Sweden Has Lost Again

Swedish Foreign Minister Maria Stenergard repeated Sweden's steadfast support to Ukraine/NATO in its war with Russia made clear in the previous post, by today expressing:

• Sweden stands strong.
• It is important to secure ourselves.
• This is a genuine Swedish interest.
• We take on our responsibility.
• We invest in our military defence. 
• We are now NATO members and we feel more safe than ever before.
• A prime interest of Sweden is to strengthen our support to Ukraine in order the get an agreement which prevents any further Russian aggression.   

Swedish Prime Minister Ulf Christerson adds that Sweden may send troops to Ukraine.

Both ministers have missed that Ukraine/NATO has lost the war and that talks between Russia and USA are now under way towards a peace deal where the winner sets the conditions. 

This means that the massive Swedish military support to Ukraine has been meaningless and has only contributed to destruction. A harsh reckoning by the people of those responsible for this disaster, now will start. 

Swedish Prime Minister Febr 19:

• More weapons to Ukraine.
• I do not think Russia wants any peace negotiations.
• Sweden is in a dangerous situation (since we are at war with Russia).

Sweden vs Russia New Match

In the Foreign Policy Debate today in the Swedish Parliament Swedish Foreign Minister Maria Stenergard opened by forcefully declaring:

• Sweden is in the midst of long term confrontation with Russia.
• Support to Ukraine is the Government's top foreign policy priority. 
• Our task is inescapable. 
• Sweden never stands alone.
• We will constrain Russias ability to do us harm, particularly through our support to Ukraine.
• We are guided by our belief in a free and strong Ukraine.
• Sweden has increased its support to Ukraine each year of the war, totalling 70 billion SEK.
• Russia's goal is to impose a sphere of influence with series of vassals and satellites.
• The war is determined on the battle field.
• For Sweden support to Ukraine is a moral obligation and an indispensable investment in our own security.
• Swedish land forces are now part of Nato's forward forces in Latvia.
• It is up to Ukraine if and when negations will start.
• Our unequivocal support....to make Ukraine's position stronger.
• Pressure on Russia's war economy must increase.
• Our military support to Ukraine must be strengthened... now its 18th military support package the largest to date.
• The only sustainable peace is one that Ukraine achieves through strength.
• Negotiations from a position of weakness would only further Russian aggression.
• Sweden will continue military support to Ukraine as long as it takes.

So Sweden is again at war again with Russia to be added to this long list of wars from 1164 to 1809 when Sweden lost Finland in the Finnish War between the Kingdom of Sweden and Russian Empire.  

But this is forgotten by the Swedish Government, which instead lives in a fantasy of repeating the glorious victory in the battle at Narva in 1700 when a Swedish army of 10.000 soldiers lead by warrior King Karl XII overpowered a Russian army of 40.000. And Sweden won Hockey WM against Russia in 1957 and 1987.

Short story: The Swedish Government is not well informed, because of lack of Intelligence?

Computational Chemistry vs Solid Mechanics

Computational Chemistry CC may take up to 50 % of supercomputer resources, while Computational Solid Mechanics CSM may take less than 10%. 

The world of molecules built from atoms and electrons can be seen as a microscopic analog of a macroscopic world built from beams of steel such as a bridge,  and so we may expect to find CC as a microscopic analog of CSM, at least in a perspective of classical continuum mechanics.  

But this is not what the above numbers show: CC requires vastly more computational work than CSM. Why is that?

Consider a macroscopic object like a bridge composed of $N$ elements, which could be the finite elements in a discretisation of the object in CSM based on Navier's equations of elasticity. The computational work can scale with $N$, that is linearly in $N$ since each element interacts with just a few neighbouring elements. Finite element codes with $N=10^6$ can be run on a lap top.

