ch121a atomic level simulations of materials and molecules

1Lecture 1Ch121a-Goddard-L02 © copyright 2011 William A. Goddard III, all rights reserved\

Ch121a Atomic Level Simulations of Materials and Molecules

William A. Goddard III, [email protected] and Mary Ferkel Professor of Chemistry,

Materials Science, and Applied Physics, California Institute of Technology

BI 115Hours: 2:30-3:30 Monday and Wednesday

Lecture or Lab: Friday 2-3pm (+3-4pm)

Teaching Assistants Wei-Guang Liu, Fan Lu, Jose Mendozq

Lecture 2, March 30, 2011QM-2: DFT


Last Time


Consider the product wavefunction

Ψ(1,2) = ψa(1) ψb(2)

And the Hamiltonian for H2 molecule

H(1,2) = h(1) + h(2) +1/r12 + 1/R

In the details slides next, we derive

E = < Ψ(1,2)| H(1,2)|Ψ(1,2)>/ <Ψ(1,2)|Ψ(1,2)>

E = haa + hbb + Jab + 1/R

where haa =<a|h|a>, hbb =<b|h|b> are just the 1 electron energies

Jab ≡ <ψa(1)ψb(2) |1/r12 |ψa(1)ψb(2)>=ʃ d3r1[ψa(1)]2 ʃd3r2[ψb(2)]2/r12 =

= ʃ [ψa(1)]2 Jb (1) = <ψa(1)| Jb (1)|ψa(1)>

Where Jb (1) = ʃ [ψb(2)]2/r12 is the Coulomb potential at 1 due to the density distribution [ψb(2)]2

Energy for 2-electron product wavefunction

Jab is the Coulomb repulsion between densities a=[ψa(1)]2 and b


The general case of 2M electronsFor the general case the HF closed shell wavefunction

Ψ(1,2….2M) = A[(φ1)(φ1)(φ2)(φ2)… )(φM)(φM)] leads to

HHFφk = k φk where we solve for k=1,M occupied orbitals

HHF = h + Σj=1,M [2Jj-Kj]

This is the same as the Hamiltonian for a one electron system moving in the average electrostatic and exchange potential, 2Jj-Kj due to the other N-1 = 2M-1 electrons

Problem: sum over 2Jj leads to 2M Coulomb terms, not 2M-1

This is because we added the self Coulomb and exchange terms

But (2Jk-Kk) φk = (Jk) φk so that these self terms cancel.

The HF equations describe each electron moving in the average potential due to all the other electrons.


The Matrix HF equations

The HF equations are actually quite complicated because Kj is an integral operator, Kj φk(1) = φj(1) ʃ d3r2 [φj(2) φk(2)/r12]The practical solution involves expanding the orbitals in terms of a basis set consisting of atomic-like orbitals,

φk(1) = Σ C Xwhere the basis functions, {XMBF} are chosen as atomic like functions on the various centers

As a result the HF equations HHFφk = k φk

Reduce to a set of Matrix equations

ΣjmHjmCmk = ΣjmSjmCmkk

This is still complicated since the Hjm operator includes exchange terms


New stuff


HF wavefunctions

Good distances, geometries, vibrational levels

But

breaking bonds is described extremely poorly

energies of virtual orbitals not good description of excitation energies

cost scales as 4th power of the size of the system.


Electron correlation

In fact when the electrons are close (rij small), the electrons correlate their motions to avoid a large electrostatic repulsion, 1/rij

Thus the error in the HF equation is called electron correlation

For He atom

E = - 2.8477 h0 assuming a hydrogenic orbital exp(-r)E = -2.86xx h0 exact HF (TA look up the energy)

E = -2.9037 h0 exact

Thus the elecgtron correlation energy for He atom is 0.04xx h0 = 1.x eV = 24.x kcal/mol.

Thus HF accounts for 98.6% of the total energy


Configuration interaction

Consider a set of N-electron wavefunctions: {i; i=1,2, ..M} where < i|j> = ij {=1 if i=j and 0 if i ≠j)

Write approx = i=1 to M) Ci i

Then E = < approx|H|approx>/< approx|approx>

E= < i Ci i |H| i Cj j >/ < i Ci i | i Cj j >How choose optimum Ci?Require E=0 for all Ci get

j <i |H| Cj j > - Ei< i | Cj j > = 0 ,which we write asHCi = SCiEi in matrix notation, ie ΣjkHjkCki = ΣjkSjkCkiEi

where Hjk = <j|H|k > and Sjk = < j|k > and Ci is a column vector for the ith eigenstate


