Dipole and Quadrupole Moments

Cohen and Tantirungrotechai97 have investigated the performance of new exchange-correlation functionals within the usual electron density schemes and compared calculated dipoles and multipoles for first and second row molecules with those obtained by established ab initio and electron density methods. The results obtained with the new functionals compare favourably with those of the previous methods and, in particular, give a value for the dipole moment of CO which is in good agreement with experiment. 

These results support the view that the CCSD(T) approximation is capable of reproducing molecular properties with a high degree of accuracy. 

In other studies using the CCSD(T) method Halkier and Coriani have calculated the quadrupole moment of C2H2 as 21.88 (in units of 10 40 C m2), to be compared with two recent experimental estimates of 20.9 and 20.5 and Halkier, Coriani and Jorgensen100 have found a value of —4.99 0.03 for N2 as compared to a recent experimental value of —4.65 + 0.08. The authors recalculate a correction term that has been applied to the experimental data and find that the correct value, for comparison with the theory, should have been —5.01 + 0.08. 

Using the same kind of augmented correlation-consistent basis sets employed in the above coupled cluster methods, but working with the complete active subspace (CASSCF and CASSCF( +1, +2)) approximations to the Cl expansion, Lawson and Harrison101 have investigated the variation with interatomic distance and spatial distribution of the quadrupole moments of P2, S2 and CI2. The a and it contributions to the quadrupole are resolved and the poor results obtained at the SCF level are attributed to the inadequate representation of the it system in the SCF approximation for P2 and S2. 

Calculations of atomic quadrupole moments are still of interest, particularly for heavier atoms where relativistic accuracy is required. Sundholm102 has reported a finite-element MCSCF method to calculate the quadrupole of Ar+ and finds a final value of —0.5271 au in comparison with a recent experimental determination of —0.5208 au. The Hartree-Fock value is —0.57213 au and the valence shell correlation correction, amounting to 0.04844 au, accounts for most of the additional contribution, a result which may be of some significance for molecular calculations involving atoms in the intermediate range of atomic numbers. 

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Bundgen P, Grein F and Thakkar A J 1995 Dipole and quadrupole moments of small molecules. An ab initio study using perturbatively corrected, multi-reference, configuration interaction wavefunctions J. Mol. Struct. (Theochem) 334 7... 

Many molecules, such as carbon monoxide, have unique dipole moments. Molecules with a center of inversion, such as carbon dioxide, will have a dipole moment that is zero by symmetry and a unique quadrupole moment. Molecules of Td symmetry, such as methane, have a zero dipole and quadrupole moment and a unique octupole moment. Likewise, molecules of octahedral symmetry will have a unique hexadecapole moment. 

In order to caleulate the force of attraction between the particle and surface. Hays had to estimate the dipole and quadrupole moments on the particle. He did this by assuming that the eharge distribution a 0) around the surface of the particle was given by... 

Hays further assumed that the dipole and quadrupole moments are given by a AnR ... 

Here are the predicted dipole and quadrupole moments for formaldehyde ... 

Figure 9.1 Generation of dipole and quadrupole moments by charges...

In the absence of an external field, the unperturbed dipole and quadrupole moments may be calculated from the electronic wave function as simple expectation values. 

Hirshfeld and Mirsky (1979) evaluated the relative contributions to the lattice energy for the crystal structures of acetylene, carbon dioxide, and cyanogen, using theoretical charge distributions. Local charge, dipole and quadrupole moments are used in the evaluation of the electrostatic interactions. When the unit cell dimensions are allowed to vary, inclusion of the electrostatic forces causes an appreciable contraction of the cell. In this study, the contributions of the electrostatic and van der Waals interactions to the lattice energy are found to be of comparable magnitude. 

To prevent misunderstanding (94), we emphasize that neither experimental hydration energies nor experimental coordination numbers are necessary for these calculations. Moreover, the coordination numbers obtained are generally not comparable to empirical hydration numbers. The only experimental quantities that enter the calculations are a) cationic radius and charge b) van der Waals radius of water c) dipole and quadrupole moment of water d) polarizabilities e) ionization potentials and f) Born repulsion exponents as well as fundamental constants (see Ref. (92)). 

Normally full valence MCSCF calculations (choosing all valence orbitals as active) represent a balanced treatment of correlation. However this is not always the case, especially not in systems containing lone-pair electrons. For example, a full valence MCSCF calculation for the water molecule yields less accurate values for the bond distance, the bond angle, and the dipole and quadrupole moment than an SCF calculation. The reason is that there are only two orbitals available for correlating the eight valence electrons (the 4ax and the 2b2 orbitals). Thus correlation is only introduced into the lone-pair orbital... 

NH2C1 has been carefully studied by microwave spectroscopy and SCF calculations of its molecular properties using an essentially DZ basis have been recently reported.110 The barrier to inversion was computed and analysed in terms of the different contributions to the barrier. The main conclusion was that the barrier is attractive in Allen s terminology.111 The dipole and quadrupole moments were also computed in this paper. Hydroxylamines have been investigated previously,112 113 but in a more recent paper, Trindle and Shillady114 have computed the potential surface for the interconversion of ammonium oxide with hydroxylamine [equation (4)]. 

Birefringences are mostly observed in condensed phases, especially pure liquids or solutions, since the strong enhancement of the effects allows for reduced dimensions (much shorter optical paths) of the experimental apparatus. Nowadays measurements of linear birefringences can be carried out on liquid samples with desktop-size instruments. Such measurements may yield information on the molecular properties, molecular multipoles and their polarizabilities. In some instances, for example KE, CME and BE, measurements (in particular of their temperature dependence) have been carried out simultaneously on some systems. From the combination of data, information on electric dipole polarizabilities, dipole and quadrupole moments, magnetizabilities and higher order properties were then obtained. 

A comparison of properties calculated by the INDO method, such as dipole and quadrupole moments of the charge distribution, with ab initio results has been given for N2, H2, CO, and HF.245 Suggested improvements in the INDO procedure were given. 

The interaction-energy curves for alkali metal-rare gas pairs are also of interest experimentally in scattering and radiation problems, and theoretically because of the expected reliability of the HF energy for this class of half-open-closed-shell systems. Calculations on LiHe and NaHe (X22+, A2U, if 22+) and their X1 + ions have been reported by Krauss et al.285 from R = 3 to 10 bohr. Both STO and GTO expansion bases were used, with comparable results except for the ASH state of NaHe. The variation of dipole and quadrupole moments with R was investigated. The X2S+ curve is... 

The calculated first hyperpolarizabilities (see Table 2-4) are surprisingly close to the experimental data, which is probably fortuitous because they were calculated without taking into account vibrational effect. These studies demonstrated also that the double-zeta basis set augmented by field-induced polarization functions, although sufficient for calculations of dipole and quadrupole moments of the studied molecules at the Kohn-Sham LDA level, is not sufficient in the case of hyperpolarizabilities. 

Further contributions to the potential energy occur when the molecule X possesses a permanent dipole and quadrupole moment. 

In the case of tetrahedral symmetry, the dipole and quadrupole moments vanish (p — 0 = 0) the first (lowest) non-zero electric moment is octupolar and can be written thus ... 

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