Experimental determination of ground state configuration

[trini]

Ok so I've been wondering, how is it that physicist's EXPERIMENTALLY determine the ground state configuration of electrons in a particular atom? In other words do they use emission spectra/thermodynamic calculations/etc?

Also, if ground state represents an atom's minimal energy state, which technically can only occur at zero kelvin as any increase in temperature above 0K results in vibrational energy being supplied to the atom from the environment and thus is NOT the minimal energy of the atom(may be close but there is certainly a difference), does this mean that these experiments involve cooling the atom in question to near 0K temperatures for analysis?

 

[alxm]

Emission spectra is enough, really. Although I don't know how 'experimental' you want to be; i.e. you need some amount of theory just to know what the different states are, and it doesn't take much more theory than that to say quite a bit about how they'll be ordered.

A single atom can't vibrate though, not any more than one hand can clap. Electronically, atoms are almost entirely in their ground state at room temperature. But in either case, there's no reason why the atom or molecule would have to be in its ground state for you to determine its ground state configuration. The levels are still there.

 

[trini]

the reason i ask is because strict theory fails above about Z=18(when 4s fills before 3d) and experimental results are used to create modified expressions, meaning that most so called ab-initio calculations of energy states are really reverse engineered answers, usually requiring modifications of the wave function to achieve desirable results. As a chemical process engineer, i find this strikingly similar to use of 'corrective' functions in thermodynamics such as the peng-robinson equation, functions which have shallow theoretical background, but are used just because they give suitable approximations. Since this is the case, how can quantum mechanics claim to be useful in predictions when there are so many cases when post-dictions are made?

To me, from an engineer's point of view, all quantum mechanics is is a framework, not a theory in the true sense of the word, in that using the schroedinger wave equation along with various other 'corrective' techniques which are determined AFTER empirical evidence is gathered WILL result in the answers we are seeking, but in so doing, we leave the concept of ab initio behind and enter a realm where abstractness is king.

now this brings me back to my orginal question, if the energy states are determined by a theory which itself depends on empirical evidence, and if the theory by itself fails to predict the correct configurations, how do we know that when the theory says a certain state is being occupied, that that is in fact the state?

 

[kanato]

What do you mean by "strict theory?" Are you referring to a theory which neglects interactions between electrons? Density functional theory correctly fills the 4s orbital before 3d for an isolated K atom (Z=19). The theory has its problems, but it's usually well known where they are and what extensions of the theory might work.

The true many body wavefunction is never calculated, except in very specific cases like H₂ or He. It's simply far too complicated to do even numerically. So some approximations are needed, and these are usually of the uncontrolled variety (cannot be systematically improved upon).

 

[alxm]

the reason i ask is because strict theory fails above about Z=18(when 4s fills before 3d) and experimental results are used to create modified expressions, meaning that most so called ab-initio calculations of energy states are really reverse engineered answers, usually requiring modifications of the wave function to achieve desirable results.

Every single thing you said here is simply false. Except that 4s does fill before 3d.

 

[alxm]

The true many body wavefunction is never calculated, except in very specific cases like H₂ or He. It's simply far too complicated to do even numerically. So some approximations are needed, and these are usually of the uncontrolled variety (cannot be systematically improved upon).

I don't know what you're talking about either. What about Møller-Plesset perturbation theory, coupled-cluster, configuration interaction, CASSCF?

 

[kanato]

Anything that is Hartree-Fock like or uses Slater determinants assumes that the wavefunction is separable: formally incorrect but usually a good approximation. Approximation via a separable wavefunction is uncontrolled as there is no systematic way to go from a separable wavefunction to a non-separable one. All those methods make some sort of expansion of the wavefunction in Slater determinants.

 

[alxm]

Anything that is Hartree-Fock like or uses Slater determinants assumes that the wavefunction is separable: formally incorrect but usually a good approximation. Approximation via a separable wavefunction is uncontrolled as there is no systematic way to go from a separable wavefunction to a non-separable one.

This isn't true at all. You've misunderstood the basics here. First off, you're not assuming the wave function is separable, you're (with Hartree-Fock) assuming a mean-field. I.e. that that the interdependence - which is still there- takes a certain form.
You are also (with HF) restricting yourself to a single determinant description.

More importantly: The Slater determinants for a system form a complete set.
You can exactly represent any system with enough determinants. Full-CI is exact.

Obviously this is systematically improvable: By including more determinants (in CI and CC), and higher orders or perturbation (in MP).

Also, I forgot to mention QMC methods which are very exact, and don't use slater determinants at all.

