Lecture 2, January 6, 2016 Quantum Mechanics-2:...
Transcript of Lecture 2, January 6, 2016 Quantum Mechanics-2:...
Lecture 1-Ch121a-Goddard-L01 © copyright 2016 William A. Goddard III, all rights reserved\ 1
Ch121a Atomic Level Simulations of Materials and
Molecules
William A. Goddard III, [email protected]
316 Beckman Institute, x3093
Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics,
California Institute of Technology
Lecture 2, January 6, 2016
Quantum Mechanics-2: DFT
Special Instructor: Julius Su <[email protected]>
Teaching Assistants:
Daniel Brooks [email protected]
Jin Qian [email protected]
Room BI 115
Lecture: Monday, Wednesday 2-3pm
Lab Session: Friday 2-3pm
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 2
Homework and Research Project
First 5 weeks: The homework each week uses generally available
computer software implementing the basic methods on
applications aimed at exposing the students to understanding how
to use atomistic simulations to solve problems.
Each calculation requires making decisions on the specific
approaches and parameters relevant and how to analyze the
results.
Midterm: each student submits proposal for a project using the
methods of Ch121a to solve a research problem that can be
completed in the final 5 weeks.
The homework for the last 5 weeks is to turn in a one page report
on progress with the project
The final is a research report describing the calculations and
conclusions
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 3
Last Time
Overview of ab initio Quantum Mechanics
Start with Schrodinger Equation
Solve for optimum Hartree Fock orbitals
This is very useful approximate wavefunction that can be
used as the starting point for improved wavefunctions
This very popular and rigorous but today we discuss a
different approach that is less rigorous but more practical
Density Functional Theory (DFT)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 4
Alternative to Hartree-Fork,
Density Functional Theory
Walter Kohn’s dream:
replace the 3N electronic degrees of freedom needed to define
the N-electron wavefunction Ψ(1,2,…N) with
just the 3 degrees of freedom for the electron density r(x,y,z).
It is not obvious that this would be possible but
P. Hohenberg and W. Kohn Phys. Rev. B 76, 6062 (1964).
Showed that there exists some functional of the density
that gives the exact energy of the system
rrr
VFV
HK ][rep-
min
Kohn did not specify the nature or form of this
functional, but research over the last 46 years has
provided increasingly accurate approximations to it.Walter Kohn (1923-)
Nobel Prize Chemistry 1998
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 5
The Hohenberg-Kohn theorem
The Hohenberg-Kohn theorem states that if N interacting
electrons move in an external potential, Vext(1..N), the
ground-state electron density r(xyz)=r(r) minimizes the
functional
E[r] = F[r] + ʃ r(r) Vext(r) d3r
where F[r] is a universal functional of r and the minimum
value of the functional, E, is E0, the exact ground-state
electronic energy.
Here we take Vext(1..N) = Si=1,..N SA=1..Z [-ZA/rAi], which is the
electron-nuclear attraction part of our Hamiltonian.
HK do NOT tell us what the form of this universal functional,
only of its existence
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 6
Proof of the Hohenberg-Kohn theorem
Mel Levy provided a particularly simple proof of Hohenberg-Kohn
theorem {M. Levy, Proc. Nat. Acad. Sci. 76, 6062 (1979)}.
Define the functional O as O[r(r)] = min <Ψ|O|Ψ>
|Ψ>r(r)
where we consider all wavefunctions Ψ that lead to the same
density, r(r), and select the one leading to the lowest expectation
value for <Ψ|O|Ψ>.
F[r] is defined as F[r(r)] = min <Ψ|F|Ψ>
|Ψ>r(r)
where F = Si [- ½ i2] + ½ Si≠k [1/rik].
Thus the usual Hamiltonian is H = F + Vext
Now consider a trial function Ψapp that leads to the density r(r)
and which minimizes <Ψ|F|Ψ>
Then E[r] = F[r] + ʃ r(r) Vext(r) d3r = <Ψ|F +Vext|Ψ> = <Ψ|H|Ψ>
Thus E[r] ≥ E0 the exact ground state energy.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 7
The Kohn-Sham equations
Walter Kohn and Lou J. Sham. Phys. Rev. 140, A1133 (1965).
Provided a practical methodology to calculate DFT wavefunctions
They partitioned the functional E[r] into parts
E[r] = KE0 + ½ ʃʃd3r1 d3r2 [r(1) r(2)/r12 + ʃd3r r(r) Vext(r) + Exc[r(r)]
Where
KE0 = Si <φi| [- ½ i2 | φi> is the KE of a non-interacting electron
gas having density r(r). This is NOT the KE of the real system.
The 2nd term is the total electrostatic energy for the density r(r).
Note that this includes the self interaction of an electron with itself.
