Famous Opinions of QM

1. Famous Opinions of QM “A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it.” (Max Planck, 1920) “All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?‘”  (Albert Einstein, 1954) “Those who are not shocked when they first come across quantum physics cannot possibly have understood it.” (Niels Bohr, 1958)  

2. Famous Opinions of QM The one great dilemma that nails us…day and night is the wave-particle dilemma. (Erwin Schrodinger, 1959)   I think I can safely say that nobody understands quantum mechanics. (Richard Feynman, 1965)  

3. Particles and Waves • EM radiation can behave as either a wave or a particle depending on the situation • “Light has properties that have no analogy at the macroscopic level, and thus, we have to combine two different ideas to describe its behavior.”

4. Wavefunctions • A wavefunction is a probability amplitude. The “square” of a wavefunction gives the probability density…the likelihood of finding the particle in region of space. • The wavefunctions and kinetic energies available to a quantum particle are quantized if the particle is subject to a constraining potential. • We can determine the wavefunctions and KEs available to our system by considering the field of force (the PE) our system is subject to.

5. The Hamiltonian • Erwin Schrodinger developed a mathematical formalism that incorporates the wave nature of matter. • H, the “Hamiltonian,” is a special kind of function that gives the energy of a quantum state, which is described by the wavefunction, Y. • H contains a KE part and a PE part: • By solving the Schrodinger equation (below) with a known Hamiltonian, we can determine the wavefunctions and energies for quantum states.

6. H-atom wavefunctions • In the H atom, we are interested in describing the regions in space where it is likely we will find the electron, relative to the nucleus…we want the wavefunction for the electron. • We can model the attraction of the H atom’s single electron to its single proton with a “Coulombic” potential curve: • The V(r) potential becomes part of the Hamiltonian for the electron.

7. 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)

8. H-atom wavefunctions (cont.) • Then the Schrodinger equation for the hydrogen atom becomes: 3-dimensional KE operator in spherical polar coordinates Radial Coulombic PE operator

9. 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 can have integer numbers ranging from 1 to infinity. The higher n, the greater the distance between the nucleus (+) and the electron (-).

10. 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 ). ml … the “z component” of orbital angular momentum. Ranges in value from - lto 0 to +l. • We can characterize the hydrogen-atom orbitals using the quantum numbers: n, l, ml

11. Orbitals and Quantum Numbers • Naming the electron orbitals is done as follows • n is simply referred to by the quantum number • l(0…n… - 1) is given a letter value as follows: • 0 = s • 1 = p • 2 = d • 3 = f - ml(- l…0…l) is usually “dropped” For example: for n = 3, l= 2  “3d orbital”

12. Quantum Mechanical Model The Bohr model is deterministic…uses fixed “orbits” around a central nucleus to describe electron structure of atoms. The QM model is probabilistic…uses probabilities to describe electron structure. A probabilistic electron structure is much more difficult to visualize. HOWEVER, the electronic energy levels are still quantized.

13. In the Bohr model, you can always find the electron in an atom, just like you can always find the moon as it orbits the earth. You can always determinethe relative location of the nucleus and electron in Bohr’s model. This is because the electron follows a particular orbitaround the nucleus. In the QM model, the electron does not travel along a particular path around the nucleus. You can never determine the electron’s exact location…you can only find where it is likely to be. The Bohr orbit is replaced by orbital which describes a volume of space in which the electron is likely to be found. Deterministic vs. Probabilistic

14. Quantum Numbers and Orbitals (cont.) • Table 7.1: Quantum Numbers and Orbitals n Orbital ml# of Orb. 0 1s 0 1 0 2s 0 1 1 2p -1, 0, 1 3 0 3s 0 1 1 3p -1, 0, 1 3 2 3d -2, -1, 0, 1, 2 5 Increasing Energy

15. Orbital Shapes (cont.) • Example: Write down the orbitals associated with n = 4. l = 0 to (n - 1) = 0, 1, 2, and 3 = 4s, 4p, 4d, and 4f Ans: n = 4 4s (1 ml sublevel) 4p (3 mlsublevels) 4d (5 mlsublevels 4f (7 ml sublevels)

16. Which of the following sets of quantum numbers (n, l, m) is not allowed? (3, 2, 2). B. (0, 0, 0). C. (1, 0, 0). D. (2, 1, 0).

17. Electron Orbitals Orbitals represent a probability space where an electron is likely to be found. All atoms have all orbitals, but many of them are not occupied. The shapes of the orbitals are determined mathematically...they are not intuitive. s orbitals p orbitals d orbitals

18. Electron Orbital Shapes • The “1s” wavefunction has no angular dependence (i.e., it only depends on the distance from the nucleus). Probability = Probability is spherical

19. Electron Orbital Shapes (cont) s (l = 0) orbitals as n increases, orbitals demonstrate n - 1 nodes. Node: an area of space where the electron CAN’T be, ever, no matter how much it wants to.

