Neuroevolution

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Neuroevolution

Neuroevolution, or neuro-evolution, is a form of artificial intelligence that uses evolutionary algorithms to generate artificial neural networks (ANN), parameters, and rules. It is most commonly applied in artificial life, general game playing and evolutionary robotics. The main benefit is that neuroevolution can be applied more widely than supervised learning algorithms, which require a syllabus of correct input-output pairs. In contrast, neuroevolution requires only a measure of a network's performance at a task. For example, the outcome of a game (i.e., whether one player won or lost) can be easily measured without providing labeled examples of desired strategies. Neuroevolution is commonly used as part of the reinforcement learning paradigm, and it can be contrasted with conventional deep learning techniques that use backpropagation (gradient descent on a neural network) with a fixed topology.

Features

Many neuroevolution algorithms have been defined. One common distinction is between algorithms that evolve only the strength of the connection weights for a fixed network topology (sometimes called conventional neuroevolution), and algorithms that evolve both the topology of the network and its weights (called TWEANNs, for Topology and Weight Evolving Artificial Neural Network algorithms).

A separate distinction can be made between methods that evolve the structure of ANNs in parallel to its parameters (those applying standard evolutionary algorithms) and those that develop them separately (through memetic algorithms).

Comparison with gradient descent

Further information: Gradient descent

Most neural networks use gradient descent rather than neuroevolution. However, around 2017 researchers at Uber stated they had found that simple structural neuroevolution algorithms were competitive with sophisticated modern industry-standard gradient-descent deep learning algorithms, in part because neuroevolution was found to be less likely to get stuck in local minima. In Science, journalist Matthew Hutson speculated that part of the reason neuroevolution is succeeding where it had failed before is due to the increased computational power available in the 2010s.

It can be shown that there is a correspondence between neuroevolution and gradient descent.

Direct and indirect encoding

Evolutionary algorithms operate on a population of genotypes (also referred to as genomes). In neuroevolution, a genotype is mapped to a neural network phenotype that is evaluated on some task to derive its fitness.

In direct encoding schemes the genotype directly maps to the phenotype. That is, every neuron and connection in the neural network is specified directly and explicitly in the genotype. In contrast, in indirect encoding schemes the genotype specifies indirectly how that network should be generated.

Indirect encodings are often used to achieve several aims:

• modularity and other regularities;
• compression of phenotype to a smaller genotype, providing a smaller search space;
• mapping the search space (genome) to the problem domain.

Taxonomy of embryogenic systems for indirect encoding

Traditionally indirect encodings that employ artificial embryogeny (also known as artificial development) have been categorised along the lines of a grammatical approach versus a cell chemistry approach. The former evolves sets of rules in the form of grammatical rewrite systems. The latter attempts to mimic how physical structures emerge in biology through gene expression. Indirect encoding systems often use aspects of both approaches.

Stanley and Miikkulainen propose a taxonomy for embryogenic systems that is intended to reflect their underlying properties. The taxonomy identifies five continuous dimensions, along which any embryogenic system can be placed:

• Cell (neuron) fate: the final characteristics and role of the cell in the mature phenotype. This dimension counts the number of methods used for determining the fate of a cell.
• Targeting: the method by which connections are directed from source cells to target cells. This ranges from specific targeting (source and target are explicitly identified) to relative targeting (e.g., based on locations of cells relative to each other).
• Heterochrony: the timing and ordering of events during embryogeny. Counts the number of mechanisms for changing the timing of events.
• Canalization: how tolerant the genome is to mutations (brittleness). Ranges from requiring precise genotypic instructions to a high tolerance of imprecise mutation.
• Complexification: the ability of the system (including evolutionary algorithm and genotype to phenotype mapping) to allow complexification of the genome (and hence phenotype) over time. Ranges from allowing only fixed-size genomes to allowing highly variable length genomes.

Molecular geometry

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Molecular_geometry

Geometry of the water molecule with values for O-H bond length and for H-O-H bond angle between two bonds

Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that determine the position of each atom.

Molecular geometry influences several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism and biological activity. The angles between bonds that an atom forms depend only weakly on the rest of a molecule, i.e. they can be understood as approximately local and hence transferable properties.

Determination

The molecular geometry can be determined by various spectroscopic methods and diffraction methods. IR, microwave and Raman spectroscopy can give information about the molecule geometry from the details of the vibrational and rotational absorbance detected by these techniques. X-ray crystallography, neutron diffraction and electron diffraction can give molecular structure for crystalline solids based on the distance between nuclei and concentration of electron density. Gas electron diffraction can be used for small molecules in the gas phase. NMR and FRET methods can be used to determine complementary information including relative distances, dihedral angles, angles, and connectivity. Molecular geometries are best determined at low temperature because at higher temperatures the molecular structure is averaged over more accessible geometries (see next section). Larger molecules often exist in multiple stable geometries (conformational isomerism) that are close in energy on the potential energy surface. Geometries can also be computed by ab initio quantum chemistry methods to high accuracy. The molecular geometry can be different as a solid, in solution, and as a gas.

