The number of [single bonds + lone pairs] on the central atom

– 4 [single bonds + lone pairs] —-> sp3 hybridization – one s orbital, 3 p orbitals
– 3 [single bonds + lone pairs] —-> sp2 hybridization – one s orbital, 2 p orbitals
– 2 [single bonds + lone pairs] =—> sp hybridization – one s orbital, 1 p orbital.

CH4 has a total of 4 single bonds around the central atom and zero lone pairs. sp3 hybridization – one s orbital, 3 p orbitals
BH3 has a total of 3 single bonds around the central atom and zero lone pairs. sp2 hybridization – one s orbital, 2 p orbitals

I was wondering what determines how many p orbitals used for hybridization. For example, CH4 use 3 three p orbitals for hybridization, while BH3 use only two.

Thank you

See pp. 88 – 104 Review. Benzene on the Basis of the Three-Electron Bond. (The Pauli exclusion principle, Heisenberg’s uncertainty principle and chemical bond). http://vixra.org/pdf/1710.0326v1.pdf

The Pauli exclusion principle and the chemical bond.

The Pauli exclusion principle — this is the fundamental principle of quantum mechanics, which asserts that two or more identical fermions (particles with half-integral spin) can not simultaneously be in the same quantum state.

Wolfgang Pauli, a Swiss theoretical physicist, formulated this principle in 1925 [1]. In chemistry exactly Pauli exclusion principle often considered as a ban on the existence of three-electron bonds with a multiplicity of 1.5, but it can be shown that Pauli exclusion principle does not prohibit the existence of three-electron bonds. To do this, analyze the Pauli exclusion principle in more detail.

According to Pauli exclusion principle in a system consisting of identical fermions, two (or more) particles can not be in the same states [2]. The corresponding formulas of the wave functions and the determinant are given in the reference (this is a standard consideration of the fermion system), but we will concentrate our attention on the derivation: “… Of course, in this formulation, Pauli exclusion principle can only be applied to systems of weakly interacting particles, when one can speak (at least approximately on the states of individual particles) “[2]. That is, Pauli exclusion principle can only be applied to weakly interacting particles, when one can talk about the states of individual particles.

But if we recall that any classical chemical bond is formed between two nuclei (this is a fundamental difference from atomic orbitals), which somehow “pull” the electrons one upon another, it is logical to assume that in the formation of a chemical bond, the electrons can no longer be regarded as weakly interacting particles . This assumption is confirmed by the earlier introduced notion of a chemical bond as a separate semi-virtual particle (natural component of the particle “parts” can not be weakly interacting).

Representations of the chemical bond given in the chapter “The Principle of Heisenberg’s Uncertainty and the Chemical Bond” categorically reject the statements about the chemical bond as a system of weakly interacting electrons. On the contrary, it follows from the above description that in the chemical bond, the electrons “lose” their individuality and “occupy” the entire chemical bond, that is, the electrons in the chemical bond “interact as much as possible”, which directly indicates the inapplicability of the Pauli exclusion principle to the chemical bond. Moreover, the quantum-mechanical uncertainty in momentum and coordinate, in fact, strictly indicates that in the chemical bond, electrons are a system of “maximally” strongly interacting particles, and the whole chemical bond is a separate particle in which there is no place for the notion of an “individual” electron, its velocity, coordinate, energy, etc., description. This is fundamentally not true. The chemical bond is a separate particle, called us “semi-virtual particle”, it is a composite particle that consists of individual electrons (strongly interacting), and spatially located between the nuclei.

Thus, the introduction of a three-electron bond with a multiplicity of 1.5 is justified from the chemical point of view (simply explains the structure of the benzene molecule, aromaticity, the structure of organic and inorganic substances, etc.) is confirmed by the Pauli exclusion principle and the logical assumption of a chemical bond as system of strongly interacting particles (actually a separate semi-virtual particle), and as a consequence the inapplicability of the Pauli exclusion principle to a chemical bond.

1. Pauli W. Uber den Zusammenhang des Abschlusses der Elektronengruppen in Atom mit der Komplexstruktur der Spektren, – Z. Phys., 1925, 31, 765-783.

2. A.S. Davydov. Quantum mechanics. Second edition. Publishing house “Science”. Moscow, 1973, p. 334.

Heisenberg’s uncertainty principle and chemical bond.

