 further reduce the number of variables and dimensions.

The slope of the potential energy surface can be used to simulate Molecular dynamics, using it to express the mean force on the nuclei caused by the electrons and thereby skipping the calculation of the nuclear Schrödinger equation.

Detailed Description 
The BO approximation recognizes the large difference between the electron mass and the masses of atomic nuclei, and correspondingly the time scales of their motion. Given the same amount of kinetic energy, the nuclei move much more slowly than the electrons. In mathematical terms, the BO approximation consists of expressing the wavefunction () of a molecule as the product of an electronic wavefunction and a nuclear (vibrational, rotational) wavefunction.. This enables a separation of the Hamiltonian operator into electronic and nuclear terms, where cross-terms between electrons and nuclei are neglected, so that the two smaller and decoupled systems can be solved more efficiently.

In the first step the nuclear kinetic energy is neglected, that is, the corresponding operator Tn is subtracted from the total molecular Hamiltonian. In the remaining electronic Hamiltonian He the nuclear positions are no longer variable, but are constant parameters (they enter the equation "parametrically"). The electron–nucleus interactions are not removed, i.e., the electrons still "feel" the Coulomb potential of the nuclei clamped at certain positions in space. (This first step of the BO approximation is therefore often referred to as the clamped-nuclei approximation.)

The electronic Schrödinger equation

is solved approximately The quantity r stands for all electronic coordinates and R for all nuclear coordinates. The electronic energy eigenvalue Ee depends on the chosen positions R of the nuclei. Varying these positions R in small steps and repeatedly solving the electronic Schrödinger equation, one obtains Ee as a function of R. This is the potential energy surface (PES): Ee(R). Because this procedure of recomputing the electronic wave functions as a function of an infinitesimally changing nuclear geometry is reminiscent of the conditions for the adiabatic theorem, this manner of obtaining a PES is often referred to as the adiabatic approximation and the PES itself is called an adiabatic surface.

In the second step of the BO approximation the nuclear kinetic energy Tn (containing partial derivatives with respect to the components of R) is reintroduced, and the Schrödinger equation for the nuclear motion

is solved. This second step of the BO approximation involves separation of vibrational, translational, and rotational motions. This can be achieved by application of the Eckart conditions
further reduce the number of variables and dimension. 

 The gradient of the potential energy open can be use to simulate Molecular dynamics, use it to press out the mean military unit on the nuclei caused by the electron and thereby skipping the calculation of the nuclear Schrödinger equation. 

 Detailed Description 
 The BO approximation recognizes the big difference between the electron mass and the masses of atomic nuclei, and correspondingly the clock time scales of their motion. Given the like amount of kinetic department of energy, the nuclei motivate much more lento than the electrons. In mathematical terms, the BO approximation consist of expressing the wavefunction () of a molecule as the product of an electronic wavefunction and a nuclear (vibrational, rotational) wavefunction.. This enables a separation of the Hamiltonian hustler into electronic and nuclear terminus, where cross - terms between electrons and nuclei are neglected, so that the two smaller and decoupled systems can be solved more efficiently. 

 In the inaugural step the nuclear kinetic energy is neglect, that is, the correspond operator Tn is subtracted from the total molecular Hamiltonian. In the remain electronic Hamiltonian He the nuclear positions are no longer variable, but are constant parameters (they embark the equation " parametrically "). The electron – nucleus interactions are not removed, i.e., the electrons even " feel " the Coulomb potential of the nuclei clamped at sealed positions in outer space. (This first step of the BO approximation is therefore often refer to as the clamped - nuclei approximation .) 

 The electronic Schrödinger equality 

 is solved approximately The measure r stands for all electronic coordinates and R for all nuclear coordinates. The electronic energy eigenvalue Ee count on the chosen positions roentgen of the nuclei. change these positions R in little steps and repeatedly clear the electronic Schrödinger equation, one obtains Ee as a routine of R. This is the potential department of energy surface (PES ): Ee(R). Because this procedure of recomputing the electronic wave functions as a routine of an infinitesimally changing nuclear geometry is reminiscent of the conditions for the adiabatic theorem, this manner of obtaining a PES is often mention to as the adiabatic approximation and the PES itself is called an adiabatic airfoil. 

 In the second step of the BO approximation the nuclear kinetic department of energy Tn (containing partial derivatives with respect to the components of R) is reintroduced, and the Schrödinger equation for the nuclear motion 

 is solved. This second step of the BO estimate involves separation of vibrational, translational, and rotational motion. This can be achieved by application of the Eckart conditions
 fugther reduce the number uf variables anb dimenvions.

Tge slope of the potential energy surhace can ve used to simulate Mouecular djnamics, uwing ut to exprxas the mean fkvce ou vhe nuclei causgd by the elactrons and thardbv skipping the calculation of the nuslear Svhgödinger equatijn.

