1. What does an atom look like?

2. Recap hydrogen atom (r)/r x Ylm(q,f) En Fnlm ~ Rnl
a 2p orbital
(r)/r x Ylm(q,f) En
Fnlm
~ Rnl
We studied the hydrogen atom spectrum in detail.
Let’s look at other elements of the periodic table

3. What changes for other atoms?
Multiple electrons, single nucleus
Consider Helium. 2 electrons, 2 protons.
Simple expectation: En = –Z2/n2 x 13.6 eV
Lowest energy eV for Z=2, n=1
n=1
n=2
Vacuum
Ionization Energy
He+ + hn  He++ + e-
54.4 eV
He+
n=1
n=2
Vacuum
Ionization Energy
He + hn  He+ + e-
25 eV He
We’re off by 30 eV !
But look at He+ instead

4. What changes for other atoms?
So el-el interaction raises energy levels by 30 eV
This is the HARD term to include in our Hamiltonian
n=1
n=2
Vacuum
Ionization Energy
He+ + hn  He++ + e-
54.4 eV
He+
n=1
n=2
Vacuum
Ionization Energy
He + hn  He+ + e-
25 eV He
We’re off by 30 eV !
But look at He+ instead

5. Electron-Electron interaction… … r3
Blue el feels an interaction from
(N-1) other electrons with a
potential that’s not spherically
symmetric U(r2,r3,…,rN)
This means we can’t separate
variables any more !!! r2 rN r1

6. Simplify … … But we must simplify things, shouldn’t we? r3
So we smear out all other electrons
into a spherically symmetric U(r)
that must be calculated
self-consistently, but scaled for
(Z-1) electrons r2 rN r1

7. Simplify
Self-consistent Field theory or
Mean-Field theory r1

8. Calculating the charge
Just like the hydrogen atom with the extra potential Uee(r)
F = fnlm(r)Ylm(q,j)/r
-ħ2/2m.d2f/dr2 + [U(r) + l(l+1)ħ2/2mr2 + Uee(r)]f = Ef
So given Uee, we can find f(r) and thus charge distribution
n(r) = |fnlm(r)|2 (over occupied states only) Σ
nlm
For every atom, need to know electronic configuration
to see which states are occupied. eg. Si 1s22s22p63s23p2

9. Energy Level diagram ∑ l = 0, 1, 2,..., n-1
m = -l, -(l-1), -(l-2),... (l-1), l
(2l+1) multiplets
s = -½, ½ (2 of them)
So one orbit can hold
2 (2l+1) = 2n2 electrons = 2, 8, 18, 32, 50,...
l = 0
l = n-1 ∑

10. Energy Level diagram ∑ Si: 14 electrons
2 in first orbit (n=1, l=0)  1s
8 in second orbit (n=2, l=0,1)  2s, 2p
4 in third orbit (n=3, l=0,1,2)  3s, 3p, 3d
2 (2l+1) = 2n2 electrons = 2, 8, 18, 32, 50,...
l = 0
l = n-1 ∑

11. Energy Level diagram Si: 14 electrons 2 in first orbit (n=1, l=0)  1s
8 in second orbit (n=2, l=0,1)  2s, 2p
4 in third orbit (n=3, l=0,1,2)  3s, 3p, 3d
Si 1s22s22p63s23p2

12. Calculating the potential
Getting Uee(r) from n(r)  Pure Coulomb electrostatics
“Hartree” approximation (ECE309)
s(r) = qn(r)
Uee(r) = (q2/4pe0)[∫s(r’)dr’/r’ + ∫s(r’)dr’/r](Z-1)/Z
Self-consistency:
Uee  f(r)  n(r)  s(r)  Uee(r) etc
This works quite well for the periodic table!

13. Need additional terms  let’s look at them now%
% MATLAB code used to generate the figures in the book: %
% "Quantum Transport: Atom to Transistor," by Supriyo Datta
% published by Cambridge University Press, May 2005
% (ISBN-10: | ISBN-13: ) %
% THIS FILE FOR: Chapter 3, Figure 3.1.4
% Copyright (c) Supriyo Datta
clear all
%Constants (all MKS, except energy which is in eV)
hbar=1.055e-34;m=9.110e-31;epsil=8.854e-12;q=1.602e-19;
%Lattice
Np=200;a=(10e-10/Np);R=a*[1:1:Np];t0=(hbar^2)/(2*m*(a^2))/q;
%Hamiltonian,H = Kinetic,T + Potential,U + Ul + Uscf
T=(2*t0*diag(ones(1,Np)))-(t0*diag(ones(1,Np-1),1))-(t0*diag(ones(1,Np-1),-1));
UN=(-q*14/(4*pi*epsil))./R;% Z=14 for silicon
l=1;Ul=(l*(l+1)*hbar*hbar/(2*m*q))./(R.*R);
Uscf=zeros(1,Np);change=1;
while change>0.1
[V,D]=eig(T+diag(UN+Uscf));D=diag(D);[DD,ind]=sort(D);
E1s=D(ind(1));psi=V(:,ind(1));P1s=psi.*conj(psi);P1s=P1s';
E2s=D(ind(2));psi=V(:,ind(2));P2s=psi.*conj(psi);P2s=P2s';
E3s=D(ind(3));psi=V(:,ind(3));P3s=psi.*conj(psi);P3s=P3s';
[V,D]=eig(T+diag(UN+Ul+Uscf));D=diag(D);[DD,ind]=sort(D);
E2p=D(ind(1));psi=V(:,ind(1));P2p=psi.*conj(psi);P2p=P2p';
E3p=D(ind(2));psi=V(:,ind(2));P3p=psi.*conj(psi);P3p=P3p';
n0=(2*(P1s+P2s+P3s))+(6*P2p)+(2*P3p);
n=n0*(13/14);
Unew=(q/(4*pi*epsil))*((sum(n./R)-cumsum(n./R))+(cumsum(n)./R));
%Uex=(-q/(4*pi*epsil))*((n./(4*pi*a*R.*R)).^(1/3));%Unew=Unew+Uex;
change=sum(abs(Unew-Uscf))/Np,Uscf=Unew;
end
[E1s E2s E2p E3s E3p]
%analytical solution for 1s hydrogen
a0=4*pi*epsil*hbar*hbar/(m*q*q);
P0=(4*a/(a0^3))*R.*R.*exp(-2*R./a0);
hold on
h=plot(R,P1s,'b');
h=plot(R,P0,'bx');
h=plot(R,P3p,'bo');
set(h,'linewidth',[2.0])
set(gca,'Fontsize',[25])
xlabel(' R ( m ) --->');
ylabel(' Probability ---> ');
axis([0 5e ]);
grid on
Si1s
Si3p
H1s
Need additional terms  let’s look at them now

14. Exchange-Correlation
Beyond Coulomb terms within SCF
“Hartree-Fock”, “Density Functional Theory”
Uee(r) = (q2/4pe0)[∫s(r’)dr’/r’ + ∫s(r’)dr’/r](Z-1)/Z
Uxc(r)  -(q2/4pe0r0), 4pr03/3 = 1/n
Self-consistency:
Uee  f(r)  n(r)  s(r)  Uee(r) etc
Correlation
“Hole” (radius r0) El
Nearby els
Coulomb Blockade