Lowdin method

The Lowdin method uses the mabix for analysis, and it is equivalent to a... 

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 Mulliken and Lowdin methods give different atomic charges, but mathematically... 

The tables below give some examples of atomic charges and bond orders calculated by various methods as a function of the basis set at the HF level of theory. It is evident that the Mulliken and Lowdin methods do not converge as the basis set is increased, and the values in general behave unpredictably. Especially the presence of diffuse functions lead to absurd behaviours, as the aug-cc-pVXZ basis sets illustrate for CH4. Note also that... 

The Mulliken and Lowdin methods give different atomic charges, but mathematically i e is nothing to indicate which of these partitionings gives the best result. There are some common problems with all population analyses based on partitioning the wave function in terms of basis functions. 

The Mulliken scheme suffers fram all of the above, while the Lowdin method solves... 

G and 6-31G basis sets [Ig], pp. 337-339). Other approaches to partitioning elecuons among orbitals and thus calculating charges and bond orders are the Lowdin method [167] and the natural atomic orbitals (NAO) population analysis of Weinhold [168]. 

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... 

The Mulliken and Lowdin methods are just particular examples of a whole family of population analysis using S" D S " matrices. The Mulliken and Ldwdin methods... 

The results of the calculations of atomic covalences are presented in Table 11.9. Again, the Mulliken and Lowdin methods demonstrate contradictory and obscure bulk-surface tendencies. For some atoms the values of the covalences, calculated according to the Mulliken scheme are close to the Lowdin ones, for the other they differ significantly. Thus, these coincidences of the Mulliken and Lowdin results can be regarded as casual. And since the nine-plane slab does not provide the bulk-like electron... 

For large molecules, computation time becomes a consideration. Orbital-based techniques, such as Mulliken, Lowdin, and NBO, take a negligible amount of CPU time relative to the time required to obtain the wave function. Techniques based on the charge distribution, such as AIM and ESP, require a sig-nihcant amount of CPU time. The GAPT method, which was not mentioned above, requires a second derivative evaluation, which can be prohibitively expensive. 

The Mulliken method is just one of a whole family of population analyses. Another commonly used is the Lowdin partitioning. The DS mabix product (eq. (9.4)) may be rewritten as... 

For two-electron systems (He, H2) the method with different orbitals for different electrons was thoroughly discussed at the Shelter Island Conference in 1951 (Kotani 1951, Taylor and Parr 1952, Mulliken 1952). A generalization of this method to many-electron systems has now been given (Lowdin 1954, 1955, Itoh and Yoshizumi 1955) and is called the method with different orbitals for different spins. 

In order to demonstrate the possibilities and limitations of this method, we will quote some recent results (Lowdin and R dei 1958) obtained by using a wave function of the type w2(l + ar12), where u is expressed either as a single exponential or as the sum of two exponentials ... 

Lowdin, P.-O., An Elementary Iteration-Variation Method for Solving the Schrodinger Equation, Technical Note from the Uppsala Quantum Chemistry Group, April 23, 1958. 

Lowdin, P.-O., Symposium on Molecular Physics atNikko, Japan, Maruzen, Tokyo, 1953, p. 13. A method of alternant molecular orbitals."... 

Lowdin, P.-O., Phys. Rev. 97, 1474, 1490, 1509, Quantum theory of many-particle systems. I. Physical interpretations by means of density matrices, natural spin-orbitals and convergence problems in the method of configuration interaction. II. Study of the ordinary Hartree-Fock approximation. III. Extension of the Har-tree-Fock scheme to include degenerate systems and correlation effects. ... 

Lowdin, P.-O., and Redei, L., Combined use of the methods of superposition of configurations and correlation factor on the ground states of He-like ions."... 

Note that for large nuclear displacements, for example, distortions exceeding about 0.3-0.4 a.u., the method based on the Lowdin transform-inverse Lowdin transform technique is not recommended. However, for smaller distortions the method discussed above appears to provide a useful approximation. 

However, the behavior of the local descriptors of reachvity is somewhat different. The local quantities are calculated using the Lowdin-based method of population analysis [47], It was observed that when the electric held is applied toward the more electronegative atom of the systems HF and HCN, the value of its nucleophilicity (CFF for electrophilic attack) decreases with increasing held strength (Figure 25.2c). The variation of FF can be well supported with the variation in the induced dipole moment, given in Table 25.1, of the species in... 

---