Lecture 13b, February 18, 2015 Energetics, Hartree...

© copyright 2015 William A. Goddard III, all rights reservedCh125-Goddard-L13b Ch125a-Goddard-

1

Elements of Quantum Chemistry with Applications to Chemical Bonding and

Properties of Molecules and Solids

William A. Goddard, III, [email protected] Beckman Institute, x3093

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

California Institute of Technology

Special Instructor: Julius Su <[email protected]>Teaching Assistants: Hai Xiao <[email protected]>

Mark Fornace <[email protected]>

Lecture 13b, February 18, 2015Energetics, Hartree Fock

Course number: Ch125a; Room 115 BIHours: 11-11:50am Monday, Wednesday, Friday

Lecture 1Ch121a-Goddard-L01 © copyright 2012 William A. Goddard III, all rights reserved\ 2

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 (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

FHK ][

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 (1923Walter 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 (xyz)=(r) minimizes the functional E[= F[+ ʃ (r) Vext(r) d3rwhere F[is a universal functional of and the minimum value of the functional, E, is E0, the exact ground-state electronic energy. Here we take Vext(1..N) = i=1,..N A=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

Lecture 1Ch121a-Goddard-L01 © copyright 2012 William A. Goddard III, all rights reserved\

Proof of the Hohenberg-Kohn theorem

4

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)] = min <Ψ|O|Ψ>

|Ψ>(r)where we consider all wavefunctions Ψ that lead to the same density, (r), and select the one leading to the lowest expectation value for <Ψ|O|Ψ>.F[ is defined as F[(r)] = min <Ψ|F|Ψ>

|Ψ>(r)

where F = i [- ½ i2] + ½ i≠k [1/rik].

Thus the usual Hamiltonian is H = F + VextNow consider a trial function Ψapp that leads to the density (r) and which minimizes <Ψ|F|Ψ> Then E[= F[+ ʃ (r) Vext(r) d3r = <Ψ|F +Vext|Ψ> = <Ψ|H|Ψ> Thus E[≥ E0 the exact ground state energy.


The Kohn-Sham equations

5

Walter Kohn and Lou J. Sham. Phys. Rev. 140, A1133 (1965).Provided a practical methodology to calculate DFT wavefunctions They partitioned the functional E[] into parts

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

Where

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

gas having density rThis is NOT the KE of the real system.The 2nd term is the total electrostatic energy for the density rNote that this includes the self interaction of an electron with itself.The 3rd term is the total electron-nuclear attraction termThe 4th term contains all the unknown aspects of the Density Functional

w

wag2 wag, 9/4/2010


Solving the Kohn-Sham equations

6

Requiring that ʃ d3r (r) = N the total number of electrons and applying the variational principle leads to

[(r)] [E[] – ʃ d3r (r) ] = 0

where the Lagrange multiplier = E[]/ = the chemical potential

Here the notation [(r)] means a functional derivative inside the integral.

To calculate the ground state wavefunction we solve

HKS φi = [- ½ i2 + Veff(r)] φi = i φi

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

where Veff (r) = Vext (r) + J(r) + Vxc(r) and Vxc(r) = EXC[]/

Thus HKS looks quite analogous to HHF


The Local Density Approximation (LDA)

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

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

KE Nuclear attraction

Coulomb repulsionExchange correlation

If the density is =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 presentAt the very minimum, Vxc needs to correct for this

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

31

xLDAx rρAρε xA = -

31

π3

43


The Local Density Approximation (LDA)

ExcLDA[rʃ d3r XC(rr

where XC(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


Generalized gradient approximations

9

The 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


LDA exchange

10

31

xLDAx rρAρε xA = -

31

π3

43

Here we say that in LDA each electron interacts with all N electrons but 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 = N/V


Generalized gradient approximations

11

cxxc EEE

drρ(r),...ρ(r)ρ(r),εE xx

sFερρ,ε LDAx

GGAx

34

31

2 ρπ24

ρs

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 5.0 1 0.0

Becke 88

X3LYP

PBEPW91

s

F(s) GGA functionals

2

11

232

1188B

sasinhsa1sasasinhsa1sF

d

521

1

2s100432

1191PW

sasasinhsa1seaasasinhsa1sF

2

34 a8110a ,

x3/1

642

5 A210a

a

, and d = 4.

9 = 0.0042 a4 and a5 zero 3

12

2 π48a , 21 βa6a , βA2

aa

x3/1

22

3 ,


adiabatic connection formalism

12

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:

1

,0xc xcE U d 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.Becke, A.D. J. Chem. Phys. (1993), 98, 1372-1377.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.


