[HN Gopher] 3D Electron Orbitals of Hydrogen
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3D Electron Orbitals of Hydrogen
Author : c0nrad
Score : 73 points
Date : 2020-06-30 16:11 UTC (6 hours ago)
(HTM) web link (blog.c0nrad.io)
(TXT) w3m dump (blog.c0nrad.io)
| ur-whale wrote:
| Very nice, but: what am I looking at? A probability density? And
| what do the control represent? Some sort of energy level? An
| isosurface of the modulus of the wave function? If so what does
| the number mean from an intuitive perspective?
| nsxwolf wrote:
| Is this really what hydrogen "looks" like in space, or is this
| just another impossible for the layperson to understand
| abstraction?
|
| I never even know how to ask the question I want to ask, just
| like it's impossible to ask if the colors in the photograph of
| a planet are "real" or not. I've just given up and decided that
| all of space is in black and white except for Earth.
| throwaway_pdp09 wrote:
| An excellent point (the first sentence I mean). I'd
| appreciate a physicist's opinion.
| antepodius wrote:
| It's a data visualisation of a property of a model of
| hydrogen. The dots are denser where the absolute value of
| the 'wavefunction' (which is just a function that takes in
| space and time coordinates and returns a complex value) is
| higher.
|
| It's not 'real' in the same way a simulation of a tennis
| ball flying through the air, rendered with dots isn't, but
| worse: that's a visualisation of a model, too, but if you
| imagine it being a simulation of possible sensory data it
| makes more sense- the world is set up so it's possible for
| a tennis ball to be seen by a conscious being, but that
| isn't true for a hydrogen atom.
|
| In a model of a tennis ball, you might have a black screen,
| and then draw a dot where the tennis ball is, according to
| the model.
|
| In this model of the hydrogen atom, you have a black
| screen, and then you draw a cloud with denser and less
| dense regions to represent where the electron 'is',
| according to the model. The problem is that electrons
| aren't point particles; in this model, an electron is a
| cloud- it's described not by some vector describing its
| position, but by the aforementioned wavefunction. It's a
| cloud in space, (except every point is complex-valued-
| they're taking the magnitude for this rendering) that
| changes (or doesn't) over time.
|
| There's layers here. To what degree is a simplified model
| 'real'? To what degree is a visualisation of a model a
| picture of a 'real' thing, even if that model were true and
| complete?
| throwaway_pdp09 wrote:
| I wasn't clear - sorry. It obviously can't be seen, and I
| (just about) get it's a probability cloud. My question
| is, is the underlying probability cloud real, in the
| sense that we can think of it existing and in it's actual
| peculiar shape, so if we could probe it we would indeed
| find something shaped like that, or is it just an
| abstraction/model 'that just works'?
|
| It's not even an easy question to ask, come to think of
| it.
| throwaway_pdp09 wrote:
| Well how about that, ask a difficult question of an
| abstract domain, and get two clear yesses. Does not
| happen often. Thanks @mncharity and @cinntaile.
| cinntaile wrote:
| Yes, the probability cloud tells you how the hydrogen
| atom (H-O-H) will be shaped. The atoms will be repelled
| by the electrons that create this probability cloud and
| this will force the atoms into a geometry with minimal
| energy (which is the shape you are probably familiar
| with). Just remember that it's a probability so the shape
| isn't set in stone, it's more like the most common shape.
| A common way to learn this is by implementing the
| Hartree-Fock algorithm.
| https://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_method
| mncharity wrote:
| As commented elsewhere, the OP renders look perhaps not
| quite right. But in general...
|
| Atoms are little balls. With electron density that's
| mostly spherically symmetric. It's very high at the
| nucleus, and falls off exponentially outward. Down by
| several orders of magnitude by the time you reach
| distances at which atoms hang out together.
|
| A 2D analogue might be a stereotypic volcano, if height
| were density. I wish I knew of a better one. Few familiar
| objects have this degree of fuzzy. Diffusing smells, but
| you can't see those.
|
| The common representations with a solidish surface at
| some large distance from the nucleus is... useful when
| doing chemistry, but badly misrepresents the physical
| object.
|
| Electron density manifests clearly and concretely. For
| example, you can poke at it with the vibrating tip of an
| Atomic Force Microscope.