 

[kanato]

Hartree-Fock has a number of approximations, one of which is a separable wavefunction. Separability means we assume \psi(x_1,x_2) \approx \phi_a(x_1)\phi_b(x_2), and then we properly symmetrize it via a Slater determinant. I could see that the Slater determinants form a complete set of separable wavefunctions, and you could represent any separable wavefunction with enough Slater determinants. But there are trivial examples of functions which cannot be systematically represented by a separable expansion of differentiable functions.

I'm more familiar with QMC Green's function methods than wavefunction methods. But AFAIK QMC wavefunction methods require additional approximations like the fixed node approximation to be computationally viable.

If you like, I can revise my earlier statement to read "the true many body wavefunction is never calculated, except in specific cases with less than a dozen or so electrons."

 

[alxm]

Hartree-Fock has a number of approximations, one of which is a separable wavefunction. Separability means we assume \psi(x_1,x_2) \approx \phi_a(x_1)\phi_b(x_2)

Maybe I didn't make myself clear enough:
Expressing the wave function as Slater determinants; as a product of single-particle wavefunctions is not an approximation.
Let this sink in. If you don't believe me, point to a textbook that says otherwise.

It's your Hamiltonian which is inseparable, you end up with cross-terms between your single-particle functions. If, and ONLY if, you ignore those cross-terms are you making an approximation. Hartree-Fock only uses a single determinant, which in effect approximates the interdependence as a mean-field. Higher-order interdependencies come in via the excited determinants.

I could see that the Slater determinants form a complete set of separable wavefunctions, and you could represent any separable wavefunction with enough Slater determinants.

So you're saying that http://en.wikipedia.org/wiki/Full_configuration_interaction" is not exact, as opposed to what everyone else thinks?

 

[trini]

the n+l rule does not fall out of the wave equation, and i apologise, i meant to refer to copper's odd configuration rather than madelung's rule. also, my argument here is that no theory in quantum mechanics has a general solution for all the elements of the periodic table, we generally find exceptions and this forces us to modify expressions. A truly ab-initio derivation would be able to account for all elements, not just some, from the theory's first principles. for example i have read here that to account for some element's energy states that's 14s and 9d(for example when solving Ni) must be introduced:

http://www.chem.ucla.edu/dept/Faculty/scerri/pdf/Howabinitio.pdf

This is flawed, as there is no evidence of energy levels higher than 8(hydrogen is the best determination of how many levels there are, 7 emission lines for 8 levels) What i am saying is that if you say ok, go ahead and use whatever numbers work to get the ordering of shell filling right, and then involve use of things either not supported by known facts(example n>8) or not related to first principles, how can u say that u are being true to your theory?

 

[alxm]

the n+l rule does not fall out of the wave equation

This is wrong, Madelung's rule can, and has been proved from 'first principles'.

no theory in quantum mechanics has a general solution for all the elements of the periodic table

There is only one theory of quantum mechanics. And it does have a general solution for all the elements in the periodic table, namely from solving the wave equation.

You imply that unless the results of these solutions can be generalized into a simple 'rule' that holds in all cases, quantum mechanics is flawed or empirical. That's not true.
More importantly, there's no reason to even assume that would be the case. The Schrödinger equation for a many-electron system is a many body problem, a nonlinear differential equation.

A truly ab-initio derivation would be able to account for all elements, not just some, from the theory's first principles.

And they do.

for example i have read here that to account for some element's energy states that's 14s and 9d(for example when solving Ni) must be introduced

That's not what it says. It says that highly accurate Ni calculations have made use of 14s, 9d, 5p and f orbitals. Scerri appears confused about basis sets, He and you are conflating two different things here - Basis functions are not orbitals. Orbitals are described by basis functions, which are arbitrary mathematical functions. You can use any (well-behaved) mathematical function for your basis set. All kinds of functions from plane waves to splines are used. Which doesn't mean every function is equally good or practical. Ultimately you want something that resembles your solution so that you can use as few functions as possible, and have the result converge as quickly as possible.

The solutions to the hydrogenic wave equation form a complete set (a mathematical concept I already alluded to in this thread), which means they can be used to describe any function. If you use these functions to describe your wave function, it's called a Slater-Type Orbital basis set. In practice, STOs are not used any more because they're mathematically difficult to integrate (as I mentioned in https://www.physicsforums.com/showthread.php?t=409019" recently). For that reason, one uses gaussian functions instead, in sets of gaussians that approximate STOs (e.g. STO-3G is a basis set of STOs where each STO is approximated by 3 Gaussians) Any introductory QC textbook will tell you all this.