Becke half and half functional

13

assume a linear model ,xcU a b

take, 0

exactxc xU E the exact exchange of the KS orbitals

approximate , 1 , 1

LDAxc xcU U

partition LDA LDA LDAxc x cE E E

set ;exactxa E LDA exact

xc xb E E

Get half-and-half functional 1 12 2

exact LDA LDAxc x x cE E E E

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


Becke 3 parameter functional

14

B31 2 3

LDA exact LDA GGA GGAxc xc x x x cE E c E E c E c E

Empirically modify half-and-half

where GGAxE is the gradient-containing correction terms to the LDA exchange

GGAcE 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

DFTxcE

Becke, A.D. J. Chem. Phys. (1993), 98, 5648-5652.Becke, A.D. J. Chem. Phys. (1993), 98, 1372-1377.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 exchangeVosko-Wilk-Nusair correlation, etc

GGA: Exchange: B88, PW91, PBE, OPTX, HCTH, etcCorrelations: 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: HCTHh, 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-GGAF PBE + VSXC


Accuracy: DFT is basis for QM on catalysts

Current flavors of DFT accurate for properties of many systemsB3LYP and M06 useful for chemical reaction mechanismsProgress is being made on developing new systems


Accuracy: DFT is basis for QM on catalysts

Current flavors of DFT accurate for properties of many systemsB3LYP 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 168K barrier G‡ = 9.3 kcal/mol.

G(173K)B3LYPM06M06L

0.00.00.0

10.85.811.4

(reductive elimination)

These calculations use extended basis

sets and PBF solvation

Reductive Elimination ThermochemistryH/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(‐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.)

Typical validation: 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)

22

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)2S(CH3)2 + O2 → (CH3)2SO2

S(CH3)2 + ½ O2 → O=S(CH3)2S(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)

ReV Cl

N

ClN

N

O

N

NN

HB

ReV Cl

N

ClN

N

O

N

NN

HC

+1 P

Ph

Ph

Ph

ReV Cl

N

ClN

N

O

N

NN

HC

+1

P

Ph

Ph

Ph

ReV Cl

N

ClN

N

O

N

NN

HB

G

PPh3

PPh3Exper: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 methodsMuch larger corrections in H2O

Fundamental philosophy of First principles predictions

QM calculations on small systems ~100 atoms get accurate energies, geometries, stiffness, mechanismsFit QM to force field to describe big systems (104 -107 atoms)Fit to obtain parameters for continuum systemsmacroscopic properties based on first principles (QM) Can predict novel materials where no empirical data available.

25

Fundamental philosophy of First principles predictions

QM calculations on small systems ~100 atoms get accurate energies, geometries, stiffness, mechanismsFit QM to force field to describe big systems (104 -107 atoms)Fit to obtain parameters for continuum systemsmacroscopic 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

26

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


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 smallMost 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

27

XYG3 approach to include London Dispersion in DFTGörling-Levy coupling-constant perturbation expansion

1

,0xc xcE U d Take initial slope as the 2nd order correlation energy:

, 2, 0

0

2xc GLxc c

UU E

where

22

2ˆˆˆ1

4i xi j eeGL

cij ii j i

fE

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

ˆee ˆxf̂

,xcU a b Substitute into with 22 GLcb E LDA exact

xc xb E E or

Combine both approaches (2 choices for b) 21 2

GL DFT exactc xc xb b E b E E

R5 21 2 3 4

LDA exact LDA GGA PT LDA GGAxc xc x x x c c cE E c E E c E c E E c E

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

DFTxE

2PTcE DFT

cEcontains the double-excitation parts of 2PT

cE2GL

cE 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

28

Solution: extend DFT to include double excitations to virtuals get London Dispersion in DFT: use Görling-Levy expansion

R5 21 2 3 4


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/molbut G2 and G3 involve far higher computational cost.

where

22

2ˆˆˆ1

4i xi j eeGL

cij ii j i

fE

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)

29

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 UM10DFTXYG3 1.02 0.75 1.38 1.42 0.98M06-2x a 1.20 1.13 1.61 1.22 0.92B2PLYP 1.94 1.81 3.06 2.16 0.73M06 a 2.13 2.00 3.38 1.78 1.69M06-La 3.88 4.16 5.93 3.58 1.86B3LYP 4.28 4.23 8.49 3.25 2.02BLYP a 8.23 7.52 14.66 8.40 3.51PBEa 8.71 9.32 14.93 6.97 3.35LDAb 14.88 17.72 23.38 8.50 5.90Ab initioHFb 11.28 13.66 16.87 6.67 3.82MP2 b 4.57 4.14 11.76 0.74 5.44QCISD(T) b 1.10 1.24 1.21 1.08 0.53

