chemistry.ipynb

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    "# Notes on Ligands, Coordination Complex, oxidation numbers\n",
    "\n",
    "\n",
    "<BR>\n",
    "<img src=\"./images/oxidation_number_rules.png\" style=\"float: left;\" alt=\"periodic table abstractly\" width=\"700\"/>\n",
    "<div style=\"clear: left\">\n",
    "<BR>\n",
    "\n",
    "\n",
    "In this case the logic for $N\\;H_4\\;Cl$ goes like this: \n",
    "* No metal so $H$ is $+1 \\; \\times \\; 4$\n",
    "* No oxygen so $Cl$ is $-1$\n",
    "* Nitrogen to balance to zero must be $+3$"
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    "[YouTube video with many examples of the above](https://www.youtube.com/watch?v=zpCy7KwqDO8)\n",
    "\n",
    "\n",
    "* From a [YouTube lecture](https://www.youtube.com/watch?v=MAwaTYjjYCQ), starting with this example...\n",
    "* We will find the coordination number and the oxidation number\n",
    "\n",
    "$\\begin{align}\n",
    "[Cu \\; {(NH_3)}_4 ] \\; SO_4\n",
    "\\end{align}$\n",
    "\n",
    "* Step 1: $SO_4$ is a poly-atomic anion with charge $2-$\n",
    "* So the coordination complex must have charge $2+$ (it is a neutral molecule)\n",
    "* Coordination number: How many ligands are coordinating to the central metal atom\n",
    "* $NH_3$ is ***mono-dentate*** one time per: Coordination number = 4\n",
    "* oxidation number of the copper\n",
    "    * Charge of $NH_3$ is 0. Add this to charge(Copper) to arrive at $2+$\n",
    "    * Copper oxidation number is $+2$.\n",
    "\n",
    "Another example:\n",
    "\n",
    "$\\begin{align}\n",
    "[Cr \\; {(OH_2)}_4 {Cl}_2 ] \\; Cl\n",
    "\\end{align}$\n",
    "\n",
    "\n",
    "* Cl anion gives us $1-$: Coordination complex will have $1+$\n",
    "* $OH_2$ is water, mono-dentate. $Cl$ is also mono-dentate.\n",
    "* Coordination number is $6$\n",
    "* Oxidation number: Water is neutral, two $Cl$ gives $-2$\n",
    "    * Oxidation number = $+3$\n",
    "\n",
    "\n",
    "## Crystal field theory [MIT Open Courseware lecture](https://www.youtube.com/watch?v=lLdPSLNxDqA)\n",
    "\n",
    "\n",
    "* Ionic bonding, simple compared to ligand theory which adds in covalent bonds\n",
    "    * simple premise: ligand lone pair (-) electrons repel d orbitals\n",
    "    * reason: I dunno... maybe the d orbitals are \"out there\" and have electrons in them\n",
    "* Octahedral case: The most common geometry for a transition metal complex\n",
    "    * What does octahedral mean?\n",
    "        * Imagine the metal is at the origin of an x-y-z coordinate system\n",
    "        * Place ligands at +1 and -1 on all 3 axes: 6 total\n",
    "        * Imagine these are vertices of a graph: Connect all close pairs with edges\n",
    "        * There are 12 such edges\n",
    "        * The resulting shape has 8 identical faces: An octahedron\n",
    "    * Now we picture the d-orbitals of the central metal in relation to these 6 ligands\n",
    "        * on-axis ligands will repel on-axis d orbitals strongly (the premise)\n",
    "    * will repel off-axis d orbitals less so\n",
    "    * off-axis = $xy$, $xz$, $yz$ less repulsion in comparison with on-axis: \"stabilized\" (3-degenerate)\n",
    "    * on-axis = $z^2$ and $x^2 - y^2$ high repulsion: \"de-stabilized\" (2-degenerate)\n",
    " \n",
    "As a result we can look at the energy diagram for the metal.\n",
    " \n",
    "<BR>\n",
    "<img src=\"./images/crystal_octahedral_energy_splitting.png\" style=\"float: left;\" alt=\"periodic table abstractly\" width=\"700\"/>\n",
    "<div style=\"clear: left\">\n",
    "<BR>\n",
    "\n",
    "\n",
    "Note: The total energy is conserved in this splitting process; so the two on-axis orbital energies\n",
    "go *up* by $3/5 \\cdot \\Delta_{o}$ ($o$ is for $o$ctahedral) and the three off-axis orbital energies\n",
    "go *down* by $-2/5 \\cdot \\Delta_{o}$. \n",
    "\n",
    "\n",
    "The final question is what determines how big $\\Delta_{o}$ gets? The answer is: This varies with the\n",
    "nature of the ligand. Some ligands have a very strong \"negative charge\" impact, others are just *meh*\n",
    "so the splitting size will vary accordingly. \n",
    "\n",
    "\n",
    "## Topics\n",
    "\n",
    "- periodic table trends\n",
    "- ionic bonds\n",
    "- chemical structure\n",
    "- covalent bonding\n",
    "    - valence bond theory\n",
    "    - Lewis structures\n",
    "    - π and σ-systems\n",
    "- VSEPR Valence Shell Electron Pair Repulsion\n",
    "- LCAO Linear Combination of Atomic Orbitals\n",
    "- hybrid orbitals\n",
    "- localized bonding\n",
    "- dipole moments\n",
    "- molecular orbital theory [(overview video: worth watching)](https://youtu.be/oWzuqLl6Ito?si=_25hRUosl5KcOq3O)\n",
    "- diatomics and triatomics\n",
    "- acids and bases\n",
    "- types of chemical reactions\n",
    "- displacement and precipitation reactions\n",
    "- redox reactions with aqueous solutions\n",
    "- metal and halogen reactivity\n",
    "\n",
    "For another day: \n",
    "- ideal gas law\n",
    "- partial pressures\n",
    "- kinetic theory of gases\n",
    "- deviations from ideality, weak forces\n",
    "- liquids and solids\n",
    "\n",
    "\n",
    "## Introduction\n",
    "\n",
    "\n",
    "In the subject of chemical bonding there are two basic approaches: Valence bond theory and \n",
    "molecular orbital theory; both intended to accommodate quantum mechanics."
