Dr. Anders: Professor Hall, one of your papers about the history of quantum chemistry closed with the sentence "... but much more remains to be achieved." Well, can you be a bit specific about that.
Prof. G.G. Hall: I see, you are putting me very much on the spot. I can't now remember what I was thinking of at that time. But it is still my opinion about the subject that, although it has developed a tremendous distance in the past thirty, forty years, it still has a long way to go. And it's not always possible to predict or to see clearly in which direction it's going to flourish. I would find myself happier if it became a bit simpler, that is to say if we could hide the messy bits a good bit more, and of course some modern programs do go a long way along those lines so as to bring two things closer together. These are the basic quantum mechanical laws of governing electrons and protons and so on and how the chemist thinks about molecules.
A: What are electrons really doing in molecules  Mullikan....
G.G. Hall: Yes, what they are really doing is important and what is important also is the other end of the story  are they giving us some insight into chemistry or biology? These subjects very much need a foundation which is not just empirical but has something more permanent and more fundamental underlying it, so that you can say, we have established this more firmly than would have been possible just by doing experiment. As you know, in chemistry we can do that  we can sometimes tell chemists that their answer is not quite correct, you should go and do it again. And theory has that inner strength from  if you like  the accuracy of the basic physics that lies behind the properties of matter. That connection between theory and experiment has got to be made as smooth and usable as possible.
A: Ab initio...?
George G. Hall in 2003 Photo and © U. Anders 
G.G. Hall: I am still in two minds, I must say, about ab initio versus what used to be called semiempirical, because I think that the two are not so different or not now so different as they used to be. The one should be able to merge into the other by making approximations which are justifiable. So that you can say, what I am looking for is a broad result, I am not looking for 20 correct decimal places. I am looking for whether the product of this reaction is this or that and twenty decimal places may be needed to do that  but I don't want to know about those twenty, I merely want to know what the answer is. It's that kind of way of superseding the detailed, the necessarily detailed, complication of ab initio the whole time. I think it has to be ab initio in the sense that the only mathematical foundations for the subject that we have are those already fixed. And we have to develop from that starting point, we are not introducing new principles in the middle, but we are developing shortcuts through it. That is why I would like to say that this result is what would interest a chemist and this is why we are saying it  without him having to wade through 100 pages of numbers.

And then you look at what relation there is between the molecular orbitals you have got and the molecular orbitals you want down in the ground state with double occupation. And you can do that calculation, you can do all of that calculation with strict theory and when you do that you can look at the integrals between the localised functions which you get, how big they are or how small they are. And the result is quite dramatic. You get the great majority of the twoelectron integrals between the six localised functions becoming small very quickly and you are left very clearly with the coulombic integrals alone. The resulting orbital equations then have really the same structure as Hückel had. My theory has one extra term in it which is the one brought in by the PariserParrPopleTheory. Or, if you like in solid state theory, it's the one used in the Hubbard theory.

Photo and © U. Anders 
But, as I say, that paper really showed how far you could justify the Hückel treatment. And one of the things that emerged from it was that, if you like, the bond energy, which Hückel had estimated from his studies of calorimetry and so on, was too small. The calculation showed that you have to use a very much larger beta integral  very much bigger than Hückel would have liked. Because we are talking about a really different definition, in my theory the two states have the same geometry so beta is defined vertically, whereas in thermo chemistry the internuclear distances are different in the states used in that definition. That source of energy variation was not part of Hückel's original theory though it had previously been pointed out by LennardJones. So he missed out a little on that.
A: Now, let's talk a bit about graph theory. Was that just a topological game or what? You have done various articles about graph theory. How would you evaluate the topic of graph theory in early quantum chemistry? What about Gutman?
G.G. Hall: Yes, I feel that I'm a heretic when it comes to graph theory, because I believe it has a solid foundation of the kind that I have been talking about so it can be justified as quite a good approximation to a version of molecular orbital theory. (The adjacency matrix of graph theory is the Hückel matrix and also my excited state MO matrix.) What has more appeal to me is to bring that connection closer and and make it more visible. But I know, I've talked this over with several graph theorists, and they have been quite strongly opposed to doing that. They say the graph is a model of the molecule and no more and no less and they want to keep it as simply a model theory which may or may not tell them something about molecules, but will always be useful as a way of characterising, describing and putting molecules into sequences and so on. And of course I don't deny any of that. I think I am a little more wary than they are of making chemical predictions because I know there are limitations in the model, I know where some of these are and I know there are applications where those limitations are not very important. So I feel that I have an insight into what they are doing that they simply don't want to have. And  as I say, I'm a heretic {laughs}. I don't make an apology for that {laughs}.
A: Have you written about this?
G.G. Hall: I don't think I have put that down so explicitely  but I think the essence of this I've expressed in several papers, yes.
A: And then one hears in the context of graph theory the names of Gutman and Trinajstic. What is the connection between the both of them?
