Hamiltonian exact 2-component

The valence correlation component of TAE is the only one that can rival the SCF component in importance. As is well known by now (and is a logical consequence of the structure of the exact nonrelativistic Bom-Oppenheimer Hamiltonian on one hand, and the use of a Hartree-Fock reference wavefunction on the other hand), molecular correlation energies tend to be dominated by double excitations and disconnected products thereof. Single excitation energies become important only in systems with appreciable nondynamical correlation. Nonetheless, since the number of single-excitation amplitudes is so small compared to the double-excitation amplitudes, there is no point in treating them separately. 

As one can see, for each set of parameters Z, k, L and d, one can select a in such a way that Eqs. (21) and (22) are fulfilled. Then, the resulting expectation values correspond to the variational minima and are equal to the appropriate exact eigenvalues of either Dirac or Schrodinger (or rather Levy-Leblond) Hamiltonian. However the corresponding functions do not fulfil the pertinent eigenvalue equations they are not eigenfunctions of these Hamiltonians. This example demonstrates that the value of the variational energy cannot be taken as a measure of the quality of the wavefunction, unless the appropriate relation between the components of the wavefunction is fulfilled [2]. 

Note that as long as the unitary transformation is exact, i.e., as long as the Taylor series is infinite and converges, the exact energy eigenvalues of the 4-component untransformed Hamiltonian are obtained. 

Fig. 95 Spin-Hamiltonian projection (level-6) of magnetic parameters on tetragonal distortion of high-spin/intermediate- and weak-field Co(II) complexes. D relates to the energy gap for the compressed bipyramid, and refers to the asymmetry parameter for the elongated one light surface - exact multiplet splitting gray area - spin Hamiltonian projection. Note the graphs for the g-components and TIP have interchanged axes...

In a second step, in order to determine the influence of the anharmonicity in the exact potential we will expand the term up to higher powers of the components of r and treat them as small perturbations to the harmonic approximation of the Hamiltonian by means of first order perturbation theory. These perturbative calculations offer insight into the effects of the anharmonic parts of the potential onto the energies and the form of the wave functions. For a discussion of the basis set method and the computational techniques used for the numerical calculation of the exact eigenenergies and eigenfunctions in the outer potential well we refer the reader to [7]. In the following we discuss the results of these numerical calculations of the exact eigenenergies and wave functions and... 

In the formalism of perturbation theory we assume the exact Hamiltonian H to be composed of two components the zero-order H0 and the perturbation V... 

Indeed, when Mr < Mt, the disconnected component of the left-hand side of Eq. (97), i.e. the expression P Ck,open Hn,oPen )i vanishes, since cluster amplitudes defining T, Eq. (41), satisfy equations (78) with n = 1,..., Mr-Equation (99) represents a generalization of the exact Eq. (88) to truncated EOMXCC schemes. Again, the only significant difference between the EOMXCC equations (98) and (99) and their EOMCC analogs (48) and (47), respectively, is the similarity transformed Hamiltonian used by both theories. As in the EOMCC theory, Eqs. (98) and (99) have the same general form (in particular, they rely on the same similarity transformed Hamiltonian) for all the sectors of Fock space. 

Unfortunately, in the presence of F(r), the unitary matrix giving the exact decoupling is not found in a closed form. A number of different approximations to the exact FW transformation have been suggested and analyzed in the literature. - With the special choice of approximations to the exact decoupling, the effective two-component ZORA Hamiltonian in the presence of electromagnetic fields is... 

In this Hamiltonian, the a/ are the shieldings, or chemical shifts, of the nuclei, J / the indirect spin-spin coupling constants between pairs of nuclei, IZi the z component of the spin operator 7, y the gyromagnetic ratic and H0 the strength of the static applied field. Within the experimental accuracy of measurements achieved thus far, no additional terms are required in the Hamiltonian to attain an exact fit of theoretical and calcu-... 

---