Roothaan

Roothaan C C J 1951 New developments in moleoular orbital theory Rev. Mod. Phys. 23 69-89... 

Roothaan C C J 1960 Self-oonsistent field theory for open shells of eleotronio systems Rev. Mod. Phys. 32 179-85... 

These are the Roothaan SCFepiiations, which clearly can be solved iteratively—guess C, form F, diagonall/e to a new C, form a new F...and so on. ... 

Ihc equations and Roothaan equations arc solved by the same tech n iques. 

HyperChcin s ah mitio calculations solve the Roothaan equations (.h9 i on page 225 without any further approximation apart from th e 11 se of a specific fin iie basis set. Th ere fore, ah initio calcii lation s are generally more accurate than semi-enipirical calculations. They certainly involve a more fundamental approach to solving the Sch riidiiiger ec nation than do semi-cmpineal methods. 

Since the first formulation of the MO-LCAO finite basis approach to molecular Ilartree-Pock 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 — arc calculated and stored on external storage. Th e second step then con sists of the iterative solution of the Roothaan equations, where the integrals from the first step arc read once for every iteration. 

Closed-shell Systems and the Roothaan-Hall Equations... 

We shall initially consider a closed-shell system with N electroris in N/2 orbitals. The derivation of the Hartree-Fock equations for such a system was first proposed by Roothaan [Roothaan 1951] and (independently) by Hall [Hall 1951]. The resulting equations are known as the Roothaan equations or the Roothaan-Hall equations. Unlike the integro-differential form of the Hartree-Fock equations. Equation (2.124), Roothaan and Hall recast the equations in matrix form, which can be solved using standard techniques and can be applied to systems of any geometry. We shall identify the major steps in the Roothaan approach. 

I he Fock matrix is a ff x ff square matrix that is symmetric if real basis functions are used. Tile Roothaan-Hall equations (2.149) can be conveniently written as a matrix equation ... 

Lei us consider how we might solve the Roothaan-Hall equations and thereby obtain the molecular orbitals. The first point we must note is that the elements of the Fock matrix, u liich appear on the left-hand side of Equation (2.162), depend on the molecular orbital oetficients which also appear on the right-hand side of the equation. Thus an iterative pi oeedure is required to find a solution. 

The Roothaan-Hall equations can now be manipulated as follows. Both sides of Equation (2.162) are pre-multiplied by the matrix... 

Tie first consideration is that the total wavefunction and the molecular properties calculated rom it should be the same when a transformed basis set is used. We have already encoun-ered this requirement in our discussion of the transformation of the Roothaan-Hall quations to an orthogonal set. To reiterate suppose a molecular orbital is written as a inear combination of atomic orbitals ... 

The cornerstone of semiempirical and ab initio molecular orbital methods is the Harhee equation and its extensions and variants, the Harhee-Fock and Roothaan-Hall equations. We have seen that the Hamiltonian for the hydrogen atom. 

Application of the variational self-consistent field method to the Haitiee-Fock equations with a linear combination of atomic orbitals leads to the Roothaan-Hall equation set published contemporaneously and independently by Roothaan and Hall in 1951. For a minimal basis set, there are as many matr ix elements as there are atoms, but there may be many more elements if the basis set is not minimal. 

The Roothaan-Hall equation set (9-57) is often written in the notation... 

For sueh a funetion, the CI part of the energy minimization is absent (the elassie papers in whieh the SCF equations for elosed- and open-shell systems are treated are C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951) 32, 179 (I960)) and the density matriees simplify greatly beeause only one spin-orbital oeeupaney is operative. In this ease, the orbital optimization eonditions reduee to ... 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the Fj y matrix elements depend on the Cy i LCAO-MO eoeffieients whieh are, in turn, solutions of the so-ealled Roothaan matrix Hartree-Foek equations- Zy Fj y Cy j = 8i Zy Sj y Cy j. One should also note that, just as F ( )i = 8i (l)j possesses a eomplete set of eigenfunetions, the matrix Fj y, whose dimension M is equal to the number of atomie basis orbitals used in the LCAO-MO expansion, has M eigenvalues 8i and M eigenveetors whose elements are the Cy j. Thus, there are oeeupied and virtual moleeular orbitals (mos) eaeh of whieh is deseribed in the LCAO-MO form with Cy j eoeffieients obtained via solution of... 

As presented, the Roothaan SCF proeess is earried out in a fully ab initio manner in that all one- and two-eleetron integrals are eomputed in terms of the speeified basis set no experimental data or other input is employed. As deseribed in Appendix F, it is possible to introduee approximations to the eoulomb and exehange integrals entering into the Foek matrix elements that permit many of the requisite Fj, y elements to be evaluated in terms of experimental data or in terms of a small set of fundamental orbital-level eoulomb interaetion integrals that ean be eomputed in an ab initio manner. This approaeh forms the basis of so-ealled semi-empirieal methods. Appendix F provides the reader with a brief introduetion to sueh approaehes to the eleetronie strueture problem and deals in some detail with the well known Hiiekel and CNDO- level approximations. 

The Roothaan equations just described are strictly the equations for a closed-shell Restricted Hartree-Fock (RHF) description only, as illustrated by the orbital energy level diagram shown earlier. To be more specific ... 

The Roothaan equations are the basic equations for closed-shell RHF molecular orbitals, and the Pople-Nesbet equations are the basic equations for open-shell UHF molecular orbitals. The Pople-Nesbet equations are essentially just the generalization of the Roothaan equations to the case where the spatials /j and /jP, as shown previously, are not defined to be identical but are solved independently. 

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