Category Archives: Willock: “Molecular Symmetry”

Going to the mat with representation, characters and group theory

Willock’s book (see https://luysii.wordpress.com/category/willock-molecular-symmetry/) convinced me of the importance of the above in understanding vibrational spectroscopy.  I put it aside because the results were presented, rather than derived.  From the very beginning (the 60’s for me) we were told that group theory was important for quantum mechanics, but somehow even a 1 semester graduate course back then didn’t get into it.  Ditto for the QM course I audited a few years ago.

I’ve been nibbling about the edges of this stuff for half a century, and it’s time to go for it.  Chemical posts will be few and far between as I do this, unless I run into something really interesting (see https://luysii.wordpress.com/2012/02/18/a-new-way-to-study-reaction-mechanisms/).  Not to worry –plenty of interesting molecular biology, and even neuroscience will appear in the meantime, including a post about article showing that just about everything we thought about hallucinogens is wrong.

So, here’s my long and checkered history with groups etc.  Back in the day we were told to look at “The Theory of Groups and Quantum Mechanics” by Hermann Weyl.  I dutifully bought the Dover paperback, and still have it (never throw out a math book, always throw out medical books if more than 5 years old).  What do you think the price was — $1.95 — about two hours work at minimum wage then.  I never read it.

The next brush with the subject was a purchase of Wigner’s book “Group Theory and its Application to the Quantum Mechanics of Atomic Spectra”  — also never read but kept.  A later book (see Sternberg later in the post)  noted that the group theoretical approach to relativity by Wigner produced the physical characteristics of mass and spin as parameters in the description of irreducible representations.  The price of this one was $6.80.

Then as a neurology resident I picked up “Group Theory” by Morton Hammermesh (Copyright 1962).  It was my first attempt to actually study the subject.  I was quickly turned off by the exposition.  As groups got larger (and more complicated) more (apparently) ad hoc apparatus was brought in to explain them — first cosets, then  subgroups, then normal subgroups, then conjugate classes.

That was about it, until retirement 11 years ago.  I picked up a big juicy (and cheap) Dover paperback “Modern Algebra” by Seth Warner — a nice easy introduction to the subject.

Having gone through over half of Warner, I had the temerity to ask to audit an Abstract Algebra course at one of the local colleges.  I forget the text, but I didn’t like it (neither did the instructor).  We did some group theory, but never got into representations.

A year or so later, I audited a graduate math abstract algebra course given by the same man.  I had to quit about 3/4 through it because of an illness in the family.  We never got into representation theory.

Then, about 3 years ago, while at a summer music camp, I got through about 100 pages of “Representations and Characters of Groups” by James and Liebeck.  The last chapter in the book (starting on p. 366) was on an application to molecular vibration.  The book was hard to use because they seemed to use mathematical terms differently than I’d learned — module for example.   I was used to.  100 pages was as far as I got.

Then I had the pleasure of going through Cox’s book on Galois theory, seeing where a lot of group theory originated  (along with a lot of abstract algebra) — but there was nothing about representations there either.

Then after giving up on Willock, a reader suggested  “Elements of Molecular Symmetry” by Ohrn.  This went well until p. 28, when his nomenclature for the convolution product threw me.

So I bought yet another book on the subject which had just come out “Representation Theory of Finite Groups” by Steinberg.  No problems going through the first 50 pages which explains what representations, characters and irreducibles are.  Tomorrow I tackle p. 52 where he defines the convolution product.  Hopefully I’ll be able to understand it an go back to Ohrn — which is essentially chemically oriented.

The math of it all is a beautiful thing, but the immediate reason for learning it is to understand chemistry better.  I might mention that I own yet another book “Group Theory and Physics” by Sternberg, which appears quite advanced.  I’ve looked into it from time to time and quickly given up.

Anyway, it’s do or die time with representation theory.  Wish me luck

Where has all the chemistry gone?

Devoted readers of this blog (assuming there are any) must be wondering where all the chemistry has gone.  Willock’s book convinced me of the importance of group theory in understand what solutions we have of the Schrodinger equation.  Fortunately (or unfortunately) I have the mathematical background to understand group characters and group representations, but I found Willock’s  presentation of just the mathematical  results unsatisfying.