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    "## Lewis Structures\n",
    "\n",
    "- Atoms can form molecules as networks of covalent bonds (very strong)\n",
    "- Lewis dot structures are a shorthand for constructing molecules with covalent bonds\n",
    "    - The process breaks down into a *basic* procedure and a more *advanced* procedure\n",
    "    - The latter comes up in the case of \"What is the Lewis structure for Sulfur Dioxide ($SO_2$)?\"\n",
    "- We are concerned with the region of the periodic table shown below, *skipping* the transition metals\n",
    "\n",
    "<BR>\n",
    "<img src=\"./images/chemistry_elements_with_valence_electrons.png\" style=\"float: left;\" alt=\"periodic table abstractly\" width=\"400\"/>\n",
    "<div style=\"clear: left\">\n",
    "<BR>"
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    "- The game is all about simple economies of valence electrons\n",
    "    - Some are present in covalent bonds\n",
    "    - Some are associated with just one atom (non-bonding)\n",
    "        - These tend to occur in twos, what are sometimes called *lone pairs* of electrons\n",
    "- Lewis structures show valence electrons as dots around an element, for example Carbon **C**\n",
    "    - Carbon ground state orbital occupancy is $1s^22s^22p^2$\n",
    "        - This means for $n=2$: Carbon has 4 electrons in its outermost shell\n",
    "        - The total capacity for $n=2$ is 8 electrons... the *octet*...\n",
    "        - ...but Lewis notation indicates what electrons are *present* in the neutral atom\n",
    "    - Around the element symbol **C** we consider four coordination sites\n",
    "        - Each site can contain one or two electrons; so this is *octet* thinking\n",
    "    - The occupancy number (for Carbon: 4) corresponds to column number, left to right\n",
    "        - Number these 1, 2, 3 (13 in the diagram below), 4, 5, 6, 7, 8\n",
    "\n",
    "<img src=\"./images/chemistry_lewis_dot_structure_periodic_table.png\" style=\"float: left;\" alt=\"periodic table with Lewis dot structures\" width=\"500\"/>\n",
    "<div style=\"clear: left\">"
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    "- Pairing electrons in a covalent bond: We have two dots between atoms\n",
    "    - These two dots are eventually redrawn as a short line\n",
    "        - The first such is a $\\sigma$ covalent bond\n",
    "        - Electrons can be \"recruited\" to create double or even triple bonds\n",
    "            - The second and third bonds are $\\pi$ bonds\n",
    "            - Bond length becomes shorter: $\\sigma$ > $\\sigma + \\pi$ > $\\sigma + \\pi + \\pi$ \n",
    "\n",
    "\n",
    "Here is a Kandinsky painting as a reminder to take study breaks.\n",
    "\n",
    "\n",
    "<img src=\"./images/kandinsky.png\" style=\"float: left;\" alt=\"Kandinsky painting\" width=\"750\"/>\n",
    "<div style=\"clear: left\">"
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    "An example might be interesting to ponder before we get to a formal statement\n",
    "of the Lewis structure rules (below). What might the Lewis structure be for\n",
    "the $HCN$ molecule?\n",
    "\n",
    "\n",
    "Solution: A $\\sigma$ bond between the Hydrogen and Carbon\n",
    "and a triple $\\sigma, \\pi, \\pi$ bond between Carbon and Nitrogen. \n",
    "\n",
    "\n",
    "Below please find the five the Lewis structure rules. Before starting that it will be\n",
    "helpful to cover *electronegativity*. There are two other topics that will be mentioned \n",
    "later as well: **Resonance structures** and **formal charge**. These can be relegated\n",
    "to down-the-road learning.\n",
    "\n",
    "\n",
    "### Electronegativity\n",
    "\n",
    "\n",
    "Context, paraphrasing from Wikipedia: Electronegativity is a *tendency* to attract or accept shared electrons \n",
    "when forming chemical bonds. Electronegativity is affected by both atomic number (in terms\n",
    "of which column of the periodic table, i.e. valence electron count) and the radial distance\n",
    "of the atomic (or molecular?) valence electrons. Higher electronegativity: stronger electron\n",
    "attraction. Electronegativity relates to bond energy. A chemical bond has a strength -- the\n",
    "energy needed to break the bond -- and this strength is related to the shared versus separated\n",
    "nature of the distribution of electrons. At the shared end of the spectrum we have covalent bonds\n",
    "and when the electrons are separated we have ionic bonds. The term electropositivity is an\n",
    "opposite of electronegativity: it characterizes an element's tendency to donate valence electrons.\n",
    "\n",
    "\n",
    "Here is the functional result of all this electronegativity context: First, the right-hand column\n",
    "of the periodic table is noble gasses that have all their outer shells filled so they are not in\n",
    "the bonding business. Ignoring the right-most column: Electronegativity increases from left to right\n",
    "in the periodic table in the column sense; and electronegativity increases from higher to lower \n",
    "energy levels, i.e. in the periodic table row sense. As a consequence, the most electronegative \n",
    "atom is ...drumroll please... Fluorine. This acts as the origin for electronegativity. \n",
    "If we are two chose the *least* electronegative atom: It will be the one *furthest* from Fluorine on \n",
    "the periodic table.\n",
    "\n",
    "\n",
    "Example: The molecule $NO_2$ features oxygen and nitrogen. Nitrogen is further from Fluorine so\n",
    "it is less electronegative than oxygen. \n",
    "\n",
    "\n",
    "### The Five Rules of Lewis structures\n",
    "\n",
    "\n",
    "These rules are the *basic set*. There are advanced rules; but I will kick that can down the road for now.\n",
    "\n",
    "\n",
