Pseudo-eigenvalue problem

The computer codes of Sambe and Felton (SF) and Dunlap et al. (DCS) are based on the choice of a Hermite-Gaussian expansion set. Applying the variational theorem with the trial function of Eq. (37) and the LSD Hamiltonian of Eq. (36) leads to the usual matrix pseudo-eigenvalue problem ... 

In the limit of a complete basis, the pseudo-eigenvalue problem (Eq. (23)) may be expressed in the form ... 

If this condition is not fulfilled, the expression for the derivatives is used to progress towards the solution. The relationship of this treatment to other methods, involving pseudo-eigenvalue problems, can also be established but the present method showed a better convergency in the case of simple polyatomic molecules as formaldehyde 88), ethylene and ethane 90). 

If one assumes the core orbitals (pc to be also eigenfunctions of the Fock operator i.e., [ v,Pc] = 0, and uses the idempotency of the projection operator = Pc ( 1)> oil recovers a simplified pseudo eigenvalue problem... 

The equation is handy and concise except for one thing. It would be even better if the right side were proportional to 0, (1) instead of being a linear eombination of all the spinoibitals. In such a case, the equation would be similar to the eigenvalue problem and we would find it quite satisfactory. It would be similar but not identical, since the operators J and K include the sought spinorbitals 4>i- Because of this, the equation would be called the pseudo-eigenvalue problem. 

First, we meet the difficulty that in order to solve the Fock equation we should first. .. know its solution. Indeed, the Fock equation is not an eigenvalue problem, but a pseudo-eigenvalue problem, because the Fock operator depends on the solutions (obviously, unknown). Regardless of how strange it might seem, we deal with this situation quite easily using an iterative approach. This is called the self-SCF iterations consistent field method (SCF). In this method (Fig. 8.6) we... 

As a matter of fact, as in the Hartree-Fock (HF) scheme, the KS equation is a pseudo-eigenvalue problem and has to be solved iteratively through a self-consistent field procedure to determine the charge density p(r) that corresponds to the lowest energy. The self-consistent solutions 4>ia resemble those of the HF equations. Still, one should keep in mind that these orbitals have no physical significance other than in allowing one to constitute the charge density. We want to stress that the DFT wavefunction is not a Slater determinant of spin orbitals. In fact, in a strict sense there is no A -electron wavefunction available in DFT. ... 

The so called Hartree-Fock equations represent a pseudo-eigenvalue problem which requires an iterative approach and for which the use of computers is ideally suited. The total energy of the helium atom calculated in this way shows an error of approximately 1.5% as compared with the experimental value. 

Because the Fock matrix depends on the one-particle density matrix P constructed conventionally using the MO coefficient matrix C as the solution of the pseudo-eigenvalue problem (Eq. [7]), the SCF equation needs to be solved iteratively. The same holds for Kohn-Sham density functional theory (KS-DFT) where the exchange part in the Fock matrix (Eq. [9]) is at least partly replaced by a so-called exchange-correlation functional term. For both HF and DFT, Eq. [7] needs to be solved self-consistently, and accordingly, these methods are denoted as SCE methods. 

Two rate-determining steps occur in the iterative SCF procedure. The first is the formation of the Fock matrix, and the second is the solution of the pseudo-eigenvalue problem. The latter step is conventionally done as a diagonalization to solve the generalized eigenvalue problem (Eq. [7]), and thus, the computational effort of conventional SCF scales cubically with system size [0(M )j. 

The optimization of the Hartree-Fock spin orbitals in Eq. (21) is a nonlinear minimization problem. By recasting Eq. (22) as a generalized eigenvalue problem, the optimization may be accomplished by repeated solution of the pseudo-eigenvalue... 

We subdivide the system into two parts, one built from metal nd orbitals (d) and another composed of valence metal ( + l)s and ( + l)p and ligand functions (v). Then the eigenvalue problem (Equation 7) can be represented in the form of the pseudo-eigenvalue Equation 8, completely restricted to the d-subspace, with the explicit form of given by Equation 9. is a A(j x Nj matrix. Thus, for a d system, for instance, Nj = 45. (H j) are rectangular... 

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 solving the problem. A set of functions that is a solution to eq. (3.42) is called self-consistent field (SCF) orbitals. 

The wave number k = 2 k/X where A = wavelength. Because the numerical value of kh must be computed from an eigenvalue problem in the vertical z coordinate, equivalence of the eigenvalue k to the wave number 2tt/X requires a pseudo-horizontal boundary condition of periodicity given by A = 2tt/X and (p x + A, 2 ) = (p x, z). It is computationally efficient to normalize the eigenseries in (2.10a) according to... 

As an approximate alternative one does not directly include the doubles corrections in the principal propagator but corrects the RPA excitation energies, obtained by solving the RPA eigenvalue problem, with a non-iterative doubles correction. This approach is called doubles corrected random-phase approximation-RPA(D) (Christiansen et al, 1998a) and is based on pseudo-perturbation theory that was described in Section 3.13. 

Here all of the I roots, a, are positive and hence physically acceptable the I constants, tp are eigenvalues to be adjusted so as to make the G, agree with their specified values, Gflame systems therefore lead to difficult multiple eigenvalue problems. If, however, the concentrations of the intermediate chemical species or free radicals are nearly equal to their pseudo-steady state values throughout the flame, then solutions of the flame equations can be obtained. 

The issue of different energies in the denominator of (19.6) is a soluble problem. Consider the whole set of exact, positive-energy solutions, whose large- and pseudo-large-component vectors are and and whose eigenvalues are collected into a diagonal matrix E. The matrix equations are... 

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