Dipole and Higher Multipole Moments

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. 

The molecular electronic polarizability is one of the most important descriptors used in QSPR models. Paradoxically, although it is an electronic property, it is often easier to calculate the polarizability by an additive method (see Section 7.1) than quantum mechanically. Ah-initio and DFT methods need very large basis sets before they give accurate polarizabilities. Accurate molecular polarizabilities are available from semi-empirical MO calculations very easily using a modified version of a simple variational technique proposed by Rivail and co-workers [41]. The molecular electronic polarizability correlates quite strongly with the molecular volume, although there are many cases where both descriptors are useful in QSPR models. 

---

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. 

There sure other quantities that are related to expectation values other than that of the Hamiltonian e.g. dipole and higher multipole moments, spin densities, field gradients). Most of the operators that come into play here are one-electron operators, and the Moller-Plesset theorem states that their expectation values (like the electron density) are affected by correlation corrections to the wave function only to second order. As a consequence, correlation does not much influence these expectation values, except when the Hartree-Fock contribution is unusually small, so that a correlation correction may even determine the sign, as is the case for the dipole moment of CO >. 

The data in Table 12.5 show that the PC corrections are of similar magnitude as the final EFG values. Clearly, the use of untransformed operators would be woefully inadequate. With PC effects being relativistic in origin, the differences are less drastic for EFGs of lighter atoms, and smaller overall, compared to EFGs, for valence-shell properties such as dipole and higher multipole moments. However, in the X2C framework where the transformation matrices to two-component form are already available there is little added... 

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. 

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. 

There are higher multipole polarizabilities tiiat describe higher-order multipole moments induced by non-imifonn fields. For example, the quadnipole polarizability is a fourth-rank tensor C that characterizes the lowest-order quadnipole moment induced by an applied field gradient. There are also mixed polarizabilities such as the third-rank dipole-quadnipole polarizability tensor A that describes the lowest-order response of the dipole moment to a field gradient and of the quadnipole moment to a dipolar field. All polarizabilities of order higher tlian dipole depend on the choice of origin. Experimental values are basically restricted to the dipole polarizability and hyperpolarizability [21, 24 and 21]. Ab initio calculations are an imponant source of both dipole and higher polarizabilities [20] some recent examples include [26, 22] ... 

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). 

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... 

Linear Dipolar Electric Polarization. Even atoms and microsystems with symmetrically distributed charges are known to exhibit polarization under the influence of an external electric field or the internal (molecular) electric field of a neighbour. Such a system is said to become endowed with an induced electric dipole (or higher multipoles). In many a case it sufiSces to assume that the induced dipole moment p E) is proportional to the field strength inducing it ... 

The dipole polarizability can be used in place of the dipole moment function, and this will lead to Raman intensities. Likewise, one can compute electrical quadrupole and higher multipole transition moments if these are of interest. 

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). 

---