Lecture 2, January 6, 2016 Quantum Mechanics-2: DFTwag.caltech.edu/home/ch121/Slides/2016/L2_2016.pdf · Becke half and half functional assume a linear model U a b xc,O O take,0 exact

Lecture 1-Ch121a-Goddard-L01 © copyright 2016 William A. Goddard III, all rights reserved\ 1

Ch121a Atomic Level Simulations of Materials and

Molecules

William A. Goddard III, [email protected]

316 Beckman Institute, x3093

Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics,

California Institute of Technology

Lecture 2, January 6, 2016

Quantum Mechanics-2: DFT

Special Instructor: Julius Su <[email protected]>

Teaching Assistants:

Daniel Brooks [email protected]

Jin Qian [email protected]

Room BI 115

Lecture: Monday, Wednesday 2-3pm

Lab Session: Friday 2-3pm

mailto:[email protected]

mailto:[email protected]

Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 2

Homework and Research Project

First 5 weeks: The homework each week uses generally available

computer software implementing the basic methods on

applications aimed at exposing the students to understanding how

to use atomistic simulations to solve problems.

Each calculation requires making decisions on the specific

approaches and parameters relevant and how to analyze the

results.

Midterm: each student submits proposal for a project using the

methods of Ch121a to solve a research problem that can be

completed in the final 5 weeks.

The homework for the last 5 weeks is to turn in a one page report

on progress with the project

The final is a research report describing the calculations and

conclusions


Last Time

Overview of ab initio Quantum Mechanics

Start with Schrodinger Equation

Solve for optimum Hartree Fock orbitals

This is very useful approximate wavefunction that can be

used as the starting point for improved wavefunctions

This very popular and rigorous but today we discuss a

different approach that is less rigorous but more practical

Density Functional Theory (DFT)


Alternative to Hartree-Fork,

Density Functional Theory

Walter Kohn’s dream:

replace the 3N electronic degrees of freedom needed to define

the N-electron wavefunction Ψ(1,2,…N) with

just the 3 degrees of freedom for the electron density r(x,y,z).

It is not obvious that this would be possible but

P. Hohenberg and W. Kohn Phys. Rev. B 76, 6062 (1964).

Showed that there exists some functional of the density

that gives the exact energy of the system

rrr

VFV

HK ][rep-

min

Kohn did not specify the nature or form of this

functional, but research over the last 46 years has

provided increasingly accurate approximations to it.Walter Kohn (1923-)

Nobel Prize Chemistry 1998


The Hohenberg-Kohn theorem

The Hohenberg-Kohn theorem states that if N interacting

electrons move in an external potential, Vext(1..N), the

ground-state electron density r(xyz)=r(r) minimizes the

functional

E[r] = F[r] + ʃ r(r) Vext(r) d3r

where F[r] is a universal functional of r and the minimum

value of the functional, E, is E0, the exact ground-state

electronic energy.

Here we take Vext(1..N) = Si=1,..N SA=1..Z [-ZA/rAi], which is the

electron-nuclear attraction part of our Hamiltonian.

HK do NOT tell us what the form of this universal functional,

only of its existence


Proof of the Hohenberg-Kohn theorem

Mel Levy provided a particularly simple proof of Hohenberg-Kohn

theorem {M. Levy, Proc. Nat. Acad. Sci. 76, 6062 (1979)}.

Define the functional O as O[r(r)] = min <Ψ|O|Ψ>

|Ψ>r(r)

where we consider all wavefunctions Ψ that lead to the same

density, r(r), and select the one leading to the lowest expectation

value for <Ψ|O|Ψ>.

F[r] is defined as F[r(r)] = min <Ψ|F|Ψ>

|Ψ>r(r)

where F = Si [- ½ i2] + ½ Si≠k [1/rik].

Thus the usual Hamiltonian is H = F + Vext

Now consider a trial function Ψapp that leads to the density r(r)

and which minimizes <Ψ|F|Ψ>

Then E[r] = F[r] + ʃ r(r) Vext(r) d3r = <Ψ|F +Vext|Ψ> = <Ψ|H|Ψ>

Thus E[r] ≥ E0 the exact ground state energy.


The Kohn-Sham equations

Walter Kohn and Lou J. Sham. Phys. Rev. 140, A1133 (1965).

Provided a practical methodology to calculate DFT wavefunctions

They partitioned the functional E[r] into parts

E[r] = KE0 + ½ ʃʃd3r1 d3r2 [r(1) r(2)/r12 + ʃd3r r(r) Vext(r) + Exc[r(r)]

Where

KE0 = Si <φi| [- ½ i2 | φi> is the KE of a non-interacting electron

gas having density r(r). This is NOT the KE of the real system.

The 2nd term is the total electrostatic energy for the density r(r).

Note that this includes the self interaction of an electron with itself.

The 3rd term is the total electron-nuclear attraction term

The 4th term contains all the unknown aspects of the Density

Functional


Solving the Kohn-Sham equations

Requiring that ʃ d3r r(r) = N the total number of electrons and

applying the variational principle leads to

[d/dr(r)] [E[r] – m ʃ d3r r(r) ] = 0

where the Lagrange multiplier m = dE[r]/dr = the chemical

potential

Here the notation [d/dr(r)] means a functional derivative inside

the integral.

To calculate the ground state wavefunction we solve

HKS φi = [- ½ i2 + Veff(r)] φi = ei φi

self consistently with r(r) = S i=1,N <φi|φi>

where Veff (r) = Vext (r) + Jr(r) + Vxc(r) and Vxc(r) = dEXC[r]/dr

Thus HKS looks quite analogous to HHF


The Local Density Approximation (LDA)

EKS = Si [<φi|- ½i2|φi >+Vext (ri)+Vxc(ri)]+½ʃʃd3r1 d3r2 [r(1)r(2)/r12]

General form of Energy for DFT (Kohn-Sham) formulation

KE Nuclear

attraction

Coulomb repulsionExchange

correlation

If the density is r =N/V then Coulomb repulsion leads to a

total of ½(N/V)2 interactions, but it should be ½(N(N-1)/V2)

Thus LDA include an extra self term that should not be

present

At the very minimum, Vxc needs to correct for this

If density is uniform then error is proportional to 1/N. since

electron density is r = N/V

( ) ( )3

1

x

LDA

x rρAρε xA = -3

1

π

3

4

3

.


