LCAO coefficients

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

Here the a, are the LCAO coefficients, which have to be determined. The formulation of HF theory where we use the LCAO approximation is usually attributed to Roothaan (1951a). His formulation applies only to electronic configurations of the type 1/ 3,...,. Following the discussion of Chapter 5, the charge... 

We now need to use the variation principle to seek the best possible values of the LCAO coefficients. To do this, I have to find Se as above, and set its first derivative to zero. I keep track of the requirement that the LCAO orbitals are... 

Roothaan actually solved the problem by allowing the LCAO coefficients to vary, subject to the LCAO orbitals remaining orthonormal. He showed that the LCAO coefficients are given from the following matrix eigenvalue equation ... 

Here, c is a column vector of LCAO coefficients and e is called the orbital energy. If we start with n basis functions, then there are exactly n different c s (and e s) and the m lowest-energy solutions of the eigenvalue problem correspond to the doubly occupied HF orbitals. The remaining n — m solutions are called the virtual orbitals. They are unoccupied. 

Once again, this doesn t help us to find the HF orbitals, because we have to know the LCAO coefficients before we know the Hamiltonian matrix. For the record, and because we will refer to them many times, the elements of the HF matrix for a closed shell system are... 

In terms of the LCAO coefficients, the procedure is very similar ... 

Suppose then that we have successfully solved the HF equations for a molecule with 2m electrons as shown in Figure 6.1, and that the occupied orbitals have energies a, b, > m- The LCAO coefficients are also collected in the column vectors a, b,..., m and the electronic energy is given by... 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

The zeros arise because (for example) atom 1 is bonded to atom 2 but not to atom 3, 4, 5 or 6. Diagonalization gives the orbital energies and normalized LCAO coefficients shown in Table 7.2. 

In each pair, provided there are no degeneracies, the LCAO coefficients for the 2pjj. orbital on each carbon atom has the same modulus. 

The two-electron integrals over LCAO orbitals have to be evaluated from the LCAO coefficients and the y,j. In general, a two-electron integral... 

Table 8.7 CNDO/2 orbital energies and LCAO coefficients for H2O...

A CNDO/2 calculation gives the HF orbital energies and LCAO coefficients shown in Table 8.7. 

Here, occ means occupied and virt means virtual. In the restricted Hartree-Fock model, each orbital can be occupied by at most one a spin and one (i spin electron. That is the meaning of the (redundant) Alpha in the output. In the unrestricted Hartree-Fock model, the a spin electrons have a different spatial part to the spin electrons and the output consists of the HF-LCAO coefficients for both the a spin and the spin electrons. 

There is very little point in trying to obtain information from the 73 x 47 z= 3431 numbers that constitute the HF-LCAO coefficients for the occupied orbitals. Mulliken population indices are given next, together with Mulliken atomic charges (Figure 10.14). 

The basis functions are kept constant throughout the calculation, as are the LCAO coefficients. 

Multiconfiguration self-consistent field (MCSCF) theory aims to optimize simultaneously the LCAO coefficients and the Cl expansion coefficients in a wavefunction such as... 

The P matrix involves the HF-LCAO coefficients and the hi matrix has elements that consist of the one-electron integrals (kinetic energy and nuclear attraction) over the basis functions Xi - Xn - " h matrix contains two-electron integrals and elements of the P matrix. If we differentiate with respect to parameter a which could be a nuclear coordinate or a component of an applied electric field, then we have to evaluate terms such as... 

For large interchain separations (8 A < R < 30 A), the LCAO coefficients of a given molecular orbital are localized on a single chain, as intuitively expected. The lowest excited state of these dimers results from a destructive interaction of the two intrachain transition dipole moments, whereas a constructive interaction prevails for the second excited stale. This result is fully consistent with the molcc-... 

In a regime of strong interaction between the chains no optical coupling between the ground slate and the lowest excited state occurs. The absence of coupling, however, has a different origin. Indeed, below 7 A, the LCAO coefficients start to delocalize over the two chains and the wavefunclions become entirely symmetric below 5 A due to an efficient exchange of electrons between the chains. This delocalization of the wavcfunclion is not taken into account in the molecular exciton model, which therefore becomes unreliable at short chain separations. Analysis of the one-electron structure of the complexes indicates that the... 

In some cases half these values are adopted as /r. The absolute value of the LCAO coefficient, [c j, serves as the measure of orbital extension, as well as the square value, 4 . ... 

Within the SCF-CI method a fixed set of molecular orbitals is used. This means that during the calculation (leading to slow convergence) the individual molecular orbitals remain unchanged. A method where the linear expansion coefficients and the LCAO coefficients are optimized simultaneously is the multi-configuration SCF (MCSCF). 

Pople-Pariser-Parr PPP (CNDO) method. MO s from LCAO SCF, i.e. electron correlation taken into account. Equations to derive the LCAO coefficients by the variational procedure were proposed by Roothaan... 

Solid and open circles represent opposite signs of the LCAO coefficients, areas being proportional to the squares of the same. 

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