Matrices pseudoeigenvalue equation

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. 

Occasionally, we shall need to solve the so called pseudosecular equation for the symmetric matrix A arising from the pseudoeigenvalue equation ... 

The one-electron Green s function G E) is the resolvent of the Liouville operator within the I-block of basis operators. A comparison ol G E) and the EOM method with its expanded P-space can be accomplished by folding the partitioned EOM equation, which has the dimensionality of the P-space, into the 1-1 subblock. The P-space repartitioned EOM pseudoeigenvalue equation (38) is written in block matrix form... 

A unitary transformation that makes the matrix of the Lagrange multiplier diagonal may again be chosen, producing a set of canonical KS orbitals. The resulting pseudoeigenvalue equations are known as the Kohn-Sham equations. 

The P3 method is generally implemented in the diagonal self-energy approximation. Here, off-diagonal elements of the self-energy matrix in the canonical, Haruee-Fock orbital basis are set to zero. The pseudoeigenvalue problem therefore reduces to separate equations for each canonical, Hartree-Fock orbital ... 

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