General Solution - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides, Slides of Chemistry

Download General Solution - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! d) General Solution for the Oscillating Perturbation Similar to spatial functions it can be shown that any periodic function of time f(t) can be expressed as a sum of oscillating functions. ( )taatf i i i ωcos)( 1 0 ∑ ∞ = += = cosine Fourier expansione.g. ti j j jeatf ω∑ ∞ = = 1 )( = Fourier expansionMore general: or any function (as the period goes to infinity) ( ) ( ) ωω ω deatf ti∫= = Fourier transform Thus, if we can solve a simple oscillating perturbation, then in principle we can actually solve the first-order time dependent perturbation theory problem for any periodic perturbation. docsity.com Fortunately, the simplest and most significant perturbation for spectroscopy is light = oscillating perturbation ( ) ( ) ( )titi eeH tHtH ωω ω −+= = )1( )1()1( ˆ cosˆ2ˆConsider therefore: ( ) ( ) ( ) ( )reeHrtH jtitiqqj rv ψψ ωω |ˆ| )1()1( −+= ( ) ( ) ( ) ( )titiqj titi jq eeH eerHr ωω ωωψψ − − += += )1( )1( |ˆ| rr Then: ( ) ( ) dteeeHita t tititi qjq qj∫ −+−=⇒ 0 )1()1( ωωω h docsity.com Note: (1-e-ia)(1-eia)=1-e-ia-eia+1 =2-[cos(a)-isin(a)]-[cos(a)+isin(a)]=2-2cos(a) =2(1-cos(a))=2(2sin2(a/2)) ( ) ( ) ( )2 2 2 )1()1( 2 1sin4 ωω ωω −       − =⇒ qj qj qjjq q tHH tP h ( ) ( ) ( ) 2 2 2 222 4 4 1 2 1 x ttxtx qj qjqj =− ⇒−=⇒−= ωω ωωωωLet ( ) ( )2 22 2 )1()1( sin x txHHtP qjjqq h =∴ ( ) ... !3 1sin 3 +−= xxxNow: the series expansion for sin(x) is: docsity.com ( ) ( )( ) ( )( ) 2 2 )1()1( lim )( 0 222 2 )1()1( 2 242 2 )1()1( 2 2 2 3 2 )1()1( ... ... ... !3 1 t HH txOt HH x txOxHH x txxHH tP qjjq resonance x qjjq qjjq qjjq q h h h h → = +−= +− =       +− =∴ ( ) !??!twhen ∞→∞→⇒ tPq Probabilities can’t be > 1 (100%) therefore this result is just an approximation and must be use with caution. The formula can be used to derive selection rules. docsity.com Selection Rules: The strongest transitions in spectroscopy arise when the electric vector of the light interacts with the dipole moment of the charge distribution of the molecule. This is called the electric dipole approximation oqjojqqj EEreH rrrr µψψ 2 1|| 2 1)1( =−=Here: Eo is the amplitude of the electric field vector and µ is the dipole moment (also a vector). 0 0 ≠ =qjµ r Electric dipole transition forbidden Electric dipole transition allowed docsity.com