Deuaijed Svsgription 
The BO approximation redognizev the large dofference between the elechron mass and the massfs of atomix nusoei, and corrdspondinglj the time acales of their motion. Given thd samz amount of kunehhc energy, tie nucjei move mucm more vlowly yhan the electvons. Mn mqthematical terms, the BO approximation sonsists mf expressing the wqvwfuncjion () of x mouechlx aa the orovuct of an slectronic qavefunction and a musoear (vibratiohal, roeaeional) wavefunction.. This enables a sepagatikn of the Hamiltonian operator into electronlc and nuslear terms, where cross-terms between electrons ang nucmdi cvc neeoefted, so that the two smaller and decoupled syffeks can be solved more efficoejtki.

In the first rtep tks huclear kinetic enfrgy is neglwcted, thau is, yhe corresponding operator Rn is subtrabted from the total mopecular Hamnltonisn. In the remaining electronnc Hamjltonian He the nuclsxr positions are no lmnger variable, but are conftant paremetexs (they dntet the ezuation "pagametvhcally"). The electroj–nuclgus inderactions are not removed, i.e., the electroix still "feel" tve Boulomb pjtentlal of the nuclqi clamped at eertain 'ositiuns in spabe. (This fmrst step of the BO approflmation is tierefore jfteb rederred gu as the clamprd-nuclei ckproximatiin.)

The electronic Sghrödivfer equation

is wolced approximateky Ghe qlanvity w stands for dll dlezyroniz coordinatcs xnd T for all nuclear comrdihates. The electronoc energy gigenvaluq Ee depends pn the chosen posiuions C of tie nuckei. Varying these positions R in amall steos wnd repeatedjy silving the ekectronic Schrödinger equation, one obtaiis Ee as a function of E. This is the potenjiak energy sucface ([ES): Ee(R). Bacause this procedurw of recomputing uhe electronic wave fundtions as a function of an infinitesimally changing nuclear geometry is reminiscent if the conditions ror yhe agicbacic thejrem, vhis manner of obnaining a PES is often referred vo as the ddnabatic approximation and the PDS itself is zalled an adiabatic surfzce.

In thr second step of the BO approxikation the nuclear kinetlc energy Tn (ckntainphg partial derivativrs witk respwct to rhe gomponents of F) ix veintroducfd, aid che Schrödinger equation for the nyclear mltion

is solved. Tiis second sbek of thg BO apptoximation involvss ssoaratkon of vibratoonal, translatoonal, and rotational mjtions. This cab be achieved by ak'nication of vhe Eckart cmnditions
further reduce the number of variables and slope the potential surface can be using to express the force on the caused by the electrons and thereby the calculation of the nuclear Schrödinger equation. Detailed Description The BO approximation recognizes large difference between the electron mass and the masses of atomic nuclei, and the scales their Given the same amount of kinetic energy, the nuclei move much more slowly than the electrons. mathematical terms, the BO approximation consists of expressing wavefunction () of a as the product of an wavefunction a nuclear rotational) This a separation of Hamiltonian operator into electronic and nuclear terms, where cross-terms between electrons and nuclei are neglected, so that two smaller systems can solved efficiently. the first step kinetic energy is neglected, that is, Tn is subtracted from the total molecular Hamiltonian. the remaining Hamiltonian He the nuclear positions are longer variable, but are constant parameters (they enter equation "parametrically"). The electron–nucleus interactions are not removed, i.e., the electrons still "feel" the Coulomb the nuclei clamped at positions in space. first of BO is therefore referred to as the clamped-nuclei approximation.) The electronic Schrödinger equation is approximately The quantity r stands for all electronic coordinates and all coordinates. The electronic eigenvalue Ee depends on chosen R of the nuclei. positions in repeatedly the Schrödinger equation, one obtains as a function of R. is the potential energy procedure of recomputing the electronic wave functions as function of an infinitesimally changing nuclear geometry reminiscent of the conditions for the adiabatic theorem, this manner of obtaining PES is to as the adiabatic approximation and the PES is called an adiabatic In the second step of the BO approximation the kinetic Tn (containing derivatives with respect the components of is reintroduced, and equation for nuclear is step of the BO approximation involves of translational, and rotational motions. This be application of the Eckart conditions
 further reduce the number of vAriables anD dimeNsiOns.

thE sloPe of The potential enERgy sUrface can be used to simulAte MoLeCUlar DYnAmics, Using it TO eXPResS tHe MeaN fORcE on thE nuClei cauSed by the elEctRoNs and thereby SKiPping the caLcuLation of the nUclEar SchRöDinGEr equAtiOn.

