Effective Hamiltonians and Perturbation Theory
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1 Effectve amltonans and Perturbaton Theory Kaml Sara.5.7
2 Computatonal Molecular Spectroscopy.5.7
3 Eds. Per Jensen and P.R. Buner Wley&Sons Chapter 8: Perturbaton Theory Effectve amltonans and Force Constants Kaml Sara and Jean Demason pp
4 Computatonal Molecular Spectroscopy Eds. Per Jensen and P.R. Buner Wley&Sons Chapter 8: Perturbaton Theory Effectve amltonans and Force Constants Kaml Sara and Jean Demason pp
5 What s an Effectve amltonan and how we construct It from the orgnal raw amltonan? The vbraton contact transformaton (good guy (For dlgent and conscentous scentsts Phenomenologcal effectve amltonans based on the prncpal symmetres of a amltonan (For less dlgent and conscentous scentsts Problems: the bloc-dagonal (rotatonal contact transformaton (bad guy. Reductons
6 What s an Effectve amltonan? J.K.G. Watson n Mol. Phys (5 np E np np n coarse spacng ( E np E n p s large m fne spacng (E np E np s small n electronc p vb-rot n vbronc p - rotaton.5.7 6
7 p P n eff (n PERT n vb PERT p rot vb rot n eff (n.5.7 7
8 .5.7 8
9 Three sets of data: Expermental data such as frequences and ntenstes of transtons Ξ ( ν ν ν 3 A A K Fundamental parameters of a molecule such as the geometrc parameters bond lengths and angles; potental constants; dpole moments etc. K ( r α τ F F V ( γ K j l mn mnl Spectroscopc constants S v such as S v J (A B D v v v ω v q t K.5.7 9
10 Ξ(ν ν ν 3.. S S S 3.5.7
11 .5.7 n j ee Ne NN e j N evr V V V m p M P ( Z Y X P h Quadratc n momenta V Coulomb nteractons
12 .5.7
13 ow a theoretcan analyzes a complex moton? V T z R r r j ej x y.5.7 3
14 . TRANSLATION z R r CENTER OF MASS x y.5.7 4
15 . ROTATION z ϕ θ χ x y.5.7 5
16 3. VIBRATION a r a r d r a r 3 d r d r 3 Center of mass m a r R r R r S ( θϕχ ( a r d r 3 3 3N 3N
17 .5.7 7
18 R V VR TRANSL ROT VIB TRANSL N T T T T T. m ( m v T translaton and rotaton V s ndependent of ( ( ( ( ( ( R R d a S d a S R R d a S R R &r &r &r &r r r & &r &r r r r r χ ϕ θ χ ϕ θ χ ϕ θ
19 pm pmd (II.Ecart(a d m (a ( ( T (I.Ecart d m (a ( ( ( (.( m ( T R V VR T r r &r r r & & & r r &r &r r r & &r d d a m f d a S d a S R χ ϕ θ χ ϕ θ χ ϕ θ χ ϕ θ
20 .5.7
21 .5.7
22 .5.7.d dsplacement from equlbrum (a. r - change of bond length (V f rr ( r.. ( ( / N vb classscal N Q P z y x Q l d m λ α α α P q p Q q / 4 / / 4 / ( / / ( 4. h h λ λ p q ± ± L 5.
23 d «a vb rot vb rot vbrot v R - ARMONIC OSCILLATOR RIGID ROTOR R CENTRIFUGAL DISTORTION V ANARMONIC INTERACTIONS VR CORIOLIS INTERACTIONS
24 vr L(vbratonal terms L(Corols terms L(rotatonal terms.5.7 4
25 ORDER OF MAGNITUDE M.R. Alev J.K.G. Watson n Molecular Spectroscopy Modern Research Vol. III -67 (985 m mn C mnq J n κ r J mn m n m n ω vb (κ 4 (D/B m n C κ ω mn vb mn κ m n ω vb ( J κ
26 L (L J J B q R p (q ω αβ β α αβ 3
27 .5.7 7
28 [ ] ( [ ] [ ] [ ] S S S ~ S S S ~ ~ ~ S λ S - ( λ λ ( S λ S ( e e S S L L λ λ λ λ n m m n n n E E m n n n E E
29 .5.7 9
30 n ~ m n m (E n E m n S m R q αβ B αβ J α J β (L L S αβ B αβ J α J β (L L / ω /( ω ω l ~ [ S ] [ S [ S ].5.7 3
31 ( [ ] ( ( ( ( l l xx l l l l l x l x z y x x B B v J v v v v v eff v B B B B d v B B J J D K C K J J C B ω ω ω ω ω ζ α α ν β β β ( / ( ( L L
32 LAB (R θ ϕ χd (JQ(q (J L QM eff.5.7 3
33 Fortunately we don t have to go through ths process every tme
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