Hartree Fock eigenvalue functions

The single nucleus state function or nuclear orbital is an eigenfunction of a Hartree-Fock eigenvalue equation for the nuclear motion... 

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. 

The Fock matrix is by design an effective one-electron Hamiltonian whose diagonalization yields a set of MOs, the canonical orbitals, from which a variationally r timized Hartree-Fock wave function may be constructed. However, since the Fock matrix (10.3.24) contains contributions from each occupied canonical orbital, we cannot solve the Hartree-Fock pseudo-eigenvalue equations (10.3.3) in a single step but must instead resort to some iterative scheme. 

It is important to realize that, although the Hartree-Fock wave function is an eigenfunction of the Fock operator with an eigenvalue equal to the sum of the orbital energies, this eigenvalue is not the same as the Hartree-Fock energy, which is the expectation value of the true Hamiltonian. Let us introduce the fluctuation potential as the difference between the two-electron part of the true Hamiltonian and the Fock potential... 

We have now succeeded in expressing the Hartree-Fock equations in the AO basis, avoiding any transformation to the MO basis. The pseudo-eigenvalue equations (10.6.16) are called the Roothaan-Hall equations [3,4]. In Exercise 10.4, the Roothaan-Hall SCF procedure is used to calculate the Hartree-Fock wave function for HeH in the STO-3G basis. 

The minimutn in the Ai singlet Hessian eigenvalue at 3.9ao arises because of an avoided crossing with a second Hartree-Fock state as indicated by the dotted line terminated by an arrow in Figure 10.3. This second RHF solution represents an excited (albeit unphysical) state of /4 symmetry. (For distances shorter than 4.0oo, the calculation of the excited RHF state becomes difficult and no plot has been attempted in this region.) Multiple RHF solutions close in energy are often found in regions where the Hartree-Fock wave function provides an inadequate description of the electronic system and where it is necessary to go beyond the Hartree-Fock model for a proper description of the electronic system. 

The Hartree-Fock equations form a set of pseudo-eigenvalue equations, as the Fock operator depends on all the occupied MOs (via the Coulomb and Exchange operators, eqs. (3.36) and (3.33)). A specific Fock orbital can only be determined if all the other occupied orbitals are known, and iterative methods must therefore be employed for determining the orbitals. A set of functions which is a solution to eq. (3.41) are called Self-Consistent Field (SCF) orbitals. 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

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... 

Slater then proposed returning to one-electron Schrodinger equations (and therefore one-electron wave functions i and corresponding eigenvalues ,) but now using not the Hartree-Fock (non-local) potential in these equations but the simplified potential... 

This representation permits analytic calculations, as opposed to fiiUy numerical solutions [47,48] of the Hartree-Fock equation. Variational SCF methods using finite expansions [Eq. (2.14)] yield optimal analytic Hartree-Fock-Roothaan orbitals, and their corresponding eigenvalues, within the subspace spanned by the finite set of basis functions. 

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. 

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