document_title: Star Wars Roleplaying Game (Fantasy Flight Games)

question_text: when does edge of the empire take place

question_tokens: when does edge of the empire take place

text: Development[edit]

text: In the 2000s, Wizards of the Coast (WotC) owned the license for all Star Wars' collectible card and roleplaying games. When the license expired in May of 2010, WotC declined the offer to renew it.[2] In August 2011 Fantasy Flight Games acquired the license to produce Star Wars related games from Lucasfilm Ltd. Upon acquisition, Fantasy Flight announced two Star Wars gaming products: the miniatures game X-Wing and the card game Star Wars: The Card Game.[3] A role-playing game was rumored to be in the works until a year later, in August 2012, when the Fantasy Flight announced the publication of Star Wars: Edge of the Empire. Edge of the Empire was to be the first standalone game in a series of three, constituting the Fantasy Flight Games Star Wars Roleplaying Game. The first installment of the "trilogy" was first sold in a beta version (Star Wars: Edge of the Empire Beta) in late 2012.[4] The completed version of Edge of the Empire, nearly double the size of the beta, was released on June 25, 2013.[5] Age of Rebellion's beta version was released in October 2013, a beginner set released on April 25, 2014,[6] and the final version of the game, the Age of Rebellion core rulebook, released on July 3, 2014.[7] The third line of products, Star Wars: Force and Destiny, for playing Jedi characters, was released in beta in September 2014.

text: Fantasy Flight initially drew criticism for releasing a beta version, making people pay twice, and for the extra expense of the unusual custom dice; but reviews after launch were enthusiastic about the dice, with Game Informer saying "In practice, this system offers tremendous flexibility to allow the players to participate in the storytelling process, rather than just waiting for the GM to respond after a die roll. The players talk together about how to interpret a roll of the dice, and shape the results to make the most exciting story. It also speaks strongly to the cinematic nature of the Star Wars universe; characters in the movies often succeed or fail along with potent side effects."[8] and Penny Arcade saying "This dice system is designed to facilitate awesome storytelling and it worked great!"[9]

text: Release timeline[edit]

text: In the following release timeline, supplements are not included, however the timeline includes dice sets and the principal rule-containing products from all four lines of standalone games, including beta versions, beginner games, and core rulebooks.

text:
August 2012: Star Wars: Edge of the Empire Beta Version
December 2012: Star Wars: Edge of the Empire Beginner Game, a boxed set including the first mass-produced official dice of the game.
January 2013: The standalone official dice sets from the game are released.
June 2013: Star Wars: Edge of the Empire core rulebook
September 2013: Star Wars: Age of Rebellion Beta Version
April 2014: Star Wars: Age of Rebellion Beginner Game
July 2014: Star Wars: Age of Rebellion core rulebook
July 2014: The official dice sets from the game are re-released, this time with the label Star Wars Roleplaying Dice.
September 2014: Star Wars: Force and Destiny Beta Version
June 2015: Star Wars: Force and Destiny Beginner Game
July 2015: Star Wars: Force and Destiny core rulebook
September 2016: Star Wars: The Force Awakens Beginner Game
July 2017: Dawn of Rebellion, the first sourcebook designed for Edge of the Empire, Age of Rebellion, and Force and Destiny, is announced.

text: The setting[edit]

text: All three installments of the Star Wars Roleplaying Game are set within the time period of the Star Wars original trilogy. Star Wars: Edge of the Empire is set shortly after the destruction of the first Death Star, and deals with characters on the fringes of galactic space. Age of Rebellion is set around the time of The Empire Strikes Back, and allows players to join the Rebellion. Like Edge of the Empire, Force and Destiny is set shortly after the destruction of the first Death Star and the death of Obi-Wan Kenobi, when the force sensitive and Jedi slowly start to re-emerge in hopes of rebuilding the Order of the Jedi.

text: Characters are created by first specifying their species, then their initial profession (a general Career with a narrow Specialization). Players can spend experience points increasing Skills or gaining Talents (special Career-based abilities) gained from their profession. As in some other career-based RPGs your current career and specialization indicates what you do and what skills you can learn easily. Characters can buy other Specializations after character creation and can use the Skills and Talents they grant without closing the character's original Career Specialization. Having a Specialization from the same or a similar career narrows the character's selection of available Career Skills and Talents, while taking a Specialization from a different career widens the field of Career Skills and Talents and rounds the character out more.

text: The Force Awakens Beginner Game[edit]

text: The Force Awakens Beginner Game is based upon the era of the 2015 movie, Star Wars: The Force Awakens, with the adventure provided taking place just before the events of the movie. This places the adventure at roughly the same time as of the movie at thirty years after the events of Return of the Jedi.

text: The system[edit]

text: Attributes and Skills[edit]

text: Attributes[edit]

text: The Attributes are Brawn (Strength), Agility (Dexterity), Intellect (Intelligence), Cunning (Wisdom), Willpower (Mental Focus), and Presence (Charisma). Attribute levels range from a minimum of 1 to a maximum of 6. Each character race has different base Attribute levels, though additional levels in each attribute can be gained during or after character creation, at the cost the value of the next level times 10. For instance, a Human who wants to upgrade their Brawn from a base level of 2 to a level of 4 during character creation will pay 30 pts. (3 x 10) for level 3, plus 40 pts. (4 x 10) for level 4 for a total cost of 70 points. Attributes can only be increased after character creation by buying the 25-point Dedication talent on the character's career specialization tree, which raises one Attribute by one level (up to the Attribute maximum of 6).

text: Strain is a derived attribute based on a Racial minimum plus the character's Will that determines how much physical, mental or emotional stress a character can take before collapsing. Wound Points are a derived attribute based on a Racial minimum plus the character's Brawn that determines how much physical damage a character can take before passing out or slipping into a coma. Soak is the protection granted by the clothing or armor a character is wearing and/or the character's natural toughness.

text: Skills[edit]

text: The game's rules assume that all characters have all the listed skills at "zero level" if they do not have a level in it. (A real-life equivalent would be how the average person vaguely knows how to give CPR even if they have never performed it before; meanwhile, a person who took an annual CPR certification course would be qualified to do it correctly and an EMT would be an expert from performing it all the time). The character's default skill level is equal to the skill's governing Attribute score. "Default" skill rolls use one green 8-sided Ability die per Attribute level.

text: Purchasing a level in a skill makes the character an expert in it. Skill levels can be purchased for five points times the level for career skills, with the additional cost of five additional points for non-career skills. For instance, buying level one in athletics skill would cost no points if it was a selected career or specialty skill during character creation, 5 points (5 x 1 = 5) if it was a non-selected career skill during or after character creation, and 10 points (5 x 1 [+5] = 10) if it was a non-career skill. Buying level two would cost 10 points (5 x 2 = 10) for a selected or non-selected career skill and 15 points (5 x 2 [+5] = 15) for a non-career skill. Skills have a maximum level of five. Regular Skill rolls use a yellow 12-sided proficiency die per Skill level in the place of a green ability die. It can also add an ability die if the skill level is higher than the character's attribute level. For instance, a character with a level of four in computers skill and a level of 5 in Intellect would roll four Proficiency dice and one Ability die. However, a character with a level of five in computers skill and a level of 4 in intellect would still roll four Proficiency dice and 1 Ability die rather than five Proficiency dice.

text: Templates[edit]

text: The player begins character design by selecting a Racial Template. Each race has different racial attribute minimums and maximums. Some races also have a free level in a racial skill (usually up to a maximum of Level 2 at creation) or have a racial talent. As an example, humans[10] have a racial template that has a score of 2 in all Attributes and can have any two different skills of the player's choice at the start of play. Characters can also pay points to increase starting wealth (a baseline of 500 credits) or racial advantages.

text: The player then picks a career, which grants career skills. Then they pick a specialization - which grants more career skills and a specialization tree (which grants specialization-based talents). Characters cannot buy a new career but may buy additional specializations.

text: Example: The Engineer Career narrows down to the Mechanic, Saboteur, and Scientist Specializations. If a character wishes to buy another Specialization, it costs less for one under their Career than for one under another career. For instance, an Engineer - Mechanic who wants to add the Engineer's Scientist Specialization would pay less than if they wanted to add the Technician's Slicer or Ace's Pilot Specialization.

text: Career Skills grant the first level free during initial character creation (but cost points later on if a newly purchased career specialization grants them). They also cost less than regular Skills when buying additional levels in them. The character's career grants four Career Skills from the Career template skill list and their career specialization grants two more career Skills from the Specialization template skill list. For instance, an Engineer-Mechanic and Technician-Mechanic have the same specialization tree but have different specialization template career skills to choose from to depict their different character concepts.

text: Talents are what gamers call "crunchy bits": advantages with cool names (like Gear Head or Hold Together, Baby) that add flavor to a character and either grant bonuses, benefit allies, remove penalties during play, or penalize adversaries. They cost experience points to buy, and must be unlocked in the order they appear on a diagram called a "Specialization Tree" (similar to those found in video games such as Star Wars: The Old Republic). The further down the diagram, the more expensive (and powerful) the Talents become. This means that sometimes a player has to buy Talents that they do not want or require in order to get to desired talents further down the same branch. However, it avoids having the character cherry-pick the more powerful talents and leaving the rest. The player can only buy a talent on the tree once (but can buy it again if it appears elsewhere on the diagram); when the tree is all filled out, the character cannot buy any more talents from that specialization tree.

text: Talents are split into two groups. "Passive" talents (which have a blue code) denote abilities that are considered always on and can be used for the entire duration of the session. "Active" talents (which have a red code) denote abilities that require a difficulty roll to turn on prior to use, and in some cases can only be used a limited number of times in a given session. Some talents have levels and can be purchased more than once. The talent's levels stack, even if they are bought at different amounts of experience points.

