162

Yes, this is a beautiful question. As you said, in lower rows of the periodic table, there are relativistic effects for the electrons. That is, for core electrons in gold, the electrons are traveling at a significant fraction of the speed of light (e.g., ~58% for Au $\ce{1s}$ electrons). This contracts the Bohr radius of the 1s electrons by ~22%. Source: ...


114

See the footnotes I've included if you would like to see more of the detail behind a specific statement. The figure below compares the reflectance spectrum for silver and gold (let's forget about aluminum, it's not relevant to this discussion; also keep in mind that where reflectance is low, absorbance is high and vice-versa). The absorption (reduced ...


49

This effect comes from the band structure of the metal, not atoms. In simplest terms (see, e.g. Ashcroft & Mermin, Solid State Physics, chapter 1) the electrons in the conduction band have a plasma frequency $\omega_{p}^{2} = 4\pi n e^{2}/m$ - this is a collective excitation of the metal. For Cu (reddish) you start exciting the plasmons around 2eV (red)...


42

I don't think that's really true anymore. Some Fortran use is historical (i.e., early codes were developed in FORTRAN because that was the best programming language for number crunching in the 70s and 80s). Heck, the name stands for "formula translation." Some Fortran use is because of performance. The language was designed to be: especially suited to ...


38

Okay, this is not so much of an answer as it is a summary of my own progress on this topic after giving it some thought. I don't think it's a settled debate in the community yet, so I don't feel so much ashamed about it :) A few of the things worthy of note are: The bond energy found by the authors for this fourth bond is 13.2 kcal/mol, i.e. about 55 kJ/...


31

This is a great question and I love it. Thank you for taking the time to not only post the question but each attempted response. I will try to address the parts of this question that I can at the moment. I will list, in order, your attempted answer along with my response. Carbene H2C: has two singlet configurations that contribute to the ground ...


29

You are using the Heisenberg uncertainty principle to relate the uncertainty in position $x$ to the uncertainty in velocity $v$. However, the quantitative version of this principle actually is $$\Delta x\cdot\Delta p\geqslant\tfrac12\hbar $$ where $\Delta x$ is the uncertainty in position $x$ and $\Delta p$ is the uncertainty in momentum $p$. Certainly, ...


29

In quantum chemistry, probably the easiest way to understand the "exchange interaction" is within the context of the Hartree-Fock model. $ \newcommand{\op}{\hat} \newcommand{\el}{_\mathrm{e}} \newcommand{\elel}{_\mathrm{ee}} \newcommand{\elnuc}{_{\mathrm{en}}} \newcommand{\core}{^{\mathrm{core}}} \newcommand{\bracket}[3]{\langle{#1}\vert{#2}\vert{#3}\rangle} ...


28

Disclaimer This post is some kind of a legacy post. Find the notation used in the question in the other answer. I added this proof as I was not entirely certain I understood the notation correctly. As it turned out, I did not. As a result I posted the complete derivation of RSPT up to second order, trying to guide anyone through it using a different notation....


27

I think it does make sense to provide a somewhat alternative view and to clarify the matter. FORTRAN vs. Fortran First off, one has to distinguish the old FORTRAN from the new Fortran, where, by convention, the name of the old language is written usually in all caps. The old FORTRAN (all the way up to FORTRAN 77) is indeed still used because of tons of ...


26

PBE The PBE functional${}^{[1]}$ belongs to the class of generalized gradient approximation (GGA) functionals for the exchange-correlation energy $E_{\mathrm{xc}}$. Considering that the dependence $E_{\mathrm{xc}}[\rho]$ may be non-local, i.e. $E_{\mathrm{xc}}$ may depend on the density $\rho$ at a given point (locality), but also on $\rho$ nearby (non-...


26

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 \begin{equation} \left(\hat{L} + \lambda \rho(\mathbf{x}) \...


25

$\newcommand{\conj}[1]{\overline{#1}{}} \newcommand{\braket}[2]{\langle{#1}\,|\,{#2}\rangle} \newcommand{\bracket}[3]{\langle{#1}\,|\,{#2}\,|\,{#3}\rangle} \newcommand{\mat}[1]{\mathbf{#1}} \newcommand{\rel}{\vec{r}_{\mathrm{e}}} \newcommand{\rnuc}{\vec{r}_{\mathrm{n}}} \newcommand{\linop}[1]{\hat{#1}} \newcommand{\Tnuc}{\linop{T}_{\mathrm{n}}} \newcommand{\...


24

First note that the acronym DFA I used in my comment originates from Axel D. Becke paper on 50 year anniversary of DFT in chemistry: Let us introduce the acronym DFA at this point for “density-functional approximation.” If you attend DFT meetings, you will know that Mel Levy often needs to remind us that DFT is exact. The failures we report at ...


