Atomic Shell Theory: Bohr-Sommerfeld model


by hbaromega
Tags: atomic, bohrsommerfeld, model, shell, theory
hbaromega
hbaromega is offline
#1
Aug8-11, 10:27 AM
P: 20
This is a very naive question. But I think, it's an important point that has been unattempted in textbooks. The question is:

How far should one trust the Bohr-Sommerfeld model or the atomic shell theory for all elements in the periodic table?

This question generally comes in mind, since we know that the Bohr's model was a kind of hypothesis or ansatz to explain the hydrogen atom spectra. And Sommerfeld modified the quantization condition in a spirit of generalization, using the analogy to the planetary motion under the central force due the sun. These all can be put together as the initial development of quantum mechanics and often regarded as theories of Old Quantum Mechanics.

Now we know that the H-atom can be exactly solved from the Schroedinger's equation. But what about atoms with higher atomic numbers (may be we can only expect hydrogen-like wavefunctions for alkali atoms)? Should Bohr-Sommerfeld model (apart from the relativistic correction) still be working ?

Then why do people say that in cuprates copper has d9 electronic state? Doesn't that sound imprecise?

Thanks.
Phys.Org News Partner Physics news on Phys.org
Modification of structural composite materials to create tailored lenses
Novel technique opens door to better solar cells
Probing metal solidification nondestructively
DiracRules
DiracRules is offline
#2
Aug8-11, 01:08 PM
P: 112
It depends on the precision you want.

First of all, you must know that Schroedinger's equation cannot be solved exactly for atoms different from hydrogen atom.

Actually, one of the best representations of the atoms of higher atomic number is given by the Hartree theory, which tries to approximate Schroedinger's equation by making hypothesis about the potential seen by the electrons.

Next, what do you mean when you say "How far should one trust the Bohr-Sommerfeld model or the atomic shell theory for all elements in the periodic table?" ?

You know, the Bohr model is an oversimplified model of the atom, and in the new quantum mechanic it is replaced by the model deduced by Schroedinger's equation, in which the electrons are replaced by waveforms.

Moreover, Bohr's model cannot account for effects related to the filling of the subshells, that is important to build the periodic table of elements or for a finer structure of the spectrum of atoms than the one studied by Bohr, and to the spin-orbit interaction: an atom immersed in a magnetic field has different levels of energy than a free atom because of the spin-orbit interaction.

Bohr's model works for rough calculations but, as I said, for finer calculations and for to have a (better) explanation of a large variety of phenomena, not explained by Bohr's model because, as you said, it is just an hypothesis (that works) built upon experimental data.

Hope it answers :D
hbaromega
hbaromega is offline
#3
Aug8-11, 02:23 PM
P: 20
I think, if you can answer my last question, it'll be clear.

How can one say Cu has d9 electronic state in cuprates? Or does it need to have d-state at all?

Let me put a few more questions?

1) Are there experiments that can see real electronic orbitals of H-atom? Note that the orbital photographs seen in most textbooks were engineeringly plotted by H. E. White (no real hydrogen atom experiments done).

Ref. http://prola.aps.org/abstract/PR/v37/i11/p1416_1

2) How can we define s, p, d orbitals for non-hydrogen atoms? We may have s',p', d' orbitals that may hold totally different kind of geometries.

DiracRules
DiracRules is offline
#4
Aug8-11, 04:04 PM
P: 112

Atomic Shell Theory: Bohr-Sommerfeld model


Actually it's the first time I hear about cuprate.
However, the fact is that the atomic configuration of an atom is not fixed, but it can change if a new configuration with lower energy is available.
I don't know (if you have some links please post me; I will search for myself) actually the structure of cuprate but, since the energies of the outer subshells narrows as you go away from hydrogen atom, it is possible that copper configuration in cuprate is more stable and less energetic if it is in 3d9 state.

As regards the photos or images of electron orbitals, last month a group of researchers at Politecnico di Milano (Italy) managed to "take a photo" of the orbitals in a molecule. I can't find the article in English, but here it is in Italian (maybe you can get it translated by google), with some photos.
I must remind you that after Schroedinger, it's almost wrong to think about orbitals as "paths": it is more correct if you think about where it is more probable to find an electron.