text: Example: If a character bought three levels of Grit (a Talent that grants a bonus of plus one per level to the character's Strain) from one specialization tree at a total cost of 45 points and bought it at one level on another Specialization Tree at a cost of 10 points, it would stack as four levels of Grit (granting a total of plus four to Strain).

text: Disadvantages can be taken during character creation to offset point costs. Edge of the Empire has Obligations, something the character is forced or compelled to do. Age of Rebellion has Duties, something the character wants to do. The number of Player Characters in the group set the base disadvantage number, as the Game Master rolls at the beginning of play to see which character's disadvantage will be used during the session. The character can pay off the disadvantage with experience points in gameplay.

text: Force and Destiny has Morality, which governs how close to slipping over to the Dark Side the Force using character is. Unlike the other two games Morality is governed by a character's actions during gameplay, with conflict being generated whenever they choose to perform a morally questionable action or choose to use the Dark Side of the Force in order to power their abilities. Morality cannot be 'bought' with XP, instead a player wanting to change their alignment must actually role play a more aggressive character to turn Dark, or perform acts of compassion to become a paragon of the Light.

text: Motivation is the character's guiding principle (a Belief, Personal Connection, or Quest). If the player uses the character's Motivation during gameplay, they get an experience point bonus.

text: The Custom Dice[edit]

text: As in Fantasy Flight Games' edition of Warhammer Fantasy Roleplay, the system requires custom polyhedral dice, or dice modified with stickers to play (the beta version came with stickers to convert ordinary dice of the right size to Star Wars dice).[11]

text: The custom dice enable the dice having results on two axes; how successful the skill check was, and how lucky the attempt was with other factors–normally only one success on the pass / fail axis is needed to succeed. There are both positive and negative types of dice, which can be added to a roll represent advantages or disadvantages in a skill check.

text: The results on the dice are Success (explosion symbol)/Failure (caltrop symbol), Advantage (a pip in a wreath)/Threat (a pip on the central facet of a faceted sphere), or Critical Success ("Triumph", a starburst in a circle)/Critical Failure ("Despair", a triangle in a circle). Blank faces confer no benefit or penalty. The result depends on subtracting the lower result from the higher result on an axis. A result of 5 Successes and 3 Failures is a Success of 2. A result of 2 Advantages and 5 Threats is a Threat of 3. However, Triumph and Despair do not cancel each other out and double as a Success or Failure result; a result of no Triumphs and 1 Despair is 1 Critical Failure / +1 normal Failure. These results mean that the character made the Skill roll with a bonus of 1 Success, but suffered 3 Threats and 1 Despair as well. The Game Master would interpret the result to indicate what problems and difficulties would happen next.

text: For example, a Rebel commando bumps into a squad of Stormtroopers turning around a corner and he shoots his blaster at them. The player rolls the 1 Success, 3 Threats, and 1 Despair from the above example. His blaster shot hits (1 Success) and does the blaster rifle's base damage +1 (from the number of Successes). The Game Master interprets the negative results to mean that the commando suffers 1 point of Strain (1 Threat), suffers 1 Black Die on the next skill roll (2 Threats), and the power cell in his blaster ran out and needs to be reloaded (1 Despair).

text: Careers and Specializations[edit]

text: Edge of the Empire[edit]

text:
Bounty Hunter: Assassin, Gadgeteer, Survivalist, Martial Artist, Operator, Skip Tracer
Colonist: Doctor, Politico, Scholar, Entrepreneur, Marshal, Performer
Explorer: Fringer, Scout, Trader, Archaeologist, Big-Game Hunter, Driver
Hired Gun: Bodyguard, Marauder, Mercenary Soldier, Demolitionist, Enforcer, Heavy
Smuggler: Pilot, Scoundrel, Thief, Charmer, Gambler, Gunslinger
Technician: Mechanic, Outlaw Tech, Slicer, Cyber Tech, Droid Tech, Modder
Universal: Force Sensitive Exile

text: Age of Rebellion[edit]

text:
Ace: Driver, Gunner, Pilot, Beast Rider, Hotshot, Rigger
Commander: Commodore, Squadron Leader, Tactician, Figurehead, Instructor, Strategist
Diplomat: Agitator, Ambassador, Quartermaster, Advocate, Analyst, Propagandist
Engineer: Mechanic, Saboteur, Scientist, Droid Specialist, Sapper, Shipwright
Soldier: Commando, Medic, Sharpshooter, Heavy, Trailblazer, Vanguard
Spy: Infiltrator, Scout, Slicer, Interrogator, Sleeper Agent
Universal: Force Sensitive Emergent, Recruit

text: Force and Destiny[edit]

text:
Consular: Healer, Niman Disciple, Sage, Arbiter, Ascetic, Teacher
Guardian: Peacekeeper, Protector, Soresu Defender, Armorer, Warden, Warleader
Mystic: Advisor, Makashi Duelist, Seer, Alchemist, Magus, Prophet
Seeker: Ataru Striker, Hunter, Pathfinder, Executioner, Hermit, Navigator
Sentinel: Artisan, Shadow, Shien Expert, Investigator, Racer, Sentry
Warrior: Aggressor, Shii-Cho Knight, Starfighter Ace

text: Official games and supplements for Star Wars Roleplaying Game[edit]

text:
Accessories

Star Wars Roleplaying Game: Characters - Critical Injury Deck
Star Wars Roleplaying Game: Ships and Vehicles - Critical Hit Deck
Star Wars Roleplaying Game: Citizens of the Galaxy Adversary Deck
Star Wars Roleplaying Game: Imperials and Rebels Adversary Deck
Star Wars Roleplaying Game: Scum and Villainy Adversary Deck
Star Wars Roleplaying Game: Creatures of the Galaxy Adversary Deck
Star Wars Roleplaying Game: Imperials and Rebels II Adversary Deck
Star Wars Roleplaying Game: Hunters and Force Users Adversary Deck

Sourcebooks

Dawn of Rebellion (Era Sourcebook)

text: Official games and supplements for Star Wars: Edge of the Empire[edit]

text:
Rulebooks / basic sets

Star Wars: Edge of the Empire - Beta
Star Wars: Edge of the Empire - Beginner Game
Star Wars: Edge of the Empire - Core Rulebook

Accessories

Edge of the Empire Game Master's Kit (includes the gamemaster's screen)

Adventures

Beyond the Rim
The Jewel of Yavin
Mask of the Pirate Queen

Rules Supplements

Enter the Unknown (Explorer Career Book)
Dangerous Covenants (Hired Gun Career Book)
Far Horizons (Colonist Career Book)
Fly Casual (Smuggler Career Book)
Special Modifications (Technician Career Book)
No Disintegrations (Bounty Hunter Career Book)

Sourcebooks

Suns of Fortune (Corellian Sector Source Book)
Lords of Nal Hutta (Hutt Space Source Book)

text: Official games and supplements for Star Wars: Age of Rebellion[edit]

text:
Rulebooks / basic sets

Star Wars: Age of Rebellion - Beta
Star Wars: Age of Rebellion - Beginner Game
Star Wars: Age of Rebellion - Core Rulebook

Accessories

Age of Rebellion Game Master's Kit (includes the gamemaster's screen)

Adventures

Onslaught at Arda I
Rescue at Glare Peak (Rebellion Day Supplemental Adventure)
Friends Like These

Rules Supplements

Stay on Target (Ace Career Book)
Desperate Allies (Diplomat Career Book)
Lead by Example (Commander Career Book)
Forged in Battle (Soldier Career Book)
Fully Operational (Engineer Career Book)
Cyphers and Masks (Spy Career Book)

Sourcebooks

Strongholds of Resistance (Alliance Worlds Source Book)

text: Official games and supplements for Star Wars: Force and Destiny[edit]

text:
Rulebooks / basic sets

Star Wars: Force and Destiny - Beta
Star Wars: Force and Destiny - Beginner Game
Star Wars: Force and Destiny - Core Rulebook

Accessories

Force and Destiny Game Master's Kit (includes the gamemaster's screen)

Adventures

Chronicles of the Gatekeeper
Ghosts of Dathomir

Rules Supplements and Sourcebooks

Keeping the Peace (Guardian Career Book)
Savage Spirits (Seeker Career Book)
Endless Vigil (Sentinel Career Book)
Disciples of Harmony (Consular Career Book)
Unlimited Power (Mystic Career Book)

Sourcebooks

Nexus of Power (Worlds Strong in the Force Source Book)

text: Official games and supplements for Star Wars: The Force Awakens[edit]

text:
Rulebooks / basic sets

Star Wars: The Force Awakens - Beginner Game


document_title: Atomic orbital

question_text: who came up with the basis of modern atomic theory

question_tokens: who came up with the basis of modern atomic theory

root

text: Atomic orbital

text: Electron properties[edit]

text: With the development of quantum mechanics and experimental findings (such as the two slits diffraction of electrons), it was found that the orbiting electrons around a nucleus could not be fully described as particles, but needed to be explained by the wave-particle duality. In this sense, the electrons have the following properties:

tag: p

text: Wave-like properties:

text:
The electrons do not orbit the nucleus in the manner of a planet orbiting the sun, but instead exist as standing waves. Thus the lowest possible energy an electron can take is similar to the fundamental frequency of a wave on a string. Higher energy states are similar to harmonics of that fundamental frequency.
The electrons are never in a single point location, although the probability of interacting with the electron at a single point can be found from the wave function of the electron. The charge on the electron acts like it is smeared out in space in a continuous distribution, proportional at any point to the squared magnitude of the electron's wavefunction.

