Quantum chemistry

Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions to physical and chemical properties of molecules, materials, and solutions at the atomic level.^[1] These calculations include systematically applied approximations intended to make calculations computationally feasible while still capturing as much information about important contributions to the computed wave functions as well as to observable properties such as structures, spectra, and thermodynamic properties. Quantum chemistry is also concerned with the computation of quantum effects on molecular dynamics and chemical kinetics.

Chemists rely heavily on spectroscopy through which information regarding the quantization of energy on a molecular scale can be obtained. Common methods are infra-red (IR) spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, and scanning probe microscopy. Quantum chemistry may be applied to the prediction and verification of spectroscopic data as well as other experimental data.

Many quantum chemistry studies are focused on the electronic ground state and excited states of individual atoms and molecules as well as the study of reaction pathways and transition states that occur during chemical reactions. Spectroscopic properties may also be predicted. Typically, such studies assume the electronic wave function is adiabatically parameterized by the nuclear positions (i.e., the Born–Oppenheimer approximation). A wide variety of approaches are used, including semi-empirical methods, density functional theory, Hartree–Fock calculations, quantum Monte Carlo methods, and coupled cluster methods.

Understanding electronic structure and molecular dynamics through the development of computational solutions to the Schrödinger equation is a central goal of quantum chemistry. Progress in the field depends on overcoming several challenges, including the need to increase the accuracy of the results for small molecular systems, and to also increase the size of large molecules that can be realistically subjected to computation, which is limited by scaling considerations — the computation time increases as a power of the number of atoms.

History

Some view the birth of quantum chemistry as starting with the discovery of the Schrödinger equation and its application to the hydrogen atom. However, a 1927 article of Walter Heitler (1904–1981) and Fritz London is often recognized as the first milestone in the history of quantum chemistry.^[2] This was the first application of quantum mechanics to the diatomic hydrogen molecule, and thus to the phenomenon of the chemical bond.^[3] However, prior to this a critical conceptual framework was provided by Gilbert N. Lewis in his 1916 paper The Atom and the Molecule,^[4] wherein Lewis developed the first working model of valence electrons. Important contributions were also made by Yoshikatsu Sugiura^[5][6] and S.C. Wang.^[7] A series of articles by Linus Pauling, written throughout the 1930s, integrated the work of Heitler, London, Sugiura, Wang, Lewis, and John C. Slater on the concept of valence and its quantum-mechanical basis into a new theoretical framework.^[8] Many chemists were introduced to the field of quantum chemistry by Pauling's 1939 text The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, wherein he summarized this work (referred to widely now as valence bond theory) and explained quantum mechanics in a way which could be followed by chemists.^[9] The text soon became a standard text at many universities.^[10] In 1937, Hans Hellmann appears to have been the first to publish a book on quantum chemistry, in the Russian ^[11] and German languages.^[12]

In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, and bonding. In addition to the investigators mentioned above, important progress and critical contributions were made in the early years of this field by Irving Langmuir, Robert S. Mulliken, Max Born, J. Robert Oppenheimer, Hans Hellmann, Maria Goeppert Mayer, Erich Hückel, Douglas Hartree, John Lennard-Jones, and Vladimir Fock.

Electronic structure

The electronic structure of an atom or molecule is the quantum state of its electrons.^[13] The first step in solving a quantum chemical problem is usually solving the Schrödinger equation (or Dirac equation in relativistic quantum chemistry) with the electronic molecular Hamiltonian, usually making use of the Born–Oppenheimer (B–O) approximation. This is called determining the electronic structure of the molecule.^[14] An exact solution for the non-relativistic Schrödinger equation can only be obtained for the hydrogen atom (though exact solutions for the bound state energies of the hydrogen molecular ion within the B-O approximation have been identified in terms of the generalized Lambert W function). Since all other atomic and molecular systems involve the motions of three or more "particles", their Schrödinger equations cannot be solved analytically and so approximate and/or computational solutions must be sought. The process of seeking computational solutions to these problems is part of the field known as computational chemistry.

