The LCAO Procedure

In fact, the HF procedure leads to a complicated set of integro-differential equations that can only be solved for a one-centre problem. If your interest lies in atomic applications, you should read the classic books mentioned above. What we normally do for molecules is to use the LCAO procedure each HF orbital is expressed as a linear combination of n atomic orbitals X . Xn 

Here the a, are the LCAO coefficients, which have to be determined. The formulation of HF theory where we use the LCAO approximation is usually attributed to Roothaan (1951a). His formulation applies only to electronic configurations of the type 1/ 3. Following the discussion of Chapter 5, the charge 

Most authors refer to the x s as basis functions. These usually overlap each other, and I will collect their overlap integrals into the n x n matrix S as in Chapter 5  

In the unlikely event that none of the basis functions overlap, then S is a unit matrix. We usually require the LCAO orbitals . to be orthonormal 

We now need to use the variation principle to seek the best possible values of the LCAO coefficients. To do this, I have to find Se as above, and set its first derivative to zero. I keep track of the requirement that the LCAO orbitals are 

In the unlikely event that none of the basis functions overlap, then S is a unit matrix. We usually require the LCAO orbitals ipA, Pb, , V m to be orthonormal and this fact can be summarized in a single matrix statement. A little manipulation will show that UTSU is then a unit matrix (with m rows and m columns), and also that 

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Notice that 1 haven t made any mention of the LCAO procedure Hartree produced numerical tables of radial functions. The atomic problem is quite different from the molecular one because of the high symmetry of atoms. The theory of atomic structure is simplified (or complicated, according to your viewpoint) by angular momentum considerations. The Hartree-Fock limit can be easily reached by numerical integration of the HF equations, and it is not necessary to invoke the LCAO method. 

Despite the obvious limitation of the LCAO procedure as revealed by the Hj and H2 problems it still is the most popular scheme used in the theoretical study of polyatomic molecules. There is a bewildering number of approximate methods, commonly distinguished in terms of cryptic acronyms, designated as either ab initio or semi-empirical, but all of them based on the LCAO construction of molecular orbitals. The precise details can be found in many books and reviews. The present summary uses the discussion of Richards and Cooper [92] as a guide. 

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... 

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

In terms of the LCAO coefficients, the procedure is very similar ... 

Atoms are special, because of their high symmetry. How do we proceed to molecules The orbital model dominates chemistry, and at the heart of the orbital model is the HF-LCAO procedure. The main problem is integral evaluation. Even in simple HF-LCAO calculations we have to evaluate a large number of integrals in order to construct the HF Hamiltonian matrix, especially the notorious two-electron integrals... 

The reason usually advanced is that whilst the occupied orbitals are determined variationally within the HF-LCAO procedure, the virtual orbitals are not. Consequently, the virtual orbitals give a very poor description of excited states. 

Implementation of the Kohn-Sham-LCAO procedure is quite simple we replace the standard exchange term in the HF-LCAO expression by an appropriate Vxc that will depend on the local electron density and perhaps also its gradient. The new integrals involved contain fractional powers of the electron density and cannot be evaluated analytically. There are various ways forward, all of which... 

Measurements by electron impact methods have led to experimental stabilization energies of a number of alkylcarbonium ions. Muller and Mulliken (1958) compared these with calculated values obtained by a procedure based on the LCAO-MO approximation, the large stabilization energies found for carbonium ions being attributed to the combined effects of hyperconjugation and charge redistribution (Table 1). 

Mulliken s formula for Nk implies the half-and-half (50/50) partitioning of all overlap populations among the centers k,l,... involved. On one hand, this distribution is perhaps arbitrary, which invites alternative modes of handling overlap populations. On the other hand, Mayer s analysis [172,173] vindicates Mulliken s procedure. So we may suggest a nuance in the interpretation [44] departures from the usual halving of overlap terms could be regarded as ad hoc corrections for an imbalance of the basis sets used for different atoms. But one way or another, the outcome is the same. It is clear that the partitioning problem should not be discussed without explicit reference to the bases that are used in the LCAO expansions. 

In the most commonly employed procedures used to solve the HF equations for non-linear molecules, the (f>i are expanded in a basis of functions according to the LCAO-MO procedure ... 

Pople-Pariser-Parr PPP (CNDO) method. MO s from LCAO SCF, i.e. electron correlation taken into account. Equations to derive the LCAO coefficients by the variational procedure were proposed by Roothaan... 

Jtfany authors refer to the HF-LCAO procedure, when discussing HF calculations made within the LCAO approximation. 

The Hartree-Fock equations are the coupled differential equations of the SCF procedure. The LCAO approximation transforms these differential equations into an ensemble of algebraic equations, which are substantially easier to solve. 

Not even the SCF procedure can overcome this problem. In the case of atoms, the central field remains a valid and good approximation. Assuming a rigid linear structure in the molecular case is clearly not good enough, although it contains an element of truth. This inherent problem plagues all LCAO-SCF calculations to an even more serious extent. 

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