tag: ol

text: Particle-like properties:

text:
There is always an integer number of electrons orbiting the nucleus.
Electrons jump between orbitals in a particle-like fashion. For example, if a single photon strikes the electrons, only a single electron changes states in response to the photon.
The electrons retain particle-like properties such as: each wave state has the same electrical charge as the electron particle. Each wave state has a single discrete spin (spin up or spin down). This can depend upon its superposition.

text: Thus, despite the popular analogy to planets revolving around the Sun, electrons cannot be described simply as solid particles. In addition, atomic orbitals do not closely resemble a planet's elliptical path in ordinary atoms. A more accurate analogy might be that of a large and often oddly shaped "atmosphere" (the electron), distributed around a relatively tiny planet (the atomic nucleus). Atomic orbitals exactly describe the shape of this "atmosphere" only when a single electron is present in an atom. When more electrons are added to a single atom, the additional electrons tend to more evenly fill in a volume of space around the nucleus so that the resulting collection (sometimes termed the atom's "electron cloud"[7]) tends toward a generally spherical zone of probability describing where the atom’s electrons will be found. This is due to the uncertainty principle.

text: Formal quantum mechanical definition[edit]

text: Atomic orbitals may be defined more precisely in formal quantum mechanical language. Specifically, in quantum mechanics, the state of an atom, i.e., an eigenstate of the atomic Hamiltonian, is approximated by an expansion (see configuration interaction expansion and basis set) into linear combinations of anti-symmetrized products (Slater determinants) of one-electron functions. The spatial components of these one-electron functions are called atomic orbitals. (When one considers also their spin component, one speaks of atomic spin orbitals.) A state is actually a function of the coordinates of all the electrons, so that their motion is correlated, but this is often approximated by this independent-particle model of products of single electron wave functions.[8] (The London dispersion force, for example, depends on the correlations of the motion of the electrons.)

text: In atomic physics, the atomic spectral lines correspond to transitions (quantum leaps) between quantum states of an atom. These states are labeled by a set of quantum numbers summarized in the term symbol and usually associated with particular electron configurations, i.e., by occupation schemes of atomic orbitals (for example, 1s2 2s2 2p6 for the ground state of neon—term symbol: 1S0).

text: This notation means that the corresponding Slater determinants have a clear higher weight in the configuration interaction expansion. The atomic orbital concept is therefore a key concept for visualizing the excitation process associated with a given transition. For example, one can say for a given transition that it corresponds to the excitation of an electron from an occupied orbital to a given unoccupied orbital. Nevertheless, one has to keep in mind that electrons are fermions ruled by the Pauli exclusion principle and cannot be distinguished from the other electrons in the atom. Moreover, it sometimes happens that the configuration interaction expansion converges very slowly and that one cannot speak about simple one-determinant wave function at all. This is the case when electron correlation is large.

text: Fundamentally, an atomic orbital is a one-electron wave function, even though most electrons do not exist in one-electron atoms, and so the one-electron view is an approximation. When thinking about orbitals, we are often given an orbital vision which (even if it is not spelled out) is heavily influenced by this Hartree–Fock approximation, which is one way to reduce the complexities of molecular orbital theory.

text: Types of orbitals[edit]

text: Atomic orbitals can be the hydrogen-like "orbitals" which are exact solutions to the Schrödinger equation for a hydrogen-like "atom" (i.e., an atom with one electron). Alternatively, atomic orbitals refer to functions that depend on the coordinates of one electron (i.e., orbitals) but are used as starting points for approximating wave functions that depend on the simultaneous coordinates of all the electrons in an atom or molecule. The coordinate systems chosen for atomic orbitals are usually spherical coordinates (r, θ, φ) in atoms and cartesians (x, y, z) in polyatomic molecules. The advantage of spherical coordinates (for atoms) is that an orbital wave function is a product of three factors each dependent on a single coordinate: ψ(r, θ, φ) = R(r) Θ(θ) Φ(φ). The angular factors of atomic orbitals Θ(θ) Φ(φ) generate s, p, d, etc. functions as real combinations of spherical harmonics Yℓm(θ, φ) (where ℓ and m are quantum numbers). There are typically three mathematical forms for the radial functions R(r) which can be chosen as a starting point for the calculation of the properties of atoms and molecules with many electrons:

text:
The hydrogen-like atomic orbitals are derived from the exact solution of the Schrödinger Equation for one electron and a nucleus, for a hydrogen-like atom. The part of the function that depends on the distance r from the nucleus has nodes (radial nodes) and decays as e−(constant × distance).
The Slater-type orbital (STO) is a form without radial nodes but decays from the nucleus as does the hydrogen-like orbital.
The form of the Gaussian type orbital (Gaussians) has no radial nodes and decays as

e

−
α

r

2

{\displaystyle e^{-\alpha r^{2}}}

.

text: Although hydrogen-like orbitals are still used as pedagogical tools, the advent of computers has made STOs preferable for atoms and diatomic molecules since combinations of STOs can replace the nodes in hydrogen-like atomic orbital. Gaussians are typically used in molecules with three or more atoms. Although not as accurate by themselves as STOs, combinations of many Gaussians can attain the accuracy of hydrogen-like orbitals.

text: History[edit]

text: The term "orbital" was coined by Robert Mulliken in 1932 as an abbreviation for one-electron orbital wave function.[9] However, the idea that electrons might revolve around a compact nucleus with definite angular momentum was convincingly argued at least 19 years earlier by Niels Bohr,[10] and the Japanese physicist Hantaro Nagaoka published an orbit-based hypothesis for electronic behavior as early as 1904.[11] Explaining the behavior of these electron "orbits" was one of the driving forces behind the development of quantum mechanics.[12]

text: Early models[edit]

text: With J.J. Thomson's discovery of the electron in 1897,[13] it became clear that atoms were not the smallest building blocks of nature, but were rather composite particles. The newly discovered structure within atoms tempted many to imagine how the atom's constituent parts might interact with each other. Thomson theorized that multiple electrons revolved in orbit-like rings within a positively charged jelly-like substance,[14] and between the electron's discovery and 1909, this "plum pudding model" was the most widely accepted explanation of atomic structure.

text: Shortly after Thomson's discovery, Hantaro Nagaoka predicted a different model for electronic structure.[11] Unlike the plum pudding model, the positive charge in Nagaoka's "Saturnian Model" was concentrated into a central core, pulling the electrons into circular orbits reminiscent of Saturn's rings. Few people took notice of Nagaoka's work at the time,[15] and Nagaoka himself recognized a fundamental defect in the theory even at its conception, namely that a classical charged object cannot sustain orbital motion because it is accelerating and therefore loses energy due to electromagnetic radiation.[16] Nevertheless, the Saturnian model turned out to have more in common with modern theory than any of its contemporaries.

text: Bohr atom[edit]

text: In 1909, Ernest Rutherford discovered that the bulk of the atomic mass was tightly condensed into a nucleus, which was also found to be positively charged. It became clear from his analysis in 1911 that the plum pudding model could not explain atomic structure. In 1913 as Rutherford's post-doctoral student, Niels Bohr proposed a new model of the atom, wherein electrons orbited the nucleus with classical periods, but were only permitted to have discrete values of angular momentum, quantized in units h/2π.[10] This constraint automatically permitted only certain values of electron energies. The Bohr model of the atom fixed the problem of energy loss from radiation from a ground state (by declaring that there was no state below this), and more importantly explained the origin of spectral lines.

text: After Bohr's use of Einstein's explanation of the photoelectric effect to relate energy levels in atoms with the wavelength of emitted light, the connection between the structure of electrons in atoms and the emission and absorption spectra of atoms became an increasingly useful tool in the understanding of electrons in atoms. The most prominent feature of emission and absorption spectra (known experimentally since the middle of the 19th century), was that these atomic spectra contained discrete lines. The significance of the Bohr model was that it related the lines in emission and absorption spectra to the energy differences between the orbits that electrons could take around an atom. This was, however, not achieved by Bohr through giving the electrons some kind of wave-like properties, since the idea that electrons could behave as matter waves was not suggested until eleven years later. Still, the Bohr model's use of quantized angular momenta and therefore quantized energy levels was a significant step towards the understanding of electrons in atoms, and also a significant step towards the development of quantum mechanics in suggesting that quantized restraints must account for all discontinuous energy levels and spectra in atoms.

text: With de Broglie's suggestion of the existence of electron matter waves in 1924, and for a short time before the full 1926 Schrödinger equation treatment of hydrogen-like atom, a Bohr electron "wavelength" could be seen to be a function of its momentum, and thus a Bohr orbiting electron was seen to orbit in a circle at a multiple of its half-wavelength (this physically incorrect Bohr model is still often taught to beginning students). The Bohr model for a short time could be seen as a classical model with an additional constraint provided by the 'wavelength' argument. However, this period was immediately superseded by the full three-dimensional wave mechanics of 1926. In our current understanding of physics, the Bohr model is called a semi-classical model because of its quantization of angular momentum, not primarily because of its relationship with electron wavelength, which appeared in hindsight a dozen years after the Bohr model was proposed.

text: The Bohr model was able to explain the emission and absorption spectra of hydrogen. The energies of electrons in the n = 1, 2, 3, etc. states in the Bohr model match those of current physics. However, this did not explain similarities between different atoms, as expressed by the periodic table, such as the fact that helium (two electrons), neon (10 electrons), and argon (18 electrons) exhibit similar chemical inertness. Modern quantum mechanics explains this in terms of electron shells and subshells which can each hold a number of electrons determined by the Pauli exclusion principle. Thus the n = 1 state can hold one or two electrons, while the n = 2 state can hold up to eight electrons in 2s and 2p subshells. In helium, all n = 1 states are fully occupied; the same for n = 1 and n = 2 in neon. In argon the 3s and 3p subshells are similarly fully occupied by eight electrons; quantum mechanics also allows a 3d subshell but this is at higher energy than the 3s and 3p in argon (contrary to the situation in the hydrogen atom) and remains empty.

