Basis atomic orbital

The complexity of the parameter-fitting procedure in the MINDO models can only be appreciated by a detailed study of the inherent assumptions. It is perhaps indicative to say the only molecular integral that is calculated exactly from the basis atomic orbitals is the overlap integral, all others being approximated or given empirical values. The repulsion potential of the atomic cores is, for example, one of the critical functions in the theory. There has to the present time been four distinct versions of the MINDO parameterization, and the latest (MINDO/3) (102) is said to remove certain deficiencies in the earlier versions (such as the prediction that HjO was linear and the underestimation of the strain energies in small ring hydrocarbons). 

Solution of the secular equation amounts to finding the roots of an iVth order equation in E. The N roots are the energies of the N molecular orbitals the forms of the orbitals in terms of the basis atomic orbitals 9are found by substituting each value of E, in turn, back into Equations A2.13 and solving for the c s using the additional condition that each MO tf)t is to be normalized,... 

Figure 3.9. (a) The diamond structure viewed as two interlocking FCC sublattices displaced by I a along 1 1 1). In this projection along the [0 01] direction, only the top face of each cube is shown, (b) The unit cell, (c) Some possible sign combinations of the basis atomic orbitals used to construct LCAO COs from two Bloch sums, (d) A qualitative CO energy-level diagram for the center of the Brillouin zone, F = k(0, 0, 0). 

Alternatively, a Bloch sum of s atomic orbitals could combine with a Bloch sum of p atomic orbitals. The symmetric (bonding) combinations of the basis atomic orbitals for the latter case are illustrated for one CC4 subunit in Figure 3.9c. The actual COs are delocalized over aU the atoms with the space group symmetry of the diamond lattice. A LCAO-CO construction from Bloch sums is thus completely analogous to a... 

We now categorize the basis atomic orbitals in any, roughly spherical, cluster according to the number of nodal planes they possess that contain the radius vector. Hence, s atomic orbitals are classified as cr-type cluster orbitals, and so are p and d 2 orbitals, where the local axes at each vertex are chosen with the X and y directions tangential to the surface of the sphere so that z points outwards along the radius vector. Radially directed hybrids, such as sp are also a-type under this classification. Examples will be illustrated below. 7r-type cluster orbitals have one intrinsic nodal plane containing the radius vector, for example, p , Py, and dy in the same axis system. A y and Aj 2 y2 functions contain two such nodal planes, and are known as 5 orbitals. ... 

Atomic orbitals are chosen as basis functions in the so-called LCAO-MO method t .1 ). However, other choices are possible, for instance, Gaussian functions, which are particularly popular nowadays (see e.g. 1>). In planar systems, it is convenient to use basis functions that are either symmetric or antisymmetric with respect to the nuclear plane, i.e. are of a or n species. If %v and %q are basis atomic orbitals with different symmetry properties, then the matrix element Fpq vanishes so that the matrix F is factorized into one o and one n blockd> ... 

The Calculation of Basis Atomic Orbitals for Molecular Wave Functions. D.P. Craig and C. Zauli. 

Added in Proofs, Combined X-ray and electron diffraction data can be used to analyze the electron density in a molecule. Identification of the square of the wave function with the electron density then gives the coefficients of the basis atomic orbitals in the various molecular orbitals, and therefore permits the evaluation of the electron populations. This method was used for instance with 1,3,5-trimethyl-benzene, and good agreement with theoretical data was obtained [B. H. O Connor,... 

E. N. Maslen, Acta Cryst. B 30, 383 (1974)]. Of course, the results depend on the basis atomic orbitals and on the definition of the population. 

We now examine how, in principle, given one of the roots of (2-29) (consider the /th—call it ef) which makes the secular determinant in (2-28) vanish, we can obtain the weighting-coefficients, cfr, r = 1,2,n, which satisfy the set of homogeneous equations (2-27) for this particular value of e. It will be recalled that in the discussion following equation (2-1) in the section concerning the LCAO-approach ( 2.1), it was stated that P in equation (2-1) should strictly be labelled with a subscript as more than one combination of the given basis atomic orbitals which will form what, on the Variation-Principle s criterion, we may regard as a suitable molecular orbital. In fact, our analysis here has confirmed that there... 

In reality, as we have seen, the basis atomic orbitals are not orthogonal Srs for adjacent carbon-2p is of the order of 0.3—which is very far from being zero. One might therefore fear that all the analysis so-far outlined would be thrown out completely if we were to include these overlap terms which have previously been neglected. 

For an arbitrary, tr-electron system made up from n basis atomic-orbitals, we have now, in principle, described how to find, by the Variation Method, the energies, e , I = 1,2and the LCAO-combinatorial weighting-coefficients, c/r, / = 1, 2,..., n, r = 1, 2,. n, of the n LCAO-molecular-orbitals, , = 1,2which describe the system. 

It was stated earlier that in the simplest assumption we were going to make we would assume that the basis atomic-orbitals were normalised, i.e.. 

H was the matrix-component of the Hiickel effective-Hamiltonian operator, effective between two basis atomic-orbitals, 4>r and 4>s, Srs was the overlap integral between 4>r and s, and H was set equal to a, H to / . This is how we developed the simple HMO-approach in Chapter Two. What Roothaan did was to show that a formally similar determinant is obtained in a full treatment of the re-electrons, but that it involves a somewhat more complicated expression for the matrix-elements, H . Furthermore, he showed that this more-complicated expression somehow had to take into account interactions between any one re-electron and all the other re-electrons. We do not go into the details of this here, except to say that, in order to find the LCAO-MO coefficients for one molecular orbital, it is necessary to know all the others, because all the others appear in the expressions for the equivalent terms, Hrs. This is a very familiar situation which mathematicians have long known how to deal with and which we encountered during our discussion of the self-consistent" Huckel-methods in 7.2—7.5 it is necessary to use an iterative scheme. An initial guess is made of all the orbitals except one and these are used to calculate the H -terms for the one orbital which has not yet... 

As for a cluster orbitals, the transformation of basis to give 7t cluster orbitals from TT-type basis atomic orbitals is not unique. To define a transformation, we must first form and reduce the representation, spanned by the... 

The discrete variational (DV) method numerically calculates the basis atomic orbitals using the following wave equation for the radial atomic orbital function Rja r) in spherical coordinates... 

Phenomena Using the Shallow Well Potential for Basis Atomic Orbitals on Anions in the Original Kohn- ham Local Density Approximation... 

Based on the consequences of the previous section, the present author supposed that the strong localization of basis atomic orbitals for Cu by the trial addition of an artificial narrow and deep well solely to the atomic potential of an isolated... 

A cyclic r-conjugated ribbon-like delocalized molecular system where the basis atomic orbitals are organized in a Mobius strip. In contrast to Hiickel systems, where ring orbitals have zero or an even number of phase inversions, Mobius systems are characterized by an odd number of nodes. The electroncounting rule for the stability of Mobius systems is opposite to the Huckel rule 4 -electron Mobius systems have a closed... 

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