Chem 442 Review for Exam 2. Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components.

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1 Chem 44 Review for Exam Hydrogenic atoms: The Coulomb energy between two point charges Ze and e: V r Ze r Exact separation of the Hamiltonian of a hydrogenic atom into center-of-mass (3D) and relative (3D) components The Schrödinger equation of a hydrogenic atom (3D) is exactly solvable Exact separation of the Schrödinger equation of a hydrogenic atom into radial (1D) and angular (D) components; the angular (D) component is further separable (two 1D) 0 The radial equation: mass Ze l( l 1) R R R ER ; is the reduced r r r 0r r The last term in the left-hand side arises from the kinetic energy of the angular part and increases with the angular momentum quantum number l The radial part of the hydrogenic wave function is proportional to (radius) l (associated / 0 Laguerre polynomial) (Slater-type orbital); a Slater-type orbital is e Zr na (spherical) The angular part of the hydrogenic wave function is the spherical harmonics: lm l, iml Y, is the product of an associated Legendre polynomial and e lm l Y ; The energy of the hydrogenic Schrödinger equation is Z E (in Ry) n The natural unit of length: a0 1 bohr 059 Angstrom (the most probable radius of the electron in hydrogen) The natural unit of energy: e0 1 Ry 136 ev (the negative of the energy of hydrogen in the ground state) The energy levels of the hydrogen atom are increasingly narrowly separated with n (the principal quantum number) The energy levels are continuous above the ionization limit 1

2 The ionization energy of the hydrogen atom is 136 ev (the negative of the ground-state energy) Quantum numbers of the hydrogenic wave function: n, l, and m l ; know the physical meaning and ranges of the quantum numbers Shells and subshells: K, L, M, etc shells; s, p, d, etc subshells; know the degrees of degeneracy The ns orbitals have n 1 nodal spheres; p orbitals have n nodal surfaces plus a nodal plane The wave function must change its sign across a nodal surface (not just vanish on the surface) The p 0 orbital is real and equal to p z orbital extending along the z axis The p +1 and p 1 orbitals are complex, but they can be linearly combined to form real p x and p y orbitals extending along x and y axes; the linear combination is justified because of degeneracy (they have the same energy) The p +1, p 0, and p 1 orbitals are eigenfunctions of the magnetic (z-component) orbital angular momentum operator with eigenvalues,0, They have the same energy and are triply degenerate The p x and p y orbitals are no longer eigenfunctions of the magnetic (z-component) orbital angular momentum operator but are still eigenfunctions of the Hamiltonian (have the same energy) The five complex d orbitals with well-defined z-component orbital angular momenta (d +, d +1, d 0, d 1, d ) can be linearly combined to form real d orbitals (d xy, d yz, d zx, d xx yy, d 3zz rr ) The average radius of the hydrogen: use the expectation value formula; use the normalized wave function; use ˆr for the radius operator; use spherical coordinates and corresponding volume element rdrsindd Radial distribution function: R r r () nl The most probable radius: find the maximum of the radial distribution function; find r where the first derivative of the radial distribution function vanishes; the zero derivatives may mean a maximum, minimum, or saddle point; also check the boundary points even if they do not have zero first derivatives The most probable point: find the maximum of the probability density

