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In molecular orbital theory bonding of two atoms can be seen as addition or subtraction of the wave functions of these atoms. Addition seems pretty logical and straight-forward - atoms overlap their electron clouds and probability to find an electron in it increases. Subtraction is the thing I don't get. How can the probability to find an electron be negative? Am I missing something?

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I think you are confusing the wavefunction $\Psi$ with the probability density function $\Psi^2$. The former is a state function describing the behavior of an electron. The latter describes the "probability of finding an electron".

Wavefunctions are usually complex functions, but even the simplest sine wave functions can be negative in the algebraic sense: the value of $\Psi=f(x)<0$. These regions where $Psi=f(x)$ has different algebraic signs are the regions corresponding to the orbital lobes of different phase, like in a p-orbital. However, $\Psi^2$ is always $\Psi^2 =f^2(x)\ge 0$. Information about the "location" of the electron is gained, information about its phase is lost.

Take as an example the two wave functions representing two atomic orbitals $AO_1$ and $AO_2$ on the graph below. $\Psi_{AO_1}=\cos{(\frac{\pi x}{2})}$. $\Psi_{AO_2}=\cos{(\frac{3\pi x}{2})}$. $\Psi_{AO_2}$ has regions where $\Psi_{AO_2}<0$ when $\frac{1}{3}<|x|<1$.

a graph of two sine-wave functions showing different phase

Now let's look at the probability density functions $\Psi^2_{AO_1}=\cos^2 {(\frac{\pi x}{2})}$ and $\Psi^2_{AO_2}=\cos^2 {(\frac{3\pi x}{2})}$. Notice that both functions are always $\ge0$.

Two probability density functions showing that they are never negative

Okay... now, when atomic orbitals combine to form molecular orbitals, one of those conservation laws (angular momentum) gets involved, and the number of orbitals has to remain constant. Thus, molecular orbitals are linear combinations of atomic orbitals. For example, if two atomic orbitals combine, two molecular orbitals are produced.

Let's go back to our two sample sine wavefunctions above and pretend they are atomic orbitals overlapping. One of the molecular orbitals could then be: $\Psi_{MO_1}=\frac{1}{2}\Psi_{AO_1}+\frac{1}{2}\Psi_{AO_2}$. This is the orbital representing constructive overlap between these two wavefunctions. To get the other orbital, we can't just add again, or we would get the same orbital. If we subtract, we get a different orbital - the one representing destructive overlap. The graphs of these two "molecular orbitals" and their probability density functions are below.

A graph of two molecular orbitals showing constructive and destructive overlap

A graph of two molecular orbital density functions

Again note the difference between the MO wavefunctions and their density functions. If these were atomic nuclei at $x=\frac{1}{3}$ and $x=-\frac{1}{3}$, then $\Psi_{MO_1}$ would be bonding and $\Psi_{MO2}$ would be antibonding (or perhaps nonbonding).

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    $\begingroup$ Teaching primarily general and organic chemistry, I spend a lot of time thinking about how to explain the fundamentals concepts of quantum mechanical bonding theories without calculus. The analogy to simpler types of waves useful because it is intuitive. The maths of quantum mechanics are more complex, but at the end of the day, QM waves behave like other types of waves. $\endgroup$ – Ben Norris Jun 11 '15 at 11:49
  • $\begingroup$ Yep, I am confusing those two. Now I am not sure what Ψ actually means. And the addition and substraction of Ψs. In a normal wave it would be the position of particles in a point in time affected by one or two waves, and in Ψ? $\endgroup$ – waterlemon Jun 11 '15 at 20:27
  • $\begingroup$ @user3116958, $\Psi$ is a mathematical object used to describe the state of a system. As we usually say, it does not have any physical meaning. The meaning of this is that it does not correspond to any physical quantity that can be measured even in principle. $\endgroup$ – Wildcat Jun 11 '15 at 22:01
  • $\begingroup$ @user3116958, the thing (which I feel from the conversation you do not quite understand yet) is that quantum waves are not classical waves, or, in other words, they are not physical waves like sound or electromagnetic waves from the classical theory. To realise that just try to answer the question, what is waving in the case of quantum waves? In classical theory wave are periodic oscillations of some physical quantity, but what oscillates for a quantum wave? $\endgroup$ – Wildcat Jun 11 '15 at 22:07
  • $\begingroup$ @Wildcat, why did chemists then decide to add (or substract) the two? $\endgroup$ – waterlemon Jun 13 '15 at 17:15

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