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

86	W. KUTZELNIGG

The analytic expression for this integral is

The limit	of (3.3) is not obvious. To get it we must expand the
first line of (3.3) in powers of	and insert the asymptotic expansion of erfc in the

second line before we collect powers of . We get for the first and second lines of (3.3) respectively

Of course,	and	as defined by (3.4) are the ’cut-off’ errors due to limitation
of the integration domain to			to

We next approximate the integral (3.3) by a numerical integration after performing the variable transformation (2.11) with . This means we first replace (3.3) by

Before we study the ’discretization errors’ let us look on how the ’cut-off errors’ and depend on the number of points chosen in (3.5c). In view of (3.5a), (3.4) and (2.13b) we have

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS		87
The minimum with respect to	(for nh fixed – and sufficiently large – ) is achieved

if

This means that one should choose roughly and that for fixed h the error decreases exponentially with n (or for fixed n exponentially with h).

The estimation of the discretization error is fortunately rather easy, relying on the results of appendix E (which contains the difficult part of the derivation). In fact the discretization error given by (2.14) is simply proportional to 1 / r . Hence

A derivation of the discretization as

is very lengthy, but leads essentially to the same result, which is not so obvious, since in appendix E we have done the phase-averaging before integrating over r, and phase averaging and integration over r need not commute.

We use again the argument that the minimum of appears close to the value of h for which the arguments of the exponential agree, i.e.

There is one difficulty insofar as (3.8) is only an estimate of the absolute value of the discretization error. It cannot be excluded that (depending on how the limit is performed, see appendix and have opposite sign. In

this case the minimum absolute error may vanish, while (3109a) is still valid.

Note that h is related to the	of an even-tempered basis	(1.3) for the H atom
ground state as
Let the smallest orbital exponent in the Gaussian basis be		and the largest
Then for sufficiently large n we have

these results, especially that for	are in good agreement with results from a purely
numerical study [21].

88 W. KUTZELNIGG

4. Conclusions

We were able to show analytically – in an unexpectedly tricky way (the mathemat- ical ingredients of which are in the appendix) – that the error of an expansion of

the function	in terms of an even-tempered Gaussian basis of dimension n goes
as	provided that the two parameters of the even-tempered basis are
optimized.

We have not shown that this is the optimum convergence, in other words whether there are other (twoor more-parameter) basis sets for which the convergence is even faster.

The examples given in the appendix give some indications on the properties which the mapping function has to satisfy that both the cut-off error and the discretization error decrease exponentially (or faster) with nh and 1 / h respectively and don't depend too strongly on r. Further studies are necessary to settle this problem.

For quantum chemistry the expansion of		in a Gaussian basis is, of course,
much more important than that of	The formalism is a little more lengthy than

for 1/r, but the essential steps of the derivation are the same. For an even-tempered

basis one has a cut-off error	and a discretization error
such that results of the type	(2.15) and (2.16) result. Of course,	is not well

represented for r very small and r very large. This is even more so for 1/r, but this wrong behaviour has practically no effect on the rate of convergence of a matrix representation of the Hamiltonian. This is very different for basis set of type (1.1).

Details will be published elsewhere.

At this point one can conjecture that the relatively rapid convergence of Gaussian geminals [22]

to describe the correlation cusp, has a somewhat similar origin as the example

studied here, and goes probably also as exp

with n the dimension of the

geminal basis.

Acknowledgement

The author thanks Stefan Vogtner for numerical studies of expansions of 1/r in a

Gaussian basis which have challenged the present analytic investigation. Discus- sions with Christoph van Wüllen and Wim K lopper on this subject have been very helpful.

This paper is dedicated to Gaston Berthier, from whom I have learned a lot. Although Berthier's publications have mostly dealt with applications of quantum mechanical methods to chemical problems, he never liked black boxes or unjustified approximations even if they appeared to work. The question why the quantum chemical machinery does so well although it often lies on rather weak grounds has concerned him very much. I am therefore convinced that he will appreciate this

excursion to applied mathematics.

