Meta tags:

Headings (most frequently used words):

exchange, of, interactions, localized, electron, in, solids, the, interaction, contents, symmetry, between, magnetic, moments, direct, see, also, notes, references, further, reading, spatial, coordinates, inclusion, spin, effects, limitations, heisenberg, hamiltonian, and, model,

Text of the page (most frequently used words):
the (259), displaystyle (107), and (74), #exchange (67), vec (57), spin (38), for (38), langle (31), rangle (31), two (30), are (30), electrons (28), frac (28), with (25), magnetic (24), phi (24), interaction (23), electron (23), mathcal (21), textrm (18), between (17), model (17), this (15), edit (15), wave (15), quantum (14), that (14), particles (14), interactions (13), can (13), from (12), hamiltonian (12), psi (12), fermions (12), heisenberg (11), antisymmetric (11), energy (11), symmetry (11), 978 (10), integral (10), states (10), localized (10), atom (10), which (10), functions (10), characteristic (10), cdot (10), when (10), isbn (9), solids (9), physical (9), not (9), function (9), same (9), one (9), value (9), symmetric (9), tfrac (9), bosons (9), terms (8), pauli (8), state (8), doi (8), system (8), orbitals (8), hydrogen (8), wikipedia (7), theory (7), have (7), but (7), only (7), ferromagnetism (7), principle (6), other (6), their (6), physics (6), direct (6), bibcode (6), repulsion (6), where (6), moment (6), values (6), molecular (6), product (6), atoms (6), has (6), first (6), spatially (6), combinations (6), space (6), text (5), may (5), non (5), page (5), different (5), chemistry (5), exclusion (5), springer (5), magnetism (5), materials (5), issn (5), electric (5), moments (5), both (5), per (5), its (5), given (5), however (5), these (5), sign (5), account (5), all (5), ising (5), 2j_ (5), called (5), right (5), spins (5), matrix (5), overall (5), position (5), combination (5), effect (5), indistinguishable (5), force (5), toggle (4), contents (4), search (4), statistics (4), description (4), press (4), introduction (4), properties (4), mechanics (4), dirac (4), 1926 (4), attraction (4), see (4), stoner (4), atomic (4), cubic (4), either (4), constant (4), sum (4), lattice (4), left (4), due (4), triplet (4), singlet (4), potential (4), each (4), solution (4), molecule (4), changes (4), distance (4), coulomb (4), overlap (4), proton (4), main (4), hide (4), move (4), sidebar (4), view (3), site (3), was (3), links (3), short (3), international (3), series (3), sciences (3), 540 (3), 1007 (3), jülich (3), pdf (3), oxford (3), metal (3), new (3), york (3), chemical (3), society (3), 120 (3), van (3), vleck (3), london (3), second (3), references (3), total (3), also (3), delocalized (3), internuclear (3), case (3), example (3), very (3), crystal (3), field (3), will (3), thus (3), orbital (3), bonding (3), over (3), limitations (3), form (3), corresponds (3), momenta (3), above (3), simple (3), heis (3), effects (3), orthogonal (3), eqs (3), written (3), corresponding (3), must (3), found (3), int (3), behave (3), spatial (3), coordinates (3), mechanical (3), exchanged (3), expectation (3), tools (3), languages (2), table (2), contact (2), about (2), privacy (2), policy (2), apply (2), using (2), last (2), categories (2), wayback (2), wikidata (2), retrieved (2), yosida (2), 1998 (2), solid (2), berlin (2), stuart (2), eds (2), 030 (2), 63101 (2), vol (2), koch (2), erik (2), 2012 (2), paul (2), clarendon (2), structure (2), further (2), reading (2), bond (2), ferromagnets (2), american (2), 353 (2), 1932 (2), susceptibilities (2), müller (2), kirsten (2), university (2), section (2), 2007 (2), griffiths (2), david (2), book (2), journal (2), possible (2), theorem (2), royal (2), 112 (2), cambridge (2), edition (2), there (2), effective (2), boson (2), confused (2), notes (2), method (2), coupling (2), intermediary (2), superexchange (2), used (2), slater (2), mechanism (2), valence (2), considered (2), while (2), particular (2), distances (2), contribute (2), systems (2), nickel (2), predicted (2), bohr (2), magnetons (2), based (2), observed (2), density (2), hence (2), expected (2), difference (2), tends (2), cannot (2), cases (2), band (2), more (2), accordingly (2), because (2), heitler (2), responsible (2), been (2), small (2), magnitude (2), determined (2), via (2), neq (2), scalar (2), dimensional (2), centered (2), related (2), quantity (2), gives (2), ranged (2), termed (2), out (2), although (2), general (2), compared (2), cause (2), consequences (2), favors (2), negative (2), rather (2), critical (2), coupled (2), therefore (2), diagonal (2), eigenvalues (2), square (2), respectively (2), obtain (2), symmetrical (2), anti (2), alpha (2), beta (2), did (2), variables (2), inclusion (2), calculation (2), integrals (2), variational (2), similar (2), results (2), requirement (2), host (2), hbar (2), nabla (2), sqrt (2), uses (2), taking (2), independently (2), occupy (2), since (2), integer (2), constraint (2), sometimes (2), give (2), increases (2), big (2), identical (2), together (2), decreases (2), distinguishable (2), subject (2), classical (2), appearance (2), upload (2), file (2), history (2), read (2), article (2), subsection (2), log (2), create (2), donate (2), menu (2), add, topic, mobile, cookie, statement, developers, code, conduct, legal, safety, contacts, disclaimers, available, under, additional, you, agree, registered, trademark, profit, organization, wikimedia, foundation, inc, use, creative, commons, attribution, sharealike, license, rendered, parsoid, edited, 2026, utc, hidden, webarchive, template, articles, https, org, index, php, title, exchange_interaction, oldid, 1354292863, yale, lux, gnd, authority, control, databases, kei, heidelberg, 60651, skomski, ralph, 2020, coey, michael, parkin, cham, publishing, 2024, 7_2, handbook, mattis, daniel, 1981, 0171, 1873, 18425, 642, 83238, statics, dynamics, pavarini, eva, anders, frithjof, jarrell, mark, schriften, des, forschungszentrums, reihe, modeling, simulation, forschungszentrum, gmbh, 89336, 796, correlated, models, mechanisms, cox, anthony, 1992, monographs, 153, 855570, 148, transition, oxides, electronic, goodenough, interscience, 1966, elliot, john, wiley, sons, 615, marshall, 1960, aps, 357, 0031, 899x, 1103, physrev, 1960phrv, 353s, review, 318, schrödinger, equation, path, harald, 2nd, world, scientific, 981, 4397, chapter, xii, robert, white, 3rd, rev, 65116, rebecca, hihinashvili, accessed, line, october, machine, 2021, archived, derivation, schroeter, darrell, 2018, 316, 99543, 1017, 9781316995433, 2018iqm, blum, alexander, 2014, 574, 2102, 6467, 1140, epjh, e2014, 50022, 2014epjh, 543b, 543, european, necessary, genesis, 762, 677, 94692, jstor, 1364, 5021, 1098, rspa, 0133, 1926rspsa, 661d, 661, proceedings, mathematical, engineering, 426, bf01397160, 1926zphy, 411h, 411, zeitschrift, für, physik, mehrkörperproblem, und, resonanz, der, quantenmechanik, girvin, steven, yang, kun, 2019, 384, 107, 13739, modern, condensed, matter, 207, 210, mullin, blaylock, 2003, 1231, 0002, 9505, 1119, 1590658, 2003amjph, 1223m, 0304067, arxiv, 1223, fermion, contribution, higher, order, chemicals, multipolar, numeric, holstein, herring, strong, cations, through, anion, build, multi, fermionic, determinant, ions, oxidation, double, generally, best, intermediate, depending, substances, rare, earth, currently, accepted, ruderman, kittel, kasuya, rkky, net, close, calculated, saturation, induction, weight, contrast, isolated, unpaired, would, measured, along, axis, observable, configuration, ferromagnet, number, majority, minority, permits, 0023, overestimate, explain, hund, mulliken, bloch, realistic, applicable, presumes, involved, context, adequate, explaining, electrically, insulating, narrow, ionic, covalent, picture, reasonable, nevertheless, theoretical, evaluations, display, metallic, conductivity, itinerant, iron, cobalt, historically, wrong, much, too, experimentally, estimated, curie, temperatures, averaged, sites, approx, 3k_, identically, except, angular, replaced, invented, 1920, solved, his, doctoral, student, 1925, defined, ernst, wilhelm, lenz, dot, fcc, 