Electron momentum operator

Kohn-Sham orbitals (18)), Vn is the external, nuclear potential, and p is the electronic momentum operator. Hence, the first integral represents the kinetic and potential energy of a model system with the same density but without electron-electron interaction. The second term is the classical Coulomb interaction of the electron density with itself. Exc> the exchange-correlation (XC) energy, and ENR are functionals of the density. The exact functional form for Exc is unknown it is defined through equation 1 (79), and some suitable approximation has to be chosen in any practical application of... 

Here p = —iV is the electron momentum operator, d, P are the standard Dirac matrices, A is the vector potential and V is the scalar potential of the external field. The wave function (r,t) is the four-component spinor. For the stationary state ... 

In (A.3) the velocity form of the dipole approximation is used. The factor of in p(E) cancels with the normalization for the plane wave, thus providing the correct continuum limit (L oo). If it is assumed that 0> is a closed-shell state, the two terms on the right-hand side of (A.2) yield identical results in (A.3). Therefore, we simplify the notation by combining the two terms and suppress the spin designations. The electronic momentum operator for our system, expressed in second quantized notation, is given by... 

The average of the first term over any electronic state is zero because of the electronic momentum operator being odd under inversion. The second term also vanishes, because... 

For example, in the first expression G = the electronic momentum operator whose time rate-of-change determines the force, the Ehrenfest force, acting on the electron density. The momentum density, whose time rate-of-change appears on the LHS, is the vector current mj (r), m times the velocity density of the electrons. The commutator in the first term on the RHS... 

The presence of a magnetic field-external, or from a nuclear spin-can be incorporated by minimal substitution for the electron momentum operator. 

Electrons and most other fiindamental particles have two distinct spin wavefunctions that are degenerate in the absence of an external magnetic field. Associated with these are two abstract states which are eigenfiinctions of the intrinsic spin angular momentum operator S... 

Now, consider the case of spinless particles not subject to external electronic and magnetic fields. We may now choose the unitai7 operator U as the unit operator, that is, T = K. For the coordinate and momentum operators, one then obtains... 

There are cases in which the angular momentum operators themselves appear in the Hamiltonian. For electrons moving around a single nucleus, the total kinetic energy operator T has the form ... 

The electronic Hamiltonian commutes with both the square of the angular momentum operator r and its z-component and so the three operators have simultaneous eigenfunctions. Solution of the electronic Schrddinger problem gives the well-known hydrogenic atomic orbitals... 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

Here, using electron field operator, momentum density is expressed as... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

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