3 Atomic spectroscopic transitions: The transition probability is proportional to transition dipole moment (amplitude of the light s electric field) if and only if the energy conservation law is satisfied The transition dipole moment = * fzˆ id for z-polarized light The excitation and deexcitation are equally likely with all other conditions being the same Selection rules for atomic transition: l l l 1; m m m 0, 1; m 0 l l l s The unit cm 1 (reciprocal centimeter or inverse centimeter or wave number) is proportional to energy unit by the coefficient of proportionality hc 1 K = 07 cm 1 ; room temperature = 00 cm 1 ; 1 ev = 8000 cm 1 ; 1 kj/mol = 80 cm 1 Helium and heavier atoms: The orbital approximation: approximate separation of variables; fill electrons in hydrogenic atomic orbitals in polyelectron atoms; exact in the absence of the electron-electron repulsion term in the Hamiltonian A wave function of more than one electron must be antisymmetric because an electron is a fermion; antisymmetric means that the function changes sign upon interchange of two electron coordinates (x, y, z, as well as spin coordinate ); related to the Pauli exclusion principle He (1s) singlet: 1s 11s (1) () (1) () r r He (1s) 1 (s) 1 singlet: 1sr1s r sr1 1sr (1) () (1) () He (1s) 1 (s) 1 triplets: 1sr1s r sr1 1sr (1) () (1) (), 1sr1sr sr11sr (1) (), 1s 1s s 11s (1) () r r r r ; they are degenerate (the same energy) because they share the identical spatial part The energy ordering if the electron-electron repulsion term is neglected: (1s) singlet << (1s) 1 (s) 1 triplets = (1s) 1 (s) 1 singlet; the energy in each case is the sum of the energies of the hydrogenic atomic orbitals The energy ordering in reality (with the electron-electron repulsion reinstated): (1s) singlet << (1s) 1 (s) 1 triplets < (1s) 1 (s) 1 singlet; the second inequality because the triplets spatial part is antisymmetric, preventing two electrons from existing at the same spatial position (spin correlation) 3

4 The electron-electron repulsion (ignored in the orbital approximation) is responsible for shielding and spin correlation Shielding explains the energy ordering s < p < d < f in the same shell; outer orbitals have smaller attraction to nucleus due to shielding; s orbitals have less shielding (lower energy) because of greater probability density at nucleus; basis of aufbau (building up) principle Spin correlation (Pauli exclusion principle or antisymmetry) explains Hund s rule; spatial part antisymmetric in triplet states, leading to lower energy with unpaired electrons The effect of the electron-electron repulsion term in the orbital approximation can be quantified by approximating the energy by an expectation value The expectation value of energy in the He wave functions in the orbital approximation contains the so-called Coulomb and exchange terms in addition to the sums of the energies of hydrogenic orbitals; the Coulomb term explains shielding; the exchange term is tied to spin correlation The total z-component spin angular momentum operator of He: Sˆ sˆ 1 sˆ The singlet spin part is an eigenfunction of S ˆz : Sˆ (1) () (1) () 0 (1) () (1) () z The triplet spin parts are eigenfunctions of S ˆz : Sˆ (1) () (1) (); ˆ z Sz(1) () (1) (); Sˆ (1) () (1) () 0 (1) () (1) () z z z z The total spin angular momentum quantum numbers in He: singlet S = 0 (M S = 0); triplet: S = 1 (M S = +1, 0, 1) Spin multiplicities of atoms: singlet (S = 0, eg, ground state He), doublet (S = 1/, eg, the hydrogen atom or a single electron), triplet (S = 1, eg, an excited He), quartet (S = 3/, eg, an excited Li) Spectroscopic transitions between different spin multiplicities are forbidden because the operator in the transition dipole moment does not act on spin and the spin wave functions with different multiplicities are orthogonal to each other Spin-orbit effect: Spin and orbital angular momenta have a weak interaction: hca coupling constant given in cm 1 ŝl ˆ ; A is the spin-orbit 4