So I’m delving into a few math books on the subject. One is  “Representations and Characters of Groups” by James and Liebeck (which provides an application to molecular vibration in the last chapter starting on p. 367).  It’s clear, and for the person studying this on their own, does have solutions to all the problems. Another is “Elements of Molecular Symmetry” by Ohrn, which I liked quite a bit.  But unfortunately I got stymied by the notation M(g)alpha(g) on p. 28. In particular, it’s not clear to me if the A in equation (4.12) and (4.13) are the same thing.

I’m also concurrently reading two books on Computational Chemistry, but the stuff in there is pretty cut and dried and I doubt that anyone would be interested in comments as I read them.  One is “Essential Computational Chemistry” by Cramer (2nd edition).  The other is “Computational Organic Chemistry” by Bachrach.  The subject is a festival of acronyms (and I thought the army was bad) and Cramer has a list of a mere 284 of them starting on p. 549. On p. 24 of Bachrach there appears the following “It is at this point that the form of the functionals  begins to cause the eyes to glaze over and the acronyms appear to be random samplings from an alphabet soup.”  I was pleased to see that Cramer still thinks 40 pages or so of Tom Lowry and Cathy Richardson’s book is still worth reading on molecular orbital theory, even though it was 24 years old at the time Cramer referred to it.  They’re old friends from grad school.   I’m also pleased to see that Bachrach’s book contains interviews with Paul Schleyer (my undergraduate mentor).  He wasn’t doing anything remotely approaching computational chemistry in the late 50s (who could?).  Also there’s an interview with Ken Houk, who was already impressive as an undergraduate in the early 60s.

Maybe no one knows how all of the above applies to transition metal organic chemistry, which has clearly revolutionized synthetic organic chemistry since the 60’s, but it’s time to know one way or the other before tackling books like Hartwig.

Another (personal) reason for studying computational chemistry, is so I can understand if the protein folding people are blowing smoke or not.  Also it appears to be important in drug discovery, or at least is supporting Ashutosh in his path through life.  I hope to be able to talk intelligently to him about the programs he’s using.

So stay tuned.

Putting Willock Aside

It’s apparent from getting through 127 Pages of “Molecular Symmetry” by David J. Willock, that he put a lot of effort into the book, and that group theory is quite important to chemistry.  But, it appears that this book is of the ‘plug and chug’ variety.  You are given formulas and codes for irreducible representations of point groups (which the book explained quite well) and then asked to apply them.  Just how the formulas are obtained hasn’t been given.  So you are expected to learn the concepts by example, which I find irritating for something basically mathematical, where concepts are clearly defined and results deduced.  I was hoping to understand characters, but I really don’t at this point.  The book made it clear how useful they are to group theory — another way to slice and dice group structure (along with subgroups, and conjugate classes).  So I’ll put the book aside, and hopefully come back to it, once I read some more group theory.

Plug and chug can even get you through a math course, but understanding is what’s crucial.  Even you haven’t looked at something for a long time (in my case decades), if you once understood something, it comes back very quickly.  Plug and chug does not, just like the combination of key strokes you need for a program you last used years ago.

Students are being cheated when plug and chug is all they get.  I did an alumni interview last night with a very bright young woman taking AP calculus (2200 on SAT’s etc. etc.).  She knew how to calculate a derivative, but had literally no idea what a derivative actually is.  She couldn’t give a geometric example of a limiting process.  It’s not because she’s stupid, because she’s far from that, it’s because she isn’t being well taught.

For those of you plowing through the book along with me, here are my comments (up to page 127 when I gave up).  Comments on earlier pages can be found in the category “Willock: “Molecular Symmetry”

*****

pp. 105 — 127

p. 105 — If you think about a vibration, you are thinking about a molecule in motion, so how can it be the subject of a symmetry element?  Because at any instant in (some) vibrations the position of the atoms are fixed, and these positions can (italics) possess varying degrees of symmetry as the example of the guitar shows.  Nice.

p. 109 — “contravenes the restrictions of the irreducible representation” — probably should add “(e.g. A1)” to avoid the idea the restrictions of all (italics) irreducible representations have somehow been given. 

p. 109 — “The molecule, as a whole, never actually moves during a vibration” — otherwise you’d violate the first law of thermodynamics, getting translational energy from nothing.

p. 110 – Further understanding of what a basis can actually be.  You can put 3 basis vectors on each atom of water, giving a total of 9 basis vectors.

p. 115 – “Property 3: The number of irreducible represntations in a point group is equal to the number of classes”.  But class was defined very incompletely on p. 93 as ‘an equivalent set of operations’ — bu how two distinct symmetry operations are equivalent is never (italics) defined, just illustrated.

