MO-LCAO

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. 

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... 

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... 

When aos are eombined to form mos, eore, bonding, nonbonding, antibonding, and Rydberg moleeular orbitals ean result. The mos (j) are usually expressed in terms of the eonstituent atomie orbitals Xa iii the linear-eombination-of-atomie-orbital-moleeular-orbital (LCAO-MO) manner ... 

Just as in the non-linear polyatomie-moleeule ease, the atomie orbitals whieh eonstitute a given moleeular orbital must have the same symmetry as that of the moleeular orbital. This means that a,7i, and 5 moleeular orbitals are formed, via LCAO-MO, from m=0, m= 1, and m= 2 atomie orbitals, respeetively. In the diatomie N2 moleeule, for example, the eore orbitals are of a symmetry as are the moleeular orbitals formed from the 2s and 2pz atomie orbitals (or their hybrids) on eaeh Nitrogen atom. The moleeular orbitals formed from the atomie 2p i =(2px- i 2py) and the 2p 1 =(2px + i 2py) orbitals are of n symmetry and have m = -1 and +1. 

I. The LCAO-MO Expansion and the Orbital-Level Sehrodinger Equation... 

In the simplest pieture of ehemieal bonding, the valenee moleeular orbitals (jti are eonstrueted as linear eombinations of valenee atomie orbitals X i aeeording to the LCAO-MO formula ... 

These integrals over mo s ean, through the LCAO-MO expansion, ultimately be expressed in terms of one- and two-eleetron integrals over the primitive atomie orbitals. It is only these ao-based integrals that ean be evaluated explieitly (on high speed eomputers for all but the smallest systems). 

One consequence of the spin-polarized nature of the effective potential in F is that the optimal Isa and IsP spin-orbitals, which are themselves solutions of F ( )i = 8i d >i, do not have identical orbital energies (i.e., 8isa lsP) and are not spatially identical to one another (i.e., (l)isa and (l)isp do not have identical LCAO-MO expansion coefficients). This resultant spin polarization of the orbitals in P gives rise to spin impurities in P. That is, the determinant Isa 1 s P 2sa is not a pure doublet spin eigenfunction although it is an eigenfunction with Ms = 1/2 it contains both S = 1/2 and S = 3/2 components. If the Isa and Is P spin-orbitals were spatially identical, then Isa Is P 2sa would be a pure spin eigenfunction with S = 1/2. 

Before addressing head-on the problem of how to best treat orbital optimization for open-shell species, it is useful to examine how the HF equations are solved in practice in terms of the LCAO-MO process. 

In the most eommonly employed proeedures used to solve the HF equations for non-linear moleeules, the d >i are expanded in a basis of funetions X i aeeording to the LCAO-MO proeedure ... 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the Fj y matrix elements depend on the Cy i LCAO-MO eoeffieients whieh are, in turn, solutions of the so-ealled Roothaan matrix Hartree-Foek equations- Zy Fj y Cy j = 8i Zy Sj y Cy j. One should also note that, just as F ( )i = 8i (l)j possesses a eomplete set of eigenfunetions, the matrix Fj y, whose dimension M is equal to the number of atomie basis orbitals used in the LCAO-MO expansion, has M eigenvalues 8i and M eigenveetors whose elements are the Cy j. Thus, there are oeeupied and virtual moleeular orbitals (mos) eaeh of whieh is deseribed in the LCAO-MO form with Cy j eoeffieients obtained via solution of... 

The basis orbitals eommonly used in the LCAO-MO-SCF proeess fall into two... 

Onee one has speeified an atomie orbital basis for eaeh atom in the moleeule, the LCAO-MO proeedure ean be used to determine the Cy,i eoeffieients that deseribe the... 

Fjj,v Cy i 8i Zy Sjj y Cy j as initial guesses for the Cy i. Using only the one-eleetron part of the Hamiltonian to determine initial values for the LCAO-MO eoeffieients may seem like a rather severe step it is, and the resultant Cy i values are usually far from the eonverged values whieh the SCF proeess eventually produees. However, the initial Cy i obtained in this manner have proper symmetries and nodal patterns beeause the one-eleetron part of the Hamiltonian has the same symmetry as the full Hamiltonian. 

Much of the development of the previous ehapter pertains to the use of a single Slater determinant trial wavefunetion. As presented, it relates to what has been ealled the unrestrieted Hartree-Foek (UHF) theory in whieh eaeh spin-orbital (jti has its own orbital energy 8i and LCAO-MO eoeffieients Cy,i there may be different Cy,i for a spin-orbitals than for P spin-orbitals. Sueh a wavefunetion suffers from the spin eontamination diffieulty detailed earlier. 

The configuration interaction (CI) method in whieh the LCAO-MO eoeffieients are determined first (and independently) via either a single-eonfiguration SCF ealeulation or an MCSCF ealeulation using a small number of CSFs. The CI eoeffieients are subsequently determined by making the expeetation value < F H F >/< F I F >... 

The MoIIer-PIesset perturbation method (MPPT) uses the single-eonfiguration SCF proeess (usually the UHF implementation) to first determine a set of LCAO-MO eoeffieients and, henee, a set of orbitals that obey F( )i = 8i (jii. Then, using an unperturbed Hamiltonian equal to the sum of these Foek operators for eaeh of the N eleetrons =... 

The simultaneous optimization of the LCAO-MO and Cl coefficients performed within an MCSCF calculation is a quite formidable task. The variational energy functional is a quadratic function of the Cl coefficients, and so one can express the stationary conditions for these variables in the secular form ... 

Expanding the MO d >i in the LCAO-MO manner, substituting this expansion into the above... 

---