CHM695Feb. 9
Does it make sense?
number of electrons+ position of nuclei+ nuclear charges
⇒ H ⇒ energy
Density has the following properties:Z
⇢(r) = n
⇢(r) has maxima at the positions of nuclei
density at the position of nuclei has information regarding nuclear charge
⇢(r)⇒ H ⇒ energy
unique
unique
Formally this is shown by Hohenberg and Kohn (1964)
H = T + Ve�e + Vn�e
E[⇢(r)] = T [⇢(r)] + Ve�e[⇢(r)] + Vn�e[⇢(r)]
Vn�e , ⇢(r)unique
This implies, ⇢(r)⇒ Hunique
or, ⇢(r) ⇒F [⇢(r)]E
E[⇢(r)] = T [⇢(r)] + Ve�e[⇢(r)] + Vn�e[⇢(r)]
universal functional
external potential
Functional obeys variational theorem
E[⇢(r)] � E[⇢0(r)]
exact density
variation of
density
Only valid for the exact density functional
Practical ComputationE[⇢(r)] = T [⇢(r)] + Ve�e[⇢(r)] + Vn�e[⇢(r)]
X
I
Zdr⇢(r)
Z
rI
Zdr⇢(r)v(r)
.
..
1
2
Z Zdr1dr2
⇢(r1)⇢(r2)
r12
??
APPROXIMATE!(see next
page)
(r1, r2, · · · , rn) = 1(r1) 2(r2) · · · (rn)For
T [⇢(r)] = �1
2
Z +1
�1 ⇤r2 d⌧
=1
8
Z 1
�1
r⇢ ·r⇢
⇢d⌧
So, the above equation has to be modified for interacting systems.
The Kohn-Sham Equation:
Idea: (slater determinant)
Advantage: Kinetic energy functional can be directly computed:
(1, · · · , n) ) ||�1 �2 · · ·�n||
T =X
i
⌧�i
�����1
2r2
�����i
�
How to obtain {�i} ?
Remember: slater determinant was constructed based on
independent electron assumption. Or for a non-interacting electronic system
Let us assume that we have a hypothetical system (which we take as our reference
system) of non-interacting electrons, which is under some effective potential Vs
Hs = �1
2
nX
i
r2i +
nX
i
Vs(ri) no e-e interaction!
Solution of this is KS
KS = ||'1 '2 · · ·'n||
fKSi 'i = ✏i'i
one electron SE like. (remember HF
equations)
fKSi = �1
2r2
i + Vs(ri)
KS orbitals
Where is the connection between this reference system and actual system (of interacting
electrons)??
We will establish that now.
reference system & actual system are connected by choosing Vs appropriately.
Choose Vs such that
⇢s(r) ⌘X
i
X
!
|'i(r,!)|2 = ⇢(r)
spins density of the actual system
Let us get back to the kinetic energy functional:
Ts =X
i
⌧'i
�����1
2r2
����'i
�approximate
Ts 6= T
J [⇢(r)] =1
2
Z Zdr1dr2
⇢(r1)⇢(r2)
r126= Ve�e[⇢(r)]
approximate
Residual contributions to T and Vee may be added separately by some other functional
As we realised earlier,
not in
T [⇢] form
EXC[⇢(r)] ⌘ (T [⇢(r)]� Ts[⇢(r)]) + (Ve�e[⇢(r)]� J [⇢(r)])
This functional is called the Exchange Correlation Functional.
KE residual e-e interaction residual
Everything that is unknown!
How to define Vs?
E[⇢(r)] = Ts[⇢(r)] + J [⇢(r)] + EXC[⇢(r)] + En�e[⇢(r)]
EXC[⇢(r)] +X
I
Zdr1 |'i(r1)|2
ZI
r1I+
For that, let us write the energy of the interacting system:
Variational minimisation of E by changing {'i}
with the constraint h'i|'ji = �ij
= �1
2
nX
i
⌦'i|r2|'i
↵+
1
2
X
i
X
j
Z Zdr1 dr2 |'i(r1)|2
1
r12|'j(r2)|2
�E
� h'i|
�1
2r2 +
Zdr2
⇢(r2)
r12+ VXC(r1)�
X
I
ZI
r1I
!|'ii
⇒
✏i |'ii=
Vs(r1) But, depends on ⇢(r)
Thus, SCF is required!
