Quantum chemistry: wave-function and density-functional...

Quantum chemistry:wave-function and density-functional methods

Trygve Helgaker

Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo

Electronic Structure e-Science Meeting,Swedish e-Science Research Centre (SeRC),

Engsholms Slott, Morko, Sweden,April 7–8, 2011

Trygve Helgaker (CTCC, University of Oslo) Quantum Chemistry Engsholms Slott, April 7–8 2011 1 / 41

Quantum chemistryWave-function vs. density-functional methods

I Quantum mechanics has been applied to chemistry since the 1920sI early accurate work on He and H2I semi-empirical applications to larger molecules

I The idea of ab-initio theory developed in the 1950sI early work in the 1950s, following the development of digital computersI Hartree–Fock (HF) self-consistent field (SCF) theory (1960s)I configuration-interaction (CI) theory (1970s)I multiconfigurational SCF (MCSCF) theory (early 1980s)I many-body perturbation theory (1980s)I coupled-cluster theory (late 1980s)

I Coupled-cluster theory is the most successful wave-function techniqueI introduced from nuclear physicsI size extensive and hierarchicalI the exact solution can be approached in systematic mannerI high cost, near-degeneracy problems

I Density-functional theory (DFT) emerged during the 1990sI Kohn–Sham theory introduced from solid-state physicsI evaluation of dynamical correlation from the densityI cost broadly similar to HF theoryI semi-empirical in characterI cannot be systematically improved


Quantum chemistryThe calculation of high-resolution NMR spectra by different electronic-structure methods

I 200 MHz NMR spectra of vinyllithium (C2H3Li)

0 100 200

MCSCF

0 100 200 0 100 200

B3LYP

0 100 200

0 100 200

experiment

0 100 200 0 100 200

RHF

0 100 200


From Hartree–Fock to coupled-cluster theoryThe Hartree–Fock approximation

I The Hartree–Fock model—the fundamental approximation of wave-function theoryI each electron moves in the mean field of all other electronsI provides an uncorrelated description: average rather than instantaneous interactionsI gives rise to the concept of molecular orbitalsI typical errors: 0.5% in the energy; 1% in bond distances, 5%–10% in other propertiesI forms the basis for more accurate treatments

I The Hartree–Fock and exact wave functions in helium:

!1.0!0.5

0.00.5

1.0

!0.50.0

0.5

!0.5

0.0

0.5

!1.0!0.5

0.00.5

.5

0.0

!1.0!0.5

0.00.5

1.0

!0.50.0

0.5

!0.5

0.0

0.5

!1.0!0.5

0.00.5

.5

0.0

I concentric Hartree–Fock contours, reflecting an uncorrelated descriptionI in reality, the electrons see each other and the contours becomes distorted


From Hartree–Fock to coupled-cluster theoryElectron correlation and virtual excitations

I For an improved description, we must describe the effects of electron correlationI in real space, the electrons are constantly being scattered by collisionsI in the orbital picture, these are represented by excitations from occupied to virtual spin orbitalsI the most important among these are the double excitations or pair excitations

I Consider the effect of a double excitation in H2:

|1σ2g〉 → (1 + tuu

gg Xuugg )|1σ2

g〉 = |1σ2g〉 − 0.11|1σ2

u〉

I the one-electron density ρ(z) is hardly affected:

-2 -1 0 1 2 -2 -1 0 1 2

I the two-electron density ρ(z1, z2) changes dramatically:

-2

0

2

-2

0

2

0.00

0.04

2

0

2

-2

0

2

-2

0

2

-2

0

2

0.00

0.04

2

0

2

-2

0

2


From Hartree–Fock to coupled-cluster theoryCoupled-cluster theory

I In coupled-cluster (CC) theory, the starting point is the HF descriptionI this description is improved upon by the application of excitation operators

|CC〉 =(

1 + tai Xai

)︸︷︷︸

singles

· · ·(

1 + tabij X abij

)︸︷︷︸

doubles

· · ·(

1 + tabcijk X abcijk

)︸︷︷︸

triples

· · · |HF〉

I with each virtual excitation, there is an associated probability amplitude tabc···ijk···I single excitations represent orbital adjustments rather than interactionsI double excitations are particularly important, arising from pair interactionsI higher excitations should become progressively less important

CCS, CCSD, CCSDT, CCSDTQ, CCSDTQ5, . . .

I The quality of the calculation depends critically on the virtual spaceI we employ atom-fixed Gaussian atomic orbitalsI virtual atomic orbitals are added in full shells at a timeI each new level introduces orbitals that recover the same amount of correlation energyI the number of virtual AOs per atom increases rapidly:

SZ DZ TZ QZ 5Z 6Z5 14 30 55 91 140

I the correlation-consistent Gaussian basis sets

cc-pVDZ, cc-pVTZ, cc-pVQZ, cc-pVTZ, cc-pV6Z, . . .


From Hartree–Fock to coupled-cluster theoryThe two-dimensional chart of quantum chemistry

I The quality of ab initio calculations is determined by the description of

1 the N-electron space (wave-function model);2 the one-electron space (basis set).