Zhao and Truhlarcompiled 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)

30

(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

Ene

rgy

(kca

l/mol

)

BLYPB3LYPXYG3CCSD(T)SVWNHF_PT2

(C)-12.00

-9.00

-6.00

-3.00

0.00

Ec_VWNEc_B3LYPEc_LYPEc_XYG3Ec_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_BEx_B3LYPEx_XYG3Ex_HFEx_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

31

Accuracy of QM methods for noncovalent interactions. Functional Total HB6/04 CT7/04 DI6/04 WI7/05 PPS5/05DFTM06-2x b 0.30 0.45 0.36 0.25 0.17 0.26XYG3 a 0.32 0.38 0.64 0.19 0.12 0.25M06 b 0.43 0.26 1.11 0.26 0.20 0.21M06-L b 0.58 0.21 1.80 0.32 0.19 0.17B2PLYP 0.75 0.35 0.75 0.30 0.12 2.68B3LYP 0.97 0.60 0.71 0.78 0.31 2.95PBE c 1.17 0.45 2.95 0.46 0.13 1.86BLYP c 1.48 1.18 1.67 1.00 0.45 3.58LDA c 3.12 4.64 6.78 2.93 0.30 0.35Ab initioHF 2.08 2.25 3.61 2.17 0.29 2.11MP2c 0.64 0.99 0.47 0.29 0.08 1.69QCISD(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 stacking complexes.

WI and PPS dominated by London dispersion.

Note: no noncovalent complexes used in fitting the 3 parameters in XYG3)

32

Accuracy (kcal/mol) of various QM methods for predicting standard enthalpies of formation

Functional MAD Max(+) Max(-)DFTXYG3 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 initioHFa 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)

33

-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

Ener

gy (k

cal/m

ol)

HFHF_PT2XYG3CCSD(T)B3LYPBLYPSVWN

HF

HF_PT2 SVWNB3LYP

BLYP

XYG3CCSD(T)

SVWN

H + CH4 H2 + CH3

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

Ene

rgy

(kca

l/mol

)Comparison of QM methods for reaction surface of

H + CH4 H2 + CH3

34

Solution: XYGJ-OS methodinclude excitations to virtual orbitals in order to describe

London Dispersion in DFTthis goes beyond using just density (occupied orbtials)

XYGJ- OS 22 ,1HF 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

Get {ex, eVWN, eLYP, ePT2} ={0.7731,0.2309, 0.2754, 0.4364}.

include only opposite spin and only local contributions

A fast doubly hybrid density functional method close to chemical accuracy: XYGJ-OS

Igor Ying Zhang, Xin Xu, Yousung Jung, William A. Goddard IIIPNAS 108 : 19896 (2011)

Density Functional Theory errors kcal/mol)

35

LDA 130.88 15.2Include density gradient (GGA)BLYP 10.16 7.9PW91 22.04 9.3PBE 20.71 9.1Hybrid: include HF exchangeB3LYP 6.08 4.5PBE0 5.64 3.9Include KE functional fit to barriers and complexesM06-L 5.20 4.1M06 3.37 2.2M06-2X 2.26 1.3

atomize barrierPopular with physicists

Popular with physicists

Popular with chemists

Include excitations to virtualsXYGJ-OS 1.81 1.0G3 (cc) 1.06 0.9

The level needed for reliable predictions

36

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)


Naphthalene 2.723 20.755 20.095

Anthracene 4.308 28.356 27.042


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)

37

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)


Naphthalene 2.723 20.755 20.095

Anthracene 4.308 28.356 27.042


Naphthalene 380.23 344.41 338.79

Anthracene 515.49 451.55 451.59




Universal Correction of Density Functional Theory to Include

London Dispersion (up to Lr, Element 103)

Hyungjun Kim, Jeong-Mo Choi, and William A. Goddard, III

J. Phys. Chem. Lett. 2012, 3, pp 360–363

Equation of States of Benzene Crystal

PBE-ulg predicts the correct cold-compression curve.


Fundamental problem in standard DFT methods

39

Use QM calculations on small systems ~100 atoms get accurate energies, geometries, stiffnessFit QM to force field to describe big systems (104 -107 atoms)Fit to obtain parameters for continuum systemsmacroscopic properties based on first principles (QM) Can predict novel materials where no empirical data available.

