Thank you for posting your algorithm. It was very interesting to read, and called to mind several concepts which I had not considered for many years. Am I correct in thinking that you have taken an ab initio molecular dynamics approach to solving the hydrogen bonding problem, by solving the interaction potentials for each time step, using the potentials to calculate the forces acting on the nuclei, and integrating the forces to arrive at the position of the nuclei for the subsequent step? If this is correct, I think the algorithm is sound. I did notice that you used the Euler method to update the velocity and position of the nuclei in your simulation (equations 13 and 14). The Euler method is known to give unstable solutions of limited accuracy.
The English Wikipedia article gives a succinct description of the instability problem:
Perhaps it would be useful to decrease the time step size or use another numerical integration method, such as the Verlet method, to improve the numerical stability of your solution. I am curious to learn whether implementing these changes affect the outcome of your simulations.
Thank you for your advice. Your are always kind.
As per your advice, I have halved the time slice. But it did not yieled significant diffirence from the original data.
I have cheked the tracks of proton if they have erroneous path. That is, I have cheked if the tracks have some zigzag path. The tracks showed no such zigzag erroneous motion except they cross the abscissa.
Euler's instability, I think, is relevant to oscillatoly system like harmonic oscillator. Our system is unpredictable when it comes to the problem whether it shows one way motion or oscillatory behavior.
My view is this. As long as we rely on the classical mechanics of the motion of the prtotons, we can never reach the true stable hydrogen molecule. That is, is my model is wrong.
I would like to know the alternative model to simulate on my personal computer.
I said that we need to execute 4-body quantum mechanical simulation in order to make a stable hydrogen melocule. Two bodies for two electrons, and two bodies for two protons. But recently, I have noticed that even such a plan can not work to make a stable hydrogen molecule. Do you know why? Answer: Some factor that can absorb an excess energy that the initial two hydrogen atoms had is neccessary in the Hamiltonian for the simulation. Otherwise, the electrons and protons will repeat the same periodic pattern of motion forever.
I am now reading a book titled "Quantum man" published from Norton and written by Lawrence M. Krauss. In that book, I have encountered the following sentence.
"Similarly, if we bring two identical atoms close together, there is not only an electric repulsion between the negatively charged electrons in one atom and those in the other atom, but the Pauli exclusion principle tells us that there is an additional repulsion because no two electrons can be in the same place in the same quantum state. Thus the electrons in one atom are pushed apart from the electrons in the neighboring atom so that they don't overlap in the same position in the same orbital configuration." (page 101, lines 15-23)
Almost of us accept the Pauli's exclusion principle at its face value. But I do not understand where the principle comes from and why and how it can work. Anyway, if the above sentence holds true, then the formation of hydrogen molecule becomes more and more difficult. Don't you think so?
I am now reading a John Gribbin's book titled "Schroedinger's Kittens." It is really an interesting read. From that book, I have known the following matter.
In classical mechanics, the two-body problem is integrable. This means that from the present knowledge of the system, we can calculate the position or velocity of the two particles both in the past and in the future. But if we add one more another third particle to this two-body system, then we will be able to integrate no more. It is not just because the relevant equation has become complex, but because it is no longer integrable in principle mathematically. H. Poincare proved this impossibility, says the book. So, except for a few special cases, the mathematical solution for the three-body problem does not exist. (But, of course, the physical solution exists.) Therefore, even if we can get any current knowledge of the parameters of the three-body system, we can neither calculate the past state nor the future state of the system.
Returning to the case of the bonding process of hydrogen atoms, if we image it in a classical mechanical way as a first guess, then we should be completely hopeless to get any idea of the trajectories of the two electrons and of the two protons. It is now four-body problem! Much more complicated than the three-body problem! And of course, needless to say, it is not integrable!
In this circumstances, how can we imagine the bonding process for a stable hydrogen molecule?
Since I myself started this discussion, I think I have the responsibility to settle the discussion down to some intermediate place from where we can further develope it to better one.
How about this idea which I would like to name a "wave collapse hypothesis." Please consider two hydrogen atoms each of which is flying nearly freely to eventually collide with the other one and make a single hydrogen molecule later. Such a couple of hydrogen atoms are actually composed from four particles, i.e., two protons and two electrons. But when the two hydrogen atoms are far apart enough yet, then the whole wave function Y may be approximated by the product of the two independent wave functions, H1(A) and H1(B).
But actually and strictly speaking, the situation is already a complicated four-body problem, the solution being described by a wave function H2(A,B) corresponding to an enormously stretched hydrogen molecule. So the more appropriate wave fucntion for the whole system must be the following.
Y=c1H1(A)H1(B) + c2H2(A,B) (2)
where the coefficient c1 is much bigger than the c2 when the interatomic distance is large. As the two atoms approach to each other, the relative magnitudes of the coefficients c1 and c2 reverse.
Here I would like to introduce a crude assumption that at the instance when the two atoms have reached the right positions relative to each other for the molecule formation to occur, the entire wave function eq(2) might suddenly collapse to the pure hydrogen molecule wave function. That is, the coefficient c1 becomes zero and at the same time the other coefficient c2 becomes unity.
Generally in the process of wave collapse, some observing agent is required to be there to cause the collapse of the wave function. So, in our case, what acts as the observing agent is the remaining problem for this "wave collapse hypothesis" to be valid. How do you think? Do you have any other idea for the hydrogen molcule formation?
