Roothaan’s method

By correct we mean equal to that given by Roothaan s method (Ref. 50). 

Two general approaches have been devised to solve this system. For a large class of excited states, the first — Roothaan s method si-82) rigorously reduces Eq. (31) and (32) to a unique eigenvalue problem by absorbing the off-diagonal Lagrange multipliers into a new effective hamiltonian. The second — Nesbet s method — defines a suitable but... 

Fock modified Hartree s SCF method to include antisymmetrization. Roothaan further modified the Hartree-Fock method by representing the orbitals by linear combinations of basis functions similar to Eq. (16.3-34) instead of by tables of numerical values. In Roothaan s method the integrodifferential equations are replaced by simultaneous algebraic equations for the expansion coefficients. There are many integrals in these equations, but the integrands contain only the basis functions, so the integrals can be calculated numerically. The calculations are evaluated numerically. This work is very tedious and it is not practical to do it without a computer. 

A Consistent Calculation of Atomic Energy Shell Corrections Strutinsky s Method in the Hartree-Fock-Roothaan Scheme... 

Formulation of Strutinsky s method in the Hartree-Fock-Roothaan scheme... 

The scientific interests of Huzinaga are numerous. He initially worked in the area of solid-state theory. Soon, however, he became interested in the electronic structure of molecules. He studied the one-center expansion of the molecular wavefunction, developed a formalism for the evaluation of atomic and molecular electron repulsion integrals, expanded Roothaan s self-consistent field theory for open-shell systems, and, building on his own work on the separability of many-electron systems, designed a valence electron method for computational studies on large molecules. 

In 1951 Roothaan and Hall independently pointed out [26] that these problems can be solved by representing MO s as linear combinations of basis functions (just as in the simple Hiickel method, in Chapter 4, the % MO s are constructed from atomic p orbitals). Roothaan s paper was more general and more detailed than Hall s, which was oriented to semiempirical calculations and alkanes, and the method is sometimes called the Roothaan method. For a basis-function expansion of MO s we write... 

Roothaan s Self-Consistent-Field Procedure.—While numerical integration techniques may be used to solve the Hartree-Fock equations in the case of atoms by the iterative method described above, the lower symmetry of the nuclear field present in molecules necessitates the use of an expansion for the determination of the molecular orbitals by a method developed by Roothaan.81 In Roothaan s approach, it is assumed that each molecular orbital may be adequately represented by a linear expansion in terms of some (simpler) set of basis functions xj, i.e. 

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) ... 

Fig. 7.8. Calculated (using ab initio SCF Hartree-Rock-Roothaan MO methods) deformation density maps for various Si202 ring-containing molecules (a) H4Si206 in the plane of the Si-O-Si-0 ring (b) H6Si20, in the Si-O-Si plane. The contour interval is 0.05 e A with negative contours dashed and the zero contour dotted (after O Keeffe and Gibbs, 1985 reproduced with the publisher s permission).

Once the basis functions have been chosen, explicit determination of the

classical method is Roothaan s self-consistent-field (S.C.F.) 23) which modem computers make easily applicable. 

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 ... 

Determinantal MO s may be obtained by a large number of computational methods based on Roothaan s self-consistent field formalism 94> for solving the Hartree-Fock equation for molecules which differ in degree of sophistication as regards the completeness and kind of the set of starting atomic wave functions, as well as the completeness of the Hamiltonian used 9S>. So a chain of various kinds of approximations is available for calculations starting from different ways of non-empirical ab initio" calculations 96>, viasemiempiricalmethods for all-valence electrons with inclusion of electronic interaction 95-97>98)... 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

The first full sets of self-consistent-field molecular orbitals (SCF-MO) were reported by dementi [20] and dementi and McLean [21] as part of a comprehensive study of linear and symmetrical ions. A value of 1.20 A (2.27 atomic units) was used for the N-N separation. The former calculation used a set of Is, 2s, and 2p Slater functions, with the exponents derived from Roothaan s [22] best free-atom values. The latter used an additional 2po Slater-type orbital (STO) and a 3c a STO on the central nitrogen. The extension of the basis improved the total energy by approximately 4.4 eV. Comparable results were obtained by Bonaccorsi and coworkers [23] using essentially the same methods and basis functions (not included in Table I). 

The momentum wave functions in various atomic models are calculated for arbitrary atomic orbitals. The nonrelativistic hydrogenic, the Hartree-Fock, the relativistic hydrogenic, and the Dirac-Fock models are considered. The momentum wave functions are obtained as a Fourier transform of the wave function in the position space. The Hartree-Fock and the Dirac-Fock wave functions in atoms are given in terms of Slater-type orbitals (STO s), i.e. the Hartree-Fock-Roothaan (HFR) method and the relativistic HFR (RHFR) method. All the wave functions in the momentum space can be expressed analytically in terms of hypergeometric functions. 

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). 

CP, Ms. Coulson 154, Box G.15, G.15.5. Letters Mulliken to Coulson, January 17, 1950, January 23, 1950. Mulliken talked about Parr s paper on butadiene using Roothaan s LCAO self-consistent-field method, to which Coulson replied that he beUeved Parr s work on butadiene to be very similar to what his group had been doing. 

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. 

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]. 

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