We can define the subshells from their energy. Dirac derived a formula that corrects Bohr's taking in account spin orbital interaction and other interactions in the second order. In this formula are present the quantum numbers n,l,j (where j is the total angular moment)



Actually, I've just read on Wikipedia that in cuprate copper is Cu++ and O--, so Cu++ is [Ar]3d9 because 2e- have been removed by the more electronegative oxygen.
hbaromega
hbaromega is offline
#5
Aug8-11, 04:44 PM
P: 20
Actually it's the first time I hear about cuprate.
However, the fact is that the atomic configuration of an atom is not fixed, but it can change if a new configuration with lower energy is available.



It's not only about cuprates.We always find the oxidation state and then decide the electronic configuration, purely from the chemistry point of view. My question is:

1) Is there any physics explanation behind the atomic configuration of non-hydrogen atoms?


As regards the photos or images of electron orbitals, last month a group of researchers at Politecnico di Milano (Italy) managed to "take a photo" of the orbitals in a molecule. I can't find the article in English, but here it is in Italian (maybe you can get it translated by google), with some photos.
I must remind you that after Schroedinger, it's almost wrong to think about orbitals as "paths": it is more correct if you think about where it is more probable to find an electron.


This nature paper has the experimental detail. By orbitals I meant the probability density distribution, not paths (see the H. E. White's paper). And as I said, I wish to see the proof of electron's probability density distribution for the H-atom (apart from the indirect way from the spectroscopy results) that we solve through Schroedinger's equation in the textbooks.

Can you see the gap between chemistry and physics? I can believe atomic number, but how can I believe the atomic configuration and hence the s,p,d,f nomenclature?


We can define the subshells from their energy. Dirac derived a formula that corrects Bohr's taking in account spin orbital interaction and other interactions in the second order. In this formula are present the quantum numbers n,l,j (where j is the total angular moment)

Did you mean subshells for the Bohr-Sommerfeld theory, i.e. the old quantum mechanics?
I know Dirac's relativistic correction and that is for the H-atom.

Actually, I've just read on Wikipedia that in cuprate copper is Cu++ and O--, so Cu++ is [Ar]3d9 because 2e- have been removed by the more electronegative oxygen.


Again, this is ridiculous. Electrons are not removed. We can say, there's no significant probability density. But in what region? I can think of chemical bonding as overlap or distribution of electronic wavefunction.

Now the questions:

1) How can I calculate the atomic wavefunction or the energy (i. e. eigenvalue) of a non-hydrogen atom?

2) Even if I succeed to do for atoms, how should I calculate the redistribution of wavefunctions when they form a molecule?
cgk
cgk is offline
#6
Aug8-11, 08:27 PM
P: 402
Quote Quote by hbaromega View Post
Now the questions:

1) How can I calculate the atomic wavefunction or the energy (i. e. eigenvalue) of a non-hydrogen atom?

2) Even if I succeed to do for atoms, how should I calculate the redistribution of wavefunctions when they form a molecule?
Both of those are done using techniques of Quantum Chemistry. This is basically hardcore numerical quantum many body physics, and there are (large and very complex) software packages which can do such calculations (e.g., Molpro, CFOUR and MRCC). Using these techniques, relative energies and properties can be calculated to around 0.1 kJ/mol ... 4 kJ/mol (depending on your patience. 1 kJ/mol is about 0.010 eV) for sufficiently friendly atoms and small molecules (e.g., energy differences between states, between products, transition states, and educts of a chemical reaction etc.). Unfortunatelly, understanding how these many-body method work require an intimate knowledge of quantum mechanics, many-body theory, and numerics, and this is not easily explained. Some terms to get you started are "Coupled cluster theory", "Multirefernce configuration interaction", "correlation consistent basis set" and "basis set extrapolation".
hbaromega
hbaromega is offline
#7
Aug9-11, 01:20 AM
P: 20
Quote Quote by cgk View Post
Both of those are done using techniques of Quantum Chemistry. This is basically hardcore numerical quantum many body physics, and there are (large and very complex) software packages which can do such calculations (e.g., Molpro, CFOUR and MRCC). Using these techniques, relative energies and properties can be calculated to around 0.1 kJ/mol ... 4 kJ/mol (depending on your patience. 1 kJ/mol is about 0.010 eV) for sufficiently friendly atoms and small molecules (e.g., energy differences between states, between products, transition states, and educts of a chemical reaction etc.). Unfortunatelly, understanding how these many-body method work require an intimate knowledge of quantum mechanics, many-body theory, and numerics, and this is not easily explained. Some terms to get you started are "Coupled cluster theory", "Multirefernce configuration interaction", "correlation consistent basis set" and "basis set extrapolation".

