The influence of quantum theory on chemistry

The beginning

The influence of quantum theory on chemistry dates back to a single publication in 1927 by Heitler and London entitled “Interaction of neutral atoms and homeopolar bonding according to quantum mechanics”. ¹ It solved a mystery that had long been the subject of speculation and scientific imagination. What kind of forces cause some atoms to attract each other strongly, while others are only weakly bound or even repel each other? It was obvious that electrical (Coulomb) forces must be responsible for this, but strong attractive forces due to electrostatic attraction were only known between species with different charges, but not between neutral atoms, for which the classical laws of physics only result in a weak attraction.

The explanation could not be given by classical physics, but only by quantum theory, which was introduced by Heisenberg and Schödinger in 1925/1926. ^{2 , 3} Chemical bonding is a quantum theoretical phenomenon that requires the description of elementary particles as quantum objects. In the case of chemical bonding, this refers to electrons, which must be regarded as waves and not as particles. According to the dual description, elementary particles can be described as waves or as particles, and experimental observations can be better explained by one of the descriptions, depending on the phenomenon. Depending on the observation, electrons can be described either as a wave function ψ(r) or as a particle ρ(r), where r are the spatial coordinates. ⁴ However, the interaction between the electrons in a molecule, which may or may not lead to a chemical bond, can only be understood with the help of the wave function ψ(r). Once the chemical bond has formed, the resulting charge distribution can just as well be described by a wave function ψ(r) or as a particle distribution ρ(r). However, while a wave function clearly defines an associated charge distribution ψ(r) → ρ(r), the reverse is not true. A given charge distribution ρ(r) can result from an infinite number of wave functions ψ(r).

It is useful to point out the central information about the physical nature of the chemical bond, which was already identified in the work of Heitler and London with the statement: “A characteristic quantum mechanical oscillation phenomenon proves to be crucial for understanding the possible behaviors between neutral atoms… (original in German, translated by the author)”. ¹ The essential features are outlined in Scheme 1, which provides a mathematically simplified expression for bond formation in H₂. A more detailed discussion in a mathematically correct form can be found in a recent review article. ⁵

Scheme 1:

Schematic representation of the classical approach for calculating the bond energy of H₂ using the electronic charge distribution ρ(r) as starting point and the quantum theoretical approach which uses the wave function ψ(r) as fundamental entity.

Scheme 1 first of all depicts the classic approach for the interaction between two hydrogen atoms where the electrons are presented by their electronic charge distribution ρ(r) as the basic entity. The sum of the charge distributions of the two hydrogen atoms, ρ(H_a) and ρ(H_b), leads to an approximate charge distribution of the hydrogen molecule, ρ(H₂) (eq. 1). The associated curve for the purely electrostatic interaction energy E _elstat (eqs. 2 and 3) posseses only a shallow energy minimum of ∼10 kcal/mol at a rather long H-H distance. This is shown in Fig. 1 taken from the original work of Heitler and London where the electrostatic interaction energy is termed E₁₁. ¹

Fig. 1:

Copy of the figure by Heitler and London from their 1927 paper which shows the potential energy curves of dihydrogen calculated classically (E₁₁) and using the wave function Ψ (E_α and E_β). Reprinted with permission from ref. 1]. Copyright 1927 Springer Nature.

The quantum theoretical approach uses the wave function ψ(r) as the fundamental quantity of electrons. The relationship between the wave function ψ(r) and the charge density ρ(r) is given by the square, ρ(r) = [ψ(r)]² (eq. 4), which reversely gives ψ(r) = ±√ρ(r). Thus, the wave function ψ(r) carries a ± phase factor which is crucial for the understanding of the electronic structure of the molecule. The quantum theoretical ansatz for H₂ uses the wave functions of the hydrogen atoms to construct the molecular wave function ψ(H₂) (eq. 5), which gives two solutions. This is a fundamental difference to the classical approach, which results from a quantum theoretical treatment of the electrons. The equation for the quantum mechanical electronic charge distribution [ψ(H₂)]² (eq. 6) leads after insertion of the binomial (eq. 5) to two solutions (eq. 7), which represent the charge distributions of the bonding and antibonding states of H₂. A comparison of equations 7 with 1 reveals that the quantum theoretical equation 1 contains a completely new contribution called “interference” or “resonance”, which is either added or subtracted. The associated energy expressions in equations 8/9 exhibit the curves for E_α and E_β in Fig. 1. A classical description of the electron-electron interaction leads to repulsion due to Coulomb’s law, while a quantum theoretical description of the electron-electron interaction using wave functions leads to significant attraction or repulsion depending on the sign of the interference term. Covalent bonding and steric repulsion are a quantum theoretical interference phenomenon.

The history of chemical bonding is full of curious developments, and the epochal work of Heitler and London is an example of this. The two authors were postdocs in the group of Schrödinger, who was in Zürich at the time. They proposed to apply the newly formulated quantum-theoretical wave function theory to the question of chemical bonding. Their mentor agreed, but he was not interested in the subject and declined to co-author the paper. And so it was that one of the great mysteries of the natural sciences was solved by two young scientists as a little-noticed side project in a famous group.

