A Different Way of Applying Quantum Mechanics to Chemistry

Time to explain what I presented at the conference, and to start, there are interpretive issues to sort. The first issue is either there is a physical wave or there is not. While there is no definitive answer to this, I argue there is, because something has to go through both slits in the two slit experiment, and the evidence is the particle always goes through only one slit. That means something else should be there. 

I differ from the pilot wave theory in two ways. The first is mathematical. The wave is taken as complex because its phase is. (Here, complex means a value includes the square root of minus 1, and is sometimes called an imginary number for obvious reasons.) However, Euler, the mathematician that really invented complex numbers, showed that if the phase evolves, as waves always do by definition of an oscillation and as action does with time, there are two points that are real, and these are at the antinodes of the wave. That means the amplitudes of the quantal matter wave should have real values there. The second difference is if the particle is governed by a wave pulse, the two must travel at the same velocity. If so, and if the standard quantum equations apply, there is a further energy, equal in magnitude to the kinetic energy of the particle. This could be regarded as equivalent to Bohm’s quantum potential, except there is a clear difference in that there is a clear value of it. It is a hidden variable in that you cannot measure it, but then again, so is potential energy; you have never actually measured that.

This gives a rather unexpected simplification: the wave behaves exactly as a classical wave, in which the square of the amplitude, which is a real number, gives the energy of the particle. This is a big advance over the probabilistic interpetation because while it may be correct,  the standard wave theory would say that A^2 = 1 per particle, that is, the probability of one particle being somewhere is 1, which is not particularly informative. The difference now is that for something like the chemical bond, the probabilistic interpretation requires all values of ψ1 to interact with all values of ψ2; The guidance wave method merely needs the magnitude of the combined amplitude.  Further, the wave refers only to a particle’s motion, and not to a state (although the particle motion may define a state). This gives a remarkable simplification for the stationary state, and in particular, the chemical bond. The usual way of calculating this involves calculating the probabilities of where all the electrons will be, then further calculating the interactions between them, recalculating their positions with the new interactions, recalculating the new interactions, etc, and throughout this procedure, getting the positional probabilities requires double integrations because forces give accelerations, which means constants have to be assigned. This is often done, following Pople, by making the calculations fit similar molecules and transferring the constants, which to me is essentially an empirical assignment, albeit disguised with some fearsome mathematics.

The approach I introduced was to ignore the position of the electrons and concentrate on the waves. The waves add linearly, but you introduce two new interactions that effectively require two new wave components. They are restricted to being between the nuclei, the reason being that the force on nucleus 1, from atom 2 must be zero, otherwise it would accelerate and there would be no bond. The energy of an interaction is the energy added to the antinode. For a hydrogen molecule, the electrons are indistinguishable, so the energy of the two new interactions are equivalent to the energy of the two equivalent ones from the linear addition, therefore the bond energy of H2 is 1/3 the Rydberg energy. This is out by about 0.3%. We get better agreement using potential energy, and introduce electric field renormalisation to comply with Maxwell’s equations. Needless to say this did not get much excitement from the conference audience. You cannot dazzle with obscure mathematics doing this.

The first shock for the audience was when I introduced my Certainty Principle, which, roughly stated, is the action of a stationary state must be quantized, and further, because of the force relationship I introduced above, the action must be that of the sum of the participating atoms. Further, the periodic time of the initial electron interactions must be constant (because their waves add linearly, a basic wave property). The reason you get bonding from electron pairing is that the two electrons halve the periodic time on that zone, which permits twice the energy at constant action, or the addition of the two extras as shown for hydrogen. That also contracts the distance to the antinode, in which case the appropriate energy can only arise because the electron – electron repulsion compensates. This action relationship is also the first time a real cause of there being a bond radius that is constant across a number of bonds has been made. The repulsion energy which is such a problem if you consider it from the point of view of electron position is self-correcting if you consider the wave aspect. The waves cannot add linearly and maintain constant action unless the electrons provide exactly the correct compensation, and the proposition is the guidance waves guide them into doing that.