I Normal distributions of errors in AEs (kJ/mol)

-200 200

HFDZ

-200 200 -200 200

HFTZ

-200 200 -200 200

HFQZ

-200 200 -200 200

HF5Z

-200 200 -200 200

HF6Z

-200 200

-200 200

MP2DZ

-200 200 -200 200

MP2TZ

-200 200 -200 200

MP2QZ

-200 200 -200 200

MP25Z

-200 200 -200 200

MP26Z

-200 200

-200 200

CCSDDZ

-200 200 -200 200

CCSDTZ

-200 200 -200 200

CCSDQZ

-200 200 -200 200

CCSD5Z

-200 200 -200 200

CCSD6Z

-200 200

-200 200

CCSD(T)DZ

-200 200 -200 200

CCSD(T)TZ

-200 200 -200 200

CCSD(T)QZ

-200 200 -200 200

CCSD(T)5Z

-200 200 -200 200

CCSD(T)6Z

-200 200

I The errors are systematically reduced by going up in the hierarchies


Coupled-cluster convergenceConvergence of the harmonic constant of N2

371.9

84.6

4.113.8

23.54.7 0.8

HF CCSD!FC CCSD"T#!FC CCSD"T# CCSDT CCSDTQ CCSDTQ5


Coupled-cluster convergenceEnergy contributions to atomization energies (kJ/mol)

I Contributions of each CC excitation level (left) and AO basis-set shell (right)

HF CCSD CCSDT CCSDTQ

1

10

100

1000

Log!Lin

DZ TZ QZ 5Z 6Z

1

10

100

1000Log!Log

I color code: HF , N2 , F2 , and CO

I The excitation-level convergence is approximately linear (log–linear plot)I each new excitation level reduces the error by about an order of magnitudeI the contributions from quintuples are negligible (about 0.1 kJ/mol)

I The basis-set convergence is much slower (log–log plot)

I each shell contributes an energy proportional to X−4 where X is the cardinal numberI a similarly small error (0.1 kJ/mol) requires X > 10I clearly, we must choose our orbitals in the best possible manner


Coupled-cluster convergenceThe exhaustion of the Schrodinger equation

Atomization energies (kJ/mol)

RHF SD T Q rel. vib. total experiment errorHF 405.7 178.2 9.1 0.6 −2.5 −24.5 566.7 566.2±0.7 0.5N2 482.9 426.0 42.4 3.9 −0.6 −14.1 940.6 941.6±0.2 −1.1F2 −155.3 283.3 31.6 3.3 −3.3 −5.5 154.1 154.6±0.6 −0.5CO 730.1 322.2 32.1 2.3 −2.0 −12.9 1071.8 1071.8±0.5 −0.0

Bond distances (pm)

RHF SD T Q 5 rel. theory exp. err.HF 89.70 1.67 0.29 0.02 0.00 0.01 91.69 91.69 0.00N2 106.54 2.40 0.67 0.14 0.03 0.00 109.78 109.77 0.01F2 132.64 6.04 2.02 0.44 0.03 0.05 141.22 141.27 −0.05CO 110.18 1.87 0.75 0.04 0.00 0.00 112.84 112.84 0.00

Harmonic constants (cm−1)

RHF SD T Q 5 rel. theory exp. err.HF 4473.8 −277.4 −50.2 −4.1 −0.1 −3.5 4138.5 4138.3 0.2N2 2730.3 −275.8 −72.4 −18.8 −3.9 −1.4 2358.0 2358.6 −0.6F2 1266.9 −236.1 −95.3 −15.3 −0.8 −0.5 918.9 916.6 2.3CO 2426.7 −177.4 −71.7 −7.2 0.0 −1.3 2169.1 2169.8 0.7


Coupled-cluster convergenceThe electron cusp and the Coulomb hole

I The wave function has a cusp at coalescence

-1.0-0.5

0.00.5

1.0

-0.5

0.00.5

-0.5

0.0

0.5

-1.0-0.5

0.00.5

.5

0.0

-

0

-1.0-0.5

0.00.5

1.0

-0.50.0

0.5

-0.10

-0.05

0.00

-1.0-0.5

0.00.5

.50.0

-

-

I It is difficult to describe by orbital expansions

-90 90

DZ

-90 90

TZ

-90 90

QZ

-90 90

5Z

I The error is inversely proportional to the number of virtual AOs

∆EX ≈ N−1 ≈ T−1/4

I Each new digit in the energy therefore costs 10000 times more CPU time!

