Monopole term

On the carbons, nitrogens and oxygens expansions up to octapole level were introduced, whereas the expansions were limited to quadrupole level for the hydrogen atoms. All atoms were given a k expansion/contraction parameter for the spherical monopole term, and all atoms except the hydrogens were given k parameters to expand or contract the non-spherical poles. The k and k values on 0(1) and 0(3), on N(l) and N(3), on C(l) and C(3) and on 0(21) and 0(22) were constrained to be equal. 

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 ... 

It has been recently shown [12] that the ELF topological analysis can also be used in the framework of a distributed moments analysis as was done for Atoms in Molecules (AIM) by Popelier and Bader [32, 33], That way, the Mo( 2) monopole term corresponds to the opposite of the population (denoted N) ... 

When the non-electrostatic terms are semiempirical, they also make up in an average way for systematic deficiencies in the treatment of electrostatics, e.g., for the truncation of the distributed multipole representation of the solute charge density at the monopole term on each center. 

The operator (2.153) is the lowest-order approximation to the monopole term. Anharmonic terms can be introduced by considering the operator... 

In this case, the dipole, quadrupole, and higher order terms may be dropped compared to the monopole term. The effective modulus when A p is given by... 

As mentioned after Equation [24], atom-centered monopoles in principle generate the higher multipoles required to describe the electronic distribution (although, of course, a finite number n of charges can give at most n nonvanishing multipole moments), and as noted by Dillet et al., the distributed monopole term provides the vast majority of the polarization effect (albeit not all). We note this only for comparative purposes, though, since calculation of the ENP terms does not actually involve the multipole moments explicitly. 

The full Coulomb interaction energy term can be expressed by combining O Eqs. (25.42) and (O 25.43) as the sum of a number of terms with multipole order 1. The term for / = 0 is the electric monopole term. The / = 2 term gives rise to electric quadrupole interactions and wiU be discussed later. (All the odd / terms vanish for reasons of symmetry. The very small even terms for Z > 2 can also be ignored.)... 

The first term in the right member of Equation 14.26 is a monopole term. It is simply the potential from the charge placed at the origin. The next term may be written... 

Since the density Xc(X2 c)%c( 2 c) localized near the Cth atom, we can use the multipole expansion to calculate its electrostatic potential at point r. The lowest order term is the monopole term of Equation 10.18, leading to... 

The same is true for the LDA correlation potential. Moreover, for neutral atoms the electrostatic potential of the nucleus cancels with the monopole term in vh, (2-3). Consequently, the total Ug also decays faster than 1/r. This implies that, within the framework of the LDA, a neutral atom does not exhibit a Rydberg series of excited states and thus is not able to bind an additional electron, i.e. to form a negative ion. 

Equation 4.2 implies that if R is large, the potential may be taken in a first approximation as just the first non-zero term in the expansion, because further terms depend on increasing inverse powers of distance. So for a non-ionic molecule with non-zero dipole the monopole term is zero, while quadrupole and higher multipole terms are much smaller than the dipole term. At large intermolecular distance, therefore, the electric potential of a molecule is to a good approximation described by just the dipole term. For condensed phases, where molecules may come very close to one another, the dipolar approximation is unsatisfactory. 

Now assume, for example, that the nuclear resonant state is an s state, and choose 0 to be 90 so that the only relevant interference term arises from the monopole term, with both P and e emerging in spherically symmetric distributions. One can then show that B = A. ... 

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