Normal-ordered electron Hamiltonian

This operator acts only on the electron-positron and electromagnetic field variables. The normal ordered form of Hm taken in the interaction representation must be added to the interaction Hamiltonian. It gives additional elements to the... 

Now we turn to lithium and three-electron lithium-like ions. Again we start with the normally-ordered no-pair Hamiltonian given in Eq. (132), and choose the starting potential to be the Hartree-Fock potential of the (Is) helium-like core. We expand the energy of an atomic state in powers of the interaction potential... 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

A full relativistic theory for coupling tensors within the polarization propagator approach at the RPA level was presented as a generalization of the nonrelativistic theory. Relativistic calculations using the PP formalism have three requirements, namely (i) all operators representing perturbations must be given in relativistic form (ii) the zeroth-order Hamiltonian must be the Dirac-Coulomb-Breit Hamiltonian, /foBC, or some approximation to it and (iii) the electronic states must be relativistic spin-orbitals within the particle-hole or normal ordered representation. Aucar and Oddershede used the particle-hole Dirac-Coulomb-Breit Hamiltonian in the no-pair approach as a starting point, Eq. (18),... 

If applied to the reference state normal order enables us immediately to recognize those terms which survive in the computation of the vacuum amplitudes. The same applies for any model function and, hence, for real multidimensional model spaces, if a proper normal-order sequence is defined for all the particle-hole creation and annihilation operators from the four classes of orbitals (i)-(iv) in Subsection 3.4. In addition to the specification of a proper set of indices for the physical operators, such as the effective Hamiltonian or any other one- or two-particle operator, however, the definition and classification of the model-space functions now plays a crucial role. In order to deal properly with the model-spaces of open-shell systems, an unique set of indices is required, in particular, for identifying the operator strings of the model-space functions (a)< and d )p, respectively. Apart from the particle and hole states (with regard to the many-electron vacuum), we therefore need a clear and simple distinction between different classes of creation and annihilation operators. For this reason, it is convenient for the derivation of open-shell expansions to specify a (so-called) extended normal-order sequence. Six different types of orbitals have to be distinguished hereby in order to reflect not only the classification of the core, core-valence,... orbitals, following our discussion in Subsection 3.4, but also the range of summation which is associated with these orbitals. While some of the indices refer a class of orbitals as a whole, others are just used to indicate a particular core-valence or valence orbital, respectively. 

The theorem is based on a perturbation of the Hamiltonian by smaU displacements of the nuclei. A high-symmetry geometry is chosen as the origin, and the nuclear displacements are described by normal modes which transform as irreps of the point group. The nuclear positions are parameters in the electronic Hamiltonian. One has, to second-order ... 

The conclusion of this analysis is that the normal-ordered QED approach as presented here, with a floating vacuum, is equivalent to the empty Dirac approach. It appears that the reinterpretation of the negative-energy states as positron states has no influence on the combination of matrix elements that results from the commutator. Normal ordering then only affects the terms involving the positron operators, and at least for the one-electron Hamiltonian this means that the reference energy will be identical in both the empty Dirac and the QED approaches, since they only have occupied electron states, and the terms that survive in the refCTence expectation value are identical in the two approaches. 

The partition of O Eq. 28.14 is based on the so-called normal ordered operators and, Qn (Cammi 2009). Specifically, Qn(A, T) is the component of the solvent reaction potential due to the correlation CC electronic density, and H(0)n is the normal ordered form of Hamiltonian of the solute in presence of the frozen Hartree-Fock reaction field Qhf-... 

However, before going into a detailed discussion of various relativistic Hamiltonians we will introduce an alternative form of the electronic Hamiltonian (3.4), which is useful for wavefunction-based correlation methods. It is obtained by switching to a particle-hole formalism and then introducing normal ordering. In the second-quantization formalism creation and annihilation operators refer to some specific set of (orthononnal) orbitals, and Slater determinants in Hilbert space translate into occupation-number veetors in Fock space. The annihilation operators in equation 3.4 by definition give zero when acting on the vacuum state... 

Since H° is the sum of hydrogenlike Hamiltonians, the zeroth-order wave function is the product of hydrogenlike functions, one for each electron. We call any one-electron spatial wave function an orbital. To allow for electron spin, each spatial orbital is multiplied by a spin function (either a or 0) to give a spin-orbital. To introduce the required antisymmetry into the wave function, we take the zeroth-order wave function as a Slater determinant of spin-orbitals. For example, for the Li ground state, the normalized zeroth-order wave function is... 

These correlated fluctuations themselves ride on a further set of coherent fluctuations taking place at a much lower frequency scale and normally attributed to the phonons, the traditional exchange Bosons associated with superconductivity. Real systems are never devoid of ionic or nuclear motion, and at the very least it is now Hamiltonian (3) (and eventually its extension to alloys) that applies for a full discussion of superconductivity density fluctuations in the nuclear coordinates are omnipresent and of course their effects on electronic ordering have been evident for quite some time. An elementary estimate of the relative importance of (monopole) polarization arising from phonons and the (multipole) equivalents arising from internal fluctuations, primarily of a dipole character, can now be easily given. 

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