(Fig.1) No exchange energies between two H2 molecules.
In fact, quantum mechanical molecular orbital (= MO ) theory failed in explaining ordinary molecular bond and intermolecular van der Waals attractions due to its serious flaws which show quantum mechanics is wrong.
As shown in in Fig.1, we think about the intermolecular interaction between H2 moleucle-1 and H2 molecule-2, and prove the quantum mechanical molecular orbital theory cannot generate any van der Waals attractions despite the fact that van der Waals attractions are actually observed between any molecules.
In the molecular orbital (= MO ) theory, each electron's wavefunction of the hydrogen molecule (= H2)-1 should be expressed as the sum of two hydrogen atomic wavefunctions (= φA and φB ) like each electron = φA + φB (= which can contain up to two electrons with the opposite spins up and down due to Pauli principle, this p.7 ) which abnormally inseparable MO wavefunctions always cause the unreal ionization ( this p.52 ). ← MO is false.
In the same way, each electron's wavefunction of H2 molecule-2 should be expressed as the sum of two hydrogen atomic wavefunctions (= φC and φD ) like each electron = φC + φD ( which wavefunction also contains up to two electrons with the opposite up and down spins due to Pauli principle ).
↑ So inside each of H2-molecule-1 and H2-molecule-2, the molecular attractive bond's exchange energy (= ∫ φAφB as H2 molecule-1 bond attractive exchange energy or ∫ φCφD as H2 molecule-2 bond attractive exchange energy ) is generated to form each H2 molecular σ bond.
Each H2 molecule consists of two electrons with the opposite up and down spins, hence, the intermolecular (exchange) energy between these two H2 molecules-1,2 must be just zero, because the molecular bond attractive exchange energy between the opposite up and down spins cancels the same number of antibonds causing Pauli repulsive exchange energy between the same up-up or down-down spins.
↑ Intermolecular exchange energies between H2 molecule-1 with ↑↓ spins ↔ H2 molecule-2 with ↑↓ spins should be zero ( ↑↓ ↔ ↑↓ = zero ).
When thinking about the intermolecular interaction between two H2 molecules, the molecular orbital (= MO ) theory has to rely on the sum (= attractive bond ) or the subtraction (= repulsive antibond ) of these two H2 molecular wavefunctions.
So in MO, the attractive intermolecular bond between two H2 molecules-1,2 is expressed as
ψ+ = ( φA + φB ) + ( φC + φD ).
And the intermolecular Pauli repulsive antibond between two H2 molecules-1,2 is expressed as
ψ- = ( φA + φB ) - ( φC + φD ).
Also in the molecular orbital (= MO ) theory, Pauli exclusion principle expressed as the conventional antisymmetric wavefunction or Slater determinant must be used to describe the total wavefunction combining multiple H2 molecules.
But the quantum mechanical Pauli antisymmetric wavefunctions have fatal flaws which only generate the unnecessary Pauli repulsive exchange energies between the same up-up or down-down spins, and cannot generate molecular attractive exchange energies between the opposite up-down spins ( this p.5-6, this p.10-lower ) due to the orthogonality between different spins.
↑ Only the attractive bond exchange integral of the opposite up and down spins becomes zero, and Pauli repulsive exchange integral of the same up-up or down-down spins are Not zero (this p.19 ). ← only unnecessary Pauli repulsion remains in MO.
In order to avoid generating the unnecessary (= unreal ) Pauli repulsive exchange energies, MO theory artificially created the ad-hoc rule where any exchange integral (= generating unphysical exchange energies ) between any two different orbitals must be zero, whether they are attractive or Pauli repulsive exchange energies. This ad-hoc rule is called "orthogonal MO orbitals ( this p.14, this p.3-lower )."
So the MO's bond (= sum ) and antibond (= subtraction ) wavefunctions must be always designed to be "orthogonal" to each other ( this p.8 ).
