The Born-Oppenheimer Approximation

Transcription

1 5.76 Lecture otes /16/94 Page 1 The Born-Oppenhemer Approxmaton For atoms we use SCF to defne 1e eff orbtals. Get V for each e n feld of e s n all other occuped () r orbtals. ψ ( r) = φ 1( r1) φ( r ) sngle antsymmetrzed product functon. Ths s a way of defnng our zero-order complete bass set. It s a bad approxmaton and accurate ab nto electronc wavefunctons are CI lnear combnaton of many confguratons (product functons). For molecules, we separate Ψ functons Φ (r;r)χ vωj ΩJM nto a product of electronc, vbratonal, and rotatonal Ths s the Born-Oppenhemer approxmaton. It s based on a good approxmaton (e move much faster than nucle) and most molecular egenstates can be well descrbed by sngle electrc*vbratonal*rotatonal product. BUT WHAT DO WE HAVE TO SLIP UDER THE RUG? ( ) How to separate H rrθφ,,,? some subtle stuff return to ths for polyatomc molecules 1. CLAMPED UCLEI T 0 get Φ ( r ; R) and V (R) by neglectng Φ Φ and Φ Φ χ ( ). ROT VIB ROT VIB. H ( R, θφ), separated nto H ( θφ, ) + H ( R ) defne ΩJM bass set neglect part of H ROT Defne V JΩ (R) = V (R) + B(R)[J(J + 1) Ω ] effectve potental Defne χ vjω (R) vbratonal bass set. R 3. EXACT Ψ use BO ψ to go beyond BO approxmaton, then put the neglected terms back nto H spectroscopc perturbatons adabatc vs. dabatc lmts (neglect of or electrostatc terms) Potental Energy Surfaces are the central organzng concept of molecular spectroscopy. Recpe: ( rr ;, θ, φ ) 1. wrte exact H. neglect nconvenent terms 3. solve the smplfed equaton to defne a complete bass set 4. put the neglected terms back n.

2 5.76 Lecture otes /16/94 Page e H = T + T + V + V + V T e e ee p = = m m e e pˆ pˆ T T R T R R m m ROT A B = + (, θ, φ ) = ( ) + H ( ; θ, φ) A B radal only KET ( R) = R µr R R ( ) defned wth respect to center of mass [Bunker, J. Mol. Spect. 8, 4 (1968)] for neglected e nduced center of mass wobble nternuclear dstance orentaton of respect to lab XYZ ROT H ( R, θφ, ) = R µr nuclear angular momentum R J L mm A B S µ = rotatonal constant hcb(r) m + m A B wth V V Ze Ze = + r RA r RB RA RB e A B Z Z e =+ R A B ee V e r > =+ spols 1e orbtal approxmaton SCF Two coordnate systems LAB XYZ both have orgn at center of mass (defnton of body frame becomes more BODY xyz complex for polyatomc molecules) related by 3 Euler angles el VIB ROT Can we separate H = H + H + H? f we could, then E evr = T + G (v) + F v (J) ψ evr =φ χ v ΩJM OT qute. e move fast, nucle slow. Take ths to extreme lmt and pretend nucle can be held fxed. CLAMPED UCLEI T 0 solve clamped nucle electrc Schrödnger Equaton at seres of fxed R : R 1, R, R n

3 5.76 Lecture otes /16/94 Page 3 manfold of egenstates, computed at grd of R n ( ) Φ ( ) = ( ) Φ( ) H rr ; n rr ; n E Rn rr ; n Rn fxed Ths defnes E (R) whch we call the potental energy functon for the -th electronc state V (R). ( ) Ths also defnes Φ rr ; R. a complete set of electronc wavefunctons whch depend parametrcally on ext: use E (R) and Equaton. ( rr ) Φ ; to defne a (non-rotatng) ( T θ,φ 0) nuclear moton Schrödnger ψ BO,v ( r; R) Φ ( r; R)χ,v (R) (no θ,φ dependence) plug nto full Schrödnger Equaton, left multply by * Φ rr ; and ntegrate over all r : r. H BO ψ = Eψ BO Φ H Φ χ = E Φ Φ χ, v r, v ndependent of r T ( R ) + E ( R ) no θ,φ ( ) r = 1 ndependent of r came from T e + V e + V + V ee only Φ T Φ χ + E ( R) χ ( R) = Eχ ( R) r, v, v, v V (R) R µ

4 5.76 Lecture otes /16/94 Page 4 Chan rule for (AB) = [ ( A)B + A B] = ( A)B + ( A)( B) + ( A)( B) + A( B) Thus Φ T Φ χ = φ T φ χ + φ µ ( φ ) χ = 1 +φ φ T χ r, v r v, r r, R r R v, keep ths, neglect the other two terms H VIBR We are left wth [ T (R) + V (R)]χ,v = E,v χ,v (R) nuclear Schrödnger Equaton So are we done yet? ope. We must reconsder T ncludng rotaton. The nuclear moton H (R,θ,φ) s not qute separable nto ĥ 1 (R) + ĥ (θ,φ) Another trck s needed to separate out θ,φ degrees of freedom. notaton s trcky here put ths back n H (,, ) ( ) R θφ = T R + R + V( R µr B(R) rotatonal constant operator - depends on R ) nuclear rotaton angular momentum depends on θ,φ BAD EWS R θ,φ couplng, therefore can't separate! m The trck s to use a standard set of angular momentum bass functons [analogous to the Y l (θ,φ) of the central force problem], then defne what we have to temporarly throw away so that we can ntegrate over θ,φ to get a new and correct rotatng molecular Schrödnger Equaton. Defne ΩJM bass functons. They are f(θ,φ) and descrbe probablty of fndng nternuclear axs (BODY z axs) pontng n θ,φ drecton (wth respect to lab) gven that the magntude of the angular momentum s [J(J + 1)] 1/ and that the proectons of J on z s M and on z s Ω

