Roothaan equations

We now furn to the problem of how the Hartree-Fock equations are solved in practice. The standard way to proceed is to expand the wavefunction in a basis. The orbital is expanded in terms of a linear combination of basis functions  

The formulation of the wavefunction in a basis leads to the so-called Roothaan equations which are obtained from the HF equations by inserting the expansion (2.7) into (2.4), multiplying from the left by the set of basis functions and integrating over the spatial coordinates. This procedure leads to the following integral matrix equation. 

It should be noted that in the way the Hartree-Fock theory is formulated, the number of integrals that has to be calculated increases rapidly with the size of the system. By inserting/(l) from eq. (2.7) into eq. (2.9), the two-electron integrals, i.e., the Coulomb and exchange integrals, are obtained in the following form  

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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. 

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. 

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. 

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

As is apparent from the above definitions, each of these effective matrices depend on basis sets and molecular orbitals of both fragments. It is also important to observe that these matrices possess a correct asymptotic behavior as at large interfragment distances they become the usual overlap and Fock matrices of the separate fragments, while the paired secular systems uncouple and converge to the separate Roothaan equations for the single monomers. Finally, as it is usual in a supermolecular approach, the interaction energy is expressed as... 

Gianinetti, E., Raimondi, M. and Tomaghi, E. (1996) Modification of the Roothaan equations to exclude BSSE from molecular interaction calculations, Int. J. Quantum Chem., 60, 157-166. 

Although the above discussion assumes that all MOs are occupied by two electrons, it turns out that the basic ideas can be extended to open-shell molecules in which there are unequal numbers of electrons in the two spin states. Without showing the complicated mathematics, we will show how the wavefunction can be determined by constructing two Fock matrices for each spin state and then solving two sets of coupled Roothaan equations ... 

The most simple approach is the Hartree-Fock (HF) self-consistent field (SCF) approximation, in which the electronic wave function is expressed as an antisymmetrized product of one-electron functions. In this way, each electron is assumed to move in the average field of all other electrons. The one-electron functions, or spin orbitals, are taken as a product of a spatial function (molecular orbital) and a spin function. Molecular orbitals are constructed as a linear combination of atomic basis functions. The coefficients of this linear combination are obtained by solving iteratively the Roothaan equations. 

Roothaan equations have been modified in a previous work with the aim of avoiding BSSE at the Hartree-Fock level of theory. The resulting scheme, called SCF-MI (Self Consistent Field for Molecular Interactions), underlines its special usefulness for the computation of intermolecular interactions. 

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. 

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). 

Ab initio calculations using direct evaluation of Roothaan equations. Financially restrictive, and thus little used... 

These O, are called Linear Combination of Atomic Orbitals Molecular Orbitals (LCAO MOs) and if they are introduced into the Hartree-Fock equations (eqns (10-2.5)), a simple set of equations (the Hartree-Fock-Roothaan equations) is obtained which can be used to determine the optimum coefficients Cti. For those systems where the space part of each MO is doubly occupied, i.e. there are two electrons in each 0, with spin a and spin respectively so that the complete MOs including spin are different, the total wavefunction is... 

Our approximations so far (the orbital approximation, LCAO MO approximation, 77-electron approximation) have led us to a tt-electronic wavefunction composed of LCAO MOs which, in turn, are composed of 77-electron atomic orbitals. We still, however, have to solve the Hartree-Fock-Roothaan equations in order to find the orbital energies and coefficients in the MOs and this requires the calculation of integrals like (cf. eqns (10-3.3)) ... 

We now consider the PPP, CNDO, INDO, and MINDO two-electron semiempirical methods. These are all SCF methods which iteratively solve the Hartree-Fock-Roothaan equations (1.296) and (1.298) until self-consistent MOs are obtained. However, instead of the true Hartree-Fock operator (1.291), they use a Hartree-Fock operator in which the sum in (1.291) goes over only the valence MOs. Thus, besides the terms in (1.292), f/corc(l) m these methods also includes the potential energy of interaction of valence electron 1 with the field of the inner-shell electrons rather than attempting a direct calculation of this interaction, the integrals of //corc(/) are given by various semiempirical schemes that make use of experimental data furthermore, many of the electron repulsion integrals are neglected, so as to simplify the calculation. 

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. 

To find the true Hartree-Fock orbitals, one must use a complete set in (1.295), which means using an infinite number of gk s. As a practical matter, one must use a finite number of basis functions, so that one gets approximations to the Hartree-Fock orbitals. However, with a well-chosen basis set, one can approach the true Hartree-Fock orbitals and energy quite closely with a not unreasonably large number of basis functions. Any MOs (or AOs) found by iterative solution of the Hartree-Fock-Roothaan equations are called self-consistent-field (SCF) orbitals, whether or not the basis set is large enough to give near-Hartree-Fock accuracy. 

Thus four of the seven lowest H20 MOs are linear combinations of the four a, symmetry orbitals listed above, and are a, MOs similarly, the two lowest b2 MOs are linear combinations of 02p and H,1j — H21.s, and the lowest bx MO is (in this minimal-basis calculation) identical with 02px. The coefficients in the linear combinations and the orbital energies are found by iterative solution of the Hartree-Fock-Roothaan equations. One finds the ground-state electronic configuration of H20 to be... 

Despite these shortcomings, the Roothaan equation has been used extensively and the Hartree-Fock energies of various small molecules have been calculated. However, the difficulties encountered in calculating the energy of large molecules are such that simplified methods are desirable in these cases. Several such methods will be discussed in the next section ... 

Now we have FC = SCe (5.57), the matrix form of the Roothaan-Hall equations. These equations are sometimes called the Hartree-Fock-Roothaan equations, and, often, the Roothaan equations, as Roothaan s exposition was the more detailed and addresses itself more clearly to a general treatment of molecules. Before showing how they are used to do ab initio calculations, a brief review of how we got these equations is in order. 

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