Lecture 19: The Hydrogen Atom • Reading: Zumdahl 12.7-12.9 • Outline – The wavefunction for the H atom • Know what wave functions look like for a particle trapped in a “box”; now we need to know what they look like for an electron attracted to a nucleus; and the energy of each wave function. – Quantum numbers and nomenclature – Orbital (i.e. wavefunction) shapes and energies • Problems (Chapter 12, 5th Ed.) – 48, 49, 50, 52, 54, 55, 56, 57, 60 1 H-atom wavefunctions • Recap: The Hamiltonian is a sum of kinetic (KE, or T) and potential (PE, or V) energy. Hˆ = Tˆ + Vˆ The ‘bar’ means average over the position of the electron. E = T + V = 12 V er P+ V (Potential E.) • The hydrogen atom potential energy is given by: r 0 −e ˆ V = V (r) = r 2 2 The Coulombic PE (V) can be generalized − Ze 2 V (r ) = ( 4πε o ) r F e=− NA e′2 = e- e2 ( 4πε o ) 2 p T= where p = mv 2m r Z + P • Z = atomic number (= 1 for hydrogen) • r is the distance between the electron and the nucleus • Only one electron allowed (for now). 3 H-atom Coordinates Frame • The radial dependence of the potential suggests that we should from Cartesian coordinates to spherical polar coordinates. r = interparticle distance (0 ≤ r ≤ ∞) e- p+ Major (azimuthal) angle θ = angle from z to“xy plane” (0 ≤ θ ≤ π) Minor angle φ = rotation in “xy plane” (0 ≤ φ ≤ 2π) 4 H-atom Allowed Energies When we solve the Schrodinger equation using the Coulomb potential, we find that the bound-state energy levels are quantized or discrete: 4 2 ⎛ ⎞ ⎛ Z me Z ⎞ En = − 2 ⎜ 2 2 ⎟ = − ⎜ 2 ⎟ ⋅ 2.178 x10−18 J n ⎝ 8ε 0 h ⎠ ⎝n ⎠ 2 • n (an integer counter) is the principal quantum number, and ranges from 1 to infinity. n=1 is the lowest energy (level) or ground state for an electron bound to a hydrogen-like nucleus. •This is the same formula Bohr gave us. •Compare and contrast these energy levels with those of the particle in a box. 5 Solve the Wave Equation for the Electron bound to the Nucleus • Set up the Schrödinger equation (SE) for the wave function in terms of x,y and z coordinates, then rewrite in polar coordinates (because V depends only on r). • Solve the SE the same way Schrödinger did: Look the answer up in a math book (Courant and Hilbert, in his case). • The solution gives a set of wave functions, and the energy of each wave function. • The wave functions (and energies) are distinct and countable (although in principle there are an infinite number of wavefunctions). • The wavefunctions are now called orbitals as they describe the probability of the electron in the vicinity of the nucleus. They are not orbits but regions of space wherein the electron orbits, hence orbitals. 6 Form of WaveFunctions (for Orbitals) • Like the particle in a box the wave function depends on the coordinate and a quantum number (like x and n). • There are three coordinates so the wave function is a product of a part that – Depends only on r and has n (and l) with it – Depends only on theta and has l (and m) with it – Depends only on phi (and has m with it) • The total wave function has the form: Ψ n ,l ,m ( r , θ , φ ) = Rn ,l ( r ) Θl ,m (θ ) Φ m (φ ) x = r sin θ cos φ Relation between Cartesian and polar coordinates. y = r sin θ sin φ z = r cos θ 7 Orbitals • Orbitals are a description of where the electron resides (like a house) • Quantum numbers are like the address of the house. • The orbital does exist even without the electron (so an empty orbital is called a virtual orbital). 