# Tag Info

54

That's a good, concise statement of Bent's rule. Of course we could have just as correctly said that p character tends to concentrate in orbitals directed at electronegative elements. We'll use this latter phrasing when we examine methyl fluoride below. But first, let's expand on the definition a bit so that it is clear to all. Bent's rule speaks to the ...

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Here's a graphic I use to explain the difference in my general chemistry courses: All electrons that have the same value for $n$ (the principle quantum number) are in the same shell Within a shell (same $n$), all electrons that share the same $l$ (the angular momentum quantum number, or orbital shape) are in the same sub-shell When electrons share the same $... 36 The inert pair effect describes the preference of late p-block elements (elements of the 3rd to 6th main group, starting from the 4th period but getting really important for elements from the 6th period onward) to form ions whose oxidation state is 2 less than the group valency. So much for the phenomenological part. But what's the reason for this ... 35 I have searched and searched, oh how I have searched. Do you know what I always tell my mom when she asks me to find something in the Internet she was not able to find herself? I ask her: "Are you sure that the thing you are looking for even exists?" I am looking for a 3 dimensional visualization of a whole (moderately complex, hydrogen is just a ball) ... 35 The meaning of covalent bonds being directional is that atoms bonded covalently prefer specific orientations in space relative to one another. As a result, molecules in which atoms are bonded covalently have definite shapes. The reason for this directionality is that covalent bonds are formed by sharing electrons between atoms, or, in other words, as you ... 28 The accepted answer has nice pictures, but has a couple of small factual inaccuracies in the last paragraph. Here's a bit more maths to illustrate what's going on. Atomic orbitals, which are one-electron wavefunctions, are split into two components: the radial and angular wavefunctions $$\psi_{nlm}(r,\theta,\phi) = R_{nl}(r)Y_{lm}(\theta,\phi)$$ so-called ... 28 Unfortunately, the sense in which orbitals are orthogonal is more or less impossible to define rigorously without recourse to functions of some kind. So, I'll give an explanation a shot using some simple, 1-D functions to illustrate the concept, followed by the pictorial orbital example you've asked for. At a basic level, in order to have any two functions ... 27 General case There is indeed a mathematical theorem that deals with the number of nodes an eigenfunction corresponding to a certain eigenvalue can possess. It was laid down by Courant$^{[1, 2]}$and it states the following: Given the self-adjoint second order (partial) differential equation \left(\hat{L} + \lambda \rho(\mathbf{x}) \... 27 If you have$n$functions (e.g. AOs) you can make a maximum of$n$new linearly independent functions (e.g. MOs). If you try to make$n+1$MOs, then any one of them can be expressed as a linear combination of the other$n$MOs. The most usual way to enforce linear independence is to enforce orthogonality, i.e. all your MOs have to have zero net overlap ... 26 When combined at a given atomic center, any two atomic orbitals add in a vectorial manner. For example, consider the orbital$\phi$defined by$\ce{p_{x}}$and$\ce{p_{y}}atomic orbitals as \begin{align} \phi = c_1 \ce{p_{x}} + c_2 \ce{p_{y}} \end{align} The orbital addition can be pictured like this for the two casesc_1 = c_2 > 0$and$c_l = -c_2 ...

26

Omitting j when alphabetically enumerating things has a long tradition. First of all, the alphabet did not always exist in the form we know it today. Quoting Wikipedia: After [...] the 1st century BC, Latin adopted the Greek letters ⟨Y⟩ and ⟨Z⟩ [...] Thus it was during the classical Latin period that the Latin alphabet contained 23 letters: [no J, V, W] [.....

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The answer is... it is not so simple. Some quantum mechanics follow, but the TL;DR version is that while $m_l=0$ corresponds to $p_z$, the orbitals for $m_l=+1$ and $m_l=-1$ lie in the $xy$-plane, but not on the axes. The reason for this outcome is that the wavefunctions are usually formulated in spherical coordinates to make the maths easier, but graphs in ...

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There is a bit of a terminology problem in the field that makes things very confusing and I will try to clarify some of this here. Part of the problem arises from the fact that sometimes only one kind of density matrix is relevant and so people drop the adjectives when talking about specific approximations. The Density Matrix We're talking electronic ...

18

Unfortunately, it only gets more complicated the deeper you dig. There is some explanation here: What exactly is an orbital?, but you should bear in mind that electronic structure theory is something that the average undergraduate student only barely touches. A strong background in QM is IMO mandatory to understand what some of these things mean. I'll see ...

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