The position of each atom is determined by the nature of the chemical bonds by which it is connected to its neighboring atoms. The molecular geometry can be described by the positions of these atoms in space, evoking bond lengths of two joined atoms, bond angles of three connected atoms, and torsion angles (dihedral angles) of three consecutive bonds.

Influence of thermal excitation

Since the motions of the atoms in a molecule are determined by quantum mechanics, "motion" must be defined in a quantum mechanical way. The overall (external) quantum mechanical motions translation and rotation hardly change the geometry of the molecule. (To some extent rotation influences the geometry via Coriolis forces and centrifugal distortion, but this is negligible for the present discussion.) In addition to translation and rotation, a third type of motion is molecular vibration, which corresponds to internal motions of the atoms such as bond stretching and bond angle variation. The molecular vibrations are harmonic (at least to good approximation), and the atoms oscillate about their equilibrium positions, even at the absolute zero of temperature. At absolute zero all atoms are in their vibrational ground state and show zero point quantum mechanical motion, so that the wavefunction of a single vibrational mode is not a sharp peak, but approximately a Gaussian function (the wavefunction for n = 0 depicted in the article on the quantum harmonic oscillator). At higher temperatures the vibrational modes may be thermally excited (in a classical interpretation one expresses this by stating that "the molecules will vibrate faster"), but they oscillate still around the recognizable geometry of the molecule.

To get a feeling for the probability that the vibration of molecule may be thermally excited, we inspect the Boltzmann factor β ≡ exp(−⁠ΔE/kT⁠), where ΔE is the excitation energy of the vibrational mode, k the Boltzmann constant and T the absolute temperature. At 298 K (25 °C), typical values for the Boltzmann factor β are:

• β = 0.089 for ΔE = 500 cm⁻¹
• β = 0.008 for ΔE = 1000 cm⁻¹
• β = 0.0007 for ΔE = 1500 cm⁻¹.

(The reciprocal centimeter is an energy unit that is commonly used in infrared spectroscopy; 1 cm⁻¹ corresponds to 1.23984×10⁻⁴ eV). When an excitation energy is 500 cm⁻¹, then about 8.9 percent of the molecules are thermally excited at room temperature. To put this in perspective: the lowest excitation vibrational energy in water is the bending mode (about 1600 cm⁻¹). Thus, at room temperature less than 0.07 percent of all the molecules of a given amount of water will vibrate faster than at absolute zero.

As stated above, rotation hardly influences the molecular geometry. But, as a quantum mechanical motion, it is thermally excited at relatively (as compared to vibration) low temperatures. From a classical point of view it can be stated that at higher temperatures more molecules will rotate faster, which implies that they have higher angular velocity and angular momentum. In quantum mechanical language: more eigenstates of higher angular momentum become thermally populated with rising temperatures. Typical rotational excitation energies are on the order of a few cm⁻¹. The results of many spectroscopic experiments are broadened because they involve an averaging over rotational states. It is often difficult to extract geometries from spectra at high temperatures, because the number of rotational states probed in the experimental averaging increases with increasing temperature. Thus, many spectroscopic observations can only be expected to yield reliable molecular geometries at temperatures close to absolute zero, because at higher temperatures too many higher rotational states are thermally populated.

Bonding

Molecules, by definition, are most often held together with covalent bonds involving single, double, and/or triple bonds, where a "bond" is a shared pair of electrons (the other method of bonding between atoms is called ionic bonding and involves a positive cation and a negative anion).

Molecular geometries can be specified in terms of 'bond lengths', 'bond angles' and 'torsional angles'. The bond length is defined to be the average distance between the nuclei of two atoms bonded together in any given molecule. A bond angle is the angle formed between three atoms across at least two bonds. For four atoms bonded together in a chain, the torsional angle is the angle between the plane formed by the first three atoms and the plane formed by the last three atoms.

There exists a mathematical relationship among the bond angles for one central atom and four peripheral atoms (labeled 1 through 4) expressed by the following determinant. This constraint removes one degree of freedom from the choices of (originally) six free bond angles to leave only five choices of bond angles. (The angles θ₁₁, θ₂₂, θ₃₃, and θ₄₄ are always zero and that this relationship can be modified for a different number of peripheral atoms by expanding/contracting the square matrix.)

$${\displaystyle 0=\left|\begin{array}{cccc}\cos {\theta }_{11}& \cos {\theta }_{12}& \cos {\theta }_{13}& \cos {\theta }_{14}\\ \cos {\theta }_{21}& \cos {\theta }_{22}& \cos {\theta }_{23}& \cos {\theta }_{24}\\ \cos {\theta }_{31}& \cos {\theta }_{32}& \cos {\theta }_{33}& \cos {\theta }_{34}\\ \cos {\theta }_{41}& \cos {\theta }_{42}& \cos {\theta }_{43}& \cos {\theta }_{44}\end{array}\right|}$$

Molecular geometry is determined by the quantum mechanical behavior of the electrons. Using the valence bond approximation this can be understood by the type of bonds between the atoms that make up the molecule. When atoms interact to form a chemical bond, the atomic orbitals of each atom are said to combine in a process called orbital hybridisation. The two most common types of bonds are sigma bonds (usually formed by hybrid orbitals) and pi bonds (formed by unhybridized p orbitals for atoms of main group elements). The geometry can also be understood by molecular orbital theory where the electrons are delocalised.

