Dear CCP4BB,
I think it prudent at this point for me to announce what could be a
very old, but serious error in the fundamental mathematics of
crystallography. To be brief, I have uncovered evidence that the "hand"
of the micro-world is actually the opposite of what we have believed
since Bijvoet's classic paper in 1951.
Those of you who know me know that I have been trying to lay down
the whole of x-ray diffraction into a single program. This is harder
than it sounds. We all know what anomalous scattering is, but a
detailed description of the math behind translating this "dynamical
theory" effect all the way to the intensity of a particular detector
pixel is hard to find all in one place. Most references in the
literature about how anomalous scattering is connected to absolute
configuration point to the classic Nature paper: Bijvoet et. al.
(1951). Unfortunately, since this is a Nature paper, it is too short to
describe the math in detail. For the calculations, the reader is
referred to another paper by Bijvoet in the Proc. Roy. Acad. Amsterdam
v52, 313 (1949). Essentially, the only new information in Bijvoet et.
al. (1951) is the assertion that Emil Fischer "got it right" in his
initial (arbitrary) assignment of the "R" and "S" reference compounds
for the absolute configuration of molecules.
I decided to follow this paper trail. The PRAA document was hard to
come by and, to my disappointment, again referenced the "real"
calculation to another work. Eventually, however, all roads lead back
to R. W. James (1946). This is the definitive textbook on scattering
theory (originally edited by Sir Lawrence Bragg himself). It is
extremely useful, and I highly recommend that anyone who wants to really
understand scattering should read it. However, even this wonderful text
does not go through the full quantum-mechanical derivation of
scattering, but rather rests on J. J. Thompson's original classical
treatment. There is nothing wrong with this because the the exact value
of the phase lag of the scattering event does not effect anything as
long as the phase lag from all the atoms is the same. The only time it
does become important is anomalous scattering. Even so, changing the
sign of the phase lag will have no effect on any of the anomalous
scattering equations as long as all the anomalous contributions have the
same sign. The only time the sign of the phase lag is important is in
the assignment of absolute configuration. Unfortunately, a full quantum
mechanical treatment of the scattering process DOES produce a phase lag
with the opposite sign of the classical treatment. This is not the only
example of this sort of thing cropping up. One you can find in any
quantum text book is the treatment of "tilting" a quantum-mechanical
spin (such as an electron). It was shown by Heisenberg that a "tilt" of
360 degrees actually only turns an electron upside-down. You have to
"tilt" it by 720 degrees to restore the initial state, or get it
"right-side-up" again. This is very counterintuitive, but true, and
unfortunately a similar treatment of scattering results in a phase lag
of +270 degrees to "restore" the electron after the scattering event,
not +90 degrees as was derived classically. To be brief, there is a
sign error.
Perhaps the reason why noone caught this until now is not just that
the quantum calculations are a pain, but that it was very tempting to
accept that the large body of literature following Fischer's convention
would not have to be "corrected" by inverting the hand of every chiral
center described up to that time. Unfortunately, we now have an even
larger body of literature (including the PDB) that must now be "corrected".
It is an under-appreciated fact in chemistry that anomalous scattering
is arguably the only direct evidence we have about the "hand" of the
micro-world. There are other lines of evidence, such as the morphology
of macroscopic crystals and some recent STEM-type microscope
observations of DNA. However, as someone with a lot of experience in
motor control I don't mind telling you how easy it is to make a sign
error in the direction of an axis. This is especially easy when the
range of motion of the axis is too small to see by eye. You end up just
swapping wires and flipping bits in the axis definitions until you "get
it right". The "right" configuration (we have all assumed) is the one
asserted in Bijvoet et. al. (1951). Apparently, the STEM observations
fell prey to such a "mistake". But can you blame them? Inverting the
"hand of the world" is going to be very hard for a lot of people to
accept. Indeed, if anyone can find an error in my math, please tell
me! I would really like to be wrong about this.
-James Holton
MAD Scientist