What a culture clash — “Property 3 can be confirmed”  – confirmation isn’t proof.  Hopefully I’ll learn something about representations and their utility in chemistry by continuing to plow forward.

My guess is that the notion of equivalence for classes is conjugate equivalence (from my knowledge of group theory) and that the classes are conjugate classes (we’ll see).

p. 116 — I never got to character theory when I studied groups (which is one reason I’m reading this book), but property 4 tells why they are important — they give another way of slicing and dicing groups (like subgroups and conjugate classes).  Again property 4 is confirmed for a variety of examples rather than proven.

p. 117 — Property 5 — Yet another way characters give you a way to slice and dice a group — orthogonality.

p. 118 –  What is a fundamental vibration of a molecule?   A vibration having some symmetry (not all vibrations do)?

p. 119 — you can simplify a reducible representation using the data for the set of irreducible representations in the standard character tables — how are these irreducible representations obtained?  Willock isn’t telling.

30 Jan ’12 — Well, it hasn’t been all in vain.  Cramer’s book on Computational Chemistry (which I’m reading concurrently) assumes that you know nomenclature for the point groups. 

Willock pp. 51 – 104

This is a continuation of my notes, as I read  Molecular Symmetry” by David J. Willock.  As you’ll see, things aren’t going particularly well.  Examples of concepts are great once they’ve been defined, but in this book it’s examples first, definitions later (if ever).

p. 51 — Note all the heavy lifting  required to produce an object with only (italics) C4 symmetry (figure 3.6)  First,  you need 4 objects in a plane (so they rotate into each other), separated by 90 degrees.  That’s far from enough objects as there are multiple planes of symmetry for 4 objects in a plane (I count 5 — how many do you get?)  So you need another 4 objects in a plane parallel to the first.  These objects must be a different distance from the symmetry axis, otherwise the object will have A C2 axis of symmetry, midway between the two planes. Lastly no object in the second plane can lie on a line parallel to the axis of symmetry which contains an object in the first plane — e.g. the two groups of 4 must be staggered relative to each other.    It’s even more complicated for S4 symmetry.  

p. 51 — The term classes of operation really hasn’t been defined (except by example).   Also this is the first example of (the heading of) a character table — which hasn’t been defined at this point.

p. 52 — Note H2O2 has C2 symmetry because it is not (italics) planar.   Ditto for 1,2 (S, S) dimethyl cyclopropane (more importantly this is true for disulfide bonds between cysteines forming cystines — a way of tying parts of proteins to each other. 

p. 55 — Pay attention to the nomenclature: Cnh means that an axis of degree n is present along with a horizontal plane of symmetry.  Cnv means that, instead, a vertical plane of symmetry is present (along with the Cn axis)

p. 57 — Make sure you understain why C4h doesn’t  have vertical planes of symmetry.

p. 59 — A bizarre pedagogical device — defining groups whose first letter is D by something they are not (italics) — which itself (cubic groups) is at present undefined.  

Willock then regroups by defining what Dn actually is.