The situation in CC is vastly different. This is because CC is based on Schrödinger's equation as the basic model of Standard Quantum Mechanics StdQM, which for a molecule with $N$ electrons involves $3N$ spatial dimensions, a full 3d space for each electron. This means that the computational work increases exponentially with $N$ which makes even $N=10$ beyond the power of any thinkable computer, and so only simplified versions of Schrödinger's equation are used in practice. Full solutions named wave functions then appear as pieces of conversation, which have no precise quantitative form. 

In Density Functional Theory DFT as the current work horse of CC, Schrödinger's equation is draconically reduced into a 3d equation in a single electrons density. The CSM analog would be to compress a complex bridge construction into one simple beam, where all element individuality is erased. 

Electron individuality is thus destroyed in DFT, asking for some form of recovery as electron exchange-correlation, which has shown to be difficult to realise.

RealQM is new methodology starting from a new form of Schrödinger's equation i terms of a collection of a non-overlapping one-electron densities keeping individuality of electrons by spatial occupation. RealQM can be seen as a refined form of DFT with one-electron densities maintaining individuality. 

RealQM in principle scales linearly with $N$ just like CSM. Below you can compare a bridge and a molecule and ask yourself why the molecule in CC with StdQM requires vastly more computational work than the bridge in CSM, and so get motivated to take a look at RealQM for which the work is comparable.

Molecule.

Bridge.

First Molecule HeH+ by RealQM and DFT

Crosscut 3d showing one electron (red) moved from He++ kernel to H+ kernel and remaining electron (yellow) around He++ kernel. Note free boundary developed between electrons starting from initial vertical cut through He++ kernel. Run code below to follow dynamics.

This is a follow up on previous post on the first molecule formed after a Big Bang when one of the two electrons of a Helium atom He joins with an approaching proton H+ to form a helium hydrid ion molecule HeH+ (or rather He+H) built by a cation He+ and a Hydrogen atom H. The energy count in Hartree is as follows:

• Energy of He atom = -2.903
• Energy of He+ and H separated = -2.000 - 0.500 = -2.500
• Energy E of HeH+ molecule = -2.592 
• Dissociation Energy of HeH+ into He+ and H = 0.092  observed
• Energy for formation FE of HeH+ from He and H+ = 0.311 

Let us compare RealQM and DFT as concerns prediction of the observed FE = 0.311. Notice the difference between He plus H+ and He+ plus H. Check by noticing that -2.903 = - 2.500 - 0.311 - 0.092. Notice that the bulk of FE is supplied by exterior forcing to make H+ approach the He++ kernel. 

RealQM code gives FE = 0.301 from E = -2.602. You can follow the transfer of one electron from He to H by running the code starting from two electron half-lobes around the He kernel with supports displayed on red and yellow. You can test a different location of H+ vs He electron split by running this code. In both cases see how one electron dynamically shifts from He to H+ forming a molecule of He+ and H starting from He and H+. 

DFT gives according to chatGPT:

DFT Functional	Predicted Dissociation Energy (Hartree)	Error Trend
LDA (Local Density Approximation)	~ -0.35	Overbinds HeH⁺ (too stable)
GGA (PBE, BLYP)	~ -0.33 to -0.34	Still overestimates bond strength
Hybrid (B3LYP, PBE0)	~ -0.30 to -0.32	Closest to exact (-0.311)
High-Accuracy (CCSD(T), FCI)	-0.311	Exact value

• LDA and GGA functionals overestimate binding, leading to a more negative dissociation energy (~ -0.34 to -0.35 Hartree).
• Hybrid functionals (B3LYP, PBE0) improve accuracy, but they still may predict a slightly too strong bond.
• Post-HF methods (CCSD(T), FCI) match experimental values (-0.311 Hartree).

We see that the precision with standard DFT is not better than RealQM, rather the opposite. It is not clear that DFT can model the dynamics of the shift of one electron from He to H+.

Notice that we are here dealing with the simplest possible problem in quantum mechanics, a molecule with only two electrons as the first molecule formed in the early universe, with H2 coming only later after dissociation of HeH+ into He+ and H (and then formation of H2 under release of energy). Would you expect that DFT after 50 years of massive investment would deliver a very convincing result? Did we get that?