5 Level splitting by the spin-orbit interaction: jmax ls, jmin l s and all j s in between separated by 1; the degree of degeneracy is j The first-order perturbation theory gives E hca j j l l s s jls,, ( 1) ( 1) ( 1) Na D line is the paradigm of spin-orbit problem: (3s) 1 state (ground state) does not split; (3p) 1 state (excited state) splits into 3p 3/ and 3p 1/ spin-orbit coupled states; be able to compute energy shift by first-order perturbation theory and the degree of degeneracy for each state The spin-orbit interaction (and the concept of spin itself) comes from special relativity; the heavier the element the greater the value of A Phosphorescence versus fluorescence Intersystem crossing versus internal conversion Born-Oppenheimer principle: The full molecular Hamiltonian: n N n N N n ˆ e ZZe I J ZIe H e i N I m m r r r i1 e I1 i N ij I 0 ij IJ 0 IJ I i 0 Ii Electronic Schrödinger equation: n n N n e Ze I e ; ; i e Ee e i1 me ij 0r I i 4 i ij 0rIi rr R rr N I J Nuclear Schrödinger equation: E E Parameters versus variables I J Potential energy surface: E IJ 0 IJ NI e n n I1 mn IJ 0rIJ I N ZZe R R R N ZZe e R r Equilibrium molecular structures; binding and dissociation energies; vibrational and rotational energy levels The dynamical degrees of freedom of a molecule: 3n electronic; 3N nuclear (3 translational, 3 or rotational, 3N 6 or 3N 5 vibrational); separation between electronic and nuclear is the BO approximation; separation between translation and the rest is exact; separation between vibration and rotation is the rigid-rotor approximation 5

6 Valence bond theory: Singlet H in VB: A(1) B() B(1) A() (1) () (1) () Triplet H in VB: A(1) B() B(1) A() ; (1) () (1) () (1) () (1) () they are triply degenerate (the same energy) because they have the identical spatial part Energies: singlet < triplets because singlet (triplets) has enhanced (have depleted) electron density in between the nuclei The and bonds in VB Covalent bonds; Lewis structure VB descriptions of N, H O, NH 3 : 90 degree HOH and HNH angles using p orbitals Promotion and sp 1, sp, and sp 3 hybridization; VB description of CH 4, ethylene, and acetylene Lone pairs The sp 3 hybridized VB description of H O and NH 3 VB description of H O isovalence series and importance of lone-pair repulsion in determining molecular structure Molecular orbital theory: LCAO MO theory; the bonding and antibonding orbital of H : Singlet (X) H in MO: X(1) X() (1) () (1) () Triplet (X) 1 (Y) 1 H in MO: X(1) Y() Y(1) X() (1) () (1) () (1) () (1) () Singlet (X) 1 (Y) 1 H in MO: X(1) Y() Y(1) X() (1) () (1) () X AB and Y A B 6

7 The energy ordering: singlet (X) << triplet (X) 1 (Y) 1 < singlet (X) 1 (Y) 1 ; the first inequality because of different orbital configurations; the second inequality due to spin correlation or Hund s rule VB versus MO; VB: A(1) B() B(1) A() (1) () (1) () ; MO: X(1) X() (1) () (1) () = A(1) B() B(1) A() A(1) A() B(1) B() (1) () (1) () ; VB has 0% ionic (too small), while MO has 50% ionic (too large); these are the simplest descriptions in respective models ; the energy is approximated by an expectation value of energy in these normalized wave functions of H + : * ˆ e j k E Hd E1 s ; know the shapes (especially the asymptotic 0R 1 S behaviors) of S, j, and k; this explains the bound potential energy curve of the bonding orbital, the unbound (repulsive) potential energy curve of the antibonding orbital, both potential energy curves converging at E 1s at R = ; antibonding orbital more antibonding than bonding orbital bonding because of different denominators LCAO MO for H + ; MO s: N A B ; 1 N S Variational theorem: simple proof based on completeness and orthonormality Variational determination of LCAO MO expansion coefficients; Lagrange s undetermined multiplier method; matrix eigenvalue equation versus operator eigenvalue equation; undetermined multiplier becomes eigenvalue and thus energy Transformation of a matrix eigenvalue equation into quadratic equation; nonexistence of matrix inverse The and bonds in MO theory The MO description of H, He, He +, O, N, and HF Covalent bonds and ionic bonds Hückel theory: < 0; applications to conjugated electrons in ethylene, butadiene, cyclobutadiene; aromaticity 7

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