The 3rd term is the total electron-nuclear attraction term
The 4th term contains all the unknown aspects of the Density
Functional
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 8
Solving the Kohn-Sham equations
Requiring that ʃ d3r r(r) = N the total number of electrons and
applying the variational principle leads to
[d/dr(r)] [E[r] – m ʃ d3r r(r) ] = 0
where the Lagrange multiplier m = dE[r]/dr = the chemical
potential
Here the notation [d/dr(r)] means a functional derivative inside
the integral.
To calculate the ground state wavefunction we solve
HKS φi = [- ½ i2 + Veff(r)] φi = ei φi
self consistently with r(r) = S i=1,N <φi|φi>
where Veff (r) = Vext (r) + Jr(r) + Vxc(r) and Vxc(r) = dEXC[r]/dr
Thus HKS looks quite analogous to HHF
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 9
The Local Density Approximation (LDA)
EKS = Si [<φi|- ½i2|φi >+Vext (ri)+Vxc(ri)]+½ʃʃd3r1 d3r2 [r(1)r(2)/r12]
General form of Energy for DFT (Kohn-Sham) formulation
KE Nuclear
attraction
Coulomb repulsionExchange
correlation
If the density is r =N/V then Coulomb repulsion leads to a
total of ½(N/V)2 interactions, but it should be ½(N(N-1)/V2)
Thus LDA include an extra self term that should not be
present
At the very minimum, Vxc needs to correct for this
If density is uniform then error is proportional to 1/N. since
electron density is r = N/V
( ) ( )3
1
x
LDA
x rρAρε xA = -3
1
π
3
4
3
.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 10
The Local Density Approximation (LDA)
ExcLDA[r(r)] ʃ d3r eXC(r(r)) r(r)
where eXC(r(r)) is derived from Quantum Monte Carlo
calculations for the uniform electron gas {DM Ceperley and BJ
Alder, Phys.Rev.Lett. 45, 566 (1980)}
It is argued that LDA is accurate for simple metals and simple
semiconductors, where it generally gives good lattice
parameters
It is clearly very poor for molecular complexes (dominated by
London attraction), and hydrogen bonding
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
LDA exchange
11
( ) ( )3
1
x
LDA
x rρAρε xA = -3
1
π
3
4
3
.
Here we say that in LDA each electron interacts with all N
electrons but really it should be N-1.
The exchange term cancels this extra term. If density is uniform
then error is proportional to 1/N. since electron density is r = N/V
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 12
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 13
Generalized gradient approximations
cxxc EEE
] drρ(r),...ρ(r)ρ(r),εE xx
( ) ( )sFερρ,ε LDA
x
GGA
x
( ) 3
4
3
12 ρπ24
ρs
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0 5.0 10.0
S
F(S
)
B88
PW91
new(mix)
PBE
Becke 88
X3LYP
PBE
PW91
s
F(s) GGA functionals
( )( )
( )2
1
1
2
32
1
188B
sasinhsa1
sasasinhsa1sF
( )( ) ( )
( ) d
52
1
1
2s100
432
1
191PW
sasasinhsa1
seaasasinhsa1sF
2
Here ( )3
12
2 π48a , 21 βa6a , βA2
aa
x
3/1
2
23 , 34 a
81
10a ,
x
3/1
64
25
A2
10aa
, and d = 4.
Becke9 = 0.0042 a4 and a5 zero
Here ( )3
12
2 π48a , 21 βa6a , βA2
aa
x
3/1
2
23 , 34 a
81
10a ,
x
3/1
64
25
A2
10aa
, and d = 4.
S is big where the density
gradient is large
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 14
Adiabatic connection formalism
] ]1
,0
xc xcE U dr r is the exchange-correlation energy at intermediate coupling strength λ.
The only problem is that the exact integrand is unknown.
Becke, A.D. J. Chem. Phys. (1993), 98, 5648-5652.
Langreth, D.C. and Perdew, J. P. Phys. Rev. (1977), B 15, 2884-2902.
Gunnarsson, O. and Lundqvist, B. Phys. Rev. (1976), B 13, 4274-4298.
Kurth, S. and Perdew, J. P. Phys. Rev. (1999), B 59, 10461-10468.
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:
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 15
Becke half and half functional
assume a linear model ,xcU a b
take , 0
exact
xc xU E the exact exchange of the KS orbitals
approximate , 1 , 1
LDA
xc xcU U
partition LDA LDA LDA
xc x cE E E
set ;exact LDA exact
x xc xa E b E E ;exact LDA exact
x xc xa E b E E
Get half-and-half functional
] ( )1 1
2 2
exact LDA LDA
xc x x cE E E Er
Becke, A.D. J. Chem. Phys. (1993), 98, 1372-1377
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 16
Becke 3 parameter functional
] ( )B3
1 2 3
LDA exact LDA GGA GGA
xc xc x x x cE E c E E c E c Er
Empirically modify half-and-half
where GGA
xE is the gradient-containing correction terms to the LDA exchange
GGA
cE is the gradient-containing correction to the LDA correlation,
1 2 3, ,c c c are constants fitted against selected experimental thermochemical data.
