Lowdin rules

As was shown in chapter three we can compute the transition densities from the Cl coefficients of the two states and the Cl coupling coefficients. Matrix elements of two-electron operators can be obtained using similar expresssions involving the second order transition density matrix. This is the simple formalism we use when the two electronic states are given in terms of a common orthonormal MO basis. But what happens if the two states are represented in two different MO bases, which are then in general not oithonormal We can understand that if we realize that equation (5 8) can be derived from the Slater-Lowdin rules for matrix elements between Slater determinants. In order to be a little more specific we expand the states i and j ... 

Assume for a moment that the expansion was made in a Slater determinant basis, and ignore the spin summation.The coupling coefficients are then given by the Slater-Lowdin rules, which in general are complicated expressions, but... 

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... 

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),... 

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. 

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. 

The evaluation of the energy integrals involving determinants of non-orthonormal orbitals were first systematically derived by P-0. Ldwdin and are known as Lowdin s Rules. 

Several points need to be made about these principles.The first principle does not, in fact, refer to the ordering of energies of atomic orbitals. What it really refers to is the order offilhng of the various orbitals. These are related but separate issues. But there is more involved in the occupation of orbitals than their individual energies, as discussed further below. The n + /rule has not yet been derived from the principles of quantum mechanics. This failure has been described as one of the outstanding problems in quantum mechanics by the leading quantum chemist Per-Olav Lowdin. ... 

Lowdin has expressed his views on the + /rule in P-O. Lowdin, Some Comments on the Periodic System of the Elements, International Journal of Quantum Chemistry, 3 Suppl., 331-334,1969. 

As mentioned in note 18, this problem has been recognized by some leading quantum chemists, such as Lowdin. Several attempts to solve the problem have been published. See, e.g.,V. Ostrovsky, What and How Physics Contributes to Understanding the Periodic Law, Foundations of Chemistry, 3,145—182,2001. Readers may also be interested in the present authors chapter in a book based on a recent international conference on the Periodic Table E.R. Scerri,The Best Representation of the Periodic System The Role of the n + / Rule and the Concept of an Element as a Basic Substance, in D. Rouvray, R.B. King (eds.). The PeriodicTable Into the 21st Century, Science Studies Press, Bristol, 2004,143—160. 

The commutation rules for operators over the nonorthogonal set % can be derived in a straightforward manner by substituting this inverse transformation and utilizing the commutation rules in the Lowdin basis ... 

As LOwdin has written, it is perhaps remarkable that, in axiomatic quantum theory, the simple energy rule (order of filling of orbitals) has not yet been derived from first principles . P. O. Ldwdin (1969), Some Comments on the Periodic System of the Elements , International Journal of Quantum Chemistry IIIS, 331-334. Neither has this situation changed since Ldwdin wrote these words. 

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