Bohr’s model seems to only work for one electron atoms, or ions. i.e. H, He +, Li 2+, …. It can’t account for electron-electron interactions properly The wave “like” properties of electron need to be explored to do job properly. Electrons as Waves Louis de Broglie : All matter has a corresponding wave character, with a wavelength determined by its momentum p (= mv) = h/p = h/mv

Electrons and Baseballs as Waves Example: Electron moving at C: = (6.626* Js)/(9.109* kg)*(0.1000*2.998*10 8 m/s) = 2.423* m = pm Example: A kg baseball moving at 150. km/hr: = (6.626* Js)/(0.1 kg)*(41.7m/s) =1.59* m v = ( m)/(3600 s) = 41.7 m/s

The Correspondence Principle Macroscopic bodies don’t feel the effect of quantum mechanics due to their large masses and slow motion Ex) The wavelength of the base ball is insignificant on the scale of the base ball Microscopic bodies do feel the effect of quantum mechanics strongly due to their small masses and fast motion Ex) The wavelength of the electron ball is very large on the scale of the size of the electron i.e m

A Little Math A function, y = g(x) describes a relationship between y with the variable x. Consider : F [g(x)] → h (x) - F is like a function acting on g(x), where g(x) is a function of x. - F is referred to as an operator, and changes g(x) into another function h(x) ex) g(x) = cx, g acts on x to giving x multiplied by some constant c ex) g(x) = cx, and F [g(x)] = [g(x)] 2 Then F[g(x)] = h(x) = [g(x)] 2 = [cx] 2 = c 2 x 2 y x g(x)g(x) h(x)

Math in Science In general, problems in science can be written as : F[g(x)] = h(x) - The operator F acting on g (x) gives back another function h (x). - The solution to this problem is the function g(x) itself. - Often there are many solutions: g 1 (x), g 2 (x), g 3 (x)…. g n (x) A special case : H[g n (x)] = c n g n (x) - The operator transforms the function to itself time some specific constant, C n -This is called and eigen equation, where g n (x) are eigen functions and c n and eigen values

The wave equation Ex) A Guitar String Only a few suitable wave forms, eigen functions, which is determined by length  = 2L, L, 2L/3 …. Wavefunction:  (x) depends on number of lobes n =1, 2, 3, … and number of nodes = n -1  (x) = sin   (x) = sin(2  )  (x) = sin(3  )  (x) n represents a series of solutions, each with a different energy E(n), which are their corresponding eigen values  =  x/2L x

A Wave in Orbit The circular path imposes a length limit on the wave function, thereby allowing for only an whole number of nodes.

H - is an operator acting on  (r) n. - This Hamiltonian operator used to calculate the total energy, E n, from  (r) n - The result ing Energy, E(n), depends on the quantum number, n, and the original wavefunction  (r) n H(  (r) n ) = E n  (r) n r = 3-D coordinates of electron  (r) n - is the n-th wavefunction corresponding to the electron The Schrödinger Equation - It includes contributions from kinetic energy, and energy from electron-electron and electron-nuclear iterations.

H(  (n, l,m, s) ) = E(n)  (n, l, m, s) The wavefunction depends on four quantum numbers, each associated with a different property of the electron. n - Principle Quantum Number l - Angular momentum Quantum Number Determines which shell the electron is in and the energy of the electron, E(n) n = 1, 2, 3, 4, … E(n) = -R y Z/n 2 Subshells exist for each shell differing in the angular momentum value. l = 0, 1 …n-1 m - Magnetic Quantum Number Related to the orientation in space that of the orbital. m = - l …+ l s - Spin Quantum Number Related to symmetry of wavefunction s = 1/2, -1/2 L = h l /2 

Wavefunctions of H n = 1m = 0 Lets for the moment ignore spin l = 0 States of l are labeled as: l = 0 S l = 1 P l = 2 D l = 3 F Therefore this state is: 1s 0 = 1s n = 2 l = 0 m = 0 n = 2 l = 1 m= 1, 0, -1 2s 2p 1, 2p 0, 2p -1 2p x, 2p y, 2p z

Wavefunctions of H n = 3 l = 0 m = 0 n = 3 l = 1 m = 1,0, -1 n = 3 l = 2 m= 2,1,0, -1,-2 3s 3p 1, 3p 0, 3p -1 3d 2, 3d 1, 3d 0, 3d -1, 3d -2 3d(xy), 3d(xz), 3d(yz), 3d(x 2 -y 2 ), 3dz 2

