Multipole moments energy

VVe therefore return to the point-charge model for calculating electrostatic interactions. If sufficient point charges are used then all of the electric moments can be reproduced and the multipole interaction energy. Equation (4.30), is exactly equal to that calculated from the Coulomb summation. Equation (4.19). 

The two-center two-electron repulsion integrals ( AV Arr) represents the energy of interaction between the charge distributions at atom Aand at atom B. Classically, they are equal to the sum over all interactions between the multipole moments of the two charge contributions, where the subscripts I and m specify the order and orientation of the multipole. MNDO uses the classical model in calculating these two-center two-electron interactions. 

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. 

Molecular energies and structures Energies and structures of transition states Bond and reaction energies Molecular orbitals Multipole moments... 

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. 

Rasmussen TD, Ren PY, Ponder JW, Jensen F (2007) Force field modeling of conformational energies importance of multipole moments and intramolecular polarization. Int J Quant Chem... 

Many authors [8-10] have demonstrated that the CP method undercorrects the BSSE. Moreover, Karlstrom and Sadlej [11] pointed out that addition of the partner orbitals to the basis set of a molecule not only lowers its energy, in accordance with the variation principle, but also affects the monomer properties (multipole moments and polarizabilities). Latajka and Scheiner [12] found that in a model ion-neutral system such as Li" -OH2, this secondary BSSE can be comparable in magnitude to the primary effect at both SCF and MP2 levels. The same authors also underlined the strong anisotropy of secondary error [13]. 

Consider tlie mutual approach of two noble gas atoms. At infinite separation, there is no interaction between them, and this defines die zero of potential energy. The isolated atoms are spherically symmetric, lacking any electric multipole moments. In a classical world (ignoring the chemically irrelevant gravitational interaction) there is no attractive force between them as they approach one another. When tliere are no dissipative forces, the relationship between force F in a given coordinate direction q and potential energy U is... 

Spectroscopic techniques have been applied most successfully to the study of individual atoms and molecules in the traditional spectroscopies. The same techniques can also be applied to investigate intermolecular interactions. Obviously, if the individual molecules of the gas are infrared inactive, induced spectra may be studied most readily, without interference from allowed spectra. While conventional spectroscopy generally emphasizes the measurement of frequency and energy levels, collision-induced spectroscopy aims mainly for the measurement of intensity and line shape to provide information on intermolecular interactions (multipole moments, range of exchange forces), intermolecular dynamics (time correlation functions), and optical bulk properties. 

The potential outside the charge distribution and due to it is simply related to the moments, as is the interaction energy when an external field is applied.14 The multipole moments are thus very useful quantities and have been extensively applied in the theory of intermolecular forces, particularly at long range where the electrostatic contribution to the interaction may be expanded in moments. Their values are related to the symmetry of the system thus, for instance, a plane of symmetry indicates that the component of n perpendicular to it must be zero. Such multipoles are worth calculating in their own right. 

In this case, and perhaps for all robust fits, if the fit is robust then its LCAO coefficients can be determined by variation of the energy. In that case the fit is said to be variational. Quantum chemists are beginning to use variational fits, but they do not yet include robust energies, in a method that they call resolution of the identity [14,15]. Equation (6), with pab replaced by PlM where L and M are the usual multipole-moment quantum numbers, can also be used to remove the first order error from fast-multipole methods [16]. 

Once the electronic multipole moments have been identified as q, net charge /t, dipole moment etc. the above expression for the electrostatic energy becomes... 

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