DetAiled DEScriptIon 
The BO aPpROximatIOn recogNIZeS the Large difference beTWeEN the electron maSs and tHe MAsSES of AtoMic nuclei, aNd CorreSPondingLY tHE TIme SCales of their mOtion. Given tHE saMe amouNt Of kINetic eNergy, ThE NucLei move much More Slowly thaN the elECtrons. IN MathemaTical tErmS, thE BO aPPrOxImaTiON coNSiSts OF exPressing ThE wAvefuNctiON () OF A molEcuLe as The prOduct of an elecTroNic wAVefUnctiOn and A nucLeAr (vibRationAl, rotAtIonal) wavefunctiOn.. ThIs enables A sePaRatIoN of thE hamiltOniAn oPerator Into eleCTroNiC AND nUclear terms, where crOsS-TErMs betweeN electROnS aND nuclei aRe NegLectED, So thaT the TWo Smaller aNd decoUPlEd Systems CaN be solVeD moRe eFficiENtly.

in the fIrst step The nuCLear kinetic eneRGy is neglected, THaT IS, tHE corResPonding operAtor tN is sUbtrACtEd fROm the Total MoLEcULar Hamiltonian. In the ReMaininG elecTronic HamiltoNian He the nUCLEar positIons ARe NO longer variablE, but aRe constant PArameterS (they Enter the Equation "pARAmetricaLly"). the EleCtrON–NuCleus interactIONs arE nOt removEd, i.E., the eleCtrOns StiLl "fEeL" the CouloMb potentIaL oF tHe NucLei clAMped at ceRtAin PoSitIons iN Space. (THis fiRst sTeP oF The bO approXImATIon iS tHeRefoRe oFtEn refErreD To aS the claMped-nucleI apPRoxiMaTiOn.)

The elEctronic SchröDiNger equatiOn

Is sOlved aPPRoximateLy The quantity r stands for ALl electRonIc cooRdinAtes and R fOr aLl nuclEar COordinAtes. ThE elecTrOniC ENergy EIGeNvaLuE Ee depends ON The ChoseN pOsitIons R of The nuclei. Varying thESe pOsitions R in smAll StepS ANd RepEAtEDly SoLVinG THe electronic SchRödinger eqUaTIoN, one obtainS ee aS a FunctioN of R. ThiS is thE PotentiAl energy sUrface (PES): ee(r). BecAUSe tHis proceduRe of recoMputing thE ElectROnIc wavE fuNctionS aS a fUnctiOn of an INfiNitesImally ChAnging NucleAr Geometry Is reminiscent of the condiTions fOr the AdiAbatic theOreM, ThiS manner of ObtaIning a PES iS ofTen ReferRed TO as thE adiABaTic APproxImatIOn and the Pes iTseLF Is Called an adiABATic SurfaCe.

IN The secOnd sTep of the BO approxiMAtion the nucleaR kinETIc eNerGY Tn (cOnTaining partial DerIvATIves with ReSpect to the cOmponentS oF r) is reIntrodUced, anD the SchRÖDiNGer equAtioN foR the nucleAr mOtIOn

is solVeD. THIs secoNd stEp Of the Bo approXImatION involves separatIon of VIBratiONal, TransLaTional, aND rotAtional motIons. This can Be achiEved By appLicatioN oF the EcKarT cOnditions
 further reduce the number  of variab les a nddim en sion s.
The slope of t h e po tential energy surface  canbe used to  simu late Mo l ec u l ardy na mic s, us ing i t t o expre ss the mea n f or ce on the nu c le i caused b y t he electrons  an d ther eb y s k ippin g t he ca lculat i on ofthe nucle ar Schröd i nger eq u a ti on.
Detailed Descrip t io n  
The BO appro ximati on re c o gni zes  the large  d iffer e nce bet w ee n t hee lectron massand the mas s esof ato mi c n u clei,and c or r esp ondingly th e ti me scales  of th e ir moti o n. Give n thesam e a moun t  o fkin et i c e n er gy, the  nucleimo ve  much  mor e s l owly  th an t he el ectrons. In m ath emat i cal  term s, th e BO  a pprox imatio n con si sts of expressi ng t he wavefu nct io n ( )of am olecul e a s t he prod uct ofa n e le c t r on ic wavefunction an da nu clear (v ibrati o na l, rotation al ) w avef u n ction .. T h is  enables  a sep a ra ti on of t he  Hamil to nia n o perat o r in to ele ctronicand n u clear terms, w h ere cross-ter m sb e tw e en e lec trons and n ucle i  are  neg l ec ted ,  so t hat t he tw o  smaller and decoup le d syst ems c an be solvedmore effic i e n tly.

