Multipole moments dipoles

Figure 2.4 Interactions between multipole moments. Dipole moments are indicated by arrows, and an element of the quadrupole tensor by the double lobe.

Approximate wave function and density functional theories provide information about the electronic structure of molecules in their electronic ground state. The information includes the electronic charge density, total energy, electric multipole moments (dipole, quadrupole, octupole, etc.), forces on the nuclei, and vibrational frequencies, which is sufficient to model a wide range of chemical phenomena. For example, equilibrium structures and transition states can be calculated from the forces, and vibrational frequencies are not only useful for the interpretation of vibrational spectra but also enable the calculation of thermo chemical data from first principles. These theories are sufficient to model experimental conditions where only the electronic ground state is significantly populated. 

Quantitatively, we expect the temporal response of a solvent to be governed by the dynamics of the translation and reorientation of its molecules. This response changes the interaction of the anisotropic charge distribution of the solute, as characterized by the multipole moments (dipole, quadrupole, etc.) of its charge distribution, with the multipole moments of the solvent molecules. In a polar solvent we can seek to relate the time scale of the solute s reorientation dynamics to the frequency dependence of the dielectric constant in the spectral region corresponding to nuclear motions. ... 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

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] ... 

The polarization interaction arises from the interaction between the ion of charge Ze and the multipole moments it induces in the atom or molecule AB. The dominant polarization interaction is the ion-mduced dipole interaction... 

The anisotropy of the product rotational state distribution, or the polarization of the rotational angular momentum, is most conveniently parametrized tluough multipole moments of the distribution [45]. Odd multipoles, such as the dipole, describe the orientation of the angidar momentum /, i.e. which way the tips of the / vectors preferentially point. Even multipoles, such as the quadnipole, describe the aligmnent of /, i.e. the spatial distribution of the / vectors, regarded as a collection of double-headed arrows. Orr-Ewing and Zare [47] have discussed in detail the measurement of orientation and aligmnent in products of chemical reactions and what can be learned about the reaction dynamics from these measurements. 

The raie gas atoms reveal through their deviation from ideal gas behavior that electrostatics alone cannot account for all non-bonded interactions, because all multipole moments are zero. Therefore, no dipole-dipole or dipole-induced dipole interactions are possible. Van der Waals first described the forces that give rise to such deviations from the expected behavior. This type of interaction between two atoms can be formulated by a Lennaid-Jones [12-6] function Eq. (27)). 

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 three moments higher than the quadrupole are the octopole, hexapole and decapoli. Methane is an example of a molecule whose lowest non-zero multipole moment is the octopole. The entire set of electric moments is required to completely and exactly describe the distribution of charge in a molecule. However, the series expansion is often truncated after the dipole or quadrupole as these are often the most significant. 

The multipole moments (charge, dipole, quadrupole) of each cell are then calculated mining over the atoms contained within the cell. The interaction between all of the 3 in the cell and another atom outside the cell (or indeed another cell) can then be lated using an appropriate multipole expansion (see Section 4.9.1). 

The Onsager model describes the system as a molecule with a multipole moment inside of a spherical cavity surrounded by a continuum dielectric. In some programs, only a dipole moment is used so the calculation fails for molecules with a zero dipole moment. Results with the Onsager model and HF calculations are usually qualitatively correct. The accuracy increases significantly with the use of MP2 or hybrid DFT functionals. This is not the most accurate method available, but it is stable and fast. This makes the Onsager model a viable alternative when PCM calculations fail. 

The molecular dipole moment is perhaps the simplest experimental measure of charge density in a molecule. The accuracy of the overall distribution of electrons in a molecule is hard to quantify, since it involves all of the multipole moments. Experimental measures of accuracy are necessary to evaluate results. The values for the magnitudes of dipole moments from AMI calculations for a small sample of molecules (Table 4) indicate the accuracy you may... 

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. 

The dipole, quadrupole etc. moments are in general not conserved, i.e. a set of population atomic charges does not reproduce the original multipole moments. 

Typical properties of the charge distribution are summarized by its various electric multipole moments. The electric dipole moment p. induced in the system by the external field is obviously... 

Eo and E (Afi(i)) are respectively the electric fields generated by the permanent and induced multipoles moments. a(i) represents the polarisability tensor and Afi(i) is the induced dipole at a center i. This computation is performed iteratively, as Epoi generally converges in 5-6 iterations. It is important to note that in order to avoid problems at the short-range, the so-called polarization catastrophe, it is necessary to reduce the polarization energy when two centers are at close contact distance. In SIBFA, the electric fields equations are dressed by a Gaussian function reducing their value to avoid such problems. 

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. 

After some algebra, the probability of jumping from one state to the other becomes proportional to the square modulus of the matrix element of the dipole moment between the bare quantum states k and j (if different from zero, or to any multipole moment having non-zero matrix elements between these two states). 

London-van der Waals forces generally are multipole (dipole-dipole or dipole-induced dipole) interactions produced by a correlation between fluctuating induced multipole (principal dipole) moments in two nearly uncharged polar molecules. Even though the time-averaged, induced multipole in each molecule is zero, the correlation between the two induced moments does not average to zero. As a result an attractive interaction between the two is produced at very small molecular distances. 

The Kirkwood-Onsager equations can be generalized to include multipole moments higher dian the dipole, leading to the expression... 

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... 

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