsteyn and I.M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York (1980)

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Quantum Chemistry in Front of Symmetry-Breakings

J.P. MALRIEU and J.P. DAUDEY

Laboratoire de Physique Quantique, Université Paul Sabatier 118 route de Narbonne, 31062 Toulouse, France

1. Introduction

Symmetry breaking is a universal phenomenon, from cosmology to the microscopic world, a perfectly familiar and daily experience which should not generate the reluctance that it induces in some domains of Physics, and especially in Quantum Chemistry. In classical physics, the symmetry breaking of an a-priori symmetrical problem is sometimes refered to as the lack of symmetry of the initial conditions. But it may be a deeper phenomenon, the symmetry-broken solutions being more stable than the symmetrical one.

Quantum chemistry experiences two types of symmetry breakings.

One is purely formal, it concerns the departure from symmetry of an approximate solution of the Schrödinger equation for the electrons (ie within the Born-Oppenheimer approximation). The most famous case is the symmetry-breaking of the solutions of the

Hartree-Fock equations[l-4]. The other symmetry-breaking concerns the appearance of non symmetrical conformations of minimum potential energy. This phenomenon of deviation of the molecular structure from symmetry is so familiar, confirmed by a huge amount of physical evidences, of which chirality (i.e. the existence of optical isomers) was the oldest one, that it is well accepted. However, there are many problems where the

Hartree-Fock symmetry breaking of the wave function for a symmetrical nuclear conformation and the deformation of the nuclear skeleton are internally related, obeying the same laws. And it is one purpose of the present review to stress on that internal link.

2. Symmetry breakings of the electronic wave function

The Schrödinger equation being linear, H commutes with the symmetry operations of space and spin, and the wave function must be symmetry-adapted. This is the basic doxa which

we transmit to our students. If they are critical, they perhaps wonder why the

atomic

orbital of the hydrogen atom is an eigenfunction, while symmetry-broken. Actually, we usually do not take time to mention that for degenerate roots, it is the projector on the stable subspace of these degenerate eigenvectors which commutes with the symmetry operators of the problem. But the drama arises when the desired state is non degenerate and when an approximate method delivers a symmetry-broken wave-function. The results is in general considered negatively as spurious, contaminated and irrelevant, despite the fact that meaningfull physics have been introduced in these solutions in a biased way, lowering the energy with respect to the symmetry-adapted description obtained at the same level of sophistication.

103

Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 103–118. © 1996 Kluwer Academic Publishers. Printed in the Netherlands.

104	J. P. MALRIEU AND J. P. DAUDEY

The most famous case concerns the symmetry breaking in the Hartree-Fock approximation. The phenomenon appeared on elementary problems, such as when the so-called unrestricted Hartree-Fock algorithms were tried. The unrestricted Hartree-Fock formalism, using different orbitals for a and electrons, was first proposed by G. Berthier [5] in 1954 (and immediately after by J.A. Pople [6] ) for problems where the number of α and electrons were different. This formulation takes the freedom to deviate from the constraints of being an eigenfunction.

For problems, where the ground state is a singlet state, the use of such a wave function appeared to give significantly lower energies than the orthodox symmetry-adapted solution in many problems, as illustrated below. Later on other types of symmetry breaking have been discovered and Fukutome [7] has given a systematics of the various HF instabilities in a fundamental paper.

2.1.ATOMIC PHYSICS

In the Be atom, the two valence electrons occupy a 2s, 2p valence shell, the 2s and 2p

Atomic Orbitals (AO) having an important "differential overlap" (ie a good coincidence of their spatial extension). The contribution of the 2p AO to the angular correlation of the

valence electrons is especially large (the Moller Plesset expansion from being poorly convergent) and the proper valence function should be written

while the RHF approximation is reduced to the component. One obtains a much

lower energy using an UHF function which looses both the space and symmetry constrainsts. The single determinant

is lower in energy than the best RHF solution			due to the inclusion of some angular
correlation through	the	component, despite the contamination by the triplet
configuration	This example illustrates wonderfully the physically suggestive

potentiality of the symmetry-broken solution. Since it tells us that when the α electrons is

on the right side of the nucleus	the	electron prefers to move into an
hybrid, ie on the left side of the nucleus. This is the best translation of the angular
correlation, and it is clear that superimposing	and	the degenerate non orthogonal

solution

into

will restore the singlet character of the wave function by eliminating the triplet contamination but still disobeying the space-symmetry constraint [8].

QUANTUM CHEMISTRY IN FRONT OF SYMMETRY BREAKINGS	105
The space symmetry would only be restored by superposing the degenerate	and

solutions in

Such phenomena do not occur in heavier alkaline earth atoms due to a poorer differential overlap between the valence s and p orbitals (smaller Ksp integrals) as explained by Kutzelnigg [9].

Another well-known atomic HF symmetry breaking is the

problem but it is more

artificial since in this unbound state, two electrons leave the atom oppositely in two diffuse orbitals [10].

2.2.THE WEAK SINGLE BOND

The most popular use of the UHF solutions concerned the single bond breaking, since it was rapidly understood that while the RHF solution of

with

imposed a constant ratio of ionic/neutral VB components whatever the interatomic distance

and therefore a spurious asymptote at (IP-EA)/2 above the dissociation into neutral atoms, the UHF solution

with

authorized one electron to concentrate on atom A while the second one concentrates an atom B. The detailed conditions for the appearance of the UHF solution have been explicited a long time ago as a special application of the Thouless' relations [2]. This relation analyzes the stability of the symmetry-adapted HF solution, using symmetry-adapted MOs [11]. The transcription of these conditions in Valence Bond terms is easy to derive, [12] and one may show that the symmetry breaking takes place when

where in the element of the Fock operator between the valence AOs a and b and is the energy difference between the neutral and the ionic VB determinants. The

solid state physicists would say that