5.1

Chapter 5

Quantum Mechanics & Atomic Structure

Prof. Myeong Hee Moon

5.2

Chapter Outline • The Hydrogen Atom • Shell Model for Many Electron Atoms • Aufbau Principle and Electron Configurations • Shells Sh ll and d th the P Periodic i di T Table bl : Photoelectron Spectroscopy • Periodic Properties & Electronic Structure

Prof. Myeong Hee Moon

5.3

5.1 Hydrogen Atom • to solve the Schrodinger Eq. more convenient to use spherical coordinates

( r ,θ , φ ) ( x, y, z )

•Solutions of the Schrodinger Eq. for one electron atom

Z 2e 4 me E = En = − 2 2 2 8ε 0 n h

n = 1,2,..

iin energy off rydberg db (1 rydberg=2.18x10-18J)

Z2 En = − 2 n

n = 1,2,.. 12

• n: p principal p q quantum number : indexes individual energy gy levels Prof. Myeong Hee Moon

5.4

Energy Levels There are four quantum numbers used to describe the electron in the hydrogen atom • • • •

n,, p principle p quantum q number,, : the size and energy gy of the orbital n = 1, 2, 3, 4, ………(only integers) l – angular momentum quantum number: the shape of the orbital. l = 0 to n-1 (only integers) ml – magnetic quantum number: the spatial orientation of the orbital. ml = -ll to 0 to +l (only integers) ms – spin quantum number: the direction and spin of the electron. ms = +1/2 or -1/2 (only two values)

Prof. Myeong Hee Moon

5.5

Quantum numbers • angular momentum L

h2 L = l (l + 1) 2 4π 2

l = 0,1,2,.. n - 1

• by different angular momentum quantum numbers (l) l=0 l=1 l=2 l=3

: s (sharp) : p (principal) : d (diffuse) : f (fundamental)

(2l + 1) degeneracy for each l

Prof. Myeong Hee Moon

5.6

Quantum Numbers

Prof. Myeong Hee Moon

5.7

Wave Functions • solution of Schrodinger Eq for each quantum state (n,l,m)

ψ nlm (r ,θ , φ ) = Rnl (r )Ylm (θ , φ ) Ylm (θ φ); spherical harmonics l (θ,φ); wave function can not be measured but the probability density for ψ 2 represents locating the electron at a particular position in the atom

(ψ nlm ) 2 dV = [Rnl (r )]2 [Ylm (θ , φ )]2 dV dV = r 2 sin i θ dr d dθ dφ

Prof. Myeong Hee Moon

5.8

Angular & Radial Parts of Wave Functions

Prof. Myeong Hee Moon

5.9

Size and Shapes of Orbitals

• S orbital bit l l=0, m=0 - spherical

• 90%of electron density

Prof. Myeong Hee Moon

5.10

Orbital Shapes: s orbital

Prof. Myeong Hee Moon

5.11

Size and Shapes of Orbitals • p orbital l 1 m=-1, l=1, 1 0 0, +1 1 same shape but diff directions, px, py, pz

Prof. Myeong Hee Moon

5.12

Orbital Shapes: 2px orbitals

Prof. Myeong Hee Moon

5.13

Size and Shapes of Orbitals • d orbital

l=2, m=-2,-1, 0, +1, +2

Prof. Myeong Hee Moon

5.14

Orbital Shapes: 3dx -y orbital 2

Prof. Myeong Hee Moon

2

5.15

Orbital Shapes: 3dz orbital 2

Prof. Myeong Hee Moon

5.16

Orbital Shapes: 3dxy orbital

Prof. Myeong Hee Moon

5.17

Orbital Shapes: 3dyz orbital

Prof. Myeong Hee Moon

5.18

Orbital Shapes: 3dxz orbital

Prof. Myeong Hee Moon

5.19

Summaries of orbitals 1. For a given l values, increase in n leads to an increase in av distance of the electron from the nucleus nucleus, thus sizes 2. Orbital with n,l has l angular nodes, n-l-1 n l 1 radial nodes Æ total n-1 nodes 3. As r approaches 0, wave function vanishes for all orbitals except s orbital. Onle an electron in s orbital can penetrate to the nucleus” nucleus . -- finite probability of being “penetrate found right at the nucleus. Average distance of e from nucleus : radius of nth Bohr orbit

n 2 a0 ⎧ 1 ⎡ l ( l + 1) ⎤ ⎫ rnl = ⎨1 + 1 − ⎬ Z ⎩ 2 ⎢⎣ n 2 ⎥⎦ ⎭

Prof. Myeong Hee Moon

5.20

Electron Spin • spin quantum numbers : ms: +1/2 1/2 (up spin) spin), -1/2 1/2 (down) • final quantum state : expressed with the combination of four quantum no. : electron must have different quantum numbers b (n, l, ml, ms) : total quantum no. = 2n2 (including spins) • spin i q.n iis nott iincluded l d d iin Schrodinger eq. It does not influence the probability density. Prof. Myeong Hee Moon