text: Modern conceptions and connections to the Heisenberg uncertainty principle[edit]

text: Immediately after Heisenberg discovered his uncertainty principle,[17] Bohr noted that the existence of any sort of wave packet implies uncertainty in the wave frequency and wavelength, since a spread of frequencies is needed to create the packet itself.[18] In quantum mechanics, where all particle momenta are associated with waves, it is the formation of such a wave packet which localizes the wave, and thus the particle, in space. In states where a quantum mechanical particle is bound, it must be localized as a wave packet, and the existence of the packet and its minimum size implies a spread and minimal value in particle wavelength, and thus also momentum and energy. In quantum mechanics, as a particle is localized to a smaller region in space, the associated compressed wave packet requires a larger and larger range of momenta, and thus larger kinetic energy. Thus the binding energy to contain or trap a particle in a smaller region of space increases without bound as the region of space grows smaller. Particles cannot be restricted to a geometric point in space, since this would require an infinite particle momentum.

text: In chemistry, Schrödinger, Pauling, Mulliken and others noted that the consequence of Heisenberg's relation was that the electron, as a wave packet, could not be considered to have an exact location in its orbital. Max Born suggested that the electron's position needed to be described by a probability distribution which was connected with finding the electron at some point in the wave-function which described its associated wave packet. The new quantum mechanics did not give exact results, but only the probabilities for the occurrence of a variety of possible such results. Heisenberg held that the path of a moving particle has no meaning if we cannot observe it, as we cannot with electrons in an atom.

text: In the quantum picture of Heisenberg, Schrödinger and others, the Bohr atom number n for each orbital became known as an n-sphere[citation needed] in a three dimensional atom and was pictured as the mean energy of the probability cloud of the electron's wave packet which surrounded the atom.

text: Orbital names[edit]

text: Orbitals are given names in the form:

text: where X is the energy level corresponding to the principal quantum number n, type is a lower-case letter denoting the shape or subshell of the orbital and it corresponds to the angular quantum number ℓ, and y is the number of electrons in that orbital.

text: For example, the orbital 1s2 (pronounced as the individual numbers and letters: "one ess two") has two electrons and is the lowest energy level (n = 1) and has an angular quantum number of ℓ = 0. In X-ray notation, the principal quantum number is given a letter associated with it. For n = 1, 2, 3, 4, 5, …, the letters associated with those numbers are K, L, M, N, O, … respectively.

text: Hydrogen-like orbitals[edit]

text: The simplest atomic orbitals are those that are calculated for systems with a single electron, such as the hydrogen atom. An atom of any other element ionized down to a single electron is very similar to hydrogen, and the orbitals take the same form. In the Schrödinger equation for this system of one negative and one positive particle, the atomic orbitals are the eigenstates of the Hamiltonian operator for the energy. They can be obtained analytically, meaning that the resulting orbitals are products of a polynomial series, and exponential and trigonometric functions. (see hydrogen atom).

text: For atoms with two or more electrons, the governing equations can only be solved with the use of methods of iterative approximation. Orbitals of multi-electron atoms are qualitatively similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, the numerical approximations must be used.

text: A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: n, ℓ, and mℓ. The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and the periodic table.

text: The stationary states (quantum states) of the hydrogen-like atoms are its atomic orbitals.[clarification needed] However, in general, an electron's behavior is not fully described by a single orbital. Electron states are best represented by time-depending "mixtures" (linear combinations) of multiple orbitals. See Linear combination of atomic orbitals molecular orbital method.

text: The quantum number n first appeared in the Bohr model where it determines the radius of each circular electron orbit. In modern quantum mechanics however, n determines the mean distance of the electron from the nucleus; all electrons with the same value of n lie at the same average distance. For this reason, orbitals with the same value of n are said to comprise a "shell". Orbitals with the same value of n and also the same value of ℓ are even more closely related, and are said to comprise a "subshell".

text: Quantum numbers[edit]

text: Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers. These quantum numbers only occur in certain combinations of values, and their physical interpretation changes depending on whether real or complex versions of the atomic orbitals are employed.

text: Complex orbitals[edit]

text: In physics, the most common orbital descriptions are based on the solutions to the hydrogen atom, where orbitals are given by the product between a radial function and a pure spherical harmonic. The quantum numbers, together with the rules governing their possible values, are as follows:

text: The principal quantum number n describes the energy of the electron and is always a positive integer. In fact, it can be any positive integer, but for reasons discussed below, large numbers are seldom encountered. Each atom has, in general, many orbitals associated with each value of n; these orbitals together are sometimes called electron shells.

text:
The azimuthal quantum number ℓ describes the orbital angular momentum of each electron and is a non-negative integer. Within a shell where n is some integer n0, ℓ ranges across all (integer) values satisfying the relation

0
≤
ℓ
≤

n

0

−
1

{\displaystyle 0\leq \ell \leq n_{0}-1}

. For instance, the n = 1 shell has only orbitals with

ℓ
=
0

{\displaystyle \ell =0}

, and the n = 2 shell has only orbitals with

ℓ
=
0

{\displaystyle \ell =0}

, and

ℓ
=
1

{\displaystyle \ell =1}

. The set of orbitals associated with a particular value of ℓ are sometimes collectively called a subshell.

text:
The magnetic quantum number,

m

ℓ

{\displaystyle m_{\ell }}

, describes the magnetic moment of an electron in an arbitrary direction, and is also always an integer. Within a subshell where

ℓ

{\displaystyle \ell }

 is some integer

ℓ

0

{\displaystyle \ell _{0}}

,

m

ℓ

{\displaystyle m_{\ell }}

 ranges thus:

−

ℓ

0

≤

m

ℓ

≤

ℓ

0

{\displaystyle -\ell _{0}\leq m_{\ell }\leq \ell _{0}}

.

text:
The above results may be summarized in the following table. Each cell represents a subshell, and lists the values of

m

ℓ

{\displaystyle m_{\ell }}

 available in that subshell. Empty cells represent subshells that do not exist.

text:
ℓ = 0
ℓ = 1
ℓ = 2
ℓ = 3
ℓ = 4
…

n = 1

m

ℓ

=
0

{\displaystyle m_{\ell }=0}

n = 2
0
−1, 0, 1

n = 3
0
−1, 0, 1
−2, −1, 0, 1, 2

n = 4
0
−1, 0, 1
−2, −1, 0, 1, 2
−3, −2, −1, 0, 1, 2, 3

n = 5
0
−1, 0, 1
−2, −1, 0, 1, 2
−3, −2, −1, 0, 1, 2, 3
−4, −3, −2, −1, 0, 1, 2, 3, 4

…
…
…
…
…
…
…

tag: table

text:
Subshells are usually identified by their

n

{\displaystyle n}

- and

ℓ

{\displaystyle \ell }

-values.

n

{\displaystyle n}

 is represented by its numerical value, but

ℓ

{\displaystyle \ell }

 is represented by a letter as follows: 0 is represented by 's', 1 by 'p', 2 by 'd', 3 by 'f', and 4 by 'g'. For instance, one may speak of the subshell with

n
=
2

{\displaystyle n=2}

 and

ℓ
=
0

{\displaystyle \ell =0}

 as a '2s subshell'.

text: Each electron also has a spin quantum number, s, which describes the spin of each electron (spin up or spin down). The number s can be +1/2 or −1/2.

text: The Pauli exclusion principle states that no two electrons in an atom can have the same values of all four quantum numbers. If there are two electrons in an orbital with given values for three quantum numbers, (n, l, m), these two electrons must differ in their spin.

text: The above conventions imply a preferred axis (for example, the z direction in Cartesian coordinates), and they also imply a preferred direction along this preferred axis. Otherwise there would be no sense in distinguishing m = +1 from m = −1. As such, the model is most useful when applied to physical systems that share these symmetries. The Stern–Gerlach experiment — where an atom is exposed to a magnetic field — provides one such example.[19]

text: Real orbitals[edit]

text: An atom that is embedded in a crystalline solid feels multiple preferred axes, but often no preferred direction. Instead of building atomic orbitals out of the product of radial functions and a single spherical harmonic, linear combinations of spherical harmonics are typically used, designed so that the imaginary part of the spherical harmonics cancel out. These real orbitals are the building blocks most commonly shown in orbital visualizations.

text: In the real hydrogen-like orbitals, for example, n and ℓ have the same interpretation and significance as their complex counterparts, but m is no longer a good quantum number (though its absolute value is). The orbitals are given new names based on their shape with respect to a standardized Cartesian basis. The real hydrogen-like p orbitals are given by the following[20][21]

text: where p0 = Rn 1 Y1 0, p1 = Rn 1 Y1 1, and p−1 = Rn 1 Y1 −1, are the complex orbitals corresponding to ℓ = 1.

text:
The equations for the px and py orbitals depend on the phase convention used for the spherical harmonics. The above equations suppose that the spherical harmonics are defined by

Y

ℓ

m

(
θ
,
φ
)
=
N

e

i
m
φ

P

ℓ

m

(
cos

θ

)

{\displaystyle Y_{\ell }^{m}(\theta ,\varphi )=Ne^{im\varphi }P_{\ell }^{m}(\cos {\theta })}