It is very useful for the present topic to consider the time prior to the publication of Heitler and London in 1927. Quantum phenomena were known since 1900 when Max Planck reported the results of the black-body radiation. ⁶ The time from 1900 to 1925/26, before Heisenberg and Schrödinger introduced the description of matter in terms of quantum theoretical equations using two different mathematical methods, may be considered as the more or less fruitless period of attempts to combine classical physics with the experimental observations of quantum phenomena. A pivotal publication about the nature of the chemical bond appeared in 1916, when Lewis suggested that “the chemical bond is at all times and in all molecules merely a pair of electrons held jointly by two atoms”. ⁷ This was a bold suggestion, because there was no physical fundament for the electron-pair model. It is important to note that Lewis based his model on the molecules known at that time, which were mainly main group compounds of the first octal row of the periodic table of the elements. Another model was suggested at the same year by Kossel, ⁸ which is less known because it focused mainly on ionic bonding. Lewis was aware of the missing physical explanation of the electron-pair model and he speculated in his work that perhaps “Electric forces between particles which are very close together do not obey the simple law of inverse squares which holds at greater distances”. ⁷ In 1923 he published the book “Valence and the Structure of Atoms and Molecules” where he outlined and discussed the electron-pair model in detail. ⁹ His model was in the meatime elaborated by Irving Langmuir, who recognized the potential of the Lewis approach for explaining the structure of molecules and extended it significantly. ¹⁰ In fact, the term “covalence” and the octet rule are due to Langmuir ¹⁰ and Kossel ⁸ but not to Lewis. ⁹

Lewis discusses in his book the possibility of quantum effects on chemical bonding, where he openly states that he dislikes quantum theory which he calls “the entering wedge of scientific bolshevism”. ¹¹ But he also recognized the limits of the human intellect, which is rooted in classical physics. The final sentence of his book calls for “the necessity of maintaining an opening of mind; so that, when the solution of these problems, which now seem so baffling, is ultimately offered, its acceptance will not be retarded by the conventions and the inadequate mental abstractions of the past”. ¹² The solution came in 1927 with the paper by Heitler and London. Lewis published in 1933 his little known third paper solely devoted to the nature of the chemical bond. ¹³ It is yet another curiosity in the series of publications on the topic. The very long work does not have a single reference. It is the desperate attempt to save a classical explanation of the chemical bond when quantum theory had been recognized as the fundament of chemical bonding. The final sentence reads “ I am afraid I have not contributed much to this difficult subject of the incomplete shells, but I believe that the study of such systems is the greatest field for the future development of chemistry and spectroscopy alike”. ¹⁴

The early years

The paper by Heitler and London did not have an immediate impact on chemical research. This is understandable when one looks at the Nobel Prize which was awarded in 1927 to Heinrich Wieland for “his investigations of the constitution of the bile acids and related substances.” ¹⁵ Figure 2 shows the formulas of the molecules studied, which were quite complicated at the time. The experimental results were obtained by elaborate chemical reactions in which the chemical bonds were simply sketched by lines without knowing the nature of the interatomic interaction. This is in contrast to the Schrödinger equation, whose application to chemical bonds could only just explain the chemical bond in H₂. Why should chemical researchers bother with complicated mathematical formulas for simple diatomic molecules when chemists were able to synthesize complex molecules without knowledge of the physical nature of chemical bonding? The heuristic models of the past, in particular Lewis’ electron pair model, had proved sufficient to carry out chemical research with enormous complexity. Chemistry can be seen as a science that can be practiced with great success without knowing the fundamental forces of the discipline. The rather crude attempts to apply the Schrödinger equation to chemical problems did not give much hope that quantum chemistry could ever be useful for actual chemical research. With the advent of computers and the development of software, the situation changed significantly.

Fig. 2:

Left: Sketch of the molecular structure of a bile acid for which the Nobel Prize in chemistry was awarded in 1927 to Heinrich Wieland. Right: Description of the chemical bond in H₂ by Heitler and London using the Schrödinger equation.

In the early days of quantum chemistry, however, there were some fundamental findings that solved long-standing questions and formed the basis for an understanding of chemical bonding that is still of great importance today. One example is the X³Σ_g ⁻ triplet ground state of O₂, which had been a puzzle for Lewis and is a failure of his electron-pair model. Lennard-Jones showed in 1929 ¹⁶ that the triplet ground state with two unpaired electrons can be explained using an alternative method to the approach which was used by Heitler and London. ¹ The latter method was later named VB (Valence Bond) while Lennard-Jones used the MO (Molecular Orbital) approach introduced by Robert Mulliken ¹⁷ and Friedrich Hund. ¹⁸ Hund used MO theory to explain the electron spectra of diatomic molecules, for which he also introduced among others the now-commonly used symmetry symbols σ, π, δ indicating the change in sign of an MO when mirrored at one or two mirror planes, ¹⁹ which are often mistakingly identified with bond multiplicity. Mathematically, MO theory ²⁰ rests on the product of sums while VB theory ²¹ is based on the sum of products of atomic orbitals. The final results of MO and VB calculations are the same when all terms are considered, but the MO wave function Ψ ₀ ^MO is generally delocalized over the entire molecule and appears incompatible with Lewis’ localized electron pair model of chemical bonding, which was proved to be very powerful in explaining the structures, reactions and bonding situation of molecules. This is the reason why the VB method was favored by chemists in the early days of quantum chemistry, although the MO theory later proved to be more powerful in explaining chemical observations due to the symmetry of the wave function.