The Local Density Approximation (LDA)

ExcLDA[r(r)] ʃ d3r eXC(r(r)) r(r)

where eXC(r(r)) is derived from Quantum Monte Carlo

calculations for the uniform electron gas {DM Ceperley and BJ

Alder, Phys.Rev.Lett. 45, 566 (1980)}

It is argued that LDA is accurate for simple metals and simple

semiconductors, where it generally gives good lattice

parameters

It is clearly very poor for molecular complexes (dominated by

London attraction), and hydrogen bonding

Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\

LDA exchange

11

( ) ( )3

1

x

LDA

x rρAρε xA = -3

1

π

3

4

3

.

Here we say that in LDA each electron interacts with all N

electrons but really it should be N-1.

The exchange term cancels this extra term. If density is uniform

then error is proportional to 1/N. since electron density is r = N/V


Generalized gradient approximations

The most serious errors in LDA derive from the assumption that

the density varies very slowly with distance.

This is clearly very bad near the nuclei and the error will depend

on the interatomic distances

As the basis of improving over LDA a powerful approach has been

to consider the scaled Hamiltonian

cxxc EEE ] drρ(r),...ρ(r)ρ(r),εE xx


Generalized gradient approximations

cxxc EEE

] drρ(r),...ρ(r)ρ(r),εE xx

( ) ( )sFερρ,ε LDA

x

GGA

x

( ) 3

4

3

12 ρπ24

ρs

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 5.0 10.0

S

F(S

)

B88

PW91

new(mix)

PBE

Becke 88

X3LYP

PBE

PW91

s

F(s) GGA functionals

( )( )

( )2

1

1

2

32

1

188B

sasinhsa1

sasasinhsa1sF

( )( ) ( )

( ) d

52

1

1

2s100

432

1

191PW

sasasinhsa1

seaasasinhsa1sF

2

Here ( )3

12

2 π48a , 21 βa6a , βA2

aa

x

3/1

2

23 , 34 a

81

10a ,

x

3/1

64

25

A2

10aa

, and d = 4.

Becke9 = 0.0042 a4 and a5 zero

Here ( )3

12

2 π48a , 21 βa6a , βA2

aa

x

3/1

2

23 , 34 a

81

10a ,

x

3/1

64

25

A2

10aa

, and d = 4.

S is big where the density

gradient is large


Adiabatic connection formalism

] ]1

,0

xc xcE U dr r is the exchange-correlation energy at intermediate coupling strength λ.

The only problem is that the exact integrand is unknown.

Becke, A.D. J. Chem. Phys. (1993), 98, 5648-5652.

Langreth, D.C. and Perdew, J. P. Phys. Rev. (1977), B 15, 2884-2902.

Gunnarsson, O. and Lundqvist, B. Phys. Rev. (1976), B 13, 4274-4298.

Kurth, S. and Perdew, J. P. Phys. Rev. (1999), B 59, 10461-10468.


Perdew, J.P. Ernzerhof, M. and Burke, K. J. Chem. Phys. (1996), 105, 9982-

9985.

Mori-Sanchez, P., Cohen, A.J. and Yang, W.T. J. Chem. Phys. (2006), 124,

091102-1-4.

The adiabatic connection formalism provides a rigorous way to define Exc.

It assumes an adiabatic path between the fictitious non-interacting KS

system (λ = 0) and the physical system (λ = 1) while holding the electron

density r fixed at its physical λ = 1 value for all λ of a family of partially

interacting N-electron systems:


Becke half and half functional

assume a linear model ,xcU a b

take , 0

exact

xc xU E the exact exchange of the KS orbitals

approximate , 1 , 1

LDA

xc xcU U

partition LDA LDA LDA

xc x cE E E

set ;exact LDA exact

x xc xa E b E E ;exact LDA exact

x xc xa E b E E

Get half-and-half functional

] ( )1 1

2 2

exact LDA LDA

xc x x cE E E Er

Becke, A.D. J. Chem. Phys. (1993), 98, 1372-1377


Becke 3 parameter functional

] ( )B3

1 2 3

LDA exact LDA GGA GGA

xc xc x x x cE E c E E c E c Er

Empirically modify half-and-half

where GGA

xE is the gradient-containing correction terms to the LDA exchange

GGA

cE is the gradient-containing correction to the LDA correlation,

1 2 3, ,c c c are constants fitted against selected experimental thermochemical data.

The success of B3LYP in achieving high accuracy demonstrates that errors of

for covalent bonding arise principally from the λ 0 or exchange limit, making

it important to introduce some portion of exact exchange

DFT

xcE



Perdew, J.P. Ernzerhof, M. and Burke, K. J. Chem. Phys. (1996), 105, 9982-

9985.

Mori-Sanchez, P., Cohen, A.J. and Yang, W.T. J. Chem. Phys. (2006), 124,

091102-1-4.


LDA: Slater exchange

Vosko-Wilk-Nusair correlation, etc

GGA: Exchange: B88, PW91, PBE, OPTX, HCTH, etc

Correlations: LYP, P86, PW91, PBE, HCTH, etc

Hybrid GGA: B3LYP, B3PW91, B3P86, PBE0,

B97-1, B97-2, B98, O3LYP, etc

Meta-GGA: VSXC, PKZB, TPSS, etc

Hybrid meta-GGA: tHCTHh, TPSSh, BMK, etc

Some popular DFT functionals


Truhlar’s DFT functionals

MPW3LYP, X1B95, MPW1B95, PW6B95, TPSS1KCIS, PBE1KCIS, MPW1KCIS,

BB1K, MPW1K, XB1K, MPWB1K, PWB6K, MPWKCIS1K

MPWLYP1w,PBE1w,PBELYP1w, TPSSLYP1w

G96HLYP, MPWLYP1M , MOHLYP

M05, M05-2xM06, M06-2x, M06-l, M06-HF

Hybrid meta-GGAM06 HF tPBE + VSXC

Band Gaps:

Critical property in designing new materials.