The next piece of “shock” comes with other atoms. The standard theory says their wave functions correspond to the excited states of hydrogen because they are the only solutions of the Schrödinger equation. However, if you do that, you find that the calculated values are too small. By caesium, the calculated energy is out by an order of magnitude. The standard answer – the electrons penetrate the space occupied by the inner electrons and experience stronger electric field. If you accept the guidance wave principle, that cannot be because the energy is determined at the major antinode. (If you accept Maxwell’s equations I argue it cannot be either, but that is too complicated to be put here.) So my answer is that the wave is actually a sum of component waves that have different nodal structures, and an expression for the so-called screening constant (which is anything but constant, and varies according to circumstances) is actually a function of quantum numbers, and I produced tables that shows the functions give good agreement for the various groups, and for excited states.

Now, the next shock. Standard theory has argued that the properties of heavy elements like gold have unusual properties because these “penetrations” lead to significant relativistic corrections. My guidance wave theory requires such relativistic effects to be very minor, and I provided  evidence that showed the properties of elements like francium, gold, thallium, and lead were as good or better fitting with my functions as any of the other elements.

What I did not tell them was that the tables of data I showed them had been published in a peer reviewed journal over thirty years before, but nobody took any notice. As someone said to me, when I mentioned this as an aside, “If it isn’t published in the Physical Review or J. Chem Phys., nobody reads it.” So much for publishing in academic journals.  As to why I skipped those two, I published that while I was starting my company, and $US 1,000 per page did not strike me as a good investment.So, if you have got this far, I appreciate your persistence. I wish you a very Merry Christmas and the best for 2020. I shall stop posting until a Thursday in mid January, as this is the summer holiday season here.

The Sociodynamics of Science

The title is a bit of an exaggeration as to the importance of this post, nevertheless since I was at what was probably my last scientific conference (NZ Institute of Chemistry, at Christchurch) I could not resist looking around at behaviour as well as the science. I also gave two presentations. Speaking to an audience gives the speaker an opportunity to order the presentation so as to give the most force to the surprising parts of it, not that many took advantage of this. Overall, very few, if any (apart from yours truly) seemed to want to provide their audience with something that might be uncomfortable for their preconceived notions.

First, the general part provided great support for Thomas Kuhn’s analysis. I found most of the invited speakers and keynote speakers to illustrate an interesting aspect: why are they speaking? Very few actually wished to educate or convince anyone of anything in particular, and personally, I found the few that did to be by far the most interesting. Most of the presentations from academics could be summarised as, “I have a huge number of research students and here is what they have done.” What then followed was a very large amount of results, but there was seldom an interesting unifying principle. Chemistry tends to be susceptible to this, as a very common student research program is to try to make a variety of related compounds. This may well have been very useful, but if we do not see why this approach was taken, it tends to feel like filling up some compendium of compounds, or, as Rutherford put it rather acidly, “stamp collecting”. These types of talks are characterised by the speaker trying to get in as many compounds as they can, so they keep talking and use up the allocated question time. I suspect that one of the purposes of these presentations is to say, “Look at what we have done. This has given our graduate students a good number of scientific publications, so if you are thinking of being a grad student, why not come here?” I can readily understand that line of thinking, but its relevance for older scientists is questionable. There were a few presentations where the output would be of more general interest, though. I found the odd presentation that showed how to do something new, where it could have quite wide applications, to be of particular interest.

Now to the personal. My first presentation was a summary of my biogenesis approach. It may have had too much information across too wide a field, but the interesting point was that it generated a discussion at the end relating to my concept of how homochirality was generated. My argument is that reproduction depends on it because the geometry prevents the formation of a second strand if the first strand is not either entirely left-handed or right-handed in its pitch. So the issue then was, it was pure chance that D-ribose containing helices predominated, in part because the chance of getting a long-enough homochiral strand is very remote, and when one arises, then it takes up all the resources and predominates. The legitimate question then is, why doesn’t the other handed helix eventually arise? It may be slower to do so, but it is not necessarily impossible. My partial answer to that is the mer units are also used to bind to some other important units for life to give them solubility, and the wrong sort gets used up and does not build up concentration. Maybe that is so, but there is no evidence.

It was my second presentation that would be controversial, and it was interesting to watch the expressions. Part of the problem for me was it was the last such presentation (there were some closing speakers after me, and after morning tea) and there is something about conferences at the end – everyone is busy thinking about how to get to the airport, etc, so they tend to lose concentration. My first slide put up three propositions: the wave functions everyone uses for atomic orbitals are wrong; because of that, the calculation of the chemical bond requires the use of a hitherto unrecognised quantum effect (which is a very specific expression involving only universally recognised quantum numbers) and finally, the commonly held belief that relativistic effects on the inner electrons make a major effect on the valence electron of the heaviest elements is wrong. 

