LCAOs

Molecules. The electronic configurations of molecules can be built up by direct addition of atomic orbitals (LCAO method) or by considering molecular orbitals which occupy all of the space around the atoms of the molecule (molecular orbital method). 

LCAO method A method of calculation of molecular orbitals based upon the concept that the molecular orbital can be expressed as a linear combination of the atomic orbitals. 

In the prooedures most oonnnonly applied to nonlinear moleoules, the ( ) are expanded in a basis aooording to the linear oombinations of AOs to fonn moleoular orbitals (LCAO-MO) [36] prooedure ... 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the matrix elements depend on the. LCAO-MO coefficients which are, in turn, solutions of the so-... 

The basis orbitals coimnonly used in the LCAO-MO process fall into two primary classes ... 

To overcome the primary weakness of GTO fimetions (i.e. their radial derivatives vanish at the nucleus whereas the derivatives of STOs are non-zero), it is coimnon to combine two, tliree, or more GTOs, with combination coefficients which are fixed and not treated as LCAO-MO parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of tight, medium, and loose GTOs are multiplied by contraction coefficients and suimned to produce a CGTO, which approximates the proper cusp at the nuclear centre. 

A double-zeta (DZ) basis in which twice as many STOs or CGTOs are used as there are core and valence AOs. The use of more basis functions is motivated by a desire to provide additional variational flexibility so the LCAO-MO process can generate MOs of variable difhiseness as the local electronegativity of the atom varies. 

Once one has specified an AO basis for each atom in the molecule, the LCAO-MO procedure can be used to... 

In this approach [ ], the LCAO-MO coefficients are detemiined first via a smgle-configuration SCF calculation or an MCSCF calculation using a small number of CSFs. The Cj coefficients are subsequently detemiined by making the expectation value ( P // T ) / ( FIT ) stationary. 

Some initial guess is made for the LCAO-KS expansion coefficients C. a a... 

Csizmadia I G, Flarrison M C, Moscowitz J Wand Sutcliffe B T 1966 Commentationes. Non-empirical LCAO-MO-SCF-Cl calculations on organic molecules with Gaussian type functions. Part I. Introductory review and mathematical formalism Theoret. Chim. Acta 6 191-216... 

Almidf J, Faegri K and Korsell K 1982 Principles for a direct SCF approach to LCAO-MO ah initio calculations J. Comput. Chem. 3 385-99... 

Parr R G 1983 Density functional theory Ann. Rev. Phys. Chem. 34 631 -56 Salahub D R, Lampson S FI and Messmer R P 1982 Is there correlation in Xa analysis of Flartree-Fock and LCAO Xa calculations for O3 Chem. Phys. Lett. 85 430-3... 

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. 

Slater P C and Koster G F 1954 Simplified LCAO method for the periodic potential problem Phys. Rev. 94 1498-524... 

Seifert G, Eschrig FI and Bieger W 1986 An approximate variation of the LCAO-Xa method Z. Phys. Chem. 267 529... 

Ravenek W and Geurts EMM 1986 Hartree-Fock-Slater-LCAO implementation of the moderately large-embedded-cluster approach to chemisorption. Calculations for hydrogen on lithium (100) J. Chem. Phys. 84 1613-23... 

HMO theory is named after its developer, Erich Huckel (1896-1980), who published his theory in 1930 [9] partly in order to explain the unusual stability of benzene and other aromatic compounds. Given that digital computers had not yet been invented and that all Hiickel s calculations had to be done by hand, HMO theory necessarily includes many approximations. The first is that only the jr-molecular orbitals of the molecule are considered. This implies that the entire molecular structure is planar (because then a plane of symmetry separates the r-orbitals, which are antisymmetric with respect to this plane, from all others). It also means that only one atomic orbital must be considered for each atom in the r-system (the p-orbital that is antisymmetric with respect to the plane of the molecule) and none at all for atoms (such as hydrogen) that are not involved in the r-system. Huckel then used the technique known as linear combination of atomic orbitals (LCAO) to build these atomic orbitals up into molecular orbitals. This is illustrated in Figure 7-18 for ethylene. 

Figure 7-18. Schematic representation of the LCAO scheme in a, T-only calculation for ethylene, The AOs

Note that the sums of the squares of the coefficients in a given MO must equal 1 (e.g., 0.3717 + 0.6015 + 0.3717 + 0.6015 = 1.0 for Pi) because each of the AOs represents a probability distribution of finding the electron at a given point in space. The total probability of finding an electron in all space for an MO must be unity, exactly as for its constituent AOs. We now can see that the LCAO approximation is only one of many possibilities to describe the electron density (= probability) for MOs. We do not have to express the electron density as a linear combination of the electron densities of AOs centered at the atoms. We could also... 

Population anaiysis methods of assigning charges rely on the LCAO approximation and express the numbers of electrons assigned to an atom as the sum of the populations of the AOs centered at its nucleus. The simplest of these methods is the Coulson analysis usually used in semi-empirical MO theory. This analysis assumes that the orbitals are orthogonal, which leads to the very simple expression for the electronic population of atom i that is given by Eq. (53), where Natomic orbitals centered... 

Because of the LCAO-MO approximation, ah iniiio and semi-empirical calculation s produce occupied and unoccupied (viriual) orbitals. The Aufban or building up" principle determines the... 

ITyperChem uses the Linear Combination of Atomic Orhilals-Molecular Orbital (LCAO-MO) approximation for all of itsah initio sem i-empirical melh ods. If rg, represen is a molecular orbital and... 

All m oleciilar orbitals are com biiiations of the same set of atom ic orbitals they differ only by their LCAO expansion coefficients. HyperC hem computes these coefficients, C p. and the molecular orbital energies by requiring that the ground-state electronic energy beat a minimum. That is, any change in the computed coefficients can only increase the energy. 

Solving the previous matrix equation tor the coefficients C describing the LCAO expan sion of th e orbitals and orbital energies n requires a matrix dia.i>onaliz(ition. If the overlap matrix were a... 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

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. 

In our treatment of molecular systems we first show how to determine the energy for a given iva efunction, and then demonstrate how to calculate the wavefunction for a specific nuclear geometry. In the most popular kind of quantum mechanical calculations performed on molecules each molecular spin orbital is expressed as a linear combination of atomic orhilals (the LCAO approach ). Thus each molecular orbital can be written as a summation of the following form ... 

Linear Combination of Atomic Orbitals (LCAO) in Hartree-Fock Theory... 

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