Orthogonality constraint

If there is more than one constraint, one additional multiplier term is added for each constraint. The optimization is then performed on the Lagrange function by requiring that the gradient with respect to the x- and A-variable(s) is equal to zero. In many cases the multipliers A can be given a physical interpretation at the end. In the variational treatment of an HF wave function (Section 3.3), the MO orthogonality constraints turn out to be MO energies, and the multiplier associated with normalization of the total Cl wave function (Section 4.2) becomes the total energy. 

In the present implementation, the unperturbed functions are not subject to any orthogonality constraint nor are required to diagonalize any model hamiltonian. This freedom yields a faster convergence of the variational expansion with the basis size and allows to obtain the phaseshift of the basis states without the analysis of their asymptotic behaviour. 

As a consequence, such examples show that the orthogonality relations (between vectors in different subspaces) alone, do not fix the S subspace. To do so, one would need some previous additional information on the basis which spans S and Sk That is to say, one would need to constrain the set of recovered O s to form a basis of the occupied subspace. This would then make additional orthogonality constraints within the subspace to take into account in the search for a K formula,... 

In order to solve the electronic structure problem for a single geometry, the energy should be minimized with respect to the coefficients (see Eq. (5)) subject to the orthogonality constraints. This leads to the eigenvalue equation ... 

In a recent paper, Glushkov and Levy [78] have presented an OPM algorithm that takes into account the necessary orthogonality constraints to lower states. One has to solve the problem... 

We now substitute (9) and (10) into (1) and (2). It is at once obvious that, unless we impose the so-called strong orthogonality constraint... 

In the first section of this work, in order to obtain maximum simplicity of interpretation, we chose to impose the strong orthogonality constraint on our model wave functions any two separate-group functions will be constrained by Eq. (11) ... 

In this section we examine this orthogonality constraint in order to evaluate its consequences for a theory of valence. Is it a substantive formal constraint on the type of model we may use does it restrict the type of physical phenomenon we can describe or is it simply a technical constraint on the method of calculation or what In fact we shall find that the strong orthogonality constraint is central to any orbital basis theory of molecular electronic structure. It has a bearing on the applicability of the model approximations we use, on the validity of most numerical approximations used within these models and (apart from the simplest MO model) has a dominant effect on the technical feasibility of the methods of solution of the equations generated by our models. Thus, it is of some importance to try to separate these various effects and attempt to evaluate them individually. 

We have already mentioned that the question of technical implementation is one of the main reasons for the use of the orthogonality constraint. If this constraint is not imposed the huge number of terms in the energy expression (the n problem) effectively prohibits quantitative calculations on any but the smallest molecular systems. The one-electron-group MO method is an exception to this general rule since... 

The resulting weights Wj and scores tj are stored as columns in the matrices W and T, respectively. Note that the matrix W differs now from the previous algorithms because the weights are directly related to X and not to the deflated matrices. Step 2 accounts for the orthogonality constraint of the scores tj to all previous... 

Earlier, it was mentioned that due to the orthogonality constraints of scores and loadings, as well as the variance-based criteria for their determination, it is rare that PCs and LVs obtained from a PC A or PLS model correspond to pure chemical or physical phenomena. However, if one can impose specihc constraints on the properties of the scores and or loadings, they can be rotated to a more physically meaningful form. The multivariate curve resolution (MCR) method attempts to do this for spectral data. 

On the other hand, no such invariance of G1 or HLSP functions occurs, so the orthogonality constraint has a real impact on the calculated energy. 

This is not due, however, to the orthogonality constraint between 4 f and 5 f orbitals, contrary to a general belief (private communication from J. P. Desclaux)... 

ORTHOGONALITY CONSTRAINTS FOR SINGLE DETERMINANTAL WAVE FUNCTIONS... 

Before deriving the Hartree-Fock equations for the excited state orbitals, we consider the orthogonality constraints imposed on these orbitals. 

We see the the Hartree-Fock wave functions, 4>o and i, do not, in general, satisfy orthogonality constraints analogous to those obeyed by the exact wave functions. However, we may impose constraints upon the Hartree-Fock function without loss of generality so that, for example. 

The imposition of the orthogonality constraint (14) to an approximate lower state wave function, such as the Hartree-Fock function, does not, in general yield an excited state energy which is an upper bound to the exact excited state energy. An upper bound to the excited state energy is obtained if we impose the additional constraint... 

It is well known that the orthogonality constraint for functions (17) and (18)... 

This restriction can be also written in terms of orthogonality constraints... 

We have presented a practical Hartree-Fock theory of atomic and molecular electronic structure for individual electronically excited states that does not involve the use of off-diagonal Lagrange multipliers. An easily implemented method for taking the orthogonality constraints into account (tocia) has been used to impose the orthogonality of the Hartree-Fock excited state wave function of interest to states of lower energy. 

The trivalent rare-earth crystal structure sequence from hep - Sm type -> La type -> fee, which is observed for both decreasing atomic number and increasing pressure, is also determined by the d-band occupancy. Figure 8.11(a) shows the self-consistent LDA energy bands of fee lanthanum as a function of the normalized atomic volume fi/Q0, where Q0 is the equilibrium atomic volume. We see that the bottom of the NFE sp band moves up rapidly in energy in the vicinity of the equilibrium atomic volume as the free electrons are compressed into the ion core region from where they are repelled by orthogonality constraints (cf eqn (7.29)). At the same time the d band widens, so that the number of d electrons increases under pressure... 

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