An understanding of the wavelike behavior of electrons in atoms and molecules is the subject of quantum chemistry.

Isomers

Isomers are types of molecules that share a chemical formula but have difference geometries, resulting in different properties:

• A pure substance is composed of only one type of isomer of a molecule (all have the same geometrical structure).
• Structural isomers have the same chemical formula but different physical arrangements, often forming alternate molecular geometries with very different properties. The atoms are not bonded (connected) together in the same orders.
  • Functional isomers are special kinds of structural isomers, where certain groups of atoms exhibit a special kind of behavior, such as an ether or an alcohol.
• Stereoisomers may have many similar physicochemical properties (melting point, boiling point) and at the same time very different biochemical activities. This is because they exhibit a handedness that is commonly found in living systems. One manifestation of this chirality or handedness is that they have the ability to rotate polarized light in different directions.
• Protein folding concerns the complex geometries and different isomers that proteins can take.

Types of molecular structure

A bond angle is the geometric angle between two adjacent bonds. Some common shapes of simple molecules include:

• Linear: In a linear model, atoms are connected in a straight line. The bond angles are set at 180°. For example, carbon dioxide and nitric oxide have a linear molecular shape.
• Trigonal planar: Molecules with the trigonal planar shape are somewhat triangular and in one plane (flat). Consequently, the bond angles are set at 120°. For example, boron trifluoride.
• Angular: Angular molecules (also called bent or V-shaped) have a non-linear shape. For example, water (H₂O), which has an angle of about 105°. A water molecule has two pairs of bonded electrons and two unshared lone pairs.
• Tetrahedral: Tetra- signifies four, and -hedral relates to a face of a solid, so "tetrahedral" literally means "having four faces". This shape is found when there are four bonds all on one central atom, with no extra unshared electron pairs. In accordance with the VSEPR (valence-shell electron pair repulsion theory), the bond angles between the electron bonds are arccos(−⁠1/3⁠) = 109.47°. For example, methane (CH₄) is a tetrahedral molecule.
• Octahedral: Octa- signifies eight, and -hedral relates to a face of a solid, so "octahedral" means "having eight faces". The bond angle is 90 degrees. For example, sulfur hexafluoride (SF₆) is an octahedral molecule.
• Trigonal pyramidal: A trigonal pyramidal molecule has a pyramid-like shape with a triangular base. Unlike the linear and trigonal planar shapes but similar to the tetrahedral orientation, pyramidal shapes require three dimensions in order to fully separate the electrons. Here, there are only three pairs of bonded electrons, leaving one unshared lone pair. Lone pair – bond pair repulsions change the bond angle from the tetrahedral angle to a slightly lower value. For example, ammonia (NH₃).

Artificial life

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Artificial_life

A selection of simulated "swimbots"

Artificial Life, also referred to as ALife, is a field of study wherein researchers examine systems related to natural life, its processes, and its evolution, through the use of simulations with computer models, robotics, and biochemistry. The discipline was named by Christopher Langton, an American computer scientist, in 1986. In 1987, Langton organized the first conference on the field, in Los Alamos, New Mexico. There are three main kinds of artificial life, named for their approaches: soft, from software; hard, from hardware; and wet, from biochemistry. Artificial life researchers study traditional biology by trying to replicate aspects of biological phenomena.

Overview

Artificial life studies the fundamental processes of living systems in artificial environments in order to gain a deeper understanding of the complex information processing that defines such systems. These topics are broad, but often include evolutionary dynamics, emergent properties of collective systems, biomimicry, as well as related issues about the philosophy of the nature of life and the use of lifelike properties in artistic works.

Philosophy

The modeling philosophy of artificial life strongly differs from traditional modeling by studying not only "life as we know it" but also "life as it could be".

A traditional model of a biological system will focus on capturing its most important parameters. In contrast, an ALife modeling approach will generally seek to decipher the most simple and general principles underlying life and implement them in a simulation. The simulation then offers the possibility to analyse new and different lifelike systems.

Vladimir Georgievich Red'ko proposed to generalize this distinction to the modeling of any process, leading to the more general distinction of "processes as we know them" and "processes as they could be".

At present, the commonly accepted definition of life does not consider any current ALife simulations or software to be alive, and they do not constitute part of the evolutionary process of any ecosystem. However, different opinions about artificial life's potential have arisen:

• The strong ALife (cf. Strong AI) position states that "life is a process which can be abstracted away from any particular medium" (John von Neumann). This view is rooted in von Neumann's work on cellular automata and universal constructors, which demonstrated that self-reproduction could be achieved by logic-based machines regardless of their physical substrate. Notably, Tom Ray declared that his program Tierra is not simulating life in a computer but synthesizing it.
• The weak ALife position denies the possibility of generating a "living process" outside of a chemical solution. Its researchers try instead to simulate life processes to understand the underlying mechanics of biological phenomena.

