Hartree-Fock molecular orbital eigenvalue

In the unrestricted Hartree-Fock method, a single-determinant wave function is used with different molecular orbitals for a and jS spins, and the eigenvalue problem is solved with separate F and F matrices. With the zero differential overlap approximation, the F matrix elements (25) become... 

It should be noted that by moving to a matrix problem, one does not remove the need for an iterative solution the F >v matrix elements depend on the Cv,i LCAO-MO coefficients which are, in turn, solutions of the so-called Roothaan matrix Hartree-Fock equations- Zv F >v Cv,i = Zv S v Cvj. One should also note that, just as F (f>j = j (f>j possesses a complete set of eigenfunctions, the matrix Fp,v, whose dimension M is equal to the number of atomic basis orbitals used in the LCAO-MO expansion, has M eigenvalues j and M eigenvectors whose elements are the Cv>i- Thus, there are occupied and virtual molecular orbitals (mos) each of which is described in the LCAO-MO form with CV)i coefficients obtained via solution of... 

The Hartree-Fock equations (5.47) (in matrix form Eqs. 5.44 and 5.46) are pseudoeigenvalue equations asserting that the Fock operator F acts on a wavefunction i//, to generate an energy value ,-, times i/q. Pseudoeigenvalue because, as stated above, in a true eigenvalue equation the operator is not dependent on the function on which it acts in the Hartree-Fock equations F depends on i// because (Eq. 5.36) the operator contains J and K, which in turn depend (Eqs. 5.29 and 5.30) on i//. Each of the equations in the set (5.47) is for a single electron ( electron 1 is indicated, but any ordinal number could be used), so the Hartree-Fock operator F is a one-electron operator, and each spatial molecular orbital i// is a one-electron function (of the coordinates of the electron). Two electrons can be placed in a spatial orbital because the, full description of each of these electrons requires a spin function 7 or jl (Section 5.2.3.1) and each electron moves in a different spin orbital. The result is that the two electrons in the spatial orbital i// do not have all four quantum numbers the same (for an atomic Is orbital, for example, one electron has quantum numbers n= 1, / = 0, m = 0 and s = 1/2, while the other has n= l,l = 0,m = 0 and s = —1/2), and so the Pauli exclusion principle is not violated. 

The wave eigenfunctions and energy eigenvalues were obtained by real space electronic state calculations, which were performed by the use of the program code SCAT of the DV-Xx molecular orbital (MO) cluster method with the Hartree-Fock-Slater approximation [8,9]. In the method, the exchange-correlation term Vxc in the one-electron Hamiltonian was expressed in terms of the statistical local potential (1),... 

Thus the spectrum which arises when Eq. (8) is Fourier transformed consists of a set of -functions at the energies corresponding to the stationary states of the ion (which via the theorem of Koopmans) are the one-electron eigenvalues of the Hartree-Fock equations). The valence bond description of photoelectron spectroscopy provides a novel perspective of the origin of the canonical molecular orbitals of a molecule. Tlie CMOs are seen to arise as a linear combination of LMOs (which can be considered as imcorrelated VB pairs) and coefficients in this combination are the probability amplitudes for a hole to be found in the various LMOs of the molecule. 

Hartree-Fock eigenvalue of the highest occupied molecular orbital the electron affinity should approximate to the eigenvalue of the lowest unoccupied molecular orbital. According to Pople s theory, the mean of these two quantities should be the same for all alternant aromatic hydrocarbons and equal to the work function of graphite Pople and Hush showed that this relationship is verified by experiment. Brickstock and Pople1 extended the theory to radicals and ions. 

Use of the LCAO expansion leads to the Hartree-Fock-Roothaan equations Fc = See. Our job is then to find the LCAO coefficients c. This is achieved by transforming the matrix equation to the form of the eigenvalue problem, and to diagonalize the corresponding Hermitian matrix. The canonical molecular orbitals obtained are linear combinations of the atomic orbitals. The lowest-energy orbitals are occupied by electrons, those of higher energy are called virtual and are left empty. 

Interpretation of the UPS spectra of molecular systems discussed below commonly relies on the one-electron picture of the neutral molecules in this case there exists a one-to-one correspondence between the major peaks in the photoelectron spectrum and the one-electron molecular orbitals (see Fig. 23.4). Usually the numerical values of the calculated binding energies of the peaks are set to the Hartree-Fock eigenvalues of the molecular orbitals in the neutral ground state of the molecule, e.g., employing the Koopman theorem. Important effects to consider when relying on quantum-chemical calculations are the various relaxation effects that occur during a photoelectron emission event. It is usually necessary to put in by hand (or in some more sophisticated theoretical fashion) corrections for the relaxation phenomena that account for differences between the molecular orbitals of the neutral molecule and the molecular ion. 

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