The success of B3LYP in achieving high accuracy demonstrates that errors of
for covalent bonding arise principally from the λ 0 or exchange limit, making
it important to introduce some portion of exact exchange
DFT
xcE
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.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 17
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 18
Truhlar’s DFT functionals
MPW3LYP, X1B95, MPW1B95, PW6B95, TPSS1KCIS, PBE1KCIS, MPW1KCIS,
BB1K, MPW1K, XB1K, MPWB1K, PWB6K, MPWKCIS1K
MPWLYP1w,PBE1w,PBELYP1w, TPSSLYP1w
G96HLYP, MPWLYP1M , MOHLYP
M05, M05-2xM06, M06-2x, M06-l, M06-HF
Hybrid meta-GGAM06 HF tPBE + VSXC
Band Gaps:
Critical property in designing new materials.
Problem standard DFT: big errors
19
2 eV
Clearly inadequate for in
silico design
What is solution?
Band Gaps:
Critical property in designing new materials.
Problem standard DFT: big errors
20
2 eV
Solutions:
Empirical: add Hubbard U and
adjust until get experimental
value
Problem: not useful for design
Rigorous theory: GW
calculate rigorous quasiparticle
excitation spectrum using
Green’s function (many-body
perturbation theory)
Problem
full GW: 10**4 cost of PBE
G0W0 (non-self-consistent)
10**3 cost of PBE
Are results worth the cost?
Band Gaps:
Critical property in designing new materials.
Results from various flavors of GW/PBE
21
2 eV
What can we do?
GW improves PBE from 1.09
eV to 0.36 eV at a cost of
10**3 to 10**4
Clearly NOT a solution for
materials design
Hybrid DFT theory
Include some exact HF exchange
22
2 eV
B3PW acceptable accuracy
but at what cost?
(12,6) SWCNT 168 atoms all
carbon, 12 k points
VASP (PBE) time 6,000 sec
memory 1900 MB
CRYSTAL (PBE) time 700 sec,
memory 300 MB
Conclusion: CRYSTAL PBE is
9 times faster than VASP
With VASP B3PW ~ 1000
times the cost of PBE
With CRYSAL B3PW is 3.2
times cost of PBE
CRYSTAL B3PW ~ 3 times
faster than VASP PBE
Topological Insulators
23
Material that is an insulator in its interior but whose surface
contains conducting states so electrons move along the surface of the
material.
On the surface of a topological insulator there are special states that
fall within the bulk energy gap and allow surface metallic
conduction. These surface state carriers have spins locked at 90° to
their momentum
At a given energy the only other available electronic states have
different spin, so surface is highly metallic and states cannot be
removed by surface passivation
Angular Resolved Photoelectron Spectrosopy
(ARPES) showed that Bi2Se3 is topological
insulator for 6 quintuple layers and larger
PBE says no band inversion not top. Insul.
G0W0 bulk Bi2Se3 has band inversion but not
possible for finite number layers
I. Aguilera, C. Friedrich, and S. Blügel, Phys. Rev. B 88, 165136 (2013)
B3PW CPU cost = 2.1 X PBE
Band gaps:
B3PW: 0.26 eV
G0W0: 0.20 eV
EXP: ~0.2 – 0.35 eV
Bulk Bi2Se3: B3PW and
G0W0/PBE both give band
inversion topological
insulator
I. Aguilera, C. Friedrich, and S. Blügel, Phys. Rev. B 88, 165136 (2013)
B3PW CPU cost = 1.8 X PBE
Band gaps:
B3PW: 0.28 eV
G0W0: 0.19 eV
EXP: ~0.13 – 0.17 eV
Bi2Te3 band gap with number quintuple layers
26Clear formation of metallic topological
insulator state by 5 QL, no experiment
Bi2Se3 band gap with number quintuple layers
27
Bi2Se3 Band gap vs number quint layers
28
PBE
B3PW
exper
Exper TI for 6 layers
PBE Gap < 0.1 eV
for 6 layers
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 systems
B3LYP and M06 useful for chemical reaction mechanisms
Progress is being made on developing new systems
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 30
Accuracy: DFT is basis for QM on catalysts
Current flavors of DFT accurate for properties of many systems
B3LYP and M06 useful for chemical reaction mechanisms
• B3LYP and M06L perform well.
• M06 underestimates the barrier.
Example: Reductive elimination of CH4 from (PONOP)Ir(CH3)(H)+
Goldberg exper at 168Kbarrier G‡ = 9.3kcal/mol.
