Population analysis Lowdin

The Lowdin population analysis scheme was created to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme, which it does. It is different in that the atomic orbitals are first transformed into an orthogonal set, and the molecular orbital coefficients are transformed to give the representation of the wave function in this new basis. This is less often used since it requires more computational work to complete the orthogonalization and has been incorporated into fewer software packages. The results are still basis-set-dependent. 

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A simple and robust quantitative MO-type approach (as opposed to density approaches) is the ubiquitous Mulliken population analysis [40]. The key concept of this easily programmed and fast method is the distribution of electrons based on occupations of atomic orbitals. The atomic populations do not, however, include electrons from the overlap populations, which are divided exactly in the middle of the bonds, regardless of the bonding type and the electronegativity. As a consequence, differences of atom types are not properly accommodated and the populations per orbital can be larger than 2, which is a violation of the Pauli principle a simple remedy for this error is a Lowdin population analysis that... 

As with other schemes of partitioning the electron density in molecules, Mulliken population analysis is arbitrary and is strongly dependent on the particular basis set employed. However, the comparison of population analyses for a series of molecules is useful for a quantitative description of intramolecular interactions, chemical reactivity and structural information. In another approach, the Lowdin population analysis, the atomic orbitals are first transformed to an orthogonal set, as are the molecular orbital coefficients [Lowdin, 1970]. 

Lovasz-Pelikan index spectral indices (0 eigenvalues of the adjacency matrix) LOVIs = LOcal Vertex Invariants local invariants Lowdin population analysis quantum-chemical descriptors Lowest-Observed-Effect Level biological activity indices (0 toxicological indices) lowest unoccupied molecular orbital quantum-chemical descriptors lowest unoccupied molecular orbital energy quantum-chemical descriptors LUDI energy function scoring functions Lu index —> hyper-Wiener-type indices... 

The Lowdin population analysis [Lbwdin, 1970] is similar to that proposed by Mulliken, but the atomic orbitals are first transformed into an orthogonal set, as are the molecular orbital coefficients. 

Spin density calculations analyze the difference between each atom s population of a- and [3-electrons using various techniques. Mulliken population analysis140 is still often used for determining spin densities despite its deficiencies experienced sometimes when large basis sets are used. The Lowdin population analysis Scheme141 tries to circumvent some of the unreasonable orbital populations predicted by the Mulliken scheme however it is somewhat still basis set dependent. The natural bond orbital (NBO) methodology142 is a whole set of analysis techniques, including natural population analysis (NPA), which is less basis set dependent than the Mulliken scheme and often considered one of the most reliable methods for population analysis. 

Lowdin population analysis avoids the problem of negative populations or populations greater than 2. Some quantum chemists prefer the Lowdin approach to that of Mulliken as the charges are often closer to chemically intuitive values and are less sensitive to basis set. 

Now we may take any x. and for this value, construct the corresponding partition of N electronic charges into atoms. If x = 0 or 1, then one has the Miilliken population analysis, if X = 2 then we have the so-called Lowdin population analysis, etc. 

The diagonal elements of F are commonly used for a Lowdin population analysis... 

A Mulliken population analysis can be obtained from the diagonal elements of PS. Such a population analysis associates 1.53 electrons with 0i and 0.47 electrons with 02- The net charge is then -f 0.47 on He and +0.53 on H. The formal charge of +1 is thus predicted to be divided more or less equally between the two atoms. A Lowdin population analysis can be obtained from the primed matrices, that is, the matrices associated with the orthonormal basis 0J,. The number of electrons associated with the hydrogen orbital is... 

An improvement on Mulliken population analysis (MPA) is natural population analysis (NPA) [A. E. Reed, R. B. Weinstcx k, and F. Weinhold, J. Chem. Phys., 83, 735 (1985)], which uses ideas related to natural orbitals (Pilar, Section 10-7). Here, one first calculates a set of orthonormal natural atomic orbitals (NAOs) from the AO basis set Xr The NAOs are then used to compute a set of orthonormal natural bond orbitals (NBOs), where each occupied NBO is classifiable as a core, lone pair, or bond orbital. Using these NBOs, one carries out a population analysis. NPA net atomic charges show less basis-set dependence than those from Mulliken population analysis. Other methods of assigning net atomic charges are discussed in the next section. Still another method of population analysis that yields net atomic charges is Lowdin population analysis (Cramer, Section 9.1.3.2). 

Z Ij is the effective nuclear charge of atom M. The corrections are implemented in a self-consistent manner, since the net charges q/f" are calculated from the cluster density matrix P using a Lowdin population analysis. The electrostatic interactions of the ion M with the ions in its WSC must be subtracted from the Madelung potential rnD> because these interactions are aheady considered in the unmodified Fock matrix F P). To avoid double counting of interactions the factor 1/2 is introduced in (6.75). 

This implementation requires simple modifications in the property part of the CRYSTAL program [23]. A Lowdin population analysis is introduced for self-consistent DM and the bond-order sums are calculated for atoms of the crystal. The lattice summation in (6.76) is made over the same part of the lattice that has been used in the integrals calculation for the self-consistent procedure (the lattice summation field is defined by the most severe tolerance used in the two-electron exchange integrals calculation). ... 

It was shown [602] that the calculations made using a valence-atomic basis without polarizing functions in the Lowdin population analysis agree better with the expected values of atomic valencies. The population analysis by Mulliken made in a nonorthogonal basis was found to be less sensitive to the inclusion of polarizing functions into the... 

The numerical values of the indices dependence on the basis-set choice requires additional investigation. It is evident that use of Lowdin atomic populations instead of Mulliken ones can change the localization of WFs, especially in the case when diffuse AOs are included in the basis set. From this point of view the WTAOs use is preferable. As was demonstrated above, the numerical values of local properties are close in Mulliken and Lowdin population analysis made with WTAOs. 

LCAOM(L) - traditional Mulliken (Lowdin) population analysis with basis set used in LCAO calculations WTAOM(L) - Mulliken (Lowdin) population analysis with WTAO basis... 

Many properties may be evaluated, among them atomic charges (derived from MuUiken and Lowdin population analysis, natural population analysis (Read and Weinhold 1983 Read et al. 1985), and electrostatic potentials), polarizabilities, hyperpolarizabUities, and magnetizabUities. 

This is precisely the leading idea of Lowdin population analysis. It can be observed that the mathematical use of the trace of P S, i.e., Tr(P S) = Tr(P S -"S") = Tr(S"P S -"), yields an infinite number of possibilities, among which the approaches of Mulliken and Ldwdin are particular cases. In... 

London Atomic Orbitals (LAO), for calculating magnetic properties, 252 London force, 19 Lone pairs, in force fields, 23 Lorentz transformation, 204 Lowdin population analysis, 218 Lowest Unoccupied Molecular Orbital (LUMO), 348... 

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