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Table of Contents

\n", "
" ] }, { "cell_type": "markdown", "id": "static-phenomenon", "metadata": {}, "source": [ "\n", "![5f+1 orbital](https://upload.wikimedia.org/wikipedia/commons/thumb/d/dd/Hydrogen_eigenstate_n5_l3_m1.png/1024px-Hydrogen_eigenstate_n5_l3_m1.png \"A 5f orbital with +1 angular momentum around the z axis, m=1. CC-SA3 license by Geek3\")\n", "\n", "# One-Electron Atoms\n", "Just as the touchstone of chemistry is the periodic table, the touchstone of quantum chemistry is the atomic wavefunction. As we shall see, our intuition about many-electron atoms is built up from our knowledge of 1-electron atoms, mainly because many-electron atoms are mathematically intractable, while 1-electron atoms are not appreciably more complicated than an electron confined to a spherical ball. A detailed mathematical exposition on 1-electron atoms--which are often called hydrogenic atoms--is provided as a [pdf](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/Hatom.pdf?raw=true). This is only a brief summary." ] }, { "cell_type": "markdown", "id": "overall-consortium", "metadata": {}, "source": [ "## Schrödinger Equation for One-Electron Atoms\n", "Denoting the mass of the electron as $m_e$, the charge of the electron as $=e$, and the [permittivity of free space](https://en.wikipedia.org/wiki/Vacuum_permittivity) as $\\epsilon_0$, the Hamiltonian for the attraction of an electron to a atomic nucleus with atomic number $Z$ (and charge $+Ze$) at the origin, $(x,y,z) = (0,0,0)$, is:\n", "$$\n", "\\hat{H}_{\\text{1 el. atom}} = -\\frac{\\hbar^2}{2m_e} \\nabla^2 - \\frac{Z e^2}{4 \\pi \\epsilon_0 r}\n", "$$\n", "where $r = \\sqrt{x^2+y^2+z^2}$ is the distance of the electron from the nucleus. In [atomic units](https://en.wikipedia.org/wiki/Hartree_atomic_units), the Hamiltonian is:\n", "$$\n", "\\hat{H}_{\\text{1 el. atom}} = -\\tfrac{1}{2} \\nabla^2 - Zr^{-1}\n", "$$\n", "\n", "Just as we did for the electron confined to a spherical ball, we rewrite the Schrödinger equation in spherical coordinates,\n", "\n", "$$\n", "\\left(-\\frac{1}{2} \\left( \\frac{d^2}{dr^2}\n", "+ \\frac{2}{r} \\frac{d}{dr}\\right) \n", " + \\frac{\\hat{L}^2}{2r^2} - \\frac{Z}{r} \\right)\n", " \\psi_{n,l,m_l}(r,\\theta,\\phi) \n", "= E_{n,l,m_l}\\psi_{n,l,m_l}(r,\\theta,\\phi)\n", "$$\n", "and use the fact the eigenvalues of the squared-magnitude of the angular momentum, $\\hat{L}^2$ are the [spherical harmonics](https://en.wikipedia.org/wiki/Spherical_harmonics)\n", "$$\n", "\\hat{L}^2 Y_l^{m} (\\theta, \\phi) = l(l+1)Y_l^{m} (\\theta, \\phi) \\qquad l=0,1,2,\\ldots m=0, \\pm 1, \\ldots, \\pm l\n", "$$\n", "and the technique of separation of variables to deduce that the wavefunctions of one-electron atoms have the form\n", "$$\n", "\\psi_{n,l,m}(r,\\theta,\\phi) = R_{n,l}(r) Y_l^{m}(\\theta,\\phi) \n", "$$\n", "where the radial wavefunction, $R_{n,l}(r)$ is obtained by solving the radial Schrödinger equation\n", "\n", "$$\n", "\\left(-\\frac{1}{2} \\left( \\frac{d^2}{dr^2}\n", "+ \\frac{2}{r} \\frac{d}{dr}\\right) \n", " + \\frac{l(l+1)}{2r^2} - \\frac{Z}{r} \\right)\n", " R_{n,l}(r) \n", "= E_{n,l}R_{n,l}(r)\n", "$$" ] }, { "cell_type": "markdown", "id": "breeding-sharing", "metadata": {}, "source": [ "## The Radial Equation for One-Electron Atoms\n", "To solve the radial Schrödinger equation, we rewrite it as a [homogeneous linear differential equation](https://en.wikipedia.org/wiki/Homogeneous_differential_equation)\n", "\n", "$$\n", "\\left( \\frac{d^2}{dr^2}\n", "+ \\frac{2}{r} \\frac{d}{dr}\n", " - \\frac{l(l+1)}{r^2} + \\frac{2Z}{r} + 2E_{n,l}\\right)\n", " R_{n,l}(r) = 0\n", "$$\n", "It is (quite a bit) more