Roothaan Method

In the Roothaan method, within the CNDO approximation the elements of the F matrix and of various matrices contributing to the F matrix are (68) ... 

The description of configuration interaction given for rr-electron methods is also valid for all-valence-electron methods. Recently, two papers were published in which the half-electron method was combined with a modified CNDO method (69) and the MINDO/2 method was combined with the Roothaan method (70). Appropriate semiempirical parameters and applications of all-valence-electron methods are most probably the same as those reviewed for closed-shell systems (71). 

Theoretical quantum-chemical study of pyridine adsorption at Hg electrode (including its charged surface) has been described by Man ko et al. [137,138]. An ab initio Hartree-Fock-Roothaan method has been employed. The electrode was modeled as a planar seven-atomic Hg-7 cluster. The deepest minimum of the total energy of the adsorption system was found for positive charge density and Py interacting with the metal through the lone electron... 

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... 

Quantum-Chemical Dynamics with the Slater-Roothaan Method... 

The Slater-Roothaan method uses solid-harmonic Gaussians [18] to fit the molecular orbitals and five other nonnegative quantities, namely the total density, the cube root of the partitioned density for both spins, and the 2/3 power of the partitioned density for both spins. They are treated as five additional orbitals of the totally symmetric irreducible representation. Changing them slightly does not affect the robust energy at all. 

Lee, C.M. (1974). Spectroscopy and collision theory. III. Atomic eigenchannel calculation by a Hartree-Fock-Roothaan method, Phys. Rev. A 10, 584-600. 

The idea to employ a finite basis set of AOs to represent the MOs as linear combinations of the former apparently belongs to Lennard-Jones [68] and had been employed by Hiickel [37] and had been systematically explored by Roothaan [38]. That is why the combination of the Hartree-Fock approximation with the LCAO representation of MOs is called the Hartree-Fock-Roothaan method. 

Solution of the Roothaan equations calls for laborious computations which become more and more so as the basis broadens, i.e. the number of AOs increases. In the last two decades theoreticians have invented a great number of clever tricks in their attempts to find how to calculate molecules the Roothaan way . However, the system of equations (4) turned out to be very tough to handle. Despite the ingenuity of researchers and the advancement in computer technique practical application of the Roothaan method is substantially limited by the size of molecular systems. 

This is closely analogous to the Hartree equations (Eq. (1.7)). The Kohn-Sham orbitals are separable by definition (the electrons they describe are noninteracting) analogous to the HF MOs. Eq. (1.50) can, therefore, be solved using a similar set of steps as was done in the Hartree-Fock-Roothaan method. 

Table 3.5. Comparison of metal-water bond lengths and binding energies calculated using (SCF) Hartree-Fock-Roothaan methods with...

Hartree-Fock-Roothaan methods have often been quite successful in the calculation of properties despite the fact that the variational principle upon which they are based ensures only the best total energy. In particular, other energetic properties such as force constants and charge-distribution properties such as electron-density distributions and electric-field gradients are well reproduced. 

Table 4.1. Comparison of orbital ionization potentials (in eV) for SiO observed experimentally from uv photoelectron spectroscopy with those calculated by the MS-SCF-Ya method, ab initio Hartree-Fock-Roothaan method, incorporating the ASCF approach (HFR ASCF), and with perturbation theory (HFR + PT)...

Fig. 4.8. Total deformation electron densities (Ap) in 4-membered silicate rings (a) experimental Ap in 0(3)-0(4) ring of coesite (b) theoretical Ap for HjSi O, calculated using Hartree-Fock-Roothaan method with a 6-31G basis set and (c) experimental Ap in 0(3)-0(5) ring of coesite (after Geisinger et al., 1987 reproduced with the publisher s permission). 

Table 5.18. Calculated and experimental geometries in borate polyhedra. Bond lengths and bond angles for the borate clusters shown calculated using ab initio SCF Hartree-Fock-Roothaan methods with various basis sets, and the modified electron-gas and modified neglect of differential overlap method, and compared with experimental values (see text for data sources)...

