Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
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The Real Generators of the Unitary Group
P. CASSAM-CHENAI
Equipe d'Astrochimie Quantique, Laboratoire de Radioastronomie
EMS., 24 rue Lhomond, F-75231 Paris Cedex 05, France
This note is dedicated to G. Berthier who has always emphasized the importance of a rigorous use of the language in scientific papers. I would like to expose here an "abus de langage" regarding "the generators of the unitary group U(n) ", usually denoted by which dates back to their introduction in quantum chemistry [1]. As a matter of fact, in
the original paper, the author concedes that they are not the generators of U(n) |
but those |
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of the linear group |
; however, as far as I am aware, none of his followers has |
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ever mentioned this point. |
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The |
generators |
which are chosen such that: |
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are Hermitian only for |
They generate, using complex numbers, the Lie algebra of |
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. This algebra contains the Lie algebra of U(n) , but it is indeed much larger. |
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The Lie algebra of U(n) |
can be generated more specifically, using real numbers, with |
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Hermitian generators denoted |
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The |
generators |
, and the generators |
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are related in the same way as the angular moment operators" |
and |
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where |
is the Kronecker symbol, and |
It is convenient to extend these relations to all couples (i,j), and to write compactly :
The structure constants for Hermitian generators are purely imaginary :
Y. Ellinger and M. Defranceschi (eds.). Strategies and Applications in Quantum Chemistry, 77–78.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.
78 |
P. CASSAM-CHENAÏ |
The fundamental representation of the generators as n x n matrices is easily obtained; the matrix elements have the following expressions :
is that it decomposes on with real numbers, even when the integrals are complex (case of an electromagnetic field, of a molecule whose symmetry group has irreducible representations which are not realizable over real numbers...) :
with |
w (respectively v ) one-electron (respectively two-electrons) Hermitian operator |
and |
Re(x) (respectively Im(x)) real part (respectively imaginary part) of the complex |
number x.
So the genuine generators of the unitary group have original properties and do not deserve to be forgotten. It would seem weird to build the theory of angular momentum using only with no mention of and . It is equally surprising that only the appear in the theory of the unitary group. In short, in the traditional approach, one builds the Lie
algebra of the linear group but uses only the Lie subalgebra corresponding to the unitary group. A more satisfactory approach would consist in generating the Lie algebra of the unitary group only, using its real generators, then to define in this algebra with Eq.(3) the rising and lowering operators
References
1. J. Paldus, J. Chem. Phys. 61, 5321 (1974).
Convergence of Expansions in a Gaussian Basis
W. KUTZELNIGG
Lehrstuhl für Theoretische Chemie, RuhrUniversität Bochum, Universitätsstr. 150, D-4630 Bochum, Germany
1. Introduction
Few papers have had as much impact on the progress of ab-initio quantum chemistry as that of Boys [1] where he proposed to use Gaussians (GTOs) as basis sets. The great breakthrough of ab-initio theory would never have been possible without the
invention of Gaussians. Nevertheless, even nowadays |
it is difficult to explain to |
a beginner why one should rely on Gaussians, which |
have the wrong behaviour |
both near the nuclei and very far from them. The ease; with which two-electron integrals over GTOs can be computed is certainly an argument. However, if one has thought a little bit on the importance of choosing basis sets with the right behaviour at the singularities of the Hamiltonian [2], one cannot but be deeply surprised that expansions in GTOs converge decently well in spite of their failure at the singularities of the Hamiltonian.
To appreciate this point somewhat better it is useful to compare three types of
Gaussian basis sets, (a) a set of Gaussians with common orbital exponents (for one l) but a sequence of principle quantum-numbers
(We consider here only the case of a single center), (b) the same set (1.1) but with n – l = 1,2,3,4, ..., (c) a set of Gaussians with the lowest possible n for each l, but with a sequence of orbital exponents
Sets of orbital exponents have been proposed mainly by Huzinaga [3], van Duijneveldt [4], Pople et al. [5]. A systematic construction of basis sets of arbitrary dimension is possible in terms of the ’even tempered’ concept of Ruedenberg et al. [6,7 ], or of some more sophisticated generalizations [8,9,10]. For a recent comprehensive review on basis sets see Feller and Davidson [11].
It does not make a significant difference that in practice one uses ’cartesian Gaussians’ rather than Gaussians with explicit inclusion of spherical harmonics. One
79
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 79–101.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.
80 W. KUTZELNIGG
should also mention that there is a fourth type of basis sets (d), namely that of Gaussian lobes [12,13] i.e. functions of type (1.2) with only but with centers spread over the molecule, not only at the position of the nuclei. These don't differ basically from case (c).
