Skip to main content
main-content

Über dieses Buch

The aim of these notes is to offer a modern picture of the pertur­ bative approach to the calculation of intermolecular forces. The point of view taken is that a perturbative series truncated at a low order can provide a valuable way for ~valuating interaction energies, especial­ ly if one limits oneself to the case of intermediate- and long-range distances between the interacting partners. Although the situation corresponding to short distances is essen­ tially left out from our presentation, the problems which are within the range of the theory form a vast and important class: a large var­ iety of phenomena of matter, in fact, depends on the existence of in­ teractions among atoms or molecules, which over a substantial range of distances should be classified as weak in comparison to the interactions occurring inside atoms or molecules. We are aware of the omission of some topics, which in principle could have been included in our review. For instance, a very scarce at­ tention has been paid to the analysis of problems involving interacting partners in degenerate states, which is of particular relevance in the case of interactions between excited atoms (only a rather quick presen­ tation of the formal apparatus of degenerate perturbation theory is in­ cluded in Chap. III). Interactions involving the simultaneous presence of more than two atoms (or mOlecules) have not been considered, with the consequent non-necessity of considering nonadditive effects which characterize the general N-body problem.

Inhaltsverzeichnis

Frontmatter

I. Introduction

Abstract
The extremely important role of intermolecular forces in determining equilibrium and non-equilibrium properties and phenomena of matter is by now a well established point. In spite of this recognition and the awareness that quantum mechanics is the natural framework to be used for a proper understanding of intermolecular interactions, a fully rigorous approach for these is not yet well established, especially if one pursues the ambitious program of dealing with the whole range of separations between the interacting partners.
Paolo Arrighini

II. Symmetry: An Excursion through Its Formal Apparatus

Abstract
Symmetry and invariance properties permeate quantum mechanics in a fundamental way. The relevant symmetry, that of the Hamiltonian operator Ĥ, is to be traced back, as well known, to the existence of one or more operators commuting with Ĥ itself.
Paolo Arrighini

III. Symmetry-Adapted Perturbation Theory: A General Approach

Abstract
Perturbation theory plays a fundamental role in the determination of quantum mechanical states. Here we shall be concerned with stationary states, so our attention will be restricted only to the time-independent variant of such theory. Besides conventional treatments, described in any textbook on quantum mechanics, a renewed interest in general perturbation theory is to be recognized in more recent years [18,19], In a remarkable series of papers, entitled “Studies in Perturbation Theory” [10,20–32], Löwdin has emphasized the partitioning technique as a tool for solving secular equations associated with eigenvalue problems, putting in evidence its connection with various perturbation expansions.
Paolo Arrighini

IV. Why Symmetry-Adapted Perturbation Theories are Needed?

Abstract
Now that we have presented some symmetry-adapted perturbation schemes, we would like to understand the reasons that suggest or motivate their introduction for the ab-initio prediction of the interaction energy ΔE = E(ABC…) – (EA + EB + EC + ….) between a number of atomic or molecular partners A, B, C, …, in a given configuration.
Paolo Arrighini

V. Symmetry-Adapted Perturbation Theories at Low Orders: From H 2 + to the General Case

Abstract
In the previous chapter we have been able to conclude that a perturbation procedure founded on the “polarization approximation” is an inadequate tool for the evaluation of intermolecular interaction energies, its utility being possibly confined to “monomer” separations corresponding to very weakly interacting subsystems (long-range interactions).
Paolo Arrighini

VI. The Calculation of the 1-st Order Interaction Energy

Abstract
In order to pave the way for further developments, we begin this section by writing down an equivalent expression for the Coulombic portion E01 of the 1-st order perturbation energy, eq. (V.3.8), which is of value for both practical and conceptual reasons.
Paolo Arrighini

VII. The Second-Order Contribution to the Interaction Energy

Abstract
After having learnt in the last chapter how to evaluate the first-order HL energy, our attention will be turned to the next order contribution to the interaction energy. The importance of making allowance for the 2-nd order contribution to the interaction energy should have clearly been recognized on the basis of our analysis of the system H…H+ in Chap. V. The inspection of Table V.1, in fact, shows that the first-order HL energy ε (1) for the state 1sσg of H 2 + exhibits, as a function of the internuclear separation R, a minimum at the approximately correct distance, but deviates considerably from the exact interaction energy value over the whole R range, notably at rather large distances. In the case of the state 2pσu, whose behaviour is of particular relevance for us because it mimics the expected behaviour of interacting many-electron partners along the lowest potential energy surface, ε (1) fails to reproduce the presence of the (shallow) minimum at R ≃ 12.5 a.u. and the (weak) attractive region from such R value to larger distances, both of which characteristics require the 2-nd order polarization contribution to be taken into account.
Paolo Arrighini

VIII. Epilogue

Abstract
In the last two chapters the reader has been brought into contact with various methods by which the calculation of the 1-st and 2-nd order contributions to the interaction energy between two atoms or molecules has most frequently been carried out.
Paolo Arrighini

Backmatter

Weitere Informationen