As you might expect, this was greeted initially with yawns and disinterest: this was going to be wrong. At least that seemed to be written over their faces. I then diverted to explain my guidance wave interpretation, which is essentially the de Broglie pilot wave concept, but with two additions: an application of Euler’s complex number theory that everyone seems to have missed, and secondly, I argued that if the wave really causes diffraction in the two-slit-type experiment, it has to travel at the same speed as the particle. These two points lead to serious simplifications in the calculation of properties of chemical bonds. The next step was to put up a lot of evidence for the different wave functions, with about 70 data points spanning a selection of atoms, of which about twenty supported the absence of any significant relativistic effect. (This does not say relativity is wrong, but merely that its effects on valence electrons are too small to be noticed at this level of analysis.) What this was effectively saying was that most of the current calculations only give agreement with observation when liberal use is made of assignable constants, which conveniently can be adjusted so you get the “right” answer.So, question time. One question surprised me: Does my new approach do anything new? I argued that the fact everyone is using the wrong wave functions, there is a quantum effect that nobody has recognised, and everyone is wrong with those relativistic effects could be considered new. Yes, but have you got a prediction? This was someone difficult to satisfy. Well, if you have access to a good physics lab, I suggested, here is where you can show that, assuming my theory is correct, make an adjustment to the delayed choice quantum eraser experiment (and I outlined the simple change) then you will reach the opposite conclusion. If you don’t agree with me, then you should do the experiment to prove I am wrong. The stunned expressions were worth the cost of going to the conference. Not that anyone will do the experiment. That would show interest in finding the truth, and in fairness, it is more a job for a physicist.

Science in Action – or Not

For my first post in 2019, I wish everyone a happy and prosperous New Year. 2018 was a slightly different year than most for me, in that I finally completed and published my chemical bond theory as an ebook; that is something I had been putting off for a long time, largely because I had no clear idea what to do with the theory. There is something of a story behind this, so why not tell at least part of it in my first blog post for the year? The background to this illustrates why I think science has gone slightly off the rails over the last fifty years.

The usual way to get a scientific thought over is to write a scientific paper and publish it in a scientific journal. These tend to be fairly concise, and primarily present a set of data or make one point. One interesting point about science is that if it is not in accord with what people expect, the odds on are it will be ignored, or the journals will not even accept it. You have to add to what people believe to be accepted. As the great physicist Enrico Fermi once said, “Never underestimate the joy people derive from hearing something they already know.” Or at least think they know. The corollary is that you should never underestimate the urge people have to ignore anything that seems to contradict what they think they know.

My problem was I believed the general approach to chemical bond theory was wrong in the sense it was not useful. The basic equations could not be solved, and progress could only be made through computer modelling, together with as John Pople stated in his Nobel lecture, validation, which involved “the optimization of four parameters from 299 experimentally derived energies”. These validated parameters only worked for a very narrow range of molecules; if they were too different the validation process had to be repeated with a different set of reference molecules. My view of this followed another quote from Enrico Fermi: I remember my friend Johnny von Neumann used to say, “with four parameters I can fit an elephant and with five I can make him wiggle his trunk.” (I read that with the more modern density functional theory, there could be up to fifty adjustable parameters. If after using that many you cannot get agreement with observation, you should most certainly give up.)

Of course, when I started my career, the problem was just plain insoluble. If you remember the old computer print-out, there were sheets of paper about the size of US letter paper, and these would be folded in a heap. I had a friend doing such computations, and I saw him once with such a pile of computer paper many inches thick. This was the code, and he was frantic. He kept making alterations, but nothing worked – he always got one of two answers: zero and infinity. As I remarked, at least the truth was somewhere in between.

The first problem I attacked was the energy of electrons in the free atoms. In standard theory, the Schrödinger equation, when you assume that an electron in a neutral atom sees a charge of one, the binding energy is far too weak. This is “corrected”througha “screening constant”, and each situation had its own “constant”. That means that each value was obtained by multiplying what you expect by something to give the right answer. Physically, this is explained by the electron penetrating the inner electron shells and experiencing greater electric field.