    "0. Draw the individual atoms with their dots (this is optional... 'know your actors')\n",
    "    - ...then follow the rules below on a clean slate\n",
    "1. Find the total number of valence electrons for a given molecule.\n",
    "    - For water we should get 8 valence electrons: 6 from oxygen and one each from the hydrogens\n",
    "    - For a negative ion we have an atom or molecule carrying additional charge\n",
    "        - Example: $CN^-$: 4 valence electrons for Carbon, 5 for Nitrogen, 1 for the ionic charge: 10 total\n",
    "2. Establish the \"center\" atom of the Lewis structure\n",
    "    - For 2-atom molecules there is no center\n",
    "    - For three or more atoms there is a center\n",
    "        - Place the least electronegative atom at the center\n",
    "        - Place the other atoms on the outside\n",
    "        - Caveat rule: Hydrogen is a special animal\n",
    "            - Hydrogen is always placed on the outside of a Lewis structure\n",
    "        - Example: $P\\;Cl_3$ will have a phosphorous atom at the center (check this)\n",
    "3. From this arrangement of atoms: Put 2 electrons between atoms to make covalent $\\sigma$ bonds\n",
    "    - Simple example: $H\\;Cl$ has 8 valence electrons (Hydrogen: 1, Chlorine: 7)\n",
    "        - It has two atoms so there is no center\n",
    "        - Place the first two valence electrons like **H:Cl** to make the $\\sigma$ bond\n",
    "            - Not now but at the conclusion: Re-draw each two dot bond ':' as a dash '-'\n",
    "    - We are working from a valence electron budget of 8 so 6 remain\n",
    "4. Complete the \"8 electron\" octets: First for the outside atoms, then for the center atom \n",
    "    - Hydrogen doesn't have an octet. Its outer shell can carry 2 electrons.\n",
    "        - In the above **H:Cl** example Hydrogen has 2 from the covalent bond: All set\n",
    "        - The Chlorine gets the remaining 6 electrons as 3 lone pairs: 8 total: All set\n",
    "        - We are done with the HCl structure\n",
    "\n",
    "```\n",
    "    ..\n",
    "H - Cl :\n",
    "    ..\n",
    "```\n",
    "        \n",
    "\n",
    "5. If the central atom does not have an octet: Move electrons from outer atoms to form double or triple bonds\n",
    "    - This happens when we run out of valence electrons without filling the octets\n",
    "        - There could be the alternative issue of too *many* remaining electrons\n",
    "    - The example directly following shows Rule 5 in action\n",
    "\n",
    "\n",
    "### Lewis structure building in practice\n",
    "\n",
    "\n",
    "#### Simple example: $O_2$\n",
    "\n",
    "\n",
    "- Rule 1: This is two Oxygen atoms, same electronegativity, 12 total valence electrons\n",
    "- Rule 2: Two atoms so no 'center atom' as such\n",
    "- Rule 3: One $\\sigma$ bond (2 electrons: 10 remaining)\n",
    "- Rule 4: Complete the octet for one Oxygen (6 electrons: 4 remaining)\n",
    "    - Try to complete the octet for the second Oxygen (4 electrons: 0 remaining)\n",
    "    - Now the second oxygen is 6 out of 8 octet electrons and rule 5 applies\n",
    "- Rule 5: Move two electrons from the first oxygen to create another valence bond\n",
    "    - This would be a first $\\pi$ bond\n",
    "    - Using dots we have `O::O`. Using lines we have `O=O`.  \n",
    "- Now the octets are full for both oxygen atoms and the Lewis structure is complete\n",
    "    - It is best to include all the valence electrons in the diagram, as in\n",
    "    \n",
    "    \n",
    "```\n",
    "  ..     ..\n",
    ": O   =  O :\n",
    "```\n",
    "\n",
    "(My diagrams in fixed-width font are a little klunky.)\n",
    "\n",
    "\n",
    "#### A more complicated example: Sulfur dioxide, $SO_2$.\n",
    "\n",
    "- Rule 1: 18 valence electrons\n",
    "- Rule 2: Sulfur is farther from Fluorine than Oxygen so it goes in the center\n",
    "- Rule 3: The covalent bonds use up 4 electrons, produce O:S:O, 14 valence electrons remain\n",
    "- Rule 4: 6 electrons to each Oxygen to complete outer octets and 2 electrons remain\n",
    "    - The sulfur atom has 4 electrons in the two $\\sigma$ bonds\n",
    "    - Add the 2 remaining valence electrons to sulfur for a total of 6\n",
    "- Rule 5: An asymmetric steal: Choose the left oxygen and move two electrons to the sulfur bond\n",
    "        - This is now a double ($\\pi$) bond and we have O::S:O\n",
    "        - The sulfur now has a completed octet, as do the two oxygens\n",
    "        - ...so we might imagine we are done...\n",
    "        - ...and this is where the advanced stuff comes in...\n",
    "        - ...but that is for another day"
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    "### Applying the five rules to nine Lewis structure problems\n",
    "\n",
    "\n",
    "1. $OF_2$\n",
    "2. $CH_4$\n",
    "3. $H_2$\n",
    "4. $CO_2$\n",
    "5. $BH_3$\n",
    "6. $O_3$\n",
    "7. ${NH_4}^{+}$\n",
    "8. ${SO_4}^{2-}$\n",
    "9. ${PO_4}^{3-}$"
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    "#### Oxygen and two fluorine \n",
    "\n",
    "- $OF_2$\n",
    "- 20 valence electrons\n",
    "- F:O:F\n",
    "- (6)F:O:F(6) leaves 4\n",
    "- No problem: Both F's get 3 lone pairs plus the bond; Oxygen gets two bonds plus 2 lone pairs\n",
    "\n",
    "```\n",
    "  ..  ..  ..\n",
    ": F : O : F :\n",
    "  ..  ..  ..\n",
    "```\n",
    "\n",
    "\n",
    "- Done"
   ]
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    "#### Methane \n",
    "\n",
    "\n",
    "- $CH_4$\n",
    "- 8 valence electrons\n",
    "- This draws itself\n",
    "\n",
    "```\n",
    "     H\n",
    "     |\n",
    "   H-C-H\n",
    "     |\n",
    "     H\n",
    "```\n",
    "\n",
    "- Each Hydrogen is 'duet-filled' (not octets!) and the carbon has an octet. \n",
    "- Done."
