# 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 wanna 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?

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#### TeethWhitener

Gold Member
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 wanna 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?

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#### TeethWhitener

Gold Member
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

[removed by user]

#### 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.

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#### TeethWhitener

Gold Member
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.

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

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