No Correlated Electron Left Behind
Electronic Structure Policies for the
Common Chemist and Condensed Matter Physicist
Sarom Sok
April 29th, 2011
From the Killing Fields to the Self‐Consistent Field:
The Adventures of a Cambodian Refugee in Quantum Chemistry
•
•
•
•
General Introduction
Hydrolysis of 1‐substituted Silatranes
Solvent‐induced shift of p‐nitroaniline
Benchmarking time‐dependent density functionals
• The combined coupled‐cluster/effective fragment potential approach
• Conclusions
From the Killing Fields to the Self‐Consistent Field:
The Adventures of a Cambodian Refugee in Quantum Chemistry
•
•
•
•
General Introduction
Hydrolysis of 1‐substituted Silatranes
Solvent‐induced shift of p‐nitroaniline
Benchmarking time‐dependent density functionals
• The combined coupled‐cluster/effective fragment potential approach
• Conclusions
Introduction
Time‐Independent Schrödinger Equation (TISE)
Ĥ Ψ = E Ψ
Electronic TISE within the Born‐Oppenheimer Approx. Hˆ elec Ψ elec = Eelec Ψ elec
Electronic Hamiltonian
Hˆ elec
∇i2 N M Z A
1
= −∑
− ∑∑
+∑
i =1 2
i =1 A=1 riA
i < j rij
N
= ∑ h ( i ) + Vee
i
Introduction
Electronic Hamiltonian ‐ Independent Particle Model
Hˆ elec = ∑ h ( i )
i
Hartree‐Fock Method
• Independent Particle Model
• Variational Principle
• Slater Determinants
Electron (i) feels an averaged repulsion field
Hˆ HF = ∑ h ( i ) + ∑ VHF ( i )
i
i
Introduction
Hartree‐Fock Method (Independent Particle Model)
• Neglects instantaneous repulsion of electrons (too close)
• Motion of electrons are uncorrelated
Ecorrelation = Ψ HF | Hˆ HF | Ψ HF − Eexact
Why is it important
• Liquefaction of noble gases
• According to Hartree‐Fock, the F2 molecule does not exist
• Interaction of biologically important molecules
Formal Scaling
CBS Limit*
Not Graduating
O(n8)
O(n7)
O(n6)
O(n5)
O(n4)
EHF
Eexact
Increase in Correlation Energy/Accuracy
Problem: Electrons get too close in Hartree‐Fock method.
Placing electrons in spatially different one‐electron functions (excitations into unoccupied orbitals).
*complete basis set limit
EFull CI
Formal Scaling
CBS Limit*
Not Graduating
O(n8)
O(n7)
O(n6)
O(n5)
O(n4)
EHF
Eexact
Increase in Correlation Energy/Accuracy
Full Configuration Interaction (CI)
occ virt
occ virt
Ψ CI = Φ ref + ∑∑ Cia Φ ia + ∑∑ Cijab Φ ijab +
i
i < j a <b
a
single
excitation
*complete basis set limit
double
excitation
occ
virt
∑ ∑
i < j < k a <b < c
triple
excitation
Bartlett, R. J.; Stanton, J. F. Rev. Comp. Chem. 1994, 5, 65
abc
Cijkabc Φ ijk
+ ...
N‐tuple
excitation
EFull CI
Formal Scaling
CBS Limit*
Not Graduating
O(n8)
O(n7)
O(n6)
O(n5)
O(n4)
ECISD
EHF
Eexact
Increase in Correlation Energy/Accuracy
CI with Singles and Doubles Excitations (CISD)
occ virt
occ virt
Ψ CI = Φ ref + ∑∑ Cia Φ ia + ∑∑ Cijab Φ ijab
i
*complete basis set limit
a
i < j a <b
Bartlett, R. J.; Stanton, J. F. Rev. Comp. Chem. 1994, 5, 65
EFull CI
Formal Scaling
CBS Limit*
Not Graduating
O(n8)
O(n7)
O(n6)
O(n5)
O(n4)
ECISD
EMP2
EHF
Eexact
Increase in Correlation Energy/Accuracy
Møller‐Plesset Second‐Order Perturbation Theory
E
*complete basis set limit
( 2)
⎡⎣( ij | ab ) − ( ia | jb ) ⎤⎦
= ∑∑
εi + ε j − ε a − εb
i < j a <b
occ virt
Bartlett, R. J.; Stanton, J. F. Rev. Comp. Chem. 1994, 5, 65
2
EFull CI
Formal Scaling
CBS Limit*
Not Graduating
O(n8)
O(n7)
O(n6)
O(n5)
O(n4)
ECCSD(T)
ECISD
ECCSD
EMP2
EHF
Eexact
Increase in Correlation Energy/Accuracy
Coupled‐Cluster Theory
( )
1 ˆ2 1 ˆ3
⎛
⎞
ˆ
ˆ
Ψ CC = exp T Φ ref = ⎜1 + T + T + T + ... ⎟ Φ ref
2!