Problem standard DFT: big errors

19

2 eV

Clearly inadequate for in

silico design

What is solution?

Band Gaps:


Problem standard DFT: big errors

20

2 eV

Solutions:

Empirical: add Hubbard U and

adjust until get experimental

value

Problem: not useful for design

Rigorous theory: GW

calculate rigorous quasiparticle

excitation spectrum using

Green’s function (many-body

perturbation theory)

Problem

full GW: 10**4 cost of PBE

G0W0 (non-self-consistent)

10**3 cost of PBE

Are results worth the cost?

Band Gaps:


Results from various flavors of GW/PBE

21

2 eV

What can we do?

GW improves PBE from 1.09

eV to 0.36 eV at a cost of

10**3 to 10**4

Clearly NOT a solution for

materials design

Hybrid DFT theory

Include some exact HF exchange

22

2 eV

B3PW acceptable accuracy

but at what cost?

(12,6) SWCNT 168 atoms all

carbon, 12 k points

VASP (PBE) time 6,000 sec

memory 1900 MB

CRYSTAL (PBE) time 700 sec,

memory 300 MB

Conclusion: CRYSTAL PBE is

9 times faster than VASP

With VASP B3PW ~ 1000

times the cost of PBE

With CRYSAL B3PW is 3.2

times cost of PBE

CRYSTAL B3PW ~ 3 times

faster than VASP PBE

Topological Insulators

23

Material that is an insulator in its interior but whose surface

contains conducting states so electrons move along the surface of the

material.

On the surface of a topological insulator there are special states that

fall within the bulk energy gap and allow surface metallic

conduction. These surface state carriers have spins locked at 90° to

their momentum

At a given energy the only other available electronic states have

different spin, so surface is highly metallic and states cannot be

removed by surface passivation

Angular Resolved Photoelectron Spectrosopy

(ARPES) showed that Bi2Se3 is topological

insulator for 6 quintuple layers and larger

PBE says no band inversion not top. Insul.

G0W0 bulk Bi2Se3 has band inversion but not

possible for finite number layers

I. Aguilera, C. Friedrich, and S. Blügel, Phys. Rev. B 88, 165136 (2013)

B3PW CPU cost = 2.1 X PBE

Band gaps:

B3PW: 0.26 eV

G0W0: 0.20 eV

EXP: ~0.2 – 0.35 eV

Bulk Bi2Se3: B3PW and

G0W0/PBE both give band

inversion topological

insulator

I. Aguilera, C. Friedrich, and S. Blügel, Phys. Rev. B 88, 165136 (2013)

B3PW CPU cost = 1.8 X PBE

Band gaps:

B3PW: 0.28 eV

G0W0: 0.19 eV

EXP: ~0.13 – 0.17 eV

Bi2Te3 band gap with number quintuple layers

26Clear formation of metallic topological

insulator state by 5 QL, no experiment

Bi2Se3 band gap with number quintuple layers

27

Bi2Se3 Band gap vs number quint layers

28

PBE

B3PW

exper

Exper TI for 6 layers

PBE Gap < 0.1 eV

for 6 layers


Accuracy: DFT is basis for QM on catalysts

Current flavors of DFT accurate for properties of many systems

B3LYP and M06 useful for chemical reaction mechanisms

Progress is being made on developing new systems


Accuracy: DFT is basis for QM on catalysts

Current flavors of DFT accurate for properties of many systems

B3LYP and M06 useful for chemical reaction mechanisms

• B3LYP and M06L perform well.

• M06 underestimates the barrier.

Example: Reductive elimination of CH4 from (PONOP)Ir(CH3)(H)+

Goldberg exper at 168Kbarrier G‡ = 9.3kcal/mol.

G(173K)B3LYPM06M06L

0.00.00.0

10.85.8

11.4(reductive elimination)

These calculations

use extended basis

sets and PBF

solvation


Reductive Elimination Thermochemistry

H/D exchange was measured from 153-173K by Girolami (J . Am. Chem. Soc., Vol. 120, 1998 6605) by NMR to have a barrier of G‡ = 8.1 kcal/mol.

G(173K)B3LYPM06

0.00.0

8.79.5(reductive elimination)

4.65.3(s-bound complex)

6.45.2(site-exchange)

Mu-Jeng Cheng

QM allows first principles predictions on new ligands, oxidation states, and

solvents. But there are error bars in the QM having to do with details of the

caculations (flavor of DFT, basis set). We use the best available methods and

compare to any available experimental data on known systems to assess the

accuracy for new systems. Some examples here and on the next slides

M06 and B3LYP functionals both consistent with experimental barrier site exchange.

These calculations use extended basis sets and PBF solvation


Reductive Elimination Thermochemistry

• B3LYP greatly underestimates the barrier since its repulsive non-bonding interactions underestimate the Pt-phosphine bond strength.

• M06L performs well and M06 underestimates the barrier.

Reductive elimination of ethane from (dppe)Pt(CH3)4 was observed from 165-205˚C in benzene by Goldberg (J . Am. Chem. Soc., Vol. 125, 2003 9444) with a barrier of G‡ = 36 kcal/mol (S‡ = 15 e.u.).

(As carbons are constrained to approach each other, the trans phosphine dissociates automatically.)