Kohn- Sham Eqn.
IMPORTANT: No approximation is yet invoked!
If we know Exc exact ground state energy can be computed.
(Note: in HF, it was assumed that n-electron wfn. is a Slater Determinant; and HF equations uses mean-field approach)
KS equation {'i}{'i}
guess
Energy⇢(r)
The performance of DFT is thus dependent solely on the choice of Exc
Integrals involved in the computation of Vs are trivial compared to HF (no exchange and coloumb integrals which
makes HF scales ~K4)
Due to diagonalization of KS equations, it is ~K3
Thus, larger system and more accurate computations compared to HF.
Computational cost:
Approximate Exchange Correlation Density Functionals
Local Density Approximation (LDA):
VXC[⇢(r)] = VX[⇢(r)] + VC[⇢(r)]
V LDAX [⇢(r)] = �3
4
✓3
⇡
◆1/3
⇢1/3(r)exact for uniform
(density) electron gas
V
LDAC [⇢(r)] = A
✓ln
x
2
X(x)+
2b
Q
tan�1
✓Q
2x+ b
◆� b
x0
X(x0)
ln
(x� x0)2
X(x)+
2(b� 2x0)
Q
tan�1 Q
2x+ b
�◆
x =
✓3
4⇡⇢
◆3/2
X(x) = x
2 + bx+ c Q =p
4c� b2
A, x0, b, c parameters
correlation part: Vosko, Wilk, and Nusair (VWN) functional
LDA performance was often poor than RHF.
Breakthrough by Axel Becke:GGA functionals (Generalised Gradient Approximation)
V BX (r) = V LDA
X (r)� � ⇢1/3(r)
✓y2(r)
1 + 6�y(r) sinh�1 (y(r))
◆
y(r) =|r⇢(r)|⇢4/3(r)
The above is called the Becke-88 functional
V LYPC ⌘ VC
⇣⇢�1/3, ⇢8/3, |r⇢|2 ,r2⇢; a, b, c, d
⌘Lee, Yang, Parr (LYP)
BLYP functional = Becke88 Exchange + LYP correlation
Another example of GGA is the PBE functional (Perdew Burke Ernzerhof; it contains no parameters)
GGA functionals showed excellent improvement over HF
Hybrid Functionals
EB3LYPXC = (1� a)ELDA
X + aEHFX + b�EB
X + (1� c)ELDAC + cELYP
C
Exact exchange computed by HF calculationsa, b, and c are parameters here.
Meta GGA functionalsr2⇢functional contains too.
E.g. B3LYP
E.g. M06-L
Gaussian Input Style:
http://www.gaussian.com/g_tech/g_ur/k_dft.htm
#BLYP/6-31G(d)
#B3LYP/6-31G(d)
#M06L/6-31G(d)
Examples:
Jacob’s Ladder
LDA
GGA
hybrid-GGA
meta-GGA
hybrid meta-GGA
accu
racy
co
mp.
tim
e
exact functional
Notes:
Variational in density only for exact functional! (practically, DFT energy is higher than the exact GS energy)
Meant for the ground state density and energy. Require special care for the excited states
KS orbitals have no physical meaning in principle; yet it is found to have high predictive power!
No Koopman’s theorem; For the exact functional, ✏HOMO ⇡ �I.E.
Self-Interaction Error:
Let us think of one electron system. Here no e-e interaction is present
Here, we still compute J [⇢]
We would like J [⇢] + EXC[⇢] = 0
But this doesn’t happen, and results in “self-interaction error”This error appears when unpaired electrons are present
Dispersion Correction
Poorly described at the LDA, GGA level.
Empirical corrections to functionals: Grimme’s correction (DFT+D functionals)
Specially parameterised hybrid functionals perform better (E.g. M06, M05-2X)