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**Lecture 17: The Hydrogen Atom**

Reading: Zuhdahl Outline The wavefunction for the H atom Quantum numbers and nomenclature Orbital shapes and energies

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Schrodinger Equation Erwin Schrodinger develops a mathematical formalism that incorporates the wave nature of matter: Kinetic Energy The Hamiltonian: d2/dx2 x The Wavefunction: E = energy

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**Potentials and Quantization (cont.)**

• What if the position of the particle is constrained by a potential: “Particle in a Box” Potential E = 0 for 0 ≤ x ≤ L = all other x

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**Potentials and Quantization (cont.)**

• What does the energy look like? n = 1, 2, … Energy is quantized E y y*y

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**Potentials and Quantization (cont.)**

• Consider the following dye molecule, the length of which can be considered the length of the “box” an electron is limited to: L = 8 Å What wavelength of light corresponds to DE from n=1 to n=2? (should be 680 nm)

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H-atom wavefunctions Recall that the Hamiltonian is composite of kinetic (KE) and potential (PE) energy. • The hydrogen atom potential energy is given by:

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**H-atom wavefunctions (cont.)**

• The Coulombic potential can be generalized: Z • Z = atomic number (= 1 for hydrogen)

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**H-atom wavefunctions (cont.)**

• The radial dependence of the potential suggests that we should from Cartesian coordinates to spherical polar coordinates. r = interparticle distance (0 ≤ r ≤ ) e- = angle from “xy plane” (/2 ≤ ≤ - /2) p+ = rotation in “xy plane” (0 ≤ ≤ 2)

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**H-atom wavefunctions (cont.)**

• If we solve the Schrodinger equation using this potential, we find that the energy levels are quantized: • n is the principle quantum number, and ranges from 1 to infinity.

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**H-atom wavefunctions (cont.)**

• In solving the Schrodinger Equation, two other quantum numbers become evident: l, the orbital angular momentum quantum number. Ranges in value from 0 to (n-1). m, the “z component” of orbital angular momentum. Ranges in value from -l to 0 to l. • We can then characterize the wavefunctions based on the quantum numbers (n, l, m).

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Orbital Shapes Let’s take a look at the lowest energy orbital, the “1s” orbital (n = 1, l = 0, m = 0) a0 is referred to as the Bohr radius, and = Å 1 1

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Orbital Shapes (cont.) Note that the “1s” wavefunction has no angular dependence (i.e., Q and F do not appear). Probability = • Probability is spherical

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Orbital Shapes (cont.) Radial probability (likelihood of finding the electron in each spherical shell

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**Orbital Shapes (cont.) Naming orbitals is done as follows**

n is simply referred to by the quantum number l (0 to (n-1)) is given a letter value as follows: 0 = s 1 = p 2 = d 3 = f - ml (-l…0…l) is usually “dropped”

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**Orbital Shapes (cont.) • Table 12.3: Quantum Numbers and Orbitals**

n l Orbital ml # of Orb. s s p , 0, s p , 0, d -2, -1, 0, 1,

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Orbital Shapes (cont.) • 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 4s (1 ml sublevel) 4p (3 ml sublevels) 4d (5 ml sublevels 4f (7 ml sublevels)

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**Orbital Shapes (cont.) s (l = 0) orbitals • r dependence only**

• as n increases, orbitals demonstrate n-1 nodes.

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**Orbital Shapes (cont.) 2p (l = 1) orbitals • not spherical, but lobed.**

• labeled with respect to orientation along x, y, and z.

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**Orbital Shapes (cont.) 3p orbitals**

• more nodes as compared to 2p (expected.). • still can be represented by a “dumbbell” contour.

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**Orbital Shapes (cont.) 3d (l = 2) orbitals**

• labeled as dxz, dyz, dxy, dx2-y2 and dz2.

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**Orbital Shapes (cont.) 4f (l = 3) orbitals**

• exceedingly complex probability distributions.

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**Orbital Energies • 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.

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