. However some quantum physicists[22][23] include a phase factor (-1)m in these definitions, which has the effect of relating the px orbital to a difference of spherical harmonics and the py orbital to the corresponding sum. (For more detail, see Spherical harmonics#Conventions).

text: Shapes of orbitals[edit]

text: Simple pictures showing orbital shapes are intended to describe the angular forms of regions in space where the electrons occupying the orbital are likely to be found. The diagrams cannot show the entire region where an electron can be found, since according to quantum mechanics there is a non-zero probability of finding the electron (almost) anywhere in space. Instead the diagrams are approximate representations of boundary or contour surfaces where the probability density | ψ(r, θ, φ) |2 has a constant value, chosen so that there is a certain probability (for example 90%) of finding the electron within the contour. Although | ψ |2 as the square of an absolute value is everywhere non-negative, the sign of the wave function ψ(r, θ, φ) is often indicated in each subregion of the orbital picture.

text: Sometimes the ψ function will be graphed to show its phases, rather than the | ψ(r, θ, φ) |2 which shows probability density but has no phases (which have been lost in the process of taking the absolute value, since ψ(r, θ, φ) is a complex number). | ψ(r, θ, φ) |2 orbital graphs tend to have less spherical, thinner lobes than ψ(r, θ, φ) graphs, but have the same number of lobes in the same places, and otherwise are recognizable. This article, in order to show wave function phases, shows mostly ψ(r, θ, φ) graphs.

text: The lobes can be viewed as standing wave interference patterns between the two counter rotating, ring resonant travelling wave "m" and "−m" modes, with the projection of the orbital onto the xy plane having a resonant "m" wavelengths around the circumference. Though rarely depicted the travelling wave solutions can be viewed as rotating banded tori, with the bands representing phase information. For each m there are two standing wave solutions ⟨m⟩+⟨−m⟩ and ⟨m⟩−⟨−m⟩. For the case where m = 0 the orbital is vertical, counter rotating information is unknown, and the orbital is z-axis symmetric. For the case where ℓ = 0 there are no counter rotating modes. There are only radial modes and the shape is spherically symmetric. For any given n, the smaller ℓ is, the more radial nodes there are. Loosely speaking n is energy, ℓ is analogous to eccentricity, and m is orientation. In the classical case, a ring resonant travelling wave, for example in a circular transmission line, unless actively forced, will spontaneously decay into a ring resonant standing wave because reflections will build up over time at even the smallest imperfection or discontinuity.

text: Generally speaking, the number n determines the size and energy of the orbital for a given nucleus: as n increases, the size of the orbital increases. When comparing different elements, the higher nuclear charge Z of heavier elements causes their orbitals to contract by comparison to lighter ones, so that the overall size of the whole atom remains very roughly constant, even as the number of electrons in heavier elements (higher Z) increases.

text: Also in general terms, ℓ determines an orbital's shape, and mℓ its orientation. However, since some orbitals are described by equations in complex numbers, the shape sometimes depends on mℓ also. Together, the whole set of orbitals for a given ℓ and n fill space as symmetrically as possible, though with increasingly complex sets of lobes and nodes.

text:
The single s-orbitals (

ℓ
=
0

{\displaystyle \ell =0}

) are shaped like spheres. For n = 1 it is roughly a solid ball (it is most dense at the center and fades exponentially outwardly), but for n = 2 or more, each single s-orbital is composed of spherically symmetric surfaces which are nested shells (i.e., the "wave-structure" is radial, following a sinusoidal radial component as well). See illustration of a cross-section of these nested shells, at right. The s-orbitals for all n numbers are the only orbitals with an anti-node (a region of high wave function density) at the center of the nucleus. All other orbitals (p, d, f, etc.) have angular momentum, and thus avoid the nucleus (having a wave node at the nucleus). Recently, there has been an effort to experimentally image the 1s and 2p orbitials in a SrTiO3 crystal using scanning transmission electron microscopy with energy dispersive x-ray spectroscopy.[24] Because the imaging was conducted using an electron beam, Coulombic beam-orbital interaction that is often termed as the impact parameter effect is included in the final outcome (see the figure at right).

text: The shapes of p, d and f-orbitals are described verbally here and shown graphically in the Orbitals table below. The three p-orbitals for n = 2 have the form of two ellipsoids with a point of tangency at the nucleus (the two-lobed shape is sometimes referred to as a "dumbbell"—there are two lobes pointing in opposite directions from each other). The three p-orbitals in each shell are oriented at right angles to each other, as determined by their respective linear combination of values of mℓ. The overall result is a lobe pointing along each direction of the primary axes.

text: Four of the five d-orbitals for n = 3 look similar, each with four pear-shaped lobes, each lobe tangent at right angles to two others, and the centers of all four lying in one plane. Three of these planes are the xy-, xz-, and yz-planes—the lobes are between the pairs of primary axes—and the fourth has the centres along the x and y axes themselves. The fifth and final d-orbital consists of three regions of high probability density: a torus with two pear-shaped regions placed symmetrically on its z axis. The overall total of 18 directional lobes point in every primary axis direction and between every pair.

text: There are seven f-orbitals, each with shapes more complex than those of the d-orbitals.

text: Additionally, as is the case with the s orbitals, individual p, d, f and g orbitals with n values higher than the lowest possible value, exhibit an additional radial node structure which is reminiscent of harmonic waves of the same type, as compared with the lowest (or fundamental) mode of the wave. As with s orbitals, this phenomenon provides p, d, f, and g orbitals at the next higher possible value of n (for example, 3p orbitals vs. the fundamental 2p), an additional node in each lobe. Still higher values of n further increase the number of radial nodes, for each type of orbital.

text: The shapes of atomic orbitals in one-electron atom are related to 3-dimensional spherical harmonics. These shapes are not unique, and any linear combination is valid, like a transformation to cubic harmonics, in fact it is possible to generate sets where all the d's are the same shape, just like the px, py, and pz are the same shape.[25][26]

text: Although individual orbitals are most often shown independent of each other, the orbitals coexist around the nucleus at the same time.

text: Orbitals table[edit]

text: This table shows all orbital configurations for the real hydrogen-like wave functions up to 7s, and therefore covers the simple electronic configuration for all elements in the periodic table up to radium. "ψ" graphs are shown with − and + wave function phases shown in two different colors (arbitrarily red and blue). The pz orbital is the same as the p0 orbital, but the px and py are formed by taking linear combinations of the p+1 and p−1 orbitals (which is why they are listed under the m = ±1 label). Also, the p+1 and p−1 are not the same shape as the p0, since they are pure spherical harmonics.

text:
s (ℓ = 0)
p (ℓ = 1)
d (ℓ = 2)
f (ℓ = 3)

m = 0
m = 0
m = ±1
m = 0
m = ±1
m = ±2
m = 0
m = ±1
m = ±2
m = ±3

s
pz
px
py
dz2
dxz
dyz
dxy
dx2−y2
fz3
fxz2
fyz2
fxyz
fz(x2−y2)
fx(x2−3y2)
fy(3x2−y2)

n = 1

n = 2

n = 3

n = 4

n = 5

. . .
. . .
. . .
. . .
. . .
. . .
. . .

n = 6

. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .

n = 7

. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .

text: Qualitative understanding of shapes[edit]

text: The shapes of atomic orbitals can be qualitatively understood by considering the analogous case of standing waves on a circular drum.[27] To see the analogy, the mean vibrational displacement of each bit of drum membrane from the equilibrium point over many cycles (a measure of average drum membrane velocity and momentum at that point) must be considered relative to that point's distance from the center of the drum head. If this displacement is taken as being analogous to the probability of finding an electron at a given distance from the nucleus, then it will be seen that the many modes of the vibrating disk form patterns that trace the various shapes of atomic orbitals. The basic reason for this correspondence lies in the fact that the distribution of kinetic energy and momentum in a matter-wave is predictive of where the particle associated with the wave will be. That is, the probability of finding an electron at a given place is also a function of the electron's average momentum at that point, since high electron momentum at a given position tends to "localize" the electron in that position, via the properties of electron wave-packets (see the Heisenberg uncertainty principle for details of the mechanism).

text: This relationship means that certain key features can be observed in both drum membrane modes and atomic orbitals. For example, in all of the modes analogous to s orbitals (the top row in the animated illustration below), it can be seen that the very center of the drum membrane vibrates most strongly, corresponding to the antinode in all s orbitals in an atom. This antinode means the electron is most likely to be at the physical position of the nucleus (which it passes straight through without scattering or striking it), since it is moving (on average) most rapidly at that point, giving it maximal momentum.

text: A mental "planetary orbit" picture closest to the behavior of electrons in s orbitals, all of which have no angular momentum, might perhaps be that of a Keplerian orbit with the orbital eccentricity of 1 but a finite major axis, not physically possible (because particles were to collide), but can be imagined as a limit of orbits with equal major axes but increasing eccentricity.

text: Below, a number of drum membrane vibration modes and the respective wave functions of the hydrogen atom are shown. A correspondence can be considered where the wave functions of a vibrating drum head are for a two-coordinate system ψ(r, θ) and the wave functions for a vibrating sphere are three-coordinate ψ(r, θ, φ).

text:
s-type drum modes and wave functions

Drum mode

u

01

{\displaystyle u_{01}}

Drum mode

u

02

{\displaystyle u_{02}}

Drum mode

u

03

{\displaystyle u_{03}}

Wave function of 1s orbital (real part, 2D-cut,

r

m
a
x

=
2

a

0

{\displaystyle r_{max}=2a_{0}}

)

Wave function of 2s orbital (real part, 2D-cut,

r

m
a
x

=
10

a

0

{\displaystyle r_{max}=10a_{0}}

)

