In computational physics and chemistry, the Hartree–Fock method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state. The Hartree–Fock method assumes that the exact, N-body wave function of the system can be approximated by a single Slater determinant or by a single permanent of N spin-orbitals. By invoking the variational method, one can derive a set of N-coupled equations for the N spin orbitals. A solution of these equations yields energy of the system. In the older literature, the Hartree–Fock method is called the self-consistent field method. In deriving what is now called the Hartree equation as an approximate solution of the Schrödinger equation, Hartree required the final field as computed from the charge distribution to be "self-consistent" with the assumed initial field. Thus, self-consistency was a requirement of the solution; the solutions to the non-linear Hartree–Fock equations behave as if each particle is subjected to the mean field created by all other particles and hence, the terminology continued.
The equations are universally solved by means of an iterative method, although the fixed-point iteration algorithm does not always converge. This solution scheme is not the only one possible and is not an essential feature of the Hartree–Fock method; the Hartree–Fock method finds its typical application in the solution of the Schrödinger equation for atoms, molecules and solids but it has found widespread use in nuclear physics.. In atomic structure theory, calculations may be for a spectrum with many excited energy levels and the Hartree–Fock method for atoms assumes the wave function is a single configuration state function with well-defined quantum numbers and that the energy level is not the ground state. For both atoms and molecules, the Hartree–Fock solution is the central starting point for most methods that describe the many-electron system more accurately; the rest of this article will focus on applications in electronic structure theory suitable for molecules with the atom as a special case.
The discussion here is only for the Restricted Hartree–Fock method, where the atom or molecule is a closed-shell system with all orbitals doubly occupied. Open-shell systems, where some of the electrons are not paired, can be dealt with by one of two Hartree–Fock methods: Restricted open-shell Hartree–Fock Unrestricted Hartree–Fock The origin of the Hartree–Fock method dates back to the end of the 1920s, soon after the discovery of the Schrödinger equation in 1926. In 1927, D. R. Hartree introduced a procedure, which he called the self-consistent field method, to calculate approximate wave functions and energies for atoms and ions. Hartree was guided by some earlier, semi-empirical methods of the early 1920s set in the old quantum theory of Bohr. In the Bohr model of the atom, the energy of a state with principal quantum number n is given in atomic units as E = − 1 / n 2, it was observed from atomic spectra that the energy levels of many-electron atoms are well described by applying a modified version of Bohr's formula.
By introducing the quantum defect d as an empirical parameter, the energy levels of a generic atom were well approximated by the formula E = − 1 / 2, in the sense that one could reproduce well the observed transitions levels observed in the X-ray region. The existence of a non-zero quantum defect was attributed to electron-electron repulsion, which does not exist in the isolated hydrogen atom; this repulsion resulted in partial screening of the bare nuclear charge. These early researchers introduced other potentials containing additional empirical parameters with the hope of better reproducing the experimental data. Hartree sought to do away with empirical parameters and solve the many-body time-independent Schrödinger equation from fundamental physical principles, i.e. ab initio. His first proposed method of solution became known as the Hartree Hartree product. However, many of Hartree's contemporaries did not understand the physical reasoning behind the Hartree method: it appeared to many people to contain empirical elements, its connection to the solution of the many-body Schrödinger equation was unclear.