3!
⎝
⎠
Tˆ = Tˆ + Tˆ + ... + Tˆ
Includes contributions from 1
TˆN Φ ref =
*complete basis set limit
N
2
occ
virt
∑ ∑
i < j < k <… a <b < c <…
t
abc…
ijk…
Φ
abc…
ijk…
higher‐order disconnected excitations. T12T2, T22, T14 appears in CCSD.
Bartlett, R. J.; Stanton, J. F. Rev. Comp. Chem. 1994, 5, 65
EFull CI
Formal Scaling
CBS Limit*
Not Graduating
O(n8)
O(n7)
O(n6)
O(n5)
O(n4)
ECCSD(T)
ECISD
ECCSD
EMP2
EHF
Eexact
Increase in Correlation Energy/Accuracy
Density Functional Theory
Hˆ DFT = ∑ h ( i ) + ∑ VHartree ( i ) + cHFexchange ∑ VExchange ( i ) + ∑ Vxc ( i )
i
i
i
i
δ Exc
Vxc ( r ) =
; Exc = ∫ f ( ρα , ρ β , γ αα , γ αβ , γ ββ ,τ α ,τ β ) dr
δρ ( r )
*complete basis set limit
Summary
• Electron correlation is important
• Different approaches to account for electron correlation
CAN WE CORRELATE ELECTRONS?
Summary
• Electron correlation is important
• Different approaches to account for electron correlation
CAN WE CORRELATE ELECTRONS?
YES, WE CAN.
Solvent‐induced shift of the lowest singlet π→π* charge‐transfer excited state of p‐nitroaniline
Background
• p‐nitroaniline
– Prototypical donor‐π‐acceptor chromophore
– Two mesomeric structures
– Strong solvent‐dependent π→π* absorption band in near UV‐Vis region
• Charge transfer from amino to nitro group across phenyl ring
– Benchmark solvent models
Neutral
Zwitterionic
Purpose
• Test the performance and applicability of the combined time‐dependent density functional theory/effective fragment potential (TDDFT/EFP1) method
TDDFT/EFP1
Time‐dependent Density Functional Theory
⎡A B ⎤ ⎡X ⎤
⎡ 1 0 ⎤ ⎡ X ⎤ V ⎡ ρ ⎤ ( r, t ) ≈ V ⎡ ρ ⎤ ( r )
xc ⎣ ⎦
xc ⎣ t ⎦
⎢
⎥⎢ ⎥ = ω⎢
⎥⎢ ⎥
⎢ B A ⎥ ⎢Y ⎥
⎢ 0 −1 ⎥ ⎢ Y ⎥
Adiabatic Approximation
⎢
⎥
⎣
⎦ ⎣⎢ ⎦⎥
⎣⎢
⎦⎥ ⎣ ⎦
Use ground‐state density functionals
Effective Fragment Potential Method (EFP1)
Einteraction = ECoulomb + Epolarization + Eremainder
H system = H abinitio + Vinteraction
region
EFP
EFP
region
ab initio
region
Computational Details
• Gas phase: B3LYP/DH(d,p)
– Optimized and characterized
• Condensed phase: B3LYP/DH(d,p)
– NVT simulation with 150 EFP waters at 300K using Nosé‐Hoover thermostat
• 20 ps equilibration followed by 20 ps production run
• 2000 configurations sampled
• Excitation energies: TD‐B3LYP/DH(d,p)
TD‐B3LYP/EFP1 Simulated π→π* Excited‐State Spectrum for pNA in Water
0.75
0.50
0.25
0.00
4.25
Calculated gas‐phase excitation energy
Normalized Intensity
1.00
4.00
HOMO
π
3.75
LUMO
π*
3.50
Vertical Excitation Energy (eV)
3.25
3.00
pNA + 150 EFP Waters
Varying the Density Functional
TDDFT/EFP1/DH(d,p)//B3LYP/EFP1/DH(d,p)
eV
Exptl.a, b, c
TD‐
B3LYP
TD‐
PBE0
TD‐
CAM
CIS(D)d
Gas phase
4.24
3.97
4.11
4.40
4.65
Condensed phase
3.26
3.37
3.44
3.50
3.65
‐0.98
‐0.60
‐0.67
‐0.90
‐1.00
0.60
0.22
0.22
0.20
0.46
Solvent Shift π→π*
Shift (condensed – gas)
Line Width (FWHM)
a
Millefiori et al. Spectrochim. Acta 1977, 33A, 21‐27
b Thomsen et al. J. Phys. Chem. A 1998, 102, 1062‐1067
c Kovalenko et al. Chem. Phys. Lett. 2000, 323, 312‐322
d Kosenkov, D.; Slipchenko, L. V. J. Phys. Chem. A. 2011, 115, 392‐401
pNA + 150 EFP Waters
The Bulk! Are We There Yet?