Metal-oxo Oxidations

• M06 performs well

• B3LYP overestimates bimolecular barriers involving bulky or polarizable species

Experiment:M06:B3LYP:

H‡(25C)13.4 kcal/mol

11.817.1

Phosphine oxidation by (Tp)Re(O)Cl2 and (Tpm)Re(O)Cl2+ was observed from 15-50˚C in 1,2-dichlorobenzene by Seymore and Brown (Inorg. Chem., Vol. 39, 2000, 325):

Experiment:M06:B3LYP:

H‡(25C)17.1 kcal/mol

16.624.1


Methods matter (must use the correct flavor

DFT and the correct basis set)

34

Commonly used methods (B3LYP, triple zeta basis set ) are insufficient for oxidation of main group elements. (Martin, J. Chem. Phys. 1998, 108(7), 2791.) B3LYP disfavors oxidation of main group elements by >10 kcal/mol

Experimental H (kcal/mol) -27

-80.1

M066311G**++

-22.0-70.7

B3LYP6311G**++

-17-58.1

M066311++G-

3df(S)-29.2-82.2

Bad, but typical in publications

Simple example

S(CH3)2 + ½ O2 → O=S(CH3)2

S(CH3)2 + O2 → (CH3)2SO2

S(CH3)2 + ½ O2 → O=S(CH3)2

S(CH3)2 + O2 → (CH3)2SO2

OK


Methods matter (for reactions in polar media,

must include solvation)Phosphine oxidation by (Tp)Re(O)Cl2 and (Tpm)Re(O)Cl2

+ observed from 15-50˚C in 1,2-dichlorobenzene Seymore and Brown; Inorg. Chem., Vol. 39, 2000, 325)

Exper:M06:

B3LYP:

H‡(25C)With solvation

17.1 16.624.1

Barrier withNo solvation

16.9

Exper:M06:

B3LYP:

H‡(25C)With solvation

13.4 11.817.1

Barrier withNo solvation

2.4

Most QM publications ignore solvation or use unreliable methods

Much larger corrections in H2O


Fundamental philosophy of First principles

predictionsQM calculations on small systems ~100 atoms get accurate

energies, geometries, stiffness, mechanisms

Fit QM to force field to describe big systems (104 -107 atoms)

Fit to obtain parameters for continuum systems

macroscopic properties based on first principles (QM)

Can predict novel materials where no empirical data available.


Fundamental philosophy of First principles

predictionsQM calculations on small systems ~100 atoms get accurate

energies, geometries, stiffness, mechanisms

Fit QM to force field to describe big systems (104 -107 atoms)

Fit to obtain parameters for continuum systems

macroscopic properties based on first principles (QM)

Can predict novel materials where no empirical data available.

General Problem with DFT: bad description of vdw attraction

Graphite layers not

stable with DFT

exper


DFT bad for all Crystals dominated by

nonbond interactions (molecular crystals)

Molecules PBE PBE-ℓg Exp.

Benzene 1.051 12.808 11.295

Naphthalene 2.723 20.755 20.095

Anthracene 4.308 28.356 27.042


Benzene 511.81 452.09 461.11

Naphthalene 380.23 344.41 338.79

Anthracene 515.49 451.55 451.59

Sublimation energy (kcal/mol/molecule)

Cell volume (angstrom3/cell) PBE 12-14% too large

PBE 85-90% too small

Most popular form of DFT for crystals – PBE (VASP software)

Reason DFT formalism not include London Dispersion (-C6/R6)

responsible for van der Waals attraction.

All published QM calculations on solids have this problem


XYG3 approach to include London Dispersion in DFT

Görling-Levy coupling-constant perturbation expansion

] ]1

,0

xc xcE U dr r Take initial slope as the 2nd order correlation energy:

, 2

, 0

0

2xc GL

xc c

UU E

where

22

2

ˆˆˆ1

4

i xi j eeGL

c

ij ii j i

fE

e e e e e e

where is the electron-electron repulsion operator, is the local exchange operator,

and is the Fock-like, non-local exchange operator.

ˆee ˆ

x

f̂

,xcU a b Substitute into with22 GL

cb E ;exact LDA exact

x xc xa E b E E or

Combine both approaches (2 choices for b) ( )2

1 2

GL DFT exact

c xc xb b E b E E

] ( ) ( )R5 2

1 2 3 4

LDA exact LDA GGA PT LDA GGA

xc xc x x x c c cE E c E E c E c E E c Er

a double hybrid DFT that mixes some exact exchange into while also introducing a

certain portion of into

DFT

xE2PT

cEDFT

cE

contains the double-excitation parts of 2PT

cE

22

2

ˆˆˆ1

4

i xi j eeGL

c

ij ii j i

fE

e e e e e e

This is a fifth-rung functional (R5) using information from both occupied and virtual KS

orbitals. In principle can now describe dispersion

Sum over virtual orbtials

40

Solution: extend DFT to include double excitations to virtuals

get London Dispersion in DFT: use Görling-Levy expansion

] ( ) ( )R5 2

1 2 3 4

LDA exact LDA GGA PT LDA GGA

xc xc x x x c c cE E c E E c E c E E c Er

Get {c1 = 0.8033, c2 = 0.2107, c3 = 0.3211} and c4 = (1 – c3) = 0.6789

XYG3 leads to mean absolute deviation (MAD) =1.81 kcal/mol,

B3LYP: MAD = 4.74 kcal/mol.

M06: MAD = 4.17 kcal/mol

M06-2x: MAD = 2.93 kcal/mol

M06-L: MAD = 5.82 kcal/mol .

G3 ab initio (with one empirical parameter): MAD = 1.05

G2 ab initio (with one empirical parameter): MAD = 1.88 kcal/mol

but G2 and G3 involve far higher computational cost.

where

22

2

ˆˆˆ1

4

i xi j eeGL

c

ij ii j i

fE

e e e e e e

Problem 5th order scaling with size

Doubly hybrid density functional for accurate descriptions of nonbond interactions,

thermochemistry, and thermochemical kinetics; Zhang Y, Xu X, Goddard WA; P. Natl.

Acad. Sci. 106 (13) 4963-4968 (2009)


Thermochemical accuracy with size

G3/99 set has 223 molecules:

G2-1: 56 molecules having up to 3 heavy atoms,

G2-2: 92 additional molecules up to 6 heavy atoms

G3-3: 75 additional molecules up to 10 heavy atoms.

B3LYP: MAD = 2.12 kcal/mol (G2-1), 3.69 (G2-2), and 8.97 (G3-3) leads to

errors that increase dramatically with size

B2PLYP MAD = 1.85 kcal/mol (G2-1), 3.70 (G2-2) and 7.83 (G3-3) does not

improve over B3LYP

M06-L MAD = 3.76 kcal/mol (G2-1), 5.71 (G2-2) and 7.50 (G3-3).