However, in 1928 J. C. Slater and J. A. Gaunt independently showed that the Hartree method could be couched on a sounder theoretical basis by applying the variational principle to an ansatz as a product of single-particle functions. In 1930, Slater and V. A. Fock independently pointed out that the Hartree method did not respect the principle of antisymmetry of the wave function; the Hartree method used the Pauli exclusion principle in its older formulation, forbidding the presence of two electrons in the same quantum state. However, this was shown to be fundamentally incomplete in its neglect of quantum statistics, it was shown that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the antisymmetric property of the exact solution and hence is a suitable ansatz for applying the variational principle. The original Hartree metho
MOLCAS is an ab initio computational chemistry program, developed as a joint project by a number of international institutes. Focus in the program is placed on methods for calculating general electronic structures in molecular systems in both ground and excited states. In September 2017 the bulk of the MOLCAS code was released as open source, under the name OpenMolcas. Ab initio Hartree–Fock, Density functional theory, second order Møller–Plesset perturbation theory, MCSCF, MRCI, CC, CASPT2 wavefunctions and energies Analytic gradient geometry optimization based on HF, DFT, CASSCF, RASSCF wavefunctions Cholesky decomposition and Resolution of the identity techniques for HF, DFT, CASSCF, CC, MBPT2, CASPT2. Analytical gradients and non-adiabatic coupling vectors. On-the-fly auxiliary basis function technique, aCD and acCD. CD/RI gradients for DFT functionals. Numerical gradient geometry optimization based on CASPT2 wavefunctions. Excited state energies for all wavefunctions, excited optimized geometries from state averaged CASSCF wavefunctions.
Transition properties in excited states calculated at the CASSCF/RASSCF level, using a unique RASSCF State Interaction Method. Solvent effects can be treated by the Onsager spherical cavity Polarizable continuum model. Combined QM and molecular mechanics calculations for systems such as proteins and molecular clusters; the NEMO procedure for creating intermolecular force fields for MC/MD simulations. Graphical selection of the active space Tully Surface Hopping Molecular Dynamics Method for localization and characterization of conical intersections and seams MOLCAS homepage OpenMolcas project page Graphical interface
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, the probabilities for the possible results of measurements made on the system can be derived from it; the most common symbols for a wave function are the Greek letters ψ or Ψ. The wave function is a function of the degrees of freedom corresponding to some maximal set of commuting observables. Once such a representation is chosen, the wave function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, correspondingly the domain of the wave function is not unique. For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; some particles, like electrons and photons, have nonzero spin, the wave function for such particles includes spin as an intrinsic, discrete degree of freedom.
When a system has internal degrees of freedom, the wave function at each point in the continuous degrees of freedom assigns a complex number for each possible value of the discrete degrees of freedom – these values are displayed in a column matrix. According to the superposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space; the inner product between two wave functions is a measure of the overlap between the corresponding physical states, is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products. The Schrödinger equation determines how wave functions evolve over time, a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation; this explains the name "wave function", gives rise to wave–particle duality.
However, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves. In Born's statistical interpretation in non-relativistic quantum mechanics, the squared modulus of the wave function, |ψ|2, is a real number interpreted as the probability density of measuring a particle's being detected at a given place – or having a given momentum – at a given time, having definite values for discrete degrees of freedom; the integral of this quantity, over all the system's degrees of freedom, must be 1 in accordance with the probability interpretation. This general requirement that a wave function must satisfy is called the normalization condition. Since the wave function is complex valued, only its relative phase and relative magnitude can be measured—its value does not, in isolation, tell anything about the magnitudes or directions of measurable observables. In 1905, Einstein postulated the proportionality between the frequency f of a photon and its energy E, E = h f, in 1916 the corresponding relation between photon's momentum p and wavelength λ, λ = h p, where h is the Planck constant.
In 1923, De Broglie was the first to suggest that the relation λ = h p, now called the De Broglie relation, holds for massive particles, the chief clue being Lorentz invariance, this can be viewed as the starting point for the modern development of quantum mechanics. The equations represent wave -- particle duality for massive particles. In the 1920s and 1930s, quantum mechanics was developed using linear algebra; those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, others, developing "wave mechanics". Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, others, developing "matrix mechanics". Schrödinger subsequently showed. In 1926, Schrödinger published the famous wave equation now named after him, the Schrödinger equation; this equation was based on classical conservation of energy using quantum operators and the de Broglie relations, the solutions of the equation are the wave functions for the quantum system. However, no one was clear on.