The calculations in question do not include "14s orbitals". They include a STO-type (approximated by Gaussians) basis function which takes the same general form as the hydrogenic 14s function. This basis function is not in itself an eigenfunction of the Fock operator and therefore not an orbital of your system. It does not imply anything whatsoever about the orbital configuration that it's being used to approximate.

A direct mathematical analogy is that if you use a http://en.wikipedia.org/wiki/File:Fourier_Series.svg" , doing so accurately means including sine functions that have much higher frequencies than the square wave itself. That does not imply anything about the square wave's frequency.

What i am saying is that if you say ok, go ahead and use whatever numbers work to get the ordering of shell filling right

Have you ever even done a quantum-chemical calculation? The only numbers used to specify the state of a system is the number of electrons and the total spin.

 

[alxm]

Also, let me just say that if you want to learn quantum mechanics and quantum chemistry, then by all means get a textbook and do so! But you're not going to learn it from reading Scerri's paper(s) and making your own (false) assumptions about how it all works.

Scerri is a philosopher of science, not a quantum chemist. He states that this makes him 'impartial' and accuses quantum chemists and physicists of overstating the success of quantum theory. I would say the opposite: Nobody is more acutely aware of where (approximate!) quantum-chemical methods fail, and why and how, as quantum chemists are. Knowing that is absolutely critical to being able to use and improve the methods.

In any case, his papers are not a way to learn QC. To begin with there are a number of simple factual errors in the paper your referenced. (I already mentioned one) Suffice to say that many of his papers, including the one you linked to, have been criticized in publications. His opinion that chemistry cannot be reduced to quantum mechanics is not mainstream opinion (an otherwise very positive review of one of his books in Angewandte simply called that claim of his 'absurd'). He's correct in one sense though: Many chemical concepts cannot be derived from quantum mechanics. But not due to a failure of quantum mechanics, but rather due to the fact that many chemical concepts are vaguely defined. (E.g. 'Explain the periodic table', well - Which periodic table? There are loads of them - organized to many different and often vague criteria)

But at least Scerri knows what he's talking about most of the time, and the factual errors can be attributed to some simple misunderstandings. But what you're doing here, is repeating a distorted version of them while throwing in some additional charges which are so wrong, I can only conclude you simply made them up. Because I cannot see how anyone could read the QM and QC textbooks and draw the conclusion that 'ab initio' methods were being fitted to experimental results.

http://www.bioinfo.ufrj.br/biomod/calcHF.f" a basic Hartree-Fock program (from Szabo and Ostlund's textbook). Note there are no experimental values in there, nor were any used in deriving the method. All you have are basic constants and basis set parameters. I'll leave it as an exercise for the reader to modify the program to optimize the basis functions and re-calculate these parameters. (Not very difficult)

 

[trini]

Well as i said, I'm an engineer, and have never taken official courses in quantum, i am learning from textbooks i have downloaded and sometime's it is difficult to visualize what a textbook is saying when nobody is there to explain. So let me apologise for making statements which i did not fully understand, i am here to learn.

let me see if i get what you are saying.

from what i understand, the procedure for determining the energy of an electron in a particular energy state is done by expressing the electrostatic potential of an initial arrangement of electrons around a nucleus as a function of position of each of the electrons(within the shells and orbitals), and then using the wavefunction to describe how the positions vary over time(within a range such that the total probability of the electron's presence will be 1), and thus determine which arrangement has the lowest energy over a time interval.

the wavefunction itself is approximated using time dependant functions which mimic the behaviour of the electron's positions as a function of time, and also takes into account the total system spin. The form of the wavefunction can vary as you alluded to(splines, plane waves etc) according to the preference of the person doing the calculation to achieve desired results.

Thus quantum mechanics takes classical laws and applies them to electrons and then also factors in the possible arrangements of the electrons within a range that the electrons are statistically more likely to be found to determine all the enrgy states it may achieve.

This is the understanding i have of the topic so far, please feel free to point out what i may be missing.

 

[alxm]

from what i understand, the procedure for determining the energy of an electron in a particular energy state is done by expressing the electrostatic potential of an initial arrangement of electrons around a nucleus as a function of position of each of the electrons(within the shells and orbitals)

No see, electrons don't have a definite position. The Hamiltonian operator for the electrons, which defines the Schrödinger Equation (SE) for the electrons has a coordinate for each electron, but it is not the location of the electron, rather it's a coordinate in the wavefunction's 'configuration space'. For instance, for a single particle wave function |\psi(x)|^2 is the probability that the particle is located at the coordinate x. For two particle wave function, |\psi(x_1,x_2)|^2 is the probability particle 1 is at location x₁ and that particle 2 is at location x₂. Note that for identical particles |\psi(x_1,x_2)|^2 = |\psi(x_2,x_1)|^2, which is a boundary condition your solution to the SE must statisfy. (and even more specifically, for electrons and other fermions, you must satisfy antisymmetry, which means \psi(x_1,x_2) = -\psi(x_2,x_1). This is a consequence of the spin-statistics theorem and ulitmately, relativistic theory, but at the level of non-relativistic QM you can just regard it as a postulate)