As far as I know, the quantum chemistry methods use Hartree-Fock or post-Hartree-Fock methods that use linear combinations of atomic orbitals and again those orbitals are treated like s, p, d kind of orbitals, which are hard to accept.

Hope you got my point. Again my question is very simple:

How can we believe that s, p, d, ... orbitals exist beyond hydrogen atom? Any experimental or theoretical evidence?
abhi2005singh
abhi2005singh is offline
#8
Aug9-11, 04:07 AM
P: 64
How can we believe that s, p, d, ... orbitals exist beyond hydrogen atom? Any experimental or theoretical evidence?
Well, one can also ask: "How do we know that these orbitals exist for H atom?". The only answer which comes to mind is that they are predicted using theory and that no experimental evidence is available against it. The theory explains well the experimental facts. This is true not only in the present case but for any theory in general.

For atoms with Z > 1, as mentioned previously, HF theory is used (other approaches are also available) which assumes existence of such orbitals. The results of these calculations predict some results. If those results are proved experimentally, then the theory stands.

In short, if the experimental results can be explained using the assumption of existence of such orbitals, then those experimental results can be treated as the proof for the existence of these orbitals.
DiracRules
DiracRules is offline
#9
Aug9-11, 04:19 AM
P: 112
How can we believe that s, p, d, ... orbitals exist beyond hydrogen atom? Any experimental or theoretical evidence?
If subshells exists for multielectron atoms, it will result in atomic spectra: there are transition rules that allow only certain decay. You can look very carefully at the spectrum and calculate the energy of the gap.

I'm quite sure it happens, and they already found evidence, but I cannot be sure 100% having not my book under hand :D

Moreover, probably this is a stronger evidence: they found two types of helium, ortohelium and parahelium, with different physical properties. This is due to the fact that parahelium has both the electron in state 1s2 (a state of singulet), whilst ortohelium is in the state 1s1 2s2 (a state of triplet). The transition rules shows that is impossible for optical transition to transform a singulet state to a triplet state and vice versa.
I think this is a proof of the existence of subshells, or not?

Another proof could that the energy levels within each subshell are very narrow, while between shells are broader, and you can see this when you ionize the atom.

http://hyperphysics.phy-astr.gsu.edu...um/helium.html
http://ibchem.com/IB/ibnotes/full/ato_htm/12.2.htm

I'm sorry for my poor speech: "because 2e- have been removed by the more electronegative oxygen." Obviously electrons aren't removed in the meaning we give to this word :D

Did you mean subshells for the Bohr-Sommerfeld theory, i.e. the old quantum mechanics?
I know Dirac's relativistic correction and that is for the H-atom.
You are right, sorry :D

Can you see the gap between chemistry and physics? I can believe atomic number, but how can I believe the atomic configuration and hence the s,p,d,f nomenclature?
Actually I can't, probably because I don't like chemistry too much :D

As far as I know, the quantum chemistry methods use Hartree-Fock or post-Hartree-Fock methods that use linear combinations of atomic orbitals and again those orbitals are treated like s, p, d kind of orbitals, which are hard to accept.
When I treated Hartree-Fock method, we studied that it was based on Schroedinger's equation. That is, Hartree hypothesized the effective potential Z(r) and put it into Schroedinger's equation. Through iterations, he arrived to a waveform that was in accord with experimental evidence. In this waveform, we have subshells.
hbaromega
hbaromega is offline
#10
Aug9-11, 05:49 AM
P: 20
@DiracRules

Sorry. A severe objection. We can think of subshells. But they don't need to have the same geometry of the hydrogen atom s,p,d orbitals.

Moreover, transition doesn't show the nature of the orbitals. It just reflects the difference between two energy levels.


