Basic Physical Chemistry Lecture 2. Keisuke Goda Summer Semester 2015

Transcription

1 Basic Physical Chemistry Lecture 2 Keisuke Goda Summer Semester 2015

2 Lecture schedule Since we only have three lectures, let s focus on a few important topics of quantum chemistry and structural chemistry Lecture 1 Hydrogen atom The most basic unit of chemistry Lecture 2 Angular momentum (spin) This is what makes chemistry interesting Lecture 3 Molecular structure The origin of various unique properties of molecules

3 Chapter 4: Quantum Mechanics in Three Dimensions 4.1: Schrodinger Equation in Spherical Coordinates 4.2: The Hydrogen Atom 4.3: Angular Momentum 4.4: Spin

4 Angular momentum As we have seen, the stationary states of the hydrogen atom are labeled by three quantum numbers (n, l, m); the principal quantum number (n) determines the energy of the state while l and m are related to the orbital angular momentum Classically, the angular momentum of a particle (with respect to the origin) is given by the formula

5 Angular momentum The corresponding quantum operators are obtained by the standard substitution

6 L x and L y do not commute Commutation relations We can get the other two commutation relations by cyclic permutations of the indices (x y, y z, z x) Note that L x, L y, and L z are incompatible observables just like x and p or E and t, which means that there is a uncertainty relation between each pair of the angular momenta

7 Commutation relations The total angular momentum, L 2, is given by Then, L 2 commutes with L x Likewise, L 2 also commutes with L y and L z This can be written more compactly L 2 is compatible with each component of L, and we can hope to find simultaneous eigenstates of L 2 and L z

8 Ladder operator technique We use a ladder operator technique very similar to the one we applied to the harmonic oscillator in Chapter 2 Since Also since If f is an eigenfunction of L 2 and L z, then Therefore, L ± are the raising and lowering operators

9 Ladder of angular momentum states L ± = the raising and lowering operators L ± f is an eigenfunction of L z with the new eigenvalue The raising operator increases the eigenvalue of L z by while the lowering operator decreases the eigenvalue by

10 Top and bottom rungs The z-component of the angular momentum cannot exceed the total, such that there must exist a top rung and also a bottom rung just like before We are interested in expressing L 2 as a function of L z and L ± Suppose the top and bottom rungs meet the following eigenfunctions It follows that for the top rung; for the bottom rung

11 Eigenfunctions and eigenvalues of L 2 and L z for the top rung; for the bottom rung Because of the ladder operators, the eigenvalues of L z are m, where m goes from l to +l in N integer steps, meaning l = -l + N, such that l = N/2 (since N is an integer, l is an integer or a half-integer) The eigenfunctions are characterized by the quantum numbers (l, m) For a given value of l, there are different values of m; in other words, there are rungs on the ladder

12 Angular momentum states (l = 2) m = 2 m = 1 m = 0 m = 1 m = 2

13 Angular momenta in spherical coordinates We need to rewrite L x, L y, and L z in spherical coordinates The gradient is given by Expressing the vector r in terms of the unit vector Using

14 Unit vectors in spherical coordinates The unit vectors in spherical coordinates can be expressed in terms of the unit vectors in Cartesian coordinates; (r, θ, ϕ) in terms of (x, y, z)

15 Angular momentum operators The unit vectors in spherical coordinates can be rewritten in terms of the unit vectors in Cartesian coordinates Hence, L can be written in terms of the unit vectors Each angular momentum operator is then found to be

16 Quantum operators of L ± and L 2 We also need to find the raising and lowering operators Using,, and

17 Eigenfunctions of L ± and L 2 Finally, we are ready to determine f which is a function of and Using the equations and operators we have found so far and We find But wait a minute! These equations are precisely equivalent to the angular equation and azimuthal equation in Chapter 4.1 Spherical harmonics are eigenfunctions of L 2 and L z

18 Eigenfunctions of H, L z, and L 2 H, L z, and L 2 commute and can therefore be obtained simultaneously Simultaneous eigenfunctions of the three commuting operators (H, L z, and L 2 ) are then given by In most problems, you do not need to remember the spherical harmonics; you just have to use this set of eigenfunctions to find the eigenvalues (energy and momenta) For a given value of l, there are different values of m; in other words, there are rungs on the ladder

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20 Concept of spin In classical mechanics, a rigid object admits two kinds of angular momentum: (1) orbital angular momentum, L, associated with the motion of the center of mass, and (2) spin, S, associated with motion about the center of mass One analogy is the earth; it has (1) orbital angular momentum corresponding to its annual revolution around the sun, and (2) spin angular momentum corresponding to its daily rotation about the north-south axis

21 Concept of spin In quantum mechanics, if the analogy is applied to the hydrogen atom, the electron has (1) orbital angular momentum corresponding to its revolution around the proton, and (2) spin angular momentum corresponding to its self rotation about some axis, but this analogy is not quite true because the electron is a structureless point particle Interestingly, there is this spin in the electron, but keep in mind that it s not exactly spin although it s called spin (the electron is not spinning)