Wave function of 3s orbital (real part, 2D-cut,

r

m
a
x

=
20

a

0

{\displaystyle r_{max}=20a_{0}}

)

tag: ul

text: None of the other sets of modes in a drum membrane have a central antinode, and in all of them the center of the drum does not move. These correspond to a node at the nucleus for all non-s orbitals in an atom. These orbitals all have some angular momentum, and in the planetary model, they correspond to particles in orbit with eccentricity less than 1.0, so that they do not pass straight through the center of the primary body, but keep somewhat away from it.

text: In addition, the drum modes analogous to p and d modes in an atom show spatial irregularity along the different radial directions from the center of the drum, whereas all of the modes analogous to s modes are perfectly symmetrical in radial direction. The non radial-symmetry properties of non-s orbitals are necessary to localize a particle with angular momentum and a wave nature in an orbital where it must tend to stay away from the central attraction force, since any particle localized at the point of central attraction could have no angular momentum. For these modes, waves in the drum head tend to avoid the central point. Such features again emphasize that the shapes of atomic orbitals are a direct consequence of the wave nature of electrons.

text:
p-type drum modes and wave functions

Drum mode

u

11

{\displaystyle u_{11}}

Drum mode

u

12

{\displaystyle u_{12}}

Drum mode

u

13

{\displaystyle u_{13}}

Wave function of 2p orbital (real part, 2D-cut,

r

m
a
x

=
10

a

0

{\displaystyle r_{max}=10a_{0}}

)

Wave function of 3p orbital (real part, 2D-cut,

r

m
a
x

=
20

a

0

{\displaystyle r_{max}=20a_{0}}

)

Wave function of 4p orbital (real part, 2D-cut,

r

m
a
x

=
25

a

0

{\displaystyle r_{max}=25a_{0}}

)

text:
d-type drum modes

Mode

u

21

{\displaystyle u_{21}}

 (3d orbital)

Mode

u

22

{\displaystyle u_{22}}

 (4d orbital)

Mode

u

23

{\displaystyle u_{23}}

 (5d orbital)

text: Orbital energy[edit]

text:
In atoms with a single electron (hydrogen-like atoms), the energy of an orbital (and, consequently, of any electrons in the orbital) is determined exclusively by

n

{\displaystyle n}

. The

n
=
1

{\displaystyle n=1}

 orbital has the lowest possible energy in the atom. Each successively higher value of

n

{\displaystyle n}

 has a higher level of energy, but the difference decreases as

n

{\displaystyle n}

 increases. For high

n

{\displaystyle n}

, the level of energy becomes so high that the electron can easily escape from the atom. In single electron atoms, all levels with different

ℓ

{\displaystyle \ell }

 within a given

n

{\displaystyle n}

 are (to a good approximation) degenerate, and have the same energy. This approximation is broken to a slight extent by the effect of the magnetic field of the nucleus, and by quantum electrodynamics effects. The latter induce tiny binding energy differences especially for s electrons that go nearer the nucleus, since these feel a very slightly different nuclear charge, even in one-electron atoms; see Lamb shift.

text:
In atoms with multiple electrons, the energy of an electron depends not only on the intrinsic properties of its orbital, but also on its interactions with the other electrons. These interactions depend on the detail of its spatial probability distribution, and so the energy levels of orbitals depend not only on

n

{\displaystyle n}

 but also on

ℓ

{\displaystyle \ell }

. Higher values of

ℓ

{\displaystyle \ell }

 are associated with higher values of energy; for instance, the 2p state is higher than the 2s state. When

ℓ
=
2

{\displaystyle \ell =2}

, the increase in energy of the orbital becomes so large as to push the energy of orbital above the energy of the s-orbital in the next higher shell; when

ℓ
=
3

{\displaystyle \ell =3}

 the energy is pushed into the shell two steps higher. The filling of the 3d orbitals does not occur until the 4s orbitals have been filled.

text:
The increase in energy for subshells of increasing angular momentum in larger atoms is due to electron–electron interaction effects, and it is specifically related to the ability of low angular momentum electrons to penetrate more effectively toward the nucleus, where they are subject to less screening from the charge of intervening electrons. Thus, in atoms of higher atomic number, the

ℓ

{\displaystyle \ell }

 of electrons becomes more and more of a determining factor in their energy, and the principal quantum numbers

n

{\displaystyle n}

 of electrons becomes less and less important in their energy placement.

text:
The energy sequence of the first 35 subshells (e.g., 1s, 2p, 3d, etc.) is given in the following table. Each cell represents a subshell with

n

{\displaystyle n}

 and

ℓ

{\displaystyle \ell }

 given by its row and column indices, respectively. The number in the cell is the subshell's position in the sequence. For a linear listing of the subshells in terms of increasing energies in multielectron atoms, see the section below.

text:
s
p
d
f
g
h

1
1

2
2
3

3
4
5
7

4
6
8
10
13

5
9
11
14
17
21

6
12
15
18
22
26
31

7
16
19
23
27
32
37

8
20
24
28
33
38
44

9
25
29
34
39
45
51

10
30
35
40
46
52
59

text: Note: empty cells indicate non-existent sublevels, while numbers in italics indicate sublevels that could (potentially) exist, but which do not hold electrons in any element currently known.

text: Electron placement and the periodic table[edit]

text: Several rules govern the placement of electrons in orbitals (electron configuration). The first dictates that no two electrons in an atom may have the same set of values of quantum numbers (this is the Pauli exclusion principle). These quantum numbers include the three that define orbitals, as well as s, or spin quantum number. Thus, two electrons may occupy a single orbital, so long as they have different values of s. However, only two electrons, because of their spin, can be associated with each orbital.

text: Additionally, an electron always tends to fall to the lowest possible energy state. It is possible for it to occupy any orbital so long as it does not violate the Pauli exclusion principle, but if lower-energy orbitals are available, this condition is unstable. The electron will eventually lose energy (by releasing a photon) and drop into the lower orbital. Thus, electrons fill orbitals in the order specified by the energy sequence given above.

text: This behavior is responsible for the structure of the periodic table. The table may be divided into several rows (called 'periods'), numbered starting with 1 at the top. The presently known elements occupy seven periods. If a certain period has number i, it consists of elements whose outermost electrons fall in the ith shell. Niels Bohr was the first to propose (1923) that the periodicity in the properties of the elements might be explained by the periodic filling of the electron energy levels, resulting in the electronic structure of the atom.[28]

text: The periodic table may also be divided into several numbered rectangular 'blocks'. The elements belonging to a given block have this common feature: their highest-energy electrons all belong to the same ℓ-state (but the n associated with that ℓ-state depends upon the period). For instance, the leftmost two columns constitute the 's-block'. The outermost electrons of Li and Be respectively belong to the 2s subshell, and those of Na and Mg to the 3s subshell.

text: The following is the order for filling the "subshell" orbitals, which also gives the order of the "blocks" in the periodic table:

text: The "periodic" nature of the filling of orbitals, as well as emergence of the s, p, d and f "blocks", is more obvious if this order of filling is given in matrix form, with increasing principal quantum numbers starting the new rows ("periods") in the matrix. Then, each subshell (composed of the first two quantum numbers) is repeated as many times as required for each pair of electrons it may contain. The result is a compressed periodic table, with each entry representing two successive elements:

text:
1s
2s                                                  2p  2p  2p
3s                                                  3p  3p  3p
4s                              3d  3d  3d  3d  3d  4p  4p  4p
5s                              4d  4d  4d  4d  4d  5p  5p  5p
6s  4f  4f  4f  4f  4f  4f  4f  5d  5d  5d  5d  5d  6p  6p  6p
7s  5f  5f  5f  5f  5f  5f  5f  6d  6d  6d  6d  6d  7p  7p  7p

text: Although this is the general order of orbital filling according to the Madelung rule, there are exceptions, and the actual electronic energies of each element are also dependent upon additional details of the atoms (see Electron configuration#Atoms: Aufbau principle and Madelung rule).

text: The number of electrons in an electrically neutral atom increases with the atomic number. The electrons in the outermost shell, or valence electrons, tend to be responsible for an element's chemical behavior. Elements that contain the same number of valence electrons can be grouped together and display similar chemical properties.

text: Relativistic effects[edit]

text: For elements with high atomic number Z, the effects of relativity become more pronounced, and especially so for s electrons, which move at relativistic velocities as they penetrate the screening electrons near the core of high-Z atoms. This relativistic increase in momentum for high speed electrons causes a corresponding decrease in wavelength and contraction of 6s orbitals relative to 5d orbitals (by comparison to corresponding s and d electrons in lighter elements in the same column of the periodic table); this results in 6s valence electrons becoming lowered in energy.

text: Examples of significant physical outcomes of this effect include the lowered melting temperature of mercury (which results from 6s electrons not being available for metal bonding) and the golden color of gold and caesium.[29]

text:
In the Bohr Model, an n = 1 electron has a velocity given by

v
=
Z
α
c

{\displaystyle v=Z\alpha c}

, where Z is the atomic number,

α

{\displaystyle \alpha }

 is the fine-structure constant, and c is the speed of light. In non-relativistic quantum mechanics, therefore, any atom with an atomic number greater than 137 would require its 1s electrons to be traveling faster than the speed of light. Even in the Dirac equation, which accounts for relativistic effects, the wave function of the electron for atoms with Z > 137 is oscillatory and unbounded. The significance of element 137, also known as untriseptium, was first pointed out by the physicist Richard Feynman. Element 137 is sometimes informally called feynmanium (symbol Fy).[30] However, Feynman's approximation fails to predict the exact critical value of Z due to the non-point-charge nature of the nucleus and very small orbital radius of inner electrons, resulting in a potential seen by inner electrons which is effectively less than Z. The critical Z value which makes the atom unstable with regard to high-field breakdown of the vacuum and production of electron-positron pairs, does not occur until Z is about 173. These conditions are not seen except transiently in collisions of very heavy nuclei such as lead or uranium in accelerators, where such electron-positron production from these effects has been claimed to be observed. See Extension of the periodic table beyond the seventh period.