Another important thing is the variational theorem: Any normalized function, if you insert it into the SE for a system and calculate the energy, will result in an energy that is higher or equal to the ground-state energy. In other words, if you're approximating the solution to the S.E., the lower the energy the better the approximation. This theorem is a consequence of the solutions to the S.E. forming a complete set. (Descriptive proof: If the solutions form a complete set, any function can be expressed as a linear combination of the solutions, multiplied by some set of constants. Meaning the energy corresponding to that function, if inserted into the S.E., is the sum of the energies of the respective solutions, multiplied by the same constants squared. So the energy for the function cannot be lower than the ground-state, and is equal to the ground-state energy if and only if the function is identical to the ground-state solution)

Now, shells and orbitals; These are derived results from quantum mechanics. They don't actually exist in themselves, they're an interpretation of a particular mathematical description. Typically you start with the solving the SE for the hydrogen atom (which has an analytical solution). You have a bunch of different energy states, corresponding to different linear and angular momenta (n and l eigenvalues). The different states for a single electron are dubbed 'orbitals' and we call the different states with the same n 'shells' and with different l values 'sub-shells'.

Now if electrons didn't interact, you could simply separate the SE into single-electron equations and write your total wavefunction as a product of these single-electron functions. (basic separation-of-variables approach to solving differential equations, which an engineer should know about). Actually you'd have to create an "anti-symmetrized" product, so that you satisfied the condition mentioned above. Such an anti-symmetrized product is called a Slater determinant (SD), and the single-electron functions are orbitals.

But for a many-electron system, the SE is inseparable and you cannot divide it into independent single-electron states and have an accurate solution; since they interact they're not independent. This is what the earlier discussion was about, but I'll get back to that. In any case, let's just assume that this is fine and write your SD of N/2 functions (orbitals). The 1/2 comes in because of spin, which allows you to put two electrons in each orbital (for the sake of simplicity I'm leaving that bit out). If you stick this SD into the electronic SE, and do a lot of math (covered in most QC textbooks), you end up with the Hartree-Fock equations. In this approximation, the electrons aren't totally independent; they just move in an averaged field of every other electron.

But you still have a bunch simultaneous differential equations to solve! The way this is solved is by expressing each orbital as a linear combination of basis functions (again, these are abitrary), and solving it using the Self-Consistent-Field (SCF) approach. You can view it as starting from a guess (for which you usually use the totally non-interacting system, which can be solved directly), then minimizing the energy of an orbital without changing any other orbital, and then minimizing the next orbital and so on, and then starting over, and (with a little luck) this will converge to the correct solution.

The Hartree-Fock/SCF method is generally the simplest method of solving the S.E., and it gives qualitively correct results for a large amount of chemistry (including, say, the potassium ground-state configuration). It gets about 95% or so of the ground-state energy, and works as the starting point for most QC methods. Configuration Interaction, for instance, is essentially just an expanded HF method which has the important characteristic that it's exact, in principle. (the drawback with full-CI is that the amount of calculating needed grows factorially with the number of electrons) With CI you use mutliple SDs (ground state and excited state SDs) to describe the system - the works because the SDs are solutions to the SE for 'non-interacting electrons', and thus form a complete set. While you get an accurate solution, it's no longer true that a single electron is fully described by a single orbital.

Finally, I should say that orbitals are not a necessary part of a quantum-mechanical description of an atom or molecule. It's a convenient way to do so because:
1) Given that HF gives you 95% or so accuracy, it's easier to start with HF and find corrections to it
2) Mathematically describing the kinetic energy is very difficult without assuming electrons move independently, at least as a starting approximation
3) It becomes difficult to satisfy the anti-symmetry condition above. If you ignore it, you end up with another error in your energy (dubbed "exchange energy").

That said, there are methods of calculating the energy that do not use orbitals. The Thomas-Fermi model for instance, and other 'orbital-free' density-functional methods. (Which have problems with 2 and 3)
The first really accurate QM calculation of a many-electron system, Hylleraas calculation of helium (1929), did not use orbitals. By using a clever ansatz and a lot of algebra, he arrived at an equation that could be solved variationally, and got the ground-state energy with 98% accuracy with two parameters, and 99.9% accuracy with six. (IIRC)