Non-Overlapping Wave Functions/Charge Densities

This is a follow up of the recent post on The Secret of Covalent Bonding with further computations comparing the non-overlapping wave functions/charge densities of RealQM meeting with (i) continuity and zero derivative, with a hypothetical case of instead (ii) zero density. 

To pin-point the essential aspect, we consider the following 1d model problem of an atom with N electrons: 

Find the function $\Psi (x)$ on the interval $[0,1]$ which minimises the energy $E=E_k + E_p$ with

• $E_p=\frac{1}{2}\int_0^1D\psi^2dx$             (kinetic energy)
• $E_k =-\int_0^1P(x)\psi (x)$           (potential energy)

over wave functions $\psi (x)$ with $D\psi =\frac{d\psi}{dx}$ the derivative, with total charge 

• $\int_0^1\psi^2dx = N$,

where $-P(x)$ is a given potential.

The Ansatz of RealQM is to seek $\Psi (x)$ on the form 

•  $\Psi (x) = \Psi_1(x) + \Psi_2(x) + .... + \Psi_N(x)$,

where the $\Psi_i(x)$ are one-electron wave functions with disjoint supports which meet on a Bernoulli free boundary with continuity and zero derivative. Running this code in a case with $N=4$ and $P(x)$ the potential from two kernels, we get the following result for case (i): 

We see in red 4 non-overlapping wave functions meeting at a free boundary marked by green with continuity and zero derivative. The total energy is -247 with kinetic energy 30. 

We run the same code but require wave-functions to vanish at meeting points and get for case (ii):

We see total energy larger total energy -231 with much larger kinetic energy 308. 

We understand that requiring wave functions to vanish as in (ii) when meeting, gives much larger kinetic energy than asking only for continuity and zero derivate as in (i), and that the net effect is larger total energy.

The purpose of this exercise is to exhibit the possibility opened by RealQM of electron concentration between kernels decreasing potential energy without balancing increase of kinetic energy, which appears as the holy grail of covalent bonding. 

In StdQM wave functions have global support, which may be overlapping, and so like the functions in (ii) above have to pay a price of added kinetic energy from charge concentration which is less favourable for binding than RealQM.

The finding of this post agrees with the real case considered in previous posts.

 

Does Quantum Mechanics Explain the Periodic Table?

Eric Scerri is a leading expert on the Periodic Table as the foundation of chemistry. This is what chatGPT says when asking the simple question to what extent the Periodic Table, formed before the emergence of quantum mechanics in the 1920s, has been explained by quantum mechanics during the now 100 years since then:

Eric Scerri, a philosopher and historian of chemistry, is known for his critical stance on the idea that quantum mechanics fully explains the periodic table. While he acknowledges the importance of quantum mechanics in understanding atomic structure and trends in the periodic table (such as electron configurations, periodicity, and chemical behavior), Scerri argues that the explanation is incomplete or limited if we rely solely on quantum mechanics.

Here are the key points of Scerri's position:

1. Quantum mechanics is useful but not fully explanatory: Scerri believes that while quantum mechanics (especially the Schrödinger equation and quantum numbers) helps rationalize some features of the periodic table, it does not provide a complete or entirely satisfactory explanation, especially for the actual order of elements.

2. Electron configurations are not derived solely from quantum mechanics: He points out that the Aufbau principle, which is used to predict electron configurations, is not a strict consequence of quantum theory but rather a semi-empirical rule. Some observed configurations (especially for heavier elements) deviate from the predicted ones.

3. Many-body problem: Scerri highlights the complexity of solving the many-body Schrödinger equation for atoms with more than one electron. Since exact solutions don't exist for such systems, approximations are required, which limits the explanatory power of quantum mechanics.

4. Philosophical perspective: He often argues from a philosophy of science perspective, suggesting that science often progresses through approximations, models, and partial explanations, rather than final, complete theories. Thus, he sees the periodic table as something emergent, not entirely reducible to quantum mechanics.

5. Chemistry's autonomy: Scerri also defends a degree of autonomy for chemistry, pushing back against the idea that chemistry is just applied physics. He sees the periodic system as a chemical construct that has its own foundations and logic, not fully subsumed under physics.

