MCHF equation

With respect to the above, I note that following the publication by Froese Fischer [18] and by McCullough [19] of codes for the numerical solution of HF (or MCHF) equations for atomic and for diatomic states respectively, it has been demonstrated on prototypical unstable states (neutral, negative ion, molecular diabatic) that the state-specific computation of correlated wave-functions representing the localized component of states embedded in the continuous spectrum can be done economically and accurately, for example, [9,10,17, 20-22] and references there in. 

This became possible not only by the state-specific nature of the computations but also by the realization that the natural orbitals produced from hydrogenic basis sets were the same as the MCHF orbitals that are computable for the intrashell states up to about N = 10 - 12. Therefore, for DES with very high N, instead of obtaining the multiconfigurational zero-order wavefunction from the solution of the SPSA MCHF equations (which are very hard to converge numerically if at all), we replaced the MCHF orbitals by natural orbitals obtained from the diagonalization of the appropriate density matrices with hydrogenic orbitals. 

In particular, as regards its implementation, first to atomic structures and subsequently to diatomic ones, this has been done in the following way In order to secure the accuracy of the Fermi-sea orbitals, we compute via the numerical solution of the state-specific HE or, most frequently, MCHF equations. The choice of the components of 0 " that are considered relevant to the overall calculation depends on the desired level of accuracy and the property. They are expressed in terms of analytic functions, whose final optimization is done variationally to all orders. 

On the other hand, if for some type of electronic structure it is impossible to obtain valid convergence of the state-specific MCHF equations because of the presence of correlating configurations whose structure corresponds to open channels, then the calculation of 4)mchf and of should exclude them. For example, this is the case of He 2s2p which interacts with the [He(2s ) - - es] continuum. Their effect is then incorporated from principal value integrals over purely scattering function spaces. 

Here, I focus on the results of fhe most complex case, that of fhe quadruply excited states in Be of S° symmefry. In Ref. [154] we found via theory and its implementation in terms of correlated wavefunctions obtained by solving the MCHF equations for the intrashell configurations, regularities in the geometry of electron distribution and in the energy spectrum. 

For the calculations of relativistic density functions we used a multi-configuration Dirac-Fock approach (MCDF), which can be thought of as a relativistic version of the MCHF method. The MCDF approach implemented in the MDF/GME program [4, 27] calculates approximate solutions to the Dirac equation with the effective Dirac-Breit Hamiltonian [27]... 

A relativistic Dirac-Hartree-Fock calculation is somewhat more complicated than the corresponding nonrelativistic calculation due to the fact that each wavefunction has a large and a small component. Thus for the n electron problem there are 2n coupled equations in the relativistic calculation rather than n as in the nonrelativistic calculation. There is an even more severe complication however, produced by the fact that each nonrelativistic (nl) orbital corresponds to two relativistic orbitals (n,l,j = 1+ J)and (n,l,j =1 -J) (except of course, if 1= 0). Consequently, what is a one configuration Hartree-Fock (HF) calculation non-relativistically usually corresponds to a multi-configuration Hartree-Fock (MCHF) relativistically. What this implies is that a single configuration Hartree-Fock calculation is usually less likely to give accurate results in the relativistic case than in the nonrelativistic case. 

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