Relativistic corrections wavefunctions

CORRECT WAVEFUNCTIONS FOR PERTURBATIONS (SPIN-ORBIT, EXTERNAL FIELD, RELATIVISTIC, ETC.) WITHIN BORN-OPPENHEIMER APPROXIMATION... 

Martin " was the first to estimate the effects of relativity on the spectroscopic constants of Cu2. The scalar relativistic (mass-velocity and Darwin) terms were evaluated perturbatively using Hartree-Fock or GVB (Two configuration SCF (ffg -mTu)) wavefunctions. At these levels the relativistic corrections for r, cu and D, were found to be — 0.05 A, 15 cm and -h0.06eV for SCF, and —0.05 A, + 14 cm and -l-0.07eV for GVB. The shrinking of the bond length is less than half of the estimate based on the contraction of the 4s atomic orbital. 

Other interesting calculations of Be ", Ne "", Be, and Ne atom have been carried out by Kenny et al. [85] in which they evaluated perturba-tionally the relativistic corrections to the total energies. In particular, they found that the Breit correction is systematically larger in the Dirac-Fock approximation and calculated the most accurate values of relativistic corrections for the Ne atom to date. These results demonstrate another useful capability of the correlated wavefunction produced by QMC to estimate relativistic effects. Similar study within the VMC method has been done on examples of Li and LiH [86]. We expect that an important future application will be to carry out similar calculations of transition... 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

Hamiltonian is not known and, as for the nonrelativistic case, further approximations have to be introduced in the wavefunction, it is tempting to derive approximate computational schemes which are still sufficiently accurate but more efficient. Here we will only summarize those approximate methods that have been used frequently to obtain information about the electronic structure of molecules with lanthanide atoms, i.e. relativistically corrected density-functional approach, pseudopotential method, intermediate neglect of differential overlap method, extended Huckel theory, and ligand field theory. 

Even though the programs described in this chapter are referred to as non-relativistic (i.e., using single-component wavefunction), for molecules containing heavy atoms it is necessary to include at least scalar relativistic effects. A popular method of adding relativistic corrections is based on the formalism developed by Douglas and Kroll (1974) and by Hess (1986, 1989), and is usually referred to as DKHn (where n denotes the order of the method). 

Relativistic variational principles are usually formulated as prescriptions for reaching a saddle point on the energy hypersurface in the space of variational parameters. The results of the variational calculations depend upon the orientation of the saddle in the space of the nonlinear parameters. The structure of the energy hypersurface may be very complicated and reaching the correct saddle point may be difficult [14,15]. If each component of the wavefunction is associated with an independent set of nonlinear parameters, then changing the representation of the Dirac equation results in a transformation of the energy hypersurface. As a consequence, the numerical stability of the variational procedure depends on the chosen representation. 

In M0ller-Plesset theory, first-order perturbation theory does not improve on the HF energy because the zeroth-order Hamiltonian is not itself the HF Hamiltonian. However, first-order perturbation theory can be useful for estimating energetic effects associated with operators that extend the HF Hamiltonian. Typical examples of such terms include the mass-velocity and one-electron Darwin corrections that arise in relativistic quantum mechanics. It is fairly difficult to self-consistently optimize wavefunctions for systems where these tenns are explicitly included in the Hamiltonian, but an estimate of their energetic contributions may be had from simple first-order perturbation theory, since that energy is computed simply by taking the expectation values of the operators over the much more easily obtained HF wave functions. 

For the HgH system numerical wavefunctions were obtained for Hg using both relativistic (Desclaux programme87 was used) and non-relativistic hamiltonians. The orbitals were separated into three groups an inner core (Is up to 3d), an outer core (4s—4/), and the valence orbitals (5s—6s, 6p). The latter two sets were then fitted by Slater-type basis functions. This definition of two core regions enabled them to hold the inner set constant ( frozen core ) whilst making corrections to the outer set, at the end of the calculation, to allow some degree of core polarizability. The correction to the outer core was done approximately via first-order perturbation theory, and the authors concluded that in this case core distortion effects were negligible. 

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