use of visualization of the ESP is discussed inSection 5.5.6). Values of Mulliken
and L€owdin bond orders, and Mulliken, natural and ESP atom charges, are com-
pared in Table5.15, for hydrogen fluoride. We see that the Mulliken charges vary
considerably, but apart from the STO-3G values, the electrostatic charges vary very
little, and the natural charges little, with the level of calculation. Bond orders,
however, are quite sensitive to the level of calculation. The utility of charges and
bond orders lies not in their absolute values, but rather in the fact that a comparison
of, say, L€owdin charges or bond orders, calculated at the same level for aseriesof
molecules, can provide insights into a trend. For example, one might argue that the
electron-withdrawing power of a series of groups A, B, etc. could be compared by
comparing the C/C bond orders in A–CH¼CH 2 , B–CH¼CH 2 , etc. Bond orders
have been used to judge whether a species is free or really covalently bonded, and
have been proposed as an index of progress along a reaction coordinate [ 266 ].
5.5.4.5 Atoms-in-Molecules
A method of population analysis that may be less arbitrary than any of those
mentioned so far is based on the theory of atoms in molecules, designated AIM,
or the quantum theory of atoms in molecules, QTAIM. This was developed by
Bader and coworkers, and is based on the mathematical partitioning of molecules
into regions which correspond to atoms. The concept may have developed from
Berlin’s ca. 1950 work on partitioning molecules into “binding” and “antibinding”
regions [ 267 ], cited by Bader in a 1964 paper on electron distribution [ 268 ]. The
first specific assertion that atoms in molecules in a sense retain their identities rather
than dissolving into a molecular pool of nuclei and electrons seems to have been
made even before the use of the terms encapsulated in the AIM or QTAIM
acronyms: in a 1973 paper by Bader and Beddall the question “are there atoms in
molecules?” was posed and answered in the affirmative [ 269 ]. An early review
(1975) proposed “a return to....’the atoms in molecules’ approach to chemistry”
(“return” in the sense of focussing on atoms rather than on bonds, which latter had
Table 5.15 Comparing Mulliken, electrostatic potential and natural charges, and Mulliken and
L€owdin bond orders, at various levels, for hydrogen fluoride. The geometry used in each case
corresponds to the method/basis set for that charge or bond order, but any reasonable geometry
should give essentially the same results. There are no experimental data
Charge on H (¼#charge on F) Bond order
Level Mulliken Electrostatic Natural Mulliken L€owdin
HF/STO-3G 0.19 0.28 0.23 0.96 0.98
HF/3–21G() 0.45 0.49 0.5 0.78 0.93
HF/6–31G 0.52 0.45 0.56 0.72 0.82
HF/6–31G 0.39 0.45 0.56 0.86 1.07
HF/6–311G 0.32 0.46 0.54 0.95 1.32
6–31+G* 0.57 0.48 0.58 0.64 0.75
6–311++G* 0.3 0.47 0.55 0.98 1.27
MP2/6–31G 0.52 0.45 0.56 0.72 0.81
5.5 Applications of the Ab initio Method 353