5.21

5.2 Shell model for many-electron atoms • For many-electron atom system, Schrodinger Eq becomes increasingly complicated complicated. -- Numerical calculations still has defects to predict atomic properties 1) difficult to interpret physically 2) as e increases, increases more and more difficult to solve solve. Æ Needs approximate approaches ™ Hartree Orbitals Self-consistent Self consistent field(SCF) orbital approximation method by Hartree : generates a set of approximate one-electron orbital,ϕα , associated energy levels, ε α , for multielectron atom, similarly to H. Æ atomic t i shell h ll model d l

Prof. Myeong Hee Moon

5.22

Hartree Orbitals Æ three assumptions for Hartree method. 1) each electron moves in an effective field created by the nucleus nucleus. & effective field for electron i depends on position ri. 2) effective field for electron I is obtained by averaging its Coulomb potential interactions with each of other electrons 3) the effective field is spherically symmetric (no angular diff) Æ from 1st ass., each electron moves as an independent particle and i d is described ib d b by a one-electron l t orbital bit l similar i il tto H atom. t : for Li atom, wave function ψ atom = ϕα (r1 )ϕ β (r2 )ϕγ (r3 ) orbital approximation for atoms. Æ similarity to H atomic orbital g dependence p is identical to H orbital(l, ( m)) - angular - radial dependence of the orbitals in many electron atom differs from that of one-electron orbital due the diff of efffective field from Coulomb potential. Prof. Myeong Hee Moon

5.23

Size & Shapes of Hartree Orbitals • radial probability density distributions of Ar for five occupied orbitals by Hartree method.

Hartree orbitals of Ar

For Ar, 18 electrons in 1s, 2s, 2p, 3s, 3p • shell: defined as all electrons with the same n values. • subshells: within each shell, a more detailed set of orbitals with same values of both n, l. Æ determine the structure of periodic table and bond.

“shell structures”

Prof. Myeong Hee Moon

5.24

Shielding effects : energy sequence • energy level diagrams calculated by Hartree method resemble H atom except the two respects. respects 1) degeneracy of p,d,f orbitals were removed, because effective field differs from Coulomb field in H, energy levels of Hartree depends on both n n, and ll. 2) due to the increase of attractive force by nuclei (Z>1), energy values are distinctively different. eff n

V

Zeff (n )e 2 (r ) ≈ − r

Zeff: effective nuclear charge, a net reduced nuclear charge experienced by a particlular e due to the presence of other e. Shielding effect Prof. Myeong Hee Moon

5.25

Shielding effects • Ar atom (Z=18): Zeff(1)~16, Zeff(2)~8, Zeff(3)~2.5 • due to shielding effect effect, Z is replaced by Zeff

[Zeff (n )]2 εn ≈ − n2

(rydbergs)

n 2a0 ⎧ 1 ⎡ l (l + 1) ⎤ ⎫ rnl ≈ − ⎨1+ 1 − ⎬ Zeff (n ) ⎩ 2 ⎢⎣ n 2 ⎥⎦ ⎭

• in s orbital, shielding smallest, e bound most tightly. Zeff(ns) > Zeff(np) > Zeff(nd) Æ

εns < εnp < εnd

Prof. Myeong Hee Moon

5.26

5.3 Aufbau principle and Electron Configurations • ground state electronic energy configuration of an atom : adding one electron at a time from the lowest E orbital orbital. : few restrictions 1) Pauli exclusion principles - No two electrons can have the same four quantum numbers (n, l, m, ms). - Spins of electrons in an orbital must be opposite. 2) Hund’s H d’ rule l - maximizes the number of unpaired electrons in orbitals • Orbital diagram for C (z = 6) would be: (↑↓) (↑↓) (↑ ) (↑ ) ( ) 1s 2s 2p not (↑↓) (↑↓) (↑↓ ) ( ) ( ) 1s

Prof. Myeong Hee Moon

2s

2p

5.27

Pauli’s exclusion

Prof. Myeong Hee Moon

5.28

Aufbau principle

Aufbau principle

Prof. Myeong Hee Moon

5.29

Electronic configuration • paramagnetic : attracted to magnetic field. With one or more unpaired e. • siamagnetic : all electrons are paired paired. • noble gas configuration is used to express electronic configuration Si : [Ne]3s23p2

Transition-metal elements and beyond y • 3d orbital filling (d block elements) : energy gy level from aufbau : ~3p