G(173K)B3LYPM06M06L
0.00.00.0
10.85.8
11.4(reductive elimination)
These calculations
use extended basis
sets and PBF
solvation
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
Reductive Elimination Thermochemistry
H/D exchange was measured from 153-173K by Girolami (J . Am. Chem. Soc., Vol. 120, 1998 6605) by NMR to have a barrier of G‡ = 8.1 kcal/mol.
G(173K)B3LYPM06
0.00.0
8.79.5(reductive elimination)
4.65.3(s-bound complex)
6.45.2(site-exchange)
Mu-Jeng Cheng
QM allows first principles predictions on new ligands, oxidation states, and
solvents. But there are error bars in the QM having to do with details of the
caculations (flavor of DFT, basis set). We use the best available methods and
compare to any available experimental data on known systems to assess the
accuracy for new systems. Some examples here and on the next slides
M06 and B3LYP functionals both consistent with experimental barrier site exchange.
These calculations use extended basis sets and PBF solvation
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
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.)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
Methods matter (must use the correct flavor
DFT and the correct basis set)
34
Commonly used methods (B3LYP, triple zeta basis set ) are insufficient for oxidation of main group elements. (Martin, J. Chem. Phys. 1998, 108(7), 2791.) B3LYP disfavors oxidation of main group elements by >10 kcal/mol
Experimental H (kcal/mol) -27
-80.1
M066311G**++
-22.0-70.7
B3LYP6311G**++
-17-58.1
M066311++G-
3df(S)-29.2-82.2
Bad, but typical in publications
Simple example
S(CH3)2 + ½ O2 → O=S(CH3)2
S(CH3)2 + O2 → (CH3)2SO2
S(CH3)2 + ½ O2 → O=S(CH3)2
S(CH3)2 + O2 → (CH3)2SO2
OK
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
Methods matter (for reactions in polar media,
must include solvation)Phosphine oxidation by (Tp)Re(O)Cl2 and (Tpm)Re(O)Cl2
+ observed from 15-50˚C in 1,2-dichlorobenzene Seymore and Brown; Inorg. Chem., Vol. 39, 2000, 325)
Exper:M06:
B3LYP:
H‡(25C)With solvation
17.1 16.624.1
Barrier withNo solvation
16.9
Exper:M06:
B3LYP:
H‡(25C)With solvation
13.4 11.817.1
Barrier withNo solvation
2.4
Most QM publications ignore solvation or use unreliable methods
Much larger corrections in H2O
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
Fundamental philosophy of First principles
predictionsQM calculations on small systems ~100 atoms get accurate
energies, geometries, stiffness, mechanisms
Fit QM to force field to describe big systems (104 -107 atoms)
Fit to obtain parameters for continuum systems
macroscopic properties based on first principles (QM)
Can predict novel materials where no empirical data available.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 37
Fundamental philosophy of First principles
predictionsQM calculations on small systems ~100 atoms get accurate
energies, geometries, stiffness, mechanisms
Fit QM to force field to describe big systems (104 -107 atoms)
Fit to obtain parameters for continuum systems
macroscopic properties based on first principles (QM)
Can predict novel materials where no empirical data available.
General Problem with DFT: bad description of vdw attraction
Graphite layers not
stable with DFT
exper
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 38
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
Molecules PBE PBE-ℓg Exp.
Benzene 511.81 452.09 461.11
Naphthalene 380.23 344.41 338.79
Anthracene 515.49 451.55 451.59
Sublimation energy (kcal/mol/molecule)
Cell volume (angstrom3/cell) PBE 12-14% too large
PBE 85-90% too small
Most popular form of DFT for crystals – PBE (VASP software)
Reason DFT formalism not include London Dispersion (-C6/R6)
responsible for van der Waals attraction.
All published QM calculations on solids have this problem
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 39
XYG3 approach to include London Dispersion in DFT
Görling-Levy coupling-constant perturbation expansion
] ]1
,0
xc xcE U dr r Take initial slope as the 2nd order correlation energy:
, 2
, 0
0
2xc GL
xc c
UU E
where
22
2
ˆˆˆ1
4
i xi j eeGL
c
ij ii j i
fE
e e e e e e
where is the electron-electron repulsion operator, is the local exchange operator,
and is the Fock-like, non-local exchange operator.