It’s a good exercise to try to construct the D4 point group yourself. 

p. 61 — “It does form a subgroup” — If subgroup was ever defined, I missed it.  Subgroup is not in the index (neither is group !).  Point group is in the index, and point subgroup is as well appearing on p. 47 — but point subgroup isn’t defined there.  

p. 62 — Note the convention — the Z direction is perpendicular to the plane of a planar molecule.

p. 64 — Why are linear molecules called Cinfinity ? — because any rotation around the axis of symmetry (the molecule itself) leaves the molecule unchanged, and there are an infinity of such rotations.

p. 67 — Ah,  the tetrahedron embedded in a cube — exactly the way an organic chemist should think of the sp3 carbon bonds.  Here’s a mathematical problem for you.  Let the cube have sides of 1, the bonds as shown in figure 3.27, the carbon in the very center of the cube — now derive the classic tetrahedral bond angle — answer at the end of this post. 

p. 67 — 74 — The discussions of symmetries in various molecules is exactly why you should have the conventions for naming them down pat.  

p. 75 — in the second paragraph affect should be effect (at least in American English)

p. 76 — “Based on the atom positions alone we cannot tell the difference between the C2 rotation and the sigma(v) reflection, because either operation swaps the positions of the hydrogen atoms.”   Do we ever want to actually do this (for water that is)? Hopefully this will turn out to be chemically relevant. 

p. 77 — Note that the definition of character refers to the effect of a symmetry operation on one of an atom’s orbitals (not it’s position).  Does this only affect atoms whose position is not (italics) changed by the symmetry operation?  Very important to note that the character is -1 only on reversal of the orbital — later on, non-integer characters will be seen.  Note also that each symmetry operation produces a character (number) for each orbital, so there are (number of symmetry operations) * (number of orbital) characters in a character table

p. 77 – 78 — Note that the naming of the orbitals is consistent with what has gone on before.  p(z) is in the plane of the molecule because that’s where the axis of rotation is.

Labels are introduced for each of the possible standard sets of characters (but standard set really isn’t defined).  A standard set (of sets of characters??) is an irreducible representation for the group.  

Is one set of characters an irreducible representation by itself or is it a bunch of them? The index claims that this is the definition of irreducible represenation, but given the amiguity about what a standard set of characters actually is (italics) we don’t really know what an irreducible representation actually is.   This is definition by example, a pedagogical device foreign to math, but possibly a good pedagogical device — we’ll see.  But at this point, I’m not really clear what an irreducible represenation actually is.

p. 77 — In a future edition, it would be a good idea to lable the x, y and z axes (and even perhaps draw in the px, py and pz orbitals), and, if possible, put figure 4.2 on the same page as table 4.2.  Eventually things get figured out but it takes a lot of page flipping. 

p. 79 — Further tightening of the definition of a representation — it’s one row of a character table.

p. 79 — Nice explanation of orbital phases, but do electrons in atoms know or care about them?

p. 80 — Note that in the x-y axes are rotated 90 degrees in going from figure 4.4a to figure 4.4b  (why?).   Why talk about d orbitals? — they’re empty in H20 but possibly not in other molecules with C2v symmetry.  

p. 80 — Affect should be effect (at least in American English)

p. 81 — B1 x B2 = A2 doesn’t look like a sum to me.  If you actually summed them you’d get 2 for E, -2 for C2, and 0 for the other two.  It does look like the product though.

pp. 81 – 82 — Far from sure what is going on in section 4.3

p.82 — Table 4.4b does look like multiplication of the elements of B1 by itself. 

p. 82 — Not sure when basis vectors first made their appearance, possibly here.  I slid over this on first reading since basis vectors were quite familiar to me from linear algebra (see the category http://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/ ).  But again, the term is used here without really being defined.  Probably not to confuse, the first basis vectors shown first are at 90 degrees to each other (x and y), but later on (p. 85 they don’t have to be — the basis 0vectors point along the 3 hydrogens of ammonia).

p. 83 — Very nice way to bring in matrices, but it’s worth nothing that each matrix stands for just one symmetry operation.  But each matrix lets you see what happens to all (italics) the basis vectors you’ve chosen. 

p. 84 — Get very clear in your mind that when you see an expression of the form

symmetry_operation1 symmetry_operation2 

juxtaposed to each other — that you do symmetry_operation2  FIRST.

p. 87  – Notice that the term character is acquiring a second meaning here — it no longer is the effect of a symmetry operation on one of an atom’s orbitals (not the atom’s position), it’s the effect of a symmetry operation on a whole set of basis elements.

p. 88 — Notice that in BF3, the basis vectors no longer align with the bonds (as they did in NH3), meaning that you can choose the basis vectors any way you want.  

p.89 — Figure 4.9 could be markedly improved.  One must distinguish between two types of lines (interrupted and continuous), and two types of arrowheads (solid and barbed), making for confuion in the diagrams where they all appear together (and often superimposed).  