   ]
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    "#### Two Hydrogen\n",
    "\n",
    "\n",
    "- $H_2$\n",
    "- 2 valence electrons; duets only\n",
    "- `H:H`\n",
    "- Done."
   ]
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    "#### Carbon Dioxide\n",
    "\n",
    "\n",
    "- $CO_2$\n",
    "- 4 + 12 = 16 valence electrons\n",
    "- Carbon (less electronegative) in the center\n",
    "- O:C:O leaves 12\n",
    "- 6 to each oxygen to complete respective octets, no electrons remain, Carbon is at 4\n",
    "- Each Oxygen donates two electrons to create respective $\\pi$ bonds\n",
    "- `O::C::O` with 4 additional to each oxygen as 2 lone pairs\n",
    "\n",
    "```\n",
    "\n",
    "  ..           ..\n",
    "  O  : : C : : O\n",
    "  ..           ..\n",
    "```\n",
    "- Done."
   ]
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    "#### Boron, 3 Hydrogen\n",
    "\n",
    "- $BH_3$\n",
    "- 3 + 1 + 1 + 1 = 6 valence electrons\n",
    "- Hydrogen on the outside, Boron at the center\n",
    "\n",
    "```\n",
    "        H\n",
    "        ..\n",
    "  H  :  B   : H\n",
    "```"
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    "#### 3 Oxygens aka Ozone\n",
    "\n",
    "- $O_3$\n",
    "- 18 valence electrons\n",
    "- O:O:O leaves 14; 6 apiece to the outer 2 leaves 2 (outer octets are done)\n",
    "- center O gets the remaining 2 for 6 of 8\n",
    "- this suggests an asymmetry: Take 2 from one outer O to make a double bond\n",
    "- result: O::O:O\n",
    "\n",
    "\n",
    "```\n",
    "   ..   ..  ..\n",
    " : O :: O : O :\n",
    "            ..\n",
    "```\n",
    "\n",
    "\n",
    "\n",
    "- Symmetry: We could also write `O:O::O`\n",
    "- These are what are known as *resonance structures*\n",
    "\n",
    "Solution: { `O::O:O`, `O:O::O` } with two resonance structures and lone pairs as shown above"
   ]
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    "#### Nitrogen, 4 Hydrogen: A positive multiple-atom (Ammonium) ion\n",
    "\n",
    "- ${NH_4}^{+}$ is called the ammonium ion\n",
    "- The plus sign means we have dropped an electron somewhere along the way\n",
    "- Total valence electrons = 5 + 1 + 1 + 1 + 1 - 1 = 8\n",
    "- Hydrogen outside, Nitrogen in the center\n",
    "\n",
    "```\n",
    "         H\n",
    "         |\n",
    "     H - N - H\n",
    "         |\n",
    "         H\n",
    "```\n",
    "\n",
    "- Fortune smiles: This is 8 electrons and the Nitrogen octet is filled\n",
    "- Done."
   ]
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    "#### Sulfur, 4 Oxygen, a double-negative (Sulfate) ion\n",
    "\n",
    "\n",
    "- ${SO_4}^{2-}$\n",
    "- Total valence electrons is 30 (six by five) plus 2 (ionic charge) 32\n",
    "- Sulfur is further from Fluorine than Oxygen so is the less electronegative: Center\n",
    "\n",
    "```\n",
    "     O\n",
    "     |\n",
    " O - S - O\n",
    "     |\n",
    "     O\n",
    "```\n",
    "\n",
    "- Sulfur has a full octet and 24 electrons remain\n",
    "- Each of the four oxygens are 6 shy and 6 by 4 is 24\n",
    "- Again we have good fortune\n",
    "\n",
    "```\n",
    "      ..\n",
    "     :O:\n",
    "  ..  |   ..\n",
    " :O - S - O:\n",
    "  ..  |   ..\n",
    "     :O:\n",
    "      ..\n",
    "```\n",
    "\n",
    "\n",
    "- Done"
   ]
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    "#### Phosphorous, 4 Oxygen, a triple-negative (Phosphate) ion\n",
    "\n",
    "\n",
    "- ${PO_4}^{3-}$\n",
    "- Valence electron count = 5 + 6 + 6 + 6 + 6 + 3 = 32\n",
    "- Phosphorous is less electronegative than Oxygen: Goes to the center\n",
    "- Four covalent bonds and we have the same situation as above\n",
    "- In fact if we go from Sulfur to Phosphorous we lose a valence electron but gain one back with the $3-$.\n",
    "\n",
    "\n",
    "```\n",
    "      ..\n",
    "     :O:\n",
    "  ..  |   ..\n",
    " :O - P - O:\n",
    "  ..  |   ..\n",
    "     :O:\n",
    "      ..\n",
    "```\n",
    "\n",
    "\n",
    "- Done\n"
   ]
  },
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    "### Advanced material\n",
    "\n",
    "\n",
    "There are two topics that we have not incorporated yet: Formal charge and resonance structures. \n",
    "(I did mention the latter briefly for the Ozone molecule.)\n",
    "\n",
    "\n",
    "*Formal charge* in covalent bond theory is a charge assigned to an atom within a molecule.\n",
    "It is defined as a difference between the Lewis structure state of the atom and the state\n",
    "of the same atom 'by itself' (or 'free from the molecule (or 'free')).\n",
