Higher Multipole Moments

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

Here are the predicted dipole and quadrupole moments for formaldehyde  

The dipole moment is broken down into X, Y, and Z components. In this case, the dipole moment is entirely along the Z-axis. By referring to the standard orientation for the molecule, we realize that this is pointing away from the oxygen atom, which is 

Exploring Chemistry with Electronic Structure Methods 

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The multipole moment of rank n is sometimes called the 2"-pole moment. The first non-zero multipole moment of a molecule is origin independent but the higher-order ones depend on the choice of origin. Quadnipole moments are difficult to measure and experimental data are scarce [17, 18 and 19]. The octopole and hexadecapole moments have been measured only for a few highly syimnetric molecules whose lower multipole moments vanish. Ab initio calculations are probably the most reliable way to obtain quadnipole and higher multipole moments [20, 21 and 22]. 

Molecular dipole moments are often used as descriptors in QPSR models. They are calculated reliably by most quantum mechanical techniques, not least because they are part of the parameterization data for semi-empirical MO techniques. Higher multipole moments are especially easily available from semi-empirical calculations using the natural atomic orbital-point charge (NAO-PC) technique [40], but can also be calculated rehably using ab-initio or DFT methods. They have been used for some QSPR models. 

Of course, the na, o a 11, and oah NBOs of the H-bonding region are important contributors to the dipole, quadrupole, and higher-multipole moments of the monomers. Thus, certain multipoles may appear to explain the geometry through their close connections to these NBOs, but this is not an incisive way to describe the physical situation. 

Representation of the density n(r) [or, effectively, the electrostatic potential — 0(r)] near any one of the sinks as an expansion in the monopole and dipole contribution only [as in eqn. (230c)] is generally, unsatisfactory. This is precisely the region where the higher multipole moments make their greatest contribution. However, the situation can be improved considerably. Felderhof and Deutch [25] suggested that the physical size of the sinks and dipoles be reduced from R to effectively zero, but that the magnitude of all the monopoles and dipoles, p/, are maintained, by the definition... 

With this notation, the electric charge qo of a monopole equals Qoo-Cartesian dipole components px, py, pz, are related to the spherical tensor components as Ql0 = pz, Qi i = +(px ipy)/y/2, with i designating the imaginary unit. Similar relationships between Cartesian and spherical tensor components can be specified for the higher multipole moments (Gray and Gubbins 1984). 

Higher multipole moments have rarely been used but expressions of the various dipole components can be obtained in the same way. 

Although 1 is one of the best investigated molecules, there is, apart from data concerning its electron density distribution, very little information available on its one-electron properties. In principle, accurate data could be obtained by correlation-corrected ab initio methods, but almost nothing has been done in this direction, which of course has to do with the fact that experimental data on one-electron properties of 1 are also rare, and therefore, it is difficult to assess the accuracy and usefulness of calculated one-electron properties such as higher multipole moments, electric field gradients, etc. 

The Born equation is based on the simple model of a spherical ion with a single charge at its centre. Such an ion has no dipole moment and no higher multipole moments, but real molecular ions are of course much more complex. Since the electrical charge is distributed among all the atoms of the... 

The AMI-PCM model does surprisingly well, even without including specific solvent interactions, suggesting that this effect may account for only a portion of the BKO discrepancies in more polar solvents. Evidently, higher multipole moments may also be important. 

Lastly, we turn to consideration of the Rg—HF heterodimers (the atom—diatomic molecule system of the right-hand side of Figure 5.2), where a crucial role is played by the induction interaction occurring between the higher multipole moments of HF and the induced dipoles originating the polarizability of the rare gas (Magnasco et al., 1989a). 

Neither atomic charges nor bond dipoles are observables. About the only experimental data for isolated molecules that can be used as parameterization reference are molecular dipoles and higher multipole moments. Substantial effort has also been expended to find electrostatic schemes that can rationalize the behavior of condensed phases (37). However, electrostatic data may be more conveniently obtained from QM calculations. Several schemes exist for partitioning the electron density into atomic charges (38). In general, methods that reproduce the QM-calculated electrostatic field outside the molecular surface are preferred. 

In Appendix B the "" (12) referred to in this chapter are written in a more explicit and familiar manner. We remark that in Section II only dipolar interactions were considered and a simpler notation was used. However, when higher multipole moments are included, a more systematic approach is necessary, and rotational invariants are particularly convenient and theoretically useful. In the present notation the D 2) and A(12) functions of Section II are " (12) and "°(12), respectively. 

Real Liquids. Real molecules usually have both dipole and quadrupole moments. Thus the results described above strongly suggest that withoi t the inclusion of higher multipole moments, simple dipolar models will be hopelessly inadequate for most real liquids. This is illustrated qualitatively in Fig. 22, which compares experimental results for a number of common liquids ... 

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