A The Standard Model's application to competing Covalent Bond theories?

[greswd]

In the quantum mechanics of covalent bonding, there are two competing theories:
https://en.wikipedia.org/wiki/Covalent_bond#Comparison_of_VB_and_MO_theories

Both are complementary, but they don't overlap fully, they still leave some gaps.What I want to ask is, can all the literature of the Standard Model distinguish between the theories, and also fill in the gaps left by both?

Or does the problem highlight the gaps of the Standard Model?

 

[TeethWhitener]

In the quantum mechanics of covalent bonding, there are two competing theories:
https://en.wikipedia.org/wiki/Covalent_bond#Comparison_of_VB_and_MO_theories

Both are complementary, but they don't overlap fully, they still leave some gaps.What I want to ask is, can all the literature of the Standard Model distinguish between the theories, and also fill in the gaps left by both?

Or does the problem highlight the gaps of the Standard Model?

The VB and MO models aren’t really different fundamental theories (so your talk about the Standard Model doesn’t really apply here). They’re more like different approximations to answering the same problem.

VB and MO are both approaches to solving the time-independent Schrodinger equation ($\hat{H}\Psi = E \Psi$) as it applies to molecules. They both assume that the true (extremely complicated) molecular wavefunction $\Psi$ can be approximated as a series of simpler functions $\phi_i$ weighted by coefficients $c_i$:
$$\Psi = \sum_i {c_i \phi_i}$$
MO theory starts with atomic orbitals for the $\phi_i$, while VB theory starts with certain bonding/antibonding/non-bonding combinations of atomic orbitals for the $\phi_i$.

 

[greswd]

VB and MO are both approaches to solving the time-independent Schrodinger equation ($\hat{H}\Psi = E \Psi$) as it applies to molecules. They both assume that the true (extremely complicated) molecular wavefunction $\Psi$ can be approximated as a series of simpler functions $\phi_i$ weighted by coefficients $c_i$:
$$\Psi = \sum_i {c_i \phi_i}$$
MO theory starts with atomic orbitals for the $\phi_i$, while VB theory starts with certain bonding/antibonding/non-bonding combinations of atomic orbitals for the $\phi_i$.

I see, thanks. Does the wavefunction explain why there is a bonding force?

 

[TeethWhitener]

I see, thanks. Does the wavefunction explain why there is a bonding force?

Ultimately, yes. The easiest way to see this is in the Born-Oppenheimer approximation: Fix the nuclei of a molecule (let's say H₂ for a simple example) at their equilibrium bond length and solve the electronic Schrodinger equation to get a ground state energy. Then change the bond length slightly and solve the Schrodinger equation again. Do this several times and you get a potential energy curve that has a minimum at the equilibrium bond length, rather than at infinity. This indicates that a bond exists between the atoms.

 

[greswd]

VB and MO are both approaches to solving the time-independent Schrodinger equation ($\hat{H}\Psi = E \Psi$) as it applies to molecules. They both assume that the true (extremely complicated) molecular wavefunction $\Psi$ can be approximated as a series of simpler functions $\phi_i$ weighted by coefficients $c_i$:
$$\Psi = \sum_i {c_i \phi_i}$$

by the way, in theory (meaning hypothetically speaking), there ought to exist an equation, or equations, that describes the true wavefunction accurately, right?

Has this equation been given a name? Even if physicists don't know what exact form it takes, they can still give it a name first I guess.

 

[TeethWhitener]

by the way, in theory (meaning hypothetically speaking), there ought to exist an equation, or equations, that describes the true wavefunction accurately, right?

Has this equation been given a name? Even if physicists don't know what exact form it takes, they can still give it a name first I guess.

Yes, it’s the Schrödinger equation. Given a molecular Hamiltonian, the true (stationary) wavefunctions are the eigenvectors of this Hamiltonian. And as long as the basis set $\{\phi_i\}$ spans the Hilbert space of wavefunctions, the equation $\psi =\sum c_i \phi_i$ is exact. Since this Hilbert space is typically infinite-dimensional, in practice we use an incomplete basis set as an approximation.

 

[greswd]

Yes, it’s the Schrödinger equation.

I've had some concepts in my head, but it was unclear and foggy, so I focused on other topics for the time being. Now, I've got a clearer idea of what I'm looking for.

Do this several times and you get a potential energy curve that has a minimum at the equilibrium bond length, rather than at infinity. This indicates that a bond exists between the atoms.

Does the energy-distance relation curve equation have a name?