Valence bond theory

Main article: Valence bond theory

As mentioned above, Heitler and London's method was extended by Slater and Pauling to become the valence-bond (VB) method. In this method, attention is primarily devoted to the pairwise interactions between atoms, and this method therefore correlates closely with classical chemists' drawings of bonds. It focuses on how the atomic orbitals of an atom combine to give individual chemical bonds when a molecule is formed, incorporating the two key concepts of orbital hybridization and resonance.^[15]

A covalent bond is formed when there is an overlap of half-filled atomic orbitals from two atoms, which together form an electron pair. The strength and energy of the system is dependent on the amount of overlap. As the atoms move together, they begin to overlap their orbitals and the electrons begin to feel the attraction of the other's nucleus. There is also a repulsion that begins to occur, which becomes too strong when the atoms are two close together. The ideal and most stable length between the two atoms is the bond distance, which is the combined repulsive and attractive forces resulting in the lowest energy configuration.

Orientation of the orbitals can have a great affect on which bond is formed if any is formed. When there is a direct overlap of one atomic orbital from each atom, a sigma (σ) bond is formed. This can be created from two s-orbitals, an s-orbital and a p-orbital, or two p-orbitals. A pi (π) bond is formed from a side-to-side overlap of two p-orbitals. The pi bond only forms if the phases of the overlapping p-orbitals are the same.^[16]

Molecular orbital theory

An anti-bonding molecular orbital of Butadiene

Main article: Molecular orbital theory

An alternative approach to valence bond theory was developed in 1929 by Friedrich Hund and Robert S. Mulliken, in which electrons are described by mathematical functions delocalized over an entire molecule. The Hund–Mulliken approach or molecular orbital (MO) method is less intuitive to chemists, but has turned out capable of predicting spectroscopic properties better than the VB method. As opposed to VB theory, MO theory does not focus just the overlap of electron density in one area causing a bond but instead describes the whole molecule as one system. This leads to a more complex understanding of the system. This approach is the conceptual basis of the Hartree–Fock method and further post-Hartree–Fock methods.

MO calculations result in orbitals or wavefunctions and energies for a molecule, which can be filled with electrons from two different atomic orbitals. These atomic orbitals come from separate atoms resulting in molecular orbitals being linear combinations of atomic orbitals.

Bonding vs Anti-bonding Orbitals

There are always the same amount of molecular orbitals as atomic orbitals. When atomic orbitals come together, a lower energy (bonding) and a higher energy (anti-bonding) orbital are formed. Times when the molecular orbital is the same energy as the atomic orbital results in a non-bonding orbital.

Bonding orbitals are formed when atomic orbitals combine leading to constructive interference, which creates a bond. The bond results in a lower energy than the atomic orbitals. The electron density is found around the bond between the two atoms, which leads to the idea that covalent bonds "share" electrons. Ultimately, this actually means, based on solving Schrödinger equation, that the electrons have the highest probability of being between the two nuclei.

Anti-bonding orbitals are the opposite of bonding orbitals. They form when atomic orbitals combine to result in destructive interference. This destructive interference results in a node between the nuclei, which means there is an area of no probability of finding electron density. The molecular orbital is at a higher energy than the atomic orbitals, so the molecule is at a less stable state. Anti-bonding orbitals are denoted by * to distinguish them from their bonding counterparts.

Non-bonding orbitals have the same energy as the atomic orbitals. They are found in MO diagrams that combine different atomic orbitals, such as an s-orbital and a p-orbital. In order to have the same amount of molecular orbitals as atomic orbitals, there have to be one bonding orbital, one anti-bonding orbital, and two non-bonding orbitals. These orbitals often correlate to lone pairs and do not affect the bonding.