M06-2x MAD = 1.89 kcal/mol (G2-1), 3.22 (G2-2), and 3.36 (G3-3).

XYG3, MAD = 1.52 kcal/mol (G2-1), 1.79 (G2-2), and 2.06 (G3-3), leading to

the best description for larger molecules.


Accuracy (kcal/mol) of various QM methods for

predicting standard enthalpies of formationFunctional MAD Max(+) Max(-)

DFT

XYG3 a 1.81 16.67 (SF6) -6.28 (BCl3)

M06-2x a 2.93 20.77 (O3) -17.39 (P4)

M06 a 4.17 11.25 (O3) -25.89 (C2F6)

B2PLYP a 4.63 20.37(n-octane) -8.01(C2F4)

B3LYP a 4.74 19.22 (SF6) -8.03 (BeH)

M06-L a 5.82 14.75 (PF5) -27.13 (C2Cl4)

BLYP b 9.49 41.0 (C8H18) -28.1 (NO2)

PBE b 22.22 10.8 (Si2H6) -79.7 (azulene)

LDA b 121.85 0.4 (Li2) -347.5 (azulene)

Ab initio

HFa 211.48 582.72(n-octane) -0.46 (BeH)

MP2a 10.93 29.21(Si(CH3)4) -48.34 (C2F6)

QCISD(T) c 15.22 42.78(n-octane) -1.44 (Na2)

G2(1 empirical parm) 1.88 7.2 (SiF4) -9.4 (C2F6)

G3(4 empirical parm) 1.05 7.1 (PF5) -4.9 (C2F4)


-5.00

0.00

5.00

10.00

15.00

20.00

25.00

30.00

-2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50

Reaction coordinate

En

erg

y (

kca

l/m

ol)

HF

HF_PT2

XYG3

CCSD(T)

B3LYP

BLYP

SVWN

HF

HF_PT2 SVWNB3LYP

BLYP

XYG3CCSD(T)

SVWN

H + CH4 H2 + CH3

Reaction Coordinate: R(CH)-R(HH) (in Å)

Energ

y (

kcal/m

ol)

Comparison of QM methods for reaction surface of

H + CH4 H2 + CH3


Reaction

barrier

heights

19 hydrogen transfer (HT) reactions,

6 heavy-atom transfer (HAT) reactions,

8 nucleophilic substitution (NS) reactions and

5 unimolecular and association (UM) reactions.

Functional All (76) HT38 HAT12 NS16 UM10

DFT

XYG3 1.02 0.75 1.38 1.42 0.98

M06-2x a 1.20 1.13 1.61 1.22 0.92

B2PLYP 1.94 1.81 3.06 2.16 0.73

M06 a 2.13 2.00 3.38 1.78 1.69

M06-La 3.88 4.16 5.93 3.58 1.86

B3LYP 4.28 4.23 8.49 3.25 2.02

BLYP a 8.23 7.52 14.66 8.40 3.51

PBEa 8.71 9.32 14.93 6.97 3.35

LDAb 14.88 17.72 23.38 8.50 5.90

Ab initio

HFb 11.28 13.66 16.87 6.67 3.82

MP2 b 4.57 4.14 11.76 0.74 5.44

QCISD(T) b 1.10 1.24 1.21 1.08 0.53

Zhao and Truhlar

compiled benchmarks

of accurate barrier

heights in 2004

includes forward and

reverse barrier heights

for

Note: no reaction

barrier heights used

in fitting the 3

parameters in

XYG3)


(A)

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

30.00

3.0 4.0 5.0 6.0

Intermolecular distance

En

erg

y (

kca

l/m

ol)

BLYP

B3LYP

XYG3

CCSD(T)

SVWN

HF_PT2

(C)

-12.00

-9.00

-6.00

-3.00

0.00

Ec_VWN

Ec_B3LYP

Ec_LYP

Ec_XYG3

Ec_CCSD(T)

Ec_PT2

(B)

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

30.00

3.0 4.0 5.0 6.0

Ex_B

Ex_B3LYP

Ex_XYG3

Ex_HF

Ex_S

HF

HF_PT2

B3LYP

BLYP

CCSD(T)

LDA

(SVWN)

A. Total Energy (kcal/mol)

Distance (A)

XYG3

B. Exchange Energy (kcal/mol)

C. Correlation Energy (kcal/mol)

B

S

B3LYP

XYG3

PT2

B3LYP

LYP CCSD(T)

VWN

XYG3

Distance (A)

Conclusion: XYG3 provides excellent accuracy for London dispersion, as good as

CCSD(T)

Test for

London

Dispersion


Accuracy of QM methods for noncovalent interactions.

Functional Total HB6/04 CT7/04 DI6/04 WI7/05 PPS5/05

DFT

M06-2x b 0.30 0.45 0.36 0.25 0.17 0.26

XYG3 a 0.32 0.38 0.64 0.19 0.12 0.25

M06 b 0.43 0.26 1.11 0.26 0.20 0.21

M06-L b 0.58 0.21 1.80 0.32 0.19 0.17

B2PLYP 0.75 0.35 0.75 0.30 0.12 2.68

B3LYP 0.97 0.60 0.71 0.78 0.31 2.95

PBE c 1.17 0.45 2.95 0.46 0.13 1.86

BLYP c 1.48 1.18 1.67 1.00 0.45 3.58

LDA c 3.12 4.64 6.78 2.93 0.30 0.35

Ab initio

HF 2.08 2.25 3.61 2.17 0.29 2.11

MP2c 0.64 0.99 0.47 0.29 0.08 1.69

QCISD(T) c 0.57 0.90 0.62 0.47 0.07 0.95

HB: 6 hydrogen bond

complexes,

CT 7 charge-transfer

complexes

DI: 6 dipole

interaction complexes,

WI:7 weak interaction

complexes,

PPS: 5 pp stacking

complexes. WI and PPS dominated

by London dispersion.