    "\n",
    "\n",
    "$q* = V - L - B/2$\n",
    "\n",
    "\n",
    "- $V$ is the number of valence electrons for the atom\n",
    "- $L$ is the number of non-bonding electrons assigned to the atom in the Lewis structure\n",
    "- $B$ is the total number of electrons shared in bonds with other atoms in the molecule\n",
    "\n",
    "\n",
    "$B$ is divided by two as the 2-electron covalent bond is considered *shared*.\n",
    "\n",
    "\n",
    "*Resonance structures* are related to a dichotomy: On the one hand the mechanical rules \n",
    "for a Lewis structure tend to imply a *location* for an electron; and on the other hand\n",
    "the rules of quantum mechanics allow for electrons to have a distributed existence that\n",
    "defies the classical concept of an object having a location. \n",
    "\n",
    "\n",
    "Here is my attempt at an unhelpful (helpful?) conversation on this topic:\n",
    "\n",
    "\n",
    "A: I have solved the Lewis structure for Ozone; but I have a problem.\n",
    "\n",
    "\n",
    "N: What is your solution? \n",
    "\n",
    "\n",
    "A: It is `O::O:O` which you will notice features both a double and a single covalence bond.\n",
    "\n",
    "\n",
    "N: And what is your problem?\n",
    "\n",
    "\n",
    "A: I could just as easily say that the solution is `O:O::O`... so how does the molecule decide?\n",
    "\n",
    "\n",
    "N: Ah, well as a practitioner of quantum mechanics I can assure you that the molecule is under no obligation to chose.\n",
    "\n",
    "\n",
    "A: But it seems to me that the two solutions are equally valid; so perhaps the 'choice' simply happens at random.\n",
    "\n",
    "\n",
    "N: Not at all. Rather the ozone molecule exists in a dual state: A superposition of your two solutions. That is: The molecule is not in strictly state `O::O:O` or state `O:O::O`. It exists in both at once.\n",
    "\n",
    "\n",
    "A: Doesn't that make it heavier? Or set up a gateway to the Multiverse? Or something? \n",
    "\n",
    "\n",
    "N: It is a bit of a cop out, but I will just claim that the mathematical solution for the configuration of electrons in this molecule is a sort of hybridization of these two symmetric configurations; hence I say \"both\".\n",
    "\n"
   ]
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    "## Valence Shell Electron Pair Repulsion (VSEPR) theory\n",
    "\n",
    "The nice thing about VSEPR is that it starts from Lewis structures. In particular remember\n",
    "that we have lone pairs in our Lewis structures in addition to 2-electron valence bonds. \n",
    "The goal is to use some heuristic (rule of thumb) arguments to come up with molecular\n",
    "structure. This is sometimes done using a sub-method referred to as the AXE method. So\n",
    "my goal here is to describe the VSEPR rationale in an arm-waving sort of way, to describe\n",
    "a key number we calculate called the *steric* number, and then to proceed to giving the \n",
    "AXE method to arrive at some example molecular structures. \n",
    "\n",
    "\n",
    "One idea to read and ponder before jumping in here: Lewis structure theory is all about\n",
    "building molecules from an economy of electrons: *Where do the electrons go?* In contrast\n",
    "VSEPR is ultimately about the geometry of molecules containing three or more atoms, a\n",
    "geometry of a 3D tree graph where the **edges** are covalent bonds. \n",
    "\n",
    "\n",
    "### Part 1 A conception of orbitals\n",
    "\n",
    "In VSEPR theory we have a metaphor of orbitals as ballons: Volumes that are blob-like,\n",
    "that keep with the notion that electrons exist in a distributed (pdf-like) space. We can\n",
    "say that *orbitals* are regions of space where one has a high probability of \"finding\" an\n",
    "electron and VSEPR is concerned with the geometric configuration of these blobby balloons,\n",
    "these orbitals. The balloons are tied together at the atom's nucleus; each can contain two\n",
    "electrons (spin up and spin down, whatever that means) and because electrons are all \n",
    "negatively charged these orbitals repel one another. Now let's go a little further with\n",
    "a taxonomy of orbitals. \n",
    "\n",
    "\n",
    "Each lone pair orbital we presume contains two electrons; and it exists relatively close\n",
    "to the nucleus because there is no other atom drawing it out *away* from the nucleus. \n",
    "Because this lone pair orbital is compact it has a highly repulsive nature. But we try \n",
    "and be encouraging and kind despite this observation."