8 Energy levels The energy expression for the QM result is the same as Bohrs, because the Virial Relation (which is also true for planets going around the sun) is also true for Quantum Mechanics and embodies the balance between potential and kinetic energy. p2 Ze 2 KE = T = >0 PE = V = − <0 2m 4πε o r V = −2T ⇐ This is the Virial Relation 1 1 V 2 −1 V 2 E = T +V = V = = 2 2V 4 T rp = n= ⎛ Ze ⎞ 2 2 ⎜ ⎟ 2 2 2 −1 V −1 ⎝ 4πε o r ⎠ − m ⎛ Ze ⎞ − m ⎛ Ze ⎞ = = = ⎜ ⎟ = ⎜ ⎟ 2 p 4 T 4 2 ⎝ 4πε o rp ⎠ 2 ⎝ 2ε o nh ⎠ 2m 2 EBohr Bohr’s suggestion: 2 9 H-atom quantum numbers • n is called the principal quantum number. In solving the Schrodinger Equation, two other quantum numbers become evident: A is the orbital angular momentum quantum number. Ranges in value from 0 to (n-1). This tell us how much energy is in the rotating part of the electron. mA is the “z component” of orbital angular momentum. Ranges in value from −A to + A Because of the range of ml values there are a possible 2A + 1 different values of m for a given value of A 10 H-atom Quantum Numbers (cont.) • We can then characterize the wavefunctions based on the three quantum numbers: n, A, mA We went from Cartesian to polar coordinates now we go from polar coordinates to integer counters, we call quantum numbers ⎧ x ⎫ ⎧ r ⎫ ⎧ n ⎫ ⎧ principal ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ y ⎬ ⇒ ⎨θ ⎬ ⇒ ⎨ A ⎬ = ⎨ angular ⎬ ⎪ z ⎪ ⎪φ ⎪ ⎪m ⎪ ⎪ magnetic ⎪ ⎩ ⎭ ⎩ ⎭ ⎩ A⎭ ⎩ ⎭ When we “look at” orbitals, the angular quantum number is the best one to see the shape. A value 0 1 symbol s 2 3 p d f 11 Orbital Numbers • Naming orbitals using QNs is done as follows – n is simply referred to as the quantum number – l (0 to (n-1)) is given a letter value (named by the spectroscopists) as follows: name A value symbol 0 s sharp 1 p principal 2 d diffuse 3 f fine - ml (-l…0…l) is usually “dropped” The Payoff: the three QNs help us understand the structure of the periodic table and the chemistry of the elements (Aufbau 12 Principle). These QNs are the numbers of chemistry. 1s Orbital Shapes • For each set of 3 quantum numbers there is a specific wave function or orbital with a unique shape. • Let’s take a look at the lowest energy orbital (or ground state), the “1s” orbital (n = 1, l = 0, m = 0) 3 3 1 ⎛Z⎞ 1 ⎛ Z ⎞ 2 −σ ψ 1s = = ⎜ ⎟ e ⎜ ⎟ e π ⎝ ao ⎠ π ⎝ ao ⎠ Z σ = r A reduced or dimensionless distance a0 • 2 Z − r a0 a0 is referred to as the Bohr radius, and = 0.529 Å 1 1 2⎞ ⎛ Z −18 E n = −2.178x10 J⎜ 2 ⎟ = −2.178x10−18 J ⎝n ⎠ 13 1s Orbital Shapes • Note that the “1s” wavefunction has no angular dependence (i.e., Θ and Φ do not appear). 