Knowledge of the bonding and anti-bonding orbitals can show information on bond order, HOMO-LUMO gap, and paramagnetic vs. diamagnetic molecules. Bond order is equal to one half the number of electrons in the bonding orbitals minus the number of electrons in the anti-bonding orbitals and characterizes bonding of a molecule. The HOMO-LUMO gap is the energy difference between the highest energy occupied orbital (HOMO) and the lowest energy unoccupied orbital (LUMO). This gap gives information about material properties, including wavelengths a compound can absorb to excite an electron. Paramagnetic molecules have unpaired electrons and diamagnetic molecules have paired electrons. This results in different properties when placed in a magnetic field.^[17]

Density functional theory

Main article: Density functional theory

The Thomas–Fermi model was developed independently by Thomas and Fermi in 1927. This was the first attempt to describe many-electron systems on the basis of electronic density instead of wave functions, although it was not very successful in the treatment of entire molecules. The method did provide the basis for what is now known as density functional theory (DFT). Modern day DFT uses the Kohn–Sham method, where the density functional is split into four terms; the Kohn–Sham kinetic energy, an external potential, exchange and correlation energies. A large part of the focus on developing DFT is on improving the exchange and correlation terms. Though this method is less developed than post Hartree–Fock methods, its significantly lower computational requirements (scaling typically no worse than n³ with respect to n basis functions, for the pure functionals) allow it to tackle larger polyatomic molecules and even macromolecules. This computational affordability and often comparable accuracy to MP2 and CCSD(T) (post-Hartree–Fock methods) has made it one of the most popular methods in computational chemistry.

Chemical dynamics

A further step can consist of solving the Schrödinger equation with the total molecular Hamiltonian in order to study the motion of molecules. Direct solution of the Schrödinger equation is called quantum dynamics, whereas its solution within the semiclassical approximation is called semiclassical dynamics. Purely classical simulations of molecular motion are referred to as molecular dynamics (MD). Another approach to dynamics is a hybrid framework known as mixed quantum-classical dynamics; yet another hybrid framework uses the Feynman path integral formulation to add quantum corrections to molecular dynamics, which is called path integral molecular dynamics. Statistical approaches, using for example classical and quantum Monte Carlo methods, are also possible and are particularly useful for describing equilibrium distributions of states.

Adiabatic chemical dynamics

Main article: Born–Oppenheimer approximation

In adiabatic dynamics, interatomic interactions are represented by single scalar potentials called potential energy surfaces. This is the Born–Oppenheimer approximation introduced by Born and Oppenheimer in 1927. Pioneering applications of this in chemistry were performed by Rice and Ramsperger in 1927 and Kassel in 1928, and generalized into the RRKM theory in 1952 by Marcus who took the transition state theory developed by Eyring in 1935 into account. These methods enable simple estimates of unimolecular reaction rates from a few characteristics of the potential surface.

Non-adiabatic chemical dynamics

Main article: Vibronic coupling

Non-adiabatic dynamics consists of taking the interaction between several coupled potential energy surfaces (corresponding to different electronic quantum states of the molecule). The coupling terms are called vibronic couplings. The pioneering work in this field was done by Stueckelberg, Landau, and Zener in the 1930s, in their work on what is now known as the Landau–Zener transition. Their formula allows the transition probability between two adiabatic potential curves in the neighborhood of an avoided crossing to be calculated. Spin-forbidden reactions are one type of non-adiabatic reactions where at least one change in spin state occurs when progressing from reactant to product.

See also

• Chemistry portal
• Physics portal

• Atomic physics
• Computational chemistry
• Condensed matter physics
• Car–Parrinello molecular dynamics
• Electron localization function
• International Academy of Quantum Molecular Science
• Molecular modelling
• Physical chemistry
• Quantum computational chemistry
• List of quantum chemistry and solid-state physics software
• QMC@Home
• Quantum Aspects of Life
• Quantum electrochemistry
• Relativistic quantum chemistry
• Theoretical physics
• Spin forbidden reactions