TDDFT/EFP1/DH(d,p)//B3LYP/EFP1/DH(d,p)
TDDFT/EFP1‐CPCM/DH(d,p)//B3LYP/EFP1/DH(d,p)
eV
Exptl.a, b, c
TD‐
B3LYP
TD‐
B3LYP
TD‐
PBE0
TD‐
CAM
TD‐
CAM
CIS(D)d
Gas phase
4.24
3.97
3.97
4.11
4.40
4.40
4.65
Condensed phase
3.26
3.37
3.37
3.44
3.50
3.51
3.65
‐0.98
‐0.60
‐0.60
‐0.67
‐0.90
‐0.89
‐1.00
Solvent Shift π→π*
Shift (condensed – gas)
a
Millefiori et al. Spectrochim. Acta 1977, 33A, 21‐27
b Thomsen et al. J. Phys. Chem. A 1998, 102, 1062‐1067
c Kovalenko et al. Chem. Phys. Lett. 2000, 323, 312‐322
d Kosenkov, D.; Slipchenko, L. V. J. Phys. Chem. A. 2011, 115, 392‐401
Increase in Zwitterionic Character
pNA
(electrons)
q(N9)
q(H10)
q(H11)
q (N14)
q(O15)
q(O16)
Dipole
(Debye)
Gas phase
‐0.64
0.32
0.32
0.58
‐0.40
‐0.40
7.6
Condensed phase
‐1.02
0.54
0.54
0.65
‐0.56
‐0.56
16.2
Structural Parameter
(angs. and degs.)
Exptl.1 MP22
B3LYP/
B3LYP EFP1
R(C2‐C1) 1.41
1.41
1.42
1.43
R(C3‐C2)
1.37
1.39
1.39
1.38
R(C4‐C3)
1.39
1.40
1.40
1.42
R(C1‐N9)
1.35
1.38
1.36
1.35
R(C4‐N14)
1.45
1.47
1.46
1.41
R(N14‐O15)
1.23
1.25
1.24
1.26
θ(C2C1N9H10)
19.9
10.0
θ(C5C4N14O15)
0.0
7.9
Neutral
Zwitterionic
Solvent‐Shift Contributions
TDDFT/EFP1/DH(d,p) (eV)
Contribution
TD‐B3LYP
TD‐CAM‐B3LYP
Solute Relaxation
‐0.09
‐0.12
‐0.22
Coulomb
‐0.41
‐0.42
‐0.39
Polarization
‐0.07
‐0.10
‐0.26
Remainder
‐0.03
‐0.03
‐0.03
Total
‐0.60
‐0.67
‐0.90
0.0
TD‐B3LYP
-0.2
Energy (eV)
TD‐PBE0
-0.4
-0.6
-0.8
-1.0
Solute Relaxation
Coulomb
Polarization
Remainder
Total Shift
-0.09 eV
-0.41 eV
-0.07 eV
-0.03 eV
-0.60 eV
Solute relaxation and polarization are sensitive to the choice of density functional Performance of TDDFT/EFP1
Single MD Snapshot
TD‐B3LYP/DH(d,p)
150 DFT Waters 150 EFP1 Waters
3.33
3.49
Total Time (minutes)
25783.6
7.6
Total Memory
19.8 GB
0.06 GB
π→π* (eV)
Summary
• TD‐B3LYP/EFP1 – Reproduces the experimental red shift and – Within ~ 0.1 eV of the experimental condense phase value for the π→π* excitation energy
• Solvent shift values and contributions (solute relaxation/polarization) are density functional sensitive
• Increase in zwitterionic character of the ground‐
state upon solvation
• Largest contribution to the solvent shift is from Coulomb interactions
Benchmarking the performance of time‐dependent density functionals
Excitation Types
Excited
T0
Tv
Te
vertical
adiabatic
Te with ZPE corrections
Ground
Benchmarking the Calculation of Vertical Excitations