So it seems that we are still using the Bohr's hypothesis (a bit modification due to Sommerfeld) to decide the electronic configuration of elements. So it's an empirical fitting of the spectroscopy results to the Bohr-Sommerfeld energy spectra formula, which is true only for the hydrogen atom.

It's strange to see that shell theory (old quantum mechanics) still works for many elements. And there exists no true quantum theory to support or deny the theory.

So I can say, we still trust the old quantum theory (since in modern quantum mechanics, non-hydrogen atom problem is non-trivial). In that sense it's as good/bad as we have the Boyle's law to describe gases.

Hope you'll agree with my point.
hbaromega
hbaromega is offline
#11
Aug9-11, 06:00 AM
P: 20
Quote Quote by abhi2005singh View Post
Well, one can also ask: "How do we know that these orbitals exist for H atom?". The only answer which comes to mind is that they are predicted using theory and that no experimental evidence is available against it. The theory explains well the experimental facts. This is true not only in the present case but for any theory in general.

See, the Bohr's model as an ansatz gives correct values for the hydrogen atom spectra (apart from the fine-structure correction). And solving Schroedinger equation for the H-atom reproduces Bohr's formula. In that sense, we have a microscopic theory that supports Bohr's formula for the H-atom. But there's no such quantum theory that support Bohr-Sommerfeld formula for other atoms. However, Bohr-Sommerfeld formula seems to work well (I hope so though I never have tested). It's similar to the shell theory that works in nuclear physics.

So my point is that we still use an ansatz (which is not true quantum mechanics) to infer
the atomic orbitals and hence the elctronic configurations.
abhi2005singh
abhi2005singh is offline
#12
Aug9-11, 06:57 AM
P: 64
However, Bohr-Sommerfeld formula seems to work well (I hope so though I never have tested).
No, it doesn't work well. There are many things which are not explained: http://en.wikipedia.org/wiki/Bohr_model#Shortcomings

So my point is that we still use an ansatz (which is not true quantum mechanics) to infer the atomic orbitals and hence the elctronic configurations.
This is also not correct. We are not using the atomic orbitals because of the BS model. These orbitals follow from quantum mechanics. Please read what @DiracRules have written in this regard.

Moreover, transition doesn't show the nature of the orbitals. It just reflects the difference between two energy levels.
Transition do show the nature of the orbitals. They tell you about the symmetry of the orbitals by utilizing the selection rules, which in most cases is good enough to infer the initial and final state orbitals.

So I can say, we still trust the old quantum theory (since in modern quantum mechanics, non-hydrogen atom problem is non-trivial). In that sense it's as good/bad as we have the Boyle's law to describe gases.
No ways. Spectroscopy cannot be described using the BS model and no one actually use it in the field of spectroscopy, apart from obtaining some initial guesses. This is specially true for elements with Z > 1.
hbaromega
hbaromega is offline
#13
Aug9-11, 11:31 AM
P: 20
Quote Quote by abhi2005singh View Post
No, it doesn't work well. There are many things which are not explained: http://en.wikipedia.org/wiki/Bohr_model#Shortcomings

This is also not correct. We are not using the atomic orbitals because of the BS model. These orbitals follow from quantum mechanics. Please read what @DiracRules have written in this regard.
I think, s,p,d orbitals were defined only for the hydrogen or hydrogen-like (alkali) atoms.
How do you determine probability densities for other atoms?

I guess, the wiki link tells about shortcoming of the Bohr's model, not about the BS model.

Transition do show the nature of the orbitals. They tell you about the symmetry of the orbitals by utilizing the selection rules, which in most cases is good enough to infer the initial and final state orbitals.
How can you be sure there are no other orbitals 'except' s,p,d,f,g, etc? Note that all these orbital geometries are defined from hydrogen atom wavefunctions. Also remember that in the hydrogen atom problem in quantum mechanics, we can actually separate the differential equation into radial and spherical harmonics part. It may not be possible to do the same for other atoms.

No ways. Spectroscopy cannot be described using the BS model and no one actually use it in the field of spectroscopy, apart from obtaining some initial guesses. This is specially true for elements with Z > 1.
In atomic spectroscopy, I guess, we look at emission lines. Or may be absorption in some cases, I'm not sure. Now probably I can calculate the wavelengths of those lines and hence differences between energy levels. Now could you tell me, what formula I'm going to use next and how I connect to symmetry of orbitals from these?
DiracRules
DiracRules is offline
#14
Aug9-11, 12:12 PM
P: 112
Quote Quote by hbaromega View Post
I think, s,p,d orbitals were defined only for the hydrogen or hydrogen-like (alkali) atoms.
How do you determine probability densities for other atoms?