text: There are no nodes in relativistic orbital densities, although individual components of the wave function will have nodes.[31]

text: Transitions between orbitals[edit]

text: Bound quantum states have discrete energy levels. When applied to atomic orbitals, this means that the energy differences between states are also discrete. A transition between these states (i.e., an electron absorbing or emitting a photon) can thus only happen if the photon has an energy corresponding with the exact energy difference between said states.

text: Consider two states of the hydrogen atom:

text: State 1) n = 1, ℓ = 0, mℓ = 0 and s = +1/2

text: State 2) n = 2, ℓ = 0, mℓ = 0 and s = +1/2

text: By quantum theory, state 1 has a fixed energy of E1, and state 2 has a fixed energy of E2. Now, what would happen if an electron in state 1 were to move to state 2? For this to happen, the electron would need to gain an energy of exactly E2 − E1. If the electron receives energy that is less than or greater than this value, it cannot jump from state 1 to state 2. Now, suppose we irradiate the atom with a broad-spectrum of light. Photons that reach the atom that have an energy of exactly E2 − E1 will be absorbed by the electron in state 1, and that electron will jump to state 2. However, photons that are greater or lower in energy cannot be absorbed by the electron, because the electron can only jump to one of the orbitals, it cannot jump to a state between orbitals. The result is that only photons of a specific frequency will be absorbed by the atom. This creates a line in the spectrum, known as an absorption line, which corresponds to the energy difference between states 1 and 2.

text: The atomic orbital model thus predicts line spectra, which are observed experimentally. This is one of the main validations of the atomic orbital model.

text: The atomic orbital model is nevertheless an approximation to the full quantum theory, which only recognizes many electron states. The predictions of line spectra are qualitatively useful but are not quantitatively accurate for atoms and ions other than those containing only one electron.


document_title: Journals (album)

question_text: when did flatline by justin bieber come out

text: Journals (album)

text: Background and recording[edit]

text: While touring with his Believe Tour throughout 2012 and 2013, Justin Bieber managed to write new songs everyday. According to his manager, Scooter Braun, the writing process had him "writing one to two songs a day for the entire tour and [...] label[ing] them by the city he wr[o]te them in."[4] In January 2013, Bieber confirmed he was writing for a new album,[5] as well as American R&B singer R. Kelly that confirmed he was working with Bieber on the album.[6] Meanwhile, in June of the same year, he confirmed a new album, the rest of the Believe Tour and a movie for 2013.[7] In July, Braun gave an interview for MTV News, where he discussed the album, claiming: "For this project, we want to do things a little bit differently. And there are more songs than 10 or 12, so if you have more music and you have more things you want to express, you gotta think out the box, and [think like] ’How do I get this out there in a unique way where it gets directly to my fans and I can express myself through music directly to them?’ And I think that’s what it’s about when you have people who are incredibly creative, things happen."[8] In August, Bieber confirmed he was working with American rapper Future. It was also reported he was working with Juicy J, Ludacris and Big Sean.[9] In December, he also confirmed he was working with Chance the Rapper and Lil' Wayne.[10]

text: As executive producer, Justin enlisted previous collaborators such as Rodney Jerkins, Maejor Ali, Diplo and Sir Nolan, while working with a range of new R&B producers such as The Audibles, Andre Harris, D.K. the Punisher, Soundz, Chef Tone, T-Minus, Jason "Poo Bear" Boyd and others. Boyd became one of the most frequent collaborators on the album, writing with Bieber several songs as well as producing some. According to the producer, Bieber suggested they to "flip those Craig David chords and [...] create something." By doing so, he came up with the song "Recovery", a song that was finished at Cherry Beach Sound in Toronto. As he recalled, the process had Bieber hearing Boyd songs and selected them to sing, and many songs from the producer turned into songs off Journals. Boyd continued: "He [Bieber] just felt them so much, so strongly. It was just a matter of me letting go of the fact that I don't care about me being a singer more so than I care about putting great music out into the universe. Then I was able to do it. The songs that I felt like were too mature, we changed them up to fit him like a custom suit."[11]

text: Musical style and influences[edit]

text: Composition[edit]

text: An R&B and soul album,[12][13][14][15] Journals steps away from Bieber's more teen-pop sound of his previous efforts, while also having pop influences. As claimed by Spin's Brandon Soderberg, "[s]onically, it’s up there with Beyoncé in terms of holding tight to the patience of ’80s and ’90s R&B while never forgetting the sugar-rush rewards of pop." Soderberg also reflected that Bieber was inspired by many R&B artists such as "Aaliyah, Drake, Maxwell, Miguel, Sade, and Usher out of EDM mode."[12] According to one of its producers, Jason Boyd, "Justin grew up listening to R&B. Journals was trying because it was going against the grain [of what he had been doing sonically], but it served so much of a purpose in him growing up."[11] Chris Martins of Billboard also noted that the album "drew on Boyd’s experience writing for 112, Usher and Chris Brown."[16]

text: The album is composed by many "slow numbers", such as "Hold Tight," "All Bad" and "PYD", which Carl Williot of Idolator considered "racy, restrained jams in the vein of The Weeknd’s mixtapes (minus the danger), each utilizing the bare minimum in terms of percussion." "One Life" was also noted for having influences by The Weeknd and Frank Ocean.[17] Williot also noted that "'Roller Coaster' took the disco-pop craze of 2013 and marries it to trap stomps,[14] as well as "freestyle-meets-Sonic the Hedgehog roller-skating grooves", completed Soderberg.[12] "Bad Day" and "Swap It Out" flirt with late ’90s R&B/bling rap sweet spot, while "Confident" and closer "Memphis" channel Timbaland's prime with their brittle, syncopated percussion and background yelps."[14] "Change Me', as noted by Soderberg, has a "The Tony Rich Project" feel to it, while "Backpack" is a futuristic pop song with a guitar solo.[12] Meanwhile, "Recovery" features a "Craig David-sampling swoon" of "Fill Me In", "All that Matters" pushes Bieber's "earnest vocal into the limelight over an instrumental that relies on little more than swirling guitar strums and plush bass," and "Heartbreaker" features "chords [that] swim in and out of focus over a low-slung groove and vocals [that] float menacingly somewhere over your head" as Fact's Aimee Cliff noticed.[15]

text: Themes and lyrical content[edit]

text: Songs[edit]

text: Journals main themes deal with heartbreak and forgiveness in a romantic relationship, with many critics noting that it was influenced by Bieber's relationship with former girlfriend Selena Gomez.[18][19][20] It finds the singer apologizing, taking responsibility for bad behavior and pledging lifelong faithfulness, while also talking about sex in some songs.[20] Adam R. Holz of Plugged In noted that "'Journals' is a mournfully desperate, frequently "freaky" set of songs that finds the 19-year-old Canadian singer begging for a second chance with an ex. And it's tempting to buy in to Justin Bieber's repeated pleas for one last loop-around with his beloved. That's because the painfully plaintive nature of his repeated pleas gives him an air of wounded authenticity. It's not hard to believe that he really has lost the one girl (Selena Gomez?) who means everything to him, and that he'll do anything to win her back."[20]

text: The first song, "Heartbreaker", finds Bieber begging for his lover to not break his heart, "suggest[ing] that while he's got real flaws, those shouldn't completely disqualify him" and "pledg[ing] love through ups and downs, good times and bad." "I know the seasons may change/And sometimes love goes from sunshine to rain/ … I still believe in us/I still believe in love," he sings.[20] "All That Matters" continues in the same vein, with lines like: "Oh, oh, whenever you're not in my presence/It feels like I'm missing my blessings, yeah."[20] "Hold Tight" "focuses almost exclusively on sex," while "Recovery" finds Bieber " taking responsibility for mistakes and praying for a second chance."[20] "Bad Day" and "All Bad" shared similarities, with the first talking about a painful break-up, while the latter has him "assuring a lover that the haters have him all wrong"[18] and, according to Holz, making a "meanspirited and sexist accusation" about how "females like to run their mouths."[20] "PYD" features Bieber and R. Kelly singing about "having sex with a woman in just about as many locations as they can brainstorm. The titular acronym stands for "put you down," a slang reference to sex."[20]

text: The eighth track, "Roller Coaster", which references "Ohio Players' 1970s hit "Love Rollercoaster", talks about the loneliness of long-distance loving,[15] while the piano ballad "Change Me" is about "accepting responsibility before proposing a solution to a problem he didn't even have."[18] "Confident" returns talking about sex, with Bieber talking about someone he's apparently just spied, while Chance the Rapper also talks about the song's main subject.[20] "One Life", co-written by Drake, "offers another take on those sorts of faithfulness pledges",[20] having a "YOLO [you only live once] with a conscience" theme.[12] "Backpack" finds Bieber "fall[ing] in love with an alien girl whom he desires to 'stay in my backpack forever'."[17] "What's Hatnin'" focuses on conflict resolution and forgiveness,[20] "Swap It Out" "looks into the complicated feelings that go into a relationship,"[13] and "Memphis" "finds Bieber exploring a relationship, wherein he wants to be with his butterfly-giving girl 24/7."[13] The bonus song, "Flatline", "explores a failed relationship, due to his busy schedule and the girl in question always contacting him at the wrong times, then not returning the phone calls he attempts to make."[21]

