Pauli exclusion and covalent bonds

[BWV]

Curious about the wave functions of elections shared in covalent bonds. If you have two hydrogen atoms w elections in the lowest energy state with the same spin and they join together to form a molecule H₂ The spin of one of the elections will change (randomly?)

The electons of the two separate H atoms have different wave functions that become (entangled or superimposed - not clear on the right term here) once the H₂ molecule is formed?

In more complex molecules the shared elections would be subject to the exclusion principle relative to the other electrons of the covalently bonded atoms?

 

[mfb]

The electons of the two separate H atoms have different wave functions that become (entangled or superimposed - not clear on the right term here) once the H₂ molecule is formed?

You should write the system as a single wave function, even if the two atoms are separated. Electrons are indistinguishable fermions, and their wave function should always be asymmetric.
If the distance between the atoms is large, it is a good approximation to look at both as independent objects - but it is just an approximation.
The same is true for more than 2 electrons.

 

[BWV]

You should write the system as a single wave function, even if the two atoms are separated. Electrons are indistinguishable fermions, and their wave function should always be asymmetric.
If the distance between the atoms is large, it is a good approximation to look at both as independent objects - but it is just an approximation.
The same is true for more than 2 electrons.

but electrons in the two separate atoms could theoretically be in the same state, and if the atoms came together to share an electron one of the states would have to change, correct?

also if you were to do something ridiculous like write down wave functions for all the electrons in a DNA molecule, it would be a single wave function?

 

[mfb]

but electrons in the two separate atoms could theoretically be in the same state

If you want to do it properly, you cannot look at individual orbitals of individual atoms. You would have to consider the setup as one system with two potential wells and two electrons inside. This system has multiple eigenstates which include both potential wells, and "an electron at atom 1" is then given by a superposition of these states.

also if you were to do something ridiculous like write down wave functions for all the electrons in a DNA molecule, it would be a single wave function?

You could do this in theory. And if you neglect interpretation issues, you can even do that for the whole universe. However, the approximation of multiple indepedent systems with their own electrons is usually easier.

 

[M Quack]

but electrons in the two separate atoms could theoretically be in the same state, and if the atoms came together to share an electron one of the states would have to change, correct?

also if you were to do something ridiculous like write down wave functions for all the electrons in a DNA molecule, it would be a single wave function?

The two electrons of two isolated H atoms would be at different positions, thus satisfying the Pauli principle.

To get into the ground state of the H2 molecule, one electron would have to flip its spin.

Even for large molecules there is one common multi-electron wave function for all electrons. According to the Pauli principle the WF has to change sign whenever you exchange two electrons. This is the basis of the Hartree-Fock method that is used by quantum chemists to calculate molecular structures and properties. (Actually, today's methods go well beyond HF, but it is still the basis of most of them).

http://en.wikipedia.org/wiki/Hartree–Fock_method

Common proteins or a DNA chain are (still) too big to calculate a complete wave function, though. I am not sure what the limit is with today's computers, but probably somewhere near a few 1000 electrons or so.

At some point or distance, however, I would guess that the electrons become essentially independent from each other. In crystal physics (e.g. superconductors) there is a coherence length which depends on temperature, impurities and so on, and which basically tells you at what distance one electron's wave function can still interfere with the wave function of another electron.

 

[Einstein Mcfly]

but electrons in the two separate atoms could theoretically be in the same state, and if the atoms came together to share an electron one of the states would have to change, correct?

also if you were to do something ridiculous like write down wave functions for all the electrons in a DNA molecule, it would be a single wave function?

If you write down the wave function of the system as a superposition of the two atomic s orbitals, you'll get two orbitals that satisfy the conditions. One will be of lower energy (paired electrons) and if both electrons want to be in this MOLECULAR orbital, they will have to obey Pauli. If one of them wants to be the same spin, it can go into the higher energy molecular orbital, but this orbital is higher in energy relative to the separate s orbitals than the lower one is lower than the separate s orbitals, so this system is unstable and the molecule will fall apart. If you think about this in reverse as suggested with the molecular orbitals, you'll see that the molecule will only form if they are paired.

 

[BWV]

Thanks for the responses

Still not clear about what happens in a system with two hydrogen atoms with electons in the same spin and in the lowest energy state - will one of the electrons change it's spin and form a covalent bond or will the exclusion principle keep a molecule of H2 from forming, even though it's Potentially a higher entropy state? Does it take energy to change the spin of an electron?

 

[M Quack]

The H2 molecule in its ground state has a lower energy than two isolated H atoms. Upon forming the bond, one electron will change its spin, releasing some energy in the form of a photon or kinetic energy.

The entropy of two isolated atoms is lower than that of the H2 molecule in the ground state. But then again the energy of an unbound proton and electron is higher still. Because of that you get a plasma at very very high temperatures.

 

[BWV]

but with two H atoms in the ground state with electrons in the same spin then there would be one less quanta of energy in the formation of a covalent bond than the formation of a H2 molecule with two H atoms in the ground state with electrons in opposite spin states? Would that not then imply that one spin state had a higher energy level than its opposite within the same orbital?

 

[M Quack]

The total energy of two isolated H atoms with parallel and antiparallel spins is identical.

The energy of the bound singlet state is always the same, hence the energy released in the forming of the bond must be identical.

What may change are the selection rules for the polarization and direction of the photon relative to the initial spins.