The Hartree-Fock-Roothaan method SCF LCAO MO

Clemens C.J. Roothaan (b. 1916), American physicist, professor at the University of Chicago. Me became interested in this topic, after recognizing that in the literature people write about the effective one-electron operator, but he could not find its mathematical expression. 

George G. Mall (b. 1925), Irish physicist, professor of Mathematics at the University of Nottingham. Mis scientific achievements are connected to localized orbitals, ionization potentials, perturbation theory, solvation and chemical reactions. 

The Hartree-Fock (HF) equations are nonlinear differential-integral equations, which can be solved by appropriate numerical methods. For example, in the case of atoms and diatomics the orbitals may be obtained in a numerical form. High accuraty at long distances from the nuclei is their great advantage. Flowever, the method is very difficult to apply for larger systems. 

A solution is the use of the LCAO MO method (algebraization of the Fock equations). It leads to simplification of the computational scheme of the Hartree-Fock method. If the LCAO expansion is introduced to the rapression for the total energy, then formula (8.41) (together with i = (i f i)) gives  

In the SCF LCAO MO method, the Fock equations (complicated differential-integral equations) are solved in a very simple way. From (8.49) and (8.30) we have 

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The methods under the category of nonempirical fall into two subclasses. The first consists of the well known Hartree-Fock-Roothaan [7, 8] LCAO-MO-SCF (self-consistent field) methods. The second is an even more rigorous... 

In full analogy with molecules, we can formulate the SCF LCAO CO Hartree-Fock-Roothaan method (CO instead MO). Each CO is characterized by a vector k e FBZ and is a linear combination of the Bloch functions (with the same k). 

If RCI expansions are used or orbitals are subdivided into inactive and active groups, or both, then variation of the orbitals themselves may lead to an essential energy decrease (in contrast to the FCI method where it does not happen). Such combined methods that require both optimization of Cl coefficients and LCAO coefficients in MOs are called MCSCF methods. Compared with the Cl method, the calculation of the various expansion coefficients is significantly more complicated, and, as for the Hartree-Fock-Roothaan approximation, one has to obtain these using an iterative approach, i.e. the solution has to be self-consistent (this gives the label SCF). 

Calculations of the electronic structure of the phosphorus molecule were carried out by ab initio MO LCAO Hartree-Fock-Roothaan SCF method in the restricted (RHF), restricted-open (ROHF) or unrestricted Hartree-Fock (UHF)... 

Semi-empirical calculations for the simple vinyl cation C2H3+ have been reported by Hoffmann (1964) and by Yonezawa et ad., (1968). More rigorous calculations by Sustmann et ad. (1969) are based on a semi-empirical method based on the neglect of diatomic differential overlap (NDDO) calibrated to results of ab initio Hartree-Fock-Roothaan SCF calculations. Recent work by Hopkinson et al. (1971) is entirely based on a non-empirical LCAO-MO-SCF method. 

This method should lead to results which are just as accurate as the results of the methods described in the previous sections, and can be used as a check on the computed potential-energy minimum E(R ) at R = Re if fl is determined from curve-fitting of the Morse potential with the computed R and De and this leads to a wrong we and/or w, then it can be assumed that De and/or Rg are/is wrong. It is to be emphasized (12) that the Morse curve can mostly not be used with essentially ionic compounds like NaF because the attraction given by the Coulomb term extends out in space to greater distances than the Morse exponential part for these compounds many other types of potential have been postulated (e.g. the Hellmann-potential or the Bom-Landd potential (77)). The reader can try to calculate cog, etc. of NaF from the SCF— LCAO—MO calculation of Matcha (72) in the Roothaan-Hartree-Fock approximation, using the Morse curve (E = —261.38 au, R =3.628 au experimental values Rg = 3.639 au, a)g=536 cm i, >g g=3.83 cm-i). 

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