1 minute → 1 week → 200 years


Coupled-cluster convergenceSolutions to slow basis-set convergence

1 Use explicitly correlated methods!

I Include interelectronic distances rij in the wave function:

ΨR12 =∑K

CKΦK + CR r12Φ0

50 100 150 200 250

-8

-6

-4

-2

CI

CI-R12

Hylleraas

2 Use basis-set extrapolation!

I Exploit the smooth convergence E∞ = EX + AX−3 to extrapolate to basis-set limit:

E∞ =X 3EX − Y 3EY

X 3 − Y 3

mEh DZ TZ QZ 5Z 6Z R12plain 194.8 62.2 23.1 10.6 6.6 1.4extr. 21.4 1.4 0.4 0.5


Density-functional theoryThe work horse of quantum chemistry

I The traditional wave-function methods of quantum chemistry are capable of high accuracy

I nevertheless, most calculations are performed using density-functional theory (DFT)


Density-functional theoryThe universal density functional

I The electronic energy is a functional E [v ] of the external potential

v(r) =∑

KZKrK

Coulomb potential

I Traditionally, we determine E [v ] by solving (approximately) the Schrodinger equation

E [v ] = infΨ〈Ψ|H[v ]|Ψ〉 variation principle

I However, the negative ground-state energy E [v ] is a convex functional of the potential

E(cv1 + (1− c)v2) ≥ cE(v1) + (1− c)E(v2), 0 ≤ c ≤ 1 convexity

x1 x2

f!x1"f!x2"

c f!x1"!!1"c" f!x2"f!x2"

c x1!!1"c"x2

f!c x1!!1"c"x2"

I The energy may then be expressed in terms of its Legendre–Fenchel transform

F [ρ] = supv

(E [v ]−

∫v(r)ρ(r) dr

)energy as a functional of density

E [v ] = infρ

(F [ρ] +

∫v(r)ρ(r)dr

)energy as a functional of potential

I the universal density functional F [ρ] is the central quantity in DFT


Density-functional theoryConjugate functionals

I As chemists we may choose to work in terms of E [v ] or F [ρ]:

F [ρ] = supv

(E [v ]−

∫v(r)ρ(r) dr

)Lieb variation principle

E [v ] = infρ

(F [ρ] +

∫v(r)ρ(r) dr

)Hohenberg–Kohn variation principle

I the relationship is analogous to that between Hamiltonian and Lagrangian mechanics

I The potential v(r) and the density ρ(r) are conjugate variables

I they belong to dual linear spaces such that∫v(r)ρ(r) dr is finite

I they satisfy the reciprocal relations

δF [ρ]

δρ(r)= −v(r), δE [v ]

δv(r)= ρ(r)

I In molecular mechanics (MM), we work in terms of E [v ]

I parameterization of energy as a function of bond distances, angles etc.I widely used for large systems (in biochemistry)

I In density-functional theory (DFT), we work in terms of F [ρ]

I the exact functional is unknown but useful approximations existI more accurate the molecular mechanics, widely used in chemistry

I Neither method involves the direct solution of the Schrodinger equation


Kohn–Sham theoryThe noninteracting reference system

I The Hohenberg–Kohn variation principle is given by

E [v ] = min ρ(F [ρ] +

∫v(r)ρ(r) dr

)I the functional form of F [ρ] is unknown—the kinetic energy is most difficult

I A noninteracting system can be solved exactly, at low cost, by introducing orbitals

F [ρ] = Ts[ρ] + J[ρ] + Exc[ρ], ρ(r) =∑

i φi (r)∗φi (r)

where the contributions are

Ts[ρ] = − 12

∑i

∫φ∗i (r)∇2φi (r)dr noninteracting kinetic energy

J[ρ] =

∫∫ρ(r1)ρ(r2)r−1

12 dr1dr2 Coulomb energy

Exc[ρ] = F [ρ]− Ts[ρ]− J[ρ] exchange–correlation energy

I In Kohn–Sham theory, we solve a noninteracting problem in an effective potential[− 1

2∇2 + veff(r)

]φi (r) = εiφi (r), veff(r) = v(r) + vJ(r) + δExc[ρ]

δρ(r)

I veff(r) is adjusted such that the noninteracting density is equal to the true densityI it remains to specify the exchange–correlation functional Exc[ρ]


Kohn–Sham theoryThe exchange–correlation functional

I The exact exchange–correlation functional is unknown and we must rely on approximations

I Local-density approximation (LDA)

I XC functional modeled after the uniform electron gas (which is known exactly)

ELDAxc [ρ] =

∫f (ρ(r)) dr local dependence on density

I widely applied in condensed-matter physicsI not sufficiently accurate to compete with traditional methods of quantum chemistry

I Generalized-gradient approximation (GGA)

I introduce a dependence also on the density gradient

EGGAxc [ρ] =

∫f (ρ(r,∇ρ(r)) dr local dependence on density and its gradient

I Becke’s gradient correction to exchange (1988) changed the situationI the accuracy became sufficient to compete in chemistryI indeed, surprisingly high accuracy for energetics

I Hybrid Kohn–Sham theory

I include some proportion of exact exchange in the calculations (Becke, 1993)I it is difficult to find a correlation functional that goes with exact exchangeI 20% is good for energetics; for other properties, 100% may be a good thing

I Progress has to a large extent been semi-empirical

I empirical and non-empirical functionals


Kohn–Sham theoryA plethora of exchange–correlation functionals

exchange, Slater local exchange, and the nonlocal gradientcorrection of Becke88. Thus,

ExcB3LYP ! a0Ex

exact " !1 # a0"ExSlater " ax#Ex

B88 " acEcVWN

" !1 # ac"EcLYP. [11]

Becke obtained the hybrid parameters {a0, ax, ac} $ {0.20, 0.72,0.19} (3) from a least-squares fit to 56 atomization energies, 42IPs, and 8 proton affinities (PAs) of the G2-1 set of atoms andmolecules (4). B3LYP leads to excellent thermochemistry (0.13eV MAD) and structures for covalently systems but does notaccount for London dispersion (all noble gas dimers are pre-dicted unstable).