   ]
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    "Moving on to $\\sigma$-bond orbitals: These are also occupied by two electrons. They are\n",
    "the longest type of covalent bond, the opposite of lone pair orbitals being compact. We\n",
    "can say that they are the weakest of the three valence bonds we learned about in Lewis \n",
    "structures. \n",
    "\n",
    "\n",
    "So far so good; now for the twist. \n",
    "\n",
    "\n",
    "In VSEPR theory: Double $\\pi$-bonds are counted as a single chemical bond with capacity \n",
    "for four electrons. Notice this is not the same as a four-electron *orbital* because I\n",
    "am not sure that would be permitted language. The reason is that orbitals are related\n",
    "more closely to quantum mechanics and quantum mechanics is very strict about electron\n",
    "double occupancy. So instead of using the word *orbital* I will stick with a (double)\n",
    "valence bond which is stronger (and therefore shorter -- rubber band analogy) than\n",
    "a single bond. Respectively $\\pi$ and $\\sigma$ bonds. \n",
    "\n",
    "\n",
    "Finally for VSEPR we also have triple $\\pi$-bonds which are stronger/shorter still. \n",
    "But these are not as compact as lone pair orbitals. So the respective repulsion\n",
    "of other bonds/orbitals is maximum for lone pairs, then triple-$\\pi$, then double-$\\pi$,\n",
    "then single $\\sigma$ valence bonds. This comes into play as a secondary factor on\n",
    "the molecular geometry we are attempting to conjure up. \n",
    "\n",
    "\n",
    "If all of the above business about **E**lectron **P**air **R**epulsion was *secondary*:\n",
    "What is the *primary* thing? Answer: The *steric number* which we now propose to \n",
    "define. "
   ]
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    "### Part 2 Steric number\n",
    "\n",
    "\n",
    "The steric number is calculated for a central atom in a molecule. The leaf atoms are\n",
    "important in this counting process but they do not have their own steric numbers.\n",
    "\n",
    "\n",
    "Counting the steric number: Single $\\sigma$ valence bonds count 1. A double $\\sigma, \\pi$ bond counts as 1. \n",
    "A triple $\\sigma, \\pi, \\pi$ bond counts as 1. A lone pair counts as $1. Considering a central (not leaf)\n",
    "atom: The sum steric number dictactes the bond geometry where each bond/lone pair repels the others.\n",
    "Hence a carbon atom surrounded by four valence bonds to Hydrogen will have a tetrahedral shape. "
   ]
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    "## Linear Combinations of Atomic Orbitals (LCAO)\n",
    "\n",
    "\n",
    "This topic originates from the notion that orbitals describe the space occupied (in a cloudy or probabilistic sense)\n",
    "by an electron; and in fact any such orbital can carry 1 or 2 electrons (of opposite spin) and no more.\n",
    "This can be stated another way: We describe an electron in an atom or molecule in terms of three-dimensional\n",
    "volumes called orbitals. We begin with atoms where orbitals are centered on the nucleus; but we find when atoms\n",
    "bind together that orbitals can be distributed across two or even more nuclear 'origin points'. \n",
    "\n",
    "\n",
    "These notes are concerned with the atomic orbitals specifically of Hydrogen. Because hydrogen is\n",
    "a simple enough system its orbitals can be solved for mathematically. Chemists then go on to say\n",
    "'suppose all atoms in a given molecule have at their disposal Hydrogen orbitals centered on their\n",
    "respective nuclei... albeit maybe scaled because of the larger mass / more protons in the nucleus /\n",
    "presence of inner shell electrons. But the basic concept is the Hydrogen orbitals act like basis\n",
    "vectors in linear algebra, like i-hat, j-hat, k-hat: They can be combined using coefficients into\n",
    "a linear combination. Very well: Let's take a further leap supposing that *molecular* orbitals are\n",
    "formed as linear combinations of atomic orbitals: Again centered on the multiple atomic nuclei. \n",
    "This is the notion of LCAO.\n",
    "\n",
    "\n",
    "In terms of first-year chemistry: Since we see the phrase \"linear combinations\" we imagine that the\n",
    "rules of linear algebra might apply. This is 100% correct: The primary method for solving an LCAO \n",
    "problem is called the **Hartree-Fock method** and it involves diagonalizing a matrix of coefficients\n",
    "in a repetition cycle that will eventually minimize the energy of the molecule. The rationale is \n",
    "that in nature things tend to head towards a minimal energy state. In atomic and molecular systems\n",
    "this is called the ground state. "
   ]
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    "### Notes from [this overview video](https://www.youtube.com/watch?v=oWzuqLl6Ito)\n",
    "\n",
    "\n",
    "There is a lot of material here... this is just some light notes from\n",
    "a first watch. I would have to watch this a couple more times plus \n",
    "the referenced related videos.\n",
    "\n",
    "\n",
    "* Positive features\n",
    "    * Good narrative style\n",
    "    * Touches on a lot of the ideas; context\n",
    "        * Specifically: Introduces MO bonding / anti-bonding orbitals\n",
    "        * Shows how single electron orbitals can distribute across *many* atoms\n",
    "    * Provides links to related videos\n",
    "    * Contrasts MO theory with valence bond theory and Lewis structures\n",
    "* Negative features\n",
    "    * This is not a thorough, logical progression of steps\n",
    "        * It is an overview so it can feel too ambitious\n",
    "    * Makes it seem like MO theory is all some sort of magical mystical art\n",
    "\n",
    "\n",
    "\n",
    "* Period 1 and Period 2 Homonuclear diatomic molecules H-H and so forth: A practical starting topic in LCAO-MO theory\n",
    "* Why do some atoms break the octet rule, others don't?\n",
    "* Orbital hybridization for example $sp^3$ creates orbitals better suited to molecular bonds (valence bond theory)\n",
    "    * Motivation in history: Linus Pauling, 1931\n",
    "        * observation: symmetric structure of methane\n",
    "        * concept: should be asymmetric: 3 p-bonds plus an s-bond\n",
    "        * resolve this by proposing hybridized orbitals\n",
    "    * For Hydrogen: 2s and 2p orbitals represented an excited state relative to the 1s ground state\n",
    "        * Hydrogen's contribution to valence bonds is through the 1s (spherical) orbital\n",
