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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
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 Quantum Mechanics, Hydrogen Atom, etcPlease review again to make sure that you are comfortable with the concepts, which you should have seen before
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 4
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 5
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 6
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 7
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 8
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 9
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 10
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 11
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]
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 12
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 13
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 14
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.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 15
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 16
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 17
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 18
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.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 19
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 20
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**
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 21
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 22
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 23
Software packages
Jaguar: Good for organometallicsQChem: very fast for organicsGaussian: many analysis toolsGAMESSHyperChemADFSpartan/Titan
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 24
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 25
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 26
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.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 27
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 28
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 29
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.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 30
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\ 31
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\ 32
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 33
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:
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 34
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 35
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.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 36
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\ 37
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 39
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
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
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 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
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:
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+
H(M06) = x kcal/molH(B3LYP) = x kcal/mol
S(B3LYP) = x + y = z e.u.
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
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)
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
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)
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
H(M06) = x kcal/molH(B3LYP) = x kcal/mol
S(B3LYP) = x + y = z e.u.
(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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\
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.
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 51
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 52
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 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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 53
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)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 55
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\ 56
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)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 57
-5.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
-2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50
Reaction coordinate
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 58
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\ 59
(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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 60
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\ 61
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 62
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 64
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 65
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 66
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 67
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 68
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 70
examples
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 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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 72
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\ 73
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)
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 74
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
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\ 76
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\ 77
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\ 78
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 79
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\ 80
Benzene Dimer
T-
shape
d
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 81
Benzene Dimer
Sand-
wich
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 82
Benzene Dimer
Parallel-
dis-
placed
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 83
Parameter OptimizationImplemented in VASP 5.2.11
0.701
2
0.696
6
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 84
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\ 85
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\ 86
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\ 87
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\ 88
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\ 89
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\ 90
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
Lecture 02 Ch121a 2012 © copyright 2012 William A. Goddard III, all rights reserved\ 91
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.
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 93
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 94
• Software packages– Jaguar– GAMESS– TurboMol– Gaussian– Spartan/Titan– HyperChem– ADF
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 95
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 96
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 97
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.
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 98
Actual jaguar input:
&genigeopt=1ifreq=1dftname=b3lyp basis=6-31g**&&zmat
O1 H2 O1 0.95H3 O1 0.95 H2 120.00&
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 99
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 100
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 101
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--
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 102
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 103
frequencies 1666.01 3801.19 3912.97
No negative frequencies!
(Compare IR spectra for gas-phase water)
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 104
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 105
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 106
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 107
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 108
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 109
A transition state should have one negative (imaginary) frequency!!!
(and ONLY one)
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 110
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!!!
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 111
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 112
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 113
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 114
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 115
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 116
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.
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 117
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”.
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 118
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)
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 119
Reaction energetics and barrier heights
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
Chem 121 - Applied Quantum Chemistry
Lecture 1Lecture 2 120
Reaction energetics and barrier heights
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