 

[TeethWhitener]

Does the energy-distance relation curve equation have a name?

It’s just what you get when you solve the Schrodinger equation in the Born-Oppenheimer approximation at varying nuclear geometries. It’s unclear what terms you’re looking for above and beyond this. The curve itself is generally called a potential energy curve, or a potential energy surface when generalized to more than one dimension. You’ll find the term “potential energy surface” throughout the literature, if you’re looking for keywords to search. There are also a number of empirical potentials that approximate the potential energy curve for atoms in a molecule, the most famous of which is probably the Morse potential.

 

[greswd]

It’s just what you get when you solve the Schrodinger equation in the Born-Oppenheimer approximation at varying nuclear geometries. It’s unclear what terms you’re looking for above and beyond this. The curve itself is generally called a potential energy curve, or a potential energy surface when generalized to more than one dimension. You’ll find the term “potential energy surface” throughout the literature, if you’re looking for keywords to search. There are also a number of empirical potentials that approximate the potential energy curve for atoms in a molecule, the most famous of which is probably the Morse potential.

thanks, I see. The force given by this potential, dU/dr, is it considered a quantum mechanical version of the electromagnetic force described by Maxwell's equations?

 

[TeethWhitener]

thanks, I see. The force given by this potential, dU/dr, is it considered a quantum mechanical version of the electromagnetic force described by Maxwell's equations?

There seems to be a lot of confusion with this question such that a short answer won’t suffice (like “Is the present king of France bald?” Or the perennial favorite, “Have you stopped beating your wife?”)

First, the Lorentz force (I’m assuming that’s what you mean when you invoke Maxwell’s equations) isn’t directly relevant here. The relevant variable is the nuclear separation, and the behavior of the system, while ultimately (ignoring the Pauli principle) a function of the electrostatic forces involved, is probably not best modeled as an electromagnetic interaction. If you’re interested in force, it’s much easier just to consider the problem as purely mechanical.

That said, force as a concept doesn’t hold a central position in quantum mechanics like it does in classical mechanics. At least not so straightforwardly. However, there is an elegant connection between force in classical mechanics and the Heisenberg equation in quantum mechanics. We see this by considering Hamilton’s equations of motion in classical mechanics, in particular:
$$\dot{p} = -\frac{\partial H}{\partial q}=-\{H,p\}$$
Depending on who you ask, force is defined as either $\dot{p}$ or $-\nabla V$; I personally prefer the first definition more, because it can sometimes be difficult to disentangle kinetic and potential energies, whereas momentum is pretty straightforward. Also, it’s certainly more straightforward to make the correspondence between the classical momentum variable and the quantum momentum operator. Regardless, Hamilton’s equation is suggestive of a way forward. Using canonical commutation, promoting $H$ and $p$ to operators, and promoting the Poisson brackets to commutators, we get:
$$\frac{d\hat{p}}{dt}=\hat{F}=\frac{i}{\hbar}[\hat{H},\hat{p}]$$
which is the Heisenberg equation for momentum (where I’ve denoted the quantum mechanical force operator by $\hat{F}$). If we consider a Hamiltonian of the form
$$\hat{H}=\frac{\hat{p}^2}{2m}+V(\hat{q})$$
and further, if we remember that the momentum operator in position space is $-i\hbar\nabla$, then plugging these values into the equation of motion gives
$$\frac{d\hat{p}}{dt}=-\nabla V(\hat{q})$$

 

[greswd]

There seems to be a lot of confusion with this question such that a short answer won’t suffice (like “Is the present king of France bald?” Or the perennial favorite, “Have you stopped beating your wife?”)

First, the Lorentz force (I’m assuming that’s what you mean when you invoke Maxwell’s equations) isn’t directly relevant here. The relevant variable is the nuclear separation, and the behavior of the system, while ultimately (ignoring the Pauli principle) a function of the electrostatic forces involved, is probably not best modeled as an electromagnetic interaction. If you’re interested in force, it’s much easier just to consider the problem as purely mechanical.

ok, so is it possible to mechanically model covalent bonding?

 

[TeethWhitener]

ok, so is it possible to mechanically model covalent bonding?

What, precisely, do you mean?

 

[greswd]

sorry, let me try to clarify

the Lorentz force (I’m assuming that’s what you mean when you invoke Maxwell’s equations) isn’t directly relevant here. The relevant variable is the nuclear separation, and the behavior of the system, while ultimately (ignoring the Pauli principle) a function of the electrostatic forces involved, is probably not best modeled as an electromagnetic interaction. If you’re interested in force, it’s much easier just to consider the problem as purely mechanical.

