Lecture #8: The Born-Oppenheimer Approximation

Lecture #8: The Born-Oppenheimer Approximation For atoms we use SCF to define 1e– orbitals. Get Veff (r) for each e– in field of e–’s in all other occupied orbitals. ψ(r) = φ1 (r1 )…φ N (rN ) single antisymmetrized product function.  This is a way of defining our zero-order complete basis set. It is a bad approximation and accurate ab initio electronic wavefunctions are CI — linear combination of many configurations (product functions). For molecules, we separate Ψ ( r;R,θ, φ ) into a product of electronic, vibrational, and rotational  functions Φi(r;R)χivΩJ|ΩJM〉. This is the Born-Oppenheimer approximation. It is based on a good approximation (e– move much faster than nuclei) and most molecular eigenstates can be well described by single electronic*vibrational*rotational product. BUT WHAT DO WE HAVE TO SLIP UNDER THE RUG?  ( r, R,θ, φ ) ? How to separate H  1. some subtle stuff — return to this for polyatomic molecules CLAMPED NUCLEI  2 Φ and TN → 0 get electronic Φi( r ; R) and nuclear Vi(R) by neglecting Φ i ∇ j  Φ ∇  χ(R) . Φi ∇ j 2. 3.  ROT−VIB ( R,θ,φ ) separated into H  ROT (θ, φ ) + H  VIB (R) For the i-th electronic state, H define |ΩJM〉 basis set neglect part of HROT Define ViJΩ(R) = Vi(R) + Bi(R)[J(J + 1) – Ω2] effective potential Define χivJΩ(R) vibrational basis set. EXACT ψ — use BO ψ° to go beyond BO approximation, then put the neglected terms back into  H spectroscopic perturbations adiabatic vs. diabatic limits (neglect of either ∇2 or electrostatic terms) Potential Energy Surfaces are the central organizing concept of molecular spectroscopy. Recipe:  1. write exact H 2. neglect inconvenient terms 3. solve the simplified equation to define a complete basis set 4. put the neglected terms back in. =T e + T  N + VeN + VNN + Vee H 2 2 p  i  = T ∑ 2m = − 2m ∑ ∇2i i e e i e 2 A 2 B Defined with respect to center of mass. See [Bunker, J. Mol. Spect. 28, 422 (1968)] for neglected e– induced center of mass wobble.  = p̂ + p̂ → T  ( R,θ,φ) = T  (R) + H  T 2m A 2m B N N N internuclear distance ROT ( R; θ,φ ) orientation of with respect to lab XYZ 2  N (R) = −  ⎡ ∂ ⎛⎜ R 2 ∂ ⎞⎟ ⎤ radial only KE T 2µR 2 ⎢⎣ ∂R ⎝ ∂R ⎠ ⎥⎦ 2  ROT ( R,θ,φ) = −  (  −2 R 2 ) H 2 2µR  nuclear angular momentum mAmB  ≡ J − L  − S R µ= rotational constant hcB(R) mA + mB ⎡ Z e2 Z e2 ⎤ VeN = −∑ ⎢ A + B ⎥ ri − R A ri − R B ⎥ i ⎢        ⎢⎣ R Ai ⎥⎦ R Bi V NN Z A Z Be 2 =+ R Vee = +∑ e 2 rij – spoils 1e orbital approximation → SCF i> j Two coordinate systems LAB XYZ ⎫ ⎬ both have origin at center of mass (definition of body frame becomes more BODY xyz ⎭ complex for polyatomic molecules) related by 3 Euler angles (need only 2 angles to locate internuclear axis, 3rd angle chosen implicitly = phase convention) Can we separate if we could, then  =H  el + H  VIB + H  ROT ? H Eevr = Ti + Gi(v) + Fiv(J) ψevr = φ i χ iv ΩJM NOT quite. e– move fast, nuclei slow. Take this to extreme limit and pretend nuclei can be held fixed. CLAMPED NUCLEI TN → 0 solve clamped nuclei electronic Schrödinger Equation at grid of fixed R : R1, R2, … Rn manifold of eigenstates, computed at grid of Rn points  ( r;R ) Φ ( r;R ) = E ( R ) Φ ( r;R ) H n i n i n i n   grid   R n fixed point This defines Ei(R) which we call the potential energy function for the i-th