Approximation Methods - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides, Slides of Chemistry

Download Approximation Methods - Advanced Quantum Chemistry and Spectroscopy - Lecture Slides and more Slides Chemistry in PDF only on Docsity! 2.1 Approximation Methods General approach to perturbation theory Ĥ is complicated and difficult to solve. Therefore write pHHH ˆˆˆ )0( += Major portion which can be solved by itself Minor part which is treated as a perturbation Use solutions to H(0) in H to get approximate solutions to H Approximations differ mathematically depending on whether solutions to H(0) give states with degenerate (equal) energies or not, or if the perturbation is time dependent docsity.com Three cases I: non-degenerate, time independent perturbation theory II: degenerate, time-independent perturbation theory III: time-dependent perturbation theory Yields approximate Φi Yields |ci (t)|2 docsity.com Will get Ψq and Eq as Taylor series ...)3(3)2(2)1()0( ++++= qqqqq ψλψλλψψψ ...)3(3)2(2)1()0( ++++= qqqqq EEEEE λλλ Ψq(n) and Eq(n) are the nth-order corrections to Ψq(0) and Eq(0), respectively. Substitute expansions into Eq. (1) and collect terms proportional to λn ( )( ) ( )( )...... ...ˆˆ )3(3)2(2)1()0()3(3)2(2)1()0( )3(3)2(2)1()0()1()0( ++++++++= +++++ qqqqqqqq qqqq EEEE HH ψλψλλψψλλλ ψλψλλψψλ Derivation ahead docsity.com Collect terms in λ [ ] [ ] [ ] 0 .... ˆˆ ˆˆ ˆ )2()0()1()1()0()2()1()1()2()0(2 )1()0()0()1()1()0()0()1(1 )0()0()0()0(0 = + −−−++ −−++ − qqqqqqqq qqqqqq qqq EEEHH EEHH EH ψψψψψλ ψψψψλ ψψλ or ( )[ ] ( ) ( )[ ] ( ) ( )[ ] 0 ... ˆˆ ˆˆ ˆ )0()2()1()1()1()2()0()0(2 )1()0()0()0()1()1(1 )0()0()0(0 = + −−+−+ −+−+ − qqqqqq qqqq qq EEHEH EHEH EH ψψψλ ψψλ ψλ docsity.com Set λ = 1 and each term in the expansion =0; that is, define the nth order perturbed Ψas the solution to the differential equation obtained by setting the coefficient of each power of λ=0. ( ) 0ˆ)( )0()0()0( =− qqEHi ψ Defines Ψq(0) and Eq(0) = zeroth order problem = unperturbed problem. known ( ) ( ) 0ˆˆ)( )1()0()0()0()1()1( =−+− qqqq EHEHii ψψ Defines Ψq(1) and Eq(1) known known known known unknown unknown Since (i) is known, use (ii) to get Eq(1) docsity.com