Born-Oppenheimer Approximation. Kaito Takahashi
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1 o-oppehee ppoato Kato Takahah
2 toc Ut Fo quatu yte uch a ecto ad olecule t eae to ue ut that ft the=tomc UNT Ue a of ecto (ot kg) Ue chage of ecto (ot coulob) Ue hba fo agula oetu (ot kg - ) Ue 4pe 0 fo pettvty (ot C kg - -3 )
3 Pote Molecule O You alway wte whee the ucleu but you eve wte the ecto o the ecto ae wtte a a le!!! YOU LDY SSUMNG ON-OPPNM PPOXMTON 3
4 Solve the hydoge ato pe 4 0 e a e e h e pe e p e / 0 / 3 ep!! a L a a a l l l l l p co!! ep / / l l l l l l P l l l Y 4
5 Full Poble V M V T N e 4 0 pe e V Z Y X z y z y z y e z y e e e Z Y e e X 5
6 o-oppehee ppoato wod Ma of ecto veu a of ucleu <<< 830 (at leat) 6
7 O ppoato equato N N e M T V T T V T V T T V M V T 0 0 7
8 O ppoato * 0 * * d T d T d 0 8
9 O ppoato 3 d 0 * d M d T N * * d d * * * Ug a-ket Notato 9
10 O ppoato 4 d M N * d M N * N N M M N N M M 0
11 O ppoato * N C M d T C N C M 0
12 O ppoato 6 o-oppehee ppoato goe C 0 N M Nuclea wavefucto gve by the epao coeffcet! v v N V M The ucleu ovg a potetal that the eult of aveagg the cotbuto cog fo the ecto at a gve uclea geoety! eece you have epaated the oto of the ecto ad ucleu.
13 Sepaato of Vaable 0 by Te Scale v v Now you ca ay uclea wave fucto o the -th ectoc tate You ca wte the eegy of the ecto a a fucto of the uclea coodate ad code t a a potetal that the ucleu fe. v 3
14 dabatc ppoato clude dagoal couplg te C N N M M C 0 0 * * * * * d d d d d C 4
15 LCO ppoato fo Datoc Molecule Kato Takahah
16 + Mot Sple Dato Solve th poble:. Ue eact oluto. Ue Ft fd way to olve the ectoc wave fucto whe gve dtace betwee to hydoge ucle
17 toc Obtal evew
18 Wavefucto 3
19 p Wavefucto p 3p 4p
20 d Wavefucto 5d 3d 4d
21 LCO To ake the olecula obtal (the ectoc wavefucto fo the olecule) let jut add up each atoc obtal (ectoc wave fucto fo the ato akg the olecule)
22 Vaatoal Theoy tal eact tal c j j j j j j j j j j j j tal c S c c c d c c d c c * * * * * * d S d j j j j * *
23 Vaatoal Theoy j c S 0 * c j j tal j Take devatve wth c * Take devatve wth c j Due to tatoay codto of the oluto c tal j tal 0 c * 0
24 Vaatoal Theoy 0 j j tal j j S c 0 * j tal j S c tal tal tal tal tal tal tal tal tal S S S S S S S S S
25 + C C C C S S 0 S S S
26 S Ovap tegal S e 3
27 toc tegal S
28 chage tegal
29 Solve Secula quato S S 0
30 odg ad tbodg Obtal
31 lecto Dety Dffeece
32 Depedece of Mat leet Dotted le ae the potetal eegy cuve
33 f S=0 the D=0 S>0 the D>0 g oe tha loweg! egy Loweg ad g Loweg t equlbu of + 0 S 0 S S S g S S S
34 odg Obtal Copae Wth act
35 tbodg Obtal
36 ddto of Obtal C ddg the cotbuto fo S C C C C z z z z p C p C p C p C C C ddg the cotbuto fo p z
37 ddtoal Obtal
38 a Set f you ue oe atoc obtal to defe the olecula obtal uually the eegy get cloe to the eact oluto Ug a bgge ba et to decbe the yte oweve bgge ba et you eed oe te to calculate.
39 Datoc Molecule
40 bove equato two + ecto wth ecto ecto epulo
41 Sp Obtal ad Spacal Obtal Whe you code oe tha oe ecto you have to code ot oly the pacal coodate but alo the p agula oetu ad the Fe pcple: Defe a the ued coodate fo ad a P atee Poduct j
42 Slate Deteat j j j j yetc wavefucto afte echage of ecto coodate Geealzato fo ecto yte: Slate Deteat k j k j k j !...
43 What appe f we ue dect S poduct of + oluto tal! altoa doe ot clude ay p te o we could obta the depedece of the eegy ug oly the patal pat of the ectoc wavefucto
44 Potetal egy Cuve coect Docato!
45 What the Poble of coect docato Ft two te have ecto o oe of the ato: ONC Lat two te have oe ecto o each oe of the ato: Valace od V
46 Soluto: Cofguato teacto Two S obtal ca ake TWO olecula obtal Why ot ue the two ad ake cobato C C C3 C C 4 C6
47 Syety of Spacal Obtal f you echage the poto/p of ecto ad ecto ad tay the ae g ad 4 ad 5 ad 6 3 vet the ae g altoa vaat ove echage of ecto o oly wth
48 Soluto: Cofguato teacto Two S obtal ca ake TWO olecula obtal Why ot ue the two ad ake cobato C C Cofguato : two ecto bodg obtal Cofguato : two ecto atbodg obtal
49 Cofguato teacto C C C C C
50 Potetal Cuve evted
51 Depedece of pato Coeffcet
52 Ue of oe obtal p C p C z z
Computational Material Chemistry. Kaito Takahashi Institute of Atomic and Molecular Sciences,
Coputatoal ateal Chesty Kato Takahash sttute of Atoc ad olecula Sceces kt@gate.sca.edu.tw A Udestad the basc theoy behd quatu chesty calculato Lea ug quatu chesty poga Udestad what the output s sayg Get
Atomic units The atomic units have been chosen such that the fundamental electron properties are all equal to one atomic unit.
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