Approximate LCAO Methods

It has been customary to classify methods by the nature of the approximations made. In this sense CNDO, INDO (or MINDO), and NDDO (Neglect of Diatomic Differential Overlap) form a natural progression in which the neglect of differential overlap is applied less and less fully. It is now clearer that there is a deeper division between methods, related to their objectives. On the one hand are approximate methods which set out to mimic the ab initio molecular orbital results. The objective here is simply to find a more economical method. On the other hand, some workers, recognizing the defects of the MO scheme, aim to produce more accurate results by the extensive use of parameters obtained from experimental data. This latter approach appears to be theoretically unsound since the formalism of the single-determinant wavefunction and the Hartree-Fock equations is retained. It can be argued that the use of the single-determinant wavefunction prevents the consistent achievement of predictions better than those obtained by the ab initio scheme where no further 

Klopman and B. O Leary, Fortschr. Chem. Forsch, (Topics Current Chem.), 1970,15,445. 

Pople and D. L. Beveridge, Approximate Molecular Orbital Theory , McGraw-Hill, New York, 1970. 

Murrell and A. J. Harget, Semi-empirical Self-consistent Molecular Orbital Theory of Molecules , Wiley, London, 1972. 

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The approximate LCAO methods of quantum chemistry can be divided on empirical (semiquantitative) and semiempirical grounds. 

The accuracy of most TB schemes is rather low, although some implementations may reach the accuracy of more advanced self-consistent LCAO methods (for examples of the latter see [18,19 and 20]). However, the advantages of TB are that it is fast, provides at least approximate electronic properties and can be used for quite large systems (e.g., thousands of atoms), unlike some of the more accurate condensed matter methods. TB results can also be used as input to detennine other properties (e.g., photoemission spectra) for which high accuracy is not essential. 

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT methods... 

The coefficients indicate the contribution of each atomic orbital to the molecular orbital. This method of representing the molecular orbital wave function in terms of combinations of atomic orbital wave functions is known as the linear combination of atomic orbitals approximation (LCAO). The combination of atomic orbitals chosen is called the basis set. 

These properties of the d-shell chromophore (group) prove the necessity of the localized description of d-electrons of transition metal atom in TMCs with explicit account for effects of electron correlations in it. Incidentally, during the time of QC development (more than three quarters of century) there was a period when two directions based on two different approximate descriptions of electronic structure of molecular systems coexisted. This reproduced division of chemistry itself to organic and inorganic and took into account specificity of the molecules related to these classical fields. The organic QC was then limited by the Hiickel method, the elementary version of the HFR MO LCAO method. The description of inorganic compounds — mainly TMCs,— within the QC of that time was based on the crystal field... 

Several valence-bond (VB) treatments of heterocyclic compounds were reported in the thirties and forties.1, 2 The known difficulty in applying the VB method to complicated molecules has made an overwhelming majority of authors use the molecular orbital (MO) method. In most cases its simplest version, the naive MO LCAO method, has been used. This approximation differs from the well-known Hiickel... 

Of the various methods of approximating the correct molecular orbitals, we shall discuss only one- the linear combination of atomic orbitals (LCAO) method. We assume that we can approximate the correct molecular orbitals by combining the atomic orbitals of the atoms that form the molecule. The rationale is that most of the time the electrons will be nearer and hence controlled by oneor the other of the two nuclei, and when this is so, the molecular orbital should be very nearly the same as the atomic orbital for that atom. The basic process is the same as the one wc employed in constructing hybrid atomic orbitals except that now we are combining orbitals on different atoms to form new orbitals that are associated with the entire molecule. We... 

A crystalline solid can be considered as a huge, single molecule subsequently, the electronic wave functions of this giant molecule can be constructed with the help of the molecular orbital (MO) methodology [19]. That is, the electrons are introduced into crystal orbitals, which are extended along the entire crystal, where each crystal orbital can accommodate two electrons with opposite spins. A good approximation for the construction of a crystal MO is the linear combination of atomic orbitals (LCAO) method, where the MOs are constructed as a LCAO of the atoms composing the crystal [19]. 

The 7r-electron structures and energies of the singlet tt-tt transitions for a number of l//-pyrrolo[l,2-a jimidazoles (39), l/f-pyrrolo[l,2-f>]-s-triazoles (40) and l//-pyrrolo[2,l-c]-s-triazoles (41) were calculated by the MO LCAO method within the semiempirical self-consistent field (SCF) approximation. A comparison of the data shows that the maximum... 

