Quantum Chemistry: Our Agenda (along with Engel) Postulates in quantum mechanics (Ch. 3) Schrödinger equation (Ch. 2) Simple examples of V(r) Particle in a box (translation, etc.) (Ch. 4-5) Harmonic oscillator (vibration) (Ch. 7-8) Particle on a ring or a sphere (rotation) (Ch. 7-8) Extension to chemical systems (electronic structure) Hydrogen(-like) atom (one-electron atom) (Ch. 9) Many-electron atoms (Ch. 10-11) Diatomic molecules (Ch. 12-13) Polyatomic molecules (Ch. 14) Computational chemistry (Ch. 16)
Lecture 1. The Simplest Chemical System. Hydrogen Atom. Part 1. References for Part 1 (Atoms) Quantum Chemistry, Engel (3rd ed. 2013) Quantum Mechanics in Chemistry, Ratner & Schatz (2001) Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005) Quantum Chemistry, D. A. McQuarrie Elementary Quantum Chemistry, F. L. Pilar (2003) Introductory Quantum Mechanics, R. L. Liboff (4th ed. 2004) A Brief Review of Elementary Quantum Chemistry http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html
An atom has translational and electronic degrees of freedoms. A molecule has translational, vibrational, rotational, and electronic degrees of freedoms. In the case of atoms… An atom has translational and electronic degrees of freedoms. To good approximation, degrees of freedom are not coupled. separation of variables! N N N N + Helec(elec) Eigenvalue (energy) = sum Htrans(rCM) N Enuc,elec Eigenfunction (wave function) = product relec (rCM as origin) rCM relec (rCM as origin) N elec(elec) rCM (~rN)
Separation of Internal Motion from External Motion H atom made up of proton and electron (2-body problem) Potential energy V in H atom Electron coordinate Nucleus Hamiltonian (2-body) Separation of Internal Motion from External Motion Full Schrödinger equation can be separated into two equations: 1. Atom as a whole through the space (rCM ~ rNucleus); 2. Motion of electron around the nucleus (relec with nucleus at origin). “Electronic” structure (1-body problem): Forget about nucleus!
V depends only on r. (”Central Potential”) where (3-dim.) Hamiltonian in spherical coordinates (r,,) Schrödinger equation in spherical coordinates r2 r is not coupled with (,). Separation of variables
Compare with a particle-on-a-sphere case (MS5118) constant radius r0 from the origin (“rigid” rotor) Schrödinger equation in cartesian coordinates constant (r0) Spherical polar coordinates in 3D Schrödinger equation in spherical polar coordinates spherical harmonics constant radius
Separation of Variables - where solved Angular part (spherical harmonics) Radial part (Radial equation) principal quantum no. , n-1 angular momentum quantum no. magnetic quantum no. (Laguerre polynom.)
Separation of variables Radial Schrödinger equation is spherical harmonic functions. Only function not known is Radial Schrödinger equation Effective potential Veff(r)
Effective potential
Separation of Variables - where Angular part (spherical harmonics) Radial part (Radial equation) principal quantum no. , n-1 angular momentum quantum no. magnetic quantum no. (Laguerre polynom.)
Eigenvalues (AO Energy levels) & Ionization energy Total energy eigenvalues are negative by convention. (Bound states) IE (1 Ry for H) length Minimum energy required to remove an electron from the ground state atomic units energy 1 Ry depend only on the principal quantum number.
Atomic Units (a.u.)
Engel says in p. 12 that “Bohr (1913) next introduced wave-particle duality which is equivalent to asserting that the electron had the de Broglie wavelength (1924).” Bohr Atom Model (1911-1913)
1885 – Johann Balmer – Line spectrum of hydrogen atoms 1913 – Niels Bohr – Theory of atomic spectra
Shells, subshells, and AO energy diagram n = 1 (K), 2 (L), 3 (M), 4(N), … Sub-shells (for each n): l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1 m = 0, 1, 2, …, l Number of orbitals in the nth shell: n2 (n2 –fold degeneracy) Examples : Number of subshells (orbitals) n = 1 : l = 0 → only 1s (1) → 1 n = 2 : l = 0, 1 → 2s (1) , 2p (3) → 4 n = 3 : l = 0, 1, 2 → 3s (1), 3p (3), 3d (5) → 9 Question: Is this AO energy diagram the same as what you have known?
