Lowdin orbitals

In order to preserve the invariance of charge distributions under rotation of the local coordinate axes of each atom, the integrals K% eSPq and (pp qq) are assumed to be independent of the azimuthal quantum number of atomic orbitals, i.e. the same value is used for any 2 s and 2 p orbitals. Finally, it should be noted that in the case of a electrons the zero-differential-overlap approximation cannot be justified as completely as for n electrons by arguing about orthogonalized Lowdin orbitals, because the expression of the S la matrix cannot be limited to first-order terms 70,71,72). 

In the Lowdin approach to population analysis [Ldwdin 1970 Cusachs and Politzer 1968] the atomic orbitals are transformed to an orthogonal set, along with the molecular orbital coefficients. The transformed orbitals in the orthogonal set are given by ... 

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. 

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. 

In the case of the second excerpt I think I can safely say that Lowdin is wrong. The simple energy rule regarding the order of filling of orbitals in neutral atoms has now entered every textbook of chemistry, although his statement may have been partly true in 1969 when he wrote his article.1 Although Lowdin can be excused for not knowing what was in chemistry textbooks I think it is also safe to assume that he is correct in his main claim that this important rule has not been derived. Nor as I have claimed in a number of brief articles has the rule been derived to this day (Scerri, 1998). 

But I want to return to my claim that quantum mechanics does not really explain the fact that the third row contains 18 elements to take one example. The development of the first of the period from potassium to krypton is not due to the successive filling of 3s, 3p and 3d electrons but due to the filling of 4s, 3d and 4p. It just so happens that both of these sets of orbitals are filled by a total of 18 electrons. This coincidence is what gives the common explanation its apparent credence in this and later periods of the periodic table. As a consequence the explanation for the form of the periodic system in terms of how the quantum numbers are related is semi-empirical, since the order of orbital filling is obtained form experimental data. This is really the essence of Lowdin s quoted remark about the (n + , n) rule. 

As a result of this way of counting nodes the 4s orbital has a lower total number of nodes, that is, 4 when compared with 5 in the case of the 3d orbital. Moreover, this order agrees with the experimentally observed order whereby 4s has lower energy than 3d.10 However, whether this is a satisfactory first principles explanation of the n + t rule, which meets the Lowdin challenge, is something that seems rather unlikely given the ad hoc nature of the manner in which nodes have been counted. 

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. 

Calculations of the ground state of He (Shull and Lowdin 1958) have given the occupation numbers for the first natural spin orbitals as shown in Table V. These results show that the... 

TABLE V. Occupation Numbers for the First Natural Spin Orbitals in the Ground State of the He atom (Shull and Lowdin... 

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 Shull, H., Phys. Rev. 101, 1730, Natural orbitals in the quantum theory of two-electron systems/ ... 

Hirschfelder, J. O., and Lowdin, P.-O., "Long range interaction of two Is hydrogen atoms expressed in terms of natural spin-orbitals."... 

Molecular Orbitals in Chemistry, Physics, and Biology, p. 513 (ed. by P.-O. Lowdin and B. Pullman). New York Academic Press 1964. 

As shown by P.-O. Lowdin,40 the complete information content of y can be obtained from its eigenorbitals, the natural orbitals 9t, and the corresponding eigenvalues n,... 

The new orthogonal hybrids have 60% d character (sdL5 hybridization) as a result of the Lowdin procedure. However, because these expressions involve three atomic orbitals, there must be one other hybrid (in addition to the two bonding hybrids) affected by the orthogonalization transformation. This hybrid, denoted n(h), belongs... 

We begin by using only the linear degrees of freedom contained in U to optimise our model and numerical approximations. We shall see that a careful consideration of these linear transformations suggests a natural generalisation of the usual AO basis in a way which enables us to use some of the conclusions of earlier sections where we discussed molecular symmetry. For a given set of non-orthogonal orbitals it is well-known from the work of Lowdin (14) that the simplest solution to equation (34),... 

Overlapping Ion Model. The ground-state wave function for an individual electron in an ionic crystal has been discussed by Lowdin (24). To explain the macroscopic properties of the alkali halides, Lowdin has introduced the symmetrical orthogonaliz tion technique. He has shown that an atomic orbital, x//, in an alkali halide can be given by... 

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