As was emphasized before (cf. Chapter 3), a molecule is not simply a collection of its constituting atoms. Rather, it is a system of atomic nuclei and a common electron distribution. Nevertheless, in describing the electronic structure of a molecule, the most convenient way is to approximate the molecular electron distribution by the sum of atomic electron distributions. This approach is called the linear combination of atomic orbitals (LCAO) method. The orbitals produced by the LCAO procedure are called molecular orbitals (MOs). An important common property of the atomic and molecular orbitals is that both are one-electron wave functions. Combining a certain number of one-electron atomic orbitals yields the same number of one-electron molecular orbitals. Finally, the total molecular wave function is the... 

Molecular orbital theory originated from the theoretical work of German physicist Friederich Hund (1896-1997) and its apphcation to the interpretation of the spectra of diatomic molecules by American physical chemist Robert S. MuUiken (1896-1986) (Hund, 1926, 1927a, b Mulliken, 1926, 1928a, b, 1932). Inspired by the success of Heitler and London s approach, Finklestein and Horowitz introduced the linear combination of atomic orbitals (LCAO) method for approximating the MOs (Finkelstein and Horowitz, 1928). The British physicist John Edward Lennard-Jones (1894-1954) later suggested that only valence electrons need be treated as delocalized inner electrons could be considered as remaining in atomic orbitals (Lennard-Jones, 1929). 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

Table 3.7. Calculated equilibrium structural properties of H,0 and NHj [bond lengths, / (0-H) and 7 (N-H) in A, bond angles

There are a number of band-structure methods that make varying approximations in the solution of the Kohn-Sham equations. They are described in detail by Godwal et al. (1983) and Srivastava and Weaire (1987), and we shall discuss them only briefly. For each method, one must eon-struct Bloch functions delocalized by symmetry over all the unit cells of the solid. The methods may be conveniently divided into (1) pesudopo-tential methods, (2) linear combination of atomic orbital (LCAO) methods (3) muffin-tin methods, and (4) linear band-structure methods. The pseudopotential method is described in detail by Yin and Cohen (1982) the linear muffin-tin orbital method (LMTO) is described by Skriver (1984) the most advanced of the linear methods, the full-potential linearized augmented-plane-wave (FLAPW) method, is described by Jansen... 

The LCAO methods can treat all electrons and need not make shape approximations to the potential. However, as for Hartree-Fock band calculations, there is a very large number of electron-electron repulsion integrals, and care must be taken in truncating their sums. A number of... 

The use of non-empirical LCAO methods for computing wave-functions for small molecules, including N3 and HN3, has been recently reviewed by Clark and Stewart . An earlier valuable review is that ofNesbet on approximate Hartree-Fock calculations on small molecules . Some of the basic ideas of modem LCAO methods will be summarized here. (For a lucid discussion on this subject see reference 29.) A more advanced treatment has recently appeared in this series . 

Usually, the. spatial function ilt is constructed from the summation of one-electron spatial orbitals (atomic orbitals) 4>. known as the basis set. u.sed to construct a MO. This approach is known as the LCAO method (/inear combination of utomic orbitals). It is an approximation of the accurate many-elcetron wave function (Eq. 28-54). The atomic orbital contributions are weighted by coefficients c,. The summation is truncated, so the ip function is not complete, which has consequences when. solving for E. 

The LCAO method extends to molecules the description developed for many-electron atoms in Section 5.2. Just as the wave function for a many-electron atom is written as a product of single-particle AOs, here the electronic wave function for a molecule is written as a product of single-particle MOs. This form is called the orbital approximation for molecules. We construct MOs, and we place electrons in them according to the Pauli exclusion principle to assign molecular electron configurations. 

What functions shall we use for the single-electron MOs We could try the exact MOs for shown in Fignre 6.2. However, these exact MOs are not described by simple eqnations, and they are inconvenient for applications. Therefore, we introdnce the LCAO method to construct approximate MOs directly from the Hartree AOs for the atoms in the molecule, guided by molecular symmetry and chemical intnition. The essential new feature compared with the atomic case is that the (multicenter) approximate MOs are spread around all the nuclei in the molecnle, so the electron density is de-localized over the entire molecule. The approximate MOs therefore differ considerably from the (single-center) AOs nsed in Section 5.2. Constructing the approximate MOs and nsing them in qnalitative descriptions of bonding are the core objectives of this section. 

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