Charge density 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) ... 

Taking the Fourier transform of the charge density operator... 

In these formulas /3(r) is the charge density operator and J(r) is the current density operator. For a discrete n-electron charge distribution... 

In all of these considerations cany the Ml conelations internal to their individual subsystems, A, B, respectively, with no relation to the other subsystem, B, A, respectively. As the operators internal to the statistical trace matrix elements of Gf in Eq.(5) involve the charge density operators of subsystem B,, it is... 

A system of many identical particles is most conveniently described by the if) field operators (5). The generic form of the electronic charge density operator in a semiconductor is... 

When the orthonormal set us(C) is now employed for representations of the hamiltonian and other relevant operators, it is in approximate theories taken as a justification to neglect in a first approximation integrals involving the product density of spin orbitals associated with different atomic centers. This is the so-called Zero Differential Overlap (ZDO) approximation. The charge density operator, for instance, would then become... 

A more drastic approximation than the NDDO is the Complete Neglect of Differential Overlap (CNDO), where all cross products in Eq. (10.62) are omitted and the atomic charge density operator is simply... 

Both the NDDO and the CNDO assumptions apply only to matrix elements involving the charge density operator. Other arguments have to be used for matrix elements of other operators, such as the kinetic energy and the current density. 

This leads to the following nonlinear Schrodinger equation provided that G r, r ) = G r r) and the charge density operators appearing from the left and right of the interaction kernel in Eq. (5.12) are the same ... 

These general equations are not limited by any multipolar approximation. In practical implementations, as mentioned above, the reaction field response function can be calculated from atomic or group dipolar polarizabilities, while the solvent charge density operator can be approximated by atomic multipoles. 

Longuet-Higgins [43] first drew attention to the fact that the dispersion interaction between two molecules could be calculated directly in terms of charge density functions, without making the usual multipole expansion of the interaction terms in the Hamiltonian. The charge density operator for molecule A at point r may be defined as (using a for particle index, a for nuclei, i for electrons)... 

More precisely, the Longuet-Higgins charge density operator is the FT of k) ... 

The interaction Hamiltonian describes now the coupling of the solute charge density operator with the electrostatic potential created by the surroundings at fix X V (r) = J dr < P T(r — r ) O (r ) This expression can be cast into ... 

The key step here for constructing a simple SCRF Hamiltonian is to replace the m-averaged solute charge density 2s(r )> ) by the expectation value of the solute charge density operator with respect to the wavefunction i.e, =... 

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