Electronic wave functions electrostatic interactions

One area where the concept of atomic charges is deeply rooted is force field methods (Chapter 2). A significant part of the non-bonded interaction between polar molecules is described in terms of electrostatic interactions between fragments having an internal asymmetry in the electron distribution. The fundamental interaction is between the Electrostatic Potential (ESP) generated by one molecule (or fraction of) and the charged particles of another. The electrostatic potential at position r is given as a sum of contributions from the nuclei and the electronic wave function. 

The form of this equation makes explicit the fact that intermolecular forces do depend upon their vibrational states as well as on their electronic states. Due to the antisymmetrization of the global electronic wave function, Vaia2(R R12) contains Coulomb exchange terms and a direct term formed by the Coulomb multipole interactions and the infinite order perturbation electrostatic effects embodied in the reaction field potential [21, 22],... 

T. Korona, B. Jeziorski, One-electron properties and electrostatic interaction energies from the expectation value expression and wave function of singles and doubles coupled cluster theory. 

The theorem has the important implication that intramolecular interactions can be calculated by the methods of classical electrostatics if the electronic wave function (or charge distribution) is correctly known. The one instance where it can be applied immediately is in the calculation of cohesive energies in ionic crystals. Taking NaCl as an example, the assumed complete ionization that defines a (Na+Cl-) crystal, also defines the charge distribution and the correct cohesive energy is calculated directly by the Madelung procedure. 

The question now arises how all the above-mentioned phenomenologically classified interactions can be quantified. Of course, theory can yield unambiguous results if the additive decomposition of the overall interaction pattern into individual contributions is a suitable approximation in a certain case. It is, however, clear from the outset that many-body effects make a decomposition difficult, although this may be circumvented by a direct reference to the electronic wave function, which automatically adjusts to a given nuclear configuration, i.e. to a given arrangement of atoms. In Ref. [214], for example, an attempt is made to monitor the cooperative action of electrostatics in crown ether hydration via maps of the electrostatic potential. 

In atoms with more than one electron, wave functions should include the coordinates of each particle, and a new term representing the electrostatic interactions between electrons. Even for the case of only two electrons, such a wave equation is so complex that it has never been solved exactly. To analyse multielectron atoms some approximations have to be made. The most practical one is to assume that the electron considered moves in an electrical potential that is a combination of all other electrons and the nucleus, and that this potential has spherical symmetry. This approximation has proven very useful, as it allows a description of energy states in a similar manner to that employed for the H atom by using a comparable set of four quantum numbers. An important, additional condition appears no two electrons can have the same set of quantum numbers in other words, no more than one electron can occupy the same energy state. This is Pauli s exclusion principle. 

Next, in Equation (2.3), there is the Coulomb attraction between the electrons (charged —e) and the nuclei (charged Za) being ViA apart from each other, and this is what brings the system into "motion". In addition, we find the electrostatic repulsion between the electrons (fourth term) and also the electrostatic repulsion between the nuclei (fifth term) separated by Rab i r>ofe that we have excluded the double-counting of these interactions (i.e., / > i, B > A). We must also keep in mind that the electronic coordinates of this nonrelativistic H only contain spatial and no spin coordinates, and a spin-dependent description is eventually achieved by requiring a certain symmetry property for the many-electron wave function Y (see Sections 2.9 and 2.11.3). 

One of the earliest approximations for calculating an electronic wave-function is due to Hartree. Rather than treat the Coulomb interaction between each pair of electrons explicitly, this method assumes that each electron interacts with the positive Coulomb field due to the nuclei and the negative Coulomb field from the smeared average total charge density of the electrons. The latter electrostatic field, called the Hartree potential V (r), is given by the classical formula... 

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