Molecular Geometry & Bonding
what the shape decides
Series

Dipole — the series

14 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. What the group settles. For each molecule, the point group found from its coordinates and the two properties that follow from the group alone. Neither column required knowing anything about the bonds.

    The dipole is not a sum of bonds

    Adding bond dipoles as vectors gets the easy cases right and rests on a quantity with several incompatible definitions. The symmetry argument is exact, needs no electronegativities, and says when the answer must be zero.

    part 1 · wrong
  2. One dihedral, three point groups. Hydrogen peroxide built at 13 values of its dihedral angle, with the point group searched for from the coordinates at each. The group is C2v when the hydrogens eclipse, C2h when they are anti, and C2 at every angle strictly between. The rails below are what the group settles on its own: the molecule may be polar except at the anti arrangement, and it is chiral except at the two ends. No energy is computed anywhere, and the marked angle is the measured one rather than a minimum found here.

    One coordinate, three point groups

    Hydrogen peroxide has four atoms and one soft internal coordinate. Turning it from nought to a hundred and eighty degrees takes the molecule through C2v, C2 and C2h — so it is chiral at every angle but two, and forbidden a dipole at exactly one of them.

    part 2 · shape
  3. The moment that does not depend on where the origin is. For each molecule, the dipole and the largest quadrupole component about the centre, and the same two about an origin moved 1.6 bohr away. The lowest non-vanishing moment is unchanged in every row and the one above it moves in every row. The 4 non-polar molecules here have a quadrupole that is a property of the molecule; the polar ones have one that is a property of a choice.

    What a dipole cannot tell apart

    A dipole moment is three numbers extracted from a whole charge distribution, and enormously many distributions give the same three. The moment above it is not even a property of the molecule unless the one below it vanishes — which is why a quadrupole is quoted with an origin and a dipole is not.

    part 3 · shape
  4. The bond sum, the measurement, and what is left over. For each pyramid: the vector sum of three bond moments estimated from the electronegativity difference and the measured geometry, the measured dipole, and the difference between them. Positive is towards the lone pair. Ammonia's bonds point that way and nitrogen trifluoride's point the other, which is why the trifluoride's far more polar bonds give it a dipole six times smaller. The leftover is 0.58 D for ammonia and at least 1.48 D for the trifluoride, so it is not one lone pair's property.

    The lone pair is not the missing term

    Ammonia's dipole is 1.47 debye and nitrogen trifluoride's is 0.235, although the N–F bonds are far more polar than the N–H ones. The bond sums explain the reversal exactly and point in opposite directions — and the lone pair that is supposed to make up the difference has to be worth 0.58 debye in one molecule and at least 1.48 in the other.

    part 4 · shape
  5. Face to face repels, edge to face attracts. Two benzene molecules 5 Å apart, one turned against the other, with only the quadrupole interaction between them. Stacked, the two negative faces meet and the energy is +8.8 kJ/mol; perpendicular, one molecule's positive rim meets the other's negative face and it is -4.6. The sign changes on the way, so the preference is not that one arrangement is weaker — it is that they are opposite. This is a molecule with no dipole moment at all.

    Zero dipole is not no interaction

    Benzene's dipole moment is exactly zero at every origin, and its quadrupole moment is large enough to decide a crystal structure. Two benzenes face to face repel by nine kilojoules a mole; edge to face they attract, and the sign changes on the way between.

    part 5 · shape
  6. The dipole moment, and how many bands there are. For each of five molecules: the dipole moment of the point-charge model, the number of modes whose dipole derivative does not vanish, and the largest derivative. The molecules with no dipole at all have the most active bands, which is the whole of the argument.

    A dipole is not what an infrared spectrum sees

    Carbon dioxide has no dipole moment at all and three of its four modes are infrared active. Methane has none and six of nine; boron trifluoride none and five of six. Water, which has the largest dipole of the five, has three modes and three bands — and its dipole predicted neither number.

    part 6 · shape
  7. What boron trifluoride's bands are strong in, and what they move. Every infrared-active mode of boron trifluoride, with its band strength and the root-mean-square displacement of its atoms in the zero point, each scaled to its own largest. The two do not order the modes the same way — the rank correlation between them is 0.2 — and the strongest band belongs to the mode at 719 cm⁻¹, in which 89.93 per cent of the motion is the lightest atom's. Every mode moves the same weighted amount of mass, exactly, so that is not what separates them either.

    The mode that moves least radiates most

    Boron trifluoride's strongest infrared band is the one in which the fluorines barely move: ninety per cent of the motion belongs to the boron, which is a fifth of the molecule's mass. The mode that moves the most mass is nine and a half times weaker. Across five molecules the rank correlation between band strength and how far the atoms actually go runs from +1 to −0.66, and every normal mode carries exactly the same weighted motion by construction.

    part 7 · shape
  8. Four tables, one answer, and a reason it could not be otherwise. The intensity–motion correlation — infrared intensity against how far the atoms move — computed with charges from four published electronegativity tables. Every molecule gives the same number on all four, to machine precision, because each is made of two elements: its charges are one number times a fixed pattern, a change of table changes only that number, and a rank correlation does not notice a rescaling. The dipole moments beside them do notice, which is the check that the tables are genuinely different.