4j_, face, bcc, body, factor, introduced, counted, twice, performing, sums, note, yet, another, serves, ferromagnetic, material, relationship, dependent, parameter, stiffness, generalization, taken, hamiltonians, pairs, many, normally, confined, intra, nearest, neighbor, longer, occur, splittings, elusive, calculate, large, analytical, formulae, worked, herein, ion, nature, primarily, pair, negligibly, positive, parallel, primary, severe, antiparallel, potentially, causing, essentially, relative, sizes, deduced, expression, energies, antiferromagnetism, below, labeled, namely, follows, named, older, literature, pointed, features, could, obtained, elementary, way, neglecting, hand, side, thereby, considering, simply, having, aforementioned, relations, seen, alternatively, represented, say, elements, after, converted, now, resultant, matrices, presented, following, analysis, resulting, vice, versa, determinants, equations, include, down, suggested, some, ratio, mean, radial, extension, detailed, including, evaluation, application, minimum, radical, 272, 292, ground, yields, diagonalized, interpreted, repulsive, point, created, distributed, probability, includes, interpretation, shown, arise, entirely, mathrm, denote, remaining, protons, quantities, assumed, real, kinetic, treat, problem, decomposed, into, unperturbed, interacting, perturbing, accounts, full, then, perturbatively, like, attempt, assuming, eigenstate, enforce, indistinguishability, constructed, classified, demands, multiple, they, means, interchanged, respect, explained, neglect, half, significant, geometrical, increasing, curvature, wavefunctions, prevent, occupied, intuitive, intuition, incorrect, 291, imply, term, reduces, physically, 258, fundamentally, demonstrated, type, coordinate, labels, wavefunction, ways, exchanging, plus, minus, those, wolfgang, were, discovered, physicists, werner, makes, them, bunch, takes, causes, closer, bose, einstein, condensation, arises, creates, hole, vicinity, tend, avoid, associated, align, analogue, gas, remains, unchanged, alters, always, ideas, experience, mediated, carrier, redirected, free, encyclopedia, item, projects, printable, version, download, print, export, switch, legacy, parser, get, shortened, url, cite, information, permanent, link, what, here, actions, english, talk, tiếng, việt, oʻzbekcha, ўзбекча, українська, svenska, русский, português, polski, 한국어, қазақша, 日本語, עברית, français, فارسی, eesti, español, deutsch, català, български, беларуская, العربية, top, personal, special, pages, recent, community, portal, learn, help, random, current, events, navigation, jump, content,

Text of the page (random words):
ic moments subsection 2 1 exchange of spatial coordinates 2 2 inclusion of spin 2 3 effects of exchange 3 direct exchange interactions in solids toggle direct exchange interactions in solids subsection 3 1 limitations of the heisenberg hamiltonian and the localized electron model in solids 4 see also 5 notes 6 references 7 further reading toggle the table of contents exchange interaction 21 languages العربية беларуская български català deutsch español eesti فارسی français עברית 日本語 қазақша 한국어 polski português русский svenska українська oʻzbekcha ўзбекча tiếng việt 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia redirected from exchange energy not to be confused with a boson exchange interaction mediated by a force carrier quantum mechanical effect in chemistry and physics the exchange interaction is a quantum mechanical constraint on the states of indistinguishable particles while sometimes called an exchange force or in the case of fermions pauli repulsion its consequences cannot always be predicted based on classical ideas of force 1 both bosons and fermions can experience the exchange interaction the wave function of indistinguishable particles is subject to exchange symmetry the wave function either changes sign for fermions or remains unchanged for bosons when two particles are exchanged the exchange symmetry alters the expectation value of the distance between two indistinguishable particles when their wave functions overlap for fermions the expectation value of the distance increases and for bosons it decreases compared to distinguishable particles 2 the exchange interaction arises from the combination of exchange symmetry and the coulomb interaction for an electron in an electron gas the exchange symmetry creates an exchange hole in its vicinity which other electrons with the same spin tend to avoid due to the pauli exclusion principle this