LCAO calculation

Not only that, there are usually a lot of them. An HF-LCAO calculation with n basis functions requires the calculation and manipulation of about such integrals. 

Clementi and Raimondi refined these results by performing atomic HF-LCAO calculations, treating the orbital exponents as variational parameters. A selection of their results for H through Ne is given in Table 9.3. 

The next step on the road to quality is to expand the size of the atomic orbital basis set, and I hinted in Chapters 3 and 4 how we might go about this. To start with, we double the number of basis functions and then optimize their exponents by systematically repeating atomic HF-LCAO calculation. This takes account of the so-called inner and outer regions of the wavefunction, and Clementi puts it nicely. 

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... 

Of interest is the SCF=Direct option. There are three ways of dealing with two-electron integrals over the basis functions in ab initio HF-LCAO calculations. The Conventional way is to calculate them once and store them on a... 

Then comes the HF-LCAO calculation (Figure 10.12). The procedure starts with an INDO run (Chapter 8) for the initial estimate of the electron density Notice once again the internal use of molecular symmetry. In early packages such as POLYATOM, the use of molecular symmetry was essential for fast execution but had to be explicitly included by the user. 

The HF-LCAO calculation follows the usual lines (Figure 11.10) and the frozen core approximation is invoked by default for the CISD calculation. CISD is iterative, and eventually we arrive at the improved ground-state energy and normalization coefficient (as given by equation 11.7) — Figure 11.11. 

As a simple example, let s return to the dineon problem discussed above. Here are the salient points from a Gaussian run at 300 pm. Figure 11.12 shows the standard HF-LCAO calculation. 

If the neon-neon interaction were a pure dispersion one, then the HF-LCAO calculation would give a fully repulsive curve. The fact that the HF-LCAO calculation gives a shallow minimum implies an element of covalency. 

It is also a common experience that traditional Cl calculations converge very poorly, because the virtual orbitals produced from an HF (or HF-LCAO) calculation are not determined by the variation principle and turn out to be very poor for representations of excited states. 

In this particular example, the Xa orbital energies resemble those produced from a conventional HF-LCAO calculation. It often happens that the Xa ionization energies come in a different order than HF-LCAO Koopmans-theorem ones, due to electron relaxation. 

As mentioned above, a KS-LCAO calculation adds one additional step to each iteration of a standard HF-LCAO calculation a quadrature to calculate the exchange and correlation functionals. The accuracy of such calculations therefore depends on the number of grid points used, and this has a memory resource implication. The Kohn-Sham equations are very similar to the HF-LCAO ones and most cases converge readily. 

Here is a KS-LCAO calculation on water at the experimental geometry of 95.7pm and 104.5°. I chose the BLYP functional this comprises Becke s 1988... 

Then Figure 13.3 shows what appears to be a standard HF-LCAO calculation. 

To show the principles involved in finding an analytical gradient expression consider an HF-LCAO calculation where the electronic energy comes to... 

Basis set dependence is important. The results in Table 16.1 were obtained for HF-LCAO calculations on pyridine. In each case, the geometry was optimized As a general rule, ab initio HF-LCAO calculations with small basis sets tend to underestimate the dipole moment, whilst extended basis sets overestimate it A treatment of electron correlation usually brings better agreement with experiment. 

HF-LCAO calculations on molecules with small electric dipoles need to be treated with caution. The classic case is CO. Burrus (1958) determined the magnitude of the vector from a Stark experiment as 0.112 0.005 D (0.374 0.017 x 10-30 Cm). 

Early ab initio HF-LCAO calculations using minimal basis sets were at first thought to have been unsuccessful, in that they gave the polarity C-O", whilst HF-LCAO calculations with extended basis sets gave the polarity C O-. That the polarity is indeed C-O" " was shown by Rosenblum, Nethercott and Townes (1958) from a molecular beam resonance experiment it goes against all chemical inmition. 

If we add a perturbation A then the self-consistency is destroyed and we need to re-do the iterative HF-LCAO calculation. The idea of self-consistent perturbation theory is to seek solutions of the perturbed HF-LCAO equations... 

There are a few interesting points about the treatment. First of all, there is no variational HF-LCAO calculation (because every available x is doubly occupied) and so the energy evaluation is straightforward. For a wavefunction comprising m doubly occupied orthonormal x s the normalizing factor N is... 

This strong dependence of the conductivity on the composition can be understood qualitatively considering the densities-of-states. The DOS s from the LCAO calculation, splitted into their atomic contributions (partial DOS s), are given in Fig. 1 and can be discussed as follows ... 

A number of theoretical calculations are available for comparison with the experimental results on Be metal. The increase of the valence density in the tetrahedral holes is well reproduced by both the early augmented plane wave (APW) calculation of Inoue and Yamashita (1973), and the all-electron HF-LCAO calculation of Dovesi et al. (1982), but the latter gives somewhat better agreement with the experimental results. 

A great variety of seemingly unrelated organic compounds have been demonstrated to transfer two electrons in a stepwise fashion, if they can be derived from the general structural types A, B or C. The intermediate oxidation level SEM thereby represents radical cations, radical anions or neutral radicals Their thermodynamic stability can be understood within a general theory of polymethines X—(CH)n 2—X containing Nil TT-electrones for which MO-LCAO calculations have been develope l... 

Quantitative MO-LCAO calculations as well as treatments as a one dimensional electron gas have been advanced. Special parameters, however, have to be introduced to account for different end groups and branching of the rr-system. Empirically a linear correlation between and n is verified in all cases so far investigated. That is, violenes behave like cyanines. The vinylene shift amounts to 100—150 nm in contrast to that of the corresponding forms OX and RED with 20-40 mm ... 

Hiickel MO calculations of the 7r-electron density for pyrazolo[3,4-d]-pyrimidine 290 reveal N-3 to be the most electron rich (69CJC1129). The same conclusion was reached with simple linear combination of atomic orbitals (LCAO) calculations (291). LCAO data for electron densities on pyrazolo[4,3-LCAO calculations exaggerate electronegativities of nitrogen atoms (see 293)... 

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