Molecular Electronic-Structure Theory: Some Current...

Molecular Electronic-Structure Theory:Some Current Trends

Trygve Helgaker

Centre for Theoretical and Computational Chemistry (CTCC),Department of Chemistry, University of Oslo, Norway

International CUTE Seminar on Computational Science for Ultimate NanoelectronicsDevices,

Mie University Research Center, Mie University, JapanSeptember 8, 2011

T. Helgaker (CTCC, University of Oslo) Molecular Electronic-Structure Theory Mie University, Japan, Sept. 8, 2011 1 / 42

Chemistry and computation

“The more progress sciences make, the more they tend to enter the domain of

mathematics, which is a kind of center to which they all converge. We may even judge

the degree of perfection to which a science has arrived by the facility with which it

may be submitted to calculation.”

Adolphe Quetelet, 1796–1874

“Every attempt to employ mathematical methods in the study of chemical

questions must be considered profoundly irrational. If mathematical analysis should

ever hold a prominent place in chemistry—an aberration which is happily

impossible—it would occasion a rapid and widespread degradation of that science.”

August Comte, 1748–1857

I Nowadays, quantum-chemical simulations are routinely carried out by nonspecialists

I we have become number crunchers

I Quantum chemistry has generated many qualitative models in chemistry

I these are useful but do not constitute the bread and butter of quantum chemistry

I Quantum chemists must provide numerical tools that compete with experiment

I ideally, our results should be as accurate as experiment: chemical accuracyI if we cannot consistently provide high accuracy, we will be out of business


Chemistry: a quantum-mechanical many-body problem

I At the deepest level, molecules are simple

I charged particles in motion, governed by the laws of quantum mechanics

I In quantum chemistry, we solve the Schrodinger equation for molecular systems

I can such simulations replace experiment?I the large number of particles makes such calculations difficult: the many-body problem

“The underlying laws necessary for the mathematical treatment of a large part of physics and thewhole of chemistry are thus completely known and the difficulty is only that the exact applicationof these laws leads to equations that are too complicated to be soluble.” Paul Dirac (1927)


The computer—the tool of quantum chemistry

I Help came from unexpected quarters. . .

I ENIAC (Electronic Numerical Integrator and Computer) (1946)

I the world’s first general-purpose electronic computerI designed to calculate artillery firing tablesI 27 metric tons, 17468 vacuum tubes, 385 multiplies per secondI “Giant Brain”: thousand times faster than mechanical computers

I the first four programmers


Playstation 3

I Since then computers have developed at an amazing speed

I The computer industry is no longer driven by military needs. . .


Moore’s law (1964)

I Computers improve by a factor of two every 18 months

I Computers are today are one million times more powerful than a generation ago

I This is a development no one could have foreseen in the 1920s

I Quantum-chemical calculations have become routine


Quantum chemistry

I In quantum chemistry we perform quantum-mechanical simulations of chemical systems

I we solve the Schrodinger equation for molecules and the condensed phase

I Such simulations are performed in most areas of modern chemical research

I 40% of all articles Journal of American Chemical Society make use of computationI this is a remarkable development for an experimental science

I In 1998, the Nobel Prize in Chemistry was awarded to Walter Kohn and John Pople

I to Kohn “for his development of the density-functional theory”I to Pople “for his development of computational methods in quantum chemistry”


Computation: “the third way”

I Numerical simulation are performed in many areas of science and engineering

I physics, astrophysics, chemistry, geology, climate modelingI weather forecasting, reservoir simulations, flight simulations, car designI predictive modeling reduces testing: simulation-based science and engineering

I Simulations are often regarded as the ‘third way’, in addition to theory and experiment

I simulations have become an important part of discovery

I Quantum-chemical simulations are used for many purposes

I experiments can be expensive, difficult or dangerous to carry outI observations can be difficult to understand or interpretI computation can substitute or complement experiment


Example: molecular structure

I An important property of molecules is their three-dimensional structure

I many experimental methods have been developed to determine molecular structure

I Today, structures of small molecules are typically determined by computation

I a comparison of calculated (blue) and measured (black) structuresExamples

139.11

108.00

150.19

107.86(20)

114.81º 132.81

128.69(10)

108.28(10)

121.27º

benzene

cyclopropane

propadienylidene

139.14(10)