decreases the energy associated with the coulomb interactions between the electrons with same spin 3 since two electrons with different spins are distinguishable from each other and not subject to the exchange symmetry the effect tends to align the spins the exchange interaction is the main physical effect responsible for ferromagnetism and has no classical analogue for bosons the exchange symmetry makes them bunch together and the exchange interaction takes the form of an effective attraction that causes identical particles to be found closer together as in bose einstein condensation exchange interaction effects were discovered independently by physicists werner heisenberg and paul dirac in 1926 4 5 exchange symmetry edit quantum particles are fundamentally indistinguishable wolfgang pauli demonstrated that this is a type of symmetry states of two particles must be either symmetric or antisymmetric when coordinate labels are exchanged 6 in a simple one dimensional system with two identical particles in two states ψ a displaystyle psi _ a and ψ b displaystyle psi _ b the system wavefunction can therefore be written two ways ψ a x 1 ψ b x 2 ψ a x 2 ψ b x 1 displaystyle psi _ a x_ 1 psi _ b x_ 2 pm psi _ a x_ 2 psi _ b x_ 1 exchanging x 1 displaystyle x_ 1 and x 2 displaystyle x_ 2 gives either a symmetric combination of the states plus or an antisymmetric combination minus particles that give symmetric combinations are called bosons those with antisymmetric combinations are called fermions the two possible combinations imply different physics for example the expectation value of the square of the distance between the two particles is 7 258 x 1 x 2 2 x 2 a x 2 b 2 x a x b 2 x a b 2 displaystyle langle x_ 1 x_ 2 2 rangle _ pm langle x 2 rangle _ a langle x 2 rangle _ b 2 langle x rangle _ a langle x rangle _ b mp 2 big langle x rangle _ ab big 2 the last term reduces the expected value for bosons and increases the value for fermions but only when the states ψ a displaystyle psi _ a and ψ b displaystyle psi _ b physically overlap x a b 0 displaystyle langle x rangle _ ab neq 0 the physical effect of the exchange symmetry requirement is not a force rather it is a significant geometrical constraint increasing the curvature of wavefunctions to prevent the overlap of the states occupied by indistinguishable fermions the terms exchange force and pauli repulsion for fermions are sometimes used as an intuitive description of the effect but this intuition can give incorrect physical results 1 7 291 exchange interactions between localized electron magnetic moments edit quantum mechanical particles are classified as bosons or fermions the spin statistics theorem of quantum field theory demands that all particles with half integer spin behave as fermions and all particles with integer spin behave as bosons multiple bosons may occupy the same quantum state however by the pauli exclusion principle no two fermions can occupy the same state since electrons have spin 1 2 they are fermions this means that the overall wave function of a system must be antisymmetric when two electrons are exchanged i e interchanged with respect to both spatial and spin coordinates first however exchange will be explained with the neglect of spin exchange of spatial coordinates edit taking a hydrogen molecule like system i e one with two electrons one may attempt to model the state of each electron by first assuming the electrons behave independently that is as if the pauli exclusion principle did not apply and taking wave functions in position space of φ a r 1 displaystyle phi _ a r_ 1 for the first electron and φ b r 2 displaystyle phi _ b r_ 2 for the second electron the functions φ a displaystyle phi _ a and φ b displaystyle phi _ b are orthogonal and each corresponds to an energy eigenstate to enforce the indistinguishability of the two electrons two wave functions for the overall system in position space can be constructed one uses an antisymmetric combination of the product wave functions in position space ψ a r 1 r 2 1 2 φ a r 1 φ b r 2 φ b r 1 φ a r 2 displaystyle psi _ rm a vec r _ 1 vec r _ 2 frac 1 sqrt 2 phi _ a vec r _ 1 phi _ b vec r _ 2 phi _ b vec r _ 1 phi _ a vec r _ 2 1 the other uses a symmetric