A very important work in the early days of quantum chemistry are the series of four papers by Erich Hückel about the electronic structure of aromatic compounds. ^{22 , 23} After reading the Heitler/London paper on H₂, Hückel became first interested in explaining the double bond in organic compounds and he published in 1930 two papers on the topic. ²⁴ This led him tackle the question why aromatic compounds possess an unusual stability toward addition reactions and why do they exhibit a puzzling dependency on the number of π electrons. Hückel tackled this question twice, with method one (now VB theory) and method two (now MO theory). The four papers published between 1931 and 1933 were the basis of understanding the peculiar bonding situation in aromatic compounds. Hückel also extended the study of benzene to related compounds which were partly not yet known at that time. The now often used “Hückel rule” 4n+2 was actually later introduced in 1951 by von Doering. ²⁵

Among the early pioneers of the emerging field of quantum chemistry was Hans Hellman, who in spite of his very short career due to his tragic death at the age of 35 made epochal contributions to the field which provided the basis of insights and methods that are relevant still today. ²⁶ Hellmann fled from Nazi Germany in 1934 to the Soviet Union and worked at the Karpow Institute in Moscow, before he became the victim of Stalin terror and was shot as alleged German spy in 1938. His name is known for the Hellman-Feynman theorem, which was independently suggested in 1939 by Feynman ²⁷ after it was introduced in 1937 by Hellmann when it was published in the very first text book (first in German, then in Russian) on quantum chemistry, which became little known due to his origin. ²⁸ It has recently been re-published. In his book Hellman introduced the so-called “Zusatzpotentiale”, which are nowadays known as pseudopotentials or effective core potentials. Hellmann was also the first to recognize that covalent bonding is caused by the decrease of the kinetic energy of the electrons and not by the decrease of the potential energy, which appeared to be the physical origin of the chemical bonding by inspecting the virial theorem at the equilibrium distance. ²⁹ After much controversy, it was shown in 1962 to be correct by Ruedenberg. ^{30 , 31} Although Hellman worked in quantum theory for only four years his ideas left deep footmarks in quantum chemistry.

Nearly all of the early work in quantum chemistry was carried out by physicist, not by chemists. The towering exception is Linus Pauling, who strongly influenced the understanding and the description of the chemical bond until today. Unlike most other scientists in the field, Pauling had an enormous knowledge of many areas of chemistry. He sensed the relevance of the upcoming quantum theory that was developed in Europe and he travelled and visited the central places during two extended periods. He spent some time with Nils Bohr at Copenhagen, Arnold Sommerfeld at Munich and Werner Heisenberg at Göttingen and he was at Zürich in the group of Schrödinger at the same time as Heitler and London. ³² When he returned to the USA he had a first-hand experience with the emerging quantum theory, which he applied and connected to chemical bonding. He realized the formally close connection between the electron-pair bonding model of Lewis and the VB approach of Heitler and London, and he set out to systematically built a bridge between quantum theory and chemistry. After publishing a series of papers on various topics of chemical bonding, he published in 1939 the book “The Nature of the Chemical Bond” ³³ which along with the textbook “General Chemistry: An introduction to Descriptive Chemistry and Modern Theory” ³⁴ in 1947 became the most important reference books of chemical bonding for a long time.

Pauling showed that it is possible to explain a wealth of experimental information of chemistry with the abstract mathematical equations of quantum chemistry and he introduced bonding models such as resonance structures which are still used. It was an amazing achievement, given the fact that quantum theory was still in an infant stage and computational chemistry using computers had not appeared. But where there is much light there is much shadow. Pauling based his quantum theoretical reasoning solely on VB method and he rejected all suggestions to use the MO approach. When Charles Coulson published in 1952 the textbook on quantum chemistry “Valence” ³⁵ where he compared the essential features of VB and MO theory, it received a very hostile review by Pauling. ^{36 .}Another early pioneer of the molecular electronic structure was Nevil Sidgwick, who appears to be forgotten in the literature. He wrote in 1927 ³⁷ and 1933 ³⁸ two textbooks about chemical bonding and valency, where he suggested using the symbol of an arrow for a bond A→B ³⁹ where both bonding electrons come from the same atom A., Such a bonding situation was introduced by Lewis and is the basis for the definition of acids and bases named after him, ⁴⁰ Sidgwick proposed the name “co-ordinate bond” ⁴¹ for what is now commonly called a dative bond. Pauling rejected the model of dative bonding saying, “We shall not find it convenient to make use of these names or of these symbols”. ⁴² A critical view on the influence of Linus Pauling on the present understanding of chemical bonding highlights his achievements as well as his shortcomings. ⁴³

The advent of computers and the triumph of MO theory

The application of quantum theoretical methods on interatomic interaction leads to formidable mathematical problems, which was recognized in 1929 by Dirac in his famous statement “The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that application of these laws leads to equations that are too complex to be solved.” ^{44 , 45} The statement was made before computers were available, and the development of computers and the associated software changed the situation dramatically. The strong influence of quantum theory on chemistry would not have been possible without computers and the associated programs for solving the Schrödinger equation.

Much of the early work focussed on approximate solutions to the Schrödinger equation based on MO theory. The starting point of MO calculations is the Hartree–Fock method, where the electrons are treated in the mean field of all the other electrons. It was independently developed by Hartree in 1928 ⁴⁶ and Fock in 1930 ⁴⁷ who showed that the Hartree method does not consider the principle of antisymmetry of the wave function. This was also reported in the same year by Slater, ⁴⁸ another pioneer of quantum chemistry, and the name Hartree-Fock-Slater is sometimes referred to. The coupled integro-differential calculations are usually approximated by the LCAO (Linear Combination of Atomic Orbitals) method suggested by Roothaan ⁴⁹ and Hall ⁵⁰ in 1951 due to the more efficient computer algorithms. Most quantum chemical progams make use of Gaussian orbitals as basis functions instead of the more physical Slater-type orbitals proposed by Boys in 1950, ⁵¹ because the integral evaluation for electron interactions is much faster.

