The application of density functional response theory to large molecular systems
Trygve Helgaker, University of Oslo, Norway
Michal Jaszuński, Peter Macak, Pawel Salek, Olav Vahtras, Mark A. Watson, Hans Ågren
Molecular Quantum Mechanics
The No Nonsense Path to Progress
July 24–29, 2004
St. John’s College—Cambridge University, Cambridge, England
Overview
1. computational techniques for large molecular systems
the Coulomb problem
response theory
2. applications to large molecular systems
excitation energies and polarizabilities
indirect nuclear magnetic spin–spin coupling constants
two-photon absorption (TPA) cross sections
1
Braithwaite, Keswick, Lake District
2
The application of density functional response theory to large molecular systems
Trygve Helgaker, University of Oslo, Norway
Michal Jaszuński, Peter Macak, Pawel Salek, Olav Vahtras, Mark A. Watson, Hans Ågren
Molecular Quantum Mechanics
The No Nonsense Path to Progress
July 24–29, 2004
St. John’s College—Cambridge University, Cambridge, England
Overview
1. computational techniques for large molecular systems
the Coulomb problem
response theory
2. applications to large molecular systems
excitation energies and polarizabilities
indirect nuclear magnetic spin–spin coupling constants
two-photon absorption (TPA) cross sections
3
Time spent in a single Kohn–Sham iteration
• Linear polyethylene chains at the LDA/3-21G level of theory (Dalton)
mins
100
– small systems dominated by
the Coulomb evaluation
80
SCF
– large systems dominated by
the SCF diagonalization
60
Coulomb
40
– the XC evaluation takes
relatively little time
20
LDA XC
200
400
600
800
1000
atoms
` ´
• For large systems, an alternative to O n3 diagonalization must be sought
– many proposals have been made (Scuseria, Head-Gordon, Challacombe, and others)
– exponential parametrization of the AO density matrix:
D (X) = exp (−XS) D0 exp (SX) ,
XT = −X
direct minimization of the energy with respect to X, without generating MOs
• However, all applications presented here use traditional diagonalization
• We shall here discuss in some detail only our implementation of the Coulomb step
– the Coulomb problem is the first obstacle that must be overcome for large systems
4
The fast multipole method (FMM) for Coulomb interactions
• Let us consider the evaluation of the Coulomb contribution to the Kohn–Sham matrix:
8
< n significant orbital products ab and cd
X
Fab =
(ab|cd) Dcd ←
: n2 significant integrals (ab|cd)
cd
• For fast evaluation, the n2 significant integrals are traditionally divided into two classes:
– the nonclassical integrals between overlapping
charge distributions; their number scales as n
– the classical integrals between nonoverlapping
charge distributions; their number scales as n2
5104
electron repulsion
n4
2104
n2
n1
3
5
7
9
11
13
15
• The nonclassical contribution to Fab is calculated by standard techniques in linear time
• The contribution from the n2 classical integrals is evaluated by multipole methods
– linear scaling is achieved by the fast multipole method (FMM)
(Head-Gordon, Scuseria)
• The FMM was developed for point particles, partitioned into a hierarchy of boxes
– finite Gaussian distributions require a generalization (White & Head-Gordon, 1994)