Following B3LYP, we introduce the extended hybrid func-tional, denoted as X3LYP:

ExcX3LYP ! a0Ex

exact " !1 # a0"ExSlater " ax#Ex

X " acEcVWN

" !1 # ac"EcLYP. [12]

We determined the hybrid parameters {a0, ax, ac} $ {0.218,0.709, 0.129} in X3LYP just as for XLYP. Thus, we normalizedthe mixing parameters of Eq. 10 and redetermined {ax1, ax2} ${0.765, 0.235} for X3LYP. The FX(s) function of X3LYP (Fig.1) agrees with FGauss(s) for larger s.

Results and DiscussionWe tested the accuracy of XLYP and X3LYP for a broad rangeof systems and properties not used in fitting the parameters.Table 1 compares the overall performance of 17 different flavorsof DFT methods, showing that X3LYP is the best or nearly best

Table 1. MADs (all energies in eV) for various level of theory for the extended G2 set

Method

G2(MAD)

H-Ne, Etot TM #E He2, #E(Re) Ne2, #E(Re) (H2O)2, De(RO . . . O)#Hf IP EA PA

HF 6.47 1.036 1.158 0.15 4.49 1.09 Unbound Unbound 0.161 (3.048)G2 or best ab initio 0.07a 0.053b 0.057b 0.05b 1.59c 0.19d 0.0011 (2.993)e 0.0043 (3.125)e 0.218 (2.912)f

LDA (SVWN) 3.94a 0.665 0.749 0.27 6.67 0.54g 0.0109 (2.377) 0.0231 (2.595) 0.391 (2.710)GGA

BP86 0.88a 0.175 0.212 0.05 0.19 0.46 Unbound Unbound 0.194 (2.889)BLYP 0.31a 0.187 0.106 0.08 0.19 0.37g Unbound Unbound 0.181 (2.952)BPW91 0.34a 0.163 0.094 0.05 0.16 0.60 Unbound Unbound 0.156 (2.946)PW91PW91 0.77 0.164 0.141 0.06 0.35 0.52 0.0100 (2.645) 0.0137 (3.016) 0.235 (2.886)mPWPWh 0.65 0.161 0.122 0.05 0.16 0.38 0.0052 (2.823) 0.0076 (3.178) 0.194 (2.911)PBEPBEi 0.74i 0.156 0.101 0.06 1.25 0.34 0.0032 (2.752) 0.0048 (3.097) 0.222 (2.899)XLYPj 0.33 0.186 0.117 0.09 0.95 0.24 0.0010 (2.805) 0.0030 (3.126) 0.192 (2.953)

Hybrid methodsBH & HLYPk 0.94 0.207 0.247 0.07 0.08 0.72 Unbound Unbound 0.214 (2.905)B3P86l 0.78a 0.636 0.593 0.03 2.80 0.34 Unbound Unbound 0.206 (2.878)B3LYPm 0.13a 0.168 0.103 0.06 0.38 0.25g Unbound Unbound 0.198 (2.926)B3PW91n 0.15a 0.161 0.100 0.03 0.24 0.38 Unbound Unbound 0.175 (2.923)PW1PWo 0.23 0.160 0.114 0.04 0.30 0.30 0.0066 (2.660) 0.0095 (3.003) 0.227 (2.884)mPW1PWp 0.17 0.160 0.118 0.04 0.16 0.31 0.0020 (3.052) 0.0023 (3.254) 0.199 (2.898)PBE1PBEq 0.21i 0.162 0.126 0.04 1.09 0.30 0.0018 (2.818) 0.0026 (3.118) 0.216 (2.896)O3LYPr 0.18 0.139 0.107 0.05 0.06 0.49 0.0031 (2.860) 0.0047 (3.225) 0.139 (3.095)X3LYPs 0.12 0.154 0.087 0.07 0.11 0.22 0.0010 (2.726) 0.0028 (2.904) 0.216 (2.908)Experimental — — — — — — 0.0010 (2.970)t 0.0036 (3.091)t 0.236u (2.948)v