It has been shown [14] for both types of basis sets (1.1) and (1.2) that a given set
of dimension n can be regarded as a member |
of a family of basis sets that in |
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the limit |
become complete both in the ordinary sense and with respect to |
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a norm in the |
Sobolev space – which is the condition for the eigenvalues and |
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eigenfunctions of a Hamiltonian to converge to the exact ones. However, as to the speed of convergence the two basis sets (1.1) and (1.2) differ fundamentally.
In a careful study of basis sets of type (1.1) applied to the ground state of the hydrogen atom Klahn and Morgan [15] were able to show that the error of the energy
goes |
as |
(n being the dimension of the basis) for fixed |
By optimization |
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of |
one can |
achieve [1.6] that the error goes as |
. |
Anyhow this rate of |
convergence is as bad as one can imagine and it makes basis set (1.1) absolutely useless. Convergence as an inverse-power law with a small exponent generally prevents accurate calculations, as is known from the slow convergence of the partialwave expansion for the interelectronic coordinate (equivalently the convergence of a CI for an atom with the highest angular equantum number l in the basis set included), where the error goes as Inclusion of a single term with the right behaviour at the Coulomb singularity (a ’comparison function’ [2]) improves
the rate of convergence, such that the error goes as |
for the expansion of the |
H-atom ground state in basis (1.1) [16] or as |
for the convergence of a CI |
[17]. |
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If one includes functions with n – l even in (1.1) (i.e. one uses set b) the basis is formally overcomplete. However the error decreases exponentially with the size of the basis [2,16]. Unfortunately for this type of basis the evaluation of the integrals is practically as difficult as for Slater type basis functions, such that basis sets of type (b) have not been used in practice.
The rate of convergence of expansions in the basis (1.2) has received little attention except for purely numerical studies [3,7,8,9,16] which indicated that the convergence is at least (unlike for bais set of type) not frustratingly slow. Rather detailed studies were performed for the even-tempered basis set, i.e. for exponents constructed from
two parameters |
and |
(for each l) |
In a numerical study of basis sets (a), (b) and (c) for the H atom ground state W.
Klopper and the present author [16] found that for the basis (c) the error goes as
i.e. the convergence is not exponential (which would be ideal, i.e. generally the case for a basis that describes the singularities correctly) but almost so. This does not only hold for the energy, but for other properties as well. However there
are properties for which the limit |
does not yield the correct result, e.g. |
CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS |
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81 |
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which is |
for the exact H ground state wave function, |
but which |
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vanishes for the expansion in (1.2) for all finite n. |
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Similarly |
is equal to |
while this second derivative is negative for |
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any finite expansion with an apparent divergency to |
for |
Some prop- |
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erties like the density at the nucleus and the variance of the energy converge very slowly to the exact values. These are, nevertheless, relatively minor defects.
Again by adding to the basis at least one function that has the correct behaviour at r = 0, e.g.
the convergence can be speeded up – and the last-mentioned defects can be removed [10,16]. However, the improvement is much less spectacular than for basis (1.1) – unless one is interested in the density at the nucleus or the variance of the energy.
There are hints [9,10,18] that the rate of convergence for basis sets of type (1.2) is even better than (1.4), if one uses better optimized basis sets than those of even tempered type (1.3),
and that the same convergence pattern is found for the expansion of |
as for |
There is no doubt that the convergence behaviour of standard Gaussians is much better than one should have expected in view of their failure at
What is the fundamental difference of basis sets of type (1.1) and (1.2)? Without claiming to give a definite answer we can say that the expansion in the basis (1.1) is closely related to the expansion in terms of Laguerre functions, i.e. in a typical orthogonal basis and that a theory much like that for Fourier series applies. There it generally holds that the singularities of the function to be expanded determine the rate of convergence [19]. An expansion in the basis (1.2) can hardly be traced back to something like a Fourier series. It must rather be viewed as a discretization of the integral representation of an exponential (or another exponential-like) function.
and entirely different features determine the error. (As to a direct application of a numerical discretization of the integral transformation (1.7) see ref. 20).
To get analytic results for the convergence behaviour of an expansion in a Gaussian basis we shall proceed in two steps.
1.We replace the integral (1.7) by an integral from s1 to s2 rather than from 0 to The errors due to this restriction of the integration domain – the cut-off errors
– can easily be estimated.
82 |
W. KUTZELNIGG |
2. |
We replace the integral from s1 to s2 by a sum over a regular grid. We do |
this by applying first a variable transformation (to be specified by some criteria) such that after this transformation an equidistant grid can be used. An estimate of the discretization error is possible by means of tricky and non-trivial application of analysis. Details on this are given in the appendix, which is a rather important part of this paper.
