Wavefunction Chemistry

[Raina Giorno]

I am aware that the square of the Wavefunction gives the probability density of finding an electron at a particular point in space. I have also heard that it's a complex number but since it's a function I am unsure as to how that could be the case (perhaps someone could please clear that up for me as a sub-question). Moreover, I have seen it described as the amplitude of something but I literally have no idea what is meant by that. However, the crux of the question is what is the Wavefunction itself (i.e not what the square of it is). I have read in the book "Why Chemical Reactions Happen" that it is essentially synonymous with the word/concept: orbital. I have also seen in various places w(x,y,x) or w(r,theta,thi) [where "w" represents the sign for a wavefunction] so, from what I can gather it is a function in three dimensions that represents the shape of a particular orbital where the function of (x,y,z) or (r,theta,thi) would, of course, be different for the s,p,d and f orbitals. However, in the book "Why Chemical Reactions Happen" it goes onto say that wavefunctions of different atoms interfere with each other to form molecular orbitals and the atomic orbitals somehow have a phase?! For my idea of a wavefunction to be correct, the idea of interference of these wavefunctions must be an idea that was thought of to try and rationalize the process of the formation of molecular bonding and anti-bonding orbitals from atomic ones to make it easier to understand and conceptualize. If I am right about the wavefunction being a purely mathematical representation of the space where an electron is likely to reside then the molecular wavefunctions are just as curious in terms of their shape and suchlike as the shapes of the atomic orbitals they came from (i.e nobody really knows why they are that shape but when atoms or molecules are arranged in a particular way, orbitals "just are" that shape). Also the idea of phase seemed to bring my argument down but then I thought that perhaps phase is simply defined as being opposite relative to another part of an orbital if it is the other side of an angular node (I suspect this is wrong but could someone please shed some light on this?)

Going back to when I said - "the molecular wavefunctions are just as curious in terms of their shape and suchlike as the shapes of the atomic orbitals they came from" - I thought this because I found it bizarre that empty orbitals could "interfere" with an orbital from another atom. For example one p orbital is empty yet all three interact with the four hydrogen s orbitals. Unless, the electrons can jump between the the degenerate p orbitals at such a frequency that all can be considered occupied I find it incomprehensible that just a portion of space can interfere with an orbital. As a result, I came to the conclusion that I quoted at the start of the paragraph. Basically meaning that when atoms come together to form molecules the electrons find the lowest possible energy spaces to reside; or in other words, a molecuar orbital / molecular wavefunction forms which is equally as curious as the atomic wavefunctions with their strange shapes.

Any help with this concept would be greatly appreciated - I am off to Oxford university this October and Molecular Orbital Theory is a small part of the suggested reading list before arrival into the first year. Also, on that note, please refrain from heavily mathematical answers because I simply won't understand as I have only done A-level Maths and A-level Chemistry. Thanks.

One problem with books on introductory quantum mechanics is that, put simply, the language of quantum mechanics is math. Specifically, most people use the Schrdinger equation which involves second derivatives and differential equations.

As you mention, the most widely used interpretation (the Copenhagen interpretation) of the wave function centers on the square of the wave function, or rather $\psi^*\psi$ as a probability density. As you mention, the wave function could be imaginary or complex, so this notation indicates a mathematical way of getting a real number for a probability density.

One thing that's amazing about quantum mechanics is that we put in some relatively simple equations to describe the motion of an electron in a hydrogen atom, and the solutions are exactly these atomic orbitals (s, p, d, f, etc.).

So I disagree that "no one knows" why atomic orbitals have particular shapes. The answer is that they are the mathematical solution to the Schrdinger equation. That is, we want to know the energies of the system and there are only certain quantized solutions.

I'll leave you with this. Without the math, it can be exceptionally hard to understand quantum mechanics. When you do understand the math, I think the results are quite beautiful. The shapes of orbitals aren't arbitrary. This is actually just a reflection of the mathematical nature of quantum mechanics.