Note: no

noncovalent

complexes used

in fitting the 3

parameters in

XYG3)


Problem with XYG3 : scales as N**5 with size

(like MP2)

] ]1

,0

xc xcE U dr r Take initial slope as the 2nd order correlation energy:

, 2

, 0

0

2xc GL

xc c

UU E

where

22

2

ˆˆˆ1

4

i xi j eeGL

c

ij ii j i

fE

e e e e e e

where is the electron-electron repulsion operator, is the local exchange operator,

and is the Fock-like, non-local exchange operator.

ˆee ˆ

x

f̂

Sum over virtual orbtials

XYG3 approach to include London Dispersion in DFT

Görling-Levy coupling-constant perturbation expansion

EGL2 involves double excitations to virtuals, scales as N5 with

size

MP2 has same critical step

Yousung Jung (KAIST) figured out how to get N3 scaling for

MP2 and for XYGJ-OSYousung

Jung


Solve scaling problem: XYGJ-OS; include only opposite spin

and only local contributions

] ( ) ( )XYGJ- OS 2

2 ,1

HF S VWN LYP PT

xc x x x x VWN c LYP c PT c osE e E e E e E e E e E

r

XYG4-OS and XYG4-LOS timings

0.0

40.0

80.0

120.0

160.0

200.0

0 20 40 60 80 100 120

alkane chain length

CP

U (

ho

urs

)

XYG4-LOS

XYG4-OS

B3LYP

XYG3


0.0

40.0

80.0

120.0

160.0

200.0

0 20 40 60 80 100 120

alkane chain length

CP

U (

ho

urs

)XYG4-LOS

XYG4-OS

B3LYP

XYG3


0.0

40.0

80.0

120.0

160.0

200.0

0 20 40 60 80 100 120

alkane chain length

CP

U (

ho

urs

)

XYG4-LOS

XYG4-OS

B3LYP

XYG3

XYGJ-OS

XYGJ-LOS


0.0

40.0

80.0

120.0

160.0

200.0

0 20 40 60 80 100 120

alkane chain length

CP

U (

ho

urs

)XYG4-LOS

XYG4-OS

B3LYP

XYG3

XYGJ-LOS


0.0

40.0

80.0

120.0

160.0

200.0

0 20 40 60 80 100 120

alkane chain length

CP

U (

ho

urs

)

XYG4-LOS

XYG4-OS

B3LYP

XYG3

XYGJ-OS

{ex, eVWN, eLYP, ePT2} ={0.7731,0.2309, 0.2754,

0.4264}.

A fast doubly hybrid

density functional

method close to

chemical accuracy:

XYGJ-OS

Igor Ying Zhang,

Xin Xu, Yousung

Jung, and wag

PNAS in press

XYGJ-OS

same accuracy

as XYG3 but

scales like N3

not N5.

Density Functional Theory errors kcal/mol)

49

LDA 130.88 15.2

Include density gradient (GGA)

BLYP 10.16 7.9

PW91 22.04 9.3

PBE 20.71 9.1

Hybrid: include HF exchange

B3LYP 6.08 4.5

PBE0 5.64 3.9

Include KE functional fit to barriers and complexes

M06-L 5.20 4.1

M06 3.37 2.2

M06-2X 2.26 1.3

atomize barrier

Popular with physicists


Popular with chemists

Include excitations to virtuals

XYGJ-OS 1.81 1.0

G3 (cc) 1.06 0.9The level needed for

reliable predictions


Accuracy of Methods (Mean absolute deviations MAD, in eV)

HOF IP EA PA BDE NHTBH HTBH NCIE All Time

Methods

(223) (38) (25) (8) (92) (38) (38) (31) (493) C100H202 C100H100

DFT methods

SPL (LDA) 5.484 0.255 0.311 0.276 0.754 0.542 0.775 0.140 2.771

BLYP 0.412 0.200 0.105 0.080 0.292 0.376 0.337 0.063 0.322

PBE 0.987 0.161 0.102 0.072 0.177 0.371 0.413 0.052 0.562

TPSS 0.276 0.173 0.104 0.071 0.245 0.391 0.344 0.049 0.250

B3LYP 0.206 0.162 0.106 0.061 0.226 0.202 0.192 0.041 0.187 2.8 12.3

PBE0 0.300 0.165 0.128 0.057 0.155 0.154 0.193 0.031 0.213

M06-2X 0.127 0.130 0.103 0.092 0.069 0.056 0.055 0.013 0.096

XYG3 0.078 0.057 0.080 0.070 0.068 0.056 0.033 0.014 0.065 200.0 81.4

XYGJ-OS 0.072 0.055 0.084 0.067 0.033 0.049 0.038 0.015 0.056 7.8 46.4

MC3BB 0.165 0.120 0.175 0.046 0.111 0.062 0.036 0.023 0.123

B2PLYP 0.201 0.109 0.090 0.067 0.124 0.090 0.078 0.023 0.143

Wavefunction based methods

HF 9.171 1.005 1.148 0.133 0.104 0.397 0.582 0.098 4.387

MP2 0.474 0.163 0.166 0.084 0.363 0.249 0.166 0.028 0.338

G2 0.082 0.042 0.057 0.058 0.078 0.042 0.054 0.025 0.068

G3 0.046 0.055 0.049 0.046 0.047 0.042 0.054 0.025 0.046

HOF = heat of formation; IP = ionization potential,

EA = electron affinity, PA = proton affinity,

BDE = bond dissociation energy,

NHTBH, HTBH = barrier heights for reactions,

NCIE = the binding in molecular clusters


Problem cannot do XYGJ-OS for crystals

Strategy: use XYGJ-OS to get accurate London

Dispersion on small cluster use to obtain

parameter for doing crystals (PBE-ulg)


Benzene 1.051 12.808 11.295

Naphthalene 2.723 20.755 20.095

Anthracene 4.308 28.356 27.042


Benzene 511.81 452.09 461.11

Naphthalene 380.23 344.41 338.79

Anthracene 515.49 451.55 451.59


Cell volume (angstrom3/cell)PBE-lg 0 to 2% too small,

thermal expansion

PBE-lg 3 to 5% too high

(zero point energy)


Problem cannot do XYGJ-OS for crystals

Strategy: use XYGJ-OS to get accurate London

Dispersion on small cluster use to obtain

parameter for doing crystals (PBE-ulg)


Benzene 1.051 12.808 11.295

Naphthalene 2.723 20.755 20.095

Anthracene 4.308 28.356 27.042


Benzene 511.81 452.09 461.11

Naphthalene 380.23 344.41 338.79

Anthracene 515.49 451.55 451.59



thermal expansion


(zero point energy)


Major challenge for DFT calculations of molecular solids

• Current implementations of DFT describe geometries and energies of strongly bound solids, but fail to describe the long range van der Waals (vdW) interactions.

