Huckel calculation

TABLE 3 Calculated Huckel molecular-orbital spin densities for substituted benzosemiquinone anions... 

TT-Electron calculations. Huckel-type molecular orbital (HMO) method Bases—first calculations Base pairs—first calculations 56CR(243)380 59BBA(36)343... 

The energy of atomization for phosgene has been calculated (HUckel) to be 1444 kJ mol" [1441] force constant calculations give a range of 1340-1370 kJ mol" [2034]. 

One shall notice that contrary to ab initio calculations, where fully localized orbitals are necessarily non-orthogonal and lead to heavy calculations, Huckel orbital localization does not bring any complication, neither to the code, nor to the computational effort. The only slight complication arises from open shell covalently paired electrons, when the two electrons are not in the same n orbital but belong to two different orbitals, say a and b. We shall represent such a singlet coupling with a plain arc that links the two electrons dots (Fig. 13.3). The wave function associated to such a case contains the determinants ab + ba. ... 

Figs. 3a and 3c show calculated Huckel spin densities for the dimer cation radical, given by the squares of the moleculttr orbital coefficients on the different atoms in the half-filled molecular orbital for the two coupling cases... 

For dilute solutions, the Debye-Huckel equation by calculations based on these Coulombic interactions is... 

In spite of the success of this method it was later felt that the calculation of the charge distribution in conjugated r-systems should be put on a less empirical basis. To achieve this, a modified Huckel Molecular Orbital (HMO) approach (Section 7.4) was developed. Again, the charge distribution in the r-skeleton is first calculated by the PEOE method. 

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. 

The logical order in which to present molecular orbital calculations is ab initio, with no approximations, through semiempirical calculations with a restricted number of approximations, to Huckel molecular orbital calculations in which the approximations are numerous and severe. Mathematically, however, the best order of presentation is just the reverse, with the progression from simple to difficult methods being from Huckel methods to ab initio calculations. We shall take this order in the following pages so that the mathematical steps can be presented in a graded way. 

The simplest molecular orbital method to use, and the one involving the most drastic approximations and assumptions, is the Huckel method. One str ength of the Huckel method is that it provides a semiquantitative theoretical treatment of ground-state energies, bond orders, electron densities, and free valences that appeals to the pictorial sense of molecular structure and reactive affinity that most chemists use in their everyday work. Although one rarely sees Huckel calculations in the resear ch literature anymore, they introduce the reader to many of the concepts and much of the nomenclature used in more rigorous molecular orbital calculations. 

Three major approximations are made to separate the Schroedinger equation into a set of smaller equations before carrying out Huckel calculations. 

The Bom-Oppenheimer approximation is not peculiar to the Huckel molecular orbital method. It is used in virtually all molecular orbital calculations and most atomic energy calculations. It is an excellent approximation in the sense that the approximated energies are very close to the energies we get in test cases on simple systems where the approximation is not made. 

Polynomial root finding, as in the previous section, has some technical pitfalls that one would like to avoid. It is easier to write reliable software for matrix diagonalization (QMOBAS, TMOBAS) than it is for polynomial root finding hence, diagonalization is the method of choice for Huckel calculations. 

Repeat each calculation after having inserted a counter" into Program QMOBAS to count the number of iterations. The statement ITER ITER 1 p I ac ed be fore th e G OTO 340 s tate m e ti t i n c I e tn e n ts th e co n te n ts of memory location ITHR, starting from zero, on each iteration. The statement PRINT ITER", ITHR prints out the accumulated numbei of itei ations at the etid of the progratn run, Cotnment on the number of itei atiotis needed to satisfy the htial nonn V I for tbe different Huckel MO calculations. 

In the Mathcad calculation of eigenvalues and eigenvectors of the Huckel matrix for ethylene (j g), the eigenvalues are given in the order upper followed by lower. The matr ix E for this order is... 

Hoffman s extended Huckel theory, EHT (Hoffman, 1963), includes all bonding orbitals in the secular matrix rather than just all n bonding orbitals. This inclusion increases the complexity of the calculations so that they are not practical without a computer. The basis set is a linear combination that includes only valence orbitals... 

Refer to Computer Project 7-2. Calculate p in units of electron volts using Wheland s extension of Huckel molecular orbital theory. 