20. Aside: What’s a node?? • Remember the guitar-string standing wave analogy? • A standing wave is a motion in which translation of the wave doesnot occur. • In the guitar string analogy (illustrated), note that standing waves involve nodes in which no motion of the string occurs.

21. Electron Orbital Shapes (cont.) 2p (l = 1) orbitals 2pz 2py 2px not spherical, but lobed. labeled with respect to orientation along x, y, and z.

22. Electron Orbital Shapes (cont.) 3p (l = 1) orbitals • more nodes as compared to 2p (expected.). • still can be represented by a “dumbbell” contour.

23. Orbital Shapes (cont.) 3d (l = 2) orbitals labeled as dxz, dyz, dxy, dx2-y2 and dz2.

24. Orbital Shapes (cont.) 4f (l = 3) orbitals We will not show the exceedingly complex probability distributions associated with f orbitals.

25. Electron Orbital Energies in the H-atom • energy increases as 1/n2 • orbitals of same n, but different lare considered to be of equal energy (“degenerate”). • the “ground” or lowest energy orbital is the 1s.

26. Orbital Summary • Orbital E increases with n. • At higher n, the electron is farther away from the nucleus...this is a higher energy configuration. • Orbital size increases with n. • There is a larger area of space where you are likely to find an electron at higher E’s. • Orbital shape is the same no matter the value of n. • 3s looks like 1s, except it’s bigger and has more nodes. Same for p, d, f, etc. • Number of nodes in an orbital goes as n - 1. • 1s has zero nodes, 2s has one node, 3s has two nodes... • 2px, 2py, 2pz each have one node, 3px, 3py, 3pz each have two nodes • the 3d orbitals each have two nodes, 4d have three, etc. • Note that number of nodes indicates relative energy! • All atoms have all orbitals…but in an unexcited atom, only those closest to the nucleus will be occupied by electrons

27. Orbital Quiz F • The shape of a given type of orbital changes as n increases. • The number of types of orbitals in a given energy level is the same as the value of n. • The hydrogen atom has a 3s orbital. • The number of lobes on a p-orbital increases as n increases. That is, a 3p orbital has more lobes than a 2p orbital. • The electron path is indicated by the surface of the orbital. T T F F

28. Electron Spin • Experiments demonstrated the need for one more quantum number. • Specifically, some particles (electrons in particular) demonstrated inherent angular momentum… • Basically, this means that electrons have two ways of interacting with an applied magnetic field. Interpretation: the electron is a bundle of “spinning” charge “spin up” “spin down”

29. Electron Spin (cont.) • The new quantum number is ms(analogous to ml). • For the electron, ms has two values: +1/2 and -1/2 ms = 1/2 ms = -1/2

30. Pauli Exclusion Principle Defn: No two electrons may occupy the same quantum state simultaneously. In other words: electrons are very territorial. They don’t like other electrons horning in. In practice, this means that only two electrons may occupy a given orbital, and they must have opposite spin.

31. Quantum Number Summary • n: principal quantum number • index of size and energy of electron orbital • can have any integral value: 1, 2, 3, 4, … • l: angular momentum quantum number • related to the shape of the orbitals • can have integral values 0 … n - 1 • ml: magnetic quantum number • related to orbital orientation (relative to the other l-level orbitals) • can have integral values –l … 0 … +l • ms: electron spin quantum number • related to the “magnetic moment” of the electron • can have half-integral values –1/2 or +1/2

32. Polyelectronic Atoms • For polyelectronic atoms, a direct solution of the Schrodinger Eq. is not possible. • When we construct polyelectronic atoms, we use the hydrogen-atom orbital nomenclature to discuss in which orbitals the electrons reside. • This is an approximation (and it is surprising how well it actually works). No solution for polyelectronic atoms!!

33. The Aufbau Principle • When placing electrons into orbitals in the construction of polyelectronic atoms, we use the Aufbau Principle. • This principle states that in addition to adding protons and neutrons to the nucleus, one simply adds electrons to the hydrogen-like atomic orbitals • Pauli exclusion principle: No two electrons may have the same quantum numbers. Therefore, only two electrons can reside in an orbital (differentiated by ms).