A central goal in the philosophy and modeling of artificial life is achieving Open-Ended Evolution (OEE). This refers to the capacity of a system to continually produce novel, complex, and adaptive behaviors or entities without reaching a stable equilibrium or predefined end-point. Researchers argue that OEE is a hallmark of natural life that current artificial systems have yet to fully replicate.

Software-based ("soft")

Techniques

• Cellular automata were used in the early days of artificial life, and are still often used for ease of scalability and parallelization. ALife and cellular automata share a closely tied history.
• Artificial neural networks are sometimes used to model the brain of an agent. Although traditionally more of an artificial intelligence technique, neural nets can be important for simulating population dynamics of organisms that can learn. The symbiosis between learning and evolution is central to theories about the development of instincts in organisms with higher neurological complexity, as in, for instance, the Baldwin effect.
• Neuroevolution

Program-based

Further information: programming game

Program-based simulations contain organisms with a "genome" language. This language is more often in the form of a Turing complete computer program than actual biological DNA. Assembly derivatives are the most common languages used. An organism "lives" when its code is executed, and there are usually various methods allowing self-replication. Mutations are generally implemented as random changes to the code. Use of cellular automata is common but not required. Another example could be an artificial intelligence and multi-agent system/program.

Module-based

A Braitenberg vehicle, able to navigate by light detection

Individual modules are added to a creature. These modules modify the creature's behaviors and characteristics either directly, by hard coding into the simulation (leg type A increases speed and metabolism), or indirectly, through the emergent interactions between a creature's modules (leg type A moves up and down with a frequency of X, which interacts with other legs to create motion). Generally, these are simulators that emphasize user creation and accessibility over mutation and evolution.

Parameter-based

Organisms are generally constructed with pre-defined and fixed behaviors that are controlled by various parameters that mutate. That is, each organism contains a collection of numbers or other finite parameters. Each parameter controls one or several aspects of an organism in a well-defined way.

Neural net–based

These simulations have creatures that learn and grow using neural nets or a close derivative. Emphasis is often, although not always, on learning rather than on natural selection.

Complex systems modeling

Mathematical models of complex systems are of three types: black-box (phenomenological), white-box (mechanistic, based on the first principles) and grey-box (mixtures of phenomenological and mechanistic models). In black-box models, the individual-based (mechanistic) mechanisms of a complex dynamic system remain hidden.

Mathematical models for complex systems

Black-box models are completely nonmechanistic. They are phenomenological and ignore a composition and internal structure of a complex system. Due to the non-transparent nature of the model, interactions of subsystems cannot be investigated. In contrast, a white-box model of a complex dynamic system has ‘transparent walls’ and directly shows underlying mechanisms. All events at the micro-, meso- and macro-levels of a dynamic system are directly visible at all stages of a white-box model's evolution. In most cases, mathematical modelers use the heavy black-box mathematical methods, which cannot produce mechanistic models of complex dynamic systems. Grey-box models are intermediate and combine black-box and white-box approaches.

Logical deterministic individual-based cellular automata model of single species population growth

Creation of a white-box model of complex system is associated with the problem of the necessity of an a priori basic knowledge of the modeling subject. The deterministic logical cellular automata are necessary but not sufficient condition of a white-box model. The second necessary prerequisite of a white-box model is the presence of the physical ontology of the object under study. The white-box modeling represents an automatic hyper-logical inference from the first principles because it is completely based on the deterministic logic and axiomatic theory of the subject. The purpose of the white-box modeling is to derive from the basic axioms a more detailed, more concrete mechanistic knowledge about the dynamics of the object under study. The necessity to formulate an intrinsic axiomatic system of the subject before creating its white-box model distinguishes the cellular automata models of white-box type from cellular automata models based on arbitrary logical rules. If cellular automata rules have not been formulated from the first principles of the subject, then such a model may have a weak relevance to the real problem.

Logical deterministic individual-based cellular automata model of interspecific competition for a single limited resource

Notable simulators

This is a list of artificial life and digital organism simulators:

List of notable simulators

Name	Driven By	Started	Ended
Polyworld	neural net	1990	ongoing
Tierra	evolvable code	1991	2004
Avida	evolvable code	1993	ongoing
TechnoSphere	modules	1995
Framsticks	evolvable code	1996	ongoing
Creatures	neural net, simulated biochemistry & genetics	1996	2001
3D Virtual Creature Evolution	neural net	2008	NA
EcoSim	Fuzzy Cognitive Map	2009	ongoing
OpenWorm	Geppetto	2011	ongoing
Lenia	continuous cellular automata	2019	ongoing

Hardware-based ("hard")

Further information: Robot

Hardware-based artificial life mainly consist of robots, that is, automatically guided machines able to do tasks on their own.