For further analysis of chemical bond, let us consider the Compton wavelength of an electron:

λc.е. = h/(me*c)= 2.4263 * 10^(-12) m

The Compton wavelength of an electron is equivalent to the wavelength of a photon whose energy is equal to the rest energy of the electron itself (the standard conclusion is given below):

λ = h/(m*v), E = h*γ, E = me*c^2, c = γ*λ, γ = c/λ

E = h*γ, E = h*(c/λ) = me*c^2, λc.е. = h/(me*c)

where λ is the Louis de Broglie wavelength, me is the mass of the electron, c, γ is the speed and frequency of light, and h is the Planck constant.

It is more interesting to consider what happens to an electron in a region with linear dimensions smaller than the Compton wavelength of an electron. According to Heisenberg uncertainty in this area, we have a quantum mechanical uncertainty in the momentum of at least m*c and a quantum mechanical uncertainty in the energy of at least me*c^2 :

Δp ≥ mе*c and ΔE ≥ me*c^2

which is sufficient for the production of virtual electron-positron pairs. Therefore, in such a region the electron can no longer be regarded as a “point object”, since it (an electron) spends part of its time in the state “electron + pair (positron + electron)”. As a result of the above, an electron at distances smaller than the Compton length is a system with an infinite number of degrees of freedom and its interaction should be described within the framework of quantum field theory. Most importantly, the transition to the intermediate state “electron + pair (positron + electron)” carried per time ~ λc.е./c

Δt = λc.е./c = 2.4263 * 10^(-12)/(3*10^8) = 8.1*10^(-20) s

Now we will try to use all the above-mentioned to describe the chemical bond using Einstein’s theory of relativity and Heisenberg’s uncertainty principle. To do this, let’s make one assumption: suppose that the wavelength of an electron on a Bohr orbit (the hydrogen atom) is the same Compton wavelength of an electron, but in another frame of reference, and as a result there is a 137-times greater Compton wavelength (due to the effects of relativity theory):

λc.е. = h/(me*c) = 2.4263 * 10^(-12) m

λb. = h/(me*v)= 2*π*R = 3.31*10^(-10) m

λb./λc.е.= 137

where R= 0.527 Å, the Bohr radius.

Since the De Broglie wavelength in a hydrogen atom (according to Bohr) is 137 times larger than the Compton wavelength of an electron, it is quite logical to assume that the energy interactions will be 137 times weaker (the longer the photon wavelength, the lower the frequency, and hence the energy ). We note that 1 / 137.036 is a fine structure constant, the fundamental physical constant characterizing the force of electromagnetic interaction was introduced into science in 1916 year by the German physicist Arnold Sommerfeld as a measure of relativistic corrections in describing atomic spectra within the framework of the model of the N. Bohr atom.

To describe the chemical bond, we use the Heisenberg uncertainty principle:

Δx * Δp ≥ ћ / 2

Given the weakening of the energy interaction 137 times, the Heisenberg uncertainty principle can be written in the form:

Δx* Δp ≥ (ћ * 137)/2

According to the last equation, the quantum mechanical uncertainty in the momentum of an electron in a chemical bond must be at least me * c, and the quantum mechanical uncertainty in the energy is not less than me * c ^ 2, which should also be sufficient for the production of virtual electron-positron pairs.

Therefore, in the field of chemical bonding, in this case, an electron can not be regarded as a “point object”, since it (an electron) will spend part of its time in the state “electron + pair (positron + electron)”, and therefore its interaction should be described in the framework of quantum field theory.

This approach makes it possible to explain how, in the case of many-electron chemical bonds (two-electron, three-electron, etc.), repulsion between electrons is overcome: since the chemical bond is actually a “boiling mass” of electrons and positrons, virtual positrons “help” overcome the repulsion between electrons. This approach assumes that the chemical bond is in fact a closed spatial bag (a potential well in the energy sense), in which “boiling” of real electrons and also virtual positrons and electrons occurs, and the “volume” of this potential bag is actually a “volume” of chemical bond and also the spatial measure of the quantum-mechanical uncertainty in the position of the electron.

Strictly speaking, with such a consideration, the electron no longer has a certain energy, momentum, coordinates, and is no longer a “point particle”, but actually takes up the “whole volume” of chemical bonding. It can be argued that in the chemical bond a single electron is depersonalized and loses its individuality, in fact it does not exist, but there is a “boiling mass” of real electrons and virtual positrons and electrons that by fluctuate change each other. That is, the chemical bond is actually a separate particle, as already mentioned, a semi-virtual particle. Moreover, this approach can be extended to the structure of elementary particles such as an electron or a positron: an elementary particle in this consideration is a fluctuating vacuum closed in a certain spatial bag, which is a potential well for these fluctuations.