Excited
T0
Tv
Te
vertical
adiabatic
Te with ZPE corrections
Ground
Time‐Dependent Kohn‐Sham Equations
Adiabatic Approximation
⎡ 1 2
⎤
∂
K
K
K
i ϕi ( r , t ) = ⎢ − ∇ + VTDDFT ( r , t ) ⎥ ϕi ( r , t )
⎢⎣ 2
⎥⎦
∂t
K
ρ (r ,t ) K
K
K
VTDDFT ( r , t ) = v ( t ) + ∫ K K dr ' + vXC [ρ ]( r , t )
r −r '
δE XC ⎡⎣ ρt ⎤⎦
K
K
⎡
⎤
vXC ( r , t ) ≈
K = vXC ⎣ ρt ⎦ ( r )
δρ ( r )
t
Adiabatic Approximation
Uses the density and density functional from the ground‐state
Linear Response TDDFT
Time domain
K
δρ ( r , t )
Fourier Transform
Frequency domain
K
δρ ( r , ω )
⎡A B⎤ ⎡X⎤
⎡1 0 ⎤ ⎡X⎤
⎢
⎥⎢ ⎥ = ω⎢
⎥⎢ ⎥
⎢ B A⎥ ⎢Y⎥
⎢ 0 −1 ⎥ ⎢ Y ⎥
⎢⎣
⎥⎦ ⎢⎣ ⎥⎦
⎢⎣
⎥⎦ ⎢⎣ ⎥⎦
Linear Response TDDFT
•
•
•
Excitation energies are determined as poles of the response function
Non‐Hermitian eigenvalue problem
Requires second functional derivative with respect to the electron density (non‐trivial for certain functional types)
Linear Response TD‐DFT
Classification of Density Functionals
Classification
Density Functional Dependence
Local Density Approximation (LDA)
f ( ρα , ρ β )
Generalized Gradient Approximation (GGA)
f ( ρα , ρ β , γ αα , γ αβ , γ ββ )
f ( ρα , ρ β , γ αα , γ αβ , γ ββ ,τ α ,τ β )
meta‐Generalized Gradient Approximation (mGGA)
Classification
Amount of Hartree‐Fock Exchange
Pure
none
Global Hybrid (GH)
constant (all range)
Range Separated Hybrid (RSH)
varies (short‐ vs. long‐ range)
Purpose
• Study the performance of different time‐
dependent density functionals for the calculation of singlet, triplet, valence (π→π*, n→π*, n→σ*, σ→π*), and Rydberg vertically excited states
• Offer recommendations for density functional usage for vertical excited‐estate calculations using time‐dependent density functional theory within the adiabatic approximation
Computational Details
• Ground state
– Optimized and characterized using the 6‐31++G(3df,3dp) basis set and the respective functional used for vertical excited state calculations
• Excited states
– Vertical excitations performed on ground‐state optimized geometries using the same basis set and functional
• Estimator of Performance
– Mean Signed Error (MSE)
– Mean Absolute Error (MAE)
– Root Mean Square Error (RMS)
Benchmark Set
• 14 small‐ to medium‐size compounds with 101 total experimental excited state energies
– 101 excited states can be broken down into:
• 63 singlets and 38 triplets states or