The quantum-mechanical description of minerals containing transition metals is at a less advanced stage. The accuracy of simple Hartree-Fock-Roothaan methods has not been fully determined for such systems. Local-density-functional methods have been successful for calculating the structural properties of high-symmetry materials, but excitation energies are still poorly reproduced. Local-density-functional cluster calculations have so far been restricted mostly to model potentials (e.g., muffin-tin potentials) so that their full power has not been utilized. We need to determine the accuracy and efficiency of Hartree-Fock-Roothaan (or Har-... 

In the expression for xj/n, we have taken into account that there is no reason whatsoever that the coefficients c were k-independent, since the expansion functions 0 depend on k. This situation does not differ from what we encountered in the Hartree-Fbck-Roothaan method (cf. p. 431), with one technical exception instead of the atomic orbitals, we have symmetry orhitals (in our case Bloch functions). 

In full analogy with molecules, we can formulate the SCF LCAO CO Hartree-Fock-Roothaan method (a CO instead of an MO). Each CO is characterized by a vector k e FEZ and is a linear combination of the Bloch functions with the same k. 

Hartree-Fock-Roothaan method, 548, 552 Heisenberg uncertainty... 

In the previous section, the Roothaan method was introduced as a method for speeding up the Hartree-Fock calculation by expanding molecular orbitals with a basis set. The accuracy and computational time of the Roothaan calculations depend on the quality and number of basis functions, respectively. Therefore, it is necessary for reproducing accurate chemical reactions and properties to choose basis functions that give highly accurate molecular orbitals with a minimum number of functions. 

These equations are called the UHF equation or the Pople-Nesbet equation (Pople and Nesbet 1954). From the beginning, these equations have the form of the Roothaan method given by simultaneous equations,... 

Although the solution method of this equation is similar to that of the Roothaan method for closed-shell molecules, it includes pairs of Fock matrices, their diagonalizations, and density matrices for the a- and j6-spins. 

The Kohn-Sham equation is also transformed into a matrix equation on the basis of the Roothaan method in Sect. 2.5. Similar to the Hartree-Fock equation, the Kohn-Sham-Roothaan equation is written as... 

Terms other than the exchange-correlation potential are the same as those in Sect. 2.5. In this method, the total electronic energy is separately calculated as different to that in the Roothaan method. 

In the coupled perturbed Kohn-Sham method, the first wavefunction derivatives are given by calculating the first derivatives of the orbitals in terms of perturbations. The Kohn-Sham method is based on the Slater determinant. Therefore, since the Kohn-Sham wavefunction is represented with orbitals, the corresponding first wavefunction derivatives are also described by the first derivatives of the orbitals. For simplicity, let us consider the Kohn-Sham-Roothaan equation in Eq. (4.13), which is a matrix equation using basis functions based on the Roothaan method. 

The relativistic correction for the kinetic energy in the Dirac equation is naturally applicable to the Kohn-Sham equation. This relativistic Kohn-Sham equation is called the Dirac-KohnSham equation (Rajagopal 1978 MacDonald and Vosko 1979). The Dirac-Kohn-Sham equation is founded on the Rajagopal-Callaway theorem, which is the relativistic expansion of the Hohenberg-Kohn theorem on the basis of QED (Rajagopal and Callaway 1973). In this theorem, two theorems are contained The first theorem proves that the four-component external potential, which is the vector-potential-extended external potential, is determined by the four-component current density, which is the current-density-extended electron density. On the other hand, the second theorem establishes the variational principle for every four-component current density. See Sect. 6.5 for vector potential and current density. Consequently, the solution of the Dirac-Kohn-Sham equation is represented by the four-component orbital. This four-component orbital is often called a molecular spinor. However, this name includes no indication of orbital, which is the solution of one-electron SCF equations moreover, the targets of the calculations are not restricted to molecules. Therefore, in this book, this four-component orbital is called an orbital spinor. The Dirac-Kohn-Sham wavefunction is represented by the Slater determinant of orbital spinors (see Sect. 2.3). Following the Roothaan method (see Sect. 2.5), orbital spinors are represented by a linear combination of the four-component basis spinor functions, Xp, ... 

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