electronic state Vi(R). This also defines Φ i ( r; R ) a complete set of electronic wavefunctions which depend parametrically on  R. Next: use Ei(R) and Φ i ( r; R ) to define a (non-rotating) ( TθN,φ → 0 ) nuclear motion Schrödinger  Equation. ψBO i,v ( r; R ) ≡ Φ i ( r; R ) χ i,v (R) (no θ,φ dependence) plug into full Schrödinger Equation, left multiply by Φ*i ( r; R ) and integrate over all r : denoted as    ψBO = EψBO H  Φχ Φi H i i,v r = E Φ i Φ i χ i,v independent of r =1  N (R) + E (R) T i no θ,φ r independent of r came from Te + VeN + VNN + Vee only  N Φ χ + E (R)χ (R) = Eχ (R) Φi T i r i,v i i,v i,v − 2 2 ∇R 2µ Vi(R) eigenvalue equation, χi,v(R) ↔ Ei,v r . Chain rule for ∇2 ∇2(AB) = ∇[ ( ∇A)B + A∇B] = (∇2A)B + (∇A)(∇B) + (∇A)(∇B) + A(∇2B) Thus  Φ χ Φi T i r i,v N 2 ⎛   φ χ + 2 − ⎞ φ (∇ φ ) ∇ χ = φi T i r i,r R i r R i,v ⎜⎝ 2µ ⎠⎟ i  N do we expect φi to depend on R? =1  N  + φi φi r T χ i,v    keep this, neglect the other two terms  VIBR H  N(R) + V (R)]χ = E χ (R) nuclear Schrödinger Equation We are left with [ T i i,v i,v i,v So are we done yet? Nope. We must reconsider neglected terms from TN including rotation.  (R,θ,φ) is not quite separable into ĥ (R) + ĥ (θ,φ) (worse for polyatomics) The nuclear motion H 1 2 Another trick is needed to separate out θ,φ degrees of freedom. notation is tricky here 2  ⎡⎣ −2  2 ⎤⎦ N  H i ( R,θ, φ ) = T (R) +  R + Vi (R) 2 2µR nuclear rotation angular put this back in B(R) rotational constant operator - depends on R momentum — depends on θ,φ BAD NEWS R ↔ θ,φ coupling, therefore can't separate! The trick is to use a standard set of angular momentum basis functions [analogous to the Ym (θ,φ) of the central force problem], then define what we have to temporarily throw away so that we can integrate over θ,φ to get a new and correct rotating molecular Schrödinger Equation. Define |ΩJM〉 basis functions. Eigenfunctions of Jz, J2, JZ They are f(θ,φ) and describe probability of finding internuclear axis (BODY z axis) pointing in θ,φ direction (with respect to lab) given that the magnitude of the angular momentum is [J(J + 1)]1/2 and that  the projection of J on Z is M and on z is Ω i.e.  M J 2 ΩJM =  2 J(J +1) ΩJM ( ) cos J, Z = 1/2 DIRECTION COSINES [ J(J + 1)]  Ω J z ΩJM = M ΩJM cos ( J, z ) = 1/2 [ J(J +1)] J z ΩJM = Ω ΩJM +L  + S J = R  , J 2 ] = 0. Total angular momentum is conserved, so it must be true that [ H J is a rigorously good quantum number. What about Ω and M? Why?  2, but J does not appear in H  (R, θ, φ). ∴ better to use J 2 than R  2 to this We are going to temporarily throw away some stuff. (Some clever algebra needed to reduce R simple form.)  = J – L  – S R  2 = ( J 2 − J 2 ) + ⎡( S2 − S2 ) + ( L2 − L2 ) R z z z ⎣ –2 ( J x L x + J y L y ) L-uncoupling –2 ( J xSx + J ySy ) S-uncoupling +2 ( L xSx + L ySy ) ⎤⎦ temporarily get rid of all stuff in [ ] ( J 2 − J 2z ) ΩJM =  2 [ J(J +1) − Ω2 ] ΩJM  (R, θ, φ) Now we can get rid of θ,φ part of H i Express unknown χi,v(R, θ, φ) as product of radial and angular factors, χ i,v,J,Ω,M (R) = R ivJΩM and ΩJM  = θ,φ ΩJM θ,φ χ i,v (R,θ, φ) ≡ ∑ χ i,v,J,Ω,M (R) ΩJM (usual schizophrenic approach: vibration as wavefunction, rotation JΩM as state vector).  ( R,θ,φ ) by 〈ΩJM| and integrate over θ,φ. left multiply Schrödinger Equation expressed in terms of H  ( R,θ,φ ) ΩJM H ∑ Ω′J′M′ χ ivΩ′J′M′ (R) Ω′J′M′ = ΩJM E ivJΩ θφ ∑ Ω′J′M′ χ ivΩ′J′M′ Ω′J′M′ θφ LHS = { TN (R) + Vi (R) + B(R) [ J(J +1) − Ω 2 ]} ΩJM ∑ Ω′J′M′ Ω′J′M′    orthonormality χ iv Ω′J′M′ φθ (All of this comes out of the θ,φ integral because the terms are independent of θ, φ or because we used |ΩJM〉 basis functions.) 2⎤ ⎡  + ΩJM ⎣neglected stuff from R ⎦ ∑ Ω′J′M′ χ ivΩ′J′M′ Ω′J′M′ some non-zero ∆Ω = ±1 matrix elements. Neglect for now. Perturbations and L,S uncoupling later! Simplifies to: LHS ≅ ⎧⎪ TN (R) + Vi (R) + B(R)[ J(J + 1) − Ω2 ]⎫⎪ χ ivΩJM  ⎬ ⎨ call this Vi,JΩ (R ) ⎪⎩ “effective potential curve” ⎭⎪ RHS = EivJΩ χivΩJM(R) None of the operators on the LHS depend on M or JZ, drop this index. Now at last we have a simple Requation. ⎡ N ⎤ ⎣T (R) + Vi,JΩ (R)⎦ χ ivJΩ = E ivJΩ χ ivJΩ different set of vibrational χ’s for each J,Ω (we can avoid this by Van Vleck transformation, later) So we are almost done. We have defined a complete basis set. ψ BO ivJΩ ( r;R,θ,φ ) = Φ i (r;R)χ ivJΩ (R) θφ ΩJM – T  N (R,q,f) ( T  N is removed because we clamped the nuclei) * Φi(r;R) is an eigenfunction of H  N (R,θ,φ) - T  N (R) * |ΩJM〉 is eigenfunction of J2, Jz, JZ and approximate eigenfunction of T 2 2 −2  −L  − S ]2 ≈ B(R)[J 2 − J 2z ] =  R = B(R)[J 2µR 2  N (R) + V (R) * χ (R) is eigenfunction of T ivJΩ iJΩ All we need now is the exact ψ mixing coefficient ψ exact = J ∑ Born-Oppenheimer ψ°ivJΩ c ivJΩ i,v,Ω The Born-Oppenheimer approximation is a good approximation when only one term in summation is important. IN THIS SPECIAL CASE EevJ = Ti + Gi(v) + Fi,v(J) and it is straightforward to go in either direction EevJ ↔ ViJ(R) Sometimes a few mixing coefficients are important — must “go beyond the Born-Oppenheimer approximation” — “PERTURBATIONS” (local vs. global) Perturbation Theory The “nominal” k, v, J state is denoted by putting it between ‘ ’. ψ EXACT ‘kvJ’ = ψ° Born-Oppenheimer kvJ +∑ H iv′J;kvJ Born −Oppenheimer ψ° E okvJ − E ivo ′J iv′J i,v′ 1st order corrections to ψ. If one or more of these correction terms is too large, must diagonalize a matrix.  cause trouble? What terms in H explicitly included in definition of ψ°BO which is also called the “adiabatic” wavefunction. This keeps potential curves for states of same symmetry from crossing. Non-crossing rule. TN(R) ignored effect on Φi( r ;R)  HROT ignored effects of stuff in [ ]. These effects can be turned off by going to J = 0. Two convenient limits Vee = e2/rij 1. Adiabatic or Born-Oppenheimer N  define Φ ad i electronic basis functions by exactly diagonalizing H − T (R) treat TN(R) as a perturbation ∂ BO ψ j ≠ 0 because ψ’s are R-dependent — especially rapid change near e.g. ψ BO i ∂R avoided crossings R ≈ Rc get non-crossing potential energy curves 2. DIABATIC Φ di exclude some undefinable part of Vee in order to define “single configuration” electronic basis states.  el (that undefinable part of Vee) as a perturbation Treat H ∂ d Φ j = 0 (we refuse to let Φd depend on R) e.g. Φ di ∂R ∂ d  el Φ d ≠ 0 because Φ j ≡ 0 but Φ di H j ∂R get crossing curves Two limiting cases Weakly avoided crossing — diabatic basis is preferable because i ∂ j is very large for R near ∂R crossing.  j would be large relative to all Strongly avoided crossing — adiabatic basis is preferable because i H vibrational level spacings. el