text: Release and promotion[edit]

text: On October 3, 2013, Bieber announced that he would release one new song every Monday for 10 weeks, leading up to the release of his Believe theatrical film, which saw a limited one-week release on Christmas Day 2013. Each of the singles released has its own original artwork, featuring a white and purple theme.[22] The album and single's artwork was designed by Cooper Sebastian. The first song issued during "Music Mondays", as Bieber had touted the weekly releases, "Heartbreaker", was issued on October 7, 2013.[23] The song reached the top of the Denmark charts, the top-ten in other three countries and peaked inside the top-twenty on the US Billboard Hot 100, the UK Singles Chart and other five territories.[24] The second song, "All That Matters", was released on October 14, 2013, and was the first to receive a music video treatment, released on December 2, 2013.[25] It was another number-one single in Denmark, while reaching the top-twenty in eight countries, including the United Kingdom. In the United States, it reached number 24.[26] "Hold Tight" was the third, released on October 21, 2013, and also his third consecutive number-one in Denmark; elsewhere it reached the top-forty in over ten countries,[27] followed by "Recovery" and "Bad Day", released on October 28, 2013 and November 4, 2013, respectively, with both reaching the top-forty in many countries, while the latter becoming the album's fourth number-one single in Denmark.[28][29] The other five subsequent singles, "All Bad", "PYD", "Roller Coaster", "Change Me" and "Confident", were released each a week and attained moderate impact on the charts worldwide,[30][31][32] with "Roller Coaster" and "Confident" also reaching the top of Denmark charts,[33][34] and the latter also receiving a music video treatment.[35]

text: On December 9, 2013, Bieber announced that the ten Music Monday releases would be packaged with an additional five new songs in a compilation entitled Complete My Journals. Although the album was initially set for release on December 16, 2013, the date was pushed back one week to December 23, as Bieber intended to include one more song on the compilation. Though it does not appear on the album itself, the bonus track, "Flatline", was available for a free download on the iTunes Store for a limited time.[36] Journals was available on iTunes from January 2, 2014, and all sixteen songs are available for purchase individually.[1] According to one of its producers and Bieber's personal friend, Jason "Poo Bear" Boyd, the album was supposed to receive a full release, as well as its singles to be promoted on radio, however the label didn't support it because it wasn't the direction they wanted Justin to go.[11] It was, however, eventually released on LP in 2016.[3]

text: Song "Looking For You", which features hip hop trio, Migos, was possibly going to appear on the album, but never made the final cut due to never being released.

text: Critical reception[edit]

text:
Professional ratings

Review scores

Source
Rating

Absolute Punk
(6.6/10)[37]

Chicago Tribune
[38]

Fact
[15]

The Los Angeles Times
[18]

Milwaukee Journal Sentinel
(mixed)[39]

New York Daily News
[19]

Newsday
B-[40]

Spin
(positive)[12]

Toronto Star
[17]

Washington Post
(negative)[41]

text: Journals was met with lukewarm reviews, with most music critics noting his artistic growth. In a positive review, Mikael Wood of Los Angeles Times praised the "tunes that take up his evolving reputation with surprising candor" as well as the fact that "he's using that voice to confront the awkward aging-in-public process head-on." He also called the album "a come-to-Jesus moment that has nothing to do with Christmas."[18] Jim Farber of New York Daily News noted that Journals "has a cohesion nowhere evident on the Bieb’s three previous studio CDs," while also praising the album for "find[ing] an energy in Bieber’s deepening personality" and the music that "finds him more fully engaged."[19] Aimee Cliff wrote for Fact that the album is "a resounding achievement" and "trumps at every turn with its secret weapon: subtlety,"[15] while Brandon Soderberg of Spin called it "a cohesive, well-paced collection of narcotic R&B," claiming that the album's "elegance is conveyed via denser production choices and our host’s artful crooning."[12] Ben Rayner of Toronto Star applauded Bieber's voice, calling it "sufficiently well suited to the material that he can almost sell his new persona." Rayner also acknowledged that " tunes are often surprisingly interesting from a production standpoint," however he pointed out that "there’s too much of the same."[17]

text: Niko John of Absolute Punk went on to write that "'Journals' is undoubtedly his greatest work to date and shows a more mature, sexual side to the Canadian pop-star. However, it is an experiment that at times works and at others feels like Bieber is trying way too hard to discard his boy image."[37] Sandeep Singh of Verdens Gang wrote that Journals is "neither a profound diary or a studio album it could have been [...] But this is undoubtedly the best Bieber has bubbled up to now."[42] The Washington Post's Alison Stewart offered a negative outlook on the project, writing that "Justin Bieber ends a bad year with a bad album."[41] In his review written for the Milwaukee Journal Sentinel, Piet Levy wrote that Bieber is "less an eager child than a confident adult, although when Bieber's flat come-ons are juxtaposed with seasoned R&B star R. Kelly on "PYD," it's clear that as a performer, he still has a lot of growing up to do."[39]

text: Track listing[edit]

text:
Journals

No.
Title
Writer(s)
Producer(s)
Length

1.
"Heartbreaker"

Justin Bieber
Brandon Green
Tony Scales
Xavier Smith
Tyler Williams

T-Minus
Maejor Ali
Bieber
Chef Tone

4:21

2.
"All That Matters"

Bieber
Andre Harris
Jason Boyd
Donovan Knight

Harris
Bieber
D.K. the Punisher[a]

3:08

3.
"Hold Tight"

Bieber
Dominic Jordan
James Giannos
Boyd
Jamal "Mally Mall" Rashid

The Audibles
Poo Bear[a]
Rashid[a]

4:15

4.
"Recovery"

Bieber
Jordan
Giannos
Boyd
Rashid
Craig David
Mark Hill

The Audibles
Poo Bear[a]
Rashid[a]

3:00

5.
"Bad Day"

Jordan
Giannos
Boyd
Rashid
Ernest Isley
Marvin Isley
O'Kelly Isley Jr.
Ronald Isley
Rudolph Isley
Christopher Jasper

The Audibles
Poo Bear[a]
Rashid[a]

2:26

6.
"All Bad"

Harris
Ryan Toby
Boyd
Bieber

Harris
Bieber
Poo Bear

3:01

7.
"PYD" (featuring R. Kelly)

Jordan
Giannos
Bieber
Sasha Siri ta
Boyd
Robert Kelly
Rashid

The Audibles
Sirota
Poo Bear[a]
Rashid[a]

5:17

8.
"Roller Coaster"

Rodney "Darkchild" Jerkins
Bieber
Julian Swirsky
Josh Gudwin

Jerkins
Julkeyz

3:20

9.
"Change Me"

Bieber
Harris
Boyd

Harris
Bieber

2:43

10.
"Confident" (featuring Chance the Rapper)

Bieber
Kenneth Coby
Maurice Simmonds
Chancelor Bennett

Soundz
4:07

11.
"One Life"

Bieber
Green
Scales

Ali
Tone

4:02

12.
"Backpack" (featuring Lil Wayne)

Bieber
Nasri Atweh
Adam Messinger
Adrian Eccleston
Dwayne Carter

Messinger
Nasri
Nolan Lambroza
Kuk Harrell[b]

4:11

13.
"What's Hatnin'" (featuring Future)

Bieber
Coby
Nayvadius Wilburn

Soundz
3:28

14.
"Swap It Out"

Bieber
Jordan
Giannos
Boyd

The Audibles
Poo Bear

4:01

15.
"Memphis" (featuring Big Sean)

Bieber
Thomas Wesley Pentz
Green
Sean Anderson
Friedrich Moritz

Diplo
Ali[a]

3:43

Total length:
55:03

text:
[show]Journals — Amazon Music edition (bonus video)

No.
Title
Length

16.
"All That Matters"
3:40

Total length:
58:43

text:
[show]Journals — iTunes Store edition (bonus videos)

No.
Title
Length

16.
"All That Matters"
3:40

17.
"Believe"
2:23

18.
"Guatemala Pencils of Promise Journal"
3:02

Total length:
64:08

text:
^a signifies a co-producer
^b signifies a vocal producer
"Recovery" contains interpolations of "Fill Me In", written by Craig David and Mark Hill.
"Bad Day" contains interpolations of "Footsteps in the Dark", written by Ernest Isley, Marvin Isley, O'Kelly Isley Jr., Ronald Isley, Rudolph Isley and Christopher Jasper.
"Memphis" contains interpolations of "Nights Off", written by Friedrich Moritz.
Graphic Design by Cooper Sebastian

text: Charts[edit]

text: Weekly charts[edit]

text: {{albumchart|Australiab

text:
Chart (2013)
Peak
position

Finnish Albums (Suomen virallinen lista)[43]
33

French Albums (SNEP)[44]
174

Irish Albums (IRMA)[45]
64

Italian Albums (FIMI)[46]
38

Norwegian Albums (VG-lista)[47]
2

UK Albums (OCC)[48]
46

UK R&B Albums (OCC)[49]
3

text: Year-end charts[edit]

text:
Chart (2013)
Position

Danish Albums (Hitlisten)[50]
78

text: Certifications[edit]

text:
Region
Certification
Certified units/Sales

Denmark (IFPI Denmark)[51]
Gold
10,000^

^shipments figures based on certification alone

text: Release history[edit]

text:
Country
Date
Format
Label
Ref.

Canada
December 23, 2013
Digital download
The Island Def Jam Music Group
[52]

United States
[53]

United States
February 12, 2016
LP
Universal Music Enterprises
[3]

Canada
February 26, 2016
[54]

Germany
March 25, 2016
[55]

France
[56]

Italy
[57]

United Kingdom
[58]

Spain
April 1, 2016
[59]