From: rparson@spotNO.SPAMColorado.edu (Robert Parson)
Newsgroups: sci.chem
Subject: Re: Various theories of QM representation of an atom/molecule/etc.
Date: 18 Apr 1998 16:46:51 GMT

In article <35354d92.0@oasis.idirect.com>,
Michael Milirud <mmichael@idirect.com> wrote:
>In many replies I've seen quite a few people mentioning some different
>theories concerning the atoms and molecules in the Quantum Mechanics. Why
>are all of those theories used? Isn't there only one correct and others
>false? Why to choose among several (also means that you have to know more
>than one)?

 There is one correct quantum mechanics, the exact solution of the
 Schrodinger equation (neglecting relativistic corrections, see below).
 However this is too difficult to do for all but the smallest molecules.
 So you need to make approximations. Different approximations are
 useful for different purposes, and if you are not careful you can
 get complete nonsense. For example, the Hartree Fock approximation
 is often quite good at predicting molecular structures, but it
 completely useless for describing molecular dissociation. To apply
 quantum mechanics appropriately in a chemical context, you need to
 understand chemistry as well as math and physics.

>P.S. I've also read that Quantum Mechanics is not good at all in describing
>the situation of atoms with high Z. While exact for light elements, it is
>very wrong for heavy elements. The anomaly is known as "Lamb Shift" and that
>a new theory - Quantum ElectroDynamics has been developed to compensate.
>Just how much more copmlex is that QED and why the QM is still being used if

 There are two different things here - ordinary relativistic corrections
 and radiative (QED) corrections. Ordinary relativistic corrections
 are important when the electrons are moving at speeds that approach that
 of light, which happens regularly in the heavy atoms. They give rise
 to such effects as contraction of the core electrons and spin-orbit
 coupling. The methods for handling these effects are well developed
 and quantum chemistry calculations include them when necessary.

 Radiative corrections are more subtle. They arise from the fact that
 the interactions between the electrons and protons are not really
 static coulomb forces, but are mediated by the electromagnetic field,
 so that you have to apply quantum mechanics to the field as well as
 to the particles. The Lamb shift is, as you say, an example. Quick
 description: in nonrelativistic quantum mechanics, the 2s and 2p
 orbitals of hydrogen have exactly the same energy. Ordinary
 relativistic corrections (spin-orbit coupling) splits the 2p orbitals
 into two sets, called 2p(1/2) and 2p(3/2); the 2p(1/2) orbitals
 have the same energy as the 2s, while 2p(3/2) is higher by about
 11 GHz, or 4.5x10^{-5} eV. Radiative (QED) effects remove the
 remaining degeneracy between 2s and 2p(1/2), producing a splitting
 of about 1 GHz, or 4x10^{-6} eV - the famous "Lamb Shift". Since
 chemical bond energies are of order 1-5 eV, these effects are tiny
 for most chemical purposes. They do get bigger with increasing
 nuclear charge, and the ordinary relativistic corrections are
 very important in the 4th and 5th row elements. QED effects
 won't get big until we've added a couple of more rows: the
 critical parameter is the fine structure constant ~1/137, and
 if anybody starts making elements with Z>137 you can expect to
 see some really weird shit like spontaneous production of
 electron-positron pairs in the neighborhood of the nucleus.

 ------
 Robert

From: rparson@spotNO.SPAMColorado.edu (Robert Parson)
Newsgroups: sci.chem
Subject: Re: Various theories of QM representation of an atom/molecule/etc.
Date: 20 Apr 1998 19:35:13 GMT

In article <3538E022.5AE2@ix.netcom.com>,
Uncle Al  <UncleAl0@ix.netcom.com> wrote:
>Robert Parson wrote:
>>
>
>[snip]
>
>>  There is one correct quantum mechanics, the exact solution of the
>>  Schrodinger equation (neglecting relativistic corrections, see below).
>
>Whoa!  You cannot have it both ways.  If it is correct, why does it need
>corrections?

 All theories are wrong, some theories are useful. Newtonian mechanics
 is the correct theory for the solar system unless you are interested
 in the precession of the perihelion of Mercury.  Relativistic
 corrections are well-defined, their size can be estimated ahead
 of time, so you can decide whether or not you need them, and you
 know what to do when you _do_ need them. This contrasts with
 the approximations made in solving the Schroedinger equation itself,
 which are often ad-hoc.

>The Schroedinger equation (correct or corrected) gives a nice closed
>solution to the hydrogen atom.  Would you vouchsafe with the same giddy
>confidence for, say, a calcium atom or a unit cell of YBCO?  You might
>get an (arbitarily) "accurate" solution, but it would be meaningless in
>physical terms of structure vs function.

 It certainly can be, and usually is. Far too many theoretical chemists
 are content to grind out numbers with little interest in interpreting
 them in chemical terms. Sturgeon's law applies here as everywhere.
 But it doesn't have to be that way - take a look at the work of Bill
 Goddard and his students on Generalized Valence Bond theory.

 There *is* a real problem here, though: the approximation methods
 that are most convenient for computation are, unfortunately, not the
 ones that best lend themselves to physical understanding. This goes
 right back to the simplest approximations: Valence Bond Theory gives
 a much more sensible description of a chemical bond than Molecular
 Orbital Theory, but it's easier to add whistles and bells to MO.
 Slater-Type Orbitals give a better description of where electrons
 are in atoms than Gaussians, but gaussians make for much nicer integrals.
 So a lot of quantum chemistry amounts to starting with a rather
 poor description of the atoms, let along the molecules, and then
 compensating for it with raw computational power.

>Accuracy in QM is paid for with understanding.  When you finally get a
>good answer it is meaningless.  That hints that the philosophical model
>is fundamentally flawed.

 Accuracy in celestial mechanics is paid for with understanding - take
 a look at Delaunay's analysis of the orbit of the moon, carried out
 in the mid-19th century - perturbation theory carried to 8th order,
 over 500 separate canonical transformations. Or calculations of the
 orbits of asteroids, or the constituents of the rings of Saturn.
 Actually, getting chemistry out the Schrodinger equation is quite a lot
 like trying to understand Saturn's rings using Newton's laws applied
 to individual rocks. (In fact, many of the techniques used to solve the
 Schroedinger equation, such as perturbation theory, were borrowed from
 celestial mechanics).

 Probably every reasonably intelligent theoretical chemist has thought
 at one time or another "There has got to be a better way!" And quite
 a few have tried, but there's lots of dead bodies all over that field.
 One that seems to have survived is Density Functional Theory, but
 only by mimicking its rival to the point that it's gotten just as
 acronym-ridden as conventional CI methods.

>Indeed, the calculus with its dependence upon
>a function's delta-epsilon decomposability may be fundamentally improper
>to analyze recursive (positive feedback, formally chaotic) systems.
>That is why weather and economics are so dismal.  Alas, we don't have
>anything to replace differential equations.

 Uncle Al, Put Down thy _Mandelbrot_, Pick Up thy _Poincare_.
 Differential equations are the natural language of chaotic systems.
 What you need is a mathematical framework for qualitative analysis
 of systems of differential equations. That's what "chaos theory"
 does for small systems of ordinary differential equations - allows
 you to understand whole sheaves of solutions at once. Would that
 a similar theory existed for the PDES of QM. Lots of dead bodies
 on that field, too.

 ------
 Robert