22 Concept of spin (s = ½)

23 Fundamental relations of spin The fundamental commutation relations of spin are similar to the commutation relations of orbital angular momentum It follows (as before) that the eigenvectors of S 2 and S z satisfy Here don t get confused about m although the same notation is used to describe orbital angular momentum The raising and lowering operators of spin can also be defined just like in orbital angular momentum The proton, neutron, and electron (also quarks and leptons) have spin ½

24 Spin ½ By far the most important case is s = ½ because ordinary particles (protons, neutrons, and electrons) as well as all quarks and leptons have s = ½ Since the spin is ½, there are only two eigenstates 1. Spin up ( ): 2. Spin down ( ): Therefore, using these eigenstates as basis vectors, the general state of a spin-½ particle can be expressed as a two-element column matrix (called a spinor) Spin-up state Spin-down state

25 Expressing spin states in terms of matrices The spin operators become 2 x 2 matrices, which we can work out by noting their effect on X+ and X- Similarly, we can find the matrix representation of S z Moreover, we can find the raising and lowering operators

26 Pauli spin matrices Likewise, we can find S x and S y from the ladder operators S x, S y, and S z all carry a factor of /2 and can hence be normalized Eigenspinors of S z Eigenspinors of S x Pauli spin matrices

27 Electron in a magnetic field A spinning charged particle constitutes a magnetic dipole; the magnetic dipole moment of the particle is proportional to its spin angular momentum Here γ is called the gyromagnetic ratio When a magnetic dipole is placed in a magnetic field B, it experiences a torque which tends to line it up parallel to the field (just like a compass needle) Discovery of spin in the electron

28 Addition of angular momenta Suppose that we have two spin-½ particles; for example, the electron and the proton in the ground state of hydrogen Each can have spin up or spin down, so there are four possibilities in all The total angular momentum of the atom is given by the operator Apply this operator to the two-spin state This doesn t look right because m is supposed to advance in integer steps In fact, there is an extra state

29 Triplet and singlet states In fact, there are three states with s = 1 (called the triplet) In fact, there is an additional state with s = 0 (called the singlet) The existence of these states can be verified by calculating the eigenvalues of S 2

30 21 cm line

31 Linear combination of spin states In general, if you combine spin s 1 and spin s 2, the total spin you expect to get is given by Roughly speaking, the highest total spin occurs when the individual spins are aligned parallel to each other, and the lowest occurs when they are anti-parallel, leaving entangled states The combined states with total spin s and z-component m will be some linear combination of the composite states Here the constants are called Clebsch-Gordan coefficients which can be calculated, but are typically available in a table

32 Clebsch-Gordan coefficients Example 1 Example 2

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34 Chapter 5: Identical Particles 5.1: Two-Particle Systems 5.2: Atoms

35 Wave function of a two-particle system For a single particle, ᴪ(r, t) is a function of the spatial coordinates and time (let s ignore spin for the moment), and for a two-particle system, the wave function is a function of the coordinates of particle one (r 1 ) and the coordinates of particle two (r 2 ) Its time evolution is determined as always by the Schrodinger equation where H is the Hamiltonian for the whole two-particle system given by As always, the normalization requirement is given by

36 Schrodinger equation for a two-particle system For time-independent potentials, we obtain a complete set of solutions by separation of variables (just as always) where the spatial wave function satisfies the time-independent Schrodinger equation Here E is the total energy of the system Important comment: The hydrogen atom is a two-particle system that consists of one proton and one electron, but we often ignore the presence of the proton and deal only with the electron (why?) The proton is ~1,000 times more massive (heavier) than the electron and then the first/second term vanishes

37 Indistinguishable particles Suppose particle 1 is in the state and particle 2 is in the state and then the wave function for the two-particle system is given by However, this is based on the assumption that we can tell the particles part otherwise it wouldn t make any sense to claim that particle 1 is in the a state and particle 2 is in the b state Important discussion All we could say about the system is that one of them is in the a state and the other is in the b state We wouldn t know which is which (this is classically silly, but quantum mechanically, this is valid) The fact is that all electrons are identical (we can t mark them in red or blue because this action alters the wave function of the system) Therefore, there is no such thing as this electron or that electron So, we have to consider this possibility into the wave function

38 Bosons and fermions Quantum mechanics neatly accommodates the existence of particles that are indistinguishable in principle by using superposition of states = c 1 + c 2 Since the two composite states have equal probabilities, then c 1 = c 2 Mathematically, this equation admits two kinds of identical particles 1. Bosons: particles with the plus sign in the above equation or with integer spin (photons, mesons, etc.) 2. Fermions: particles with the minus sign in the above equation or with half integer spin (electrons, protons, quarks, etc.) Bosons and fermions have quite different statistical properties