108.02(20)

150.30(10)

107.81

114.97º

108.37 132.80(5)

128.79

121.2(1)º

CCSD(T)/cc-p(C)VQZ calculations empirical re geometry T. Helgaker (CTCC, University of Oslo) Molecular Electronic-Structure Theory Mie University, Japan, Sept. 8, 2011 9 / 42

Example: vibrational spectroscopy

I Molecules vibrate at characteristic infrared frequencies

I these frequencies are useful for identifying molecules

I For new compounds, computation can be of great help

I the new compound C2H2Si had been isolated, but what is its structure?silapropadienylidene (top) or silacyklopropyne (bottom)

Sherrill et al.14 Recently, frozen-core CCSD!T"/cc-pVTZcalculations of six C2H2Si isomers were reported by Ikuta etal.15 to determine equilibrium structures, ionization poten-tials, and electron affinities. The authors also identified an-other silacyclopropenylidene 4 as a minimum on the poten-tial energy surface !PES". In case of the isoelectronic CNHSifamily, Maier et al.9 found the bent chain HSiCN !7" to bethe global minimum on the PES followed by the chain mol-ecule HSiNC !8" and cyclic azasilacyclopropenylidene !9", aresult confirmed by more recent theoretical studies.16,17

The present paper reports results of high-level ab initiocalculations of selected molecular properties of singletC2H2Si and CNHSi structural isomers. In detail the studycomprises the determination of molecular equilibrium struc-tures obtained at the CCSD!T" level of theory18 using Dun-ning’s hierarchy of correlation consistent basis sets19–22 andevaluation of harmonic and anharmonic force fields at se-lected levels of theory to determine centrifugal distortionconstants, vibration-rotation interaction constants, and har-monic as well as fundamental vibrational frequencies. Formolecules for which sufficient laboratory isotopic data areavailable, i.e., c-C2H2Si !1" and H2CCSi !2", the combina-

tion of experimental ground state rotational constants androtation-vibration interaction constants !i

A,B,C from theoryhas been used to evaluate empirical equilibrium structuresre

emp. Additionally, for energetically higher lying C2H2Si andCNHSi structural isomers the combination of theoreticalequilibrium rotational and rotation-vibration interaction con-stants is used to predict accurate ground-state rotational con-stants. The determination of fundamental vibrational fre-quencies permitted a qualitative comparison against infrareddata from matrix isolation experiments for more than half adozen molecules of the sample. Additionally, spectroscopi-cally important parameters such as dipole moments and 14Nnitrogen quadrupole coupling constants have been calcu-lated.

The molecules studied here are promising candidates forfuture laboratory spectroscopic studies in particular byFTMW spectroscopy.

II. THEORETICAL METHODS

Quantum chemical calculations were performed with the2005 Mainz-Austin-Budapest version of ACESII

23 employingcoupled-cluster !CC" theory24 in its variant CCSD!T".18

Some calculations were performed using the developmentversion of CFOUR

25 at Mainz with its recent parallel imple-mentation of CC energy and first- and second-derivativealgorithms26 and calculations at the CCSDT!Q"27,28 levelwere performed with the string-based many-body codeMRCC29 which has been interfaced to CFOUR. In the frozen-core !fc" approximation, Dunning’s d augmented correlationconsistent basis sets cc-pV!X+d"Z19 with X=T and Q wereemployed for the silicon atom and standard basis setscc-pVXZ20 for hydrogen, carbon, and nitrogen #denoted asCCSD!T" /cc-pV!X+d"Z in the following$. For calculationscorrelating all electrons the basis sets cc-pCVXZ21,22 andtheir weighted variants cc-pwCVXZ22 with X=T, Q, and 5were used, the former type, however, only for the structuraloptimization of a subsample of molecules — c-C2H2Si !1",H2CCSi !2", HSiCN !7", and HSiNC !8" — to study differ-ences in its performance against the weighted basis sets.