– in the continuous FMM (CFMM), Gaussians are divided into branches based on size
– branches add an extra level of complexity to the FMM code
5
The branch-free FMM (BFMM)
• In BFMM, we avoid branches altogether by partitioning the contributions differently:
– each integral is decomposed into a point-charge term and a size-correction term:
“ pq ”3/2 Z Z exp(−pr2 ) exp(−qr2 )
1P
2Q
−1
sc
dr
dr
=
R
+
J
1
2
pq
PQ
π2
r12
−1
– the n2 point-charge contributions RP
Q are treated by FMM, without branches
sc are evaluated explicitly, in linear time:
– the n local size corrections Jpq
”h
”i
“
“
pq
1
−2
−4
sc
2
Jpq = − √
exp −αRP Q RP Q + O RP Q , α =
πα
p+q
• Observe: Coulomb integrals with RP Q ≈ 0 cannot be decomposed in this manner
– such neighbour interactions are in any case not treated by multipoles in FMM
mins
60
• LDA/3-21G polyethylenes
– CFMM:
contracted basis, 10−7 Eh
CFMMun2,25
50
40
CFMM8,20
30
– CFMMun and BFMMun:
internally decontracted basis, 10−8 Eh
BFMMun2,25
20
10
200
6
400
600
800
1000
atoms
Response theory
• The expectation value of  in the presence of a perturbation V̂ ω of frequency ω:
D ˛ ˛ E D ˛ ˛ E Z
˛ ˛
˛ ˛
t ˛Â˛ t = 0 ˛Â˛ 0 + Â; V̂ ω ω exp (−iωt) dω + · · ·
• The linear response function is evaluated in the following manner:
(
Â; V̂ ω ω = −A† (E − ωS)−1 Vω
|
{z
}
←
E electronic Hessian
S metric matrix
linear equations
the linear equations are solved iteratively in a reduced space
key step: the multiplication of the Hessian with trial vectors a = Eb
quadratic response functions are obtained in a similar manner
• We have implemented DFT linear and quadratic
response theory in Dalton
linear-scaling formation of Coulomb and XC
contributions to Hessian products
sparsity not exploited in one-electron matrices
linear scaling up to about 150 carbons due to
small prefactor of linear algebra
7
Review and preview
• Large-system code conversion:
– Coulomb FMM removes the first bottleneck for large systems
for maximum efficiency, density fitting is mandatory
– wave-function optimization is often more difficult for large systems
diagonalization expensive
SCF convergence often troublesome
– response theory poses relatively few problems
iterative subspace algorithms used from the beginning
convergence of the linear equations is easy (compared with SCF iterations)
for large systems, most of the time is spent doing (nonsparse) linear algebra
• Large-system applications:
– excitation energies and polarizabilities of linear hydrocarbon chains
– indirect nuclear spin–spin coupling constants in large molecules
– two-photon absorption (TPA) cross sections of large systems
8
Excitation energies of linear hydrocarbons and of D6h graphite sheets
• The lowest 20 singlet and 20 triplet LDA/4-31G excitation energies as functions of n−1
nalkanes Cn H2 n2
polyenes Cn Hn2
0.6
0.6
0.3
0.3
200
12
6
4
2
3
200
0.6
0.3
0.3
12
6
4
4
6
2
3
D6 h graphite sheets
polyynes Cn H2
0.6
200
12
2
3
200
12
4
6
2
3
• In the linear chains, the lowest excitation energy decreases roughly as a + bn−1 .
BLYP/6-31G
2
20
200
∞
99%
2
20
200
∞
99%
polyethylenes
369
259
251
251
60
348
257
251
250
60
polyenes
301
81
34
32
110
162
43
29
29
90
polyynes
264
66
41
41
80
194
51
36
36
80
• In the graphite sheets, the lowest excitation energy is proportional to n−1/2 .