• 60 valence and 41 Rydberg states
– 60 valence states can be broken down into:
• 30 π→π*, 26 n→π*, 3 n→σ*, 1 σ→π*
benzene, butadiene, cyclopentadiene, ethylene, formaldehyde, furan, methylenecyclopropene, pyrazine, pyridine, pyrrole, s‐tetrazine, s‐trans acrolein, s‐trans glyoxal, water
Benchmarked Density Functionals
Functional Year
SVWN
BLYP
PW91
PBE
OLYP
BHHLYP
B3LYP
PBE0
X3LYP
CAM‐B3LYP
1980
1988
1992
1997
2001
1993
1994
1997
2004
2004
Type
LDA
GGA
GGA
GGA
GGA
GH‐GGA
GH‐GGA
GH‐GGA
GH‐GGA
RSH
%HF
50
20
25
21.8
19 – 65
Functional Year
VS98
PKZB
TPSS
M06‐L
TPSSm
revTPSS
TPSSh
M05
M05‐2X
M06
M06‐2X
M06‐HF
M08‐HX
M08‐SO
1998
1999
2004
2006
2007
2009
2004
2005
2006
2006
2006
2006
2008
2008
Type
%HF
mGGA
mGGA
mGGA
mGGA
mGGA
mGGA
GH‐mGGA
GH‐mGGA
GH‐mGGA
GH‐mGGA
GH‐mGGA
GH‐mGGA
GH‐mGGA
GH‐mGGA
10
28
56
27
54
100
52.23
56.79
Convergence Issues
Excited State Type
Count
non‐mGGA mGGA
Singlet
63
35
Triplet
38
25
Excited State Type
Count
non‐mGGA mGGA
Valence
60
28
Rydberg
40
32
Valence Excited State Type
Count
non‐mGGA mGGA
π→π*
30
12
n→π*
26
16
n→σ*
3
0
σ→π*
1
0
Performance for Singlet and Triplet
Exchange-Correlation (XC) Functional
Triplet Total
Singlet Total
Total
M08-SO
M08-HX
M06-HF
M06-2X
M06
M05-2X
M05
TPSSh
revTPSS
TPSSm
M06-L
TPSS
PKZB
VS98
CAMB3LYP
X3LYP
PBE0
B3LYP
BHHLYP
OLYP
PBE
PW91
BLYP
SVWN
GH-mGGA
mGGA
RSH
GH-GGA
GGA
LDA
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
Mean Absolute Error (eV)
Singlets
LDA, GGA, GH‐GGA
• PBE0 (0.25 eV)
• CAM (0.28 eV)
• OLYP (0.89 eV)
mGGA, GH‐GGA
• M06‐2X (0.21 eV)
• M06 (0.81 eV)
Triplets
LDA, GGA, GH‐GGA
• B3LYP, X3LYP, PBE0 (0.32 eV)
• CAM (1.10 eV)
mGGA, GH‐GGA
• M06‐2X (0.24 eV)
• PKZB (0.57 eV)
SVWN better than GGA for singlets and triplets!
Performance for Valence and Rydberg
Exchange-Correlation (XC) Functional
Rydberg Total
Valence Total
Total
M08-SO
M08-HX
M06-HF
M06-2X
M06
M05-2X
M05
TPSSh
revTPSS
TPSSm
M06-L
TPSS
PKZB
VS98
CAMB3LYP
X3LYP
PBE0
B3LYP
BHHLYP
OLYP
PBE
PW91
BLYP
SVWN
Valence
LDA, GGA, GH‐GGA
• B3LYP, X3LYP, PBE0 GH-mGGA
(< 0.30 eV)
• CAM (0.84 eV)
mGGA, GH‐GGA
• M06 (0.25 eV)
• M06‐HF (0.48 eV)
mGGA
• M06‐L (0.29 eV)
Rydberg
LDA, GGA, GH‐GGA
RSH
• CAM (0.22 eV)
GH-GGA
• PBE0 & BHHLYP (0.25 eV)
mGGA, GH‐GGA
GGA
• M06‐2X (0.17 eV)
• M06 (1.08 eV)
LDA
SVWN better than GGA 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 for Rydberg!