It is especially worth noting that in this consideration, electrons are strongly interacting particles, and therefore the Pauli principle is not applicable to chemical bond (for more details, see the section “The Pauli Principle and the Chemical Bond”) and does not prohibit the existence of the same three-electron bonds with a multiplicity of 1.5.

The above is easy to demonstrate with the example of a chemical bond of 1 Å length. Then the wavelength of Broglie is written in the form (the length of the chemical bond is L = 2 * Δx):

λ = 2*π*Δx

and the Heisenberg uncertainty ratio takes the form:

Δx * Δp ≥ (ћ * 137 * 2 * π) / 2

from which we get:

L * Δp ≥ ћ * 137 * 2 * π

where L is the length of the chemical bond, and Δp is the quantum mechanical uncertainty of the momentum of each electron in a given chemical bond.
Whence, we obtain a formula for determining the uncertainty of the momentum in a chemical bond:

Δp ≥ (ћ * 137 * 2 * π) / L

Having made the necessary calculations for a length of 1 Å, we obtain:

Δp ≥ (ћ * 137 * 2 * π) / 10 ^ (-10)

Δp ≥ 9.078 * 10 ^ (-22) kg * m / s

That is, the uncertainty in the pulse is greater than me * c

(me * c = 2.73 * 10 ^ (-22) kg * m / s)

(it is clear that the uncertainty of the electron velocity will be greater than the speed of light), which should be based on our assumptions.

See pp. 88 – 104 Review. Benzene on the Basis of the Three-Electron Bond. (The Pauli exclusion principle, Heisenberg’s uncertainty principle and chemical bond). http://vixra.org/pdf/1710.0326v1.pdf

Benzene on the basis of the three-electron bond:

Review. Benzene on the Basis of the Three-Electron Bond. (The Pauli exclusion principle, Heisenberg’s uncertainty principle and chemical bond). http://vixra.org/pdf/1710.0326v1.pdf

1. Structure of the benzene molecule on the basis of the three-electron bond.
http://vixra.org/pdf/1606.0152v1.pdf

2. Experimental confirmation of the existence of the three-electron bond and theoretical basis ot its existence.
http://vixra.org/pdf/1606.0151v2.pdf

3. A short analysis of chemical bonds.
http://vixra.org/pdf/1606.0149v2.pdf

4. Supplement to the theoretical justification of existence of the three-electron bond.
http://vixra.org/pdf/1606.0150v2.pdf

5. Theory of three-electrone bond in the four works with brief comments.
http://vixra.org/pdf/1607.0022v2.pdf

6. REVIEW. Benzene on the basis of the three-electron bond. http://vixra.org/pdf/1612.0018v5.pdf

7. Quantum-mechanical aspects of the L. Pauling’s resonance theory.
http://vixra.org/pdf/1702.0333v2.pdf

8. Quantum-mechanical analysis of the MO method and VB method from the position of PQS.
http://vixra.org/pdf/1704.0068v1.pdf

9. Review. Benzene on the Basis of the Three-Electron Bond. (The Pauli exclusion principle, Heisenberg’s uncertainty principle and chemical bond). http://vixra.org/pdf/1710.0326v1.pdf

Bezverkhniy Volodymyr (viXra):http://vixra.org/author/bezverkhniy_volodymyr_dmytrovych

Another thing that has always bugged me, which is also related to this topic, is the hybridization of Oxygen (especially explaining the bond angle of 104.5° for water).
I have spoken to a lot of people and professors and it seems that the opinions on this are pretty much split.

On the one hand, there is people saying that Oxygen hybridizes just as Carbon does and that the two lone pairs push the angle from 109.5° to 104.5°.

On the other hand, people claim that the s and p-orbitals of Oxygen are energetically too far apart to hybridize and this model is just an easy but inaccurate explanation.
According to this, the two Hydrogens bond to two p-orbitals at an angle of 90°, but due to repulsion don’t sit exactly on the p-axes but are slightly off (104.5°), resulting in a poorer orbital overlap (unfavorable, but more favorable than hybridization).

A third approach i have heard is bent’s rule (Henry bent), which says that in an sp^n hybrid-orbital, n does not have to be a whole integer. So the orbitals for the lone pairs would have more s-character (because closer to the nucleus) and for the Hydrogens more p-character (because further away from nucleus).

Maybe some day, if you have the time and resources you could discuss this topic.
I would really be interested in your take on this.

Best regards from Austria.