At first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where the wave function is large. This was shown to be incompatible with the elastic scattering of a wave packet off a target. While a scattered particle may scatter in any direction, it does not break up and take off in all directions. In 1926, Born provided the perspective of probability amplitude; this relates calculations of quantum mechanics directly to probabilistic experimental observations. It is ac
Chemistry is the scientific discipline involved with elements and compounds composed of atoms and ions: their composition, properties and the changes they undergo during a reaction with other substances. In the scope of its subject, chemistry occupies an intermediate position between physics and biology, it is sometimes called the central science because it provides a foundation for understanding both basic and applied scientific disciplines at a fundamental level. For example, chemistry explains aspects of plant chemistry, the formation of igneous rocks, how atmospheric ozone is formed and how environmental pollutants are degraded, the properties of the soil on the moon, how medications work, how to collect DNA evidence at a crime scene. Chemistry addresses topics such as how atoms and molecules interact via chemical bonds to form new chemical compounds. There are four types of chemical bonds: covalent bonds, in which compounds share one or more electron; the word chemistry comes from alchemy, which referred to an earlier set of practices that encompassed elements of chemistry, philosophy, astronomy and medicine.
It is seen as linked to the quest to turn lead or another common starting material into gold, though in ancient times the study encompassed many of the questions of modern chemistry being defined as the study of the composition of waters, growth, disembodying, drawing the spirits from bodies and bonding the spirits within bodies by the early 4th century Greek-Egyptian alchemist Zosimos. An alchemist was called a'chemist' in popular speech, the suffix "-ry" was added to this to describe the art of the chemist as "chemistry"; the modern word alchemy in turn is derived from the Arabic word al-kīmīā. In origin, the term is borrowed from the Greek χημία or χημεία; this may have Egyptian origins since al-kīmīā is derived from the Greek χημία, in turn derived from the word Kemet, the ancient name of Egypt in the Egyptian language. Alternately, al-kīmīā may derive from χημεία, meaning "cast together"; the current model of atomic structure is the quantum mechanical model. Traditional chemistry starts with the study of elementary particles, molecules, metals and other aggregates of matter.
This matter can be studied in isolation or in combination. The interactions and transformations that are studied in chemistry are the result of interactions between atoms, leading to rearrangements of the chemical bonds which hold atoms together; such behaviors are studied in a chemistry laboratory. The chemistry laboratory stereotypically uses various forms of laboratory glassware; however glassware is not central to chemistry, a great deal of experimental chemistry is done without it. A chemical reaction is a transformation of some substances into one or more different substances; the basis of such a chemical transformation is the rearrangement of electrons in the chemical bonds between atoms. It can be symbolically depicted through a chemical equation, which involves atoms as subjects; the number of atoms on the left and the right in the equation for a chemical transformation is equal. The type of chemical reactions a substance may undergo and the energy changes that may accompany it are constrained by certain basic rules, known as chemical laws.
Energy and entropy considerations are invariably important in all chemical studies. Chemical substances are classified in terms of their structure, phase, as well as their chemical compositions, they can be analyzed using the tools of e.g. spectroscopy and chromatography. Scientists engaged in chemical research are known as chemists. Most chemists specialize in one or more sub-disciplines. Several concepts are essential for the study of chemistry; the particles that make up matter have rest mass as well – not all particles have rest mass, such as the photon. Matter can be a mixture of substances; the atom is the basic unit of chemistry. It consists of a dense core called the atomic nucleus surrounded by a space occupied by an electron cloud; the nucleus is made up of positively charged protons and uncharged neutrons, while the electron cloud consists of negatively charged electrons which orbit the nucleus. In a neutral atom, the negatively charged electrons balance out the positive charge of the protons.
The nucleus is dense. The atom is the smallest entity that can be envisaged to retain the chemical properties of the element, such as electronegativity, ionization potential, preferred oxidation state, coordination number, preferred types of bonds to form. A chemical element is a pure substance, composed of a single type of atom, characterized by its particular number of protons in the nuclei of its atoms, known as the atomic number and represented by the symbol Z; the mass number is the sum of the number of neutrons in a nucleus. Although all the nuclei of all atoms belonging to one element will have the same