Coefficients of the atomic orbitals

The calculation proceeds as illustrated in Table 2.2, which shows the variation in the coefficients of the atomic orbitals in the lowest-energy wavefunction and the energy for the first four SCF iterations. The energy is converged to six decimal places after six iterations and the charge density matrix after nine iterations. 

The raw output of a molecular structure calculation is a list of the coefficients of the atomic orbitals in each LCAO (linear combination of atomic orbitals) molecular orbital and the energies of the orbitals. The software commonly calculates dipole moments too. Various graphical representations are used to simplify the interpretation of the coefficients. Thus, a typical graphical representation of a molecular orbital uses stylized shapes (spheres for s-orbitals, for instance) to represent the basis set and then scales their size to indicate the value of the coefficient in the LCAO. Different signs of the wavefunctions are typically represented by different colors. The total electron density at any point (the sum of the squares of the occupied wavefunctions evaluated at that point) is commonly represented by an isodensity surface, a surface of constant total electron density. 

The dipolar parts of the analyzed hfs tensors have been compared with calculated values obtained from first order expressions of the electron-nuclear dipole interaction (5.3)57. The coefficients of the atomic orbitals used in this computation, which considers all two- and three-center contributions, are obtained from an extended Huckel calculation (ethyl groups replaced by protons). It has been found that almost 100% of the unpaired electron is located on the CuS4 fragment so that the replacement of the ethyl groups by protons is of minor importance for the calculation of the atomic orbital coefficients. The experimental and theoretical hfs data, summarized in Table 8, are found to... 

With these considerations in mind, the process of constructing the correct linear combinations of the subsets proceeds exactly as in the case of the carbocyclic systems. The correct coefficients of the atomic orbitals are simply the characters of the representations. For the E orbitals we will obtain some imaginary coefficients, but these may be eliminated by taking the appropriate linear combinations. We can thus write, almost by direct inspection of the character table of the C4 group ... 

The calculations of the molecular orbital energies e (which we employ in the same manner as is usual in the semiempirica] methods for organic molecules) and the evaluation of the coefficients of the atomic orbitals and sets of atomic orbitals in Table I in the final molecular orbital requires the solution of secular determinants (one for each irreducible representation) of the form 6 c =0, where Htj has its usual... 

The diazomethane-imine addition can be interpreted in terms of the FMO theory.51 Calculations on diazoalkanes and benzylideneanilines lead to two orbital interactions, a HOMOdiazoalkane-LUMOimine and a LUMOdiazoalkane-HOMOiminc, but regardless of which one is the favored interaction, the coefficients of the atomic orbitals of the bonding atoms favor the formation of a 1,2,3- and not a 1,2,4-triazoline.338 And indeed, the 1,2,4-triazolines proposed by Mustafa327 have never been observed. 

Because of the electron interaction terms, the matrix elements in this approximation are themselves a function of the coefficients of the atomic orbitals. Initially, therefore, a reasonable guess is made as to the electronic distribution in the molecule and the calculation carried out until... 

The solution of the secular equation Fy —F5y = 0 requires the evaluation of the constituent matrix terms Fy. The Fy s are, however, themselves functions of the coefficients of the atomic orbitals amt through Pjel and therefore can only be evaluated by solving the secular equation. The Hartree-Fock procedure thus requires that a preliminary guess be made as to the values of the molecular population distribution terms Pici these values are then used to calculate the matrix elements Fy and thence solve the secular determinant. This, in turn, provides a better approximation to the wave function and an. .improved set of values of Pm. The above procedure is repeated with this first improved set and a second improved set evaluated. The process is repeated until no difference is found between successive improved wave functions. Finally, it may be shown that when such a calculation has been iterated to self-consistency the total electronic energy E of a closed shell molecule is given by... 

The coefficients of the atomic orbitals making up the molecular orbitals will change because oxygen is more electronegative than carbon and so has a greater share of electrons... 

Step 5. Beside each energy level, draw the orbitals, showing the signs of the coefficients of the atomic orbitals. All the % bonds are straightforward, but we meet a problem with the two a bonds in the product, which appear to be independent entities. In the next step we have to identify the symmetry these orbitals have with respect to the plane of symmetry maintained through the reaction, and it is not possible to do this for a pair of independent orbitals. The answer is to combine them they are held one bond apart, and they must interact in a % sense. The interaction of the two bonding a orbitals (Fig. 6.17a) and the two antibonding a ... 

Another special case, in which the unsymmetrical diene is part of a conjugated system which cannot easily be placed in any of the categories, C-, Z- or X-substituted, is tropone 6.173 when it reacts as a diene. On account of its symmetry, we have to work out the coefficients of the atomic orbitals by some other means than by the simple arguments used above and in Chapter 2 (pp. 60-65). The coefficients of the HOMO and LUMO are shown in Fig. 6.28. The numbers for the HOMO are not easily guessed at, and we must be content, in this more complicated situation, to accept the calculation which led to them. 

The coefficients of the atomic orbitals in the retype molecular orbitals of the extended Hiickel treatment are reasonably close to those of both the Hiickel and Pople-SCF 7r-electron calculations on NF2, cis- and ran -N2F2. The deviations tend to increase with increasing orbital energy. 

Using this HOMO-LUMO pair, estimate the relative sizes of the coefficients of the atomic orbitals on the atoms at which bonding is to occur. 

The second technique which both confirms some of our deductions and provides useful quantitative data for frontier orbital analysis is electron spin resonance spectroscopy9,10 (ESR). This technique detects the odd electron in radicals the interaction of the spin of the electron with the magnetic nuclei (1H, 13C, etc.) gives rise to splitting of the resonance signal, and the degree of splitting is proportional to the electron population at the nucleus. Since we already know that the coefficients of the atomic orbitals, c, are directly related... 

In discussing pericyclic reactions so far, we have only been looking at the denominator of the third term of equation 2-7. However, the coefficients of the atomic orbitals also play their part. They particularly influence the regio-selectivity, the site-selectivity and the periselectivity of cycloaddition reactions. The former term refers to the orientation of a cycloaddition for example, methoxybutadiene (204) gives166 the ortho adduct (206) rather than the meta adduct (203) with acrolein (205). Site-selectivity and periselectivity,... 

Table 4-1 The coefficients of the atomic orbitals in the molecular orbitals of conjugated systems7...

The )3 value depends on the geometrical conditions of the interaction, and Cq and Cl are the coefficients of the atomic orbitals in contact in the expression of... 

The output from a molecular orbital calculation, regardless of the specific computational method used, will always report a number of results. These will include the geometry of the system, the total energy for ab initio calculations, heat of formation for semiempirical calculations, the energy of each molecular orbital and the coefficients of the atomic orbitals for each atom. 

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