ˆee ˆ
x
f̂
,xcU a b Substitute into with22 GL
cb E ;exact LDA exact
x xc xa E b E E or
Combine both approaches (2 choices for b) ( )2
1 2
GL DFT exact
c xc xb b E b E E
] ( ) ( )R5 2
1 2 3 4
LDA exact LDA GGA PT LDA GGA
xc xc x x x c c cE E c E E c E c E E c Er
a double hybrid DFT that mixes some exact exchange into while also introducing a
certain portion of into
DFT
xE2PT
cEDFT
cE
contains the double-excitation parts of 2PT
cE
22
2
ˆˆˆ1
4
i xi j eeGL
c
ij ii j i
fE
e e e e e e
This is a fifth-rung functional (R5) using information from both occupied and virtual KS
orbitals. In principle can now describe dispersion
Sum over virtual orbtials
40
Solution: extend DFT to include double excitations to virtuals
get London Dispersion in DFT: use Görling-Levy expansion
] ( ) ( )R5 2
1 2 3 4
LDA exact LDA GGA PT LDA GGA
xc xc x x x c c cE E c E E c E c E E c Er
Get {c1 = 0.8033, c2 = 0.2107, c3 = 0.3211} and c4 = (1 – c3) = 0.6789
XYG3 leads to mean absolute deviation (MAD) =1.81 kcal/mol,
B3LYP: MAD = 4.74 kcal/mol.
M06: MAD = 4.17 kcal/mol
M06-2x: MAD = 2.93 kcal/mol
M06-L: MAD = 5.82 kcal/mol .
G3 ab initio (with one empirical parameter): MAD = 1.05
G2 ab initio (with one empirical parameter): MAD = 1.88 kcal/mol
but G2 and G3 involve far higher computational cost.
where
22
2
ˆˆˆ1
4
i xi j eeGL
c
ij ii j i
fE
e e e e e e
Problem 5th order scaling with size
Doubly hybrid density functional for accurate descriptions of nonbond interactions,
thermochemistry, and thermochemical kinetics; Zhang Y, Xu X, Goddard WA; P. Natl.
Acad. Sci. 106 (13) 4963-4968 (2009)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 41
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.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 42
Accuracy (kcal/mol) of various QM methods for
predicting standard enthalpies of formationFunctional MAD Max(+) Max(-)
DFT
XYG3 a 1.81 16.67 (SF6) -6.28 (BCl3)
M06-2x a 2.93 20.77 (O3) -17.39 (P4)
M06 a 4.17 11.25 (O3) -25.89 (C2F6)
B2PLYP a 4.63 20.37(n-octane) -8.01(C2F4)
B3LYP a 4.74 19.22 (SF6) -8.03 (BeH)
M06-L a 5.82 14.75 (PF5) -27.13 (C2Cl4)
BLYP b 9.49 41.0 (C8H18) -28.1 (NO2)
PBE b 22.22 10.8 (Si2H6) -79.7 (azulene)
LDA b 121.85 0.4 (Li2) -347.5 (azulene)
Ab initio
HFa 211.48 582.72(n-octane) -0.46 (BeH)
MP2a 10.93 29.21(Si(CH3)4) -48.34 (C2F6)
QCISD(T) c 15.22 42.78(n-octane) -1.44 (Na2)
G2(1 empirical parm) 1.88 7.2 (SiF4) -9.4 (C2F6)
G3(4 empirical parm) 1.05 7.1 (PF5) -4.9 (C2F4)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 43
-5.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
-2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50
Reaction coordinate
En
erg
y (
kca
l/m
ol)
HF
HF_PT2
XYG3
CCSD(T)
B3LYP
BLYP
SVWN
HF
HF_PT2 SVWNB3LYP
BLYP
XYG3CCSD(T)
SVWN
H + CH4 H2 + CH3
Reaction Coordinate: R(CH)-R(HH) (in Å)
Energ
y (
kcal/m
ol)
Comparison of QM methods for reaction surface of
H + CH4 H2 + CH3
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 44
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)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 45
(A)
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
3.0 4.0 5.0 6.0
Intermolecular distance
En
erg
y (
kca
l/m
ol)
BLYP
B3LYP
XYG3
CCSD(T)
SVWN
HF_PT2
(C)
-12.00
-9.00
-6.00
-3.00
0.00
Ec_VWN
Ec_B3LYP
Ec_LYP
Ec_XYG3
Ec_CCSD(T)
Ec_PT2
(B)
-5.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
3.0 4.0 5.0 6.0
Ex_B
Ex_B3LYP
Ex_XYG3
Ex_HF
Ex_S
HF
HF_PT2
B3LYP
BLYP
CCSD(T)
LDA
(SVWN)
A. Total Energy (kcal/mol)
Distance (A)
XYG3
B. Exchange Energy (kcal/mol)
C. Correlation Energy (kcal/mol)
B
S
B3LYP
XYG3
PT2
B3LYP
LYP CCSD(T)
VWN
XYG3
Distance (A)
Conclusion: XYG3 provides excellent accuracy for London dispersion, as good as
CCSD(T)
Test for
London
Dispersion
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 46
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)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 47
Problem with XYG3 : scales as N**5 with size
(like MP2)
] ]1
,0
xc xcE U dr r Take initial slope as the 2nd order correlation energy:
, 2
, 0
0
2xc GL
xc c
UU E
where
22
2
ˆˆˆ1
4
i xi j eeGL
c
ij ii j i
fE
e e e e e e
where is the electron-electron repulsion operator, is the local exchange operator,
and is the Fock-like, non-local exchange operator.
