Imperial College London

Henry Rzepa

**[Moore's
law](http://en.wikipedia.org/wiki/Moore's_law){target="_blank"}** describes
a long-term trend in the evolution of computing hardware, and it is
often interpreted in terms of processing speed. Here I chart this rise
in terms of the size of computable molecules. By computable I mean
specifically how long it takes to predict the geometry of a given
molecule using a quantum mechanical procedure.

::: {#attachment_5292 .wp-caption .aligncenter style="width: 149px"}
[![](http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2011/10/LSD.jpg "LSD"){.size-full
.wp-image-5292 loading="lazy" decoding="async"
aria-describedby="caption-attachment-5292" width="139"
height="192"}](http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2011/10/LSD.jpg)

LSD, the 1975 benchmark for computable molecules.
:::

The geometry (shape) of a molecule is defined by 3N-6 variables, where N
is the number of atoms it contains. Optimising the value of variables in
order to obtain the minimum value of a function was first conducted by
chemical engineers, who needed to improve the function of chemical
reactor plants. The mathematical techniques they developed were adopted
to molecules in the 1970s, and in **1975** a milestone was reached with
the molecule above. Here, N=49, and 3N-6=141. The function used was one
describing its computed enthalpy of formation, using a quantum
mechanical procedure known as MINDO/3. The computer used was what passed
then for a supercomputer, a CDC 6600 (of which a large well endowed
university could probably afford one of). It was almost impossible to
get exclusive access to such a beast (its computing power was shared
amongst the entire university, in this case of about 50,000 people), but
during a slack period over a long weekend, the optimised geometry of LSD
was obtained (it's difficult to know how many hours the CDC 6600 took to
perform this feat, but I suspect it might have been around 72). The
result was announced by Paul Weiner to the group I was then part of (the
Dewar research group), and Michael immediately announced that this
deserved an unusual Monday night sojourn to the Texas Tavern, where
double pitchers of beer would be available. You might be tempted to ask
what the reason for the celebration was. Well, LSD was a "real molecule"
(and not a hallucination). It meant one could predict for the first time
the geometry of realistic molecules such as drugs and hence be taken
seriously by people who dealt with molecules of this size for a living.
And if you could predict the energy of its equilibrium geometry, you
could then quickly move on to predicting the barriers to its reaction. A
clear tipping point had been reached in computational simulation.

In 1975, MINDO/3 was thought to compute an energy function around 1000
to 10,000 faster than the supposedly more accurate *ab initio* codes
then available (in fact you could not then routinely optimise geometries
with the common codes of this type). With this in mind, one can subject
the same molecule to a modern ωB97XD/6-311G(d,p) optimisation. This
level of theory is probably closer to 10^4^ to 10^5^ times slower to
compute than MINDO/3. On a modest "high performance" resource (which
nowadays runs in parallel, in fact on 32 cores in this case), the
[calculation](http://hdl.handle.net/10042/to-9679){target="_blank"}
takes about an hour (starting from a 1973 X-ray structure, which turns
out to be quite a poor place to start from, and almost certainly poorer
than the 1975 point). In (very) round numbers, the modern calculation is
about a million times faster. Which (coincidentally) is approximately
the factor predicted by Moore's law.

I will give one more example, this time for an example dating from
around **2003**, 28 years on from the original benchmark.

::: {#attachment_5296 .wp-caption .aligncenter style="width: 338px"}
[![](http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2011/10/lactide.jpg "lactide"){.size-full
.wp-image-5296 loading="lazy" decoding="async"
aria-describedby="caption-attachment-5296" width="328"
height="288"}](http://www.ch.imperial.ac.uk/rzepa/blog/wp-content/uploads/2011/10/lactide.jpg)

Transition state for lactide polymerisation.
:::

This example has 114 atoms, and hence 3N-6 =336, or 2.42 times the 1975
size. It is a transition state, which is a far slower calculation then
an equilibrium geometry. It is also typical of the polymerisation
chemistry of the naughties. Each run on the computer (B3LYP/6-31G(d),
with the alkyl groups treated at STO-3G) now took about 8-10 days (on a
machine with 4 cores), and probably 2-4 runs in total would have been
required per system (of which four needed to be studied to derive
meaningful conclusions). Let us say 1000 hours per transition state.
Together with false starts etc, the project took about 18 months to
complete. Move on to **2010**; added to the model was a significantly
better (= slower) basis set and a solvation correction, and a single
calculation now took [67
hours](http://hdl.handle.net/10042/to-4950){target="_blank"}. In
**2011**, it would be reduced to \~10 hours (by now we are up to 64-core
computers).

In 2011, calculations involving \~250 atoms are now regarded as almost
routine, and molecules with up to this number of atoms cover most of the
discrete (*i.e.* non repeating) molecular systems of interest nowadays.
But the 1975 LSD calculation still stands as the day that realistic
computational chemistry [came of
age](http://www.ch.imperial.ac.uk/rzepa/blog/?p=4647){target="_blank"}.

::: {.wp_orcid_field}
[https://orcid.org/0000-0002-8635-8390](https://orcid.org/0000-0002-8635-8390){target="_blank"
rel="author"}
:::


<strong>
 Moore’s law
</strong>
describes a long-term trend in the evolution of computing hardware, and it is often interpreted in terms of processing speed. Here I chart this rise in terms of the size of computable molecules. By computable I mean specifically how long it takes to predict the geometry of a given molecule using a quantum mechanical procedure. LSD, the 1975 benchmark for computable molecules.

Messaggi di Rogue Scholar

Computers 1967-2011: a personal perspective. Part 4. Moore’s Law and Molecules.