Given the orbitals as combinations of two basis vectors, the character of symmetry operation and a basis vector, acquires yet another meaning — how much of the original orbital is left after the symmetry operation. 

p. 91 — A definition of irreducible representations as the ‘simplest’ symmetry behavior.  Simplest is not defined.  Also for the first time it is noted that symmetries can be of orbitals or vibrations.  We already know they can be of the locations of the atoms in a molecule.  

Section 4.8 is extremely confusing.

p. 92 — We now find out that what was going on with a character sum of 2 on p. 81 — The sums  were 2 and 0 because the representations were reducible.  

 

p. 93 (added 29 Jan ’12) — We later find out (p. 115) that the number of reducible representations of a point group is the number of classes.  The index says that class is defined an ‘equivalent set of operations’ — but how two distinct operations are equivalent is never defined, just illustrated.

p. 100 — Great to have the logic behind the naming of the labels used for irreducible representations (even if they are far from intuitive)

p. 101 — There is no explanation of the difference between basis vector and basis function. 

All in all, a very difficult chapter to untangle.  I’m far from sure I understand from p. 92 – 100.  However, hope lies in future chapters and I’ll push on.  I think it would be very difficult to learn from this book (so far) if you were totally unfamiliar with symmetry.  

Answer to the problem on p. 67.  Let the sides of the cube be of length 1.  The bonds are all the same length, so the carbon must be in the center of the cube.  Any two of the bonds point to the opposite corners of a square of length 1.  Therefore the ends of the bonds are sqrt(2) apart.   Now drop a perpendicular to the middle of this line to get to the carbon in the center.  This has length 1/2.  So we have a right triangle of side 1/2 and ( sqrt(2))/2.  So the answer is 2 * arctan(1.414).  Arctan(1.414 is) 54.731533 degrees giving the angle as 109.46 degrees.

Willock pp. 1 – 50

These are some very detailed comments, and (hopefully) helpful hints for you as you read  “Moleular Symmetry” by David J. Willock.  For why I’m reading it,  why you should too, and what a group actually is see the first post in this series — http://luysii.wordpress.com/2011/12/11/chemistry-helps-you-understand-group-theory-and-not-vice-versa/

p. 5 — Distinguish in carefully in your mind the difference between a symmetry element and a symmetry operation as you read the book. 

      A symmetry element  is a geometric structure (point, line, plane) about which a molecule is symmetric.  On p. 23 This definition is clarified — a symmetry element is the set of points which aren’t moved when a symmetry operation takes place.  Points of what? The points of the space in which the molecule is embedded.

      A symmetry operation is an action carried out using a symmetry element which leaves the shape of the molecule unchanged (although no atom of the molecule may wind up where it was after the operation).  Ammonia has a 3 fold axis of rotation (e.g. the symmetry element is a line), but 3 distinct symmetry operations about the symmetry element (rotation by 120, 240 and 360 == 0 degrees). 

p.  7 — Principal axis — the line of symmetry (axis) with the largest (highest) order of rotation.  Always aligned with the Z axis (vertical).  One of the many important conventions you’ll need to remember is that the highest (largest) symmetry axis defines  the vertical direction.  You’ll need to know what vertical is to appreciate the labelling of the symmetry planes of water on p. 10, and how to distinguish it from horizontal in the case of BF3 (p. 11).

p. 7. — Also very important to keep straight — after a symmetry operation the atoms of a molecule may move, but the axes do not move – why is this important?  – because it’s very easy to get confused about how to do the a second symmetry operation after you’ve done the first. Remembering this will save you later on (see figure 2.4 p. 30). 

p. 9 — Another convention — the Greek letter sigma is always used for a plane of symmetry.  If you’re reading this on your own — start yourself a symbol table with the page the symbol is first defined (not the same as when Willock first mentions the term, definitions sometimes come later — this takes some getting used to if you’re used to math books where everything is defined before being used).  Molecular Symmetry should have a symbol table  (newer math books do) but it doesn’t.