That is the root of your problems. But do not be afraid, you are not alone! As I pointed out in comment to Geoff Hutchison answer, many chemistry students (and let us be honest here, not only students) are strongly reluctant to the following two simple very well established and no longer questioned facts:

So, the right way to proceed is to learn some required math first (at least calculus), then physics (at least classical and quantum mechanics in the form of wave mechanics), and only then quantum chemistry. Thus, my advice is: do not put the cart before the horse! If you have some free time, learn some math.

Wavefunction is quantum mechanical conterpart to trajectory in classical mechanics. Unfortunately, since we interact with the world in a way exposing to us mostly classical newtonian physics, there is no way to find pleasant, easily understandable everyday analogy. However, you may consider atomic orbitals as building blocks for 'trajectory' of electrons. They don't have to be filled to be used as building blocks.

AFAIK, wikipedia provides basic knowledge on the topics I mentioned. It is OK to have very vague understanding of concepts I mentioned until 3rd year in university. There is no need to have deep understanding of quantum mechanics to successfully apply MO theory.

Ordinary wavefunctions are complex functions defined in a Hilbert Space (HS), corresponding to pure quantum states. There is not a one-to-one correspondence between pure states and wavefunctions. Forinstance if $\Psi$ is a valid wavefunction in the energy basis (withHamiltonian operator $\hatH$), then the wavefunction $\Psi = e^i\phi\Psi$ represents the same state because $$E = \int d^3x \> (\Psi)^* \hatH \Psi = \int d^3x \> \Psi^* \hatH \Psi$$

Wavefunctions are used in the wavefunction formulation of quantummechanics, which is only one of many available formulations. Thereare other formulations of quantum mechanics, where wavefunctions areused for different purposes and no longer represent a quantum state,and there are formulations where wavefunctions are not used at all.You can find a basic review of nine formulations of quantum mechanicsin this article and a new formulation of quantum mechanicswithout wavefunctions in this article.

Wavefunctions do not describe mixed quantum states. Mixed quantumstates are outside the scope of the wavefunction formulation ofquantum mechanics. For instance, an atom in a molecule does not havewavefunction associated to its quantum state, because atoms inmolecules are open systems and quantum correlations with neighboratoms prohibit the description of the properties of the atom withwavefunctions. A heat bath at non-zero temperature does not havewavefunction describing its quantum state either. Mixed quantumstates are properly described by other formulations of quantummechanics.

Ordinary wavefunctions cannot be used to rigorously describe instablequantum states. Ordinary wavefunctions --sometimes named "Dirac Kets"in the abstract representation of wavefunctions-- can be applied tostable particles, but not to decaying particles or resonances.

There are attempts to extend the ordinary wavefunction formulation toinstable systems. Those generalized states are sometimes named "Gamowkets" and are defined in a Rigged Hilbert Space (RHS). You can findthis formalism in this book.

From the above, it looks like if wavefunctions are a kind of analogof the phase space densities $\rho$ used in classical mechanics. Thisis more than a mere formal analogy and we can develop a phase-spaceformulation of quantum mechanics where wavefunctions are replaced byquantum phase-space $\rho$ and operators are replaced by phase-spacefunctions, for instance the energy of a quantum system in thephase-space formulation is computed as

$$ i\hbar \frac\partial \psi_\mathrmclass\partial t = \left[ -\hbar^2\frac\nabla^22m + V + \hbar^2\frac\nabla^2 \sqrt\psi_\mathrmclass^*\psi_\mathrmclass2m \sqrt\psi_\mathrmclass^*\psi_\mathrmclass \right] \psi_\mathrmclass$$

sometimes named the classical Schrdinger equation in the literature.Note that the last term between brackets is absent in quantummechanics. This classical equation is nonlinear and this nonlinearitykills the superposition principle and other aspects associated toquantum mechanics.

I think that @permeakra's last comments are very important, which is that you don't have to have a deep knowledge of quantum mechanics to be able to apply it and also that a lot of the maths detail can be left until your last undergraduate year.

One other point not mentioned above is that any area has its own often obscure language and notation. Quantum is one of the worst in this respect, e.g. basis sets, orbitals, wavefunctions, operators, hamiltonians, eigenvalues, eigenfunctions, bra's, ket's, stationary states and so on. Often textbooks (and lecturers) assume that one is already familiar with these concepts and if you are not (the general case) this jargon just gets in the way when you start to learn.

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