*(Fig.1) ↓ No Coulomb force between two quantum wavefunctions.*

In fact, quantum mechanical probability wavefunction is unphysical and unable to use ordinary Coulomb electric force for forming covalent molecular bonds.

As shown in the upper figure-right, in the realistic atomic orbit such as successful Bohr's atom, an electron is actually moving around, approaching the other positive nucleus and avoiding the other negative electron to form stable molecular bond by ordinary Coulomb electric force.

On the other hand, quantum mechanical unphysical probability wavefunction or electron cloud must always spread out symmetrically around the nucleus to infinity.

This symmetrically-spreading quantum mechanical wavefunction or electron cloud is like the symmetrical spherical charge which average electron's position is at the same position as the central nucleus, which generates the zero net charge by the negative electron and positive nucleus canceling each other.

If the quantum mechanical spreading wavefunction tries to shrink only to one position, the localized shrunk wavefunction or de Broglie wave becomes sharper (= de Broglie wavelength is shorter ), and the electron's kinetic energy drastically increases, which disagrees with experimental atomic energies.

As a result, the unphysical quantum mechanical wavefunction, which cannot use the real Coulomb force for forming molecular bonds, has to rely on another unphysical energy called exchange energy allegedly caused by lower kinetic energy.

*(Fig.2) ↓ An electron-1 must exist in two different H atoms-A and B simultaneously.*

To generate covalent molecular bond energy (or strong Pauli repulsive energy ), each quantum mechanical electron must exist in all different atoms in a molecule simultaneously.

For example, in the upper hydrogen H2 molecule consisting of two H atoms-A and B, both the electron-1 and electron-2 must exist in two different H atoms-A and B simultaneously ( this p.3, this p.11, this p.4 ).

φ_{A} ( r1 ) means the electron-1 existing in H-atomic wavefunction φ_{A}.

φ_{B} ( r1 ) means the electron-1 existing in H-atomic wavefunction φ_{B} ( this p.18 ).

H2 molecular Hamiltonian H or total energy consists of two electrons' kinetic energy expressed as space derivative (= de Broglie wave ) and Coulomb electric energy terms among two H atomic nuclei-A, B and two electrons-1, 2, as shown above.

*(Fig.3) ↓ Coulomb energy integral of two same H atomic wavefunctions ∫φ _{A}φ_{A} or ∫φ_{B}φ_{B}, and unphysical exchange energy of two different H atomic wavefunctions ∫φ_{A}φ_{B} or ∫φ_{B}φ_{A}*

Quantum mechanics unrealistically demands each electron-1 and 2 must exist in two different H atoms-A and B to form fictitious H2 molecular covalent bonds.

To form molecular attractive covalent bonds, symmetric wavefunctions (= spin part is up-down antisymmetric or bonding ) are necessary ( this p.3-4 ).

By putting H2 molecular total energy Hamiltonian (= H ) between these two symmetric wavefunctions (= ψ_{+} ) where each electron-1 and 2 exists in two different H atoms-A and B, we can get two different integrals called Coulomb energy integral and (unphysical) exchange energy integral ( this p.3-4, this p.14 ).

Coulomb energy integral is the normal integral where each electron-1 (or electron-2 ) consists of two same H atomic wavefunctions like ∫**φ _{A}**(r1)

Unphysical exchange (= resonance ) energy integral consists of two **different** H atomic wavefunctions like ∫**φ _{A}**(r1)

This unphysical exchange energy integral is necessary to cause fictitious quantum mechanical molecular bonds (or Pauli repulsive antibonding ), and
the ordinary Coulomb energy (integral) has **No** power to form molecular attractive bonds ( this p.3, this p.3-4, this-lower-Fig.2, this p.4-5, this p.4-5-Fig.1 ) or Pauli repulsive antibonding.

In this symmetric wavefunction, the total wavefunction becomes gentler, which means the total de Broglie wavelength becomes longer, and electron's kinetic energy decreases.

This decrease in electron's kinetic energy is the driving force of forming molecular attractive covalent bonds caused by unphysical exchange energy ( this p.4 ).

We often see the wrong explanation that higher electron's density between two H atoms may cause decrease in Coulomb electric energy and molecular attractive bonds.

But even if Coulomb potential energy decreases, kinetic energy increases to conserve the total energy, the molecular attractive bonds cannot be formed, so the electron's lower kinetic energy even in the lower Coulomb potential energy region is the reason of quantum mechanical unphysical exchange molecular bonds.

This exchange energy is unphysical and **wrong** ( this p.11 ).

Because if the real attractive force is working between two atoms, electrons are attracted and **accelerated** toward the other atoms, so electrons' kinetic energies should increase instead of decrease (= when two atoms repeling each other, electrons decelerate, and kinetic energy should decrease ).

↑ Quantum mechanical exchange energy based on kinetic energy **violates** basic physical principle, so wrong.

This contradictory quantum mechanical exchange energy due to kinetic energy change lacks real exchange force or physical reality ( this p.9-upper, this p.8-last, this p.5, this p.8-right-Discussion ).

