Operators, Hamiltonian, correction

Operators, Hamiltonian, correction for parametric model for actinide... 

The irreducible tensor product between two (spherical) vectors is defined in Eq. (37). An important feature of this Hamiltonian is that it explicitly describes the dependence of the coupling constants J, Am, and T, on the distance vectors rPP between the molecules and on the orientations phenomenological Hamiltonian (139). Another important difference with the latter is that the ad hoc single-particle spin anisotropy term BS2y, which probably stands implicitly for the magnetic dipole-dipole interactions, has been replaced by a two-body operator that correctly represents these interactions. The distance and orientational dependence of the coupling parameters J, A, , and Tm has been obtained as follows. 

Hund s third rule is a relativistic correction to the first two rules, introducing a splitting of the terms given by the previous rules. The energy operator (Hamiltonian) commutes with the square of the total angular momentum J = L - - S, and therefore, the energy levels depend rather on the total momentum Jp = J J + This means that they depend on the mumal orientation of L and S (this is a relativistic effect due to the spin-orbit coupling in the Hamiltonian). The vectors L and S add in quantum mechanics in a specific way (see... 

Although in the Dirac-Coulomb Hamiltonian the one-particle operator is correct to all orders in a, the two-particle interaction is only correct to a°. The Dirac-Coulomb Hamiltonian is not invariant under Lorentz transformations, however it can be considered as the leading term of a yet unknown relativistic many-electron Hamiltonian which fulfills this requirement. An operator which also takes into account the leading relativistic corrections for the two-electron terms is the Coulomb-Breit term (Breit 1929,1930,1932, 1938),... 

After the initial transformation Uq, the new odd part Op in fx needs to be deleted by the next transformation Ux, and all that we require is that new odd terms are of higher order in the scalar potential V. Now, we do not restrict any of the subsequent unitary transformations to any specific choice of unitary transformation and employ the most general form of Eq. (11.57) without specifying the expansion coefficients that satisfy the xmitarity conditions Eqs. (11.60)-(11.66). Because the first term of each expansion of the imitaiy matrices according to Eq. (11.57) is the unit operator 1, a transformation of any operator yields exactly this operator plus correction terms. Hence, the first free-particle transformation already produced the final zeroth- and first-order even terms. So, we already have all even terms up to 1st order for the DKH Hamiltonian collected,... 

Si is now of order 3. The commutator relations derived above hold for the new operators as well. The commutator with produces a term that cancels the odd operator of order 1. The next commutator, with Oi, generates a term of order 4, which we neglect since we are concerned with an expansion to order 2. The commutator with Si produces terms of which the highest order is 3, which again we neglect. The Hamiltonian correct to order 2 is therefore given by the lowest terms of Si,... 

The scalar operator is called the Darwin operator and the spin-dependent operator the spin-orbit operator. We will meet these again in the chapter on perturbation theory (chapter 17). The Hamiltonian correct to order 2 in 1/c can then be written as... 

A Moeller-Plesset Cl correction to v / is based on perturbation theory, by which the Hamiltonian is expressed as a Hartree-Fock Hamiltonian perturbed by a small perturbation operator P through a minimization constant X... 

Since commuting operators have simultaneous eigenfunctions, it follows that correct eigenfunctions of the BO Hamiltonian must also be eigenfunctions of and Sz with eigenvalues S (S + 1) andM5 = S,S 1,., —S. All 2S + 1 members of a... 

In the above derivation, we have made no explicit assumption about the total electron spin quantum number S so that the results should be correct for S = 1 /2 as well as higher values. However, the fine structure term is not usually included in spin Hamiltonians for 5=1/2 systems. The fine structure term can be ignored since in that case the results of operating on a spin-1/2 wave function is always zero ... 

The actual form of the Hamiltonian operator hp does not have to be defined at this moment. As in standard perturbation theory, it is assumed that the solution of the electronic structure problem of the combined Hamiltonian HKS +HP can be described as the solution y/(0) of HKS, corrected by a small additional linear-response wavefunction /b//(,). Only these response orbitals will explicitly depend on time - they will follow the oscillations of the external perturbation and adopt its time dependency. Thus, the following Ansatz is made for the solution of the perturbed Hamiltonian HKS +HP ... 

The perturbation method is a unique method to determine the correlation energy of the system. Here the Hamiltonian operator consists of two parts, //0 and H, where //0 is the unperturbed Hamiltonian and // is the perturbation term. The perturbation method always gives corrections to the solutions to various orders. The Hamiltonian for the perturbed system is... 

Molecules are described in terms of a Hamiltonian operator that accounts for the movement of the electrons and the nuclei in a molecule, and the electrostatic interactions among the electrons and the electrons and the nuclei. Unlike the theory of the nucleus, there are no unknown potentials in the Hamiltonian for molecules. Although there are some subtleties, for all practical purposes, this includes relativistic corrections, [2] although for much of light-element chemistry those effects are... 

The above operators apply only to primitive basis functions that have the spin degree of freedom included. In the current work we follow the work of Matsen and use a spin-free Hamiltonian and spin-free basis functions. This approach is valid for systems wherein spin-orbit type perturbations are not considered. In this case we must come up with a different way of obtaining the Young tableaux, and thus the correct projection operators. 

---