ch121a atomic level simulations of materials and molecules

Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 1

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

316 Beckman Institute

Room BI 115Lecture: Monday, Wednesday Friday 2-3pm

Lab Session:

Lecture 2, April 2, 2014QM-2: DFT

TA’s Caitlin Scott and Andrea KirkpatrickSpecial Advice and Help: Julius Su/SKIES


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 Quantum Mechanics, Hydrogen Atom, etcPlease review again to make sure that you are comfortable with the concepts, which you should have seen before


The Hartree Fock Equations

General concept: there are an infinite number of possible orbitals for the electrons. For a system with 2M electrons we will put the electrons into the M lowest orbitals, with two electrons in each orbital (one up or a spin, the other down or b spin)

M occ orb2M elect


The wavefunction is written as Ψ(1,2,3,4, ..N-1,N) = A[(φaa)(φab)(φba)(φbb)---------(φza)(φzb)]

Where the A is the antisymmetrizer or determinant operator where the 1st column is φaa(1), φaa(2), φaa(3), etcThe 2nd column is φab(1), φab(2), φab(3), etcThus there are N! termsThis guarantees that the wavefunction changes sign if any 2 electrons are interchanged (Pauli Principle)Properties of determinant: if two columns are identical get zero. Thus can never have 2 electrons in same orbital with same spin Can take every column to be orthogonal; thus <φa|φb>=0Also can recombine any two orbitals and the wavefunction does a = (cos) φa + (sin) φb not changeb = (-sin) φa + (cos) φb

The Hartree Fock EquationsClosed shell M occ orb

N=2M elect

1 2 N

ab

z


The energy (closed shell)

Hel (1,2,---N) = Si h(i) + Si<j 1/rij

where h(i) = - ½ 2 + Si ZA/Rai is the interaction of all nuclei A with electron 1 plus the kinetic energy, a total of N termsand the other term is the Coulomb interaction between each pair of electrons, a total of N(N-1)/2 termsIf we ignore the antisymmetrizer, so that the wavefunction is a Hartree productΨ(1,2,3,4, ..N-1,N) = [(φaa)(φab)(φba)(φbb)---------(φza)(φzb)]Then the energy is

Eproduct = Sa 2<a|h|a> + Sa Jaa + Sa<b 4Jab

Thus N=2M 1e terms and M+2M(M-1)=M(2M-2+1)= N(N-1)/2 2e terms

The electronic Hamiltonian is

M=N/2 terms M=N/2 terms M(M-1)/2 terms


The Coulomb energy

Jab= <Φa(1)Φb(2) |1/r12 |Φa(1)Φb(2)>

= ʃ1,2 [a(1)]2 [b(2)]2/r12

=ʃ1 [a(1)]2 Jb(1)

where Jb(1) = ʃ [b(2)]2/r12 is the coulomb potential evaluated at point 1 due to the charge density [b(2)]2 integrated over all space

Thus Jab is the total Coulomb interaction between the electron density ra(1)=|a(1)|2 and rb(2)=|b(2)|2

Since the integrand ra(1) rb(2)/r12 is positive for all positions of 1 and 2, the integral is positive, Jab > 0


Consider the effect of the Antisymmetrizer

Two electrons with same spinΨ(1,2)) = A[(φaa)(φba)]= (φaa)(φba) - (φba)(φaa)

1 2 1 2New term in energy is the exchange term-<(φaa)(φba)|Hel(1,2)|(φba)(φaa)> is a sum of 3 terms<(φaa)(φba)|h(1)|(φba)(φ1a)> = <φaa|h(1)|φba><φba|φaa>

<(φaa)(φba)|h(2)|(φba)(φ1a)>=<φaa|φba><φba|h(2)|φaa>

<(φaa)(φba)|1/r12|(φba)(φ1a)>=Kab

Thus the only new term is -Kab note that it is negative because one side is exchanged but not the otherThus the total energy becomesE = <a|h|a> + <b|h|b> + Jab – Kab

0

0


The Exchange energy

Kab= <Φa(1)Φb(2) |1/r12 |Φb(1)Φa(2)>

= ʃ1 [a(1)b(1)] ʃ2 [b(2)a(2)]/r12

= ʃ1 [a(1) {P12 b(2)] ʃ2[b(2)]/r12 } a(1)] = ʃ1 [a(1) Kb(1) a(1)]

No simple classical interpretation, but we have written it in terms of an integral operator Kb(1) so that is looks similar to the Coulomb case


Relationship between Jab and Kab

The total electron-electron repulsion part of the energy for any wavefunction Ψ(1,2) = A[(φaa)(φba)] must be positive