Biochemical-based ("wet")

Further information: Artificial cell, Synthetic biology, and Xenobiology

Biochemical-based life is studied in the field of synthetic biology. It involves research such as the creation of synthetic DNA. The term "wet" is an extension of the term "wetware". Efforts toward "wet" artificial life focus on engineering live minimal cells from living bacteria Mycoplasma laboratorium and in building non-living biochemical cell-like systems from scratch.

In May 2019, researchers reported a new milestone in the creation of a new synthetic (possibly artificial) form of viable life, a variant of the bacteria Escherichia coli, by reducing the natural number of 64 codons in the bacterial genome to 59 codons instead, in order to encode 20 amino acids.

In 2020, Sam Kriegman and Douglas Blackiston reported the creation of a biological robot aided by artificial intelligence.

In 2021, the same team that developed Xenobots reported a further breakthrough: the first biological robots capable of kinematic self-replication. Unlike traditional biological reproduction (growth/birth), these synthetic organisms spontaneously collect loose cells in their environment to assemble new copies of themselves, a process previously seen only at the molecular level.

Open problems

How does life arise from the nonliving?

• Generate a molecular proto-organism in vitro.
• Achieve the transition to life in an artificial chemistry in silico.
• Determine whether fundamentally novel living organizations can exist.
• Simulate a unicellular organism over its entire life cycle.
• Explain how rules and symbols are generated from physical dynamics in living systems.

What are the potentials and limits of living systems?

• Determine what is inevitable in the open-ended evolution of life.
• Determine minimal conditions for evolutionary transitions from specific to generic response systems.
• Create a formal framework for synthesizing dynamical hierarchies at all scales.
• Determine the predictability of evolutionary consequences of manipulating organisms and ecosystems.
• Develop a theory of information processing, information flow, and information generation for evolving systems.

How is life related to mind, machines, and culture?

• Demonstrate the emergence of intelligence and mind in an artificial living system.
• Evaluate the influence of machines on the next major evolutionary transition of life.
• Provide a quantitative model of the interplay between cultural and biological evolution.
• Establish ethical principles for artificial life.

Related subjects

1. Agent-based modeling is used in artificial life and other fields to explore emergence in systems.
2. Artificial intelligence has traditionally used a top down approach, while ALife generally works from the bottom up.
3. Artificial chemistry started as a method within the ALife community to abstract the processes of chemical reactions.
4. Evolutionary algorithms are a practical application of the weak ALife principle applied to optimization problems. Many optimization algorithms have been crafted which borrow from or closely mirror ALife techniques. The primary difference lies in explicitly defining the fitness of an agent by its ability to solve a problem, instead of its ability to find food, reproduce, or avoid death. The following is a list of evolutionary algorithms closely related to and used in ALife:
   • Ant colony optimization
   • Bacterial colony optimization
   • Genetic algorithm
   • Genetic programming
   • Swarm intelligence
5. Multi-agent system – A multi-agent system is a computerized system composed of multiple interacting intelligent agents within an environment.
6. Evolutionary art uses techniques and methods from artificial life to create new forms of art.
7. Evolutionary music uses similar techniques, but applied to music instead of visual art.
8. Abiogenesis and the origin of life sometimes employ ALife methodologies as well.
9. Quantum artificial life applies quantum algorithms to artificial life systems.

History

Main article: History of artificial life

Criticism

Artificial life has had a controversial history. John Maynard Smith criticized certain artificial life work in 1994 as "fact-free science". Mario Bunge criticized the ideas of strong artificial life as part of his wider critique of computationalism. He wrote that proponents of strong ALife are mistakenly erasing the distinction between a simulation and the process that is being simulated. He had no such objections to the weak ALife program.

Orbital hybridisation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Orbital_hybridisation

In chemistry, orbital hybridisation (or hybridization) is the concept of mixing atomic orbitals to form new hybrid orbitals (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form chemical bonds in valence bond theory. For example, in a carbon atom which forms four single bonds, the valence-shell s orbital combines with three valence-shell p orbitals to form four equivalent sp³ mixtures in a tetrahedral arrangement around the carbon to bond to four different atoms. Hybrid orbitals are useful in the explanation of molecular geometry and atomic bonding properties and are symmetrically disposed in space. Usually hybrid orbitals are formed by mixing atomic orbitals of comparable energies.

History and uses

Chemist Linus Pauling first developed the hybridisation theory in 1931 to explain the structure of simple molecules such as methane (CH₄) using atomic orbitals. Pauling pointed out that a carbon atom forms four bonds by using one s and three p orbitals, so that "it might be inferred" that a carbon atom would form three bonds at right angles (using p orbitals) and a fourth weaker bond using the s orbital in some arbitrary direction. In reality, methane has four C–H bonds of equivalent strength. The angle between any two bonds is the tetrahedral bond angle of 109°28' (around 109.5°). Pauling supposed that in the presence of four hydrogen atoms, the s and p orbitals form four equivalent combinations which he called hybrid orbitals. Each hybrid is denoted sp³ to indicate its composition, and is directed along one of the four C–H bonds. This concept was developed for such simple chemical systems, but the approach was later applied more widely, and today it is considered an effective heuristic for rationalizing the structures of organic compounds. It gives a simple orbital picture equivalent to Lewis structures.