As shown in the upper figure, these two bond (= ψ+ ) and antibond (= ψ- ) wavefunctions always cause the zero exchange energy like
∫ψ+ψ- = 0 and ∫ (ψ+)2 + (ψ-)2 ← No intermolecular exchange energy like
∫ φBφC = ∫ φAφC = ∫ φBφD = ∫ φAφD = 0
↑ This zero exchange energy (= hence, neither molecular attractive nor Pauli repulsive exchange energies) due to the use of the ad-hoc bond and antibond orbitals is seen in no exchange molecular bond energy between two helium atoms (= He-He ). You should consider replacing each H2 molecule in the above figure by a He atom ( this p.3 ) where the numbers of attractive bonds and repulsive antibonds are the same canceling each other.
Strictly speaking, the coefficients of the attractive bond is slightly smaller than the repulsive antibond ( this 9.3-11-12 ), so the MO theory always causes a small amount of unnecessary Pauli repulsive energies between any two molecules or two helium atoms, which contradicts the fact that van der Waals attraction is always dominant between any two molecules or helium atoms at some distance.
The problem is a pair of an attractive bond (= which can contain up to two electrons with up-down spins ) and a repulsive antibond (= which can also contain up to two electrons with up-down spins ) contains the maximum four electrons, so when there are more than four electrons or molecules, this MO theory collapses.
(Fig.2) Unnecessary Pauli repulsion between four H2 molecules-1,2 and H2 molecules-3,4
Quantum mechanical molecular orbital (= MO ) theory tries to avoid generating unnecessary Pauli repulsive exchange energies using the ad-hoc attractive bond and repulsive antibond wavefunctions which are orthogonal to each other (= using the fact that exchange energy between bond and antibond wavefunctions is always zero ).
The problem is a pair of a bond ( with up to two electrons of up-down spins ) and an antibond (= with up to two electrons of up-down spins ) can contain up to four electrons (= cannot contain more than four electrons ! )
So even if physicists can manage to avoid the unreal Pauli repulsion between two H2 molecules-1 and 2 (= four electrons ), they cannot avoid unreal Pauli repulsion between H2 molecules-1,2 and H2 molecules-3,4 (= the total is eight electrons = more than four electrons ), hence quantum mechanics is false in the actual multi-electron system containing more than four electrons and many molecules.
As shown in Fig.2, a pair of H2 molecule-1 and H2 molecule-2 contains two electrons with the same up-up spin and two electrons with the same down-down spin (= the total is four electrons with two up spins and two down spins ).
And another pair of H2 molecule-3 and H2 molecule-4 also contains two electrons with the same up-up spin and two electrons with the same down-down spin (= total is four electrons ).
Any exchange energies (= exchange integrals ) between two electrons with the opposite up-down spins become zero due to the orthogonal integral of up-down spins, so we think about the intermolecular interaction between two electrons with the same up-up spin of H2 molecules-1,2 and two electrons with the up-up spin of H2 molecules-3,4 (= down-down spins in H2-1,2 and down-down spins in H2-3,4 gives the same exchange energy as this ).
As shown in Fig.2, the total sum of bond (= ψ+ ) and antibond (= ψ- ) with the same up-up spin in H2 molecules-1,2 becomes ψ+ + ψ- = 2( φA + φB ).
In the same way, the total sum of bond (= ψ'+ ) and antibond (= ψ'- ) with the same up-up spin in H2 molecules-3,4 becomes ψ'+ + ψ'- = 2( φ'A + φ'B ).
↑ All these φA, φB, φ'A, φ'B mean the same hydrogen atomic 1s wavefunctions existing in different positions.
The total wavefunction (= sum ) of H2 molecules-1,2 = 2( φA + φB ) is clearly Not orthogonal to the total wavefunction (= sum ) of H2 molecules-3,4 = 2( φ'A + φ'B ).
↑ Hence, the unreal Pauli repulsive exchange energies between the total H2 molecules-1,2 (= 2( φA + φB ) ) and the total H2 molecules-3,4 (= 2( φ'A + φ'B ) ) are Not zero, as shown in Fig.2
↑ The total exchange energy between H2 molecules-1,2 and H2 molecules-3,4 is equal to the sum of four exchange energies of ∫ ψ+ψ'+, ∫ ψ+ψ'-, ∫ ψ-ψ'+, ∫ ψ-ψ'-, which is Not zero = unnecessary Pauli repulsion remains. → No van der Waals or molecular bond attractions between H2 molecules-1,2 and H2 molecules-3,4 which four molecules must always push and move farther and farther away from each other by the unnecessary (= unreal ) Pauli repulsion in MO !
