Coulomb multipole expansion

One recent development in DFT is the advent of linear scaling algorithms. These algorithms replace the Coulomb terms for distant regions of the molecule with multipole expansions. This results in a method with a time complexity of N for sufficiently large molecules. The most common linear scaling techniques are the fast multipole method (FMM) and the continuous fast multipole method (CFMM). 

Another approach to reducing the cost of Coulombic interactions is to treat neighboring interactions explicitly while approximating distant interactions by a multipole expansion. In Eigure la the group of charges, i) at positions = (ri ... 

The original FMM has been refined by adjusting the accuracy of the multipole expansion as a function of the distance between boxes, producing the very Fast Multipole Moment (vFMM) method. Both of these have been generalized tc continuous charge distributions, as is required for calculating the Coulomb interactioi between electrons in a quantum description. The use of FMM methods in electronic structure calculations enables the Coulomb part of the electron-electron interaction h be calculated with a computational effort which depends linearly on the number of basi functions, once the system becomes sufficiently large. 

That is, in the singlet-singlet transition both Coulomb interaction and exchange interaction are involved. However, when the distance between D and A is large, the exchange term can be ignored, and we can use the multipole expansion for e2/rij, that is,... 

The first terms of the power series obtained by the multipole expansion of the Coulomb intermolecular potential account for dipole-dipole interactions prevailing in systems of polar molecules. As an adequate approximation for ensembles of... 

Theoretical chemists learn about a number of special functions, the Hermite functions in connection with the quantisation of the harmonic oscillator, Legendre and associated Legendre functions in connection with multipole expansions, Bessel functions in connection with Coulomb Greens functions, the Coulomb wave functions and a few others. All these have in common that they are the solutions of second order linear equations with a parameter. It is usually the case that solutions of boundary value problems for these equations only exist for countable sets of values of the parameter. This is how quantisation crops up in the Schrddinger picture. Quantum chemists are very comfortable with this state of affairs, but rarely venture outside the linear world where everything seems to be ordered. 

Analogously, a Coulomb field can be expressed as the gradient of a scalar potential that obeys the Laplace equation in a source-free region such as the vacuum in conventional electrostatics. To find the general form of B(3> in a multipole expansion, we therefore solve the Laplace equation for 3>g, and evaluate the gradient of this solution... 

Starting from a multipole expansion of intramolecular Coulomb interactions, we present an efficient configuration interaction calculation for the electron terms = 2,3, 4, and the hole terms (hf)", n = 2-5. We have studied magnetic moments for the electron and hole terms. The coupling of spin and orbital momenta differs from the Lande g-factor scheme of atoms. The magnetic moments do not depend on the orientation of the molecule with respect to an external magnetic field. 

The problem is also a challenge from both group theoretical [6,7] and experimental [8] point of view. In the following we will use a method which is based on a multipole expansion of the Coulomb interaction between electrons on a same molecule [9,10]. Thereby we systematically include electronic transitions which go beyond the usual Hartree-Fock scheme and hence our approach is equivalent to a full configuration interaction calculation. The details of our technique are given in Ref. [10]. 

We will use the basis vectors (1) where > i2 and apply equation (4) when needed. For two tlu electrons, our basis (1) consists of 15 different state vectors I/) (for two holes, the fivefold hu degeneracy leads to 45 states). In the following we will study the intramolecular correlations of electrons (holes) within a multipole expansion of the two-body Coulomb potential V(r, f) = 1/lr —1 (charge e = unity). In terms of real spherical harmonics YJ, where r stands for m = 0,... 

When D and A are substantially separated in space, FCoul reduces to the dipole approximation discussed above. If the center-to-center distance R is comparable with the size of the molecules involved, then the dipole approximation may not provide an accurate representation of VCoul. This is because information regarding the shape of the interacting molecules is discarded when the dipole approximation, or even a low-order multipole expansion, is invoked [7,11,12], In particular, molecules with extended or asymmetric transition densities tend to exaggerate errors in the dipole-dipole approximation. In this case, information on transitions of each molecule should be retained as a transition density, rather than a dipole transition moment [36,37]. The Coulombic coupling... 

The equilibrium state is determined by a minimization of the free energy. The total interaction in colloidal systems is mainly determined by electrostatic interactions and can be divided in three components. The first component occurs at interactions between net charged molecules or molecules with asymmetric charged distribution. These charge distributions can often be described by multipole expansions, i.e., a combination of monopole, dipole, quadrupole, etc., and is a fruitful approach if each multipole expansion can be described by one or two terms. The interaction is given by the sum of interactions between the terms, where the first contribution is the interaction between ions and is given by Coulomb s law and is the main contribution in systems with... 

Schulten238 outlined the development of a multiple-time-scale approximation (distance class algorithm) for the evaluation of nonbonded interactions, as well as the fast multipole expansion (FME). efficiency of the FME was demonstrated when the method outperformed the direct evaluation of Coulomb forces for 5000 atoms by a large margin and showed, for systems of up to 24,000 atoms, a linear dependence on atom number. 

An example of the scalar product that is used in electron—atom collision theory is the multipole expansion of the two-electron Coulomb potential. 

Due to the non-local character of the Coulomb operator, the decomposition for the electrostatic energy is more complex. In order to distinguish between local and global terms, we need to introduce atom-dependent screening densities, (hard) and (soft), that generate the same multipole expansion as the local density ha — nA A n, where is the nuclear charge of atom A. 

The point multipole expansion of the Coulombic matrix element governing the first-order electric dipole moment in Eq. (3) gives the selection rules of the ligand polarization model through Eq. (4),... 

The multipole expansion has already been used in certain quantum chemical calculations [59-65]. As localized orbitals are concentrated in certain spatial region, they can also be represented by their multipole moments. In the following we investigate whether the Coulomb integrals in terms of localized orbitals can be substituted by the multipole expansion of electric moments. 

As it can be seen Coulomb integrals cannot always be regularly approximated by using multipole expansion. The above values resulted in our calculations are aimed to demonstrate their transferable properties. 

Note that a distinction is made between electrostatic and polarization energies. Thus the electrostatic term, Ue e, here refers to an interaction between monomer charge distributions as if they were infinitely separated (i.e., t/°le). A perturbative method is used to obtain polarization as a separate entity. The electrostatic and polarization contributions are expressed in terms of multipole expansions of the classical coulomb and induction energies. Electrostatic interactions are computed using a distributed multipole expansion up to and including octupoles at atom centers and bond midpoints. The polarization term is calculated from analytic dipole polarizability tensors for each localized molecular orbital (LMO) in the valence shell centered at the LMO charge centroid. These terms are derived from quantum calculations on the... 

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