combination of the product wave functions in position space ψ s r 1 r 2 1 2 φ a r 1 φ b r 2 φ b r 1 φ a r 2 displaystyle psi _ rm s vec r _ 1 vec r _ 2 frac 1 sqrt 2 phi _ a vec r _ 1 phi _ b vec r _ 2 phi _ b vec r _ 1 phi _ a vec r _ 2 2 to treat the problem of the hydrogen molecule perturbatively the overall hamiltonian is decomposed into an unperturbed hamiltonian of the non interacting hydrogen atoms h 0 displaystyle mathcal h 0 and a perturbing hamiltonian which accounts for interactions between the two atoms h 1 displaystyle mathcal h 1 the full hamiltonian is then h h 0 h 1 displaystyle mathcal h mathcal h 0 mathcal h 1 where h 0 ℏ 2 2 m 1 2 ℏ 2 2 m 2 2 e 2 r a 1 e 2 r b 2 displaystyle mathcal h 0 frac hbar 2 2m nabla _ 1 2 frac hbar 2 2m nabla _ 2 2 frac e 2 r_ a1 frac e 2 r_ b2 and h 1 e 2 r a b e 2 r 12 e 2 r a 2 e 2 r b 1 displaystyle mathcal h 1 left frac e 2 r_ ab frac e 2 r_ 12 frac e 2 r_ a2 frac e 2 r_ b1 right the first two terms of h 0 displaystyle mathcal h 0 denote the kinetic energy of the electrons the remaining terms account for attraction between the electrons and their host protons r a 1 b 2 displaystyle r_ a1 b2 the terms in h 1 displaystyle mathcal h 1 account for the potential energy corresponding to proton proton repulsion r a b displaystyle r_ ab electron electron repulsion r 12 displaystyle r_ 12 and electron proton attraction between the electron of one host atom and the proton of the other r a 2 b 1 displaystyle r_ a2 b1 all quantities are assumed to be real two eigenvalues for the system energy are found e e 0 c j e x 1 s 2 displaystyle e_ pm e_ 0 frac c pm j_ rm ex 1 pm mathcal s 2 3 where the e displaystyle e_ is the spatially symmetric solution and e displaystyle e_ is the spatially antisymmetric solution corresponding to ψ s displaystyle psi _ rm s and ψ a displaystyle psi _ rm a respectively a variational calculation yields similar results h displaystyle mathcal h can be diagonalized by using the position space functions given by eqs 1 and 2 in eq 3 c displaystyle c is the two site two electron coulomb integral it may be interpreted as the repulsive potential for electron one at a particular point φ a r 1 2 displaystyle phi _ a vec r _ 1 2 in an electric field created by electron two distributed over the space with the probability density φ b r 2 2 displaystyle phi _ b vec r _ 2 2 s displaystyle mathcal s a is the overlap integral and j e x displaystyle j_ mathrm ex is the exchange integral which is similar to the two site coulomb integral but includes exchange of the two electrons it has no simple physical interpretation but it can be shown to arise entirely due to the anti symmetry requirement these integrals are given by c φ a r 1 2 1 r a b 1 r 12 1 r a 1 1 r b 2 φ b r 2 2 d 3 r 1 d 3 r 2 displaystyle c int phi _ a vec r _ 1 2 left frac 1 r_ ab frac 1 r_ 12 frac 1 r_ a1 frac 1 r_ b2 right phi _ b vec r _ 2 2 d 3 r_ 1 d 3 r_ 2 4 s φ b r 2 φ a r 2 d 3 r 2 displaystyle mathcal s int phi _ b vec r _ 2 phi _ a vec r _ 2 d 3 r_ 2 5 j e x φ a r 1 φ b r 2 1 r a b 1 r 12 1 r a 1 1 r b 2 φ b r 1 φ a r 2 d 3 r 1 d 3 r 2 displaystyle j_ rm ex int phi _ a vec r _ 1 phi _ b vec r _ 2 left frac 1 r_ ab frac 1 r_ 12 frac 1 r_ a1 frac 1 r_ b2 right phi _ b vec r _ 1 phi _ a vec r _ 2 d 3 r_ 1 d 3 r_ 2 6 although in the hydrogen molecule the exchange integral eq 6 is negative heisenberg first suggested that it changes sign at some critical ratio of internuclear distance to mean radial extension of the atomic orbital 8 9 10 the detailed calculation including evaluation of the above integrals with ground state hydrogen atom wave functions and application of the variational principle to obtain the minimum energy in both cases of the hydrogen molecule for atomic orbitals and the hydrogen radical i e with only one electron for molecular orbitals can be found in the book of müller kirsten pp 272 292 only in the second edition 11 inclusion of spin edit the symmetric and antisymmetric combinations in equations 1 and 2 did not include the spin variables α spin up β spin down there are also antisymmetric and symmetric combinations of the spin variables α 1 β 2 α 2 β 1 displaystyle alpha 1 beta 2 pm alpha 2 beta 1 7 to obtain the overall wave function these