Configuration interaction upper bound theorm

Consider the M solutions of the CI equationsHCi = SCiEi ordered as i=1 lowest to i=M highest

Then the exact ground state energy of the systemSatisfies Eexact ≤ E1

Also the exact first excited state of the system satisfiesE1st excited ≤ E2 etcThis is called the Hylleraas-Unheim-McDonald Theorem


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

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-)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 (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


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)] = 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 + Vext

Now 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

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 term

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

wag


Solving the Kohn-Sham equationsRequiring 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)

We approximate Exc[ras

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

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

3

1

xLDAx rρAρε

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

cxxc EEE

drρ(r),...ρ(r)ρ(r),εE xx

sFερρ,ε LDAx

GGAx

3

4

3

12 ρπ24

ρs

Becke 88

X3LYP

PBEPW91

s

F(s) GGA functionals

2

11

232

1188B

sasinhsa1

sasasinhsa1sF

d

521

1

2s100432

1191PW

sasasinhsa1

seaasasinhsa1sF

2


adiabatic connection formalism

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

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 ;exact LDA exactx xc xa E b E E ;exact LDA exact

x xc xa E b E E

Get half-and-half functional 1 1

2 2exact LDA LDA

xc x x cE E E E

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


Becke 3 parameter functional

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 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: 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-GGAFPBE + VSXC


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

4

i xi j eeGLc

ij 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 ˆx

f̂

,xcU a b Substitute into with22 GL

cb E ;exact LDA exactx xc xa E b 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

DFTxE2PT

cEDFTcE

contains the double-excitation parts of 2PTcE

22

2ˆˆˆ1

4

i xi j eeGLc

ij ii j i

fE

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


Final form of XYG3 DFT

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) = 0.6789

Use 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

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

Functional 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(1 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

Ene

rgy

(kca

l/mol

)

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)

Ene

rgy

(kca

l/mol

)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

Ene

rgy

(kca

l/mol

)

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 stacking complexes.

WI and PPS dominated by London dispersion.

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


Problem

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

4

i xi j eeGLc

ij 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 ˆx

f̂

Sum over virtual orbtials

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

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


XYGJ-OS method most accurate DFT(including London Dispersion) at modest cost

R5 21 2 3 4


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

DFTxE2PT

cEDFTcE

contains the double-excitation parts of 2PTcE

22

2ˆˆˆ1

4

i xi j eeGLc

ij ii j i

fE

c4 = (1 – c3) 3 parameters

Ying Zhang , Xin Xu, Goddard; P. Natl. Acad. Sci. 106 (13) 4963-4968 (2009)

Use Görling-Levy coupling-constant perturbation expansion

XYG3 most accurate DFT, but costs too high for large systems

Yousung Jung figured out how to dramatically reduce the costs while retaining the accuracy

XYGJ-OS

Yousung Jung


0.0

40.0

80.0

120.0

160.0

200.0

0 20 40 60 80 100 120

alkane chain length

CP

U (

hours

)

XYG4-LOS

XYG4-OS

B3LYP

XYG3

Timings XYGJ-OS and XYGJ-LOS for long alkanes

0.0

40.0

80.0

120.0

160.0

200.0

0 20 40 60 80 100 120

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

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

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

XYG4-LOS

XYG4-OS

B3LYP

XYG3

XYGJ-OS

XYGJ-LOS

XYGJ-LOS

XYGJ-OS


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


Comparison of speeds

NCIE All Time (31) (493) C100H202 C100H100

0.140 2.771 0.063 0.322 0.052 0.562 0.049 0.250 0.041 0.187 2.8 12.3 0.031 0.213 0.013 0.096 0.014 0.065 200.0 81.4 0.015 0.056 7.8 46.4 0.023 0.123 0.023 0.143

0.098 4.387 0.028 0.338 0.025 0.068 0.025 0.046

HOF

Methods

(223) DFT methods SPL (LDA) 5.484 BLYP 0.412 PBE 0.987 TPSS 0.276 B3LYP 0.206 PBE0 0.300 M06-2X 0.127 XYG3 0.078 XYGJ-OS 0.072 MC3BB 0.165 B2PLYP 0.201 Wavefunction based methods HF 9.171 MP2 0.474 G2 0.082 G3 0.046


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)

G2-1G2-2G3-3

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)

HAT12NS16UM10HT38

Reaction barrier heights (kcal/mol)

Truhlar NHTBH38/04 set and HTBH38/04 set


0. 01. 0

2. 03. 0

4. 05. 0

6. 07. 0

8. 0

B3LY

P

BLYP PBE

LDA HF MP2

QCIS

D(T)