Electron Affinity RealQM vs DFT

Chart of electron affinity from 0 to 0.133 Hartree with grey zero affinity.

Electron affinity is a measure of the drop in total energy in energy when a neutral atom A captures an  electron under release of energy forming a negatively charges anion A-  named negative affinity.

An atom with zero affinity has no tendency to capture another electron.  

We consider two basic cases one with zero and one with negative affinity:

• Helium with 2 electrons filling the 1st shell as a noble gas with zero electron affinity.
• Fluorine with negative electron affinity by filling the 2nd shell from 7 to 8 electrons.  

Observed negative electron affinities range from 0.1- 0.3 Hartree with 0.12 for Fluorine. The total energy of Fluorine is -99.7 Hartree, and so to recover a change of 0.1 Hartree in computation requires a precision of 4 correct decimals.  

Here you can run RealQM as essentially a 3-line parameter-free code in 3d with only input the kernel charge giving the following total energies in Hartree:

• Helium    -2.903
• Helium-   -2.906
• Fluorine   -99.4
• Fluorine-  -99.8

We see that RealQM recovers zero affinity for Helium and the trend of negative affinity for Fluorine, if not the exact value with the present resolution of a $50^3$ grid. 

Density Functional Theory DFT as a very complex code, typically gives positive affinity for Helium, and can give values in the range of 0.12 for Fluorine under suitable adjustments of the code. 

PS This is what chatGPT has to say about the role of DFT in years to come:

• DFT will continue to be the dominant method for simulating chemical systems in the foreseeable future. Despite its limitations, it offers the best trade-off between accuracy, computational cost, and scalability. However, machine learning (ML) is emerging as a potential competitor—and in some cases, it might even surpass DFT.

The question is if RealQM can take over this role as a new form of DFT with a collection of one-electron densities instead of just one common density. The investment in DFT has been massive over a period of 50 years, while  RealQM is a spin-off of computational mechanics realised with little manpower. It is thus of interest to compare RealQM and DFT on basic tasks.

Self-Interaction in DFT vs RealQM

A main difficulty of Density Functional Theory DFT as working with a single electron density $\rho (x)$ depending on a 3d spatial coordinate $x$ representing all electrons, is that electron self-interaction is present and has to be eliminated. 

Without correction DFT gives a much too small effective net electric potential outside a neutral atom as the net potential from kernel and electrons, for which the true net potential is $-\frac{1}{r}$ with $r$ the distance to the kernel for any atom, the same for all atoms as that of the Hydrogen atom. This is the case without van der Waal dipole effects.    

Real Quantum Mechanics RealQM is a new alternative to StandardQM StdQM with DFT, works with a collection of non-overlapping one-electron charge densities. 

In RealQM  there is no self-interaction since electron Coulomb potentials contribute to the total energy only for pairs of distinct electrons. 

RealQM thus gives the correct effective potential $-\frac{1}{r}$ simply because for an atom with kernel charge $Z$ and $Z$ electrons, each electron interacts with $Z-1$ other electrons with net 1 as the charge in the effective potential $-\frac{1}{r}$ of the Hydrogen atom.

In StdQM the effective potential of $-\frac{1}{r}$ is viewed to be the result of incomplete shielding of the kernel by the surrounding electrons always leaving a net potential of $-\frac{1}{r}$ even if the total net charge is $0=Z-Z$. But why the shielding effect is precisely $Z-1$ is not so obvious with the typical overlapping electron orbitals used in Hartree-Fock and DFT based on Hartree-Fock. 

StdQM/DFT: 

• works with globally overlapping electron densities without boundaries,
• has to struggle to remove effects of non-physical electron self-interaction.

RealQM: 

• works with nonoverlapping electron densities meeting at a free boundary,
• has no electron self-interaction. 

