Atomic Charges

The concept of atomic charge--the net electronic and nuclear charge on each atom-- is frequently used by chemists to rationalize observed chemical behavior. In reality, this is not a measurable physical property, since the electrons are a diffuse charge distribution that can arbitrarily be assigned to any atomic center. Nevertheless, because of the general utility of atomic charges, a number of methods have been developed for calculating them from quantum chemical wave functions.

These methods fall into two broad categories--methods based on the orbital occupancy and methods involving spatial decomposition of the overall electron distribution. The latter method has extensively developed by Bader (see bibliography), but orbital-based methods are the most prevalent methods.

The most widely used charge partitioning scheme are Mulliken Populations which assign charge to an atomic center on the basis of the total electron density in basis functions located on that center.

This partitioning is often undermined by the large spatial extent of basis functions included in many modern basis sets. E.g. the diffuse p function on center A actually has the centroid of its density near center B, but all charge in that function is assigned to A.

A related problem is that Mulliken Populations are extremely basis set dependent. Seemingly innocuous basis set changes can lead to very large shifts in Mulliken Populations, even when the basis sets are very large.

Finally, Mulliken Populations can show large, spurious changes as the molecular configuration is changed.

Natural Population Analysis (NPA) was developed by Reed, et al. to address these and other problems with Mulliken Populations. The NPA algorithm involves partitioning the charge into atomic orbitals on each center, constructed by dividing the electron density matrix into sub-blocks with the appropriate symmetry. NPA is much less basis set dependent than Mulliken Populations.

An extension of NPA is Natural Bonding Analysis which partitions the NPA charge into core orbitals, bonding orbitals, lone pairs, and Rydberg states.

Example Hydrogen Fluoride NPA and NBO results (HF/6-31G*):

			(------	Natural	Population	------)
Atom	No	Natural Charge	Core	Valence	Rydberg	Total
F H	1 2	-0.55552 0.55552	1.99999 0.00000	7.54644 0.44406	0.00909 .00042	9.55552 0.44448
* Total	->	0.00000	1.99999	7.99051	0.00951	10.00000

Natural Bond Orbitals (Summary):

NBO	Occupancy	Energy
1. BD (1) F 1-H 2	2.00000	-26.09745
2. CR (1) F 1	1.99999	-26.9745
3. LP (1) F 1	2.00000	-0.62916
4. LP (1) F 1	2.00000	-0.62916
5. LP (1) F 1	1.99959	-1.40890
6. RY* (1) F 1	0.00000	1.54280
7. RY* (1) F 1	0.00000	4.02307

Basis Set Dependence of Atomic Charge Methods

Example: Chlorine atomic charge in CH3Cl, HF Mulliken and NPA charges calculated with many basis sets (at the HF/6-31G* optimized geometry).

NPA Charges usually corroborate qualitative chemical concepts. E.g. electron donating and withdrawing moieties on benzene (HF/6-31G** NPA):

Sometimes atomic charges can lead to novel and/or ambiguous interpretations of chemical phenomena. E.g. what role does resonance play in the 19 kcal/mole C-N rotational barrier in formamide?

Method	O	N	C	O	N	C
Mulliken Pop.	-0.58	-0.73	.57	-0.48	-0.73	0.48
NPA	-0.71	-0.95	.67	-0.63	-1.01	0.67
CHELPG	-1.39	-1.48	1.98	-1.34	-1.22	1.76

For more detailed overview see review by Bachrack, 1994; and for more on the use of atomic charges for interpreting resonance effects, see refs by Wiberg, et al, and Streitweiser, et al.

The Hartree-Fock procedure produces a set of occupied and unoccupied molecular orbitals:

These orbitals can provide chemical insight, but must be used with some considerations:

1) The canonical orbitals produced by the HF procedure tend to be very delocalized and hence difficult to relate to simple VB models.

2) The overall wave function is invariant to unitary transformations of the occupied or virtual molecular orbital sets, so that the shape of individual MO's is in some sense arbitrary (which is not true for the electron density).

This is the basis of creating localized orbitals which more closely correspond to our chemical intuition.

3) The simple picture of 1-electron MO's is lost in more sophisticated multi-configuration wave functions.

Because of the difficulties in calculating accurate reaction barriers, there is a long history of relating reaction rates to the electronic properties of the reactants and products.

I.E. to relate [[Delta]]G^{[[daggerdbl]]} to some properties of reactants A and B:

The best known of these strategies is Fukui's Frontier Molecular Orbital (FMO) method which perturbatively considers interactions of filled MO's on one reactant with empty MO's on the other. This is usually simplified to a simple equation of the HOMO and LUMO orbitals:

FMO theory has been used in many studies to predict reactivities and stereofacial selectivities, although it has been seriously criticized by Dewar (M.J.S. Dewar, J. Mol. Struct. (Theochem), 200, pp. 301-323 (1989)).

The HF molecular orbital energies ([[epsilon]]) are the total energies for an electron in an MO interacting with the nuclei and the electrons in the other MO's:

Koopman's theorem is that [[epsilon]]i is a good approximation to the IP of an electron in occupied orbital i. Note that this estimates the vertical IP--that is, it does not account for relaxation of the (n-1) electron state. If i is a virtual orbital, [[epsilon]]i is an approximation to the electron affinity of that orbital

Example: Ionization energies for the first ten elements (e.v.)

Atom	Koopman's I.P.	RHF [[Delta]] E	UHF [[Delta]] E	Exp. I.P.	Koopman's E.A.	Exp. E.A.
H	13.60	13.60	13.60	13.60	-1.53	0.747
He	24.95	23.45	23.45	24.59	-11.02	--
Li	5.34	5.34	5.34	5.39	-0.47	0.62
Be	8.40	8.05	8.05	9.32	-1.49	--
B	8.66	7.94	8.04	8.30	-1.50	0.3
C	11.91	10.80	10.81	11.26	-1.13	1.12
N	15.46	13.99	13.91	14.53	-4.22	--
O	14.16	12.00	12.04	13.62	-3.38	1.466
F	18.38	15.74	15.68	17.42	-2.19	3.448
Ne	22.90	19.83	19.70	21.56	-38.36	--

Like any measurable property, NMR shifts can be calculated from the ab initio wave functions; however, only with the development of the individual gauge for localized orbitals (IGLO) method and an improved version of the earlier gauge independent atomic orbitals (GIAO) have reliable NMR shift calculations become possible.

Some examples from a recent review*:

¹¹⁹Sn chemical shifts (ppm)

Compound	IGLO/DZ	IGLO/DZ(P)	Exp.
SnH4	-374	-380	-493
CH3SnH3	-275	-273	-346
(CH3)2SnH2	-169	-162	-225
(CH3)3SnH	-76	--	-104

BH3NH3 Chemical shifts: (shift method//geometry opt. method)

"PISA" is a method for approximating solvation effects.

Atom	IGLO //HF	IGLO //MP2	IGLO //PISA	PISA-IGLO//MP2	PISA-IGLO//PISA	Exp.
d(11B)-13.2	-13.2	-15.0	-17.8	-16.8	-19.5	-23.8
d15N)2.7	2.7	8.0	4.9	3.0	-0.8	-13.0

*D. Cremer, L. Olsson, F. Reichel, E. Kraka, "Calculation of NMR Chemical Shifts--The Third Dimension of Quantum Chemistry" Israel Journal of Chemistry, 33, pp. 369-385 (1993).