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