A simple test of consistency of any atom model is to check if the net potential outside the atom is $-\frac{1}{r}$. RealQM directly passes this test, while basic DFT has to be modified to pass the test by eliminating non-physical effects of electron self interaction.

 

Standard Quantum Mechanics as Classification without Physics vs RealQM

This is a continuation of the previous post but can be enjoyed independently.

Science and religion both arise from human minds seeking to find mental interpretations of the Word as the creation of superhuman mind. Quantum Mechanics QM as the mechanics of atoms and molecules came out from a perceived shortcoming of classical Newtonian physics in the late 19th century, so immensely successful in describing the macroscopic world, to capture the microscopic world of atoms and molecules. 

The German physicist Erwin Schrödinger in 1926 took the first step out of a deadlock into the modern physics of QM by formulating a mathematical model of the Hydrogen atom with one electron surrounding a proton kernel in the form of an eigenvalue problem for a partial differential equation for a negative charge density subject to Coulomb attraction from a positive kernel. 

The success was complete since the  eigenvalues precisely agreed with the observed spectrum of excited states of the Hydrogen atom with corresponding eigenfunctions as wave functions representing vibrational spatial modes of the electron taking the following form:

to be compared with those of e g a circular membrane    

The complete success of the Schrödinger equation capturing the spectrum of the Hydrogen atom with one electron demanded generalisation to atoms with more than one electron, which was accomplished  by formally adding a new set of 3d coordinates for each electron into a linear Schrödinger equation (S) in $3N$ spatial dimensions for an atom/molecule with $N>1$ electrons like a Gold atom with 79 electrons in $237$ spatial dimensions, as a partial differential equation of a completely new form with solutions named wave functions for which the physics had to be invented as probabilities of electron configurations. 

(S) has come to serve as the foundation of the modern physics in the form of Standard QM StdQM filling text books of modern physics. 

The first task for (S) was to explain the "Aufbau" of the  Periodic Table in terms of wave functions and then the following strange idea came up: 

• Describe ground state electron configurations of atoms/molecules with $N>1$ as linear combinations of excited states of the one electron of the Hydrogen atom.                                  

To see the strangeness compare with an idea of

• describing a complex system as a copy of a simple component building the system
• preformatism of the 17th century as a tiny human ("homunculus") inside the egg of a female. 

We understand that this is illogical: A complex system cannot be copy of its simple parts. Nevertheless it serves as a fundamental principle of StdQM: Excited states of a Hydrogen atom describe all atoms/molecules. It defies logic but is state of the art, still 100 years after conception. It does not help to recall that the eigenfunctions of the Hydrogen atom form a complete orthonormal system and so can describe anything. 

The fact that StdQM comes out from a formal extension from one to many electrons, means that StdQM appears as an (ad hoc) classification system rather than as model of real physics. This is evident by looking at the rich classification imposed by StdQM culminating in the Standard Model:

• symmetric and antisymmetric wave functions as bosons and fermions
• electrons with spin-up and spin-down
• paired and unpaired electrons 
• Hund's rule, Madelung rule, octet rule
• s, p, d, f states in atoms 
• quarks, leptons, gluons, weak force, strong force...

It is natural to compare with the plant classification system of Linné (18th century) based on the number of stamens and pistils in flowers, which is artificial and not reflective of natural relationships, but anyway still is used.  

RealQM offers an alternative to StdQM as a atom/molecule model based on Coulomb interaction between atomic kernels and non-overlapping electron densities as a natural system without need of elaborate classification. 

We can put StdQM ve RealQM into a broader perspective of idealism and realism with StdQM imposing atomic form and RealQM uncovering natural atomic form. RealQM gives a different rationale of the Periodic Table as an electron density packing problem. 

If you force a metal through a square die in an extrusion process, it will come out with quadratic cross section. If you force Hydrogen eigenfunction form upon general atomic states, everything will look Hydrogenic but you will violate natural physics.