Hamiltonian operator space representation

The Spin adapted Reduced Hamiltonian SRH) is the contraetion to a p-electron space of the matrix representation of the Hamiltonian Operator, 2 , in the N-electron space for a given Spin Symmetry [17,18,25,28], The basis for the matrix representation are the eigenfunctions of the operator. The block matrix which is contracted is that which corresponds to the spin symmetry selected In this way, the spin adaptation of the contracted matrix is insnred. 

By performing a partial Wigner transform with respect to the coordinates of the environment, we obtain a classical-like phase space representation of those degrees of freedom. The subsystem coordinate operators are left untransformed, thus, retaining the operator character of the density matrix and Hamiltonian in the subsystem Hilbert space [4]. In order to take the partial Wigner transform of Eq. (1) explicitly, we express the Liouville-von Neumann equation in the Q representation,... 

When considering relaxation, a Liouville space representation is typically used in which the Hamiltonian and density matrix are represented as superoperators in addition to the relaxation operator being represented as a superoperator. Once a... 

Once again, the operational application of this development to CASSCF wave functions means that the states (kets) are replaced by the configuration-interaction vectors, and the Hamiltonian operator by its matrix representation in the space of the configuration-state functions (see Appendix). 

Vi = d/dri. In deriving Eq. (B.106) we employed the space representation of the Hamiltonian operator [see Eq. (2.95)], and the fact that the classic Hamiltonian function can be split into kinetic- and potential-energy contributions iiccording to Eq. (2.100). Terms proportional to in Eq. (B.106) arise from the kinetic part of // applied to the product of terms on the right side of Eq. (B.104) (using, of course, the product rule of conventional calculus). 

Now the DK Hamiltonian may be calculated to the desired level of accuracy. Within our finite basis set approximation the multiple integral expressions occurring at the evaluation of the momentum space operators are reduced to simple matrix multiplications, which are computationally not very demanding. As soon as /foKHn has been evaluated within the chosen p -representation, it can be transformed back to the usual configuration space representation by applying the inverse transformation This Hamiltonian is then available for every variational procedure without any further modifications. 

However, these benefits come at a price. Both Vgg and Vxc and their contributions to the transformations obviously change at each self-consistent iteration so the net effect is that some very complicated operator products, involving both momentum and direct space representations, must be done at every iteration. What Rosch and co-workers noticed [44] was that the singular part of the Hamiltonian Vxe of course does not change from iteration to iteration, so they attempted an incomplete DKH transformation which retained only V g and incorporated, therefore, the bare electron-electron interactions in the transformed Hamiltonian. 

So far only the position-space formulation of the (stationary) Dirac Eq. (6.7) has been discussed, where the momentum operator p acts as a derivative operator on the 4-spinor Y. However, for later convenience in the context of elimination and transformation techniques (chapters 11-12), the Dirac equation is now given in momentum-space representation. Of course, a momentum-space representation is the most suitable choice for the description of extended systems under periodic boundary conditions, but we will later see that it gains importance for unitarily transformed Dirac Hamiltonians in chapters 11 and 12. We have already encountered such a situation, namely when we discussed the square-root energy operator in Eq. (5.4), which cannot be evaluated if p takes the form of a differential operator. 

Moments play a crucial role in the multiscale structure of Continuous Wavelet Transform (CWT) theory. The generalized scale-translation dependent moments, designated as scalets in this work, play an immediate role in transforming Schrodinger-Hamiltonian operators (with rational fraction potentials) into an explicit space-scale representation, which in turn directly leads to a CWT-basis representation. 

First-quantization operators conserve the numbCT of electrons. Following the discussion in Section 1.3, such operators are in the Fock space represented by linear combinations of operators that contain an equal number of creation and annihilation rqierators. The explicit form of these number-conserving operators depends on whether the first-quantized operator is a one-electron operator or a two-electron operator. One-electron operators are discussed in Section 1.4.1 and two-electron operators in Section 1.4.2. Finally, in Section 1.4.3 we consider the second-quantization representation of the electronic Hamiltonian operator. 

The Hamiltonian again has the basic form of Eq. (63). The system is described by the nuclear coordinates, Q, which are relative to a suitable nuclear configuration Q. In conbast to Section in.C, this may be any point in configmation space. As a diabatic representation has been assumed, the kinetic energy operator matrix, T, is diagonal with elements... 

In the case of a perfect crystal the Hamiltonian commutes with the elements of a certain space group and the wave functions therefore transform under the space group operations accorc g to the irreducible representations of the space group. Primarily this means that the wave functions are Bloch functions labeled by a wave vector k in the first Brillouin zone. Under pure translations they transform as follows... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

Since it can be shown that "( ), like the original Hamiltonian H, commutes with the transformation operators Om for all operations R of the point group to which the molecule belongs, the MOs associated with a given orbital energy will form a function space whose basis generates a definite irreducible representation of the point group. This is exactly parallel to the situation for the exact total electronic wavefunctions. 

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