MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5.80 Lecture #1 Fall, 2008 Page 1 Lecture #1: Matrices are Useful in Spectroscopic Theory For next time, read handout on the Van Vleck transformation and look at the notes on coupled harmonicoscillators. Outline * Correspondences bra ⎛ ⎞ row column i = ⎜ ⎟ N × 1 column vector ⎝ ⎠ ψi → ket ψ*i → i = 1 × N row vector operator O → O N × N square matrix * Solution of Schrödinger Equation corresponds to solving a determinantal secular equation for the{Ek} ψ  ;* The {Ek} are the Hkk in the special “diagonal” representation of H* The {ψk} are obtained from the basis states {φk} as columns of the unitary matrix T from the similarity transformation T†HφT = Hψ that “diagonalizes” Hφ; * Matrix representation of arbitrary f(Q). Usual procedure to obtain a fit model. e.g. Born-expressedVan Vleck a finite matrix Oppenheimerin terms of transformation expressed incombine and normal structure ↓ terms of linearlymode parametersintroduces microscopicdependentseparationse.g. fij force many smallexact H a a completeexact H truncate H molecular constants Fit model* finite simplifiedmatrixOR* algebraicformulasexpressed interms ofeffectivemolecularconstantsDunhamconstants ωe, ωexeApproximate H perturbation theory differential set of basis operator functions an infinite matrix model of variables constants terms parameters not necessarilythe same even if they have same name5.80 Lecture #1 Fall, 2008 Page 2 Matrices are useful because: * they display all necessary information; * can be “read” and simplified by perturbation theory “order sorting” via EioH− ij Ej ;o * labor saving tricks for avoiding the evaluation of unnecessary integrals, such as formulas for all matrix elements of Q and P in the Harmonic Oscillator basis set. No integrals actually evaluated. No functions actually looked at. Quantum Mechanics Operators follow the rules of matrix multiplication. e.g. ∫ ψ*i (AB ) ψjdτ = ∑ ( ∫ ψ*i Aψkdτ)( ∫ ψ*kBψjdτ)k = ∑ AikBkj =(AB)ij k This is very useful because we can generate many matrices by simple operations on one matrix. E.g. Q = R – Re V(Q) = ∑ cn Qn n matrix of Qn(Qn) = (Q)n Q × Q × … Q so instead of evaluating Q1, Q2, Q3, etc. we just evaluate Q and derive all the rest by matrix operations. completenessof {ψ} ↑ formulas, not integrals! ⎡* shortcuts to selection rules There is even some diagrammatic insight ⎢⎣* calculations of a specific element of Qn5.80 Lecture #1 Fall, 2008 Page 3v = ±1 non-zero matrix elements of Qv = 0, ±2 selection rulesAt the end of this lecture we will see that we arenot restricted to integer powers of Q.Q =000000 00Q2()=00000000 000 00etc.Suppose we have a convenient and complete (orthonormal) basis set {i}.Any arbitrary function (including an eigenfunction of H) can be expanded in terms of the {}.wavefunction pictureN =kak *  i i i kd = akii=1can be matrix picture = U  kk0=10* kkU is a transformation that converts { } into {}. ii= 0…1…00 =10 * i iN  1= 0…1…0column matrix 5.80 Lecture #1 Fall, 2008 Page 4 N We know ψk = ∑ aki φi i=1 We want ψ = Uφ where U is N × N matrix that transforms φ into ψ. By this we mean ⎛ 0 Uk1 ⎛ ⎞ 0 ⎛ ⎞ 0 ⎛ ⎞ ⎞  1  ⎜ ⎜ ⎜ ⎜ ⎜⎜⎝ ⎟ ⎟ ⎟ ⎟ ⎟⎟⎠   ⎜ ⎜ ⎜ ⎜ ⎜⎜⎝ ⎟ ⎟ ⎟ ⎟ ⎟⎟⎠ ⎜ ⎜ ⎜ ⎜ ⎜⎜⎝   ⎟ ⎟ ⎟ ⎟ ⎟⎟⎠ ⎜ ⎜ ⎜ ⎜ ⎜⎜⎝    ⎟ ⎟ ⎟ ⎟ ⎟⎟⎠ 0 Uk 2 ∑ Ukiφiψk = + + … = = i 0 0 0 UkNψ φ φ φ k-th row of U ⎞ ⎛ 0 Uk1   ⎛ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜⎜⎝  1  ⎟ ⎟ ⎟ ⎟ ⎟⎟⎠ ⎜ ⎜ ⎜ ⎜ ⎜⎜⎝  UkN ⎟ ⎟ ⎟ ⎟ ⎟⎟⎠ = 0 ψ φ Uki = aik = ∫ φ*i ψkdτ We can go in the opposite directionU–1ψ = φ and show that U–1 = U† U−ij 1 = U*ji φiis the i-th row of U–1 or the i-th column of U*.5.80 Lecture #1 Fall, 2008 Page 5 Derivation of Secular Determinant Now for every operator there is a matrix representation. O11  O1N   ON1  ONN ⎛ ⎞ ⎟ ⎟⎟⎠ ⎜ ⎜⎜⎝ O → Oφ = Oij ∫ φ*iOφjdτ= Schrödinger Equation in ψ eigenbasis picture Hψk = Ek ψk ∞ ψk = ∑ ai kφi mixing coefficient basis function - convenient usually defined aseigenfunctions of a parti=1 of H called H° H°φi = E°i φi Matrix notation: Hij φ = ∫φ*i Hφjdτ = φ i H j φ5.80 Lecture #1 Fall, 2008 Page 6 H11 H1N HN1 ⎜H  ⎜21 ⎜  ⎜ ⎜⎜⎜⎜ φφφ φ ⎜⎜⎜⎜ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛  a1k a1k ⎞ ⎟⎟⎟⎟ ⎠ ⎛ ⎜⎜⎜⎜ ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟⎟⎠ initially unknownmixing coefficients initially unknownenergies⎞ φ ⎟⎟⎟⎟ = Ek ⎟⎟⎟⎟ ⎟⎟⎟⎟ ⎠ ⎝ φ⎠ ∑ i ⎛ ⎜ ⎜ ⎜ ⎜ ⎜⎜⎝ ⎞ ⎟⎟⎟⎟ ⎠ = ⎞ a1k  ⎜⎜⎜⎜  aNk ⎛ ⎝ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ ⎝ φ⎠ aikH1i H2i aik φ∑ ∑ i i ⎛ representation⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝ ∑ i ∑ ∑ i i ⎝ ⎟⎟⎟⎟⎟⎟⎟⎠⎜⎜⎜⎜⎜⎜⎜⎝⎜ ⎟ A system of N linear homogeneous equations in N unknowns (aki i = 1…N) . A nontrivial solution aikexists if the determinant of coefficients of the unknown { } is zero. 0 0  0 H11 − Ek H12 H1N unit matrix 0 = H21 H22 − Ek = |Hφ HNN − Ek –1Ek| = ) aNk aik H1i − δ1iEk etc. ( = Ek  HNi aik move everything into one column matrix aNk (H2 iaik − δ2 iEkaik ) (H1iaik − δ1iEkaik ) etc. convenient LHS =5.80 Lecture #1 Fall, 2008 Page 7 This is the secular determinant. Must solve for special values of Ek which satisfy the requirement0 = |Hφ – 1Ek|. These are eigenvalues of H . {Ek} k = 1, 2, … NCOMPUTERS! 1. start with complete set {φ}2. compute all Hφ matrix elements 3. “diagonalize” Hφ time required ∝ N3 aik4. solve for { } : one set of N coefficients for each of the N eigenvalues. How to