General Problem with DFT: bad description of vdw attraction

(London dispersion)

Invalidates multiscale paradigm



40

1


, 2, 0

0

2xc GLxc c

UU E

where

22

2ˆˆˆ1

4i xi j eeGL

cij ii j i

fE


ˆee ˆxf̂

,xcU a b Substitute into with 22 GLcb E LDA exact

xc xb E E or

Combine both approaches (2 choices for b) 21 2

GL DFT exactc xc xb b E b E E

R5 21 2 3 4


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

DFTxE

2PTcE DFT

cEcontains the double-excitation parts of 2PT

cE2GL

cE This is a fifth-rung functional (R5) using information from both occupied and virtual KS orbitals. In principle can now describe dispersion



Final form of XYG3 DFT

41

R5 21 2 3 4


we adopt the LYP correlation functional but constrain c4 = (1 – c3) to exclude compensation from the LDA correlation term. This constraint is not necessary, but it eliminates one fitting parameter.Determine the final three parameters {c1, c2, c3} empirically by fitting only to the thermochemical experimental data in the G3/99 set of 223 molecules:

Get {c1 = 0.8033, c2 = 0.2107, c3 = 0.3211} and c4 = (1 – c3) = 0.6789Use 6-311+G(3df,2p) basis set

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/molbut G2 and G3 involve far higher computational cost.


Thermochemical accuracy with size

42

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 formation

43

Functional MAD Max(+) Max(-)DFTXYG3 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 initioHFa 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)


Comparison of QM methods for reaction surface ofH + CH4 H2 + CH3

44

-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

Ener

gy (k

cal/m

ol)


HF

HF_PT2 SVWNB3LYP

BLYP

XYG3CCSD(T)

SVWN

H + CH4 H2 + CH3


Ene

rgy

(kca

l/mol

)


Reaction barrier heights

45

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 UM10DFTXYG3 1.02 0.75 1.38 1.42 0.98M06-2x a 1.20 1.13 1.61 1.22 0.92B2PLYP 1.94 1.81 3.06 2.16 0.73M06 a 2.13 2.00 3.38 1.78 1.69M06-La 3.88 4.16 5.93 3.58 1.86B3LYP 4.28 4.23 8.49 3.25 2.02BLYP a 8.23 7.52 14.66 8.40 3.51PBEa 8.71 9.32 14.93 6.97 3.35LDAb 14.88 17.72 23.38 8.50 5.90Ab initioHFb 11.28 13.66 16.87 6.67 3.82MP2 b 4.57 4.14 11.76 0.74 5.44QCISD(T) b 1.10 1.24 1.21 1.08 0.53

Zhao and Truhlarcompiled 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

Ene

rgy

(kca

l/mol

)

BLYPB3LYPXYG3CCSD(T)SVWNHF_PT2

(C)-12.00

-9.00

-6.00

-3.00

0.00

Ec_VWNEc_B3LYPEc_LYPEc_XYG3Ec_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_BEx_B3LYPEx_XYG3Ex_HFEx_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.

47

Functional Total HB6/04 CT7/04 DI6/04 WI7/05 PPS5/05DFTM06-2x b 0.30 0.45 0.36 0.25 0.17 0.26XYG3 a 0.32 0.38 0.64 0.19 0.12 0.25M06 b 0.43 0.26 1.11 0.26 0.20 0.21M06-L b 0.58 0.21 1.80 0.32 0.19 0.17B2PLYP 0.75 0.35 0.75 0.30 0.12 2.68B3LYP 0.97 0.60 0.71 0.78 0.31 2.95PBE c 1.17 0.45 2.95 0.46 0.13 1.86BLYP c 1.48 1.18 1.67 1.00 0.45 3.58LDA c 3.12 4.64 6.78 2.93 0.30 0.35Ab initioHF 2.08 2.25 3.61 2.17 0.29 2.11MP2c 0.64 0.99 0.47 0.29 0.08 1.69QCISD(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 stacking complexes.

WI and PPS dominated by London dispersion.