3 1 ⎛Z⎞ ψ1s = ⎜ ⎟ e π ⎝ ao ⎠ 2 −Z r a0 Probability Density = 3 1 ⎛ Z ⎞ 2 −σ = ⎜ ⎟ e π ⎝ ao ⎠ ψψ * Probability is spherical, depends only on r EVERY orbital has the factor: (Z12.55), why 90% not 100% e −σ n 14 Counting Orbitals Three Rules: n = 1, 2,3" 0≤A<n − A ≤ mA ≤ A • Table 12.3: Quantum Numbers and Orbitals n l Orbital 1 0 2 0 1 3 0 1 2 1s 2s 2p 3s 3p 3d ml 0 0 -1, 0, 1 0 -1, 0, 1 -2, -1, 0, 1, 2 2 # of Orb. n 1 1 3 1 3 5 15 Naming Orbital • Example: Write down the orbitals associated with n = 4. Ans: n = 4 l = 0 to (n-1) = 0, 1, 2, and 3 = 4s, 4p, 4d, and 4f Number of orbitals, or degeneracy: 4s (1 ml sublevel) 4p (3 ml sublevels) 4d (5 ml sublevels 4f (7 ml sublevels) Total number of orbitals for any n: n2 (=16 for n=4) 16 Identify Orbital Names • Z12.49: Which designations (Orbital Addresses) are incorrect (or correct) and why? – 1s, 1p, 7d, 9s, 3f, 4f, 2d • Z12.52 How many orbitals can have the Designation 5 p,3d z 2 , 4d , n = 5, n = 4 • Z12.50: Which of the follwing QNs are not allowed for H atom? What is wrong? n A m Answer a 2 1 −1 2p b 1 1 c d e 8 7 −6 X 8j 1 0 3 2 2 2 X 3d f 4 3 4 X g 0 0 0 X 0 17 S Family of Orbital Shapes S or (l = 0) orbitals; increasing n • r dependence only Nodes: Zeros of Polynomial As n increases, orbitals demonstrate n-1 nodes. No node at the origin (?) What is an orbital? What are we seeing? 18 All 3 are identical in shape and size they just point along x,y and z (respectively) P Orbital Shapes 2p (l = 1) orbitals All have a planar node through the middle, normal to the direction x = r sin θ cos φ • not spherical, but lobed. ψ2p z y = r sin θ sin φ 1 = 4 2π z = r cosθ 3 ⎛ Z ⎞ 2 −σ 2 ⎜ ⎟ e ⋅ σ cos θ ⎝ ao ⎠ • Labeled with respect to orientation along x, y, and z. •Orbitals have spatial direction (nodes help describe that) 19 P Orbital Shapes increase n (Z12.54, 56) 3p orbitals; contrast with 2p Why no 1p orbitals? ψ3p z 2 ⎛Z ⎞ = ⎜ ⎟ 81 π ⎝ a o ⎠ 3 2 (6σ − σ )e 2 −σ 3 z = r cos θ σ = Zr a o Where does this node occur? σ = ? (Z12.60) • more nodes (one planar and now one radial) as compared to 2p (expected.). Why is one nodal surface Planar and the other spherical? (Z12.57) • still can be represented by a “dumbbell” contour: (angular part stays the same) 20 cos θ D Orbital Shapes 3d (l = 2) orbitals xy = r 2 sin 2 θ cos φ sin φ • labeled as dxz, dyz, dxy, dx2-y2 and dz2. e.g. think about dxy, it literally is x time y. 2 planar nodes21 F Orbital Shapes 4f (l = 3) orbitals • exceedingly complex probability distributions. 22 3 planar nodes How to look at Orbital Shapes Begin with a ball (sphere); Don’t cut it at all: s orbital (l=0), only one types Cut it (in half) with a single plane: This generates a p orbital (l=1), three different ones; generated by cutting on three different planes. To see it better sometimes we color two halves differently (y/b). Cut it (into quarters) with two perpendicular (orthogonal) planes. This generates a d orbital (l=2), 5 different ones; Cut it (into eights) with three perpendicular planes. (eg. One in xy, one in xz and one in yz plane) This generates f orbitals (l=2), 7 different ones 23 Orbital Energies (H Atom) Continuum 0 ------------------------------------ • energy increases as -1/n2 • orbitals of same n, but different l are considered to be of equal energy (“degenerate”). • the “ground” or lowest energy orbital is the 1s. All discrete or bound states are below the E=zero line. 24