Chem 373- Lecture 19: Many Electron Atoms

Lecture 19: Many Electron Atoms The material in this lecture covers the following in Atkins. The structure of many-electron atoms 13.4 The orbital approximation (a) The helium atom (b) The Pauli principle (c) penetration and shielding (d) the building-up principle (e) The configuration of ions (f) Ionization energies and electron affinities 13.5 Self-consistent field method Lecture on-line Many electron atoms (PDF Format) Many electron atoms (PowerPoint) Handout for this lecture

Electronic Structure of Many - Electron Atoms

He − atom The Helium atom and Helium like ions Or The two electron atom

We shall consider an atom made up of two electrons and a nucleus of charge Z

e

2

→ r 12 → r 2

X

Z e 1 → r Z 1

Y

He − atom We might also describe the two electrons in spherical coordinates

Electronic Structure of Many - Electron Atoms

e

2

θ → r 2 φ X

Z

→ r 12

2

2

φ

θ

e 1

1 → r Z 1 1 Y

Electron #1 : (r1,θ1,φ1) Electron #2 : (r2,θ2,φ2) The Hamiltonian of the system is given by  − h2 2 Ze2   − h2 2 e2 Ze2  ∇2 − ∇1 − + + ∴H=   4 πε or1  2me 4 πε or2  4πε or12  2me

Electronic Structure of Many - Electron Atoms He − atom Here  − h2 2 Ze2  ∇1 − ho(1) =  4 πε or1  2me is the Hamiltonian of a single electron in the field of a nucleus of charge Z Z

Z e θ

e 1

2 θ

1 → r Z 1

→ r 2 φ

X

Y

Same for  − h2 2 Ze2  ∇2 − ho(2) =  4 πε or2   2me

X

2 Z

2 Y

Electronic Structure of Many - Electron Atoms

He − atom

e2 The term represents repulsion between 4πε or12 two electrons at the distance r 12 .

e

2

→ r 12

Z e

→ r 2

1

→ r 1 Z

Y

Y

Electronic structure of many - electron atoms

For a many electron atoms we have the exact Hamiltonian 2 n − h2 n n ee Ze H = ∑[ ∇i2 − ]+ ∑ ∑ 4 πε ori i=1 2me j i> j 4 πε orij

ee H = ∑ h(Z, i) + ∑ ∑ i =1 j i > j 4 πε o rij n

n

n

Here Ze 2 −h2 2 h(Z, i) = ∇i − 2me 4 πε ori

Electron − electron repulsion between electron i and electron j

Electronic Structure of Many - Electron Atoms

Shielding We can approximately add this potential to the attractive potential

due to the nucleus as Vi (r) to modify ho (i) as

 −h2 2  Ze 2 h'(Z, i) =  ∇i − + Vi (ri ) i = 1, 2, 4,.... 4πε ori  2me  Re pulsion from other electrons N

H ≈ H' = ∑ h' i (Z, i) i =1

Electronic Structure of Many - Electron Atoms Orbital approximation For the approximate Hamiltonian H' = ∑ h'(Z, i) i

and the corresponding Schrödinger equation is H' Ψ(1, 2, 3, 4.., n) = EΨ(1, 2, 3, 4.., n) We shall now show that the the many - electron wavefunction Ψ(1, 2, 3,..) that is a solution to the above Schrödinger equation can be written as r r r r r Ψ(1,2,3,..,n) = ψ 1(r1 )ψ 2 (r2 )ψ 3 (r2 ).ψ 4 (r4 )....ψ n (rn )

Where ψ i is a solution to : for each i = 1, 2,..n, and  −h2 2 Ze 2 h' (Z, i) =  ∇i − 4 πε ori  2me

hi' (ri )ψ i (ri ) = ε i ψ i (ri )  + Vi (ri ) 

Electronic Structure of Many - Electron Atoms Orbital approximation We have r r r r r H' Ψ(1, 2, 3, .., n) = {∑ h'i (i)}ψ 1( r1)ψ 2 ( r2 )ψ 3 ( r2 ).ψ 4 ( r4 )....ψ n ( rn ) i

r r r r r = ∑ h'i (i)ψ 1( r1)ψ 2 ( r2 )ψ 3 ( r2 ).ψ 4 ( r4 )....ψ n ( rn ) i

r r r r r = ∑ ε i ψ 1(r1 )ψ 2 (r2 )ψ 3 (r2 ).ψ 4 (r4 )....ψ n (rn ) i

r r r r r = {∑ εi } ψ 1( r1)ψ 2 ( r2 )ψ 3 ( r2 ).ψ 4 ( r4 )....ψ n ( rn ) i

Thus H' Ψ(1, 2, 3, .., n) = EΨ(1, 2, 3, .., n) n

E = { ∑ εi } i=1

Electronic Structure of Many - Electron Atoms

Shielding

The solutions to the equations  − h2 2  Ze2 ∇i − + Vi (ri ) ψ i (ri ) = εi ψ i (ri ) h'i (i)ψ i (i) =  4 πε ori   2me can again be written on the form ψ i (ri ) = Ylm (θ .φ)Rnl (ri ) However, now Rnl (ri ) is a solution to :