• Get volumes ~ 10% too large

• XYGJ-OS solves this problem but much slower than standard methods

Nlg,

lg 6 6,

-ij

ij i j ij eij

CE

r dR

DFT D DFT dispE E E

C6 single parameter from QM-CC

d =1

Reik = Rei + Rek (UFF vdW radii)

DFT-low gradient (DFT-lg) gives accurate description of the long-range 1/R6 attraction of the London dispersion but at cost of standard DFTAdd the low-gradient 1/R6 one parameter fitted to XYGJ-OS


PBE-lg for benzene dimer

T-shaped Sandwich Parallel-displaced

PBE-lg parameters

Nlg,

lg 6 6,

-ij

ij i j ij eij

CE

r dR

Clg-CC=586.8, Clg-HH=31.14, Clg-HH=8.691

RC = 1.925 (UFF), RH = 1.44 (UFF)

First-Principles-Based Dispersion Augmented Density Functional Theory: From

Molecules to Crystals’ Yi Liu and wag; J. Phys. Chem. Lett., 2010, 1 (17), pp

2550–2555


DFT-lg description for graphite

graphite has AB stacking (also show AA eclipsed graphite)

Exper E

0.8, 1.0, 1.2

Exper c 6.556

PBE-lg

PBE

Bin

din

g e

ne

rgy (

kca

l/m

ol)

c lattice constant (A)


DFT-lg description for benzene

PBE-lg predicted the EOS of benzene crystal (orthorhombic phase I) in good agreement with

corrected experimental EOS at 0 K (dashed line).

Pressure at zero K geometry: PBE: 1.43 Gpa; PBE-lg: 0.11 Gpa

Zero pressure volume change: PBE: 35.0%; PBE-lg: 2.8%

Heat of sublimation at 0 K: Exp:11.295 kcal/mol; PBE: 0.913; PBE-lg: 6.762


Graphite Energy Curve

BE = 1.34 kcal/mol (QMC: 1.38, Exp: 0.84-1.24)c =6.8 angstrom (QMC: 6.8527, Exp: 6.6562)


Hydrocarbon Crystals – Get excellent results

for PBE-lg


Benzene 1.051 12.808 11.295

Naphthalene 2.723 20.755 20.095

Anthracene 4.308 28.356 27.042


Benzene 511.81 452.09 461.11

Naphthalene 380.23 344.41 338.79

Anthracene 515.49 451.55 451.59



thermal expansion


(zero point energy)

Most popular form of DFT for crystals – PBE (VASP software)

Strategy: use XYGJ-OS to get accurate London Dispersion on small

clusters PBE-lg parameters. Use PBE-lg for large systems


DFT-ℓg for accurate Dispersive Interactions for Full

Periodic Table

Hyungjun Kim, Jeong-Mo Choi, William A. Goddard, III1Materials and Process Simulation Center, Caltech

2Center for Materials Simulations and Design, KAIST


Universal PBE-ℓg Method

UFF, a Full Periodic Table Force Field for Molecular Mechanics and Molecular

Dynamics Simulations; A. K. Rappé, C. J. Casewit, K. S. Colwell, W. A. Goddard

III, and W. M. Skiff; J. Am. Chem. Soc. 114, 10024 (1992)

Derived C6/R6 parameters from scaled atomic polarizabilities for Z=1-103 (H-

Lr) and derived Dvdw from combining atomic IP and C6

Universal PBE-lg: use same Re, C6, and De as UFF, add a single new

parameter slg


blg Parameter Modifies Short-range Interactions

blg =1.0 blg =0.7

12-6 LJ potential (UFF parameter)

lg potentiallg potential

When blg =0.6966,ELJ(r=1.1R0) = Elg(r=1.1R0)


Benzene Dimer

T-shaped


Benzene Dimer

Sandwich


Benzene Dimer

Parallel-

displaced


Parameter Optimization

Implemented in VASP 5.2.11

0.7012

0.6966


Graphite Energy Curve

BE = 1.34 kcal/mol (QMC: 1.38, Exp: 0.84-1.24)c =6.8 angstrom (QMC: 6.8527, Exp: 6.6562)


Hydrocarbon Crystals

• Sublimation energy (kcal/mol/molecule)

• Cell volume (angstrom3/cell)


Benzene 1.051 12.808 11.295

Naphthalene 2.723 20.755 20.095

Anthracene 4.308 28.356 27.042


Benzene 511.81 452.09 461.11

Naphthalene 380.23 344.41 338.79

Anthracene 515.49 451.55 451.59


Simple Molecular Crystals


Average error: 3.86 (PBE) and 0.96 (PBE-ℓg)

Maximal error: 7.10 (PBE) and 1.90 (PBE-ℓg)


F2 0.27 1.38 2.19

Cl2 2.05 5.76 7.17

Br2 5.91 10.39 11.07

I2 8.56 14.47 15.66

O2 0.13 1.50 2.07

N2 0.02 1.22 1.78

CO 0.11 1.54 2.08

CO2 1.99 4.37 6.27


Simple Molecular Crystals

• Cell volume (angstrom3/cell)


F2 126.47 126.32 128.24

Cl2 282.48 236.23 231.06

Br2 317.30 270.06 260.74

I2 409.03 345.13 325.03

O2 69.38 69.35 69.47

N2 180.04 179.89 179.91

CO 178.96 178.99 179.53

CO2 218.17 179.93 177.88


Inert Gas Crystals


Average error: 1.70 (PBE) and 0.74 (PBE-ℓg)

Maximal error: 3.14 (PBE) and 1.68 (PBE-ℓg)


Ne 0.40 0.69 0.46

Ar 0.45 1.38 1.85

Kr 0.48 1.62 2.66

Xe 0.63 2.09 3.77


Heavy Atom Fluorides


aSpin-orbit coupling term is corrected.