How can you be sure there are no other orbitals 'except' s,p,d,f,g, etc? Note that all these orbital geometries are defined from hydrogen atom wavefunctions. Also remember that in the hydrogen atom problem in quantum mechanics, we can actually separate the differential equation into radial and spherical harmonics part. It may not be possible to do the same for other atoms.
I have to correct what I said previously and say that you are right saying that we cannot do the same for multielectron atoms if we want to solve it exactly. The problem is the presence of the potential interaction between electrons.
However, this problem is, in a certain way, overcome by making hypothesis about the potential that affects the electrons ( Z(r) ). By making this hypothesis, we can split the equation in an angular and radial part.
I realize this cannot satisfy you, but it's the way things are going since 1930! If it was wrong, it would have been corrected :D

If you do not think that approximation is a sort of trickery that hides true reality, think of this: physicists approximated the potential interaction between electrons in the way I told you also to treat identical particles, and to take into account the spin. And it works very well too :D


But I think that the one of the best way to "believe" into the existence of subshells comes from spectroscopy.

Quote Quote by hbaromega View Post
In atomic spectroscopy, I guess, we look at emission lines. Or may be absorption in some cases, I'm not sure. Now probably I can calculate the wavelengths of those lines and hence differences between energy levels. Now could you tell me, what formula I'm going to use next and how I connect to symmetry of orbitals from these?
From spectroscopy, you can see that each level is split into other sublevel: for example, transition rules says that you can't have a transition 3s->2s but 3p->2s or 3d->2p.
Now, since [itex]\Delta E_{3p\rightarrow 2s}[/itex] is different from [itex]\Delta E_{3d\rightarrow 2p}[/itex], you can't explain this if you don't consider subshells into your model.

Now you can say, why do we not consider subshells as proper shells?
The mathematical answer can be "because they share the same principal quantum number n".
The experimental answer can be "because going up through the periodic table, we see that removing certain electrons requires more energy than the trend. We say thus that we passed into another shell".
hbaromega
hbaromega is offline
#15
Aug9-11, 02:50 PM
P: 20
Now the discussion has become pretty interesting.

Quote Quote by DiracRules View Post
I have to correct what I said previously and say that you are right saying that we cannot do the same for multielectron atoms if we want to solve it exactly. The problem is the presence of the potential interaction between electrons.
However, this problem is, in a certain way, overcome by making hypothesis about the potential that affects the electrons ( Z(r) ). By making this hypothesis, we can split the equation in an angular and radial part.
I realize this cannot satisfy you, but it's the way things are going since 1930! If it was wrong, it would have been corrected :D

If you do not think that approximation is a sort of trickery that hides true reality, think of this: physicists approximated the potential interaction between electrons in the way I told you also to treat identical particles, and to take into account the spin. And it works very well too :D
If an approximation works, then there should be arguments for it. If we lack reasoning, then it's just a fitting formula.

I find no reasoning to extend the same hydrogen atomic orbitals (electronic probability densities) and corresponding energy eigen values for other atoms.

It may be satisfactory for chemists, but not for physicists.

But I think that the one of the best way to "believe" into the existence of subshells comes from spectroscopy.



From spectroscopy, you can see that each level is split into other sublevel: for example, transition rules says that you can't have a transition 3s->2s but 3p->2s or 3d->2p.
Now, since [itex]\Delta E_{3p\rightarrow 2s}[/itex] is different from [itex]\Delta E_{3d\rightarrow 2p}[/itex], you can't explain this if you don't consider subshells into your model.
Question is : While calculating the energy of the levels (s, p, d, whatever), are we using the hydrogen-atom energy eigenvalue formula? Don't we require to modify that since many electrons are talking to each other now?

Now you can say, why do we not consider subshells as proper shells?
The mathematical answer can be "because they share the same principal quantum number n".
The experimental answer can be "because going up through the periodic table, we see that removing certain electrons requires more energy than the trend. We say thus that we passed into another shell".
Again filling the shells by putting electrons one by one doesn't make any physics sense.
Electrons are always interacting and they are in a cloud, not in isolated islands.

