Dirac Hamiltonian one-electron

One of the major fundamental difference between nonrelativistic and relativistic many-electron problems is that while in the former case the Hamiltonian is explicitly known from the very beginning, the many-electron relativistic Hamiltonian has only an implicit form given by electrodynamics [13,37]. The simplest relativistic model Hamiltonian is considered to be given by a sum of relativistic (Dirac) one-electron Hamiltonians ho and the usual Coulomb interaction term ... 

Of course, what has just been stated for the one-electron Dirac Hamiltonian is also valid for the general one-electron operator in Eq. (11.1). However, the coupling of upper and lower components of the spinor is solely brought about by the off-diagonal ctr p operators of the free-partide Dirac one-electron Hamiltonian and kinetic energy operator, respectively. We shall later see that the occurrence of any sort of potential V will pose some difficulties when it comes to the determination of an explicit form of the unitary transformation U. A universal solution to this problem will be provided in chapter 12 in form of Douglas-Kroll-Hess theory. 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

A systematic development of relativistic molecular Hamiltonians and various non-relativistic approximations are presented. Our starting point is the Dirac one-fermion Hamiltonian in the presence of an external electromagnetic field. The problems associated with generalizing Dirac s one-fermion theory smoothly to more than one fermion are discussed. The description of many-fermion systems within the framework of quantum electrodynamics (QED) will lead to Hamiltonians which do not suffer from the problems associated with the direct extension of Dirac s one-fermion theory to many-fermion system. An exhaustive discussion of the recent QED developments in the relevant area is not presented, except for cursory remarks for completeness. The non-relativistic form (NRF) of the many-electron relativistic Hamiltonian is developed as the working Hamiltonian. It is used to extract operators for the observables, which represent the response of a molecule to an external electromagnetic radiation field. In this study, our focus is mainly on the operators which eventually were used to calculate the nuclear magnetic resonance (NMR) chemical shifts and indirect nuclear spin-spin coupling constants. 

We start from the Dirac one-particle Hamiltonian for an electron bound to a point nucleus... 

The relativistic DV-Xa calculations are based on the one-electron Hamiltonian for the Dirac-Slater MO method which is given as... 

Even though the spin orbitals obtained from (2.23) in general do not have the full symmetry of the Hamiltonian, they may have some symmetry properties. In order to study these Fukutome considered the transformation properties of solutions of (2.24) with respect to spin rotations and time reversal. Whatever spatial symmetry the system under consideration has, its Hamiltonian always commutes with these operators. As we will see, the effective one-electron Hamiltonian (2.25) in general only commutes with some of them, since it depends on these solutions themselves via the Fock-Dirac matrix. 

Thus in general the elements of the group G = S x T do not commute with the effective one-electron Hamiltonian (2.25). Some of these operations may, however, commute with 9tefr and in such a case they form a subgroup of G. That subgroup characterizes the GHF solution under study in the sense that the corresponding Fock-Dirac matrix is invariant under the elements g of the invariance group of the GHF solution,... 

The deficiencies of this procedure have been carefully analysed by Boring and Wood [62] who worked with an approximate treatment of the Dirac equations, due to Cowan and Griffith [63]. In this method the spin-orbit operator is omitted from the one-electron Hamiltonian but the mass-velocity... 

Matrix theory for Dirac one-electron problems was set up in the last section, and we shall now generalize this, first for closed-shell atoms and then for the general open-shell case. We use the effective Hamiltonian of (95) as the starting... 

Just to see what kind of problems are associated with application of tiie Dirac equation, let us consider two non-interacting Dirac electrons. Let us assume that a two-electron Hamiltonian is simply a sum of tiie two one-electron Hamiltonians. Such two-electron problem can be... 

In the preceding two chapters, we dealt with general unitary transformation schemes to produce a one-electron Hamiltonian valid for only the positive-energy part of the Dirac spectrum that governs the electronic bound and continuum states. Evidently, these unitary transformation schemes are elegant but involved. Developments in quantum chemistry always focus on efficient approximations in a sense that the main numerical contribution of some physical effect is reliably captured for any class of molecule or molecular aggregate. The so-called elimination techniques have been very successful in this sense and are therefore discussed in the present chapter. 

In chapter 10, we have already discussed how the size of the small-component basis set can be made equal to that of the large-component basis set by absorbing the kinetic-balance operator into the one-electron Hamiltonian. In this chapter, we have elaborated on this by introducing a pseudo-large component that has led to the modified Dirac equation. 

MRCI, CCSD(T), and DMRG) compared to the choice of the one-electron Hamiltonian (DKH and effective core potential) with a special emphasis on the order of the DKH Hamiltonian. The calculated results are compared to data from experiment. The effect of the one-particle basis set can be seen for CsH from the entries for the contracted bases of quality [lls9pSd] and [Ils9p8rf2/1 ], respectively. Note also the importance of the approximation for the total wave function (as, for example, highlighted in the case of SnO for uncomelated Dirac-Hartree-Fock or even nonrelativistic Hartree-Fock compared to the coupled-cluster results). 

Dirac-Hartree-Fock (DHF) calculations (the relativistic elaboration of Hartree-Fock theoryS3) of an appropriate generator state for the fourteen lanthanides (Ce to Lu) are performed. The program written by Desclaux is used for numerical DHF calculations.3= The one-electron Hamiltonian is fully... 

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 modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

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