Equilibrium geometries were calculated using analyticgradient techniques.30 Harmonic and anharmonic force fieldswere calculated using analytic second-derivativetechniques31,32 followed by additional numerical differentia-tion to calculate the third and fourth derivatives needed forthe anharmonic force field.32,33 While the CCSD!T"/cc-pVQZ level of theory has been found to yield molecularforce fields of very high quality and is hence often used asthe level of choice in these calculations !e.g., Refs. 34–36", itis computationally !rather" demanding for larger moleculesand/or molecules carrying second row elements such asthose studied here. At the same time it has been shown on anumber of occasions, that accurate empirical equilibriumstructures are obtained also with zero-point vibrational cor-rections computed using smaller basis sets such as cc-pVTZ!see, e.g., Refs. 37–41". As a consequence, the force fields inthe present study were calculated at the CCSD!T" /cc-pV!T+d"Z level of theory in the fc approximation for the totalsample of 12 molecules. For a subsample of those, the two

Si

C

C

Si

C C

r1 r2

r3

a1

a2

r1

r2

r3a1

a2

r1

r2

r3

a2

a1

1

2

3

4

5

6

7

r1 r2

r3

a1

a2

r1

r2

r3a1

a2

r1r2 r3

a2

a1

Si C C

SiCC8

9

10

11

12

r1

r2 r3

a1

r1

a1

r2a2

r1a1

r2a2

a3

r3 r4

r1

r2r3

r4

a1

a2a3

r1

r2

a1

a2

r1

r3r2

a1

Si C C

Si

C C

Si C N

Si CN

Si

C N

SiC N

CN Si

Si

N C

FIG. 1. !Color online" C2H2Si !left column" and CNHSi !right column"structural isomers investigated here. c-C2H2Si !1" and HSiCN !7" are theglobal minima on the C2H2Si and CNHSi potential energy surfaces, respec-tively. The isomers are ordered from bottom to top according to their rela-tive stability. All molecules are planar except for 5 where the HSiH andCSiC planes are arranged perpendicularly.

214303-2 S. Thorwirth and M. E. Harding J. Chem. Phys. 130, 214303 !2009"

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

0

20

40

60

80

100

0 500 1000 1500 2000 2500

Inte

nsity

Frequency (1/cm)

Spectra of SiC2H2 isomers: DZP CCSD(T)

SilapropadienylideneSilacyclopropyne

Experiment

I a comparison with computed spectra shows that the structure is cyclic


Example: chemistry in astrophysics

I Conditions in the universe are typically very different from those on earth

I these conditions cannot be created in laboratoriesI molecules under such conditions must be studied on computers

I Molecular clouds are cool dense regions of the interstellar medium

I contain molecular hydrogen, helium and small amounts of other moleculesI low density allow very reactive molecules to existI H+

3 , H–C≡C–C≡C–C≡C–C≡C–C≡C–C≡N, C=C=C, C3Si (cyclic)

I Neutron stars are remnants of the gravitational collapse of massive stars

I extreme densities and magnetic fields 1012 stronger than on earthI chemistry is dominated by magnetic rather than electric interactionsI oblong atoms, long chains of hydrogen atoms and helium molecules


Historical overview: wave-function and density-functional methods

I Quantum mechanics has been applied to chemistry since the 1920s

I early accurate work on He and H2I mostly semi-empirical calculations on larger molecules

I The concept of ab-initio theory developed in the 1950s

I no reliance on empirical parametersI calculations reproducible in different laboratories

I Hartree–Fock theory dominated in the 1960s

I uncorrelated mean-field theoryI qualitative description of chemical systems

I Configuration-interaction (CI) and related theories developed in the 1970s

I first attempt at electron correlationI lacks size-extensivity

I Coupled-cluster (CC) and related theories emerged in the 1980s

I size-extensive treatment of correlationI the exact solution can be approached in systematic mannerI imported from nuclear physics (and then re-exported)

I Density-functional theory (DFT) emerged during the 1990s

I evaluation of dynamical correlation from the density (rather than from the wave function)I Kohn–Sham theory introduced from solid-state physicsI semi-empirical in character and cannot be systematically improved


Historical development: an illustration

I Experimental and calculated 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


Hartree–Fock theory

I Hartree–Fock theory provides the fundamental approximation of wave-function theory

I the best single-determinantal approximation to the exact wave functionI 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 properties

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


Electron correlation by virtual excitations

I For an improved description, we must describe the effects of electron correlation

I 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 X uugg )|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