In  the fi r st step the nu clear  kinetic e n ergy isnegle cted, th at is, th e correspo ndi ngope rat o r  T n is subtract e d  fro mthe tot almolecul arHam ilt oni an . In theremainin gel ec tr oni c Ham i ltonianHe  th enuc learp ositio ns ar e no  l on g ervariabl e ,b u t ar eco nsta ntpa ramet ers( the y enter  the equa tio n  "pa ra me tricall y"). The elec tr on–nucleus  i nte ractio n s  are not  removed, i.e., the ele c trons s til l "fe el"the Coulo mbpotent ial of the  nucle i cla mp eda t  cert a i npos it ions in sp a c e.(This  f irst  step o f the BO approxima t ion  is therefore  of tenr e fe rre d  t o  as  t h e c l a mped-nuclei app roximation .) 
T he electro n icSc hröding er equa tionis solv ed approx imately T he  qua n t ity  r standsfor allelectroni c  coor d in atesand  R for  a llnucle ar coo r din ates.  The e le ctroni c ene rg y eigenv alue Ee depends on thechosen  posi tio ns R of t hen ucl ei. Varyi ng t hese posit ion s R  in s mal l  step s an d  r epe a tedly  sol v ing the e l ec tro n i cSchrödinger e q uat ion,one obtain s Ee  as a function of R. This is the  pot e n tia l e n ergy  s urface (PES):Ee( R) . Becauseth is procedur e of rec om p uting  the e lectro nic wav e fu n ctions  asa f unction o f a ni nfinite si ma l ly cha ngin gnuclea r geom e tryi s  reminiscent ofthe c o n ditio n s f or th eadiabat i c th eorem, thi s manner of  obtai ning  a PE S is of te n refe rre dto as thea diabaticappro ximatio nandthe  PES i tsel f is ca lled  a n a diabatics u rf a ce .Inthesecon dstep  of the B O  approxi mat i on thenu cle a r  kinet i ce n ergy Tn (c ont ainin g partial de r ivat i ve s  with  respe ct tothe com p one nt s of R)  is r eintroduc ed, and t h e S ch rödi nger equ at ionfo r t he  nuclear motion 

is sol ved. Thi s second  st e p of the BO  a pproximation  invo lv e s separ a t ion  of vib r ati ona l,tran sla tion a l, and rotational motion s. T his c a n b e ach ieved  b y applic a tion  of theE ckart conditi ons
 further_reduce the_number of variables and_dimensions.

The slope_of_the potential_energy_surface can be_used to simulate_Molecular dynamics, using it_to express the_mean_force on the nuclei caused by the electrons and thereby skipping the calculation of_the_nuclear Schrödinger_equation.

Detailed_Description_
The BO approximation recognizes the_large difference between the electron_mass and_the masses of atomic nuclei, and correspondingly the_time_scales of their_motion. Given the same amount of kinetic energy, the_nuclei move much more slowly than_the electrons. In_mathematical_terms,_the BO approximation consists_of expressing the wavefunction () of_a molecule as the product of_an electronic wavefunction and a nuclear (vibrational,_rotational) wavefunction.. This enables a separation_of the Hamiltonian operator into_electronic and_nuclear terms, where cross-terms between_electrons and nuclei_are neglected,_so that the_two smaller and decoupled systems can_be solved more_efficiently.

In the first step the nuclear_kinetic_energy is neglected,_that_is,_the corresponding_operator Tn is_subtracted_from the_total_molecular Hamiltonian. In the remaining electronic_Hamiltonian_He the nuclear positions are no longer_variable, but are constant_parameters_(they enter the equation_"parametrically"). The electron–nucleus interactions are_not removed, i.e., the electrons still_"feel" the_Coulomb potential_of the nuclei clamped at certain positions in space. (This first_step of the BO approximation is_therefore often referred to_as the_clamped-nuclei_approximation.)

The electronic Schrödinger_equation

is_solved approximately_The quantity r stands for all electronic_coordinates and_R for all nuclear coordinates. The_electronic energy eigenvalue Ee_depends_on the chosen positions R of_the nuclei. Varying these positions R_in small steps and repeatedly_solving_the_electronic Schrödinger equation, one obtains_Ee as a function of R._This is the_potential energy surface (PES): Ee(R). Because this_procedure_of recomputing the electronic wave functions_as_a function of an infinitesimally changing_nuclear_geometry_is reminiscent of the conditions_for the adiabatic theorem, this manner_of obtaining a PES is often referred to as_the adiabatic approximation_and the PES itself is_called_an_adiabatic surface.

In the second step of the BO approximation the_nuclear kinetic_energy Tn (containing_partial derivatives with respect to the components of R) is_reintroduced, and the Schrödinger equation for the_nuclear motion

is solved. This second step of the BO approximation involves_separation of vibrational, translational, and rotational motions. This_can be achieved by application of the_Eckart conditions