Multipole translation operator

Note that the inheritance of the field vectors from parent to child (and vice versa) is crucial for the overall linear scaling of the algorithm. If this could not be done, forming a V vector would involve a summation over the higher level boxes, which would lead to an C>(M log M) scaling. Finally we note that inheritance is only possible through the multipole translation operator, because the field experienced at a parent s center must be shifted to the child s center. 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

Upward pass. Sweep up from the smallest cells to the largest cell to obtain the multipole moments for cells at each subdivision level. This is done by using the translation operators that allow us to obtain the multipole moment of the parent cell from multipole moments of children cells (see for details [123,128,132]). 

The keys to FMM are the so-called translation operators. Let us briefly illustrate what they are. Suppose that the multipole expansion of density p is centered at R, so that its coefficients are given (see equation 68) by... 

Next, we will see how this translation operator can improve the performance of TC. When creating the tree of a TC program, we have to compute the multipole expansions of a density, and those of its children, grandchildren, etc. TC does... 

We can think of equation (76) as the equivalent of equations (66) and (67). The usefulness of local expansions comes fi"om two facts. First, the coefficients P/.m) of a local expansion can be obtained from the coefficients [Qi.m] of the respective multipole expansion, via a second translation operator. Again, the cost of this transformation is 0(L ), and does not depend on N. Second, often several local expansions W, W [,... have to interact with a given partition pj, via equation (74). Instead of evaluating equation (74) for each of the local expansions, we can create the sum expansion W with coefficients... 

A vitally important aspect of ion trap operation is the ability to impart translational energy selectively to ions via resonance absorption of alternating current (ac) voltages (10-450 kHz) applied to the endcap. Unlike for linear quadrupole (or other multipole) collision cells, the absorption of energy is m/z specific as each m/z in the trap precesses at a specific set of frequencies, the most important of which for MS/MS is ooz, the fundamental frequency of motion in the z dimension, which is defined by... 

For neutral molecules, the dipole polarizabilities and hyperpolarizabilities are invariant to the choice of the moment center. Other multipole polarizabilities may be invariant in certain cases of high molecular symmetry. The changes that may occur in P , P ,. . . upon shifting an evaluation center are determined by the changes in the moments or moment operators. If a particular origin translation leads to... 

Previously, Lazzeretti et al had shown that, within the CTOCD-DZ approach, the multipole polarizabilities (of any order) of nuclear magnetic shieldings are invariant to a gauge translation irrespective of the quality of the basis set employed. On the other hand, they depend upon the origin of the eoordinate system for multipole higher than dipole, because of the intrinsie origin dependence of the related operators. 

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