Hybridisation theory is an integral part of organic chemistry, one of the most compelling examples being Baldwin's rules. For drawing reaction mechanisms sometimes a classical bonding picture is needed with two atoms sharing two electrons. Hybridisation theory explains bonding in alkenes and methane. The amount of p character or s character, which is decided mainly by orbital hybridisation, can be used to reliably predict molecular properties such as acidity or basicity.

Overview

Orbitals are a model representation of the behavior of electrons within molecules. In the case of simple hybridization, this approximation is based on atomic orbitals, similar to those obtained for the hydrogen atom, the only neutral atom for which the Schrödinger equation can be solved exactly. In heavier atoms, such as carbon, nitrogen, and oxygen, the atomic orbitals used are the 2s and 2p orbitals, similar to excited state orbitals for hydrogen.

Hybrid orbitals are assumed to be mixtures of atomic orbitals, superimposed on each other in various proportions. For example, in methane, the C hybrid orbital which forms each carbon–hydrogen bond consists of 25% s character and 75% p character and is thus described as sp³ (read as s-p-three) hybridised. Quantum mechanics describes this hybrid as an sp³ wavefunction of the form ${\displaystyle N(s+\sqrt{3}p\sigma )}$, where N is a normalisation constant (here 1/2) and pσ is a p orbital directed along the C-H axis to form a sigma bond. The ratio of coefficients (denoted λ in general) is ${\displaystyle \sqrt{3}}$ in this example. Since the electron density associated with an orbital is proportional to the square of the wavefunction, the ratio of p-character to s-character is λ² = 3. The p character or the weight of the p component is N²λ² = 3/4.

Types

sp³

Four sp³ orbitals.

See also: tetrahedral molecular geometry

Hybridisation describes the bonding of atoms from an atom's point of view. For a tetrahedrally coordinated carbon (e.g., methane CH₄), the carbon should have 4 orbitals directed towards the 4 hydrogen atoms.

Carbon's ground state configuration is 1s² 2s² 2p² or more easily read:

C	↑↓	↑↓	↑	↑
	1s	2s	2p	2p	2p

This diagram suggests that the carbon atom could use its two singly occupied p-type orbitals to form two covalent bonds with two hydrogen atoms in a methylene (CH₂) molecule, with a hypothetical bond angle of 90° corresponding to the angle between two p orbitals on the same atom. However the true H–C–H angle in singlet methylene is about 102° which implies the presence of some orbital hybridisation.

The carbon atom can also bond to four hydrogen atoms in methane by an excitation (or promotion) of an electron from the doubly occupied 2s orbital to the empty 2p orbital, producing four singly occupied orbitals.

C*	↑↓	↑	↑	↑	↑
	1s	2s	2p	2p	2p

The energy released by the formation of two additional bonds more than compensates for the excitation energy required, energetically favoring the formation of four C-H bonds.

According to quantum mechanics, the lowest energy is obtained if the four bonds are equivalent, which requires that they are formed from equivalent orbitals on the carbon. A set of four equivalent orbitals can be obtained that are linear combinations of the valence-shell (core orbitals are almost never involved in bonding) s and p wave functions, which are the four sp³ hybrids.

C*	↑↓	↑	↑	↑	↑
	1s	sp3	sp3	sp3	sp3

In CH₄, four sp³ hybrid orbitals are overlapped by the four hydrogens' 1s orbitals, yielding four σ (sigma) bonds (that is, four single covalent bonds) of equal length and strength.

The following:

translates into:

sp²

Three sp² orbitals.

Ethylene structure

See also: trigonal planar molecular geometry

Other carbon compounds and other molecules may be explained in a similar way. For example, ethylene (C₂H₄) has a double bond between the carbons. For this molecule, carbon sp² hybridises, because one π (pi) bond is required for the double bond between the carbons and only three σ bonds are formed per carbon atom. In sp² hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals, usually denoted 2p_x and 2p_y. The third 2p orbital (2p_z) remains unhybridised.

C*	↑↓	↑	↑	↑	↑
	1s	sp2	sp2	sp2	2p

forming a total of three sp² orbitals with one remaining p orbital. In ethylene, the two carbon atoms form a σ bond by overlapping one sp² orbital from each carbon atom. The π bond between the carbon atoms perpendicular to the molecular plane is formed by 2p–2p overlap. Each carbon atom forms covalent C–H bonds with two hydrogens by s–sp² overlap, all with 120° bond angles. The hydrogen–carbon bonds are all of equal strength and length, in agreement with experimental data.

sp

Two sp orbitals

See also: linear molecular geometry

The chemical bonding in compounds such as alkynes with triple bonds is explained by sp hybridization. In this model, the 2s orbital is mixed with only one of the three p orbitals,

C*	↑↓	↑	↑	↑	↑
	1s	sp	sp	2p	2p

resulting in two sp orbitals and two remaining p orbitals. The chemical bonding in acetylene (ethyne) (C₂H₂) consists of sp–sp overlap between the two carbon atoms forming a σ bond and two additional π bonds formed by p–p overlap. Each carbon also bonds to hydrogen in a σ s–sp overlap at 180° angles.