p. 12 — The description of when a plane is dihedral and when it is horizontal is ambiguous at best, and confusing at worst.  You’ll need the actual book to follow the following.  In figure 1.15 there are horizontal C2 axes passing through the C-C bond center, but they are not in the mirror planes  shown.  However, there is a third mirror plane which isn’t shown.  When this axis is put in one sees that the dihedral mirror plane splits all 3 C2 rotation axes (which is why it’s called a dihedral plane).   Then another example is given of a dihedral plane which splits the angle with two other planes (called v — for vertical I guess, because the plane contains the vertical (Z) axis).  A horizontal plane is perpendicular to the principal axis.  This could have been made more explicit, so a clear understanding of sigma(v), sigma(h) and sigma(d) emerges.  On p. 23 Willock does make all this explicit.  It was probably better to do this at the outset.  

p. 21 “The absorption in an NMR experiment takes a short, but finite time”.  It is truly maddening to find out how long this time is using Google or any of my NMR books.   I’ve speculated about the time it takes for absorption of a wavelength of light in an earlier post.  Here it is again — if any physicist types are reading this, please correct me if I’m wrong, or enlighten me further. 

   ***** The penultimate chapter of Anslyn and Dougherty “Modern Physcal Organic Chemistry” contains an excellent discussion of photochemistry, with lots of physics clearly explained but it leaves one question unanswered which has always puzzled me.  How long does it take for a photon of a given wavelength to be absorbed?  
       On p. 811 there is an excellent discussion of the way the quantum mechanical operator for kinetic energy (-hBar/2m * del^2) is related to kinetic energy.  The more the wavefunction changes in space, the higher the energy.  It’s like cracking a whip, the faster you move the handle up and down (e.g. the faster the frequency), the more energy you impart to the whip.   Note that the kinetic energy operator applies to particles (like protons, neutrons, electrons) with mass.

          Nonetheless, in a meatball sort of way, apply this to the (massless) photon.  Consider light from the classical point of view, as magnetic and electrical fields which fluctuate in time and space.  The fields of course exert force on charged particles, and one can imagine photons exerting forces on the electrons around a nucleus, changing their momentum, hence doing work on them.  Since energy is conserved (even in quantum mechanics), it’s easy to see how the electrons get a higher energy as a result.  The faster the fields fluctuate, the more energy they impart to the electrons.

         Now consider a photon going past an atom, and being absorbed by it.  It seems that a full cycle of field fluctuation (e.g. one wavelength) must pass the atom.  So here’s a back of the envelope calculation, which seems to work out.  Figure an atomic diameter around 1 Angstrom (10^-10 meters) for the average atom.  The chapter is about photochemistry, which is absorption of light energetic enough to change electronic energy levels in an atom or a molecule.  All the colored things we see, are colored because changes between their electronic energy levels are absorb photons of visible light — the colors actually result from the photons NOT absorbed.  So choose light of 6000 Angstroms — which has a wavelength of 6 * 10^-7 meters.  It will appear orange to you.

        In one second, light moves 3 * 10^8 meters, regardless of how many many wavelengths it contains. If the wavelength were 1 meter it would move past a point in 1/(3 * 10^8) seconds But wavelength of the visible  light  I chose is 6 * 10 ^-7 meters, so the wavelength moves past in 6*10^-7/3 * 10^8 = 2 x 10^-15 seconds, which (I think) is how long it takes visible light to be absorbed.  Have I made a mistake?  Are there any cognoscenti out there to tell me different?

p. 26 — Note that sigma(v’) is in the plane of benzene and water here, while in the case of BF3 (p. 11) the mirror plane in the plane of BF3 is called sigma(h) — this is because the latter is perpendicular to the principal axis of rotation (p. 7), while the former two are parallel to it.   It’s best to have these things clear in your mind when reading further.

p. 26 — Note which of the two operations in C2sigma(v’) is done first — it is always the rightmost — this is standard in the mathematical literature, and probably comes from the way functions within functions are applied — sin(3x) means that you do 3x first and then apply sine to it.  Yet another convention to keep straight and crucial in what follows.