Early methods of MO theory with empirically fitted integral parameters were the HMO (Hückel MO) method ⁵² and the PPP (Pariser-Parr-Pople) approach, ⁵³ which considered only π electrons. An important extension was the EHT (Extended Hückel Theory) program developed by Roald Hoffman, which included also σ electrons. ⁵⁴ Further semiempirical methods are variants of the ZDO (Zero Differential Overlap) approach which neglected multicenter integrals of the Fock matrix and where other integrals were neglected or replaced by simple terms. Popular methods of that period are the CNDO (Complete Neglect of Differential Orbitals), INDO (Intemediate Neglect of Differential Orbitals) and NDDO (Neglect of Diatomic Differential Orbitals) which were introduced by Pople. ⁵⁵ In the late 1960s, the author turned to ab initio methods and he developed the now very popular series of Gaussian programs, the first version of which became available in 1970. ⁵⁶ The Gaussian program has been continuously developed to date and has become the leading ab initio program with new quantum chemical methods many of which introduced by Pople.

Very important variants of the semiempirical methods were the programs MINDO (Modified INDO) ⁵⁷ and MNDO (Modified NDDO) ⁵⁸ developed by Michael Dewar where parameters are part of the energy terms, which were optimized with regard to experimental values such as heat of formation, ionization potentials and dipole moments. The accuracy of the methods was often sufficiently high to be used as reference for synthetic chemistry. The program MNDO and later variants ⁵⁹ were the most often used quantum chemical program in the 1980s. A very convenient feature of the MNDO method was the ability to optimize molecular geometries using different versions of the Newton-Raphson algorithm. Another important step was the development of methods for calculating transition states of chemical reactions (number of imaginary frequencies 1) rather than equilibrium structures (number of imaginary frequencies 0) which was pioneered by McIver and Komornicki. ⁶⁰

Quantum chemical methods for calculating equilibrium geometries and transition states are nowadays routine. The ground breaking work in the field goes back to Peter Pulay, who derived the equations for the first and second derivaties of the molecular energy with respect to nuclear coordinates. ⁶¹ They form the mathematical fundament for all algorithms aiming at geometry optimizations. The derivation of mathematical equations and the casting into efficient algorithms of dedicated computer programs during the first decades after the appearance of computers were an integral part of quantum chemistry. The importance of quantum chemical methods was recognised early on by Ken Houk and by Paul Schleyer in collaboration with John Pople, whose systematic studies and application of computational methods to countless organic and later also inorganic topics led to a gradual acceptance of quantum chemistry by experimental chemists.

A very important role for the broader use of quantum chemical programs that were developed by different groups was played by the QCPE (Quantum Chemistry Program Exchange) center at Bloomington Indiana, which started operating in 1963. ⁶² Its purpose was to provide an inexpensive mechanism for theoretical chemists and other scientists to exchange software. Most of the computer programs were distributed as source code, so scientists, if they wanted to, could learn from or improve upon the inner workings of the algorithms. QCPE reached its zenith in the 1980s when computational chemistry was growing rapidly and becoming widely recognized by the scientific community. The service was convenient and much used by experts, students, and experimentalists who wanted to perform research calculations in the study of molecules. QCPE also played an educational role by conducting workshops and providing on-call help to countless beginners. The introduction of the Internet in the 1990s diminished the role of QCPE.

Along with the development of quantum chemical programs came the triumph of MO theory over the VB approach. It became soon apparent that the equations of MO theory could be programmed in computer algorithms that were much faster than VB methods. Geometry optimizations with the VB approach are drastically more time consuming than with MO programs. But there is also a conceptual reason why MO methods were recognized in the 1950s to be superior to the VB approach. This is the symmetry of the wave function, which provides information that is highly relevant for explaining the reaction mechanism of chemical reactions.

The person who first recognized the role of symmetry of particularly the HOMO (Highest Occupied Molecular Orbital) and the LUMO (Lowest Unoccupied Molecular Orbital) was Kenichi Fukui. ⁶³ He published his first work in the field in 1952 ⁶⁴ and systematically developed what became known as FMO (Frontier Molecular Orbital) model. ⁶⁵ The relevance of the FMO model was not immediately recognized, because it was introduced with complicated mathematical formulas. The situation changed in the 1960s when Woodward and Hoffmann suggested the orbital symmetry rules, which were shown to explain the reaction course of a variety of organic reactions. ⁶⁶ In particular the pericyclic reactions, whose mechanism has been an enigma in organic chemistry and were already termed as reactions without mechanism, could now easily be explained and predictions could be made for thermal as well as as photochemical reactions. Fukui and Hoffman shared the Nobel Prize in 1981. Woodward, who had already received a Nobel Prize for his synthetic work in 1965, deceased in 1979.

The spatial symmetry of molecular orbitals turned out to be a key feature for explaining all kinds of chemical reactions in organic well as inorganic chemistry. Several textbooks appeared in the 1960s and 1970s where MO theory was introduced in chemistry. ⁶⁷ Further developments of VB theory ⁶⁸ did not alter this development, because the advantages of MO theory became clearly apparent. The difficulty to connect MO theory with the electron-pair model of Lewis was removed in 1960 when Boys showed that the delocalized canonical MOs can be transferred via unitary transformation to localized orbitals, which closely resemble Lewis structures. ⁶⁹ MO wave functions are much more flexible than VB structures and CMOs (Canonical MOs) are just one set of solutions of the Hartree–Fock equations. On the other hand, the eigenvalues of CMOs may be used to explain the results of photoelectron spectroscopy, which became another area where MO theory turned out to be very useful. ⁷⁰ The connection between MO theory and Lewis structures has become routine nowadays with the help of the NBO (Natural Bond Orbital) method by Weinhold. ⁷¹ It is based on the concept of natural MOs which was introduced by Löwdin, another pioneer of quantum chemistry, in 1955. ⁷² Löwdin founded the International Summer Institutes (1958–1987) held in Sweden–Norway, and the International Winter Institutes (1960–1988) taking place in Florida, which decisively affected the careers of many of today’s leading quantum chemists. ⁷³ He also initiated the yearly Sanibel Symposia on Quantum Chemistry, which started in 1961 that still takes place today.