If the total sum of bond and antibond of H2 molecules-3,4 becomes 2( φ'A - φ'B ) instead of 2( φ'A + φ'B ), it becomes orthogonal to the total sum of H2 molecules-1,2 of 2( φA + φB ) ( × 2( φ'A - φ'B ) ), and can cancel the exchange energies between H2 molecules-1,2 and H2 molecules-3,4.
↑ But in this case, H2 molecular bond inside H2 molecule-3 ( or 4 ) becomes Pauli repulsive (= φ'A - φ'B ), hence, H2 molecule-3 ( and 4 ) itself is broken and unable to exist. ← As a result, it's impossible to eliminate unnecessary Pauli repulsive exchange energies in MO theory which resultantly contradicts the experimental observation.
This quantum mechanical molecular orbital (= MO ) theory's unnecessary unreal Pauli repulsions among molecules, which unnecessary Pauli repulsion does Not allow any molecules with more than two single (= σ ) bonds to be formed, disagree with the observed results.
In conclusion, quantum mechanical molecular orbital theory is proven to be wrong.
This is why Huckel theory representing MO can only deal with π bond based on the p orbitals such as C=C=C, and Huckel cannot deal with the ordinary σ single bond such as C-C-C-C ( or H-H -- H-H -- H-H van der Waals attraction ).
↑ π bonds with px, py, pz orbitals containing + and - phases can be easily made to be orthogonal (= p ± opposite phases could cancel each other and zero easily ) compared to σ single bond with s orbital with no ± phases unlike p orbital.
Even any other molecular orbital theories such as the latest extended Huckel theory allegedly dealing with σ single bonds also have to depend on various artificial rules and empirical parameters which must be artificially adjusted from experimental observations instead of deriving them from the useless quantum mechanical theory, because even the extented Huckel MO theory cannot calculate the exchange energies based on spins ( this p.2 2nd-last paragraph ).
Even if you try to make a new pair of bond and antibond combining all four H2 molecules-1,2,3,4, each bond or antibond wavefunction can contain up to two electrons, so you need to prepare four different wavefunctions which are Not orthogonal (= hence, unreal Pauli repulsive exchange occurs ) to contain all eight electrons of four H2 molecules, or even if you can prepare four different orthogonal wavefunctions, it must break some of the original H2 molecular bonds. ← Unreal Pauli repulsion cannot be avoided in quantum mechanical MO theory after all.
(Fig.3) Four molecular orbitals (= MO ) contain four electrons with up-spin
We show how unrealistic the quantum mechanical molecular orbital (= MO ) is, using its typical example in the system of four H2 molecular interaction.
Each H2 molecule consists of two electrons with up-spin and down-spin, so the total number of electrons of four H2 molecules-1,2,3,4 is eight with four up-spins and four down-spins.
As I said, quantum mechanical Pauli antisymmetric wavefunction has fatal flaws which only generate excessive Pauli repulsive antibond exchange energies between the same spin (= up-up or down-down spins ), and can Not generate the molecular bond attractive exchange energies between different up-down spins (= so No molecules in quantum mechanics ! ) due to the orthogonality between different spins (= integral of different spins is always zero like ∫ ↑↓ = 0, while integral of the same spin is Not zero ∫ ↑↑ = 1 ).
So we think about the interaction (= excessive Pauli repulsion ) between four electrons with the same up spins contained in four H2 molecules, as shown above (= interaction among four electrons with the same down-down spins is equal to this up-up spin case ).
The most symmetric form of the (unphysical) molecular orbital hybrid wavefunctions (= ψ1 ~ ψ4 ) of four H2 molecules becomes like Fig.3 where each single electron must always spread all over the four H2 molecules simultaneously, no matter how far away these four molecules are from each other, using quantum mechanical parallel worlds.
For example, one of four ( unscientific ) hybrid wavefunctions ψ3 is the sum or subtraction of four H1-H4 molecular wavefunctions like
ψ3 = ( H1 - H2 ) + ( H3 - H4 )
where H1 is the sum of two hydrogen atomic wavefunctions ( H1 = φA + φB ).
