gravenewworld
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schrodinger equation for the He atom
I know that the shrodinger equation is unsolvable for anything other than the hydrogen and hydrogen like atoms. My question is, does anyone know how
the orbital functions are approximated for the helium atom and above?
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tryptic
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This is not my speciality, although I study theoretical physics. Lots of ways are used for this kind of approximate computations, as far as I
know..For example, the Finite Element Method, a general numerical method for partial differential equations, can be used, but then there are lots of
more specialized methods that can probably yield much better accuracy in some situations. So in general I guess the method to use depends on the
atom/molecule in question. Helium is relatively simple so there are probably lots of specialized approximations for it that can be found in scientific
literature. Some amount of math will probably be involved, but if you want to study orbitals and stuff, you'll need it anyway.
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Marvin
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Its a result of the so called "3 body problem" where exact solutions are difficult if not impossible to find by differential methods. The
most common solution to the problem in this case was developed by Hartree and Fock and is called the Hartree-Fock (HF) or Self Consistant Field (SCF)
method.
The idea is that you solve for one electron at a time, treating the wavefunctions of the other electrons as static. This is a little white lie at
first because changing one wavefunction, will of course, affect the others, however the result should be more accurate than the guess you started
with. When youve done all electrons and thus obtained a better aproxamation you go back to the first electron and do it again. The aproximations get
closer and closer to a solution.
This is iteration, and computers are very good at it. They arnt good at algebra though and the equations produced would get rapidly out of hand and
complicated very quickly done this way, so the problem is first turned from an algebraic one to a numerical one.
This is done by (if you are still interested/need to know) constructing a set of ideal orbitals (the set must be dimensonally seperate - that is to
say zero integral overlap (understanding fourier analysis is a good way to get this concept)). The computer then iterates the SCF until things stop
changing much and thats the solution.
This method lends its way to doing calculations for molecules, you can choose a nice simple basis set, like the polar sin series aproxamating the
hydrogenic orbitals and iterating gives you a contribution of each electron in a classical orbital. This is useful because if we solved the real
system and got an equation it would make no sense to chemists anyway. This tells us what atoms are hybridised. It also gives us the energy levels of
the molecules (which is the real goal).
Ok in a nutshell. You pretend the atom uses the hydrogenic orbitals (or something more complex) and solve for one electron at a time, iterating until
you get the required accuracy. That then gives you a contribution of each electron to an orbital and a set of energy levels.
The other electrons distort the hydrogenic orbitals in a real atom, so the accuracy of the method depends not only on how many iterations you do, but
how flexable your orbital set is.
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chemoleo
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How do you know all this stuff Marvin?
So essentially you are saying that the problem of solving the Schroedinger equation for atoms higher than H reside in its electrons not acting
independently, but constantly influencing each other? Like, one electron 'moves' slightly and thereby influences the other's moves,
which in turn acts on the first one... and so on?
Isn't this exactly the case in galaxies/solar systems, where gravitational influences constantly change everything around? Yet, positions can be
predicted with very high accuracies, particularly in the latter case...
I don't know though whether the planets are considered independently of each other (in the calculations), i.e. whether the
earth's/moon's influence on Mars is included in any predictive calculations of the Mars orbit... anyone?
Never Stop to Begin, and Never Begin to Stop...
Tolerance is good. But not with the intolerant! (Wilhelm Busch)
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tryptic
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Quantum mechanics is really really different from classical mechanics, chemoleo. You can't really think about this stuff in classical mechanics
terms. Sure there are probably other approximation methods, but that Hartree-Fock method seems to make a lot of sense.
Now that I think of it, stuff like the finite element method possibly wouldn't make sense with multiparticle quantum problems, it would just take
too much computation power and memory..
[Edited on 5-8-2004 by tryptic]
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vulture
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| Quote: |
So essentially you are saying that the problem of solving the Schroedinger equation for atoms higher than H reside in its electrons not acting
independently, but constantly influencing each other?
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Yes, every electron represents an electric charge and has a magnetic spin, so they clearly influence eachother.
The Hartree central field method can be used to approximate the potential energies of the electrons and from thereon one can substitute this into
quantummechanical models/calculations.
EDIT: Tryptic beat me to it.
[Edited on 5-8-2004 by vulture]
One shouldn't accept or resort to the mutilation of science to appease the mentally impaired.
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chemoleo
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Thanks, I am aware that classical mechanics is differnet from quantum mechanics; spins, wave functions, probability and such making the whole matter
radically different.
I just wanted confirmation that the problem indeed resides in one electron affecting the other, affecting the first one in turn.
Sure the field equations etc are radically different, but as a very crude analogy (i know no wave functions apply etc), is this three body problem not
also found in planetary bodies? Wouldnt it also be excruciatingly hard to take account of the gravitational fields influencing each other? Presumably
the effec t is very small, as the mass /graviation of the sun dominates everythign...