Note: no noncovalent complexes used in fitting the 3 parameters in XYG3)


Problem

48

1


, 2, 0

0

2xc GLxc c

UU E

where

22

2ˆˆˆ1

4i xi j eeGL

cij ii j i

fE


ˆee ˆxf̂



EGL2 involves double excitations to virtuals, scales as N5 with size

MP2 has same critical step

Yousung Jung (KAIST) has figured out how to get linear scaling for MP2

XYGJ-OS and XYGJ-OS


A solution: XYGJ-OS: include excitations to virtual orbitals in order to describe London Dispersion in DFT

Goes beyond using just density (occupied orbitals)Scales as (size)**3 just as B3LYP (CCSD scales as (size)**7

49

XYGJ- OS 22 ,1HF 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

Get {ex, eVWN, eLYP, ePT2} ={0.7731,0.2309, 0.2754, 0.4364}.

include only opposite spin & only local contributions N**3 scaling

A fast doubly hybrid density functional method close to chemical accuracy: XYGJ-OS

Igor Ying Zhang, Xin Xu, Yousung Jung, WAGPNAS (2011) in press

Xin Xu


0.0

40.0

80.0

120.0

160.0

200.0

0 20 40 60 80 100 120alkane chain length

CP

U (

hour

s)

XYG4-LOS

XYG4-OS

B3LYP

XYG3

Timings XYGJ-OS and XYGJ-LOS for long alkanes

XYG4-OS

XYG4-LOSB3LYP

XYG3

XYGJ-OS

XYGJ-LOS

XYGJ-LOS

XYGJ-OS


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

51

HOF IP EA PA BDE NHTBH HTBH NCIE All Methods

(223) (38) (25) (8) (92) (38) (38) (31) (493) DFT methods SPL (LDA) 5.484 0.255 0.311 0.276 0.754 0.542 0.775 0.140 2.771BLYP 0.412 0.200 0.105 0.080 0.292 0.376 0.337 0.063 0.322PBE 0.987 0.161 0.102 0.072 0.177 0.371 0.413 0.052 0.562TPSS 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 0.187PBE0 0.300 0.165 0.128 0.057 0.155 0.154 0.193 0.031 0.213M06-2X 0.127 0.130 0.103 0.092 0.069 0.056 0.055 0.013 0.096XYG3 0.078 0.057 0.080 0.070 0.068 0.056 0.033 0.014 0.065XYGJ-OS 0.072 0.055 0.084 0.067 0.033 0.049 0.038 0.015 0.056 MC3BB 0.165 0.120 0.175 0.046 0.111 0.062 0.036 0.023 0.123B2PLYP 0.201 0.109 0.090 0.067 0.124 0.090 0.078 0.023 0.143Wavefunction based methods HF 9.171 1.005 1.148 0.133 0.104 0.397 0.582 0.098 4.387MP2 0.474 0.163 0.166 0.084 0.363 0.249 0.166 0.028 0.338G2 0.082 0.042 0.057 0.058 0.078 0.042 0.054 0.025 0.068G3 0.046 0.055 0.049 0.046 0.047 0.042 0.054 0.025 0.046HOF = 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


Comparison of speeds

52

All Time (493) C100H202 C100H100

2.771 0.322 0.562 0.250 0.187 2.8 12.3 0.213 0.096 0.065 200.0 81.4 0.056 7.8 46.4 0.123 0.143

Methods

DFT methods SPL (LDA)

BLYP

PBE

TPSS B3LYP

PBE0

M06-2X

XYG3

XYGJ-OS

MC3BB

B2PLYP


Density Functional Theory errors kcal/mol)

53

LDA 130.88 15.2Include density gradient (GGA)BLYP 10.16 7.9PW91 22.04 9.3PBE 20.71 9.1Hybrid: include HF exchangeB3LYP 6.08 4.5PBE0 5.64 3.9Include KE functional fit to barriers and complexesM06-L 5.20 4.1M06 3.37 2.2M06-2X 2.26 1.3

atomize barrierPopular with physicists

Popular with physicists

Popular with chemists

Include excitations to virtualsXYGJ-OS 1.81 1.0G3 (cc) 1.06 0.9

Accuracy needed for predictions

More rigorous foundation

No exact exchange, fastwag uses for catalysis

Does well not well founded

G3 CC (4 semiempirical parameters) cannot be used for potential curves


0.01.02.03.04.05.06.07.08.09.0

10.0

B3LY

P

M06

M06-

2x

M06-

L

B2PL

YP

XYG3

XYG4

-OS G2 G3

MAD

(kca

l/mo

l)

G2-1

G2-2

G3-3

Heats of formation (kcal/mol)

Large molecules

small molecules


0.0

5.0

10.0

15.0

20.0

25.0

B3LY

P

BLYP PBE

LDA HF MP2

QCIS

D(T)

XYG3

XYG4

-OS

MAD

(kca

l/mo

l)

HAT12

NS16

UM10

HT38

Reaction barrier heights (kcal/mol)

Truhlar NHTBH38/04 set and HTBH38/04 set


0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

B3LY

P

BLYP PBE

LDA HF MP2

QCIS

D(T)

XYG3

XYG4

-OS

MAD

(kca

l/mo

l)

