Understanding Atom Orbital Calculations: Wave Equations, Calculus, and Matrices

Click For Summary

Discussion Overview

The discussion revolves around the calculations of atomic orbitals, focusing on the mathematical frameworks such as wave equations, calculus, and matrices used in quantum mechanics. Participants explore the nature of atomic orbitals, the Schrödinger equation, and the implications of these calculations in understanding atomic structure.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the certainty of calculated atomic configurations and inquires about the mathematical tools used in these calculations.
  • Another participant explains that quantum mechanics describes atomic systems through wavefunctions, which are solutions to the Schrödinger equation, requiring multivariable calculus and knowledge of partial differential equations.
  • A later reply emphasizes that while quantum theory is a well-established framework, absolute certainty in scientific knowledge is elusive, citing historical shifts in understanding due to quantum mechanics.
  • One participant provides a resource for equations describing hydrogen orbitals, noting that while these solutions may not perfectly describe nature, they align with experimental predictions.
  • Another participant clarifies that the s, p, d orbitals are solutions for one-electron systems, while multi-electron systems lack exact solutions but can be approximated using linear combinations of hydrogenic orbitals (LCAO).
  • One participant raises a question about the electron shell structure, expressing curiosity about the observed patterns in maximum electron capacity per shell.
  • A participant discusses the radial and angular components of the Schrödinger equation solutions, mentioning spherical harmonics and their relation to atomic orbitals.

Areas of Agreement / Disagreement

Participants express a range of views on the certainty of atomic orbital calculations, the applicability of quantum mechanics, and the nature of electron shell configurations. There is no consensus on the exact nature of atomic orbitals in multi-electron systems, and the discussion remains open-ended regarding the electron shell capacity patterns.

Contextual Notes

Limitations include the dependence on the definitions of atomic orbitals and the unresolved complexities in multi-electron systems. The discussion also highlights the historical context of scientific understanding in quantum mechanics.

Quantom
Messages
25
Reaction score
0
I know they must use some sort of wave equation to calculate the exact orbitals of an atom, but I'm interested how they can be so absolutely sure that their calculated configurations are not wrong, what do they use calculus...matrices...etc?
 
Physics news on Phys.org


They use the tools of... quantum mechanics.

Quantum mechanics states that all systems (an atom being a simple system with protons and neutrons surrounded by a collection of electrons) can be described by a mathematical function (the wavefunction). All wavefunctions must be solutions to a partial differential eigenfunction equation: the Schrödinger equation. Different "orbitals" are, for the most part, different solutions to this equation.

As for the math required to solve it... You'll need multivariable calculus to set up Schrödinger's equation in the coordinate system that it is usually solved in, and you'll need a knowledge of partial differential equations. Actually, most undergraduate quantum chemistry courses don't solve the hydrogen atom from start o finish completely =/ It's pretty involved...

As for your question: "I'm interested how they can be so absolutely sure that their calculated configurations are not wrong"
Well, we can't be absolutely sure of anything. Scientists in the 1900's thought that they knew absolutely everything that there is to know about nature, but the discovery of quantum mechanics proved them wrong. Keep in mind that quantum theory is just that... a theory.

However, current experiments, especially in spectroscopy, establish the possible reality of atomic orbitals beyond a reasonable doubt.
 


Physicist always like to see chemist who are interested in learning some underlying physics on the side!

Here is a basic page with the equations that describe various hydrogen orbitals:

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydwf.html

These solutions are derived from the laws of quantum mechanics. These might not be an exact description of nature, but they make the correct predictions for all the experiments that have been done so far.
 


It should be noted that the s,p,d... etc. orbitals you refer to are solutions to the one-electron Schroedinger eqn. only. For atoms and molecules with more electrons, multi-electron systems, no exact closed solutions are known. Nonetheless, multi-electron systems can be approximated using linear combinations of hydrogenic orbitals (LCAO). Moreover, many properties of atomic and molecular systems can be explained, if not exactly predicted, using either AO's or LCAO's. Some of the strongest evidence of the fundamental role of atomic orbitals may be found in VSEPR theory. This qualitative theory predicts the structure of molecules of all kinds by considering repulsions between electron pairs occupying valence shell orbitals and hybrids. (This theory is normally described in any modern first year college chemisty text.) So, the existence of atomic orbitals in multi-electron systems has proven quite useful for a wide range of predictions, even if their existence hasn't been mathematically proved.

Regards,
Jim Ritchie
 


why do the shells of atoms contain at max 2,8,8 electrons etc? seems a strange pattern
it would seem more logical to be something like 2,8,18 ... or something that increased as n squared or even n cubed??
 


Quantom: the solutions to the Schrödinger Equation for the Coulomb potential have a radial component and an angular component (the composite wavefunction is found by just multiplying both solutions). The angular solutions to the Schrödinger Equation are called spherical harmonics. The spherical harmonics are a family of functions defined by the quantum numbers l and m. You can look them up online, and if you have Mathematica, you can use it to plot them. Try plotting the absolute value squared for a couple of spherical harmonics and you'll see that the look exactly like the atomic orbitals.
 

Similar threads

  • · Replies 11 ·
Replies
11
Views
3K
  • · Replies 9 ·
Replies
9
Views
3K
  • · Replies 36 ·
2
Replies
36
Views
9K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 16 ·
Replies
16
Views
3K
  • · Replies 4 ·
Replies
4
Views
1K
  • · Replies 22 ·
Replies
22
Views
3K
  • · Replies 6 ·
Replies
6
Views
2K
  • · Replies 38 ·
2
Replies
38
Views
7K
  • · Replies 1 ·
Replies
1
Views
2K