Coupled-cluster (CC) theory

I The CC state is obtained from the HF state by applying all possible excitation operators

|CC〉 =(

1 + tai X a

i

)︸︷︷︸

singles

· · ·(

1 + tabij X ab

ij

)︸︷︷︸

doubles

· · ·(

1 + tabcijk X abc

ijk

)︸︷︷︸

triples

· · ·(

1 + tabcdijkl X abcd

ijkl

)︸︷︷︸

quadruples

· · · |HF〉

I with each excitation, there is an associated probability amplitude tabc···ijk···

I we must also provide a set of n virtual orbitals for such excitations

I We expect lower-order excitations to be more important than higher-order ones

I This classification provides a hierarchy of ‘truncated’ CC wave functions:

I CCSD: CC up to double excitations (n6)I CCSDT: CC up to triple excitations (n8)I CCSDTQ: CC up to quadruple excitations (n10)I CCSDTQ5: CC up to quintuple excitations (n12)

I Errors are typically reduced by a factor of three to four at each new level

HF CCSD CCSDT CCSDTQ

1

10

100

1000

Log!Lin


Coupled-cluster convergence

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


The 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


The 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 atomization energies (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


Bond distances

HF MP2 CCSD CCSD(T) emp. eks.H2 RHH 73.4 73.6 74.2 74.2 74.1 74.1HF RFH 89.7 91.7 91.3 91.6 91.7 91.7H2O ROH 94.0 95.7 95.4 95.7 95.8 95.7HOF ROH 94.5 96.6 96.2 96.6 96.9 96.6HNC RNH 98.2 99.5 99.3 99.5 99.5 99.4NH3 RNH 99.8 100.8 100.9 101.1 101.1 101.1N2H2 RNH 101.1 102.6 102.5 102.8 102.9 102.9C2H2 RCH 105.4 106.0 106.0 106.2 106.2 106.2HCN RCH 105.7 106.3 106.3 106.6 106.5 106.5C2H4 RCH 107.4 107.8 107.9 108.1 108.1 108.1CH4 RCH 108.2 108.3 108.5 108.6 108.6 108.6N2 RNN 106.6 110.8 109.1 109.8 109.8 109.8CH2O RCH 109.3 109.8 109.9 110.1 110.1 110.1CH2 RCH 109.5 110.1 110.5 110.7 110.6 110.7CO RCO 110.2 113.2 112.2 112.9 112.8 112.8HCN RCN 112.3 116.0 114.6 115.4 115.3 115.3CO2 RCO 113.4 116.4 115.3 116.0 116.0 116.0HNC RCN 114.4 117.0 116.2 116.9 116.9 116.9C2H2 RCC 117.9 120.5 119.7 120.4 120.4 120.3CH2O RCO 117.6 120.6 119.7 120.4 120.5 120.3N2H2 RNN 120.8 124.9 123.6 124.7 124.6 124.7C2H4 RCC 131.3 132.6 132.5 133.1 133.1 133.1F2 RFF 132.7 139.5 138.8 141.1 141.3 141.2HOF ROF 136.2 142.0 141.2 143.3 143.4 143.4


Density-functional theory: the 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)


The 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


Kohn–Sham theory: the noninteracting reference system

I The Hohenberg–Kohn variation principle is given by

E [v ] = minρ

(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 (XC) 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 theory: the 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)

E LDAxc [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

E GGAxc [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


A 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


A comparison of Kohn–Sham and coupled-cluster theories

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)


Deficiencies of DFT

I DFT is a short-sighted theory: electron interactions are calculated locallyI this gives rise to a number of serious deficiencies such as incorrect long-range potentialsI inability to describe van der Waals interactions (important in biology)I inability to describe charge-transfer excitations

I Example: tripeptide excitations

NN

O

H

N

O

OH

H

excitation type PBE B3LYP CAM exp.

n2 → π∗2 local 5.58 5.74 5.92 5.61

n1 → π∗1 local 5.36 5.57 5.72 5.74

n3 → π∗3 local 5.74 5.88 6.00 5.91

π1 → π∗2 CT 5.18 6.27 6.98 7.01

π2 → π∗3 CT 5.51 6.60 7.68 7.39

n1 → π∗2 CT 4.61 6.33 7.78 8.12

n2 → π∗3 CT 5.16 6.83 8.25 8.33

π1 → π∗3 CT 4.76 6.06 8.51 8.74

n1 → π∗3 CT 4.26 6.12 8.67 9.30

I Other deficiencies:I Coulomb self-interaction, poor bond dissociation


Towards larger and more complicated molecules. . .