The integral (1.7), which is the starting point for the expansion of a hydrogen-like
1s function in a Gaussian basis, is rather complicated. There is a much simpler counterpart of (1.7) which is relevant for the expansion of the Coulomb potential 1/r in a Gaussian basis, namely
It has, in fact, been found in a numerical study [21] that this type of expansion has a
very similar convergence behaviours as that of |
, i.e. that the error also goes as |
. The origin of this behaviour is essentially the same for the expansion of the two functions. Since (1.8) is formally much simpler, it is recommended to study the expansion of 1/r first.
In fact only the expansion of 1/r will be treated here in detail, while a full study
of the expansion of |
will be published elsewhere. |
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The key feature is – |
both for the expansion of 1/r or |
in terms |
of ’even- |
tempered' Gaussians – that, for large n, the cut-off error goes as |
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with h the step size and that the discretization errors goes as |
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with |
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a and b constants. While – for fixed n – a small h is good for the discretization error, it is bad for the cut-off error and vice versa. The best compromise is that
which implies that the overall error goes as
The similarity between 1/r and |
, as far as the expansion in a Gaussian basis |
is concerned, leads to another interesting aspect. In many-electron quantum mechanics we have in principle to solve both Schrödinger and Poisson equations. We don't realize this usually because the Poisson equations are first solved in closed form – which is not possible for the Schrödinger equation. This procedure destroys the equivalence between the matter field and the electromagnetic field and one may want to consider an approach in which one solves the Poisson equations numerically in a basis of Gaussians rather than solving it exactly. Work on these lines is in progress [21].
2. Expansion of 1/r in a Gaussian basis
We proceed in two steps. Starting point is the identity (1.8) or equivalently
CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS |
83 |
In doing so we make two ’cut-off ’ errors
The error function erfx has a power series expansion for small x and an asymptotic expansion for large x
and the following inequalities hold
which allow us to estimate |
and |
in two alternative ways. |
We have indicated the order of errors of these estimates after the semicolons. We
see that |
(2.6a) is a close estimate for |
if |
while (2.6c) is a close estimate |
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for |
. On the other hand the relative error |
approaches 1, i.e. |
100% |
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for |
and |
for |
. Note that the cut-off error never exceeds |
100%. |
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The range of r-value for which f ( r ) is a good approximation to 1/r is |
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In this range the total cut-off error |
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is determined by the |
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’lower-cut-off’ error |
(2.6a), with respect to which the ’upper-cut-off error’ (2.6c) is |
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84 |
W. KUTZELNIGG |
negligible. In a wide range of r ’flat’ gaussian are more important than ’steep’ ones, which only matter for small r.
The next step on the way to an expansion of 1/r in a Gaussian basis is to replace the integral (2.2) by a sum. Before we divide the range between into n intervals, we apply a variable transformations, such that after this transformation an equidistant grid can be used.
Let us define the normalized functions (with respect to square integration over r)
Then (2.8b) becomes
Obviously we must choose p(x) such |
that the domain between |
and – |
which |
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have different orders of magnitude |
– |
is covered in a balanced way. One may fur- |
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ther require that all |
g(r, x) have |
about the same weight in the |
sum. The |
latter |
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requirement leads to |
the condition |
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Obviously an exponential mapping looks also good in the sense of the first criterion.
One |
sees |
easily that f(r) is independent of the choice of such that we may |
as |
well |
take |
Of course, this is only a plausiblity argument and we need |
a |
rigorous criterion for the optimum mapping. We come back to this problem in the conclusions.
We hence have
We now approximate (2.12a) as a sum (with |
the discretization error). |
CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS |
85 |
Estimates for the discretization error are derived in the appendix. Unlike the esti- mates (2.6) these are not obtained as strict inequalities, but rather as leading terms of asymptotic expansions. For the integral (2.12a) with the integration limits
to |
the discretization error is (for large n and sufficiently small h, see appendix |
E) |
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To arrive from (E.2) and (E.7b) at (2.14) one must identify a of appendix E with and realize that (E.2) or equivalently f ( x ) in (C.1) is normalized to 1. To
establish the relation to (2.1) one must multiply (E.2) by The relative discretization error happens to be independent of r (at least as far as its dominant term is concerned). Using the arguments of the appendix one finds for the optimum interval length as function of dimension n of the basis
and for the overall error (for that range of r values for which |
and |
are suffiently |
small). |
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3. Estimation of the error of an expectation value of 1/r
In practice one will – in fact – not be interested in the accuracy of f(r) as a function of r, but rather in the error of matrix elements like that over a hydrogenlike 1s function
as
To estimate this error we insert (2.2) into (3.1b) and integrate first over r such that