Molecule shape

Shapes of the different types of hybrid orbitals

Hybridisation helps to explain molecule shape, since the angles between bonds are approximately equal to the angles between hybrid orbitals. This is in contrast to valence shell electron-pair repulsion (VSEPR) theory, which can be used to predict molecular geometry based on empirical rules rather than on valence-bond or orbital theories.

sp^x hybridisation

As the valence orbitals of main group elements are the one s and three p orbitals with the corresponding octet rule, sp^x hybridization is used to model the shape of these molecules.

Coordination number	Shape	Hybridisation	Examples
2	Linear	sp hybridisation (180°)	CO2
3	Trigonal planar	sp2 hybridisation (120°)	BCl3
4	Tetrahedral	sp3 hybridisation (109.5°)	CCl4
Interorbital angles	${\displaystyle \theta =\arccos \left(-\frac{1}{x}\right)}$

sp^xd^y hybridisation

As the valence orbitals of transition metals are the five d, one s and three p orbitals with the corresponding 18-electron rule, sp^xd^y hybridisation is used to model the shape of these molecules. These molecules tend to have multiple shapes corresponding to the same hybridization due to the different d-orbitals involved. A square planar complex has one unoccupied p-orbital and hence has 16 valence electrons.

Coordination number	Shape	Hybridisation	Examples
4	Square planar	sp2d hybridisation	PtCl42−
5	Trigonal bipyramidal	sp3d hybridisation	Fe(CO)5
	Square pyramidal		MnCl52−
6	Octahedral	sp3d2 hybridisation	Mo(CO)6
7	Pentagonal bipyramidal	sp3d3 hybridisation	ZrF73−
	Capped octahedral		MoF7−
	Capped trigonal prismatic		TaF72−
8	Square antiprismatic	sp3d4 hybridisation	ReF8−
	Dodecahedral		Mo(CN)84−
	Bicapped trigonal prismatic		ZrF84−
9	Tricapped trigonal prismatic	sp3d5 hybridisation	ReH92−
	Capped square antiprismatic

sd^x hybridisation

In certain transition metal complexes with a low d electron count, the p-orbitals are unoccupied and sd^x hybridisation is used to model the shape of these molecules.

Coordination number	Shape	Hybridisation	Examples
3	Trigonal pyramidal	sd2 hybridisation (90°)	CrO3
4	Tetrahedral	sd3 hybridisation (70.5°, 109.5°)	TiCl4
5	Square pyramidal	sd4 hybridisation (65.9°, 114.1°)	Ta(CH3)5
6	C3v Trigonal prismatic	sd5 hybridisation (63.4°, 116.6°)	W(CH3)6
Interorbital angles	${\displaystyle \theta =\arccos \left(\pm \sqrt{\frac{1}{3}\left(1-\frac{2}{x}\right)}\right)}$

Hypervalent molecules

Main article: Hypervalent molecule

Octet expansion

In some general chemistry textbooks, hybridization is presented for main group coordination number 5 and above using an "expanded octet" scheme with d-orbitals first proposed by Pauling. However, such a scheme is now considered to be incorrect in light of computational chemistry calculations.

Coordination number	Molecular shape	Hybridisation	Examples
5	Trigonal bipyramidal	sp3d hybridisation	PF5
6	Octahedral	sp3d2 hybridisation	SF6
7	Pentagonal bipyramidal	sp3d3 hybridisation	IF7

In 1990, Eric Alfred Magnusson of the University of New South Wales published a paper definitively excluding the role of d-orbital hybridisation in bonding in hypervalent compounds of second-row (period 3) elements, ending a point of contention and confusion. Part of the confusion originates from the fact that d-functions are essential in the basis sets used to describe these compounds (or else unreasonably high energies and distorted geometries result). Also, the contribution of the d-function to the molecular wavefunction is large. These facts were incorrectly interpreted to mean that d-orbitals must be involved in bonding.

Resonance

In light of computational chemistry, a better treatment would be to invoke sigma bond resonance in addition to hybridisation, which implies that each resonance structure has its own hybridisation scheme. All resonance structures must obey the octet rule.

Coordination number	Resonance structures
5	Trigonal bipyramidal
6	Octahedral
7	Pentagonal bipyramidal

In computational VB theory

Further information: Modern valence bond theory

While the simple model of orbital hybridisation is commonly used to explain molecular shape, hybridisation is used differently when computed in modern valence bond programs. Specifically, hybridisation is not determined a priori but is instead variationally optimized to find the lowest energy solution and then reported. This means that all artificial constraints, specifically two constraints, on orbital hybridisation are lifted:

• that hybridisation is restricted to integer values (isovalent hybridisation)
• that hybrid orbitals are orthogonal to one another (hybridisation defects)

This means that in practice, hybrid orbitals do not conform to the simple ideas commonly taught and thus in scientific computational papers are simply referred to as sp^x, sp^xd^y or sd^x hybrids to express their nature instead of more specific integer values.