involved than the previous cases we have considered, but the same basic technique reveals that the eigenenergies are \n", "$$\n", "E_n = -\\frac{Z^2}{2n^2}\n", "$$\n", "and the radial wavefunctions are the product of an [associated Laguerre polynomial](https://en.wikipedia.org/wiki/Laguerre_polynomials#Generalized_Laguerre_polynomials) and an exponential,\n", "$$\n", "R_{n,l}(r) \\propto \\left(\\frac{2Zr}{n}\\right)^l L_{n-1-l}^{2l+1}\\left(\\frac{2Zr}{n}\\right) e^{-\\frac{Zr}{n}}\n", "$$\n", "with\n", "$$\n", "n=1,2,3,\\ldots \\\\\n", "l=0,1,\\ldots,n-1 \\\\\n", "m = 0,\\pm 1, \\pm2, \\ldots, \\pm l\n", "$$" ] }, { "cell_type": "markdown", "id": "laden-senior", "metadata": {}, "source": [ "![Hydrogenic orbitals](https://upload.wikimedia.org/wikipedia/commons/b/b0/Atomic_orbitals_n1234_m-eigenstates.png \"Hydrogenic orbitals; the colors indicate the complex phase. CC-SA4 by Geek3.\")\n", "\n", "## Eigenenergies and Wavefunctions for One-Electron Atoms\n", "The eigenenergies of the Hydrogenic wavefunctions do not depend on $m$ or $l$. So there are $(n+1)^2$ degenerate eigenfunctions, with energies\n", "$$\n", "E_n = -\\frac{Z^2}{2n^2}\n", "$$\n", "The energy eigenfunctions are: \n", "$$\n", "\\psi_{nlm}(r,\\theta,\\phi) \\propto \\left(\\frac{2Zr}{n}\\right)^l L_{n-1-l}^{2l+1}\\left(\\frac{2Zr}{n}\\right) e^{-\\frac{Zr}{n}} Y_l^{m} (\\theta, \\phi)\n", "$$\n", "These eigenfunctions are complex-valued, because the spherical harmonics are complex-valued. Like all other one-electron wavefunctions, these eigenfunctions are referred to as *orbitals*. For historical reasons, orbitals are labelled by their principle quantum number $n$ (which specifies their energy), their total angular momentum quantum number $l$, and the quantum number that specifies their angular momentum around the $z$ axis, $m$,\n", "$$\n", "\\hat{L}_z Y_l^{m} (\\theta, \\phi) = \\hbar m Y_l^{m} (\\theta, \\phi)\n", "$$\n", "The $l$ quantum number is stored by a letter code that dates back to the pre-history of quantum mechanics, where certain spectral lines were labelled as **s**harp ($l=0$ indicated no spatial degeneracy that could be broken by an external field), **p**rinciple ($l=1$ lines were still relatively sharp), **d**iffuse ($l=2$ lines were quite diffuse due to the 5-fold degeneracy of d orbitals), and **f**undamental ($l=3$). \n", "\n", "Note that the orbital images that appear above do not look that much like the usual orbital pictures, with the exception of the $m=0$ orbitals. This is because of the complex-valuedness. We often instead use the *real* spherical harmonics, which are defined simply as:\n", "\\begin{align}\n", "S_l^{m>0}(\\theta,\\phi) &= \\frac{1}{\\sqrt{2}} \\left(Y_l^{-m} (\\theta, \\phi) + (-1)^{m} Y_l^{m} (\\theta, \\phi) \\right) \\\\\n", "S_l^{m=0}(\\theta,\\phi) &= Y_l^{m=0} (\\theta, \\phi) \\\\\n", "S_l^{m<0}(\\theta,\\phi) &= \\frac{i}{\\sqrt{2}} \\left(Y_l^{-m} (\\theta, \\phi) - (-1)^{m} Y_l^{m} (\\theta, \\phi) \\right) \n", "\\end{align}\n", "The following animations shows one can take linear combinations of the (complex) spherical harmonics to form the $p_x$, $p_y$, etc. orbitals one generally uses in chemistry.