In summary, Scerri does not reject quantum mechanics, but he questions the common claim that it fully explains the periodic system. He calls for a more nuanced understanding of how different scientific disciplines interrelate, especially in the case of chemistry and physics.

We learn that Scerri as an expert in the field does not see that the Periodic Table can satisfactorily be explained by quantum mechanics and so calls for something more. But there is nothing more to grab. Quantum mechanics is supposed to be the best scientific model/explanation of the physics of atoms forming molecules forming chemistry. Quantum mechanics has fully replaced older atomic theories. If quantum mechanics cannot explain the Periodic Table, nothing can it would seem. 

The basic feature of the Periodic Table are the periods 2, 8, 18 and 32 of the form $2n^2$ with $n=1,2,3,4$, which appear as the number of atoms/elements in the rows of the table as 2, 8, 8, 18, 18, 32 and 32 thus with repetitions, as the number of electrons of the noble gases at the end of the row. What is the quantum mechanical explanation of these periods? 

The story is that this is because the eigenfunctions of the Schrödinger equation for the Hydrogen atom with one electron appear with a multiplicity of $n^2$ for $n=1,2,3,4$, which is doubled to fit with $2n^2$ by counting each electron twice. 

But the logic appears to be missing: Why expect the multiplicity of eigenfunctions of the Hydrogen atom with one electron to give the periodicity of atoms with many electrons? Why double the count of the number of electrons? Why is the periodicity repeated?

As concerns the sequence $2n^2$, there are many ways to this result. In RealQM it comes out as the solution to an electron packing problem partitioned into shells $n=1,2,3,...$ of increasing radius divided into half-shells filled with up to $n^2$ electrons allowing repetition of periods. This explanation carries the physical logic of packing of electrons of occupying different volumes depending on distance to the kernel.   

Stupid Swedish Foreign Policy

All political parties in Sweden from left to right have come to a historic consensus on foreign policy:

1. Trump is bad even if what he now does is to continue the politics of Biden: Genocide in Gaza, proxy war against Russia in Ukraine and bombing of the Houtis. 
2. Putin is bad and the only way to stop him from invading Sweden is to participate in the US/NATO proxy war against Russia in Ukraine with maximal military support.  

1. means that Trump will not come to help Sweden in case of Russian retaliation because of 2. Worst possible combination.

This is leading into catastrophe for the Swedish people, but opposition to this stupid policy is suppressed by very effective propaganda.

A better alternative is to come to a peaceful agreement with Putin by stoping sending Swedish weapons to Ukraine. This would open to rightful criticism of Trump without ruining Swedish society.

Form Classical to Modern to Post-Modern Physics

Classical physics in the form Newtonian mechanics emerged during the scientific revolution of the 18th and 19th centuries from an Enlightenment of rational logic/mathematics combined with observation of reality in a fundamental shift away from religion scholastics. The basic idea was rational mechanics as physics, which was (more or less) understandable and not only magical.  

In the late 19th century classical physics incorporated electro-magnetics made understandable through Maxwell's equations.  

But the modern physics of relativity theory and quantum mechanics emerging in the beginning of the 20th century signified a return to magical thinking. 

All the great physicists of the 20th century Einstein, Bohr, Schrödinger, Dirac, Feynman, Gell Mann, Weinberg admitted that quantum mechanics cannot be understood, while suggesting that this only adds to its beauty as the prime achievement of human intellect. 

To teachers of quantum mechanics this created a problem: How to teach a subject that is not understood by even the sharpest minds? The only way out for the average teacher was to pretend to understand and refer to the admissions of the top physicists of non understanding, as only a sort of teasing: Of course quantum mechanics is well understood and so can be presented in text books to students expected to come to understanding by diligent study. 

The computer is often presented as an off-spring of quantum mechanics thus showing its power, but the first realisations of the idea of a computer was made in rational mechanical terms as the Analytical Engine by Babbage with all its wheels and gears. A realisation in terms of quantum mechanics would be a quantum computer but the question is if there will ever be such a thing. 

Is it possible to imaging a form of post-modern physics with a return to the rationality of classical physics. My answer in all modesty is RealQM.  