5 5.76 Lecture otes /16/94 Page 5.e. ( ) DIRECTIO COSIES cos J, Z J = R + L+ S Total angular momentum s conserved, so t must be true that [ H, J ] = 0. J s a rgorously good quantum number. better to use J than R but J does not appear n We are gong to temporarly throw away some stuff. R = J L S R = ( J Jz ) + ( S SZ) + ( L LZ) JL+ JL L-uncouplng ( x x y y) ( JS x x + JS y y) ( LS x x + LS y y) S-uncouplng + temporarly get rd of all stuff n [ ] J ( J z )ΩJM = h [ J(J +1) Ω ]ΩJM ow we can get rd of θ,φ part of H (R, θ, φ) [ J( J + 1) ] 1/ Express unknown χ,v (R, θ, φ) as product of χvj,,, Ω, M( R) and Ω J M θφω, JM RvJΩM θφ, cos, ( J z) = = M Ω [ J( J + 1) ] 1/ J Ω JM = J( J + 1) ΩJM Jz Ω JM = M ΩJM J Ω JM = Ω ΩJM z H (R, θ, φ). χ,v (R,θ,φ) χ,v,j,ω,m (R) ΩJM JΩM left multply Schrödnger Equaton n H ( R, θ, φ ) by ΩJM and ntegrate over θ,φ. ( ) ΩJM H R, θ, φ χ ( R) Ω J M = ΩJM E χ Ω J M vω J M vω J M Ω JM Ω JM θφ θφ

6 5.76 Lecture otes /16/94 Page 6 { ( ) ( ) ( ) ( 1) } LHS = T R + V R + B R J J + Ω ΩJM Ω J M χv Ω J M Ω JM orthonormalty All of ths comes out because t s ndependent of θ, φ or because we used ΩJM bass functons. φθ + ΩJM neglected stuff from R ΩJ M χvω J M Ω JM some non-zero Ω = ±1 matrx elements. eglect for now. Perturbatons and L,S uncouplng later! Smplfes to: LHS T ( R) + V( R) + B( R) J ( J + 1) Ω χv ΩJM RHS = E χ vωjm (R) call ths V,JΩ ( R) effectve potental curve one of the operators on the LHS depend on M or J z, drop ths ndex. ow at last we have a smple R- equaton. T ( R) V ( R) + E, Ω χ = χ Ω Ω J vj vj vj dfferent set of vbratonal χ s for each J, Ω(we can avod ths by Van Vleck transformaton, later) Ω So we are almost done. We have defned a complete bass set. BO Ψ vjω ( r;r,θ, φ)=φ (r;r)χ vjω (R) θφ ΩJM

7 5.76 Lecture otes /16/94 Page 7 * Φ (r;r) s an egenfuncton of H T (R,θ,φ) * Ω JM s egenfuncton of J, J z, J Z and approxmate egenfuncton of T (R,θ,φ) T (R) = R = B( R)[ J L S] B( R)[ J J ] z µr * χ vj (R) s egenfuncton of T (R) + V J (R) Ω All we need now s the exact Ψ Ω Ψ J exact = Ψ Born-Oppenhemer vjω c vjω,v,ω mxng coeffcent The Born-Oppenhemer approxmaton s a good approxmaton when only one term n summaton s mportant. I THIS SPECIAL CASE E evj = T + G (v) + F,v (J) and t s straghtforward to go n ether drecton E evj V J (R) Sometmes a few mxng coeffcents are mportant must go beyond the Born-Oppenhemer approxmaton PERTURBATIOS Perturbaton Theory The nomnal k, v, J state s denoted by puttng t between. Ψ EXACT kvj =Ψ Born-Oppenhemer kvj + H v J;kvJ o E, v E kvj Born Oppenhemer Ψ o vj vj 1st order correctons to ψ. If one or more of these correcton terms s too large, must dagonalze a matrx. What terms n H cause trouble? V ee = e /r T (R) H ROT explctly ncluded n defnton of Ψ BO whch s also called the adabatc wavefuncton. Ths keeps potental curves for states of same symmetry from crossng. gnored effect on Φ ( r ;R) gnored effects of stuff n [ ]. These effects can be turned off by gong to J = 0.

8 5.76 Lecture otes /16/94 Page 8 Two convenent Lmtng Treatments 1. Adabatc or Born-Oppenhemer defne Φ ad electronc bass functons by exactly dagonalzng H T ( R) treat T (R) as a perturbaton BO e.g. Ψ R Ψ BO 0 because Ψ s are R-dependent especally rapd change near avoded crossngs R R c get non-crossng potental energy curves d. DIABATIC Φ exclude some undefnable part of V ee n order to defne sngle confguraton electronc bass states. Treat H el (that undefnable part of V ee ) as a perturbaton d e.g. Φ R Φ d = 0 (we refuse to let Φ d depend on R) because get crossng curves R Φ d el d 0 but H d Φ Φ 0 Two lmtng cases Weakly avoded crossng dabatc bass s preferable because crossng. R s very large for R near Strongly avoded crossng adabatc bass s preferable because all vbratonal level spacngs. H el would be large relatve to