|
| Electron states, orbitals, seem less often encountered
| that directly. Rather than at one step remove - seeing
| density, or some other phenomena, and explaining it with
| states.
|
| Though there's a fun STM image I'm just now failing to
| quickly find. An STM scans a tip across a sample,
| measuring and mapping the tunneling current between them.
| Usually with a boringly symmetric ball of a tip, so the
| interestingness is all sample. In this case however, the
| sample had a grid of boring s states, and tip conduction
| was through a tilted f state. So the sample repeatedly
| mapped the tip. The image is thus a grid of little images
| of the tip's f state, laid out like rows of little buoys.
| EDIT: Maybe this is it: https://arxiv.org/pdf/cond-
| mat/0305103.pdf fig 6 page 12 (though it doesn't entirely
| match what I'm remembering).
|
| Just to be clear, representing density by dots, rather
| than say by a color gradient on voxels, is purely a data-
| visualization rendering choice. Like old newspapers using
| halftone images. Which I mention only because there are
| misconceptions around the individual dots themselves
| representing something about the electron.
| ars wrote:
| The concept of "look" doesn't apply to hydrogen. It doesn't
| have a fixed, defined, shape.
|
| To define "look" you must say "look how", i.e. with what
| sense are you looking.
|
| For atoms you can use gravity to look at them, or the
| electromagnetic force, or the strong force. And the atoms
| will look different each way.
|
| I suppose you could define look as "how strongly will this
| test particle interact with the atom at this distance, using
| this force". But notice "how strongly" - there is no fixed
| boundary, the interaction just gets weaker (or less likely)
| as things get farther.
| 8bitsrule wrote:
| Heisenberg (1958): "What we observe is not nature itself,
| but nature exposed to our method of questioning."
| ypcx wrote:
| Seems like the relevant blog entry is here:
| https://blog.c0nrad.io/posts/hydrogen-pt2/
|
| Edit: the Reddit discussion has more info:
| https://www.reddit.com/r/Physics/comments/gt1set/interactive...
| plus wrote:
| What's being plotted is the amplitude of a "hydrogenic"
| orbitals Ps_{n,l,m}(x, y, z)^2. Hydrogenic orbitals are the
| eigenfunctions (wavefunctions) of the 1-electron 1-nucleus
| Hamiltonian (many-electron wavefunctions are visually similar
| to this but due to electron-electron correlation become waaaaay
| more mathematically complex).
|
| This is a nice system because the result is analytic (ignoring
| relativistic effects and assuming a point-like nucleus).
| Specifically, Ps_{n,l,m}(r, th, ph) = L_n(r) * Y_l^m(th, ph),
| where L_n(r) is the n-th order Laguerre polynomial and
| Y_l^m(th, ph) is the m-th spherical harmonic with angular
| momentum l.
|
| From a chemical perspective, n indicates which electron "shell"
| you are in, l indicates which type of orbital (l=0 is an
| s-orbital, l=1 is a p-orbital, l=2 is a d-orbital, etc.), and m
| indicates which of the different orbitals within that shell
| having that angular momentum (e.g. p_x vs p_y vs p_z).
| ur-whale wrote:
| I think you misunderstood my question:
|
| 1. The fact that the function is easy to compute because
| there is an analytical solution to the ODE when the atom is
| simple enough tells precious little about what the picture
| actually represents.
|
| 2. The fact that the function you talk about has _6_
| parameters and this is a 3D visualization (3 degrees of
| freedom) is confusing.
|
| 3. The chemistry lesson about orbitals is also an interesting
| fact but still not properly correlated to the interactive
| depiction. Notoriously missing: where are m,n,l actually
| depicted in the story? Am I looking at one specific choice
| for those? What are the menu entries?
|
| I think there is something that would truly help: if one
| would take a volume integral over a infinitesimal cube of the
| 3D interactive representation, what physical units would the
| result be in?
| antepodius wrote:
| The 3-d space shown is just physical space.
|
| n,l,m are the triplet of numbers you can select in the top
| right. They're called quantum numbers, and they describe
| the state of this particular system, in particular the
| state of the electron.