39 Pauli exclusion principle It follows from the equation that two identical fermions (for example, two electrons) cannot occupy the same state: Pauli exclusion principle Proof If the a state and b state are identical ( ), We are left with no wave function; in other words, the wave function vanishes or does not exist (two identical fermions cannot share the same state) Chemistry version of the Pauli exclusion principle No two electrons can have the same state; in other words, no two electrons can share the same quantum numbers (n, l, m l, m s )

40 Effect of the Pauli exclusion principle No two fermions (e.g., electrons) can occupy the same state while bosons can occupy the same state

41 Effect of distinguishability What is the effect of the plus and minus signs? Suppose that particle 1 is in state a and particle 2 is in state b (1) If the particles are distinguishable, then (2a) If the particles are identical bosons (2b) If the particles are identical fermions Let s calculate the expectation value of the square of the separation distance between the two particles in these different cases

42 (1) Distinguishable particles

43 (2) Identical particles: bosons (+), fermions (-)

44 (2) Identical particles: bosons (+), fermions (-)

45 Let s compare the two cases (1) Distinguishable particles Exchange force (2) Identical particles: bosons (+), fermions (-) Identical bosons tend to be somewhat closer together, and identical fermions somewhat farther apart, than distinguishable particles If there is some overlap of the wave functions, the system behaves as if there were a force of attraction between identical bosons, pulling them closer together, and a force of repulsion between identical fermions, pushing them apart (called an exchange force)

46 Covalent bond So far we have been ignoring spin we need to consider the complete state of the electron that consists of not only its position wave function, but also the orientation of its spin (spinor) The singlet combination of electrons is antisymmetric whereas three triplet states are all symmetric The singlet state leads to bonding (called the covalent bond), meaning that covalent bonding requires the two electrons to occupy the singlet state with total spin zero

47 Covalent bond It is more stable to share two electrons and close the shell; in other words, the total energy of the two-hydrogen system (hydrogen molecule) is lower than the sum of two individual hydrogen atoms stable

48 Early Atomic Models

49 Hamiltonian for a general atom Consider a neutral atom of atomic number Z that consists of a heavy nucleus with electric charge Ze, surrounded by Z electrons (mass m and charge e) and the Hamiltonian for this system is given by The first term in the bracket represents the sum of the kinetic and potential energies of the jth electron in the electric field of the nucleus, and the second sum is the potential energy associated with the mutual repulsion of the electrons with the factor of ½ correcting for the fact that the summation counts each pair twice The complete wave function can be found by solving the Schrodinger equation, but as it is easy to guess, it s impossible to solve this analytically because it s too complicated

50 Shells of electron states Chemistry is exciting because electrons are fermions, not bosons if they were bosons, all the electrons would occupy the ground state (n, l, m l ) = (1, 0, 0) and chemistry would be very boring Fortunately, electrons are identical fermions and hence subject to the Pauli exclusion principle, making only two electrons occupy any given orbital (n, l, m l ) with one spin up (s = ½, m s = +½) and the other one spin down (s = ½, m s = -½) in the singlet configuration The n = 1 shell has room for 2 electrons l = 0, m = 0 (singlet) 2 states (spin: ) The n = 2 shell has room for 8 electrons l = 1, m = -1, 0, 1 (triplet) 6 states (spin: ) l = 0, m = 0 (singlet) 2 states (spin: ) The n = 3 shell has room for 18 electrons l = 2, m = -2, -1, 0, 1, 2 10 states (spin: ) l = 1, m = -1, 0, 1 (triplet) 6 states (spin: ) l = 0, m = 0 (singlet) 2 states (spin: )

51 Subshells (s, p, d, f,..) l = 0: s shell l = 1: p shell l = 2: d shell l = 3: f shell

52 Aufbau principle Aufbau principle ( aufbau in German means building-up ) Electrons fill orbitals starting at the lowest available energy levels before filling higher levels

53 Hund s rule For example, the configuration tells us that two electrons in the orbital (1, 0, 0), two in the orbital (2, 0, 0), and two in some combination of the orbitals (2, 1, 1), (2, 1, 0), and (2, 1, -1) Hund s rule S = total spin quantum number L = total orbital angular momentum J = total angular momentum (orbital angular momentum + spin)

54 Ground state electron configurations

55 Periodic table (most familiar version)

56 Periodic table in terms of subshells

57 Periodic table in terms of subshells

58 Ionization energy The ionization energy of an atom is the amount of energy required to remove an electron (in the lowest possible energy state) from the atom in the gas state

59 Ionization energy

60 Third report Pick one of the papers on the course webpage and write a 1-page report about it in English. The theme of your report is about the novelty of the work demonstrated in the paper. The report should be written in a A4-sized paper. Only single spacing is acceptable (no double spacing). Font size: 11 pt Font type: Arial, Times New Roman, or Calibri Write down your name and ID number at the top of your report. Do not forget to write the title for your report. Submission deadline: 4/28 Put your report on the lecturer s desk before the lecture begins.