I believe, this is a serious issue which has been neglected since 1930. We should try to solve two, three, four atoms problems numerically and figure out true atomic orbitals for them.

And even if a large community accepts a theory, without a justification, the theory cannot be regarded as the correct or true theory.
abhi2005singh
abhi2005singh is offline
#16
Aug9-11, 07:15 PM
P: 64
I guess, the wiki link tells about shortcoming of the Bohr's model, not about the BS model.
There is not much difference between the two.
How can you be sure there are no other orbitals 'except' s,p,d,f,g, etc?
It has been working so far. Do not freak out if someone gives you this answer. This is true with every theory/model.
Now could you tell me, what formula I'm going to use next and how I connect to symmetry of orbitals from these?
This has already been answered by @DiracRules
If an approximation works, then there should be arguments for it. If we lack reasoning, then it's just a fitting formula.
Most of the theories (if not all) are "fitting formula". If the fitting works for all/most cases, then it is a good theory/model.
Again filling the shells by putting electrons one by one doesn't make any physics sense.
This is done in all the multi-fermion cases. If this does not make sense to you in atomic theory, then this should not make sense to you in all branches of Physics involving fermions.
We should try to solve two, three, four atoms problems numerically and figure out true atomic orbitals for them.
People have tried, they do not know enough mathematics.

Again, answer to all your questions seems to be that "it works".
cgk
cgk is offline
#17
Aug9-11, 08:06 PM
P: 402
hbaromega.
you are confusing lots of things. (I admit it's simple with this topic, as there is lots of information around which is misleading or plainly wrong, even in textbooks). Let's clarify some things:

1) The spherical harmonics (s,p,d,f,...) have /nothing/ to do with the hydrogen atom. These are the solutions of the homogenous angular Laplace equation, and they form (exactly) the angular part of any Laplace problem with a spherical potential. If a potential is slightly non-spherical (think of a atom in a molecule), it is often a good idea to expand the solutions into spherical harmonics anyway, because these form a convenient complete orthonormal set of functions on the sphere (i.e., for angular functions). Note that this is /NOT/ an approximation as long as the expansion is not truncated (complete orthonormal set!). In practice that means for example that the orbitals of the O atom in H2O have not only s and p character, but also a bit of d, of f, of g etc character, with increasingly negligible weight. This is realized in practical calculations by using systematic series of basis sets for expaning the one-particle wave functions (for example, the mentioned correlation consistent basis sets). By doing that you can approach the infinite basis limit to any degree you like.

2) There are no "true orbitals" for systems with more than one electron. Orbitals are, by definition, ONE PARTICLE wave functions. They are DEFINED in terms of some kind of Hartree-Fock or Kohn-Sham mean field picture (there are also "natural orbitals", which are something different, but let's ignore them for now). Orbitals are not true wave functions either[1]. Rather, they are used to BUILD wave functions by plugging them into Slater determinants or configuration state functions.

3) Of course 3--6 electron systems can be calculated effectively exactly, and this has been done long time ago. Some people are still doing it now. The most accurate methods for that are typically of the variational monte carlo or iterative complent class. For systems with more than 6 electrons the mentioned quantum chemistry methods come into play. Note that also these methods are accurate to more than 0.01% in total energies, often much more (1 kJ/mol is a VERY VERY SMALL energy compared to the total energies of the systems calculated! Even Hartree-Fock typically gets total energies right to around 1%).

4) The aufbau principle and shell filling you mention is based on the Hund rules, which were originally empirically derived. Nevertheless, the theoretical reasoning behind them is sound (go look up Hund rules in Wikipedia), and almost always works. If you don't like the reasoning, there is no reason to believe it: You can /calculate/ the energies of the different states (using quantum chemistry) and see that it works. You don't have to believe it, you can test it.

5) Ab initio quantum chemistry is not an empirical science. It calculates properties of molecules using /nothing/ as input execept for the Schroedinger equation and fundamental constants (hbar, electron mass, electron charge etc.). It can calculate true wave functions to any desired degree of accuracy, the limits are only given by the computational power which you are willing to put into it.

