Hamiltonian normal order

The Hamiltonian and the cluster operators in our formulation will also be written in normal order and hence will involve normal-ordered generators. The Hamiltonian, normal ordered with respect to a core 0), has the structure ... 

The component can be represented over the initial I) and intermediate J) states of the system defined by Wn, which is a normal-ordered Hamiltonian with respect to the I) state ... 

The Hamiltonian in normal order with respect to its own exact eigenfunction (of full-CI type) is... 

We start from the Hamiltonian (in normal order with respect to the genuine vacuum)... 

Turning now back to the single-root MR BWCC approach, we derive the basic equations for the effective Hamiltonian and cluster amplitudes in the spin-orbital form without the use of the BCH formula. We limit ourselves to a complete model space which implies that amplitudes corresponding to internal excitations (i.e. excitations within the model space) are equal to zero. In our derivation we shall work with the Hamiltonian in the normal-ordered-product form, i.e. 

As concerns cluster amplitudes, if we employ the exact Hamiltonian in the normal-ordered-product form (31) with the /i-th configuration as a Fermi vacuum, the basic equation for the single-root wave operator (25) takes the form... 

The Dirac-Coulomb-Breit Hamiltonian H qb 1 rewritten in second-quantized form [6, 16] in terms of normal-ordered products of spinor creation and annihilation operators r+s and r+s+ut, ... 

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... 

For the analysis of the various formalisms, manipulation of the equations, generating normal product of terms via Wick s theorem, and particularly for indicating how the proofs of the several different linked cluster theorems are achieved, we shall make frequent use of diagrams. For the sake of uniformity, we shall mostly adhere to the Hugenholtz convention/1/. All the constituents of the diagrams will be operators in normal order with respect to suitable closed-shell determinant taken as the vacuum. We shall refer to the creation/annihilation operators with respect to this vacuum after the h-p transformation.The hamiltonian H will also be taken to be in normal order with respect to... 

Note that the first and second terms on the right-hand side of this equation are simply the spin-orbital Fock operator (in normal-ordered form), and the last two terms are the Hartree-Fock energy (i.e., the Fermi vacuum expectation value of the Hamiltonian). Thus, we may write... 

This result is easily generalized the normal-ordered form of an operator is simply the operator itself minus its reference expectation value. For the Hamiltonian example, above, the normal-ordered Hamiltonian is just the Hamiltonian minus the SCF energy (i.e., may be considered to be a correlation operator). Owing to its considerable convenience for coupled cluster and many-body perturbation theory analyses, this conventional form of f given in Eq. [105] is adopted for the remainder of this chapter. 

The concepts of normal ordering and Wick s theorem provide the mathematical tools needed to derive programmable coupled cluster equations from the more formal expressions given in Eqs. [50] and [51]. If we truncate the cluster operator such that T = Tj + T2 insert it into the similarity-transformed normal-ordered Hamiltonian, H = e lij e, we obtain... 

Using the connected cluster form of H defined above, as well as the techniques of Wick s theorem and normal ordering, we may derive a programmable form of the energy expression in the CCSD approximation. In accord with Eq. [50] and the normal-ordered Hamiltonian, the energy is given by... 

For all other terms, we may use the advantage of normal-ordering of the operators to determine all the fully contracted terms of the operator product. For example, for the second term on the right-hand side of Eq. [122], insertion of the definition of the normal-ordered Hamiltonian gives... 

For the contribution of Hjq to the T2 amplitude equation, on the other hand, we must evaluate the matrix elements of the normal-ordered Hamiltonian between doubly excited determinants and Oq, namely,... 

The second form, the normal order of the generator product, shows that the operator also preserves and eigenvalues since it is constructed from operators that do so. The expansion of the Hamiltonian operator in this spinpreserving operator basis shows that the Hamiltonian operator itself must preserve the and eigenvalues of the wavefunctions on which it acts. The definition of the operator results in the identities... 

The QED radiative effects are treated perturbatively by the inclusion of Hrad-Accordingly, the unperturbed Hamiltonian Ho reduces to the external-field problem (normal ordering is indicated by )... 

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... 

For the following it is furthermore crucial that normal order in (8, 9) refers to the creation and annihilation operators of the (renormalized) asymptotic fields of the homogeneous system, i.e. with zero external four-potential A, so that this intrinsic Hamiltonian density is independent of the external four-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),... 

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