Orthonormalization Lowdin

The Lowdin orthonormalized basis of atomic orbitals A is obtained by the matrix transformation... 

At this point, if one were to consider only the axial properties of the system, the analysis would be equivalent to the extended onedimensional analysis discussed above. The energies of the bands—now effectively one dimensional—are the same as given by eq (44) for the simple collection of harmonic wells above. The full analysis, as can be seen by considering the consequences of Lowdin orthonormalization, mixes character of the x and y coordinates into that of the axial modes, although the smallness of the off-diagonal matrix elements essentially preserves the block-diagonal character of the problem. 

The overlap integrals can be used to construct the matrix to carry out Lowdin orthonormalization [42-44]. The last contribution, C of the table, is probably not as important as the first two. For simplicity, therefore, it is reasonable to consider the current due to the first two terms. The problem then reduces to one analogous to the electron transfer theory [69,70]. The transfer matrix element needed in the expression for the current is... 

Solution Let us first choose a reference orthonormal set (j>, jp) to be used consistently in displaying the various matrices and vectors under discussion. For simplicity, we choose (pi, (p2 to be the ( Lowdin-orthogonalized ) functions that are closest to xa and Xb >n the mean-squared-deviation sense. The non-orthogonal functions xa and Xb (with(xJXb) = S ) can then be expressed in terms of reference orthonormal functions as... 

The best NSOs are those that minimize the electronic energy subject to orthonormality constraints (4), and hence satisfy Lowdin s Eqs. (45) and (46). For the energy functional, Eq. (91), these equations become the spatial orbital Euler... 

There are numerous ways in which to orthonormalize a basis. Here we choose to employ the symmetric orthonormalization procedure described by Lowdin (51), which has the benefit over other orthogonality procedures that the new basis is as close as possible, in a least squares sense, to the original basis (52)... 

To alleviate a number of these problems, Lowdin proposed that population analysis not be carried out until the AO basis functions tp were transformed into an orthonormal set of basis functions / using a symmetric orthogonalization scheme (Lowdin 1970 Cusachs and Politzer 1968)... 

As was shown in chapter three we can compute the transition densities from the Cl coefficients of the two states and the Cl coupling coefficients. Matrix elements of two-electron operators can be obtained using similar expresssions involving the second order transition density matrix. This is the simple formalism we use when the two electronic states are given in terms of a common orthonormal MO basis. But what happens if the two states are represented in two different MO bases, which are then in general not oithonormal We can understand that if we realize that equation (5 8) can be derived from the Slater-Lowdin rules for matrix elements between Slater determinants. In order to be a little more specific we expand the states i and j ... 

The precise connection with finite dimensional matrix formulas obtains simply from Lowdin s inner and outer projections [21, 22], see more below, or equivalently from the corresponding Hylleraas-Lippmann-Schwinger-type variational principles [24, 25]. For instance, if we restrict our operator representations to an n-dimensional linear manifold (orthonormal for simplicity) defined by... 

The spin-orbitals 17,) iGA + f form a basis set for the supersystem A —B. One possible procedure of realizing the transformation (218) as well as (219) is the Lowdin symmetric orthonormalization method137. ... 

This new Hamiltonian is with respect to an orthonormal basis. If the square-root is that of Lowdin orthogonalization [ 117,118], then the new basis is [117,119] as similar as possible (in a well-defined mathematical sense) to the initial basis subject to the constraint of orthogonality. 

Equation (5-9) can be regarded as a canonical (Lowdin) orthonormalisation of the set of vectors 0 , or equivalently as a polar decomposition of the operator 3l (J0rgensen 9) Thus the Schrodinger equation for the n-electron Hamiltonian, H, Eq. (2-2), can always be formally transformed to the eigenvalue problem (5-10 a) for the effective Hamiltonian,, acting in the subspace S sparmed by a finite set of orthonormal vectors 0 the ligand field Hamiltonian (1-5) must therefore be an approximation to this object. 

Lowdin s symmetric orthogonalization method is an often employed technique for the generation of orthonormal molecular basis sets. Since within most LCAO MO methods, density matrices are determined by AO basis set coefficients, and idempotency of density matrices is a property easily controlled on an orthonormal basis, Lowdin s transforma-... 

The same Lowdin-type orthonormalization-deorthonormalization method can also be applied for the restoration of the idempotency of geometry-dependent macromolecular density matrices within the context of the recently introduced ADMA macromolecular density matrix technique, reviewed in the following text. 

If nuclear geometry variations are small, they can be treated the same way as in the ALDA method. In particular, the orthonormalization-deorthonormalization method described in the context of the SALDA method can be adapted for the ADMA technique. Employing two Lowdin-type transformations using relatively inexpensive macromolecular overlap matrices S( ) and S(K ) for two, slightly different nuclear geometries K and K, a satisfactory approximation P(/, [A ]) of the density matrix P(K ) can be obtained in terms of density matrix P(K). Motivated by the involvement of transformed overlap matrices, this technique is referred to as the SADMA method. ... 

The procedure to orthonormalize basis functions isn t unique. But the degrees of mixture between basis functions accompanied with the orthonormalization should be equalized for all basis functions to preserve characteristics of original ones. From this viewpoint, well-known Schmidt s method is unsuitable here. Lowdin developed the systematic way to orthogonalize basis functions and emphasized its usefulness for the bond analysis(7). He orthogonalized into, using the infinite series... 

The computations were carried out using programs based upon the Reeves algorithm s. This requires as input orthonormal spatial orbitals, and for this purpose the Lowdin procedure s was used, which minimizes the... 

Klein [65] notes that the des Cloizeaux transformation is a different formulation of the symmetric transformation of Lowdin [99] and uses this fact to prove that orthonormal functions in Hq which differs minimally (in a least squares sense) from the d true eigenfunctions of H. This is often referred to as a maximum... 

One sees from the functional dependence of the energy, eq (44), on trigonometric functions that the energies of the bands are of the standard form. Because of the one-dimensional nature of the problem and the 2x2 dimensions of the secular matrix, it is possible simply to include account of overlap automatically without specifically using methods of orthonormalization, such as Lowdin s method [42-44]. 

We should note here, that by construction, the matrix T is neither orthogonal nor normalized deviations from orthonormality are particularly pronounced in complexes with highly covalent metal-ligand bonds, as for example in Fe-S clusters. However, denoting the overlap matrix S = Tt-T, Lowdin s procedure (75) is used to obtain an orthogonal matrix C. Using this matrix U yf(p) is transformed into ///yr( pj of (5) as given by (76). 

In the background, we have the so-called Lowdin orthogonalization. The introduction of the orthonormal orbitals produces an effective changing of the Hamiltonian. 

The evaluation of the energy integrals involving determinants of non-orthonormal orbitals were first systematically derived by P-0. Ldwdin and are known as Lowdin s Rules. 

Because this orbital transformation is not properly unitary (the < are normalized and orthogonal only through first order in the X ), the set i must, in each iteration, be orthonormalized (by, for example, the Schmidt or Lowdin procedure). 

Like the familiar one-particle Sturmians of Shull and Lowdin, generalized Sturmians obey potential-weighted orthonormality relations. To see this, we move the term in Vq to the right-hand side of equation (3), multiply by a conjugate function in the basis set, and integrate over the coordinates. This gives us the relation... 

Lowdin, P.-O. (1993). Some remarks on the resemblance theorems associated with various orthonormalization procedures. Int. J. Quantum Chem. 48,225-232. 

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