Ab Initio Chemistry: StdQM vs RealQM

Discovering the World from a desk.

The ideal mathematical model of some physics is according to Einstein a model without parameters. Newton's Law of Gravitation is the prime example within classical physics/mechanics taking the from

• $\Delta\phi (x)  = \rho (x)$

where $\phi (x)$ is gravitational potential, $\rho$ mass density and $\Delta$ the Laplacian differential operator with respect to a Euclidean coordinate $x$. 

The prime example within modern physics/quantum mechanics is Schrödinger's equation for the hydrogen atom taking the form of an eigenvalue problem with eigenvalue $E$ 

• $-\frac{1}{2}\Delta\psi (x) -\frac{\psi (x)}{\vert x\vert} = E\psi (x)$  

where $\psi (x)$ is a wave-function and $-\frac{1}{\vert x\vert}$ is a kernel potential. No parameter appears if units are properly chosen.

The amazing property of a parameter-free model is to give information about physics without other physics input than a mathematical model without parameters. This allows the theoretical physicist at his desk to make predictions about the world without any input from the world. Such a prediction is said to be ab initio. Amazing! A priori knowledge in the sense of Kant.

In general, however, mathematical models of some physics contain parameters which have to be known from experiment or other models, in order to get anything out of the model. The Standard Model of particle physics contains 19 parameters which cannot be normalised by choice of units. It is very difficult to determine these parameters by experiment. 

Let us now compare StdQM with RealQM as concerns ab initio simulation of chemical bonding. 

For StdQM we turn to chatGPT and get this response stating that this is impossible because the Schrödinger equation of StdQM, which is parameter free, is uncomputable and thus has to be modified or prepared to deliver any result and the preparation builds on massive input of know-how and parameters/experiments.

In contrast the new Schrödinger equation of RealQM is parameter-free and readily computable and thus delivers ab initio simulation capability of e g chemical bonding.  See posts on Real Quantum Chemistry with links to everything about RealQM.

PS Here is what Eric Scerri says about ab initio StdQM in Selected Papers on the Periodic Table:

• Whereas most chemists and educators seem to believe that all is well, I think that there is some benefit in pursuing the question of how much is strictly explained from the theory. 
• It is indeed something of a miracle that quantum mechanics explains the periodic table to the extent that it does at present. 
• But we should not let this fact seduce us into believing that it is a completely ab initio explanation. We have not yet arrived at the super - ab initio phase of quantum chemistry and nor are we even close. 
• If anything, the compromises that have been struck with the acceptance of parametrization as well as the mixing of wavefunction and DFT approaches begin to question the earlier promise of ab initio quantum chemistry.

The Secret of Covalent Chemical Bonding

Recent posts have exhibited the fact that the physics of chemical bonding still is debated as a fundamentally unresolved problem. Consensus appears to be that covalent bonding results from some form of "sharing of electrons", which decreases kernel potential energy by electron charge concentration between kernels without full compensation of increase of kinetic energy.

It is further agreed that this picture can be given support by quantum mechanics with the caveat that full solution of Schrödinger's equation for systems with several electrons is impossible. The idea is that somehow "delocalisation" of electrons over an entire molecule as a purely quantum mechanical effect, will make electron charge concentration possible without full compensation of increase of kinetic energy. But the quantitative details appear evasive.

RealQM offers a different account of the physics of covalent bonding which we here illustrate in a generic 1d molecule with two atomic kernels and two electrons. The crucial feature of RealQM is decomposition of the total electron wave function U(x) = U1(x) + U2(x) into one-electron wave functions with non-overlapping supports  meeting at a free boundary X with continuity and zero (normal) derivative. (Bernoulli condition). Running this code we get this result with kernel potential in blue and electron wave functions in red and green: 

We see a concentration of electron charge densities between the kernels meeting with non-zero joint value and zero (normal) derivates at the free boundary X. The total energy is -65.258 with kinetic energy 26.771.

We compare with a model with overlapping electron densities as in StdQM using this code:

We find a higher total energy -63.144 with substantially higher kinetic energy 35.111. 