9
BLYP vs. LDA
• In general, BLYP performs in much the same manner as LDA for excitation energies:
– the BLYP excitation energies are typically within 1 or 2 mEh of the LDA energies
– the differences between LDA and BLYP are smaller for longer chains
– BLYP increases the triplet–singlet separation slightly
• Plots of the LDA/6-31G and BLYP/6-31G excitation energies in the polyynes:
0.09
0.09
singlet
triplet
0.07
0.07
0.05
0.05
50
100
150
50
100
150
• In general, LDA and GGA functionals underestimate excitation energies
– for HF, CO, and H2 O, the underestimation is 15% to 20%
– this problem is exacerbated in extended conjugated systems
– the introduction of nonlocality (beyond GGA) may be very important in providing
quantitatively correct excitation energies, in particular in conjugated systems
10
Polarizabilities and group polarizabilities
• The LDA STO-3G and 4-31G polarizabilities as functions of the number of carbons n
3
50
2
30
1
200
300
150
8
200
8
300
200
450
4
polyynes Α
600
150
210
30
100
140
20
70
polyynes Αn
polyenes Αn
150
300
– simple model: α(n) = anb ,
300
200
450
600
graphite Α
1000
150
50
alkanes Αn
100
80
polyenes Α
10
12
4
12
40
alkanes Α
100
120
10
300
450
graphite Αn32
1000
150
300
450
1≤b≤3
– the polyethylene polarizability depends linearly on n, even for very small systems
– the polyyne and polyene polarizabilities increase first quadratically and then linearly,
they are an order of magnitude larger than the polyethylene polarizability
– the graphite polarizability is proportional to n3/2
• Polarizability model: plots of the inverse excitation energies times n:
6
60
60
300
4
40
40
200
20
2
50
100
150
100
20
alkenes
alkanes
50
100
150
11
graphite
alkynes
50
100
150
100
200
300
400
Indirect nuclear spin–spin coupling constants
• With each nucleus in a molecule, there is an associated magnetic moment MP :
– their direct interactions vanish in isotropic media
– the residual indirect interaction arises from hyperfine
interactions with the electrons ≈ 10−16 Eh ≈ 1 Hz
• The indirect nuclear spin–spin coupling constants are calculated as the second
derivatives of the total electronic energy (i.e., by linear response theory)
– for each nucleus, 3 singlet and 7 triplet response equations are solved
• The introduction of DFT has created something of revolution in the calculation of
spin–spin coupling constants, greatly expanding the application range of theory
• The accuracy of DFT is similar to that of wave-function theory:
LDA
HF
30
30
30
CAS
30
30
BLYP
30
30
RAS
30
30
B3LYP
30
SOPPA
30
12
30
30
30
CCSD
30
30
Simulated 200 MHz NMR spectra of vinyllithium
experiment
0
RHF
100
0
200
MCSCF
0
100
200
100
200
B3LYP
100
0
200
13
Valinomycin C54 H90 N8 O18
• DFT can be applied to large molecular systems such as valinomycin (168 atoms)
– there are a total of 7587 spin–spin couplings to the carbon atoms in valinomycin
– below, we have plotted the magnitude of the reduced LDA/6-31G coupling constants
on a logarithmic scale, as a function of the internuclear distance:
1019
1016
1013
500
1000
1500
– the coupling constants decay in characteristic fashion, which we shall examine
– most of the indirect couplings beyond 500 pm are small and cannot be detected
14
Valinomycin LDA/6-31G spin–spin couplings to CH, CO, CN, CC greater than 0.01 Hz
100
30
10
3
1
0.3
0.1
0.03
100
200
300
400
15
500
600
Valinomycin LDA/6-31G spin–spin couplings to CH, CO, CN, CC greater than 0.01 Hz
100
30
10
3
1
0.3
0.1
0.03
100
200
300
400
16
500
600
Valinomycin LDA/6-31G spin–spin couplings to CH, CO, CN, CC greater than 0.01 Hz
100
30
10