*(Fig.4) Even when an electron-1 approaches the other H-atom-B's positive nucleus decreasing Coulomb potential energy, its kinetic energy keeps decreasing, violating energy conservation = trick of unphysical exchange energy. *

No quantum mechanical Schrödinger equations are solvable except for one-electron hydrogen (= H ) atom ( this p.21-upper ).

In this only solvable H atom, the total energy E, which is the sum of electron's kinetic energy expressed as derivative and Coulomb potential energy, must be always constant and conserved in any electron's positions ( this-p.2-upper, this p.2-1st-paragraph ).

So in H atom, as the electron-1 is farther away from the H atom-A's positive nucleus, it **increases** Coulomb potential energy and **decreases** electron's kinetic energy to keep the constant and conserved total energy E.

In molecules such as hydrogen molecule (= H2 ) and hydrogen molecular H+ ion whose Schrödinger equations are unsolvable ( this p.4-lower ), quantum mechanics must artificially choose fake trial wavefunctions or basis sets.

In H2 molecule (or H+ molecule ), two hydrogen atomic 1s wavefunctions (= H-atom-A and B ) are used in the chosen trial wavefunctions ( this p.4, this p.6-7, this p.3 ).

As shown in the upper figure, in H2 molecule, even when the electron-1 approaches the other H atom-B's positive nucleus **decreasing** Coulomb potential energy, the electron-1's kinetic energy keeps **decreasing** due to the use of the H-atom-A's 1s wavefunction.

This is clearly **violation** of total energy conservation law (= **both** kinetic and Coulomb potential energies **decrease** ! ), so quantum mechanical molecules are false and unphysical.

So decrease in the kinetic energy is enhanced in exchange energy where the lower kinetic energy region of H atom-A (or H atom-B ) is expanded by larger probability amplitude of the other H atom-B (or H atom-A ) wavefunction near the other H atomic nucleus.

This is the trick of how the unphysical quantum mechanical exchange energy decreases the total energy (= by decreasing kinetic energy and violating energy conservation law ) to form molecular attractive bonds.

*(Fig.5) The single electron-1 (or electron-2 ) must unrealistically exist in both H atom-A and H atom-B simultaneously to generate Pauli repulsive exchange energy.*

When choosing antisymmetric wavefunction where the signs of two terms of exchanging two H atom A and B wavefunctions are different (= spin is symmetric or triplet like up-up or down-down ), we can obtain Pauli repulsive exchange energy or antibonding.

The sign of this antisymmetric wavefunction's exchange energy integral is **negative** = opposite to the molecular bond attractive exchange energy integral.

So in this (negative) Pauli repulsive exchange energy integral with the opposite sign of molecular attractive bond exchange energy caused by decreased kinetic energy, electron kinetic energy increases (= without decreasing Coulomb potential energy, so total energy is Not conserved in this, either ).

This **increase** in electron's **kinetic energy** is the driving force of unphysical quantum mechanical Pauli repulsion ( this p.9-10, this p.6-4th-paragraph, this-2.2, this-introduction-3rd-paragraph, this p.3-left-middle ).

↑ Pauli antisymmetric wavefunctions have steeper slope that increases kinetic energy by shorter de Broglie wavelength ( this p.13-2nd-paragraph ).

There is **No** real repulsive force or force carrier in this quantum mechanical Pauli repulsive exchange energy caused by increased kinetic energy.

If the real repulsion is exerted between two atoms, electrons must be **decelerated** or slow down (= decrease kinetic energy ) by repulsion from the other atom, instead of increasing kinetic energy.

So quantum mechanical Pauli repulsive exchange energy allegedly caused by the increased kinetic energy **disagrees** with physical principle, so unreal.

*(Fig.6) In normal Coulomb energy integral, only Coulomb repulsion between H atom-A's electron and H atom-B's electron exists, but in abnormal exchange energy, unphysical repulsion between electrons existing in the same H atom-A (or H atom-B ) appears, which is self-contradictory.*

When integrating the (anti-)symmetric wavefunctions with H2 molecular Hamitonian energy, we can obtain the normal Coulomb electric repulsive energy between the electron-1 existing in H atom-A (or H atom-B ) and the electron-2 existing in H atom-B (or H atom-A ).

Coulomb energy: ∫**φ _{A}**(r1)

↑ In the normal Coulomb energy integral, when the electron-1 exists in H-atom-A's wavefunction, the other electron-2 must exist in H-atom-B's wavefunction.

But in the **abnormal** exchange energy integral,

Exchange energy: ∫**φ _{A}**(r1)

↑ As seen here, in the abnormal exchange energy integral, even when the electron-1 exists in H atom-A's wavefuntion, the electron-2 can exist in the **same H** atom-A's wavefunction, which **contradicts** the normal Coulomb electric energy.

This is clearly self-contradiction, and shows quantum mechanical exchange energy is **Not** a real thing.

2024/5/16 updated. Feel free to link to this site.