The results of quantum chemical methods that were published until the end of the 1970s were considered as qualitative or semi-quantitative at best and experimental data were considered as indicator of the error in the calculations. The principle accuracy of the computed values was known since in 1968 Kolos and Wolniewicz reported a theoretical value for the bond dissociation energy of H₂, ⁷⁴ which led Herzberg to reexamine the experimental value that was slightly corrected in agreement with the calculated result. ⁷⁵ But larger molecules appeared to be too complicated to be accurately calculated. The situation changed gradually when Henry (“Fritz”) Schaefer systematically studied the remaining error in the approximate solutions of the Schrödinger equation using newly developed CI (Configuration Interaction) techniques for estimating electron correlation. A famous case was the geometry and the singlet-triplet splitting of CH₂. Experimental studies had suggested that methylene is linear ⁷⁶ and that the ¹A₁ singlet state is 19 kcal/mol ⁷⁷ above the ³B₁ triplet state. In contrast, quantum chemical calculation predicted that CH₂ is bent with a bending angle of 135^{o 78} and that the singlet-triplet split is 11 ± 2 kcal/mol. ⁷⁹ The theoretical values were proven to correct by later studies which gave a bending angle of 136^{o 80} and a value data of 9.01 kcal/mol ⁸¹ for the singlet-triplet split, which perfectly agree with more sophisticated calculations that suggested 9.0 kcal/mol, ⁸² This was the beginning of the situation when theoretical results became accurate enough to critically examine experimentally derived values for increasingly larger molecules. It culminated in the introduction of the CCSD(T) method, ⁸³ which uses coupled-cluster theory ⁸⁴ with single and double excitations and perturbative estimate of triple excitation for the correlation energy, which is presently the gold standard for single-reference systems. At the same time atomic basis sets were developed for all atoms of the periodic systems of the elements. ⁸⁵ This development was only possible by the advent of ever more powerful computers and the introduction of sophisticated programs, which had a pivotal role for the impact of quantum chemistry on chemical research.

Futher developments and the advent of DFT

The development and introduction of quantum chemical methods, which followed the advent of computers, was not restricted to finding approximate solutions to the Schrödinger equation. A very important extension was the incorporation of relativistic effects on molecular properties and chemical reactions. Relativistic effects were initially considered as irrelevant for chemistry, which is due to another famous statement by Dirac: “The general theory of quantum mechanics is now almost complete, the imperfections that still remain being in connection with the exact fitting in of the theory with relativity ideas. These give rise to difficulties only when high-speed particles are involved, and are therefore of no importance in the consideration of atomic and molecular structure and ordinary chemical reactions, in which it is, indeed, usually sufficiently accurate if one neglects relativity variation of mass with velocity and assumes only Coulomb forces between the various electrons and atomic nuclei”. ^{43 , 44} This statement has proven to be wrong for heavy elements. The direct effect of relativity involves mainly the electrons in inner core orbitals of the heavier elements due to their high kinetic energy, but the alteration of the core orbital electrons does not leave the electrons in the outer core and finally the valence orbital electrons unaffected. The energy changes associated with chemical reactions are very small compared with the atomic total energies and thus, energy changes of the core electrons can have a strong influence on the energy of the valence orbitals. Relativistic effects become important for structures and properties of atoms of the fourth row of the periodic system of the elements (for the third-row atoms already for Cu) and the chemistry of heavier atoms cannot reliably be calculated without considering relativity.

Early pioneering work in relativistic quantum chemistry was carried out by Pekka Pyykkö and Kenneth Pitzer, who published in the 1970s ground breaking studies in the field. ^{86 , 87} Approximate solutions of the Dirac equations were developed and programs became available in the 1980s where the properties of heavy-atom molecules could be calculated with increasing accuracy. The introduction of ever more sophisticated relativistic methods in quantum chemistry is an ongoing field of research in computational chemistry. ⁸⁸ Molecules containing all atoms of the periodic table can nowadays be calculated with sufficient accuracy to complement experimental studies.

There was another important method development which greatly facilitates the calculation of heavy-atom molecules. The electrons in the core orbitals are hardly affected by chemical processes that occur in the valence shell. The replacement of the core orbital electrons by parametrized mathematical functions in PP (Pseudopotentials) or ECP (Effective Core Potentials) strongly reduces the computational costs of the calculations, because the number of explicitly calculated electrons remains the same for atoms in the same group of the periodic system. ⁸⁹ Pioneering work started in the 1950s in the group of Preuss ⁹⁰ using the early fundamental studies of Hellmann. ⁹¹ The accuracy of relativistic PP or ECP methods is usually sufficiently high for questions of synthetic chemistry. ⁹² Two- and four-component methods with various degree of approximation of the Dirac equation are now available for calculating molecules with atoms from all over the periodic system including superheavy elements.

