Roothaan procedure

This restriction is not demanded. It is a simple way to satisfy the Pauli exclusion principle, but it is not the only means for doing so. In an unrestricted wavefunction, the spin-up electron and its spin-down partner do not have the same spatial description. The Hartree-Fock-Roothaan procedure is slightly modified to handle this case by creating a set of equations for the a electrons and another set for the p electrons, and then an algorithm similar to that described above is implemented. 

In order to solve for the energy and wavefunction within the Hartree-Fock-Roothaan procedure, the AOs must be specified. If the set of AOs is infinite, then the variational principle tells us that we will obtain the lowest possible energy within the HF-SCF method. This is called the HF limit, This is not the actual energy of the molecule recall that the HF method neglects instantaneous electron-electron interactions, otherwise known as electron correlation. 

When the Schrodinger equation is solved in the Hartree-Fock— Roothaan procedure, the coefficients c,> are obtained and the wavefunction is at hand.2 Unfortunately, all the chemical information is contained in this wave-function, and it is expressed as a (very) long list of coefficients. As an example, a restricted Hartree-Fock calculation of benzene using the 6-31G basis set will have 102 atomic orbitals and 21 doubly occupied MOs for a total of 2142 coefficients. For the chemist, the interesting and pertinent data are entangled in a series of numbers, and the question becomes how to extract the chemical concepts from these numbers. 

With radicals there is no convenient method like the Hartree-Fock-Roothaan procedure commonly used for closed-shell systems. In contrast, the open-shell theory is typical of a number of methods suggested which differ in accuracy from the viewpoint of true SCF theory, in range of applicability, complexity, and computing feasibility. A critical survey of open-shell SCF methods reported by Berthier D covers the literature up to 1962. We shall not duplicate that review here we propose rather to note some features of open-shell methods relevant to their computation feasibility and to mention procedures published after 1962. The unrestricted treatments that assume different space orbitals for different spins will be disregarded here because the restricted wave functions... 

When discussing open-shell methods, it is convenient to use the formalism of the well-known Roothaan procedure 3). Hence, consider an open-shell configuration for which the total energy can be expressed as... 

The position of the benzene radical ions among the other benzenoid systems is somewhat exceptional. The presence of the degenerate frontier molecular orbitals makes it difficult to perform standard open-shell calculations because the LHP method is inherently incapable of accommodating systems of this type and the Roothaan procedure diverges here The simple Cl treatments based on the HMO computational... 

Then, we calculate the elements Fpq for all atomic orbitals p, g for unit cells 7 — 0,1,2,... Jmax- What is 7 iax The answer is certainly non-satisfactory ymax = 00. In practice, however, we often take 7 ,ax as being of the order of a few cells most often, we take jn,ax = 1. For each from the FEZ, we calculate the elements Fpq and Spq of Eqs. (9.56) and (9.58), and then solve the secular equations within the Haitiee-Fock-Roothaan procedure. This step requires diagonalization (see Appendix K available at booksite.elsevier.com/978-0-444-59436-5, p. el05). As a result, for each k we obtain a set of coefiicients c for the crystal orbitals and the energy eigenvalue Sn(k)-... 

For each k from the FEZ we calculate the elements Fpq and Spq and then solve the secular equations within the Hartree-Fock-Roothaan procedure. This step requires diagonalization" (see Appendix K, p. 982). As a result, for each k we obtain a set of coefficients c. 

What takes place in an ab initio calculation is pretty complicated. However, because all the mathematics is done by a computer, ab initio calculations are not difficult to perform. So, let s briefly review what is done in ab initio methods. This approach is generally called the Roothaan procedure, or Roothaan s formulation of the Hartree-Fock method (minus step 7, below). 

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 simplicity of the standard SCF procedure has been preserved. The closed shell Roothaan equations and the Guest and Saunders open shell equations have been modified at the cost of a negligible complication with respect to the usual algorithm. 

Our general procedure is to represent the atoms in a molecule using the Hartree-Fock orbitals of the individual atoms occurring in the molecule. (We will also consider the interaction of molecular fragments where the Hartree-Fock orbitals of the fragments are used.) These are obtained with the above bases in the conventional way using Roothaan s RHF or ROHF procedure[45], extended where necessary. 

The proper way of dealing with periodic systems, like crystals, is to periodicize the orbital representation of the system. Thanks to a periodic exponential prefactor, an atomic orbital becomes a periodic multicenter entity and the Roothaan equations for the molecular orbital procedure are solved over this periodic basis. Apart from an exponential rise in mathematical complexity and in computing times, the conceptual basis of the method is not difficult to grasp [43]. Software for performing such calculations is quite easily available to academic scientists (see, e.g., CASTEP at www.castep.org CRYSTAL at www.crystal.unito.it WIEN2k at www.wien2k.at). 

Pople-Pariser-Parr PPP (CNDO) method. MO s from LCAO SCF, i.e. electron correlation taken into account. Equations to derive the LCAO coefficients by the variational procedure were proposed by Roothaan... 

Rather than working with the Hartree-Fock differential equations (1.290) (with i=l,2,. ..) one usually uses the following procedure, due to Roothaan. Each orbital , is expanded in terms of some chosen complete set of one-electron basis functions gk ... 

If the basis set used is finite and incomplete, solution of the secular equation yields approximate, rather than exact, eigenvalues. An example is the linear variation method note that (2.78) and (1.190) have the same form, except that (1.190) uses an incomplete basis set. An important application of the linear variation method is the Hartree-Fock-Roothaan secular equation (1.298) here, basis AOs centered on different nuclei are nonorthogonal. Ab initio and semiempirical SCF methods use matrix-diagonalization procedures to solve the Roothaan equations. 

Using the Roothaan-Hall Equations to do ab initio Calculations - the SCF Procedure... 

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