\n", "\n", "![animation of 2p orbital](https://github.com/PaulWAyers/IntroQChem/blob/main/linkedFiles/Orbital_p1-px_animation.gif?raw=true \"animation of 2p real and complex orbitals; by Geek3 CC-SA4 license\")\n", "\n", "![animation of 3p orbital](https://github.com/PaulWAyers/IntroQChem/blob/main/linkedFiles/Orbital_3p1-3px_animation.gif?raw=true \"animation of 3p real and complex orbitals; by Geek3 CC-SA4 license\")\n", "\n", "Using the [orbitron](https://winter.group.shef.ac.uk/orbitron/atomic_orbitals/7i/index.html), you can visualize the (real, Cartesian) spherical harmonics and the [radial wavefunctions](https://winter.group.shef.ac.uk/orbitron/atomic_orbitals/7f/7f_wave_function.html) for hydrogenic orbitals. Most orbitals have very complicated formulas, but a few have simple equations, including:\n", "\n", "$$\n", "\\psi_{\\text{1s}}(r) = \\psi_{100}(r) = \\sqrt{\\frac{Z^3}{\\pi}}e^{-Zr} \\\\\n", "\\psi_{n,n-1,m} \\propto r^{n-1} e^{\\tfrac{-Zr}{n}} Y_l^{m_l} (\\theta,\\phi)\n", "$$\n", "\n" ] }, { "cell_type": "markdown", "id": "different-perception", "metadata": {}, "source": [ "### 📝 What is the expectation value of $r^k$ for the $l=n-1$ orbital of a hydrogenic atom. " ] }, { "cell_type": "markdown", "id": "healthy-understanding", "metadata": {}, "source": [ "## 🪞 Self-Reflection\n", "- Using the conversion from atomic units to traditional chemical units of kJ/mol, what is the energy of the Hydrogen atom? How accurately, in atomic units, must one determine the energy of a one-electron atom in order to attain \"chemical accuracy\" of ~1 kJ/mol?\n", "- Write a small Python script to evaluate the expectation value of the radius, $r$, for a one-electron atom. \n", "- Test to confirm that the Heisenberg uncertainty principle for position and momentum holds for the ground state of a Hydrogenic atom.\n", "- To what extent is the shape of the spherical harmonic intuitive, especially the doughnut shapes associated with an electron's angular momentum around the z axis.\n", "\n", "## 🤔 Thought-Provoking Questions\n", "- In one-electron atoms, the eigenenergies depend only on the principle quantum number, $n$, and not the angular momentum quantum number, $l$. Why are $s$ orbitals lower in energy than $p$ orbitals in real multielectron atoms, but not one-electron atoms? It turns out this is *not* an accidental degeneracy, but a hidden symmetry of the Hydrogen atom.\n", "- Suppose electrons did not repel each other. Can you write the wavefunction for a many-electron atom in that case?\n", "- Why do you think solving the Schrödinger equation for the one-electron molecule is more complicated than solving the Schrödinger equation for the one-electron atom?\n", "- For what $Z$ is the energy of a one-electron atom comparable to the rest-mass energy of an electron, $mc^2$? For atomic numbers close to this value, relativistic effects become extremely important.\n", "\n", "## 🔁 Recapitulation\n", "- What are eigenfunctions for the $n=l+1$ state of a one-electron atom? \n", "- What are the energy eigenfunctions and eigenvalues for a one-electron atom?\n", "- How does the energy increase as the atomic number increases? \n", "\n", "## 🔮 Next Up...\n", "- Multielectron systems\n", "- Approximate methods.\n", "\n", "## 📚 References\n", "My favorite sources for this material are:\n", "- [Randy's book](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/DumontBook.pdf?raw=true)\n", "- D. A. MacQuarrie, Quantum Chemistry (University Science Books, Mill Valley California, 1983)\n", "- [One-electron atoms](https://github.com/PaulWAyers/IntroQChem/blob/main/documents/Hatom.pdf?raw=true) (my notes).\n", "\n", "There are also some excellent wikipedia articles:\n", "- [Hydrogen-like atoms](https://en.wikipedia.org/wiki/Hydrogen-like_atom)\n", "- [Atomic orbitals](https://en.wikipedia.org/wiki/Atomic_orbital)" ] }, { "cell_type": "code", "execution_count": null, "id": "civil-roots", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.3" }, "toc": { "base_numbering": "6", "nav_menu": {}, "number_sections": true, "sideBar": false, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": true, "toc_position": {}, "toc_section_display": false, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 5 }