Lowdin orthogonalization method

Firstly, the function (70) is invariant under a linear transformation of the m doubly occupied orbitals amongst themselves. A proof of this statement seems hardly necessary as, in the case m = N, equation (70) is equivalent to a Slater determinant, and this property of a determinant is well-known. The m orbitals m may therefore be orthogonalized amongst themselves by a linear transformation, without altering the total wavefunction. This, of course, may be done in several ways, by transforming to MOs for example, but perhaps the most convenient method is to employ the Lowdin symmetric orthogonalization method 73... 

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 Mulliken scheme suffers from all of the above, while the Lowdin method solves problems (1), (2) and (3). In the orthogonalized basis all off-diagonal elements are 0, and the diagonal elements are restricted to values between 0 and 2. 

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 Lowdin method is equivalent to a population analysis of the density matrix in the orthogonalized basis set (Section 13.2) formed by transforming the original set of functions by... 

Lowdin now makes a detour to discuss the method of Cl. He expands the wave function in the set of Slater determinants that can be formed from the one-electron basis, j/. He introduces the word complete Cl to emphasize that the expansion is, in principle, exact. We have since learned to use the expression full Cl in the case where the one-electron basis is finite (Lowdin made a sarcastic comment about our use of the word complete in the complete active space (CAS) concept at a Sanibel Conference in 1980). He computes the matrix elements between the Slater determinants without assuming orthogonality between the MOs. These are the famous Lowdin matrix elements, which generalized the Slater formulae to the nonorthogonal case. 

The formal properties of operator L eq 2.18 (known as the symplectic structure ) allow the introduction of a variational principle eq D3, " a scalar product (eq Bl), and ultimately to reduce the original non-Hermitian eigenvalue problem (eq 2.18) to the equivalent Hermitian problem which may be solved using standard numerical algorithms (Appendices B—E). For example, F is a Hermitian operator. Lowdin s symmetric orthogonalization procedure " " leads to the Hermitian eigenvalue problem as well (eq E5), which may be subsequently solved by Davidson s algorithm (Appendix E). The spectral transform Lanczos method developed by Ruhe and Ericsson is another example of such transformation. 

The analytical determination of the derivative dEtotldfir of the total energy Etot with respect to population n, of the r-th molecular orbital is a very complicated task in the case of methods like the BMV one for three reasons (a), those methods assume that the atomic orbital (AO) basis is non-orthogonal (b), they involve nonlinear expressions in the AO populations (c) the latter may have to be determined as Mulliken or Lowdin population, if they must have a physical significance [6]. The rest of this paper is devoted to the presentation of that derivation on a scheme having the essential features of the BMV scheme, but simplified to keep control of the relation between the symbols introduced and their physical significance. Before devoting ourselves to that derivation, however, we with to mention the reason why the MO occupation should be treated in certain problems as a continuous variable. 

The most important feature of the SINDOl method is that an orthogonalization transformation of the basis functions is taken into account exphcitly in solving the HF LCAO equations. The one-electron integral matrix H is transformed to Lowdin symmetrically orthogonahzed [225] AOs = S / x... 

The PRDDO method employs three mathematically equivalent basis sets. The first (p) consists of pure Slater orbitals. The second (( >) is a Schmidt orthogonalized, principal axes basis (see below). The third basis (X) consists of orthogonal atomic orbitals (OAOs), which are obtained from the 4> basis via a LOwdin symmetric orthogonalization ... 

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