Quantum Chemistry

a branch of theoretical chemistry that considers problems of structure and reactivity of chemical compounds as well as problems of bonding from the point of view of the concepts and methods of quantum mechanics. In principle, quantum mechanics permits calculation of the properties of atomic and molecular systems on the basis of the Schrödinger equation, Pauli exclusion principle, and universal physical constants. Various physical characteristics of the molecule (for example, energy, electric and magnetic dipole moments) may be obtained as eigenvalues of the operators of the corresponding quantities if the exact form of the wave function is known. However, it has yet proved impossible to obtain exact analytic solutions for the Schrödinger equation for systems containing two or more electrons. The use of functions with an extremely large number of variables has made it possible to obtain approximate solutions, which numerically approach the ideally exact solutions to any desired degree. Nevertheless, in spite of the use of modern electronic computers with operating speeds of hundreds of thousands, or even millions, of operations per second, such “direct” solutions of the Schrödinger equation have been achieved only for systems with several electrons, for example, for the molecules H₂ and LiH. Inasmuch as chemists are interested in systems containing tens or hundreds of electrons, simplifications become necessary. For this reason, various approximate quantum-mechanical theories have been proposed for the description of such systems. These theories have proved to be more or less satisfactory, depending on the nature of the problems under consideration: the valence bond theory, formulated in 1927 by W. Heitler and F. London in Germany and developed by J. Slater and L. Pauling in the USA; the crystal field theory, proposed by the German scientist H. Bethe in 1929 and subsequently developed by the American scientist J. Van Vleck (this theory was applied to chemical problems in the 1950’s as the ligand field theory after the studies of the British scientist L. Orgel and the Danish scientists C. Jorgensen and C. Ballhausen). The molecular orbital (MO) theory appeared toward the end of the 1920’s. Worked out by J. Lennard-Jones (Great Britain), R. Mulliken (USA), and F. Hund (Germany), this theory was subsequently developed by many other investigators.

These approximate theories have coexisted and even complemented each other for a long time. However, at the present time, the great successes achieved in the synthesis of molecules and in the determination of their structure and the broad development of computer technology have caused researchers to favor the MO theory. This has happened because only the MO theory has developed a universal language, which in principle is suitable for the description of any molecule with greatly varied structure and complexity. The MO theory includes the most general physical concepts of the electron structure of molecules and (which is no less important) makes use of the mathematical apparatus, which is highly suited to quantitative calculations using electronic computers.

The MO theory is based on the assumption that each electron in the molecule is located in the field of all of its atomic nuclei and remaining electrons. The atomic orbital (AO) theory, describing the electron structure of atoms, is included in the MO theory as a special case, in which the system contains only one atomic nucleus. Furthermore, the MO theory considers all chemical bonds as multicenter bonds (in accordance with the number of atomic nuclei in the molecule) and thus fully delocal-ized. From this point of view, every type of predominant localization of electron density in the vicinity of a certain portion of the atomic nuclei is an approximation, the validity of which must be verified in each concrete case. The ideas of W. Kossel regarding the generation of separated ions in chemical compounds (these ions being isoelectronic with the noble gases) or the views of G. Lewis (USA) concerning the two-center, two-electron chemical bonds (symbolically denoted by the valence line) are naturally included in the MO theory as certain special cases.

The MO theory is based on the one-electron approximation, in which each electron is considered as a quasi-independent particle described by its own wave function. Another frequently used approximation is the formulation of one-electron MO’s from the linear combination of AO’s (LCAO-MO approximation).

If the above approximations are accepted and if only the universal physical constants are used without the introduction of any experimental data (except perhaps the equilibrium inter-nuclear distances, which are being increasingly less used at the present time), then purely theoretical calculations (ab initio calculations) can be carried out according to the scheme of the self-consistent field (SCF) method of Hartree and Fok. Such SCF-LCAO-MO calculations have now become possible even for systems containing several tens of electrons. In this case, the main difficulties arise from the need to evaluate a large number of integrals. Although such calculations are unwieldy and expensive, the results obtained, in any case, are not always satisfactory from the quantitative point of view. This arises since, in spite of various improvements in the SCF scheme (for example, introduction of the configurational interaction and other methods of taking the electron correlation into account), researchers are ultimately limited by the possibilities of the one-electron LCAO-AO approximation.

