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Quantum Dynamics (NWI-SM295) Quantum dynamics (NWI-SM295) Prof. dr. ir. Gerrit C. Groenenboom Theoretical Chemistry, Institute for Molecules and Materials Radboud University Nijmegen http://www.theochem.ru.nl May 11, 2021 Contents 1 Born-Oppenheimer approximation 3 1.1 Molecular Hamiltonian . .3 1.2 Mass weighted coordinates . .4 1.3 The Born-Oppenheimer approximation . .4 1.4 Validity of the Born-Oppenheimer approximation . .6 1.5 Second derivative nonadiabatic coupling . .7 2 Diatomic molecules 8 2.1 Harmonic oscillator approximation . 10 2.2 Anharmonic oscillator, variational approach . 11 2.3 Rotational levels of diatomic molecules . 12 2.4 Morse oscillator . 12 2.5 Numerical approach to one-dimensional bound states . 13 3 Time-dependent Schr¨odingerequation 15 3.1 Time-dependent Schr¨odingerequation in arbitrary basis . 16 3.2 Matrix exponentiation . 18 3.3 Free particles in one dimension . 19 3.4 Gaussian wave packets . 20 3.5 Expectation values of Gaussian wave packets . 23 3.6 Middle of the wave packet . 23 3.7 Flux in one dimension . 24 3.8 The Ehrenfest theorem . 26 4 Time-independent scattering in one dimension 28 4.1 Some properties and examples of flux . 28 4.2 Boundary conditions for 1-D scattering problems . 29 4.3 Numerical solution for 1d tunneling problem . 30 5 Multichannel collinear scattering 33 6 Elastic scattering 37 6.1 Classical equations of motion . 37 1 CONTENTS Page 2 7 Elastic scattering in quantum mechanics 41 7.1 Free particles in 3D . 41 7.1.1 Flux in spherical polar coordinates . 42 7.2 Boundary conditions for elastic scattering . 43 7.3 Partial wave expansion . 45 7.4 Numerical method for elastic scattering . 46 7.5 Alternative boundary conditions . 47 8 Inelastic scattering 48 8.0.1 Properties of the S-matrix . 51 8.1 Coupled channels equation . 52 9 Angular momentum theory 53 9.1 Unitary transformations in place and time . 53 9.2 Rotation operator . 54 3 9.3 Rotations in R ................................... 55 9.4 Summary of elementary angular momentum theory . 57 9.5 Wigner rotation matrices . 58 9.6 Wigner D-matrix elements as wave functions . 59 9.7 Euler angles . 60 9.8 Spherical harmonics addition theorem . 61 Lecture notes on Quantum Dynamics, version May 11, 2021 Chapter 1 Born-Oppenheimer approximation 1.1 Molecular Hamiltonian We consider a molecular system consisting of n electrons and N nuclei. The notation of the Cartesian coordinates, linear momenta, masses, and charges are shown in Table 1.1. The Hamiltonian is given by H^ = T^e + T^N + V^e;e + V^e;N + V^N;N ; (1.1) with the kinetic energy of the electrons n n X p^2 2 X T^ = i = − ~ r2; (1.2) e 2m 2m i i=1 e e i=1 the kinetic energy of the nuclei N N X P^2 X 2 T^ = α = − ~ r2 ; (1.3) N 2m 2M α α=1 α α=1 α the Coulomb repulsion between the electrons X e2 1 V^ = ; (1.4) e;e 4π jr − r j i<j 0 i j the Coulomb attraction between the electrons and nuclei n N 2 X X e −Zα V^ = ; (1.5) e;N 4π jr − R j i=1 α=1 0 i α Particles coordinates momenta masses charges ~ Electrons (i = 1; : : : ; n) ri p^i = i ri me −e ^ ~ Nuclei (α = 1;:::;N) Rα Pα = i rα Mα Zαe Table 1.1: Notation of coordinates, linear momenta, masses, and charges. 3 CHAPTER 1. BORN-OPPENHEIMER APPROXIMATION Page 4 and the Coulomb repulsion between the nuclei 2 X e ZαZβ V^N;N = : (1.6) 4π0 jRα − Rβj α<β 1.2 Mass weighted coordinates Before we proceed, we introduce mass weighted coordinates. This is not essential, but it sim- plifies the equations. Coordinate transformations are frequently used in quantum dynamics to simplify a problem. For N nuclei, we define 3N mass weighted coordinates by 0Q 1 0Q 1 0Q 1 1 p 4 p 3N−2 p @Q2A = M1R1; @Q5A = M2R2; ::: ; @Q3N−1A = MN RN : (1.7) Q3 Q6 Q3N We will use the column vector Q to denote all 3N nuclear mass weighted coordinates. The nuclear kinetic energy operator (Eq. 1.3) now takes the form 3N 2 X @2 2 T^ = −~ = −~ r2 : (1.8) N 2 @Q2 2 Q i=1 i The 3n electronic coordinates are denoted by the column vector q, and the total Coulomb interaction will be denoted by V^ (q; Q) ≡ V^e;e + V^e;N + V^N;N : (1.9) 1.3 The Born-Oppenheimer approximation The Born-Oppenheimer approximation relies on the nuclei being much heavier than the electrons, Mα me. Already the