We understand that the reason the non-overlapping wave functions above has smaller kinetic energy than the overlapping below, is that the non-overlapping meet at a free boundary with non-zero value and zero derivative allowing dU1/dx*dU1/dx and dU2/dx*dU2/dx to be smaller,  since overlapping wave functions have to meet zero end point values. The overlap region of StdQM is thus in RealQM replaced by a free boundary effectively decreasing the kinetic energy while keeping the potential energy and thus decreasing the total energy into bonding. 

Conclusion: 

RealQM offers an explanation in clear physical terms of covalent chemical bonding emerging from concentration of charge density between kernels without compensating increase of kinetic energy thus with lower total energy than separated atoms. The above experience in the basic model problem is reflected in the full molecule models reported in these posts. If this analysis is correct, it asks for major revision of the standard physics of covalent bonding.

Why Philosophy of Chemistry?

The recent post on the talk The Nature of Chemical Bonding by Eric Scerri, connects to an emerging field of Philosophy of Chemistry promoted by him. Asking chatGPT about the role of philoposphy in science we get the message that 

• Philosophy of Quantum Mechanics is a very active field since 100 years. 
• Philosophy of Newtonian Mechanics has played out its role since "science is settled".
• Philosophy of Chemistry is a new field formed during the last 20 years. 

We are led to an idea that if a philosophy of a certain scientific discipline is currently an active topic, like philosophy of quantum mechanics and chemistry, it expresses that "science is not yet settled".  It signals that there are foundational questions which cannot be resolved within the discipline itself using the tools of the discipline, and a resort to some form of metaphysics or philosophy is brought in to help.  

Beginning in the later 19th century and culminating in the 1930s, a search for the foundations of mathematics engaged many famous mathematicians like Cantor, Russell, Hilbert and Brouwer without ever coming to any agreement and the interest then slowly faded away when computer science gave its resolution, leaving pure mathematicians to explore new ideas without worrying about the true nature of in particular the infinite set of real numbers and the set of all square integrable functions of real numbers. Giving them names was decided to be enough. Today philosophy of mathematics is largely viewed as "settled science".

2nd Trump Talk to Putin

Who has the upper hand?

On March 18 Trump called Putin for the 2nd time since he won the November election on a promise to end the US/NATO proxy war against Russia in Ukraine in 24 hours. See also previous post.

To get a into a position of strength before the 2nd talk, T resumed military support to the proxy war, after a short stop to show good will to peace. In the talk P repeated his condition for progress towards peace as a return to a halt of military support from the US. 

P agreed to a proposal from T to have a 30-day cease-fire limited to energy infrastructure, and directly ordered Russian implementation, as a sign of good will. T did not say he was willing to stop military support, which could have been a sign of good will.

T suggested that the war possibly could be settled in ice-hockey matches between US and Russia, which was met positively by P.  But T did not say he was willing to stop military support, which would  have been welcomed by P as a sign of good will. 

T prides himself to be a man of quick action, but the talks with P follow a very slow schedule. It is still unclear if T really wants  peace or just seeks a way to prolong the war indefinitely giving in to neo-cons.

Swedish Defense Minister Pål Jonsson declares that Sweden will continue to send weapons to Ukraine independent of any peace talks between US and Russia, and that peace will only create even bigger problems for Sweden and Europe. The stocks of Saab sky-rocket. Anything which is good for Russia like peace, is bad for Sweden. Russia won over Sweden in Ice-hockey WC in 1967, 1969, 1970, 1973, 1981 and 1986.

Former Swedish Prime Minister Magdalena Andersson says that it absurd that T and P talk peace over the head of Sweden/Ukraine.

Sweden stands strong in its war against Russia. Finland lost to the Russian Empire in 1809 must be brought back under the Swedish Kingdom.

The Still Unknown Nature of Chemical Bonding

Eric Scerri named the 2nd most influential chemist for the decade 2010-2020 gave a plenary talk at 26th Conference of the ISPhil of Chemistry in 2022 on The Nature of Chemical Bonding starting out presenting two views according to Robin Hendry:

• Structural as actual "bonds".
• Energetic as "bonding". 

Scerri continues recalling Coulson's words (1955):

• A bond does not exist: no-one has ever seen it, no-one ever can. It is a figment of our own imagination.