3
1
0.3
0.1
0.03
100
200
300
400
17
500
600
Valinomycin LDA/6-31G spin–spin couplings to CH, CO, CN, CC greater than 0.01 Hz
100
30
10
3
1
0.3
0.1
0.03
100
200
300
400
18
500
600
Valinomycin LDA/6-31G spin–spin couplings to CH, CO, CN, CC greater than 0.01 Hz
100
30
10
3
1
0.3
0.1
0.03
100
200
300
400
19
500
600
Valinomycin LDA/6-31G spin–spin couplings to CH, CO, CN, CC
100
1
0.01
0.0001
500
1000
20
1500
Spin and orbital hyperfine coupling mechanisms
• Hyperfine coupling to the spin of the electrons
˛
˛ E
˛
˙ ˛ FC
¸D
˛
˛
SD
FC
T
SD
T
X 0 ˛hP + hP ˛ nT nT ˛(hQ ) + (hQ ) ˛ 0
= −2
KFC+SD
PQ
EnT − E0
n
T
– the Fermi-contact (FC) and spin–dipole (SD) operators:
hFC
P
8πα2
=
δ(rP ) s,
3
hSD
P
=α
T
2 3rP rP
2I
− rP
3
5
rP
s
– there are no mixed FC/SD contributions to isotropic coupling constants
• Hyperfine coupling to the orbital motion of the electrons
˙ ˛ PSO ˛ ¸ D ˛˛ PSO T ˛˛ E
˛ E
D ˛
X 0 ˛hP ˛ nS nS ˛(hQ ) ˛ 0
˛
˛
DSO
KSO
P Q = 0 ˛hP Q ˛ 0 − 2
EnS − E0
n =0
S
– the diamagnetic spin–orbit (DSO) and paramagnetic spin–orbit (PSO) operators:
hDSO
PQ
=
rT r I − rP rT
Q
4 P Q 3
α
,
3
3
rP rQ
hPSO
= −α2
P
rP × i∇
3
rP
• The many terms make the calculation of spin–spin coupling constants a difficult task
21
Relative importance of the contributions to spin–spin coupling constants
• The isotropic indirect spin–spin coupling constants can be uniquely decomposed as:
PSO
FC
SD
KP Q = KDSO
P Q + KP Q + KP Q + KP Q
• The short-range spin–spin coupling constants are often dominated by the FC term.
– since the FC term is easy to calculate, it is tempting to ignore the other terms
– however, none of the contributions can be a priori neglected (N2 and CO):
200
PSO
PSO
100
SD
PSO
FC
FC
FC
FC
FC
FC
FC
FC
0
FC
FC
SD
SD
-100
H2
HF
H2O NH3 CH4 C2H4 HCN
OH NH CH CC NC
N2
CO C2H2
CC
– the DSO contribution, in particular, is very small
• We shall see that the situation is rather different for large internuclear separations.
22
The different long-range decays of the Ramsey terms
• Letting M be the center of the product Gaussian Ga Gb , we obtain
˛
˛
˛
˛
E
“
”
D
E
D
˛ FC ˛
˛ SD ˛
2
−3
Ga ˛hP ˛ Gb ∝ RP M ,
Ga ˛hP ˛ Gb ∝ exp −µRP M ,
˛
˛
˛
˛
D
E
E
D
˛ DSO ˛
˛ PSO ˛
−2
−2
−2
Ga ˛hP ˛ Gb ∝ RP
Ga ˛hP Q ˛ Gb ∝ RP M RQM ,
M
• Insertion in Ramsey’s expression gives (red positive, blue negative)
1019
FC decays exponentially
SD decays as R3
mixed signs
1019
mixed signs
1016
1016
1013
1013
500
1000
500
1500
DSO decays as R2
negative
1019
1016
1013
1013
500
1000
1500
500
23
1500
PSO decays as R2
positive
1019
1016
1000
1000
1500
The long-range orbital contributions
• At large separations, the couplings are dominated by the orbital contributions:
−2
JDSO
P Q ∝ RP Q ,
−2
JPSO
P Q ∝ RP Q
← large separations
• Moreover, in this limit, the DSO contributions all become negative:
˛ E
D ˛
˛ −3 −3
˛
DSO
2α4
0 ˛rP rQ rP · rQ ˛ 0 < 0 ← large separations
KP Q = 3
• Also, the PSO contributions become positive, nearly cancelling the DSO contributions.
– use of Taylor expansion, the virial theorem, and the resolution of identity give:
−3
PSO
JDSO
P Q + JP Q ∝ RP Q
← large separations, large basis
• However, the PSO contributions converge very slowly to the basis-set limit:
103
500
1000
103
1500
1
1
103
103
107
LDA631G
107
24
500
LDAHII
1000
1500
Basis-set convergence
• At large separations, the PSO term is
by far the most sensitive to the basis set.
1
• As the basis-set increases, the positive PSO
term increases, reducing the negative total
spin–spin coupling constants in magnitude.
• We have here plotted the four spin–spin terms
in the STO-3G, 6-31G, and HII basis sets.
