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High-Accuracy Quantum Chemistry

Trygve Helgaker

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

Laboratoire de Chimie Theorique,Universite Pierre et Marie Curie, Paris, France

November 6, 2012

T. Helgaker (CTCC, University of Oslo) High-Accuracy Quantum Chemistry LCT, UPMC, November 6 2012 1 / 40

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, 1798–1857

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) High-Accuracy Quantum Chemistry LCT, UPMC, November 6 2012 9 / 40

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 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


High-precision quantum-chemical calculations

I Dramatically improved ability to treat molecular systems accurately over the last decades

I development of techniques for systematic convergence towards the exact solutionI extensive benchmarking on small and light molecular systems

I In many cases, we can now confidently confirm or reject experimental observations

I Many black-box methods have been developed

I well-defined levels of theoryI relatively easy to use by the nonspecialist

I Still, the exact solution can be approached in infinitely many ways

I at many (incomplete) levels of theory, agreement with experiment may be obtainedI such agreement always arises by error cancellationI error cancellation is treacherous: the right answer for the wrong reason

I We must be wary of the pitfalls of error cancellation

I we shall here review high-precision quantum chemistry,I we shall be paying special attention to the problem of error cancellation


Hartree–Fock theory

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


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

· · · |HF〉

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

I the important point is to parameterize the excitations rather than the resulting statesI we must also provide a set of virtual orbitals for such excitations (vide infra)

I We expect the 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


Many-body perturbation theory: approximate coupled-cluster theory

I Coupled-cluster amplitudes may be estimated by perturbation theory

I Caveat: the resulting perturbation series is frequently divergent, even in simple cases

I here are some examples for the HF molecule (10 electrons):

-0.030

-0.010

1020

3040

cc-pVDZ at 2.5Re

-0.300

0.300

5

10

15

aug-cc-pVDZ at 2.5Re

-0.006

-0.002

0.002

5

10

15

20

cc-pVDZ at Re

-0.006

-0.002

0.002 5

10

15

20

25

30aug-cc-pVDZ at Re

I However, to lowest order, perturbational corrections are very useful and popular

I MP2 (approximate CCSD) and CCSD(T) (approximate CCSDT)I correlation effects are typically overestimated, leading to fortuitously good results


Basis-set convergence

I In CC theory, the wave function is generated by virtual excitations:

|CC〉 =(

1 + tai X a

i

)· · ·(

1 + tabij X ab

ij

)· · ·(

1 + tabcijk X abc

ijk

)· · · |HF〉

I In all examples up to now, we have worked in a complete AO basis

I However, the overall quality is determined by the description of

one-electron basis sets

wave-

functi

on

models

exac

tso

lution

1 the wave-function model2 the AO basis set

I We shall now consider basis-set convergence

I new sources of errorsI new opportunities for error cancellation

I We begin by investigating convergence in the helium atom


One-electron basis sets for virtual excitations

I In CC theory, the wave function is generated by virtual excitations:

|CC〉 =(

1 + tai X a

i

)· · ·(

1 + tabij X ab

ij

)· · ·(

1 + tabcijk X abc

ijk

)· · · |HF〉

I we need to provide a basis of N one-electron functions for virtual excitationsI operation count is at least (and typically) proportional to N4

I The quality of the calculation depends critically on the virtual space

I we employ atom-fixed Gaussian atomic orbitals (harmonic-oscillator functions)I virtual atomic orbitals are added in full shells at a timeI each shell contains 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 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

I What is the reason for this excruciatingly slow convergence?


AO convergence: the helium atom

I The helium atom contains only two electrons

I ideal system to study basis-set convergenceI pair interactions dominate also molecules

I Historical interest: the first many-body system treated with quantum mechanics

I experimental ionization potential of helium: 24.59 eV

I Unsold 1927: 20.41 eV

I first-order perturbation theory—not much better than Bohr theory

I Hylleraas 1928: 24.47 eV

I expansion in antisymmetric orbital productsI excruciatingly slow AO convergence

I Hylleraas 1929: 24.58 eV

I introduced the interelectronic coordinate r12 to arbitrary powersI the discrepancy of 0.01 eV due to relativistic correctionsI full agreement between experiment and quantum mechanics

I Hylleraas discovered both the slow CI convergence and the efficacy of introducing r12

I the question of CI expansions vs. explicit correlation is still with us today


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 local kinetic energy

I Consider the local energy of the helium atom

Eloc = (HΨ)/Ψ ← constant for exact wave function

I The electronic Hamiltonian has singularities at points of coalescence

H = −1

2∇2

1 −1

2∇2

2 −2

r1−

2

r2+

1

r12

I the infinite potential terms must be canceled by infinite kinetic terms at coalescence

I Local kinetic energy in the helium atomI positive around the nucleusI negative around the second electron

I Negative kinetic energy counterintuitiveI classical forbidden regionI internal “tunneling”I w. f. decays towards the singularityI the Coulomb hole

-0.5

0.0

0.5

1.0

0.0

0.5

-100

0

100

200

300

-0.5

0.0

0.5

I The difficulty Hylleraas ran into was the description of the Coulomb hole!