Ze h 2 δ 2Rnl (r) 2 δRnl (r) h 2 l(l + 1) − { + + ) + {− }Rnl (r) 2 2 r δr 2µ 4 πε or 2µmr δ r + Vi (ri )Rnl (r) = ERnl (r)

Electronic Structure of Many - Electron Atoms

Shielding

The final solution for a many - electron atom with nuclear charge Z can to a good approximation be written as : r ψ i ( ri ) = Ylimi (θi .φi )Rnili (ri ) where Rnl (ri ) has the same analytical form as for the hydrogenic one - electron atom with atom charge Z. Only throughout Z is replaced by Z - ρnl = Znl Shielding constant

With the energy  Znl 2µ Ze 4  ε Z,n = −   2h 2 

Effective nuclear charge

Electronic Structure of Many - Electron Atoms Shielding Znl = Z - ρnl 1. Electrons in orbitals with quantum number n = n * will be shielded from the nucleus by electrons of lower n - quantum number that are closer to the nuclei. Thus n=1 n=2 n=3 n=4

σ nl increases with n and Znl (the effective nuclear charge) will decrease with n

Electronic Structure of Many - Electron Atoms

Shielding

For a given n The shielding σ nl increases with l and the effective nuclear charge Znl decreases with l.

3s En = −

3p

3d

Z2µe 4 32π 2 ε o2 h 2n2

3s

En ≅ −

3p 3d (Zinl )2 µe 4

32π 2 ε o2 h 2n2

Orbitals with th same n - quantum number have different energy if l is different in many - electron atoms but the same energy in hydrogenic atoms

Electronic Structure of Many - Electron Atoms

Z

eff

=

' Znl

=

Z Z − σnl

Shielding

Electronic Structure of Many - Electron Atoms Building − up principle He − atom To construct the wavefunction we use the Pauli exclusion principle :

Each orbital ψ nlm (different n, l, m) can only contain two electrons .The electrons must have different spins (α (i)/β(i) ) Thus : Ψ(1, 2) = 1s(1)α (1)1s(2)α (2)

E = ε1s + ε1s

1s(1)

σHe 1s = .32; Z' = 2 - .32 Z=2

Z= 1.68

ε= -

(Z'nl )2 µe 4 32π 2 ε o2 h 2n2

Z=1.0

r/a

Electronic Structure of Many - Electron Atoms Building − up principle Li − atom

Each orbital ψ nlm (different n, l, m) can only contain two electrons .The electrons must have different spins (α (i)/β(i) ) According to the Pauli exclusion principle the K - shell (1s)2 is full and the next electron must go into 2p or 2s orbitals of L - shell In many - electron shells orbitals of same n (shell) but different l do not have the same energy. The energy increases with l as the orbitals penetrates less and less to the nuclei Thus within a subshell : s < p < d < f

Electronic Structure of Many - Electron Atoms Building − up Thus the electron configuration principle og Li is 1s2 2s

Ψ(1, 2, 3) = 1s(1)α (1)1s(1)β(1)2s(1)α (1) or Ψ(1, 2, 3) = 1s(1)α (1)1s(1)β(1)2s(1)β(1) 1s

(Z=2.68)

For Be we have 1s2 2s2

2s (Z= 3.0) 2s(Z=1.7) 1s (Z=3)

Ψ(1, 2, 3, 4) = 1s(1)α (1)1s(1)β(1) ×2s(1)β(1)2s(1)β(1)

the radial functions for 1s and 2s of Li have exponents of 2.68 and 1.7, respectively, rather than 3

Electronic Structure of Many - Electron Atoms Building − up principle and configuration 2 2 1 B : 1s 2s 2p x of atoms py px pz

C : 1s2 2s2 2p1x 2p1y

N : 1s2 2s2 2p1x 2p1y 2p1z

px

px

py

py

pz

pz

Hunds rule : Electrons occupy different orbitals of sub - shell before double occupation

Parallel spin prefered

Electronic Structure of Many - Electron Atoms Building − up principle and configuration 2 2 2 1 1 O : 1s 2s 2p x 2p y 2pz of atoms py px pz F : 1s2 2s2 2p2x 2p2y 2p1z

Ne : 1s2 2s2 2p2x 2p2y 2p2z

px

px

py

py

pz

pz

Electronic Structure of Many - Electron Atoms

Occupation of the sub - shells follow the order 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s with some exceptions

Electronic Structure of Many - Electron Atoms  Znl 2µ Ze 4  Ionization potentials ε Z,n = −  2  2h 

Electronic Structure of Many - Electron Atoms

 Znl 2µ Ze 4  ε Z,n = −  2  2h 

Electron affinities

What you should know from this lecture You should unders tan d the Be able to construct the concept of a shielding cons tan t electron configuration for σ. an atom using the Pauli You should also know that exclusion principle and σ depends on a particular Hundsbuild - up principle element as well as the n and l Realize that orbitalswith the same quantum number (σ Z,n,l ) and n - quantumnumber but different l - quatum number have different that the effective charge energies in many electron atoms is given by eff = Z − σ Zn,l Z, n , l

Be aware that the orbitals in a many - electron atom can be written as ψ Z n,l,m(r,θ,ϕ )=Rn,l (R)Yl,m (θ,ϕ )

Understand the concept of ionization potentials and electron affinities and how they are related to shielding and the effective nuclear charge