Mean Absolute Error (eV)
Performance for π→π* and n→π*
Exchange-Correlation (XC) Functional
Valence n-π*
Valence π-π*
Valence Total
M08-SO
M08-HX
M06-HF
M06-2X
M06
M05-2X
M05
TPSSh
revTPSS
TPSSm
M06-L
TPSS
PKZB
VS98
CAMB3LYP
X3LYP
PBE0
B3LYP
BHHLYP
OLYP
PBE
PW91
BLYP
SVWN
GH-mGGA
mGGA
RSH
GH-GGA
GGA
LDA
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
Mean Absolute Error (eV)
n→π*
LDA, GGA, GH‐GGA
• All GH‐GGAs (0.25 – 0.32 eV)
• CAM (0.56 eV)
mGGA, GH‐GGA
• M06 (0.22 eV)
• M06‐HF (0.58 eV)
π→π*
LDA, GGA, GH‐GGA
• All LDA, GGA, GH‐GGA (0.27 – 0.36 eV)
• CAM (1.14 eV)
mGGA, GH‐GGA
• M06‐L (0.25 eV)
• M05 (0.50 eV)
Summary
Excited State Type
Best non‐mGGA (MAE eV)
Best mGGA (MAE eV)
Singlet
PBE0 , CAM (<0.28)
M06‐2X (0.21)
Triplet
PBE0, B3LYP, X3LYP (<0.32)
M06‐2X (0.24)
Valence
PBE0, B3LYP, X3LYP (<0.30)
M06‐L, M06, M06‐2X (<0.30)
Rydberg
PBE0, BHHLYP, CAM (<0.25)
M06‐2X (0.17)
π→π*
X3LYP, B3LYP (<0.27)
M06‐L (0.25)
n→π*
PBE, B3LYP, X3LYP (0.28)
M06 (0.22)
PBE0 (0.28)
M06‐2X (0.22)
Overall
•
•
•
•
•
•
SVWN does better than all GGAs overall DO NOT RECOMMEND any GGAs
For pure density functionals use SVWN (0.54) or M06‐L (0.39)
For GH‐GGAs use PBE0
For GH‐mGGAs use M06‐2X
Avoid using CAM‐B3LYP for triplets
The combined coupled‐
cluster/effective fragment potential approach
Background
• A majority of interesting chemical and all biological processes do not take place in a vacuum.
• Such processes often occur in solution, on surfaces, and at active sites of biochemical systems.
• Use of high level ab initio electronic structure methods is computationally prohibitive for large systems.
Effective Fragment Potential (EFP)
• A discrete approach to treating environmental effects.
EFP
• Effect of fragment molecules
region
contribute to the one‐electron
ab initio
region
terms in ab initio Hamiltonian.
H system = H abinitio + Vinteraction
region
Effective Fragment
Potential
Effective Fragment Potential (EFP1)
Einteraction =ECoulomb + Epolarization + Eexchange repulsion/charge transfer
Vinteraction =
K
L
k =1
l =1
Elec
Pol
V
μ
,
s
+
V
(
)
∑ k
∑ l ( μ, s ) +
Distributed
Multipolar expansion
LMO
polarizability expansion
Jensen, J.H.; Gordon, M.S. Mol. Phys. 1996 , 89, 1313 Day, P.N.; Jensen, J.H.; Gordon, M.S.; Webb, S.P.; Stevens, W.J.; Krauss, M.;
Garmer, D.; Basch, H.; Cohen, D. J. Chem. Phys. 1996, 105, 1968
Adamovic, I.; Freitag M.A.; Gordon, M.S. J. Chem. Phys. 2003 ,118, 6725
M
Rep
V
∑ m ( μ, s )
m =1
Fit to Functional Form
Effective Fragment Potential (EFP1)
• Interfaced with several ab initio methods
–
–
–
–
–
–
–
a
Hartree‐Fock Theory (HF)a
Density Functional Theory (DFT)b
Second‐Order Møller‐Plesset Perturbation Theory (MP2)c
Multi‐Configurational Self‐Consistent Field Theory (MCSCF)d
Single Excited Configuration Interaction (CIS)e
Time‐Dependent Density Functional Theory (TD‐DFT)f
Multi‐Reference Perturbation Theory (MRPT)g
Day, P.N.; Jensen, J.H.; Gordon, M.S.; Webb, S.P.; Stevens, W.J.; Krauss, M.; Garmer, D.; Basch, H.; Cohen, D. J. Chem. Phys. 1996, 105, 1968
b Adamovic, I.; Freitag M.A.; Gordon, M.S. J. Chem. Phys. 2003 ,118, 6725
c Song, J.; Gordon, M.S.; Unpublished work.
d Krauss, M.; Webb, S. P. J. Chem. Phys. 1997, 107, 5771
e Arora, P.; Gordon, M.S. J. Chem. Phys. A. 2010, 114, 6742‐6750.
f Yoo, S.; Zahariev, F.; Sok, S.; Gordon, M.S. J. Chem. Phys. 2008, 129, 144112
g DeFusco, A.; Gordon, M.S. J. Phys. Chem. A. 2011, ASAP, April 14, 2011.
