Integrals, overlap

Similar procedures can be used to evaluate other group overlap integrals. The results for tetrahedral and octahedral geometries are given in Tables 8-9 and 8-10. 

The radial functions for atoms through argon tabulated by dementi are particularly useful for simple MO calculations. Other sources are Watsonand Richardson et al. for transition metal atoms, and Slater for simple single-exponential-term functions. 

Bond distances are needed for overlap integral calculations. Standard sources are Sutton,Pauling, and Wells. 

Evaluation of two-atom overlap integrals has been discussed by Mulliken et al. Tables of overlap integrals are given in Mulliken et al., Jaffe et al., Leifer et al., and Craig et al. Lofthus gives some additional master formulas.  

In Sections 8-16 and 8-17 the example calculations of MnO and CrF - outline in some detail the use of the VSIE s given in Tables 8-11 and 8-12. 

Operations and representations are merely theoretical constructs. What is actually observed are the interactions. In quantum mechanics, interactions are expressed as matrix elements of operators in a function space. When the operator is the unit operator, the matrix elements are just overlap integrals. These are the simplest form of interactions. 

We start our analysis by examining symmetry selection rules for overlap integrals. Consider the overlap integral between the ith component of a function space which transforms according to the irrep F, and the fcth component of another function space transforming as F. The overlap integral, %, is a scalar quantity and thus must be invariant under the action of linear symmetry operators acting on the functions. 

An integral being an infinite sum, the operator can be brought inside the bracket and then transform the bra and ket parts directly. 

By summing over all e G and dividing by the group order one obtains a form to which the GOT can be applied. 

We now rewrite this result in terms of elements of the overlap matrix S  

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A more elaborate theoretical approach develops the concept of surface molecular orbitals and proceeds to evaluate various overlap integrals [119]. Calculations for hydrogen on Pt( 111) planes were consistent with flash desorption and LEED data. In general, the greatly increased availability of LEED structures for chemisorbed films has allowed correspondingly detailed theoretical interpretations, as, for example, of the commonly observed (C2 x 2) structure [120] (note also Ref. 121). 

Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

The last factor, the square of the overlap integral between the initial and final vibrational wavefunctions, is called the Franck-Condon factor for this transition. 

Next, we consider the simple overlap integral of two such basis functions with different powers of Cartesian coordinates and different Gaussian width, centered at different points. Let nuclei 1 locate at the origin, and let nuclei 2 locate at —R, then... 

It can be observed that we can separate the overlap integral into the product of three independent spatial integrals... 

In the MNDO rnelluKi Ui e resonance integral, is proportion al Lo the overlap integral, S y ... 

S.-i is the overlap integral, and are ionisation potentials for the appropriate orbitals and /J. a is a parameter dependent upon both of the two atoms A and B. 

If we could evaluate ot, p, and S, which are called the coulomb, exchange, and overlap integrals respectively, we could compute E. 

One restriction imposed by Huckel theory that is rather easy to release is that of zero overlap for nearest-neighbor interactions. One can retain a — as the diagonal elements in the secular matrix and replace p by p — EjS as nearest-neighbor elements where S is the overlap integral. Now,... 

Millikan has shown that the overlap integral for hydrogen-like p orbitals in linear hydrocarbons is about 0.27 (Millikan, 1949). 

What is the energy separation E2 — E of the bonding and antibonding orbitals in ethylene, assuming that the overlap integral S is 0.27 ... 

If the eonstituent atomie orbitals Xj have been orthonormalized as diseussed earlier in this ehapter, the overlap integrals [Pg.195]

Two Slater type orbitals, i andj, centered on the same point results in the following overlap integrals ... 

The first term in this expansion, when substituted into the integral over the vibrational eoordinates, gives ifj(Re) , whieh has the form of the eleetronie transition dipole multiplied by the "overlap integral" between the initial and final vibrational wavefunetions. The if i(Rg) faetor was diseussed above it is the eleetronie El transition integral evaluated at the equilibrium geometry of the absorbing state. Symmetry ean often be used to determine whether this integral vanishes, as a result of whieh the El transition will be "forbidden". 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

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