#Hf, heat of formation at 298 K; PA, proton affinity; Etot, total energies (H-Ne); TM #E, s to d excitation energy of nine first-row transition metal atoms andnine positive ions. Bonding properties [#E or De in eV and (Re) in Å] are given for He2, Ne2, and (H2O)2. The best DFT results are in boldface, as are the most accurateanswers [experiment except for (H2O)2].aRef. 5.bRef. 19.cRef. 4.dRef. 35.eRef. 38.fRef. 34.gRef. 37.hRef. 7.iRef. 10.j1.0 Ex (Slater) % 0.722 #Ex (B88) % 0.347 #Ex (PW91) % 1.0 Ec (LYP).k0.5 Ex (HF) % 0.5 Ex (Slater) % 0.5 #Ex (B88) % 1.0 Ec (LYP).l0.20 Ex (HF) % 0.80 Ex (Slater) % 0.72 #Ex (B88) % 1.0 Ec (VWN) % 0.81 #Ec (P86).m0.20 Ex (HF) % 0.80 Ex (Slater) % 0.72 #Ex (B88) % 0.19 Ec (VWN) % 0.81 Ec (LYP).n0.20 Ex (HF) % 0.80 Ex (Slater) % 0.72 #Ex (B88) % 1.0 Ec (PW91, local) % 0.81 #Ec (PW91, nonlocal).o0.25 Ex (HF) % 0.75 Ex (Slater) % 0.75 #Ex (PW91) % 1.0 Ec (PW91).p0.25 Ex (HF) % 0.75 Ex (Slater) % 0.75 #Ex (mPW) % 1.0 Ec (PW91).q0.25 Ex (HF) % 0.75 Ex (Slater) % 0.75 #Ex (PBE) % 1.0 Ec (PW91, local) % 1.0 #Ec (PBE, nonlocal).r0.1161 Ex (HF) % 0.9262 Ex (Slater) % 0.8133 #Ex (OPTX) % 0.19 Ec (VWN5) % 0.81 Ec (LYP).s0.218 Ex (HF) % 0.782 Ex (Slater) % 0.542 #Ex (B88) % 0.167 #Ex (PW91) % 0.129 Ec (VWN) % 0.871 Ec (LYP).tRef. 27.uRef. 33.vRef. 32.

Xu and Goddard PNAS ! March 2, 2004 ! vol. 101 ! no. 9 ! 2675

CHEM

ISTR

Y


Kohn–Sham theoryA comparison with coupled-cluster theory

I Reaction enthalpies (kJ/mol) calculated using the DFT/B3LYP and CCSD(T) models

B3LYP CCSD(T) exp.CH2 + H2 → CH4 −543 1 −543 1 −544(2)C2H2 + H2 → C2H4 −208 −5 −206 −3 −203(2)C2H2 + 3H2 → 2CH4 −450 −4 −447 −1 −446(2)CO + H2 → CH2O −34 −13 −23 −2 −21(1)N2 + 3H2 → 2NH2 −166 −2 −165 −1 −164(1)F2 + H2 → 2HF −540 23 −564 −1 −563(1)O3 + 3H2 → 3H2O −909 24 −946 −13 −933(2)CH2O + 2H2 → CH4 + H2O −234 17 −250 1 −251(1)H2O2 + H2 → 2H2O −346 19 −362 3 −365(2)CO + 3H2 → CH4 + H2O −268 4 −273 −1 −272(1)HCN + 3H2 → CH4 + NH2 −320 0 −321 −1 −320(3)HNO + 2H2 → H2O + NH2 −429 15 −446 −2 −444(1)CO2 + 4H2 → CH4 + 2H2O −211 33 −244 0 −244(1)2CH2 → C2H4 −845 −1 −845 −1 −844(3)


Kohn–Sham excitation energiesExcitation energies of CO

I HF (grey), CCSD (red), CC3 (black) LDA (yellow), BLYP (green), B3LYP (blue)

!!

!

!! !

!!

"

" ""

" "

"

"

#

# # ## #

#

#

$

$ $$ $ $

$

$

%

% %% % %

%%

&

& & & & &

&

&

11 2 3 4 5 6 7 8

2

4

6

8

10

12

14

I Statistics for errors for HF, CO, and H2O (%)

HF CCSD LDA BLYP B3LYP

∆ 8.6 0.2 −17.9 −20.3 −12.3∆std 4.9 1.1 8.2 8.3 5.5


Kohn–Sham excitation energiesThe asymptotic behaviour and importance of exact exchange

I DFT represents local excitations well

I excitations to outer valence and charge-transfer (CT) excitations less well describedI the potential falls off too fast—the asymptotic behaviour should be

limr→∞

vxc(r) = −1

r

I This can be corrected by the inclusion of exact exchange

I exact exchange can be introduced in different manners

2 4 6 8

0.2

0.4

0.6

0.8

1.0HF

B3LYP

CAM!B3LYP

LC

I the proportion of exact exchange as a function of r


Kohn–Sham excitation energiesCharge–transfer excitations in tripeptide

NN

O

H

N

O

OH

H

excitation type PBE B3LYP CAM exp.

n2 → π∗2 local 5.58 5.74 5.92 5.61n1 → π∗1 local 5.36 5.57 5.72 5.74n3 → π∗3 local 5.74 5.88 6.00 5.91π1 → π∗2 CT 5.18 6.27 6.98 7.01π2 → π∗3 CT 5.51 6.60 7.68 7.39n1 → π∗2 CT 4.61 6.33 7.78 8.12n2 → π∗3 CT 5.16 6.83 8.25 8.33π1 → π∗3 CT 4.76 6.06 8.51 8.74n1 → π∗3 CT 4.26 6.12 8.67 9.30


Kohn–Sham excitation energiesDiagnostic for TDDFT excitation energies

I The quality of excitation energies declines with increasing degree of charge transferI when can excitation energies be trusted?