p. 26 — Point group — all the symmetry operations a molecule can undergo. None of the amino acids making up our proteins (except glycine) have more than 1 (the identity operation).

p. 27 — Note the convention for the X, Y and Z axes — Z points in the direction of the principal axis.  Y in the plane of the molecule (assuming there is one) and X perpendicular to Y and Z.  None of this tells you which is positive and negative on each axis — but (without saying so) the X,Y and Z axes are a right handed coordinate system (remember the old right hand rule — use the fingers of your right hand to coil the positive X into the positive Y, and your thumb will point in the positive Z direction.

p. 27 -  I think the bottom diagram of sigma(v’) is incorrect H1 and H2 aren’t changed by it.

p. 27 — The following is also very important to note (and not especially clear),  Distinguish the global coordinate axes (capital X, Y, Z) which aren’t changed by the symmetry operation and the vectors attached to atoms (small x, y, z) which   are (italics) changed by the symmetry operations.  So capital X, Y, Z means global coordinate axis, while small x, y, z means vector.  Easy to confuse the two.

p. 28 — Another important convention — when faced with a table for the multiplication of symmetry operations, do the operation in the row of operations at the top of the table first and the operation in the column of operations on the left side of the table second. 

Remember all this advice and you’ll do a lot less looking back. 

p. 31 — the Sn operation.  Note that in the example given (staggered ethane), that neither S6 nor C2 in the plane horizontal to it is an actual symmetry element of the molecule. 

p. 36 — The term basis is mentioned but not really defined.  It appears to be an arrow attached to an atom which follows the atom through various symmetry operations.  Apparently not the same as vectors attached to the oxygen of water (p. 27) which are labeled x, y and z. 

p. 37 — Perhaps the improper rotation on BF3 which requires applications (with a total of 720 degrees) to get things back to where they started, is relative to some rotations in particle space that I recall reading about, that took 720 degrees of rotation to get things back together.  

p. 38 — Requiring that an  inverse of a symmetry operation exist is a crucial property of groups, but here it is introduced by fiat without saying why.  For what a group is see the first post in the series. 

End of chapter 2 — The fact that the set of symmetry operations of a molecule must be closed, along with the fact that there are only finitely many of them, means that for any symmetry operation (S), applying it over and over eventually gets you back to the original operation.  This means there is some n in the non-negative integers such that for every symmetry operation S, S^n = S (S^n is a Mathematica convention meaning S to the nth power — multiplying S by itself n times — multiplication of symmetry operations just means applying one after the other — rightmost first .  A bit of thought then shows that S^(n-1) = E (the identity).  A bit more thought shows that the powers of S form a group by themselves (they contain 1, and have an inverse).  

      This means that they are a subgroup of group of all symmetry operations.  Subgroup isn’t in the index, as this is a chemistry book, not a math book.  It’s simple enough concept — a subset of a group of symmetry operations which is also a group (e.g. it contains E, the identity, every element has an inverse and a more obscure characteristic called associativity — which it inherits from the parent group).

       This is true for each symmetry operation.  What molecular geometry allows you to do, which isn’t obvious when group theory is studied abstractly, is to see how the powers of each element combine with each other. E.g. For water what is C2*sigma(v)?  The symmetries of water are an example of a very peculiar group called the viergruppe (German — four group).   Nice ! ! ! ! 

p. 45 — Introduces the character table, and notes that not all the symbols in it have been defined yet.  The term character also isn’t defined at this point.  It’s an interesting teaching technique, and very different from those used in math books. 

p. 46 — Pasteur was lucky to find tartaric acid which crystalizes in two form each containing a pure enantiomer.   How often does this happen? 

p. 48 — Watch out when reading about point groups. Lots of molecules contain a 2 fold rotation axis (e.g. C2), but the C2 point group contains only that symmetry operation and none other (except the identity) — this is why it must have at least 4 points (because 3 points define a plane, which is a different symmetry element). 

p. 48 — The drawing of the C(s) symmetry group is truly terrible and should be improved in a second edition.  Notice that the symmetry plane sigma is labelled sigma(h) even though it appears vertical. 

Hopefully this was helpful — I wish I’d written all this down the first time I went through these pages.