I An important task is to develop methods applicable to large molecules

I 30 years ago we were able to study molecules consisting of a few atoms

I today molecules containing hundreds of atoms are routinely studied

I Molecules of biological interest often contain thousands of atoms

I to have impact on biology, quantum chemistry must be applicable to such systems

I these will be important for rational drug design


Linear-scaling methods

I Our goal is to make molecular studies on thousands of atoms routineI redesign algorithms to curb cost and utilize new computer architecturesI methods must be developed whose cost scales linearly with system size

I Such methods are being developed for Kohn–Sham DFTI 642-atom crambin protein using BP86/6-31G theoryI energy in 3 h, forces in 26 min (2.4 s per atom) on a single Intel Xeon 2.66 GHz processor

I Our next step is massive parallelism. . .


CTCC: Theory and Modeling

I Centre of excellence established in 2007 for an initial period of 5 yearsI 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 institutionI extended until 2017 following a midterm evaluation in 2010

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.”


CTCC in numbers

I Annual financial supportI from the Norwegian Research Council (NRC): 1.4 MEURI from home institutions in 2009: 1.0 MEUR

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 PublicationsI more than 200 papers published June 2007 – November 2010

I Computer resourcesI provided by NOTUR (the Norwegian Metacenter for Computational Science)I 2000 CPU years in 2009


Leadership and organization

I Center LeadershipI Prof. Kenneth Ruud, Director, UiTI Prof. Trygve Helgaker, Vice-Director, UiO

I Board of DirectorsI 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 BoardI 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


WP1: Large periodic and nonperiodic systems

I Efficient linear-scaling methods for large molecular and periodic systems

I new integration and optimization techniques

I Prof. Trygve Helgaker

I Department of Chemistry, UiO

I electron structure

I highly accurate methods

I benchmarking

I large systems

I density-functional theory

I response theory


WP2: Fragment approach for large systems

I Large molecular systems constructed from accurately calculated subsystems

I Perturbed Atoms in Molecules and Solids (PATMOS)

I Prof. Inge Røeggen

I Department of Physics, UiT

I electron correlation

I intramolecular interactions

I computational chemistry

I chemical bonding


WP3: Multiscale methods with wavelets

I Different regions of space treated at different resolutions and accuracies

I use of scaling and detail (wavelet) functions

I Prof. Tor Fla

I Department of Mathematics, UiT

I density-functional theory

I wavelets

I bioinformatics


WP4: Properties and spectroscopy

I Modeling of spectroscopic techniques by computation

I linear and nonlinear optics, effects of solvation

I Ass. Prof. Luca Frediani

I Department of Chemistry, UiT

I linear and nonlinearmolecular properties

I solvation

I multiwavelets


WP5: Dynamics and time development

I The modeling of chemical reactions by on-the-fly dynamics

I application to metal clusters, water clusters and organic reactions

I Prof. Einar Uggerud


I mass spectrometry


I reaction mechanisms

I molecular clusters


WP6: Bioinorganic chemistry

I Applications of quantum chemistry to problems in metallobiochemistry

I in conjunction with experimental work in synthesis, spectroscopy and electrochemistry

I Prof. Abhik Ghosh

I Department of Chemistry, UiT


I bioinorganic chemistry

I porphyrin chemistry

I metal complexes


WP7: Catalysis and organometallic chemistry

I Organo- and organometallic catalysis

I application of methods for large systems and dynamics

I Prof. Mats Tilset


I organometallic chemistry

I reaction mechanisms

I homogeneous catalysis

I C-H activation

I electron transfer


WP8: Gas-phase reactions and photochemistry

I The study of complex gas-phase reactions

I chemical processes of atmospheric relevance

I Prof. Claus Jørgen Nielsen


I atmospheric chemistry

I spectroscopy

I gas-phase chemistry

I aerosols


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 120 participants


CTCC visitors

I The CTCC provides two separate 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 Visitor statistics: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 countries


Molecular Electronic-Structure Theory: Some Current...

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