G.G. Hall: I'm not quite sure that I can answer your question. There was a school of graph theory .... {G.G.Hall is reflecting}
I think they both started at that point  that is in Croatia, in Zagreb. And Gutman later moved from there to Serbia. Gutman I know much better than Trinajstic, he spent several nights in our house and we have collaborated before and after that. So I feel, I know Gutman considerably better. Gutman is a very vigorous researcher in the subject and has a good deal of technical skills, more so than, I think, most of other people who are working in the field.
A: Email of Gutman?
G.G. Hall: I don't have an Email but an University address: Faculty of Science, University of Kragujevac, P.O.Box 60, Yu34000 Kragujevac. He did have a big problem, as you know, the University was in the war area and he could not return to it, he could correspond and his students could still work but he couldn't go himself, he had several years as a kind of exile. So he has had a very difficult period personally but I think he is now fully returned and fully active.
A: By the way, do you recall what OPIT stands for  you used the term in your recollections?
G.G. Hall: {laughs heartily} OPIT was our ab initio program which used a modest number of floating Gaussians to get ideas about molecular structure. The OP refers to the optimisation of the positions and exponents of the Gaussians and IT to the fact that the MO had to be iterated to become selfconsistent.
A: You are our time witness  now, about McWeeny and the Gaussians. You are a theoretician and you can evalutate McWeeny's claim much better than I can.
G.G. Hall: I think the position as I see it, is this: McWeeny certainly wrote an earlier short paper, it was v ery brief, pointing out that Gaussians did have an integral advantage and really that was it. He didn't make anything very significant out of that.
.... The context of McWeeny's paper was cristallography where Gaussians are a more natural thing to use because they are already come into treatments of a lot of lattice sums and things like that. The whole translation business of the Gaussians had already been pointed out in relation to cristallography, so McWeeny also makes that point. What Boys was doing and had done in his papers was to use Gaussians systematically. He had looked in detail at all the possible integrals for a molecule and showed exactly how they can be integrated using, at worst, one new function.
A: How many of Boys' papers would you put in, four or five.
G.G. Hall: Oh, you ask the most awkward questions.... Yes, he had a paper prior to the first Gaussian paper which was purely on atoms where he showed how you could superimpose electronic configurations and improve on the state splitting ratios for atoms. You know it's a very old calculation, which you can get in quite a few textbooks, for oxygen where you couple two p electrons and look at the three lowest states, triplet P, the singlets S and D. If you look at the spacings between the energies of these they come out as a multiple of one of the integrals. You get a ratio for these spacings which is not right because, with spectroscopy, you can see exactly what it should be.
So what Boys did in his paper was to take not just the ground state wave function but an excited state wave function which interacted with the ground state to show that you got a different ratio much closer to the correct one. From one point of view understanding oxygen is probably not very important, but from the point of view of him developing his understanding of quantum chemistry, I think it was probably very important because he showed that you could actually take different wave functions, approximate wave functions, and allow them to interact meaningfully. And that was very much a part then of what he did later with his Gaussians. He knew his Gaussians were going to be bad wave functions, so he had to know that he could improve them which meant doing this superposition.
He had his own terminology. Part of Boys you may not understand well is this, he invented his own terms for things and some of them have stuck and some haven't so these wave functions were codetors (coupled determinants of orbitals) and he was superimposing codetors which was a very powerful idea at that time.
A: But he had big enemies. So wasn't LongetHiggins one of them and then Pople. I heard that he wasn't well liked among his colleagues. You wrote about it as well.
G.G. Hall:You really want to know?? So let's switch that off...
G.G. Hall: Now, Boys was appointed by LennardJones. Boys had been in his division during the war, doing research for him. He knew him as a former postdoc. He knew very well all about Boys and he was quite happy to appoint him. And their point of view about the subject was not 100 % in agreement at all either. But LJ was very happy that Boys was doing something different from what he was doing. And he rather appreciated the complementarity more than the contrast. Whereas LonguetHiggins couldn't make that jump?
A: He was too young still, LH.
G.G. Hall: {laughs}.
A: What about Boys getting sick during the Oxford conference
G.G. Hall: I think you may have got this story a little bit wrong. Yes, this was the first quantum conference in England that Coulson organized at Oxford. Boys was given a privileged position of speaking on the first day. And there was some criticism of his talk certainly, but he was genuinely ill because I was the one trying to act as the middle man, I saw him every day, reported about the discussions and heard what the doctor reported. I made sure the college authorities were giving him enough food and whatever. But he was certainly quite ill, a gastric upset.
I wouldn't like Boys to be thought of ungracious to colleagues. He was not, and he was genuinely ill in Oxford, I can tell you that. He went to the Boulder conference, in America, but he didn't get a great deal out of that. But when we ran conferences in Nottingham for English theorists he often came to those, so that his absence wasn't a general thing. And I think, he sometimes wanted to change them. He didn't like the standard format of the conferences, he thought a great deal of time was wasted this way so he would rather have a set theme to a conference to narrow it down and then got everyone deliberately to focus on that particular theme. We tried that once and it didn't really work but in a way we tried it for bim.
A: Boy's unfortunately did not get old enough to see the growth of his ideas. What did he die of, cancer?
G.G. Hall: I don't think it is incorrect to say yes but I don’t remember the details.
A: Did Boys have any direct students who went on in this field?