Because of its severe approximations, in using the Huckel method (1932) one ignores most of the real problems of molecular orbital theory. This is not because Huckel, a first-rate mathematician, did not see them clearly they were simply beyond the power of primitive mechanical calculators of his day. Huckel theory provided the foundation and stimulus for a generation s research, most notably in organic chemistry. Then, about 1960, digital computers became widely available to the scientific community. 

One of the things illustrated by this calculation is that a surprisingly good approximation to the eigenvalue can often be obtained from a combination of approximate functions that does not represent the exact eigenfunction very closely. Eigenvalues are not vei y sensitive to the eigenfunctions. This is one reason why the LCAO approximation and Huckel theory in particular work as well as they do. 

In PPP-SCF calculations, we make the Bom-Oppenheimer, a-rr separation, and single-electron approximations just as we did in Huckel theor y (see section on approximate solutions in Chapter 6) but we take into account mutual electrostatic repulsion of n electrons, which was not done in Huckel theory. We write the modified Schroedinger equation in a form similar to Eq. 6.2.6... 

Having filled in all the elements of the F matr ix, we use an iterative diagonaliza-tion procedure to obtain the eigenvalues by the Jacobi method (Chapter 6) or its equivalent. Initially, the requisite electron densities are not known. They must be given arbitrary values at the start, usually taken from a Huckel calculation. Electron densities are improved as the iterations proceed. Note that the entire diagonalization is carried out many times in a typical problem, and that many iterative matrix multiplications are carried out in each diagonalization. Jensen (1999) refers to an iterative procedure that contains an iterative procedure within it as a macroiteration. The term is descriptive and we shall use it from time to time. 

HMO method. This is because electron repulsion is taken into account in the SCF calculation whereas it is not taken into account in the Huckel calculation. 

Note that agreement with Pariser and Parr s empirical value is better for Y13 than for Yn ) Use Salem s values to calculate election densities on the three carbon atoms of the allyl anion for one iteration beyond the initial Huckel values, as was done in Exercise 8.9.1. Comment on the results you get, as to the qualitative picture of the anion, the influence of election repulsion on the charge densities, and agreement or lack of agreement with the results already obtained with the Pariser and Parr parameters. 

The Huckel method and is one of the earliest and simplest semiempirical methods. A Huckel calculation models only the 7t valence electrons in a planar conjugated hydrocarbon. A parameter is used to describe the interaction between bonded atoms. There are no second atom affects. Huckel calculations do reflect orbital symmetry and qualitatively predict orbital coefficients. Huckel calculations can give crude quantitative information or qualitative insight into conjugated compounds, but are seldom used today. The primary use of Huckel calculations now is as a class exercise because it is a calculation that can be done by hand. 

An extended Huckel calculation is a simple means for modeling the valence orbitals based on the orbital overlaps and experimental electron affinities and ionization potentials. In some of the physics literature, this is referred to as a tight binding calculation. Orbital overlaps can be obtained from a simplified single STO representation based on the atomic radius. The advantage of extended Huckel calculations over Huckel calculations is that they model all the valence orbitals. 

The primary reason for interest in extended Huckel today is because the method is general enough to use for all the elements in the periodic table. This is not an extremely accurate or sophisticated method however, it is still used for inorganic modeling due to the scarcity of full periodic table methods with reasonable CPU time requirements. Another current use is for computing band structures, which are extremely computation-intensive calculations. Because of this, extended Huckel is often the method of choice for band structure calculations. It is also a very convenient way to view orbital symmetry. It is known to be fairly poor at predicting molecular geometries. 

Molecular mechanics and semiempirical calculations are all relativistic to the extent that they are parameterized from experimental data, which of course include relativistic effects. There have been some relativistic versions of PM3, CNDO, INDO, and extended Huckel theory. These relativistic semiempirical calculations are usually parameterized from relativistic ah initio results. 

Sandstrom et al. (65) evaluated the Kj value for 4,5-dimethyl-A-4-thiazoline-2-thione (46) in water (Scheme 19) K-j= 10. A-4-Thiazoline-2-thiones are less basic in the first excited state (61) than in the ground state, so application of Forster s cycle suggests that the thione form is even more favored in the first excited state. Huckel molecular orbital (HMO) calculations suggest that electronic effects due to substitution in... 

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