I We have developed an inexpensive diagnostic Λ =∑

ia κ2ia〈|φi ||φa|〉/(

∑ia κ

2ia)

I the PBE functional (left) is erratic for Λ < 0.6I the CAMB3LYP functional (right) is uniformly reliable

I local excitations, Rydberg excitations, charge-transfer excitations

I Peach et al., J. Chem. Phys. 128, 044118 (2008)


Benchmarking by the Lieb variation principleThe adiabatic connection

I Let Eλ[v ] be the ground-state electronic energy at interaction strength λ

I We may then calculate the energy by expansion about the noninteracting system λ = 0:

Fλ[ρ] = F0[ρ] +

∫ λ

0F ′λ[ρ]dλ = Ts[ρ] + λ (J[ρ] + Ex[ρ]) + Ec,λ[ρ]

I The adiabatic connection: F ′λ[ρ] plotted against the interaction strength λ

F ′λ[ρ] = J[ρ] + Ex[ρ] + E ′c,λ[ρ] ← AC integrand

0.0 0.2 0.4 0.6 0.8 1.0

!9.5

!9.4

!9.3

!9.2

!9.1

!9.0

"

WXC,"!a.u.

CCSD

CCSD"T#

Ec$#%

!Tc$#%0.0 0.2 0.4 0.6 0.8 1.0

!0.025

!0.020

!0.015

!0.010

!0.005

0.000

"

WCCSD"T# $#%

!WCCSD$#%!

a.u.

WCCSD "T#$#% ! WCCSD$#%


Benchmarking by the Lieb variation principleFrom dynamical to static correlation: dissociation of H2

I As H2 dissociates, correlation changes from dynamical to static

0.0 0.2 0.4 0.6 0.8 1.0

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

1.4 bohr

5 bohr

10 bohr


Benchmarking by the Lieb variation principleBLYP cannot do static correlation

I The BLYP functional treats correlation as dynamical at all bond distances

R = 1.4 bohr

BLYP

FCI

R = 3.0 bohr

BLYP

FCI

R = 5.0 bohr

BLYP

FCI

R = 10.0 bohr

BLYP

FCI


Benchmarking by the Lieb variation principleBLYP manages by error cancellation

I The improved BLYP performance arises from an overestimation of exchangeI error cancellation between exchange and correlation reduces total error to about one third

R = 1.4 bohr

FCI

BLYP

HF

R = 3.0 bohr

FCI

BLYP

HF

R = 5.0 bohr

FCI correlation

BLYP correlation

BLYP exchange

FCI exchange

R = 10.0 bohr


Methods for large moleculesConstruction of Kohn–Sham matrices and molecular gradients

I Quantum chemistry is well developed for small and medium-sized moleculesI well-established levels of theory of high accuracy and reliabilityI wide variety of phenomena amenable to a rigorous treatmentI however, often a more realistic modeling requires studies on very large systems

I Our goal is to make molecular studies on thousands of atoms routineI redesign algorithms to curb cost and utilize new computer architecturesI different requirements may be necessary for electronic-structure models

I Fast integral evaluation for Kohn–Sham matrices and molecular gradientsI expansion of solid harmonics in Hermite rather than Cartesian GaussiansI density fitting and fast-multipole methods: linear complexity and fastI timings for BP86/6-31G** molecular gradients in linear polyene chains:

0

20

40

60

80

100

0 50 100 150 200 250

Tim

e (s

)

Number of carbon atoms

XC

FF-J

NF-J

1el


Methods for large moleculesExamples of energy optimizations and force evaluations

I Single Intel Xeon 2.66 GHz processor

I 392-atom titin fragment BP86/6-31G(*)

I energy in 50 min; gradient in 9 min (1.4 s per atom)

I 642-atom crambin protein BP86/6-31G

I energy in 3 h, gradient in 26 min (2.4 s per atom)

I Exact exchange is one to two orders of magnitude slower


Methods for large moleculesDifficulties with large systems

I For more than 1000 atoms, standard optimization techniques become problematic

I Roothaan diagonalization combined with DIIS averaging may oscillate or divergeI diagonalization is inherently of nonlinear complexity