|
| Basically: n is how much energy the electron has (the
| higher, the further from the nucleus). l and m further
| describe which orbital the electron is in; l is related to
| the electron's angular momentum, m is related to its angle
| to the x-y plane in this visualisation. Only certain
| combinations of these numbers are allowed by physics
| (angular momentum, l, has to be smaller than energy, n, for
| example). n,l,m together describe the state of the electron
| inside the hydrogen atom.
|
| So what are the dots? Basically, they're meant to represent
| a cloud. Where the cloud is denser, the electron has more
| measure; the electron is more there than in other places.
| Practically speaking, if you made a measurement to see
| where the electron was, your results would
| probabilistically correlate with the density of the cloud.
| The process whereby the electron goes from being a
| probabilistic cloud to a point particle interacting with
| your test particle back to a probabilistic cloud is called
| 'wave function collapse' in the Copenhagen interpretation,
| or, more generally, 'magic'.
|
| (Or it's just how the universal wavefunction's branches
| look from the inside.)
|
| A volume integral would be unitless, by definition: the
| value of the square-absolute-value of the wavefunction at
| any point (what's represented by this graph) is the
| probability of finding the electron at that point per cubic
| metre. A volume integral from negative to positive infinity
| in x, y, and z gives 1 (no units).
| plus wrote:
| > n is how much energy the electron has (the higher, the
| further from the nucleus)
|
| I disagree with this description. It is true that higher
| values of n correspond to wave functions with higher
| energy, but higher values of l also correspond to wave
| functions with higher energy. n indicates the number of
| _radial nodes_.
|
| > A volume integral would be unitless
|
| The volume integral has units of "number of electrons".
| Calling this unitless is unnecessarily misleading I feel,
| even if it is technically correct, since in physics we
| tend not to give units to quantities like this.
| plus wrote:
| It is not a 6-parameter function, it is a family of
| 3d-functions each characterized by 3 parameters. Those
| parameters can be modified using the dropdown in the top-
| right corner of the page.
|
| The Hamiltonian is an operator that describes the energy of
| a system. Eigenfunctions of the Hamiltonian are quantum
| states, referred to as wave functions. The squared
| amplitude of a wave function is a probability distribution
| function. When discussing the wave functions of electrons,
| the probability amplitude is sometimes referred to as the
| electron density. You are looking at a sampling from the
| electron density of the wave functions of the 1-electron
| 1-nucleus Hamiltonian operator. There are different wave
| functions (different entries in the dropdown box at the
| top-right corner of the screen) because the Hamiltonian
| operator has more than one eigenfunction. Each
| eigenfunction is characterized by the 3 "quantum numbers":
| n, l, and m. "n" indicates the number of radial nodes --
| areas of a given distance from the nucleus where the
| electron density is 0. "l" indicates the number of angular
| nodes -- areas arranged in a certain angular pattern around
| the nucleus where the electron density is 0.
|
| > I think there is something that would truly help: if one
| would take a volume integral over a infinitesimal cube of
| the 3D interactive representation, what physical units
| would the result be in?
|
| Number of electrons (possibly fractional, if you aren't
| sampling the whole space). For this particular Hamiltonian,
| the integral over all space should be numerically 1 for any
| given eigenfunction, since we are looking at the 1-electron
| Hamiltonian.
| ur-whale wrote:
| > It is not a 6-parameter function, it is a family of
| 3d-functions each characterized by 3 parameters. Those
| parameters can be modified using the dropdown in the top-
| right corner of the page.
|
| Sorry to be pedantic, but these two things are the exact
| same thing.
| plus wrote:
| You were asking how it is to be interpreted, and it is as
| I said: a family of 3D functions each characterized by 3
| parameters. What you are saying is technically correct,
| but misses the point of what I was trying to convey.
|
| Edit: also, three of the parameter (x, y, z or r, th, ph
| depending on whether you are using a Cartesian or
| spherical coordinate system) are continuous, real, and
| unbounded (well, unbounded in a Cartesian sense anyway).
| In contrast, n, l, and m are discrete integer-valued
| quantum coefficients that obey the relations n > 0, 0 <=
| l < n, -l <= m <= l.
| _Microft wrote:
| > 2. The fact that the function you talk about has 6
| parameters and this is a 3D visualization (3 degrees of
| freedom) is confusing.
|
| Why is that a problem?