Other issues; Large core pseudopotential (U: 14 electrons, Np: 15

electrons).


UF6 1.78 3.76 11.96

NpF6 -- 3.52 --

XeF2 5.71 9.51 (9.82a) 12.3

XeF4 5.42 10.03 (10.34a) 15.3


Density Functional Theory errors kcal/mol)

72

LDA 130.88 15.2

Include density gradient (GGA)

BLYP 10.16 7.9

PW91 22.04 9.3

PBE 20.71 9.1

Hybrid: include HF exchange

B3LYP 6.08 4.5

PBE0 5.64 3.9

Include KE functional fit to barriers and complexes

M06-L 5.20 4.1

M06 3.37 2.2

M06-2X 2.26 1.3

atomize barriersPopular with physicists


Popular with chemists

replace the N-electron wavefunction Ψ(1,2,…N) with just the 3

degrees of freedom for the electron density r(x,y,z).

E = Functional not known, but have accurate approx.

rrr

VFV

HK ][rep-

min

Acceptable

errors

Calculate Solvent Accessible Surface of the solute by rolling a

sphere of radius Rsolv over the surface formed by the vdW

radii of the atoms.

Calculate electrostatic field of the solute based on electron

density from the orbitals

Calculate the polarization in the solvent due to the

electrostatic field of the solute (need dielectric constant e)

This leads to Reaction Field that acts back on solute atoms,

which in turn changes the orbitals. Iterated until self-

consistent.

Calculate solvent forces on solute atoms

Use these forces to determine optimum geometry of solute in

solution.

Can treat solvent stabilized zwitterions

Difficult to describe weakly bound solvent molecules

interacting with solute (low frequency, many local minima)

Short cut: Optimize structure in the gas phase and do single

point solvation calculation. Some calculations done this way

Essential issues: must include Solvation effects in the QM

Solvent: e = 99

Rsolv= 2.205 A

PBF Implementation in

Jaguar (Schrodinger Inc):

pK organics to ~0.2 units,

solvation to ~1 kcal/mol

(pH from -20 to +20)

The Poisson-Boltzmann Continuum Model in Jaguar/Schrödinger is extremely accurate

6.9 (6.7)

-3.89 (-52.35)

6.1 (6.0)

-3.98 (-55.11)

5.8 (5.8) -

4.96 (-49.64)

5.3 (5.3)

-3.90 (-57.94)

5.0 (4.9) -

4.80 (-51.84)

pKa: Jaguar (experiment)

E_sol: zero (H+)

Comparison of PBF (Jaguar) pK with experiment

Protonated Complex

(diethylenetriamine)Pt(OH2)2+

PtCl3(OH2)1-

Pt(NH3)2(OH2)22+

Pt(NH3)2(OH)(OH2)1+

cis-(bpy)2Os(OH)(H2O)1+

Calculated (B3LYP) pKa(MAD: 1.1)

5.5

4.1

5.2

6.5

11.3

Experimental pKa

6.3

7.1

5.5

7.4

11.0

cis-(bpy)2Os(H2O)22+

cis-(bpy)2Os(OH)(H2O)1+

trans-(bpy)2Os(H2O)22+

trans-(bpy)2Os(OH)(H2O)1+

cis-(bpy)2Ru(H2O)22+

cis-(bpy)2Ru(OH)(H2O)1+

trans-(bpy)2Ru(H2O)22+

trans-(bpy)2Ru(OH)(H2O)1+

(tpy)Os(H2O)32+

(tpy)Os(OH)(H2O)21+

(tpy)Os(OH)2(H2O)

Calculated (M06//B3LYP) pKa

(MAD: 1.6)9.18.86.2

10.913.015.211.013.95.66.3

10.9

Experimental pKa

7.911.08.2

10.28.9

>11.09.2

>11.56.08.0

11.0

PBF (Jaguar) predictions of Metal-aquo pKa’s

-40

-30

-20

-10

0

10

20

30

40

50

0 5 10 15 20

G (

kca

l/m

ol)

pH

32.6

34.6 40.0

37.9

34.6

Resting states

Insertion

transition states

Use theory to predict optimal pH for each

catalyst

Optimum pH is 8

LnOsII(OH2)(OH)2

is stable

LnOsII(OH)3-

is stable LnOsII(OH2)3

+2

is stable

Predict the relative free energies of

possible catalyst resting states and

transition states as a function of pH.

Predict Pourbaix Diagrams to determine the

oxidation states of transition metal complexes as

function of pH and electrochemical potential

Black experimental data from Dobson

and Meyer, Inorg. Chem. Vol. 27, No.19, 1988.

Red is from QM calculation (no fitting) using M06 functional, PBFimplicit solvent

Max errors: 200 meV, 2pH units

Trans-(bpy)2Ru(OH)2

This is essential in using theory to

predict new catalysts

78

80

81

82

83

84

CANDLE SOLVATION FOR QM CALCULATIONS

charge-asymmetric nonlocally-determined local-electric Field (CANDLE) Solvation model; Sundararaman &WAG; J. Chem. Phys., 142 (6): 064107 (2015)

Solvents: H2O, CH3CN, CHCl3, CCl4)

CANDLE SOLVATION FOR QM CALCULATIONS

charge-asymmetric nonlocally-determined local-electric Field (CANDLE) Solvation model; Sundararaman &WAG; J. Chem. Phys., 142 (6): 064107 (2015)

Solvents: H2O, CH3CN, CHCl3, CCl4)

Documents

Lecture 2, January 6, 2016 Quantum Mechanics-2: DFTwag.caltech.edu/home/ch121/Slides/2016/L2_2016.pdf · Becke half and half functional assume a linear model U a b xc,O O take,0 exact