Coupled‐Cluster Theory (CC)
The solution of the electronic Schrödinger equation,
Ĥ Ψ = E Ψ
is approximated by,
Tˆ
Ψ ≈ Ψ CC = e Φ ref
leading to,
Tˆ
Tˆ
ˆ
He Φ ref = E e Φ ref
J. Čížek J. Chem. Phys.; 1966, 45, 4256
J. Čížek Adv. Chem. Phys.; 1969, 14, 35
J. Čížek J. Paldus Int. J. Quant. Chem.; 1971, 5, 359
Crawford, T. D.; Schaefer III, H. F. Rev. in Comp. Chem.; 2000, 14, 33
Formal Scaling of CC Methods
CCSD
N6
CCSD(T)
N7
CR‐CC(2,3)
N7
EOM‐CCSD
N6
EOM‐CCSD(T)
N8
CC/EFP1
• Perform coupled‐cluster calculation using a HF/EFP1 reference wavefunction,
ˆ
ˆ
Ψ CC = eT Φ ref = eT Φ HF/EFP1
• This can be extended to the excited‐state using equation of motion coupled‐cluster†,
ˆ Tˆ Φ
Ψ EOM-CC = Rˆ Ψ CC = Re
HF/EFP1
1
abc…
ˆ
ˆ
ˆ
ˆ
ˆ
R = R0 + R1 + R2 + … , Rn Ψ = ∑ rijkabc…… Ψ ijk
…
n!
†Stanton, J. F.; Bartlett, R. J. J. Chem. Phys. 1993, 98, 7029 and references therein
Application of CC/EFP1
NO3‐ (H2O)15 Solvation • Nitrates
– Most abundant naturally occurring ion in our environment (sea salt aerosols in troposphere)
– Photochemistry (Surface vs. interior)a
• MD simulations with Polarizable Force Field Methods: cross over from surface to interior 300 – 500 waters
• QM/MM (MP2/EFP) Monte Carlo/Simulated Annealing:
cross over from surface to interior ≈ 32 waters
• MP2/DH(d,p) optimized geometries at 32 waters show E(interior) – E(surface) = 0.5 kcal/mol
a
Miller, Y.; Thomas, J. L.; Kemp, D. D.; Finlayson‐Pitts, B. J.; Gordon, M. S.; Tobias, D. J.; Gerber, R. B. J. Phys. Chem. A. 2009, 113, 12805‐12814
Computational Details
NO3‐ (H2O)15 Solvation
• Lowest surface and interior structures
– MP2/EFP1/DH(d,p) Monte Carlo/Simulated Annealing simulations
• CCSD(T)/EFP1/6‐31+G(d) single‐point energy
– Calculation: 26 minutes on 1 cpu w/ 6.3 GB memory
• Compare with fully ab initio calculations CCSD(T)/6‐31+G(d)
– EFP1 waters are converted to ab initio waters
– Calculation: 2 days on 72 cpus w/ 52.5 GB dist. mem.
