The concepts of an atom and chemical bond in physics and chemistry: the role of approximations

Semina  Scientiarum Nr 10 2011 Andrzej Koleżyński The concepts of an atom and chemical bond in physics and chemistry: the role of approximations Apart from the theoretically driven evolution of the meaning of some of the important physical and chemical concepts, there is yet another often forgotten source of important problems arising in physics and chemistry. It results from the approximations that are necessary for any application of quantum mechanics to practical problems within these two areas of scientific inquiry. As a side effect of such approximations, the fundamental differences in the understanding of some concepts by a physicist and a chemist may appear. This problem is best demonstrated by the analysis of the concept of an atom and chemical bond as an example. The picture of atom and chemical bond in classical chemistry and quantum mechanics For chemists the atom consists of atomic core with positive effective charge (due to the screening effect of bare nuclear charge from valence electrons by the electrons occupying inner shells) and surrounding valence electrons (i.e. electrons belonging to outermost shells). The valence electrons are entirely responsible for the behavior and properties of particular atom as well as for the 32 | Andrzej Koleżyński properties of chemical bonds created with other atoms and these bonds are entities possessing well defined properties like bond length, bond order, bond strength or bond character. Such a view of atoms and bonds is extremely useful and fruitful in analyzing, explaining and predicting the behavior and properties of chemical species in many practical cases. What is more – such „chemical” picture of atoms and bonds prevails in chemical education and scientific research although a quite different view of these concepts emerges from extremely successful application of quantum mechanical formalism to many chemical problems. This fact is intriguing and it is worth to take a closer look at the picture emerging from this physical theory. In quantum mechanics, the atom is treated as a system of interacting particles – positively charged nucleus surrounded by negatively charged electrons. As a result, isolated atoms are electrically neutral. The state of such system and its evolution is described by respective state vector (wave function) obeying well known Schrödinger equation. The solution of this equation (with the assumption that we know the exact form of wave function) should in principle give us the entire knowledge about the system and its behavior. In practice it is not possible, since for every system of more than two particles (similarly as for famous three body problem formulated by Poincare), we cannot solve this equation analytically and the exact solution is not available (it is possible to obtain the analytical solution in case of hydrogen atom and singly ionized helium only). Almost all interesting systems, however, consists of heavier atoms or molecules and we are forced to make some approximations to make it possible to retrieve any useful information about the quantum system under study. We wish to focus on the two major approximations commonly used in computational physics and chemistry, namely adiabatic approximation (or Born-Oppenheimer approximation – the two names are often used interchangeably, despite small difference between these ap- The concepts of an atom. . .  33 proximations) and independent electrons approximation (self consistent field approach). Since the forces exerted on electrons and nuclei constituting a molecule or a crystal, due to their electric charge are of the same order of magnitude, the resulting changes in their momenta must be the same. Consequently, it is rational to expect that the actual momenta of the electrons and nuclei are of a similar magnitude. Thus, since the nuclei are much heavier than the electrons, the electrons must move much faster than the nuclei. On a typical time-scale of the nuclear motion, the electron cloud will adjust instantly to changes in the nuclear configuration (they will relax to the instantaneous ground-state spatial configuration). As a result, the solution of the time-independent Schrödinger equation for the system of interest may rest on the assumption that the nuclei are stationary. In such a case one can solve this equation independently for the electronic ground-state first, then calculate the energy of the system in that configuration and at the end solve for the nuclear motion. Such separation of electronic and nuclear motion is known as the Born-Oppenheimer approximation. The second approximation mentioned above, i.e. self consistent field approximation, is an approach to the problem of interacting electrons proposed by Hartree1 , Fock2 and Slater3 . The electrons are treated independently of each other, moving in and interacting with self consistent electrostatic field taken as the spatial average over the positions of all other electrons. Thus the N electron problem is effectively reduced to much simpler one of solving N independent one-electron problems. While ignoring electron correlation effects, such an approach simplifies the N-electron problem considerably and allows us to perform the calculations for more complex systems. Unfortunately, the correlation effects are not negligible in many cases and additional steps including in some way 1 D.R. Hartree, Proc. Cam. Phil. Soc., 24, 426 (1928). V. Fock, Z. Physik, 61, 126 (1930). 3 J.C. Slater, J. Chem. Phys., 1, 687 (1933). 2 34 | Andrzej Koleżyński the electron correlation potential into the Hamiltonian are necessary. This led to the development of two different sets of computational methods. The first one consists of Heitler-London4 derivatives as a strong correlation limit and the second one incorporates various molecular orbital methods (ab initio Hartree-Fock and DFT Kohn-Sham5 ,6 derivatives, where the correlation contribution can be included into the self consistent field). Although the detailed analysis of this problem is beyond the scope of this study, it should be mentioned that, according to these two sets of methods, two different roads towards a qualitative as well as a quantitative understanding of a chemical bond have been pursued. Based on the concept of the resonance and the resonance structures and essentially using Heitler-London approach, Pauling proposed the Valence Bond (VB) method. The second road, the Molecular Orbital approach, was first developed by Hund and Mulliken and extended later among others by Slater, Lennard-Jones and Hückel7 and is basically founded on the assumption of independent electrons. When formulated broadly enough, both these roads, lead to identical result, but in practice only few lowest order corrections can be calculated for these two limiting cases. While MO method has been (and in fact still is) very successful, since most chemical bonds are relatively weekly correlated, it simply fails in cases with strong cor (...truncated)