XYG3

XYG4

-OS

MAD

(kca

l/mo

l)

HB6CT7DI 6WI 7PPS5

Nonbonded interaction (kcal/mol)

Truhlar NCIE31/05 set


-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

Ene

rgy

(kca

l/mol

)

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)

Ene

rgy

(kca

l/mol

)Comparison of QM methods for reaction

surface of H + CH4 H2 + CH3


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


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

• 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 dimerT-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

din

g e

ne

rgy

(kca

l/mol

)

c lattice constant (A)


Universal PBE-ℓg MethodUFF, 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-

shape

d


Benzene Dimer

Sandwi

ch


Benzene Dimer

Parallel-

displac

ed


Parameter OptimizationImplemented in VASP 5.2.11

0.701

2

0.696

6


Hydrocarbon Crystals

• Sublimation energy (kcal/mol/molecule)

• Cell volume (angstrom3/cell)

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


Simple Molecular Crystals

• Sublimation energy (kcal/mol/molecule)

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

Lecture 1Lecture 2 60

Chem 121 - Applied Quantum ChemistryMethod:· 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.


Chem 121 - Applied Quantum Chemistry

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



Running an actual calculation– Determine the starting geometry of the

molecule you wish to study

– Determine what you’d like to find out

– Determine what methods are suitable and/or affordable for the above calculation

– Prepare input file

– Run job

– Evaluate result



Example: Good ol’ water

Starting geometry: water is bent, (~104º), a normal O-H bond is ~0.96 Å. For illustration, however, we’ll start with a pretty bad guess.

Simple Z-matrix:O1 H2 O1 1.00H3 O1 1.00 H2 110.00

1.00 Å 1.00 Å

110º



What do we wish to find out?

How about the IR spectra?

What is a suitable method for this calculation? Well, any, really, since it is so small. But 99% of the time the answer to this question is “B3LYP/6-31G**” – a variant of density functional theory that is the main workhorse of applied quantum chemistry, with a standard basis set. Let’s go with that.



Actual jaguar input:

&genigeopt=1ifreq=1dftname=b3lyp basis=6-31g**&&zmat

O1 H2 O1 0.95H3 O1 0.95 H2 120.00&



Running time!

Jaguar calculates the wave function for the atomic coordinates we provided

From the wave function it determines the energy and the forces on the current geometry

Based on this, it determines in what direction it should move the atoms to reach a better geometry, i.e. a geometry with a lower energy



1.00 Å 1.00 Å

110º

0.96 Å 0.96 Å

104º

Our horrible guess Target geometry

Think elastic springs: The bonds are too long, so there will be a force towards shorter bonds

Forces



Optimization – minimization of the forces. When all forces are zero the energy will not change and we have the resting geometry

O1 H2 O1 0.9500000000 H3 O1 0.9500000000 H2 120.0000000000 SCF energy: -76.41367730925-- O1 H2 O1 0.9566666804 H3 O1 0.9566666820 H2 106.8986301461 SCF energy: -76.41937497895-- O1 H2 O1 0.9653619358 H3 O1 0.9653619375 H2 103.0739287925 SCF energy: -76.41969584939 -- O1 H2 O1 0.9653155294 H3 O1 0.9653155310 H2 103.6688074046 SCF energy: -76.41970381840--



0.9653155294 Å

103.6688074046º

Computer accuracy

0.96 Å 0.96 Å

103.7º

“actual” accuracy

Accuracy

0.9653155294 Å

Accuracy is a relative concept



frequencies 1666.01 3801.19 3912.97

No negative frequencies!

(Compare IR spectra for gas-phase water)



Vibrational levels

“zero” level

Zero Point Energy (ZPE)

Zero Point Energies

Optimized energy is at the zero level, but in reality the molecule has a higher energy due to populated vibrational levels.

At 0 K, all molecules populate the lowest vibrational level, and so the difference between the “zero” level and the first vibrational level is the Zero Point Energy (ZPE)

From our calculation:The zero point energy (ZPE): 13.410 kcal/mol



Thermodynamic data at higher temperatures

T = 298.15 K

U Cv S H G --------- --------- --------- --------- --------- trans. 0.889 2.981 34.609 1.481 -8.837 rot. 0.889 2.981 10.503 0.889 -2.243 vib. 0.002 0.041 0.006 0.002 0.000 elec. 0.000 0.000 0.000 0.000 0.000 total 1.779 6.003 45.117 2.371 -11.080

Most thermodynamic data can be computed with very good accuracy in the gas phase. Temperature dependant



Transition states

ReactantProduct

Transition State (TS)

CH3Br + Cl- CH3Cl + Br- TS

Reaction coordinate

Line represents the reacting coordinate, in this case the forming C-Cl and breaking C-Br bonds

Stationary points: points on the surface where the derivative of the energy = 0



CH3Br + Cl- CH3Cl + Br- TS

Reaction coordinate

Not a hill, but a mountain pass

Transition state = stationary point where all forces except one is at a minimum.