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS

89

Appendix

Estimation of the discretization error

A. GENERAL CONSIDERATIONS

We want to approximate the integral

by dividing the integration domain

into n intervals of the same length h and by approximating f(x) in each interval by its value at the center of the interval. The discretization error is then

To estimate	(in a more traditional way)	we make a Taylor expansion of f(x)
around	in the k-th interval. We write	(assuming that f ( x ) is differentiable

an infinite number of times, which is the case for the functions that we study here)

We express

and proceed similarly with	in a next step and so on such that
finally

The	are Bernoulli numbers.

The expansion coefficients in (A.4) are essentially those of cosech(x/2).

The equality sign in (A.4) only holds if the series converges. Otherwise the series

is at least asymptotic in the sense that the sum truncated at some k differs from

90	W. KUTZELNIGG

the exact by

This also holds if f is only (2k – 1) times differentiable,

such that one has has to truncate the expansion anyway.

The discretization studied here is related to that of the Euler-McLaurin method well-known in numerical mathematics (see e.g. [23]). The difference is that in this method one approximates the mean value of f ( x ) in the interval by the average of the values at the boundaries of the interval, while we approximate it by its value at the center of the interval. This choice is more closely related to the expansion of a function in a basis.

For the Euler-McLaurin discretization an error formula similar to (A.4) holds,

namely without the factor	which corresponds to the expansion co-
efficients of coth(x /2).

An equidistant integration grid may not be the best choice. Let us therefore consider that we perform a variable transformation in the integral before we discretize.

To define the error by (A.1) and to apply the error formula (A.4) we must replace

and

and

respectively

We are mainly interested in the transformation

Eqn. (A.4) or	its counterpart with h replaced by and		allows
us to estimate	for small h	it is less convenient for	so large that

the Taylor series within an interval converges slowly or diverges.

There is an alternative – and for our purposes more powerful – way to estimate the discretization error, namely in terms of the Fourier expansion of a periodic

function. We write see (A.1), as [24]

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS

91

Only the cosine terms contribute, because the sine terms vanish at

The larger l and the smaller h the more rapidly oscillating is the cosine factor in

(A.9) and the smaller is the contribution to	For sufficiently small h usually the
term with l = 1 dominates in the sum.

A very popular method of numerical integration is that of		[23]. It has the
advantage that with n points in a	integration one gets the same accuracy

as with 2n points on an equidistant grid – provided that the integrand is well approximated as a polynominal of degree n, or is expandable in an orthogonal basis like in Laguerre polynomials. For the examples that we study here this condition

is far from beeing satisfied, and therefore the	integration is not supposed to
be helpful.

We now study some special examples that are closely related to those that we are interested in.

B. THE EXPONENTIAL FUNCTION WITH AN EQUIDISTANT GRID

For the example

a closed expression for the truncation error can be obtained

In this case the relative error is the same for all intervals and one gets

We write	to indicate that this is a discretization error.
If one expands (B.3) in powers of		one gets the same result as from (A.4) namely

noting that

92	W. KUTZELNIGG
The series (A.4) has here the radius of convergence	but it can be continued

analytically beyond its radius of convergence.

Let us now argue that we are actually interested in the integral

and that the first approximation step is to replace

by y and the second one the

discretization, then the total error consists of the cut-off-error

and the discretization error (B.4).

The limit

of the discretization error (B.4) is

while from the Fourier expansion (A.9) we get

The identity between (B.8) and (B.9) is not immediately recognized. One sees at least easily that for small h one gets from (B.9)

in agreement with what one gets from the Taylor expansion of (B.8) or immediately from (A.4). The agreement of (B.8) and (B.9) is confirmed in terms of a relation familiar in the theory of the digamma function

together with

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS

93

and

which implies

from which one is immediately led to the equivalence of (B.8) and (B.9)

If one limits the sum (B.9) to the term with l = 1 and expands in powers of h, the

coefficient of the leading term in

instead of the correct value

(see B.10). Convergence with l for small h is pretty (though not extremely)

fast.