[1] (and never probability densities. There abs square gives a density, but not the orbitals themselves),
hbaromega
hbaromega is offline
#18
Aug10-11, 04:49 PM
P: 20
Quote Quote by cgk View Post
hbaromega.
you are confusing lots of things. (I admit it's simple with this topic, as there is lots of information around which is misleading or plainly wrong, even in textbooks). Let's clarify some things:

1) The spherical harmonics (s,p,d,f,...) have /nothing/ to do with the hydrogen atom. These are the solutions of the homogenous angular Laplace equation, and they form (exactly) the angular part of any Laplace problem with a spherical potential. If a potential is slightly non-spherical (think of a atom in a molecule), it is often a good idea to expand the solutions into spherical harmonics anyway, because these form a convenient complete orthonormal set of functions on the sphere (i.e., for angular functions). Note that this is /NOT/ an approximation as long as the expansion is not truncated (complete orthonormal set!).
I think, hydrogen atom and one electron atom are equivalent. Now once you put one extra electron, trouble enters. Now the Schroedinger equation has two coordinates: [itex]r_1[/itex] and [itex]r_2[/itex], and due to Coulomb repulsion they are coupled, i.e. we have a term like [tex]e^2/|r_{1}-r_{2}|[/tex]. So unless we get rid of electronic repulsion, we cannot do separation of variables and hence cannot have the spherical harmonics for the angular part.

In practice that means for example that the orbitals of the O atom in H2O have not only s and p character, but also a bit of d, of f, of g etc character, with increasingly negligible weight. This is realized in practical calculations by using systematic series of basis sets for expaning the one-particle wave functions (for example, the mentioned correlation consistent basis sets). By doing that you can approach the infinite basis limit to any degree you like.
Before we go to H2O, I am curious about knowing orbitals of O (oxygen).


2) There are no "true orbitals" for systems with more than one electron. Orbitals are, by definition, ONE PARTICLE wave functions. They are DEFINED in terms of some kind of Hartree-Fock or Kohn-Sham mean field picture (there are also "natural orbitals", which are something different, but let's ignore them for now). Orbitals are not true wave functions either[1]. Rather, they are used to BUILD wave functions by plugging them into Slater determinants or configuration state functions.
I agree. But again the single-body wavefunction in the Hartree-Fock or the Slater determinants are ambiguous (even the shape).

3) Of course 3--6 electron systems can be calculated effectively exactly, and this has been done long time ago. Some people are still doing it now. The most accurate methods for that are typically of the variational monte carlo or iterative complent class. For systems with more than 6 electrons the mentioned quantum chemistry methods come into play. Note that also these methods are accurate to more than 0.01% in total energies, often much more (1 kJ/mol is a VERY VERY SMALL energy compared to the total energies of the systems calculated! Even Hartree-Fock typically gets total energies right to around 1%).
I must underline the word effectively. I'm just curious to know, did the results show (effective) orbitals? How do they look like? Could you provide some references where I can see that?

4) The aufbau principle and shell filling you mention is based on the Hund rules, which were originally empirically derived. Nevertheless, the theoretical reasoning behind them is sound (go look up Hund rules in Wikipedia), and almost always works. If you don't like the reasoning, there is no reason to believe it: You can /calculate/ the energies of the different states (using quantum chemistry) and see that it works. You don't have to believe it, you can test it.
I agree that it was originally empirically derived. But probably Schroedinger or Hartree-Fock equation cannot take care of that.

5) Ab initio quantum chemistry is not an empirical science. It calculates properties of molecules using /nothing/ as input execept for the Schroedinger equation and fundamental constants (hbar, electron mass, electron charge etc.). It can calculate true wave functions to any desired degree of accuracy, the limits are only given by the computational power which you are willing to put into it.
How does the ab initio method distinguish a hydrogen and a oxygen atom? Note that I'm only interested about orbitals in an atom, not in a molecule.

[1] (and never probability densities. There abs square gives a density, but not the orbitals themselves),
I didn't get what you wanted to mean here.


Register to reply

Related Discussions
on bohr- sommerfeld theory Special & General Relativity 0
integers in Bohr's atomic theory Quantum Physics 5
Bohr's Atomic Model General Physics 7
Bohr's atomic model Introductory Physics Homework 6