Hamiltonian operator wave-function based calculations

The two-component methods, though much simpler than the approaches based on the 4-spinor representation, bring about some new problems in calculations of expectation values of other than energy operators. The unitary transformation U on the Dirac Hamiltonian ho (Eq.4.23 is accompanied by a corresponding reduction of the wave function to the two-component form (Eq.4.26). The expectation value of any physical observable 0 in the Dirac theory is defined as ... 

We have already seen in section 8.1 that (i) a Dirac electron with electromagnetic potentials created by all other electrons [cf. Eq. (8.2)] cannot be solved analytically, which is the reason why the total wave function as given in Eq. (8.4) cannot be calculated, and also that (ii) the electromagnetic interactions may be conveniently expressed through the 4-currents of the electrons as given in Eq. (8.31) for the two-electron case. Now, we seek a one-electron Dirac equation, which can be solved exactly so that a Hartree-type product becomes the exact wave function of this system. Such a separation, in order to be exact (after what has been said in section 8.5), requires a Hamiltonian, which is a sum of strictly local operators. The local interaction terms may be extracted from a 4-current based interaction energy such as that in Eq. (8.31). Of course, we need to take into account Pauli exchange effects that were omitted in section 8.1.4, and we also need to take account of electron correlation effects. This leads us to the Kohn-Sham (KS) model of DFT. 

Current relativistic electronic structure theory is now in a mature and well-developed state. We are in possession of sufficiently detailed knowledge on relativistic approximations and relativistic Hamiltonian operators which will be demonstrated in the course of this book. Once a relativistic Hamiltonian has been chosen, the electronic wave function can be constructed using methods well known from nonrelativistic quantum chemistry, and the calculation of molecular properties can be performed in close analogy to the standard nonrelativistic framework. In addition, the derivation and efficient implementation of quantum chemical methods based on (quasi-)relativistic Hamiltonians have facilitated a very large amount of computational studies in heavy element chemistry over the last two decades. Relativistic effects are now well understood, and many problems in contemporary relativistic quantum chemistry are technical rather than fundamental in nature. 

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