We want to make the overall error minimal for fixed n. We express the total error in terms of h and n

We want to minimize ε as function of h for fixed n. Since the discretization error only depends on h, it is obvious that one should make h as small as possible, in order to minimize it. We can therefore assume that h is so small that

Asymptotically for large n the solution of this transcendental equation is

Since lnn is a slowly varying function of n, the error goes essentially as

This is

the typical behaviour of a discretization error for a numerical integration [23], but is atypical for the examples that we want to study.

C. THE GAUSSIAN WITH AN EQUIDISTANT GRID

Our next example is

94	W. KUTZELNIGG

At first glance this looks similar to (B.1). However, there are two differences between (B.1) and (C.1) that have spectacular consequences.

1. While the function f(x) in (B.1) is convex for all x, the f(x) in (C.1) is concave

from x = 0 to the inflection point			and convex from	to	This
means that the discretization error is negative for intervals between 0 and					and
positive between	and	such that a partial cancellation of the error is possible.

2. While for f ( x ) in (B.1) all derivatives at x = 0 are non-zero, the odd-order

derivatives of the f ( x ) in (C.1) vanish at x = 0. Since these enter the error formula (A.4) there is no contribution of the boundary at x = 0 to the given by

(A.4), whereas for (appendix B) the derivatives at x = 0 determine the error.

Prom (A.4) we conclude that for sufficiently small h

Not only is this error negative, meaning that we overestimate the integral (C.1), but it also appears that the error decreases very rapidly with y, such that one is

tempted to conclude that in the limit		(and hence	vanishes,
independently of h.
In fact for	the odd-order derivatives of f ( x ) vanish at either boundary such

that (A.4) gives the result zero. Of course (A.4) only holds for h smaller than

the radius of convergence	of the series. There is no reason why		should be
independent of y, and we shall, in fact see that		This makes the

estimate (C.2) rather useless because its range of validity is too limited (unlike for the example of appendix B).

The explicit expression for the discretization error is

Unlike for the example of appendix B a closed summation is not possible. However, (C.3) allows us to discuss the behaviour of for large h, where the sum is dominated by the first term

For large h one cannot reduce the error significantly by increasing n. There is obviously a limiting function for which for large h is given by (C.4). For small h (C.3) is not convenient because it is slowly convergent.

Fortunately the Fourier expansion method helps us for small and intermediate h

but large n. We get in the limit

CONVERGENCE OF EXPANSIONS IN A GAUSSIAN BASIS

95

This is (at variance with C.3) a rapidly converging series for

For h suficiently small the first term with l = 1 is a good approximation to the sum (C.5a).

If the upper integration limit in (C.5a) is y = nh rather than i.e. for finite n, a simple closed expression is not obtained. However, one can estimate the leading term in an expansion in powers of such that

The asymptotic expansion of (C.5b) in powers of h agrees with (C.2). In fact the

first term neglected in (C.5b) starts with terms of an expansion in powers of h vanish. h=0.

In the limit of course, all has an essential singularity at

From this asymptotic expansion in powers of			no conclusions on the radius
of convergence of	are possible, but there are some hints that the radius of
convergence is that of cosech		i.e.the series (A.4) probably converges for

This conjecture is consistent with the result that for			the radius of conver-
gence reduces to 0.
At	the arguments of the exponential functions in (C.5a) and (C.5b) agree,
which implies that near		goes through zero. Between h = 0 and
is slightly negative and rather well approximated by			(C.2), while for
increases rapidly and soon approaches 1.
Near	the cut-off error

and the discretization error have the same order of magnitude, hence the minimum

of

is also close to

The minimum error therefore goes as

The prefactor of the exponential in (C.7) is less easily obtained. To get it one has to solve the transcendental equation for h and insert this into Numerically one obtains that this factor is close to 1/2.