All of these molecular quantum effects ultimately manifest themselves as classical forces in everyday macroscopic life.

so earlier, I was wondering how these molecular forces are related to the Maxwellian E and B forces.

 

[TeethWhitener]

You’ll have to clarify further

All of these molecular quantum effects ultimately manifest themselves as classical forces in everyday macroscopic life.

Which molecular quantum effects in particular? And which classical forces in particular? $\mathbf{E}$ and $\mathbf{B}$ in Maxwell’s equations are fields, not forces. The potentials associated with those fields affect the motion of particles on the quantum level, it it’s unclear what you mean by molecular forces relating to those fields.

 

[greswd]

my apologies, what I'm thinking of you might find odd.

Which molecular quantum effects in particular? And which classical forces in particular?

For example, the quantum covalent bonding that is the premise of this post.
The molecular bondings, and repulsions, which allow objects to be solid in the macroscopic realm.
Like when you tug a weight with a rope, and the normal reaction force, like when you place a bottle on a table which prevents it from falling to the ground. These seem like pretty classical forces.

The potentials associated with those fields affect the motion of particles on the quantum level, it it’s unclear what you mean by molecular forces relating to those fields.

Maxwell's equations are very comprehensive, but, at the quantum level, they have to be combined with the Schrodinger equation.

Maybe one might call it "Maxwell-Schrodingerian electromagnetism".

It seems, at least to me, that this type of electromagnetism manifests itself at the macroscopic level.
So I'm wondering what the description of "Maxwell-Schrodingerian electromagnetism" at the macroscopic level is.

 

[TeethWhitener]

Maxwell's equations are very comprehensive, but, at the quantum level, they have to be combined with the Schrodinger equation.

For static or slowly-varying potentials, it's straightforward to do this by simply including the potential in the Hamiltonian. This is known in some circles as the semiclassical approximation. In intense or high-frequency EM fields, or in situations where high accuracy is needed, the situation is more complicated. For a fully accurate description, you have to start with quantum electrodynamics, which contains Maxwell's equations in the classical limit (as $\hbar \to 0$).

It seems, at least to me, that this type of electromagnetism manifests itself at the macroscopic level.
So I'm wondering what the description of "Maxwell-Schrodingerian electromagnetism" at the macroscopic level is.

Again, given the examples you've articulated, it's just not clear what you mean. A bottle on a table is not described by Maxwell's equations; it's described by Newton's equations (assuming the bottle and table are both elecrically neutral and not magnetic). Ditto for the rope. Maxwell's equations are never invoked to describe this force.

If you're asking how macroscopic mechanical forces arise from properties of materials at the microscopic level, it's generally simplest to do what I said above: take the semiclassical approximation (QED corrections will be vanishingly small) where you insert the proper potentials (namely nucleon-nucleon, electron-nucleon, and electron-electron interactions) directly into the Hamiltonian. This will give you the electronic structure of the material. In the case of the rope, you can use the Born-Oppenheimer approximation to get the bulk modulus of the rope (how far the molecules must move to break the bonds, which gives you the ultimate strength of the rope). For the bottle on the table, the relevant force is electron-electron repulsion, in particular the overlap integral of the wavefunctions of the bottle atoms and the wavefunctions of the table atoms.

 

[greswd]

If you're asking how macroscopic mechanical forces arise from properties of materials at the microscopic level, it's generally simplest to do what I said above: take the semiclassical approximation (QED corrections will be vanishingly small) where you insert the proper potentials (namely nucleon-nucleon, electron-nucleon, and electron-electron interactions) directly into the Hamiltonian. This will give you the electronic structure of the material. In the case of the rope, you can use the Born-Oppenheimer approximation to get the bulk modulus of the rope (how far the molecules must move to break the bonds, which gives you the ultimate strength of the rope). For the bottle on the table, the relevant force is electron-electron repulsion, in particular the overlap integral of the wavefunctions of the bottle atoms and the wavefunctions of the table atoms.

thanks for the comprehensive description, would it be correct to say that this demonstrates macroscopic electromagnetic phenomena which are beyond the scope of Maxwell's equations?

 

[TeethWhitener]

thanks for the comprehensive description, would it be correct to say that this demonstrates macroscopic electromagnetic phenomena which are beyond the scope of Maxwell's equations?

No. Where did you get that from?