Never Stop to Begin, and Never Begin to Stop...
Tolerance is good. But not with the intolerant! (Wilhelm Busch)
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gravenewworld
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Thanks for answering. Yes the 3 body problem is found in planetary bodies. Thats how the 3 bodied problem originated in the first place. The 3
bodied problem for the planetary bodies has still not been solved. The shrodinger eq can not be solved for the He atom because it is an absolute
nightmare, it is a 6 dimensional problem. Beyond the He atom you can forget about trying to solve the SE.
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I am a fish
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| Quote: | Originally posted by chemoleo
Sure the field equations etc are radically different, but as a very crude analogy (i know no wave functions apply etc), is this three body problem not
also found in planetary bodies? Wouldnt it also be excruciatingly hard to take account of the gravitational fields influencing each other? Presumably
the effec t is very small, as the mass /graviation of the sun dominates everythign... |
Imagine two planets orbiting a (much more massive) star. The problem in solving the equations of motion exactly, is that the force between the
planets changes the orbit of each planet, which in turn alters the force between them. This change in force leads to a another change in the orbit,
which leads to a further change in force, and so on. However, if the interplanetary force is very small (in comparison to the force exerted by the
star), the change in force due to the change of orbit will be very small squared, and so can (to a good approximation) be ignored.
Using a similar technique for a helium atom is more difficult, because the electron-electron and electron-nucleus forces are comparable. Therefore,
the higher order corrections (i.e. the changes due to changes and so on) are still large.
1f `/0u (4|\\| |234d 7|-|15, `/0u |234||`/ |\\|33d 70 937 0u7 /\\/\\0|23.
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Marvin
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Classical mechanics does not differ from the Schrodinger equation in essence. Spin etc only enters into it when you construct the potential energy
for the atom you are creating. You could create one for an object roating around a star that only used gravity.
Its difference to predicting where the planets are arises because you want to know where a planet is at a given time. The Schrodinger equation is the
answer to the different question, how can I describe the motion of these objects independantly of time.
The planets do affect one another, and I bilieve 2 planets and one major planetoid were discovered because of variations in the others orbits. Taking
these into account together with all the moons is a huge computational task, and one in which any errors tend to accumulate. I understand a good
aproach for this sort of problem is the 'Fast Multipole Method' .
What makes the Schrodinger equation more down to earth for me is that it was neerly discovered a century before by a mathematician called Hamilton.
He knew nothing about the structure of atoms of course, he was dealing with the motion of objects in gravity, but the basic method if carried through
ends up virtually identical to Heisenburgs 'Matrix mechanics', developed independly of Schrodinger. Matrix mechanics and Schrodingers wave
mechanics were later shown to be identical.
The whole esoteric thing of wave partical duality didnt happen until wave mechanics and quantum theory were nailed together as quantum mechanics.
Before that, the value of the 'wave' was just considered to be a measure of the probability of finding the partical at that point at any
instant.
The Schrodinger cat experiment, now commonly used to explain and confuse students in equal measure, was actually intended as a refutation.
Schrodinger, by forcing the same atomic state on the cat was saying, if you accept the new intepretation of the wave as *real* for atoms, you also
have to accept it for cats being alive and dead at the same time (instead of meerly having a probability they are alive or dead) and this is clearly
absurd.
There is no dividing line between classical mechanics and the Shrodinger equation, though it is a radically different way of going about it. A lot of
books do like to foster the myth that it was created by him out of nothing.
chemleo, I am utterly rubbish at every other aspect of physical chemistry. This is what stopped me failing phys chem.
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Geomancer
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IIRC, the newtonian three body problem was recently "solved", that is, an infinite series was found to express position as a function of
time. I think it converged really slowly or something, so is of theoretical interest only. Interestingly, there was a prize for solving this problem.
Poincare won it, although he never actually found a solution as above. He did, though, lay the groundwork for chaos theory.
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shivgan3
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good work you are great really great service to mankind
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unionised
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"Classical mechanics does not differ from the Schrodinger equation in essence"
Except that one expressly talks about wave behaviour and the other doesn't.
In classical mechanics it makes sense to talk about the location of an object - like the earth- in quantum mechanics it doesn't make sense to do this.
What, in classical mecahnics, is responsible for the quantisation of the energy of a "particle in an infinitely deep box". I chose this because it's
one of the few cases where an exact answer to Schrodinger's equation can be found.
There really is a difference between classical mechanics and Schrodinger's work; to be honest I wonder how you did get through physical chem.
I think the 3 body problem has been proved to be (at least in some instances) chaotic.
In other instances (for example 3 identical planets in an equilateral triangle, all orbiting their combined centre of gravity) it can be solved
exactly.
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