hydrogen nucleus has 1836.15 times the mass of an electron. This allows a two-step approach to solving the Schr¨odingerequation. In the first step the kinetic energy of the nuclei is neglected and the electronic Hamiltonian is defined, (el) H^ ≡ T^e + V^ (q; Q): (1.10) (el) For a given choice of the positions of the nuclei (Q), the electronic wave functions Φi (q; Q) (i = 0; 1;::: ) are the solutions of the electronic Schr¨odingerequation, ^ (el) (el) (el) (el) H Φi (q; Q) = Ei (Q)Φi (q; Q): (1.11) (el) The electronic energies Ei (Q) depend on the positions of the nuclei, and they are usually referred to as adiabatic or Born-Oppenheimer potential energy surfaces. We assume that the electronic wave functions are orthonormal Z Z (el) (el) (el);∗ (el) hΦi jΦj iq ≡ ··· Φi (q; Q)Φj (q; Q)dq1dq2 ··· dq3n = δi;j: (1.12) Lecture notes on Quantum Dynamics, version May 11, 2021 CHAPTER 1. BORN-OPPENHEIMER APPROXIMATION Page 5 In the second step of the Born-Oppenheimer approximation we use the electronic wave functions as a basis to expand the total wave function X (el) Ψ(q; Q) = Φi (q; Q)χi(Q): (1.13) i To find an equation for the nuclear wave functions χi(Q) we first substitute the expansion into the time-independent Schr¨odingerequation ^ ^ (el) X (el) [TN + H − E] Φj (q; Q)χj(Q) = 0: (1.14) j Next, we project from the left with electronic wave functions, integrating over the electronic coordinates (el) ^ ^ (el) X (el) hΦi (q; Q)jTN + H − Ej Φj (q; Q)χj(Q)iq = 0: (1.15) j When evaluating the matrix elements, one has to remember that the nuclear kinetic energy operator acts the nuclear as well as the electronic wave function since 2 (el) rQΦj (q; Q)χj(Q) = (1.16) 2 (el) (el) (el) 2 [rQΦj (q; Q)]χj(Q) + 2[rQΦj (q; Q)] · rQχj(Q) + Φj (Q)rQχj(Q): In this equation gradient rQ is the vector operator 0 @ 1 @Q1 B . C rQ = @ . A : (1.17) @ @Q3N To evaluate the matrix elements of the electronic Hamiltonian we use Eqs. (1.11) and (1.12) and find (el) ^ (el) X (el) (el) hΦi (q; Q)jH − Ej Φj (q; Q)χj(Q)iq = [Ei (Q) − E]χi(Q): (1.18) j Combining the last four equations we arrive at a set of coupled equations X 1 [T^ + E (Q) − E]χ (Q) = 2 G (Q) + F (Q) · r χ (Q); (1.19) N i i ~ 2 i;j i;j Q j j where the non-adiabatic coupling matrix elements are given by (el) 2 (el) Gi;j(Q) ≡ hΦi (q; Q)jrQjΦj (q; Q)iq (1.20) (el) (el) Fi;j(Q) ≡ hΦi (q; Q)jrQjΦj (q; Q)iq: (1.21) In the Born-Oppenheimer approximation all non-adiabatic coupling matrix elements are ne- glected and Eq. (1.19) can be solved separately for each electronic state (i) [T^N + Ei(Q)]χi;v(Q) = Ei;vχi;v(Q): (1.22) The total wave function is then a product of an electronic and a nuclear wave functions (el) Ψi;v(q; Q) = Φi (q; Q)χi;v(Q): (1.23) Lecture notes on Quantum Dynamics, version May 11, 2021 CHAPTER 1. BORN-OPPENHEIMER APPROXIMATION Page 6 1.4 Validity of the Born-Oppenheimer approximation In general, the Born-Oppenheimer approximation is accurate when the separation of the electronic energies is large compared to the nuclear kinetic energy. To show this, we first observe that ∗ Fi;j(Q) = −Fj;i(Q); (1.24) which follows from taking the first derivative of Eq. (1.12). Dropping the subscript q for the scalar products we have (el) @Φ(el) @ (el) (el) @Φi (el) (el) j @ hΦi jΦj i = h jΦj i + hΦi j i = δi;j = 0; (1.25) @Qα @Qα @Qα @Qα which shows that @Φ(el) (el) (el);∗ (el) j @Φi (el) (el) @Φi hΦi j i = −h jΦj i = −hΦj j i: (1.26) @Qα @Qα @Qα For non-degenerate real electronic wave functions we have Fi;j(Q) = −Fj;i(Q) (1.27) and hence Fi;i(Q) = 0. To derive and expression for i 6= j we taking the first derivative with respect to nuclear coordinate Qα of (el) ^ (el) (el) hΦi jH jΦj i = 0; (1.28) which gives ^ (el) @ (el) ^ (el) (el) (el) @H (el) (el) ^ (el) @ (el) h Φi jH jΦj i+hΦi j jΦj i+hΦi jH j Φj i = 0; (for i 6= j): (1.29) @Qα @Qα @Qα Again restricting ourselves to the real, non-degenerate case we derive, using Eqs. (1.11), (1.27), and (1.29) h i (el) ^ (el) (el) hΦi (q; Q)j rQ; H jΦj (q; Q)i F (Q) = : (1.30) i;j (el) (el) Ej (Q) − Ei (Q) This result shows that the first derivative couplings Fi;j(Q) are small when the electronic (el) (el) energies are well separated, i.e., when jEj (Q) − Ei (Q)j is large.
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