spin combinations have to be coupled with eqs 1 and 2 the resulting overall wave functions called spin orbitals are written as slater determinants when the orbital wave function is symmetrical the spin wave function must be anti symmetrical and vice versa accordingly e displaystyle e_ above corresponds to the spatially symmetric spin singlet solution and e displaystyle e_ to the spatially antisymmetric spin triplet solution j h van vleck presented the following analysis 12 the potential energy of the interaction between the two electrons in orthogonal orbitals can be represented by a matrix say e ex displaystyle e_ textrm ex from eq 3 the characteristic values of this matrix are c j ex displaystyle c pm j_ textrm ex the characteristic values of a matrix are its diagonal elements after it is converted to a diagonal matrix that is eigenvalues now the characteristic values of the square of the magnitude of the resultant spin s a s b 2 displaystyle langle vec s _ a vec s _ b 2 rangle is s s 1 displaystyle s s 1 the characteristic values of the matrices s a 2 displaystyle langle vec s _ a 2 rangle and s b 2 displaystyle langle vec s _ b 2 rangle are each 1 2 1 2 1 3 4 displaystyle tfrac 1 2 tfrac 1 2 1 tfrac 3 4 and s a s b 2 s a 2 s b 2 2 s a s b displaystyle langle vec s _ a vec s _ b 2 rangle langle vec s _ a 2 rangle langle vec s _ b 2 rangle 2 langle vec s _ a cdot vec s _ b rangle the characteristic values of the scalar product s a s b displaystyle langle vec s _ a cdot vec s _ b rangle are 1 2 0 6 4 3 4 displaystyle tfrac 1 2 0 tfrac 6 4 tfrac 3 4 and 1 2 2 6 4 1 4 displaystyle tfrac 1 2 2 tfrac 6 4 tfrac 1 4 corresponding to both the spin singlet s 0 displaystyle s 0 and spin triplet s 1 displaystyle s 1 states respectively from eq 3 and the aforementioned relations the matrix e ex displaystyle e_ textrm ex is seen to have the characteristic value c j ex displaystyle c j_ textrm ex when s a s b displaystyle langle vec s _ a cdot vec s _ b rangle has the characteristic value 3 4 i e when s 0 displaystyle s 0 the spatially symmetric spin singlet state alternatively it has the characteristic value c j ex displaystyle c j_ textrm ex when s a s b displaystyle langle vec s _ a cdot vec s _ b rangle has the characteristic value 1 4 i e when s 1 displaystyle s 1 the spatially antisymmetric spin triplet state therefore e e x c 1 2 j e x 2 j e x s a s b 0 displaystyle e_ rm ex c frac 1 2 j_ rm ex 2j_ rm ex langle vec s _ a cdot vec s _ b rangle 0 8 and hence e e x c 1 2 j e x 2 j e x s a s b displaystyle e_ rm ex c frac 1 2 j_ rm ex 2j_ rm ex langle vec s _ a cdot vec s _ b rangle 9 where the spin momenta are given as s a displaystyle langle vec s _ a rangle and s b displaystyle langle vec s _ b rangle dirac pointed out that the critical features of the exchange interaction could be obtained in an elementary way by neglecting the first two terms on the right hand side of eq 9 thereby considering the two electrons as simply having their spins coupled by a potential of the form 2 j a b s a s b displaystyle 2j_ ab langle vec s _ a cdot vec s _ b rangle 10 it follows that the exchange interaction hamiltonian between two electrons in orbitals φ a displaystyle phi _ a and φ b displaystyle phi _ b can be written in terms of their spin momenta s a displaystyle vec s _ a and s b displaystyle vec s _ b this interaction is named the heisenberg exchange hamiltonian or the heisenberg dirac hamiltonian in the older literature h h e i s 2 j a b s a s b displaystyle mathcal h _ rm heis 2j_ ab langle vec s _ a cdot vec s _ b rangle 11 j ab displaystyle j_ textrm ab is not the same as the quantity labeled j ex displaystyle j_ textrm ex in eq 6 rather j ab displaystyle j_ textrm ab which is termed the exchange constant is a function of eqs 4 5 and 6 namely j a b 1 2 e e j e x c s 2 1 s 4 displaystyle j_ ab frac 1 2 e_ e_ frac j_ rm ex c mathcal s 2 1 mathcal s 4 12 however with orthogonal orbitals in which s displaystyle mathcal s 0 for example with different orbitals in the same atom j ab j ex displaystyle j_ textrm ab j_ textrm ex effects of exchange edit if j ab displaystyle j_ textrm ab is positive the exchange energy favors electrons with parallel spins this is a primary cause of ferromagneti...