Roothaan Approach

Following the Dirac s quote, once the Schrodinger equation (4.249) was established The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known (Dirac, 1929). Unfortunately, the molecular spectra based on the eigen-problem Equation (4.249) is neither directly nor completely solved without specific atoms-in-molecule and/or symmetry constraints 

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

However, when written as a linear combination over the atomic orbitals the resulted MO-LCAO wave-function  

Worth mentioning that while Eq. (4.267) unfolds under the eigen-function equations 

It corresponds with the effective Hamiltonian called as Fock operator driving the self-consistent equation (4.267), under the current form 

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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. 

Hartree-Fock theory as constructed using the Roothaan approach is quite beautiful in the abstract. This is not to say, however, that it does not suffer from certain chemical and practical limitations. Its chief chemical limitation is the one-electron nature of the Fock operators. Other than exchange, all election correlation is ignored. It is, of course, an interesting question to ask just how important such correlation is for various molecular properties, and we will examine that in some detail in following chapters. 

In this work we show the importance of exponent optimization on the Roothaan approach to the solution of the Hartree-Fock equations for the lowest singlet and triplet state of the confined helium atom. By using this procedure, we found for the lowest triplet state the Hartree-Fock method as an appropriate method to describe it. By contrasting the exchange-only... 

The orthogonalization transformation [Eq. (14)] is also a very important consideration in semiempirical calculations as will be discussed in later sections. In semi-empirical methods, the Hartree-Fock-Roothaan approach is simplified. 

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. 

If the basic set is chosen to consist of atomic orbitals, this relation forms the fundament for the MO-LCAO method in molecular and crystal theory. In its SCF form this approach was first used by Coulson (1938), and later it has been systematized by Roothaan (1951). More details about the SCF results within molecular theory will be given later in a special section. 

Assuming the same molecular geometry and the same MO s for both the parent and ionized systems, the first ionization potential can be expressed in the SCF approach (Longuet-Higgins and Pople or Roothaan) (106) as... 

The fundamental tool for the generation of an approximately transferable fuzzy electron density fragment is the additive fragment density matrix, denoted by Pf for an AFDF of serial index k. Within the framework of the usual SCF LCAO ab initio Hartree-Fock-Roothaan-Hall approach, this matrix P can be derived from a complete molecular density matrix P as follows. 

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... 

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 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. 

As presented, the Roothaan SCF process is carried out in a fully ab initio manner in that all one- and two-electron integrals are computed in terms of the specified basis set no experimental data or other input is employed. As described in Appendix F, it is possible to introduce approximations to the coulomb and exchange integrals entering into the Fock matrix elements that permit many of the requisite F, v elements to be evaluated in terms of experimental data or in terms of a small set of fundamental orbital-level coulomb interaction integrals that can be computed in an ab initio manner. This approach forms the basis of so-called semi-empirical methods. Appendix F provides the reader with a brief introduction to such approaches to the electronic structure problem and deals in some detail with the well known Htickel and CNDO- level approximations. 

When it comes to the analysis of similar approaches stemming from the DFT the numerous attempts to cope with the multiplet states must be mentioned [113,154]. In these papers attempts are made to construct symmetry dependent functionals capable to distinguish different multiplet states in a general direction proposed by [99,100]. It turns out, however, that the result [113] is demonstrated for the lower multiplets of the C atom which are all Roothaan terms. It is not clear whether this methodology is going to work when applied to the d-shell multiplets which may be either non-Roothaan ones or even nontrivially correlated multiple terms. 

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. 

Beginning in the 1960s, Richard Bader initiated a systematic study of molecular electron density distributions and their relation to chemical bonding using the Hellmann-Feynman theorem.188 This work was made possible through a collaboration with the research group of Professors Mulliken and Roothaan at the University of Chicago, who made available their wave-functions for diatomic molecules, functions that approached the Hartree-Fock limit and were of unsurpassed accuracy. 

To continue with the Roothaan-Hall approach, we substitute the expansion (5.52) for the i// s into the Hartree-Fock equations 5.47, getting (we will work with in, not n, HF equations since there is one such equation for each MO, and our m basis functions will generate m MO s) ... 

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