Lowdin normalization

The N particle (and its p-reduced companions) representable density matrix T(p) can be defined as follows in the Lowdin normalization (x, is a combined space-spin coordinate)... 

In quantum chemistry file Lowdin normalization appears natural, but other choices are often made depending on the circumstances. 

This is the normalization adopted for example by McWeeny, 1967, 1992. One also finds 1/2 N(N-1) as prefactor, which corresponds to a normalization to the distinct number of pairs, e. g. Lowdin, 1959 or Parr and Yang, 1989. 

We employ Lowdin s normalization convention [82] in which the trace of the 1-RDM equals the number of electrons,... 

Further properties of the model arise from interpretations based on mutually orthogonal and normalized atomic orbitals. The common Lowdin symmetrically orthogonalized basis gives... 

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). 

Table 4 presents the proton affinities PA and deprotonation enthalpies DPE of thiouracils calculated at the B3LYP/6-31+G(d,p) computational level. Inspecting this Table, we find that first, the thio substitution of uracil systematically increases its PAs by 3-6 kcal/mol and second, it decreases the DPEs of uracil by 6-13 kcal/mol. As far as the base pair A thioU is considered, it particularly implies a lowering, on the one hand, of the potential well at the Sio atom of thiouracil corresponding to the proton transfer from the N8 atom of adenine to the Sio of thiouracil and, on the other one, a raising of the potential well at the N3 atom of thiouracil involved in the proton transfer from the N3-H bond to the Ni atom of adenine. Therefore, summarizing, the thio substitution of uracil is in a favor to the double proton transfer mechanism of the occurrence of the spontaneous point mutations proposed by Lowdin [19]. Table 4 also includes the PAs of the T2 tautomer of uracil and thiouracils. A comparison with the corresponding PAs of the parental normal nucleobases shows that their tautomerization to the T2 form is accompanied by an increase of the proton affinity by 20-23 kcal/mol. 

U. As U is neither normalized nor orthogonal, we introduce an overlap matrix S defined in Equation 15 and use Lowdin s symmetric orthogonalization scheme to... 

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). 

Having now completed our treatment of RSPT, let us return to the problem of finding a mechanism for explicitly computing, in any given order, only those terms in or /" > that arc size consistent. Recall that RSPT, as normally expressed, contains size-inconsistent terms that cancel when grouped together properly, but that nevertheless appear in the formal RSPT expression. Recall also that the size-inconsistent terms could be characterized by a factorization into products of two or more terms that did not share common summation indices. In MBPT, the formal cancellation of size-inconsistent terms in RSPT is carried out explicitly (Brueckner, 1955a,b Bartlett and Silver, 1975 Kelly, 1969 Lowdin, 1968 Brandow, 1977). Formally, we may thus write as... 

Fig. 3.2. The symmetric (or Lowdin s) orthogonalization of the normalized vectors u and v. The vectors are pushed off by the same angle in such a way as to assure u and v to become orthogonal.

It is well known that for singlet states, the UHF solutions with p pp are really possible when electron correlations become sufficiently strong. More exactly, the spin-polarized HF determinant < ) appear only under the non-singlet (triplet) instability which was defined by Cizek and Paldus in [63]. At the same time, solutions of the spin-projected variational HF method (the Lowdin s extended HF scheme) always exist [19]. The wave functions of this type will be signified by 4> ). This is usually defined by (apart from a normalization factor)... 

Before ending this section on relativity theory, we reflect on a remark made by Lowdin [27] regarding the perihelion motion of Mercury. Describing a gravitational approach within the consmiction of special relativity, he demonstrated that the perihelion moved but that the effect was only half the correct value. The problem here is the fundamental inconsistency between the force-, momentum and the energy laws, while the discrepancy for so-called normal distances are almost impossible to observe directly since (1 - x(r)) 1. However using the present method to the classical constant of motion... 

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