HB6

CT7

DI6

WI7

PPS5

Nonbonded interaction (kcal/mol)

Truhlar NCIE31/05 set


Comparison of QM methods for reaction surface of H + CH4 H2 + CH3

57-5.00

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

Ener

gy (k

cal/m

ol)


HF

HF_PT2 SVWNB3LYP

BLYP

XYG3CCSD(T)

SVWN

H + CH4 H2 + CH3


Ene

rgy

(kca

l/mol

)


DFT-ℓg for accurate Dispersive Interactions for Full Periodic Table

58

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

2Center for Materials Simulations and Design, KAIST


Current challenge in DFT calculation for energetic Current challenge in DFT calculation for energetic materialsmaterials

59

• Current implementations of DFT describe well strongly bound geometries and energies, 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

• DFT‐low gradient (DFT‐lg) model accurate description of the long‐range1/R6 attraction of the London dispersion but at same cost as standard DFT

Nlg,

lg 6 6,

- ij

ij i j ij eij

CE

r dR

DFT D DFT dispE E E

C6 single parameter from QM-CCd =1Reik = Rei + Rek (UFF vdW radii)


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 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 GpaZero 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


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

ding

ene

rgy

(kca

l/mol

)

c lattice constant (A)


Universal PBE-ℓg Method

63

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)


DFT D DFT dispE E E

Reik = Rei + Rek (UFF vdW radii)

Problem cannot yet do XYGJ-OS for crystals

Solution: use XYGJ-OS or CCSD to get accurate London Dispersion on small vdW clusters. Use to modify PBE for doing crystals by adding low gradient correction (PBE-lg) (also B3LYP-lg) for accurate description of the long-range 1/R6 attraction of the London dispersion


Universal low gradient (ulg) method for DFT-ulg

Universal force field (UFF): Rappé, Goddard JACS 114, 10024 (1992)Generic approach to force fields for whole periodic table (to Z=103 Lr)For each atom: 6 rule based parameters 618 to describe all molecules for all atoms up to Z=103UFF has two vdw parameters: D0 and R0 per atom based on•atomic polarizability from HF QM•ionization potential from experiment•atom size from experiment

Problem with DFT-lg: need a C6 parameter for every pair of atoms. Can get from XYGJ-OS or CCSD calculation on small <100 atom complexes, but for atoms up to Lr (Z=103) would need 5356 parameters, far too tedious

ulg strategy: base C6 term in DFT-ulg on the C6 from UFF

wag962. Universal Correction of Density Functional Theory to Include London Dispersion (up to Lr, Element 103); HJ Kim, JM Choi, wag; J. Phys. Chem. Lett. 2012, 3, 360−363


Universal low gradient (ulg) method

1. Match long R

2. Match at mid-range regime (r = 1.1R0):

3. Then introduce a single general scaling parameter for whole periodic table (slg),

ulg method: use van der Waal’s parameters from Universal Force-Field

With 1 parameter, DFT-ulg defined for Z=1 to 103

UFF vdw terms (up to Lr, Z=103) DFT-ulg


Determine the single parameter in DFT-ulg from Benzene dimer interactions

J-M Choi, HJ Kim, WAG

DFT-ulg fit a single parameter slg to benzene dimer CCSD(T)Get Slg = 0.7012


Parameter OptimizationImplemented in VASP 5.2.11

0.7012

0.6966


70

Validation: C6 parameters for lg fit to PBE for benzene dimer does excellent job on crystals


Naphthalene 2.723 20.755 20.095

Anthracene 4.308 28.356 27.042


Naphthalene 380.23 344.41 338.79

Anthracene 515.49 451.55 451.59





Crystals: Polyaromatic Hydrocarbons

V0 (Å3) Sublimation E (kcal/mol)

Compres.B0 (GPa)

PBE 511.8 1.05 1.3PBE-ulg 452.1 12.81 8.8

PBE-Grimme 420.3 13.33 10Exp. 461.8 11.3 ~8

Heat Vapor. PBE PBE-ulg Exp.Naphthalene 0.89 18.93 18.4‐23.5Anthracene 1.75 25.80 24.6‐30.0

Phenantracene 1.52 24.39 23.6‐26.5

Benzene crystal:

Volume PBE PBE-ulg Exp.Naphthalene 380.2 344.4 342.3Anthracene 515.5 451.6 455.2

Phenantracene 524.5 461.7 459.5


Equation of States of Benzene Crystal

PBE-ulg predicts the correct cold-compression curve.