What I came up with is too complicated to go into here, but basically the concept was that since the Schrödinger equation (the basis of quantum mechanics) is a wave equation, assume there was a wave. That is at odds with standard quantum mechanics, but there were two men, Louis de Broglie and David Bohm, who had argued there was a wave that they called the pilot wave. (In a recent poll of physicists regarding which interpretation was most likely to be correct, the pilot wave got zero votes.) I adopted the concept (well before that poll) but I had two additional features, so I called mine the guidance wave.

For me, the atomic orbital was a linear sum of component waves, one of which was the usual hydrogen-like wave, plus a wave with zero nodes, and two additional waves to account for the Uncertainty Principle. It worked to a first order using only quantum numbers. I published it, and the scientific community ignored it.

When I used it for chemical bond calculations, the results are accurate generally to within a few kJ/mol, which is a fraction of 1% frequently. Boron, sodium and bismuth give worse results.  A second order term is necessary for atomic orbital energies, but it cancels in the chemical bond calculations. Its magnitude increases as the distance from a full shell increases, and it oscillates in sign depending on whether the principal quantum number is odd or even, which results when going down a group of elements, that the lines joining them zig zag.

Does it matter? Well, in my opinion, yes. The reason is that first it gives the main characteristics of the wave function in terms only of quantum numbers, free f arbitrary parameters. More importantly, the main term differs depending on whether the electron is paired or not, and since chemical bonding requiresthe pairing of unpaired electrons, the function changes on forming bonds. That means there is a quantum effect that is overlooked in the standard calculations. But you say, surely they would notice that? Recall what I said about assignable parameters? With four of them, von Neumann could use the data to calculate an elephant! Think of what you could do with fifty!

As a postscript, I recently saw a claim on a web discussion that some of the unusual properties of gold, such as its colour, arise through a relativistic effect. I entered the discussion and said that if my paper was correct, gold is reasonably well-behaved, and its energy levels were quite adequately calculated without needing relativity, as might be expected from the energies involved. This drew almost derision – the paper was dated, an authority has spoken since then. A simple extrapolation from copper to silver to gold shows gold is anomalous – I should go read a tutorial. I offered the fact that all energy levels require enhanced screening constants, therefore Maxwell’s equations are not followed. These are the basic laws of electromagnetism. Derision again – someone must have worked that out. If so, what is the answer? As for the colour, copper is also coloured. As for the extrapolation, you should not simply keep drawing a zig to work out where the zag ends. The interesting point here was that this person was embedded in “standard theory”. Of course standard theory might be right, but whether it is depends on whether it explains nature properly, and not on who the authority spouting it is.

Finally, a quote to end this post, again from Enrico Fermi. When asked what characteristics Nobel prize winners had in common: “I cannot think of a single one, not even intelligence.”

Science and the afterlife.

Science is not about a collection of facts, but rather it is a way of analyzing what we observe. The concept is, if you have a good theory it will predict things, then you can go out, do experiments, and see if you are right. Of course, in general it is not so easy to form a theory without resorting to nature, which means that most theories largely explain what is known. Nevertheless, the objective is to make a limited number of predictions of what we do not know. Someone then goes out and carries out the experiments, tests or whatever and if the theory is correct, the observations are just what the theory predicted. That is great news for the theoretician.

That is all very well, but what happens when there is a phenomenon that cannot be the subject of an experiment. In a previous post (http://wp.me/p2IwTC-6q ) I remarked that my Guidance Wave interpretation of quantum mechanics permitted, but by no means required, life after death. This cannot be experimented on and reported, however there are a number of reports of people who claim to have “almost died” or momentarily died and who have been revived and then given quite strange and explicit stories. What can science say about them?

One comes from a book I was given to read. Written by a pastor Todd Burpo, it tells of what his son, who was just short of four years old at the time, reported after he had nearly died in hospital. The son made several statements, all of which entailed an “out of body experience” and most involved a short visit to heaven to sit on Jesus’ lap. However, two of the descriptions were of more interest to me, because they described how the “out of body” boy saw his parents, each in a different room, and described what they did. In principle, there is no way he could have this information. The story also has a very frustrating element. The boy described the marks on Jesus’ hands, from the crucifixion. The pastor took that as clear evidence, because how would the boy know where the marks were? The most obvious reply is, he lived in a religious house and there probably were images around. Further, and this is the frustrating part, the standard Roman crucifixion did not have nails through the hands, so the boy was wrong, right? The problem is, Jesus did not have a standard crucifixion. What usually happened was that the victim was left on the cross until the flesh had more or less rotted away or had been picked by the crows, then the residue was disposed of, but not given a burial. To cut the body down and give it for burial was never done, except this time. Accordingly, if it were non-standard, it may have been the soldiers put the nails where it would be easier to get them out later. In short the killer evidence essentially ends up as useless. There is then the added complication that if true, Jesus may have given the image the pastor wanted.