Thank you for reading
This is not a quantum mechanical problem but a problem in kinetics. Consider:
H + H -->H2* (where * indicates the internal kinetic energy of H2 is greater than the barrier to dissociation)
H2* + M --> H2 + M (collisional deactivation)
H2* --> H+ H (dissociation of the energized H2 competes with dissociation)
Thank you for your response very much, Sir.
Your wrote that the two hydrogen atoms that are approaching to each other will make a temporary energized combination which you denoted as H2*. And then the fate of this combination H2* will bifurcate into two ways, one being to be a complete hydrogen molecule and the other being to be two separated hydrogen atoms depending on the internal kinetic energy of the H2*. And you wrote that the problem is not a quantum mechanical one but a kinetic one. Is my understanding correct?
Sir, I think that there is a difference in the stance for viewing even such a simple case of hydrogen-hydrogen reaction between you chemists and us physicists. You chemists, Sir, are trained to think the chemical reacton by using the simple symbolic chemical reaction equation like
H2* + M --> H2 + M (4)
But we physicists are trained to consider the atomic scale phenomena by using quantum mechanical philosophy. Everthing we physiciats are concerned is the quantum mechanical state of the electrons,the state of the nuclei. Even though the concept of energy gap of energy barrier for a chemical reaction is still useful for chemists, it is already an old idea for quantum physicists.
Your word "kinetics" sounds to me to mean the theory of equilibrium of velocities of chemical reaction. Am I correct? No, we physicists are not interested in the equilibrium theory, but we are uneasy as to what is going on at the moment of a chemical reaction. We physicists wish to know how do the quantum mechenical states of the each electron and the states of the each nucleus change during the "instant of the chemical reaction."
Now I would like to ask you a question.
Is the fate of a consequence of a comination of H2* of hydrogen atoms deterministic as to whether to make a complete hydrogen molecule or to dissociate from each other again? And I wish to know the physical meaning and definition of the symbol M in the equation (4).
Please feel completely free to rebut my argument, since I myself am uncertain about what is the point of your opinion, sorry. If you have time to write a comment, then I will absolutely waiting for your further response, Sir.
Thank you for your time
You write 'No, we physicists are not interested in the equilibrium theory, but we are uneasy as to what is going on at the moment of a chemical reaction. We physicists wish to know how do the quantum mechanical states of the each electron and the states of the each nucleus change during the "instant of the chemical reaction." '
There is no "instant of reaction." The total energy of H2* is, necessarily, greater than that of two well-separated hydrogen atoms at rest with respect to each other. Without some loss of energy the H2* will fall apart to two hydrogen atoms. No quantum mechanical analysis can get around conservation of energy.
You ask "Is the fate of a consequence of a comination of H2* of hydrogen atoms deterministic as to whether to make a complete hydrogen molecule or to dissociate from each other again? And I wish to know the physical meaning and definition of the symbol M in the equation (4)."
M is another molecule (or the wall of a reaction vessel) that can accept some of the excess energy of the H2*. Note that H2* is a short-hand for a wide range of ro-vibrational states of a pair of hydrogen atoms in the vicinity of the attractive well. The fate of H2* in a collision is not deterministic, as there are state-to-state transition probabilities to take into account even for a given geometry and velocity of the collision of H2* with M. More prosaiclly, as the collisions of H2* with M occur stochastically, the fate of H2* depends also on the time between formation of H2* and subsequent collisions.
From your explanation, I have imagined the process as follows.
First, two well-separated hydrogen atoms start to approach each other. In this stage, the wave function of the electron in each atom is almost the pure single hydrogenic one. As they come closer to each other, I think, we must think the two atomic system in terms of a single whole wave function. Now, it is not a merely simple sum of the separated atoms but is a system composed from two protons and two electrons that must be described by the above whole wave function.
When the distance between the two approaching hydrogen atoms become less than a critical value D*, the entire system falls in a rotational-vibrational state in the vicinity of the attractive well. This is the instant(?) of the formation of the intermediate H2*, isn't it? Or should we assume the existence of the H2* from the first? From this stage, the situation diverges to two fates.
In one fate, the intermediate H2* will at some time later encounter with the external molecule M on which the H2* dumps its excess energy to become a stable hydrogen molecule. Since the timing of the encounter is stochastic, it requires a finite time to complete the reaction to become the hydrogen molecule. Therefotre, there is no instant of reaction.
In other fate, the intermediate H2* will fall apart by itself. Because it could not find or encounter the convenient molecule M to deal the internal excess energy with.
Thus your theory depends deterministically on the formation of the intermediate H2* and the existence of the molecule M for H2* to become a stable hydrogen molecule. Is my understanding correct?
Here, may I pose you some questions?
Q1. This is purely a thought experiment. If a pure hydrogen "atom" gas is enclosed in an adibatic vessel, then no hydrogen molecule will be formed because of the requirement of energy conservation?
Q2. Can we work out the necessary conditions for the H2* to be formed? Do you think there is such a critical distance D*? Can we calculate them by only classical physics?
Q3. Is the process of the transfer of the excess energy of the H2* to the external molecule M so easy? Can we describe it by only the classical physics?
Q4. Is the process of the transfer of the excess energy of the H2* to the external molecule M gradually continuous or instantaneous? How should we think?
I am sorry, if the focus of my argument is out of the course of this discussion.
Thank you for your time