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"# Table of Contents

\n",
""
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"\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."
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"## 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",
"$$\n",
"\\left(-\\frac{1}{2} \\left( \\frac{d^2}{dr^2}\n",
"+ \\frac{2}{r} \\frac{d}{dr}\\right) \n",
" + \\frac{l(l+1)}{2r^2} - \\frac{Z}{r} \\right)\n",
" R_{n,l}(r) \n",
"= E_{n,l}R_{n,l}(r)\n",
"$$"
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"## 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",
"$$\n",
"n=1,2,3,\\ldots \\\\\n",
"l=0,1,\\ldots,n-1 \\\\\n",
"m = 0,\\pm 1, \\pm2, \\ldots, \\pm l\n",
"$$"
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"![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"
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"### 📝 What is the expectation value of $r^k$ for the $l=n-1$ orbital of a hydrogenic atom. "
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"## 🪞 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)"
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