↑ A single electron with up-spin in this unphysical ψ3 hybrid wavefunction must always spread over all four distant H2 molecules. ← Impossible !
For simplicity, we think about the case where only H2 and H3 molecules are close to each other, and other H1 and H4 molecules are much more distant from other H2 molecules.
Hence, only the exchange interactions (= exchange integral ) between H2 and H3 molecules are not zero, and all other exchange interactions are almost zero and negligible in this position.
In this positional relationship, only two exchange energies between hybrid wavefunctions are not zero like
∫ ψ2ψ3 =2S ∫ ψ1ψ4 = -2S
where S denotes the exchange integral between the whole H2 molecule-2 wavefunction (= H2 = φC + φD ) and the whole H2-molecule-3 wavefunction (= H3 = φ'A + φ'B )
All other combinations of exchange energies are zero (= due to longer intermolecular distances ) like
∫ ψ1ψ2 = ∫ ψ3ψ4 = ∫ ψ1ψ3 = ∫ ψ2ψ4 = 0
Their total Pauli repulsive exchange energies based on antisymmetric wavefunctions become
-4S2 -4S2 = -8S2
↑ After all, unnecessary and excessive Pauli repulsion (= repulsive exchange energies ) always happens between quantum mechanical molecules (= because there are No molecular bond attractive exchange enegies in MO ), which contradicts the fact that van der Waals intermolecular attraction is always dominant between any molecules including H2 molecules.
In the upper four hybrid MO wavefunctions (= ψ1 - ψ4 ), only when four H2 molecules happen to be arranged just symmetrically at vertices of tetrahedral distribution (= in the special case when all distances between any two H2 molecules are equal ), this unnecessary Pauli repulsion becomes zero (= no repulsion or no van der Waals attraction ).
But in all other positional relationships, the unnecessary unreal Pauli repulsion is always generated in this irrational quantum mechanical molecular orbitals.
↑ Except when all exchange energies of exchange integrals between any combinations of the above four hybrid orbitals become just zero (= ∫ ψ1ψ2 = ∫ ψ2ψ3 = ∫ ψ3ψ4 = ∫ ψ1ψ3 = ∫ ψ1ψ4 = ∫ ψ2ψ4 = 0 = strict "orthogonal" rule ), the unnecessary (= unreal ) Pauli repulsions are always generated and these four molecules must be moving farther and farther away from each other, contrary to the fact that the actual intermolecular van der Waals attractions are always dominant and preventing molecules from disintegrating in any positional relationships of molecules.
In Fig.3, we think about the special case of four H2 molecules.
If we think about the cases of an odd number or an odd number × 2 of H2 molecules such as 3, 5, 6, 7, 9, 10, 11, 12.. × H2 molecules or asymmetric molecules, it is impossible to eliminate excessive Pauli repulsions in any symmetric positions. ← Quantum mechanics is false.
In the actual world where there are far more than four molecules, there are always some molecules closer to a molecule-1 and there are some other molecules more distant from a molecule-1, which various distance differences between molecules break the above symmetry (= special tetrahedral distribution is possible only in four molecules, and broken in more than four molecules ) and always cause strong unrealistic Pauli repulsion among any molecules (= not only H2 molecules but also all σ bond molecules such as C-C-C ), which unrealistic MO molecules with excessive Pauli repulsion contradict the actual molecules bound by van der Waals attraction.
Actual molecules are always moving around randomly and colliding with each other (= like the upper figure case of H2 and H3 molecules ), and even solid and liquid molecules can be arranged to almost arbitrary different positions (= different intermolecular distances ) around other molecules without generating excessive Pauli repulsion.
So if quantum mechanical molecular orbital theory was right, all molecules including solids and liquids are always moving farther and farther away from each other (← all stable liquids would evaporate into gas by the unreal MO Pauli repulsion ! ), while fluctuating thermally, and our body would easily disintegrate by this excessive Pauli repulsion in quantum MO !
Actual molecules in almost all positions (= almost all positional relationships ) always experience van der Waals attraction from other molecules, so quantum mechans and its ad-hoc molecular orbital (= MO ) theory disagree with the actual molecules, hence wrong.
2022/6/18 updated. Feel free to link to this site.