G.G. Hall: Nick Handy and Izzy Shavitt are two. Several of Boys' students moved into computing science and did very well in that. I know three of them who became professors of
computing science, Geoff Cook, Colin Reeves and Vic Price.
A: Professor Hall, you also had a Japanese interview, I tried to find it. I found the journal (a history of science journal) in its English version, but I could not discover the interview article.
G.G. Hall: I probably have a copy of that at home. One is in Kagaku (Chemistry) 39, 244, 1984, another is in Kagaku (Science) 54, 724, 1984. You could also see New Scientist of 20 Sept 1984 , 53.
G.G. Hall: Yes, there is also an article about my Japanese experience in Mathematics Today, a journal for the IMA institute, for members. Maybe it can be bought by anybody but it's designed primarly for members of the Institute so it's sort of a inhouse journal. However, it is refereed.
A: And what I would like an access to is your ideas now about mathematics and education.
G.G. Hall:
On Mathematical Education I've done about 53 short articles for that journal {laughs}., well, I did one last year, I sent another one off just before I left for here, so there are several coming out each year. I attempt two things in these: yes, certainly they are intended for, say a high school teacher, to give an example of thinking mathematically about something that could occur in every day life. I pride myself that these are all things that had occured to me in that way, they are not artifically manufactured examples, they are real examples. And I would like to think also that they could be tackled by, if not a high school student, then at least by a first year university student because the mathematical level should be kept down to the level where it becomes possible to do that, which means there are some complicated things that I can't possibly put into an article.
Now, I've been interested in mathematical education partly because of my job. I think it probably goes back further than that; I enjoyed doing mathematics in school, we had some good teachers and I felt that I wanted to expand and share that kind of experience with others. And it has been, as things have worked out, a job that I had to do in Nottingham university: that is to say we had this huge change in what they called the new mathematics that came into schools and we suddenly found that the nearby local teachers were very frightened by this, they didn't understand it, it had not been part of their training and they didn't know what to do.
So we decided we would set up a Centre to try to rectify this  which we did with a lot of outside help from Shell. And that has been running ever since, although it has changed its character several times in the meantime. It is still growing, it's main business, the modern mathematics problem, was of course a relatively short term problem. But the problems the teachers have in High School continued to go on and continued to multiply in their complexity. And I would say that it is still part of the Shell Centre philosophy that they need a great deal of help in developing the best techniques for teaching subjects. So part of our idea has been to not just to provide material, or look at material others provided, but actually to have a lot of feedback from teachers on it and put it through various editions so that what comes out is a thoroughly tested piece that can be taken into any classroom by any teacher. It is an engineering approach to what is normally an arts subject and that of course has it's own diplomatic problems.
A: Yes, I asked you, as mathematican, is there enough being done mathematically or is it too much number crunching?
G.G. Hall: There is always a problem of balance between number crunching and pure theory making, and I'm not sure I know the right point of that balance, because I think what I'm doing differs from what a pure mathematican would do is  I'm trying to create new things, new ideas, new types of calculations, new applications, new situations that have never been looked at before, new in that sense. And that is a different kind of experience from doing mathematics itself, doing mathematics is a logical stepbystep process. But applied mathematics is saying: perhaps I'm here  how do I get there, can I actually put steps in between. And that is a diffent kind of thinking. I find that a very interesting subject in itself to try to understand the kind of thought processees that go into this. Number crunching is a little bit like that, it is sort of moving along the pedestrian‘s way to do something and maybe at the end you reach somewhere. But it is also possible that you can make a single jump and reach the same place because you see an application of some other theory which the person who was doing it didn't see. And this is why new things like graph theory and group theory and so on have a part which is different part from the computing part.
A: And just one more question, Professor Hall. You are a theoretical mathematican or you are a mathematican. And you got such a wide application of your mathematics in solid state physics and quantum chemistry. Now, how about the final word of a mathematican about mathematics and science. Can you enclose all this into mathematics or .... you know you do so many topics, how do you view science  from a mathematical point of view or applied mathematics?
G.G. Hall: {laughs}. Okay, my point of view as a mathematican is that I believe strongly in applied mathematics that is mathematics for a purpose not just for its intrinsic beauty which is a perfectly valid approach but is not for me. And I'm looking at other subjects, other situations and trying to be as logical and systematic about them as I can. Which is basically what mathematics is doing. The strange consequence of that for me is this  this is what I discovered at the beginning  I started research not knowing anything about quantum chemistry as such, quantum mechanics yes, I'd done courses, quite a few courses in fact, but quantum chemistry no  when I first looked at quantum chemistry, I saw an awfull mess, and I said, the right place for us mathematicans is in the middle of a mess where there are things to be sorted out, logically, consistently, systematically, to bring into this mess some of the order which it obviously needs. So that is why I stuck with it, and this subject now is nothing like the mess that it was then. It has discarded some of the erroneous ideas and theories, it has focused on things that work, it has a lot of solid calculations that have changed the subject and have given us a great deal of insight into the subject. So the amount of input that there has been into science is tremendous.
A: Professor Hall, thank you very much for the interview.