Eee =∫ (d3r1)((d3r2)|Ψ(1,2)|2/r12 > 0

This follows since the integrand is positive for all positions of r1 and r2 then

Thus Jab – Kab > 0 and hence Jab > Kab > 0

Thus the exchange energy is positive but smaller than the Coulomb energy

Note that Kaa = <Φa(1)Φa(2) |1/r12 |Φa(1)Φa(2)> = Jaa


E = Sa 2<a|h|a> + Sa [2Jaa – Kaa] + Sa<b (4Jab – 2Kab)

E = Sa 2<a|h|a> + Sa,b (2Jab – Kab)There are M2 terms, so it appears that we have 2M2 = 2(N/2)(N/2) = N2/2 terms, but we should have N(N-1)/2 = N2 –N/2This is because we added N/2 fake terms, Jaa that must be cancelled by the N/2 fake Kaa terms.

Also note Sa,b 2Jab = (½)ʃ1,2 [r(1)] [r(2)]2/r12

where (1)=r Sa [Φa(1)]2 is the total electron density, the classical electrostatic energy for this charge density

The final energy for closed shell wavefunction

The total energy is

E = Sa 2<a|h|a> + Sa Jaa + Sa<b (4Jab – 2Kab)

One from aa and one from bb

[2Jaa – Kaa]


The Hartree Fock Equations

Variational principle: Require that each orbital be the best possible (leading to the lowest energy) leads to

HHF(1)φa(1)= ea φa(1)

where we solve for the occupied orbital, φa, to be occupied by both electron 1 and electron 2

Here HHF(1)= h(1) + 2[Ja(1) - Ka(1)]

This looks like the Hamiltonian for a one-electron system in which the Hamiltonian has the form it would have neglecting electron-electron repulsion plus the average potential due the electron in the other orbital

Thus the two-electron problem is factored into M=N/2 one-electron problems, which we can solve to get φa, φb, etc


Self consistency

However to solve for φa(1) we need to know 2[Ja(1) - Ka(1)]

which depends on all M orbitals

Thus the HHFφa= ea φa equation must be solved iteratively until it is self consistent

But after the equations are solved self consistently, we can consider each orbital as the optimum orbital moving in the average field of all the other electrons

In fact the motions between these electrons would tend to be correlated so that the electrons remain farther apart than in this average field

Thus the error in the HF energy is called the correlation energy


He atom one slater orbital

If one approximates each orbital as φ1s = N0 exp(-zr) a Slater orbital then it is only necessary to optimize the scale parameter z

In this case

He atom: EHe = 2(½ z2) – 2Z + z (5/8)z

Applying the variational principle, the optimum z must satisfy dE/dz = 0 leading to 2 z - 2Z + (5/8) = 0Thus z = (Z – 5/16) = 1.6875KE = 2(½ z2) = z2

PE = - 2Z + z (5/8)z = -2 z2 E= - z2 = -2.8477 h0

Ignoring e-e interactions the energy would have been E = -4 h0

The exact energy is E = -2.9037 h0 Thus this simple approximation of assuming that each electron is in a H1s orbital and optimizing the size accounts for 98.1% of the exact result.


The Koopmans orbital energy

The next question is the meaning of the one-electron energy, ea in the HF equationHHF(1)φa(1)= ea φa(1)

Multiplying each side by φa(1) and integrating leads toea <a|a> = <a|HHF|a> = <a|h|a> + 2Sb<a|Jb|a> - Sb<a|Kb|a>

= <a|h|a> + Jaa + Sb≠a<a|2Jb-Kb|a>

Thus in the approximation that the remaining electron does not change shape, ea corresponds to the energy to ionize an electron from the a orbital to obtain the N-1 electron system Sometimes this is referred to as the Koopmans theorem (pronounced with a long o). It is not really Koopmans theorem, which we will discuss later, but we will use the term anyway


The ionization potential

There are two errors in using the ea to approximate the IPIPKT ~ - ea First the remaining N-1 electrons should be allowed to relax to the optimum orbital of the positive ion, which would make the Koopmans IP too largeHowever the energy of the HF description is leads to a total energy less negative than the exact energy, Exact = EHF – Ecorr Where Ecorr is called the electron correlation energy (since HF does NOT allow correlation of the electron motions. Each electron sees the average potential of the other)which would make the Koopmans IP too smallThese effects tend to cancel so that the ea from the HF wavefunction leads to a reasonable estimate of IP

(N-1)e

exactHF from Ne

Ne

exactHFexact IP

Koopmans IP


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) = Σm Cm X , m where the basis functions, {X , =1, m mMBF} are chosen as atomic like functions on the various centers

As a result the HF equations HHFφk = lk φk

Reduce to a set of Matrix equations

ΣjmHjmCmk = ΣjmSjmCmklk

This is still complicated since the Hjm operator includes exchange terms

We still refer to this as solving the HF equations


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.


