Lowdin

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. 

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... 

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 sixth article in this collection takes up the story from where paper 4 left off. If the n + (Madelung) rule can be fully reduced, then it might rightly be claimed that the periodic table reduces fully to quantum mechanics. This is a question that has been asked in a much-quoted paper by Per-Olov Lowdin, the influential quantum chemist who for many years led the Quantum Chemistry project at the University of Florida. 

The paper resulted from an invitation to contribute to what eventually became a three-volume dedication to the memory of Lowdin. In addition to addressing the question of whether the n + rule has been derived, I used this opportunity to explore the reduction of the periodic table in more general terms. 

The final paper in this collection is very recent and appeared in the International Journal of Quantum Chemistry where articles on Madelung s rule have previously been published. It was in this journal that its founder Per Olov Lowdin first drew attention to the fact that the rule had not yet been derived from quantum mechanics.24 More recently Allen and Knight published what they claimed provided just such a long-awaited derivation.25... 

P.-0. Lowdin, Some Comments on the Periodic System of the Elements, International Journal of Quantum Chemistry, (Symposium) 11 IS, 331-334, 1969. 

L. C. Allen, E. T. Knight, The Lowdin Challenge, International Journal of Quantum Chemistry, 90 80-88, 2000. 

W. H. E. Schwarz, Towards a Physical Explanation of the Periodic Table (PT) of Chemical Elements, in Fundamental World of Quantum Chemistry A Tribute to Per-Olov Lowdin, Vol. 3, E. Brandas, E. Kryachko (eds.), Springer, Dordrecht, pp. 645-669, 2004. Also see S.-G. Wang, W. H. E. Schwarz, Icon of Chemistry The Periodic System of Chemical Elements in the New Century, Angewandte Chemie International Edition, 2009 (in press). 

Lowdin, P-O. 1969. Some comments on the periodic system of elements. International Journal of Quantum Chemistry. Symposium 3 331-334. 

The discrepancy between the two sequences of numbers representing the closing of shells and the closing of periods occurs, as is well known, due to the fact that the shells are not sequentially filled. Instead, the sequence of filling follows the so-called Madelung rule, whereby the lowest sum of the first two quantum numbers, n + 1, is preferentially occupied. As the eminent quantum chemist Lowdin (among others) has pointed out, this filling order has never been derived from quantum mechanics (2),... 

LOWDIN S REMARKS ON THE AUFBAU PRINCIPLE AND A PHILOSOPHER S VIEW OF AB INITIO QUANTUM CHEMISTRY. 

It is indeed a great honor to be invited to contribute to this memorial volume. I should say from the outset that I never met Lowdin but nevertheless feel rather familiar with at least part of his wide-ranging writing. In 1986 I undertook what I believe may have been the first PhD thesis in the new field of philosophy of chemistry. My topic was the question of the reduction of chemistry to quantum mechanics. Not surprisingly this interest very soon brought me to the work of Lowdin and in particular his analysis of rigorous error bounds in ab initio calculations (Lowdin, 1965). 

I later discovered a short article in which Lowdin made some interesting remarks that resonated with me (Lowdin, 1969). 

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). 

The examination of this idea has subsequently formed an integral part of my research in the philosophy of chemistry. This has also led to a certain amount of disagreement with other authors who appear to interpret Lowdin s remark in a somewhat different manner (Ostrovsky, 2001). I now deeply regret not having contacted Lowdin directly in order to seek his own clarification. In the present contribution I intend to revisit this question and to take the opportunity to respond to some critics as well as hopefully injecting some new ideas into the discussion. 

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. 

Lowdin, P-O., Studies in Perturbation Theory. X. Bounds to Energy Eigenvalues in Perturbation Theory Ground State, Physical Review, 1965 139A 357-364. 

One of the authors to ask whether a theoretical derivation of the n + rule might be found was the late Per Olov Lowdin, who wrote... 

There have been a number of attempts to meet the "Lowdin challenge," as it has been called. Allen and Knight published an explanation in the International Journal of Quantum Chemistry, which has turned out to be rather problematic as I have recently argued [25-27], In addition, Ostrovsky has published an account in which he claims to explain the n + ( rule, but this account is far from transparent, or convincing, at least to this author [28],... 

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. 

PER-OLOV LOWDIN, Quantum Chemistry Group, University of... 

---