Multipole integrals

The first and second terms are the contributions of the atomic nuclei and the electrons, respectively. The calculation of the multipole integrals in the second term is simpler and faster than that of the potential integrals. After rearrangement of Eq. (12) a molecular multipole moment can be expressed as a sum of the atomic contributions (ukvlwm i ... 

This guarantees that higher order multipole integrals will integrate to integer numbers, for example, for the dipole integral in the z direction, d ... 

We notice that because each multipole vector has 0 L ) components [n = 0,..., L for each n, there are 2L -I-1 m-components, thus a total of L(2L -I-1)], the total cost of evaluating a single integral using the multipole expansion has complexity. The spherical multipole integrals may be... 

The two-electron integrals are for our purposes real entities, so it is clear that using complex terms (soHd harmonics) in the multipole expansion is unnecessary and only makes the computer implementation slower and more difficult. A reformulation in terms of real multipole integrals and interaction matrix elements is possible by splitting each term into a real and a complex part and dealing with them separately. After some algebra, one can see that the imaginary part drops out and one obtains the multipole expansion in terms of real-valued multipole moments and interaction matrix elements. The realvalued multipole expansion may be cast into exactly the same form as that of the complex series. As both formulations are formally very similar, we do not introduce the real formulation here but instead refer the interested reader to the literature, e.g.. Ref. 13. 

Here, RapyS is the distance vector between the primitive centers aap and byg. Using contracted multipole integrals instead of primitive integrals, we obtain the following expansion for contracted integrals ... 

The second consideration is the geometry of the molecule. The multipole estimation methods are only valid for describing interactions between distant regions of the molecule. The same is true of integral accuracy cutoffs. Because of this, it is common to find that the calculated CPU time can vary between different conformers. Linear systems can be modeled most efficiently and... 

One of the major selling points of Q-Chem is its use of a continuous fast multipole method (CFMM) for linear scaling DFT calculations. Our tests comparing Gaussian FMM and Q-Chem CFMM indicated some calculations where Gaussian used less CPU time by as much as 6% and other cases where Q-Chem ran faster by as much as 43%. Q-Chem also required more memory to run. Both direct and semidirect integral evaluation routines are available in Q-Chem. 

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. 

With only s- and p-functions present, the two-centre two-electron integrals can be modelled by multipoles up to order 2 (quadrupoles), however, with d-functions present multipoles up to order 4 must be included. In MNDO/d all multipoles beyond order 2 are neglected. The resulting MNDO/d method typically employs 15 parameters per atom, and it currently contains parameters for the following elements (beyond those already present in MNDO) Na, Mg, Al, Si, P, S, Cl, Br, 1, Zn, Cd and Hg. 

The additional integrals are just expectation values of x,y and z, and their inclusion requires very little additional computational effort. Generalization to higher-order multipoles is straightforward. 

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

Each coefficient of the multipole expansion is computed by a numerical integration - after aligning and normalizing the found orientation distribution. 

The atomic electrostatic moments of an atom are obtained by integration over its charge distribution. As the multipole formalism separates the charge distribution into pseudoatoms, the atomic moments are well defined. 

If the moments are referred to the nuclear position, only the electronic part of the charge distribution contributes to the integral. According to the multipole formalism of Eq. (3.32),... 

Since the spherical core- and valence-scattering factors in the multipole expansion are based on theoretical wave functions, expressions for the corresponding density functions are needed in the analytical evaluation of the integrals in Eqs. (8.35)... 

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