I Large systems are typically more difficult to converge

100 150 200 250 300 3500

0.1

0.2

0.3

0.4alanine residue peptides

HF HOMO!LUMO gap

B3LYP HOMO!LUMO gap

lowest HF Hessian eigenvalue

B3LYP eigenvalue


Methods for large moleculesDensity-matrix energy optimization

I We have developed a density-matrix minimization based on the parameterization

D(X) = exp(−XS)D0 exp(SX), XT = −X

I Avoids diagonalization (of cubic cost) and provides linear scaling by sparsity

200 400 600 800 1000 1200

2000

4000

6000

8000

10000dens

sparse

time in RH Newton equations against the number of atomsalanine residue peptides HF!6!31G


Methods for large moleculesAugmented Roothaan–Hall (ARH) method

I Optimization based on D(X) may be carried out in many ways

I conjugate gradient, quasi-Newton, Newton methods

I The full Newton equations are given by

(Fvvn − Foo

n )X + X (Fvvn − Foo

n )︸︷︷︸large 2nd-order F term

+Gov([Dn,X])− Gvo([Dn,X])︸︷︷︸small 2nd-order G term

= Fvon − Fov

n︸︷︷︸1st-order F term

I dominant part of Hessian treated exactly, the remainder by update

I Comparison of ARH (left) and standard RH–DIIS (right)

I 51-molecule water cluster (full triangles), insulin (full squares), vitamin B12 (empty circles)

10-6

10-4

10-2

100

102

104

0 10 20 30 40

En

erg

y e

rro

r /

Ha

rtre

e

Iterations

(a)

10-6

10-4

10-2

100

102

104

0 10 20 30 40

Iterations

(b)


Methods for large moleculesThree-level ARH scheme

I The ARH scheme has subsequently been implemented within a three-level (3L) scheme:I grand-canonical atomic optimization: starting guessI valence-basis molecular optimization: crude molecular optimizationI full-basis molecular optimization: final adjustments

I Comparison with other methods for 23 transition-metal complexes

van Lenthe ARH-3L QCP1 QCP2100 76 109 157

I A water droplet containing 736 water molecules (diameter 35 A)I BP86/6-31G* level of theory (0+5+9 iterations)I 2208 atoms, 7360 electrons, 25760 primitive and 13248 contracted GaussiansI wall time 76 h, CPU time 270 h (4 IBM Power6 4.7 GHz cores)


Methods for large moleculesComputers and hardware

I Requirements for our new linear-scaling code

I not only linear scaling but fastI easily adaptable to new computational methods and computer architectures

I Hardware technology and platforms change rapidly

I we will never have routine access to largest and fastest computersI this gives us some time to adapt code to emerging technologiesI early adaptation is difficult; late adaptation is dangerous

I Moderns computers combine many nodes (1000s) with many cores (4,8,16)

I the use of OpenMP for many cores is fairly straightforwardI the use of MPI for many nodes is much more difficultI our current code uses OpenMP, we aim for a hybrid MPI/OpenMP solution


Centre for Theoretical and Computational Chemistry (CTCC)Theory and Modeling

I Centre of excellence established in 2007 for a period of 5 (10) years

I one of 21 Norwegian centres of excellence, the only one in chemistryI shared between the Universities of Tromsø (UiT) and Oslo (UiO), with UiT as host institution

Theory and modelling

Bioinorg

anic chem.

Organic and organo-

metallic chemistry

Solid-state systemsSpect

roscop

y

Heterogeneous andhomogeneous catalysisChemical biology

Materials science Atmospheric chem.

I Experimentalists and theorists from chemistry, physics, and mathematics

“The vision of the CTCC is to become a leading international contributor tocomputational chemistry by carrying out cutting-edge research in theoretical andcomputational chemistry at the highest international level.”


Centre for Theoretical and Computational Chemistry (CTCC)CTCC in numbers

I Financial support

I from the Norwegian Research Council (NRC) in 2009: 11.1 MNOKI from home institutions in 2009: 7.9 MNOK

I Staff (total and UiT + UiO)

I 10 senior members (5+5)I 3.5 researchers (2+1.5)I 20 postdocs (11+9)I 13 PhD students (6+7)I 5 master students (2+3)I 3 affiliates (1+2)I 4 adjunct professors in 20% position (3+1)I 1.6 administrative staff (1+0.6)

I Publications

I more than 220 papersI more than 1200 citations

I Computer resources

I provided by NOTUR (the Norwegian Metacenter for Computational Science)I 20 million CPU hours annually


Centre for Theoretical and Computational Chemistry (CTCC)Senior and affiliate members

I Principal investigators at the University of TromsøI Tor Fla, multiscale methods with waveletsI Luca Frediani, properties and spectroscopyI Abhik Ghosh, bioinorganic chemistryI Kenneth Ruud, directorI Inge Røeggen, fragment approach for large systems

I Principal investigators at the University of OsloI Knut Fægri, clusters, surfaces and solidsI Trygve Helgaker, large periodic and nonperiodic systemsI Claus Jørgen Nielsen, gas-phase reactions and photochemistryI Mats Tilset, catalysis and organometallic chemistryI Einar Uggerud, dynamics and time development