The second is also interesting. Harvard neurosurgeon Dr Eben Alexander was in a coma for several days caused by severe bacterial meningitis. During his coma, he too had a vivid journey, first into other rooms, from which he described people’s actions that he had no possibility of knowing about through his physical body, and then into what he knew to be the afterlife. Now he had previously been a skeptic about this, and considered such accounts to be hallucinations, but in his own case his neocortex was non-functional during his coma, and furthermore, he gives nine different scientific reasons why what he experienced cannot be due to such hallucinogens or imagination. Since I am not an expert in brain function, I cannot comment usefully on his analysis. On one hand he is a prominent neurosurgeon, and should be an expert on brain function, so his analysis should be taken seriously, nevertheless, as Richard Feynman remarked about science, the easiest person to fool is yourself.

Perhaps the most spectacular accounts have been presented by Elisabeth Kübler-Ross, MD, a psychiatrist. She had noted that if children have this experience, they always see their mother and father if the parent is dead, but never if they are still alive. Christian children often see Jesus; Jewish ones never do. One particularly unusual account came from a woman who described what people were doing trying to resuscitate her after an accident. She claimed to have had this out of body experience and had watched everything. What is unusual about this is the woman was blind; the out of body “her” could see everything, but when she was resuscitated, she reverted to being blind. Another unusual report was from someone who met one of his parents in this “afterlife”, who confirmed being dead. This parent had died only one hour previously, five hundred miles or so away.

At this point we should look at the structure of a scientific proposition. There are two conditional forms for a statement that apply to a proposition under a given set of conditions:
(a) If the hypothesis is correct, then we shall get a certain set of observations.
(b) If and only if the hypothesis is correct, then we shall get a certain set of observations.
The difference lies here. In (a) there may be a multiplicity of different hypotheses that could lead to the observation, such as a hallucination, or a memory dump. This would apply to observations that the person could have recalled or imagined. In (b) there is only one explanation possible, therefore the hypothesis must be correct. Obviously, it is difficult to assert there is only one possible explanation, nevertheless, seeing something in another room when nearly dead seems to only being explained by part of the person (the soul, say) travelling out of the body into the other room.

So, where does this leave us? Essentially, in the position that there can be no proof until you die. Before that it is all a matter of faith. Nevertheless, as I argued, my guidance wave interpretation of quantum mechanics at least makes this possible within the realms of physics, but it does not require it. Accordingly, you either believe or you do not. The one clear fact though, is that if you do believe, it will almost certainly make dying easier, and that in itself is no bad thing.

Life after death

The issue of whether there is life, or consciousness, after death is one of those questions that can only be answered by dying. If there is, you find out. My wife was convinced there is, and she was equally convinced that I, as a scientist, would quietly argue the concept was ridiculous. However, as she was dying of metastatic cancer we had a discussion of this issue, and I believe the following theory gave her considerable comfort. Accordingly, I announced this at her recent funeral, in case it helped anyone else, and I have received a number of requests to post the argument. I am doing two posts: one with the mathematics, and one where I merely assert the argument for those who want a simpler account. The more mathematical post is at (http://my.rsc.org/blogs/84/1561 ).

First, is there any evidence at all? There are numerous accounts of people who nearly die but do not, and they claim to see a tunnel of light, and relations at the other end. There are two possible explanations:
(1) What they see is true,
(2) When the brain shuts down, it produces these illusions.
The problem with (2) is, why does it do it the same way for all? There was also an account recently of someone who died on an operating table, but was resuscitated, and he then gave an account of what the surgeons were doing as viewed from above. The following study may be of interest (http://rt.com/news/195056-life-after-death-study/ ) One can take this however one likes, but it is certainly weird.