ˆee ˆ
x
f̂
Sum over virtual orbtials
XYG3 approach to include London Dispersion in DFT
Görling-Levy coupling-constant perturbation expansion
EGL2 involves double excitations to virtuals, scales as N5 with
size
MP2 has same critical step
Yousung Jung (KAIST) figured out how to get N3 scaling for
MP2 and for XYGJ-OSYousung
Jung
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 48
Solve scaling problem: XYGJ-OS; include only opposite spin
and only local contributions
] ( ) ( )XYGJ- OS 2
2 ,1
HF S VWN LYP PT
xc x x x x VWN c LYP c PT c osE e E e E e E e E e E
r
XYG4-OS and XYG4-LOS timings
0.0
40.0
80.0
120.0
160.0
200.0
0 20 40 60 80 100 120
alkane chain length
CP
U (
ho
urs
)
XYG4-LOS
XYG4-OS
B3LYP
XYG3
XYG4-OS and XYG4-LOS timings
0.0
40.0
80.0
120.0
160.0
200.0
0 20 40 60 80 100 120
alkane chain length
CP
U (
ho
urs
)XYG4-LOS
XYG4-OS
B3LYP
XYG3
XYG4-OS and XYG4-LOS timings
0.0
40.0
80.0
120.0
160.0
200.0
0 20 40 60 80 100 120
alkane chain length
CP
U (
ho
urs
)
XYG4-LOS
XYG4-OS
B3LYP
XYG3
XYGJ-OS
XYGJ-LOS
XYG4-OS and XYG4-LOS timings
0.0
40.0
80.0
120.0
160.0
200.0
0 20 40 60 80 100 120
alkane chain length
CP
U (
ho
urs
)XYG4-LOS
XYG4-OS
B3LYP
XYG3
XYGJ-LOS
XYG4-OS and XYG4-LOS timings
0.0
40.0
80.0
120.0
160.0
200.0
0 20 40 60 80 100 120
alkane chain length
CP
U (
ho
urs
)
XYG4-LOS
XYG4-OS
B3LYP
XYG3
XYGJ-OS
{ex, eVWN, eLYP, ePT2} ={0.7731,0.2309, 0.2754,
0.4264}.
A fast doubly hybrid
density functional
method close to
chemical accuracy:
XYGJ-OS
Igor Ying Zhang,
Xin Xu, Yousung
Jung, and wag
PNAS in press
XYGJ-OS
same accuracy
as XYG3 but
scales like N3
not N5.
Density Functional Theory errors kcal/mol)
49
LDA 130.88 15.2
Include density gradient (GGA)
BLYP 10.16 7.9
PW91 22.04 9.3
PBE 20.71 9.1
Hybrid: include HF exchange
B3LYP 6.08 4.5
PBE0 5.64 3.9
Include KE functional fit to barriers and complexes
M06-L 5.20 4.1
M06 3.37 2.2
M06-2X 2.26 1.3
atomize barrier
Popular with physicists
Popular with physicists
Popular with chemists
Include excitations to virtuals
XYGJ-OS 1.81 1.0
G3 (cc) 1.06 0.9The level needed for
reliable predictions
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 50
Accuracy of Methods (Mean absolute deviations MAD, in eV)
HOF IP EA PA BDE NHTBH HTBH NCIE All Time
Methods
(223) (38) (25) (8) (92) (38) (38) (31) (493) C100H202 C100H100
DFT methods
SPL (LDA) 5.484 0.255 0.311 0.276 0.754 0.542 0.775 0.140 2.771
BLYP 0.412 0.200 0.105 0.080 0.292 0.376 0.337 0.063 0.322
PBE 0.987 0.161 0.102 0.072 0.177 0.371 0.413 0.052 0.562
TPSS 0.276 0.173 0.104 0.071 0.245 0.391 0.344 0.049 0.250
B3LYP 0.206 0.162 0.106 0.061 0.226 0.202 0.192 0.041 0.187 2.8 12.3
PBE0 0.300 0.165 0.128 0.057 0.155 0.154 0.193 0.031 0.213
M06-2X 0.127 0.130 0.103 0.092 0.069 0.056 0.055 0.013 0.096
XYG3 0.078 0.057 0.080 0.070 0.068 0.056 0.033 0.014 0.065 200.0 81.4
XYGJ-OS 0.072 0.055 0.084 0.067 0.033 0.049 0.038 0.015 0.056 7.8 46.4
MC3BB 0.165 0.120 0.175 0.046 0.111 0.062 0.036 0.023 0.123
B2PLYP 0.201 0.109 0.090 0.067 0.124 0.090 0.078 0.023 0.143
Wavefunction based methods
HF 9.171 1.005 1.148 0.133 0.104 0.397 0.582 0.098 4.387
MP2 0.474 0.163 0.166 0.084 0.363 0.249 0.166 0.028 0.338
G2 0.082 0.042 0.057 0.058 0.078 0.042 0.054 0.025 0.068
G3 0.046 0.055 0.049 0.046 0.047 0.042 0.054 0.025 0.046
HOF = heat of formation; IP = ionization potential,
EA = electron affinity, PA = proton affinity,
BDE = bond dissociation energy,
NHTBH, HTBH = barrier heights for reactions,
NCIE = the binding in molecular clusters
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 51
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)
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
Molecules PBE PBE-ℓg Exp.
