Roothaans Self-Consistent-Field Procedure

Roothaan s Self-Consistent-Field Procedure.—While numerical integration techniques may be used to solve the Hartree-Fock equations in the case of atoms by the iterative method described above, the lower symmetry of the nuclear field present in molecules necessitates the use of an expansion for the determination of the molecular orbitals by a method developed by Roothaan.81 In Roothaan s approach, it is assumed that each molecular orbital may be adequately represented by a linear expansion in terms of some (simpler) set of basis functions xj, i.e. 

Hellmann, Einfuhrung in die Quanten-chemie , Franz Deuticke, Leipzig, Germany, 1937, p. 283. 

Use of a complete (necessarily infinite) set of basis orbitals in the expansion for the molecular orbitals would insure absolute convergence to the Hartree-Fock limit. In practice, this is both impossible and unnecessary, and only a finite number of basis functions is employed in the expansion. The selection of the finite basis set is, therefore, of crucial importance in determining how closely one approximates the true Hartree-Fock solution. 

A complete review of the characteristics of various types of basis sets has been given recently by Schaefer.44 The radial form of STO s is similar to the (nodeless) hydrogenic atomic orbitals, rn -1e - r where n is the principal quantum number and C is a variable exponent.45 Their angular dependence is described by multiplication by a spherical harmonic The use of (jTO s in molecular calculations was 

The minimum size basis set which can be used in an SCF calculation includes one STO for each occupied atomic orbital with distinct n and l values as determined by the electronic configuration of the atom in question. While of conceptual and 

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Now, in the Hartree-Fock method (the Roothaan-Hall equations represent one implementation of the Hartree-Fock method) each electron moves in an average field due to all the other electrons (see the discussion in connection with Fig. 53, Section 5.23.2). As the c s are refined the MO wavefunctions improve and so this average field that each electron feels improves (since J and K, although not explicitly calculated (Section 5.2.3.63) improve with the i// s ). When the c s no longer change the field represented by this last set of c s is (practically) the same as that of the previous cycle, i.e. the two fields are consistent with one another, i.e. self-consistent . This Roothaan-Hall-Hartree-Fock iterative process (initial guess, first F, first-cycle c s, second F, second-cycle c s, third F, etc.) is therefore a self-consistent-field procedure or SCF procedure, like the Hartree-Fock procedure... 

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

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. 

Quantum mechanical calculations are carried out using the Variational theorem and the Har-tree-Fock-Roothaan equations.t - Solution of the Hartree-Fock-Roothaan equations must be carried out in an iterative fashion. This procedure has been called self-consistent field (SCF) theory, because each electron is calculated as interacting with a general field of all the other electrons. This process underestimates the electron correlation. In nature, electronic motion is correlated such that electrons avoid one another. There are perturbation procedures whereby one may carry out post-Hartree-Fock calculations to take electron correlation effects into account. " It is generally agreed that electron correlation gives more accurate results, particularly in terms of energy. 

The best possible wavefunction of the form of (10) is called the Hartree-Fock wavefunction. For molecules it is difficult to solve (11) numerically. The most widely used procedure was proposed by Roothaan.28 This involves expressing the molecular orbitals t/> (.x) as a linear combination of basis functions (normally atomic orbitals) and varying the coefficients in this expansion so as to find the best possible solutions to (11) within the limits of a given basis set. This procedure is called the self-consistent field (SCF) method. As the size and flexibility of the basis set is increased the SCF orbitals and energy approach the true Hartree-Fock ones. 

Since the Fock matrix is dependent on the orbital coefficients, the Roothaan equations have to be repeatedly solved in an iterative process, the self-consistent field (SCF) procedure. One important step in the SCF procedure is the conversion of the general eigenvalue equation (7) into an ordinary one by an orthogonalization transformation... 

The PPP method is the first semiempirical method presented here where the Fock matrix does depend on the MO coefficients C [via the density matrix elements P, see Eq. (11)]. Therefore, the Roothaan equations (by definition due to the ZDO approximation) in the orthogonal basis, Eq. (13), have to be solved in an iterative process until self-convergence is achieved [self-consistent field (SCF) procedure]. As starting coefficients C°, usually the orbitals of an HMO calculation are used. 

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). 

Ab initio calculations are based on iterative procedures and provide the basis for self-consistent field-molecular orbital SCF-MO) methods. Electron-electron repulsion is specifically taken into account. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved in the Roothaan method by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved. 

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