The size of molecules which can be calculated with good accuracy was greatly extended when quantum chemical methods were combined with force-field approaches for those parts of the system whose interactions can be modelled by classical energy terms. QM/MM (Quantum Mechanics/Molecular Mechanics) methods were suggested in the 1970s ⁹³ and programs with suitable parameters were developed in the course of time, which made it possible to perform calculations on very large systems which give insight into the structure and reactivity of proteins. ⁹⁴ QM/MM methods have become an integral part of biochemical research. The Nobel Prize in chemistry was awarded in 2013 to Martin Karplus, Michael Levitt and Arieh Warshel for the development of multiscale models for complex chemical systems. ⁹⁵

The introduction of QM/MM methods was closely connected to the development of MD (Molecular Dynamics) approaches, which calculate the alteration of the molecular structure under the influence of temperature and pressure at a given time scale. ⁹⁶ MD methods are not confined to QM/MM approaches and there are various methods available for investigating the time evolution of molecular structures. Important alternatives which do not rely on empirical parameters are AIMD (Ab Initio MD) approaches ⁹⁷ such as the Car-Parinello method ⁹⁸ and BOMD (Born Oppenheimer MD) procedures.

Further important developments of quantum chemical methods included approaches for calculating the interaction of molecules with light and the influence of electric and magnetic fields on their electronic structure, which are useful for all kinds of spectroscopy. The most important method developments were approaches for calculating NMR (Nuclear Magnetic Resonance) spectra, which requires the derivative of the energy with respect to an external magnetic field. ⁹⁹ NMR spectroscopy and X-ray crstallography are the most important analytical tools for determining the structure of molecules. The development of theoretical methods and programs has significantly enhanced the relevance of quantum chemistry for this experimental research. The first methods for practical use were the IGLO (Individual Gauge for Localized Orbitals) approach of Kutzelnigg ¹⁰⁰ and the related LORG (Localized Orbitals/Local Origin) method by Hansen and Bouman. ¹⁰¹ More recent methods are based on the GIAO (Gauge Invariant Atomic Orbitals) approach. ¹⁰² Programs for calculating NMR chemical shifts have become a routine tool of quantum chemistry.

A major shift occurred during the 1990s when the wave function (WF) based methods, which are based on approximate solutions of the Schrödinger or Dirac equations, were no longer the dominant quantum chemical methods but were replaced by DFT (Density Functional Theory). DFT is also an approximate quantum chemical method, which uses the electron density ρ(r) instead of the wave function ψ(r) as starting point of the calculation of the energy. Density functional theory has its roots in the Thomas–Fermi model ¹⁰³ for the electronic structure of materials, which was useful for solid state systems but not accurate enough for molecules. The foundation for present DFT methods was provided by the Hohenberg-Kohn (HK) theorems. The first HK theorem demonstrates that the ground-state properties of a many-electron system such as the energy are uniquely determined by an electron density that depends on only three spatial coordinates. If the electron density ρ(r) is known, one obtains the energy of the system. The second HK theorem defines an energy functional for the system and proves that the ground-state electron density minimizes this energy functional. The HK theorem was further developed by Kohn and Sham to produce the working equations of present DFT methods where the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of noninteracting electrons moving in an effective potential. Mathematicaly the Hartree–Fock (HF) eq. (10) are replaced by the formally analogous Kohn-Sham (KS) eq. (11):

In the HF eq. (10), the Fock operator H ^HF acts on the one-electron orbitals φ ^HF to give the lowest energy orbitals of the system φ _i ^HF and the associated eigenvalues ε _i ^HF. Eq. (10) give the Hartree–Fock energy which lacks the contribution due to electron (Coulomb) correlation. In contrast, the Kohn–Sham operator H ^KS in eq. (11) has terms for the electron correlation and various other terms giving the Kohn–Sham orbitals φ _i ^KS which aim at the correct electron density ρ(r) that provides the total energy E inclusive of electron correlation. The holy grain of DFT is to find proper functions in the Kohn–Sham operator H ^KS, which leads to accurate values for E _pot.

The computational simplicity of KS-DFT and the development of improved functionals that provide numerical results accurate enough for many chemical problems have made DFT the dominant quantum chemical method over the last two decades. The Noble Prize in chemistry was awarded in 1998 to Walter Kohn for his founding of the density-functional theory along together with John A. Pople for his development of computational methods in quantum chemistry. ¹⁰⁶

Early attempts to estimate the correct electron density ρ(r) were based on the LDA (Local Density Approximation) which assumes that the density is the same everywhere. The accuracy of DFT methods was significantly improved by the GGA (Generalized Gradient Approximation) approach, where not only the density but also the gradient of the density is considered in order to account for the non-homogeneity of the true electron density. ¹⁰⁷ Two examples are the BP86 and BLYP functionals introduced in the 1980s, which provided reasonably accurate geometries and energies for molecules and which are still used these days. ¹⁰⁸ Both are examples of pure functionals where only DFT terms for the exchange correlation energy are employed. In contrast, hybrid functionals use a mixture of Hartree–Fock exchange and DFT exchange where the contributions of both terms are parameterized with respect to experimental values. A prominent example is B3LYP, which became the leading functional for some time. ¹⁰⁹

There is an ongoing development of new functionals, which are often particularly accurate for some classes of molecules. ¹¹⁰ The drawback of DFT methods is that the level of theory cannot be systematically improved like WF approaches. It requires the experience of the theoretician and comparison with other functionals to determine if the calculated results are trustworthy. A frequently employed approach uses DFT for geometry optimization and CCSD(T) energy calculation at the DFT optimized structures to obtain more accurate energies which indicate if the data provided by the functional is reliable.