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-lg 0 to 2% too small,
thermal expansion
PBE-lg 3 to 5% too high
(zero point energy)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 52
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)
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
Molecules PBE PBE-ℓg Exp.
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-lg 0 to 2% too small,
thermal expansion
PBE-lg 3 to 5% too high
(zero point energy)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
Major challenge for DFT calculations of molecular solids
• Current implementations of DFT describe geometries and energies of strongly bound solids, but fail to describe the long range van der Waals (vdW) interactions.
• Get volumes ~ 10% too large
• XYGJ-OS solves this problem but much slower than standard methods
Nlg,
lg 6 6,
-ij
ij i j ij eij
CE
r dR
DFT D DFT dispE E E
C6 single parameter from QM-CC
d =1
Reik = Rei + Rek (UFF vdW radii)
DFT-low gradient (DFT-lg) gives accurate description of the long-range 1/R6 attraction of the London dispersion but at cost of standard DFTAdd the low-gradient 1/R6 one parameter fitted to XYGJ-OS
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 54
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 55
DFT-lg description for graphite
graphite has AB stacking (also show AA eclipsed graphite)
Exper E
0.8, 1.0, 1.2
Exper c 6.556
PBE-lg
PBE
Bin
din
g e
ne
rgy (
kca
l/m
ol)
c lattice constant (A)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 56
DFT-lg description for benzene
PBE-lg predicted the EOS of benzene crystal (orthorhombic phase I) in good agreement with
corrected experimental EOS at 0 K (dashed line).
Pressure at zero K geometry: PBE: 1.43 Gpa; PBE-lg: 0.11 Gpa
Zero pressure volume change: PBE: 35.0%; PBE-lg: 2.8%
Heat of sublimation at 0 K: Exp:11.295 kcal/mol; PBE: 0.913; PBE-lg: 6.762
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
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)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 58
Hydrocarbon Crystals – Get excellent results
for PBE-lg
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
Molecules PBE PBE-ℓg Exp.
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-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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 59
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 60
Universal PBE-ℓg Method
UFF, a Full Periodic Table Force Field for Molecular Mechanics and Molecular
Dynamics Simulations; A. K. Rappé, C. J. Casewit, K. S. Colwell, W. A. Goddard
III, and W. M. Skiff; J. Am. Chem. Soc. 114, 10024 (1992)
Derived C6/R6 parameters from scaled atomic polarizabilities for Z=1-103 (H-
Lr) and derived Dvdw from combining atomic IP and C6
Universal PBE-lg: use same Re, C6, and De as UFF, add a single new
parameter slg
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 61
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)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 62
Benzene Dimer
T-shaped
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 63
Benzene Dimer
Sandwich
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 64
Benzene Dimer
Parallel-
displaced
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 65
Parameter Optimization
Implemented in VASP 5.2.11
0.7012
0.6966
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 66
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)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 67
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
Molecules PBE PBE-ℓg Exp.
Benzene 511.81 452.09 461.11
Naphthalene 380.23 344.41 338.79
Anthracene 515.49 451.55 451.59
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 68
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)
Molecules PBE PBE-ℓg Exp.
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 69
Simple Molecular Crystals
• Cell volume (angstrom3/cell)
Molecules PBE PBE-ℓg Exp.
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 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 70
Inert Gas Crystals
• Sublimation energy (kcal/mol/molecule)
Average error: 1.70 (PBE) and 0.74 (PBE-ℓg)
Maximal error: 3.14 (PBE) and 1.68 (PBE-ℓg)
Molecules PBE PBE-ℓg Exp.
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 71
Heavy Atom Fluorides
• Sublimation energy (kcal/mol/molecule)
aSpin-orbit coupling term is corrected.
Other issues; Large core pseudopotential (U: 14 electrons, Np: 15
electrons).
Molecules PBE PBE-ℓg Exp.
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
Density Functional Theory errors kcal/mol)
72
LDA 130.88 15.2
Include density gradient (GGA)
BLYP 10.16 7.9
PW91 22.04 9.3
PBE 20.71 9.1
Hybrid: include HF exchange
B3LYP 6.08 4.5
PBE0 5.64 3.9
Include KE functional fit to barriers and complexes
M06-L 5.20 4.1
M06 3.37 2.2
M06-2X 2.26 1.3
atomize barriersPopular with physicists
Popular with physicists
Popular with chemists
replace the N-electron wavefunction Ψ(1,2,…N) with just the 3
degrees of freedom for the electron density r(x,y,z).