Hobza S22 database

Twenty-two prototypical small molecular complexes for non-covalent interactions in biological molecules (h-bonded, dispersion dominated, and mixed)7 hydrogen bonded Mean average error (MAE)PBE-ulg: 0.53 kcal/molPBE-Grimme: 1.01 kcal/molvdw-DF: 0.59 kcal/mol (lundqvist, PRL 2004)

8 dispersion dominated MAEPBE-ulg: 1.26 kcal/molPBE-Grimme: 0.58 kcal/molvdw-DF: 1.86 kcal/mol

Overall:PBE-ulg: 0.70 kcal/molPBE-Grimme: 0.65 kcal/molvdw-DF: 1.20 kcal/molXYGJ-OS: 0.46 kcal/mol

Mean average error (MAE)PBE-ulg: 0.22 kcal/molPBE-Grimme: 0.38 kcal/molvdw-DF: 1.06 kcal/mol


Big Challenge for DFT

74

Proper description of spin statesOrganometallic reaction barriers depend strongly on spinAntiferromagnetsCuprate superconductorsGround states of Mn, Fe, Co, Ni metals

Current optimization of DFT methods focus mainly on 1st

and 2nd row compounds (H-Ar) but applications involve transition metals, lanthanides, actinides where local d and f orbitals can lead to magnetically complex systems

Example of the challenge:Group 10: s2d8 (3F) vs. s1d9 (3D) vs. s0d10 (1S)


Ground state configurations for group 10

75

Ni Pd Pt

wag206-Theoretical Studies of Oxidative Addition and Reductive Elimination. II. Reductive Coupling of H-H, H-C, and C-C Bonds from Pd and Pt Complexes J. J. Low and W. A. Goddard III; Organometallics 5, 609 (1986)

Exper GVB-CI HF Exper GVB-CI HF Exper GVB-CI HF


Ab initio methods Ni atom (all electron)

76

method s1d9 (3D) s0d10 (1s) Ni atomexper -0.69 39.43HF(wag 1986) 15.30 114.80HF(G3 basis, Yu 2012) 1.72 55.17HF (numerical nonrelativistic) 29.29 126.14 Cowan-GriffinHF (numerical relativistic) 37.59 139.29 Cowan-GriffinGVB-CI (wag 1986) -14.20 26.20MP2(G3 basis, Yu 2012) -30.92 -44.60 using s2d8 stateCCSD(G3 basis, Yu 2012) not conv 47.07 using s2d8 state

Basis set issues

Basis set issues: G3 basis: contraction of 3s and 3p core functions (overlaps 3d)

Reference state issues for MP2 and CCSD: used s2d8


Compare DFT methods Ni atom (all electron)

77

method s1d9 (3D) s0d10 (1s)exper -0.69 39.43HF(G3 basis, Yu 2012) 1.72 55.17GVB-CI (wag 1986) -14.20 26.20PBE(G3 basis, Yu 2012) -12.30 29.37PBE0(G3 basis, Yu 2012) -9.18 85.06B3LYP(G3 basis, Yu 2012) -9.11 22.65M06-L(G3 basis, Yu 2012) 36.92 -51.31M06(G3 basis, Yu 2012) -10.33 19.43M06-HF(G3 basis, Yu 2012) -14.01 49.11M06-2X(G3 basis, Yu 2012) -3.59 48.36XYGJ-OS (G3 basis, Yu 2012) 0.03 -2.12

Highlight s1d9 < 3 kcal/mols0d10 <10 kcal/mol

Need to use multiple spin states in DFT optimization


Pt atom using LANL Core Effective Potential

78

method s0d10 (1s) s2d8 (3F) Pt atomexper 11.07 14.76HF(wag 1986) 31.40 8.40HF(Yu 2012) 25.41 5.95HF (numerical nonrelativistic) -32.52 75.64 Cowan-GriffinHF (numerical relativistic) 20.75 9.22 Cowan-GriffinGVB-CI (wag 1986) 12.20 14.20PBE(Yu 2012) 14.39 -0.05PBE0(Yu 2012) 15.03 9.08B3LYP(Yu 2012) 14.67 6.84M06-L(Yu 2012) 14.01 0.86M06(Yu 2012) 0.40 19.17M06-HF(Yu 2012) 24.95 21.07M06-2X(Yu 2012) 11.72 15.24XYGJ-OS na na

