1. What does an atom look like?

2. The strange world of Quantum Mechanics
“It is impossible for both The Ultimate Answer and
the Ultimate Question to be known about in the same
universe, as they will cancel each other out and take
the Universe with them, to be replaced by something
even more bizarre..”

3. 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

4. 3D analogue of drum modes2p 3p 4p 3d 4d 5d

5. 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

6. 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

8. 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

9. 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

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

11. 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

12. 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 ∑

13. Energy Level diagram (n+l) rule (6Z) 1/3 = n+l E ~ Z2
Coulomb repulsion ‘screens’ electrons

14. Energy Level diagram ∑ Si: 14 electrons
2 (2l+1) = 2n2 electrons = 2, 8, 18, 32, 50,...
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
l = 0
l = n-1 ∑

15. 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

16. Calculating the potential
Getting Uee(r) from n(r)  Pure Coulomb electrostatics
“Hartree” approximation
E.4pr2 = Qenc/e0
= q∫s(r’)dr’/e0 r
Gauss’ Law
V(r) = -∫Edr’ = -q∫dr’/4pe0r’2∫s(r’’)dr’’
= q∫s(r’)dr’/4pe0r +∫dr’s(r’)q/4pe0r’ ∞ r r r' ∞ r r ∞
U = qV

17. Calculating the potential
Getting Uee(r) from n(r)  Pure Coulomb electrostatics
“Hartree” approximation
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!

18. % Code to calculate Radial functions and energies of silicon clear all
close 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=(5e-10Np);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); % Only count upto 3p2
n=n0*(13/14); % Count 13 out of 14 electrons
Unew=(q/(4*pi*epsil))*((sum(n./R)-cumsum(n./R))+(cumsum(n)./R));
change=sum(abs(Unew-Uscf))/Np,Uscf=Unew;
End
[E1s E2s E3s E2p E3p]
Quantum transport: Atom to Transistor (Supriyo Datta)

19. % --------------------------------------------------------------------------------
% 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

20. Screening gives Aufbau principle
-Zq
VCoul ≈ l(l+1)ħ2/2mr2
4pe0r[1 + r/l]2
Screening length long for
weak Z (Thomas Fermi)
l = 1.65a0/Z1/3
Madelung, Klechkowski, n+l rule
To get larger (n+l) need higher Z
Large n+l has smaller l, more screening, higher energy

21. Screening gives Aufbau principle

22. Screening in solids (Thomas Fermi)
Molecule
Solids
-Zq
VCoul ≈
-Zq
e-r/l
4pe0r[1 + r/l]2
4pe0r
l = 1.65a0/Z1/3
l = √(e/q2D)
2V = -qn/e
2dV = -qdn/e
n = n(qV)
dn = (∂n/∂E)qdV = qDdV
2dV = -q2DdV/e = -dV/l2

23. Variational treatment of He
<H> = <YHe|-ħ2/2m(12+22 )
–2q2/4pe0r1
–2q2/4pe0r2
+q2/4pe0r12|YHe>/<YHe|YHe>
YH1s(r1) = 1/√(pa03) e-Zr1/a0
YHe(r1,r2) = Z3/(pa03) e-Z|r1+r2|a0
Treat Z as a variational parameter

24. Estimate for screening
<H> = <YHe|-ħ2/2m(12+22 )
–2q2/4pe0r1
–2q2/4pe0r2
+q2/4pe0r12|YHe>/<YHe|YHe>
= q2/4pe0a0[Z2–4Z+5Z/8]
Zmin = 27/16 ≈ 1.7 (15% screening/electron)
<H> = 77.5eV
Exact = = 79.4eV

25. Total Energy For He, added energy of 1st electron in
presence of 2nd, and 2nd in isolation
Can’t just some SCF energies of 1s, 2s,
3s, 2p etc. Must subtract ½ the interaction
energy
(ie, only include it for 1 member of
each pair)

26. A nonlocal correction (‘Exchange’)
F = fnlm(r)Ylm(q,j)/r
-ħ2/2m.d2f(r)/dr2 + [U(r) + l(l+1)ħ2/2mr2 + UH(r)]f(r)
∫UF(r,r’)f(r’)dr’ = Ef(r)

27. Exchange-Correlation
Beyond Exchange terms
“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