The exception is at its maximum



ReactantProduct

TS

Derivative of the energy = 0

Second derivative: For a minimum > 0For a maximum < 0

So a TS should have a negative second derivative of the energy

Second derivative of the energy = force



A transition state should have one negative (imaginary) frequency!!!

(and ONLY one)



ReactantProduct

TS

Optimizing transition states:

Simultaneously optimize all modes (forces) towards their minimum, except the reacting mode

But for the computer to know which mode is the reacting mode, you must have one imaginary frequency in your starting point

Inflection points

Region with imaginary frequency

Must start with a good guess!!!



Example:CH3Br + Cl- CH3Cl + Br-

What do we know about this reaction? It’s an SN2 reaction, so the Cl- must come in from the backside of the CH3Br. The C-Cl forms at the same time as the C-Br forms. The transition state should be five coordinate



2.0 2.2Cl Br

H H

H

C

Initial guess: C-Cl = 2.0 Å, C-Br = 2.2 Å

Single point frequency on the above geometry: frequencies 98.64 99.58 109.11 310.66 1339.10 1348.64

frequencies 1349.46 1428.45 1428.73 2838.52 3017.70 3017.93

No negative frequencies! Bad initial guess



Refinement :Initial guess most likely wrong because of erronous C-Br and C-Cl bond lengths

Let the computer optimize the five-coordinate structure

Frozen optimizations: Just like a normal optimization, but with one or more geometry parameters frozen

In this case, we optimize the structure with all the H-C-Cl angles frozen at 90º



Result:

2.32 2.62Cl Br

C-Cl and C-Br bonds quite a bit longer in the new structure

Frequency calculation: frequencies -286.26 168.54 173.32 173.43 874.16 874.76 frequencies 976.23 1413.99 1414.65 3220.91 3420.84 3421.80

One negative frequency! Good initial guess



Time for the actual optimization:

Jaguar follows the negative frequency towards the maximum

Geometry optimization 1: SCF Energy = -513.35042353681Geometry optimization 2: SCF Energy = -513.34995058422Geometry optimization 3: SCF Energy = -513.35001640704Geometry optimization 4: SCF Energy = -513.34970196448Geometry optimization 5: SCF Energy = -513.34968682825Geometry optimization 6: SCF Energy = -513.34968118535

Final energy higher than starting energy (although only 0.5 kcal/mol)

Frequency calculation frequencies -268.67 162.64 174.22 174.31 848.15 848.24 frequencies 960.97 1415.75 1415.96 3220.77 3420.80 3421.15

One negative frequency! We found a true transition state



2.46 2.51Cl Br

Final geometry: C-Cl = 2.46 ÅC-Br = 2.51 Å Cl-C-H = 88.7ºBr-C-H = 91.3º

Structure not quite symmetric, the hydrogens are bending a little bit away from the Br.



Solvation calculations

Explicit solvents: Calculations where solvent molecules are added as part of the calculation

Implicit solvents: Calculations where solvation effects are added as electrostatic interactions between the molecule and a virtual continuum of “solvent”.



Reaction energetics and barrier heights

Collect the absolute energies of the reactants, products and transition states

CH3Br + Cl- TS CH3Cl + Br- -53.078938 + -460.248741 -513.349681 -500.108371 + -13.237607

Sum each term

CH3Br + Cl- TS CH3Cl + Br- -513.327679 -513.349681 -513.345978

Define reactants as “0”, and deduct the reactant energy from all terms

CH3Br + Cl- TS CH3Cl + Br- 0 -.022002 -.018299

Convert to kcal/mol (1 hartree = 627.51 kcal/mol)




Convert to kcal/mol (1 hartree = 627.51 kcal/mol)

CH3Br + Cl- TS CH3Cl + Br- 0 -13.8 -11.5

But this doesn’t make sense




CH3Br + Cl- TS CH3Cl + Br- 0 -13.8 -11.5

Solvation not included!

Include solvation corrections!

CH3Br + Cl- TS CH3Cl + Br- 0 9.2 -6.4

ch121a atomic level simulations of materials and molecules

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