NO3‐ Solvation
15 Waters
Surface A
0 (0) [0]
15 Waters
Surface B
0.3 (1.6) [‐1.3]
CCSD(T)‐EFP1/6‐31+G(d)//MP2‐EFP1/DH(d,p) (CCSD(T)/6‐31+G(d)//MP2‐EFP1/DH(d,p))
[Difference: EFP – Full ab initio]
15 Waters
Interior
6.7 (1.0) [ 5.6]
Computational Details
π→π* Solvent‐Shift of pNA
• Condensed phase: B3LYP/DH(d,p)
– NVT simulation with 150 EFP waters at 300K using Nosé‐Hoover thermostat
• 20 ps equilibration followed by 20 ps production run
• Excitation Energies
– EOM‐CCSD/EFP1/DH(d,p)
– EOM‐CCSD/EFP1/6‐31+G(d)
π→π* Solvent Shift of p‐nitroaniline
Excitation Energy (eV)
Method‐EFP1/Basis Set//B3LYP‐EFP1/DH(d,p)
EOM‐CCSD/
DH(d,p)
EOM‐CCSD/
6‐31+G(d)
CIS(D)/
6‐31+G(d)a
Gas
4.24
3.97
4.73
4.57
4.65
Condensed
3.26
3.37
3.45
3.41
3.65
‐0.98
‐0.60
‐1.28
‐1.16
‐1.00
0.60
0.22
0.26
0.27
0.46
π→π*
Shift
Line Width (FWHM)
a
Kosenkov, D.; Slipchenko, L. V. J. Phys. Chem. A. 2011, 115, 392‐401
Millefiori et al. Spectrochim. Acta 1977, 33A, 21‐27
c Thomsen et al. J. Phys. Chem. A 1998, 102, 1062‐1067
d Kovalenko et al. Chem. Phys. Lett. 2000, 323, 312‐322
b
Normalized Intensity
Exptl.b, c, d
TD‐B3LYP/
DH(d,p)
1.0 EOM‐CCSD/DH(d,p)
Condensed‐phase
0.8 Spectrum of
π→π* state of pNA
0.6
0.4
0.2
0.0
4.2
4.0
3.8
3.6
3.4
3.2
Vertical Excitation Energy (eV)
3.0
Summary
• Nitrate solvation
– Largest error is on the order of 5 kcal/mol
– Delocalized system may be difficult to treat with the CC/EFP1 method
• pNA solvent shift
– Calculated shift is less than 0.18 eV from experimental shift
• The CC/EFP1 method offers a significant step towards a full treatment of solvent effects with CC theory
Acknowledgements
•
•
•
•
•
•
•
•
•
Iowa State University ‐Summer REU w/ Kai‐Ming Ho
Mark Gordon – exp ( thanks ) = 1 + thanks + 1 thanks 2 + 1 thanks3 + ...
2!
3!
Mike Schmidt – GAMESS and coding Dr. Federico Zahariev – all things dealing with ρ ( r )
D. Soohaeng Yoo –MD simulations and TDDFT/EFP1
Dr. Albert Defusco – EFP energy decomposition
Dr. Lyudmila Slipchenko – CC/EFP1
Dr. Tony Smith and Dr. Luke Roskop
Leo C. DeSesso; personal attorney
Acknowledgements
It takes a village research group to raise a child scientist kidult.
[Ancient]
[Older]
[Old]
[Current]
[REU @ISU]
;
Equations: Time‐Independent Schrödinger
Ĥ Ψ = E Ψ
Hˆ = Tˆe + Tˆn + Vˆen + Vˆee + Vˆnn
2
2
M
N M
N
M
∇
Z
Z AZB
∇
1
i
A
A
ˆ
H =∑
−∑
− ∑∑
+∑ +∑
i =1 2
A =1 2 M A
i =1 A=1 riA
i < j rij
A< B rAB
N
Hˆ elec Ψ elec = Eelec Ψ elec
Hˆ elec
within the Born‐Oppenheimer Approximation
N M
∇
ZA
1
= −∑
− ∑∑
+∑
i =1 2
i =1 A =1 riA
i < j rij
N
2
i
E = Eelec + Enucl
Equations: Independent Particle Model (IPM)
Hˆ IPM Ψ IPM = EIPM Ψ IPM
N
Hˆ IPM = ∑ h ( i )
i
N
Ψ IPM = ∏ φi
i
N
EIPM = ∑ ε i
i =1
h ( i ) φi = ε iφi
Equations: Basis Set
φi =
basis
functions
∑
α
cα i χ i
Linear Combination of Atomic Orbitals (LCAO)
Used in this talk:
DH(d,p)
6‐31G(d)
6‐31+G(d)
6‐31++G(d)
6‐311++G(3df,3pd)
EFP Equations and Density Matrix
E = E + ∑ Pμ ,ν Vμν + VN
( efp )
μ ,ν
Vμν = ∫ dr1φμ (1) V
*
( efp )
VN
= ∑ Z AV
( efp )
( efp )
(1) φν (1)
( A)
A
V
( efp )
K
= ∑V
k
Elec
k
L
M
(1) + ∑ Vl (1) + ∑ V (1)
l
Pol
m
Rep
m
Line Width
w/ EFP Fragments
w/ EFP Fragments Removed
Difference
• Rotation about –NO2 group during solvation in the ground state contributes to redshift.
Farztdinov, V. M.; Schanz, R.; Kovalenko, S. A.; Ernsting, N. P.; J. Phys. Chem. A; 2000, 104, 11486