The development of quantum chemical methods comprised not only methods for calculating geometries, energies and other observable quantities like spectroscopic values. An important ingredient of chemical research are models for describing the structure, reactivity and bonding of molecules, which serve as useful ordering scheme in the infinite complexity of chemical objects. Chemistry considers the physical world as composed of atoms and the molecular and solid state compounds. There are only ∼100 different atoms which built the basis of the chemical universe and the complexity comes from the chemical bonds which are formed between them. Heuristic bonding models helped as a guidance to make experimental findings accessible to scientific approaches.

The results of experimental studies and measurements may be considered as the fundament upon which the science of chemistry is built as a multifaced and ever increasing house where models provide rules and ordering schemes. They connect the different areas in the huge building in order to not get lost. Prior to quantum theory, heuristic bonding models had been developed which culminated in the electron-pair model of Lewis ⁷ and the closed-shell model of Kossel. ⁸ The usefulness of these models and their use until today suggests that quantum theoretical information is unknowingly hidden therein. The task of quantum chemists is not to dismiss the concepts but to create well defined quantum theoretically rooted definitions of models, which provide a more solid compass for orientation in the chemical universe. The numerical values, which are provided by these models are not observable quantities and they are somewhat fuzzy due to the used method. Chemistry may therefore be considered as a scientific discipline of fuzzy concepts, in so far as the characteristic features of chemical research comprises non-observable entities. They are based on strictly physical measurements, but the simple listing of measured values would lead to a mindless stamp collection without understanding of ordering principles and without help in the design of future experiments.

Chemically relevant information comes from the charge distribution in heteroatomic molecules, where some atoms carry more electronic charge than others. Pauling proposed in 1932 a useful scale of dimensional numbers for electronegativity, ¹¹¹ a concept proposed by Berzelius long before quantum theory came to the fore. ¹¹² It is still a widely used measure of the relative strength of atoms to polarise a chemical bond. Mulliken suggested in 1955 a population analysis where atomic partial charges and bond orders can be calculated using MO theory. ¹¹³ It turned out that the values significantly depend on the size of the basis set and other approaches for calculating partial charges were developed. The most important method is the NBO (Natural Bond Orbital) scheme, which was introduced by Weinhold in 1980 ¹¹⁴ and has been further developed until today. ⁷¹ The NBO method can be used in conjunction with WF or DFT calculations. It transforms the delocalized wave functions into two-center localized orbitals which closely resemble the Lewis model of chemical bonds and it provides information about the polarity of the bond as well as the hybridization of the atoms and their partial charges. There are also variants to account for multi-center bonds. ¹¹⁵ The NBO method is presently the most popular approach for the bonding analysis of chemical bonds.

Another important method for analyzing the electronic structure of molecules is the QTAIM (Quantum Theory of Atoms in Molecules) developed by Bader, which is based on a topological analysis of the electron density ρ(r) that divides it into atomic basins. ¹¹⁶ Using the first derivatives (gradient field) ∇ρ(r) and second derivatives (Laplacian) ∇²ρ(r) of the electron density, physically well-defined atomic regions and interatomic bond paths may be identified, which typify a molecular structure by the zero-flux surfaces that separate the atomic basins. The QTAIM method belongs to the group of real-space partitioning approaches where the charge density ρ(r) rather than the wave function ψ(r) is used for the bonding analysis. The IQA (Interacting Quantum Atoms) model, ¹¹⁷ is based on Bader’s QTAIM definition of separated atomic basins and provides information about the nature of the chemical bond. Another important topological approach is the ELF (Electron Localization Function) method of Becke and Edgecombe introduced in 1990. ¹¹⁸ ELF calculations provide a measure of the likelihood of finding an electron in the neighborhood space of a reference electron located at a given point and with the same spin. Physically, this measures the extent of spatial localization of the reference electron and provides a method for the mapping of electron pair probability in multielectronic systems. Finally, the responses of the electron density to external perturbations became an important offspring of DFT termed Conceptual DFT (CDFT). CDFT connects chemical reactivity with various descriptors deduced from derivatives of the energy and response functions. CDFT provides physically derived mathematical expressions for electronegativity, chemical hardness, Fukui functions and numerous other descriptors, which are useful for understanding molecular structures and reactivity. The starting point of CDFT may be traced back to a work by Parr and coworkers where they derived an expression for electronegativity based on DFT. ¹¹⁹ The concept was further developed by Parr and Young who connected CDFT with the FMO model of Fukui. ¹²⁰ The continuous development of CDFT has recently been described in a review article. ¹²¹

All of the above methods analyze the electronic structure of the finally formed chemical bond in molecules. There are also methods which focus on the formation of the chemical bond in a molecule between chosen fragments. Various EDA (Energy Decomposition Analysis) methods have been developed with the goal to understand the interatomic interactions along the path to the bond formation. The first approach was introduced by Kitaura and Morokuma in 1976 ¹²² where the interatomic interactions are divided into physically meaningful terms. It was restricted on Hartree–Fock calculations and it is therefore hardly used anymore. A related partitioning scheme based of DFT calculations termed ETS (Extended Transition State) method was suggested by Ziegler and Rauk in 1977 ¹²³ and further developed by the Baerends group. ¹²⁴ An important addition was made in 2008 when the NOCV (Natural Orbitals for Chemical Valency) method of Mitoraj and Michalak ¹²⁵ was merged to the ETS-NOCV (EDA-NOCV) model where the orbital (covalent) interactions are broken down into pairwise orbital interactions. ¹²⁶ The numerical results of EDA-NOCV calculations are a quantitative bridge to the orbital interaction models of Woodward and Hoffmann and the FMO model of Fukui. ¹²⁷ Two other terms which are quantitively estimated in the EDA-NOCV method are the electrostic (Coulomb) interactions, which are typically attractive, and the Pauli (exchange) repulsion, which are often neglected although they are very important forces that significantly influence the equilibrium bond lengths and the bond dissociation energy of chemical bonds. ¹²⁸ An alternative EDA scheme based on absolutely localized molecular orbitals has been developed by Head-Gordon. ¹²⁹

