Solving for the Molecular Orbitals

The ground-state tt-electron configuration of the allyl system is built up by putting electrons in pairs into the MOs, starting with those of lowest energy. Thus far, we have been describing our system as the allyl radical. However, since we have as yet made no use of the number of tt electrons in the system, our results so far apply equally well for the allyl cation, radical, or anion. 

Configurations and total tt energies for these systems in their ground states are depicted in Fig. 8-4. The total tt-electron energies are obtained by summing the one-electron energies, as indicated earlier. 

EXAMPLE 8-2 For a planar, unsaturated hydrocarbon having formula C H, where all the carbons are part of the unsaturated framework, how many pi MOs are there  

SOLUTION Each carbon atom brings one AO into the basis set, so there are x basis AOs. These x independent AOs mix to form x independent MOs.  

We still have to find the coefficients that describe the MOs as linear combinations of AOs. Recall from Chapter 7 that this is done by substituting energy roots of the secular determinant back into the simultaneous equations. For the allyl system, the simultaneous equations corresponding to the secular determinant (8-16) are 

Solving the previous matrix equation for the coefficients C describing the LC AO expansion of the orbitals and orbital energies 8 requires a matrix diagonalization. If the overlap matrix were a unit matrix would simply diagonalize the 

Because of the presence of the overlap matrix, however, you must first diagonalize the overlap matrix  

Finally, you must diagonalize S H S to obtain the molecular orbitals and orbital energies  

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From these overlap integrals and the experimental orbital reduction factors, it is possible to solve for the molecular orbital coefficients. For k we have... 

Now that you know the mathematical form, you can solve the independent-electron Schrodinger equation for the molecular orbitals. First substitute the LCAO form above into equation (47) on page 193, multiply on the left by and integrate to represent... 

Since P depends on the solution of the secular equation, which in turn depends on P, it is clear that we must solve iteratively for the molecular orbitals. In general, we will consider only the first few iterations and start the first iteration with = ZM, where is the effective charge of the nuclear core of the pth orbital (for more than one orbital per atom we have ZA = EM(y4) Zfi). The potential surface of the system is then approximated by... 

The proper way of dealing with periodic systems, like crystals, is to periodicize the orbital representation of the system. Thanks to a periodic exponential prefactor, an atomic orbital becomes a periodic multicenter entity and the Roothaan equations for the molecular orbital procedure are solved over this periodic basis. Apart from an exponential rise in mathematical complexity and in computing times, the conceptual basis of the method is not difficult to grasp [43]. Software for performing such calculations is quite easily available to academic scientists (see, e.g., CASTEP at www.castep.org CRYSTAL at www.crystal.unito.it WIEN2k at www.wien2k.at). 

When wc attempt to solve the Schrodinger equation to obtain the various molecular orbitals, we run into the same problem found earlier for atoms heavier than hydrogen. We are unable lo solve the Schrodinger equation exacljy and therefore must make some approximations concerning the form of the wave functions for the molecular orbitals. 

The molecular orbital approach to describing hydrogen also starts with two hydrogen nuclei (a and b) and two electrons (1 and 2), but we make no initial assumption about the location of the two electrons. We solve (at least in principle) the Schrodinger equation for the molecular orbitals around the pair of nuclei, and we then write a wave equation for one electron in a resulting MO ... 

This approach used in solving for n molecular orbitals within a self-consistent field is known as the Hartree-Fock solutionl l The molecular orbitals are the individual electronic states that describe the spatial part of the molecular spin orbitall l Electrons are... 

All the parts of the HMOT analysis of fulvene. A. The secular determinant with numbering as given in the picture of fulvene. B. The secular determinant after dividing by 3 and substituting (a - ) / 3 = -x. C. The energy diagrams derived from solving the sixth-order polynomial from the secular determinant. D. The secular ec uations for fulvene. E. The wavefunctions for the molecular orbitals. F. Pictures of all the Hiickel MOs. 

The problem has now become how to solve for the set of molecular orbital expansion coefficients, c. . Hartree-Fock theory takes advantage of the variational principle, which says that for the ground state of any antisymmetric normalized function of the electronic coordinates, which we will denote H, then the expectation value for the energy corresponding to E will always be greater than the energy for the exact wave function ... 

The first calculations on a two-electron bond was undertaken by Heitler and London for the H2 molecule and led to what is known as the valence bond approach. While the valence bond approach gained general acceptance in the chemical community, Robert S. Mulliken and others developed the molecular orbital approach for solving the electronic structure problem for molecules. The molecular orbital approach for molecules is the analogue of the atomic orbital approach for atoms. Each electron is subject to the electric field created by the nuclei plus that of the other electrons. Thus, one was led to a Hartree-Fock approach for molecules just as one had been for atoms. The molecular orbitals were written as linear combinations of atomic orbitals (i.e. hydrogen atom type atomic orbitals). The integrals that needed to be calculated presented great difficulty and the computations needed were... 

The parameters a and 02 are arbitrary parameters to be solved by minimizing the molecular energy. The solutions are called the molecular orbitals, which are responsible for bonding atoms A and B together. The equilibrium distance between A and B and the bonding energy are the most important results from this calculation. 

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