500
103
1000
1500
FC red
SD green
DSO yellow
PSO blue
107
The Dirac vector model
• The Dirac vector model predicts sign alternation
with increasing number of intervening bonds.
75
• The plot shows the distribution of signs in
valinomycin (LDA/6-31G) as function of the
number of intervening bonds.
negative
50
25
• In the limit of many bonds, all couplings become
negative because of the dominant DSO term.
25
positive
5
10
15
20
The Karplus relation
• There are 282 vicinal coupling constants among
the 7578 carbon couplings in valinomycin.
• As predicted by the Karplus relation, these
couplings vanish for dihedral angles close to 90◦ .
10
8
6
4
2
• The Karplus curve arises from the FC term
(blue); the other terms (red) do not contribute.
45
90
135
180
-2
Large long-range couplings
• To the right, we have plotted the maximum,
mean, and minimum couplings as functions of
the number of intervening bonds.
• There are peaks at 11, 13, and 15 intervening
bonds (greater than 1 Hz in magnitude).
• All peaks correspond to distances less than
350 pm, respectively.
26
1
103
106
5
10
15
20
Two-photon absorption (TPA)
• As an example of a nonlinear process, we consider two-photon absorptions (TPAs)
– TPA cross sections depend quadratically on the intensity of light
– excitations can be reached by applying half the wavelength
• TPA cross sections can be extracted from quadratic response theory:
TfAB =
X
k>0
0|B̂|kk|Â − A0 |f 0|Â|kk|B̂ − B0 |f +
ωk − ωf + ω0
ωk − ω0
!
– we have implemented TPA and other quadratic-response properties in DFT
• For small systems, DFT gives reasonably accurate TPA cross sections:
– excitation energies (left, eV) and TPA cross sections (right, au) for HF (Sadlej basis)
15
6000
10
3000
5
1
–
2
3
4
5
1
6
RHF , CCSD , LDA , BLYP , B3LYP
27
2
3
4
5
6
TPA in extended systems
• LDA/6-31G TPA cross section of the lowest excited state in polyethylenes Cn H2n+2
TPA
DOS
9000
60000
DOSTPA
15
6000
40000
3000
20000
10
5
40
80
40
120
80
40
120
80
120
– for a given state, the TPA cross section decreases with increasing n
– at the same time, the density of state (DOS) increases quadratically with n
– in a given energy interval, the total cross section increases linearly for n > 100
• Corresponding plots for the lowest excited state in polyenes Cn Hn+2
50000
1400
TPA
6 10 7
5 10 7
DOS
1200
40000
1000
30000
800
20000
600
10000
20
40
60
80
100 120 140
DOSTPA
4 10 7
400
3 10 7
2 10 7
200
1 10 7
20
40
60
80
100 120 140
20
40
60
80
100 120 140
– for polyenes, the TPA cross sections are much larger and increases with increasing n
– extended conjugated systems are not well served by local density theory
28
AF455 (C137 H172 N6 )
• Studies in nonlinear optics
frequently involve large molecules
– ideally suited for DFT
– better functionals are needed
• AF455 is a system of particular
interest
• LDA/4-31G TPA cross sections
of the ten lowest excitations (eV)
400000
300000
200000
100000
1.8
2
2.4
2.6
29
Conclusions
• We have considered the calculation of molecular properties of large systems by DFT
– the first bottleneck is removed by introducing fast Coulomb evaluation
exact exchange and density fitting
– the single largest obstacle is wave function optimization
diagonalization expensive, convergence slow
– by contrast, the application of response theory is relatively straightforward
• We have considered the evaluation of some properties
– excitation energies and polarizabilities of long hydrocarbon chains
unsaturated systems may require exact exchange or current DFT
– indirect nuclear spin–spin coupling constants in large molecular systems
long-range couplings are dominated by the orbital contributions DSO and PSO
−2
−3
;
together,
as
R
separately, they decay is RP
Q
PQ
at large separations, the PSO contribution is most sensitive to basis set
– two-photon cross sections
quadratic response theory applicable to large systems
– extended conjugated systems
ultranonlocality requires approaches beyond LDA and GGA
30