Minimal Basis set – STO-3G

For benzene the smallest possible basis set is to use a 1s-like single exponential function, exp(-zr) called a Slater function, centered on each the 6 H atoms and

C1s, C2s, C2pz, C2py, C2pz functions on each of the 6 C atoms

This leads to 42 basis functions to describe the 21 occupied MOs

and is refered to as a minimal basis set.

In practice the use of exponetial functions, such as exp(-zr), leads to huge computational costs for multicenter molecules and we replace these by an expansion in terms of Gaussian basis functions, such as exp(-ar2).

The most popular MBS is the STO-3G set of Pople in which 3 gaussian functions are combined to describe each Slater function


Double zeta + polarization Basis sets – 6-31G**

To allow the atomic orbitals to contract as atoms are brought together to form bonds, we introduce 2 basis functions of the same character as each of the atomic orbitals:Thus 2 each of 1s, 2s, 2px, 2py, and 2pz for CThis is referred to as double zeta. If properly chosen this leads to a good description of the contraction as bonds form.Often only a single function is used for the C1s, called split valenceIn addition it is necessary to provide one level higher angular momentum atomic orbitals to describe the polarization involved in bondingThus add a set of 2p basis functions to each H and a set of 3d functions to each C. The most popular such basis is referred to as 6-31G**


6-31G** and 6-311G**

6-31G** means that the 1s is described with 6 Gaussians, the two valence basis functions use 3 gaussians for the inner one and 1 Gaussian for the outer function

The first * use of a single d set on each heavy atom (C,O etc)

The second * use of a single set of p functions on each H

The 6-311G** is similar but allows 3 valence-like functions on each atom.

There are addition basis sets including diffuse functions (+) and additional polarization function (2d, f) (3d,2f,g), but these will not be relvent to EES810


Effective Core Potentials (ECP, psuedopotentials)

For very heavy atoms, say starting with Sc, it is computationally convenient and accurate to replace the inner core electrons with effective core potentials

For example one might describe: • Si with just the 4 valence orbitals, replacing the Ne core with

an ECP or • Ge with just 4 electrons, replacing the Ni core • Alternatively, Ge might be described with 14 electrons with the

ECP replacing the Ar core. This leads to increased accuracy because the

• For transition metal atoms, Fe might be described with 8 electrons replacing the Ar core with the ECP.

• But much more accurate is to use the small Ne core, explicitly treating the (3s)2(3p)6 along with the 3d and 4s electrons


Software packages

Jaguar: Good for organometallicsQChem: very fast for organicsGaussian: many analysis toolsGAMESSHyperChemADFSpartan/Titan


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

VF

V 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) d3rwhere 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)

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 [ (1) (2r r )/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

wag


Solving the Kohn-Sham equationsRequiring 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 [ (1) (2r r )/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 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 r = N/V

3

1

xLDAx 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


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 LDAx

GGAx

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

PBEPW91

s

F(s) GGA functionals

2

11

232

1188B

sasinhsa1

sasasinhsa1sF

d

521

1

2s100432

1191PW

sasasinhsa1

seaasasinhsa1sF

2

Here 312

2 π48a, 21βa6a, βA2

aa

x3/1

22

3 , 34 a81

10a ,

x3/1

642

5 A2

10aa

, and d = 4.

Becke9 = 0.0042 a4 and a5 zero

Here 312

2 π48a, 21βa6a, βA2

aa

x3/1

22

3 , 34 a81

10a ,

x3/1

642

5 A2

10aa

, and d = 4.

S is big where the density gradient is large


Adiabatic connection formalism

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.

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

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: 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-GGA06 = M HF + tPBE + VSXC

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

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 DG‡ = 9.3 kcal/mol.

NO O

P(t-Bu)2(t-Bu)2P IrIII

CH3

NO O

P(t-Bu)2(t-Bu)2P Ir

CH3

H

H

DG(173K)B3LYPM06M06L

0.00.00.0

10.85.8

11.4(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 DG‡ = 8.1 kcal/mol.