    The table that could not have mattered

    Charges taken from one of four electronegativity tables invite a worry, because the tables disagree with each other. They do disagree — hydrogen cyanide's dipole runs over a factor of eleven between them — and for the molecules in question the worry could not have applied, because a molecule of two elements has charges that are one number times a fixed pattern and a rank correlation does not notice a rescaling.

    part 8 · shape
  9. Twelve hydrogen bonds, four tables. The bond dipole of hydrogen against each partner, in debye, on each of the four tables after all four are anchored to the same hydrogen–fluorine separation. Positive is hydrogen at the positive end. A bond whose marks straddle the axis is one the tables disagree about the direction of, and there are 3 of them.

    Four tables and one molecule to disagree about

    A molecule of two elements is provably safe from the choice of electronegativity table, and a series down a group should land where the tables do disagree. It does. All four agree that hydrogen iodide is the exception — and they disagree about what its dipole is by 1.245 debye, which is two and a half times the 0.448 that was measured.

    part 9 · shape
  10. One number separates the disputed bonds from the agreed ones. Every bond in the collection, ordered by how big its electronegativity difference is as a fraction of each table's range. The six the tables disagree about are all below 0.0507; the 25 they agree about are all above 0.0584. Nothing lies between, and which elements a bond joins does not enter — a dispute is what happens when the difference is small enough that the tables' own disagreement about it is larger.

    A dispute is a small difference

    Four electronegativity tables disagree about the direction of a quarter of a set of hydrogen bonds, and the interhalogens look like the natural test: five compounds with no hydrogen in them, to say whether the disagreement is about one awkward element or about the whole idea. All four tables agree about every interhalogen. They agree because the differences are large, and that is not a fact about halogens.

    part 10 · shape
  11. The gap closed when the other hundred and twenty-two pairs arrived. Every pair sorted by the size of its normalised electronegativity difference: the 31 common bonds on the left, all 153 pairs the four tables cover on the right. Filled marks are pairs the four tables put on different sides of zero. On the small set they are the 6 smallest with nothing in between; on the full set 27 uncontested pairs sit below the largest contested one.

    A gap that was a choice of bonds

    One number separates every disputed bond from every agreed one with nothing in between — on thirty-one bonds, which is enough to see a gap and not enough to know it is real. It is not real. The line is in the same place on all one hundred and fifty-three pairs the four tables cover, and twenty-seven agreed pairs now sit below the largest disputed one.

    part 11 · shape
  12. Two quantities, and the best straight line between the two kinds of pair. Every pair the four electronegativity tables all cover, by how much the two elements differ and by how much the tables disagree about them. Filled points are pairs whose sign the tables dispute. The line is the boundary that misclassifies fewest — 3 of 153, against 6 for the best rule using the difference alone. It slopes upward, which is the mechanism: more dispute buys a larger difference and still leaves the sign in doubt.

    Two numbers caught what one could not

    No single threshold on the electronegativity difference separates the bond pairs whose polarity the four tables dispute from those they agree on — the best misses six of a hundred and fifty-three. Adding how much the tables disagree halves that to three and catches every disputed pair. And three is what four coin-flips produce: the rule flags nineteen pairs, an eighth of which should look agreed for no reason at all, which is 2.375 against the three observed.

    part 12 · shape
  13. Correlated tables stop cancelling each other's luck. The number of the 19 flagged pairs expected to show all tables agreeing by chance, against the number of tables, for four correlations between them. Independent tables halve it with every table added, and it falls below the 0.0513 at which one agreement would be significant at ten tables. Correlated tables share part of their luck, and the expectation falls only as a power of the panel size: at a correlation of a half it is still 0.095 with four hundred.

    No panel of this kind can find an exception

    Four electronegativity tables leave three exceptions to a rule for which bond polarities they dispute, and four independent coins would produce two and a half. How many tables before one exception would mean something? Ten, if tables were coins. Near the boundary they are not: they correlate at 0.32, two of the four are nearly one table, and the requirement becomes seventy-three — with a plausible range running past any panel that could exist.

    part 13 · wrong
  14. Letting disputed pairs through narrows the rule and barely moves the panel. Over 61,353 straight boundaries in the plane of difference and dispute, the fewest pairs any boundary flags for each number of disputed pairs it lets through (bars), and the number of tables that flagged count needs at the measured correlation of 0.325 (dots). Missing none, nineteen is the least — the published rule. Each miss saves one or two flags and two to five tables; at eight misses, half the disputed pairs, the rule flags 8 and still needs 46 tables.

    The rule is not the lever

    One exception among nineteen flagged bond pairs would need seventy-three electronegativity tables to mean anything, and a rule that flagged fewer pairs looked like the way to need fewer. Searched over sixty-one thousand boundaries, nineteen is already the fewest that misses no disputed pair; each missed pair buys a table or five; and a rule flagging a single pair would still need fourteen. The requirement lives in the correlation between the tables, which moves it nine times as far as any rule can.

    part 14 · wrong

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