Review: Separation of Variables - where Angular part (spherical harmonics) Radial part (Radial equation) principal quantum no. , n-1 angular momentum quantum no. magnetic quantum no. (Laguerre polynom.)
Eigenfunctions (Atomic orbitals): Electronic states wave functions (eigenfunctions) nlm nl Radial wave functions Rnl Bohr Radius
Review: particle-on-a-sphere solutions (MS5118)
Review: particle-on-a-sphere solutions (MS5118)
Review: particle-on-a-sphere solutions (MS5118)
Eigenfunctions (Atomic orbitals): Electronic states shell shape symmetry
Shells, subshells, and AO energy diagram n = 1 (K), 2 (L), 3 (M), 4(N), … Sub-shells (for each n): l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1 m = 0, 1, 2, …, l Number of orbitals in the nth shell: n2 (n2 –fold degeneracy) Examples : Number of subshells (orbitals) n = 1 : l = 0 → only 1s (1) → 1 n = 2 : l = 0, 1 → 2s (1) , 2p (3) → 4 n = 3 : l = 0, 1, 2 → 3s (1), 3p (3), 3d (5) → 9 Question: Is this AO energy diagram the same as what you have known?
Three quantum numbers, nlm n: Principal quantum number (n = 1, 2, 3, …) Determines the energies of the electron Shells l: Angular momentum quantum number (l = 0, 1, 2, …, n1) Determines the angular momentum of the electron Subshells Ll = (s, p, d, f,…) m: magnetic quantum number (m = 0, 1, 2, …, l) Determines z-component of angular momentum of the electron Lz, m =
Let’s focus on the radial wave functions Rnl. How far the shell is apart from the nucleus 1s 2s 2p 3s 3p 3d 1 1 1 radial node 2 radial nodes 1 1 1 1 *Bohr Radius *Reduced distance 1 radial node (ρ = 4, )
Radial wave functions (l = 0, m = 0): s orbitals
Wave function Probability density
Radial distribution function (RDF) Probability density. Probability of finding an electron at a point (r,θ,φ) Integral over θ and φ Radial distribution function. Probability of finding an electron at any radius r Wave function (1s) Radial distribution function (1s) Q: Derive it! Bohr radius
Atomic Units (a.u.)
Question:
Radial wave functions for l 0 : np orbitals (l = 1) and nd orbitals (l = 2) p orbital for n = 2, 3, 4, … ( l = 1; ml = -1, 0, 1 ) 2p 3p 3d d orbital for n = 3, 4, 5, … (l = 2; ml = -2, -1, 0, 1, 2 )
Radial wave functions (l 0) Probability density
Separation of variables Radial Schrödinger equation is spherical harmonic functions. Only function not known is Radial Schrödinger equation Effective potential Veff(r)
Effective potential
Spectroscopic transitions and Selection rules Transition (Change of State) hcRH n1, l1,m1 Photon n2, l2,m2 All possible transitions are not permissible. Photon has intrinsic spin angular momentum : s = 1 d orbital (l=2) s orbital (l=0) (X) forbidden (Photon cannot carry away enough angular momentum.) Selection rule for hydrogen atom
Review: Shells, subshells, and AO energy diagram n = 1 (K), 2 (L), 3 (M), 4(N), … Sub-shells (for each n): l = 0 (s), 1 (p), 2 (d), 3(f), 4(g), …, n1 m = 0, 1, 2, …, l Number of orbitals in the nth shell: n2 (n2 –fold degeneracy) Examples : Number of subshells (orbitals) n = 1 : l = 0 → only 1s (1) → 1 n = 2 : l = 0, 1 → 2s (1) , 2p (3) → 4 n = 3 : l = 0, 1, 2 → 3s (1), 3p (3), 3d (5) → 9 Selection rule for hydrogen atom
Spectra of hydrogen atom (or hydrogen-like atoms) Balmer, Lyman and Paschen Series (J. Rydberg) n1 = 1 (Lyman), 2 (Balmer), 3 (Paschen) n2 = n1+1, n1+2, … RH = 109667 cm-1 (Rydberg constant) hcRH Electric discharge is passed through gaseous hydrogen. H2 molecules and H atoms emit lights of discrete frequencies.
Engel says in p. 12 that “Bohr (1913) next introduced wave-particle duality which is equivalent to asserting that the electron had the de Broglie wavelength (1924).” Bohr Atom Model (1911-1913)
1885 – Johann Balmer – Line spectrum of hydrogen atoms 1913 – Niels Bohr – Theory of atomic spectra