I Affiliate membersI Bjørn Olav Brandsdali, University of TromsøI Harald Møllendal, University of OsloI Svein Samdal, University of Oslo

I Adjunct professors (20% positions)I Sonia Coriani, University of TriesteI Odile Eisenstein, University of MontpellierI Benedetta Mennucci, University of PisaI Magdalena Pecul, University of WarsawI Trond Saue, University of Strasbourg


Centre for Theoretical and Computational Chemistry (CTCC)Organization and management

I Center Leadership

I Prof. Kenneth Ruud, Director, UiTI Prof. Trygve Helgaker, Co-director, UiO

I Senior Researcher Forum

I all senior CTCC members

I Administrative staff

I Stig Eide, Head of Administration, UiTI Anne Marie Øveraas, Office Manager, UiO (60% + 20% position)

I Board of Directors

I Prof. Fred Godtliebsen, chairman, vice dean of research, Faculty of Science, UiTI Prof. Anne-Brit Kolstø, vice chairman, UiOI Dr. Nina Aas, StatoilI Prof. Knut J. Børve, University of BergenI Prof. Aslak Tveito, Simula Research Center

I Scientific Advisory Board

I Prof. Emily Carter, Princeton UniversityI Prof. Odile Eisenstein, University of MontpellierI Prof. Kersti Hermansson, Uppsala UniversityI Prof. Mike Robb, Imperial College LondonI Prof. Per-Olof Astrand, Norwegian University of Science and Technology


Centre for Theoretical and Computational Chemistry (CTCC)CTCC seminars and workshops

I Biannual CTCC meetings

I joint Tromsø–Oslo meetings of all members twice a yearI informal presentations of students and postdocsI one meeting on conjunction with Norwegian Chemical Society (NKS) meeting

I CTCC group seminars

I 74 in Oslo since October 2007I 101 in Tromsø since July 2007

I 6 workshops with UiO research groups

I Catalysis Seminar, with inGAP1 April 17 2008I Organic Quantum Chemistry, November 25 2008I From Ab Initio Methods To Density-Functional Theory, with CMA2, January 13 2009I Mini-Seminar on Computational Inorganic Chemistry, with Fermio3, April 29 2010I Mini-Seminar on Computational Materials Science, May 4 2010I Workshop On Computational Quantum Mechanics, with CMA, June 18–19 2010

1Innovative Natural Gas Process and Products, a Centre for Research-based Innovation, UiO

2Centre of Mathematics for Applications, a Centre of Excellence, UiO

3Functional Energy Related Materials in Oslo, UiO


Centre for Theoretical and Computational Chemistry (CTCC)CTCC visitors

I CTCC visitors programs

I 12 months for visiting professors each yearI 24 months for graduate and postgraduate visitors each year

I Visiting professors

I Swapan Chakrabarti, University of Calcutta (3 months 2008, UiT)I Daniel Crawford, Virginia Tech (6 months 2009, UiT + UiO)I Pawe l Kozlowski, University of Louisville (3 months 2010, UiT)I Ludwik Adamowicz, University of Arizona (5 weeks 2010, UiO)I Taku Onishi, Mie University (10 months 2010–2011, UiO)I Mark Hoffmann, University of North Dakota, (6 months 2010–2011, UiT + UiO)I Wim Klopper, Universitat Karlsruhe (6 months 2010–2011, UiO)

I Short visits

I 2007: 21 visits of 20 unique visitors from 11 countriesI 2008: 56 visits of 45 unique visitors from 21 countriesI 2009: 32 visits of 30 unique visitors from 17 countriesI 2010: 56 visits of 46 unique visitors from 18 countries

I Gropen–Almlof Lectures

I annual lecture series established by the CTCC in 2008I Bjorn Roos (2008), Tom Ziegler (2009), Michele Parrinello (2010)

I Division for Computational Chemistry of the Norwegian Chemical Society (NKS)

I established in 2008 following an initiative of the CTCCI Kongsvinger 2008, Bergen 2009, Trondheim 2010, Lillstrøm 2011


Centre for Theoretical and Computational Chemistry (CTCC)CTCC meetings and conferences

I Coastal Voyage of Current Density Functional TheoryI September 19–22 2007I Coastal Express between Tromsø and TrondheimI 43 participants form 15 countriesI 22 talks and 11 posters

I Molecular Properties 2009I June 18–21 2009I Hotell Vettre, Asker (Oslo)I satellite symposium to the 13th ICQC in HelsinkiI 117 participants from 21 countriesI 35 talks and 54 posters

I Quantum Chemistry beyond the Arctic CirclePromoting Female Excellence in Theoretical and Computational Chemistry

I June 23–26 2010I Sommarøy and TromsøI 75 participants from 20 countriesI 29 talks and 25 posters

I XVth European Seminar on Computational Methods in Quantum ChemistryI June 16–19 2011I Oscarsborg, Drøbak (Oslo)I about 100 participants


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Quantum chemistry: wave-function and density-functional...

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