Isovalent hybridisation

Main article: Isovalent hybridisation

Although ideal hybrid orbitals can be useful, in reality, most bonds require orbitals of intermediate character. This requires an extension to include flexible weightings of atomic orbitals of each type (s, p, d) and allows for a quantitative depiction of the bond formation when the molecular geometry deviates from ideal bond angles. The amount of p-character is not restricted to integer values; i.e., hybridizations like sp^2.5 are also readily described.

The hybridization of bond orbitals is determined by Bent's rule: "Atomic s character concentrates in orbitals directed towards electropositive substituents".

For molecules with lone pairs, the bonding orbitals are isovalent sp^x hybrids. For example, the two bond-forming hybrid orbitals of oxygen in water can be described as sp^4.0 to give the interorbital angle of 104.5°. This means that they have 20% s character and 80% p character and does not imply that a hybrid orbital is formed from one s and four p orbitals on oxygen since the 2p subshell of oxygen only contains three p orbitals.

Hybridisation defects

Hybridisation of s and p orbitals to form effective sp^x hybrids requires that they have comparable radial extent. While 2p orbitals are on average less than 10% larger than 2s, in part attributable to the lack of a radial node in 2p orbitals, 3p orbitals which have one radial node, exceed the 3s orbitals by 20–33%. The difference in extent of s and p orbitals increases further down a group. The hybridisation of atoms in chemical bonds can be analysed by considering localised molecular orbitals, for example using natural localised molecular orbitals in a natural bond orbital (NBO) scheme. In methane, CH₄, the calculated p/s ratio is approximately 3 consistent with "ideal" sp³ hybridisation, whereas for silane, SiH₄, the p/s ratio is closer to 2. A similar trend is seen for the other 2p elements. Substitution of fluorine for hydrogen further decreases the p/s ratio. The 2p elements exhibit near ideal hybridisation with orthogonal hybrid orbitals. For heavier p block elements this assumption of orthogonality cannot be justified. These deviations from the ideal hybridisation were termed hybridisation defects by Kutzelnigg.

However, computational VB groups such as Gerratt, Cooper and Raimondi (SCVB) as well as Shaik and Hiberty (VBSCF) go a step further to argue that even for model molecules such as methane, ethylene and acetylene, the hybrid orbitals are already defective and nonorthogonal, with hybridisations such as sp^1.76 instead of sp³ for methane.

Photoelectron spectra

One misconception concerning orbital hybridization is that it incorrectly predicts the ultraviolet photoelectron spectra of many molecules. While this is true if Koopmans' theorem is applied to localized hybrids, quantum mechanics requires that the (in this case ionized) wavefunction obey the symmetry of the molecule which implies resonance in valence bond theory. For example, in methane, the ionised states (CH₄⁺) can be constructed out of four resonance structures attributing the ejected electron to each of the four sp³ orbitals. A linear combination of these four structures, conserving the number of structures, leads to a triply degenerate T₂ state and an A₁ state. The difference in energy between each ionized state and the ground state would be ionization energy, which yields two values in agreement with experimental results.

Two distinct states for CH₄⁺ exist (A₁ and T₂), both of which result from the ionization of CH₄. This gives rise to the two unique peaks on the photoelectron spectrum of methane.

Localised vs canonical molecular orbitals

Main articles: Localised molecular orbitals and Natural bond orbital

Bonding orbitals formed from hybrid atomic orbitals may be considered as localised molecular orbitals, which can be formed from the delocalised orbitals of molecular orbital theory by an appropriate mathematical transformation. For molecules in the ground state, this transformation of the orbitals leaves the total many-electron wave function unchanged. The hybrid orbital description of the ground state is therefore equivalent to the delocalised orbital description for ground state total energy and electron density, as well as the molecular geometry that corresponds to the minimum total energy value.

Two localised representations

Main article: Sigma-pi and equivalent-orbital models

The symmetry-adapted and hybridised lone pairs of H₂O

Molecules with multiple bonds or multiple lone pairs can have orbitals represented in terms of sigma and pi symmetry or equivalent orbitals. Different valence bond methods use either of the two representations, which have mathematically equivalent total many-electron wave functions and are related by a unitary transformation of the set of occupied molecular orbitals.

For multiple bonds, the sigma-pi representation is the predominant one compared to the equivalent orbital (bent bond) representation. In contrast, for multiple lone pairs, most textbooks use the equivalent orbital representation. However, the sigma-pi representation is also used, such as by Weinhold and Landis within the context of natural bond orbitals, a localised orbital theory containing modernised analogs of classical (valence bond/Lewis structure) bonding pairs and lone pairs. For the hydrogen fluoride molecule, for example, two F lone pairs are essentially unhybridised p orbitals, while the other is an sp^x hybrid orbital. An analogous consideration applies to water (one O lone pair is in a pure p orbital, another is in an sp^x hybrid orbital)