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)