Roothaan s equations

Roothaan s equations. In 1951 using Fock s method the American physicist C. C. Roothaan worked out a system ol nonlinear algebraic equations providing the AO coefficients of Eq. (1) ... 

The solution of Roothaan s equations, therefore, demands a knowledge of the following types of integral. First, overlap integrals between atomic orbitals, which are relatively easily calculated. Secondly, the matrix elements of He between pairs of atomic orbitals. These correspond to the Coulomb and the resonance integrals in the Hiickel theory. Thirdly various types of electron repulsion integral typified by... 

The Pariser-Parr-Pople Method and Roothaan s Equations... 

As we have just seen, the PPP-method does properly take care of 1 /rl7-interactions between pairs of re-electrons. This philosophy had in fact also been approached by Roothaan1136 (the publication sequence is a little confusing because Roothaan s work was done about ten years before he published it and, in the meantime, various other people had also made attempts at the problem). Roothaan formulated a set of equations which have become known as Roothaan s Equations . These are a generalisation to this full re-electron-Hamiltonian of the secular equations which we encountered (during Chapter Two) in Hiickel theory. In 2.3, we wrote the secular determinant arising in Hiickel theory in the form ... 

Table II. Comparison of Matrix Eiements Calculated from Roothaan s Equations with Elements Which Are Proportional to Overlap for the Case of Linear Hj...

In these expressions Vi, % xe molecular orbitals of the usual LCAO type. Pariser and Parr used for their calculations molecular orbitals obtained by solving the Hiickel equations which are not, in general, consistent with Roothaan s equations. Two points should be noted at this stage. First of all it is assumed that doubly excited configurations such as unimportant in the wave... 

The next step in the SCF MO calculation is to choose explicit forms for the seven AOs. The orbital energies and the coefficients of the symmetry orbitals are then found using Roothaan s equations. 

The basis sets that are used in molecular calculations are not orthonormal sets. The basis functions are normalized, but they are not orthogonal to each other. This gives rise to the overlap matrix in Roothaan s equations. In order to put Roothaan s equations into the form of the usual matrix eigenvalue problem, we need to consider procedures for orthogonalizing the basis functions. 

In the last subsection we described two ways of orthogonalizing the basis set or deriving a transformation X, which enables one to solve Roothaan s equations by a diagonalization. The use of X = is conceptually simple, and only in unusual situations, where linear dependence... 

To solve the unrestricted Hartree-Fock equations (3.312) and (3.313), wc need to introduce a basis set and convert these integro differential equations to matrix equations,just as we did when deriving Roothaan s equations. We thus introduce our set of basis functions = 1, 2,..., X and... 

We will begin this chapter by constructing determinantal trial functions from the Hartree-Fock molecular orbitals, obtained by solving Roothaan s equations. It will prove convenient to describe the possible N-electron functions by specifying how they differ from the Hartree-Fock wave function Fo Wave functions that differ from Tq by w spin orbitals are called n-tuply excited determinants. We then consider the structure of the full Cl matrix, which is simply the Hamiltonian matrix in the basis of all possible N-electron functions formed by replacing none, one, two,... all the way up to N spin orbitals in Section 4.2 we consider various approximations to the full Cl matrix obtained by truncating the many-electron trial function at some excitation level. In particular, we discuss, in some detail, a form of truncated Cl in which the trial function contains determinants which differ from To by at most two spin orbitals. Such a calculation is referred to as singly and doubly excited Cl (SDCI). 

For the sake of simplicity, we assume in this chapter that our molecule of interest has an even number of electrons and is adequately represented, to a first approximation, by a closed-shell restricted HF determinant, j Fo)- Suppose we have solved Roothaan s equations in a finite basis set and obtained a set of 2K spin orbitals The determinant formed from the N lowest energy spin orbitals is I Fq)- As we have seen in Chapter 2, we can form, in addition to Po> large number of other N-electron determinants from the 2K spin orbitals. It is convenient to describe these other determinants by stating how they differ from I Fq) Thus the set of possible determinants include Fo> the singly excited determinants (which differ from I Fq) in having the spin orbital Xa replaced by xX the doubly excited determinants etc., up to and including N-tuply excited determinants. We can use these many-electron wave functions as a basis in which to expand the exact many-electron wave function Oo>. If I Fq) is a reasonable approximation to Oo>, then we know from the variation principle that a better approximation (which becomes exact as the basis becomes complete) is... 

Thus a mathematical problem must be solved which leads to Roothaan s equations. In the case of ammonia these equations have been solved by Kaplan who obtained the foUowing results. Table 1 contains the explicit form of the molecular orbitals obtained with the values of the associated energies e, (in atomic units). 

This adds meaning to the concept that we stand on the shoulders of intellectual giants We have tried to present an understandable derivation of Roothaan s equation in Appendix B leaning on Slater s short form [1], and there is the final SCF equation (in Pople s notation [3]) ... 

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