Yu basis: LACV3P**++f,


Ground state s1d9 (3D)


method s1d9 (3D) s2d8 (3F) Pd atomexper 21.91 77.94HF(wag 1986) -12.70 41.80HF(Yu 2012) 1.72 55.17HF (numerical nonrelativistic) -17.29 86.71 Cowan-GriffinHF (numerical relativistic) 2.30 50.50 Cowan-GriffinGVB-CI (wag 1986) 19.60 82.20MP2 13.41 5.43CCSD 16.83 0.92PBE( Yu 2012) 9.43 67.17PBE0(Yu 2012) 19.99 88.40B3LYP(Yu 2012) 19.98 84.79M06-L(Yu 2012) 30.66 97.18M06( Yu 2012) 38.63 114.45M06-HF(Yu 2012) 10.09 89.50M06-2X(Yu 2012) 26.07 97.61XYGJ-OS na naYu basis:LACVP in Qchem

Pd atom using LANL

Core Effective Potential



States for Ni d8 atom (real orbitals)

80

Hole type expHF PBE PBE0 B3LYP M06-L M06 M06-HF M06-2X

z2, x2-y2 σδ 0 0 0 0 0 0 0 0 0

xy, z2 σδ 0 0 0.08 0.06 0.05 -0.42 -0.27 -0.53 -0.83xz, yz ππ 9.004 11.3 10.2 10.4 10.25 16.22 10.4 -7.93 4.13

xz, x2-y2 πδ 9.004 11.6 9.39 9.79 9.8 13.17 7.5 0.52 6.05yz, x2-y2 πδ 9.004 11.6 9.39 9.79 9.8 13.17 7.5 0.52 6.05xz, xy πδ 9.004 11.6 10.4 10.5 10.42 16.38 10.5 -7.7 4.29yz, xy πδ 9.004 11.6 10.4 10.5 10.42 16.38 10.5 -7.7 4.29xz, z2 σπ 27.01 34.3 26.6 28.2 28.33 35.85 18.1 10.28 19.9yz, z2 σπ 27.01 34.3 26.6 28.2 28.33 35.85 18.1 10.28 19.9

xy, x2-y2 δδ 36.02 45.4 36.7 38.3 38.44 48.36 23.8 14.84 26.92ok exc exc exc ok

Bottom line: for transition metal systems, current levels of DFTbased on foundation of sand: must address in next generation DFT

Lecture 1Ch121a-Goddard-L01 © copyright 2012 William A. Goddard III, all rights reserved\ 81Lecture 1Lecture 2

Method:· Semi-Empirical, used for very big systems, or for rough approximations of geometry (extended Huckel theory, CNDO/INDO, AM1, MNDO)

· HF (Hartree Fock). Simplest Ab Initio method. Very cheap, fairly inaccurate· MP2 (Moeller-Plasset 2). Advanced version of HF. Usually not as cheap or as accurate as B3LYP, but can function as a complement.· CASSCF (Complete Active Space, Self Consisting Field). Advanced version of HF, incorporating excited states. Mainly used for jobs where photochemistry is important. Medium cost, Medium Accuracy. Quite complicated to run…· QCISD (Quadratic Configuration Interaction Singles Doubles). Very advanced version of HF. Very Expensive, Very accurate. Can only be used on systems smaller than 10 heavy atoms. · CCSD (Coupled Cluster Singles Doubles). Very much like QCISD. Density Functional Theory

LDA (local density approximation)PW91, PBE

· B3LYP (density functional theory). Cheap, Accurate.

Generally, B3LYP is the method of choice. If the system allows it, QCISD or CCSD can be used. HF and/or MP2 can be used to verify the B3LYP results.


Basis Set: What mathematical expressions are used to describe orbitals. In general, the more advanced the mathematical expression, the more accurate the wavefunction, but also more expensive calculation.

· STO-3G - The ‘minimal basis set’. Not particularly accurate, but cheap and robust. · 3-21G - Smallest practical Basis Set. · 6-31G - More advanced, i.e. more functions for both core and valence. · 6-31G** - As above, but with ‘polarized functions’ added. Essentially makes the orbitals look more like ‘real’ ones. This is the standard basis set used, as it gives fairly good results with low cost. · 6-31++G - As above, but with ‘diffuse functions’ added. Makes the orbitals stretch out in space. Important to add if there is hydrogen bonding, pi-pi interactions, anions etc present. · 6-311++G** - As above, with even more functions added on… The more stuff, the more accurate… But also more expensive. Seldom used, as the increase in accuracy usually is very small, while the cost increases drastically. · Frozen Core: Basis sets used for higher row elements, where all the core electrons are treated as one big frozen chunk. Only the valence electrons are treated explicitly


• Software packages– Jaguar– GAMESS– TurboMol– Gaussian– Spartan/Titan– HyperChem– ADF

Lecture 13b, February 18, 2015 Energetics, Hartree...

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