    "        * Heavier atoms (C and so on) use hydrogen 2s and 2p for hybridization\n",
    "\n",
    "\n",
    "\n",
    "* Number of atomic orbitals before molecule = number of molecular orbitals\n",
    "* Systems with less potential energy are more stable: They have less energy available to change form\n",
    "* Introducing the anti-bonding orbital\n",
    "\n"
   ]
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    "### Anti-bonding orbitals\n",
    "\n",
    "\n",
    "[Wikipedia overview](https://en.wikipedia.org/wiki/Antibonding_molecular_orbital)\n",
    "\n",
    "\n",
    "* Anti-bonding weakens the chemical bond between two atoms.\n",
    "* It is not a sure thing to break the molecule apart; but it does affect the total energy.\n",
    "* Anti-bonding (high energy) orbitals fill only after bonding (lower energy) orbitals fill\n",
    "\n",
    "\n",
    "Consider two hydrogen atoms far apart: They have one electron each, occupying respective 1s orbitals. \n",
    "Now they move close together and the two atomic orbitals transform into two molecular orbitals: 1s and 1s*.\n",
    "These two new orbitals conserve the total orbital count. \n",
    "\n",
    "\n",
    "The two electrons can both drop down into the 1s molecular orbital. This lowers the total energy\n",
    "in comparison to that of the two free atoms; so $H_2$ is a stable bound state of molecular hydrogen.\n",
    "The 1s* orbital contains no electrons. \n",
    "\n",
    "\n",
    "Now do the same thing with Helium: Both 1s and 1s* are occupied (4 electrons total). The anti-bonding\n",
    "1s* has a big energy gap relative to the energy of two separate Helium atoms. The bonding 1s orbital\n",
    "has a less-big negative energy gap relative to the energy of two separate Helium atoms. In sum the \n",
    "$He_2$ molecule has more energy than He + He: The Helium molecule is unstable and breaks apart.\n",
    "\n",
    "\n",
    "\n",
    "### $\\sigma$ and $\\pi$ bonds\n",
    "\n",
    "\n",
    "Note: nodal planes are planes that cut orbitals at the point where they become zero. A p orbital\n",
    "has a nodal plane passing through the nucleus, for example.\n",
    "\n",
    "\n",
    "From wikipedia: \n",
    "\n",
    "#### $\\sigma$ bonds\n",
    "\n",
    "\n",
    "* $\\sigma$ bonds are the strongest type of covalent bond\n",
    "    * This is due to a maximization of overlap between bonding orbitals along the bond axis\n",
    "    * This is in contrast to $\\pi$ bonds that overlap (less) in a side-by-side manner\n",
    "* $\\sigma$ bonds involve head-on matching of orbitals from two atoms along the inter-nuclear (bond) axis\n",
    "    * This axis is often labeled the z-axis\n",
    "* $\\sigma$ bonds are symmetric (unchanged by / tolerant of) rotation about the bond axis\n",
    "    * This is, again, in contrast to $\\pi$ bonds\n",
    "* $\\sigma$ bonds are commonly of type **s+s**, **pz+pz**, **s+pz** and (invoking d-orbitals) **dz2+dz2**\n",
    "    * These are hybridized orbitals\n",
    "* $\\sigma$ bond hybridized molecular orbitals do not have a nodal plane (where they go to zero)\n",
    "* $\\sigma^*$ anti-bonding molecular orbitals *do* have a nodal plane\n",
    "    * This resides between the two nuclei and is perpendicular to the z-axis\n",
    "\n",
    "\n",
    "#### $\\pi$ bonds\n",
    "\n",
    "* $\\pi$ bonds are built from overlap of parallel p orbitals: Side by side dumbells\n",
    "    * Another good comparison is two boats rafted together\n",
    "    * Wikipedia has a helpful remark here:\n",
    "        * \"Molecular fragments joined by a pi bond cannot rotate about that bond without breaking the $\\pi$\n",
    "        * bond because rotation involves destroying the parallel orientation of the constituent p orbitals.\"\n",
    "* $\\pi^*$ anti-bonding orbitals have two nodal planes\n",
    "    * one is a nodal plane through both nuclei perpendicular to the atomic p orbitals\n",
    "    * one is in the xy-plane between the two nuclei\n",
    "\n",
    " \n",
    "#### Delocalization and resonance\n",
    "\n",
    "\n",
    "Delocalization refers to electrons that are not associated with strictly one or two atoms; \n",
    "but may have an orbital that extends across three or more atoms. This is related to the concept\n",
    "of resonance and is seen in benzene (6-Carbon) rings, schematically as a circle inside the hexagon.\n",
    "\n",
    "\n",
    "Note: An isomer is an alternative structure for a molecule using the same atoms. This is *not* the\n",
    "same thing as resonance structures. Resonance structures vary from one another in placement of \n",
    "electrons in the bond assembly *without* changing the arrangement of the atoms i.e. the graph of \n",
    "the atoms within the molecule. "
   ]
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    "## Molecular Orbital Theory\n",
    "\n",
    "\n",
    "[MIT Lecture](https://www.youtube.com/watch?v=O192jrR80oo)\n",
    "\n",
    "\n",
    "* wave function is another name for orbital\n",
    "* we know atomic orbitals come together to form molecular orbitals\n",
    "    * this is the LCAO concept of course\n",
    "    * electrons become delocalized\n",
    "* we create two types of molecular orbitals: bonding (g) and anti-bonding (u)\n",
    "* the number of orbitals is *conserved*: $n$ atomic orbitals become $n$ MOs\n",
    "    * anti-bonding is key to this conservative $n = n$ picture\n",
    "    * 'constructive' and 'destructive' interference g and u respectively\n",
    "* Simple starting example (Hydrogen like): $1s_a + 1s_b = \\sigma_{1s}$\n",
    "    * Constructive, bonding\n",
    "    * Defines a bonding axis\n",
    "\n",
    "\n",
    "I interrupt these notes to tell you a concept that brings together some\n",
    "things you already know. This concerns the statement \"The square of the\n",
    "wave function is the probability density function (pdf).\" This is not\n",
    "going to be rigorous but it may help a bit. The pdf defines \"where the\n",
    "electron is likely to be\" as a volume of space associated with the \n",
    "molecule as \"the origin\". But how can two atomic orbitals be added\n",
    "together destructively to create nodal points where the wave function\n",
    "goes to zero? The answer is that the wave function is ***complex valued***.\n",
    "Let's suppose that at a particular point the \"a\" atomic orbital or\n",
    "wave function has the value $0.3 e^{i\\frac{\\pi}{6}}$ and the \"b\" atomic\n",
    "orbital or wave function has the value $0.3 e^{i\\frac{-5\\pi}{6}}$. \n",