The Coulomb hole: the forbidden region

I Each electron is surrounded by a classically forbidden region: the Coulomb hole

I without a good description of this region, our results will be inaccurate

-1 -0.5 0 0.5 1

-0.4

-0.2

0

0.2

0.4


Solutions to slow basis-set convergence

1 Use explicitly correlated methods!

I Include interelectronic distances rij in the wave function:

ΨR12 =∑

K

CK ΦK + 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


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


Atomization energies (AEs)

I The atomization energy is the difference between atomic and molecular energies:

De =∑

A E A(

2S+1L)− E (Re)

I Statistics based 20 closed-shell organic molecules (kJ/mol)

T

6 extr.

-75

25

mean errors

T Q 5 6 extr.

20

40

standard deviations

I AEs increase with cardinal number.I AEs increase with excitation level in the coupled-cluster hierarchy:

HF < CCSD < CCSD(T) < MP2

I MP2 overestimates doubles contribution

I benefits from error cancellation at the MP2/TZ level

I CCSD(T) performs excellently, but DZ and TZ are inadequate:

I DZ and TZ basis are inadequate for CCSD(T)


The (in)adequacy of CCSD(T)

CCSD(T) CCSDT CCSDTQ experimentcc-pCV(56)Z cc-pCV(Q5)Z cc-pVTZ De D0

CH2 757.9 −0.9 758.9 0.1 759.3 0.5 758.8 714.8±1.8H2O 975.3 0.1 974.9 −0.3 975.7 0.5 975.2 917.8±0.2HF 593.2 0.0 593.0 −0.2 593.6 0.4 593.2 566.2±0.7N2 954.7 −1.6 951.3 −5.0 955.2 −1.1 956.3 941.6±0.2F2 161.0 −2.4 159.6 −3.8 162.9 −0.5 163.4 154.6±0.6CO 1086.7 0.0 1084.4 −2.3 1086.7 0.0 1086.7 1071.8±0.5

I The excellent performance of CCSD(T) for AEs relies on error cancellation:

I relaxation of triples from CCSD(T) to CCSDT reduces the AEs;I inclusion of quadruples from CCSDT to CCSDTQ increases the AEs.

I The error incurred by treating the connected triples perturbatively is quite large (about 10%of the full triples contribution) but canceled by the neglect of quadruples.

I The rigorous calculation of AEs to chemical accuracy requires CCSDTQ/cc-pCV6Z!


Bond distances

I Statistics based on 28 bond distances at the all-electron cc-pVX Z level (pm):

-2

-1

1

HF

MP2

MP3

MP4

CCSD

CCSD(T)

CISD

pVDZ

pVTZpVQZ

I bonds shorten with increasing basis: DZ > TZ > QZI bonds lengthen with increasing excitations: HF < CCSD < MP2 < CCSD(T)I considerable scope for error cancellation: CISD/DZ, MP3/DZI CCSD(T) mean errors: DZ: 1.68 pm; TZ: 0.01 pm; QZ: −0.12 pm


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


Error cancellation and the Pauling point

I Convergence 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


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


Conjugate functionals and conjugate variables

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 finiteI they satisfy the reciprocal relations

δF [ρ]

δρ(r)= −v(r),

δE [v ]

δv(r)= ρ(r)

I In molecular mechanics (MM), we model 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 model 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 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)


Conclusions

I Quantum chemistry provides a set of well-defined levels of approximation

I excitation-level expansionI basis-set expansion

I The “exact” result can be approached in a systematic manner

I convergence is typically slow but smooth

I In practice, we have to be content with low levels of theory

I low excitation levels, approximated by perturbation theoryI small basis sets

I Sometimes complicated interplay between different approximations

I error cancellations often occur, discipline is needed

I Useful and reliable (balanced) standard levels of theory have been implemented in codes

I can be usefully applied without deep knowledge of quantum chemistry

I DFT is much cheaper and more generally applicable

I can be fruitfully used in connection with manybody methods


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