Chapter 5. Quantum chemistry in Molecular Modeling


5.10 Properties derived from the wavefunction

The electronic wavefunction which is computed in ab initio as well as semi-empirical quantum chemical methods can be used to derive observable quantities of a molecule, but it can also be analyzed and used to rationalize certain chemical phenomena.

electrical properties

The electric dipole moment of a molecule can be calculated directly from the positions of the nuclei and the electronic wavefunction [6]:


The dipole moment can be viewed as the first term of an expansion of the electric field due to the molecule, the next higher term being the quadrupole moment. It is also possible to obtain the dipole moment and polarizabilities directly as derivatives of the energy with respect to a uniform electric field [21]. The electrostatic potential of the molecule represents the interaction between the charge distribution of the molecule and a unit point charge located at some position p :


Calculation of the molecular electrostatic potential at the surface of the molecule (described by the total electron density) can indicate how the molecule will interact with polar molecules or charged species. Visualization of this can be nicely accomplished using color coding [6].

Atomic charges

Although concepts like atomic point charges or bond dipoles are widely used in molecular mechanics, there is no unique definition of atomic charge in a molecule. All ways to attribute a part of the electron density to individual atoms are to a certain extent arbitrary. As a first analysis, or as a way to compare related systems, Mulliken Population Analysis can be applied. The electron density distribution (the probability of finding an electron in a volume element dr) is :


Integrated over entire space this gives the total number of electrons (Sv is the overlap):


This can be separated into diagonal and off-diagonal terms, where the former represent the net population of the basis orbitals and the latter are make up the overlap population.



In the Mulliken scheme the overlap population is simply shared between the contributing atoms, which leads to the following charge for each basis orbital :


Summing of the charges in the orbitals associated with each atom gives the atomic charge.
An important disadvantage of the Mulliken population analysis is that extended basis sets can lead to unphysical results, e.g. charges of more than 2e, which result from the fact that the basis orbitals centered at one atom actually describe electron density close to another nucleus. Population Analysis based on Natural Atomic Orbitals does not have this problem.
An approach which may be physically more relevant is to fit charges at the atomic positions to the molecular electrostatic potential measured at a grid of points. This still leaves some arbitrariness in the choice of the grid, and the procedure is computationally much more demanding than the other types of population analysis.

References:

[1] Levine
Quantum Chemistry, 4th ed., 1991
[2] Reviews in Computational Chemistry,
Volumes 1 to 4, K.B. Lipkowitz and D.B. Boyd, eds., VCH publishers.
[3] G. Nàray-Szabò, P.R. Surjàn and J.G. Angyàn
Applied Quantum Chemistry, Reidel, 1987
[4] W.J. Hehre, L. Radom, P. von R. Schleyer and J.A. Pople
Ab initio molecular orbital theory, Wiley, 1986
[5] Szabo and Ostlund
Modern Quantum Chemistry, McGraw-Hill, 1989.
[6] Spartan User's Guide, version 3.0,
Wavefunction, Inc., 1993.
[7] Foresman, J.B.; Frisch, A.
Exploring Chemistry with Electronic Structure Methods: A Guide to Using Gaussian, Gaussian Inc., 1993.
[8] D. Feller and E.R Davidson,
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[11] Palmer, I.J.; Ragazos, I.N.; Bernardi, F.; Olivucci, M.; Robb, M.A.,
J. Am. Chem. Soc. 1993,115, 673 - 682
[12] Olivucci, M.; Ragazos, I.N.; Bernardi, F.; Robb, M.A.,
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[13] Roos, B.O.; Andersson, K.; Fülscher, M.P.,
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[14] Pople, J.A.; Head-Gordon, M.; Fox, D.J.; Raghavachari, K.; Curtiss, L.A.,
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[15] Curtiss, L.A.; Raghavachari, K.; Trucks, G.W.; Pople, J.A.,
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[16a] Allinger, N.L.; Schmitz, L.R.; Motoc, I.; Bender, C.; Labanowski, J.,
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[16b] L.R. Schmitz et al.
J. Phys. Org. Chem., 1993, 6, 551;
Heteroatom Chem. 1992, 3, 69;
J. Comput. Chem., 1992, 13, 838;
J. Phys. Org. Chem., 1992, 5, 225;
J. Am. Chem. Soc., 1992, 114, 2880.
[17] Boyd, D.B.,
in Reviews in Computational Chemistry, K.B. Lipkowitz and D.B. Boyd, eds., VCH, Vol. 1, 1990, 321 - 354.
[18] Stewart, J.J.P.,
in Reviews in Computational Chemistry, K.B. Lipkowitz and D.B. Boyd, eds., VCH, Vol. 1, 1990, 45 - 82.
[19] Suzuki, H.
Electronic Absorption Spectra and Geometry of Organic Molecules, Academic Press, 1967.
[20] Cramer, C.J.; Truhlar, D.G.,
Science, 1992, 256, 213 - 217.
[21] Dykstra, C.E.; Augspurger, J.D.; Kirtman, B.; Malik, D.J.,
in Reviews in Computational Chemistry, K.B. Lipkowitz and D.B. Boyd, eds., VCH, Vol. 1, 1990, 83 - 118.


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Chapter 5 MM Syllabus 1995 MODIFIED November 8, 1995
Fred Brouwer, Lab. of Organic Chemistry, University of Amsterdam.