DG(173K)B3LYPM06

0.00.0

8.79.5(reductive elimination)

4.65.3(s-bound complex)

6.45.2(site-exchange)

Os

P

CH2

PMe

Me

Me

Me

CH3

H

+1

Os

P

CH2

PMe

Me

Me

Me

H3C

H

+1

Os

P

CH2

PMe

Me

Me

Me

CH3

H

+1

Os

P

CH2

PMe

Me

Me

Me

CH2

H

+1H

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 DG‡ = 36 kcal/mol (DS‡ = 15 e.u.).

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

PtPPh2

Ph2P CH3

CH3

CH3

CH3

PtPPh2

Ph2P CH3

CH3H3C

PtPPh2

Ph2P CH3

CH3H3C

PtPPh2

Ph2P CH3

CH3

H3C

CH3

G(M06) = 0.0G(B3LYP) = 0.0G(M06L) = 0.0

G(M06) = 19.1G(B3LYP) = 13.9

G(M06) = 31.6G(B3LYP) = 27.6G(M06L) = 37.6(S = 10.9 e.u.)

H3CCH3

x A

x x


Metal-oxo Oxidations

• M06 performs well• B3LYP overestimates bimolecular barriers involving bulky or

polarizable species

Experiment:M06:B3LYP:

DH‡(25C) 13.4 kcal/mol11.817.1

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

(Tp)Re(O)Cl2G(exp,298) = 23.0 kcal/mol

H(exp) = 17.1 kcal/molS(exp) = -19.7 e.u.

(Tpm)Re(O)Cl2+

G(exp,298) = 19.1 kcal/molH(exp) = 13.4 kcal/mol

S(exp) = -19.0 e.u.

PPh3

PPh3(Tp)Re(O)Cl2

H(M06) = x kcal/molH(B3LYP) = x kcal/mol

S(B3LYP) = x + y = z e.u.

(Tpm)Re(O)Cl2+



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:

DH‡(25C) 17.1 kcal/mol16.624.1


Methods matter (must use the correct flavor DFT and the correct basis set)

43

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 DH (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)

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

(Tp)Re(O)Cl2G(exp,298) = 23.0 kcal/mol

H(exp) = 17.1 kcal/molS(exp) = -19.7 e.u.

(Tpm)Re(O)Cl2+

G(exp,298) = 19.1 kcal/mol

H(exp) = 13.4 kcal/mol

S(exp) = -19.0 e.u.

PPh3

PPh3(Tp)Re(O)Cl2



(Tpm)Re(O)Cl2+

H(M06) = x kcal/mol

H(B3LYP) = x kcal/molS(B3LYP) = x + y = z e.u.

Exper:M06:

B3LYP:

DH‡(25C)With solvation

17.1 16.624.1

Barrier withNo solvation

16.9

Exper:M06:

B3LYP:

DH‡(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

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 )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 atomsUse these forces to determine optimum geometry of solute in solution.Can treat solvent stabilized zwitterionsDifficult 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: = 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)5.54.15.26.511.3

Experimental pKa

6.37.15.57.411.0

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

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

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

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

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

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

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

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

0 2 4 6 8 10 12 14 16 18 20-40

-30

-20

-10

0

10

20

30

40

50

pH

G (

kca

l/mo

l)

32.6

34.6 40.0

37.9

34.6

Resting states

Insertiontransition states

Use theory to predict optimal pH for each catalyst

LnOsII

OH2

H3C

OH

H

LnOsII

OH

H3C

OH

H

Optimum pH is 8

Os

OH

OHN

N

NOH

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, PBF implicit solventMax errors: 200 meV, 2pH units

Trans-(bpy)2Ru(OH)2

This is essential in using theory to predict new catalysts


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.


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


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 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


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

DFTxE

2PTcE DFT

cE contains the double-excitation parts of 2PT

cE

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

54

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

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

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

2

ˆˆˆ1

4

i xi j eeGLc

ij 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)


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(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

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 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

,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) figured out how to get N3 scaling for MP2 and for XYGJ-OS

Yousung Jung


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

XYGJ- OS 2

2 ,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

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

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

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-LOS

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

{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 wagPNAS in press

XYGJ-OS same accuracy as XYG3 but scales like N3

not N5.

63

Density Functional Theory errors kcal/mol)

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


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

Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 69-5.00

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


examples


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 largeXYGJ-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-CCd =1Reik = 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/mol

)

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 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


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


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

PBE-lg 3 to 5% too high (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 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

Sand-

wich


Benzene Dimer

Parallel-

dis-

placed


Parameter OptimizationImplemented in VASP 5.2.11

0.701

2

0.696

6


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


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)

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 barriersPopular with physicists

Popular 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.

VF

V HK ][rep-

min

Acceptable

errors


Old slides

Chem 121 - Applied Quantum Chemistry

Lecture 1Lecture 2 92

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



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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