E = Functional not known, but have accurate approx.
rrr
VFV
HK ][rep-
min
Acceptable
errors
Calculate Solvent Accessible Surface of the solute by rolling a
sphere of radius Rsolv over the surface formed by the vdW
radii of the atoms.
Calculate electrostatic field of the solute based on electron
density from the orbitals
Calculate the polarization in the solvent due to the
electrostatic field of the solute (need dielectric constant e)
This leads to Reaction Field that acts back on solute atoms,
which in turn changes the orbitals. Iterated until self-
consistent.
Calculate solvent forces on solute atoms
Use these forces to determine optimum geometry of solute in
solution.
Can treat solvent stabilized zwitterions
Difficult to describe weakly bound solvent molecules
interacting with solute (low frequency, many local minima)
Short cut: Optimize structure in the gas phase and do single
point solvation calculation. Some calculations done this way
Essential issues: must include Solvation effects in the QM
Solvent: e = 99
Rsolv= 2.205 A
PBF Implementation in
Jaguar (Schrodinger Inc):
pK organics to ~0.2 units,
solvation to ~1 kcal/mol
(pH from -20 to +20)
The Poisson-Boltzmann Continuum Model in Jaguar/Schrödinger is extremely accurate
6.9 (6.7)
-3.89 (-52.35)
6.1 (6.0)
-3.98 (-55.11)
5.8 (5.8) -
4.96 (-49.64)
5.3 (5.3)
-3.90 (-57.94)
5.0 (4.9) -
4.80 (-51.84)
pKa: Jaguar (experiment)
E_sol: zero (H+)
Comparison of PBF (Jaguar) pK with experiment
Protonated Complex
(diethylenetriamine)Pt(OH2)2+
PtCl3(OH2)1-
Pt(NH3)2(OH2)22+
Pt(NH3)2(OH)(OH2)1+
cis-(bpy)2Os(OH)(H2O)1+
Calculated (B3LYP) pKa(MAD: 1.1)
5.5
4.1
5.2
6.5
11.3
Experimental pKa
6.3
7.1
5.5
7.4
11.0
cis-(bpy)2Os(H2O)22+
cis-(bpy)2Os(OH)(H2O)1+
trans-(bpy)2Os(H2O)22+
trans-(bpy)2Os(OH)(H2O)1+
cis-(bpy)2Ru(H2O)22+
cis-(bpy)2Ru(OH)(H2O)1+
trans-(bpy)2Ru(H2O)22+
trans-(bpy)2Ru(OH)(H2O)1+
(tpy)Os(H2O)32+
(tpy)Os(OH)(H2O)21+
(tpy)Os(OH)2(H2O)
Calculated (M06//B3LYP) pKa
(MAD: 1.6)9.18.86.2
10.913.015.211.013.95.66.3
10.9
Experimental pKa
7.911.08.2
10.28.9
>11.09.2
>11.56.08.0
11.0
PBF (Jaguar) predictions of Metal-aquo pKa’s
-40
-30
-20
-10
0
10
20
30
40
50
0 5 10 15 20
G (
kca
l/m
ol)
pH
32.6
34.6 40.0
37.9
34.6
Resting states
Insertion
transition states
Use theory to predict optimal pH for each
catalyst
Optimum pH is 8
LnOsII(OH2)(OH)2
is stable
LnOsII(OH)3-
is stable LnOsII(OH2)3
+2
is stable
Predict the relative free energies of
possible catalyst resting states and
transition states as a function of pH.
Predict Pourbaix Diagrams to determine the
oxidation states of transition metal complexes as
function of pH and electrochemical potential
Black experimental data from Dobson
and Meyer, Inorg. Chem. Vol. 27, No.19, 1988.
Red is from QM calculation (no fitting) using M06 functional, PBFimplicit solvent
Max errors: 200 meV, 2pH units
Trans-(bpy)2Ru(OH)2
This is essential in using theory to
predict new catalysts
78
80
81
82
83
84
CANDLE SOLVATION FOR QM CALCULATIONS
charge-asymmetric nonlocally-determined local-electric Field (CANDLE) Solvation model; Sundararaman &WAG; J. Chem. Phys., 142 (6): 064107 (2015)
Solvents: H2O, CH3CN, CHCl3, CCl4)
CANDLE SOLVATION FOR QM CALCULATIONS
charge-asymmetric nonlocally-determined local-electric Field (CANDLE) Solvation model; Sundararaman &WAG; J. Chem. Phys., 142 (6): 064107 (2015)
Solvents: H2O, CH3CN, CHCl3, CCl4)