Lowdin orthogonalization

We have presented above the derivation of eqns 38 and 39 in great detail because it includes expressions of general utility, in particular the variation of the eigenvectors (eqns 7 and 24) of an MO problem after Lowdin orthogonalization and the resulting variation of the population matrix P. The generalization to a Hamiltonian more complicated than that of eqn 19 is possible by following step by step the above derivation. 

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... 

Two orthogonal orbitals that interact, yAB 0, are not completely localized they correspond to Lowdin orthogonal orbitals. Two symmetrical wavefunctions mix ( lA2 + B2) and AB)) giving rise to S0 and S2 while antisymmetric wavefunction A2 -B2) remains an eigenstate ... 

The Process that effects the transformation is called orthogonalization, since the result is to make the basis functions orthogonal. The favored orthogonalization procedure in computational chemistry, which I will now describe, is Lowdin orthogonalization (after the quantum chemist Per-Olov Lowdin). 

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. 

A virtue of the simplicity of this example is the fact that Lowdin orthogonalization [42-44,48] can be applied by hand. Let L+ stand for the eigenvalues of the upper left hand block of S... 

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

A pair-function calculation, where each electron pair is associated to a quasi-localized bond function built from Lowdin orthogonalized Is orbitals. This is equivalent to the treatment of ethane mentioned above 14) ... 

Thus the orthogonalizing Process of (4.99) (or rather one possible orthogonalization process, Lowdin orthogonalization) is the use of an orthogonalizing matrix to transform Hby pre- and postmultiplication (Eq. 102) into H. H satisfies the standard eigenvalue equation (Eq. (4.103)), so... 

In the case of the two-electron integrals, this provides the traditional justification for the NDDO approximation [42] Numerical studies have shown that the three- and four-center as well as certain two-center two-electron integrals are indeed small in a Lowdin-orthogonalized basis and may therefore be neglected. For the remaining two-center two-electron NDDO integrals, the orthogonalization transformation leads to a moderate... 

The solution of the matrix equation (41) may be unfolded through the Lowdin orthogonalization procedure [166, 167], involving the diagonalization of the overlap matrix via a given unitary matrix ((/), U) U) = (1), using the following procedure ... 

The new orbitals (pj, as linearly independent combinations of the occupied canonical orbitals (pj, span the space of the canonical occupied HF orbitals (pj). They are in general non-orthogonal, but we may apply the Lowdin orthogonalization procedure (symmetric orthogonalization, see Appendbt J, p. 977). 

As an illustration, consider the familiar problem of combining the two (Lowdin-orthogonalized) AO, A(r) and B(r) (say, two b orbitals centered on nuclei A and B, respectively), which contribute a single electron each to form the chemical bond A-B in this 2-AO model. The two basis functions x = (A, B) then form the bonding (p and antibonding MO combinations q> = ([Pg.167]

Using the variation condition one can derive a linear equation for Xq which, in turn, can be used to evaluate the first-order correction to the density matrix. This leads to an iterative scheme involving matrix manipulations in the local space. It is important to note that Kirtman s treatment describes not only inductive but delocalization effects, as well. A practical limitation is that one should work with orthogonal AO basis sets therefore the calculations have been done with semiempirical ZDO model Hamiltonians [91], or with explicitly Lowdin-orthogonalized basis sets in ab initio calculations. 

The coefficients Z , etc. determine the direction and s-character of the hybrids. The optimal calculation of these parameters is a difficult task and several procedures are known in the literature for this purpose [221, 222]. For the present purposes it is sufficient to invoke the chemical intuition and start with a set of hybrids with standard s-characters and directed along the bonds. Since such hybrids are not orthogonal on each atomic center (except in the case of some special bond angles), a consecutive Lowdin-orthogonalization allows one to readjust their 5-characters and bond directions. 

The most important change in MSINDO with respect to SINDOl is a modification of the approximate Lowdin orthogonalization of the basis [253]. Only the first-order terms in overlap are retained in (6.38). In SINDOl the expansion was to second order. If only first-order terms are taken into account, no transformation of the two-electron integrals is necessary. The one-electron integrals are transformed... 

Turning first to the convergence behavior of the crystal orbital calculations (see tables 1 and 2) we observe that our best value for the total energy per C2 is about 1 kcal/mol above the value extrapolated from cluster calculations. The main source of this discrepancy lies in the unavoidable use of a threshold value in the Lowdin orthogonalization in order to escape near linear dependence problems caused by the very small intercell distances in polyyne. The agreement between CO- calculations on polyyne and MO- calculations on oligoynes is, however, sufficiently close to reject a recent criticism by Teramae et al. (7) on earlier polyyne studies by the author (6). 

An interesting many-body approach was developed by Basilevsky and Berenfeld (1972) and by Kvasnicka et al. (1974). These authors turned to a Lowdin-orthogonalized basis and wrote the dimer Hamiltonian as ... 

Fig. 15.3. Change in the block structure of one-electron integrals in Lowdin basis upon R cx). The labels a, b refer to the original overlapping basis set, while i, k denote Lowdin-orthogonalized orbitals. The two bases are in a one-to-one correspondence...

The LCAO version of the Bom-Oppenheimer Hamiltonian has a standard form in terms of the creation and annihilation operators referring to the Lowdin-orthogonalized basis [16,17] ... 

Matrix above facilitates to construct the Lowdin-orthogonalized counterpart of vectors as... 

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