What I told Claire arises from my interpretation of quantum mechanics, which is significantly different from most others’. First, some background. (If you have no interest in physics, you can skip this and go to the last three paragraphs.) If you fire particles such as electrons one at a time through a screen with two slits, each electron will give a point reading on a detector screen, but if you do this for long enough, the points give the pattern of wave diffraction. This is known as wave-particle duality, and at the quantum level, an experiment either gives properties of a particle or those consistent with a wave, depending on how you do it. So, how is that explained? Either there is a wave guiding the particles or there is not. Most physicists argue there is not and the electrons just happen to give that distribution. You ask, why? They tend to say, “Shut up and compute!” Einstein did not agree, and said, “God does not play dice.” What we know is that computations based on a wave equation give remarkably good agreement with observation, but nobody can find evidence for the wave. All we detect are the particles, but of course that is what the detectors are set up to detect. It is generally agreed that the formalism that enables calculations is sufficient. For me, that is not sufficient, and I think there must be something causing this behaviour. Suppose you cannot see ducks but you here a lot of quacking, why do you assume the quacks are just the consequence of your listening, and there are no ducks? There is a minority who believe there is a wave, and the pilot wave concept was formed by de Broglie.

Modern physics states the wave function is complex. In general, this is true, but from Euler’s theory of complex numbers, once (or twice) a period (which is defined as the time from one crest, say, to the next) the wave becomes momentarily real. My first premise is
The physics of the system are determined only when the wave becomes real.
From this, the stability of atoms, the Uncertainty Principle and the Exclusion Principle follow. Not that that is of importance here, other than to note that this interpretation does manage to do what standard theory effectively has as premises. My next premise is
The wave causes the wave behaviour.
At first sight, this seems obvious, but recall that modern quantum theory does not assert this. Now, if so, it follows that the wave front must travel at the same velocity as the particle; if it did not, how could it affect the particle? But if it travels at the same velocity, the energy of the system must be twice the kinetic energy of the particle. This simply asserts that the wave transmits energy. Actually, every other wave in physics transmits energy, except for the textbook quantal matter wave, which transmits nothing, it does not exist, but it defines probabilities. (As an aside, since energy is proportional to mass, in general this interpretation does not conflict with standard quantum mechanics.) For this discussion, the most important consequence is that both particle and wave must maintain the same energy. The wave sets the particle energy because the wave is deterministic, which means that once the wave is defined, it is defined for every future with known conditions. The particle, however, suffers random motion and has to be guided by the wave in my theory.

Now, what is consciousness? Strictly speaking, we do not know exactly, but examination of brains that are conscious appear to show considerable ordered electrical activity. But if electrical activity is occurring, that is the expenditure of energy. (The brain uses a remarkably high fraction of the body’s energy.) But since the movement of electrons is quantum controlled, then the corresponding energy must be found in an associated set of waves. Moreover, it is the associated wave that is causal, and it alone can overcome the randomness that may arise through the uncertainty of position of any particle. The wave guides the particle! Another important feature of these Guidance Waves is they are linear, which means they are completely separable. This is a general property of waves, and is not an ad hoc addition. It therefore follows that when we are conscious and living “here”, there is a matrix of waves with corresponding energy “there”.

Accordingly, if this Guidance Wave interpretation of quantum mechanics is correct, then the condition for life after death is very simple: death occurs because the body cannot supply the energy required to match the Guidance Waves that are organizing consciousness, and the random motion of particles in the brain, due to heat, overpower the order that bodily consciousness requires. The body now is no longer conscious, and hence is dead, and useful brain activity ceases. But if at the point where the brain can no longer provide its energy contribution for consciousness, the energy within the Guidance Wave can dissociate itself from the body and maintain itself “there”, and recall that the principle of linearity is that other waves do not affect it, then that wave package can continue, and since it represents the consciousness of a person, that consciousness continues. What happens next depends on the conditions applicable “there”, and for that we have no observations.

Is the Guidance Wave interpretation correct? As far as I am aware, there is no observation that would falsify my alternative interpretation of quantum mechanics, while my Guidance Wave theory does make two experimental predictions that contradict standard quantum mechanics. It also greatly simplifies the calculation of some chemical bond properties. However, even if it is correct, that does not mean there is life after death, but at least in my interpretation of quantum mechanics it is permitted. That thought comforted Claire in her last days, and if it comforts anyone else, this post is worth it.