24

There is nothing trivial about MCSCF calculations because it is hard to predict a priori how long a calculation will take. There are well-defined equations for calculating how many determinants $$ D(n,N,S) = \binom{n}{N/2+S} \binom{n}{N/2-S} $$ or configuration state functions (CSFs) $$ D(n,N,S) = \frac{2S+1}{n+1} \binom{n+1}{N/2-S} \binom{n+1}{N/2+S+1} $$...


24

There is no fundamental law preventing simple chemical reactions: things are complex because of the combinatorial complexity of chemical compounds The complexity of many chemical reactions is a byproduct of the fact that there is a very, very large variety of possible chemicals. Much of that complexity happens because of the almost infinite way even some ...


23

When you rip, tear or mechanically deform a polymer (for example a piece of plastic) you are putting energy into the material. The energy from this deformation causes the polymer chains in the vicinity of the deformation to attempt to align. To some degree this partial alignment makes continued deformation easier. To continue tearing the polymer apart you ...


22

Unfortunately, nothing in the bonding situation in carbon monoxide is easily explained, especially not the dipole moment. According to the electronegativities of the elements, you would expect the partial positive charge to be at the carbon and a partial negative charge at oxygen. However, this is not the case, which can only be explained by molecular ...


21

Black powder is a mixture of solids. As solids, they are not particularly inclined toward fast reactions between each other. Elevated temperature generally liven things up, because it allows diffusion of solids into each other. However, both carbon (charcoal) and potassium nitrate are relatively stable compounds, so even when potassium nitrate is melted, ...


21

$\newcommand{\el}{_\mathrm{e}}$In quantum chemistry, when a nomenclature in which one distinguishes between "static" and "dynamic" correlation is used, "correlation" referrers to all the deficiencies of the Hartree-Fock (HF) single-determinantal approach. For instance, the the correlation energy is defined as the difference between the exact (non-...


20

This is actually an active area of research for water clusters. In principle, for $\ce{(H2O)_{n}}$ there should be a "melting" phase transition, much like for ice to liquid water. So, in principle, if you had an accurate enough theoretical method, you could do molecular dynamics and see when the melting point of the cluster matches bulk water. Defining a ...


19

The reaction coordinate is the progress of a reaction from reactants to products with various intermediates and transition states in between. It is an abstraction. It has no relation to time. Rather it is the progress of bond-forming and bond-breaking reaction steps. The free energy change of partially formed and partially broken bonds cannot be measured. ...


19

tl;dr The next in the series is called φ bond. There is even a tiny Wikipedia article about it. Nicolau pointed me to the Wikipedia article, that had at the time a tiny section about the φ symmetry of the bond. Ben also kindly agreed with my naming proposition. I'd like to back up just a little bit an quote one sentence of this article: The type ...


19

No, this is not possible. Actually, if I would have to think of the most unlikely chemical conceivable, that would be it. Let's see why: Krypton is a noble gas that doesn't bond to anything. All of the known krypton compounds can be counted on one hand, and most of them contain fluorine. Putting krypton in a large molecule like this just can't be. This is ...


19

Very interesting question, and it kept me up despite daylight saving time cheating me of one hour of sleep last night... A good reference is Albright, Burdett and Whangbo, Orbital Interactions in Chemistry 2nd ed. pp 282ff. which explains this in much greater detail than I can. (In general, that book is fantastic.) I will try my best to summarise what they ...


19

we assume for no particular reason that dipole moments must be behaving like vectors Ah, but there is a reason. Consider the interaction of a molecule with the scalar potential $$ E_{\text{int}} = \int \rho(\mathbf{r})\phi(\mathbf{r}) \, \mathrm{d}^3 \mathbf{r} $$ where the integral over all space is turned into an expansion: $$ \int \rho(\mathbf{r})\phi(...


18

The controversy surrounding the $\mathrm{p}K_\mathrm{a}$ of hydronium mostly arises from the definition of $K_\mathrm{a}$ or lack thereof. There is no IUPAC definition of $\mathrm{p}K_\mathrm{a}$ or $K_\mathrm{a}$. The closest IUPAC defined term is the standard equilibrium constant, which can be denoted as $K^\circ$ or just $K$. Physical chemistry ...


18

TL;DR VB theory treats atomic orbitals (including hybridized orbitals) as providing a good mathematical/physical description of the true form of the molecular wavefunction. MO theory uses atomic orbitals (with Gaussian radial functions) as a tool of computational convenience in an effort to define a molecular wavefunction that in its final form often bears ...


18

Method Most of the time CCSD(T) would indeed be a huge overkill for geometry optimisation. DFT and MP2 have way better performance/cost ratio. Note that I said DFT and MP2 above, not or MP2; this is well known procedure to compare DFT geometries with MP2 ones to be on the safe side: there should be no big differences. On the DFT side hybrids might be ...


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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