Wavefunction of H n = 4 l = 0 m = 0 n = 4 l = 1 m = 1,0, -1 n = 4 l = 2 m = 2,1,0, -1,-2 4s 4p 1, 4p 0, 4p -1 4d 2, 4d 1, 4d 0, 4d -1, 4d -2 n = 4 l = 3 m= 3,2,1,0, -1,-2,-3 4f 3, 4f 2, 4f 1, 4f 0, 4f -1, 4f -2, 4f -3

Heisenberg Uncertainty Principle p Observation p ? Measurement effects the state of the system. There is a limit imposed on the degree of certainty to which you can know the position (r) and momentum (p) of a particle r r1r1 r2r2 r ?  p  r ≥ h/4   p – error in p  r – error in r

Exercise An electron is traveling between C and C. What is the smallest error in the position you can expect? What is the error in position if it were a proton? Need error in momentum  p? We know that  v = C Therefore  p = m  v  p = (9.109* kg)*( * 2.998*10 8 m/s)  p = 1.36* kg m/s For an electron

Recall that  p  r ≥ h/4   r ≥ h/(4  p   r ≥ (6.626* Js)/[(4* )*(1.36* kg m/s)]  r ≥ 3.88* m  r ≥ (6.626* Js)/[(4* )*(2.51* kg m/s)] For an proton  p = (1.674 × kg)*( * 2.998*10 8 m/s)  p = 2.51* kg m/s  p = m  v  r ≥ 2.10* m

Probability Distribution A particle position and momentum cannot be known exactly Therefore a particle is characterized by a probability distribution function The probability distribution is determined by the wavefunction: P α r 2   (n,l,m,s;r)  (x) P(x)

Hydrogen Orbitals The hydrogen orbitals are determined from the wavefunctions Ex) 1s    ( 1,  0, 0  r ) 2s    ( 2, 0, 0; r ) The S orbitals with n > 0, and l = 0 - spherical, ie. identical in all directions The probability distribution can be graphed as a function of the radius as P(r) = r 2  2

radial probability density plot Notice the shell structure as n increases Notice the nodes in the wavefunction For 1s 0 nodes For 2s 1 nodes For 3s 2 nodes

P Orbitals In order to visualize the electron densities in 3D space associated with the p orbitals, they are constructed from combinations of the hydrogen wavefunctions with n > 1, and l = 1. These  are complex valued, and are recombined to make them real valued. The resulting functions p x and p y are aligned along the x and y axis. The remaining function  (2,1,0) is aligned along z axis and is referreedd to as p z. ie.    (2,1,0),  (2,1,-1) The p x and p y electron density functions do not correspond to m = 1 and -1, as p z does to m =0.

P Orbitals These dumbbell shaped orbital are referred to as the p (polar) orbitals, which are labeled according to their orientation, 2p x, 2p y, 2p z The number of nodes increases with n as n-1, ie. 1 for 2p The p orbital has one nodal plane where the sign changes when crossed. The orbitals are plotted as the boundary enclosing total of 90% probability PxPx PyPy PzPz

D Orbitals The d orbitals are constructed from the hydrogen wavefunctions with n > 2, and l = 2.  (3,2,2) and  (3,2,-2) a well as  (3,2,1) and  (3,2,-1), are combined to make them real valued functions. The four corresponding distribution functions are have four lobes in the xy, xz and yz plane. (in between the axes) ie.  (3,2,2)  (3,2,1),  (3,2,0),  (3,2,-1),  (3,2,-2)

D Orbitals A fourth orbital exists in the xy plane aligned on the axes. The remaining fifth orbital, d z2, resembles a P z orbital with a donut like shape in the xy plane (z 2 -x 2 and z 2 -y 2 are superimposed) Sign changes when nodal plane (cone) is crossed There are 2 nodes 3d

F Orbitals Constructed from seven H wavefunctions to make them real valued Composed of 8 lobes There are 3 nodes for 4f

The Orbitals of the Hydrogen Atom 0 nodes 1 node 2 nodes Radial nodes 1 planar node 2 planar nodes

Concepts Properties of waves (wavelength, frequency, amplitude, speed) Electromagnetic spectrum, speed of light Planck’s equation and Planck’s constant Wave-particle duality (for light, electrons, etc.) Atomic line spectra and relevant calculations Ground vs. excited states Heisenberg uncertainty principle Bohr and Schrödinger models of the atom Quantum numbers (n, l, ml) Shells (n), subshells (s,p,d,f) and orbitals Different kinds of atomic orbitals (s, p, d, f) and nodes