Chemistry helps you understand group theory and not vice versa

Back in the day, we were told that group theory was important in quantum mechanics, because it simplified the Schrodinger equation, and ultimately the interpretation of spectra.  We really didn’t get into groups in the graduate QM course I took in ’61, but were told to look at Weyl’s book “The Theory of Groups and Quantum Mechanics” which didn’t get into representation theory until p. 120 after a lot of linear algebra and physics.  What chemist had the time? I didn’t.

In retirement, I’ve indulged my taste for math, and audited a graduate algebra course, which had a fair amount of group theory, but no representation theory.  In addition the instructor dumped all over chemistry and physics as part of his schtick of being a pure mathematician.  He avoided all applications to chemistry.  He did hold up for derision a book written by a chemist with several large groups explicitly written out.

So now that I’m reading chemistry again and am up to Ch. 14 of Anslyn and Dougherty  on computation of orbitals, I thought it was time to give group theory another shot.

The postulates of group theory really couldn’t be simpler, and as you delve into the subject, it’s amazing how much structure can arise from them.

Here they are.

A group is just a set G, which can be finite or infinite.  Chemical symmetry uses only finite sets, but in physics groups with an infinite number of elements are possible.

Just one operation is defined on G.  It is a binary operation, which is to say it takes any 2 elements of the group, and produces a third element of the group.  Examples include addition and multiplication, which take any two numbers and give you a third.    In mathematical terms, the operation is said to be closed.  Call the operation *.

* : G x G –> G ;

* : g *  h |–> i, where g, h and i are elements of G not necessarily distinct.

There is only one other requirement on the operation.  The operation must be associative — so that given 3 elements of the group, the order on which you subject them to the group operation doesn’t matter.  e.g.

a * (b * c) = (a * b) * c ; where you do the operation inside the parentheses first

The group must have one special element (usually called e for the German einheit meaning identity), but 1 will do.

For any element of a of the group 1 * a = a * 1 = 1

Lastly, every element of the group must have an inverse (written a’ ) such that

a * a’ = 1  and a’ * a = 1, so 1 is its own inverse, and other elements can be as well.

So the positive integers are NOT a group under addition (no inverse).  The integers are not a group under multiplication ( 0 has no inverse).

That’s all there is to it.  Yet the classification of all finite groups took 30 years, and 10,000 pages in journals, and people aren’t really sure they’ve got it done. The classification of the quasithin groups (don’t ask) was the subject of a 1221 page paper.

Representation theory is the correspondence of the elements of a group  to a set of matrices.  The group operation of a representation is always matrix multiplication.  You can find a gentle introduction to matrices in the 9 posts  in the category — https://luysii.wordpress.com/category/linear-algebra-survival-guide-for-quantum-mechanics/.

The math books don’t have any chemistry, and while groups are worth studying for their own sweet selves, I want to see what they offer the chemist (well, more realistically- HOW they offer what they do to the chemist).  So I’ve begun reading “Molecular Symmetry” by D. J. Willock.  Lots of chemical structures, schematic diagrams (some not particularly intuitive), and an approach to groups through symmetry.

What is really exciting about all this, is that chemistry lets you explore the structure of groups by moving a molecule around so that it superimposes on itself.  The book shows how the set of symmetry operations you can perform on a molecule form the elements of a group (showing that the elements of a group aren’t limited to just numbers).   The group operation is simply doing one symmetry operation after another. So instead of screwing around with matrices, which have no visual content (although their effects do), you can play around with simple molecules like water, ammonia, benzene and watch a group in action.

Technically what you are really doing when you do this is looking at what is called a group action — the action of a group on another set (e.g. the distribution of a collection of atoms in space in this case).  But, to my mind it’s really a way (and a better one) of representing a group.  Water is an example of the Viergruppe (4 group), which has only 4 elements.  The instructor said it’s a very strange group.

Parenthetically it is worth noting that Slater (of the Slater orbitals, and Slater determinant of quantum mechanics) hated groups — calling them the gruppenpest.  Anslyn has a lot about both in  pp. 817 – 822.

It’s too early to fully recommend the book, but the first 50 pages are quite fine.  Stay tuned

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