    "If we square these and take the absolutely they both give us 0.09\n",
    "which is a nice positive number. But if we superimpose the two \n",
    "wave functions first by adding the two values: They happen to be\n",
    "\"out of phase\" so their sum is zero and the resulting square has\n",
    "magnitude zero. So wave functions are complex valued and that is how\n",
    "the math gets us into *constructive* and *destructive* forms of \n",
    "interference.\n",
    "\n",
    "\n",
    "It is a philosophical matter to ask whether the atoms decided to play\n",
    "dice using complex numbers or if the atoms are doing something else and\n",
    "this complex number business is simply a convenient method invented\n",
    "by the humans to describe that behavior. \n",
    "\n",
    "\n",
    "Ok back to our notes on this lecture...\n",
    "\n",
    "\n",
    "* constructive interference continued\n",
    "    * stable binding lower energy sweet spot distributed charge happy\n",
    "    * you must add energy to $H_2$ to dissociate the two atoms\n",
    "* Now under the flag of conservation of orbitals: $1s_a - 1s_b = {\\sigma_{1s}}^*$\n",
    "    * This produces a node (zero amplitude) between the two nuclei\n",
    "        * The sidebar remark example above illustrates a node\n",
    "    * Now with the antibonding orbital we have a *higher* energy\n",
    "    * to be continued\n",
    " \n",
    "#### Problem 6 MO energy level diagrams A and B\n",
    "\n",
    "\n",
    "Preparation note: If all electrons in a molecule are paired: Diamagnetic. If there are unpaired electrons: Paramagnetic.\n",
    "\n",
    "\n",
    "Preparation note 2: This seems rather esoteric... but B-B and C-C \n",
    "molecules feature s-p orbital hybridization... somehow this implies\n",
    "that a B-C+ molecule will favor $\\pi$ bonds.\n",
    "\n",
    "\n",
    "* a) Answer is Y\n",
    "    * As we go across a period (as in the n = 2):\n",
    "    * Added protons in the nucleus give more apparent positive charge\n",
    "    * This binds outer shell electrons more closely\n",
    "    * Fluorine has lower $s2$ energy than Oxygen so \"Y\" is the answer\n",
    "* b) Answer is 2\n",
    "    * OF+ Lewis structure shows no unpaired electrons\n",
    "    * That is: We can satisfy the octet rule with $O=F$\n",
    "    * But this is incorrect from the MO perspective\n",
    "    * MO Perspective\n",
    "        * We have 12 valence electrons in OF+\n",
    "        * Look in the center of both diagrams A and B\n",
    "        * Here we have a succession of orbitals by energy\n",
    "        * $\\sigma_{2s}$ gets 2 electrons\n",
    "        * Next up: $\\sigma^*_{2s}$ gets 2 electrons!\n",
    "        * Next: Two $\\pi$ plus one $\\sigma$ bonding: 6 electrons\n",
    "        * This leaves us with 2 electrons yet to add\n",
    "        * The next  orbital up in the sequence are two $pi^*$ orbitals\n",
    "            * These are anti-bonding orbitals\n",
    "            * They have equivalent energy levels\n",
    "            * They fill according to \"Hund's rule\" one electron in each\n",
    "            * So the answer is: Two unpaired electrons according to MO\n",
    "* c) Answer is X\n",
    "    * Same rationale as part a)\n",
    "    * Boron has fewer protons; less electronegative\n",
    "    * Equivalently: Less effective nuclear charge apparent in Boron\n",
    "    * Therefore its 2p and 2s energies will be higher\n",
    "* d) Answer is B\n",
    "    * BN is diamagnetic meaning there are unpaired electrons\n",
    "    * There are 8 valence electrons total\n",
    "    * Drop these into diagrams A and B to see which one yields unpaired electrons\n",
    "    * Result: Diagram B\n",
    "* e) Answer is Paramagnetic\n",
    "    * BC+ has six valence shell electrons\n",
    "    * Diagram A would give us 2 unpaired electrons: Paramagnetic\n",
    "    * Diagram B would give us no unpaired electrons: Diamagnetic\n",
    "    * Prep note 2 indicates Diagram A more likely, hence the answer"
   ]
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    "## Types of Reactions\n",
    "\n",
    "\n",
    "For our purposes (avoiding nuclear chemistry) a chemical reaction will involve a redistribution of \n",
    "electrons (and implicitly chemical bonds) together with a change in *energy*. The chemical reaction\n",
    "starts with some *reactants* (aka *reagents*) and arrives at some *products*. \n",
    "\n",
    "\n",
    "Reactions occur at some rate which is governed by quantity of reactants and temperature... and \n",
    "by other things, particularly presence of a *catalyst*. \n",
    "\n",
    "\n",
    "Reactions that produce heat are *exothermic*. \n",
    "\n",
    "\n",
    "Reactions that require heat are *endothermic*. \n",
    "\n",
    "\n",
    "**I interrupt to say: More properly *energy*, usually in the form of *heat* ($\\Delta$) \n",
    "or *light* ($h\\nu$).**\n",
    "\n",
    "\n",
    "Reactions in which oxidation and reduction occur are called *redox* reactions. Antonym: *non-redox*.\n",
    "\n",
    "\n",
    "Most simple redox reactions are in one of three sub-categories: *Combination*, *Decomposition*, or *Single Replacement*.\n",
    "\n",
    "\n",
    "Chemical reactions may involve any combination of four basic characteristics:\n",
    "\n",
    "\n",
    "- Evolution of a gas\n",
    "- Precipitation\n",
    "- Change in temperature\n",
    "- Change in state, for example melting\n",
    "\n",
    "\n",
    "Equilibrium reactions are chemical reactions that can proceed in both directions:\n",
    "Reactants to products and vice versa. \n",
    "\n",
    "\n",
    "A reaction equation features stoichiometric balance: The quantities of each atom seen\n",
    "on opposite sides of the reaction arrow are equal. This can be achieved by solving for integer \n",
    "coefficients in the manner of a Diophantine problem."
   ]
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   "source": [
    "## Acids and Bases"
   ]
  },
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   "source": [
    "## Displacement and Precipitation Reactions"
   ]
  },
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   "source": [
    "## Redox Reactions"
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   "metadata": {},
   "source": [
    "## Metal and Halogen Reactivity"
   ]
  },
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   "execution_count": null,
   "id": "48aa9bec-e3a1-4b5e-b4ef-c13bb1632885",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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