|
| It's no different than defining a family of linear
| functions _f_a,b(x) = a x + b_ and then letting the user
| pick values for the parameters _a_ and _b_ before plotting
| it onto the xy - plane.
| o_v_o wrote:
| 2. Basically it has six parameters because the wave
| function changes based on the first three parameters
| (n,l,m) and can then be solved w.r.t. x, y, and z, though
| it'll be easier to do in spherical coordinates.
|
| 3. This hydrogen atom has a nucleus and one electron. Think
| of n as the energy level of that electron - electrons have
| discrete energy levels, so as n increases the electron
| occupies the next discrete energy available to it.
|
| l is another quantized value which corresponds to what we
| call the orbital angular momentum of the electron, which
| partially determines the shape of the orbital. This is a
| big part of the visualization you see - as you change the
| value of l, we see different shapes, and if you increase
| the number of particles in the visualization, you get
| changes in those shapes. These different shells have names
| - s, p, d, etc - that correspond to the integer value of l
| - 0, 1, 2, etc.
|
| Importantly, what's being graphed in the visualization is a
| solution to the specified wave function. It's a 3D
| probability map, effectively. Where there is a higher
| chance of the electron being located, the particles are
| more concentrated, whereas lower chance regions have lower
| populations of particles.
|
| m is called the magnetic quantum number and can have
| integer values from -l to +l, and further specifies the
| particular state of the electron in its "shell" - s, p, d,
| etc again. If the wave function has n=2 and l=2, then it's
| in the d shell, and can have values of m from -2 to +2. The
| actual value of m determines the final "shape" of the
| orbital, again depicted as a probability map - every dot
| you see plotted can be a location of the electron, so
| plotting a lot of them based on the probability
| distribution gives you a visualization of the regions
| available to that electron.
|
| So the menu entries are just values of n,l,m that aren't
| separated by commas.
|
| I hope that clarifies some things!
| steerablesafe wrote:
| If this is supposed to be the probability density then it looks
| way off. I suspect that there is a mistake in calculating the
| absolute value of the wave function.
|
| Edit: In this base the absolute value of the wave function is
| supposed to be rotational symmetric around the z axis.
| raverbashing wrote:
| Why? The PDE should be Psi^2 no? (which is what it's being
| plotted)
|
| > In this base the absolute value of the wave function is
| supposed to be rotational symmetric around the z axis.
|
| For the D orbital?
| evanb wrote:
| If the wavefunction is supposed to have n,l,m quantum numbers
| (as suggested by the interface) then yes, it should be
| rotationally symmetric around the z axis.
| steerablesafe wrote:
| The phi-dependence of the wave-functions are exp(i m phi), it
| has magnitude 1. |Psi|^2 = conj(Psi)*Psi, the phi-dependency
| cancels out.
| Denvercoder9 wrote:
| Yes, judging from a quick look at the code (and not being
| familiar with the GiNaC library used) this seems to square the
| wave function instead of squaring the absolute value.
|
| [1]
| https://github.com/c0nrad/hydrogen/blob/master/hydrogen.cpp#...
| ssivark wrote:
| At a first glance, there seems to be something wonky with the
| quantum numbers: http://hyperphysics.phy-
| astr.gsu.edu/hbase/qunoh.html
| mncharity wrote:
| Some similar work: https://www.willusher.io/webgl-volume-
| raycaster/#Hydrogen%20... , though caveat, that while labeled
| "Hydrogen Atom", the density shown is of just one state of the
| electron;
| http://phelafel.technion.ac.il/~orcohen/DFTVisualize.html .
|
| There are many more visualizations that show total electron
| density rather than individual states. Though arguably far far
| too few, given how pervasively unsuccessfully these topics are
| taught.
|
| If you'd like to explore, GPAW https://wiki.fysik.dtu.dk/gpaw/
| can be useful. Here's a random example of use:
| https://www.brown.edu/Departments/Engineering/Labs/Peterson/...
| supernova87a wrote:
| It's a great visualization --
|
| I would suggest, though, that some contour lines (or translucent
| shells?) might help make the point more apparent to someone
| trying to learn about the shapes (which I suppose is the point).
|
| After all, the point is to grasp something visual about it, and
| just vaguely discernible clouds of points don't probably convey
| that sufficiently, although they are accurate of course.
