Coulomb interaction operator

The interaction operator Vsm= V(rs,rm,Rs,Rm)is defined in terms of the Coulomb interaction operator l/ r-r1 = T(r-r) and the charge density operators of the solute Ws(r) and the surrounding medium QmCr1) ... 

For a quasi-relativistic framework as relevant to chemistry (21), we may neglect the magnetic retardation between the electrons and the nuclei and therefore employ standard Coulombic interaction operators for the electrostatic interaction. The interaction between the electrons and the nuclei is not specified explicitly but we only describe the interactions by some external 4-potential. For the sake of brevity this 4-potential shall comprise all external contributions. Explicit expressions for the interaction between electrons and nuclei will be introduced at a later stage. Furthermore, we can neglect the relativistic nature of the kinetic energy of the nuclei and employ the non-relativistic kinetic energy operator denoted as hnuc(I),... 

The operator P (second term in (1.15)) is a complete scalar and does not require any changes. The Coulomb interaction operator (the last item in Eq. (1.15)), after expanding l/rn in terms of spherical functions, in an irreducible form is equal to... 

Making use of the properties of the eigenvalues of Casimir operators, mentioned in Chapter 5, we are in a position to find a number of interesting features of the matrix elements of the Coulomb interaction operator. Thus, it has turned out that for the pN shell there exists an extremely simple algebraic expression for this matrix element... 

Let r stand for the radius vector of the ith electron with respect to the center of mass of the molecule. Assuming that all r, are much less than b, we can expand the Coulomb interaction operator into a multipole series. Keeping only the dipole terms, we get the following expression for P0n(b)... 

The averaging of the Coulomb interaction operator can be easily performed following the prescription given in [29,30] ... 

The electron-electron interaction is usually supposed to be well described by the instantaneous Coulomb interaction operator l/rn. Also, all interactions with the nuclei whose internal structure is not resolved, like electron-nucleus attraction and nucleus-nucleus repulsion, are supposed to be of this type. Of course, corrections to these approximations become important in certain cases where a high accuracy is sought, especially in computing the term values and transition probabilities of atomic spectroscopy. For example, the Breit correction to the electron-electron Coulomb interaction should not be neglected in fine-structure calculations and in the case of highly charged ions. However, in general, and particularly for standard chemical purposes, these corrections become less important. 

Things become very complex when we try to switch on the electronic interaction. First, what should we use as the interaction eneigy operator Nobody knows, but one idea might be to add to the sum of the Dirac one-electron operators the Coulombic interaction operators of all the particles. This is what is known as the Dirac-Coulomb (DC) model... 

By substituting in Eq.4.18 the Coulomb interaction operator by its relativistic extension (4.19) one obtains the so-called Dirac-Breit many electron Hamiltonian... 

It is also appropriate to mention that the picture change problem is by no means restricted to one-electron operators, although the way of phrasing it in the case of the Coulomb repulsion terms is usually somewhat different from the one presented here. For instance, Sucher [61] considered the modification of the Coulomb repulsion operator caused by the no-pair approximation. The expression he derived is simply the interaction operator in the very approximate two-component formalism. This form of the Coulomb interaction operator was also investigated by Samzow et al. [62]. They found that for valence orbitals of heavy systems the corresponding modification of two-electron integrals is relatively small. 

By contrast to the transformation of standard Coulomb interaction operators to spherical coordinates, this task is much more involved in the case of the Breit interaction discussed in section 8.1. The Breit operator is symmetric with respect to an exchange of the full electron coordinate sets of electrons 1 and 2. This symmetry is partially lost if the radial terms are considered separately. 

Here, H k) is a one-electron operator that describes the motion of an electron in the crystal and is equal to the sum of the kinetic-energy operator and the Coulomb interaction operator between the electron and fixed atomic nuclei and J k) and X k) are the Coulomb and exchange operators, respectively, which describe the interaction of the given electron with the other electrons of the crystal. 

Each electron in the system is assigned to either molecule A or B, and Hamiltonian operators and for each molecule defined in tenns of its assigned electrons. The unperturbed Hamiltonian for the system is then 0 = - A perturbation XH consists of tlie Coulomb interactions between the nuclei and... 

Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

The f operators are the usual kinetic energy operators, and the potential energy V(r,R) includes all of the Coulomb interactions ... 

If classical Coulombic interactions are assumed among point charges for electrostatic interactions between solute and solvent, and the term for the Cl coefficients (C) is omitted, the solvated Eock operator is reduced to Eq. (6). The significance of this definition of the Eock operator from a variational principle is that it enables us to express the analytical first derivative of the free energy with respect to the nuclear coordinate of the solute molecule R ,... 

Modification of the potential operator due to the finite speed of light. In the lowest order approximation this corresponds to addition of the Breit operator to the Coulomb interaction. 

Asm is an antisymmetrizer operator between electrons from these two groups s and m which is usually expressed as a sum of the identity operator (1) and normalized permuting operator Pms Asm =l+pms. The total Hamiltonian is symmetric to any electron permutation. The interaction energy Vsm can be cast in terms of a direct Coulomb interaction and an exchange Coulomb interaction ... 

Of course, the Coulomb interaction appears in the Hamiltonian operator, H, and is often invoked for interpreting the chemical bond. However, the wave function, l7, must be antisymmetric, i.e., must satisfy the Pauli exclusion principle, and it is the only fact which explains the Lewis model of an electron pair. It is known that all the information is contained in the square of the wave function, 1I7 2, but it is in general much complicated to be analyzed as such because it depends on too many variables. However, there have been some attempts [3]. Lennard-Jones [4] proposed to look at a quantity which should keep the chemical significance and nevertheless reduce the dimensionality. This simpler quantity is the reduced second-order density matrix... 

For allowed transitions on D and A the Coulombic interaction is predominant, even at short distances. For forbidden transitions on D and A (e.g. in the case of transfer between triplet states (3D + 3A —> 1D + 3A ), in which the transitions Ti —> S0 in D and So —> Ti in A are forbidden), the Coulombic interaction is negligible and the exchange mechanism is found, but is operative only at short distances (< 10 A) because it requires overlap of the molecular orbitals. In contrast, the Coulombic mechanism can still be effective at large distances (up to 80-100 A). 

The classical potential energy term is just a sum of the Coulomb interaction terms (Equation 2.1) that depend on the various inter-particle distances. The potential energy term in the quantum mechanical operator is exactly the same as in classical mechanics. The operator Hop has now been obtained in terms of second derivatives with respect to Cartesian coordinates and inter-particle distances. If one desires to use other coordinates (e.g., spherical polar coordinates, elliptical coordinates, etc.), a transformation presents no difficulties in principle. The solution of a differential equation, known as the Schrodinger equation, gives the energy levels Emoi of the molecular system... 

This paper considers the hyperspherical harmonics of the four dimensional rotation group 0(4) in the same spirit ofprevious investigations [2,11]), where the possibility is considered of exploiting different parametrizations of the 5" hypersphere to build up alternative Sturmian [12] basis sets. Their symmetry and completeness properties make them in fact adapt to solve quantum mechanical problems where the hyperspherical symmetry of the kinetic energy operator is broken by the interaction potential, but the corresponding perturbation matrix elements can be worked out explicitly, as in the case of Coulomb interactions (see Section 3). A final discussion is given in Section 4. 

---