Regardless of which approach is used to analyze the electronic structure and bonding situation, the user should be aware of and critically examine the mathematical steps and physical assumptions as well as the computational algorithms of the method. Different approaches may come to different conclusions about the nature of the chemical bond, which becomes understandable when the approximations of the approach are analysed. Their use as black box methods without detailed knowledge of their fundamentals by merely using a keyword leads to meaningless numbers and confusing statements.

Summary and comments

This article describes the most important aspects on the influence of quantum theory on chemistry. It started with the epochal paper by Heitler and London in 1927, ¹ which showed that the chemical bond is a quantum theoretical phenomen whose physical mechanism could only be explained after quantum theory was introduced by Heisenberg and Schrödinger in 1925/1926. ^{2 , 3} The immediate impact on chemistry was very small, because chemists had developed very helpful bonding models such as the electron-pair model which enabled them to synthesize complex molecules. Most quantum chemical studies in the early years were carried out by physicists, until Linus Pauling published numerous studies where a bridge was built between the mathematical formulation of quantum theory and the experimental observation of chemistry. It culminated in the book “The Nature of the Chemical Bond” published in 1939 which became the most important reference book of chemical bonding for a long time. ³³

The further development of quantum chemistry was directly connected to the invention of computers and the accompanying progress of hardware architecture as well as computer programs, which started in the 1960s. At the same time VB theory was replaced by MO theory as the leading quantum theoretical approach for calculating molecules and for the explanation of molecular structures and reactivities. This was due the much faster computer algorithms and the conceptual advantage of MO theory where the symmetry of the orbitals proved to be crucial for the explanation of experimental findings. Electron density based DFT methods became prevailant in the new millenia when GGA functionals and parameterized hybrid functionals were introduced, which have sufficient accuracy for many chemical problems with much lower computational costs than correlated WF based approaches. Highly correlated MO methods remain still important for obtaining reliable values for energies and other physical properties.

The history of quantum chemistry in a nutshell ¹³⁰ reads like a success story, but there are also critical aspects. Much of the development is devoted to methodical progress which serves to explain experimental findings and which is helpful to design new experiments. But the transformation of the numerical results into a scientific understanding of matter and their behaviour during a chemical reaction is rarely considered. This article is mainly concerned with the development of electronic structure theory because this is the area where quantum theoretical insights have had their strongest and most visible impact on chemical research. There are countless publications reporting the development or application of quantum chemical methods, and there are also countless experimental studies supported by quantum chemical calculations. However, while the practical aspects of modern chemical research have been significantly influenced by the development of electronic structure theory, the impact of quantum theory on chemistry as a science discipline has not changed that much. This is in contrast to physics, where the understanding of matter and the fundamental laws of nature has been severely shaken by the advent of quantum theory. Physics and chemistry are two ways of looking at the world, but while quantum physics has become a lively area of ongoing debate that raises essential philosophical questions, the development of quantum chemistry is largely a development of the theory of electronic structure. Philosophical aspects of quantum chemistry are an exotic topic with little influence on chemistry so far. ^{130 , 131}

It must be remembered that chemistry is first and foremost a scientific discipline that aims to help people understand the world through the acquisition of knowledge. The translation of this knowledge into applications in the chemical industry is an important but secondary aspect. Chemistry means the understanding of the physical world at a molecular scale. Chemical research is mainly concerned with the discovery of new molecules and new reactions as well as their analysis and behavior in experimental setups. This gives chemistry more the character of an engineers discipline than a scientific category aiming at an understanding in terms of physical principles. Chemical models are very important as ordering schemes for practical use, but they are often mistaken as the physical origin of the experimental findings. This becomes obvious by the discussion of the chemical bond, where the electron pair model of Lewis is wrongly stated to be the driving force for the bond formation. The undiscriminate mixing of model and physical reality is a pertaining issue of chemical research where insights of quantum theory is still missing.

There are presently two technical developments which may have an impact on the progress of quantum chemistry. If quantum computers become available as reliable tools for computation, the accuracy of the numerical results and the size of molecules that can be calculated would dramatically increase. This would have a strong impact in areas such as biochemistry and material sciences. The second aspect is the use of AI (Artifical Intelligance) which has already started. There is big hype on the influence of AI techniques in all aspects of human life. It should be kept in mind that AI methods are a perfect tool for the interpolation within the known universe of values and data. But true progress is an extrapolation from existing knowledge rasther than the interpolation of that knowledge provided by AI. It requires the human mind and human creativity to break new grounds. Quantum theory could never be developed by AI if only classical physics would be given as input.

In the last section of the book “Valence and the Structure of Atoms and Molecules”, ⁹ which was completed in 1923 before the beginning of quantum chemistry and which was entitled “The Future of Quantum Theory”, Lewis writes his vision, which still applies today: “In that old American institution, the circus, at the end of the performance the majority of the spectators are satiated by the thrill and ready to turn again to quieter pursuits, but there are always some who not only remain in their seats but continue to pay to see the still bloodier feats of the supplementary performance.” ¹³²