In this connection, semiempirical quantum-mechanical calculations have received extensive development. These calculations likewise go back to the Schrödinger equation, but instead of calculating an enormous number (millions) of integrals, most of them are omitted (on the basis of their small order of magnitude), whereas the remaining ones are simplified. The loss of accuracy is compensated by the corresponding calibration of parameters, which are derived experimentally. Semiempirical calculations enjoy great popularity owing to the optimum combination of simplicity and precision in solving a variety of problems.

The calculations described above cannot be compared directly with purely theoretical (nonempirical) calculations, since both types offer different possibilities and, hence, different problems. Owing to the characteristics of the parameters used in the semiempirical approach, one should not expect to obtain a wave function that would satisfactorily describe different (let alone all) one-electron properties. This constitutes the basic difference between the semiempirical calculations and the nonempirical calculations, which are capable, at least in principle, of providing a universal wave function. For this reason, the power and attractiveness of semiempirical calculations lie not in the ability to provide quantitative information as such, but in the opportunity for interpreting the results obtained in terms of physicochemical concepts. Only this type of interpretation leads to a true understanding, since without it the calculations provide merely some quantitative characteristics of phenomena, which can be determined more reliably by experiments. It is this specific feature of semiempirical calculations, which makes them invaluable and permits them to compete successfully with completely nonempirical calculations, which are increasingly realizable owing to the continuing development of computer technology.

As far as the accuracy of semiempirical quantum-mechanical calculations is concerned, it depends (as in any semiempirical approach) more on the skillful calibration of parameters than on the theoretical validity of the computation scheme. Thus, if parameters are selected from the empirical spectra of certain molecules and then used for calculation of the optical spectra of related compounds, excellent agreement is easily obtained with experiment; however, such an approach is not of general value. For this reason, the primary problem in empirical calculations is not related to the determination of parameters in general but to the use of one group of parameters (for example, those derived from optical spectra) for the calculation of other molecular characteristics (for example, thermodynamic characteristics). Only then it becomes certain that one is dealing with physically meaningful quantities, which possess general significance and are useful in conceptual thinking.

In addition to quantitative and semiquantitative calculations, modern quantum chemistry also includes a large group of results derived from qualitative considerations. It is frequently possible to obtain very convincing information concerning the structure and properties of molecules without any cumbersome calculations by using various fundamental concepts that are based mainly on the consideration of symmetry.

Considerations of symmetry play an important role in quantum chemistry, since they make it possible to check the physical meaning of the results of the approximate analysis of multielectron systems. For example, by proceeding from the point group of symmetry of a molecule, it is possible to arrive at a single-valued solution to the problem of the orbital degeneracy of the electron levels, regardless of the selection of the calculated approximation. Knowledge of the degree of orbital degeneracy is frequently sufficient for inferences concerning many important molecular properties, such as ionization potentials, magnetism, and configurational stability. The principle of conservation of orbital symmetry is the basis for the modern approach to the mechanisms of concerted reactions (Woodward-Hofmann rules). The above principle may be ultimately derived from a general topological analysis of the bonding and antibonding lobes in the molecule.

It should be kept in mind that modern chemistry deals with millions of compounds and that its scientific foundations are not monolithic. In some cases, success is achieved by using merely qualitative considerations of quantum chemistry, whereas its entire arsenal becomes inadequate in other cases. For this reason, in appraising the current state of quantum chemistry, it is always possible to cite many examples of the strength as well as the weakness of modern quantum-chemical theory. Only one thing is clear, however. If the level of quantum chemistry studies has been previously judged on the basis of the technical complexity of the computational apparatus employed, the availability of electronic computers has moved the physicochemical meaning-fulness of the studies into the foreground. From the point of view of the internal interests of quantum chemistry, attempts at going beyond the boundaries of the one-electron approximation represent the most valuable studies. At the same time, the one-electron approximation still contains many unused opportunities for a variety of practical purposes in various branches of chemistry.