 

[greswd]

No. Where did you get that from?

hmmm...maybe they're still within the scope of Maxwell's equations but beyond the scope of Coulomb's law, the Biot-Savart law, Jefimenko's equations and the like? would that be correct?

 

[TeethWhitener]

hmmm...maybe they're still within the scope of Maxwell's equations but beyond the scope of Coulomb's law, the Biot-Savart law, Jefimenko's equations and the like? would that be correct?

All of those laws follow from Maxwell's equations. But the Coulomb potential provides a good example. In fact, for most materials under most conditions relevant at the macroscale, the Coulomb law suffices. The Hamiltonian describing the nuclear-nuclear, electron-nuclear, and electron-electron interactions includes the Coulomb potential, as seen in this Wiki entry:
https://en.wikipedia.org/wiki/Molecular_Hamiltonian
(link because I'm too lazy to write all that out in LaTeX). Under quasi-adiabatic conditions, this suffices. In conditions where the electron-nuclear motion is strongly coupled, corrections must be made, but they can generally be treated within this semiclassical framework. Maxwell's equations only really cease to apply in strong fields or when the quantum nature of light is manifest (absorption of photons by matter springs to mind).

 

[greswd]

Maxwell's equations only really cease to apply in strong fields or when the quantum nature of light is manifest (absorption of photons by matter springs to mind).

well, the photoelectric effect and photoluminescence can be demonstrated at a macro level, I don't know if that means electromagnetic macro phenomena beyond the scope of Maxwell's equations.

the Coulomb potential provides a good example. In fact, for most materials under most conditions relevant at the macroscale, the Coulomb law suffices. The Hamiltonian describing the nuclear-nuclear, electron-nuclear, and electron-electron interactions includes the Coulomb potential

but its the Coulomb potential applied to the Schrodinger equation, that model is quite different from the simple Coulomb's law right?

 

[TeethWhitener]

well, the photoelectric effect and photoluminescence can be demonstrated at a macro level, I don't know if that means electromagnetic macro phenomena beyond the scope of Maxwell's equations.

Yes, absorption and emission of photons is a quantum phenomenon. Maxwell's equations can't deal adequately with that.

but its the Coulomb potential applied to the Schrodinger equation, that model is quite different from the simple Coulomb's law right?

The phenomena predicted by letting $V=q_1q_2/R$ in the Schrodinger equation are different from what is predicted for charged classical particles, but the form of the Coulomb potential is the same whether it's a quantum operator or a classical potential. You could certainly add in QED corrections at this level (and for some very high accuracy spectroscopic work, people do), but in general, the semiclassical approximation works just fine.

 

[greswd]

Yes, absorption and emission of photons is a quantum phenomenon. Maxwell's equations can't deal adequately with that.

The phenomena predicted by letting $V=q_1q_2/R$ in the Schrodinger equation are different from what is predicted for charged classical particles, but the form of the Coulomb potential is the same whether it's a quantum operator or a classical potential. You could certainly add in QED corrections at this level (and for some very high accuracy spectroscopic work, people do), but in general, the semiclassical approximation works just fine.

so earlier you mentioned that the Lorentz force isn't directly relevant, in this case, what category of force do the macroscopic mechanical forces from materials fall under?

 

[TeethWhitener]

so earlier you mentioned that the Lorentz force isn't directly relevant, in this case, what category of force do the macroscopic mechanical forces from materials fall under?

What are categories of force?

 

[greswd]

What are categories of force?

The different categories which the forces of nature fall under. Perhaps "non-Lorentz electromagnetic force"?

 

[TeethWhitener]

The different categories which the forces of nature fall under. Perhaps "non-Lorentz electromagnetic force"?

You’ll need to be more specific. What categories do the forces of nature fall under?

 

[greswd]

You’ll need to be more specific. What categories do the forces of nature fall under?

For example, the four known forces would be the four main categories. Maybe with other sub-categories.

What do you think of the term "non-Lorentz electromagnetic force"?
Or maybe there are other better, more specific terms

Some of the nuclear forces hardly manifest on a macroscopic level even, in terms of being able to push or pull a macroscopic object, unlike the macroscopic mechanical forces of materials, which form the vast majority of our everyday experience with forces.

 

[PeterDonis]

What do you think of the term "non-Lorentz electromagnetic force"?

There is no such thing. The Lorentz force law is the force law for electromagnetic force at the classical level.

The questions you are asking do not seem consistent with an "A" level (i.e., graduate level) understanding of the subject matter being discussed.

Your original question has been answered, so I am closing this thread.