34. 4s 3s 3p 3d 2s 2p 1s Orbital Energies H has only one electron, so all of the sublevels in a given principal level have the same energy...they are degenerate. Energy In many-electron atoms, a given electron is simultaneously attracted to the nucleus and repelled by other electrons, causing the energies of the sublevels to change relative to H. When we put electrons in orbitals, we fill them in order of increasing energy, not n.

35. Let’s fill some orbitals RULES • Orbitals are filled starting from the lowest energy. • The two electrons in an orbital must have opposite spin. • Example: Hydrogen (Z = 1) 1s1 1s 2s 2p • Example: Helium (Z = 2) 1s2 1s 2s 2p

36. Let’s fill some more orbitals • Lithium (Z = 3) 1s22s1 1s 2s 2p • Berillium (Z = 4) 1s22s2 1s 2s 2p • Boron (Z = 5) 1s22s22p1 1s 2s 2p

37. Filling Orbitals (cont.) • Carbon (Z = 6) 1s22s22p2 REVISED RULES • Orbitals are filled starting from the lowest energy. • The two electrons in an orbital must have opposite spin. • Hund’s Rule: the orbitals in degenerate series (such as 2p in the example above) must each have an electron before any of them can have two. 1s 2s 2p Hund’s Rule: Lowest energy configuration is the one in which the maximum number of unpaired electrons are distributed amongst a set of degenerate orbitals.

38. Filling Orbitals (cont.) • Carbon (Z = 6) 1s22s22p2 1s 2s 2p • Nitrogen (Z = 7) 1s22s22p3 1s 2s 2p

39. Filling Orbitals (cont.) • Oxygen (Z = 8) 1s22s22p4 1s 2s 2p • Fluorine (Z = 9) 1s22s22p5 1s 2s 2p • Neon (Z = 10) 1s22s22p6 full 1s 2s 2p

40. Ne 3s • Compare to Neon (Ne) (Z = 10) 1s22s22p6 full 1s 2s 2p Filling Orbitals (cont.) • Sodium (Z = 11) 3p 1s 2s 2p 3s 1s22s22p63s1 [Ne]3s1

41. Filling Orbitals (cont.) • Sodium (Z = 11) [Ne]3s1 Ne 1s22s22p63s1 3s • Phosphorus (P) (Z = 17) [Ne]3s23p3 Ne 3s 3p • Argon (Z = 18) [Ne]3s23p6 Ne 3s 3p

42. Filling Orbitals (cont.) We now have the orbital configurations for the first 18 elements. Elements in same column have the same # of valence electrons! Valence Electrons: The total number of s and p electrons in the highest occupied energy level.

43. The Aufbau Principal (cont.) • Similar to Sodium, we begin the next row of the periodic table by adding electrons to the 4s orbital. • Why not 3d before 4s? • 3d is closer to the nucleus • 4s allows for closer approach; therefore, is energetically preferred.

44. Back to Filling Orbitals • Elements Z=19 and Z= 20: Z= 19, Potassium: 1s22s22p63s23p64s1 = [Ar]4s1 Ar 4p 4s 1s22s22p63s23p64s2 = [Ar]4s2 Z= 20, Calcium: Ar 4p 4s

45. 3d 3d 4p 4p Filling Orbitals (cont.) • Elements Z=21 to Z=30 have occupied d orbitals: Z= 21, Scandium: 1s22s22p63s23p64s23d1 = [Ar] 4s23d1 Ar 4s 1s22s22p63s23p64s23d10 = [Ar] 4s23d10 Z= 30, Zinc: Ar 4s

46. The Aufbau Principal (cont.) • Elements Z=19 and Z= 20: Z= 19, Potassium: 1s22s22p63s23p64s1 = [Ar]4s1 Z= 20, Calcium: 1s22s22p63s23p64s2 = [Ar]4s2 • Elements Z = 21 to Z = 30 have occupied d orbitals: Z= 21, Scandium: 1s22s22p63s23p64s23d1 = [Ar] 4s23d1 Z = 24, Chromium: [Ar] 4s13d5 exception Z= 30, Zinc: 1s22s22p63s23p64s23d10 = [Ar] 4s23d10

47. Write down the orbitals for each n on separate lines. Arrows drawn as shown will give you the order in which the orbitals should be filled. Note that this scheme fills 4s before 3d, as expected. What if you forget the orbital-filling order?

48. + Polyelectronic Atoms e- “Screening”: The presence of other electrons means a given electron does not feel the attraction of the nucleus as strongly as it would in hydrogen. “Penetration”: Orbitals that have some probability density close to the nucleus will be energetically favored over orbitals that do not.

49. Periodic Table This orbital filling scheme gives rise to the modern periodic table.

50. Periodic Table After Lanthanum ([Xe]6s25d1), we start filling 4f.