Scerri adds a statement by his colleague Seifert from an email conversation:

• Neither the structural nor the energetic provide "suitable characterization of chemical bonding". 

This is not a good start in 2022 in view of Gilbert Lewis clarification in 1916: 

• In the mind of the organic chemist the chemical bond is no mere abstraction; it is a definite physical reality. a something that binds atom to atom.  

Scerri then proceeds to scrutinise of how chemical bonding is described in text books of chemistry concluding that there is no satisfactory explanation of in particular the covalent bond somehow emerging from "sharing of electrons" as the dominating theme (together with the ionic bond as simply Coulomb attraction between ions). 

Scerri struggles to make sense of the textbook explanations following the very natural idea that somehow a bond can form if electrons can find a configuration close to kernels with decreasing potential energy which is not balanced by increase of kinetic energy supposedly emerging from two conflicting actions: 

• concentration of electron charge distribution decreasing potential energy
• delocalisation of electrons charge distribution avoiding increase of kinetic energy.   

But Scerri stops there: The Nature of Chemical Bonding is still a mystery, since it is the title of hus talk in 2022. But it must have a nature in physical terms, it cannot just be imagination. The World consists of molecules as atoms held together by chemical bonds.

It seems to me that Scerri is saying that only modest progress has been made over more than 100 years, despite the giant progress of modern physics in the form of quantum mechanics supposedly describing all of the physics of atoms and molecules including chemical bonding. How can this be? How is it possible that the nature of chemical bonding is still basically a mystery?

Here RealQM comes in offering a physical explanation how electrons can concentrate between atomic kernels to form a "glue" from decrease of potential energy without increase of kinetic energy. The secret is revealed in this plot as print-out of this code from a previous post from a collection of posts on H2:

We see two H kernels in black and a cross-cut along a line through the kernels of two electron wave functions with non-overlapping supports meeting at a plane orthogonal to the line midway between the kernels (as a free boundary) in yellow with plots of characteristic functions of the supports in black.

Computations are 3d on a fixed mesh with 1 million mesh points with mesh sise 0.1 in atomic units. The code is fully explicit consisting of 3 lines for update pf wave functions, potentials and free boundary. The computed energy  (-1.146) is in fair agreement with observation (-1.17) with further improvement under additional iterations. 

The essence is that in RealQM electron wave functions can meet with non-zero joint values at a free boundary which makes concentration of electron charge between the kernels possible without increase of kinetic energy. 

RealQM offers a new theoretical explanation of chemical bonding which agrees with previous intuition without theory.  Here are posts on Real Quantum Chemistry.

Geometry of the H2O Molecule by RealQM

This is a follow up of the previous post with now special attention to the non-symmetric geometry of the  H2O (or HOH) molecule with the two H atoms on one side the O atom, thus forming a dipole with positive charge on the side of the H atoms and negative on the other. 

An O atom has 8 electrons with 2 electrons in a first shell and the remaining 6 with the following thinkable electron distributions in 2nd + 3rd shell:

1.  6
2.  4 + 2
3.  3 + 3
4.  2 + 4   

We compute using RealQM in spherical symmetry using this code to find that 3 matches the observed ground state energy. We conclude that the O atom has an electron shell distribution as 2+3+3 and understand that 3 electrons in the outer shell invites to non-symmetry. 

We now turn to RealQM in 3d with 3 electrons in a third shell out of which 2 bind to the 2 electrons of the H atoms on one side leaving 1 electron free on the other side. We compute dissociation energy using this code with output: 

We see the 2 valence electrons of O (red and yellow) bonding with the 2 electrons of the H atoms (green) leaving 1 O electron on the other side (blue). We see that a somewhat bigger angle than 90 degrees of the HOH molecule is compatible with the 3 O electrons (red, yellow and blue), in qualitative agreement with observed 104.5 degrees.  

We see that RealQM offers an explanation of the non-symmetric geometry of the HOH molecule in direct physical terms of electron charge distribution.  This is different from that offered by StdQM in terms of 2 2 valence electrons and 2 lone-pair electrons in a shell with 6 electrons without direct physics.