| phonon wrote:
| https://daugerresearch.com/orbitals/index.shtml is a much nicer
| (and I believe more accurate) visualization.
| vertbhrtn wrote:
| Is there a way to make these shapes evolve with time?
| ars wrote:
| They don't. They exist in all the shapes at the same time,
| which is what the plot is showing, a sort of combination of all
| the shapes, with more dots where the electron "exists" more
| often.
| plus wrote:
| It's possible to simulate the time-evolution of the electron
| density using the time-dependent Schrodinger equation. That
| said, if the initial state is chosen to be an eigenfunction
| of the Hamiltonian (read: any of the things being plotted on
| this page), there will be no time dependence -- the electron
| density will remain static. However, if the initial state is
| chosen as a superposition of states (read: any configuration
| that is _not_ given by the square amplitude of an
| eigenfunction of the Hamiltonian), you can simulate the time
| evolution.
| dnautics wrote:
| a bit of lay-hn-reader explanation from a chemistry/math major-
| now-dev (my physics might be a bit wrong, apologies in advance).
|
| These diagrams show the probability density of the electrons
| around a hydrogen nucleus, which is the simplest (and a pretty
| good in general) model for how electrons live around atom nuclei.
| The more dots, the denser the probability, aka: how likely or not
| one might find an electron in this particular position.
|
| In the upper right corner, there's the psi(n, l, m) selector
| which lets you pick the geometry.
|
| n is the "principal quantum number" which corresponds to "the
| gross energy level/frequency" of electron. The way to think about
| this (I think) is this: If you are plucking a string on a guitar,
| or play a wind instrument, the more nodes that it has, the higher
| the energy of the vibration. Similarly for electrons around
| atoms. As you pick diagrams with a higher n, you'll see more
| nodes (internal regions with zero density) in the distribution.
| These are also higher energy states. Generally, if you look
| carefully you should be able to find (n - 1) surfaces, though for
| the (n, 0, 0) diagrams some of these node surfaces are tiny
| spheres close to the nucleus, so you might not see them.
|
| l is the angular quantum number. This number determines how many
| of those nodal surfaces are "not spherical". So in a (n, 1, X)
| diagram, you should eventually see a plane cutting through if you
| play around with the orientation; In an (n, 2, x) you should see
| two intersecting planes cutting through, or in some cases a cone
| (more on that later).
|
| m is the magnetic quantum number, and presumes that the atom is
| sitting in a nonzero magnetic field, and selects for different
| energies that relative orientations in that magnetic field have.
| This splits the different possibilities based on direction
| relative to magnetic field, and not curve qualities (number of
| nodes; shape of nodes).
|
| There's another quantum number, which is the "spin quantum
| number" that has to do with the Pauli Exclusion principle, that
| two electrons can share an orbit simultaneously. This doesn't
| really change the shape of the orbital, so I presume that's why
| it's not there.
|
| (1, 0, 0) is possible, but probably not shown because it's
| boring.
|
| As for why you could have a "plane" or a "cone"; the display
| coordinate systems are somewhat arbitrary, and as with most
| quantum mechanics, "reality" is actually a weighted linear sum
| (superposition) of all of these possibilities; so a "plane" and a
| "cone" are roughly equivalently "surfaces", but the cone is a
| linear combination of a bunch of planes rotated around a line but
| is selected because it's a convenient and easy basis component
| with the other "planes" to generate coverage of the vector space
| of all possibilities. To really butcher the explanation: It turns
| out that you have to play that "rotate trick" because the space
| of "all possible probability distributions" has a fixed
| dimension, and you run out of ways to chop up three dimensional
| spaces with planes, so you have to mash them together to get
| correct coverage of the space of distributions.
|
| How this corresponds to the periodic table. The S block (left
| side) elements are mostly filling their (row, 0, 0) orbitals,
| then P block (right side) elements are filling their (row - 1, 1,
| _) orbitals. The transition metals are filling their (row - 2, 2,
| _) orbitals, and the inner transition metals are filling their
| (row - 3, 3, _) orbitals . Although it seems elegant, reasoning
| for the "row - X" and not "row" is a bit complicated, empirical
| and not theoretical, and if you'd like to understand why, look up
| "aufbau principle".
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