Hartree-Fock-Roothaan LCAO method

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)... 

The idea to employ a finite basis set of AOs to represent the MOs as linear combinations of the former apparently belongs to Lennard-Jones [68] and had been employed by Hiickel [37] and had been systematically explored by Roothaan [38]. That is why the combination of the Hartree-Fock approximation with the LCAO representation of MOs is called the Hartree-Fock-Roothaan method. 

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. 

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... 

At each point r, the electronic density p(r,K) of a molecule of nuclear conformation K can be computed by the Hartree-Fock-Roothaan-Hall SCF LCAO ab initio method. Using a basis set cp(K) of atomic orbitals (pj(r,K)... 

Table 3.7. Calculated equilibrium structural properties of H,0 and NHj [bond lengths, / (0-H) and 7 (N-H) in A, bond angles

Each of these methods is based on the AFDF approach. Within the framework of the conventional Hartree-Fock-Roothaan-Hall self-consistent field linear combination of atomic orbitals (LCAO) ab initio representation of molecular wave functions built from molecular orbitals (MOs), the AFDF principle can be formulated using fragment density matrices. For a complete molecule M of some nuclear configuration K, using an atomic orbital (AO) basis of a set of n AOs density matrix P can be determined using the coefficients of AOs in the occupied MOs. The electronic density p(r) of the molecule M, a function of the three-dimensional position variable r, can be written as... 

Whereas the concepts and method described in this contribution are equally applicable to various approximate and more advanced quantum-chemical representations, the basic concepts will be discussed and illustrated within the framework of the conventional Hartree-Fock-Roothaan-Hall SCF LCAO ab initio representation of molecular wave functions and electronic densities, as can be computed, for example, using the Gaussian family of computer programs of Pople and co-workers. The essence of the shape analysis methods will be discussed with respect to some fixed nuclear arrangement K note, however, that the generalizations will involve changes in the nuclear arrangement K. 

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

LCAO (p. 360) atomic basis set (p. 363) Hartree-Fock-Roothaan method (p. 364) bonding orbital (p. 371) antibonding orbital (p. 371) instabihty (p. 372)... 

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). 

Only basis orbitals of the same symmetry can be included in any one LCAO molecular orbital if it is to be an eigenfunction of the symmetry operators. The a basis function can combine with the Is, 2s, and 2p functions on the oxygen. The 2 basis function can combine with the 2py function on the oxygen, and the,2px function on the oxygen cannot combine with any of the other basis functions. Table 21.2 contains the values of the coefficients determined by the Hartree-Fock-Roothaan method for the seven canonical molecular orbitals, using Slater-type orbitals (STOs) as basis functions, with... 

As a result of spin orthogonality, only (1 /2) of the exchange terms are nonzero but they are there This was first pointed out by Fock [6] and was added as a correction to the method then developed by Hartree [7]. The combined method is now called the Hartree-Fock method if tabulated numerical orbitals are used, but the Hartree-Fock-Roothaan method in an LCAO basis. Today this method is... 

Since the first formulation of the MO-LCAO finite basis approach to molecular Hartree-Fock calculations, computer applications of the method have conventionally been implemented as a two-step process. In the first of these steps a (large) number of integrals — mostly two-electron integrals — are calculated and stored on external storage. The second step then consists of the iterative solution of the Roothaan equations, where the integrals from the first step are read once for every iteration. 

The Hartree-Fock orbitals are expanded in an infinite series of known basis functions. For instance, in diatomic molecules, certain two-center functions of elliptic coordinates are employed. In practice, a limited number of appropriate atomic orbitals (AO) is adopted as the basis. Such an approach has been developed by Roothaan 10>. In this case the Hartree-Fock differential equations are replaced by a set of nonlinear simultaneous equations in which the limited number of AO coefficients in the linear combinations are unknown variables. The orbital energies and the AO coefficients are obtained by solving the Fock-Roothaan secular equations by an iterative method. This is the procedure of the Roothaan LCAO (linear-combination-of-atomic-orbitals) SCF (self-consistent-field) method. 

The formal analysis of the mathematics required incorporating the linear combination of atomic orbitals molecular orbital approximation into the self-consistent field method was a major step in the development of modem Hartree-Fock-Slater theory. Independently, Hall (57) and Roothaan (58) worked out the appropriate equations in 1951. Then, Clement (8,9,63) applied the procedure to calculate the electronic structures of many of the atoms in the Periodic Table using linear combinations of Slater orbitals. Nowadays linear combinations of Gaussian functions are the standard approximations in modem LCAO-MO theory, but the Clement atomic calculations for helium are recognized to be very instructive examples to illustrate the fundamentals of this theory (67-69). 

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). 

The LCAO approximation was introduced to the Hartree-Fock method, independently, by C.CJ. Roothaan and G.G. Hall. 

In the late 1920s and early 1930s, a team led by Hartree [9] formulated a self-consistent-field iterative numerical process to treat atoms. In 1930, Fock [10] noted that the Hartree-SCF method needed a correction due to electron exchange and the combined method was known as the Hartree-Fock SCF method. It was not until 1951 that a molecular form of the LCAO-SCF method was derived by Roothaan [11] as given in Appendix B but we can give a brief outline here. The Roothaan method allows the LCAO to be used for more than one atomic center and so the path was open to treat molecules Now, all we have to do is to carry out the integral for the expectation value of the energy as... 

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